Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Innovative Food Processing Technologies: Advances in Multiphysics Simulation Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg EDITORS

A John Wiley & Sons, Inc., Publication

The IFT Press series reflects the mission of the Institute of Food Technologists—to advance the science of food contributing to healthier people everywhere. Developed in partnership with Wiley-Blackwell, IFT Press books serve as leading-edge handbooks for industrial application and reference and as essential texts for academic programs. Crafted through rigorous peer review and meticulous research, IFT Press publications represent the latest, most significant resources available to food scientists and related agriculture professionals worldwide. Founded in 1939, the Institute of Food Technologists is a nonprofit scientific society with 22,000 individual members working in food science, food technology, and related professions in industry, academia, and government. IFT serves as a conduit for multidisciplinary science thought leadership, championing the use of sound science across the food value chain through knowledge sharing, education, and advocacy. IFT Press Advisory Group Casimir C. Akoh Christopher J. Doona Jung Hoon Han David B. Min Ruth M. Patrick Syed S.H. Rizvi Fereidoon Shahidi Christopher H. Sommers Yael Vodovotz Mark Barrett Karen Nachay Margaret Kolodziej IFT Press Editorial Board Malcolm C. Bourne Dietrich Knorr Theodore P. Labuza Thomas J. Montville S. Suzanne Nielsen Martin R. Okos Michael W. Pariza Barbara J. Petersen David S. Reid Sam Saguy Herbert Stone Kenneth R. Swartzel

A John Wiley & Sons, Inc., Publication

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2011

Titles in the IFT Press series • Accelerating New Food Product Design and Development (Jacqueline H. Beckley, Elizabeth J. Topp, M. Michele Foley, J.C. Huang, and Witoon Prinyawiwatkul) • Advances in Dairy Ingredients (Geoffrey W. Smithers and Mary Ann Augustin) • Bioactive Proteins and Peptides as Functional Foods and Nutraceuticals (Yoshinori Mine, Eunice Li-Chan, and Bo Jiang) • Biofilms in the Food Environment (Hans P. Blaschek, Hua H. Wang, and Meredith E. Agle) • Calorimetry in Food Processing: Analysis and Design of Food Systems (Gönül Kaletunç) • Coffee: Emerging Health Effects and Disease Prevention (YiFang Chu) • Food Carbohydrate Chemistry (Ronald E. Wrolstad) • Food Ingredients for the Global Market (Yao-Wen Huang and Claire L. Kruger) • Food Irradiation Research and Technology (Christopher H. Sommers and Xuetong Fan) • Foodborne Pathogens in the Food Processing Environment: Sources, Detection and Control (Sadhana Ravishankar, Vijay K. Juneja, and Divya Jaroni) • High Pressure Processing of Foods (Christopher J. Doona and Florence E. Feeherry) • Hydrocolloids in Food Processing (Thomas R. Laaman) • Improving Import Food Safety (Wayne C. Ellefson, Lorna Zach, and Darryl Sullivan) • Innovative Food Processing Technologies: Advances in Multiphysics Simulation (Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg) • Microbial Safety of Fresh Produce (Xuetong Fan, Brendan A. Niemira, Christopher J. Doona, Florence E. Feeherry, and Robert B. Gravani) • Microbiology and Technology of Fermented Foods (Robert W. Hutkins) • Multivariate and Probabilistic Analyses of Sensory Science Problems (Jean-François Meullenet, Rui Xiong, and Christopher J. Findlay • Nanoscience and Nanotechnology in Food Systems (Hongda Chen) • Natural Food Flavors and Colorants (Mathew Attokaran) • Nondestructive Testing of Food Quality (Joseph Irudayaraj and Christoph Reh) • Nondigestible Carbohydrates and Digestive Health (Teresa M. Paeschke and William R. Aimutis) • Nonthermal Processing Technologies for Food (Howard Q. Zhang, Gustavo V. Barbosa-Cánovas, V.M. Balasubramaniam, C. Patrick Dunne, Daniel F. Farkas, and James T.C. Yuan) • Nutraceuticals, Glycemic Health and Type 2 Diabetes (Vijai K. Pasupuleti and James W. Anderson) • Organic Meat Production and Processing (Steven C. Ricke, Michael G. Johnson, and Corliss A. O’Bryan) • Packaging for Nonthermal Processing of Food (Jung H. Han) • Preharvest and Postharvest Food Safety: Contemporary Issues and Future Directions (Ross C. Beier, Suresh D. Pillai, and Timothy D. Phillips, Editors; Richard L. Ziprin, Associate Editor) • Processing and Nutrition of Fats and Oils (Ernesto M. Hernandez and Afaf Kamal-Eldin) • Processing Organic Foods for the Global Market (Gwendolyn V. Wyard, Anne Plotto, Jessica Walden, and Kathryn Schuett) • Regulation of Functional Foods and Nutraceuticals: A Global Perspective (Clare M. Hasler) • Resistant Starch: Sources, Applications and Health Benefits (Yong-Cheng Shi and Clodualdo Maningat) • Sensory and Consumer Research in Food Product Design and Development (Howard R. Moskowitz, Jacqueline H. Beckley, and Anna V.A. Resurreccion) • Sustainability in the Food Industry (Cheryl J. Baldwin) • Thermal Processing of Foods: Control and Automation (K.P. Sandeep) • Trait-Modified Oils in Foods (Frank T. Orthoefer and Gary R. List) • Water Activity in Foods: Fundamentals and Applications (Gustavo V. Barbosa-Cánovas, Anthony J. Fontana Jr., Shelly J. Schmidt, and Theodore P. Labuza) • Whey Processing, Functionality and Health Benefits (Charles I. Onwulata and Peter J. Huth)

Contents

Preface, ix Contributors, xiii 1.

Introduction to Innovative Food Processing Technologies: Background, Advantages, Issues, and Need for Multiphysics Modeling, 3 Gustavo V. Barbosa-Cánovas, Abdul Ghani Albaali, Pablo Juliano, and Kai Knoerzer

2.

The Need for Thermophysical Properties in Simulating Emerging Food Processing Technologies, 23 Pablo Juliano, Francisco Javier Trujillo, Gustavo V. Barbosa-Cánovas, and Kai Knoerzer

3.

Neural Networks: Their Role in High-Pressure Processing, 39 José S. Torrecilla and Pedro D. Sanz

4.

Computational Fluid Dynamics Applied in High-Pressure Processing Scale-Up, 57 Cornelia Rauh and Antonio Delgado

5.

Computational Fluid Dynamics Applied in High-Pressure High-Temperature Processes: Spore Inactivation Distribution and Process Optimization, 75 Pablo Juliano, Kai Knoerzer, and Cornelis Versteeg

6.

Computer Simulation for Microwave Heating, 101 Hao Chen and Juming Tang

7.

Simulating and Measuring Transient Three-Dimensional Temperature Distributions in Microwave Processing, 131 Kai Knoerzer, Marc Regier, and Helmar Schubert

8.

Multiphysics Modeling of Ohmic Heating, 155 Peter J. Fryer, Georgina Porras-Parral, and Serafim Bakalis

9.

Basics for Modeling of Pulsed Electric Field Processing of Foods, 171 Nicolás Meneses, Henry Jaeger, and Dietrich Knorr

10.

Computational Fluid Dynamics Applied in Pulsed Electric Field Preservation of Liquid Foods, 193 Nicolás Meneses, Henry Jaeger, and Dietrich Knorr vii

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11.

Novel, Multi-Objective Optimization of Pulsed Electric Field Processing for Liquid Food Treatment, 209 Jens Krauss, Özgür Ertunç, Cornelia Rauh, and Antonio Delgado

12.

Modeling the Acoustic Field and Streaming Induced by an Ultrasonic Horn Reactor, 233 Francisco Javier Trujillo and Kai Knoerzer

13.

Computational Study of Ultrasound-Assisted Drying of Food Materials, 265 Enrique Riera, José Vicente García-Pérez, Juan Andrés Cárcel, Victor M. Acosta, and Juan A. Gallego-Juárez

14.

Characterization and Simulation of Ultraviolet Processing of Liquid Foods Using Computational Fluid Dynamics, 303 Larry Forney, Tatiana Koutchma, and Zhengcai Ye

15.

Multiphysics Modeling of Ultraviolet Disinfection of Liquid Food—Performance Evaluation Using a Concept of Disinfection Efficiency, 325 Huachen Pan

16.

Continuous Chromatographic Separation Technology—Modeling and Simulation, 335 Filip Janakievski

17.

The Future of Multiphysics Modeling of Innovative Food Processing Technologies, 353 Peter J. Fryer, Kai Knoerzer, and Pablo Juliano

Index, 365 Color plate section appears between pages 208 and 209.

Preface

The food industry is an increasingly competitive and dynamic arena, with consumers now more aware of what they eat and, more importantly, what they want to eat. Important food quality attributes such as taste, texture, appearance, and nutritional content are strongly dependent on the way the foods are processed. In recent years, with the aim to improve, or replace, conventional processing technologies in order to deliver higher-quality and better consumertargeted food products, a number of innovative technologies, also referred to as “emerging” or “novel” technologies have been proposed, investigated, developed, and in some cases, implemented. These technologies take advantage of other physics phenomena such as high hydrostatic pressure, electric and electromagnetic fields, and pressure waves. Some of the most promising innovative technologies, in various stages of development and adoption, are discussed in this book, namely high-pressure processing (also in combination with heat), microwave processing, ohmic heating, pulsed electric field processing, ultrasound processing (liquid- and airborne), ultraviolet light (UV) processing, and enhanced continuous separation. These innovative technologies provide the opportunity not only for the development of new foods but also for improving the safety and quality of conventional foods through milder processing. Different physical phenomena, utilized by these technologies, can potentially reduce energy and water consumption and therefore assist in reducing the

carbon and water footprint of food processing, thus playing an important role toward environmental sustainability and global food security. Apart from the underlying thermo- and fluiddynamic principles of conventional processing, these innovative technologies incorporate additional Multiphysics dimensions, for example, pressure waves, electric and electromagnetic fields, among others. To date, some of them still lack an adequate, complete understanding of the basic principles of intervening in temperature and flow evolution in product and equipment during processing. Their proper application, development and optimization of suitable equipment and process conditions still require a significant amount of further knowledge. Computational Fluid Dynamics (CFD) is already established as a tool for characterizing, improving, and optimizing traditional food processing technologies. Innovative technologies, however, provide additional complexity and challenges for modelers because of the concurrent interacting Multiphysics phenomena. In order to differentiate Multiphysics modeling from CFD modeling, the word “Multiphysics” will be capitalized throughout the book. Four symposia were organized at two consecutive Annual Meetings of the Institute of Food Technologists (IFT) in 2008 and 2009 (New Orleans and Anaheim, respectively) to gather Multiphysics modeling experts in innovative technologies to present and discuss the latest advances in their respective fields. These symposia highlighted the ix

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Preface

importance and key role of Multiphysics modeling to further advance the development of each innovative technology and facilitate their introduction into the food industry. Written by international experts from world-class research centers, academia, and industry, this book explains and discusses how Multiphysics modeling—that is, the simulation of the entire process comprising the actual equipment, varying process conditions, and the thermophysical properties of the food to be treated—can be applied in the development, optimization, and scale-up of innovative food processing technologies. The most recent research outcomes are shown to demonstrate benefits to process efficiency and the impact on scalability, safety, and quality. The first part of this book includes two chapters introducing the rationale of the book and some common themes to all chapters. Chapter 1 is the introductory chapter outlining the range of innovative processing technologies covered, briefly describing the technologies and making the case for the necessity of Multiphysics modeling for their design, development, and application. Chapter 2 discusses the importance of determining the relevant (common and technology-specific) thermophysical properties and their essential role for accurate model prediction. The second part of the book is an extensive collection of chapters devoted to the various case studies on the modeling of innovative food processing technologies. For clarity and convenience, they are divided into subsections focusing on high-pressure processing (Chapters 3–5), technologies utilizing electric and/or electromagnetic effects (microwave, ohmic heating, and pulsed electric field processing; Chapters 6–11), processes using ultrasound waves (in liquids or air) (Chapters 12 and 13), ultraviolet light (UV) processing (Chapters 14 and 15), and finally, one chapter on innovative chromatographic separation technologies (Chapter 16). Chapter 3 discusses two fundamentally different modeling approaches to characterize high-pressure (low-temperature) systems. It introduces the reader to the very promising modeling technique known as artificial neural networks (ANN), as well as the more

generalized visual programming approach referred to as macroscopic modeling. In Chapters 4 and 5, “conventional” CFD modeling approaches for highpressure processes at both low and high temperatures are discussed and their application for equipment design, scale-up, and optimization are highlighted. Also described is their application to present the process outcomes in terms of safety and quality of the processed foods. Chapter 6 and 7 covers the extension of classical CFD with a further Multiphysics dimension, electromagnetic radiation, and the implementation for designing and characterizing microwave heating processes. Chapter 7 also discusses various temperature mapping techniques and introduces the use of magnetic resonance imaging (MRI) for the determination of microwaveinduced three-dimensional heating patterns. In Chapter 8, historical and new developments of Multiphysics modeling applied to ohmic heating are presented. Chapters 9, 10, and 11 are devoted to modeling of pulsed electric fields processing, covering the basics of the technology, its application for predicting liquid food pasteurization, and the “multiobjective” optimization of the technology for liquid food processing. Chapters 12 and 13 present two distinctly different ultrasound applications. Chapter 12 covers liquid-borne ultrasound, including a review on its use in food processing, followed by an extensive review of the mathematics and physics involved in this technology, and this is concluded with a novel approach of modeling ultrasound-induced streaming. Chapter 13 details the use of airborne ultrasound for the improvement of drying processes at low temperatures. The complex mathematics is described and the chapter is concluded by experimental studies, highlighting the advantages and commercial potential of this innovative drying technology. Chapters 14 and 15 both describe UV processing for liquid food disinfection/pasteurization as an effective alternative to thermal treatments. Chapter 14 focuses on the characterization of several alternative reactor designs by Multiphysics modeling, whereas Chapter 15 compares the performance of different commercially available reactors using Multiphysics modeling and the introduction of the concept of “disinfection efficiency.” The final technology chapter (Chapter

Preface

16) introduces an innovative continuous separation process based on the chromatographic simulated moving bed principle. It outlines the procedure of modeling these types of technologies and highlights the advantages over conventional column or bedbased separation processes. Chapter 17 is the take-home message of this book, which concludes with a summary on what was presented in the chapters before and provides an outlook on future trends in Multiphysics simulation of innovative food processing technologies. Three questions are posed: (1) What can be usefully modeled today?; (2) What extra data is needed?; and (3) How much detail is needed, or Where shall we stop? This chapter is not intended to provide definitive answers to these questions, but it suggests some future research directions and places where research ought to or is expected to arrive.

xi

The editors wish to thank all collaborators in this book for their excellent contributions, and the time and effort they have devoted to making this book a comprehensive interdisciplinary reference source for engineers, technologists and scientists, and researchers from academia and industry alike. We believe that the value of this book is not limited to food engineering; it is also useful for other branches of process and chemical engineering. We would also like to thank the Institute of Food Technologist’s Nonthermal Processing Division, the International Division, and the Food Engineering Division for sponsoring the session symposia that led to the development of this book. Kai Knoerzer Pablo Juliano Peter Roupas Cornelis Versteeg

Contributors

Víctor M. Acosta Grupo de Ultrasonidos de Potencia Consejo Superior de Investigaciones Científicas (CSIC) Serrano, 144, E28006 Madrid, Spain

Hao Chen Department of Biological Systems Engineering Washington State University Pullman, WA 99164-6120 (Currently with Microsoft, Redmond, WA)

Abdul Ghani Albaali Princess Sumaya University for Technology P.O. Box 1438 Al-Jubaiha 11941 Jordan

Antonio Delgado Institute of Fluid Mechanics Friedrich-Alexander University Erlangen-Nuremberg Cauerstrasse 4, D-91058 Erlangen Germany

Serafim Bakalis Centre for Formulation Engineering School of Chemical Engineering University of Birmingham Birmingham B15 2TT United Kingdom

Özgür Ertunç Institute of Fluid Mechanics Friedrich-Alexander University Erlangen-Nuremberg Cauerstrasse 4, D-91058 Erlangen Germany

Gustavo V. Barbosa-Cánovas Department of Biological Systems Engineering Washington State University Pullman, WA 99164-6120

Larry Forney School of Chemical and Biomolecular Engineering Georgia Institute of Technology 311 Ferst Drive, N.W. Atlanta, GA 30332

Juan Andrés Cárcel Grupo de Análisis y Simulación de Procesos Agroalimentarios (ASPA) Departamento de Tecnología de Alimentos Universidad Politécnica de Valencia Camí de Vera s/n, E46022, Valencia Spain

Peter J. Fryer Centre for Formulation Engineering School of Chemical Engineering University of Birmingham Birmingham B15 2TT United Kingdom xiii

xiv

Contributors

Juan A. Gallego-Juárez Grupo de Ultrasonidos de Potencia Consejo Superior de Investigaciones Científicas (CSIC) Serrano, 144, E28006, Madrid Spain José Vicente García-Pérez Grupo de Análisis y Simulación de Procesos Agroalimentarios (ASPA) Departamento de Tecnología de Alimentos Universidad Politécnica de Valencia Camí de Vera s/n, E46022, Valencia Spain Henry Jaeger Department of Food Biotechnology and Food Process Engineering Technische Universität Berlin Koenigin-Luise-Str. 22 D-14195 Berlin Germany Filip Janakievski CSIRO Food and Nutritional Sciences 671 Sneydes Road Werribee, VIC 3030 Australia Pablo Juliano CSIRO Food and Nutritional Sciences 671 Sneydes Road Werribee, VIC 3030 Australia Dietrich Knorr Department of Food Biotechnology and Food Process Engineering Technische Universität Berlin Koenigin-Luise-Str. 22 D-14195 Berlin Germany Tatiana Koutchma Guelph Food Research Centre, Agriculture and Agri-Food Canada 93 Stone Road West Guelph, ON, N1G 5C9 Canada

Jens Krauss Institute of Fluid Mechanics Friedrich-Alexander University Erlangen-Nuremberg Cauerstrasse 4, D-91058 Erlangen Germany Kai Knoerzer CSIRO Food and Nutritional Sciences 671 Sneydes Road Werribee, VIC 3030 Australia Nicolás Meneses Department of Food Biotechnology and Food Process Engineering Technische Universität Berlin Koenigin-Luise-Str. 22 D-14195 Berlin Germany Huachen Pan Institute of Mechatronic Engineering Hangzhou Dianzi University 310018 Hangzhou China Georgina Porras-Parral Centre for Formulation Engineering School of Chemical Engineering University of Birmingham Birmingham B15 2TT United Kingdom Cornelia Rauh Institute of Fluid Mechanics Friedrich-Alexander University Erlangen-Nuremberg Cauerstrasse 4, D-91058 Erlangen Germany Marc Regier Fachhochschule Trier University for Applied Sciences Schneidershof, 54293 Trier Germany

Contributors

Enrique Fernando Riera Franco de Sarabia Grupo de Ultrasonidos de Potencia Consejo Superior de Investigaciones Científicas (CSIC) Serrano, 144, E28006, Madrid Spain

José S. Torrecilla Department of Chemical Engineering Universidad Complutense de Madrid Avenida Complutense s/n 28040 Madrid Spain

Pedro D. Sanz Malta Consolider Team Department of Processes ICTAN, CSIC c/ José Antonio Novais, 10 28040 Madrid Spain

Francisco Javier Trujillo CSIRO Food and Nutritional Sciences 11 Julius Avenue North Ryde, NSW 2113 Australia

Helmar Schubert Universitaet Karlsruhe (TH)/Karlsruhe Institute of Technology (KIT) Institute of Engineering in Life Sciences Dept. I: Food Process Engineering Karlsruhe, Germany Juming Tang Department of Biological Systems Engineering Washington State University Pullman, WA 99164-6120

Cornelis Versteeg CSIRO Food and Nutritional Sciences 671 Sneydes Road Werribee, VIC 3030 Australia Zhengcai Ye Bechtel Oil, Gas and Chemicals, Inc. 3000 Post Oak Blvd Houston, TX 77056

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Chapter 1 Introduction to Innovative Food Processing Technologies: Background, Advantages, Issues, and Need for Multiphysics Modeling Gustavo V. Barbosa-Cánovas, Abdul Ghani Albaali, Pablo Juliano, and Kai Knoerzer

1.1. Introduction In a world that is demanding environmental sustainability and food security, innovation is a key requirement for the sustained growth of the food industry. Furthermore, product innovation is the response to the growing demand for value addition along with more sophisticated and diverse food products. Modern food technology provides a handful of novel processing options to explore, which could provide more diverse food industry products and more competitive and efficient processes. Many of these innovative technologies can provide new opportunities for the development of new foods and for the improvement of safety and quality of more conventionally manufactured foods through milder processing. This book discusses innovative technologies that take advantage of physical forces and phenomena such as high hydrostatic pressure, electric and electromagnetic fields, and pressure waves, for example, high-pressure processing (also in combination with heat), microwave processing, ohmic heating, pulsed electric field (PEF) processing, ultrasound processing (liquid and airborne), and ultraviolet light (UV) processing. Innovative processing technologies present a number of hurdles that need to be addressed from

concept development to implementation. In particular, proper application, development, and optimization of suitable equipment and process conditions require a significant amount of further knowledge and understanding. In this book, the basic principles, current research, challenges, and commercial applications of the respective technologies, as well as the development and application of computational fluid dynamics (CFD) and, more broadly, Multiphysics modeling as a tool for characterizing, improving, and optimizing innovative food processing technologies are covered. Most innovative processing technologies have a common challenge, that is, to achieve a sufficient uniformity of the treatment or the process. This challenge is often already an issue at laboratory scale and it can become progressively worse when scaling up to pilot plants and, subsequently, to commercial equipment. Among other potential technologyspecific issues, nonuniformity of the treatment is most commonly encountered. In fact, the nonuniformities of the process and the lack of process validation of innovative processes are the greatest limitations for industrial uptake. Nonuniform treatment is, however, not specific to innovative processing technologies; conventional

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

processing technologies often encounter the same problem. For example, in conventional heat treatment processes such as canning, the temperature at the product surface is significantly higher than at the product center during most of the processing time, and only after prolonged holding times are temperature gradients throughout the product diminished. Another clear example of nonuniformity in conventional processing is the drying process of particulates. In this case, spatial and temporal heterogeneities in temperature and water content in the food product can be even more pronounced. The product goes (1) through an initial linear drying phase with water removal from the product surface, (2) over the falling rate period with moisture flux from the inside of the product to its surface, and (3) to a stage of product and drying medium (moisture) equilibrium with almost no further change in water content. In drying food products other important factors often come into play, increasing the degree of nonuniformity: product shrinkage and reduced moisture transport (increasing viscosity of contained liquids) up to a stage where pores are blocked. In the case of many innovative processing technologies as described throughout this book, nonuniformities may be reduced through technology-specific effects. However, these nonuniformities may be more pronounced due to increased complexities influenced by additional Multiphysics phenomena. This introductory chapter outlines the range of innovative food processing technologies covered in this book and gives a short overview of their benefits and advantages over traditional technologies. Some additional background information on the technologies, not covered in the respective technologyspecific chapters, is provided. Furthermore, this chapter makes a case for the need for applying Multiphysics modeling in these technologies for their design, including scale-up and optimization. The chapter summarizes the problems and challenges faced by the modelers, particularly with respect to the prediction of temperature, flow and technology-specific field distributions (e.g., sound intensity and electric or electromagnetic fields), and the extent of microbial or enzymatic inactivation and their distribution in equipment and products.

1.2. Multiphysics Modeling 1.2.1. Definition Multiphysics modeling is an extension of classical CFD. By definition, CFD is one of the branches of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. The geometry of the modeled scenario, including all components, is discretized into finite cells on which the governing partial differential equations (PDEs), namely the continuity, momentum, and energy conservation equations, are solved. This is detailed in the chapters specific to the respective technologies. Because these are PDEs, they cannot be solved analytically. Numerical techniques, such as finite differences, finite volumes, or finite element methods, must be applied to achieve an approximated solution (Sun 2007). Multiphysics modeling is based on the same principles as conventional CFD, that is, geometry discretization, and solving the PDEs is performed in a similar manner. However, Multiphysics modeling comprises additional physical phenomena such as electromagnetic waves, electrical fields, and acoustic waves related to the innovative technologies discussed further in this chapter. These phenomena can also be described by physically based PDEs (specific to each innovative technology), which have to be solved simultaneously with the ones from classical CFD. In some cases, the expression of the process outcome based on the attributes of the processed food, that is, the remaining microbial load, enzyme activity, and chemical reaction products, is required. Within Multiphysics modeling, reaction kinetics (i.e., microbial inactivation, quality degradation, chemical reaction, and structural responses) can be coupled with the specific differential equations to provide the spatial distributions of reaction response. Multiphysics models that concurrently solve the PDEs of classical CFD and the additional technologyspecific physical phenomena and the differential equations describing the reaction response require significantly greater computational resources. The increase in affordable computational power in recent years has allowed the simulation of innovative processes.

Chapter 1

Introduction to Innovative Food Processing Technologies

5

Figure 1.1. Number of commercial high-pressure equipment units around the world as of 2009 (Tonello 2010).

1.3. Innovative Food Processing Technologies 1.3.1. Background This section presents a brief description of each technology covered in this book. The major design problems and application limitations of these technologies are highlighted as an introduction to subsequent chapters. Ways in which Multiphysics modeling of innovative food processing technologies can assist in their development will be discussed. 1.3.1.1. High-Pressure Processing (HPP) and High-Pressure Thermal Sterilization (HPTS) HPP has demonstrated wide applicability for producing high-quality foods. HPP has become accepted as an attractive alternative to traditional preservation methods utilizing preservatives or thermal processing (Hernando Saiz et al. 2008, Chapters 3–5). HPP is commonly referred to as a nonthermal process of liquid and solid foods through application of high pressure in the order of 100–800 MPa (1,000 to 8,000 bar) and holding times of several minutes. HPP of foods is of increasing interest because it allows the inactivation of vegetative organisms at low or moderate temperature with minimum degradation (Abdul Ghani and Farid 2007). HPP offers opportunities for increased shelf life and preservative-free stabilization of meats, seafood, vegetable products, and juices. HPP can be

used not only for preservation, but also for modifying the physical and functional properties of some foods. More than 70 companies currently utilize HPP, producing more than 170,000 tons of products (Tonello 2010). Several HPP-treated food products, including juices, jams, jellies, yogurts, ready-to-eat meat, and oysters, are already widely available in the United States, Europe, Japan, New Zealand, and Australia. These successful applications have led to a pronounced increase in commercial-scale HPP units around the world during the past 10 years, as shown in Figure 1.1. In addition to inactivation of microorganisms and some spoilage enzymes (Seyderhelm et al. 1996; Yen and Lin 1996), promising results have been obtained with respect to the application on gelation of food proteins (Ohshima et al. 1993), improvement of digestibility of proteins, and tenderization of meat products (Ohmori et al. 1991; Jung et al. 2000a, 2000b; Buckow et al. 2010b). These changes in proteins have been used successfully in fish meat; in Carpaccio and Carpaccio-like products, high pressure allows the “processing” of the product, while still maintaining its raw characteristics. However, because of the application of high pressures, these products have retained “fresh-like” qualities and texture compared with heat-processed food, are microbiologically safe, and have an extended shelf life compared with raw food. Gomez-Estaca et al.

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

(2009) investigated HPP on fish products (such as salmon, tuna, and cod), showing superior sensory results. If the aim of the process is the inactivation of microbial spores, high pressure alone is not sufficient. However, a combination of high pressure and elevated temperatures, also referred to as HPTS or pressure-assisted thermal sterilization, can result in synergistic inactivation of these spores at potentially lower temperatures or shorter processing times, thus improving the quality of the processed foods while potentially reducing energy consumption (Bull et al. 2009). In this application, the increase in pressure is used as a means to increase the temperature evenly and fast in the product. There are two approaches to achieve highpressure conditions. In the direct approach, a piston is utilized, which compresses the content of the high-pressure chamber. In the indirect approach, a pressure-transmitting liquid (e.g., water) is pumped into the treatment chamber (high-pressure vessel) using a high-pressure pump followed by a “pressure intensifier.” Liquids at extremely high pressures are compressible, requiring extra fluid to be pumped into the vessel. During compression, the temperature of the processed food and the pressure-transmitting fluid increases due to the compression force working against intermolecular forces. The magnitude of the adiabatic temperature increase depends on a number of factors, such as the pressure medium and food product thermophysical properties (density, thermal expansion coefficient, and specific heat capacity) and initial temperature (see, e.g., Chapters 2, 4, and 5). Higher fat content of the food and higher initial temperature, for example, lead to an increase in compression heating. The phenomenon of increasing compression heating at elevated initial temperatures is important; for example, in HPTS, the product and the pressure medium are preheated to achieve higher process temperatures, which in turn allows inactivation of microbial spores (Wilson et al. 2008). In HPP, the greater the pressure level and time of application, the greater the potential for changes in the structure and appearance of the treated foods. This is especially true for raw high-protein foods,

where pressure-induced protein denaturation may be visually evident. High pressures can also induce significant structural changes (or damages) in some sensitive foods, such as strawberries or lettuce. Cell deformation and cell membrane damage can result in softening and cell serum loss. Usually, these changes are undesirable because the food will appear to be processed and no longer fresh or raw. Limitations of HPP and HPTS Although great progress has been made in the development of economically viable high-pressure applications, the scientific community and the food industry recognized in the early 2000s that engineering fundamentals, including CFD models, were required to design, evaluate, optimize, and scale up high-pressure processes of foods (Hendrickx and Knorr 2001). The limitation of HPP to date mainly lies in the limited throughput and, relative to heat processing, the high cost of equipment, labor (HPP is not yet a fully automated process), and maintenance. High maintenance costs are caused mainly by the extreme processing conditions. Furthermore, there are only a few large-scale commercial high-pressure equipment suppliers worldwide that have expertise in the food industry, including Avure Technologies, Inc. (Kent, WA), Kobelco (Kobe Steel Ltd., Kobe, Japan), and NC Hyperbaric (Burgos, Spain). A common issue in both HPP and HPTS is the nonuniformity of some aspects of the treatment. HPP generates pressure waves in liquids, which travel at the speed of sound (sound in water travels at 1,500 m/s). Therefore, pressure is commonly assumed to be transmitted instantaneously and uniformly. However, treatment nonuniformities can occur during HPP not only as a result of different compressibilities of the various substances in the food product, including trapped air (also headspace), but also because of the food packaging material. In addition, if the purpose of the process is the inactivation of the vegetative microorganisms, a nonuniform treatment can occur because some microorganisms are supposedly more resistant to the pressure when embedded in a fat matrix. Foods with higher fat or oil content may, therefore, protect the microorganisms in some areas in the food where fat is contained.

Chapter 1

Introduction to Innovative Food Processing Technologies

In the case of processing above room temperature (initial temperature), for example, in HPTS, nonuniform treatment temperature is likely to be more pronounced. In addition to pressure, temperature is an important process variable. In heterogeneous food materials, with the contents exhibiting differences in compression heating, temperatures may not be uniformly distributed in the food products. Furthermore, the packaging material, the material of the product carrier, and the steel of the high-pressure vessel are not heated to the same extent as the food; therefore, temperature gradients are developed throughout the system, leading to heat flux from the products to the cooler areas (which are mainly the steel walls). These spatial temperature heterogeneities increase over the process time. Although, theoretically, the preheated product heats up uniformly during compression to sterilization temperatures, during pressure holding time temperatures may decrease in certain areas of the vessel. This can affect spore inactivation, and spores may survive the process if temperature loss is not prevented. Product carriers have been developed as a means of retaining heat throughout the vessel during both pressure come-up and holding times (Chapter 5). Multiphysics modeling can greatly assist in the characterization of temperature distribution, subsequent microbial distributions, and other quality changes as a result of temperature inhomogeneities. These models can also be applied to the redesign and optimization of equipment and determination of adequate processing conditions for optimum process/product performance. 1.3.1.2. Microwave and Radio Frequency Processing Microwave heating refers to the use of electromagnetic waves of certain frequencies to generate heat in a material (Metaxas and Meredith 1983; Roussy and Pearce 1995; Metaxas 1996). Typically, microwave food processing uses frequencies of 2,450 and 915 MHz. In domestic ovens, 2,450 MHz frequency is commonly utilized, while in industrial heating application both frequencies are used, depending on the product to be treated, that is, product size and composition, associated with the relevant thermophysical properties (Chapters 2, 6, and 7).

7

Microwave heating has been proposed as an alternative to traditional heating methods in many food manufacturing processes, such as (re)heating, baking, (pre)cooking, tempering of frozen food, blanching, pasteurization, sterilization, and dehydration (Metaxas and Meredith 1983; Decareau 1985; Buffler 1993; Metaxas 1996; Schubert and Regier 2005; Tang et al. 2008). Microwave and radio frequency heating for pasteurization and sterilization are rapid; therefore, less time is required for come-up to the desired process temperature compared with conventional heating. This is particularly true for solid and semisolid foods that depend on slow thermal diffusion process in conventional heating. Microwave and radio frequency heating can approach the benefits of hightemperature short-time (HTST) processing, whereby bacterial destruction is achieved, while thermal degradation of the desired components is reduced. Heating with microwaves primarily involves two mechanisms. Water in the food is often the main component responsible for dielectric heating. Due to their dipolar nature, water molecules follow the alternating electric field associated with electromagnetic radiation. The second major mechanism is through the oscillatory migration of ions in the food under the influence of the alternating electric field. Such oscillatory motion of water molecules and ions and the associated intermolecular friction lead to a conversion of electromagnetic energy to thermal energy. The dielectric properties, namely the dielectric constant and the loss factor (Chapter 2), determine the strength of the electric field inside the food and its conversion into heat. These properties strongly depend on the composition (or formulation) of the food, with moisture and salt being the two primary determinants of interest (Mudgett 1985, 1986; Sun et al. 1995; Nelson and Datta 2001). The subsequent temperature rise in the food depends on the duration of heating, the location in the food, convective heat transfer at the surface, and the heat conduction and extent of evaporation of water inside the food and at its surface. Although the final objective of each process differs, an increase in product temperature is seen as

8

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

a common theme. There has also been some speculation on the so-called nonthermal effects of electromagnetic waves in the microwave frequency range. Four theories have been proposed to explain “nonthermal” or nondirect thermal effects of microwaves on, for example, microorganisms: selective heating, electroporation, cell membrane rupture, and magnetic field coupling (Kozempel et al. 1998). The selective heating theory states that solid microorganisms are heated more effectively by microwaves than the surrounding medium and are thus killed more readily. Electroporation is caused when pores form in the membrane of the microorganisms due to electrical potential across the membrane, resulting in leakage (this is similar to one of the theories on the effect of PEF processing for cold pasteurization). Cell membrane rupture is related to the voltage drop across the membrane, which causes it to rupture, which is also a theory in PEF processing. In the fourth theory, cell lysis occurs due to coupling of electromagnetic energy with critical molecules within the cells, disrupting vitally important internal cell components. Although researchers have repeatedly reported nonthermal effects of microwave processing, the general consensus (Heddleson and Doores 1994; Heddleson et al. 1994) is that the reported nonthermal effects are likely to be due to the lack of precise measurements of the time–temperature history and its spatial variations. A number of studies have shown that thermal effect is the essential contributor to the destruction of microorganisms (Goldblit and Wang 1967; Rosen 1972; Fujikawa et al. 1992). Therefore, to date, it is presumed that only thermal effects on microbial inactivation are effective, and microbial inactivation caused by microwave processing is essentially the same as in conventional thermal processing. Of course, the rates of heating and temperature distributions are quite different. Limitations of Electromagnetic Heating Volumetric microwave and radio frequency heating is theoretically more uniform than conventional heating (Datta and Hu 1992). There are, however, a number of microwave-specific factors that induce nonuniform heating patterns. First, electromagnetic field distribution inside a microwave cavity is, in most cases, not

uniform. Placing dielectrics (i.e., food products) into the microwave field leads to a change in the field distribution. Therefore, differences in the products, for example, product size, shape, and particularly composition with varying dielectric properties, will almost certainly lead to changes in process outcomes. However, not only do the field variations in the cavity cause nonuniform processing, the field characteristics inside the product are also heterogeneous. The heterogeneous composition of the different food components (and different dielectric properties) is an important factor in the heating of foods. Differences in dielectric properties lead to differences in temperature increases, even in a perfectly homogeneous microwave field. As these properties are in most cases strongly temperature-dependent, changes in temperature may compensate or may increase the nonuniformity. In particular, in cases where increasing temperatures lead to increasing loss factors (the imaginary part of the complex dielectric permittivity; Chapter 2), a so-called thermal “runaway” phenomenon can occur. With increasing temperature the rate of converting the electromagnetic energy into thermal energy increases as well; therefore, the gradients between hot and cold areas in the product become more pronounced. Another important factor in heating is the socalled focusing effect of the microwaves into specific areas in the product. This phenomenon is strongly dependent on the geometrical properties of the product. For example, a spherical product that does not exceed a certain size (due to limited penetration) can exhibit a pronounced hot spot in its geometrical center. Other phenomena causing uneven heating patterns include edge and corner overheating (caused by the penetration and absorption of the microwaves from more than one direction) and the development of standing waves inside the product (which is mainly dependent on the dielectric constant (the real part of the complex dielectric permittivity; Chapter 2). The time–temperature history at the coldest point for a conventional thermal process is generally predictable for a food that is all solid or all fluid. For example, for a conduction-heated (solid) food, it is usually the geometric center. In microwave heating,

Chapter 1

Introduction to Innovative Food Processing Technologies

even for a solid food, it is less straightforward to predict the coldest point and it can change during the heating process depending on temperaturedependent material properties and oven characteristics (Fleischman 1996; Zhang et al. 2001). A number of approaches have been proposed to improve the uniformity associated with microwave heating. These include rotating and oscillating the food in the microwave cavity (Geedipalli et al. 2007), providing an absorbing medium (such as hot water) surrounding the product (Chen et al. 2008; Chapter 6), equilibrating after heating (Fakhouri and Ramaswamy 1993), and cycling the power (Chapter 7). Success to date is limited due to the dependence of the materials’ properties on temperature and the nonuniform distribution of the electromagnetic field inside the food and the microwave cavity. Utilizing a lower microwave frequency of 915 MHz and radio frequencies to improve uniformity of heating have the potential to improve the evenness of heating (Chen et al. 2008), as the penetration depth into the food is greater and the field nonuniformities are less pronounced. Combinations of microwave and conventional technologies in many different configurations (e.g., hot air, vacuum, or infrared heating) have also been used to improve treatment uniformity; (Contreras et al. 2008; Turabi et al. 2008; Abbasi and Azari 2009; Kowalski and Mierzwa 2009; Kowalski and Rajewska 2009; Seyhun et al. 2009; Uysal et al. 2009). These approaches can be successful for some applications, especially where the cold spot is located at the food surface (Chapter 7); however, in food products with high salt or sugar content, the cold spot is usually within in the food, as the penetration depth of the electromagnetic waves is reduced. It remains a challenge to uniformly treat food products with microwaves and to achieve the targeted process outcomes; Multiphysics models, however, will greatly assist in designing microwave processes by evaluating process performance and developing appropriate control strategies (Chapters 6 and 7). Accordingly, Multiphysics models (including temperature-dependent properties of foods) need to be developed and subsequently validated to ascertain the location of the point of lowest integrated time–temperature history (Chapter 7).

9

1.3.1.3. Ohmic Heating Ohmic heating is defined as a process wherein electric currents are passed through foods or other materials with the primary purpose of heating them. The heating occurs in the form of internal electric energy dissipation within the material. Ohmic heating is distinguished from other electrical heating methods by the presence of electrodes contacting the food, the frequency of the current, or the waveform. The main purpose for the development of ohmic heating processes was to allow for HTST sterilization of solid–liquid mixtures (Chapter 8). Applications of ohmic heating in the food industry to date are scarce, although there are a number of advantages over other (conventional) heating methods. The main advantages for ohmic heating are the associated rapid and relatively uniform heating of the food product, depending on the electrical conductivity of the food components. This is expected to reduce unwanted thermal effects on the product that often occur in conventional heating applications, caused by the need to heat the product by the transfer of thermal energy from a heating medium to a low temperature product, where excessive treatment times are necessary for sufficient heat penetration from the surface of a solid product to its core. Potential applications for ohmic heating include its use in blanching, evaporation, dehydration, fermentation, and extraction. At present, the primary type of application is a heat treatment for microbial control, for example, for the pasteurization of milk, and also for processing of sauces, fruits, and tomatoes (Chapter 8). The principal mechanisms of microbial inactivation in ohmic heating are thermal in nature. Recent literature, however, indicates that a mild electroporation mechanism may occur during ohmic heating (similar to the effects utilized in PEF processing (Lebovka et al. 2005; Kulshrestha and Sastry 2006). The principal reason for the additional microbial inactivation effect to heating of ohmic treatment may be its low frequency (50–60 Hz), which allows cell walls to build up charges and form pores. This is in contrast to high-frequency methods such as microwave or radio frequency heating,

10

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

where the electric field is essentially reversed before sufficient charge buildup occurs at the cell walls. Nevertheless, temperature is the principal critical process factor in ohmic heating. As in conventional thermal processes, the key issue is identifying the slowest heating zone. Fundamentally, there is only one critical factor: the temperature–time history of the coldest point. Since the primary critical process factor is the thermal history and location of the cold spot, the effects on microbial inactivation are the same as for thermal processes. Locating the slowest heating zones during ohmic heating, however, cannot be extrapolated from current knowledge of conventional heating, and requires special consideration. Several factors significantly affect the temperature within an ohmic process. The critical parameters in continuous flow ohmic heating systems include electrical conductivities of the respective phases of the food, temperature dependence of the electrical conductivity, design of the heating device (e.g., location and orientation of the electrodes), extent of interstitial fluid motion, residence time distribution, thermal properties of the food, and electric field strength (Chapter 8). Limitations of Ohmic Heating The main limitation of ohmic heating is the heterogeneous nature (in composition) of the food products and their corresponding electrical conductivities that leads to differences in the conversion of the electrical current into thermal energy. As in microwave heating, in ohmic heating, thermal runaway can also occur, because electrical conductivity, which is the property that influences electrical energy dissipation, usually increases with increasing temperature. Therefore, especially in stationary (i.e., not moving in a stream) solid products, there may be areas that are very hot (usually areas close to the electrodes), which in some instances may even be burned, while in other areas (with initially lower electrical conductivities, or farther away from the electrodes) almost no heating occurs. Uniform heating with ohmic processing is theoretically possible, but at the same time challenging due to the various factors impacting on the slowest heating zone and the time–temperature history

throughout the product. Multiphysics modeling (including the temperature-dependent properties of the foods: mainly the electrical conductivity) can greatly assist the evaluation and optimization of ohmic heating systems to achieve heating uniformity (Chapter 8). 1.3.1.4. PEF PEF processing is an innovative nonthermal processing technology mainly for liquid and pumpable foods (including emulsions, suspensions, and semisolids such as sausage meat), predominantly used for the inactivation of microorganisms at ambient or mild temperatures, thereby preserving the fresh flavor, color, functional properties, and integrity of heat-sensitive compounds (Chapters 9–11). PEF can also be used to enhance extraction yield of juices and bioactives from plant sources. PEF is one of the most appealing nonthermal technologies for preservation of liquid foods due to reduced heating effects compared with traditional pasteurization methods (Barbosa-Cánovas et al. 1999). In PEF processing, a liquid or other pumpable material is passed through an electrode arrangement where the PEF is applied. For microbial inactivation, foods are processed by means of brief pulses of a strong electric field with field strengths of around 15–40 kV/cm. For extraction of plant materials and pretreatment of meat for processing, only about 0.7 to 3 kV/cm is required (Toepfl et al. 2006). The utilization of PEF leads to the formation of pores (the so-called electroporation [temporary or permanent]), in the membranes of microbial or plant cells, which disturbs and damages the membrane’s functionality, leading to inactivation of the cells and the partial release of the cell contents to make extraction or other processing more efficient. Membrane disruption occurs when the induced membrane potential exceeds a critical value of 1 V in many cellular systems, which, for example, corresponds to an external electric field of about 10 kV/ cm for Escherichia coli (Castro et al. 1993). The most relevant factor affecting microbial inactivation and extraction enhancement by PEF is, therefore, the electric field intensity. The combination of electric field intensity, total treatment time during PEF and pulse shapes, and the associated temperature

Chapter 1

Introduction to Innovative Food Processing Technologies

increase determine the extent of membrane disruption in bacterial and plant cells (Hamilton and Sale 1967). Other factors affecting the performance of the PEF process include the microbial entity to be inactivated (type, concentration, and growth stage of microorganism) and the treatment media (pH, antimicrobials, and ionic compounds, electric conductivity, and medium ionic strength). PEF produces products with slightly different properties from conventional pasteurization treatments. Most enzymes are not affected by PEF. The fact that the maximum temperature reached is lower than in thermal pasteurization means that some of the flavors associated with the raw material are not destroyed. Spores, with their tough protective coats, and dehydrated cells are mostly able to survive PEF processing. The survival of spores and enzymes means that products have to be refrigerated after passing through PEF processing in order to slow the action of the enzymes and keep pathogens from growing; PEF alone is generally not capable of producing ambient shelf-stable products. However, acidic well-packaged products may have a useful ambient shelf life. As indicated before, another potential application of PEF, which is gaining increasing interest, is the utilization of the technology for enhanced extraction of plant cell material. Because PEF induces electroporation in cell walls at relatively low energy inputs, allowing the cell contents to leak out, it holds promise as an efficient way of getting useful components out of cells and cell membranes (Corrales et al. 2008; Lopez et al. 2009a, 2009b; Loginova et al. 2010; Puertolas et al. 2010). To date, however, PEF has been mainly researched to preserve the quality of foods, such as to improve the shelf life of orange juice, apple juice, milk, and liquid eggs, as well as the fermentation properties of brewer ’s yeast. Martín-Belloso and SolivaFortuny (2010) have summarized the work of several researchers on food-borne pathogenic microorganisms in different food products. Limitations of PEF Processing Issues that may arise with PEF include electric arcing, dielectric breakdown of the treated food, and a pronounced

11

temperature increase (caused by ohmic heating). Several factors play a role here, including the material’s electrical conductivity, the frequency of the pulses, their duration (width), adequacy of deaeration, back pressure, and the flow rate of the liquid (laminar or turbulent flow regime; residence time in the treatment chamber). Because the pulse duration is only in the range of microseconds and, therefore, the overall treatment time is short, temperature increases during treatment are often assumed to be minimal and temperature effects neglected in inactivation studies. In processing liquids with PEF, a nonuniformity of the treatment can be a result of the interaction between the flow, heat transfer, electric field phenomena, and effects on microbial or plant cells. Predictions of the increase in temperature caused by the electric field are similar to ohmic heating and less complicated compared with the dissipation of electromagnetic energy in microwave processing. Moreover, the property influencing this dissipation effect, that is, the electrical conductivity, is easier to measure, and usually shows a less complex behavior with temperature than the two dielectric properties in microwave processing, that is, the dielectric constant and the loss factor (Chapter 2). However, the purpose of the pulsed (potentially alternating) electric field is, unlike in microwave processing, not an increase in temperature. The temperature increase should be minimized in most PEF applications. The main aim is a nonthermal inactivation of vegetative microorganisms for cold pasteurization or a nonmechanical means of opening cells for enhanced extraction. In particular, for the purpose of cold pasteurization, a great degree of electric field uniformity is needed to ensure a similar treatment of the entire liquid product. Ideally, the same number of electric pulses and electric field strength is applied to all microorganisms present in the liquid. Typically, pasteurization requires inactivation of up to 99.999%, that is, 5 log of the target organism. If only a small fraction of microorganisms bypass proper treatment through regions of low electric field strength, it is not possible to reach the required extent of inactivation.

12

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Achieving this uniformity, however, is very challenging; the electric field distribution is strongly dependent on the configuration of the treatment chamber (and to a lesser extent on the electrical conductivity and other thermophysical properties of the processed media). PEF chamber designs such as co-field, coaxial, or colinear electrode arrangements (Chapters 9–11) exhibit pronounced nonuniformities in flow, temperature, and electric field distributions. Uniform fields can be achieved in parallel plate configurations, which are mainly applied for batch processing. If the field is not uniform, the induced temperature increase is also uneven across the volume of the treatment chamber. Often, several treatment cells are arranged to process in series, which reduces the effects of imperfections in single treatment cells. Thus, in processes for inactivation of specific microorganisms that show synergistic effects of temperature and electric field on inactivation, temperature nonuniformities will lower the performance of the process. Nonuniformities can be minimized, but to some extent will always occur. To enable a comparable treatment history of the entire product, the flow pattern is very important. Laminar flow conditions, which can be found in low-throughput laboratory-scale systems, are to be avoided. In laminar flow, each microorganism follows a more or less straight path through the treatment chamber; therefore, pronounced differences in exposure to varying electric field strengths and temperatures will occur. Modifying the treatment chamber with grids (Chapter 10) or increasing the flow rate to give turbulent flow (Buckow et al. 2010a) can improve the uniformity of exposure of the product to the important treatment variables (e.g., temperature and electric field strength) and furthermore improve temperature uniformity due to increased (turbulent) thermal conduction and convective flows. For characterizing process performance, information on the field distributions is essential. However, such local information inside the chambers is difficult and near to impossible to obtain experimentally. For further development of the PEF technology,

numerical simulations can be applied to improve the fundamental understanding of the physical phenomena in the process and to optimize it with respect to the chamber design and operating conditions (Gerlach et al. 2008; Chapters 9–11). 1.3.1.5. Ultrasound Processing This technology is based on pressure waves at frequencies exceeding 20 kHz, that is, more than 20,000 vibrations per second. It is considered as another innovative process that has been investigated for many different purposes over the last decades. While in the earlier work mainly the lower frequencies of around 20 kHz were studied, research and applications currently include frequencies of several hundred kHz, to several MHz (Chapter 12). Ultrasound systems consist of a generator for turning electrical energy into high-frequency alternating current, a transducer for converting the alternating current into mechanical vibrations, and a delivery probe for conveying the sonic vibrations into a medium to couple sonic vibrations to the treated material. The transducers may take the shape of a rod, plate, bar, or sphere, and are usually manufactured from titanium, aluminum, or steel. The ultrasonic transducer can be mounted outside on the wall of a vessel or flow cell and be in indirect contact with foods, or it can be inserted into a treatment chamber or flow cell of specified geometry to transmit energy directly into a food system with better energy efficiency (Feng and Yang 2005). There are also transducers that are designed for effective transmission into air (Chapter 13). Ultrasound has attracted considerable interest in the food industry due to its useful effects in food structure modification (e.g., emulsification, extraction, crystallization, and viscosity alteration), food preservation, and enzyme modulation (Patist and Bates 2008). As one of the innovative and advanced food processing technologies, it can be applied to develop gentle but targeted processes to improve the quality and safety of processed foods and, thus, offers the potential for improving existing processes as well as for developing new process options.

Chapter 1

Introduction to Innovative Food Processing Technologies

Ultrasound alone has some effects on the inactivation of vegetative organisms in liquid food products. The bactericidal effect of ultrasound is generally attributed to intracellular cavitation (Hughes and Nyborg 1962). It is proposed that micro-mechanical shocks and jet streaming are created by microscopic cavitation bubbles induced by the fluctuating pressures under the ultrasonication process (Chapter 12). These shocks and microjets disrupt cellular structural and functional components up to the point of cell lysis. Positive effects have been observed when ultrasound is used in combination with temperature (thermo-sonication) or pressure (mano-sonication) or both (mano-thermo-sonication) in the inactivation of pathogenic bacteria, spoilage microorganisms, and enzymes (Cameron et al. 2009; Demirdoven and Baysal 2009; Lee et al. 2009). The use of temperature and ultrasound together has been successful in reducing the enzymatic activity in some target products such as juices, providing better stability during storage (Terefe et al. 2009). Sonicated milk is the most explored product; it shows positive results in pasteurization standards, better homogenization and color, as well as new physical properties for the development of dairy products (Chouliara et al. 2010). Most developments of ultrasound for food applications are nonmicrobial in nature, that is, their main aim is not inactivation of microorganisms (Hoover 1997). High frequencies in the range of 0.1 to 20 MHz, pulsed operation, and low power levels (100 mW) are used for nondestructive testing (Gunasekaran and Ay 1994). These industrial applications include texture, viscosity, and concentration measurements of many solid and fluid foods; composition determination of eggs, meats, fruits and vegetables, dairy, and other products; thickness, flow level, and temperature measurements for monitoring and control of several processes; and nondestructive inspection of egg shells and food packages. Apart from testing applications, process improvements have been observed in applications such as cleaning surfaces (Tolvanen et al. 2009), enhance-

13

ment of dewatering, drying and filtration, inactivation of microorganisms and enzymes, disruption of cells, degassing of liquids, emulsification, accelerating heat transfer and extraction processes (Patist and Bates 2008; Vilkhu et al. 2008), enhancement of processes dependent on diffusion (e.g., enzyme activity, targeted infusion of small compounds into porous food matrices), and also targeted movement of two-phase systems, such as oil droplets or particles dispersed in a continuous aqueous phase (Doblhoffdier et al. 1994; Hawkes et al. 1997; Groschl 1998). It is evident that ultrasound technology has a wide range of actual and future applications in the food industry. More recently, research activities related to the sonochemistry in certain foods products have gained interest, involving the reactions that ultrasound generates in food during processing. Jambrak et al. (2009) show that these chemical reactions can be used to generate new compounds in food for specific purposes such as the modification of proteins. Hydroxylation of phenolic compounds to enhance their antioxidant properties has also been studied by Ashokkumar et al. (2008). Another interesting application is the use of airborne ultrasound for enhanced drying of food products. Difficulties in the propagation of ultrasound waves in air and the impedance mismatch at the transducer/air interface have led to the development of especially adapted transducers that have been applied, for example, to drying of carrots, lemon peel, and other food products (Garcia-Perez et al. 2009; Chapter 13). Limitations of Ultrasound Processing Although potential applications of ultrasound processing are many and diverse, the uptake by industry to date is not widespread. Reasons for this include a lack of knowledge of ultrasound intensity distribution in tank systems and, particularly, in flow-through systems, where the forced convection disturbs the ultrasound field, as well as the effect of the pressure waves on the food product. Depending on the equipment, with the generators, the transducers, treatment cells, the frequency and power of the ultrasound

14

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

waves, and the product properties, the effect of ultrasound can differ significantly. Ultrasound processing comprises another Multiphysics phenomenon: the acoustic field. Several factors must be considered, for example, ultrasound frequency, intensity, the associated speed of sound and sound absorption, and impact on the acoustic field. Although the speed of sound in a homogeneous medium is independent of the sound wave frequency and intensity, varying composition of the treated product strongly impacts the speed. The sound absorption is dependent on the composition as well as the frequency and intensity of the ultrasound waves. In addition to this, occurring cavitation (Chapter 12) significantly influences the speed of sound, the sound absorption and, therefore, the acoustic field distribution. The speed of sound in a cavitating medium can, for example, decrease from a value of 1,500 m/s to values as low as 20 m/s. As discussed in Chapter 12, the ultrasound waves in a cavitating medium can be completely absorbed by the cavitation bubbles in the close vicinity of the ultrasound transducer and, therefore, large parts of the sonoreactors may not undergo ultrasound treatment. Although this pronounced absorption leads to the conversion of the sound energy into motion and the formation of a turbulent jet, which could in turn result in the treated liquid being well-mixed and, therefore, undergoing a similar treatment over time, the presence of solid products, which due to size and different densities cannot completely follow the flow, may induce further treatment nonuniformity. During ultrasound processing, standing waves (the so-called bands) can occur. This formation of bands can be an intended desirable effect, for example, for separating multiphase products such as emulsions. In other cases, where the sound waves are meant to induce other effects, such as cell disruption, sono- or biochemical reaction, the standing waves can unintentionally impair the process performance. Hence, generic Multiphysics models, including acoustics, heat and fluid flow and, potentially, coupling to the kinetics of food transformation, enhanced diffusion, microbial interaction, and enzyme modulation need to be developed. Such models can assist

in process design, scale-up and optimization and subsequent uptake of the technology by the food industry.

1.3.1.6. UV Processing UV light for food processing has been investigated for many years but is still considered as an innovative technology in food processing. In this technology, UV-C light (wavelength of 254 nm) is predominantly being used as a disinfection method to inhibit or inactivate foodborne microorganisms, mainly in liquid food products (Chapters 14 and 15). Fresh produce can be processed using UV light, which has a germicidal effect on many types of microorganisms (bacteria, viruses, protozoa, molds, and yeasts). However, the effect of UV light on microorganisms in liquids depends on variables such as density of the liquid, types of microorganisms, UV-C absorptivity of the liquid, and the solids (suspended or soluble) in the liquid. Although the use of UV light is well established for air and water treatment and surface decontamination, its use for treating liquid foods is still limited. Recently, interest in using UV has increased as a viable alternative to thermal pasteurization for a range of liquid foods and ingredients (fresh juices, fruit purees, soft drinks, raw milk, liquid eggs, liquid sugars and sweeteners, etc.) (Koutchma 2009). Pumpable fruit and vegetable products are generally very suitable for processing by UV light to reduce the microbial load (Guerrero-Beltran and BarbosaCánovas 2004) as long as sufficient fluid mixing allows the entire product to be exposed to a certain required dose of UV radiation. The germicidal properties of UV irradiation are mainly due to DNA damage induced through absorption of UV light by DNA molecules. This mechanism of inactivation results in a sigmoidal curve of microbial population reduction (Bolton 1999). UV treatment can be used for primary disinfection or as a backup for other purification methods such as carbon filtration, reverse osmosis, or pasteurization. As UV has no residual effect, the best position for a treatment system is immediately prior to the point of use. This ensures that any incoming microbiological

Chapter 1

Introduction to Innovative Food Processing Technologies

contaminants are destroyed and that there is little chance of post-treatment contamination. In addition to UV-C light, UV light with wavelengths other than 254 nm can also be used as a radiation source to inactivate microorganisms in foods (liquids or solids). In general, wavelengths ranging from 100 (UV-V, vacuum UV light) to 400 nm (UV-A) are suitable for UV light processing (Bintsis et al. 2000; Sastry et al. 2000). UV disinfection has many advantages over alternative methods. Unlike chemical treatment, UV does not introduce toxins or residues into the process and mostly does not alter the chemical composition, taste, odor, or pH of the water or liquid being disinfected. As a physical method, UV irradiation has a positive consumer image and is of interest to the food industry as a low-cost nonthermal method of preservation. Recent advances in the science and engineering of UV light irradiation have demonstrated that this technology holds considerable promise as an alternative method to traditional thermal pasteurization for liquid foods and ingredients, fresh juices, soft drinks, and beverages. Limitations of UV Processing Compared with water, liquid foods have a range of optical and physical properties, diverse chemical compositions, and solid-phase characteristics (particle size and size distribution, shape and volume fraction), influencing UV light transmittance (UVT), dose delivery, momentum transfer (laminar or turbulent flow), and consequently microbial inactivation (Koutchma 2009; Chapter 14). As there is no practical method for evaluating the spatially resolved performance experimentally and predictions of the process performance are not straightforward for liquid foods (compared with water), Multiphysics modeling is essential for evaluating particles and fluid velocities in the UV reactor, particle mixing, particle location, residence times, UV fluence rate (irradiance) distribution and resulting changes in bacterial count. As mentioned, UV is mainly useful for surface decontamination (e.g., on fresh produce) and for disinfection of liquids transparent to the UV light to a certain extent. Although consisting of electromagnetic waves, the penetration of UV light into opaque

15

substances is limited at these wavelengths. Therefore, microorganisms on the surface of products can be protected by the so-called shadowing effect, caused, for example, by overlapping parts of the products. Treating opaque liquids is impossible under laminar flow conditions as the product flowing through the center of a UV transparent glass tube will not “see” the UV light. Providing a highly turbulent flow, however, can allow sufficient treatment uniformity, as all particles will likely be close to the glass walls at least for a certain period of time. Residence time in such a flow reactor must be sufficiently long to ensure similar treatment histories of the entire liquid product. Multiphysics modeling can assist in the UV chamber design and optimizing process conditions according to the absorptivity and other properties of the fluid, while assuring a similar treatment history of all portions of the liquid or dispersion. In all technologies and their associated specific issues regarding nonuniformity discussed in the previous sections, Multiphysics modeling can assist in providing insights into the internal distribution of processes in treatment chambers and products. It can be utilized to improve the systems design, performance, optimization, and scale-up to commercial applications by reducing inhomogeneities and for the process to become acceptable and viable.

1.4. Modeling Challenges Previous sections have described a number of limitations encountered in innovative food processing technologies and how Multiphysics modeling can assist in overcoming them. However, there are practical complexities in modeling and validating models for these technologies that will be covered in this section.

1.4.1. Modeling Complexity in Innovative Processing As mentioned earlier in this chapter, modeling innovative processing involves additional physics phenomena to conventional CFD. This implies that the fundamental conservation equations from thermofluiddynamics need to be coupled with the PDEs to

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

describe the respective field phenomena (i.e., electromagnetic, electric, and acoustic fields), thereby providing considerably increased complexity. Developing a Multiphysics model requires the same steps as developing a CFD model (Sun 2007), for example, the geometry definition, where all objects in the model scenario are constructed and assembled in the modeling software package or imported from a computer-aided design (CAD) drawing of the system (treatment chamber, peripheral devices, piping, food, packages, etc.). In particular, the materials forming the complete computational domain (i.e., the processing system), commonly referred to as subdomains, may include solids, liquids, and gases. The next step is discretization, that is, approximating the computational domain by finite cells. Next, material specific thermophysical properties need to be allocated to each subdomain as functions of the process variables (e.g., temperature and pressure; Chapter 2) and initial conditions and boundary conditions need to be defined. Furthermore, the model needs to specify time dependency, whether they are transient or stationary. The main differences between conventional CFD and Multiphysics modeling are as follows: • The geometry discretization step often needs more details than in conventional CFD. For example, in electromagnetics modeling with the finite difference method 15 cubic cells per wavelength have been recommended (QWED Sp.z o.o. 2003). In microwave processing at 2.45 GHz, with a wavelength in vacuum of around 12 cm, the maximum mesh cell size should therefore be below 1 mm (edge length). Furthermore in modeling an acoustic field with the finite element method, the resolved three-dimensional (3D) mesh should have at least 12 degrees of freedom per wavelength for each possible direction of the wave (i.e., the degrees of freedom of the complete mesh in three dimensions should be 1,728 times the model volume in wavelengths). Higher frequencies, with shorter wavelengths, therefore, limit the feasible volume of the model scenario (COMSOL Multiphysics 2007). • As discussed in Chapter 2, there is a lack of thermophysical property data of foods, in particular

expressed as a function of the process variables needed for model accuracy. As will be shown, more properties are needed to model innovative processes than conventional processes. • Each boundary condition will have to be defined for each Multiphysics phenomenon as a requirement to solve PDEs at the interface of each subdomain. • In most cases, when relatively high frequencies are involved (PEF, ultrasonic processing, ohmic heating, microwave, UV), time resolution of the respective waves is not feasible due to the short time scales (i.e., a higher frequency gives smaller time scales) across the domain. Therefore, an integrated value from the steady-state solution of the wave equation is often used as a source term in the conservation equations for momentum and energy. Multiphysics models are highly nonlinear mathematical problems. With each additional PDE, the degree of nonlinearity increases. Depending on the mesh, the thermophysical properties (as functions of the process variables), and the number of physics phenomena being coupled, the difficulty for convergence of the model may increase. When that is the case, models may not be as robust as those developed utilizing classical CFD PDEs.

1.4.2. Validation of Multiphysics Models Validation is an essential step to complete the modeling process. Models used for prediction of process variables and their distributions may converge, suggesting solutions that might be plausible, but in fact are not accurate (Nicolaï et al. 2001). Therefore, particularly in the case of highly nonlinear Multiphysics problems, the numerical solutions must always be validated before using them for further studies, such as equipment and process redesign, optimization, or scale-up. The validation process involves the comparison of predicted data (i.e., temperature, velocities, inactivation extent, and chemical or physical change) with measured data. This can be done using two approaches: (1) direct validation of process variables; or (2) indirect

Chapter 1

Introduction to Innovative Food Processing Technologies

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Table 1.1. Tools for the validation of Multiphysics models. Technology High-pressure processing

Process variable or outcome to validate

Direct (D)/ Indirect (I)

Temperature

D I

Microwave, ohmic heating

Pulsed electric field (PEF)

Ultrasound

Ultraviolet (UV)

Method

Chapter or reference

Thermocouples, wireless temperature logger Enzymatic or other temperature time integrator (TTI), liquid crystals

Chapter 5

Fluid velocity

D

High-pressure PIV

Indicator

D/I

Temperature

D I

Fluid velocity Indicator

D D/I

Temperature

D

Indicator

I D/I

Enzymatic, microbial, colorimetric determination methods among others Thermocouples, fiber-optic probes MRI, infrared thermography, microwave radiometry, time temperature integrators, liquid crystals PIV (particle tagging), LDA Enzymatic, microbial, colorimetric determination methods among others Thermocouples (not in the area of high-electric field strength), fiber-optic probes Enzymatic TTI Enzymatic, microbial, colorimetric determination methods among others Thermocouples (type K) PIV, LDA Hydro- and microphones Qualitative visual (cavitation fields, band formation of particles), chemical markers Enzymatic, microbial, colorimetric, determination methods among others PIV, LDA Microbial (also referred to as biodosimetry), colorimetric, determination methods among others

Temperature Fluid velocity Acoustic intensity

D D D I

Indicator

D/I

Fluid velocity Indicator

D D/I

validation by means of a (bio)chemical or microbial indicator. Table 1.1 classifies the indirect and direct validation tools to determine the process variables or outcomes for each technology. Direct measurements include: • temperature measurements by utilizing resistance thermometers, thermocouples, or fiber-optic sensors • flow measurement by means of Laser Doppler Anemometry (LDA) or Particle Imaging Velocimetry (PIV)

(Pehl et al. 2000; Grauwet et al. 2010a, 2010b) (Pehl and Delgado 1999) (Denys et al. 2000) Chapters 7 and 8 Chapters 6–8

Chapter 8 Chapter 6 Chapter 10; (Buckow et al. 2010a) — Chapter 11 — Chapter 12 Chapter 13 (Klima et al. 2007; Sutkar and Gogate 2010) — (Hofman et al. 2007) Chapters 14, 15

• sound intensity by means of hydro- or microphones • particle (size) change by measuring particle size distribution, for example, with a Focused Beam Reflectance Measurement (FBRM) device. The direct measurement of other process variables, such as electric and electromagnetic field distribution, including, for example, microwave and UV light, is often not feasible in the constrained space of the respective processing equipment. However, the process outcomes, such as microbial inactivation,

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

nutrient degradation, structural changes, and material separation, are the true process indicators. Unless they are spatially resolved, these only give overall outcomes and are of no or limited value to validate local variations in a treatment cell or zone. An indirect measurement involves the evaluation of enzymatic, microbial, color, chemical, or any other biological or physical change that represents a change in temperature or any other process variable when typical measurement devices cannot be utilized. For example, temperature distribution changes after microwave sterilization can be measured using whey protein and measuring Maillard reaction components at different locations (Chapter 6). Another example is the use of magnetic resonance imaging (MRI) to measure the change in proton resonance frequency, referred to as chemical shift, to establish 3D temperature distribution changes (Chapter 7). Methods for validation will be covered in more detail in the following chapters. Once the data of the measured process variables or process outcomes are gathered, there are different ways of comparing simulated with measured data. A common method to validate transient simulations is the comparison of profiles at specific points of the modeling domain, which are measurable. Another approach is to compare model and measured data at several specific time and location coordinates (in a 2D or 3D grid) throughout the process period in a parity plot, for example, represent measured temperature versus simulated ones at identical locations and selected times in a plot (e.g., Knoerzer et al. 2007, 2008). Comparison of 2D or 3D distributions requires working in matrices. Distributions of inactivation or chemical or physical changes cannot be validated through direct measurements at selected points. The study of a certain volume in packages containing an initial amount of substance can partially resolve this problem. By this method overall averages for the whole vessel or vessel areas where packages are located are calculated from the predictions in the model. For example, Chapter 5 shows a case where a relative activity ratio (Eq. 5.16) is utilized to determine enzyme inactivation distributions after HPP.

1.5. Concluding Remarks After examining the literature, we note that only some of the Multiphysics models developed to describe innovative processing technologies have been thoroughly validated. The overarching aim of the models is to represent a process outcome that will provide certain design or optimization aids. However, in practice, not all of the validation variables or outcomes in Table 1.1 have been measured to match a certain model. In the case of HPP, PEF, ultrasound, and ohmic heating, more outcome-related models need adequate validation to establish, for example, accurate predictions of microbial, enzymatic or chemical reaction distributions, or other outcomerelated parameters. On the other hand, more complete outcome-related validated models have been developed for microwave processing. For example, the Multiphysics models presented in Chapter 6 have assisted in the filing of microwave sterilization processing in the U.S. Food and Drug Administration. As such, Multiphysics models will mainly be useful when reaching a stage of predicting process outcomes that leverage technologies to industrial levels. In order to achieve this, a more direct collaboration between processing equipment manufacturers, interested industry partners, and researchers is needed for successful design and implementation of innovative food processing technologies.

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Chapter 2 The Need for Thermophysical Properties in Simulating Emerging Food Processing Technologies Pablo Juliano, Francisco Javier Trujillo, Gustavo V. Barbosa-Cánovas, and Kai Knoerzer

2.1. Introduction Multiphysics modeling of any engineering problem comprises simultaneously solving partial differential equations (PDEs) of mathematical expressions representing different types of physical phenomena, that is, in a coupled form (Chen 2006). To solve the corresponding PDE, it is essential that the underlying equations, thermophysical properties, and boundary conditions are individually set up for each physical phenomenon. As such, the correct expression of the thermophysical properties as functions of the process variables affecting them is important for accurate model prediction (Knoerzer et al. 2007, 2008, 2010a; Juliano et al. 2009). In turn, accurate model prediction is challenged by the lack and the uncertainty (variability) of thermophysical property values for many materials, including foods and expressions of the specific process variables, such as temperature, pressure, and concentration (Knoerzer et al. 2010a). This chapter is an overview of the thermophysical properties required for Multiphysics modeling, and the dependence of these properties on technology-specific variables for accurate prediction of physical phenomena manifested during novel processing of foods.

Working with thermophysical properties in Multiphysics modeling of innovative food processing can result in a number of challenges, mainly: (1) understanding the nature of the food materials, (2) accurately determining the properties of the food materials under specific process conditions, and (3) understanding the functional dependence of a food material on the process variables (e.g., temperature, pressure, electric field strength, and (ultra)sound intensity). Food materials not only have varied composition and structure, but can also change due to processing conditions and during storage (e.g., biochemical reactions in fresh and living products, chemical changes provoked by process variables, changes in moisture content, micro- and macrostructural changes, and rheological changes). Thus, the overarching challenge is to identify expressions for each food material, including the composition variables (e.g., moisture, ash content, lipids, carbohydrates, and protein), structural variables (e.g., porosity and tortuosity), and processing variables (e.g., temperature and pressure). This chapter will show specifically that thermophysical properties do not vary significantly with process variables other than temperature and pressure. Seasonal variations in living food materials such as fruits, vegetables,

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

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and meats can hardly be predicted. Therefore, the variability of food materials may be unpredictable and difficult to include in a single model, which might affect the end result of a Multiphysics model or its predictability for different materials. However, as long as the composition of the food is included in the expression of thermophysical properties, and models are properly validated, these variations may be negligible.

2.2. Definitions and Methods to Determine Thermophysical Properties This section will define the thermophysical properties required to establish Multiphysics models for innovative food processes. In particular, suitable properties will be selected for processes requiring the use of temperature, elevated pressure, electric fields, ultrasound irradiation, microwave processing and ultraviolet light, and the eventual inclusion of material and process variables. When food components or other materials are included in the model, empirical expressions that have been adjusted to measured data may include other material variables (e.g., composition and porosity) as they intervene in

the mass, momentum, and energy balance processes. Fundamental governing equations included in CFD models comprise the following properties: density (mass, momentum, and energy conservation equation), specific heat capacity and thermal conductivity (energy conservation equation), and viscosity (momentum conservation equation). As shown in Table 2.1, equations for other (multi)physics phenomena to be coupled with CFD models require further thermophysical properties such as the compression heating properties (relevant in high-pressure processing), dielectric properties (radio frequency and microwave processing), electrical conductivity (ohmic heating and pulsed electric fields), sound absorption coefficient and velocity of sound (ultrasound processing), and absorptivity (ultraviolet processing).

2.2.1. Density, Porosity, and Related Properties The density, ρ, of a material is defined as mass per unit volume (SI unit of density is kg/m3). Indeed, there are different forms of density that can be used, such as true, material, particle, apparent, and bulk

Table 2.1. Summary of essential properties for emerging food processing technologies. Technology

Property

Equation

All food processing technologies

Density Specific heat capacity Thermal conductivity Viscosity Thermal expansion coefficient Compressibility Compression heating coefficient Dielectric constant Loss factor Electrical conductivity Electrical conductivity

Mass, momentum, Energy conservation equation Momentum conservation equation

High-pressure processing (HPP) and High-pressure thermal sterilization (HPTS) Microwave and radio frequency Ohmic heating and pulsed electric fields Ultrasound

Ultraviolet

Sound absorption coefficient Velocity of sound Absorptivity Absorption coefficient (or spectral absorption coefficient) Optical density

Compression heating (Eq. 2.10; Chapters 4 and 5) Maxwell’s equations including the constitutive relations (Chapters 6 and 7) and energy conservation including source term Charge conservation and energy conservation including source term (Chapters 8, 9, 11) Wave equation, Helmholtz equation, momentum, and energy conservation including source terms (Chapter 12) Radiation intensity (Chapter 14)

Chapter 2

Thermophysical Properties in Simulating Food Processing Technologies

density, depending on its application in process calculations or product characterization. However, the apparent density is more commonly used as input into the model equations. The volume measurement method is what determines the difference between them. True and material densities are calculated by excluding volumes occupied by internal and external pores within the food, while particle, apparent, and bulk densities are determined from less accurate measurement methods that include pore volume (Barbosa-Cánovas et al. 2005). In most engineering designs, solids and liquids are assumed to be incompressible—in other words, density changes moderately with changes in temperature and pressure. In food engineering, however, the density of solid and liquid foods changes with temperature and pressure, and composition changes as well. In the case of liquid foods, no generic equations exist to predict the density. In the literature most of the density data are correlated empirically as a function of temperature, pressure, water, solids, and fat content. Different types of nonlinear correlation such as exponential, quadratic, and cubic are used to relate density and moisture content (Lozano 2007). Another way of accounting for a material’s structure is in its porosity, which indicates the volume fraction of void space or air space inside the material. Volume determination is relative to the amount of internal (or closed) or external (or open) pores present in the food structure. Therefore, like density, different forms of porosity are also used in food processing studies, namely open pore, closed pore, apparent, bulk, and total porosities (Rahman 1995; Barbosa-Cánovas et al. 2005, 2007). Porosity in foods is mainly predicted from empirical correlations, which are valid for individual foods under given processing conditions. Fundamental models exist that are based on the conservation of mass and volume, as well as a number of other terms that account for interaction of components and formation or collapse of air or the void phase during processing (Lozano 2007). Shrinkage or the reduction in volume or geometric dimensions during processing is also important in solid materials. During post-processing the volume of the material is larger than its initial

25

volume, and is termed “expansion.” Two types of shrinkage, isotropic and anisotropic, are usually observed in the case of food materials. Isotropic shrinkage is described as the uniform shrinkage of materials under all geometric dimensions, whereas anisotropic (or nonuniform) shrinkage develops in different geometric dimensions. The former is common in fruits and vegetables, while the latter is known in animal tissue, as in meat and fish (Sahin and Sumnu 2006). Most of the density, shrinkage, and porosity prediction models for liquid and solid foods are empirical in nature. Recent models have been developed to predict porosity during air drying based on drying temperature, moisture content, initial porosity, and product type (Lozano 2007). Volume change and porosity are important parameters in estimating diffusion coefficients for shrinking systems. Furthermore, porosity and tortuosity are used to calculate effective diffusion during the mass transfer process (Knoerzer et al. 2004). A material’s volume can be measured by buoyant force, liquid, gas or solid displacement, gas adsorption, or by estimating the material’s geometric dimensions. The buoyant force method for apparent or particle volume determination utilizes sample weight differences in air and water, while the liquid displacement method measures the increase in liquid volume (material is immersed in a non-wetting fluid such as mercury or toluene). A gas pycnometer is a gas displacement device that uses air pressure differences in a sample cell connected to a manometer to determine material volume. Apparent or particle density can be determined by coating particles in order to include internal pores in the volume measured. For solid displacement, sand or glass beads can be used instead. Porosity can be measured by direct and microscopic methods, or can be estimated from density data.

2.2.2. Viscosity The viscosity ( μ ) is a physical property of fluid materials (gases, liquids, or semisolid foods) and represents the internal friction of a fluid or its resistance to flow (the SI unit of dynamic viscosity is Pa·s) (Bourne 2002). Similar to the friction that occurs between moving solids, viscosity transforms the

26

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

kinetic energy of motion of the fluid into heat. For instance, highly viscous food materials like honey offer higher resistance to flow than lower viscosity liquids such as water. Thus, more power is consumed by the pump during pumping of honey to achieve the same flow rate of pumping water. In addition, honey exhibits a higher temperature increase under flowing conditions than water due to higher viscous heat dissipation (caused by friction). Modeling food-processing operations often involves solving the momentum transport equation, which accounts for the balance between the forces (e.g., pressure gradients and stress tensor) applied to a differential volumetric element of a fluid, and the resulting acceleration of the fluid. Viscosity is usually incorporated into the rheological model as representing the stress tensor, which accounts for the shear stress acting upon the fluid. Shear stress (τ ) is a force per unit of area acting parallel or tangential to the fluid element:

τ=

F A

(2.1)

Simple liquids and gases such as water and air are called Newtonian fluids, given that they exhibit an ideal correlation between the shear stress (force per unit area acting tangential to the fluid) and the resulting velocity gradient perpendicular to the direction of shear:

τ=μ

dv dy

(2.2)

For a Newtonian fluid, viscosity represents a linear correlation between shear stress and velocity gradient. However, foods are structurally complex materials that frequently exhibit non-Newtonian behavior (Tabilo-Munizaga and Barbosa-Cánovas, 2005; Welti-Chanes et al. 2005). Non-Newtonian fluids are usually divided into the following general classes: (1) those with properties independent of shear rate and (2) those with properties dependent on shear rate (Steffe 1992). Some common rheological models of liquid foods are Power law model

τ = K (γ )

n

(2.3)

Bingham model

τ = τ o + η pγ

(2.4)

Herschel-Bulkley model

τ = τ o + K (γ )

n

(2.5)

where γ = dv dy is the shear rate and η p is apparent viscosity. For a pure substance, the viscosity is highly dependent on temperature and to a lesser extent on pressure. For complex materials like food, the viscosity also depends on composition. Viscosity is usually determined by measuring the resistance to flow in a capillary tube, or the torque produced by the movement of an element through the fluid. There are two main categories of viscometers applicable to foodstuffs: capillary, and falling ball, as well as commercial rheometers able to provide viscosity according to rheological models. For Newtonian liquid foods it is sufficient to measure μ at a single value of γ . In order to describe a non-Newtonian food, additional properties must be measured by attaining flow curves with a rheometer and determining yield stress. Viscoelastic and semisolid foods have been extensively studied during the last decades. Rheological characterizations of nonNewtonian foods have been in the form of τ versus γ curves, dynamic characteristics, time effect on viscosity at constant temperature, and others. Values of these parameters have been compiled by different authors (Kokini 1992; Steffe 1992; Rao 2007). Liquid foods, such as beer, tea, coffee, clarified fruit juice, wines, cola drinks, vegetable oils, and milk exhibit Newtonian behavior. As an approximation, viscosity of Newtonian foods can be estimated as the weighted average between the viscosity of water ( μ w ) and that of the prevalent soluble substance. Different empirical equations relating liquid food viscosity with both soluble solids and temperature have been published (Rao 2007). The viscosity of salt and sugar solutions (two major food solutes) are also available (Kubota et al. 1981). Vitali and Rao (1984) reported that the effect of concentration on viscosity of fruit juices at constant temperature can be represented by an exponential-type relationship. Hydrolytic enzymes present in natural fruit and

Chapter 2

Thermophysical Properties in Simulating Food Processing Technologies

vegetable juices, or purees, degrade polysaccharide chains and therefore alter the viscosity.

2.2.3. Specific Heat Capacity The specific (isobaric) heat capacity, Cp, is the amount of heat (in joules) needed to raise the temperature of 1 kg of matter by 1 K at a given temperature. The SI unit for Cp is therefore J/kg/K. Specific heat capacity of solids and liquids depends on temperature but does not generally exhibit pronounced pressure dependence. It is common to use the constant pressure specific heat, Cp, which thermodynamically represents the change in enthalpy H (kJ/kg) for a given change in temperature T when it occurs at pressure P: ⎛ ∂H ⎞ CP = ⎜ ⎝ ∂T ⎟⎠ P

(2.6)

Assuming there is no phase change, the amount of heat Q that must be added to a unit mass M (kg of mass or specific weight kg/m3) to raise the temperature from T1 to T2 can be calculated using the following equation: Q = M ⋅ CP ⋅ (T2 − T1 )

(2.7)

The specific heat of foods is drastically influenced by water content. For example, specific heat has been found to vary exponentially with water content in fruit pulps above ambient temperatures. Furthermore, nonaqueous components show lower Cp (Barbosa-Cánovas et al. 2007; Lozano 2007). The specific heat of oils and fats is usually about one-half the specific heat of water, while the specific heat of dry materials in grains and powders is approximately one-third to one-fourth that of water (Rahman 1995). As a result of solute–water interactions, the Cp of each individual component in a food differs from the Cp of a pure component, and usually changes with the concentration of soluble solids. Cp has been measured at different temperatures in fresh and dried fruits, meats, cereal grains and cereal products, oils and fats, powders, and other dry foods (Lozano 2007). Although linear correlations of Cp with concentration are known in liquid foods, variations are often neglected for engineering calculations at near room temperature.

27

Several methods are known for measuring specific heat capacity experimentally. Cp can be determined by methods of mixtures and differential scanning calorimetry (DSC). For methods of mixtures, a calorimeter of known specific heat is used and CP is determined from a heat exchange balance. In the DSC method, the sample is put in a special cell where the temperature is increased at a constant heating rate. The specific heat of the food is obtained from a single heat thermogram, which records heat flow as a function of time or temperature.

2.2.4. Thermal Conductivity and Diffusivity Thermal conductivity, k, is the property of a material indicating its ability to conduct heat. It represents the quantity of heat Q that flows per unit time through a food of a certain thickness and area with a specific temperature difference between faces; the SI unit for k is W/m/K. The rate of heat flow Q through a material by conduction can be predicted by Fourier ’s law of heat conduction. A simplified approximation follows: k ⋅ A ⋅ (T2 − T1 ) Q = x

(2.8)

where A is the surface area of the food, x is its thickness, T1 is the temperature at the outer surface where heat is absorbed, and T2 is the temperature at the inner surface. Thermal diffusivity α (SI unit, m2/s) defines the rate at which heat diffuses by conduction through a food composite and is related to k and Cp through density ρ as follows:

α=

k ρ ⋅ CP

(2.9)

Thermal diffusivity establishes the speed of heat of three-dimensional propagation or diffusion through the material. It is represented by the rate at which temperature changes in a certain volume of food material, while transient heat is conducted through it in a certain direction in or out of the material (depending on whether the operation involves heating or cooling). Equation 2.9 shows that α is directly proportional to the thermal conductivity at a given

28

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

density and specific heat. Physically, it relates the ability of the material to conduct heat to its ability to store heat (Barbosa-Cánovas and Rodríguez 2005). The thermal conductivity of food materials is greatly influenced by the water content. Water shows greater relative magnitudes in comparison to other food constituents. Thus, k increases with increased moisture content. It is common to find a linear relationship between thermal conductivity and moisture content at ambient conditions, but also a quadratic relationship, as well as multiple correlations of moisture, temperature, and composition can be found for k in food materials (Lozano 2007). Some models consider that different components of foods (e.g., fibers) are arranged in layers either parallel or perpendicular to the heat flow (Salvadori et al. 1997). For example, in products such as meat, heat is usually transferred parallel to the fibers and k is dependent on the direction of the heat flow. More general in nature are the randomly distributed models, which consider that the food is composed of a continuous phase with a discontinuous phase dispersed within (solid particles being in either regular or irregular arrangement; Mattea et al. 1989; Lozano 2007). In porous materials, porosity must be included in the model because air has a thermal conductivity much lower than that of other food components. Models including density or porosity, and pressure, have been developed for fruits and vegetables, meat and meat products, dairy products, cereals, and starch (Lozano 2007). Several models for predicting α in foods have also appeared in the literature; however, most are product-specific and a function of water content (sometimes water activity) or temperature. Although the influence of carbohydrates, proteins, fat, and ash on thermal diffusivity has also been investigated, it was found that temperature and water content are the major factors affecting α (Rahman 1995). Experimental methods used to determine k are, for example, the Fitch method and the line source method. In the Fitch method, a solid slab of a certain food receives heat from one layer and conducts it to a copper plug. Conductivity k is obtained from the food’s temperature as a function of heat conduction time. The line source method is based on the

use of a thermal conductivity probe to measure a temperature–time relation on a thin cylindrical food piece to which constant heat is applied. Thermal diffusivity α is usually either found by direct experimental methods or estimated through Equation 2.9. Several direct methods for α determination can be based on a one-dimensional heat conduction equation where geometrical boundary conditions are defined. For instance, an apparatus can be used where the sample is located in a special cylinder and immersed in a water bath at constant temperature. Thermocouples located at the center of the sample (axis) and surface of cylinder measure temperature at different heating times. Transient temperature variations are used for the analytical solution. Indirect methods, although they might yield more accurate diffusivity values, require more time and instrumentation for the three-parameter determination (ρ, k, and CP; Lozano 2007). It is worth mentioning the role of the surface heat transfer coefficient, as it is one of the important parameters necessary for design and control of food processing and associated equipment where fluids (air, nitrogen, steam, water, or oil) participate. Although the surface heat transfer coefficient is not a property of food, it is used to quantify the transfer rate of heat by convection from a liquid or a gas (especially boiling liquids and condensing vapors) to the surface of foods. It plays an important role when evaluating the effectiveness of heat transfer in processes where hot water or steam is applied through the evaluation of the overall resistances during heat transfer (Juliano et al. 2008).

2.2.5. Compression Heating Coefficient and Related Properties All materials change their volume when subjected to temperature or pressure change. For food processing operations it is commonly assumed that pressure does not appreciably affect the volume of a liquid or the solid objects; however, in the case of food processing under high hydrostatic pressure up to several hundred MPa, this assumption does not hold true. A significant compression of fluids and some solids (especially polymeric materials used, e.g., as food

Chapter 2

Thermophysical Properties in Simulating Food Processing Technologies

packaging) occurs when materials are subjected to these pressure levels. The adiabatic temperature change of an isotropically compressed or a decompressed material can be expressed as follows: dT αP = ⋅ T = kC ⋅ T dP ρ ⋅ CP

(2.10)

where α P is the thermal expansion coefficient and kC is the compression heating coefficient. The coefficient of thermal expansion describes how the volume of an object changes with change in temperature. In particular, it measures the fractional change in volume per degree change in temperature at a constant pressure; the SI unit is K−1. Likewise, the compressibility is a measure of the relative volume change of a fluid or solid as a response to a pressure (or mean stress) change; the SI unit is Pa−1. As indicated earlier, all materials expand or contract when their temperature or pressure changes, and the expansion or contraction always occurs in all directions. This change in volume occurs at the same rate in any direction in isotropic materials. Some materials expand when cooled, such as freezing water, and therefore have negative thermal expansion coefficients (Knoerzer et al. 2010a). The adiabatic compressibility β S is a thermodynamic property and can be written at constant entropy as:

βS = −

1 ⎛ ∂V ⎞ V ⎜⎝ ∂p ⎟⎠ S

(2.11)

The thermal expansion coefficient α P is a thermodynamic property and can be written at a constant pressure as:

αP =

1 ⎛ ∂V ⎞ ⎜ ⎟ V ⎝ ∂T ⎠ P

(2.12)

where V is the volume of the material and ∂V/∂T is the rate of change of that volume with pressure and temperature, respectively. In general, the thermal expansion coefficients increase from solids over liquids to gases (Harvey et al. 1996). The variation of this parameter has been studied for water from the NIST database by Juliano et al. (2008). However, very limited information on

29

the thermal expansion coefficient of food materials is available in the literature. For example, values for sunflower and olive oils, as well as tomato paste and pressure-transmitting fluids, have been reported by Guignon et al. (2009, 2010) and Aparicio et al. (2010), respectively. Min et al. (2010) have determined the compressibility (and density) of selected liquid and solid foods as they vary with increasing pressure. As shown in Equation 2.10, all compressible materials undergo a change in temperature when subjected to pressure. The degree of temperature change is hereby dependent on a complex interaction of the thermal expansion coefficient, density, and specific heat capacity of the material. It is challenging to determine these properties separately under highpressure conditions. Therefore, for predicting the extent of compression heating during a high-pressure process the pressure–temperature-dependent properties can be combined into one pressure–temperaturedependent parameter, referred to as the compression heating coefficient kC (Knoerzer et al. 2010a, 2010b) with the SI unit Pa−1. For modeling high-pressure processes, knowledge of these properties is imperative for all materials involved in the modeled scenario (see Chapters 4 and 5 for further information). The variation of compression heating of water based on thermophysical properties from the NIST database for water and steam (Harvey et al. 1996) has been summarized in the literature (Ardia et al. 2004; Knoerzer et al. 2007; Mathys and Knorr 2009). Limited information is available on the compression heating for food materials. Studies on selected food materials have been published in the last few years (Otero et al. 2000, 2006; Rasanayagam et al. 2003; Ardia et al. 2004; Patazca et al. 2007; Shao et al. 2007; Zhu et al. 2007). Until recently, the extent of compression heating of nonfood solid materials in high-pressure processing was unknown and assumed negligible. This assumption holds true for metals, which not only have lower thermal expansion coefficients, but at the same time exhibit significantly higher densities, making the kC values small compared with those of liquids. However, it was shown by Knoerzer et al. (2010b) that some polymeric insulating plastics undergo pronounced heating under pressure, often

30

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

exceeding the adiabatic heating of water. This can be explained by the fact that these materials often have lower densities, but particularly lower specific heat capacities, and exhibit greater compressibility, associated with greater thermal expansion capacity. Also for liquids, pronounced differences in kC values can be found, as shown in Knoerzer et al. (2010a), associated with differences in molecular bonds, that is, hydrogen and van der Waal’s bonds affecting the thermal expansion coefficients. Water, for example, shows less pronounced heating than nonpolar liquids, such as propylene-glycol, with the difference being that the values are increasing for water with increasing pressures and temperatures; whereas nonpolar liquids show greatest compression heating at low pressures, and almost no temperature dependency (Knoerzer et al. 2010a, 2010b). The compression heating coefficient of food materials can be derived from the temperature– pressure profiles obtained in an adiabatic high-pressure system. There are certain types of thermocouples (i.e., K and T types), which can reliably measure temperature at high-pressure conditions. A detailed description of this method can be found elsewhere (Knoerzer et al. 2010a, 2010b). Thermodynamic equations are able to predict the thermal expansion coefficient and the compressibility of pure water and gases. However, food materials are more complex and their values cannot be derived theoretically. As long as density and heat capacity are known as functions of temperature and pressure, the thermal expansion coefficient can be calculated from the compression heating of the food material according to Equation 2.10. Very recently, a volume piezometer has been integrated into a high-pressure system to provide in situ data for compressibility and density (Min et al. 2010).

2.2.6. Dielectric Properties The dielectric permittivity, ε, is a complex number used to explain interactions of foods with electric fields. It determines the interaction of electromagnetic waves with matter and defines the charge density under an electric field. In solids, liquids, and gases the complex permittivity comprises two values:

• the real part, dielectric constant, ε', related to the capacitance of a substance and its ability to store electrical energy; and • the imaginary part, dielectric loss factor, ε", related to the attenuation of the electromagnetic energy, dissipated into thermal energy when the food is subjected to an alternating electrical field (i.e., dielectric relaxation and ionic conduction). Both parameters are dimensionless because they are relative values (Chapters 6 and 7). Dielectric properties (ε', ε") are primarily determined by the food’s chemical composition (presence of mobile ions and permanent dipole moments associated with water and other molecules) and, to a much lesser extent, by their physical structure (Barbosa-Cánovas et al. 2007). The influence of water and salt (or ash) content largely depends on the manner in which they are bound or restricted in movement by other food components. Free water and dissociated salts have high values of the dielectric properties, while bound water, associated salts, and colloidal solids exhibit lower values. Power dissipation is directly related to the dielectric loss factor ε" (see also Chapters 6 and 7) and temperature increase during microwave processing (as per the energy conservation equation) further depends on the specific heat of the food, the thermal conductivity, and the density of the material. Permittivity also strongly depends on the frequency of the applied alternating electric fields. Frequency contributes to the polarization of (polar) molecules such as water. In general, the permittivity increases with temperature, whereas the loss factor may either increase or decrease depending on the operating frequency (Mohsenin 1984; Regier and Schubert 2005). In microwave processing at frequencies of either 915 MHz or 2.45 GHz, the loss factor decreases with increasing temperature, caused by a greater mobility of ions and dipoles due to lower viscosities (Regier and Schubert 2005). Comprehensive tabulations of electrical property data are available for foods (Zhang 2007). The electrical field inside the food is determined by the dielectric properties and geometry of the load and the food processing chamber configuration.

Chapter 2

Thermophysical Properties in Simulating Food Processing Technologies

Dielectric properties are of great importance in measuring and heating applications, and also in the selection of proper packaging materials and in the design of microwave and radio frequency heating equipment since they influence how the material interacts with the electromagnetic waves. They furthermore determine the penetration depth of electromagnetic waves into foods. Known methods for measuring dielectric properties are the cavity perturbation, open-ended coaxial probe, and transmission line methods. Since modern microwave network analyzers have become available, the methods of obtaining dielectric properties over wide frequency ranges have become more efficient. Computer control of impedance and network analyzers has facilitated the automatic measurement of dielectric properties; special calibration methods have also been developed to eliminate errors caused by unknown reflections in the coaxial line systems. Distribution functions and empirical relationships can be used to express the temperature dependence of dielectric properties.

2.2.7. Electrical Conductivity The electrical conductivity, σ, is a measure of how well electric current flows through a material with a certain cross-sectional area A, length L, and resistance R. It is the inverse value of electrical resistivity (measure of resistance to electric flow), as expressed in the following equation (Regier and Schubert 2005; Zhang 2007):

σ=

L A⋅ R

(2.13)

The electrical conductivity (SI unit, S/m) of foods has been found to increase with temperature and also with water and ionic content. Mathematical relationships have been developed to predict the electrical conductivity of food materials (Buckow et al. 2010). Below freezing temperatures, electrical conductivity shows a pronounced decrease, since ice conducts less well than water. Phase transitions in foods (such as starch gelatinization) and cell structural changes also affect electrical conductivity. As

31

in thermal properties, the porosity of the food plays an important role in the conduction of electrons through the food. Electrical properties are important when processing foods with pulsed electric fields, ohmic heating, induction heating, radio frequency, and microwave heating (Chapters 6–11). Electrical conductivity plays a fundamental role in ohmic heating, a process in which electricity is transformed into thermal energy when an alternating current (AC) is applied to the food. Ohmic heating has potential use in fluid pasteurization; hence, knowing the effective electrical conductivity or the overall resistance of liquid–particle mixtures is important. Liquids and liquid–particle mixtures can also be pasteurized with pulsed electric fields technology. In this case, products with low electrical conductivity are better and more energy-efficient to process, unless a synergy of heat and electric field strength is assumed; then higher electric conductivities assist in heating up the liquid from moderate initial temperatures to the process target temperature. The electrical conductivity of a material is generally measured by passing a known current at constant voltage through a known volume of the material and by determining resistance. The total conductivity is then calculated simply by taking the inverse of the total resistivity. Basic measurements involve bridge networks (such as the Wheatstone bridge circuit) or a galvanometer. There are other devices that measure electrical conductivity of foods under ohmic or conventional heating conditions, using thermocouples and voltage and current transducers to measure voltage across and current through samples (Zhang 2007).

2.2.8. Acoustic Properties Modeling of sound fields is gaining importance in the food industry given several new applications of power ultrasound, such as enzyme activity modulation and enhanced extraction. The most important properties of a fluid, utilized to model sound fields, are speed of sound ( c ) and attenuation coefficient ( α ).

32

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

2.2.8.1. Speed of Sound The speed of sound is the rate at which an acoustic wave travels through a fluid. This thermodynamic property of the fluid depends on the following equilibrium conditions: ⎛ ∂p ⎞ c2 = ⎜ ⎟ ⎝ ∂ρ ⎠ adiabatic

(2.14)

For an ideal gas, Equation 2.14 takes the form: c 2 = γ RT

(2.15)

where γ = ( c p cv ) , R is the universal constant, and T is the absolute temperature (Lighthill 1978). For an ideal gas, c depends only on temperature. For liquids, the speed of sound can be expressed as: c2 =

1 βρ

(2.16)

where β is the adiabatic compressibility. β and ρ vary with the equilibrium temperature and pressure of the liquid. Hence, the speed of sound of pure liquids depends on temperature and pressure. Since there is no simple theory for predicting these variables, this dependence must be measured experimentally; the resulting values are usually expressed as empirical equations (Shoitov and Otpushchennikov 1968). As any other thermodynamic property, the speed of sound is a function of composition for mixtures. In nondispersive media, acoustic waves travel at the same speed of sound regardless of frequency (Leighton 1997). However, in a dispersive medium the speed of sound can vary with frequency (McClements and Povey 1989). This may occur in food emulsions where a secondary phase is dispersed in a continuous phase. In an ideal case, that is, a non-scattering two-phase system, the volume average values for compressibility and density can be used (McClements et al. 1990):

β = (1 − φ ) β1 + φβ2

(2.17)

ρ = (1 − φ ) ρ1 + φρ2

(2.18)

where φ is the dispersed phase volume fraction and the subscripts 1 and 2 represent the continuous and dispersed phases, respectively. Then, the mixture values of β and ρ can be used to calculate the speed

of sound. More realistic approaches include the effect of sound scattering, a deviation or reflection of sound at the phase boundary. The two most important sources of sound scattering, in the long wavelength limit (particle radius of dispersed phase r << λ wavelength of ultrasonic wave), are viscoinertial and thermal scattering (McClements et al. 1990). Visco-inertial scattering occurs due to differences of density between the phases, whereas thermal scattering is caused by differences in compressibility, heat capacity, and thermal diffusion between the phases. This will be explained further in Section 2.2.8.2. Ament (1953) developed an equation of sound propagation through emulsions, accounting for visco-inertial scattering. The Ament formula can be plotted in a three-dimensional figure to visualize the dependence of the speed of sound with the volume fraction and Γr . The latter is the product of the shear wave skin depth ( Γ ), at which the amplitude of the wave has diminished to 1/e of its original value, and particle radius of the dispersed phase (r). It has been shown that for the region where Γr ≈ 1 the speed of sound is frequency-dependent and in those regions where this condition does not apply ( Γr ≠ 1) the speed of sound is frequencyindependent (Povey and McClements 1998). The speed of sound in liquids can be strongly influenced by the presence of bubbles, especially for cavitating systems, where bubbles are generated by pressure changes during the propagation of highintensity ultrasonic waves in liquids. For example, the speed of sound in a cavitating liquid may decrease from 1,500 m/s (in the case of pure water) to 20 m/s, which is even less than the velocity of the speed of sound in air (340 m/s; Servant et al. 2001). Systems containing bubbles may exhibit resonant scattering, where attenuation is sharply increased near the resonant frequency. In resonant scattering the magnitude of the scattered wave is of the same order as the incident wave. Multiple scattering of the bubbles has a pronounced effect on the speed of sound. In this case, the speed of sound and the attenuation are a complex function of the frequency and number and size of bubbles (McClements and Povey 1989). The speed of sound is generally determined by measuring the acoustic transit time over a known

Chapter 2

Thermophysical Properties in Simulating Food Processing Technologies

path length. For instance, a receiver acoustic interferometer allows measuring the speed of sound by knowing the distance traveled by the sound and the phase shift between received and transmitted signals (Barthel and Nolle 1952; McCartney and Drouin 1977). The pulse-echo-overlap (PEO) method sends a pulsed sound from the transmitter to a reflector. The signal on its way back again excites the transducer and a new echo is sensed. The process is repeated until the wave is completely damped. The speed of sound is obtained by measuring the time between the maximum of two consecutive echoes (Junquera et al. 2002). Broadband resonance techniques have also been widely used to determine speed of sound and to characterize liquids (Eggers and Kaatze 1996). Unfortunately, in spite of current technologies capable of accurately determining the speed of sound in fluids, and the increased use of ultrasound to characterize food materials, there is a lack of reported data on speed of sound of food materials in the public domain. 2.2.8.2. Sound Attenuation Coefficient Sound attenuation is the loss occurring during the propagation of sound waves. The attenuation coefficient represents the damping of the initial pressure amplitude P0 of an ultrasonic wave propagating along a distance r (Lamberti et al. 2009): P = P0 e −α r

(2.19)

The sound attenuation at ultrasonic frequencies in a pure liquid is caused by absorption mechanisms that convert energy from the ultrasonic wave into heat (McClements et al. 1990). In liquid, the most important mechanisms are viscosity, thermal conduction, and molecular relaxation. As explained in Section 2.2.2, viscosity transforms the fluid kinetic energy of motion into heat. Hence, viscous losses occur when there is a relative motion between adjacent portions of the fluid upon passage of sound waves. Besides that, the temperature of the fluid locally increases during compression and decreases during expansion. Thermal conduction losses occur when heat flows from the compressed and “hot” parts of the medium to the adjacent and expanded “cold” parts. Molecular relaxation losses are caused by

33

molecular level processes, such as conversion of kinetic energy of molecules into stored potential energy formed by structural rearrangement of clusters, and rotational and vibrational energies of molecules (Kinsler et al. 2000). For multiphase systems such as emulsions, the energy of the ultrasound wave may be scattered at the interface between primary and secondary phases. When ultrasound is scattered, the ultrasound energy is redirected away from the incident wave (McClements and Povey 1989). A first ideal approach assumes that there is no interaction between the phases. Thus, the attenuation of a twophase non-scattering system is equal to the volume average of the absorption coefficients:

α = (1 − φ ) α1 + φα 2

(2.20)

However, in real emulsions, scattering of ultrasound has a pronounced effect on both speed of sound and attenuation (McClements et al. 1990). The difference between the overall attenuation ( α ) and that caused by absorption alone ( α o ) is termed “excess attenuation” ( α exc ), which is further defined as a combination of various scattering mechanisms (McClements and Povey 1989):

α = α o + α exc

(2.21)

Overall attenuation is usually described with the simplified equation below (Eggers and Kaatze 1996):

α ( f ) = Bf 2 + α exc ( f )

(2.22)

where f is frequency, Bf = α o is the attenuation due to absorption alone, and B is a constant. As mentioned before, the two main sources of scattering in the long wavelength limit ( r << λ ) are visco-inertial ( α vis ) and thermal scattering ( α th ). Visco-inertial scattering occurs when droplets have a different density compared with the surrounding fluid. In the presence of an ultrasonic wave a net force causes the droplet to oscillate; the oscillation is damped by the viscosity of the surrounding fluid. In thermal scattering, differences in compressibility, heat capacity, and thermal diffusion between the droplets of the dispersed phase and the continuous primary phase cause the ultrasound wave to be scattered (Povey and McClements 1998). When the compressibility of the 2

34

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

particles differs from the surrounding liquid, the particles and the liquid expand differently under the influence of the ultrasonic wave. This difference creates a temperature gradient at the particle–liquid interface. Energy flowing at this boundary is irreversible according to thermodynamic principle (Dukhin et al. 2005). The thermal scattering of a dispersion is a function of the thermal expansion, heat capacity, and thermal conductivity of both phases, and also depends on the size distribution of the particles. As explained in Section 2.2.8.1, systems containing bubbles may exhibit multiple strong scattering, which not only has a pronounced effect on the speed of sound, but also affects the overall attenuation of the sound wave. Therefore, sound fields can be strongly attenuated a few millimeters from the source in high-power horn reactors due to the presence of cavitation bubbles (see Chapter 12; Trujillo and Knoerzer 2009). Similar to the speed of sound, attenuation can also be experimentally determined via standard methods such as ultrasonic pulse techniques (McSkimin 1957). Chapter 12 also explains a method of determining the absorption coefficient (only absorption without scattering) based on measuring acoustic streaming. However, also for the absorption coefficients, there is a lack of experimental data reported in the public domain. Hence, attenuation measurements are needed for future modeling of ultrasoundbased processes. For cavitating systems, there are still many complexities that need to be solved, given the strong scattering produced by bubbles formed during cavitation.

absorber of electromagnetic radiation (c, mol/L), and path length of light (d, m).

ε=

A c⋅d

(2.23)

In spectroscopy, the absorbance A (optical density) is defined as: A = − log

I I0

(2.24)

where I is the intensity of light at a specified wavelength that has passed through a sample (transmitted light intensity) and I0 is the intensity of the light before it enters the sample (incident light intensity). The liquid itself and the concentration of the suspended units can be transparent if A << 1, opaque if A >> 2 or semitransparent for anything in between these extremes. In a majority of cases, liquid foods will absorb UV radiation. The absorptivity of a pure substance is determined by means of spectrophotometric analysis. In particular, since this property is utilized to model UV processing, the absorptivity is determined according to the operational frequency. Generally, a standard curve of the absorbance versus concentration of a pure substance is determined and the slope of the curve defines the absorptivity. Chemical handbooks such as the Merck Index (Merck Publishing Group, Rahway, NJ) also provide values of absorptivity for standard materials and food additives. In the case of liquids containing heterogeneous mixtures or several substances absorbing at the same frequency, an overall absorptivity can be determined either empirically or by application of mixture rules.

2.2.9. Molar Absorptivity The molar absorptivity of a substance has a particular application in the modeling of UV light processing. It is the main optical characteristic observed in all liquid foods undergoing UV treatment. The molar absorptivity (ε, L/mol/m) of the absorbing species is a measure of the amount of light absorbed per unit concentration, or optical density and forms part of the Lambert-Beer ’s law (Eq. 2.23) in a linear relationship between absorbance A, concentration of an

2.3. Final Remarks and Future Recommendations Accurate model prediction requires implementation of the thermophysical properties of all materials involved in a Multiphysics model scenario. The use of constant values for the respective properties generally yields only poor agreement with predictions from actual measured data. Therefore, including the functional dependence of the properties on

Chapter 2

Thermophysical Properties in Simulating Food Processing Technologies

temperature, pressure, and product-specific parameters and variables (e.g., composition and structure) is essential for accurate model prediction. It has been shown that during processing (by technologies explored in this book) thermophysical properties are mainly affected by temperature, pressure, and changes in the process and product parameters, although product parameters such as composition and structural porosity, as a matter of simplification, are generally assumed to be constant. When modeling these processing technologies the governing equations of mass, momentum, and energy conservation have a number of properties in common, including density, specific heat capacity, thermal conductivity, and viscosity. For high-pressure processing the main variables acting are pressure, temperature, and initial composition of the chamber components, which affect the compression heating properties such as density, specific heat capacity, and thermal expansion coefficient, as well as viscosity and thermal conductivity. This has been covered in Chapters 4 and 5, which summarize the changes in these properties at high pressure under low and high temperatures. During microwave processing, properties included in the governing conservation equations and the dielectric properties are only dependent on the temperature, the material, and the selected frequency; they are not affected by the electromagnetic field strength. Chapters 6 and 7 present models wherein these properties vary as a function of temperature and initial product composition. Similarly, for ohmic heating and pulsed electric fields processing, the governing conservation equation properties and the electrical conductivity are only affected by the temperature of a specific material without changing with the variable electric field. Chapters 9 and 11 provide more details on the variation of these thermophysical properties with temperature and composition. Ultrasound processing models (Chapter 12) require governing conservation equation properties, as well as the speed of sound and the sound attenuation coefficient, which are both dependent on temperature and pressure, and frequency in dispersive media. In the case of mixtures they also depend on composition. For dispersions (suspensions and

35

emulsions), the speed of sound and attenuation depend on size and distribution of particles or secondary phase droplets. For systems under cavitation, the strong scattering caused by bubbles dramatically changes both the speed of sound and attenuation. Therefore, these two properties are a complex function of frequency, and cavitation extent. Given that cavitation bubbles are produced (grow and collapse) close to the acoustic source where sound intensity is high, the speed of sound and the attenuation become dependent on the sound field. In addition, the relevant properties in UV processing included in governing conservation equations vary with temperature and material composition, while the absorptivity is only dependent on material composition. Chapters 14 and 15 provide some of the values of absorptivity (absorption coefficient) for various liquid foods. A number of publications have tabulated data and empirical equations describing the variation of thermophysical properties, including the governing conservation equations at atmospheric conditions as a function of temperature and composition of specific food materials and polymers (product and component related). Properties related to other physics phenomena have not been determined to the same extent. As pointed out in this chapter, the main hurdle in developing accurate Multiphysics models is the lack of thermophysical property values. In addition, little information exists in the literature on thermophysical properties at high-pressure conditions. It would be highly desirable that sufficient property data be determined in the form of a database, one that at least shows the values for carbohydrates, proteins, lipids, fiber, and ions present in different foods, as well as values for selected food materials, which may also represent a group of materials. Once this information becomes available, the rule of mixtures can be applied to most of the thermophysical properties reviewed in this chapter, and overall values for particular foods or food blends can be determined. However, it should be pointed out that the speed of sound and sound attenuation coefficients in dispersions do not necessarily obey simple rules of mixtures, given that they may also depend on particle size and distribution of the dispersed phase.

36

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

To satisfy the need for these properties, empirical equations as functions of temperature and pressure are essential for obtaining reliable Multiphysics models. Furthermore, these models will need to account for the composition and structure of the food materials. A way of overcoming this challenge is by providing an open platform for the collection of property data, which will enable the development of accurate models for various food categories. A web site approach similar to the Wikipedia principle could assist in developing an extensive database accessible to the general public. Inter-laboratory validation would then be needed to assure data consistency and accuracy.

Notations Latin Symbols A A B CP c c d f F H I, I0 kC k K L M n P, p Q Q R R r r S T t u, v

Absorbance Area Constant Specific heat capacity Concentration Speed of sound Path length Frequency Force Enthalpy Light intensity Compression heating coefficient Thermal conductivity Constant Length Mass and specific mass Exponential constant Pressure Heat/energy Heat/energy flow Electrical resistivity Gas constant = 8.3142 Distance Radius of dispersed particles Entropy Temperature Time Velocity

— m2 m−1/Hz2 J/kg/K mol/L m·s−1 m Hz N J/kg W/m2 Pa−1 W/m/K m kg or kg/m3 Pa J J/s m/S J/K/mol m m J/K K s m/s

V x y y

Volume Thickness Direction of shear Distance

m3 m — m

Greek Symbols αP α α αo

α exc α th α vis β ρ ε ε ε ε' ε"

φ γ γ η ηp λ τ σ

Thermal expansion coefficient Thermal diffusivity Sound attenuation coefficient Sound attenuation coefficient due to absorption alone Excess attenuation Thermal scattering sound attenuation Visco-inertial sound attenuation Compressibility Density Porosity Dielectric permittivity Molar absorptivity Real part of complex permittivity; dielectric constant Imaginary part of complex permittivity, dielectric loss factor Volumetric fraction = ( c p cv ) Ratio between constant pressure and constant volume heat capacities Shear rate Viscosity Apparent viscosity Wavelength Shear stress Electrical conductivity

K−1 m2/s m−1 m−1 m−1 m−1 m−1 Pa−1 kg/m3 — — L/mol/m — —

— —

s−1 Pa·s Pa·s m N/m2 S/m1

Abbreviations AC PDE CFD SI

Alternating current Partial differential equation Computational fluid dynamics International System of Units (Système international d’unités)

Chapter 2

DSC NIST UV

Thermophysical Properties in Simulating Food Processing Technologies

Differential scanning calorimetry National Institute for Standards and Technology Ultraviolet

Operators d ∂

Differential Partial differential

References Ament WS. 1953. Sound propagation in gross mixtures. J Acoust Soc Am 25:638–641. Aparicio C, Otero L, Sanz PD, Guignon B. 2010. Specific volume and compressibility measurements of tomato paste at moderately high pressure as a function of temperature. J Food Eng. DOI: 10.1016/j.jfoodeng.2010.10.021. Ardia A, Knorr D, Heinz V. 2004. Adiabatic heat modelling for pressure build-up during high-pressure treatment in liquidfood processing. Food Bioproducts Proc 82:89–95. Barbosa-Cánovas GV, Rodríguez JJ. 2005. Thermodynamic aspects of high hydrostatic pressure food processing. In: GV Barbosa-Cánovas, MS Tapia, MP Cano, eds., Novel Processing Technologies, 183–206. New York: CRC Press. Barbosa-Cánovas GV, Ortega-Rivas E, Juliano P, Yan H. 2005. Food Powders. Physical Properties, Processing and Functionality. New York: Springer-Verlag. Barbosa-Cánovas GV, Juliano P, Peleg M. 2007. Engineering properties of foods. In: GV Barbosa-Cánovas, ed., Food Engineering. Encyclopedia of Life Support Systems, 25–44. Oxford: EOLSS Publishers Co Ltd. Barthel R, Nolle AW. 1952. A precise recording ultrasonic interferometer and its application to dispersion tests in liquids. J Acoust Soc Am 24:8–15. Bourne MC. 2002. Texture, viscosity, and food. In: Bourne MC, ed., Food Texture and Viscosity, 1–32. London: Academic Press. Buckow R, Schroeder S, Berres P, Baumann P, Knoerzer K. 2010. Simulation and Evaluation of Pilot-Scale Pulsed Electric Field (PEF) Processing. J Food Eng 101(1):67–77. Chen XD. 2006. Modeling thermal processing using computational fluid dynamics (CFD). In: DW Sun, ed., Thermal Food Processing, 133–151. Boca Raton: Taylor & Francis. Dukhin AS, Goetz PJ, Travers B. 2005. Use of ultrasound for characterizing dairy products. J Dairy Sci 88:1320–1334. Eggers F, Kaatze U. 1996. Broad-band ultrasonic measurement techniques for liquids. Meas Sci Technol 7:1–19. Guignon B, Aparicio C, Sanz PD. 2009. Volumetric properties of sunflower and olive oils at temperatures between 15 and 55C under pressures up to 350MPa. High Press Res 29:38–45. Guignon B, Aparicio C, Sanz PD. 2010. Volumetric properties of pressure-transmitting fluids up to 350 MPa: Water, ethanol,

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ethylene glycol, propylene glycol, castor oil, silicon oil, and some of their binary mixture. J Chemical Eng Data 55(9): 3017–3023. Harvey AH, Peskin AP, Sanford AK. 1996. NIST/ASTME— IAPSW Standard Reference Database 10, version 2.2. Juliano P, Knoerzer K, Barbosa-Cánovas GV. 2008. High pressure thermal processes: thermal and fluid dynamic modeling principles 35. In: R Simpson, ed., Engineering Aspects of Thermal Processing, 91–158. Boca Raton, FL: CRC Press/ Taylor & Francis. Juliano P, Knoerzer K, Fryer P, Versteeg C. 2009. C. botulinum inactivation kinetics implemented in a computational model of a high pressure sterilization process. Biotechnol Prog 25: 163–175. Junquera E, Ruiz M, Lopez S, Aicart E. 2002. A technique and a method for the continuous, simultaneous, and automatic measurement of density and speed of sound in pure liquids and solutions. Rev Sci Instrum 73:416–421. Kinsler LE, Frey AR, Coppens AB, Sanders JV. 2000. Fundamentals of Acoustics. New York: John Wiley & Sons. Knoerzer K, Regier M, Erle U, Pardey KK, Schubert H. 2004. Development of a model food for microwave processing and the prediction of its physical properties. J Microw Power Electromagn Energy 39:167–177. Knoerzer K, Juliano P, Gladman S, Versteeg C, Fryer P. 2007. A computational model for temperature and sterility distributions in a pilot-scale high-pressure high-temperature process. AIChE J 53:2996–3010. Knoerzer K, Regier M, Schubert H. 2008. A computational model for calculating temperature distributions in microwave food applications. Innov Food Sci Emerg Technol 9:374–384. Knoerzer K, Buckow R, Sanguansri P, Versteeg C. 2010a. Adiabatic compression heating coefficients for high-pressure processing of water, propylene-glycol and mixtures—A combined experimental and numerical approach. J Food Eng 96:229–238. Knoerzer K, Buckow R, Versteeg C. 2010b. Adiabatic compression heating coefficients for high pressure processing—A study of some insulating polymer materials. J Food Eng 98:110–119. Kokini JL. 1992. Measurement and simulation of shear and shear free (extensional) flows in food rheology. Kubota K, Kurisu S, Suzuki K, Matsumoto T, Hosaka H. 1981. Study on the flow equations of sugar, salt and sugar-salt skim milk solutions. J Jpn Soc Food Sci Technol-Nippon Shokuhin Kagaku Kogaku Kaishi 28:186–193. Lamberti N, Ardia L, Albanese D, Di Matteo M. 2009. An ultrasound technique for monitoring the alcoholic wine fermentation. Ultrasonics 49:94–97. Leighton TG. 1997. The Acoustic Bubble. New York: Academic Press. Lighthill J. 1978. Waves in Fluids. Cambridge: Cambridge University Press. Lozano JE. 2007. Thermal properties of foods. In: GV BarbosaCánovas, ed., Food Engineering. Encyclopedia of Life Support Systems, 45–64. Oxford: EOLSS Publishers Co Ltd.

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Mathys A, Knorr D. 2009. The properties of water in the pressuretemperature landscape. Food Biophys 4:77–82. Mattea M, Urbicain MJ, Rotstein E. 1989. Effective thermalconductivity of cellular tissues during drying—Prediction by a computer-assisted model. J Food Sci 54:194. McCartney ML, Drouin G. 1977. Moderate-accuracy interferometer for speed of sound measurement in liquids. Rev Sci Instrum 48:214–218. McClements DJ, Povey MJW. 1989. Scattering of ultrasound by emulsions. J Phys D Appl Phys 22:38–47. McClements DJ, Povey MJW, Jury M, Betsanis E. 1990. Ultrasonic characterization of a food emulsion. Ultrasonics 28:266–272. McSkimin HJ. 1957. Ultrasonic pulse technique for measuring acoustic losses and velocities of propagation in liquids as a function of temperature and hydrostatic pressure. J Acoust Soc Am 29:1185–1192. Min S, Sastry SK, Balasubramaniam VM. 2010. Compressibility and density of select liquid and solid foods under pressures up to 700 MPa. J Food Eng 96:568–574. Mohsenin NN. 1984. Electromagnetic Radiation Properties of Food and Agricultural Products. New York: Gordon and Breach Science. Otero L, Molina-Garcia A, Sanz P. 2000. Thermal effect in foods during quasi-adiabatic pressure treatments. Innov Food Sci Emerg Technol 1:119–126. Otero L, Ousegui A, Guignon B, Le Bail A, Sanz PD. 2006. Evaluation of the thermophysical properties of tylose gel under pressure in the phase change domain. Food Hydrocolloids 20:449–460. Patazca E, Koutchma T, Balasubramaniam VM. 2007. Quasiadiabatic temperature increase during high pressure processing of selected foods. J Food Eng 80:199–205. Povey MJW, Mcclements DJ. 1998. Ultrasonics in food engineering. Part I: Introduction and experimental methods. J Food Eng 8:217–245. Rahman S. 1995. Food Properties Handbook. Boca Raton, FL: CRC Press. Rao MA. 2007. Rheology of Fluid and Semisolid Foods: Principles and Applications. New York: Springer. Rasanayagam V, Balasubramaniam VM, Ting E, Sizer CE, Bush C, Anderson C. 2003. Compression heating of selected fatty food materials during high-pressure processing. J Food Sci 68:254–259.

Sahin S, Sumnu SG. 2006. Physical Properties of Foods. New York: Springer. Salvadori VO, DeMichelis A, Mascheroni RH. 1997. Prediction of freezing times for regular multi-dimensional foods using simple formulae. Food Sci Technol Lebensmitt Wiss Technol 30:30–35. Schubert H, Regier M. 2005. Introducing microwave processing of food: principles and technologies. In: H Schubert, M Regier, eds., The Microwave Processing of Foods. Cambridge: Woodhead Publishing Limited. Servant G, Laborde JL, Hita A, Caltagirone JP, Gerard A. 2001. Spatio-temporal dynamics of cavitation bubble clouds in a low frequency reactor: Comparison between theoretical and experimental results. Ultrason Sonochem 8:163–174. Shao YW, Zhu SM, Ramaswamy H, Marcotte M. 2007. Compression heating and temperature control for high pressure destruction of bacterial spores: An experimental method for kinetics evaluation. Food Bioprocess Technol. doi: 10.1007/ s11947-008-0057-y Shoitov YS, Otpushchennikov NF. 1968. Pressure dependence of the speed of sound in liquid. Izvestiya VUZ Fisika 11: 137–139. Steffe JF. 1992. Rheological Methods in Food Process Engineering. East Lansing, MI: Freeman Press. Tabilo-Munizaga G, Barbosa-Cánovas GV. 2005. Rheology for the food industry. J Food Eng 67:147–156. Trujillo FJ, Knoerzer K. 2009. CFD modelling of the acoustic streaming induced by an ultrasonic horn reactor. In: Seventh International Conference on CFD in the Minerals and Process Industries. Melbourne, Australia. Vitali AA, Rao MA. 1984. Flow properties of low-pulp concentrated orange juice—Serum viscosity and effect of pulp content. J Food Sci 49:876–881. Welti-Chanes J, Vergara-Balderas F, Bermudez-Aguirre D. 2005. Transport phenomena in food engineering: Basic concepts and advances. J Food Eng 67:113–128. Zhang H. 2007. Electrical properties of foods. In: GV BarbosaCánovas, ed., Encyclopedia of Life Support Systems (EOLSS). Developed under the Auspices of the UNESCO, 115–125. Oxford, UK: Eolss Publishers. Zhu S, Ramaswamy HS, Marcotte M, Chen C, Shao Y, Le Bail A. 2007. Evaluation of thermal properties of food materials at high pressures using a dual-needle line-heat-source method. J Food Sci 72:E49–E56.

Chapter 3 Neural Networks: Their Role in High-Pressure Processing José S. Torrecilla and Pedro D. Sanz

The main objective of a mathematical model is to reproduce the corresponding process performance and/or estimate important variable values. Because of this, the modelization of processes is useful mainly as a previous stage in the design of equipment. With this aim, although detailed knowledge of the process is not always required, the input and output variables of the system to be studied should be known. Here, two such methods usually used to design this type of models are described, viz. models based on neural networks (NNs) and macroscopic models. Although successful applications will be shown here, the presented models have to be adjusted prior to applying them to new systems.

3.1. Brief History of NNs In the nineteenth century, Alexander Bain and William James established the foundation of modern NNs, which later, in the middle of the twentieth century, were developed by Warren McCulloch and Walter Pitts (McCulloch and Pitts 1943). These algorithms were further developed during the ensuing two decades, but in the 1960s, as a result of the exaggeration of successful results, bitter critique focused mainly on the capability of NNs was published and

research tasks were delayed for nearly a decade. In 1969, Minsky and Papert published a book titled Perceptrons, which was partially responsible for the reinitiation of NN research activity (Minsky and Papert 1969). Shortly afterwards, the backpropagation (BP) algorithm, one of the most important components of NN algorithms, was presented by Werbos and developed by Rumelhart and his colleagues among other research groups (Werbos 1974; Rumelhart et al. 1986). In the 1980s, Holpfield’s publications stimulated NN investigations (Hopfield 1982, 1984). In the last two decades, algorithms that handle and process information in a way similar to the human brain such as NNs and fuzzy logic have been developed (Chen 1996). In particular, the rapid development of NN algorithms and their numerous applications in commercial and scientific sectors are mainly based on the notable growth in computer technology. Now, NN technology is used in food science, chemical engineering, process control, medical diagnosis, forensic analysis, weather forecasting, financial applications, investment analysis, and others. In all these applications, depending on their requirements, NNs can be applied to tasks related to classification and/or prediction activities (Huang et al. 2007).

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

39

40

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

INPUTS

WEIGHTS

0.5

1

ACTIVATION FUNCTION

x

I iWi i

0.7

x= –0.9

TRANSFER FUNCTION

f ( x)

1 1 e

x

OUTPUTS

f(x) = –0.289

–2

Figure 3.1. Schema and an example of the calculation procedure of an artificial neuron (x, I, i, and f(x) are the activation function, input value, layer, and sigmoid function as transfer function, respectively).

3.2. Basis of NNs Although biological neurons are much more complicated than artificial neurons, the design and development of artificial neurons have been inspired by biological cells. A biological neuron consists of dendrites, soma, axon, and synapses. The signal is received by dendrites, processed in the soma or cell body and the output is emitted by the axon and transmitted to other neurons by synapses. An artificial neuron receives signals, which are processed in the neurons by mathematical algorithms (a combination of linear or nonlinear algorithms), and the output information is then sent to another neuron as input or is used directly as a network result. A schematic of an artificial neuron and an example of a calculation procedure are shown in Figure 3.1. The connectivity of artificial neurons emulates the biological neurological system, which is in most cases to design an NN; artificial neurons are joined and depending on the connections between them, different types of NNs can be defined, for example, neurons may or may not receive information back from the next layers, forming NNs called feedback or feedforward networks, respectively, whereas the recurrent networks, once the signal is received in a neuron from a given layer, are communicated to other neurons in the same layer before sending it onto the next or last layer. In most cases, the artificial neurons are arranged in layers. Each connection is controlled by parameters called weights. As some input signals may be more important than others, to obtain the best result (the maximum accuracy possible, vide infra), the weight

values must be optimized using real data from the system to be modeled (Huang et al. 2007). The optimization of the weights is commonly called the learning process, which is the most important process in the design of NN. Depending on the NN type, their weights can be optimized by supervised or unsupervised methods. The most common learning process used is the supervised method, which requires a database comprising input and target output values. For instance, to optimize the weights of a feedforward network, a supervised learning process such as a BP algorithm is normally used. The unsupervised learning process requires only input information. Kohonen’s selforganization map is a network that follows an unsupervised learning process (vide infra). Throughout the chapter, every statistical result has been calculated using estimated values worked out by the NN models and the real values from the external validation samples.

3.2.1. BP Algorithm The BP (also called the retro-propagation of error) is a common method used to optimize the weights of supervised networks. These algorithms can be used to optimize the weights of NNs with fully or partially interlayer connections (no intra-layer connection). Basically, the information is inputted into an NN and is fed forward throughout hidden and output layers to calculate a response. The calculation process in each neuron of the hidden and output layers consists of activation and transfer functions.

Chapter 3

Neural Networks: Their Role in High-Pressure Processing

41

The activation function, Equation 3.1, means that the input data to each neuron are multiplied by weight; the result, xk, is fed into a transfer function. The sigmoid or hyperbolical tangent function is the most commonly used transfer function, Equations 3.2 and 3.3, respectively. The calculated value, yk, is the output of the considered neuron. This calculation procedure is shown in Figure 3.1.

calculated. When every data set of the learning sample has been used, a cycle is finished and another begins with the first data set of the aforementioned database (Torrecilla et al. 2009a). A more detailed explanation can be found elsewhere (Fine 1999; Arbib 2003).

∑w

The networks optimized by the unsupervised learning process are able to learn from the input signals to ascertain their regularity without knowing in advance what the desired outputs are. Although unsupervised learning may provide more effective solutions than supervised learning (Torrecilla et al. 2009b), unsupervised learning processes are still in an early trial stage, which can only be used for some simple applications (Huang et al. 2007). Due to the fact that one of the most representative, interesting, and influential unsupervised networks is Kohonen’s self-organization map (Kohonen 1987), the learning process of this type of NN is exposed here. Self-organizing maps (SOMs), Kohonen’s selforganization map, or Kohonen’s NN can learn to detect irregularities and correlations in their input data and adapt their future responses to that input accordingly, that is, they are able to recognize groups with similar characteristics (Kohonen 1987; Demuth et al. 2005). The architecture of SOM models is shown in Figure 3.3. This is composed of two layers, viz. input and competitive layers. Every circle and arrow represents a neuron and weight, respectively, that is, there are as many weights as arrows and the number of neurons is equal to the product of the width and length of the competitive layer. In order to represent the whole input database, the neurons, defined by weight vectors, look for the best place in the competitive layer during the learning process (Figure 3.3). The weights of the SOM model optimization can be summarized in five stages:

xk =

jk

⋅ yj

(3.1)

j =1

⎛ 1 ⎞ yk = f ( x k ) = ⎜ ⎝ 1 + e − xk ⎟⎠

(3.2)

⎛ 1 − e −2 xk ⎞ yk = f ( x k ) = ⎜ ⎝ 1 + e −2 xk ⎟⎠

(3.3)

where i, j, and k subscripts are the input, hidden, and output layers, respectively. Given that the optimum weights are calculated by differential calculations, all algorithms that compose the BP algorithm (activation, transfer function) must be differentiable (vide infra). Then, this output is compared with the real or known value (rk) and the mean square error value (MSE) is calculated, Equation 3.4. MSE =

1 N

N

∑ (r − y ) k

k

2

(3.4)

k

where N is the number of neurons in the output layer. Once the MSE has been calculated, the optimization of the weight begins. The weights of the hidden and output layer connections are first optimized. An example of the forward calculation procedure and the equations required to optimize every weight are shown in Figure 3.2. As can be seen, two different groups of equations are used to optimize the weights from hidden-output layers and inputhidden layers. The determination of the level of generalization of the NN is performed once the weights have been optimized. For that purpose, the NN is tested by new databases commonly called verification. In this way a testing or an internal validation sample and an external validation sample is obtained. That is, the optimized NN calculates the output values using a new database and the prediction error is then

3.2.2. Unsupervised Learning Process

1. Assign random values to the weights. 2. A data set from the learning sample is presented to the SOM. 3. The neuron with the least Euclidean distance between its weights and data set (D) is selected (Eq. 3.5). This is called the winning neuron.

42

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Forward calculation

1

0.1

Output value

0.3 1

2

0.2

0.5

0.4

1

0.5

0.6

Weights 0.1

Weights 0.1

0.2

Mean square error

0.9

Input values

Real values

Input layer

Hidden layer

Output layer

Backpropagation Error o

y ko (1 y ko )w jk ( y ko

k

1

Error o y ij (1 y ij ) y i

w (t 1) w (t ) Error

2

o k

R)

o k

o k

y (1 y )w jk ( y

w (t 1) w (t )

o k

1

w(t 1)

w(t) µ Error y k (1 y k ) y jk

1

0.1

R)

Mean square error

Error o y ij (1 y ij ) y i

Figure 3.2. Forward calculation and backpropagation procedures (w, t, Error, Error o, μ, R, i, j, and k are weight, time, MSE, and MSE retro-propagated, learning coefficient, real values, input layer, hidden layer, and output layer, respectively).

4. The weights of the selected neuron are optimized so that they become more similar to the input vector (Eq. 3.6). 5. The weights of the neighboring neurons are also optimized but with proportionally less Euclidean distance to the winning neuron Equation 3.7. S

Input values

D j = X − Wj =

∑ (X − W ) ; 2

i

ij

j = 1 ,2 ,… , N

i

(3.5) W j (n) = W j (n − 1) + Lr ⋅ [ X (n − 1) − W j (n − 1)] (3.6)

Input layer Figure 3.3. Schematic diagram of self-organizing map model.

W j ( n) = (1 − Lr ) ⋅ W j ( n − 1) + Lr ⋅ X ( n − 1) (3.7) where W, n, X, Lr, N, and S are the weights, iteration of a given cycle, input vector, learning rate, number

Chapter 3

Neural Networks: Their Role in High-Pressure Processing

of weights of the SOM, and number of data set of the learning sample, respectively (Demuth et al. 2005; Raju et al. 2006). The process is iteratively repeated. A cycle is finished when the entire database has been presented to the SOM (Fonseca et al. 2006). Once the SOM model has been trained, it is able to extract the relevant information in order to classify new input vectors (which are interpolated in the learning range) and the model can classify the input data according to how they are grouped in the input space, that is, the input data can be adequately represented. A more detailed explanation can be found elsewhere (Arbib 2003).

3.3. How NNs Are Helping the Chemical Industry

43

algorithm are necessary. In order to validate the NN algorithms as a powerful chemometric tool, a system based on low concentrations (less than 15 ppm) of toluene, heptane, and 1-ethyl-3-methylimidazolium ethylsulfate and 1-butyl-3-methylimidazolium methylsulfate ILs in acetone was selected. The imidazolium ring of both ILs and toluene are ultraviolet (UV) active in the same region. But the UV-visible spectroscopy also fulfilled all the aforementioned conditions. For those reasons NN algorithms can be reliably tested to solve overlapping effects of quaternary mixtures. In this process, the mean prediction error (MPE; Eq. 3.8) was less than 2.5% and the mean correlation coefficient was greater than 0.95. MPE =

1 N

∑ n

rn − yn ⋅ 100 rn

(3.8)

Five applications of NNs in chemical engineering are presented in this section in order to show the wide applicability of NN models: (1) quantification of chemicals in quaternary mixtures; (2) solving overlapping effects in biosensors; (3) process control using two NNs; (4) filter of experimental signals; and (5) geographical classification of crude oils.

In Equation 3.8, N, yn, and rn are the number of observations, model estimation, and real value, respectively. Therefore, this approach can be adapted to deconvolute the contribution of each chemical. As a result, in the IL field, this approach is very interesting for further applications to digital control, or measurement devices (Torrecilla et al. 2007a, 2009a).

3.3.1. Quantification of Two Ionic Liquids (ILs), Heptane, and Toluene

3.3.2. Solving Overlapping Effects in Biosensor

ILs are a class of low-temperature molten salts consisting of an organic cation and inorganic or organic anion. In recent years, ILs have attracted increasing attention as replacements for conventional organic solvents in catalysis, separation processes, electrochemistry, and many other fields. Recently, ILs are being measured using interpolation in physicochemical properties (density, viscosity, refractive index, etc.), proton nuclear magnetic resonance, gas chromatograph, and others. However, these procedures are not adequate to measure/control on-line chemical processes (extraction, distillation, etc.) because of the time required to prepare samples. Given the importance of these processes, an analytical technique with a sample preparation time less than the sampling time of the process and a reliable

The major obstacle in amperometric detection of glucose in real samples is the interference arising from electrooxidize substances such as ascorbic and uric acids existing in a measured system. In Torrecilla et al. (2008a), an amperometric biosensor based on a colloidal gold—cysteamine—gold disk electrode with an enzyme glucose oxidase and a redox mediator, tetrathiafluvalene, co-immobilized atop the modified electrode was used for the simultaneous determination of glucose and ascorbic and uric acids in ternary mixtures. The concentrations of these chemicals were between 0 and 1 mM. In light of these results (MPE < 1.7% and R2 > 0.99), the NN model is able to solve the interferences between glucose and ascorbic and uric acids, without any chemical pretreatment.

44

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

3.3.3. Process Control Using Two NNs A control system based on a combination of two NNs was designed for the control of pH in the neutralization process of acidic liquid streams. The first NN is a plant neural model that predicts future pH values from past/present values of pH and valve stem positions and future values of valve stem positions. The second NN is a plant inverse neural model that calculates future valve stem positions from present/past values of pH and valve stem positions and future values of set points. The performance of the controller was studied first by numerical simulation. The controller was further implemented in a continuous stirred tank reactor in which the neutralization of acetic and propionic acids with sodium hydroxide was performed. The controller requires off-line and on-line learning of the direct NN and on-line learning of the inverse NN. The off-line training of the direct NN was made by numerical simulation of the neutralization process in openloop conditions. The adaptive properties of the controller were shown by means of the changes in time of the network weights. The effect of adaptation of weights when noise is present was explored and, for the range of conditions studied, the controller maintains its capacity of control for at least 100 hours. The controller can be applied to processes with a short delay time. The input layer of each NN should contain a sufficiently large number of nodes to take into account the number of past values necessary to cover the maximum delay time expected for the plant. If a maximum delay is assumed in the model but the actual delay time of the plant is zero or shorter than this maximum, the controller has practically the same performance as in the system with an actual delay equal to the maximum assumed. The experimental validation of the controller showed that the control efficiency is good only for perturbations that do not involve strong buffering changes (e.g., for small perturbations of set point, flow, and concentration of acids). Under these conditions, the NNs are able to adapt their weights on-line and to control perturbations never seen before. The controller performance was also good after successive buffering changes between the two

buffers. A detailed description of this controller can be found in the literature (Palancar et al. 1998).

3.3.4. Filter of Experimental Signals In this example, an integrated NN/thermogravimetric analyzer (TGA) is designed to filter a noisy signal from TGA equipment. The data collected (sample weight loss as function of sample temperature) from the TGA equipment were transferred into an NNtrained computer for modeling and filtering of the output. The NN/TGA filter is composed of a model (NNM) and a filter (NNF) of the system both based on feedforward networks. The NNM was designed and optimized to estimate the sample weight as a function of the temperature of the sample. Its topology consists of two input nodes (temperature of the sample and noisy signal) and output with noise. The NNF was designed to filter a noisy signal from TGA equipment. It was made up of two inputs that correspond to each variable and an output that calculates the signal without noise. To evaluate the competence of the optimized NN/TGA filter, a random noise was added to an experimental signal without noise. Such an integrated NN/TGA filter is capable of filtering the noisy TGA output signal based on the models and patterns created (MPE < 0.3%), without any previous phenomenological knowledge. This is very interesting for further applications to digital control, or measurement devices. This importance is even greater when the NN/TGA filter is applied to ILs where the knowledge of their properties is scarce. A detailed description of this filter can be found in the literature (Torrecilla et al. 2008b).

3.3.5. Geographical Classification of Crude Oils In environmental disasters caused by the spillage of crude oil, ascertaining the geographical origin of the crude oil by analysis is relatively simple. Fonseca et al. (2006) have designed an unsupervised network based on an SOM that can classify samples of crude oils on the basis of gas chromatography-mass

Chapter 3

Neural Networks: Their Role in High-Pressure Processing

spectroscopy (GC-MS) descriptors. To optimize the network, two databases were used; one of them comes from the Instituto Hidrográfico (Lisbon, Portugal) and the other one from the European Crude Oil Identification System (EUROCRUDE). Once the network was optimized, correct estimations were obtained in 70% of the tested sets in the verification process. Assembling four SOMs, this percentage increased to values higher than 94%. Equally good predictions were obtained for a small test set of weathered samples. This investigation added value to the GC-MS descriptors already in use for practical analytical work.

3.4. The Role of NNs in the Food Industry The main advantages and disadvantages related to NN applications in some food sectors such as olive oil, wine, waste treatment, food adulterations, and vitamins will be summarized here.

3.4.1. Determination of Polyphenolic Compound Concentration in Olive Oil Mill Wastewater The waste generated during the manufacture of extra virgin olive oil (EVOO) has a serious environmental impact due to its high content of organic substances (sugars, tannins, polyphenols, polyalcohols, pectins and lipids, etc.). It is known that caffeic acid (CA) and catechol (CT) are two of the major contributors to the toxicity of these wastes. Given their electrochemical characteristics, a laccase biosensor (LB) is commonly used to determine CA and CT. Because of the similarities in the oxidized species produced, the amperometric signal overlapping in the reduction voltammograms is high, and therefore, a powerful tool is required to differentiate this signal. Once the NN was optimized, it was externally validated using real concentrations taken from three different olive oil mills in Spain (Almendralejo, Badajoz; Martos, Jaén; Villarejo de Salvanes, Madrid). The MPE, Equation 3.8, was less than 0.5% and the correlation coefficient was higher than 0.99. These statistical results are even better and more selective than other non-portable commercial

45

analytical equipment. Therefore, the integrated NN/ LB system was shown to be an adequate approach to estimate simultaneously both hazardous chemicals in olive oil mill wastewater (Torrecilla et al. 2007b).

3.4.2. Discrimination of Wines Using unsupervised networks, Masoum et al. (2006) determined that polyphenol extracts from wines are not only characteristic of the grape variety but also characteristic of the year of harvest (vintage) and geographical origin. Learning vector quantization (LVQ) with orthogonal signal correction (OSCLVQ) and partial least squares discriminant analysis (PLS-DA) techniques were used. An LVQ network correctly classified the polyphenols extracted from the wines depending on three important characteristics: grape variety, type of soil, and year of harvest obtaining percentages of correct classification of 93, 89, and 85%, respectively. The OSC–LVQ method has been shown to be a slightly better pattern recognition technique in comparison with OSC–PLS-DA when applied to two-dimensional (2D) nuclear magnetic resonance spectra. This unsupervised network has not only been applied to classify wines but also to demonstrate the nonlinear relation between the aforementioned characteristics and the type of wines.

3.4.3. Identifying and Detecting Adulterations In recent decades, due to the high price of EVOO, an appreciable incidence of adulteration has been detected. The substitution or adulteration of EVOO with cheaper ingredients is not only an economic fraud but may also have severe health implications for consumers, an example being the Spanish toxic oil syndrome. Two unsupervised networks based on an SOM and LVQ network models have been tested for the identification of edible and vegetable oils and to detect adulteration of EVOO. A bibliographical database for developing learning process and internal validation, and six other different databases,

46

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

from bibliographical sources, were used to perform their external validation. The results showed that less than 3 and 5.5% of the samples were misclassified. The models’ performances were analyzed by the number of misclassifications. In the worst of the cases, the SOM and LVQ models were able to classify more than the 94% of the samples and to detect adulterations of EVOO with corn, soya, sunflower, and hazelnut oils when their oil concentrations were higher than 10, 5, 5, and 10%, respectively. In light of these results, both models showed to be adequate to classify these studied samples in 13 types of vegetable oils (Torrecilla et al. 2009b).

3.4.4. Determination of Carotenoid Concentrations in Foods Due to their antioxidant activity, carotenoids show a strong correlation between their intake and a reduced risk of some diseases, such as cancer, atherogenesis, bone calcification, eye degeneration, and neuronal damage. Given that the lycopene and β-carotene are active in the same region of UV-vis spectroscopy, their determination by linear algorithms is not suitable (Torrecilla et al. 2008c). In order to use a fast, simple analytical technique such as UV-vis spectroscopy, a nonlinear algorithm based on NN algorithms was applied. Using the absorbance of mixtures composed of lycopene and β-carotene with concentrations between 0.4 and 3.2 μg/mL and their respective concentrations distributed following an experimental design, the NN (feedforward network) was optimized. Once the NN model was optimized, NN/ UV-vis spectroscopy was applied to determine the concentration of both chemicals in foods related with tomatoes. The MPE was 50 times lower than when a linear model was used in place of the aforementioned nonlinear algorithm (MPE < 1.5%; R2 > 0.99).

3.5. NNs in High-Pressure Processes In light of the aforementioned properties and applications, a detailed description of several successful NN applications in the HP sector will be shown

highlighting their main advantages and the principal differences between NN and other known types of models. In the last decade, some research groups have investigated NN estimation capability partially connected with the food sector, for example, Si-Moussa et al. (2008) have estimated the vapor-liquid equilibrium of binary systems composed of carbon dioxide and chemicals that are applicable to the cosmetic and food sectors. But, to the best of our knowledge, in the food sector, the number of references directly connected with high pressure processing (HPP) processes and NN algorithms is scarce. Here, two successful applications of NN are detailed.

3.5.1. Estimation of Freezing Temperature of Foods at High Pressure Freezing/melting of aqueous systems is a phenomenon considered in many disciplines, not only food technologists but also in all sectors interested in the conditions of ice formation such as geologists, oceanographers, meteorologists, astrophysicists, biologists, and engineers. Compared with traditional processes such as air blast freezing, pressure-shift freezing produces smaller ice crystals with a uniform distribution in the whole product (Otero and Sanz 2000; Zhu et al. 2005). So, its texture is better preserved and drip losses upon thawing are reduced (Fuchigami et al. 2002; Le Bail et al. 2002; Urrutia-Benet et al. 2007). The main difficulties in developing this emerging technology from a laboratory to an industrial scale lie in process parameter optimization and energy cost minimization. To deal with these two problems, heat transfer should be modeled (Boillereaux et al. 2003; Otero and Sanz 2003; Pham 2006). For that purpose, the initial freezing point appears as a fundamental variable. At this point, all thermophysical properties change suddenly with the formation of ice (freezing process) or its melting (thawing process). For example, the specific heat capacity, also required for heat transfer simulation, changes from 2,046.5 J/ kg/°C in solid to 3,857.4 J/kg/°C in liquid state at the freezing point of water at 100 MPa (−8.8°C) (Otero et al. 2002a). The thermophysical properties

Chapter 3

Neural Networks: Their Role in High-Pressure Processing

depend on the ice content that can be calculated as a function of the initial freezing temperature (Fikiin 1998). Accurate knowledge of the initial freezing temperature is also essential to minimize the energy required to reach freezing/thawing conditions. Finally, its determination allows the discovery of metastable regions and explores new possibilities for process optimization (Schlüter et al. 2004; Urrutia Benet et al. 2006). Aqueous solutions and food models are often used to simulate food. In this way, the water (P, T )-phase diagram of Bridgman (1912) is routinely applied to decide on the operating conditions for high-pressure food freezing/thawing processes. In addition to the empirical equations proposed by Wagner et al. (1994), the freezing temperatures of water can be computed up to 20 GPa for different ice polymorphs. Sanz et al. (2004) have described qualitatively the phase diagram of water until about 10 GPa by means of computer simulation. For aqueous systems, the (P, T )-phase diagram has seldom been determined. The experimental initial freezing point at high pressure has been measured for sucrose and sodium chloride solutions by Guignon et al. (2005). Those authors provided and tested different equations to calculate the initial freezing temperature of food under pressure. As an alternative to traditional calculations, an NN was also designed and described in the manuscript. In addition, new experimental data were presented for gelatine gels (with and without sucrose), for sucrose solutions with hydrocolloids (guar gum or xanthan and locust bean gum mix), for tylose gel as food models, and for broccoli. These values, together with data collected from the literature, were used to evaluate the aforementioned linear and nonlinear algorithms (Guignon et al. 2008). Different models have been used in order to predict the freezing temperature as a function of pressure: empirical models such as a polynomial equations, the Simon-like equation, and the linear additive model; NN models; and theoretical models such as Raoult’s modified law and the Robinson and Stokes’ equation extended to high pressure conditions. From the results obtained it was shown that NNs are an interesting alternative to obtain the freezing temperature at any given pressure of any aqueous

47

system knowing only its water activity (within its working range). Calculated freezing temperatures differ from those measured by approximately 0.3°C. The NN is even able to reproduce literature data within a 95% confidence interval when the water activity of the substance is within the training range.

3.5.2. Application of NN on Thermal/ Pressure Processing Recently, HPP technology has been introduced in the food industry mainly to control the microbiological and/or enzymatic activity of food products. As it is known, hydrostatic pressure acts instantaneously and uniformly throughout a mass of waterrich food independently of its size, shape, or composition, which has been described as a possible advantage of HPP processing over traditional thermal processing. On the other hand, it is important to recognize that pressurization/depressurization induces a temperature change due to the work of compression/expansion in both the food and the pressure-transmitting fluid. The value and sign of these temperature changes depend on pressure and temperature conditions, the rate of compression/ decompression, the composition of the food, and the type of the pressure-transmitting fluid (Otero et al. 2000). For instance, in vegetable oils, a pressure change of 200 MPa induces temperature changes higher than 20°C (Shimizu 1992). Therefore, as each pressure treatment implies a thermal effect, pressure and temperature are not independent. Moreover, during compression, temperature changes in the sample and the pressure-transmitting fluid will be different due to their different thermophysical properties. As a result, during the holding time at high pressure, heat transfer phenomena are taking place between the sample, the pressure medium, the pressure vessel, and the surroundings leading to thermal gradients in the treated product. Hence, when high hydrostatic pressure is applied to foodstuffs to control their microbiological and/or enzymatic activity, it is important to bear in mind that the obtained results depend both on the pressure applied and the temperature during the treatment.

48

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Taking into account that typical pressure treatment times for extending the shelf life of food could range from a few minutes up to half an hour, the whole pressure treatment or—at least an important part of it—can take place at a temperature quite different from the initial temperature (Otero et al. 2002b). A pressure increase of 450 MPa may result in a sample temperature increase of almost 15°C, and nearly 15 minutes (depending on the insulation properties of the vessel) is necessary to re-reach the target temperature in the sample. If the treatment time to be used were shorter, the complete treatment would have been performed at a higher temperature than desired. Hence, it is essential to know both the thermal and the pressure evolution in the treated sample to obtain reproducible results. Unfortunately, this has not always been recognized and many researchers do not control/document temperature conditions during pressure treatments and attribute the obtained results directly to the pressure applied and the initial product temperature. Some authors have shown the importance of the thermal effects on inactivation studies at HPP conditions (De Heij et al. 2002. Ting et al. 2002) and recommended reporting the pressure and temperature history in all studies. If this is not possible, those authors suggest reporting the extreme values of pressure, temperature, and time as indicators of the pressure and temperature evolution. Another alternative is based on the modelization of the temperature and/or pressure profile. Some attempts have been made by different authors to model the thermal behavior of food during HPP (Otero and Sanz 2003). Among the many difficulties met in this way, the main problem of physically based models is the almost complete lack of appropriate thermal properties of foodstuffs and pressuretransmitting fluid at high pressure to date. Trying to overcome this obstacle, Torrecilla et al. (2004) have shown the capability of NNs in modeling the thermal behavior of food during HPP without the requirement of thermophysical properties. As a consequence of this work and in trying to overcome the aforementioned difficulties, an NN able to predict the temperature changes in a food sample after pressurization and the time needed for thermal

re-equilibration in the HPP system was designed (Torrecilla et al. 2005). Two years later, the improvement of the NN performance by varying the number of neurons in the hidden layer, selecting the architecture and the learning coefficient with the highest predictive ability, and taking into account the effect of the training algorithm was carried out (Plumb et al. 2005; Torrecilla et al. 2007c). A priori knowledge of the above-mentioned variables is very useful in designing and optimizing pressure food processes because these variables provide, together with the programmed pressure, a proper notion of the thermal evolution of the sample during the pressure treatment (Ting et al. 2002). In Torrecilla et al. (2007c), a correct selection of the training algorithm and an optimized NN for thermal/pressure food processing has been used. In this work, different NNs trained with different learning functions have been assessed with respect to their predictive ability. From this assessment, the Levenberg-Marquardt algorithm was identified as the best learning function. The gradient descent, conjugate gradient, and Bayesian regularization algorithms implemented by MATLAB™ (The MathWorks Inc., Natick, MA) and used in previous works gave poorer results. In Torrecilla et al. (2005), an NN was developed for thermal/pressure food processing without any previous selection of the most adequate training algorithm, that is, assuming the default option of the NN computational package (batch gradient descent with a user-defined learning rate). In this case, the MPEs (1.6 and 0.95% for temperature and time predictions, respectively) were significantly higher than those found in Torrecilla et al. (2007c). Also, the correlation coefficients (R2) clearly show better predictive ability of the NN trained after a correct selection of the learning function (0.99 and 0.99 vs. 0.98 and 0.99 for temperature and time predictions, respectively). Therefore, it is evident that a correct selection of the training algorithm allows maximizing the predictive ability of the NN, as Plumb et al. (2005) pointed out. Thus, not only the network architecture (as is frequently mentioned in the literature), but also the training algorithm must be optimized in order to obtain the most predictive models.

Chapter 3

Neural Networks: Their Role in High-Pressure Processing

Moreover, once the NN is optimized, a large number of combinations of processing conditions can be evaluated after a short computing time and this gives an important advantage when designing and optimizing high-pressure processes in the food industry (Torrecilla et al. 2007c).

3.6. A Macroscopic Model for Thermal Exchange in an HPP System The importance of macroscopic models is based on their applicability to many systems. For instance, macroscopic models can be used (Otero et al. 2002a) to analyze all the thermal exchanges in an HPP system, evaluate the impact of the different components, and therefore allow regulation of the temperature in a given process.

3.6.1. Simple Representation and Modeling of Thermal Processes in HPP Treatments To simplify processes and for practicality purposes, it could be assumed that thermal processes are governed by sets of linear or nonlinear algebraic equations and linear and nonlinear differential equations. In those cases there are several more or less adapted tools for its mathematical modeling. Among them there is the SIMULINK package, which is a toolbox of MATLAB™. It is an icon-driven interface for the construction of a block diagram representation of processes. A simple graphical representation of a process composed of an input, the system, and an output is shown in Figure 3.4. Each single part or component of an HPP system involved in a heat transfer process can also be represented by a block diagram. Considering a thermoregulation device as a single part, this device should be connected to an HP vessel in order to maintain a

Input

System

Figure 3.4. Block diagram of a process.

Output

49

programmed temperature in the vessel during HPP (Figure 3.5). How the fluid (water in this case) is continuously circulating through a closed circuit from the thermoregulation bath is shown in Figure 3.5. The block diagram of the thermoregulation bath working with two independent coolers and a heater is represented in Figure 3.6. Let us suppose that it is intended to reach 60°C as the temperature in the sample situated in the HPP vessel. The heat flow from the walls of the thermoregulation fluid container is also represented. In addition, the external temperature of the bath (20°C in this case) and the temperature of the sample situated in the HPP vessel are both accounted for. In order to evaluate each partial heat balance, a set of different mathematical operators is shown in Figure 3.6. Each integration step, the current temperature of the water circulating through the heat exchanger, as shown in Figure 3.5, is checked, bearing in mind the programmed temperature for the sample in the HPP vessel. Suitable controls on the thermoregulation activities have to be performed accordingly, for example, by switching the coolers “on” or “off” during the continuous running of the heater. Assuming that the steady-state temperature of the sample is the same as the corresponding one for the pressure-transmitting fluid Tf1(°C) that is situated in the HPP vessel and considering also that the vessel is long enough, the rate of heat variation q into the vessel is governed by Equation 3.9 q=

2π H (T f 1 − T f 2 ) ⎡ ⎛ ri +1 ⎞ ⎤ n ln ⎜ ⎢ 1 ⎝ ri ⎟⎠ 1 ⎥ + + ⎢ ⎥ λi r2 h2 ⎥ ⎢ r1h1 i =1 ⎢⎣ ⎦⎥

(3.9)

∑

where H is the height of the vessel (m), Tf 2(°C) the temperature outside the vessel, ri and λi the radius (m) and the thermal conductivity (Wm–1/°C) of the different layers of the vessel, respectively, h1 and h2 the inside and outside surface heat transfer coefficients (Wm–2/°C) of the vessel, respectively, and r1 and r2 the inside and outside radius of the vessel, respectively (Kakaç and Yener 1985).

50

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

High-pressure cylinder Second steel cylinder Thermal isolation 100 Thermoregulation circuit

180 300

300 340 Pressuretransmitting fluid High-pressure pump

Thermoregulation bath

Figure 3.5. Thermoregulation as a part of a high-pressure system.

On the other hand, the temperature variation due to a compression or an expansion also has to be considered. Such effects are governed by dT α ⋅ Tk ⋅ V = dP Cp

(3.10)

being α (°C–1) the thermal expansivity coefficient of the sample filling the HPP vessel, V its specific volume (m3/kg, the inverse of the density), and Cp its specific heat capacity (J/kg/°C). Taking this contribution into account (Otero et al. 2007), the initial block diagram shown in Figure 3.6 has to be extended accordingly as represented in Figure 3.7.

3.6.2. Using Macroscopic Models in HPP This model takes into account the thermal exchanges performed in the thermoregulation bath and in the pipes between the bath and the coil surrounding the HPP vessel. In fact, the model represented in Figure 3.7 (Otero et al. 2002a) corresponds to the case of a pilot plant HPP unit (ACB GEC Alsthom, Nantes, France). Its cylindrical vessel has a diameter of 100 mm, is 300 mm tall, and has a net volume of 2.35 L. In this case, the equipment was filled with water that acted simultaneously both as the sample and as the pressure-transmitting fluid.

Chapter 3

Neural Networks: Their Role in High-Pressure Processing

20 External temperature + + + –

K–

1 s

Water

Integrator

Sum 1 –K Water

– + Sum 3 –

–K Heater

+

60

Thermostat Sum 2 Cooler n° 1

Temperature of the sample 1

On/off Cooler n° 2

0 On/off

Figure 3.6. Block diagram of a thermoregulation bath.

Before solving the model represented in Figure 3.7, it is necessary to input the thermophysical properties of the different components taking part in the HPP system as a function of pressure and temperature. However, currently, to find all those thermophysical properties is not easy. In the case of food, these properties could be approximated by using the methods described in Otero et al. (2010). In the case of the pressure-transmitting fluid, it could help to use data from Otero et al. (2002a) for water and data from Guignon et al. (2008) for water and for other fluid. The last task to be performed is to integrate the differential equations taking part in the model. For this purpose, the Euler or other suitable method can be chosen. The model is able to predict the evolution of temperature versus time in different points of the HPP system such as inside the HP vessel, at the entrance, at the exit of the surrounding coil, and inside the thermoregulation bath (Otero et al. 2002a).

51

Once a macroscopic model is working, contrasting a variety of its numerical results with the behavior of the corresponding experimental results from real high-pressure processes is required. Only after accurate results are obtained from both kinds of data can the collection of an important amount of results (using the macroscopic model with conditions similar to those previously checked) become simple. In relation to this, concerns regarding the approaches related to steady-state processes and/or with linear behavior for heat transfer that have been initially adopted to build the macroscopic model are always required. In fact, results from this modeling approach have been obtained in order to provide data for building NNs in HPP (Torrecilla et al. 2004, 2005, 2007c). Apart from that, a better description about what happens during HPP is achieved if not only pure heat transfer processes are considered but also more complex processes with energy, mass, and momentum interactions (Otero et al. 2007). Unfortunately, to improve the model described in Section 3.6 by accounting for all those complex processes is not easy at all.

3.7. Conclusions In the field of numerical modeling of chemical and food processing operations, two powerful tools, namely artificial NNs and macroscopic models, have been described in this chapter. NNs can be used for the prediction of important characteristics of a process without the need for assumptions in nonlinear mathematical relations between dependent and independent variables. The examples presented in this chapter show, for example, how composition of chemicals and foods upon processing and process parameters and variables (e.g., freezing temperature during HPP) can be reliably estimated. Besides, macroscopic models also appear to be an interesting tool: Although the creation of macroscopic models requires notable assumptions and simplifications, they have proven to be useful for the prediction of important process data in HPP of foods. A characteristic example is the determination of the

52 1/z

1/s

+ – ++

-K-

Mux

1/s

+ –

+ –

++

-K-

Steel

+

+

+

+

+ -K-

Exit temp. at the coil Entrance tubeambient heat flow +

+ –

-K-

+ -K- –

-K-

1/s

Thermost.

Thermor. bath

Compressor

-K-

100

Bath-amb. flow -K-

-K-

1 on/of 1/0

– +

+ –

Refrig. circuit Exit tube-ambient heat flow + -K- –

Heat flow through the wall insulators

Heat flow at the top

Heat flow at the bottom

Entrance tube in the surrounding coil

-K

– –

Steel-coil heat flow

-K-

Medium temp. in the coil

Entrance temp. at the coil

-K-

1/s

Steel-coil heat flow Water temp. in the Steel vessel temp.

Exit temp. at the coil

-K-

Temp. at the entrance and exit of the coil

1 – z Delay

MATLAB Function dT/dp

-K-

+ +

1/s

Figure 3.7. Extended model for the simulation of thermal exchanges in high-pressure equipment.

Init. pressure Delay

Pressure

Final pressure

Pressure

Steel-water heat flow

Steel-water heat flow

Target temp.

39

Bath temp.

Tamb Ambient temp.

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Neural Networks: Their Role in High-Pressure Processing

temperature profiles inside the HPP vessel for a specific set of process conditions. Macroscopic models allow the fast and reliable generation of the resulting process variables caused by changes in the process parameters, such as the initial temperature of the vessel or the boundary temperature of the system. The combination of the aforementioned tools opens a wide spectrum of applications that could be successfully deployed not only in the academia but more importantly for industrial applications as well.

Acknowledgments This work has been supported by the National Plan of Spanish I+D+I of the Ministry of Education and Science (MEC) through the projects AGL2007-63314/ALI, the CSD2007-00045 MALTA CONSOLIDER-INGENIO 2010, and the Madrid Community through the project S2009/PPQ-1551.

Notation Cp H h i j k Lr MPE MSE N n NN P q r T V w x y α λ

Specific heat capacity, J/kg/°C Height, m Surface heat transfer coefficient, Wm−2/°C Input layer Hidden layer Output layer Learning rate Mean prediction error Mean square error Number of output neurons Iteration of a cycle of the learning process Neural network Pressure Rate of heat transfer, W Radius, m Temperature, °C Specific volume, m3/kg Weight of neural networks Input value into neuron Output value of the neuron Thermal expansivity coefficient, °C−1 Thermal conductivity, Wm−1/°C

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References Arbib MA. 2003. The Handbook of Brain Theory and Neural Networks, 2nd ed. Cambridge: The Massachusetts Institute of Technology. Boillereaux L, Cadet C, Le Bail A. 2003. Thermal properties estimation during thawing via real-time neural network learning. J Food Eng 57:17–23. Bridgman PW. 1912. Water, in the liquid and five solid forms, under pressure. Proceedings of the American Academy of Arts and Sciences, XLVIII(13):439–558. Chen CH. 1996. Fuzzy Logic and Neural Network Handbook. New York: McGraw-Hill. De Heij W, Van Schepdael L, Van Den Berg R, Bartels P. 2002. Increasing preservation efficiency and product quality through control of temperature distributions in high pressure applications. High Press Res 22(3-4):653–657. Demuth H, Beale M, Hagan M. 2005. MATLAB User ’s Guide, V 4.0: Neural Network Toolbox. Natick, MA: MathWorks Inc.. Fikiin KA. 1998. Ice content prediction methods during food freezing: A survey of the Eastern European Literature. J Food Eng 38:331–339. Fine TL. 1999. Feedforward Neural Network Methodology. New York: Springer-Verlag. Fonseca AM, Biscaya JL, Aires-de-Sousa J, Lobo AM. 2006. Geographical classification of crude oils by Kohonen selforganizing maps. Anal Chim Acta 556:374–382. Fuchigami M, Ogawa N, Teramoto A. 2002. Trehalose and hydrostatic pressure effects on the structure and sensory properties of frozen tofu (soybean curd). Innov Food Sci Emerg Technol 3:139–147. Guignon B, Otero L, Molina-García AD, Sanz PD. 2005. Liquid water-ice I phase diagrams under high pressure: Sodium chloride and sucrose models for food systems. Biotechnol Prog 21(2):439–445. Guignon B, Torrecilla JS, Otero L, Ramos A, Molina-García AD, Sanz PD. 2008. The initial freezing temperature of foods at high pressure. CRC Crit Rev Food Sci Nutr 48:328–340. Hopfield JJ. 1982. Neural networks and physical systems with emergent collective computational abilities. Proc Natl Acad Sci U S A 79:2554–2558. Hopfield JJ. 1984. Neural networks and physical systems with graded response have collective properties like those of two state neurons. Proc Natl Acad Sci U S A 81:3088– 3092. Huang Y, Kangas LJ, Rasco BA. 2007. Applications of Artificial Neural Networks (ANNs) in Food Science. CRC Crit Rev Food Sci Nutr 47:113–126. Kakaç S, Yener Y. 1985. Heat Conduction, 2nd ed. Berlin: Springer-Verlag. Kohonen T. 1987. Self-Organization and Associative Memory, 2nd ed. Berlin: Springer-Verlag. Le Bail A, Chevalier D, Mussa DM, Ghoul M. 2002. High pressure freezing and thawing of foods: A review. Int J Refrigeration 25:504–513.

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Masoum S, Bouveresse DJR, Vercauteren J, Jalali-Heravi M, Rutledge DN. 2006. Discrimination of wines based on 2D NMR spectra using learning vector quantization neural networks and partial least squares discriminant analysis. Anal Chim Acta 558:144–149. McCulloch WS, Pitts W. 1943. A logical calculus of the ideas immanent in nervous activity. Bull Math Biophys 5:115–133. Minsky ML, Papert S. 1969. Perceptrons: An Introduction to Computational Geometry. Cambridge, MA: MIT Press. Otero L, Sanz PD. 2000. High-pressure shift freezing. Part I: Amount of ice instantaneously formed in the process. Biotechnol Prog 16:1030–1036. Otero L, Sanz PD. 2003. Modelling heat transfer in high pressure food processing: A review. Innov Food Sci Emerg Technol 4:121–134. Otero L, Molina-García AD, Sanz PD. 2000. Thermal effect in foods during quasi-adiabatic pressure treatments. Innov Food Sci Emerg Technol 1:119–126. Otero L, Molina AD, Sanz PD. 2002a. Some interrelated thermophysical properties of liquid water and ice. I. A user-friendly modeling review for food high-pressure processing. CRC Crit Rev Food Sci Nutr 42(4):339–352. Otero L, Molina-García AD, Ramos AM, Sanz PD. 2002b. A model for real thermal control in high-pressure treatment of foods. Biotechnol Prog 18(4):904–908. Otero L, Ramos AM, de Elvira C, Sanz PD. 2007. A model to design high-pressure processes towards an uniform temperature distribution. J Food Eng 78(4):1463–1470. Otero L, Guignon B, Aparicio C, Sanz PD. 2010. Modeling thermophysical properties of food under high pressure. CRC Crit Rev Food Sci Nutr 50(4):344–368. Palancar MC, Aragon JM, Torrecilla JS. 1998. pH-control system based on artificial neural networks. Ind Eng Chem Res 37:2729–2740. Pham QT. 2006. Modelling heat and mass transfer in frozen foods: A review. Int J Refrigeration 29:876–888. Plumb AP, Rowe RC, York P, Brown M. 2005. Optimisation of the predictive ability of artificial neural network (ANN) models: A comparison of three ANN programs and four classes of training algorithms. Eur J Pharm Sci 25:395–405. Raju KS, Kumar DN, Duckstein L. 2006. Artificial neural networks and multicriterion analysis for sustainable irrigation planning. Comput Oper Res 33:1138–1153. Rumelhart DE, Hinton GE, Williams RJ. 1986. Learning representations by back-propagation errors. Nature 323:533–536. Sanz E, Vega C, Abascal JLF, MacDowell LG. 2004. Phase diagram of water from computer simulation. Phys Rev Lett 92(25 I):255701–255701. Schlüter O, Urrutia Benet G, Heinz V, Knorr D. 2004. Metastable states of water and ice during pressure-supported freezing of potato tissue. Biotechnol Prog 20:799–810. Shimizu T. 1992. High-pressure experimental apparatus with windows for optical measurements up to 700 MPa. High Press Biotechnol 224:525–527.

Si-Moussa C, Hanini S, Derriche R, Bouhedda M, Bouzidi A. 2008. Prediction of high-pressure vapor liquid equilibrium of six binary systems, carbon dioxide with six esters, using an artificial neural network model. Braz J Chem Eng 25(01): 183–199. Ting E, Balasubramaniam VM, Raghubeer E. 2002. Determining thermal effects in high-pressure processing. Food Technol 56(2):31–35. Torrecilla JS, Otero L, Sanz PD. 2004. A neural network approach for thermal/pressure food processing. J Food Eng 62:89–95. Torrecilla JS, Otero L, Sanz PD. 2005. Artificial neural networks: A promising tool to design and optimize high-pressure food processes. J Food Eng 69:299–306. Torrecilla JS, Fernández A, García J, Rodríguez F. 2007a. Determination of 1-ethyl-3-methylimidazolium ethylsulfate ionic liquid and toluene concentration in aqueous solutions by Artificial Neural Network/UV spectroscopy. Ind Eng Chem Res 46:3787–3793. Torrecilla JS, Mena ML, Yáñez-Sedeño P, García J. 2007b. Quantification of Phenolic Compounds in Olive Oil Mill Wastewater by Artificial Neural Network/Laccase Biosensor. J Agric Food Chem 55:7418–7426. Torrecilla JS, Otero L, Sanz PD. 2007c. Optimization of an artificial neural network for thermal/pressure food processing: Evaluation of training algorithms. Comput Electron Agric 56:101–110. Torrecilla JS, Mena ML, Yáñez-Sedeño P, García J. 2008a. A neural network approach based on gold-nanoparticle enzyme biosensor. J Chemometrics 22:46–53. Torrecilla JS, Fernández A, García J, Rodríguez F. 2008b. Design and optimisation of a filter based on Neural Networks. Application to reduce noise in experimental measurement by TGA of thermal degradation of 1-Ethyl-3-methylimidazolium ethylsulfate ionic liquid. Sens Actuators B Chem 133: 426–434. Torrecilla JS, Cámara M, Fernández-Ruiz V, Piera G, Caceres JO. 2008c. Solving the spectroscopy interference effects of βcarotene and lycopene by neural networks. J Agric Food Chem 56:6261–6266. Torrecilla JS, Rojo E, García J, Oliet M, Rodríguez F. 2009a. Determination of toluene, n-heptane, [emim][EtSO4], and [bmim][MeSO4] ionic liquids concentrations in quaternary mixtures by UV-vis spectroscopy. Ind Eng Chem Res 48:4998–5003. Torrecilla JS, Rojo E, Oliet M, Domínguez JC, Rodríguez F. 2009b. Self-organizing maps and learning vector quantization networks as tools to identify vegetable oils. J Agric Food Chem 57:2763–2769. Urrutia Benet G, Chapleau N, Lille M, Le Bail A, Autio K, Knorr D. 2006. Quality related aspects of high pressure low temperature processed whole potatoes. Innov Food Sci Emerg Technol 7:32–39. Urrutia-Benet G, Balogh T, Schneider J, Knorr D. 2007. Metastable phases during high-pressure—Low-temperature

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processing of potatoes and their impact on quality-related parameters. J Food Eng 78:375–389. Wagner W, Saul A, Pruss A. 1994. International equations for the pressure along the melting and along the sublimation curve of ordinary water substance. J Phys Chem Ref Data 23: 515–525.

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Werbos P. 1974. Beyond regression—New tools for prediction and analysis in the behavioral sciences. PhD thesis, Harvard University. Zhu S, Ramaswamy HS, Le Bail A. 2005. Ice-crystal formation in gelatine gel during pressure shift versus conventional freezing. J Food Eng 66:69–76.

Chapter 4 Computational Fluid Dynamics Applied in High-Pressure Processing Scale-Up Cornelia Rauh and Antonio Delgado

4.1. Introduction The present contribution deals with the use of computational fluid dynamics (CFD) for simulating and scaling up high-pressure processes. Thus, it appears convenient to address briefly three different thematic blocks here. First, it is the goal of this contribution to discuss high-pressure processes to the level that provides an introductory understanding of their most relevant features. This allows the exclusion of such high-pressure fields in which similar problems with scaling up occur but are considered beyond the framework of the present scholarly piece. The second thematic block is dedicated to CFD. As completeness in the treatment of such a subject is neither aimed at nor achievable, only particular topics are elucidated here. Last but not least, the third thematic block is related to the necessity of including CFD in the investigation of high-pressure processes and its role in scaling up. The effect of high pressure on matters of different constitution and origin is currently the objective of very different investigations in different fields such as physics, chemistry, applied mathematics, biology, oceanography, astronomy, material sciences, technical fabrication, medicine, and biotechnology as well as food sciences and technology. Although the basic

effects of high pressure have a great deal in common, it appears convenient to focus the present contribution on biomatter, that is, on matter of biological origin such as food and biotechnological substances. In the addressed field, treatment with media pressurized up to several tens of MPa, that is, pressure near the critical point is performed in order to exploit exceptional physicochemical properties. Among them a substantial decrease in viscosity and a drastic increase in diffusion constant could be observed. In this context, surface energies drop and extraction of hardly soluble systems can be realized even on technical scale (Eckert et al. 1996; Higashi et al. 2001; Jaeger et al. 2002; Sarrade et al. 2003). Despite its high potential, this subject is considered out of the framework of the present chapter. Also, the research activities that took place in the field of marine biology, for example, under deep-sea conditions, (see e.g., Zobell and Morita 1957; Kato et al. 1998) are not considered here. Although high-pressure treatment of biomatter has drastically gained a high interest only in the last decades, the very first research activities in this field were reported as early as in the late nineteenth century. Hite (1899) could observe a reduction of microorganisms in highly pressurized milk. A few years later, the Nobel laureate Bridgman (1914)

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

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reported on the irreversible denaturation of albumen at a pressure level of 700 MPa. However, the revival of intensive research activities in food applications started in the past decades in Japan (Deuchi and Hayashi 1992; Fuchigami et al. 1997, 1998a, 1998b; Hayakawa et al. 1998; Yoshioka et al. 1999; Suzuki 2002, 2003). It has been substantially emphasized by the increasing interest in industrial production in which the peculiarity of high pressure of affecting both molecular and cellular processes is specifically used. In the present high-pressure treatment of food and biotechnological materials, high-pressure processing is considered as an emerging method with an extremely wide spectrum of possible applications. Basically, this is due to the fact that pressure is a fundamental thermodynamic quantity, comparable to temperature or chemical potential. In fact, the results of sufficiently applying high pressures to biomatter prove to be extremely manifold as pressure influences mechanisms and kinetics as well as structures and generates novel phenomena. This reveals, in turn, pressure as a “basic tool for research, specific structure creation and process design” (Knorr 1996). Knorr (1996) compares the high-pressure treatment of food and biosubstances even with the introduction of “a new dimension.” Most recently, Delgado et al. (2007) have shown that high-pressure processes depend on the “complete process trajectory,” which, in turn, must consider not only the temporal but also the spatial dimensions. Thus, high-pressure processes can only be treated and controlled adequately by considering the available initial (i.e., the initial activity of enzymes or viability of microorganisms) and boundary conditions (temperature regime, applied pressure profile, etc.) and the physicochemical interactions of all acting components (biomatter, high-pressure pumps). Thus, system theory (Delgado et al. 2007) must be applied that allows for considering (1) spatiotemporal effects in connection with (2) unavoidable natural variations in the material constitution and the biological activity of the pressurized biomatter and (3) complex geometry of the pressurizing system, treatment chamber, and, if available, distribution of packaged food units, and, last but not least, product

and process inhomogeneities. Such a system treatment requires obviously tools such as CFD, which enable managing spatiotemporal distributions and, therefore, local changes resulting in heterogeneities. CFD has achieved a high level of recognition in engineering, natural sciences, and even medicine. This is primarily a consequence of the cross-sectional character of fluid mechanics in various disciplines such as aerospace and microgravity (Dreyer et al. 1994; Delgado et al. 1996; Kowalczyk and Delgado 2007a), aerodynamics and aviation (Becker et al. 2002), automotives engineering (Al-Hamamre and Trimis 2009), transport techniques (Khier et al. 2000), process engineering (Raufeisen et al. 2008), production technologies (Epple et al. 2009; Rauh et al. 2009), material sciences (Frohnapfel et al. 2007), environmental sciences (Diez et al. 2007), astrophysics (Peltier 1989), as well as human and animal biology (Kowalczyk and Delgado 2007b; Becker et al. 2008, 2009). On the other hand, the employed CFD has experienced a drastic increase recently due to the development of new architectured super computers and novel mathematical algorithms that allow for the first time the calculation of constitutively and geometrically very complex systems. These conditions are particularly given in highpressure processes up to several hundreds of MPa with biomatter capable of flowing and, more crucially, showing pressure-induced phase transition. In contrast to the opinion prevailing in the former literature, Delgado et al. (2007) as well as Delgado and Hartmann (2003) have shown that high-pressure treatments act, at least under nonacademic conditions, instantaneously but not homogeneously. More crucial pressurization always generates a deformation of the biomatter, that is, high-pressure treatment always induces a forced field of movement. Thus, in the case of matter capable of flowing, this has to generate forced convection. Due to the fact that during compression mechanical energy is converted into thermal energy, the temperature of the biomatter is elevated (with the exception of the most academic case of isothermal compression). Hardly controllable thermal diffusion, for example, at the wall of the treatment chamber induces therefore natural convection in addition. Thus, high-pressure

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processes expose thermofluiddynamical effects as a rule. The existence of convection phenomena has been ignored for a long period of time or even challenged in the literature. But, for the first time, Pehl and Delgado (1999) as well as Delgado and Hartmann (2003) and Delgado et al. (2007) have conclusively proven their existence experimentally, theoretically, and numerically. In addition, considerations of thermofluiddynamical phenomena and mechanisms in connection with molecular and cellular reactions as well as mass transport and phase transition are treated in the most recent literature (Yasuda and Mochizuki 1992; Sanz et al. 1997; Sanz and Otero 2000; LeBail et al. 2003; Luscher et al. 2004; Baars et al. 2007; Delgado et al. 2008a, 2008b). The appearance of transport phenomena in diffusive and convective fields makes CFD essential for making progress in understanding, simulating, controlling, and automating high-pressure processes. More than that CFD provides a suitable platform for investigating process homogeneity and scalability.

4.2. Thermofluiddynamic Phenomena under High-Pressure Conditions In general, CFD is based on mathematical balancing models that, for a given, fixed balance volume, express the temporal change of the quantity of interest such as mass, momentum, and energy due to (1) diffusive and convective transport through the surface of the balance volume and (2) generation within the volume. In addition, the constitutive equations as well as the initial and boundary conditions must be available for ensuring completeness of the mathematical model and, thus, its solvability. As a rule, balancing leads to a system of integral or differential equations that are solved classically by using numerical finite approaches such as finite differences, volumes, elements, or boundary elements (Ferziger and Peric 2002). However, when describing thermofluiddynamical phenomena and mechanisms under high pressure, the “state-of-the-art” proves to be not sufficient for formulating all balance equations required. This is the case particularly with regard to the constitution

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of the biomatter and high-pressure-induced molecular and cellular reactions. Thus, advanced approaches that enable managing other sources of knowledge (Kilimann et al. 2006) must be implemented for completing the knowledge base. The present chapter focuses on classical CFD procedures. As a consequence of this, no considerations regarding hybrid methods are discussed here that enable managing further knowledge; these methods are the subject of intensive investigations in the present (Kilimann et al. 2006). However, for the sake of achieving better access to modeling and simulation by CFD, it appears convenient to discuss first some particular high-pressure-induced phenomena. Of course, this is done from the point of view of applying CFD for simulating high-pressure treatment.

4.2.1. Transport of Mass, Momentum, and Energy Literature reviews concerning this topic have been published by Hendrickx et al. (1998) and Otero and Sanz (2003) for particular aspects, and by Delgado et al. (2008b) in a more general sense. Mass, momentum, and energy can be assumed to be conservative quantities. Thus, their corresponding fields are determined by constitutive, state, and transport parameters. Additionally, the transport of mass, momentum, and energy is often strongly interrelated in flow fields. This is especially the case in high-pressure systems in which the peculiarity of high pressure for affecting both molecular and cellular processes is specifically used. Basically, the mechanical force acting during pressurization alters the molecular equilibrium. This may generate not only a modification of the molecular and cellular interactions but also an increase in the energy level (Heremans 1982; Denys et al. 1997, 2000b; Cheftel et al. 2000; Fachin et al. 2002; Doster and Friedrich 2004; Kowalczyk et al. 2004, 2005; Kowalczyk and Delgado 2007a; Kulisiewicz et al. 2007). In turn, this could lead to both a modification of the matter pressurized and an increase in temperature, if the transfer of energy from the pressure chamber to the surroundings is inhibited by, for example, the chamber walls. Thus, in the sense of system theory

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(Delgado et al. 2007), the interaction of all components of the high-pressure system has an impact on the occurrence of thermofluiddynamical fields. This corresponds to a direct influence on product and process inhomogeneities, which have to occur almost always (Denys et al. 1997, 2000a; Sanz et al. 1997; Pehl and Delgado 1999; Otero et al. 2000; Pehl et al. 2000; Hartmann 2002; Hartmann and Delgado 2002; Hartmann et al. 2003; Otero and Sanz 2003; Kowalczyk et al. 2004). Here, exceptions are provided by the limiting case of a single, pure liquid component under isothermal and adiabatic conditions. However, the isothermal change of state requires a dissipative transfer of the complete pressurizing energy to the surroundings. But this can be performed only approximately by extreme slow compression due to the finite heat conductivity and the nonvanishing thermal capacity of the pressure chamber. Thus, isothermal pressurization would lead to an inapplicable long treatment time and to substantially reduced energy process efficiency. Furthermore, it is connected to the loss of the benefits wished in certain cases (e.g., in the inactivation of microorganisms) resulting from the synergetic action of the effect of pressure and temperature increase in general occurring during pressurization. Last but not least, nonvanishing thermal conductivity and finite thermal capacity of the pressurizing system inhibit additionally the strict occurrence of an adiabatic change and lead at the same time to the generation of temperature inhomogeneities. As shown by Delgado and Hartmann (2003), the existence of temperature gradients perpendicular to the direction of gravity represents a necessary and sufficient condition for the initiation of a flow in matter capable of flowing. As a consequence of this, process inhomogeneities with the corresponding transport of matter, momentum, and energy occur. Additionally, matter of biological origin consists, as a rule, of several components and phases with different physical and chemical behavior. This includes material heterogeneities that, in turn, exclude per se a homogeneous treatment (Delgado and Hartmann 2003). Furthermore, inhomogeneities induce timedependent equalization actions in the field that could substantially have different time and length scales

(Pehl and Delgado 1999; Pehl et al. 2000; Kowalczyk et al. 2004; Baars et al. 2007; Kowalczyk and Delgado 2007a, 2007c). As a consequence of this, the actual state of the processed biomatter depends on the whole “treatment history” and, thus, on the exchange of momentum, energy, and mass (Delgado et al. 2007). Therefore, any scaling up must aim at achieving an identical or, at least, a comparable treatment history. The high-pressure processes considered here are characterized by the flowability of the biomatter. Thus, in general, the latter property is given for a biomatter consisting of the liquid aggregate state. Hereby, the liquid component must be considered as compressible due to the high values of the pressure applied. This represents a crucial contrast to most other applications with condensed matter capable of flowing in which compressibility shows to be negligible. Thus, compressibility is considered in the literature as one of the most prominent properties of high-pressure processes; although it takes values substantially smaller than those of gases (Meschede 2006) often treated in CFD. Therefore, with respect to using CFD for simulating pressurized liquid biomatter, this represents well manageable requirements. Also, the pressure-induced changes of thermal conductivity and capacity reach only moderate values and, therefore, do not cause significant problems in the simulation. In contrast to this, high pressure may alter viscosity in a drastic way. Först (2002) has shown viscosity of edible oils and fats increasing up to four orders of magnitude. As, additionally, the pressure dependence of biomatter shows to be strongly nonlinear, complex interrelations between the mass, momentum, and energy transport must be expected (Delgado and Hartmann 2003). As an example, Song et al. (2009) have found pressure being able to alter basically the structure of turbulence and, as a result of this, the transport in the pressure chamber.

4.2.2. Influence of Pressure on Molecular and Cellular Reactions The reaction of biomatter to pressurization depends strongly on its components. It is well accepted in the

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literature that macromolecular components such as proteins or enzymes could be altered or denaturized at sufficiently high-pressure level (Heremans 1982; Denys et al. 1997, 2000; Cheftel et al. 2000a; Fachin et al. 2002; Doster and Friedrich 2004; Kowalczyk et al. 2004, 2005; Kowalczyk and Delgado 2007a; Kulisiewicz et al. 2007). This provides a suitable basis for the specific creation of new structures (Tauscher 1995; Hinrichs 2000; Kieffer et al. 2007). In contrast to this, the mechanisms induced by pressure acting on low molecular weight compounds such as vitamins are generally reversible (Tauscher 1995; Baars et al. 2004). This allows for the preservation of the beneficial properties of foods during high-pressure treatment (Takahashi et al. 1993; Cheftel 1995; Tauscher 1995; Meyer et al. 2000; Heinz and Knorr 2002). Additionally, the physicochemical occurrences connected to a high-pressure exposition may strongly affect or even inactivate the biotic function of organisms (Ludwig et al. 1992; Heinz and Knorr 1996; Gaenzle et al. 2001; Kilimann et al. 2006). It is obvious that molecular as well as cellular reactions depend on the spatial distributions of the mass, momentum, and energy fields available in the pressurizing chamber. Furthermore, the time behavior is of particular interest. From the relationship of the characteristic time of the pressure propagation (which is connected to the sound velocity) and the characteristic time of the treatment (such as the hydrodynamic and thermal time as inner scales or the pressurizing and the pressure-holding time as process scales) the impact of the pressure on the molecular equilibrium can be considered instantaneous (Delgado and Hartmann 2003). Exceptions that prove this rule are very fast molecular or cellular reactions with time scales in the order of magnitude of the pressure propagation, but these have not been investigated particularly with respect to this issue yet. Realizing the relevance of spatiotemporal distributions is of crucial importance as the treatment history determines, for example, the level of preservation of biomatter achievable including microbiological safety, the quality of the product in question, as well as the processing energy required. Thus,

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scaling up can be performed only adequately when the transport fields are known. This shows the inalienability of and the enormous potential made accessible by CFD. This compilation elucidates that discussing the approaches presented in the literature for modeling and simulating high-pressure treatments purely from the point of view of the specific treatment goals is somewhat tedious. This is especially due to the immense variety of the mechanisms, kinetics, structures, and phenomena possible that makes it difficult to find common classification attributes. However, in Delgado et al. (2008b), a generalized way of classification is proposed. It is based on the complexity and the level of simplification considered being appropriate for modeling and for reducing calculation costs to a reasonable extent. In contrast to this, in the next section the mathematical models proposed in the literature are discussed without going into details of possible simplifications.

4.3. Mathematical Modeling and Numerical Simulation of High-Pressure Processes The high-pressure treatment normally takes place in a so-called high-pressure autoclave. In many cases, tempering mechanisms are installed at the outer wall of the autoclave. The high-pressure processing itself consists of three steps: compression, pressure holding, and decompression. Thus, the use of balancing approaches requires equations suitable for describing these different phases (Hartmann et al. 2004; Kilimann et al. 2006; Rauh et al. 2009). However, most of the contributions in the literature deal with the effect of high pressure on a specific treatment step (Rademacher et al. 2002; Hinrichs and Rademacher 2005) or even the various steps are not differentiated explicitly but described by the thermodynamical conditions existing during the pressure holding time. However, in the literature balancing approaches that consider in general available spatiotemporal distribution of the physicochemical effects of high

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pressure have been employed only recently (e.g., Denys et al. 2000a; Hartmann and Delgado 2002; Kilimann et al. 2006; Delgado et al. 2008b; Rauh et al. 2009). This is obviously due to large numerical power requirements as a consequence of the complexity of the resulting balance equations (see the following equations). But most crucial, it represents a result of assumptions in the literature that simplify modeling and simulation to a substantial extent but that are also questionable (Rauh et al. 2009).

4.3.1. Basic Balancing Equations and Thermophysical Properties In summary, it can be stated that any realistic model must enable the description of spatiotemporal instead of pure time-dependent effects as done within usual pure kinetic considerations. In this sense, modeling the effect of high pressure on pure, chemically and electromagnetically inert, and newtonian substances capable of flowing within a continuum model can be done by using the wellintroduced balances of thermofluiddynamics. This requires formulations of the conservation of mass,  ∂ρ + ∇ ⋅ ( ρU ) = 0, ∂t momentum,    ⎤ DU ⎡ ∂U =⎢ + (U ⋅ ∇ )U ⎥ Dt ⎣ ∂t ⎦  1 1  1 1 1 = − ∇p + ∇ ⋅ ( μ∇U ) + Fg , ρ ρ Re 0 ρ Fr0

(4.1)

(4.2)

and energy (formulated for the temperature T ) DT α T Dp 1 1 = ΠT0 + ∇ ⋅ ( λ ∇T ) Dt cp ρ Dt cp ρ Re 0 Pr0 +

1 Ec 0 μ Φ, cp ρ Re 0

(4.3)

which are quoted in dimensionless notation here (Delgado et al. 2008b). In the sense of similarity analysis, this reduces not only the number of influence quantities but also delivers basic rules for scaling up.

Although the model Equations 4.1–4.3 are valid for the different pressure-generating systems employed, a piston system is considered. In this context, the initial dimensional chamber length L*0 and the piston velocity u0* represent obviously adequate length and velocity scales that lead to the characteristic process time τ c = L*0 / u0* for the pressurizing and depressurizing phase, in which the motion of the biomatter occurs mainly due to forced convection (Delgado and Hartmann 2003). In contrast to this, buoyancy, that is, natural convection, dominates the intermediate pressure-holding phase as shown experimentally by Pehl et al. (2000), theoretically by Delgado and Hartmann (2003), and numerically by Hartmann and Delgado (2002), Baars et al. (2007), and Rauh et al. (2009). However, for natural convection no characteristic velocity can be given a priori. Instead, the reference velocity * ubouy = g*α 0* ΔT * L*0 , which results from a balance of driving and hindering forces in the natural flow field, is often used in the literature (Spurk 2004). In this definition the dimensional parameters α 0* and ΔT * denote the isothermal expansion coefficient and a characteristic temperature difference driving the natural convection. The further state and transport parameters appearing in the model equations are obtained by self-relation. Thus, the density, thermal conductivity, thermal capacity, isothermal expansion coefficient, thermal diffusivity, as well as the dynamic viscosity are normalized by using the corresponding dimensional λ* quantities ρ0* , λ * , cp* , α * , a* = * * and μ * (as well ρ cp μ* as the kinematic viscosity ν * = * , of course) at ρ * * pressure p0 and T0 . Here, the normalization of the temperature has been performed by considering T0* as a further characteristic quantity of the thermal field. Additionally, the pressure in the momentum Equation 4.2 scales u*2 with the kinetic pressure ρ0* 0 . In contrast to this, 2 u*2 it scales with the total pressure p0* + ρ0* 0 ≈ ρ0* cp* 0T0* 2 in the energy Equation 4.3.

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Computational Fluid Dynamics in High-Pressure Processing Scale-Up

In the model Equations 4.1–4.3, the symbols Φ and Fg represent the dissipation due to friction and the gravitational force in dimensionless notation, respectively. In addition to the dimensionless simplexes obtained from the self-relation of the constitutive and kinematic parameters, the following dimensionless groups appear in the balance Equations 4.1–4.3:

ρ0*U 0* L*0 characterizes μ0* the ratio of convective and diffusive momentum; U *2 • the Froude number Fr0 = * 0 * balances the effect L0 g of gravity relative to that of convection; U 0*2 • the Eckert number Ec 0 = * * allows a statement cp 0T0 regarding the influence of flow-induced compressibility on dissipation; and μ0* cp* 0 ν * • the Prandtl number Pr0 = * = 0* expresses λ0 a0 the ratio of momentum and energy diffusion.

• the Reynolds number Re 0 =

The additional dimensionless group Π T0 = α 0*T0* appears in the energy equation and characterizes the increase of thermal energy due to the volumetric work that has to be applied for pressurization. Further details to similarity analysis are given, for example, by Stichlmair (1990). In order to examine the consequences of any spatiotemporal thermofluiddynamical distribution during the high-pressure processing onto the desired biotechnological conversion, model equations other than Equations 4.1–4.3 for as much as additional scalars than variables in question must be introduced. As a rule, the scalar variable represents the inactivation of a cellular or molecular system. In a general sense the additional variable is treated as a component with a given interaction with the dispersed phase. By nature, the implied interaction may correspond to physicochemical as well as cellular mechanisms. But, in any case, the molecular (index m) or cellular (index c) systems are transported by

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mechanisms induced by forced and natural convection as mentioned above. Furthermore, due to their small length scales ( lm* ≈ 10 −9 m for proteins and lc* ≈ 10 −6 m for microorganisms), they have to be expected to be in mechanical and thermal equilibrium. It can be shown that there is a local equilibrium, that is, the availability of a “congruent thermofluiddynamical state” of the continuous phase and the dispersed molecular or cellular systems at any time during high-pressure treatment with respect to given spatial coordinates. Fortunately, this facilitates modeling substantially as the effects of high pressure on molecular or cellular reactions can be abstracted mathematically as those on a general scalar quantity φ *. φ* In dimensionless formulation for φ = * the φ0 corresponding transport model writes as  ∂φ 1 + ∇ ⎡⎣φU ⎤⎦ = ∇ ⋅ ( Dϕ ∇φ ) + Da 0 Qϕ . (4.4) ∂t Re 0 Sc 0 The left side of this model equation coincides completely with that of the mass conservation. Thus, for a molecular biocomponent, Equation 4.4 expresses simply high-pressure-induced conversion. The latter is connected to the diffusion as well as the generation or reduction of the biocomponent in question. Diffusion effects are due to the possible existence of field inhomogeneities of the scalar in question φ . Thus, the parameter Dϕ in the first term on the right side of the model (Eq. 4.4) represents a diffusion constant of the molecular or cellular process in question. The order of magnitude of diffusion depends obviously on the reciprocal value of the product of the Reynolds number Re 0 and the Schmidt number ν* Sc 0 = 0* . The latter gives an estimation of the ratio D0 of diffusive transport of the momentum and of the scalar in question φ . A further effect related to the (positive or negative) generation of φ is given by the (dimensionless) source term Da 0 Qϕ in the model Equation 4.4, in which the strength of the source Qϕ is regulated by the value of the Damköhler-number Da 0 . The latter describes the relation of a convective time scale and the time scale of the molecular reaction.

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For example, for an n-order reaction with a concentration c = φ , the source strength is Qϕ = − kc n and L* k * c* n−1 the Damköhler number is Da 0 = 0 0 * 0 . u0 In agreement to the discussion of local congruent thermofluiddynamical states of the dispersed molecules and the continuous phase, one of the basic ideas of the model Equation 4.4 consists of considering that the molecule in question reacts as far as the required values of the thermodynamic variables pressure and temperature are available. However, it is important to understand that introducing an additive source term in the model (Eq. 4.4) postulates local homogeneity in addition to local mechanical and thermal equilibrium between the dispersed system and the continuous phase being valid. More concretely, it means that reactions in the specifically considered point of the biomatter and those in a spatial neighboring domain run in dependency of the same local thermofluiddynamical state. In other words, there is no interference such as restrictions or inhibitions between reactions occurring concurrently at places of small spatial distance.

4.3.2. Thermophysical Properties and Limitations of Balancing Equations For the use of the explained differential equations it has to be taken into account that the thermophysical properties (viscosity, thermal conductivity, thermal capacity, and density) of the treated material are pressure- and temperature-dependent. Therefore, formulations have to be given to determine this behavior. In the case of pure water, relations for the density and thermal capacity exist, for example, from Saul and Wagner (1989), and for the viscosity and thermal conductivity from the International Association for the Properties of Water and Steam (IAPWS). The validity of these models is restricted to a region from the melting lineup to 1,273 K and 25,000 MPa for the density and the thermal capacity and to temperatures between 273 and 423 K at a pressure of up to 500 MPa as well as a temperature range from 273 to 398 K up to 400 MPa for the viscosity and the thermal conductivity, respectively.

For more complex biomatter, there is a lack of precise data and corresponding models describing these thermophysical properties (compare also Otero and Sanz 2003). Torrecilla et al. (2004, 2005) overcome this by using artificial neural networks for prediction of process parameters. Some examples for examination of properties of food materials are given by Denys and Hendrickx (1999) and Denys et al. (2000b) for thermal conductivity, density, and thermal expansion coefficient of apple sauce and tomato paste under pressure as well as for thermal capacity under atmospheric conditions. Eder and Delgado (2002) and Eder et al. (2003) determine the behavior of the density of water, ethanol, glycerine mixtures, sucrose solutions, and water and sunflower oil with dissolved carbon dioxide. Research on the pressure dependency of the viscosity of sucrose solutions, whey protein solutions, and sunflower oil is done by Först (2002) and of heat conductivity of water, corn oil, sucrose solutions, and other liquids by Bridgman (1949), Lawson et al. (1959), and Werner et al. (2003). The lack of thermophysical data represents a severe restriction for the use of balancing models. More than that, constitutive complex biomatter can show complex mechanical, that is, rheological behavior. Not only pure viscous but also elastic and time-dependent phenomena can occur. This may affect drastically the stability and convergence of CFD procedures. For avoiding any misinterpretation in the following discussions, for any modeling and simulation approach presented the availability of suitable data for the biomatter in question is assumed to be given; no further comments on the model limitations are discussed.

4.4. Prediction of Process Impact and Control of High-Pressure Treatment Control of treatment represents one of the most important current research goals for high-pressure processes (Delgado et al. 2007). With respect to industrial application, scaling up is closely related to this task. First indications of adequate rules for scaling up can be directly found by considering the similarity

Chapter 4

Computational Fluid Dynamics in High-Pressure Processing Scale-Up

Table 4.1. Characteristic order of magnitude of dimensionless groups in balancing equations. Dimensionless groups Ec Re Fr Pr ΠD = Ec/Re ΠT

Order of magnitude 10−8 102 103 10 10−10 10−2

groups appearing in the basic Equations 4.1–4.3. Table 4.1 shows the typical order of magnitude of dimensionless parameters. Summarizing the values presented it can be concluded that (1) dissipative effects as expressed by Ec can be neglected, but in contrast to this (2) forced convection (influence of Re), natural convection (Fr), compression work (ΠT)as well as viscous diffusion (Pr) contribute significantly to mass, momentum, and energy transfer in the chamber. However, this means neither equality of the contributions to the transport processes in the pressurized chamber nor similarity of the transport fields. This can be directly deduced, for example, from a comparison of the momentum and energy Equations 4.2 and 4.3. Achieving similarity imposes the very restrictive conditions to be fulfilled: (1) unidirectional flow, (2) negligible pressure differences, (3) vanishing effects of gravity, (4) constant material parameters, and (5) a Prandtl number close to unity. Similar requirements can be derived when comparing the biochemical balance Equation 4.4 with the momentum or energy balance. Thus, under practical conditions strict similarity of the different fields can hardly be achieved. In turn, this increases the difficulties connected to the transfer of results from the laboratory to the production scale. In fact, when scaling up all dimensionless quantities appearing in the basic Equations 4.1–4.4, they must reach the same value in the laboratory and production scale. However, it must be taken into consideration that the scaling rules for the different similarity groups can deviate basically and, thus, deliver even contradictory results. For illustrating this statement the simple case of two systems

65

with the same biomatter as well as initial and boundary conditions should be discussed briefly. In this case only geometrical similarity is kept. But, while the forced convection scales with Re ∝ L*0 , the 1 natural convection depends on Fr ∝ * . Now, L0 these are obviously contradictory rules for scaling up. As a consequence of this it must be stated that strict similarity cannot be achieved when scaling up. Instead, only partial or asymptotic similarity (i.e., for Fr → ∞ or Re → ∞) appears to be realizable. As a consequence of this it can be concluded that the most appropriate way to achieve process homogeneity in scaled-up systems consists of a design that makes use of CFD in order to analyze a priori the transport phenomena during the high-pressure application. In this sense CFD represents also a powerful tool for realizing adaptive control strategies as basically required (Delgado et al. 2007). For elucidating this statement, results obtained by using CFD are presented and discussed. These are based on the work of Hartmann (2002), Rauh et al. (2009), and Kilimann et al. (2006), who proved by experimental validation the ability of the present numerical code to describe thermofluiddynamics and biochemical reactions under high pressure, respectively. For numerical solution of the partial differential equations that model the mass, momentum, and energy transport, discretization of the derivatives with respect to time and space was conducted accordingly to the second-order backward Euler scheme and the highresolution scheme. The discretized equations were solved by applying the finite volume method. For grid generation and for solution of differential equations, the commercial software ANSYS ICEM CFD 10.0 and CFX 10.0 (Canonsburg, PA), respectively, were applied. Additionally, FORTRAN codes developed by the present authors with about 25,000 statements were implemented. These codes describe the piston movement for pressure generation, pressureand temperature-dependent properties of water and some model foods (e.g. sucrose solutions), and the source terms (i.e., inactivation kinetics) of the transport equations. The three-dimensional computational domain is meshed with a structured grid shown in Figure 4.1.

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4.4.1. Different Phases of High-Pressure Treatment

In the vicinity of all walls the grid is refined. This can be seen in Figure 4.1B at the lateral surface. The whole mesh consists of 2.5 · 106 elements. As the bottom of the computational domain is moving, all calculations take place on a moving mesh.

In addition to the reasoning presented in the preceding section it must be taken into consideration that scaling up can only be achieved when similarity of the “treatment history” is given in the different phases of the high-pressure treatment. As mentioned above, in the literature only a minor number of publications are devoted to the study of the complete high-pressure treatment. Pehl and Delgado (1999) showed experimentally for the first time that pressurization, pressure holding, and depressurization phases lead to strongly deviating thermofluiddynamic effects. This is in agreement with the numerical results of Rauh (2009), which are depicted graphically in Figure 4.2. The simulation shows a vertical chamber pressurized by a piston. From this diagram the existence of five different phases can be concluded. In the initial time period during compression (marked by 1 in the diagram) and in the decompression phase (5) forced convection dominates. The period (2) is characterized by the development of velocity and temperature

Figure 4.1. Computational mesh: (A) perspective front view; (B) top view.

Tad 1.00 0.95 0.89 0.84 0.79 0.74 0.68 0.63 0.58 0.53 0.47 0.42 0.37 0.32 0.26 0.21 0.16 0.11 0.05 0.00

u/(L/tprocess)

T

18 16 14 12 10 8 6 4 2 0

W=0 W = 0.25 W = 0.5

0

0.2

0.4

0.6

0.8

1

t/tprocess

Figure 4.2. Development of temperature field and vertical velocity near the cylindrical wall at mid-height of the autoclave during a high-pressure process with aqueous sucrose solution containing W = 0, 0.25, 0.5 kg/kg sucrose (1, 5: forced convection; 2: development of thermal/ hydrodynamic layer; 3: fully developed thermal/hydrodynamic layer; 4: horizontal temperature stratification). See color insert.

Chapter 4

Computational Fluid Dynamics in High-Pressure Processing Scale-Up

67

boundary layers in free convection as the biomatter adheres to the wall of the chamber and, in general, exchanges thermal energy through the wall. Consequently field gradients must appear. As the inner momentum and thermal time reaches values being substantially smaller than the process time (see e.g., Pehl and Delgado 1999; Baars et al. 2007), the generation of the layers can be considered as finished in phase (3). In phase (4) a decrease of the natural convection due to the buildup of a horizontal density-driven temperature stratification in the center of the vessel occurs. From these flow features it can also be concluded that scaling up to a production scale can only be realized in part. For example, it is well known that the thickness of boundary layers increases nonlinearly along the wall. Thus, any change of the chamber length results in boundary extensions that do not scale.

Although CFD can manage inactivation of microorganisms in a similar way, only molecular, that is, biochemical reactions, are considered in the following for convenience. In Rauh et al. (2009), the uniformity of enzymatic conversion processes was examined. This was carried out by evaluation of pressure- and temperature-induced inactivation of the enzymes polyphenoloxidase (PPO), lipoxygenase (LOX), α-amylase (BSA), and β-glucanase (BGLU). The study of p-T diagrams for these reactions (not shown) indicates that at different pressure and temperature levels, covered by high-pressure processes, the sensitivity of the reactions varies significantly. As proposed by Baars et al. (2007), the inactivation rate k can be divided into two parts describing the pressure sensitivity (kP) and temperature sensitivity (α) of the reaction:

4.4.2. Examples for CFD-Based Investigations of the Impact of Process Inhomogeneities on Molecular and Cellular Systems

with

The previous sections have elucidated the missing strict scalability of mass, momentum, and energy transport in high-pressure processes and, thus, of process inhomogeneities. Consequently, identical “treatment histories” cannot be achieved in principle. This must be taken into consideration in any scale-up reasoning, but has been ignored in the literature yet. As a consequence of this, transferring data from the laboratory to the production scale has to lead to severe discrepancies and confusions as can be frequently found in the literature. This is particularly the case when considering the impact of transport phenomena, that is, of process inhomogeneities on biochemical and cellular reactions. In general, homogeneity can be assumed to be given in a lab frame, but hardly under production conditions. Thus, scaling-up process goals are subjected to severe restrictions that have been studied systematically only in single cases (Otero and Sanz 2003; Delgado et al. 2008b). However, systematic studies of transport processes cannot be performed without using CFD.

k ( p, T ) = kP (1 + α ) ⋅ (6 )

kP ( p, T0 ) = k ( p0 , T0 ) + ∫

α ( p, T ) =

p p0

( ∂k / ∂p )T dp

( ∂k / ∂T ) p (T − T0 ) kP

Table 4.2 summarizes the dimensionless values of kP* = kP t process and α for the inactivation kinetics of PPO, LOX, BSA, and BGLU. At a process pressure of 700 MPa, the temperature dependence of the inactivation reactions is quantified by the variable α in Table 4.2. For determination of the thermal sensitivity, that is, the dependence of (bio)chemical reactions on temperature variations, both kP* and α have to be considered. Values of α much greater than 1 indicate that small changes in temperature have great consequences for the rate of the inactivation reaction. However, if kP* is much smaller or larger than 1, this has a small effect on the overall process impact as the reaction is too slow or fast, respectively, to cause differences in conversion during the process time. Values of α close to −1 lead in conjunction with kP* >> 1 to remaining heterogeneities. From this, it can be concluded that for processes starting at T0 = 293 and 323 K the inactivation of LOX is the one most likely

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Table 4.2. Pressure and temperature sensitivity of inactivation reactions (PPO, LOX, BSA, BGLU) expressed by kP* = kP ⋅ tprocess and α, respectively (p = 700 MPa, T0 = 293, and 323 K, T = Tad). α (p, T)

kP* ( p, T0 ) T0 = 293 K PPO LOX BSA BGLU

<10−3 7.76 0.06 0.06

T0 = 323 K

T0 = 293 K

T0 = 323 K

0.02 0.35 0.33 0.09

12.80 −0.95 2.23 −0.26

13.53 34.25 2.31 129.61

(a) A* 1.00 0.95 0.89 0.84 0.79 0.74 0.68 0.63 0.58 0.53 0.47 0.42 0.37 0.32 0.26 0.21 0.16 0.11 0.05 0.00

PPO

LOX

to show nonuniformity due to thermal heterogeneities. The inactivation of PPO, LOX, and BGLU is expected to be more likely during a process starting at T0 = 323 than 293 K. The behavior of the inactivation of BSA is similar in a process starting at 293 K to the one starting at 323 K. For processes starting at 323 K (a) and 293 K (b), Figure 4.3 illustrates the distribution of the remaining activity of the different enzymes at the end of the holding time. Case (a) shows significant heterogeneities of PPO, LOX, BSA, and BGLU distributions and

BSA

BGLU

(b) BSA BGLU PPO LOX A* 1.00 0.95 0.89 0.84 0.79 0.74 0.68 0.63 0.58 0.53 0.47 0.42 0.37 0.32 0.26 0.21 0.16 0.11 0.05 0.00 Figure 4.3. Dimensionless enzyme activity A* (enzymes suspended in water) at the end of pressure-holding time of a highpressure process (autoclave 2.3 L; pressure ramp 400 MPa/s; pressure-holding time 120 s at 700 MPa; initial temperature of process [a] 323 K and [b] 293 K). PPO, polyphenoloxidase; LOX, lipoxygenase; BSA, bacillus subtilis α-amylase; BGLU, β-glucanase (Rauh et al. 2009). See color insert.

Chapter 4

Computational Fluid Dynamics in High-Pressure Processing Scale-Up

69

Dimensionless temperature increase

1.0 Particle 1 Particle 2

0.8

0.6

0.4

0.2

0.0

–0.2

0.0

0.2

0.4

0.6

0.8

1.0

Dimensionless pressure application time tt* Figure 4.4. Different treatment histories of two enzymes due to nonuniform temperature distributions and convective and diffusive transport phenomena during the process. Left: particle tracks of two different enzymes (Particle 1, Particle 2) during a high-pressure process; right: temperatures faced by the two enzymes (Particle 1, Particle 2) at different locations and times. See color insert.

process (b) of LOX and BSA. In both processes, convective transport of the liquid plays an important role. Thus, convective transport of the enzymes influences the treatment history of the enzymes during the process. This leads for all investigated enzymes to differences in topology of the activity fields in comparison with those of the temperature fields. Model Equation 4.4 in its completeness is currently the subject of investigation (Rauh et al. 2009). Thus, in the literature (Delgado et al. 2008b) only simplified forms for (a) pure time-dependent molecular and cellular reactions and (b) the additional effect of convection as indicated by ∇ ⎡⎣φU ⎤⎦ can be found, but no diffusion in accordance to the term 1 ∇ ⋅ ( Dϕ ∇φ ) are treated. Re 0 Sc 0 As a further example of the results reported in the literature by Kitsubun (2006), Figure 4.4 shows the results of simulation of the spatial trajectories of two different enzymes during a high-pressure process (left). The location for different times of the

enzymes is followed from their entrance into the vessel up to the end of the process. During the process the moving enzymes pass different temperature levels corresponding to the temperature distribution in the pressurized chamber. The right part of Figure 4.4 plots the temperature that each of the enzyme (Particle 1, Particle 2) is facing at its present location during the whole process. Therefore, both the spatial trajectories and the temperature distribution determine the “treatment history” of the enzymes in question.

4.5. Conclusions and Outlook This chapter elucidates the urgent necessity of considering high-pressure treatment as a process generated by a system with strong interrelated components (Delgado et al. 2007). This is particularly the case when aiming at process homogeneity and scaling up. The latter corresponds to the requirement of realizing similar “treatment history” in the laboratory and production scale.

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From the results obtained it must be stated that strict similarity cannot be achieved under practical conditions. Only asymptotic similarity appears to be performable. Using CFD represents a unique possibility for considering mass, momentum, and energy transport as well as the impact of these quantities on molecular and cellular system along the process trajectory available in the treatment chamber. Yet, the use of CFD for a complete simulation of transport processes covering all stages of the highpressure process has taken place only in single cases. Additionally, CFD suffers from the restrictions imposed by missing physicochemical material data. For overcoming this situation, in the future the design and employment of approaches other than balancing is expected. Cognitive methods (Torrecilla et al. 2004; Kilimann et al. 2006) (Chapter 3) provide the most suitable platform for adaptive strategies as required in general for biomatter due to its high natural variance (Delgado et al. 2007). Kilimann et al. (2006) used methods of multivariate statistical analysis to analyze experimental data of inactivation of Lactococcus lactis toward correlations between variables. With this knowledge they built a fuzzy logic-based model to predict lethal and sublethal injury of the microorganisms in dependence of pressure, temperature, and co-solutes. Additionally, the application of hybrid methods is expected to increase dramatically. This is due to the enormous potential connected to them. Figure 4.5 illustrates this at results obtained from a hybrid that combines CFD with the Verhulst-Perl model for the inactivation of L. lactis (Kilimann et al. 2006). Distributions of stress-resistant cell counts relative to the initial one ( log ( N N 0 ) ) of L. lactis showafter treatment for 4 minutes at 500 MPa baroprotective effects of 1.5 M sucrose (middle) and 4 M NaCl (right) in milk buffer compared with milk buffer without any additives (left). In the cases with baroprotectives the inactivation is lower. This is even true although the temperature is higher in these processes (316 K instead of 304 K) and inactivation increases with raising temperature. To minimize thermally induced spatial concentration heterogeneities in the considered high-pressure process, four possibilities arise (Baars et al. 2007).

log (N/N0) 0 –1 –2 –3 –4 –5 –6 –7 –8

x* y*

Figure 4.5. Distributions of stress-resistant cell counts N ( log ) of Lactococcus lactis (4 minutes at 500 MPa). Left: N0 treatment in milk buffer (304 K); middle: milk buffer with 1.5 M sucrose (316 K); right: milk buffer with 4 M NaCl (316 K).

1. Homogenization of the chamber content by convection (mixing). 2. Homogeneous temperature field in the chamber due to heat conduction. 3. Volume of the thermal boundary layer should be small with respect to the chamber volume. Furthermore, convective transport of fluid from the wall into the center of the pressure chamber should be prevented. 4. Avoidance of thermal gradients by adjusting the wall temperature to the temperature present under adiabatic conditions. Chapter 5 provides other alternatives to control temperature inhomogeneities during high pressure processing.

Acknowledgment Parts of this work has been carried out with financial support from the Deutsche Forschungsgemeinschaft (DFG), Bundesministerium für Bildung und Forschung, and Commission of the European Communities, Framework 6, Priority 5 “Food Quality and Safety,” Integrated Project NovelQ FP6-CT-2006-015710.

Chapter 4

Computational Fluid Dynamics in High-Pressure Processing Scale-Up

Notation

Fr0

Froude number

Fr0 =

Pr0

Prandtl number

Pr0 =

Π T0

Dimensionless group

Re 0

Reynolds number

Sc 0

Schmidt number

Latin Variables a c cp Dϕ D  Dt Fg g k L p Qϕ t T ΔT u U X Y

Thermal diffusivity Concentration Thermal capacity Diffusion constant Total derivative Gravitational force Gravity constant Reaction rate Length Pressure Source Time Temperature Characteristic temperature difference Norm of velocity vector Velocity Coordinate Coordinate

Greek Variables α ∇ ∂ ∂t φ Φ λ μ ν ρ τ ener τc

Isothermal expansion coefficient Gradient operator Partial derivative Scalar Dissipation Thermal conductivity Dynamic viscosity Kinematic viscosity Density Inner thermal time scale Characteristic process time

Dimensionless Numbers Da 0

Damköhler number

Ec 0

Eckert number

L*0 k0* c0* n−1 u0* U *2 Ec 0 = * 0 * cp 0T0

Da 0 =

71

U 0*2 L*0 g*

μ0* cp* 0 ν 0* = * λ0* a0 Π T0 = α 0*T0* ρ*U * L* Re 0 = 0 *0 0 μ0 ν 0* Sc 0 = * Dϕ 0

Indices * 0 buoy

Dimensional parameter Reference parameter Buoyancy

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Delgado A, Hartmann C. 2003. Pressure treatment of food: Instantaneous but not homogeneous effect. In: R Winter, ed., Proceedings of the 2nd International Conference on High Pressure Bioscience and Biotechnolgy. Advances in High Pressure Bioscience and Biotechnolgy, 459–464. Dortmund, 16–19 September 2002. Berlin: Springer. Delgado A, Nirschl H, Becker T. 1996. First use of cognitive algorithms in investigations under compensated gravity. Microgravity Sci Technol 9(3):185–192. Delgado A, Baars A, Kowalczyk W, Benning R, Kitsubun P. 2007. Towards system theory based adaptive strategies for high pressure bioprocesses. High Press Res 27(1):7–16. Delgado A, Rauh C, Benning R. 2008a. Thermodynamisches Modell zum Fest/flüssig-Phasenübergang von Substanzen hohen molaren Volumens. Chem Ing Tech 80(8):1185–1192. Delgado A, Rauh C, Kowalczyk W, Baars A. 2008b. Review of modeling and simulation of high pressure treatment of materials of biological origin. Trends Food Sci Technol 19:329–336. Denys S, Hendrickx ME. 1999. Measurement of the thermal conductivity of foods at high pressure. J Food Sci 64(4):709–713. Denys S, Van Loey AN, Hendrickx ME, Tobback PP. 1997. Modeling heat transfer during high-pressure freezing and thawing. Biotechnol Prog 13:416–423. Denys S, Van Loey AM, Hendrickx ME. 2000a. A modeling approach for evaluating process uniformity during batch high hydro-static pressure processing: Combination of a numerical heat transfer model and enzyme inactivation kinetics. Innov Food Sci Emerg Technol 1:5–19. Denys S, Van Loey AM, Hendrickx ME. 2000b. Modeling conductive heat transfer during high pressure thawing processes: Determination of latent heat as a function of pressure. Biotechnol Prog 16:447–455. Deuchi T, Hayashi R. 1992. High pressure treatments at sub-zero temperature: Application to preservation, rapid freezing and rapid thawing of foods. In: C Balny, R Hayashi, K Hermans, P Masson, eds., High Pressure and Biotechnology, 353–355. London: John Libbey and Co. Ltd. Diez L, Zima BE, Kowalczyk W, Delgado A. 2007. Investigation of multiphase flow in sequencing batch reactor (SBR) by means of hybrid methods. Chem Eng Sci 62(6):1803–1813. Doster W, Friedrich J. 2004. Pressure–temperature phase diagrams of proteins. In: J Buchner, T Kiefhaber, eds., Protein Folding Handbook. Part I, 99–126. Weinheim: Wiley-VCH Verlag GmbH & Co. KGaA. Dreyer M, Delgado A, Rath HJ. 1994. Capillary rise of liquid between parallel plates under microgravity. J Colloid Interface Sci 163(1):158–168. Eckert CA, Knutson BL, Debenedetti PG. 1996. Supercritical fluids as solvents for chemical and material processing. Nature 383:313–318. Eder C, Delgado A. 2002. Interferometrische Dichtemessung an flüssigen Lebensmitteln. In: A Leder, M Brede, B Ruck, D Dopheide, et al., eds, Lasermethoden in der Strömungsmesstechnik, 10, Fachtagung, Rostock, 10–12 September 2002 (Beitrag 56). Rostock: Universitätsdruckerei.

Eder C, Delgado A, Goldbach M, Eggers R. 2003. Interferometrische in-situ Densitometrie fluider hochkomprimierter Lebensmittel. In: D Dopheide, H Müller, V Strunck, B Ruck, A Leder, eds, Lasermethoden in der Strömungsmesstechnik, 11, Fachtagung, Braunschweig, 9–11 September 2003 (Beitrag 28). Braunschweig: Universitätsdruckerei. Epple P, Karic B, Ilic C, Becker S, Durst F, Delgado A. 2009. Design of radial impellers: A combined extended analytical and numerical method. Proc Inst Mech Eng C J Mech Eng Sci 223(4):901–917. Fachin D, Van Loey A, Indrawati I, Ludikhuyze L, Hendrickx M. 2002. Thermal and high-pressure inactivation of tomato polygalacturonase: A kinetic study. J Food Sci 67:1610–1615. Ferziger JH, Peric M. 2002. Computational Methods for Fluid Dynamics, 3rd ed. Berlin: Springer. Först P. 2002. In-situ Untersuchungen der Viskosität fluider, komprimierter Lebensmittel-Modellsysteme. Fortschritt-Berichte VDI. Reihe 3. Nr. 725. Düsseldorf: VDI-Verlag. Frohnapfel B, Jovanovic J, Delgado A. 2007. Experimental investigation turbulent drag reduction by surface-embedded grooves. J Fluid Mech 590:107–116. Fuchigami M, Kato N, Teramoto A. 1997. High pressure freezing effects on textural quality of carrots. J Food Sci 62:804–807. Fuchigami M, Kato N, Teramoto A. 1998a. High-pressurefreezing effects on textural quality of Chinese cabbage. J Food Sci 63:122–125. Fuchigami M, Teramoto A, Ogawa N. 1998b. Structural and textural quality of kinu-tofu frozen-thenthawed at highpressure. J Food Sci 63(6):1054–1057. Gaenzle MG, Ulmer HM, Vogel RF. 2001. High pressure inactivation of Lactobacillus plantarum in a model beer system. J Food Sci 66:1174–1181. Hartmann C. 2002. Numerical simulation of thermodynamic and fluid-dynamic processes during the high-pressure treatment of liquid food systems. Innov Food Sci Emerg Technol 3:11–18. Hartmann C, Delgado A. 2002. Numerical simulation of convective and diffusive transport effects on a high-pressure-induced inactivation process. Biotechnol Bioeng 79:94–104. Hayakawa K, Ueno Y, Kawamura S, Kato T, Hayashi R. 1998. Microorganism inactivation using high-pressure generation in sealed vessels under sub-zero temperature. Appl Microbiol Biotechnol 50:415–418. Hartmann C, Delgado A, Szymczyk J. 2003. Convective and diffusive transport effects in a high pressure induced inactivation process of packed food. J Food Eng 59:33–44. Hartmann C, Schuhholz J-P, Kitsubun P, Chapleau N, Le Bail A, Delgado A. 2004. Experimental and numerical analysis of the thermofluiddynamics in a high-pressure autoclave. Innov Food Sci Emerg Technol 5:399–411. Heinz V, Knorr D. 1996. High pressure inactivation kinetics of bacillus subtilis cells by a three state model considering distributed resistance mechanisms. Food Biotechnol 10:149–161. Heinz V, Knorr D. 2002. Effects of high pressure on spores. In: ME Hendrickx, D Knorr, eds., Ultra High Pressure Treatments of Foods, 77–114. New York: Kluwer Academic/ Plenum Publishers.

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Hendrickx M, Ludikhuyze L, Van den Broeck I, Weemaes C. 1998. Effects of high pressure on enzymes related to food quality. Trends Food Sci Technol 9:197–203. Heremans K. 1982. High pressure effects on proteins and other biomolecules. Ann Rev Biophys Bioeng 11:1–21. Higashi H, Iwai Y, Arai Y. 2001. Solubilities and diffusion coefficients of high boiling compounds in supercritical carbon dioxide. Chem Eng Sci 56:3027–3044. Hinrichs J. 2000. Ultrahochdruckbehandlung von Lebensmitteln mit Schwerpunkt Milch und Milchprodukte-Phänomene, Kinetik und Methodik. Fortschritt-Berichte VDI, Reihe 3, Nr. 656. Düsseldorf: VDI-Verlag. Hinrichs J, Rademacher B. 2005. Kinetics of combined thermal and pressure-induced whey protein denaturation in bovine skim milk. Int Dairy J 15:315–323. Hite BH. 1899. The effect of pressure in the presentation of milk. West Virginia Agricultural Experiment Station, Morgantown. Bulletin 58:15–35. Jaeger PT, Eggers R, Baumgartl H. 2002. Interfacial properties of high viscous liquids in a supercritical carbon dioxide atmosphere. J Supercritical Fluids 24:203–217. Kato C, Li L, Nogi Y, Nakamura Y, Tamaoka J, Horikoshi K. 1998. Extremely barophilic bacteria isolated from the mariana trench, challenger deep, at a depth of 11,000 meters. Appl Environ Microbiol 64(4):1510–1513. Khier W, Breuer M, Durst F. 2000. Flow structure around trains under side wind conditions: A numerical study. Comput Fluids 29(2):179–195. Kieffer R, Schurer F, Köhler P, Wieser H. 2007. Effect of hydrostatic pressure and temperature on the chemical and functional properties of wheat gluten: Studies on gluten, gliadin and glutenin. J Cereal Sci 45(3):285–292. Kilimann KV, Kitsubun P, Chapleau N, Le Bail A, Delgado A, Gänzle MG, Hartmann C. 2006. Experimental and numerical study of heterogeneous pressure-temperature-induced lethal and sublethal injury of lactococcus lactis in a medium scale high-pressure autoclave. Biotechnol Bioeng 94(4):655–666. Kitsubun P. 2006. Numerical Investigation of Thermofluiddynamical Heterogeneities during High Pressure Treatment of Biotechnological Substances. Dissertation, TU Munich. Knorr D. 1996. Advantages, opportunities and challenges of high hydrostatic pressure application to food systems. In: R Hayashi, C Balny, eds., Proceedings of the International Conference on High Pressure Bioscience and Biotechnology, High Pressure Bioscience and Biotechnology, 279–287. Kyoto: Elsevier. Kowalczyk W, Delgado A. 2007a. Numerical simulation of phase change at high hydrostatic pressure under variable-gravity environment. Numer Heat Transfer Part A 51:735–751. Kowalczyk W, Delgado A. 2007b. Simulation of fluid flow in a channel induced by three types of fin-like motion. J Bionic Eng 4(3):165–176. Kowalczyk W, Delgado A. 2007c. On convection phenomena during high pressure treatment of liquid media. High Press Res 27(1):85–92.

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Kowalczyk W, Hartmann C, Delgado A. 2004. Modeling and numerical simulation of convection driven high pressure induced phase changes. Int J Heat Mass Transfer 47(5): 1079–1089. Kowalczyk W, Hartmann C, Luscher C, Pohl M, Delgado A, Knorr D. 2005. Determination of thermophysical properties of foods under high hydrostatic pressure in combined experimental and theoretical approach. Innov Food Sci Emerg Technol 6(3):318–326. Kulisiewicz L, Baars A, Delgado A. 2007. Effect of high hydrostatic pressure on structure of gelatine gels. Bull Pol Acad Sci Tech Sci 55:239–244. Lawson AW, Lowell R, Jain AL. 1959. Thermal conductivity of water at high pressures. J Chem Phys 30:643–647. LeBail A, Boillereaux L, Davenel A, Hayert M, Lucas T, Monteau JY. 2003. Phase Transition in foods: Effect of pressure and methods to assess or control phase transition. Innov Food Sci Emerg Technol 4(1):15–24. Ludwig H, Bieler C, Hallbauer K, Scigalla W. 1992. Inactivation of microorganisms by hydrostatic pressure. In: C Balny, R Hayashi, K Hermans, P Masson, eds., High Pressure and Biotechnology, 25–32. London: John Libbey and Co. Ltd. Luscher C, Balasa A, Fröhling A, Ananta E, Knorr D. 2004. Effect of high-pressure-induced ice I-to-ice III phase transitions on inactivation of listeria innocua in frozen suspension. Appl Environ Microbiol 70(7):4021–4029. Meschede D. 2006. Gerthsen Physik, 23rd ed. Berlin: Springer. Meyer RS, Cooper KL, Knorr D, Lelieveld HLM. 2000. Highpressure sterilization of foods. Food Technol 54(11):67–72. Otero L, Sanz PD. 2003. Modeling heat transfer in high pressure food processing: A review. Innov Food Sci Emerg Technol 4:121–134. Otero L, Molina-García AD, Sanz PD. 2000. Thermal effect in foods during quasi-adiabatic pressure treatments. Innov Food Sci Emerg Technol 1:119–126. Pehl M, Delgado A. 1999. An in-situ technique to visualize temperature and velocity fields in liquid biotechnical substances at high pressure. In: H Ludwig, ed., Advances in High Pressure Bioscience and Biotechnology, 519–522. Heidelberg: Springer. Pehl M, Werner F, Delgado A. 2000. First visualization of temperature fields in liquids at high pressure. Exp Fluids 29:302–304. Peltier WR. 1989. Mantle Convection, Plate Tectonics and Global Dynamics. The Fluid Mechanics of Astrophysics and Geophysics. Montreux, Switzerland: Gordon and Breach Science Publishers. Rademacher B, Werner F, Pehl M. 2002. Effect of the pressurizing ramp on the inactivation of Listeria innocua considering thermodynamic processes. Innov Food Sci Emerg Technol 3:19–24. Raufeisen A, Breuer M, Botsch T, Delgado A. 2008. DNS of rotating buoyancy- and surface tension-driven flow. Int J Heat Mass Transfer 51:6219–6234. Rauh C. 2009. Modellierung und Simulation von Kurzzeit-UltraHochdruckprozessen. Fortschritt-Berichte VDI. Reihe 3. Nr. 901. Düsseldorf: VDI-Verlag.

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Rauh C, Baars A, Delgado A. 2009. Uniformity of enzyme inactivation in a short time high pressure process. J Food Eng 91:154–163. Sanz PD, Otero L. 2000. High-pressure freezing. Part 2. Modeling of freezing times for a finite cylindrical model. Biotechnol Prog 16:1037–1043. Sanz PD, Otero L, de Elvira C, Carrasco JA. 1997. Freezing process in high-pressure domains. Int J Refrigeration 20(5):301–307. Sarrade S, Guizard C, Rios GM. 2003. New applications of supercritical fluids and supercritical fluids processes in separation. Sep Purif Technol 32:57–63. Saul A, Wagner W. 1989. A fundamental equation for water covering the range from the melting line to 1,273 K at pressures up to 25,000 MPa. J Phys Chem Ref Data 18:1537–1564. Song K, Han Y, Al-Salaymeh A, Jovanovic J, Rauh C, Delgado A. 2009. In-situ investigation of turbulent fluid flow under high pressure conditions by means of Laser Doppler Anemometry and numerical simulation. In: A Delgado, C Rauh, H Lienhart, B Ruck, A Leder, D Dopheide, eds., 17. Fachtagung Lasermethoden in der Strömungsmesstechnik, Beitrag 45. Erlangen: GALA e.V., pp. 1–9. Spurk JH. 2004. Strömungslehre, Einführung in die Theorie der Strömungen. 5. Auflage, Berlin: Springer-Verlag. Stichlmair J. 1990. Kennzahlen und Ähnlichkeitsgesetze im Ingenieurwesen. Essen: Altos-Verlag Doris Stichlmair. Suzuki A. 2002. High pressure-processed foods in Japan and the world. In: R Hayashi, ed., Trends in High Pressure Bioscience and Biotechnology, 365–374. Amsterdam: Elsevier SC. Suzuki A. 2003. Current development of high pressure processed foods in Japan. IFT Annual Meeting, Chicago, 34–37.

Takahashi Y, Ohta H, Yonei H, Ifuku Y. 1993. Microbicidal effect of hydrostatic pressure on satsuma mandarin juice. Int J Food Sci Technol 28:95–102. Tauscher B. 1995. Review Pasteurization of food by hydrostatic high pressure: Chemical aspects. Z Lebensm Unters Forsch 200:3–13. Torrecilla JS, Otero L, Sanz PD. 2004. A neural network approach for thermal/pressure food processing. J Food Eng 62:89–95. Torrecilla JS, Otero L, Sanz PD. 2005. Artificial neural networks: A promising tool to design and optimize high-pressure food processes. J Food Eng 69:299–306. Werner M, Baars A, Delgado A. 2003. Bestimmung der Wärmeleitfähigkeit niedrigviskoser Lebensmittel unter Hochdruck mittels Hitzdrahtmethode. In: JP Baselt, G Gerlach, eds., 6. Dresdner Sensor-Symposium—Sensoren für zukünftige Hochtechnologien und Neuentwicklungen für die Verfahrenstechnik. Dresden: Universitätsverlag w.e.b., Band 20 (37/40). Yasuda A, Mochizuki K. 1992. The behavior of triglycerides under high pressure: The high pressure can stably crystallize cocoa butter in chocolate. High Press Biotechnol 224:255–259. Yoshioka K, Yamada A, Maki T, Yoshimoto C, Yamamato T. 1999. Application of high pressurisation to fish meat: the ultrastructural changes and nucleotide in frozen carp muscle under high pressure thawing. In: H Ludwig, ed., Advances in High Pressure Bioscience and Biotechnology, 501–504. Heidelberg: Springer. Zobell CE, Morita RY. 1957. Barophilic bacteria in some deep sea sediments. J Bactseriol 73(4):563–568.

Chapter 5 Computational Fluid Dynamics Applied in High-Pressure High-Temperature Processes: Spore Inactivation Distribution and Process Optimization Pablo Juliano, Kai Knoerzer, and Cornelis Versteeg

5.1. Introduction High-pressure high-temperature (HPHT) processing, also known as pressure-assisted thermal processing (PATP) or pressure-assisted thermal sterilization (PATS), is a novel emerging preservation method for the development of shelf-stable low-acid food products. HPHT involves combining pressures of 600 to 800 MPa with moderate initial chamber temperatures of 60 to 90°C. To achieve sterilization conditions, it takes advantage of the compression heating developed in the product and pressure-transmitting fluid, as well as the high pressure, to eliminate spore-forming bacteria (Matser et al. 2004; Margosch 2005; Barbosa-Cánovas and Juliano 2007). For instance, pressurization temperatures of 90 to 116°C combined with pressures of 500 to 700 MPa have been used to inactivate a number of strains of Clostridium botulinum spores (Farkas and Hoover 2000; Margosch et al. 2004). Other researchers showed that certain bacterial endospores (Clostridium sporogenes, Bacillus stearothermophilus, Bacillus licheniformis, Bacillus cereus, and Bacillus subtilis) in selected matrices like phosphate buffer, beef, vegetable cream, and tomato puree (Gola et al. 1996; Raso et al. 1998; Rovere et al. 1998; Meyer et al. 2000; Balasubramanian and Balasubramaniam

2003; Krebbers et al. 2003) can be eliminated after short-time exposure to temperatures and pressures above 100°C and 700 MPa, respectively. In February 2009, the U.S. Food and Drug Administration accepted a petition for the commercial production of a PATS-processed mashed potato-based product. This is the first process of this kind that has been filed and accepted (in a 35-L high-pressure sterilization vessel). Several HPHT combinations have been proposed for spore inactivation (de Heij et al. 2003; Leadley 2005; Barbosa-Cánovas and Juliano 2007), among which, the application of pressures greater than 600 MPa with short holding times (5 minutes or less) seems to be an appropriate balance between economical processing for industrial purposes and food safety. Shorter processing times would increase productivity and equipment lifetime, and reduce maintenance costs. Furthermore, a shorter thermal pressure process would result in a product with increased quality retention, making the process potentially more convenient than conventional incontainer sterilization (i.e., canning). This would particularly be the case for obtaining high-quality foods in large containers. In fact, in several HPHT studies, the process has been shown to produce foods with higher nutrients, color, and flavor along

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

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with other improved sensory attributes (Krebbers et al. 2002; Matser et al. 2004; Juliano et al. 2006a, 2006b, 2007; Barbosa-Cánovas and Juliano 2007). Thermal processing of foods requires the evaluation of the temperature and flow distributions involved at both the equipment and product level to assess the process performance. This information from HPHT processes can be utilized to predict the safety level achieved (i.e., the extent of C. botulinum inactivation), the product shelf stability (i.e., inactivation of spoilage organisms and enzymes), and the anticipated product quality (i.e., “cook” level and nutrition). Heat and mass transfer coupled with fluid dynamic models can greatly assist in evaluating the process performance through temperature and flow prediction. Computational fluid dynamic (CFD) models can be developed and applied to determine how a number of variables in the HPHT process affect heat generation and transfer and, therefore, temperature evolution throughout the vessel (Juliano et al. 2008b). CFD allows thermal and fluid dynamic models to be coupled to inactivation or degradation kinetic models and provide a two-dimensional (2D) or three-dimensional (3D) prediction of the safety and quality of all treated packages. A visual indication of the process performance can be given, showing vessel zones that may not be used as they do not reach the target temperatures or required inactivation, or result in undesirable overprocessing and quality degradation. Furthermore, the models can be utilized to improve process and equipment design. For example, compression heat retained in the vessel can be maximized by determining the optimum thickness of internal product carrier walls. The performance of the process at different scales, from small laboratory-scale vessels to industrialsize scales (i.e., 300–600 L capacity vessels) can also be compared. Once CFD models are validated against measured temperature data, output distributions from these models can assist regulatory authorities to approve the commercial use of the process. To achieve sterilization during pressurization, all treated food must at least reach the target compression temperature for a certain time during pressureholding time. To achieve this goal, a number of

variables must be controlled from the start. This chapter will provide an overview of the application of CFD models to predict uniformity of temperature and achieved sterility in HPHT systems.

5.2. Description of an HPHT Processing System 5.2.1. Vessel Components and Requirements A high-pressure system designed for sterilization conditions must be able to achieve high pressures of and above 600 MPa and chamber temperatures of at least 90°C. This can be accomplished by building a pressure chamber of appropriate materials and designing a high-pressure pumping system with fast pressure come-up. Today, sterilization systems range from laboratory scale (0.02–1.5 L), pilot scale (2–50 L), to industrial scale (150 L). Existing types of equipment have varied configurations (Balasubramaniam et al. 2004) that offer different levels of compression heat retention and efficiency. A typical (batch) HPHT machine system, applicable to all temperatures, consists of (1) a thick cylindrical steel vessel (often wire wound) with two end closures, (2) a means of restraining end closures, (3) a low-pressure pump, (4) an intensifier for system compression, using liquid from the lowpressure pump to generate high-pressure process fluid, and (5) necessary system controls and instrumentation (Farkas and Hoover 2000). Additionally, a preheating system with fittings for a package carrier and a post-process package cooling system are required. Stainless steel is preferred in the design of high-pressure chambers so that filtered (potable) water can be used as the compression fluid (Farkas and Hoover 2000). Other system components include the compression fluid, a product carrier basket, and the food to be processed (packed in flexible or semirigid containers). For HPHT treatment, the typical fluids used in pressure vessels include water, water with propylene-glycol, edible oils, and water/edible oil emulsions (Meyer et al. 2000). Oils assist in pump lubrication and provide additional compressionheating assistance. For commercial food processing

Computational Fluid Dynamics in High-Pressure High-Temperature Processes

purposes, the use of potable water is the most recommended compression medium for maintaining the cleanliness of product packages (Farkas and Hoover 2000). Packaging used for high-pressure treated foods, from institutional size to individual pouches, must accommodate more than a 12% reduction in volume, and be able to return to its original volume, without loss of seal integrity and barrier properties (Farkas and Hoover 2000; Caner et al. 2004). Until now, identification of suitable packages that can survive pressure sterilization with the combined highpressure and high-temperature conditions, that is, that retain seal and overall integrity, as well as adequate barrier properties against oxygen and water vapor, remains a challenge. Packaging materials tested at HPHT conditions were reviewed elsewhere (Koutchma et al. 2009; Juliano et al. 2010) The processed foods may differ in composition and, therefore, differ in their mechanical and thermophysical properties such as viscosity, density (porosity, tortuosity), compressibility, specific heat, and thermal expansivity, affecting compression

P1

5.2.2. The HPHT Process and Its Processing Variables The HPHT process consists of (1) preheating food packages in a carrier outside the vessel, (2) transferring the preheated carrier into the vessel and equilibrating up to an initial temperature, (3) pressurizing and holding at a target pressure, (4) releasing pressure, (5) removing carrier from vessel, and (6) cooling down products in the carrier and removing the products. Therefore, the temperature history inside an HPHT-processed food is determined by six main process time intervals (Juliano 2006; BarbosaCánovas and Juliano 2007; Figure 5.1): (1) product preheating to a target temperature Th, (2) product

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heating and heat transfer during processing. A common problem is the headspace initially contained in the pouches or the air released and water vapor developed as a result of preheating. For this reason, and to avoid degradative reactions during shelf storage, deaeration and a minimum headspace are recommended.

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Figure 5.1. Typical temperature profile of a pressure-assisted thermal process.

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equilibration to initial temperature Ts, (3) product temperature increase to Tp1 due to compression heating, (4) product cooling down to Tp2 due to heat removal through the chamber, (5) product temperature decrease to Tf during decompression, and (6) product cooling to Tc. The critical process variables can be divided into preheating and HPHT process variables. Preheating variables at least include the temperature (heating fluid and product) and heating time, whereas HPHT variables include pressure (level, compression rate, and decompression rate), temperature (chamber, fluid filling the vessel, incoming pressure fluid, carrier, and product) and time. The temperature of the cooling fluid and product after processing as well as cooling time also needs to be taken into account. Factors affecting these variables during preheating, equilibration, pressure come-up, pressure holding, decompression, and cooling are summarized in detail by Juliano et al. (2008b). The selected preheating method can affect initial temperature distribution within and between individual packages and potentially provide a nonuniform microbial inactivation after HPHT. Water baths, steam, steam injection in water, or dielectric heating have been suggested as preheating alternatives (Juliano et al. 2006b). Faster preheating methods provide less uniformity, and thus require a longer time for equilibration to achieve temperature uniformity. 5.2.2.1. Compression Heating and Heat Losses Fluids like air and water as well as food packages undergo compression heating when pressurized above room temperature (Ting et al. 2002). A rise between 20 and 40°C has been observed in foods and packaging materials during high-pressure treatment, depending on the product initial temperature and food material. In fact, the compression-heating rate varies according to the composition of the food, in some cases similar to that of water at different pressure-temperature combinations (Ting et al. 2002; Balasubramanian and Balasubramaniam 2003). Few researchers have reported data on compression heating rates for food products at HPHT conditions. Compression heating of water is depen-

dent on the initial temperature (Ts) and has been reported as 3.0, 4.0, 4.6, and 5.3°C/100 MPa at initial temperatures, 20, 60, 75, and 90°C, respectively (Farkas and Hoover 2000; Balasubramaniam et al. 2004). Figure 5.2 shows the almost linear increase in temperature due to compression in (1) water, (2) orange juice, and (3) a glycol-water mixture (compression-heating fluid) up to 90°C and 600 MPa (Ardia et al. 2004; Knoerzer et al. 2010b). On the other hand, compression heating of steel is almost zero maintaining the core of the vessel wall near the initial temperature (Ting et al. 2002; de Heij et al. 2003). Therefore, without insulation, or internal heating in the steel walls, a thermal gradient is developed during pressurization and holding times (Denys et al. 2000b; Otero and Sanz 2003), leading to heat loss toward the chamber wall and subsequent cooling down of the fluid/sample system. Figure 5.1 shows a typical temperature profile during HPHT treatment, indicating the cooling down experienced in the holding process. Loss of heat is reflected in the difference between initial and final temperature during holding time (Tp2 < Tp1), and temperature at the beginning and end of the pressurization-depressurization period (Tf < Ts). In this case, “cold spots” may be located close to the vessel wall as a result of heat flow from the pressureheated product and pressure fluid next to the vessel wall. Moreover, during compression a certain amount of liquid (which may be preheated to an insufficient extent) enters the vessel (about 17% of the vessel volume, depending on the pressure increase). For improved insulation, that is, to prevent heat loss through the steel wall, a material with low thermal conductivity (less than 1 W/m/K) could be used as part of the pressure vessel design (de Heij et al. 2003; Van Schepdael et al. 2003), for example, as an internal liner. Another alternative is to use a polymeric carrier in which products are preheated to the initial temperature and then carried into the vessel. Vessel materials suggested for this application are polyoxymethylene (POM), polyetheretherketone (PEEK), polytetrafluoroethylene (PTFE), polypropylene (PP), or ultrahigh-molecular-weight polyethylene (UHMWPE). Knoerzer et al. (2010c) measured temperature of selected polymers during

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Figure 5.2. Experimental and predicted compression heating curves of (a) distilled water; (b) orange juice; (c) water glycol mixture; and (d) insulating polymers and fluids (adapted from Ardia et al. 2004; Knoerzer et al. 2010b; Knoerzer et al. 2010c).

pressurization up to 90°C and 750 MPa and demonstrated that PP and Polyethylene (PE) have higher compression heating than water within the whole range (Figure 5.2d), whereas PTFE has higher compression heating up to 500 MPa. The authors did not

report a linear compression heating rate value of the polymers due to the logarithmic nature of curves, as opposed to the near linearity observed in fluids at elevated temperature (Figure 5.2a–c). Other authors have reported a compression heating of 4.5°C/100 MPa

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on PP; however, the conditions were not clearly reported (Schauwecker et al. 2002). In order to maintain temperature, some HPHT installations can use an internal cylindrical heater or “furnace” located inside the chamber near the vessel wall. The furnace then surrounds the preheated polymeric liner for carrying the food packages. The furnace temperature is set higher than target initial temperature to prevent potential heat loss through the carrier into the vessel walls.

5.3. Developing a CFD Model for an HPHT System

The increase in temperature due to pressurization depends on the change of the expansion coefficient αp, density ρ (kg/m3), and specific heat capacity Cp of both the food and liquid during compression, as summarized in Equation 5.1. The thermodynamic derivation of the compression heating equation is covered in earlier literature (Ardia et al. 2004; Knoerzer et al. 2007; Juliano et al. 2008b). (5.1)

In a high-pressure system (insulated or not), heat can be diffused throughout the vessel boundaries. Then, the balance in Equation 5.2 can be expressed using terms for unsteady-state heat conduction during the pressure come-up step by using Fourier ’s law of heat diffusion, with heat generation term Q (Davies et al. 1999; Datta 2001). ∂ ⋅ ( k∇T ) + Q ( ρC pT ) = ∇   Compression ∂t Fourier’s  Law Rate of accumulation

of unsteady heat conduction

∂P ∂t

(5.2)

heating

where k is the overall thermal conductivity of the fluid/food system inside the chamber. In this case, convective currents within the pressurizing fluid are assumed negligible and are not included in this balance. Based on Equation 5.1, Q represents the compression heating rate term expressed as:

(5.3)

where Q > 0 at ∀ t > ts (pressure come-up step) Q = 0 at ∀ t, tp1 < t < tp2 (pressure-holding step) Q < 0 at ∀ t, tp2 < t < tf (pressure-release step) Hence, Equation 5.2 also applies to the representation of all high-pressure processing steps. Thermal properties in the energy balance shown in Equation 5.2 can be rearranged so that the thermal diffusivity γ is part of the unsteady heat conduction term: ∂T Tα p ∂P = + γ ∇ 2T ∂t ρC p ∂t

5.3.1. Compression Heating Integrated in the Heat Transfer Balance

dT Tα p = . dP ρC p

Q = Tα p

(5.4)

The thermal diffusivity is directly proportional to the thermal conductivity at a given density and specific heat, that is,

γ =

k ρ ⋅Cp

(5.5)

In conventional heat transfer calculations, thermal conductivity is generally assumed to be independent of temperature (Holdsworth 1997). However, it is questionable whether or not this assumption can be made for some thermal properties in food materials and other polymeric materials, as their dependency on temperature and pressure is unknown.

5.3.2. Fundamental Equations of Fluid Motion The thermo- and fluid-dynamic behavior of the pressure medium influences temperature distribution and is described by conservation equations of mass, momentum, and energy (Hartmann et al. 2003). These equations account for the convection currents inside the vessel and are expressed as follows: • Mass conservation or equation of continuity (Chen 2006):  ∂ρ + ∇ ⋅ ( ρV ) = 0 ∂t

(5.6)

Chapter 5

Computational Fluid Dynamics in High-Pressure High-Temperature Processes

81

 where V is the velocity vector. This equation assumes that the fluid is initially at rest. Density increases with increasing pressure; therefore the first term in Equation 5.4 becomes nonzero. Since the left  side must be equal to zero, the fluid velocity V must adopt nonzero values at pressure increments, thus enforcing a fluid motion. • Energy conservation (Chen 2006):  ∂ ( ρC pT ) ∂P + ∇ ⋅ ( ρVC pT ) = + ∇ ⋅ ( k∇T ) (5.7) ∂t ∂t • Momentum conservation or Navier-Stokes Equation of Motion (Chen 2006):   ⎤  ⎡ ∂V ρ⎢ + (V ⋅∇ ) V ⎥ = −∇P + ∇ ⋅ (η ⋅∇V ) + ρg (5.8) ⎣ ∂t ⎦ where η represents the viscosity of the compressed fluid and g represents the gravity constant. These partial differential equations are solved simultaneously, or as referred to in the literature, “coupled” by using CFD software (Knoerzer et al. 2007; Juliano et al. 2008b). These equations can also be coupled with other differential equations representing inactivation or degradation kinetics, and turbulence.

5.3.3. Boundary Conditions and Possible Assumptions The complete computational domain is defined by the sample, medium, carrier, and pressure vessel setup (also called subdomains) forming the highpressure system (e.g., Figure 5.3). The initial conditions of a system can be defined in its subdomains. Provided the system is initially at thermal equilibrium (i.e., uniform temperature distribution), the heat retention during pressure holding is limited by the sample’s mass, thermal diffusivity, and heat transfer coefficient at the sample boundary. Boundary conditions represent the thermal and/or flow behavior at the system’s external or internal boundaries. These options can be determined when constructing the model in CFD software packages. The most commonly applied boundary conditions in HPHT modeling are the symmetry boundary,

Steel wall boundary

Carrier

Water subdomains

Pressure water inlet Figure 5.3. Computational domain of a rotation-symmetric high-pressure vessel including a carrier for CFD modeling (adapted from Knoerzer et al. 2007).

inflow velocity boundary, the pressure boundary, and boundaries between the different subdomains: • The symmetry boundary is defined for an axissymmetric model system, also called pseudo 3D model, and requires less computational time than a 3D model. This is mainly possible by assuming the compression fluid enters from the geometrical center of the bottom part of the vessel (Figure 5.3). (This assumption is not valid for horizontal vessels since convective motion due to gravity is not axis-symmetric; therefore a full 3D model is required.) • The inflow velocity boundary applies to indirect pressure systems, that is, systems using an external high-pressure intensifier pump, where the inlet

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

tube allows the entrance of pressure fluid from the pump. • The pressure boundary is a condition including an expression for the pressure increase during compression and a constant pressure for the holding time is required. For decompression, the release of pressure fluid through the outlet can be imposed by a normal flow pressure condition. • Other boundary conditions include the heat transfer between the liquid and the solid subdomains (i.e., compression fluid, food, packages, and carrier and steel vessel walls). Continuity of heat flux at the vessel wall can be assumed between all fluid-solid and solid-solid boundaries. If laminar flow conditions exist, the fluid velocity at the chamber walls can be assumed zero due to fluid adhesion to the walls (no slip condition). Equations for these boundary conditions are detailed elsewhere (Juliano et al. 2008b; Knoerzer et al. 2010a). Other boundary conditions have been applied to specific models, such as the use of time-dependent vessel walls (due to heat transferred into the steel mass) (Hartmann et al. 2004) and the description of boundaries for turbulent flow conditions inside the vessel (i.e., logarithmic wall function) (Knoerzer et al. 2007; Juliano et al. 2009).

5.3.4. Physical Properties of Foods and Other Materials at HPHT Conditions In order to reach good prediction accuracy, CFD models require thermophysical properties to be included as a function of pressure and temperature. For food and other polymeric materials, the thermal expansivity, specific heat, density, viscosity, and thermal conductivity at high-pressure and hightemperature conditions are mostly unknown. However, the properties have been published for water, some oils and alcohols, and some insulating polymeric materials (Harvey et al. 1996; Knoerzer et al. 2010b, 2010c). Information on the dependency of thermophysical properties for food and other materials with pressure and temperature can be found in Chapter 2 as well as other references (Juliano et al. 2008b; Knoerzer et al. 2010b, 2010c).

The above-mentioned thermal and transport properties of water were extracted from the NIST/ ASME database (Harvey et al. 1996) to assess their variation with pressure (Juliano et al. 2008b), based on a compression heating of 4.6°C/100 MPa at an initial temperature of 75°C. Figure 5.4 shows 3D representations of thermal expansivity, specific heat, and density as a function of temperature and pressure. The thermal expansion coefficient α and the specific heat capacity Cp have shown to be more sensitive to temperature than to pressure, whereas density ρ has shown to be more affected by pressure change. During come-up time, α gradually decreased by 28% when reaching 700 MPa and 105°C, in comparison to its value at atmospheric pressure and temperature 75°C. Similarly, Cp was reduced by 12% and ρ was increased by 15 to 18%. Thermal conductivity k has shown an increase of 44% at 700 MPa and 105°C, at a similar rate as temperature increases, transferring heat 1.4 times faster than water at atmospheric pressure and temperature of 75°C. Similarly, thermal diffusivity increased as water was pressurized, following the same trend as thermal conductivity and the increase in temperature due to compression. Zhu et al. (2007) also found an increase in thermal diffusivity with increased pressure, not only in water but also in potato and cheese. The term in Equation 5.4 containing the ratio αp/(Cp·ρ), or kC as referred to in Knoerzer et al. (2010b, 2010c), has shown to be less variable with pressure than the conduction term containing the thermal diffusivity (Juliano et al. 2008b) for water at initial temperatures of 75°C. Knoerzer et al. (2010b, 2010c) reported that kC, and therefore compression heating, of glycol-water mixtures and some insulating polymers is mainly dependent on pressure, whereas water and PTFE also show pronounced dependence on temperature. Viscosity showed a small relative increase with pressure increase (6%) to 700 MPa and 105°C, behaving similarly to atmospheric pressure, while showing significant decrease (25%) if heated from 75 to 99°C at atmospheric pressure. If cooling down would occur during holding time at 700 MPa (e.g., from target 105°C to the chamber temperature of

Chapter 5

Computational Fluid Dynamics in High-Pressure High-Temperature Processes

Density (kg/m3)

1200 1150 1100 1050 1000 950 360 Tempe

600 200 400 Pa) (M e ur ss Pre

380 rature (K

)

x 10–4

Expansivity (1/K)

6.5 6 5.5 5

83

75°C), properties (ρ, Cp, α, γ ) would show only small variations (1–6%) except for the viscosity, which would increase by 39% (Juliano et al. 2008a). The heat transfer coefficient h is used to quantify the transfer rate of heat by convection from a liquid to the surface of the food and pouch. It enables evaluating the effectiveness of heat transfer in processes through the evaluation of overall resistances participating in the system under study. In particular, in HPHT processing this property is used to calculate the heat transferred between the food package and the surrounding fluid. Heat transfer coefficients can be extracted from empirical relations between dimensionless Nusselt, Prandtl, and Reynolds numbers according to laminar or turbulent conditions (Perry 1997). However, not much information has been published on the validity of empirical equations, as applied to extracting heat transfer coefficients at high-pressure conditions. Different regions inside the vessel may require a particular equation (including dimensionless numbers) at the boundary to represent accurate heat transfer coefficients.

4.5 600 200 400 Pa) (M e ur ss Pre

Specific heat capacity (kJ/kgK)

360 Tempe 380 rature (K 400 ) 4.2 4.1 4 3.9 3.8

200 400 600 Tempe rature (K )

400

360 380 Pa) (M e Pressur

Figure 5.4. Response surface plots of thermophysical properties of liquid water at increasing pressure and temperature, according to compression heating rate of 4.5°C/100 MPa. Properties were calculated from NIST/ASME Properties database (Adapted from Juliano et al. 2008b). Units: temperature [°C]; density (kg/m3); specific volume (m3/kg); Cp (kJ/K·kg); compressibility (1/MPa); expansion coefficient (1/K); thermal conductivity (W/m·K); viscosity (Pa·s).

5.3.5. Other Important Assumptions Related to Components and Materials inside the HPHT Vessel In order to design a model that predicts chamber temperature profiles during pressurization, a number of assumptions need to be made. The assumptions made for a particular vessel system should provide the closest prediction of the process as influenced by factors such as the vessel design and plant system, insulation, and compression medium composition. (Juliano et al. 2008b) Initial temperatures in the vessel furnace (if present), the fluid surrounding the polymeric carrier, the polymeric carrier inner fluid, and food packages forming the system ought to be assumed homogeneous, at rest, and at thermal equilibrium. Liquids can be assumed Newtonian, compressible and chemically inert. In addition, most pressure pump systems increase the pressure inside the vessel at a constant rate; therefore, the pressurization rate as well as pressure release rate can be assumed constant (Otero et al. 2000; Carroll et al. 2003).

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

It can also be assumed that the steel vessel has a predefined constant volume, thus neglecting the expansion of the pressure vessel due to associated pumping during compression. Moreover, it should be noted that vessel expansion may add several percentage points in fluid to the vessel’s volume (Otero and Sanz 2003). Furthermore, for good accuracy, the vessel walls having variable temperature over the wall surface during pressure application need to be taken into account (Hartmann et al. 2004). The increase of mass in the pressure medium and consequent volume reduction in the samples can be neglected (i.e., volume is assumed constant and density is a function of pressure). In particular, deformation of packages should obey a prescribed dynamic. Packages are assumed to have no headspace (after preheating) and volume changes due to air compression should be negligible. The compression heat generated can be assumed dissipated by a combination of conduction and convection within the pressurizing fluid and turbulence may be included where applicable (Knoerzer et al. 2007). Furthermore, consideration of the temperature of the incoming fluid is necessary since convective currents play a major role in the temperature value achieved as well as uniformity (Hartmann et al. 2003; Knoerzer et al. 2007; Juliano et al. 2009). In addition, if unknown, the compression heating effect of the packaging material layer surrounding the food may be assumed to be similar to water. A further assumption is that the food does not change its composition, structure, and phase state between initial and termination points of the process (Farkas and Hoover 2000). Package disturbance by flow of the incoming pressure fluid may be neglected and can be assumed geometrically distributed and fixed within the vessel. The location, size, and shape of the samples should be defined to establish boundary conditions. An initial approximation due to the cylindrical shape of the vessel could be to have a cylindrical type of package for an axis-symmetric model (Hartmann and Delgado 2003b; Juliano et al. 2009). Gravity influence, however, cannot be assumed as negligible, especially during holding stage. Natural convection is mainly influenced by gravita-

tional forces (Knoerzer et al. 2007). Preheating and high-pressure processing times may be assumed constant (i.e., constant fluid inlet velocity for a single pressurization step). Laminar or turbulent conditions will depend on the inlet geometry and velocity. Input variables for the model may be the initial temperature inside the product and vessel (Ts), target pressure P1, pressurization rate, holding time (tp1 – tp2), and decompression rate (Figure 5.1).

5.4. Prediction of Temperature Uniformity and Flow by Means of CFD Modeling CFD, a computer-aided analysis of fluid conservation laws (mass, momentum, energy), is a convenient modeling approach for the prediction of temperature and flow during the HPHT processes in two or three dimensions. It allows the simulation of a process for devices (solid and semisolid structures) that interact with fluid, by solving the governing equations for fluid flow and heat transfer (Eqs. 5.4, 5.5, and 5.6). These partial differential equations describing the entire system are transformed into a system of equations and solved numerically to approximate the exact solution (Nicolaï et al. 2001). The underlying methods for the numerical analysis are the finite volume, finite element, and finite difference methods described elsewhere (Juliano et al. 2008b). These discretization methods provide a solution for a “discrete” number of points (or grid), where conservation equations are applied. In all cases, computational grids are tailored to provide a “mesh-independent” solution for the numerical approximation of the governing equations. Several authors have done extensive research into developing discrete CFD models that predict transient temperature and flow distributions, uniformity, and the loss of compression heating through the high-pressure vessel walls during all high-pressure processing steps (Denys et al. 2000a, 2000b; Hartmann and Delgado 2002a, 2003a; Hartmann et al. 2003, 2004). Some recent models include solid materials (Knoerzer et al. 2007; Otero et al. 2007; Juliano et al. 2009), whereas some predict temperature distribution in three dimensions (Ghani and

Chapter 5

Computational Fluid Dynamics in High-Pressure High-Temperature Processes

Farid 2007) and at high-pressure sterilization conditions (Knoerzer et al. 2007; Juliano et al. 2009). Due to the vertical configuration of the modeled laboratory- and pilot-scale vessels, simulations included a vertical pressure fluid inlet near the center bottom (Hartmann 2002; Hartmann and Delgado 2002b, 2003a, 2003b; Hartmann et al. 2004; Knoerzer et al. 2007; Otero et al. 2007). Thus, in most cases, 2D cross-sections were used as the computational domain (Figure 5.3) similar to that in Figure 5.3 due to rotation symmetry at the central axis. Examples described in this section use the finite elements and finite volume methods to develop CFD models. They will show how CFD models were developed and applied to evaluate the effects of varying inlet velocities, presence of packages, vessel size, and presence of sample carriers on temperature and flow distribution during and after HPHT processing.

5.4.1. Effect of the Inlet Velocity at Laminar Conditions In indirect high-pressure systems a certain amount of fluid is pumped into the already filled vessel, further increasing the amount of fluid to reach the target pressure level. The velocity at the inlet, where the pressure fluid is pumped into the vessel, determines the pressure come-up time. It has been found that the velocities used in a micro 4-mL vessel modeled by Hartmann (2002) provided laminar flow at the inlet, whereas a 35-L pilot-scale vessel modeled by Knoerzer et al. (2007) gave turbulent conditions at the inlet region, as will be shown later. A study on low-temperature conditions (Hartmann and Delgado 2002a) analyzed the thermo- and fluiddynamic effects of the pressurizing fluid (water) in a high-pressure vessel by numerical simulations. Temperature and fluid velocity profiles were modeled in a 4-mL chamber, pressurizing up to 505 MPa with an initial temperature of 15°C and pressure-holding time of 200 s at three inflow velocities (2, 4, and 8 mm/s). The process simulations were based on a numerical solution of fluid dynamics, using the finite volume method in a CFX 4.4 commercial package from AEA technologies (Pittsburgh, PA). The velocity

85

profile was maintained constant at the inlet crosssection for each case, giving a parabolic velocity profile further downstream in the inlet tube. In the inlet tube, velocity was zero at the surrounding wall, reaching a maximum value at its central axis. The flow field obtained inside the micro vessel, a low-temperature vessel, was governed by forced convection. Within a close region around the inlet, the entering fluid underwent a strong deceleration, resulting in a nonuniform temperature distribution. Temperature gradients caused a nonhomogeneous density distribution that generated a buoyancyinduced fluid motion (natural convection due to the gravitational field). Very low inlet velocities gave near isothermal conditions inside the vessel, that is, no temperature rise due to compression occurring in the system. However, a faster compression rate gave a temperature increase, which was close to the maximum adiabatic conditions. Validation was performed with a temperature probe, which was placed in the central plane near the wall. The numerical values were fitted within the error range of 0.7 K of the experimental values throughout the whole process, which attributed to the uncertainty of the manual positioning of the temperature probe (± 1 mm) and temporal resolution of the thermocouple (approximately 2 s), thus showing good agreement with experimental data.

5.4.2. Modeling at HPHT Turbulent Flow Conditions Knoerzer et al. (2007) used the finite element method (COMSOL Multiphysics, COMSOL AB, Stockholm, Sweden) to model a much higher inlet velocity of 5.7 m/s velocity for a 35-L pilot-scale vessel only filled with water, which corresponded to a Reynolds number of 60,000, creating turbulent flow in the vessel bottom region. In this case, the pronounced turbulent region with arising eddies (and therefore increased thermal conductivity, also referred to as turbulent thermal conductivity) provided significant cooling during pressure come-up and holding steps. Thus, opposite from what was observed by Hartmann (2002) in a small vessel at laminar conditions, higher inlet velocities (at turbulent conditions) in the larger

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

vessel still hindered the system from achieving maximum compression heating. To account for turbulence, the equations for energy and momentum conservation were extended by further terms, taking into account the contributions of arising eddies (increased thermal conductivity due to mixing and increased dynamic viscosity due to increased shear). To solve this flow problem, an averaged representation was necessary, which was done by employing Reynolds Averaged NavierStokes (RANS) equations (Nicolaï et al. 2001; COMSOL Multiphysics 2006), comprising terms that include average velocity and pressure values and a fluctuating term represented by the Reynolds stress tensor. In particular, Knoerzer et al. (2007) applied the k-ε model, a commonly used stochastic turbulence model for industrial applications, by including an additional “turbulent viscosity” term to the equations expressing conservation of momentum and continuity as described elsewhere (Juliano et al. 2008a). A cylindrical carrier, used to preheat the product to the initial target temperature and transport the packages into a (pilot-scale) vessel, can act as a barrier to flow of incoming colder fluid into the package area. In particular, carriers made of solid polymeric materials may prevent heat losses to the vessel walls and to the top and bottom areas of the pressure vessel. Knoerzer et al. (2007) simulated the temperature distribution obtained after placing two types of food carriers inside a 35-L pilot-scale sterilization, that is, a metal composite carrier (developed in the original design of the vessel to operate with a furnace) and a PTFE carrier. They simulated all processing steps in the 35-L vessel filled with water pressurized to 600 MPa and 415 s pressure holding (at an initial temperature of 90°C). To validate the simulated temperatures, a 3 × 3 thermocouple array placed in an axis-symmetric plane was set up in randomized form at several runs. An example of the modeling domain and temperature distribution is shown in Figure 5.3. The models included a compression heating term from Equation 5.3 and thermal properties such as density, specific heat, and thermal expansivity as functions of temperature and pressure.

The simulations showed significant cooling down (up to about 40°C) below the carrier due to the colder turbulent incoming fluid and demonstrated that both carriers provided a barrier against cooling down (Figure 5.5). However, the lower region inside the metal carrier was colder and, therefore, an uneven temperature distribution was observed and validated. An excellent correlation (R2 = 0.97) was obtained by comparing predicted and measured values in a parity plot, which included all locations of the thermocouples at all time steps throughout the process. On the other hand, the carrier made of PTFE was able to retain most of the compression heat generated even during holding time throughout its entire volume, providing better insulation than the metal carrier (Figure 5.5) at the end of pressure holding. The axis-symmetric model was further developed (Juliano et al. 2009) by including the steel vessel walls, the vessel lid, and the cylindrical packages containing a “water-like” solid. The initial temperature of the vessel lid was considered lower since for the actual process it does not include a heating source. As expected, this addition lowered the temperature at the upper region of the vessel and increased the natural convection.

5.4.3. Effect of the Vessel Size Hartmann and Delgado (2003b) developed models of different sized vessels (micro-scale [0.8 L], pilotscale [6.3 L], and semi-industrial-scale [50.3 L]) to study the effect of scale on temperature distribution. The models included five packages in each vessel containing an enzyme solution and equally distributed through the chamber height, operating at 550 MPa and 40°C. More heat retention was found in packages contained in the larger 50.3 L vessel, resulting in an average temperature difference per package of around 7 K compared with the 0.8 L vessel and 4 K compared with the 6.3 L vessel (Figure 5.6). Lower temperatures were mainly attributed to the incoming “cold” pressure medium, which was less influential in a larger scale vessel with bigger packages. Our group has compared the temperature distribution predicted by a CFD model representing to

Chapter 5

Computational Fluid Dynamics in High-Pressure High-Temperature Processes

87

Carrier opening

1.25

Time=425 Surface: Temperature (K)

1.25

1.2

1.2

1.15

1.15

1.1

1.1

1.05

1.05

1

1

0.95

0.95

0.9

0.9

0.85

0.85 0.8 0.75

0.7

0.7

0.65

0.65

0.55 0.5

Vessel wall

0.6

Central axis

0.8 0.75

0.4

0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

inlet 0 –0.13 –0.1 –0.07 –0.04 –0.01 0.02

385

380

375

370

365

0.5 0.45

0.05

390

0.55

0.4

Carrier bottom

Max: 394

0.6

0.45

0.1

Time=415 Surface: Temperature (K)

360

355

350

0.1 0.05

(a) Metallic composite carrier

0 –0.13 –0.1 –0.07 –0.04 –0.01 0.02

345 Min: 343

(b) PTFE carrier

Figure 5.5. Thermal profile of two CFD models of a 35-L vessel at the end of holding time (415 s) at 600 MPa including: (a) a metal composite carrier and (b) a PTFE carrier at the end of holding time (adapted from Knoerzer et al. 2007).

scale an Avure 35-L sterilization vessel with a model representing a Stansted 3-L sterilization vessel, both with compression fluid entering at the bottom of the vessel and containing a PTFE carrier with an opening at the top and run at the same conditions (initial

temperature of 90°C and final pressure of 600 MPa; come-up time 130 s; holding time 315 s; decompression time 15 s; Figure 5.7). The predictions show the higher heat retention provided by the 35-L unit due to a more pronounced loss of heat in the 3-L vessel

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

322 K

317.5 K 0.8 L

6.3 L

313 K

50.3 L Figure 5.6. Temperature distribution in vessels of different scales (0.8, 6.3, and 50.3 L) at the end of the process (1,200 s; adapted from Hartmann and Delgado 2003b).

(a)

(b)

T (°C) Vessel wall

110

Vessel wall

3-L unit

35-L unit

100 90 80 70 60 Carrier walls Figure 5.7. Comparison of the temperature distribution predicted by CFD models of two high-pressure sterilization vessels including a PTFE carrier at the end of 315 s holding time (initial temperature 90°C and 600 MPa): (a) a 35-L vessel; (b) a 3-L vessel (not to scale). See color insert.

during pressurization (Figure 5.8a). In this case, the model assumes no compression heating of PTFE at the carrier region, which might assist in diminishing the heat loss rate inside the 3-L vessel.

5.5. Distribution of Process Sterility by Coupling with Kinetic C. botulinum Inactivation Models CFD software packages can assist not only in calculating temperature evolution but also in coupling governing equations of fluid dynamics with kinetic inactivation equations to predict enzyme and microbial inactivation distribution from temperature distributions. Coupling with inactivation models can be done either internally (solved simultaneously within the software package) or externally (by converting the transient temperature profile predicted by the CFD model into a parameter representing the extent of inactivation). An equation describing the temporal and spatial enzyme activity or microbial inactivation distribution (Eq. 5.9) can be coupled to determine the inactivation distribution or relative retention throughout the vessel volume at different times (Hartmann et al. 2003). ∂A ∂A ∂A ∂A +u +v +w = K ( P, T ) A ∂t ∂x ∂y ∂z

(5.9)

where A is the relative enzyme activity or microbial load (actual value related to initial value); K(P,T) is the inactivation rate constant; and u, w, and v are the components of the fluid velocity vector in the x-, y-, and, z-directions, respectively. The left-hand side contains the coupling between A and the flow field, that is, the velocity of the solution. The right-hand side represents the coupling of A and the temperature distribution. The following subsections describe the prediction of C. botulinum inactivation distribution, in a 35-L sterilization vessel, accounting for some of the effects discussed in the sections above.

5.5.1. Effect of Carrier Composition Knoerzer et al. (2007) evaluated the C botulinum spore inactivation distribution inside a vessel, as shown in Figure 5.9 for three scenarios: (1) vessel without carrier, filled only with water; (2) vessel with metal carrier; and (c) vessel with PTFE carrier. The temperature component was used to calculate

130

600

120

500

110

400

100

300

90

200

Pressure (MPa)

Temperature (°C)

(a)

Pressure 35-L unit 3-L unit

100

70 0

100

200 300 Time (s)

400

(b)

0.6

1000

300

900

0.5

250

800

Height (mm)

700

Height (mm)

0 500

600 500 400 300 200

0.4

200

0.3

150

100

0.2

50

0.1

100 20 40 60 Width (mm)

20 Width (mm)

35-L unit

3-L unit

ITD = 0.50

ITD = 0.24

40

Integrated Temperature Distribution

80

0

Figure 5.8. Heat retention performance in two CFD models representing the inside of water contained in PTFE carriers in a 35-L unit and in a 3-L unit: (a) predicted temperature profiles in a central vessel point of each vessel; (b) integrated temperature distributor (ITD) distribution value throughout an axis-symmetric cross-section of the vessels at the end of the HPHT process.

89

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

log S 0 –2 –4 –6 –8 –10 –12

(between 2 and 9 log reductions). For the insulated PTFE carrier, over 94.6% of the carrier ’s length showed more than 12 (up to 14) log reductions in C. botulinum spores (Figure 5.9). Based on the increased extent of inactivation observed when using an insulated carrier, this type of model can be of great assistance in finding an optimum carrier design that produces thermal uniformity and uniform spore inactivation throughout the carrier, as well as optimum processing times. Section 5.6.2.2 will show the application of these models to determine optimum carrier wall thickness.

5.5.2. Comparison of the Model Output Based on the Selected Inactivation Kinetics

–14 (a) without

(b) metal

(c) PTFE

carrier

Figure 5.9. Predicted distribution of extent of C. botulinum log reduction extent inside a 35-L pilot-scale high-pressure vessel in three scenarios: (a) vessel without carrier, (b) vessel including a metal composite carrier, and (c) vessel including a Teflon (PTFE) carrier (from Knoerzer et al. 2007). See color insert.

the F-value, which was transformed into inactivation values by using Equation 5.10. t

F = ∫ 10

T (t )−Tref zT

dt = D@T =Tref ⋅ log S.

(5.10)

0

where F is the time the material would be held at a Tref of 121.1°C to achieve the same microbial inactivation of the process, zT is the temperature increment that changes the rate of the process by a factor of 10, S is the survival ratio N/N0 (N and N0 are the final and initial number of spores), and D@T =Tref the time at Tref to lower the number of spores by a factor of 10. The traditional first-order inactivation kinetics can also be expressed as: log S (t ) = −10

T (t )−Tref zT

⋅

t D@T =Tref

(5.11)

In the vessel without a carrier, inactivation distribution yielded less than 1 log reduction while an inhomogeneous distribution was seen in the metal carrier

Juliano et al. (2009) developed a more detailed CFD model, as described in Figure 5.10. This model was applied as a platform to compare the performance of known C. botulinum inactivation kinetic models for predicting inactivation distribution. In addition to the previously considered linear kinetic model including the F-value (Eq. 5.11), the Weibullian model (Eq. 5.12), an nth-order model (Eq. 5.13), and a combination of the nth-order model and the log linear model were selected. log S (t ) = −b [T ] ⋅ t n′[T ]

(5.12)

where b[T] and n′[T] are temperature-dependent parameters. Assuming that spore inactivation is only affected by temperature, these parameters can be expressed as function of the thermal history before, during, and after pressurization. Furthermore, inactivation of a selected thermo-baroresistant C. botulinum strain has been expressed as a function of both temperature and pressure in an nth-order model (Margosch et al. 2006). dN (t ) = − ki (T , P ) ⋅ N n dt

(5.13)

A reaction order of 1.35 was found by fitting k to curves obtained at different pressure–temperature combinations of 70–120°C and 600–1,400 MPa as well as ambient pressure. Single values of each constant at each combination were condensed into:

Chapter 5

(a)

Computational Fluid Dynamics in High-Pressure High-Temperature Processes

Z

91

(b) Time=350 Surface:Temperature [K] Max: 394 Arrow: Velocity field [m/s]

r

1.4

390

1.35

metal lid

1.3 1.25

top water entrance

380

1.2 1.15

air layer

1.1

1

370

1.05 1 0.95

2

0.9 stainless steel

0.85 3

0.8 0.75 0.65 0.6

340

0.55

5

0.5 0.45 6

330

0.4 0.35

PTFE carrier

350

0.7

4

water

360

0.3

7

320

0.25 0.2

metal valve

0.15

310

0.1 0.05 0

300

–0.05 –0.34 –0.26 –0.18 –0.1 –0.02 0.06

Min: 293 water inlet Figure 5.10. CFD model of a 35-L vessel including carrier, packages, steel walls, and metal lid: (a) computational domains of the model structure and (b) thermal and flow profile in the vessel at the end of holding time (315 s) at 600 MPa (from Juliano et al. 2009). Conditions simulated include starting temperature and pressurization rate to a final pressure of 600 MPa and a holding time of 315 s. Arrows proportional to the maximum velocity at specific time. See color insert.

ki′ (T , P ) = e A0 + A1⋅P + A2 ⋅T + A3 ⋅P

2 + A ⋅T 2 + A ⋅P⋅T + A ⋅P⋅T 2 4 5 6

(5.14)

where k ′ = ki′ (T , P ) = ki ⋅ N 0n−1 (Margosch et al. 2006). The rationale for comparing the performance of these inactivation models is due to the questionable

validity of the traditional F0 method, which assumes first-order kinetics for the inactivation of C. botulinum and the linearity of the D-value curves as a function of temperature (constant zT; Peleg, 2006). Furthermore, an HPHT process most likely requires

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

an expression that accounts for the possible combined effect of pressure and temperature. The kinetic models were expressed in the form of an ordinary differential equation (Eq. 5.15) that was externally coupled with the CFD model temperature output through the MATLAB routine also used by Knoerzer et al. (2007), which allowed predicting inactivation for non-isothermal scenarios. d [ log S (t )] = f [ log S (t )] dt

(5.15)

where f is a generic function of logS(t), S being the survival ratio N/N0, and N and N0 the final and initial number of spores. The routine fitted the temperature data to give a function T(t) in all locations inside the vessel, and was subsequently substituted in the differential equation, which was then solved for the total time of the process. The output of this routine was capable of providing: (1) log S(t) versus t plots at different locations, (2) log S(x,y) distributions at specific times, and (3) log S(x,y,t) versus t animations.

Table 5.1 summarizes the microbial inactivation models used (A: traditional first order log-linear kinetics model; B: Weibullian model; C: nth-order kinetics model; D: combined log-linear/nth-order model) expressed in Equation 5.15 and includes the respective parameter values for C. botulinum. The log-linear first order and Weibullian models originated from C. botulinum inactivation data at near atmospheric conditions and, therefore, only accounted for the temperature variation. On the other hand, the nth-order model was obtained from C. botulinum inactivation data corresponding to several combinations of temperature and pressure (Margosch et al. 2006) and thus depended on both temperature and pressure. Considering that the nth-order model was only valid above 100°C in the pressure range 0.1 to 600 MPa, a combined discrete model was tested, which solved for log-linear kinetics at temperatures equal to or less than 100°C and for nth-order kinetics at temperatures higher than 100°C (Table 5.1).

Table 5.1. Description of selected C. botulinum inactivation models and their parameter values for comparison in a CFD model platform of a 35-L high-pressure system (adapted from Juliano et al. 2009).

Model A—Linear kinetics B—Weibull distribution C—nth-order kinetics

D—combined linear nth order

d [ log S (t )] dt

Parameters for C. botulinum

T (t )−Tref

−

Tref = 121.1°C zT = 10°C D@T =Tref = 12.6 s

10 zT D@T =Tref

−b [T (t )] ⋅ t

n′[T (t )]

−10( n −1) ⋅ log S (t ) ⋅

⎧ log S (t ) ⎫ ⋅ ⎨− ⎬ ⎩ b [T (t )] ⎭

n′[T (t )]−1 n′[T (t )]

ki′[T ( t ) , P ( t )] ln (10 )

T (t ) − Tref ⎤ ⎡ k ′[T ( t ) , P ( t )] ⎥ 10 zT ⎢ if ⎢T ≤ Tc , − , − 10( n −1) ⋅ log S (t ) ⋅ i ⎥ ln (10 ) D@T = Tref ⎥⎦ ⎢⎣

b [T (t )] = ln {1 + e0.3⋅[T (t )−102.3] } 0.425 n ′ [T (t )] = 0.325 + 1 + e0.0994⋅[T (t )−101] ki′[T (t ) , P (t )] = e A0 + A1⋅ P + A2 ⋅T + A3⋅P

2 + A ⋅T 2 + A ⋅P⋅T + A ⋅P⋅T 2 4 5 6

where n = 1.35 A0 = 2.456 ; A1 = −0.023 ; A2 = −0.149 ; A3 = 2.259 × 10 −5 ; A4 = 1.462 × 10 −3 ; A5 = 1.798 × 10 −4 ; A6 = −1.806 × 10 −7 Refer to models A and C Tc = 100°C is the critical temperature when model A switches to model C

A—from (Pflug 1987); B—from (Campanella and Peleg 2001); C—from (Margosch et al. 2006).

Chapter 5

(a)

(b)

(c)

Computational Fluid Dynamics in High-Pressure High-Temperature Processes

(d)

log S 0

–2

–4

–6

–8

–10

–12

–14

–16

Figure 5.11. Predicted distribution of C. botulinum log reduction extent according to four selected kinetic inactivation models using a CFD model platform for a high-pressure 35-L sterilization system: (a) traditional log-linear kinetic model, (b) Weibull distribution model, (c) nth-order kinetic model, and (d) combined discrete log-linear and nth-order kinetic model (adapted from Juliano et al. 2009). See color insert.

The distribution of C. botulinum inactivation log reduction predicted by each model from the CFD platform for the 35-L vessel is shown in Figure 5.11. The final inactivation calculated inside each package when using model A (log-linear kinetics) was 16.5 log reductions; model C (nth-order kinetics) and model D (nth-order and combined log linear/nth order) achieved approximately 12.0 log reductions; model B (Weibull) achieved only around 9.4 log reductions at the end of the process. Hence, the conventional thermal processing kinetics (not accounting for the

93

combined effects with pressure) required shorter holding times to achieve a 12D reduction of C. botulinum spores compared with the other models. However, the temperature distribution inside the vessel resulted in a more uniform inactivation distribution when using the Weibull model or nth-order kinetics model, compared with the log-linear kinetics model (Figure 5.11). The lower inactivation and more uniform distribution provided by models B, C, and D (Table 5.1) can be explained by (1) the tailing of curves given by the model parameters and (2) the fact that only data from different C. botulinum strains were available and used in each model. Furthermore, the Weibull model (B) only accounted for the temperature variation, not pressure, which also affected the outcome. This showed how the CFD platform became quite useful in evaluating and comparing the inactivation extent and uniformity provided by different C. botulinum inactivation models in the same system. Inactivation models used in different food media and accounting for the heat transfer effect of the selected package during the preheating step, to be developed in the future, could be evaluated through such CFD platforms. These CFD platforms could be used as an aid for regulatory filing of the technology as well as in process and equipment design.

5.6. Dimensionless Parameters to Express the Process Performance In order to evaluate the performance of a highpressure process and make comparisons to others, a dimensionless parameter able to predict temperature uniformity during treatment inside the processing vessel is more convenient. It is desirable that this process performance parameter can assist in (1) providing the extent of “commercial sterility” achieved throughout the volume of prepackaged food contained in the vessel; (2) predicting the quality degradation as a result of temperature (pressure) use; and (3) HPHT process design and validation. Aspects of the HPHT process include thermal process evaluation based on container size and shape, food composition, equipment modification and optimization, scale-up studies, energy use, modification and

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

optimization of process conditions, and others. (Juliano et al. 2008b). This section will describe two performance parameters that have been able to characterize the process uniformity (Hartmann and Delgado 2003b; Knoerzer et al. 2010a).

This type of measure of process uniformity is useful for comparing whole pouches, but gives no information on uniformity inside the pouches.

5.6.1. Process Uniformity Λ

To overcome this limitation and in order to conveniently evaluate a process with respect to temperature performance, a parameter that accounts not only for temperature profiles in discrete locations (i.e., a package, a carrier, or the whole vessel), but also for the temperature variations across the entire volume has been proposed (Knoerzer et al. 2010a). The so-called ITD value is an expression for process performance that represents the temperature distributions in relation to a target temperature during HPHT throughout the whole vessel volume.

Hartmann and Delgado (2003b) conducted a proportional scale-up study through CFD modeling, but in this case, modeled the use of five packages containing enzyme solution in the 0.8-, 6.3-, and 50.3-L vessels (Figure 5.6). Equation 5.16 was used for predicting the inactivation of B. subtilis α-amylase. Process uniformity Λ was defined as the ratio of the minimum ( Α ave _ min ) and maximum ( Α ave _ max ) average activity retention per package in all five packages: Λ=

Α ave _ min Α ave _ max

(5.16)

Table 5.2 describes the uniformity values obtained for each vessel volume while varying the heat transfer coefficient hpp of the packaging material at the boundary of each package. Process uniformity for the 0.8-L vessel did not depend on hpp, whereas the other larger vessels were greatly affected. The lowest heat transfer coefficients provide more inactivation uniformity between packages because heat is retained better inside the packages for a major part of the process. For instance, a uniformity of 0.97 for the 50.3-L vessel indicates that heat is retained inside the package throughout most of the process, leading to a high degree of inactivation and uniformity.

Table 5.2. Process uniformity for simulated vessel volumes and heat transfer coefficients (adapted from Hartmann et al. 2003). hpp (W/m2·K) Λ (–) Vessel volume (L)

0.8 L 6.3 L 50.3 L

1 × –10−4

1 × –10−3

1 × –10−2

0.86 0.90 0.97

0.84 0.78 0.81

0.84 0.74 0.69

5.6.2. The Integrated Temperature Distributor (ITD)

5.6.2.1. Concept and Determination Method Commercial thermal processing is currently characterized by using the F-value (thermal death time; Eq. 5.10) to relate the temperature history of a process to a reference temperature for target bacterial spore inactivation (e.g., C. botulinum; Bacillus stearothermophilus). In this case, zT (°C) represents the thermal resistance constant using the following model: zT = −

(T − Tref )

( log Dref − log D )

(5.17)

where the reference decimal reduction time Dref (minute) is at a reference temperature Tref (°C), within the range of temperatures T (°C) used to generate experimental data. The decimal reduction time D (minute) of the target microorganism at a given temperature is expressed as: D=

t log N ref − log N

(5.18)

where N is the number of survivors (typically colony forming units [CFU] per gram or mL) at time t, and Nref is the initial number of microorganisms (Ball 1943). As stated earlier, the F-value model has been questioned due to the assumption of the linearity of

Chapter 5

Computational Fluid Dynamics in High-Pressure High-Temperature Processes

According to F-value …

Transient temperature profiles throughout carrier volume

Tdt

H

ITD(r,z) = 10

Integrated temperature distribution

Integrated profile relative to target T and holding t dz

[…] […] […]

Integration of integrated profile per row R

dr

array

Integration of integrated columns per row

Ttarget = Tmax,adiabatic thold = 300 s

ITD value

(a)

(b)

carrier height/mm

[…] […] […]

0

thold

Ttarget 10

[…] […] […] […] […] […]

ROI

95

1000 900 800 700 600 500 400 300 200 100

0.6 0.5 0.4 0.3 0.2

0.1 20 40 60 carrier radius/min (c)

Figure 5.12. ITD value determination in a high-pressure process (at discrete time): (a) depiction of region of interest (ROI, purple); (b) flowchart a MATLAB routine for determination; and (c) example output of an integrated temperature distribution (adapted from Knoerzer et al. 2010a).

the inactivation kinetics of target bacterial spores to very low numbers as well as the linearization of the temperature dependence of the D-value to obtain the zT value (Peleg, 2006). In the case of HPHT processing, a microbial kinetic expression depending on pressure and temperature would be required to provide an accurate representation of the sterility achieved. Moreover, the F-value provides a representation of the level of sterility achieved at a single point in the sterilization unit, not accounting for temperature gradients throughout the unit and, carrier, or package volumes. Therefore, the ITD value has been developed as an expression to evaluate the thermal process performance in an HPHT process throughout the vessel without requiring inputs from microbial kinetic parameters and only accounting for temperature and time data: t

∫ T ( t ) dt 0

tprocess

rmax zmax

ITD =

∫ ∫

rmin zmin

10

−Ttarget

TΔ

drdz

(rmax − rmin ) ⋅ ( zmax − zmin )

(5.19)

where rmin, rmax, zmin, and zmax cover the region of interest (ROI; in this case, the carrier volume), tprocess is the process time of interest (in this case, the holding time where most of the heat loss is expected), TΔ is a temperature gradient of 10 K, and Ttarget is the targeted hold temperature of the process. The ITD is made dimensionless by relating the integrated temperature profiles in each location to the process (holding) time and Ttarget and the surface (volume) integral to the area (volume) of the ROI. The flowchart for the determination of the ITD value is shown in Figure 5.12. Being dimensionless, this expression is universally applicable to any equipment size and process time, making it a convenient tool for comparisons between different types and scales of equipment and processes. Depending on whether the calculation is applied to an axis-symmetric or full 3D temperature distribution, the ITD value represents either the pseudovolumetric 2D or volumetric 3D thermal process performance. The following conclusions can be drawn from the calculated ITD value (in combination with the integrated temperature distribution plot):

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

• ITD = 1: perfect target temperature uniformity with respect to process time and distribution • ITD < 1: under processing in some or all parts of the carrier volume • ITD > 1: over processing in some or all parts of the carrier volume 5.6.2.2. Determination of the Optimum Insulating Carrier Wall Thickness with the ITD Value by Using Predicted Temperatures The ITD expression has been successfully utilized for predicting the optimum thickness of a PTFE carrier to be used inside a 35-L vessel (Knoerzer et al. 2010a) such that the best compromise between heat retention and usable volume could be determined. The usable volume VUsable (Eq. 5.20) was increased by decreasing the wall thickness values dwall, with a load capacity between 1.3 and 26.3 L. VUsable = π ⋅ (rcarrier − dwall ) ⋅ hcarrier _ int 2

(5.20)

where rcarrier is the carrier ’s external radius, dwall is the carrier wall thickness, and hcarrier_int is the carrier ’s internal height. By following the procedure shown in Figure 5.12, a carrier thickness value for maximum temperature performance (7 mm) was determined, as well as a compromise value for maximization of usable volume and temperature performance (4 mm). In either case, a significant decrease from the wall thickness of the PTFE carrier originally built for the 35 L vessel (dwall = 28 mm) was determined, suggesting an increase from a 12-L capacity to a 22– 24 L capacity while maintaining ITD values close 0.9 (normalized to the maximum ITD value of the scenarios investigated). Future trials including compression heating values for the polymeric materials and thermal conductivities as function of temperature and pressure will further improve the accuracy of these models.

5.7. Overview and Future Challenges This chapter has described the application of CFD in the prediction of temperature distribution and flow for the design and characterization of heat transfer phenomena occurring in HPHT processes. Particular

attention was given to the loss of compression heat through the metal steel vessel walls and its impact on sterilization performance. It was shown how CFD models can include all parts of the equipment including packages and carrier while representing all steps of the process. CFD models allowed demonstrating how the inlet temperature fluid can influence temperature distribution inside the chamber and how a product carrier can act as a thermal insulator. The importance of material thermophysical properties as a function of temperature and pressure for improved model accuracy was highlighted. Although these property functions have been determined for water and other liquids as well as some insulating plastics, more research is needed to characterize other food, packaging, or carrier materials. CFD models provide the possibility of coupling fundamental equations of fluid motion with kinetic models to determine the extent of microbial or enzyme inactivation or any other process throughout products inside the vessel as well as its impact on quality. CFD models could represent the importance of selected C. botulinum spore inactivation models, and allowed comparison of the thermal effect with the combined pressure thermal effect on temperature distribution achieved. Parameters such as the ITD value can characterize the sterilization performance throughout a volumetric section of the vessel contents and assist in design and process characterization. One challenge ahead is the accuracy of validation tools for these models. Thermocouples have assisted in measuring temperature profiles at specific locations. However, thermocouple systems in many HPHT systems tend to easily fail through their closure connections and therefore comprehensive validation experiments become very time-consuming and, in some cases may lack accuracy. Wireless temperature devices such as a “Thermo-Egg” (Knoerzer et al. 2010d) have been recently developed and are promising accurate results while allowing simultaneous temperature measurement at several vessel points. Measuring thermophysical properties of foods, process fluids, packaging, and construction materials in situ under pressure and elevated temperatures

Chapter 5

Computational Fluid Dynamics in High-Pressure High-Temperature Processes

will further enhance the accuracy and usefulness of the CFD models. The same applies to the process of interest (e.g., microbial inactivation, enzyme inactivation, or biochemical and chemical transformations). Validated kinetic models are needed to couple these to the outputs of the CFD models to predict performance. Conversely, microbial validation has also encountered challenges due to the sometimes erratic behavior of microbial spores during preheating and sterilization. Spore inactivation models considering the preheating stage, which provides conditions for spore germination, will aid in accurate validation of CFDassisted sterility predictions. As CFD software packages become more user friendly, it is foreseen that CFD models will be more frequently utilized for the characterization and design of HPHT processes.

Notation Latin Letters A b CP

D@T =Tref dwall F g h hcarrier,int k K K k n n n N0, N P

Relative activity or actual activity related to the initial activity (%) Function of process temperature and/ or pressure history Isobaric heat capacity (J· kg−1 · K−1) Cp, mixture Specific heat capacity of mixture (J · kg−1 · K−1) Decimal reduction time (minute) Carrier wall thickness (m) Thermal death time (minute) Gravity constant (9.81 m/s2) Heat transfer coefficient (W/m2/K) Internal carrier height (m) First-order kinetic constant (s−1) Inactivation rate constant (s−1) Inverse of inactivation rate constant (s) Thermal conductivity (W·m−1 · K−1) Function of process temperature and/ or pressure history Normal to surface Order of inactivation kinetic Initial and final number of microbial spores Pressure (Pa)

P0 P1, Ptarget P2 Pf prate Pref Q R r, R rcarrier R2 S T t T0 Tc Tf tf Th TL tp Tp1 tp1 Tp2 tp2 Tref ts Ts, Ti TΔ Ttarget u,v,w V Vusable vin x,y,z ZP ZT

97

Atmospheric pressure (Pa) Target pressure (Pa) Pressure at the end of the holding time (Pa) Pressure including a fluctuating term (Pa) Pressure rate (MPa · s−1) Reference pressure (Pa) Volumetric compression heating rate (J·m−3 · s−1) Universal gas constant (8.314 J · mol−1 · K−1) Radius (radial direction, m) Carrier radius (m) Coefficient of determination Survival ratio (N/N0) Temperature (K) Time (s) Initial temperature of sample (K) Temperature after cooling (K) Temperature after decompression (K) Time after decompression (s) Preheating temperature (K) Temperature of fluid (K) Process time (s) Temperature after compression heating (K) Time after compression (s) Temperature at the end of holding time (K) Time after holding stage (s) Reference temperature (K) Start time (s) Initial temperature (K)TV , Twall Wall temperature (K) Temperature gradient (10 K) Target temperature (K) Velocity components in x-,y-, and z-direction (m/s) Volume (m3) Usable volume / load capacity (m3) Inlet velocity (m/s) Spatial directions Pressure sensitivity (Pa) Thermal sensitivity (°C, K)

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Greek Letters α, αp β ηT ρ τ

Thermal expansion coefficient (K−1) Compressibility (Pa−1) Turbulent viscosity (Pa · s) Density (kg · m−3) Pressure come up time (s)

Abbreviations 2D 3D ASME CFD CFU HPHT HPP ITD NIST PATP PATS PE PEEK POM PP PTFE RANS ROI UHMWPE

Two-dimensional Three-dimensional American Society of Mechanical Engineers Computational fluid dynamics Colony forming units High-pressure high temperature High-pressure processing Integrated temperature distributor National Institute of Standards and Technology Pressure-assisted thermal processing Pressure-assisted thermal sterilization Polyethylene Polyetheretherketone Polyoxymethylene Polypropylene Polytetraflourethylene Reynolds averaged navier-stokes Region of interest Ultrahigh-molecular-weight polyethylene

Operators d Σ ∂ Δ ∇ ∇·

Differential Sum Partial differential Difference Gradient (nabla-operator) Divergence

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vegetable cream treated with high pressures. High Press Biosci Biotechnol 253–259. Hartmann C. 2002. Numerical simulation of thermodynamic and fluiddynamic processes during the high pressure treatment of fluid food systems. Innov Food Sci Emerg Technol 3(1): 11–18. Hartmann C, Delgado A. 2002a. Numerical simulation of convective and diffusive transport effects on a high-pressure-induced inactivation process. Biotechnol Bioeng 79(1):94–104. Hartmann C, Delgado A. 2002b. Numerical simulation of thermofluiddynamics and enzyme inactivation in a fluid food system under high hydrostatic pressure. Trends High Press Biosci Biotechnol 533–540. Hartmann C, Delgado A. 2003a. Numerical simulation of thermal and fluiddynamical transport effects on a high pressure induced inactivation. High Press Res 23(1–2):67–70. Hartmann C, Delgado A. 2003b. The influence of transport phenomena during high-pressure processing of packed food on the uniformity of enzyme inactivation. Biotechnol Bioeng 82(6): 725–735. Hartmann C, Delgado A, Szymczyk J. 2003. Convective and diffusive transport effects in a high pressure induced inactivation process of packed food. J Food Eng 59(1):33–44. Hartmann C, Schuhholz J, Kitsubun P, Chapleau N, Bail A, Delgado A. 2004. Experimental and numerical analysis of the thermofluiddynamics in a high-pressure autoclave. Innov Food Sci Emerg Technol 5(4):399–411. Harvey AH, Peskin AP, Sanford AK. 1996. NIST/ASTME— IAPSW Standard Reference Database 10, version 2.2. de Heij WBC, Van Schepdael LJMM, Moezelaar R, Hoogland H, Matser A, van den Berg RW. 2003. High-pressure sterilization: Maximizing the benefits of adiabatic heating. Food Technol 57(3):37–42. Holdsworth SD. 1997. Thermal Processing of Packaged Foods. New York: Blackie Academic & Professional. Juliano P. 2006. High Pressure Thermal Sterilization of Egg Products. PhD dissertation, Washington State University, Pullman, WA. Juliano P, Li BS, Clark S, Mathews JW, Dunne PC, BarbosaCánovas GV. 2006a. Descriptive analysis of precooked egg products after high-pressure processing combined with low and high temperatures. J Food Qual 29(5):505–530. Juliano P, Toldra M, Koutchma T, Balasubramaniam VM, Clark S, Mathews JW, Dunne CP, Sadler G, Barbosa-Cánovas GV. 2006b. Texture and water retention improvement in highpressure thermally treated scrambled egg patties. J Food Sci 71(2):E52–E61. Juliano P, Clark S, Koutchma T, Ouattara M, Mathews JW, Dunne CP, Barbosa-Cánovas GV. 2007. Consumer and trained panel evaluation of high pressure thermally treated scrambled egg patties. J Food Qual 30(1):57–80. Juliano P, Knoerzer K, Barbosa-Cánovas GV. 2008a. High pressure processes: Thermal and fluid dynamic modeling applications. In: R Simpson, ed., Engineering Aspects of Thermal Processing, 159–207. Boca Raton, FL: CRC Press/Taylor & Francis.

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Juliano P, Knoerzer K, Barbosa-Cánovas GV. 2008b. High pressure thermal processes: thermal and fluid dynamic modeling principles. In: R Simpson, ed., Engineering Aspects of Thermal Processing, 91–158. Boca Raton, FL: CRC Press/Taylor & Francis. Juliano P, Knoerzer K, Fryer P, Versteeg C. 2009. C. botulinum inactivation kinetics implemented in a computational model of a high pressure sterilization process. Biotechnol Prog 25(1):163–175. Juliano P, Koutchma T, Sui Q, Barbosa-Cánovas GV, Saddler G. 2010. Food plastic packaging for high pressure processing. Food Eng Rev 2(4):274–297. Knoerzer K, Juliano P, Gladman S, Versteeg C, Fryer P. 2007. A computational model for temperature and sterility distributions in a pilot-scale high-pressure high-temperature process. AIChE J 53(11):2996–3010. Knoerzer K, Buckow R, Juliano P, Chapman B, Versteeg C. 2010a. Carrier optimisation in a pilot-scale high pressure sterilisation plant—An iterative CFD approach. J Food Eng 97(2): 199–207. Knoerzer K, Buckow R, Sanguansri P, Versteeg C. 2010b. Adiabatic compression heating coefficients for high-pressure processing of water, propylene-glycol and mixtures—A combined experimental and numerical approach. J Food Eng 96(2):229–238. Knoerzer K, Buckow R, Versteeg C. 2010c. Adiabatic compression heating coefficients for high pressure processing—A study of some insulating polymer materials. J Food Eng 98(1):110–119. Knoerzer K, Smith R, Kelly M, Steele R, Sanguansri P, Versteeg C. 2010d. The Thermo-Egg: A combined novel engineering and reverse logic approach for determining temperatures at high pressure. Food Eng Rev 2(3):216–225. Koutchma T, Juliano P, Song Y, Setikaite I, Patazca E, Dunne PC, Barbosa-Cánovas GV. 2009. Packaging evaluation for high pressure/high temperature sterilization of shelf-stable foods. J Food Process Eng. DOI: 10.1111⁄j.1745−4530.2008.00328 (available online). Krebbers B, Matser AM, Koets M, Berg RW. 2002. Quality and storage-stability of high-pressure preserved green beans. J Food Eng 54(1):27–33. Krebbers B, Matser A, Hoogerwerf S, Moezelaar R, Tomassen M, Berg R. 2003. Combined high-pressure and thermal treatments for processing of tomato puree: Evaluation of microbial inactivation and quality parameters. Innov Food Sci Emerg Technol 4(4):377–385. Leadley C. 2005. High pressure sterilisation: A review. Campden Chorleywood Food Res Assoc 47:1–42. Margosch D. 2005. Behavior of bacterial endospores and toxins as safety determinants in low acid pressurized food. PhD dissertation, Technical University Muenchen-Weihenstephan, Germany. Margosch D, Ehrmann MA, Ganzle MG, Vogel RF. 2004. Comparison of pressure and heat resistance of Clostridium botulinum and other endospores in mashed carrots. J Food Prot 67(11):2530–2537.

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Margosch D, Ehrmann MA, Buckow R, Heinz V, Vogel RF, Ganzle MG. 2006. High-pressure-mediated survival of Clostridium botulinum and Bacillus amyloliquefaciens endospores at high temperature. Appl Environ Microbiol 72(5):3476–3481. Matser AA, Krebbers B, Berg RW, Bartels PV. 2004. Advantages of high pressure sterilisation on quality of food products. Trends Food Sci Technol 15(2):79–85. Meyer RS, Cooper KL, Knorr D, Lelieveld HLM. 2000. Highpressure sterilization of foods. Food Technol 54(11):67–72. Nicolaï BM, Scheerlinck N, Verboven P, Baerdemaeker JD. 2001. Stochastic finite-element analysis of thermal food processes. In: J Irudayarai, ed., Food Processing Operations Modeling. Design and Analysis. 265–304. New York: Marcel Dekker. Otero L, Sanz P. 2003. Modelling heat transfer in high pressure food processing: A review. Innov Food Sci Emerg Technol 4(2):121–134. Otero L, Molina-Garcia A, Sanz P. 2000. Thermal effect in foods during quasi-adiabatic pressure treatments. Innov Food Sci Emerg Technol 1(2):119–126. Otero L, Ramos AM, de Elvira C, Sanz PD. 2007. A model to design high-pressure processes towards an uniform temperature distribution. J Food Eng 78(4):1463–1470. Peleg M. 2006. Advanced Quantitative Microbiology for Foods and Biosystems. Models for Predicting Growth and Inactivation. Boca Raton, FL: Taylor & Francis. Perry RH. 1997. Perry’s Chemical Engineers’ Handbook. New York: Mc. Graw Hill.

Pflug IJ. 1987. Using the straight-line semilogarithmic microbial destruction model as an engineering design-model for determining the F-value for heat processes. J Food Prot 50(4): 342–346. Raso J, Barbosa-Cánovas G, Swanson BG. 1998. Sporulation temperature affects initiation of germination and inactivation by high hydrostatic pressure of Bacillus cereus. J Appl Microbiol 85(1):17–24. Rovere P, Gola S, Maggi A, Scaramuzza N, Miglioli L. 1998. Studies on bacterial spores by combined pressure-heat treatments: possibility to sterilize low acid foods. In: NS Isaacs, ed., High Pressure Food Science, Bioscience and Chemistry, 354–363. Cambridge: The Royal Society of Chemistry. Schauwecker A, Balasubramaniam VM, Sadler G, Pascall MA, Adhikari C. 2002. Influence of high-pressure processing on selected polymeric materials and on the migration of a pressuretransmitting fluid. Packaging Technol Sci 15(5):255–262. Ting E, Balasubramaniam VM, Raghubeer E. 2002. Determining thermal effects in high-pressure processing. Food Technol 56(2):31–35. Van Schepdael LJMM, de Heij WBC, Hoogland H. 2003. Method for high pressure preservation. [PCT WO 02/45528 A1.]. Zhu S, Ramaswamy HS, Marcotte M, Chen C, Shao Y, Le Bail A. 2007. Evaluation of thermal properties of food materials at high pressures using a dual-needle line-heat-source method. J Food Sci 72(2):E49–E56.

Chapter 6 Computer Simulation for Microwave Heating Hao Chen and Juming Tang

6.1. Introduction Microwave (MW) heating has been used in many food processing unit operations, including extensive uses in precooking of bacon for food service chains and restaurants, thawing of frozen meat blocks for further processing, and, to a lesser extent, in pasteurization and sterilization of prepackaged foods. Compared with conventional heating methods, MW energy is directly converted to thermal energy in foods, significantly reducing process time. However, MW heating is, unlike conventional surface heating, based on complex Multiphysics phenomena, thus the design of such systems is more complicated. MW propagation and field patterns in foods and MW heating systems are governed by Maxwell’s equations, and analytical solutions are available only for very simple cases that are rarely seen in real-world applications. For realistic problems, numerical techniques such as Finite Difference Time Domain (FDTD) method, Finite Element Method (FEM), and Method of Moment (MoM) are often used in the development of computer simulation models to provide a more accurate insight into the interaction between the heated products and MWs (see also Chapter 7 in this book).

Several early papers (McDonald and Wexler 1972; Wu and Tai 1974; Spiegel 1984) summarize computational methods to study electromagnetic (EM) field distribution. In Wu and Tai (1974), a coupled integral equation was developed to solve a two-dimensional (2D) scattering problem for an arbitrary dielectric cylinder. They employed a pair of electrical fields and normal derivatives at the surface of the scattering objects and then solved the complete set of linear equations by using MoM. In McDonald and Wexler (1972), FEM was employed to solve Helmholtz equations within the computational region for a 2D radiating antenna and a dielectrical object. More complex FEM-based computer simulation models were developed later to study 3D MW heating of stationary packaged foods in domestic ovens (Zhang and Datta 2000). Spiegel (1984) provided a good review on FDTD method for medical applications. FDTD method has several advantages over other numerical methods. It requires relatively low computer memory and computational resources. As such, the FDTD method was used extensively in early studies to solve realistic problems such as temperature response to the MWirradiated human eye (Taflove and Brodwin 1975a, 1975b), response of military crafts to electromagnetic

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

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(EM) pulses (Kunz and Lee 1978), and temperature rises in biological tissues when exposed to MW fields (Cook 1952; Michaelson 1969; Schwan 1972; Guy et al. 1974; Livesay and Chen 1974). The dielectric properties of foods are temperaturedependent. To gain a more realistic insight into MW heating of foods, it is important to consider the coupling effects between heat transfer and EM field in computer models. Ma et al. (1995) developed a 3D FDTD model to obtain the MW heating pattern of gels placed inside a domestic MW oven. Chen et al. (2007) simulated an industrial-scale MW sterilization system by a combination of EM and thermal models using the FDTD methods. Recently, the conformal FDTD technique is used in simulation models for moving packaged foods in continuous pilot-scale MW systems (Chen et al. 2008). In this chapter, we will provide a general overview of the above-mentioned methods to solve MW field equations and then present several case studies to illustrate the use of conformal FDTD method in studying electric field patterns and heating uniformity in food trays during MW sterilization in a pilotscale 915 MHz system.

6.2. EM Wave Equations 6.2.1. Maxwell’s Equations The theoretical foundation for EM wave propagation was described by Maxwell around 1873 by a set of vector equations based on results of various experiments and theoretical derivations conducted by numerous researchers (see also Chapter 7 of this book). These equations have either a differential or an integral form. The differential form is used to govern the relations and variations of the electric and magnetic field at any point any time. For the equations to be valid, it is assumed that the field vectors have continuous derivatives, and are singlevalued and continuous functions of time and space (Balanis 1989). To completely define an EM field, boundary conditions need to be incorporated to take discontinuous charges and currents along the interface between different media into consideration. The differential form of Maxwell’s equations is given in the following equations (Balanis 1989):

∇ × Ε = − μr μ 0

∂Η ∂t

∇ × H = σ E + εr ε0

∂E ∂t

(6.1) (6.2)

∇⋅D = q

(6.3)

∇⋅B = 0

(6.4)

where E is electric field vector (V/m), and H is magnetic field vector (A/m); D is electric flux density (C/m2), and D = εrε0E; B is magnetic flux density (Wb/m2), and B = μrμ0H; ε0 and εr denote the dielectric permittivity in free space and the value relative to free space, respectively; μ0 and μr denote the dielectric permeability in free space and relative value, respectively; and q is electric charge density per unit volume (coulombs/m3). εr and μr are vectors in general; they are reduced to scalar for isotropic substances. Equation 6.1 represents Faraday’s law, indicating that the circulation of electric field strength surrounded by a closed contour is determined by the rate of change of the magnetic flux density. Equation 6.2 is Ampere’s law, indicating that the circulation of magnetic field strength enclosed by a closed contour is equal to the net current through the surface. Equation 6.3 presents Gauss’s electric law, which states that net electric flux out of an enclose region is equal to the charges contained within the region. Equation 6.4 presents Gauss’s magnetic law, which requires that the net magnetic flux out of a region is zero.

6.2.2. Boundary Conditions The field discontinuities along the interface between two different media are expressed in field vectors. The boundary conditions can be derived from Gauss’s electric and magnetic law along the median interface: nˆ × ( E 2 − E1 ) = 0

(6.5)

nˆ × (H 2 − H1 ) = 0

(6.6)

nˆ ⋅ ( D2 − D1 ) = 0

(6.7)

nˆ ⋅ ( B2 − B1 ) = 0

(6.8)

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where nˆ is normal unit component of the interface, the subscripts 1 and 2 denote the regions of different media. Equations 6.5 and 6.6 state that the tangential components across an interface between two media are continuous for electrical field and magnetic field, respectively. Equations 6.7 and 6.8 show the continuity of normal component across the boundary for electric flux density and magnetic flux density, respectively. It should be noted that Equation 6.7 holds only if there is no impressed magnetic current density along the boundary, which is valid for MW heating of foods.

where f (x), g( y), and h(z) are functions to be determined from wave equations for specific boundary conditions. They represent the behavior of EM waves in each of the three directions. Here we will briefly discuss the solutions in three classical coordinate systems (rectangular, cylindrical, spherical system). Detailed procedures have been presented in numerous publications (e.g., Harrington 1961; Balanis 1989; Sadiku 2001).

6.3. Solutions to Maxwell’s Equations

E ( x, y, z ) = aˆ x E x ( x, y, z ) + aˆ y E y ( x, y, z ) + aˆ z Ez ( x, y, z )

6.3.1. Analytical Solutions Equations 6.1–6.4 are coupled partial differential equations with at least two unknown variables in each equation. To uncouple these equations, only one unknown variable can appear inside the equation, while the others need to be eliminated, leading to second-order partial differential equations. For sinusoidal time-varying field, Equations 6.1 and 6.2 can be written as (Balanis 1989): ∇ 2 E = −ω 2 με E + jωμσ E

(6.9)

∇ 2 H = −ω 2 με H + jωμσ H

(6.10)

and where σ is electrical conductivity for a lossy material. A complex variable γ is commonly used, referred to as the propagation constant: where γ 2 = jωμσ − ω 2 με , or simply γ = jβ + α . The EM fields can be obtained by solving wave Equations 6.9 and 6.10 with appropriate boundary conditions. One of the most popular methods to solve these equations is known as separation of variables (Wylie 1960; Hilderbrand 1962). With this method, the solution of a scalar function, for example, Ex(x, y, z) in Cartesian (rectangular) coordinate system, is assumed to be a multiplication of three different scalar functions each having only one independent variable. Therefore, the field component Ex(x, y, z) can be written in the following form: E x ( x, y, z ) = f ( x ) g ( y ) h ( z )

6.3.1.1. Rectangular Coordinate System In the rectangular coordinate system, a general solution for the electric field E can be written as:

(6.11)

(6.12)

where aˆ x , aˆ y , and aˆ z are unit vectors in x-, y-, and z-directions, respectively. Substituting Equation 6.12 into wave Equation 6.9, the general equation is reduced into three scalar equations that can be solved by using Equation 6.11 (separation of variables). Depending on the nature of the problem, different structures of the functions are used to represent distinct wave propagations. For example, standing waves are represented using sinusoidal functions while travelling waves are constructed by exponential functions. As an example, an EM wave is assumed to propagate inside a rectangular geometry, which has a rectangular cross-section with infinite length regarding the wave propagation direction. This is illustrated in Figure 6.1. By following the procedures of the separation of variable method, one can derive the electric field component Ex as: E x ( x, y, z ) = [ A1 cos (β x x ) + A2 sin (β x x )]

[ B1 cos (β y y ) + B2 sin (β y y )]

[C1e− jβ z + C2 e+ jβ z ] z

z

(6.13)

where A1, A2, B1, B2, C1, and C2 are constants and determined by applying appropriate boundary conditions. The waveguide is limited in x- and y-directions; the forms of the waves in these directions are described with sinusoidal functions. In z-direction, the EM wave propagates into infinity, producing traveling waves in both +z- and −z-directions. These

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y y

ρ

b

ϕ x

a

z

a

x

Figure 6.1. Rectangular geometry for wave propagation.

are shown in Equation 6.13. For other components, due to the symmetric merits of Maxwell’s equations, the general solutions can be constructed similarly by applying the duality or reciprocity theorem mentioned above. 6.3.1.2. Cylindrical Coordinate System In cylindrical systems, solutions for Maxwell’s equations can be constructed as in the rectangular coordinate system; cylindrical coordinates, however, are more convenient. The general solution for a cylindrical coordinate system can be written as: E ( x, y, z ) = aˆ ρ Eρ ( x, y, z ) + aˆφ Eφ ( x, y, z ) + aˆ z Ez ( x, y, z )

(6.14)

where aˆ ρ , aˆφ , and aˆ z are unit vectors in cylindrical coordinates. Figure 6.2 shows a cylindrical coordinate system that has a circular cross-section in the x-y-plane and infinite length in z-direction. The general solution can be constructed by using the separation of variable method:

ψ ( ρ, φ , z ) = [ A1J m ( βρ ρ ) + A2Ym ( βρ ρ )]

[ B1 cos ( mφ ) + B2 sin ( mφ )]

[C1e− jβ z + C2 e+ jβ z ] z

z

(6.15)

z Figure 6.2. Cylindrical coordinate system.

where ψ ( ρ, φ, z ) is the solution for any field in the cylindrical coordinate system. J m ( β ρ ρ ) is the first Bessel function while Ym ( β ρ ρ ) is the second. A1, A2, B1, B2, C1, and C2 are constants and are determined by applying appropriate boundary conditions. In cylindrical coordinate systems, J m ( β ρ ρ ) and Ym ( β ρ ρ ) represent standing waves. Therefore, Equation 6.15 indicates a general field solution for an EM wave that is limited by a cylindrical waveguide and propagates into infinity in z-direction. For most radiation problems, it is required to calculate the field patterns outside the cylinder. For these cases, the first and second kind of Bessel functions is replaced by Hankel function, which represents a traveling wave. Its solution is shown in the following equation:

ψ ( ρ, φ, z ) = [ A1 H m(1) ( β ρ ρ ) + A2 H m(2) ( β ρ ρ )]

[ B1 cos (mφ ) + B2 sin (mφ )]

[C1e− jβ z + C2 e+ jβ z ] z

z

(6.16)

6.3.1.3. Spherical Coordinate System The spherical coordinates are desirable when a spherical geometry is involved in computing the EM field. The spherical coordinate system is shown in the Figure 6.3:

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105

6.3.2. Numerical Solutions for EM Fields

Ψ ϕ

r

Figure 6.3. Spherical coordinate system.

The general solutions are obtained by solving the following vector wave equations in the spherical coordinate system. E (r , ϕ , ψ ) = aˆr Er (r , ϕ , ψ ) + aˆϕ Eϕ (r , ϕ , ψ ) + aˆψ Eψ (r , ϕ , ψ )

(6.17)

Similar to the solution of the rectangular and the cylindrical coordinate system, the wave Equation 6.17 is solved by using the separation of variable technique. E (r , ϕ , ψ ) = [ A1 jn (β r ) + A2 yn (β r )]

[ B1 Pnm (cos ψ ) + B2Qnm (cos ψ )] [C1 cos (mϕ ) + C2 sin (mϕ )]

(6.18)

where jn(βr) and yn(βr), which indicate the standing wave along the radius direction, are spherical Bessel functions of the first and second kind, respectively. Pnm and Qnm are referred to as the associated Legendre functions of the first and second kind, respectively. A1, A2, B1, B2, C1, and C2 are constants to be determined by appropriate boundary conditions. For the traveling wave outside the sphere along the radius, jn(βr) and yn(βr) are replaced by Hankel functions of the first and second kind, hn(1) (βr ) and hn(2) (βr ) , respectively.

For complex geometries, there is no closed-form solutions. Numerical methods are used to provide approximate solutions. The fundamental concept of numerically solving complex equations such as Maxwell’s equations and thermal equations is to discretize the geometry of interest into numerous cells with specified cell sizes. The numerical approach employs linear or polynomial approximations for each cell, thus reducing complex partial differential equations to sets of simple linear or polynomial equations. Starting from the mid-1960s, considerable efforts have been made to numerically solve practical EM-related problems by taking advantage of modern powerful computers (Sadiku 2001). The following sections describe briefly the most commonly used numerical methods to solve Maxwell’s field equations. Those methods are the Finite Difference Method (FDM), the MoM, and the FEM. 6.3.2.1. FDM The FDM was first developed by Thom (Thom and Apelt 1961) to solve hydrodynamic problems. This technique uses finite difference approximations in algebraic form to replace the differential equations and calculate the value of the dependent variable at one point based on those values at its neighboring points. Basically, the FDM follows the following three steps to obtain the numerical solutions: • Discretize the geometry into numerical grid lines with specified cell size. • Replace the differential equations with finite difference equations by appropriate approximations in their algebraic form. • Solve the linear equations along with boundary and initial conditions. Detailed procedures are slightly different depending on the nature of the problems, the solution regions, as well as boundary conditions. For EM-related problems, the finite difference technique was deployed to solve the time differentiated Maxwell’s curl equations (Yee 1966; Taflove and Brodwin 1975a, 1975b; Taflove 1980; Taflove and Umashankar

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1980, 1981, 1990; Jurgens et al. 1992; Okoniewski 1993). Thus, this technique is usually called FDTD. In brief, FDTD method is a time-marching procedure that simulates the continuous EM waves in a finite spatial region while continuously time-stepping until the desired simulation time is achieved or a stable field pattern is established (Taflove 1998). Obviously, only finite computational grids can be practically applied, although the modeled region extends theoretically to infinity. As a result, absorbing boundary conditions (ABC) are employed to terminate the computational grids that permit all the outgoing waves to propagate into infinity with negligible reflections. Another factor to be considered in employing FDTD technique is to ensure the simulation accuracy by well-resolved spatial and temporal variations using time- and space-sampling process. The sampling in space is determined by following the Nyquist theorem that requires at least 10 cells per wavelength. The sampling in time depends on the selection of spatial sampling size to ensure the overall stability of the algorithm. Yee Algorithm Kane Yee in 1966 developed an effective FDTD algorithm to simulate wave propagations inside a lossless material by solving a set of finite-difference equations for the time-dependent Maxwell’s curl equations. The Yee algorithm simultaneously calculates the coupled E and H fields. Using the Yee algorithm in a 3D network, the E component in each direction is surrounded by four H components of its vertical counterpart, while each H component is surrounded by four E components in a similar fashion. The interlinked E and H field components intrinsically keep the validity of Faraday’s and Ampere’s Laws. The algorithm naturally maintains the continuity of the tangential E and H fields across the interfaces of distinct media, requiring no extra efforts to satisfy the boundary conditions along the interfaces. The explicit secondorder central-difference equations are employed in Yee algorithm, which intuitively guarantees the two Gauss’s Law relations. The Yee algorithm calculates all E field components based on the latest H fields and stores the values in the internal memory at a particular simulation time step, and then calculates

all H field components from the newly obtained E fields. The computation continues until the required simulation time is reached. The above strategy eliminates the matrix inversion that requires lengthy computation time and large computer memory. It also naturally simulates a wave propagation without spurious decay due to artificial nonphysical wave that resulted from the algorithm. For the sake of illustration, 1D FDTD algorithms are presented here. A 3D problem is discussed in more details in the case study. Figure 6.4 shows how the FDTD simulation approaches in time (t) and space (x). The EM wave is propagated into x-direction, while only Ey and Hz are assumed to exist according to the Right Thumb Rule. For the 1D wave propagation problem shown in Figure 6.4, the Maxwell’s equations are reduced to (Taflove 1998): ∂E y 1 ⎛ ∂H z ⎞ = ⎜− − σ Ey ⎟ ⎠ ∂t ε ⎝ ∂x

(6.19)

∂H z 1 ⎛ ∂E y ⎞ = ⎜− − ρ′Hz ⎟ ⎠ ∂t μ ⎝ ∂x

(6.20)

where μ and ε are permeability and permittivity, respectively; σ is the electrical conductivity; and ρ′ is denoted as magnetic loss, which are set to 0 for simplification. The latter assumption is applicable to food applications. Equations 6.19 and 6.20 denote an EM wave propagating in TE mode, indicating that the electrical field is vertical to the wave propagation direction. The finite difference form for Equation 6.19 is obtained using the approximation: E yn +1 [iΔx ] − E yn [iΔx ] Δt ⎞ ⎛ H zn +1 / 2 [(i + 1 / 2 ) Δx ] − ⎜ n +1 / 2 H 1 [(i − 1 / 2 ) Δx ] − σ E n +1/ 2 iΔx ⎟⎟ = ⎜− z [ ] y ε⎜ Δx ⎟ ⎟ ⎜ ⎠ ⎝ (6.21) where Δx is the space resolution, i is the spatial index, and n is the time step. Here n + 1/2 is used because Hz is placed half-time step delayed from the

Chapter 6

Computer Simulation for Microwave Heating

107

t E

E

E

E

E

E

E

3Δt H

5/2Δt

H

E

E

H E

H E

H E

H E

E

2Δt H

3/2Δt

H

E

E

H E

H E

H E

H E

E

1Δt

1/2Δt E 0

H 1/2Δx

H H H H H 5/2Δx 7/2Δx 9/2Δt 11/2Δx 3/2Δx E E E E E E Δx

2Δx

3Δx

4Δx

5Δx

6Δx

x

Figure 6.4. E and H field arrangement of FDTD algorithm for one-dimensional wave propagation using central difference time-stepping procedures.

corresponding Ey from Figure 6.4. The term E yn+1/ 2 is estimated using a semi-implicit approximation: E yn+1 / 2 =

E yn+1 + E yn 2

(6.22)

step n – 1/2 and n + 1/2 is stored in the memory available for computation. As a result, to estimate the magnetic field Hz used in Equation 6.20, a semiimplicit approximation is employed: H zn =

Rearrange Equation 6.21 with Equation 6.22 and remove Δx for simplification: Δt ⎛ 1 − σΔt ⎞ ⎟ n ⎜ ε ε 2 n +1 E y [i ] = ⎜ E [i ] − Δx σΔt σΔt ⎟ y 1+ ⎟ ⎜⎝ 1 + 2ε ⎠ 2ε n +1 / 2 n +1 / 2 {H z [i + 1 / 2 ] − H z [i − 1 / 2 ]} (6.23) Similar procedures are followed to compute for Hz in Equation 6.20, except that Hz at time step n exists due to magnetic loss. From Figure 6.4, the magnetic field Hz is supposed to be placed half a time step away from the electrical field so that only Hz at time

H zn+1 / 2 + H zn−1 / 2 2

(6.24)

With Equation 6.23, we obtain the explicit finite difference equation for Hz.

H

n +1 / 2 z

⎛ 1 − ρ ′Δt ⎞ ⎜ 2μ ⎟ [i ] = ⎜ ρ ′Δt ⎟ H zn −1/ 2 ⎟ ⎜1+ ⎝ 2μ ⎠ Δt μΔx − {E n [i + 1] − Eyn [i ]} ρ ′Δt y 1+ 2μ

(6.25)

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Equations 6.23 and 6.25 can be conveniently programmed for E and H field calculation.

f ( x i + Δx , t n ) = f ( x i , t n )

Numerical Dispersion Numerical dispersion is unavoidable whenever a numerical approach is used to solve the continuous wave equation. Numerical dispersion is defined as the variation of numerical wavelength with frequency f since the wavelength is represented in terms of cell size. In this section, the 1D scalar wave equation is obtained first, followed by an asymptotic study for numerical accuracy. For simplification, assuming the nonexistence of electric and magnetic loss, decoupling the Ey and Hz in Equations 6.19 and 6.20, and the second-order partial differential equation is obtained as: ∂ 2 H z ∂ ⎛ ∂H z ⎞ ∂ ⎛ 1 ∂E y ⎞ 1 ∂ Ey = ⎜ = ⎜− =− ⎟ ⎟ 2 ∂t ∂t ⎝ ∂t ⎠ ∂t ⎝ μ ∂x ⎠ μ ∂t ∂x (6.26) 2

Similarly, according to Equation 6.19, with the assumption of σ = 0: ∂2 Ey 1 ∂2 H z =− ∂ t∂ x ε ∂x 2

(6.27)

Combining Equations 6.26 and 6.27 yields: ∂2 H z 1 ∂2 H z ∂2 H z = = u2 2 2 με ∂x ∂t ∂x 2

(6.28)

where u = 1/ με is the speed of light inside the material under consideration. Equation 6.28 is a 1D scalar wave equation for Hz. Following similar procedures, wave equations for Ey are obtained as: 2 ∂2 Ey 2 ∂ Ey = u ∂t 2 ∂x 2

(6.29)

Obviously, both Ey and Hz vary as the same function of time and space. The solution of one field component leads to another. From now on, we replace Ey and Hz with a more general function f (x, t). The solution for f leads to both solutions for Ey and Hz. In Yee’s algorithm, the partial differential equations in Equations 6.28 and 6.29 are differentiated using second-order central difference equations. In detail, by Taylor ’s expansion, at space point xi with space increment Δx and fixed time point tn:

∂ f ( x i , t n ) ( Δx ) ∂ f 2 ( x i , t n ) 2 + + O ⎡⎣( Δx ) ⎤⎦ ∂x 2 ∂x 2 (6.30) 2

+ Δx

f ( x i − Δx , t n ) = f ( x i , t n ) ∂ f ( x i , t n ) ( Δx ) ∂ f 2 ( x i , t n ) 2 + + O ⎡⎣( Δx ) ⎤⎦ 2 ∂x 2 ∂x (6.31) 2

− Δx

2 where O ⎡⎣( Δx ) ⎤⎦ is a notation for the remaining higher order terms that approach to zero much faster than the lower order of Δx. Combining Equations 6.30 and 6.31,

f ( xi + Δx, t n ) + f ( xi − Δx, t n ) = 2 f ( xi , t n ) + ( Δx )

2

∂f 2 ( xi , t n ) 2 + O ⎡⎣( Δx ) ⎤⎦ (6.32) 2 ∂x

2 with Δx small enough, O ⎡⎣( Δx ) ⎤⎦ can be neglected without significant loss of accuracy. Therefore, the second-order partial differential equation can be represented as:

∂f 2 ( xi , t n ) ∂x 2 f ( xi + Δx, t n ) + f ( xi − Δx, t n ) − 2 f ( xi , t n ) (6.33) = (Δx )2 Similarly, the second-order partial derivatives with respect to time are obtained as: ∂f 2 ( x i , t n ) ∂t 2 f ( xi , t n + Δt ) + f ( xi , t n − Δt ) − 2 f ( xi , t n ) (6.34) = ( Δt )2 For simplification, we write f ( xi , t n ) = fi n , xi +1 = xi + Δx and t n+1 = xn + Δt . With these shorthand notations, combining Equations 6.33 and 6.34, the wave equations in Equations 6.28 and 6.29 can be reduced to: fi n+1 + fi n−1 − 2 fi n fn + fn −2fn = u2 i +1 i −1 2 i 2 (Δt ) (Δx )

(6.35)

Rearranging the terms in Equation 6.35 to obtain the explicit expression of the latest value at a specified

Chapter 6

space point based on its previous value, the current value and the values of the neighboring points: ⎛ uΔt 2 ⎞ n fi n+1 = ⎜ ( fi+1 + fi−n1 − 2 fi n ) + 2 fi n − fi n−1 ⎝ Δx ⎟⎠

(6.36)

Equation 6.36 is the numerical dispersion relationship of the finite difference estimation for 1D scalar wave equation (Taflove 1998). It uses the current and previous values stored in the computer memory to calculate the latest value. For example, when evaluating the sinusoidal traveling wave in a 1D finite difference scheme: fi n = e j (ωΔt − k1Δx )

(6.37)

where ω = 2π f is the angular frequency; k1 is a numerical wave number that differs from the continuous wave number k = ω/u due to the numerical approximations. This can be observed by substituting Equation 6.37 into Equation 6.36 and factoring out the common terms on both sides of the resulting equation. Finally, the implicit form of the dispersion relation is obtained as (Taflove 1998): e jωΔt + e − jωΔt ⎛ uΔt ⎞ ⎛ e jk1Δx + e − jk1Δx ⎞ =⎜ − 1⎟ + 1 (6.38) ⎝ Δx ⎟⎠ ⎜⎝ ⎠ 2 2 2

By using Euler identity, Equation 6.38 can be reduced to: ⎛ uΔt ⎞ cos(ωΔt ) = ⎜ [cos(k1Δx ) − 1] + 1 (6.39) ⎝ Δx ⎟⎠ 2

The difference between k1 and k results in the numerical errors, which we need to depress to develop a reliable simulation algorithm. Based on Equation 6.39, two cases will be considered: (a) Δt → 0, Δx → 0: For very small arguments, one-term Taylor ’s series is applied to approximate the cosine functions in Equation 6.39: (ωΔt )2 ⎛ uΔt ⎞ 1− =⎜ ⎝ Δx ⎟⎠ 2

2

⎡ ( k1Δx )2 ⎤ − 1⎥ + 1 (6.40) ⎢1 − 2 ⎣ ⎦

Simplifying Equation 6.40 leads to k = ω / u . This is equal to the continuous wave number. Therefore, when the time and space increment

Computer Simulation for Microwave Heating

109

approach zero, the numerical results converge to the analytical solutions. (b) uΔt = Δx : This is another special case, by which Equation 6.40 again is reduced to k = ω / u. While the two special cases are considered above, the general formulation for the numerical wave number is k1 =

⎧ ⎛ Δx ⎞ 2 ⎫ 1 cos −1⎨⎜ ⎟⎠ [ cos(ωΔt ) − 1] + 1⎬ ⎝ Δx Δ u t ⎩ ⎭

(6.41)

Equation 6.41 can be used to analyze the numerical error due to finite differences by solving the numerical wave number with indistinct cell sizes in terms of the wavelength. For example, with cell size Δx = λ0/10 and Δt = Δx/2u, Equation 6.41 leads to k = 0.636 / Δx . Following the similar definition to the analog-phase velocity of the continuous wave equation as vp = ω/k, where vp is defined as phase velocity for the continuous wave equation, the numerical phase velocity is denoted as v p = ω / k . Therefore with the previous case, v p = 0.9873u, indicating that for a continuous wave propagating into 10λ0, the simulated numerical wave only propagates over a distance of 9.873λ0. This discrepancy results in a phase error of 45.72. When the cell size is reduced to Δx = λ 0 /20, the phase velocity is changed to be 0.997u. Accordingly, the phase error is reduced by three quarters to 10.8. Numerical Stability One convenient method to analyze the numerical stability is to use the Eigenvalue method, which separates the space and time derivative part of the original wave equation for analysis purposes. By setting the stable range for the complete spectrum of spatial Eigenvalues, it is guaranteed that all possible numerical modes in the grid are stable. The detailed procedure for Eigenvalue problems is presented in the following equations: ∂2 n fi = Πfi n ∂t 2

(6.42)

where Π is denoted as the operator of second-order time derivatives for a wave function. Combining the left side of Equation 6.35 and right side of Equation 6.42:

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

fi n+1 + fi n−1 − 2 fi n = Πfi n 2 Δ t ( )

(6.43)

The numerical stability requires that the absolute value of fi n+1 fi n is less than one for all possible wave propagation modes. To take this requirement into consideration, rearranging Equation 6.43 yields: 2

n +1 ⎞ ⎛ fi n+1 ⎞ 2 ⎛ fi ⎡ ⎤ 2 Π Δ t − + ( ) ⎜⎝ f n ⎟⎠ ⎣ ⎦ ⎜⎝ f n ⎟⎠ + 1 = 0 i i

(6.44)

Solving the Equation 6.44 gives: 2

2 2 fi n+1 ⎡⎣2 + Π ( Δt ) ⎤⎦ ± ⎡⎣2 + Π ( Δt ) ⎤⎦ − 4 = 2 fi n

To guarantee

(6.45)

fi n+1 fi n ≤ 1 , we need to have

2

⎡⎣2 + Π ( Δt )2 ⎤⎦ − 4 ≤ 0 , which reduces to: −4 ≤Π≤0 (Δt )2

(6.46)

Following the similar procedure, the Eigenvalue Π can be limited by the spatial resolution: −4u2 ≤Π≤0 (Δx )2

has the capability of being implemented in generic computer programs for broad range applications such as waveguide problems, and absorption of EM energy by biological bodies (Sadiku 2001). Basically four main steps are involved in the FEM: • Divide the solution region into multiple subregions or elements. • Obtain the partial differential equation in each element. • Integrate the solutions of all subregions into the solution for the entire solution space. • Solve the obtained linear equations. These procedures are illustrated in Figure 6.5. Usually, in FEM method, the computational domain is divided into numerous elements, each having a triangular or other simple shape. For each element, there are three zeniths, which are indexed by integer numbers. To illustrate the principle of FEM, we calculate the potential distribution inside the irregular geometry shown in Figure 6.5. Basically, the potential in each triangular element is assumed to be constant Ve and the overall distribution of the potential is assumed equal to the sum of the potential in each element. N

(6.47)

V ( x, y ) ≈ ∑ Ve ( x, y )

For any spatial grid inside the computational domain, to ensure the numerical stability requires that the Eigenvalues of the spatial mode fall completely into the range of the Eigenvalues based on the timestepping equations, that is, Equation 6.47 should be completely contained inside Equation 6.46. For this reason, the time step can be limited by a predetermined cell size Δx. Δt ≤

Δx c

(6.49)

e =1

3

2

6.3.2.2. FEM Another popular method to numerically solve EM problems is the FEM. This method is powerful for very complex geometries. The FEM

1

2 1

1

1

3

3 8

7

1

1

2

3

1

3 3 2

9

4

2

8

(6.48)

Equation 6.48 gives the limits for the time increment selection for the 1D scalar wave equation. The extension of Equation 6.48 to 2D and 3D problems is straightforward (Taflove 1998).

3

2

1 1 1

2 3 6

5

2 3

4 2 3

2

7 Figure 6.5. FEM discretization for irregular geometry.

6

5

Chapter 6

where N is the number of elements, V(x,y) is the desired potential distribution. Since the element is triangular, Ve(x,y) is represented by a linear equation with three constants to be determined (Sadiku 2001). Ve ( x, y ) = a + bx + cy

(6.50)

where a, b, and c are constants calculated from the potentials at each zenith. From Equation 6.50, we denote the potential at zenith i as Vei (i = 1, 2, 3), the value of a, b, and c can be computed using matrix manipulation: ⎡ a ⎤ ⎡1 x1 ⎢ b ⎥ = ⎢1 x 2 ⎢ ⎥ ⎢ ⎢⎣ c ⎥⎦ ⎢⎣1 x3

y1 ⎤ y2 ⎥ ⎥ y3 ⎥⎦

−1

⎡Ve1 ⎤ ⎢V ⎥ ⎢ e2 ⎥ ⎢⎣Ve3 ⎥⎦

(6.51)

Combining Equations 6.50 and 6.51, the potential at any point in the element is obtained as: Ve ( x, y ) − [1 x

⎡1 x1 y ] ⎢1 x2 ⎢ ⎢⎣1 x3

y1 ⎤ y2 ⎥ ⎥ y3 ⎥⎦

−1

⎡Ve1 ⎤ ⎢V ⎥ ⎢ e2 ⎥ ⎢⎣Ve3 ⎥⎦

(6.52)

Having considered the potential in each element, the potential distribution is obtained by assembling all elements in the solution region. The detailed procedures are documented in numerous publications. The equations involved are listed in this section: W=

1 T ε [V ] [C ][V ] 2

(6.53)

where W is the energy associated with the solution region, and ε is dielectric permittivity.

[V ] = [V1 V2 … Vn ]T ⎡C11 ⎢C ⎢ 21 ⎢C31 ⎢ ⎢C41 [C ] = ⎢C51 ⎢ ⎢C61 ⎢C71 ⎢ ⎢C81 ⎢C ⎣ 91

C12 C22 C32 C42 C52 C62 C72 C82 C92

C13 C23 C33 C43 C53 C63 C73 C83 C93

C14 C24 C34 C44 C54 C64 C74 C84 C94

C15 C25 C35 C45 C55 C65 C75 C85 C95

C16 C26 C36 C46 C56 C66 C76 C86 C96

C17 C27 C37 C47 C57 C67 C77 C87 C97

(6.54) C18 C19 ⎤ C28 C29 ⎥ ⎥ C38 C39 ⎥ ⎥ C48 C49 ⎥ C58 C59 ⎥ ⎥ C68 C69 ⎥ C78 C79 ⎥ ⎥ C88 C89 ⎥ C98 C99 ⎥⎦ (6.55)

Computer Simulation for Microwave Heating

111

where [C] is a global coefficient matrix that has each element denoting the coupling between different nodes, that is, Cij is the coupling between node i and j. The Cij is calculated based on the continuity of the potential distribution across the boundary of neighboring elements. For example, to calculate C12 in Figure 6.5, there are totally nine global nodes and eight elements to represent the solution region. Each element has three local nodes indexed in counterclockwise direction. [1] [2 ] C12 = C13 + C23

(6.56)

[1] 13

where C is the coupling coefficient between local [2 ] nodes 1 and 3 in local element 1. C23 is the coupling efficient between local nodes 2 and 3 in local element 2. Having obtained the global coefficient matrix from Equation 6.56, physical conditions are applied in Equation 6.53 to solve the desired potential. For example, to solve the Laplace’s equation, we require that the associated energy W be minimum. Therefore, the first partial derivatives of W are set to zero: ∂W ∂W ∂W = == =0 ∂V1 ∂V2 ∂Vn

(6.57)

Substituting Equation 6.53 into Equation 6.57, we obtain a set of linear equations. These equations are then solved by several popular methods such as iteration method and band matrix method (Sadiku 2001). FEM has been applied to solve a wide range of problems due to its capability for handling complex geometry and relative easy extension for general purposes. However, there are several disadvantages associated with this method, including intensive memory usage for matrix inversion as well as tedious data preprocessing. 6.3.2.3. MOM The MoM is another popular numerical approach applied to a wide range of EM problems. This method takes moments with corresponding weighing functions, discretizes the differential or integral equations for each of these moments, and solves the obtained matrix for parameters of interest (Harrington 1967, 1968, 1992). A detailed description of the MoM technique can be found in Richmond (1965), Mittra (1973), Tsai

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

(1978), and Jurgens et al. (1992). Here we only briefly present some of the main procedures. For illustration purposes, we use the MoM technique to calculate the dissipated power of EM radiation. The primary concern of the product’s response to an EM field is the specific absorption rate (SAR) defined as: SAR =

σ E ρ

2

(6.58)

where σ is the electrical conductivity, ρ is the mass density, and E is the electric field inside a product (Kastner and Mittra 1983). According to Equation 6.58, the SAR can be calculated from the value of electric field E. The electric field E, on the other hand, can be derived by representing Maxwell’s curl equations using a tensor integral equation (TIE) (Livesay and Chen 1974; Kong 1981). Following the procedures developed by Livesay and Chen (1974), the solution of E is obtained as: E (r ) = − jωμ0 ∫ G0 (r , r ′ ) ⋅ J eq (r ′ ) dv ′

(6.59)

where G0(r, r′ ) is free-space Green’s function, indicating the electrical field at a space point r due to an infinitesimal source at space point r′. Jeq(r′ ) is the equivalent current density, which is a singular function in each direction: J eq (r , r ′ ) = δ (r − r ′ )

(6.60)

The TIE is then obtained as: PV E (r ) = χ (r ) 1+ 3 jωε 0

The electric field E is then determined from Equation 6.62 as:

[ E ] = − [G ]−1 [ E i ]

(6.63)

A computer program was developed by Guru and Chen (1976) to compute the SAR inside a biological body using MoM following Equations 6.58 to 6.63. It is obvious from Equation 6.63 that matrix inversion, elimination, and iteration are involved in the computation. This usually requires significant computer memory and lengthy computation times as does the FEM method. For these reasons, the application of MoM is limited to solve the problems with scattering and irradiating geometries that are electrically large, in the order of λ3.

6.4. MW Heating Equations 6.4.1. Dissipated Power in Dielectric Materials For MW heating applications, the EM power is dissipated inside the dielectric material to heat the product. Therefore, it is necessary to calculate dissipated MW power associated with EM waves. This can be derived from the time-averaged Poynting vector over one period (Balanis 1989): 1 ψ av = Re [ E × H* ] 2

(6.64)

By taking the volume integral and applying the divergence theorem, the dissipated real power is achieved as:

∫ χ (r ′ ) E (r ) v

E (r ) ⋅ G (r , r ′ ) dv ′ + χ (r ) 1+ 3 jωε 0

Pd =

i

(6.61)

where PV is the principal value (Sadiku 2001), Ei(r) is the incident electrical field, which is already known. E(r) is the electrical field to be determined by MoM. Equation 6.61 can be written in a matrix form, suitable for programming.

[G ][ E ] = − [ E i ]

(6.62)

1 2 σ E dv 2 ∫∫∫V

(6.65)

where Pd is EM power dissipated inside a volume V. σ denotes electrical conductivity of the media inside a volume V. Once the power dissipation is calculated, it is then used as a source for heating the product.

6.4.2. Heat Transfer Equations For MW heating applications, it is desirable to obtain information on the temperature changes in

Chapter 6

response to the dissipated MW power. The temperature profile can be determined by applying the energy conservation requirement on a differential control volume (Incropera and DeWitt 2001). The law of energy conservation states that for a unit volume in the medium, the rate of energy transfer by conduction or convection plus the volumetric rate of thermal energy generation must be equal to the rate of change of thermal energy stored within the volume. For solid foods, this can be expressed in the following equation:

ρc p

∂T = ∇ ( K (T ) ⋅∇T ) + q ∂t

(6.66)

where ρ is the mass density (kg/m3), cp is the specific heat capacity (J/kg · K), q is power density (J/m3s or W/m3), and K(T) is thermal conductivity (W/m · K). For MW heating, q is provided by the dissipated MW power. Temperature distribution can be determined by solving Equation 6.66 with appropriate boundary and initial conditions. Three types of boundary conditions are encountered in heat transfer Equation 6.66: the Dirichlet condition, the Neumann condition, and the Convective surface condition. The Dirichlet condition corresponds to a situation that the surface is maintained at a fixed temperature Ts; the Neumann condition requires that the heat flux on the boundary be fixed; and the Convective surface condition is obtained by exchanging thermal energy between different mediums (Incropera and DeWitt 2001).

Computer Simulation for Microwave Heating

113

to reading errors. Special measures need to be taken to reduce the interference. Fiber-optic temperature sensors, on the other hand, do not interfere with EM waves, and thus they serve as a reliable means for direct temperature measurement of selected locations during MW heating. Several methods have been used to determine MW heating patterns (see also Chapter 7 of this book). These methods include infrared imaging and chemical marker methods. Infrared cameras provide direct temperature information for the surface or a cut surface of a food package after heating processes. But it is difficult to use this method to obtain temperature profile inside food packages during a heating process. Chemical marker methods, on the other hand, provide information on heating patterns based on formation of new chemical compounds during complete heating processes. For example, one chemical marker method measures color changes as a result of formation of chemical compound (M-2) due to Maillard reaction between amino acids and a reducing sugar, ribose in food matrices. The color response depends upon heat intensity at temperatures beyond 100°C and, therefore, serves as an indirect indicator of temperature distribution during MW heating for food sterilization that calls for product temperature to be above 100°C. Detailed information about the kinetics of Marker-2 formation with temperature could be found elsewhere (Kim and Taub 1993, Lau et al. 2003; Pandit et al. 2006).

6.4.3. Measurement of Temperature Responses

6.5. Computer Simulation of MW Heating

Direct or indirect means for temperature measurement are used to validate computer simulation models. Temperature can be directly measured by numerous electronic sensors such as resistance temperature detectors (RTDs), thermistors, and thermocouples. Of all these methods, thermocouples are widely used to monitor processing temperatures (Desmarais and Breuereis 2001). But when exposed to a high EM field, the metal parts of the electronic sensors may cause EM interference (EMI) leading

We have discussed analytical and numerical solutions to EM field equations applied to solve industrial MW heating problems. A particular challenge to numerically simulate complicated MW heating process is to concurrently calculate EM and thermal field with temperature-dependent dielectric properties. This section describes the use of the conformal FDTD method in the development of a computer simulation model for coupled EM and heat transfer within a food package during MW heating in a

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Ey

915 MHz water emersion MW sterilization system. A detailed description of this system is presented in Section 6.6. We first describe equation formulation, step-by-step validation for simple cases with analytical solutions, followed by experimental validation for a complicated system, and finally the applications of the model for optimization of the system process parameter to achieve desirable heating uniformity.

z

Hz Ez

Ex

Ez

6.5.1. Model Formulation Using the FDTD Method

Hy Ez Ey

Hx

(i, j, k)

∂H x 1 ⎛ ∂E y ∂E z ⎞ = ⎜ − ∂t μ t ⎝ ∂z ∂y ⎟⎠

(6.67)

∂H y 1 = ∂t μt

⎛ ∂E z ∂E x ⎞ − ⎜⎝ ⎟ ∂x ∂z ⎠

(6.68)

∂H z 1 ⎛ ∂E x ∂E y ⎞ = ⎜ − ∂t μ t ⎝ ∂y ∂x ⎟⎠

(6.69)

∂E x 1 ⎛ ∂H z ∂H y ⎞ = ⎜ − − σ t Ex ⎟ ⎠ ∂t ε t ⎝ ∂y ∂z

(6.70)

∂E y 1 ⎛ ∂Hx ∂H z ⎞ = ⎜ − − σ t Ey ⎟ ⎝ ⎠ ε t ∂z ∂t ∂x

(6.71)

∂E z 1 ⎛ ∂H z ∂H x ⎞ = ⎜ − − σ t Ez ⎟ ⎠ ∂t ∂y ε t ⎝ ∂x

(6.72)

Ez

Ey Hy

Ex

Hz

Ex

The Maxwell’s equations in a 3D rectangular coordinate system can be presented in a time harmonic form (Taflove 1998):

Ex

Hx

Ey y

x Figure 6.6. Layout of field components for FDTD algorithm.

In space, a grid point can be expressed as: (i, j, k ) = (i ∗ δ x, j ∗ δ y, k ∗ δ z )

(6.73)

where δ x, δ y, and δ z are the cell sizes in x-, y-, and z-direction, respectively. For a function having space and time dependence, it can be expressed as: g( x, y, z, t ) = g(i ∗ δ x, j ∗ δ y, k ∗ δ z, n ∗ δ t )

where μt , ε t , and σ t denote dielectric permeability, permittivity, and conductivity, respectively. The subscripts t emphasize that these constitutive parameters inside a food are dependent of temperature and thus heating time. To numerically solve Equations 6.67–6.72, we use the Yee’s method to discretize the computational region into unit cells and estimate the derivatives by second-order central finite-difference approximation. To achieve second-order accuracy, electrical field component and magnetic field component are placed half cell size away from each other, as shown in Figure 6.6.

= g n (i, j, k )

(6.74)

where g( x, y, z, t ) is a generic function and δ t is a time step. From these notations, the second-order central finite-difference forms for Equations 6.67–6.72 are derived to be suitable for computer programming (Taflove and Brodwin 1975a, 1975b; Lau and Sheppard 1986). H xn+1 / 2 (i, j + 1 / 2, k + 1 / 2) = H xn−1 / 2 (i, j + 1 / 2, k + 1 / 2) +

δt μ ∗ δ i , j +1 / 2,k +1 / 2

⎡ E (i, j + 1 / 2, k + 1) − E (i, j + 1 / 2, k ) ⎤ ⎥ ⎢ n n ⎣ + Ez (i, j, k + 1 / 2) − Ez (i, j + 1, k + 1 / 2) ⎦ n y

n y

(6.75)

Chapter 6

H yn+1 / 2 (i + 1 / 2, j, k + 1 / 2) = H yn−1 / 2 (i + 1 / 2, j, k + 1 / 2) +

δt μ * δ i +1 / 2, j ,k +1 / 2

⎡ E (i + 1, j, k + 1 / 2) − E (i, j, k + 1 / 2) ⎤ ⎢ n ⎥ n ⎣ + E x (i + 1 / 2, j, k ) − E x (i + 1 / 2, j, k + 1) ⎦ n z

n z

(6.76)

H zn+1 / 2 (i + 1 / 2, j + 1 / 2, k ) = H zn−1 / 2 (i + 1 / 2, j + 1 / 2, k ) +

δt μ ∗ δ i +1 / 2, j +1 / 2,k

⎡ E (i + 1 / 2, j + 1, k ) − E (i + 1 / 2, j, k ) ⎤ ⎥ ⎢ n n ⎣ + E y (i, j + 1 / 2, k ) − E y (i + 1, j + 1 / 2, k ) ⎦ n x

n x

(6.77)

⎛ σ i +1 / 2, j ,k δ t ⎞ n E xn+1 (i + 1 / 2, j, k ) = ⎜ 1 − E x (i + 1 / 2, j, k ) ε i +1 / 2, j ,k ⎟⎠ ⎝

+

δt ε i +1 / 2, j ,k ∗ δ i +1 / 2, j ,k

⎡ H zn+1 / 2 (i + 1 / 2, j + 1 / 2, k ) ⎤ ⎥ ⎢ n +1 / 2 ⎢ − H z (i + 1 / 2, j − 1 / 2, k ) ⎥ ⎢ + H yn+1 / 2 (i + 1 / 2, j, k − 1 / 2) ⎥ ⎥ ⎢ ⎢⎣ − H yn+1 / 2 (i + 1 / 2, j, k + 1 / 2) ⎥⎦ (6.78)

⎛ σ i , j +1 / 2,k δ t ⎞ n E yn+1 (i, j + 1 / 2, k ) = ⎜ 1 − E y (i, j + 1 / 2, k ) ε i , j +1 / 2,k ⎟⎠ ⎝

+

δt ε i , j +1 / 2,k * δ i , j +1 / 2,k

⎡ H xn+1 / 2 (i, j + 1 / 2, k + 1 / 2) ⎤ ⎢ ⎥ n +1 / 2 ⎢ − H x (i, j + 1 / 2, k − 1 / 2) ⎥ ⎢ + H zn+1 / 2 (i − 1 / 2, j + 1 / 2, k ) ⎥ ⎢ ⎥ ⎢⎣ − H zn+1 / 2 (i + 1 / 2, j + 1 / 2, k ) ⎥⎦ (6.79)

⎛ σ i , j ,k +1 / 2δ t ⎞ n Ezn+1 (i, j, k + 1 / 2) = ⎜ 1 − Ez (i, j, k + 1 / 2) ε i , j ,k +1 / 2 ⎟⎠ ⎝

+

δt ε i , j ,k +1 / 2 ∗ δ i , j ,k +1 / 2

⎡ H yn+1 / 2 (i + 1 / 2, j, k + 1 / 2) ⎤ ⎥ ⎢ n +1 / 2 ⎢ − H y (i − 1 / 2, j, k + 1 / 2) ⎥ ⎢ + H n+1 / 2 (i, j − 1 / 2, k + 1 / 2) ⎥ ⎥ ⎢ x n +1 / 2 ⎣⎢ − H x (i, j + 1 / 2, k + 1 / 2) ⎦⎥ (6.80)

From Equations 6.75–6.80, the six field components are calculated from the previous values of the specific location as well as the previous values of

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115

the neighboring field components. The computation approaches one grid at each time step and is terminated after a steady state is reached. We also assume that the permeability is isotropic and timeindependent, but the dielectric constant and electrical conductivity are dependent on both time and space. It is well known that a least 10 grids are needed in one wavelength to keep FDTD algorithm valid (Chen et al. 2007). In addition, for accuracy and stability, the time step δ t is limited by a value determined by the local media and the cell size in three directions.

δt ≤

1 (ε t )min −2 −2 −2 c (δ x )max + (δ y)max + (δ z )max

(6.81)

where c is the speed of light in free space. We also explicitly denote the subscript t of dielectric constant to emphasize its time dependence in food applications. To ensure the stability of the program everywhere and anytime in the problem space, Equation 6.81 has to be satisfied.

6.5.2. ABC When dealing with MW-heated food, the EM wave is treated as a plane wave to the object so that it is assumed that MWs approach the food package from infinity, part of its energy is absorbed by the food and part is reflected back to infinity. To allow a computer with limited memory to handle the computation, it is necessary for the model to be able to terminate the computation at particular space grids while, at the same time, keeping the scatter field effectively approaching into infinity. To meet this requirement, the lattice boundary should allow outgoing numerical wave propagation and suppress reflected waves from all directions. This lattice truncation is theoretically called ABC. The formulations of ABC were developed in Taflove (1998) by using Taylor series approximation. Using Taylor series expansion, one can obtain first-order, second-order, and higher order ABC. While the first-order ABC introduces large calculation errors, higher order ABCs increase mathematic

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

and programming overhead. Therefore, the secondorder ABC is used in our numerical model because of its relative higher accuracy compared with the first-order ABC and moderate computational demand compared with higher order ABCs. For the secondorder ABC, every node on the lattice boundary needs a 2D array to store its past and the current values of the interior nodes to facilitate calculations. A 3D lattice has six faces, each has two tangential fields so that total 72 2D arrays are needed to facilitate the calculations. For the sake of illustration, we assume there is an Ez field component tangential to the boundary x = 0. The second-order ABC is implemented in the following formulation: 2 − s − 1 / s n+1 [ Ez (2, j, k ) + Ezn−1 (0, j, k )] 2 + s + 1/ s 2(1 / s − s) ⎡ Ezn (0, j, k ) + Ezn (2, j, k ) ⎤ + 2 + s + 1 / s ⎢⎣ − Ezn+1(1, j, k ) − Ezn−1(1, j, k ) ⎥⎦ 4(1 / s + s) n Ez (1, j, k ) − Ezn−1 (2, j, k ) + (6.82) 2 + s + 1/ s

Ezn+1 (0, j, k ) =

where s = δ t μεδ 0, j ,k . Here we only present the updated equation for Ez on the lattice boundary; for the other components tangential to the boundary, the update equations are similar. The complete set of update equations for ABC is presented in (Taflove 1998).

6.5.3. Source Specification For illustration purposes, we only focus on MW heating of food packages. For those applications, the plane wave excitation instead of pulse excitation is found to be more suitable (Ma et al. 1995). Thus, we use a sinusoidal plane wave with a prescribed energy level as the excitation source. In numerical modeling to radiate the desired plane wave in the computation lattice, a simple approach is to place the excitation plane on the lattice boundary. However, this does not account for any possible outgoing fields scattered by the object, hence introducing undesired reflected wave that distorts the final results. Thus the excitation source is placed on a reference plane several cells away from the lattice

boundary. This placement introduces the desired plane wave propagating through the FDTD grids and, in the meantime, allows the reflected fields to propagate into infinity through the lattice boundary without any reflection. As an example, assuming a sine wave with the dominant TE10 mode in a rectangular waveguide far away from an object, it only has E y field component and propagates into zdirection. We then set up the excitation plane at k = 3: E yn (i, j, 3) ← E0 sin(2π fnδ t ) + E yn (i, j, 3)

(6.83)

where E0 is determined by a specified source power. In the numerical simulation, the sinusoidal wave was activated at time step n = 1 and left on until steady state is established, which usually needs four or five wave cycles.

6.5.4. EM Energy Related to MW Heating After numerically solving E and H equations, the dissipated power for each unit volume inside a food package can be calculated as a root mean square of the electric field: 2 q (r , T ) ≈ 2πε 0 fErms ε ′′(r , T )   ( E * ∗ E )dV ∫ = 2πε 0 f V ε ′′(r , T ) 2V

(6.84)

6.5.5. Finite Difference Equations for Heat Transfer In calculating temperature distributions, we consider two types of heat transfer during a MW heating process: (1) heat conduction that takes place inside food package, and (2) heat convection at the surface of food package between food and water that surrounds the food package. In this section, a thermal model is developed to numerically solve both types of heat transfer (Incropera and DeWitt 2001).

ρ(T )c p (T )

∂T = ∇ ⋅ ( K (T )∇T ) + q (r , T ) ∂t q = hA(Ts − T )

(6.85) (6.86)

Chapter 6

Equation 6.85 describes the relationship between partial derivatives of temperature with heating time. Thermal conductivity K (T ), mass density ρ(T ) and specific heat c p (T ) are functions of temperature T. Equation 6.86 calculates the temperature on the surface of food package from the heat exchange between food and water. Here h is the heat transfer coefficient of 220 Wm −2 k −1 (heat transfer value varies with different conditions). The value of 220 Wm −2 k −1 was obtained from experiments where the temperature history was adjusted to match the measured temperature at a cold spot of a food package merged into hot water and A is the surface area of food package. The heat transfer equation Equation 6.85 is solved numerically with the FDTD method by using the same 3D mesh generated for the EM field. Its finite difference form between FDTD cells is T n+1 (i, j, k ) = (1 − 12 F0 )* T n (i, j, k ) + F0 [T n (i − 1, j, k ) + T n (i + 1, j, k ) + T n (i, j − 1, k ) + T n (i, j + 1, k ) + 4T n (i, j, k − 1) q (i, j, k )Δt + 4T n (i, j, k + 1)] + (6.87) ρc p For illustration purposes, we use a cell setup of Δx = Δy = 2Δz for all finite difference equations. On the surface of the food package, besides the MW heating, there is heat convection introduced by surrounding hot water. For the nodes on the surface vertical to z-plane, to calculate the boundary condition, the thermal equation q = hΔxΔy[Ts − T (i, j, k , n)] was used to replace the conduction term [T (i, j, k + 1, n) − T (i, j, k , n)] ∗ K ∗ Δx .

6.5.6. Communication Algorithm between EM and Thermal Field The electric and magnetic fields are solved with an EM solver Quick Wave -3D (QW3D, Warsaw, Poland) that employs the conformal FDTD method. In QW3D, the physical geometry was defined in a parameterized macro as a user-defined object. In this object file, the MW heating system is represented as a metal box with a Perfect Electrical Conductor

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117

(PEC) surface. Since its operating frequency is 915 MHz, only TE10 can propagate through the rectangular waveguide on the top and bottom of the system to be simulated (Balanis 1989). Once the model is developed with the specified source type, input power, and boundary conditions, the mesh for the EM field can be generated automatically in QW3D by using conformal FDTD method. The mesh is then kept in an external file so that it can be imported into the heat transfer model to calculate temperature response caused by MW heating. The coupling between the EM and the thermal field is defined in a script that exchanges data between the EM and the thermal model and calculates the temperature response. Once the temperature is updated, the dielectric properties and the loss factor also change, which requires the temperature to be recalculated (Chen 2008). In QW3D, it is possible to extract the power density of a subdomain inside a food package, which significantly reduces the size of the data set handled by the EM and the thermal model. During MW heating, the temperature distribution is obtained from the EM power dissipated inside the food package, and then new dielectric properties are calculated with a linear interpolation technique in QW3D. So there are two steps happening in the communication algorithm between EM and thermal field. First, the dissipated power is calculated by solving the integrated Maxwell’s equations with conformal FDTD method; second, the dissipated power is used to calculate the temperature response, which results in the change of the thermal properties. Finally, once the thermal properties are updated, the EM field needs to be reevaluated as well. The steps of the algorithm are 1. Obtain dissipated power by solving Maxwell’s equations with conformal FDTD method (i.e., QW3D). 2. Initialize temperature for computation in the next time step based on the previous temperature distribution. 3. Loop through the whole computational grid. (a) For each inner node, calculate dissipated power by taking mean values of the power

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

density from the node itself and its adjacent nodes. (b) For boundary nodes, set the corresponding power from corresponding nodes to zero. (c) Conduct heat transfer calculation. 4. Update the temperature distribution and feedback to QW3D and repeat step 1.

6.5.7. Model Validation A numerical model is not very useful until it is validated. To check the validity of the FDTD technique proposed in previous sections, the electric field inside a cylindrical food model is calculated and compared with the analytical results. The cylindrical model is shown in Figure 6.7. The cylinder was assumed to have an infinite length in z-direction and circular cross section in the x-y-plane. For simplification purposes, it was assumed that the cell size was uniform in the whole computational lattice, that is, δ x = δ y = δ . The cylinder had a radius of 20δ . The incident wave was assumed to be a sinusoidal wave with an operating frequency of f0 . We investigated two cases, with frequencies of f0 = 0.915 GHz and

j * dy

(25

1 2

)

, 25 12 ,

The geometry function to describe this object was

(i − 25 ) + ( j − 25 ) 1 2 2

1 2 2

≤ 20

(6.88)

Thus, for each pair (i, j ) , if it satisfied (Eq. 6.88), we assigned dielectric parameters for the specified food package; otherwise, the dielectric properties for free space were used. Due to the 2D lattice, the plane wave source was specified as the following according to Equation 6.88: Ez (i, 3) ← Ez (i, 3) + 1000 ∗ sin(2π fnδ t )

50

(6.89)

y Cylindrical scatter x

25.5 Hy

Ez 0

f0 = 2.45 GHz , which are designated by the U.S. Federal Communications Commission (FCC) for MW applications. Because the cylinder was infinite in z-direction, there was no variation in this direction with respect to the geometry and the incident fields. Thus, this problem could be treated as a 2D scattering problem. In addition, we assumed that the incident wave was a +y-directed TM wave; therefore, it only had three field components involved in this simulation: Ez, Hx, Hy. Based on the above analysis, we built a 2D computational lattice consisting of 50 grids in each direction. Each of them contained a medium with dielectric parameters dependent on its location. The center of the cylinder was assumed to be the central point of the lattice

20d Computational domain

Hx 25.5

50

Another interesting feature of this problem was that the modeled structure was symmetric over the center line. The dielectric properties were also assumed to be symmetric over the same line. In addition, since the incident wave was in +z-direction, the field components that were normal to the propagation direction were symmetrical: E yn (26, j ) = E yn (25, j )

(6.90)

H xn (26, j ) = H xn (25, j )

(6.91)

i * dx

Figure 6.7. Cylindrical food model to verify the FDTD algorithm.

Therefore, we truncated the computational lattice on the center line and imposed the update equations for

Chapter 6

119

(a) 1300 exact calculated

1200 1100 Ez (15j)

1000 900 800 700 600 500 400 300

(b)

5

10

15

20

25 j

30

35

40

45

25 j

30

35

40

45

1400 exact calculated

1200 1000 Ez (25j)

field components to reduce computer memory storage and simulation time. As indicated earlier, we investigated two cases for validation purposes. One was for the food package exposed to MWs with a frequency of 0.915 GHz; the other one was the same food package exposed to MWs with a frequency of 2.45 GHz. The modeled food had the following dielectric parameters: ε ′f = 4ε 0, ε ′′f = 0. Therefore, for the case with a frequency of f0 = 0.915 GHz, the wavelength was λ f = c 4 f0 = c 2 f0 = 0.16 m . Using 80 discrete points per wavelength used in FDTD technique, the cell size was δ = λ f 80 = c 160 f0 = 0.002 m; the time step was chosen according to δ t = δ 2c = 3 ps. Thus, it required 320 time steps to complete the simulation for one wave cycle. Since it usually needs four or five wave cycles to reach the steady state, 1,600 time steps were used in the simulation. The maximum absolute value of the electric field E y (25, j ) and E y (10, j ) in the last half wave cycle were then recorded and compared with the analytical results, computed using series expansion method (Jones 1964). The results are presented in Figure 6.8. Both results show a good agreement. The numerical model correctly predicted the location of peak and null of the envelope of the electric field with a maximum error of δ . It was also shown that the amplitude of the induced electric field inside the cylinder was approximated with only 5% error presented. In the second case, the numerical results for the same cylindrical food exposed to MWs at a frequency of f0 = 2.45 GHz were computed using the same parameters as in the literature (Taflove and Brodwin 1975a, 1975b) for validation and comparison purposes. At this frequency, with the same deduction as the last example, we had λ f = c 2 f0 = 0.06 m. Twenty discrete points per wavelength were used. Thus, the cell size used for this case was δ = λ f 20 = 0.003 m; the time step is then δ t = δ 2c = 5 ps. These selections required 80 time steps for simulating one complete wave cycle. As before, five wave cycles are used to achieve the steady state. Results are reported in Figure 6.9.

Computer Simulation for Microwave Heating

800 600 400 200

5

10

15

20

Figure 6.8. Comparison between numerical and analytical results for 915 MHz microwave-irradiated cylindrical food package.

Again, similar results were obtained from simulated and analytical calculations. The maximum error of the amplitude of the electrical field is within 10% of the incident amplitude, while the expected location of field peaks deviated from the analytical solution less than δ . Another observation was that there were only one nulls of the envelope of Ez in Figure 6.8 but four nulls in Figure 6.9. This is because the radius of the cylinder was equal to the wavelength in the first scenario (Figure 6.8) while

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Temperature profile for hot water heating only

(a) 1400

80

exact calculated

70 Temperature (°C)

1200

Ez (15j)

1000 800 600 400

50 40 30

FDTD FEM Experimental result

20

200 0

60

10 5

10

15

20

25 j

30

35

40

45

0

0

500

1000

1500

2000

2500

Heating time (second) Figure 6.10. Validation of the heat transfer model.

(b) 2200 exact calculated

2000 1800

Ez (25j)

1600 1400 1200 1000 800 600 400 200

5

10

15

20

25 j

30

35

40

45

Figure 6.9. Comparison between numerical and analytical results for microwave-irradiated cylindrical food package at a frequency of 2450 MHz.

equal to a quarter of the wavelength in the second scenario (Figure 6.9). Therefore, a single mode EM field is achieved at 915 MHz while multimode is inevitable at 2450 MHz. For food applications, the multimode system brings up a critical problem: The cold spot in packaged foods during MW heating processes cannot be predicted with confidence. Since the temperature at the cold spot needs to reach over 121°C to ensure the safety of the food (Guan et al. 2004), it is desirable for a MW heating system

to keep the cold spot fixed. A unique design to provide 915 MHz single-mode MW field pattern in foods is described in Chen et al. (2007). Parallel to the development and validation of a numerical model for the EM field, the numerical model for the heat transfer also needs to be validated independently before it can be coupled with the EM model to simulate an MW heating process. Since the FDTD method is employed to solve the heat transfer equations, an FEM model and experimental measurements are used to validate the FDTD results for the heat transfer (Figure 6.10). The simulation result for the heat transfer calculated by the FDTD method is shown as solid red line; the dashed blue line is the simulation result calculated by using the FEM model. Directly measured temperature response from an experiment is shown in the yellow solid dashed line. The difference between the calculated temperature and the actual measurement was less than 1°C, which shows a good agreement. It is noted that the simulated heating curve varies with different heat transfer coefficient. Therefore, to achieve the best agreement between simulation and experiment results for hot water heating, the heat transfer coefficient was set to 220 Wm−2 k−1.

Chapter 6

6.6. Simulation Model for MW Sterilization 6.6.1. Introduction MW energy heats solid food products in a short time period, thus offers the potential to overcome the limitation of slow conventional heating that relies on heat conduction within food packages (DeCareau 1985; Burfoot et al. 1988; Harlfinger 1992; Meredith 1998) and significantly improves product quality (Stenstrom 1974; O’Meara et al. 1977; Ohlsson 1987; Ohlsson 1991; Giese 1992; Guan 2003). For MW sterilization processes, heating uniformity is very important. On one hand, the product temperature at the cold spots should reach a desired level (∼120°C) to kill Clostridium botulinum spores and cause a reduction of heat resistant spoilage microorganisms. On the other hand, the temperature at the hottest spot should not be too high to cause bursting of package or melting of plastic package materials. It is difficult to design MW systems that provide relative stable and uniform heating because of standing waves. The discontinuity in the EM field components at the boundary between food packages and surrounding medium also cause nonuniform distribution of the MW field within the food packages (Ayoub et al. 1974; DeCareau 1985; Ruello 1987; Schiffmann 1990; Stanford 1990). Severe corner and edge heating may result from the nonuniform heat distribution (Ohlsson 1991). Many techniques have been developed to improve heating uniformity. Instead of air, water is used as the surrounding medium of food packages to minimize the nonuniform heating effects (Stenstrom 1972; Ohlsson 1992; Lau 2000; Guan et al. 2002; Guan 2003). As mentioned before, for MW heating purposes, two frequencies are designated by the U.S. FCC: namely, 915 MHz and 2450 MHz. A frequency of 2450 MHz is used in domestic MW ovens and some of the industrial MW systems. However, the high operating frequency produces multimode EM fields inside the cavities, making heating patterns difficult to predict and control, in particular when food packages are processed on moving belts. Recent studies (Guan 2003; Chen et al. 2007) have indicated that it is possible to use single-mode 915 MHz MW

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121

heating systems to provide relatively uniform heating and more stable heating patterns within food products both in pouches and trays. The following section discusses the use of computer simulation models to study electric field distributions and temperature distributions in prepackaged mashed potato processed in a 5-kW 915 MHz MW sterilization system developed at Washington State University (WSU; Pullman, WA). The models have been validated using the chemical marker M-2 method and direct temperature measurements using fiber-optic temperature sensors (Chen et al. 2007, 2008).

6.6.2. Physical System The considered MW sterilization system consists of a rectangular cavity with one horn-shaped applicator on the top and another identical applicator on the bottom. Figure 6.11 shows a front view of the system with an exposed interior cut and a vertical central

Microwave in Rectangular waveguide Water

Horn waveguide

Microwave transparent pressure window

Air

Air Food

Horn waveguide

z Rectangular waveguide x Microwave in Figure 6.11. Pilot-scale EM heating system (front view).

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

water

food

Table 6.1. Dielectric properties of mashed potato at different salt levels (Guan et al. 2004). Salt content 0% 0.5% 1.0%

Dielectric constant (ε′)

Loss factor (ε″)

54.5 52.3 48.7

38.1 68.5 95.2

y cavity x Figure 6.12. Pilot-scale microwave heating system (top view).

plane, while Figure 6.12 shows the top view for the central section of the cavity. MW energy is provided by a 5-kW generator operating at a frequency of 915 MHz through a standard waveguide that supported only TE10 mode. After passing through a circulator, the MW energy is equally divided at a T-junction and fed to the two-horn applicators through two standard waveguides. The length of each leg of the waveguide after the T-junction could be adjusted to control the phase difference between the MWs at the entry port of the two-horn applicators, which meant the phase shift between the two waves interacting inside the cavity could be controlled to achieve the desired field distribution in foods. The MW horn applicator and the processing cavity are separated by a MW transparent window to allow circulation of pressurized water preheated by a heat exchanger to 121°C. Prepackaged mashed potato or other foods are heated by the combination of MW energy and surface water. More details about the system can be found in Chen et al. (2007).

6.6.3. Numerical Modeling for MW Sterilization Process A detailed description of a computer model for simulating MW sterilization process containing stationary and moving food packages is provided in Chen et al. (2008). This model was developed and

validated following the steps described in Section 6.5. The simulated temperature patterns agreed well with the heating patterns indicated by brown color through formation of Chemical Marker-2 (4-hydroxy-5-methy-3(2H)-furanone) in whey protein gel. Detailed information about kinetics for formation of M-2 can be found in Lau and Tang et al. (2003). The following part of the section illustrates its use to study the influence of dielectric properties of foods on heating uniformity for two different cavity configurations (cases 5 and 6), each providing a specific MW-induced heating pattern in foods. Under a confidentiality agreement with industrial partners of the WSU MW Sterilization Consortium, we cannot reveal detailed information for those two configurations. But we are confident that the simulation results demonstrate the benefits of computer simulations. In this simulation, we consider three different levels of salt content in mashed potato shown in Table 6.1. Figure 6.13 shows the meshed geometry of the modeled system (Figure 6.12). Figure 6.14a–c shows the predicted electric field distribution in different cross-sectional areas in mashed potato. It is observed that the overall general electric field intensity patterns in the central horizontal plane (top view) of the food do not vary with salt contents. But salt content significantly changes the MW penetration (front and side views). At 0% salt content, the food predominantly undergoes center heating, while at 1.0% salt content, the food mainly shows surface heating. These observations have been confirmed by our experimental results using chemical marker M-2.

Chapter 6

Computer Simulation for Microwave Heating

123

Monitored tray

(a) Case 5

(b) Case 6

Figure 6.13. Mesh for modeled geometry.

Figure 6.15 shows simulated temperatures at the cold spot and its adjacent location (8 mm apart) in the mashed potato as the package passes through two interconnected MW cavities. The results show the sensitivity of the location within a food package to temperature rise close to the point of cold spot, and indicate what would happen if a temperature sensor is placed slightly off the predetermined cold spot. Figure 6.16 shows the temperature distributions in mashed potato at a cross-sectional area at simulation time t = 100 s, t = 150 s, t = 270 s, and t = 320 s when immersed in water at 125°C while being heated by 915 MHz MW energy in a single-mode cavity. With increasing heating time, the center product temperature was increased by MW heating. We also used the simulation model to study heating uniformity in a larger MW system developed at WSU. This system has four connected MW cavities while food packages are moved on a conveyor belt through those cavities to complete a sterilization process under circulation of water at a temperature of 125°C. For the simulation, we used power settings of 16.5 KW, 7 KW, and 5 KW, respectively, for each of the four cavities. The food packages moved through the central plane of the heating cavities.

Results were obtained that show the temperature distributions over the y-z-plane where the cold spot was located. The temperature distributions were selected to represent the intermediate status during the process, which were the time steps when the food packages pass the center of the respective MW cavity. It is also observed that the temperature difference decreased along the moving direction (x-direction; Figure 6.17). As in the previous case, steeper change was found along z-direction than that found in ydirection. It should be noted that the final temperature distribution at t = 340 s was much smoother than those at previous time steps since the temperature value at any point varied only within a small range (<5°C). In this section, the temperature distributions over the vertical surface of the y-z-plane where the cold spot was found were evaluated using computer simulation. This information was used to facilitate the validation of the optical sensor measurement of the temperatures used to develop appropriate thermal process parameters (e.g., belt moving speed, holding time after MW heating). Although not presented here, smooth patterns after MW heating were found in the x-direction.

× 10–3 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5

18 16 14 12 10 8 6 4 2 2

4 2

2

4

6

8

10

12

14

16

18

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 × 10–3 Side view (y-z) 4 2

2

4

6

8

10

12

14

16

18

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 × 10–3

4 6 8 10 12 14 16 18 Top view of central plane (x-y)

Front view (x-z)

(a) Electric field distribution (density) inside mashed potato with 0% salt content. × 10–3

18

4 2

3.5

16

12

14

2.5

3

3.5 × 10–3

12

14

16

0.5 1 1.5 2 2.5 4 6 8 10 12 14 16 18 Top view of central plane (x-y) Front view (x-z) (b) Electric field distribution inside mashed potato with 0.5% salt content.

3

14

3

12

2.5

10

2

8

2 0.5

4 2

1

4

6 1

8 1.5

10 2

2

4

6

8

10

0.5

2

2

× 10–3

18

4 2

3.5

16 14

3

12

2.5

10

2

8 4

1

2

0.5 2

4

6

8

10

12

14

2

4

0.5

6

1

8

1.5

10 2

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2.5

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Side view (y-z) 4 2

1.5

6

16

18

Side view (y-z)

1.5

6

4

2 0.5

16

Top view of central plane (x-y)

4 1

6 1.5

8

10 2

12

14

2.5

3

18

18

3.5 × 10–3

16

18

3.5 × 10–3

16

18

3.5 × 10–3

Front view (x-z)

(c) Electric field distribution (density) inside mashed potato with 1.0% salt content. Figure 6.14. Electric field distribution (top, front, and side views) inside a food package containing mashed potato with different salt contents. (a) Electric field distribution (density) inside mashed potato with 0% salt content. (b) Electric field distribution inside mashed potato with 0.5% salt content. (c) Electric field distribution (density) inside mashed potato with 1.0% salt content. See color insert.

124

Temperature profile from simulation at selected point 110 Cold spot 105 One point to the 100 left of cold spot 95 90 85 80 75 70 65 60

0

1

2

3

4

5

6

Heating time (minute) Figure 6.15. Temperature profile at neighboring points from the cold spot.

115

120

110

110

105

100

100 95

90

90 85

80 70 20 15 10 y

80 5

2

0 1

3

z

5

4

6

75

125 120 115 110 105 100 95 90 85 80 20

120 115 110 105 100 95 90 15 10 y

5

120 115 110 105

10

y

5

0 0

t = 270 s

2

z

4

6

122 120 118 116 114 112 110 108 106 104

0 0

z

4

85

124 122

125 Temperature (°C)

Temperature (°C)

125

15

2

6

t = 150 s

t = 100 s

100 20

125

In this chapter, we provided a general overview of methods to calculate and validate EM field and coupled EM and thermal equations. We then used case studies to illustrate computer simulation models developed using the FDTD method for studying MW sterilization processes. It was shown that MW sterilization technology as an emerging novel technology has significant benefits over conventional methods. It significantly reduces the processing time to reach the required temperature level and thus it is possible to largely retain the quality of the heated food package. The 915 MHz single-mode sterilization technology developed at WSU uses a combination of hot water and MW heating that also improves heating uniformity.

120

130 Temperature (°C)

Computer Simulation for Microwave Heating

6.7. Conclusion

Temperature (°C)

Temperature (°C)

Chapter 6

120

120

118

115

116 114

110 105 20

112 110 15

10 y

5

0 1

2

3

4

5

6

108

z

t = 320 s

Figure 6.16. Temperature distribution at a vertical plane in mashed potato for different heating times. Mashed potato was immersed in 125°C water while heated with 915 MHz energy. See color insert.

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

120

122

125 120 115 110 105 100 95 90 85 80 20 18 16

110 105

14 12 10 5 8 6 4 y 2 0 6

3 4 z

2

1

0

100 95

120

125 Temperature (°C)

Temperature (°C)

115

118

120

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115

114

110

1 2 3 105 4 z 20 18 16 14 12 5 10 8 6 4 2 0 6 y

t = 60 s

124

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125

120

120

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115 5

6

116

Temperature (°C)

Temperature (°C)

112 110 108

t = 150 s 126

4 110 3 0 2 4 6 2 z 8 10 12 14 1618 20 0 1 y

0

130 129 128 127 126 125 124 0 2 4 6 8 10 12 14 16 18 20 0 y

129.5 129 128.5 128 127.5 127 126.5 126 6 125.5 5 4 3 125 2 z 1

t = 230 s t = 340 s Figure 6.17. Temperature distribution in a vertical plane at different simulation time corresponding to the central position of each of the four MW heating cavities.

In developing the MW system configuration, we used computer simulation models for EM field calculations, while in evaluating temperature distribution for process development we used a coupled EM field and heat transfer model. Both models were developed using the FDTD methods and validated step-by-step with analytical and experimental test results. The single-mode EM wave patterns in the system make it possible to track the heating patterns in the packaged foods during the thermal process. Without the use of computer simulation models, it is extremely expensive and time-consuming to design such complicated systems by doing trial-anderror experiments, especially for high-power industrial applications. This is due to numerous variables influencing the heating patterns in MW systems,

leading to a large number of combinations of variables that need to be considered in the design. In fact, the high labor and equipment costs keep food manufacturing companies from making adequate investment in research and development effects in this direction.

Notation Latin Symbols Boundary surface area A A1, A2, Constants B1, B2, C1, C2 aˆ Unit vector in coordinate directions

m2 –

–

Chapter 6

a, b, c B

Constants Magnetic flux (density)

C CP

Contour path Specific isobaric heat capacity Specific volumetric heat capacity Speed of light in free space (vacuum) Electric flux density Electric field Element number in FEM Frequency Heat transfer coefficient Magnetic field intensity Spatial index Space points Equivalent current density Source current density Conduction current density Imaginary unit Thermal conductivity Wave number Numerical wave number Number of elements in FEM Normal unit vector/ component of surface Time step Dissipated microwave power Dissipated power density provided by heating source Thermal energy provided by circulating water on the boundary Electric charge density

CV c D E e f h H i I, j, k Jeq Ji JC j K k k1 N nˆ n Pd q q

q r, r′ S T TS t u V

Space points Surface for computation domain Temperature Circulating water bath temperature Time Speed of light Voltage/potential

– T = V·s·A−1· m−1

Computer Simulation for Microwave Heating

Element potential in FEM Volume Phase velocity Energy

Ve V vP W

127

V m3 m·s−1 J

J·kg−1·K−1 J·kg−1·K−1 m·s−1 −2

A·s·m V·m−1 – Hz, s−1 W·m2·K−1 A·m−1 – – A·m−2 A·m−2 A·m−2 −1 W·m−1·K−1 m−1 m−1 – – – W·m−3 W·m−3 J C·m−3 = A·s·m−3 – –

Greek Symbols 8.854·10−12 A·s·V−1·m−1 Relative dielectric constant – Complex dielectric permittivity – Real part of complex permittivity – Imaginary part of complex – permittivity, dielectric loss factor Wavelength m Wavelength in free space/ m vacuum Permeability of vacuum =1/ε 0c20 = 1.257·10−6 V·s·A−1·m−1 Relative magnetic permeability – Dielectric permeability – Ludolph’s number, circular ≈ 3.14159 constant Mass density kg·m−3 Conductivity S·m−1 Propagation constant – Field solution – Angular frequency Hz = s−1

ε0 Dielectric constant of vacuum εr ε ε′ ε″ λ λ0 μ0 μr μ π ρ σ γ Ψ ω

Coordinates x, y, z ρ, ϕ, z r, φ, ψ

Cartesian coordinates Cylindrical coordinates Spherical coordinates

Operators

°C °C

∂ ∇ ∑ Δ, δ

s m·s−1 V

∇⋅ ∇× f, g, h

Partial differential Nabla operator Sum Difference (but also denoting cell sizes in mesh discretization) Divergence Curl Functions

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Functions G0(r,r′) Hm(1)(βρρ) Hm(2)(βρρ) hn(1)(βr) hn(2)(βr) Jm(βρρ) Ym(βρρ) jn(βr) yn(βr) 2 O ⎢⎣( Δx ) ⎥⎦

Green’s function First Hankel function Second Hankel function First spherical Hankel function Second spherical Hankel function First Bessel function Second Bessel function First spherical Bessel function Second spherical Bessel function Notation for higher order terms in Taylor expansion

Abbreviations 1D 2D 3D ABC EM EMI FCC FDM FDTD FEM MoM MW M-2 PEC PV RTD rms SAR TIE

One-dimensional Two-dimensional Three-dimensional Absorbing boundary condition Electromagnetic Electromagnetic interference Federal Communications Commission Finite difference method Finite difference time domain Finite element method Method of moments Microwave Chemical compound for heating pattern determination Perfect electrical conductor Principal value Resistance temperature detector Root mean square Specific absorption ratio Tensor integral equation

References Ayoub J, Berkowitz D, et al. (1974).Continuous microwave sterilization of meat in flexible pouches. J Food Sci 39: 303–313. Balanis CA. 1989. Advanced Engineering Electromagnetics. John Wiley & Sons. Burfoot D, Griffin W, et al. (1988).Microwave pasteurization of prepared meals. J Food Eng 8:145–156. Chen H. 2008. Computer simulation of combined microwave and water heating processes for packaged foods using finite

difference time domain method. PhD, Department of Biological Systems Engineering, Washington State University, Pullman, WA, USA, 97–102. Chen H, Tang J, et al. (2007). Coupled simulation of an electromagnetic heating process using the Finite Difference Time Domain method. J Microwave Power Electromagn Energy 41(3):50–68. Chen H, Tang J, et al. (2008). Simulation model for moving food packages in microwave heating processes using conformal FDTD method. J Food Eng 88:294–305. Cook HF. 1952. A physical investigation of heat production in human tissues when exposed to microwaves. Br J Appl Phys 3:1–6. DeCareau R. 1985. Microwaves in the Food Processing Industry. New York: Academic Press, Inc. Desmarais R, Breuereis J. 2001. How to select and use the right temperature sensor. Sensor 18(1). Giese J. 1992. Advances in microwave food processing. Food Technol 46:118–123. Guan D. 2003. Thermal processing of hermetically packaged low-acid foods using microwave-circulated water combination (MCWC) heating technology. Washington State University, 2003.: xix, 167 leaves, bound. Guan D, Plotka V, et al. (2002).Sensory evaluation of microwave treated macaroni and cheese. J Food Process Preserv 26:307–322. Guan D, Cheng M, et al. (2004).Dielectric properties of mashed potatoes relevant to microwave and radio-frequency pasteurization and sterilization processes. J Food Sci 69(1):E30– E37. Guru BS, Chen KM. 1976. A computer program for calculating the induced EM field inside an irradiated body. Department of Electrical Engineering and System Science Michigan State University, East Lansing, MI 48824. Guy AW, Lin JC, et al. (1974). Measurement of absorbed power patterns in the head and eyes of rabbits exposed to typical microwave sources. Proc. 1974 Conf. Precision Electromagnetic Measurements (London, England), 255–257. Harlfinger L. 1992. Microwave sterilization. Food Technol 46(12):57–60. Harrington RF. 1961. Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill. Harrington RF. 1967. Matrix methods for field problems. Proc IEEE 55(2):136–149. Harrington RF. 1968. Field Computation By Moment Methods. New York: Macmillan. Harrington RF. 1992. Origin and Development of the Method Moments for Field Computation. New York: IEEE Press, Computational electromagnetics. Hilderbrand FB. 1962. Advanced Calculus for Applications. Englewood Cliffs, NJ: Prentice-Hall. Incropera FP, DeWitt DP. 2001. Introduction to Heat Transfer, 4th ed. New York: John Wiley & Sons. Jones DS. 1964. The Theory of Electromagnetism. New York: Macmillan.450–452.

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Jurgens TG, Taflove A, et al. (1992).Finite-difference timedomain modeling of curved surfaces. IEEE Trans Antennas Propag 40(4):357–366. Kastner R, Mittra R. 1983. A new stacked two-dimensional spectral iterative technique (SIT) for analyzing microwave power deposition in biological media. IEEE Trans Microwave Theory Tech MTT-31(1):898–904. Kim HJ, Taub IA. 1993. Intrinsic chemical markers for Aseptic processing of particulate foods. Food Technol 47(1):91–97. Kong JA. 1981. Research Topics in Electromagnetic Theory. New York: John Wiley.290–355. Kunz KS, Lee KM. 1978. A Three-Dimensional Finite-Difference solution of the external response of an aircraft to a complex transient EM environment: Part I—The method and its implementation. IEEE Trans Electromagn Compat EMC-20(2). Lau MH. 2000. Microwave heating uniformity in model food systems. From Microwave pasteurization and sterilization of food products. PhD dissertation, Washington State University, USA. Lau RWM, Sheppard RJ. 1986. The modelling of biological systems in three dimensions using the time domain finitedifference method: Part I. The implementation of the model. Phys Med Biol 31(11):1247–1256. Lau MH, Tang J, et al. (2003).Kinetics of chemical marker formation in whey protein gels for studying microwave sterilization. J Food Eng 60(4):397–405. Livesay DE, Chen KM. 1974. Electromagnetic fields induced inside arbitrary shaped biological bodies. IEEE Trans Microwave Theory Tech MTT-22(12):1273–1280. Ma LH, Paul DL, et al. (1995).Experimental Validation of a Combined Electromagnetic and Thermal Fdtd Model of a Microwave-Heating Process. IEEE Trans Microwave Theory Tech 43(11):2565–2572. McDonald BH, Wexler A. 1972. Finite-element solution of unbounded field problems. IEEE Trans Microwave Theory Tech (1972 Symp, Issue) MTT-20:841–847. Meredith R. 1998. Engineer ’s Handbook of Industrial Microwave Heating. United Kingdom: The Institution of Electrical Engineers. Michaelson SM. 1969. Biological effects of microwave exposure. Proc. Biological Effects and Health Implications of Microwave Radiation Symp, Med. College Virginia (Richmond), Rep. BRH/DBE 70(2):35–58. Mittra R, ed. 1973. Computer Techniques for Electromagnetics. New York: Pergamon. Ohlsson T. 1987. Sterilization of foods by microwaves. International Seminar on New Trends in Aseptic Processing and Packaging of Food stuffs, 22 October 1987, Munich. SLK Report nr 564. Goteborg, Sweden: The Swedish Institute for Food and Biotechnology. Ohlsson T. 1991. Microwave heating uniformity. AICHE Conference on Food Engineering, Chicago. March 11–12. Ohlsson T. 1992. Development and evaluation of microwave sterilization process for plastic pouches. Paper presented at the

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AICHE Conference on Food Engineering, March 11–12, Chicago. Okoniewski M. 1993. Vector wave equation 2D-FDTD method for guided wave equation. IEEE Microwave Guided Wave Lett 3(9):307–309. O’Meara J, Farkas D, et al. (1977). Flexible pouch sterilization using combined microwave-hot water hold simulator. Contact No. (PN) DRXNM 77–120, U.S. Army Natick Research & Development Laboratories, Natick, MA 01760. Pandit RB, Tang J, et al. (2006).Kinetics of chemical marker M-2 formation in mashed potato—A tool to locate cold spots under microwave sterilization. J Food Eng 76(3): 353–361. Richmond JH. 1965. Digital computer solutions of the rigorous equations for scattering problems. Proc IEEE 53:796–804. Ruello JH. 1987. Seafood and microwaves: Some preliminary observations. Food Technol Aust 39:527–530. Sadiku MNO. 2001. Elements of Electromagnetics. New York: Oxford University Press. Schiffmann R. 1990. Problems in standardizing microwave over performance. Microwave World 11(3):20–24. Schwan HP. 1972. Microwave radiation: Biophysical considerations and standards criteria. IEEE Trans Biomed Eng BME-19:304–312. Spiegel RJ. 1984. A review of numerical models for predicting the energy deposition and resultant thermal response of humans exposed to electromagnetic fields. IEEE Trans Microwave Theory Tech MTT-32:730–746. Stanford M. 1990. Microwave oven characterization and implications for food safety in product development. Microwave World 11(3):7–9. Stenstrom L. 1972. Taming microwaves for solid food sterilization. IMPI Symposium, Ottawa, Ganada. May 24–26. Stenstrom L. 1974. US Patents 3,809,845; 3,814,889. Taflove A. 1980. Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems. IEEE Trans Electromagn Compat EMC-22(3): 191–202. Taflove A. 1998. Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method. Boston: Artech House. Taflove A, Brodwin ME. 1975a. Computation of the electromagnetic fields and induced temperatures within a model of the microwave-irradiated human eye. IEEE Trans Microwave Theory Tech MTT-23(11):888–896. Taflove A, Brodwin ME. 1975b. Numerical solution of steadystate electromagnetic scattering problems using the timedependent Maxwell’s equations. IEEE Trans Microwave Theory Tech MTT-23(8):623–630. Taflove A, Umashankar K. 1980. A hybrid moment method/finitedifference time-domain approach to electromagnetic coupling and aperture penetration into complex geometries. IEEE Trans Antennas Propag AP-30(4):617–627. Taflove A, Umashankar K. 1981. Solution of complex electromagnetic penetration and scattering problems in unbounded

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Wu TK, Tai LL. 1974. Numerical analysis of electromagnetic fields in biological tissues. Proc IEEE(Lett) 62:1167–1168. Wylie CR. 1960. Advanced Engineering Mathematics. New York: McGraw-Hill. Yee KS. 1966. Numerical solution of initial boundary-value problems involving Maxwell’s equations in isotropic media. IEEE Trans Antennas Propag AP-14:302–307. Zhang H, Datta AK. 2000. Coupled electromagnetic and termal modeling of microwave oven heating of foods. J Microwave Power Electromagn Energy 35(2):71–85.

Chapter 7 Simulating and Measuring Transient Three-Dimensional Temperature Distributions in Microwave Processing Kai Knoerzer, Marc Regier, and Helmar Schubert

7.1. Introduction Microwave ovens are commonplace in households and are established there as devices of everyday use. However, the uptake of microwave processes in the food industry is still limited, although they have several advantages over conventional thermal processing of foods. The volumetric heating avoids resistance to heat transport, and thereby enables quality improvement of heated and dried food products. The dissipation of energy within the product also avoids thermal inertia and leads to increased rate of heat transfer. Microwave heating, however, is very dependent on the product itself, its geometry and thermophysical properties. Microwave heating has been proposed as an alternative to traditional heating methods in many food manufacturing processes for several decades. Examples are (re)heating, baking, (pre)cooking, tempering of frozen food, blanching, pasteurization, sterilization, and dehydration (Metaxas and Meredith 1983; Decareau 1985; Buffler 1993; Metaxas 1996; Schubert and Regier 2005; Tang et al. 2008). Although the final objective of each process differs, an increase in product temperature is seen as a common theme.

If the final objective is product preservation, as in pasteurization, sterilization, and drying, it is very important to achieve a treatment with predefined minimum temperatures for a certain time (a minimum time–temperature treatment) or predefined minimum water content for assurance of microbial safety. In other processes, temperatures have to remain below certain values to achieve a desired product quality with a maximum preservation of color, flavor, and vitamins, among other quality attributes. In traditional heating and drying processes, heat transfer is predominantly limited by conducting the heat from the product surface to its center. The locations of the slowest heating or drying region are therefore found in the thermal center of the treated product. This means that overprocessing can occur near surface regions. As microwaves heat volumetrically, they can theoretically improve heating uniformity, leading to better final product quality. Unfortunately, there are a number of well-known microwave specific factors that cause nonuniform heating patterns during microwave processing. Examples of these include microwave focusing effects, corner and edge (over)heating, inhomogeneous electromagnetic (EM) field distributions and variation in dielectric properties in heterogeneous

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

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food materials. As a consequence of these factors, heating patterns can differ significantly from one product to another. The quality and safety of industrially microwave processed products are compromised when these factors are ignored (Mudgett 1986). It is largely for the reason that microwaveinduced heating patterns are more difficult to predict that such processes have not been widely adopted in the food industry. Product and process development of microwave processing started off as trial-and-error experimentation. To allow for appropriate process and equipment design, optimization and scale-up, while ensuring product safety and economical processing, the knowledge of temperature distributions throughout the microwave-heated products is therefore essential. Measuring transient temperature distributions in three dimensions (i.e., volumetrically), however, is very challenging and—as shown later—only the technology based on magnetic resonance imaging (MRI) can provide sufficient information noninvasively (see Section 7.3). Nowadays, in order to design thermal process operations, the use of models able to predict the temperature distribution during treatment inside a processing unit is desirable. This greatly assists not only in process design and scale-up but also in process approval from governmental regulators by illustrating that the technology and the process is well understood. Calculation of realistic EM field and temperature distributions within microwave equipment and treated product is complicated because of the interacting EM, thermo-, and fluid-dynamic phenomena and thus the necessity of coupling the governing partial differential equations, which ideally are solved in a parallel manner but can be solved sequentially to improve computational efficiency. Due to the lack of powerful computers and appropriate software during the initial introduction of microwave technology to food industry, it was nearly impossible to predict electric field distributions, let alone to accurately model the coupled problem, especially when products were present. Latest advances in computer and information technology assist in overcoming this limitation.

Numerical software packages for solving the separated problems have been commercially available for many years, but software packages coupling the partial differential equation problem, which best describes the real process, have also become available.

7.2. Microwave Thermal Modeling Electromagnetism is described by the partial differential equations introduced by J.C. Maxwell (1865), known as Maxwell’s equations, and the equations describing the interaction of microwaves with different materials, the constitutive relations. Maxwell’s equations:  (7.1) ∇⋅D = ρ  (7.2) ∇⋅ B = 0   ∂B (7.3) ∇×E = − ∂t    ∂D (7.4) ∇×H = j + ∂t Maxwell’s equations describe the coupled theory of the former separately described electric and magnetic phenomena. Equations 7.1 and 7.2 describe the source (ρ) of an electric field E without a magnetic  monopole as a source for the magnetic field H. Equations 7.3 and 7.4 describe the coupling between the electric and magnetic fields. (For a complete list of symbols, the reader is referred to the “Notation” section of this chapter.) The interaction of EM waves (e.g., microwaves) with matter is expressed by the material equations or constitutive relations (Eqs. 7.5–7.7), where the dielectric constant ε (the interaction of nonconducting matter with an electric field), the conductivity σ, and the permeability μ (the interaction with a magnetic field) dictate their behavior. Constitutive relations:   (7.5) D = ε0 εr ⋅ E   (7.6) B = μ 0 μr ⋅ H   (7.7) j = σ ⋅E

Chapter 7

Simulating and Measuring Transient Three-Dimensional Temperature Distributions

Where ε0 is the dielectric constant and μ0 the magnetic permeability in free space (vacuum) and εr and μr are their relative values. Generally, all these parameters can be complex tensors; in the case of food materials, some simplifications are possible: food behaves nonmagnetically, therefore the relative permeability can be set to μ = 1 and the permittivity tensor can be reduced to a complex constant with real (ε ′) and imaginary part (ε ″ ), which include the electrical conductivity σ, as shown in Equation 7.8.

σ 2 ⋅ π ⋅ f ⋅ ε0

(7.8)

P = 2 ⋅ π ⋅ f ⋅ ε 0 ⋅ ε r′′⋅ E 2 V

(7.9)

ε r′′ = PV =

Equation 7.9 describes the volumetrically dissipated power PV as function of microwave frequency f, dielectric properties (ε0, ε ″), and the electric field E. Depending on the problem to be modeled, the coupled phenomena can include equations for fluid flow, namely the conservation equations for mass, momentum, and energy (for these equations, refer to other chapters in this book; e.g., Chapter 5). Neglecting mass transfer, a general equation for heat transfer can be described by

ρCP

∂T − ∇ ⋅ (k ⋅∇T ) = −∇qR + Qem ∂t

(7.10)

Here ρ is the density, CP the specific heat capacity, k the thermal conductivity of the materials used in the scenario, qR the radiation power flux density, and Qem the heat source generated by the dissipated microwaves (Qem = PV; Eq. 7.9). The left side of this equation is well known from the traditional equation describing the conduction ∂T of heat ( ρCP = ∇ ⋅ (k ⋅∇T )), and the terms on ∂t the right side were added for heat transfer by radiation and for the heat source by the dielectric losses, respectively (Metaxas 1996). Assuming the product consists of a moist solid material, the radiation term has to be taken into account only at surfaces to gaseous materials, and Equation 7.10 can be simplified to

ρCP

∂T − ∇ ⋅ (k ⋅∇T ) = PV ∂t

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(7.11)

It has to be noted that the thermophysical properties are generally not constant but dependent on temperature (and moisture content of the food). In microwave modeling, as well as in modeling of any other food process alike, the knowledge of the thermophysical properties of all materials involved in the simulated scenario as function of the changing conditions during processing is imperative for accurate simulation results (see also Chapter 2). Heat transfer and EM equations are coupled extrinsically by the values of the temperature and the EM heat generation and intrinsically by the temperature dependency of the thermophysical material properties. At the product surface, boundary conditions need to take the external heat transfer into account. Boundary conditions for both EM and heat transfer application modes are given in the example in Section 7.4. Although today’s computers have tremendous numerical calculation performance, with respect to processor speed and fast accessible memory, compared with computers some years ago, it is still no match for the full complexity of these models without some simplifications. In the case of pure electromagnetism, commercial numerical software packages are available. A comparison of their potential for microwave heating has been addressed by, for example, Yakovlev (2000, 2006). Some self-developed software codes are also described in literature. Most of them originated from the telecommunication area but were developed further for microwave heating applications (Regier and Schubert 2001). The discretization of the partial differential equations or their corresponding integral equations together with the suitable boundary conditions on a calculation grid is common to all numerical techniques. In practice, the method of finite differences (FDM) and the finite element method (FEM) are most common; however, the method of moments (MOM), the transmission line matrix method (TLM),

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and the boundary element method (BEM) have also been applied. For short bursts of high microwave power densities, the heat transfer component of Equation 7.11 can be neglected, as it is much slower than the microwave heat generation. The temperature rise in a defined volume is then directly proportional to the dissipated microwave power, which can be inferred from the effective electric field value and the dielectric loss factor or the electric conductivity, respectively (see Eqs. 7.8 and 7.9).

ρCP

∂T = PV = 2 ⋅ π ⋅ f ⋅ ε 0 ⋅ ε r′′⋅ E 2 = σ ⋅ E 2 ∂t

(7.12)

Some studies using this approximation can be found, for example, in Fu and Metaxas (1994), Liu et al. (1994), Sundberg et al. (1998), Dibben and Metaxas (1994), and Zhao and Turner (1997). Since the late 1990s, and increasing since the early 2000s, a growing number of papers describe the coupled EM field and thermal model. Examples, covering both one- and two-way couplings of temperature-dependent thermophysical properties, including the heat conduction can be found in Torres and Jecko (1997), Ma et al. (1995), Kopyt and Celuch-Marcysiak (2002, 2003, 2004, 2007), Rabello et al. (2005), Zhao and Turner (2000), Sun et al. (2007), Zhu et al. (2007), Sekkak et al. (1995), Lee et al. (2005), Chen et al. (2008), Rakesh et al. (2009), Tilford et al. (2007), Sabliov et al. (2007), Al-Rizzo et al. (2005), and Cordes et al. (2007). This is by all means no complete list of references of work done on simulating microwave application; however, it is considered to form a good basis for the interested reader. In addition to the heat conduction, heat transport by radiation may be addressed by a ray-tracing algorithm (Haala and Wiesbeck 2000). Since the temperatures (and temperature differences) for most food applications are more moderate in comparison with ceramics sintering, where the latter software code originates from, the radiation term becomes negligible and only heat transport by convection or evaporation has to be taken into consideration. The studies from the above-mentioned examples were mostly based on self-developed (academic)

software codes and lack the comfort of user friendliness, full support by software vendors, and extensive manuals. Therefore, they are predominantly useful for the developers in their academic research rather than being suitable for industrial use. In addition to that, most of them occupy further simplifications, such as using water as food material, constant thermophysical properties, or calculating the temperatures only in one or two dimensions, among others. In this chapter, one example of a validated microwave heating simulation, incorporating the EM waves, their interaction with the food product, heat conduction and free convection on the food surface, with the inclusion of temperature-dependent thermophysical properties (Knoerzer 2006; Knoerzer et al. 2008), will be shown in detail in Section 7.4. Furthermore, an approach for simulating the microwave heating of arbitrarily shaped foods and the control of temperature distributions by feedbackcontrolled simulations will be presented.

7.3. Temperature Measurement (Mapping) Methods in Microwave Fields While simulation tools offer insights into processes, hardly feasible by other means, the full potential of simulations is only tangible after validation. However, this is a very challenging task; the EM fields and temperature distributions are not easily measurable without changing them by the measurement procedure itself. A relatively old bibliography covering different temperature indication methods in microwave ovens can be found in Ringle and Donaldson (1975). This section will present a review on more up-to-date temperature measurement strategies. There are a number of established methods for measuring temperature distributions in EM fields, such as thermocouples, fiber optic thermometers, infrared (IR) thermography, microwave radiometry, and temperature indicators (e.g., liquid crystal foils, thermo paper, and model substances). However, many of these methods have severe disadvantages.

Chapter 7

Simulating and Measuring Transient Three-Dimensional Temperature Distributions

7.3.1. Thermocouples Thermocouples are fabricated from electrical conducting wires made of two different metal alloys. They generate an open-circuit voltage, called the Seebeck effect, or Seebeck voltage, discovered in 1821 (Seebeck 1823), which is proportional to the temperature difference between the hot and reference junctions: ΔU = S ⋅ (Thot − Tref )

(7.13)

Here ΔU is the change in voltage, S is the Seebeck coefficient (dependent on the thermocouple’s metal composition), Thot is the temperature at the hot junction, and Tref is the temperature at the cold/reference junction. Without enormous efforts, conventional thermocouples cannot be used in microwave devices. Their metallic wires cause unacceptable disturbance of the fields, which can lead to destruction of the device, as well as damage to the product caused by arcing and plasma generation. Moreover, the method is invasive and only a poor spatial resolution is achievable; that is, the number of locations where temperatures can be measured is limited.

7.3.2. Fiber Optic Probes The basis of the fiber optic thermometry generally used is the temperature-dependent photoluminescence of optically excited phosphors. The key element of fiber optic probes is the sensor consisting, for example, of a small amount of manganeseactivated magnesium fluorogermanate mounted at the tip of the optical fiber. When the sensor material is excited with blueviolet light, the phosphor exhibits a deep red fluorescence. After the activation pulse, the intensity of fluorescent radiation decays exponentially. The decay time can be directly correlated with the temperature. More information can be found in Knoerzer et al. (2005). While the use of fiber optic sensors overcomes the problem of pronounced interaction with the EM field, the achievable spatial resolution is often not sufficient. Furthermore, the fiber optic thermometers

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are rather delicate and expensive in comparison with conventional thermocouples.

7.3.3. Model Substances A different approach to determine temperature distributions in microwave-heated products is the use of model substances. These substances change their properties when a certain temperature is reached. Model substances, however, should represent foods with regard to certain properties. For microwave heating scenarios, it can be sufficient that only the dielectric properties match the food properties in a certain temperature interval and at a frequency of 2.45 GHz (Risman et al. 1993). But often (especially at lower microwave powers or longer heating times), it is essential that also the thermal conductivity and the heat capacity of the model food match the values of real foods. The following factors can be temperature dependent and have been used for temperature mapping (Risman et al. 1993): • Texture (or viscosity) • Optical transparency • Melting temperature • Coagulation or solidification • Color change (e.g., by changing of pH value with added indicator) Often gels are used because they are easy to cut and mold to an exact shape. One published method uses the color change of the model substance (Risman et al. 1993), another one the coagulation of contained proteins (Wilhelm and Satterlee 1971). A downside of this approach is the fact that for getting information on the temperature distributions within the product, the samples have to be sliced. This method is described further in Chapter 6.

7.3.4. Infrared Thermography All objects (especially hot ones) emit thermal radiation that is in the visible but also in the ultraviolet as well as infrared portion of the EM spectrum, for example, an incandescent light bulb or the sun. The frequency and thus the intensity of this radiation, caused by vibrating atoms and molecules,

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behaving like Hertz’ dipoles, increases with increasing temperature. Thermal imaging cameras can measure the thermal radiation patterns on the surface of the radiating object, and from the frequencies of the radiation, the actual temperatures can be calculated. IR thermography provides good spatial resolutions and is established for measuring surface temperatures. Online IR sensors provide a reasonably accurate readout of the surface temperature of an object that is in the sensor ’s field of view. However, similar to conventional contact probes, IR sensors comprise metallic components and cannot be placed inside the EM fields of microwave equipment. Measurements are only possible through a shielded window. Therefore, the efficacy of the temperature measurement, which in any case represents only the surface temperature of one part of the heated product, is reduced.

7.3.5. Microwave Radiometry The principle and advantages of microwave radiometry are similar to the IR thermography (Stephan 2004), except that the thermal energy detected and used as basis for temperature estimation is in the microwave frequency range rather than the IR region. As the penetration depth of radiation in the microwave frequency range is significantly higher than in the IR frequency range, temperatures several millimeters below the surface can be measured. The measurement of surface or near-surface temperatures in volumetric heating methods like microwave heating is, as indicated before, not sufficient. Using this technique, the product has to be sliced to obtain information on the entire, inner temperature distribution.

7.3.6. Liquid Crystal Foils and Thermo Paper 7.3.6.1. Liquid Crystal Foils Liquid crystals are organic substances, which exhibit properties of both solid and liquid matter. They scatter incident light selectively, depending on their apparent molecular structure. Each liquid crystal compound possesses a

p Figure 7.1. Helical structure in cholesteric liquid crystals; p: helix pitch length (K. Hiltrop 2005, pers. comm.).

helical structure with a characteristic pitch length p (see Figure 7.1). The helix pitch lengths, which, for example, can be altered by changing temperatures, are in the range of the wavelength of visible light. Therefore, changes in temperature are directly reflected in changes of the liquid crystals’ color. Since the fundamental chemical structure is unaffected by the external stimulus, this change is reversible and a liquid crystal coating can respond repeatedly to the same physical changes. 7.3.6.2. Thermo Paper The basis of using thermo paper as a temperature indicator is its special coating. The active slice of the thermo paper contains temperature sensitive chemicals, the so-called color developer, which starts to melt after reaching a certain temperature and change their appearance from transparent to black. This color change usually happens within seconds of reaching a rated temperature. Liquid crystal foils and thermo paper are also established methods for measuring temperature distributions (Grünewald and Rudolf 1981; Feher 1997). However, both methods also provide only surface temperature information. To get information on the inner temperature of a material, the thermal paper has to be placed at the cross-section of interest. In this case, the method is invasive. Furthermore, the use of thermo paper is limited to providing information on temperatures being above or below a certain temperature.

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7.3.7. Magnetic Resonance Imaging Almost all of the problems inherent to the abovementioned approaches can be avoided by using MRI as a tool for temperature measurement (Nott and Hall 1999, 2004; Nott et al. 1999, 2000; Bows et al. 2001; Nott and Shaarani 2003; Knoerzer et al. 2005, 2009). This approach allows for noninvasive threedimensional (3D) measurements of microwaveinduced heating patterns. Since the probe is not required to be in contact with the product, temperatures within any water-containing product can be observed during or after microwave heating without destroying the product. Good spatial resolutions can be achieved not only for surface temperatures but also for temperature distributions throughout the product. The large drawback of this technology is the cost of an MRI tomograph, which exceeds the costs for all devices mentioned above by at least one order of magnitude (i.e., at least 10 times). However, as the MRI temperature mapping technique is superior in performance and 3D resolution to the methods mentioned above, this technique will be presented here in more detail. 7.3.7.1. Basics of Temperature Mapping Using MRI Imaging methods based on the resonant absorption and emission of EM energy by hydrogen nuclei placed in strong magnetic fields (MRI) are well known for their widespread clinical applications. Temperature mapping using MRI is possible as temperature shift leads to a very small shift in the resonance frequency for hydrogen nuclei in water. Nuclear Magnetic Resonance (NMR) Many protons (such as 1H, the hydrogen proton) have a  nuclear spin and thus a magnetic moment μ (Figure 7.2). Bringing the nonarranged magnetic moments (Figure 7.3a) in a magnetic field B0 causes an alignment in parallel or antiparallel orientation according to the external magnetic field and, as a result, the magnetization M (Figure 7.3b).   (7.14) μ = γ ⋅I

(a)

137

(b)

Figure 7.2. Nucleus spin (a) and magnetic moment (b) of a hydrogen proton.

(a)

(b) → M

B0

Figure 7.3. (a) Nonarranged magnetic moments; (b) alignment of magnetic moments in an external magnetic field.

  M = ∑μ

(7.15)

where μ is the magnetic moment, γ is the magnetogyric ratio (constant for 1H), I is the angular  momentum, and M is the magnetization. In a macroscopic sample, transverse nuclear magnetization, which precesses around the orientation of the external magnetic field B0 with the same angular frequency ω, may be created with an EM excitation pulse, fulfilling the resonance condition (Eq. 7.16; Figure 7.4). The external magnetic field causes  a precession of the transverse magnetization M with the socalled Larmor frequency:

ω L = γ ⋅ B0

(7.16)

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

In this equation, ωL is the precession (Larmor) frequency and B0 the external magnetic flux density. This precession induces an AC voltage in the radio frequency (RF) coil, the MR signal, which is directly proportional to the spin density, thus the H (proton) density and the water content. When the RF pulse is switched off, the phase relationship will decay with a characteristic time constant (referred to as T2) and their alignment along the z-axis will relax with a different time constant T1; both those parameters are temperature dependent. A further temperature-dependent parameter measurable by MRI is the self-diffusion coefficient. However, similar to the time constants, this parameter also shows a strong dependence on the composition of the fluid in the food.

B0

Another parameter, basically independent of the composition of the fluid in the food, and measurable by MRI, is referred to as the chemical shift (of the MR frequency). The method of measuring temperatures using this parameter is based on the temperature dependence of the water proton chemical shift: Increasing temperature leads to breaking of hydrogen bonds and thus to a stronger shielding of the hydrogen proton due to a reduced hydrogen– oxygen distance by oxygen-located electron clouds. As a result, the precession frequency decreases, and thus the phase (spin angle) of the MR signal (Δf/ΔT = 0.01 ppm/°C; Hindman 1966). From a measured phase difference between the sample with known initial temperature and the heated sample, the actual temperature can be calculated (Figure 7.5). Spatial resolution is introduced into NMR by a place-to-frequency transformation, where the static magnetic field is varied linearly across the sample with a magnetic field gradient G by using additional coils (Knoerzer et al. 2009).

M(t0) f

M(tp) Y

z y

M(t>tp)

x

Figure 7.4. Schematic representation of inverting the mag netization M (or M ) in the transverse plane.

Phase image at Tref

7.3.7.2. Equipment Basically any commercially available MRI tomograph can be used for performing temperature mapping studies; from large-sized tomographs as used in medical diagnostics for humans (or even larger ones used in veterinary diagnostics) down to small-scale high-resolution tomographs with diameters in the millimeter range. In the validation studies, as outlined in Section 7.4, a Bruker Avance 200 SWB tomograph (Bruker Biospin MRI GmbH, Ettlingen, Germany; Figure

Δ f (Hz) 60

Phase image at T

ΔT (K) 325 320

40

315 20 (a)

(b)

(c)

310 (d)

305 0 Figure 7.5. (a) MRI phase image of a model food at known initial temperature (Tref = const); (b) phase image of heated sample; (c) phase difference distribution (c = a − b); and (d) temperature distribution (calculated from phase difference with knowledge of a chemical shift of Δf/ΔT = 0.01 ppm/°C). See color insert.

Chapter 7

Simulating and Measuring Transient Three-Dimensional Temperature Distributions

(a)

Jacket with liquid nitrogen

Shim unit

139

(b)

Jacket with liquid hydrogen

Gradient unit RF coil

Magnet

Sample holder

Figure 7.6. (a) Schematic representation of an MRI tomography; (b) tomograph used for temperature mapping studies.

7.6) was used. The superconducting magnet with a 150 mm room-temperature bore produces a magnetic flux density of 4.7 T. Spatial resolution is achieved with a Bruker mini0.36 gradient system generating field gradients of up to 0.14 T/m in three orthogonal directions. An open-ended Bruker birdcage probe head with a 64-mm inner diameter was used as an antenna. Pulse sequences were used as provided by Bruker. 7.3.7.3. Method For measuring temperature or phase distribution, respectively, a gradient echo pulse sequence (Kimmich 1997) was used. Temperature distributions were measured in a variable number of slices (from 10 up to 50 slices) perpendicular to the z-axis, with a slice thickness of 1 mm, a 2D resolution of 64 × 64 pixels, which corresponds in the 64-mm birdcage probe to a 3D

resolution of 1 mm3. The overall measuring time was 12.8 s, with an accuracy of approximately ±2 K. The MRI data (binary 3D matrices) obtained were exported and analyzed with self-developed MATLAB 7.0™ (The Mathworks Inc., Natick, MA) programs. To avoid temperature equilibration between heating and measuring temperature distribution in the tomograph, a microwave device was developed together with Gigatherm AG (Grub, Switzerland) (Hermann 2004). This system allows the transmission of microwaves through a circular waveguide into the RF coil (birdcage), where the sample to be investigated is placed (Figure 7.7), making it possible to measure in situ 3D temperature distributions during microwave processing. Heating experiments with differently shaped model food samples and real foods were performed

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

appr. 2.5 m

rectangular waveguide

microwave generator

circular waveguide

magnet

sample

RF coil / birdcage

waterload

Figure 7.7. Sketch of the device for introducing microwaves into the MRI tomograph.

and temperature changes at discrete points measured by MRI were quantitatively validated using fiber optic thermometry (Knoerzer et al. 2009).

7.4. Examples of Validated Microwave Heating Models This section presents the simulation approach developed by Knoerzer et al. (2008). The methodology is based on a user-friendly interface coupling two commercial software packages to model timedependent temperature profiles not only of microwave treated products with simple shapes but also of arbitrarily shaped products in three dimensions. Validation of the model was performed using noninvasive MRI, allowing for measuring temperature distributions during the running process in three dimensions. With the coupled model, hot and cold spots in the products can be detected to test appropriate microwave treatment control strategies. A further feature of the model describes a feedback control loop in the simulation, which is used to optimize the microwave process conditions (pulse profile) to allow minimal time/temperature treatments and improved uniform temperature distributions.

7.4.1. Model Geometry The geometry of the model microwave cavity is depicted in Figure 7.8b. The model was designed in full 3D to be a good approximation of the microwave chamber in the equipment as outlined in the previous section (h = 300 mm, d = 84 mm). Samples used were predominantly simple shapes (cylinders, spheres) of a model food with known temperature-dependent thermophysical properties (Knoerzer et al. 2004). However, a method was developed to allow the introduction of models of real food structures based on a 3D spin density measurement technique (Knoerzer 2006) as shown in Figure 7.9.

7.4.2. Process, Initial, and Boundary Conditions 7.4.2.1. Process Conditions In the uncontrolled microwave heating simulation of model food cylinders, the microwave power was set to constant values of P = 19 and 23.1 W between t = 200 and 650 s, and to P = 0 at all other times. In the feedbackcontrolled simulation, the microwave power was set to 37 W as long as the temperature in the hottest spot

Chapter 7

Simulating and Measuring Transient Three-Dimensional Temperature Distributions

(a)

141

(b)

appr. 2.5 m

rectangular waveguide

microwave generator

circular waveguide sample

magnet birdcage

waterload simulated area

Figure 7.8. (a) Sketch of microwave device for introducing microwaves into MRI tomograph and (b) depiction of area of interest in the simulation process.

was below the predefined maximum temperature of 343 K, and set to P = 0 when this temperature was exceeded. A stop condition was triggered when the temperature in the coldest spot is equal or greater than the predefined target temperature (333 K) for a predefined time of 30 s. 7.4.2.2. Initial Conditions At the start of the process, both sample and surrounding air are in thermal equilibrium at T0 = 298 K. In the feedbackcontrolled simulation, the initial temperature and the temperature of the surrounding air was set to T0 = 295 K for the “pure” microwave heating process. In the combined microwave–hot air heating process, the initial temperature was also 295 K, whereas the air temperature was set to the target temperature of 333 K. 7.4.2.3. Boundary Conditions The boundary conditions for the EM interaction with the surface of the sample are described by reflection and transmission.

The boundary of the cavity is assumed to be a perfect conductor and thus   (7.17) E × n = 0 and H ⋅ n = 0 Here n is the normal vector to the surface. The boundary conditions at an interface between two dielectric materials are    E2 − E1 × n = 0 (7.18)    ε 2 E2 − ε1 E1 ⋅ n = 0 (7.19)   H 2 − H1 × n = 0 (7.20)   μ2 H 2 − μ1 H1 ⋅ n = 0 (7.21)

(

(

(

)

(

)

)

)

A heat transfer based on free convection on the boundary of the heated object (∂V) was modeled for thermal interactions. −k

∂T + h ⋅ (T − Ta ) = 0 on ∂V ∂n

(7.22)

The heat transfer coefficient was calculated taking thermal conductivity of the surrounding medium

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

(a) 50

25

0 50

25

0

0

50

25

50 25 (c)

(b) 0 50

50

25

0 0

50

50

25

25

0

25

0 50

25

0

0

25

50

50

25

0

0

25

50

Figure 7.9. The inner structure of the chicken wing as determined by MRI (middle: MRI raw data; corners: matrix subdivided into (a) bones, (b) fat, and (c) meat).

and shape of the sample into consideration (VDI Wärmeatlas 2002). A good approximation for the heat transfer coefficient was found to be h=

kair r

(7.23)

with the thermal conductivity of the surrounding air (kair) and an average radius ( r ) of the sample with H/d (height-to-diameter ratio) between 0.7 and 1.3.

7.4.3. Material Properties The EM and thermal properties, that is, loss factor ε ″, dielectric constant ε ′, thermal conductivity k, density ρ, and heat capacity CP, and their variation with water content and temperature were determined experimentally for a model food, developed for microwave heating research, and fitted with thirdorder polynomials (Knoerzer et al. 2004). These equations were implemented as material properties in the respective subdomains in the software packages used.

Chapter 7

Simulating and Measuring Transient Three-Dimensional Temperature Distributions

In the case of the chicken wing heating, thermophysical properties of fat, meat, and bone material were taken from literature (as constants).

7.4.4. Computational Methods The partial differential equations describing the EM part of the model were solved with the finite difference time domain (FDTD) method using QuickWave-3D™ (QWED Sp. z o.o., Warsaw, Poland). FEM was chosen for the calculations of heat generation and conduction and solved using COMSOL Multiphysics™ (COMSOL AB, Stockholm, Sweden). The interface between both software packages was developed in MATLAB 7.4™ (The Mathworks Inc.). The cross-software communication was enabled by transforming the different output formats (text files from QuickWave-3D™ and structural arrays from COMSOL Multiphysics™ into MATLAB matrices containing the data (see flowchart in Figure 7.10). Although the QuickWave3D™ and COMSOL Multiphysics™ are based on very different numerical schemes and associated different meshes, the data provided by the solvers could be transformed into MATLAB matrices by interpolation and in MATLAB defined as locationand time-dependent functions, for example, PV = f(x,y,z,t). A graphical user interface (GUI) was programmed to allow users not familiar with MATLAB to run the simulations. The software interface uses the data provided in the GUI to allocate material properties (see Section 7.4.3) and process conditions (e.g., microwave frequency, pulse duration, initial and external temperatures, among others) to predefined geometries in both QuickWave-3D™ and COMSOL Multiphysics™. Another option in the GUI is to define geometrical properties of simple shapes (e.g. cylinders, cubes, spheres) with subsequently solving the model. The presented approach describes a one-way simulation, that is, after calculating the dissipated power by QuickWave-3D™ with constant dielectric properties, the heat transfer is calculated with temperature- and location-dependent properties. Following Equations 7.9 and 7.12, PV is directly

143

proportional to the loss factor ( PV ∝ ε ′′ ), and thus, a modified form can be used to take the dependency of the dielectric properties and, thus, the dissipated power into consideration. PV (T , x, y, z ) = PV ,QuickWave −3 D ( x, y, z ) ⋅

ε ′′(T , x, y, z ) ε initial ′′ (7.24)

In Equation 7.24, PV,QuickWave-3D is the solution of the EM simulation, ε initial ′′ is the initial loss factor of the sample, and ε″(T,x,y,z) is the recalculated loss factor taking different temperatures at different locations in the sample into account. A self-developed MATLAB 7.4™ code was used to transform the structure data of real foods as determined by MRI into the format readable by the simulation software packages and allocating the corresponding material properties. Different materials yield different intensities in MRI spin density measurements, making it possible, by multithreshold setting, to subdivide a full 3D MRI image (3D matrix) into different matrices corresponding to the materials and allocate the matching thermophysical properties at the specific locations to these materials. Figure 7.9 represents such a matrix subdivided into submatrices, and Figure 7.10 represents the procedure of simulating the microwave heating of this food product. A further code was developed in MATLAB 7.4™ to allow for a feedback-controlled simulation. This code is capable of identifying hottest and coldest spots in the simulated heating scenario. After a predefined maximum temperature in the hot spot is reached, the dissipated power is set to zero, and then back to the predictions from QuickWave-3D™ once the hot spot temperature is falling below that value, according to the description in Section 7.4.2.1. Once the predefined temperature is reached and held for a certain time, a stop command is triggered and the resulting pulse profile (as shown in Figure 7.19) stored. In QuickWave-3D™, the computational domain was subdivided into cubic elements with 1-mm edge length (corresponding to 1.5 million cells), so that at least 15 elements per wavelength also inside the product are achieved to ensure grid independence of

Interface – complete simulation controlled by one GUI

(a)

QuickWave-3D TM – simulates the power dissipation

COMSOL TM – solves the energy balance y-dim./ × 10 m 0.060.04 5 0.02 0 TM 4 COMSOL 0.06 – Input 3 0.04 2 0.02 cube with 1 size of 0 0.04 0.06 MRI 0 0 0.02 martix x-dim./m

chicken wing water load

(b)

50 25

0 y-d 50 im 25 0 0 ./m m

z-dim./m

QuickWave -3DTM – Output

waveguide

pv/Wmm3

e′, e″ = const

z-dim./mm

–4

25 50

x-dim./mm

(c)

T/K

z-dim./mm

Interface – calculation of 3D temperature distribution as function of time solution at discrete time: theating = 400 s, PMW = 25 W, matrix size = 643 mm3 340 335 60 330 50 325 40 320 30 315 310 20 305 10 300 0 295 50 50 00 y-dim ./mm m m ./ x-dim (d) Figure 7.10. Example of flowchart of the simulation (with QuickWave-3D™ and COMSOL Multiphysics™ solution): microwave heating of a chicken wing. (a) GUI allowing communication with the interface controlling both QuickWave-3D™ and COMSOL Multiphysics™; (b) QuickWave-3D™ model; (c) COMSOL Multiphysics™ structure; (d) solution of coupled EM thermal solver.

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the predictions; the mesh in COMSOL Multiphysics™ was automatically generated and the number of mesh elements increased until no further noticeable change in model predictions could be observed. The number of elements was dependent on the size of the sample and the material properties and varied between 9,000 and 20,000 tetrahedral elements. A workstation with a dual-core processor (CPU speed 2.6 GHz) and RAM of 2 GB running a 32-bit Windows XP professional OS was used, which allowed the simulation to be completed within less than 2 h. Depending on the number of pulses needed to achieve the desired temperature distribution in the controlled simulations, the overall solution time of the model increased accordingly.

7.4.5. Validation of the Simulation Initial validation was performed qualitatively by visually comparing simulated and measured 3D temperature distributions results. Then, measured temperature profiles at selected points (the hot and cold spots) were compared with profiles predicted by the coupled model at the same locations for quantitative validation. Furthermore, a software algorithm was developed to extract measured and simulated temperature distributions from the 3D output matrices (from both the MRI measurements and the coupled models) in axial cross-sections at

discrete times. The temperature values at all locations in the resulting 2D matrices were used to obtain parity plots. The routine was also programmed to perform a statistical analysis by calculating the coefficient of determination R2 as a measure of accuracy of the model predictions.

7.4.5.1. Visual Comparison of 3D Temperature Distributions Figure 7.11 shows both simulated and measured 3D temperature distributions in a microwave-heated model food cylinder at a discrete time of 250 s and a microwave power of 19 W. Figures 7.12 and 7.13 provide a close-up view of two cross-sections of the cylinder; one in the middle (h = 16 mm) of the cylinder, the second one near the top (h = 26 mm). A good qualitative agreement of simulation and measurement was found for all modeled scenarios, with the slices from both simulation and MRI measurement showing very similar temperature distribution profiles. 7.4.5.2. Comparison of Temperature Profiles in Discrete Points (Hot and Cold Spots) Figures 7.14 and 7.15 show predicted and measured temperature profiles in the hot and cold spots of one slice (h = 16 mm) of the heated model food cylinder at microwave powers of 19 and 23.1 W, respectively.

T (K) 310

14

30 simulation

25 20

12

measurement

308

10 306

z (mm)

8 slices

15 10 5 0 60 40 x (mm

)

40 30 20 y (mm)

145

6

304

4 2 0 50 30 x (mm 20 )

302 60 40 y (mm)

300

Figure 7.11. Visual comparison between the simulated (left) and the measured (right) heating of a model food cylinder (at a discrete time): PMW = 19 W; t = 250 s; Text = 298 K. See color insert.

T (K) 308

306

304

302 simulation measurement Figure 7.12. Visual comparison between the simulated (left) and the measured (right) heating of the center cross-section (h = 16 mm) of the microwave-heated model food cylinder (at a discrete time). See color insert. T (K) 308

306

304

302 simulation measurement Figure 7.13. Visual comparison between the simulated (left) and the measured (right) heating of a cross-section near the top (h = 26 mm) of the microwave-heated model food cylinder (at a discrete time). 340

hot spot measured cold spot measured hot spot simulated cold spot simulated

335 330

hot spot

temperature (K)

325 320 315 310 cold spot 305 300 295

0

200

400

600

800

1000

time (s) Figure 7.14. Comparison of temperature profiles in a hot and a cold spot in a microwave-heated cylindrical model food sample (microwave power PMW = 19 W).

146

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hot spot measured cold spot measured hot spot simulated cold spot simulated

340

147

hot spot

temperature (K)

330

320

cold spot

310

300 0

200

400

600

800

1000

time (s) Figure 7.15. Comparison of temperature profiles in a hot and a cold spot in a microwave-heated cylindrical model food sample (microwave power PMW = 23.1 W).

Good quantitative agreement was found in all cases.

7.4.6. Simulated Heating of a Chicken Wing The simulated heating of the chicken wing, described in Section 7.4.1 and shown in Figure 7.10 was performed at a microwave power of 25 W for 600 s. Figure 7.17 shows the temperature distribution in the chicken wing after heating. It is obvious that uncontrolled process conditions lead to the

350 s 600 s bisecting line

80 temperature (predicted) (°C)

7.4.5.3. Comparison of Cross-Sections Figure 7.16 shows a confidence (parity) plot of simulated versus measured temperatures at different times of the heating process at identical locations of a crosssection of the microwave-treated sample. A statistical analysis showed a coefficient of determination R2 greater than 0.95, which means measurement and prediction are in good agreement.

100

60

40

20

0 0

20

40 60 80 temperature (measured) (°C)

100

Figure 7.16. Confidence (parity) plot of simulated temperatures versus measured temperatures of identical locations of the axial cross-section where the hottest spot is located.

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

development of unacceptably uneven temperature distributions caused by the microwave specific factors described earlier and in detail in Ringle and Donaldson (1975), Knoerzer (2006), WäpplingRaaholt and Ohlsson (2005), Ryynänen et al. (2004), Risman (1992), Sinell (1986); Ohlsson and Bengtsson (2001), and Buffler (1993).

z-dimension (mm)

T (K) 350 60

340

50

330

40

320

30 20

310

10 t = 800 s 0 60 40 x-dimen s

60 20 40 m) 20 0 0 n (m io s n e im ion (mm y-d )

300

Figure 7.17. 3D temperature distribution in a chicken wing after microwave treatment of 600 s (power was switched on after 200 s) as calculated using the new simulation procedure (flowchart in Figure 7.10). See color insert.

7.4.7. Controlled Microwave Heating of a Model Food Cylinder In modeled microwave heating scenarios, 3D temperature distributions are clearly defined at any point in time. This fact allows a feedback algorithm to control the process conditions (microwave power) in the simulated scenarios based on identifying and evaluating the hottest and coldest spot in the heated product. Such algorithms can then predict the timedependent process conditions needed to ensure the most uniform heating achievable. In practice, such a feedback control loop is difficult to accomplish in a real-life industrial microwave process, as temperature distributions have to be measured inline. Figure 7.18 shows the temperature profiles in the hottest and the coldest spot in a microwave-heated cylindrical model food sample. The microwave power was set to P = 37 W as long as the temperature in the hot spot was below the maximum temperature, and to P = 0 as soon as this temperature was exceeded. The maximum product temperature was set to 343 K, whereas a required minimum temperature of 333 K was to be reached and held for 30 s.

Figure 7.18. (a) Pure microwave heating (PMW = 37 W; external temperature Text = 295 K); (b) combined microwave–hot air heating (PMW = 37 W; Text = 333 K).

Chapter 7

Simulating and Measuring Transient Three-Dimensional Temperature Distributions

Figure 7.18a shows the example of a “pure” microwave heating; that is, the temperature outside the product was set to the initial temperature of the sample (T = 295 K). Figure 7.18b shows a combined microwave–hot air process; that is, the temperature inside the microwave oven was set to the target temperature (T = 333 K). In the case of the microwave heating alone, the maximum temperature was reached and regulated after about 200 s. However, the temperature in the coldest spot (which was found at the product surface) did not reach the target temperature, even after more than 20 minutes. The reason for this is the heat loss from the heated sample to the cooler surrounding air. When hot air is introduced in the combined process, the temperature in the hot spot again reaches maximum temperature after about 200 s. In this particular case, the temperature in the coldest spot reached the target temperature after about 10 minutes. The combined process therefore enables safe minimal time/temperature treatments. The output of the feedback-controlled simulation is shown in Figure 7.19, a microwave power pulse program that can be used in the control unit of a microwave oven and, thus, allows for repeating the

40 35

power (W)

30 25 20 15 10 5 0

0

100

200

300 400 time (s)

500

600

Figure 7.19. Output of the on/off control tool; microwave power pulse program (example of combined microwave–hot air heating as described above).

149

simulated scenario in reality by controlling or setting (no feedback) the microwave power.

7.5. Summary, Conclusions, and Outlook Microwaves heat food volumetrically and more rapidly than conventional heating methods. However, several microwave intrinsic factors associated with volumetric heating can (and most certainly will if microwave power is supplied continuously into a solid food product) lead to the development of nonuniform temperature distributions, with edge and corner overheating, as well as the development of hot and cold spots within the product. Over the last decades, research in microwave processing pushed toward the successful development of computational models to accurately predict the process, yielding 3D temperature distributions. Approaches to achieve this challenging goal varied from very simplified models to sophisticated models coupling computational fluid dynamics with solvers for EM problems. Since the early 2000s, a growing number of articles describe this coupling. In this chapter, one successfully validated simulation was described in detail, accurately predicting temperature profiles during a heating process in a lab-scale microwave system. The one-way coupling, enhanced by an approach to implement the temperature-dependent loss factor and thus dissipated power into the energy balance is computationally convenient and very time efficient compared with a two-way coupling. However, it has to be mentioned that software packages capable of performing two-way coupling are available today, which are based on FTDT and therefore superior in efficiency to codes based on FEM. The presented software procedure simulates both EM waves and heat transfer and therefore the temperature evolution in arbitrarily shaped foods. While simulation tools offer insights into processes that are hardly feasible by other means, the full potential of simulations is only tangible after validation. As described earlier, in the case of microwave processing, this is a rather challenging task, as most probes commonly used for measuring temperatures, such as thermocouples, fiber optic probes, or

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

thermal IR imaging, either interact with the microwave field or yield only poor spatial resolution. To date, there is only one methodology that provides the same data as the model, that is, transient 3D temperature distributions. This MRI methodology was highlighted in this chapter and used for validating the simulated scenario by introducing microwaves into the MRI probe where the food product was held. As shown by both the simulated predictions and the MRI measurements, significant temperature heterogeneities occurred as long as the treatment was uncontrolled, while more uniformity was achieved when a control tool was implemented in the simulation. This kind of feedback-controlled simulation is one-of-a-kind to date and allows for predicting optimal power programs, ensuring safe minimal time–temperature treatments in microwave food processing. Simulations of this type can be used both for equipment and process design and optimization. Furthermore, they will greatly assist in demonstrating the safety of the process for industry and regulatory bodies.

P PMW PV PV,QuickWave-

Notation

ε0

Latin Letters  B

Magnetic flux (density)

CP D d e E f G

Specific heat capacity Electric flux density Diameter Euler ’s number Electric field Frequency Magnetic field gradient

h, H h H  I j k  M, M n p

Height Heat transfer coefficient Magnetic field Nuclear spin Electric current density Thermal conductivity Magnetization Normal to the surface Pitch length of helical liquid crystal structure

T = V · s/ (A · m) J/(kg ·K) A · s/m2 m ≈2.71828 V/m Hz, 1/s T/m = V·s/ (A·m2) m W/(m2 · K) A/m kg · m2/s A/m2 W/(m · K) A/m — m

3D

Qem qR R2 S1 T Tref T1, T2

t U V x,y,z

Power Microwave power Power density Power density calculated by QuickWave-3D™ Electromagnetic heat source Radiation power flux density Coefficient of determination Signal intensity of luminescence Temperature Reference temperature Transversal and longitudinal relaxation times Time Voltage/potential Volume Spatial coordinates

W = J/s W = J/s W/m3 W/m3 W/m3 W/(m2 · s) — arb. units K, °C K, °C s

s V m3 m

Greek Letters

εr ε εr′ εr″ Φ μ0

Dielectric constant of vacuum Relative dielectric constant Complex dielectric constant Real part of complex permittivity Imaginary part of complex permittivity, dielectric loss factor Magnetization inversion angle Permeability of vacuum

μr, μ Relative magnetic permeability  Magnetic moment μ π Ludolph’s number, circular constant ρ Charge density ρ Mass density σ Conductivity τ Decay time

8.854 · 10−12 A · s/(V · m) — — —

— ° = 1/ε 0c20 = 1.257 · 10−6 V·s/(A·m) — A · m2 ≈ 3.14159 A · s/m3 kg/m3 S/m s

Chapter 7

Simulating and Measuring Transient Three-Dimensional Temperature Distributions

γ

Magnetogyric ratio

ω ωL

Angular frequency Larmor frequency

Hz/T = m2/ (V· s2) Hz = 1/s Hz = 1/s

Operators ∂ ∇ ∑ Δ ∇⋅ ∇×

Partial differential Nabla operator Sum Difference Divergence Curl

Abbreviations 2D 3D AC BEM CAPE CPU EM FDM FDTD FEM FPA GUI IR LED MOM MR MRI NMR OS pixel ppm RAM RF TLM

Two-dimensional Three-dimensional Alternating current Boundary element method Computer-aided process engineering Central processing unit Electromagnetic Finite difference method Finite difference time domain Finite element method Focal plane array Graphical user interface Infrared Light-emitting diode Method of moments Magnetic resonance Magnetic resonance imaging Nuclear magnetic resonance Operation system Picture element Parts per million Random access memory Radio frequency Transmission line matrix

References Al-Rizzo HM, Tranquilla JM, Feng M. 2005. A finite difference thermal model of a cylindrical microwave heating applicator

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using locally conformal overlapping grids: Part I—Theoretical formulation. J Microwave Power Electromagn Energy 40(1): 17–30. Bows JR, Patrick ML, Nott KP, Hall LD. 2001. Three-dimensional MRI mapping of minimum temperatures achieved in microwave and conventional food processing. Int J Food Sci Technol 36:243–252. Buffler CR. 1993. Microwave Cooking and Processing: Engineering Fundamentals for the Food Scientist. New York: AVI Book. Chen H, Tang J, Liu F. 2008. Simulation model for moving food packages in microwave heating processes using conformal FDTD method. J Food Eng 88(3):294–305. Cordes BG, Eves EE, Yakovlev VV. 2007. Modeling-based minimization of time-to uniformity in microwave heating systems. Proceedings 11th AMPERE Conference on Microwave and High Frequency Heating (Oradea, Romania, September 2007), 305–308. Decareau RV. 1985. Microwaves in the Food Processing Industry. Orlando, FL: Academic Press Inc. Dibben DC, Metaxas AC. 1994. Finite-element time domain analysis of multimode application using edge elements. J Microwave Power Electromagn Energy 29(4):242–251. Feher LE. 1997. Simulationsrechnungen zur verfahrenstechnischen Anwendung von Millimeterwellen für die industrielle Materialprozesstechnik. Wissenschaftliche Berichte FZ KA 5885. Karlsruhe: Forschungszentrum Karlsruhe. Fu W, Metaxas AC. 1994. Numerical prediction of threedimensional power density distribution in a multi mode cavity. J Microwave Power Electromagn Energy 29(2):67–75. Grünewald TH, Rudolf M. 1981. Messung der Temperatur und der Temperaturverteilung im Mikrowellenfeld. Zeitschrift für die Lebensmittelwirtschaft 32(3):85–88. Haala J, Wiesbeck W. 2000. Simulation of microwave, conventional and hybrid ovens using a new thermal modelling technique. J Microwave Power Electromagn Energy 35(1): 34–43. Hermann A. 2004. Aufbau und Inbetriebnahme einer Vorrichtung zur Einkopplung von Mikrowellen in einen NMRTomographen. Diplomarbeit: Universität Karlsruhe (TH). Hindman JC. 1966. Proton resonance shift of water in gas and liquid states. J Chem Phys 44:4582–4592. Kimmich R. 1997. NMR-Tomography-Diffusometry-Relaxometry. Berlin: Springer. Knoerzer K. 2006. Simulation von Mikrowellenprozessen und Validierung mittels bildgebender magnetischer Resonanz. Dissertation, Universität Karlsruhe (TH). Knoerzer K, Regier M, Erle U, Schubert H. 2004. Development of a model food for microwave processing and the prediction of its physical properties. J Microwave Power Electromagn Energy 39(3–4):167–177. Knoerzer K, Regier M, Schubert H. 2005. Measuring temperature distributions during microwave processing. In: H Schubert, M Regier, eds., The Microwave Processing of Foods. Cambridge: Woodhead Publishing Ltd.

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Knoerzer K, Regier M, Schubert H. 2008. A computational model for calculating temperature distributions in microwave food applications. Innov Food Sci Emerg Technol 9(3):374–384. Knoerzer K, Regier M, Hardy EH, Schuchmann HP, Schubert H. 2009. Simultaneous microwave heating and three-dimensional MRI temperature mapping. Innov Food Sci Emerg Technol 10(4):537–544. Kopyt P, Celuch-Marcysiak M. 2002. Review of numerical methods for solving heat conduction coupled to conformal electromagnetic FDTD solver. European Symposium Numerical Methods in Electromagnetics (JEE ‘02), Toulouse, March, 140–145. Kopyt P, Celuch-Marcysiak M. 2003. Coupled electromagnetic and thermal simulation of microwave heating process. 2nd International Workshop on Information Technologies and Computing Techniques for the Agro-Food Sector, Proceedings, Barcelona, November–December, 51–54. Kopyt P, Celuch-Marcysiak M. 2004. Coupled FDTDFEM approach to modelling of microwave heating process. Proceedings of the 5th International Conference on Computation in Electromagnetics (CEM 2004), Stratford-uponAvon, April. Kopyt P, Celuch-Marcysiak M. 2007. Coupled electromagnetic thermodynamic simulations of microwave heating problems using the FDTD algorithm. J Microwave Power Electromagn Energy 41(4):18–29. Lee HK, Li BQ, Huo Y, Tang J. 2005. Validation of 3-D electromagnetic-thermal model for microwave food processing. Ht2005: Proceedings of the ASME Summer Heat Transfer Conference 2005, Vol 3, 781–786. Liu F, Turner I, Bialkowski M. 1994. A finite difference time domain simulation of power density distribution in a dielectric loaded microwave cavity. J Microwave Power Electromagn Energy 29(3):138–148. Ma L, Paul DL, Pothecary N, Railton C, Bows JR, Barrat L, Mullin J, Simons D. 1995. Experimental validation of a combined electromagnetic and thermal FDTD model of a microwave heating process. IEEE Trans Microwave Theory Tech 43(11):2565–2571. Maxwell JC. 1865. A dynamical theory of the electromagnetic field. Phil Trans Royal Soc London Ser B Biol Sci 155:459–512. Metaxas AC. 1996. Foundations of Electroheat. Chichester: John Wiley & Sons. Metaxas AC, Meredith RJ. 1983. Industrial Microwave Heating. London: Peter Peregrinus Ltd. Mudgett RE. 1986. Microwave properties and heating characteristics of foods. Food Technol 40:84–93, 98. Nott KP, Hall LD. 1999. Advances in temperature validation of foods. Trends Food Sci Technol 10(11):366–374. Nott KP, Hall LD. 2004. Measurement of temperature and other heat induced changes in foods by magnetic resonance imaging (MRI). ICEF9—9th International Congress on Engineering and Food, Montpellier, France.

Nott KP, Shaarani SM. 2003. The effect of microwave heating on potato texture studied with magnetic resonance imaging. Magnetic Resonance in Food Science, Latest Developments, Proceedings, Paris, 38–45. Nott KP, Hall LD, Bows JR, Hale M, Patrick ML. 1999. Threedimensional MRI mapping of microwave induced heating patterns. Int J Food Sci Technol 34:305–315. Nott KP, Hall LD, Bows JR, Hale M, Patrick ML. 2000. MRI phase mapping of temperature distributions induced in food by microwave heating. Magn Reson Imaging 18:69–79. Ohlsson T, Bengtsson NE. 2001. Microwave technology and foods. Adv Food Nutr Res 43:65–140. Rabello AA, Silva EJ, Saldanha RR, Vollaire C, Nicolas A. 2005. Adaptive time-stepping analysis of nonlinear microwave heating problems. IEEE Trans Magn 41(5):1584–1587. Rakesh V, Datta AK, Amin MHG, Hall LD. 2009. Heating uniformity and rates in a domestic microwave combination oven. J Food Process Eng 32(3):398–424. Regier M, Schubert H. 2001. Microwave processing. In: P Richardson, ed., Thermal Technologies in Food Processing, 179–207. Boca Raton: CRC Press. Ringle EC, Donaldson DB. 1975. Measuring electric field distribution in a microwave oven. Food Technol 29(12):46–54. Risman PO. 1992. Metal in the microwave oven. J Microwave Power Electromagn Energy 13(1): 28–33. Risman PO, Ohlsson T, Lingnert H. 1993. Model substances and their use in microwave heating studies. SIK Rep 588: 1–10. Ryynänen S, Risman PO, Ohlsson T. 2004. Hamburger composition and microwave heating uniformity. J Food Sci 69(7): M187–M196. Sabliov CM, Salvi DA, Boldor D. 2007. High frequency electromagnetism, heat transfer and fluid flow coupling in ANSYS multiphysics. J Microwave Power Electromagn Energy 41(4):5–17. Schubert H, Regier M. 2005. The Microwave Processing of Foods. Cambridge, UK: Woodhead Publishing Ltd. Seebeck TJ. 1823. Magnetische Polarisation der Metalle und Erze durch Temperaturdifferenz. Berlin: Abhandlung der Akademischen Wissenschaften. Sekkak A, Pichon L, Razek A. 1995. 3-D non linear modelling of microwave heating process using finite element method. Electric and Magnetic Fields—From Numerical Models to Industrial Applications, 99–102. Sinell HJ. 1986. Der Einfluss der Mikrowellenbehandlung auf Mikroorganismen im Vergleich zur konventionellen Hitzebehandlung. DFG-Abschlussbericht Si/55:24–21. Stephan KD. 2004. Continuous non-contact remote temperature sensing during microwave heating with microwave radiometry. 38th Annual Microwave Symposium, Proceedings, Toronto, Canada, July 14–16. Sun H, Zhu HM, Feng HD, Xu L. 2007. Thermoelectromagnetic coupling in microwave freeze-drying. J Food Process Eng 30(2):131–149.

Chapter 7

Simulating and Measuring Transient Three-Dimensional Temperature Distributions

Sundberg M, Kildal PS, Ohlsson T. 1998. Moment method analysis of a microwave tunnel oven. J Microwave Power Electromagn Energy 33(1):36–48. Tang ZW, Mikhaylenko G, Liu F, Mah JH, Pandit R, Younce F, Tang JM. 2008. Microwave sterilization of sliced beef in gravy in 7-oz trays. J Food Eng 89(4):375–383. Tilford T, Baginski E, Kelder J, Parrot K, Pericleous K. 2007. Microwave modeling and validation in food thawing applications. J Microwave Power Electromagn Energy 41(4):30–45. Torres F, Jecko B. 1997. Complete FDTD analysis of microwave heating processes in frequency-dependent and temperaturedependent media. IEEE Trans Microwave Theory Tech 45(1):108–117. VDI Wärmeatlas. 2002. 9 Auflage. Dusseldorf: Springer. Wäppling-Raaholt B, Ohlsson T. 2005. Improving the heating uniformity in microwave processing. In: H Schubert, M Regier, eds., The Microwave Processing of Foods. Cambridge: Woodhead Publishing Limited. Wilhelm MS, Satterlee LD. 1971. A 3-dimensional method for mapping microwave ovens. Microwave Energy Appl Newsl 4(5):3.

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Yakovlev VV. 2000. Comparative analysis of contemporary EM software for microwave power industry. In: Microwaves: Theory and Applications in Material Processing V. Ceramic Transactions, Vol. 111, 551–558. Westerville, OH: The American Ceramic Society. Yakovlev VV. 2006. Examination of contemporary electromagnetic software capable of modeling problems of microwave heating. In: M. Willert-Porada, ed., Advances in Microwave and Radio Frequency Processing, 178–190. Springer Verlag. Zhao H, Turner I. 1997. A generalised finite-volume time-domain algorithm for microwave heating problems on arbitrary irregular grids. 13th Annual Review of Progress in Applied Computational Electromagnetics, Monterey, CA, USA, 352–359. Zhao H, Turner IW. 2000. The use of a coupled computational model for studying the microwave heating of wood. Appl Math Model 24(3):183–197. Zhu J, Kuznetsov AV, Sandeep KP. 2007. Mathematical modeling of continuous flow microwave heating of liquids (effects of dielectric properties and design parameters). Int J Thermal Sci 46(4):328–341.

Chapter 8 Multiphysics Modeling of Ohmic Heating Peter J. Fryer, Georgina Porras-Parral, and Serafim Bakalis

8.1. Introduction Thermal processing of foods is governed by the need to achieve the required level of sterility without overprocessing the material to the point where nutritional and organoleptic quality is unacceptably diminished. Uniformity of process is key: If it were possible to give all of the food material the same process over the same time, it would be possible to produce a material that had uniform levels of sterility and product quality. In applications where hightemperature short-time (HTST) processing is possible, food products with improved quality can be produced because the reactions that lead to microbial destruction have higher activation energies than those that result in quality loss; with appropriate heat exchangers, it is possible to produce high-quality single-phase products while heating and cooling at several degrees Celsius per second. This approach is, however, limited in its applicability to multiphase foods because of the time required to conduct heat to and from the centers of particles during a process. Conduction heating limits the timescale over which processing of solids can occur as conduction is a much slower process compared with what is possible using convective mixing.

As shown in Figure 8.1, for particles >1 cm, process times are in the order of minutes rather than seconds. This limits the timescale of processing and thus the quality level of the food that can be achieved. In practice, conventionally cooked solid–liquid mixtures are limited in the process temperatures that can be used because excessive heating of the liquid may occur while the solids have still not reached the target temperature. “Ohmic”—or “direct resistance”—heating is one of the alternative thermal processes that have been proposed based on volumetric heat generation; that is, heat is generated locally within the material rather than transferred in from outside. An electric potential is applied across the food, and current flows through the three-dimensional (3D) food matrix, which will usually be a solid–liquid mixture of a sort that cannot be processed quickly by other methods. Heat is generated as a consequence of the passage of electric current through the material. Ohmic heating has a long history, much of which is discussed by de Alwis and Fryer (1990a); electrical pasteurization of milk was quite widespread in the United States in the 1930s, and a number of processes have been developed over the last 100 years. Interest in the process over the last 25 years was spurred by

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

155

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100 90

T (°C)

80 70

As already discussed in this book, there are a number of types of volumetric heating processes:

1 cm 2.5 cm 5 cm 10 cm

60 50 40 30 20 –2 10

10–1

100 101 Time (minute)

102

103

Figure 8.1. Conduction heating of spheres of diameters 1–10 cm, initially at 20°C, immersed in a fluid at 100°C, showing different thermal responses.

the development of a process by Capenhurst in the United Kingdom, later commercialized by APV Baker. This process incorporated continuous flow through pipework connecting a series of electrodes with aseptic processing and filling, with voltage gradients of the order of 10 V/cm (much less than the thousands of V/cm found in pulsed electric field applications) and was developed to enable processing of high-solids fraction foods with solids of up to 25 mm in size, at flows of up to 3,000 kg/h. Described by Parrott (1992), the heater consisted of a vertical or near-vertical tube up to 10 m in length and 75 mm in diameter, containing a series of up to seven electrode housings, each containing a single cantilever electrode across the tube. Current density is much higher in the connecting tubes than in the electrode housings, so the majority of the heating occurs in the tube. Other manufacturers include Emmepiemme SRL (Italy), who has installed more than 20 plants (manufacturer ’s information, 2007). Commercial plants for the processing of sauces, fruits, and tomatoes have been sold, although the process is not in widespread use. The original concept was to use ohmic processes to give HTST sterilization processes to solid–liquid mixtures, although the method has been used for pasteurization (Tucker et al. 2002) and proposed for blanching (Icier et al. 2006).

• Microwave processing, in which heat is generated by molecular vibrations (Knoerzer et al. 2006, 2009, and also Chapters 6 and 7). • Radio frequency heating, where heat is generated by dielectric heating, where the energy is transferred directly to the product, targeting it and not the surrounding air (Orsat and Raghavan 2005). • High-pressure thermal sterilization, in which heat is generated volumetrically by the increase in pressure up to 700 MPa—if heat losses through the vessel walls can be prevented, it is possible to reach uniform sterilization temperatures (Knoerzer et al. 2007, and Chapter 5). All of these technologies can be limited in practice by nonhomogeneities of the fields, which result in heat generation (e.g., the field patterns shown in work such as Ayappa et al. 1992; van Remmen et al. 1997; Zhang and Fryer 1993a). Without effective understanding and mathematical models, it is difficult to design and scale up the processes other than empirically. Any inhomogeneities in the delivery of the field generate nonuniform heating, affecting the rate and the distribution of microbial destruction. As with conventional thermal processing, it becomes necessary to control any process to process the most difficult part to reach, which may well lead to quality loss for the rest of the material. Some of the inhomogeneities that arise are unavoidable, as the food itself is not uniform. Nonuniformities on the cellular scale of plant or animal tissue are probably inevitable, but at a larger scale, problems may arise due to heterogeneous food composition. The process designer must also consider the possibility of introducing nonuniformity into the process as a result of incorrect design, either of the process equipment or of the formulation of the product. In slow processes, it will be possible for the temperature differences that result from these changes to equalize themselves out as a result of thermal conduction. The larger the distance between highand low-temperature regions, the longer this process will take, resulting in more restrictions on the

Chapter 8

process. Ideally, the process should offer HTST quality on particulate foods. A number of authors have addressed the issue of modeling ohmic heating of foods, first in attempts to understand the process (de Alwis et al. 1989) and then to try to develop efficient models that can be used in process and product design. This chapter reviews the types of modeling and experimental approach that have been carried out, with the aim of explaining the key design factors that would have to be taken into account in order to model the process. The approach taken is somewhat historical, reflecting the way the models were developed.

8.2. Electrical Heating of Foods: Governing Processes 8.2.1. Electrical Heating The familiar Ohm’s law describes the relationship between voltage V, current I, and resistance R in a system: V = IR

(8.1)

which leads to the equation that describes the power generation: P = I2R = V2 R

(8.2)

where P is in Watts. This is not the most convenient way of representing heat generation; it is best to consider the field gradient that results from the distribution of materials of various electrical conductivities. The voltage distribution is given by Laplace’s equation: ∇ ⋅ (κ ⋅∇V ) = 0

(8.3)

where κ is the electrical conductivity, and then the rate of heat generation is given by Q = κ E2

(8.4)

where E =∇V

(8.5) 3

Q is a volumetric heating rate (W/m ) that can be related to temperature increase in the absence of heat losses through the volumetric heat capacity (ρCp) as

Multiphysics Modeling of Ohmic Heating

ΔT =

κ E2 ρC p

157

(8.6)

In a solid, the evolution of temperature T, as a function of time t, will be a function of conventional thermal conduction within a material as well as heat generation and is governed by:

∂ (8.7) ( ρC pT ) = ∇ (λ ⋅∇T ) + Q ∂t throughout the system with appropriate boundary conditions. The problem is complicated even in the absence of fluid flow and thermal convection by the dependence of electrical conductivity on temperature, so that the voltage and thermal fields are coupled. Solution of industrially relevant problems can be computationally expensive, so most early studies simulated simplified two-dimensional (2D) cases.

8.2.2. Combination with Flow Equations Even in a notionally static system of solids in a liquid, fluid flow will occur through convection. In industrial systems where fluid flow cannot be neglected (e.g., APV heater) and where fluid is pumped past a series of electrodes, forced convection also occurs. The number and complexity of the equations to be solved consequently increases, as does computational difficulty. The form of the equations is the same as in other volumetric heating processes such as high-pressure sterilization (Knoerzer et al. 2007, and Chapter 5). For the flows in the system, the Navier–Stokes equation must be solved: ⎤ ⎡ ∂v ρ ⎢ + v ⋅∇v ⎥ = −∇P + ∇ ⋅ ( μ ⋅ v ) + ρ g (8.8) ⎦ ⎣ ∂t where v is the velocity vector and μ the viscosity. The corresponding thermal field, assuming nonisothermal flow, is then

∂ ( ρC p T ) + v ⋅∇ ( ρC p T ) = Q + ∇ ⋅ (λ∇T ) (8.9) ∂t where the source term Q is calculated from Equations 8.3 and 8.4. Complete solution of flow, thermal, and electric fields thus requires solution of Equations

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8.3, 8.4, 8.8 and 8.9, which are strongly coupled through the temperature dependence of electrical conductivity, viscosity, and other physical properties. These dependencies can be complex and nonlinear, such as the rapid changes of electrical conductivity when cells lyse (see below, and Halden et al. 1990). These equations need to be solved with appropriate initial and boundary conditions: • For the electric field, either insulation or a constant voltage across the boundaries of the system; voltage is applied at t = 0. • For temperature, usually insulated boundaries; a uniform initial temperature across the system is also commonly assumed. • For velocity, few models have attempted anything like a realistic case; models have assumed either no motion or constant-phase velocity with slip between particle and liquid; interfacial heat transfer is then controlled by a heat transfer coefficient. The difficulty of solving this set of equations means that very few workers have attempted their solution in anything like a realistic situation. Many of the simulations have concentrated on the identification of conditions for worst or best case or design rules, which can be used for ohmic formulations.

8.2.3. Validation of Basic Approach 8.2.3.1. Electrical Conductivity The first studies of ohmic heating were carried out to confirm that the heat generation was purely resistive and to identify how different parameters can be measured. Clearly, electrical conductivity is a key component of the understanding needed to generate efficient models to predict heating rates. However, it is important to note that electrical conductivity is frequency dependent; a number of mistakes were made in the development of formulations when data measured using conventional conductivity meters (which work at very high frequency) was used to build formulations for the APV ohmic heater (which used U.K. mains frequency of 50 Hz). Commonly, this gives errors in the region of 10% (Mitchell and de Alwis 1989).

The electrical conductivity for some materials can, to a first approximation, be taken to be a linear function of temperature over the range considered in the heating process: that is, κ = κ0 + kT, where κ0 and k depend on the properties of the material. This approximation can be adopted for most simple water-based materials (i.e., brine, food gels, agar, egg albumen). In general, however, the temperature– conductivity relationship for food solids or complex liquids—such as starch solutions—will be nonlinear. This in part arises because of the structure of foods: for example, when starch gelatinizes at 70°C. A survey of different materials and the ways in which electrical conductivity can change with temperature are given in Halden et al. (1990); subsequent work has given more detail (such as Palaniappan and Sastry 1991; Wang and Sastry 1993). Changes can occur due to processes such as • Phase change, such as melting of fats or gelatinization of starch-based fluids such as sauces. • Breakdown of cells; for example, when cells lyse at ca. 70–80°C, they release their contents and increase the conductivity. • Expression of air: Foods such as mushrooms need pre-treatment to remove air before they can heat at any significant rate. A large number of studies have since been done on determining the conductivity of various materials. Pre-treatments such as blanching and salt infusion soaking treatments were proposed in order to match the electrical conductivities of the liquid and solid phases (Palaniappan and Sastry 1991; Wang and Sastry 1993). The conductivity of mixtures has been shown to vary with factors such as particle size (Zareifard et al. 2003), fat content (Sarang et al. 2008; Zell et al. 2009), and structure (Wang et al. 2001). At the low frequencies used, electrolysis can sometimes be a problem; the original APV system used platinized titanium to avoid electrolysis, but this has not always been done in subsequent studies. Electrolytic reactions will absorb some of the electrical energy dissipated in the system, and can often be detected by anomalously high “specific heat capacities” of the food—power is going in but the temperature is not changing.

Chapter 8

1. Using a potential model to predict a simple distribution of the field: For some limited cases of high symmetry, the system is analytic and the heat generation in the solid and liquid can be calculated. For example, if a cylinder or sphere of solid of conductivity κs is placed in a liquid of conductivity κl, the field distribution that results is then analytic, and the ratio of the heating rates for the two can be found. For spheres, the ratio of heating rates between the liquid at infinity and the solid is RTs =

( ρC p )l (κ s + 2κ l ) ( ρC p )s 9κ sκ l

2

=

( ρC p )l ( Rκ + 2 ) ( ρC p )s 9 Rκ

2

(8.10) and then for cylinders RTc =

( ρC p )l (κ c + κ l ) ( ρC p )s 4κ cκ l

2

=

( ρC p )l ( Rκ + 1) ( ρC p )s 4 Rκ

2

(8.11) This type of analysis identifies cases where solids overheat liquids; if the thermal capacities are equal, solids will heat faster for 1 < Rκ < 4, while the maximum for the cylinder is the same heating rate at Rκ = 1. Figure 8.2 shows experimental validation from de Alwis et al. (1989); the data suggests that the same shape of curves are seen, and thus the process is truly ohmic.

159

1.5

1.0 RT

8.2.3.2. Simple Models Preliminary work used a simple batch cell (de Alwis et al. 1989) that allowed measurements at the frequencies and field strengths that are representative of the real system. This approach has been widely followed (such as by Ye et al. 2003b; Marra et al. 2009); the ohmic process works at voltage gradients that are low enough that they can readily be generated under lab conditions. The cell developed by de Alwis et al. (1989) has an open surface, and is thus prone to evaporative losses, which affect temperature; most subsequent cells have been closed and sometimes pressurized to eliminate boiling. Two sets of preliminary work were carried out by de Alwis et al. (1989) using very simplistic models that generated the confidence to build more complete ones:

Multiphysics Modeling of Ohmic Heating

0.5

0.0 0.0

1.0

2.0

3.0

4.0

RK Figure 8.2. Variation of the differential heating rates between solid spheres and the fluid showing a plot of Equation 8.10 assuming equal volumetric heat capacities in both phases and experimental data for 4 cm dough spheres (de Alwis et al. 1989).

2. Using a circuit analogy: Single planar solids with conductivities higher and lower than the liquid were placed both parallel to the field and at right angles to it, and the field applied. Typical results for the less conductive solid are shown in Figure 8.3. When the particle is essentially in parallel to the field, the electric field will be diverted around it; as a result, it will underheat the liquid. In contrast, when it is placed in series with the fluid, all (or most) of the current has to flow through the more resistive component, which overheats as a result. That particles can overheat the fluid is significant—in conventional processing, obviously, the reverse is true. One key difference between thermal and electrical heating is the variation in the appropriate conductivity. While thermal conductivity can vary from ca. 0.1–1 W/mK for foods (and up to 400 W/mK for copper), electrical conductivity can vary many orders of magnitude, from very strong conductors (metals of the order 107 S/m) to almost total insulators (of the order of 10−10 S/m for glass). Both very high- and very low-conductivity inclusions, such as metal and plastic, will in practice have a negligible heating rate, in the first case because the

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

80

80

Temperature (°C)

(b) 100

Temperature (°C)

(a) 100

60 40 20 0

0

50

100 Time (s)

150

200

60 40 20 0

0

50

100 Time (s)

150

200

Figure 8.3. Plots of heating (heavy line: solid, thin line: liquid) of potato particles (3 × 4 × 0.75 cm) with the large phase (a) parallel and (b) perpendicular to the field (de Alwis et al. 1989).

conductivity is so low that heating is low (although the current flow may be high) and in the second case because no current flows through the solid. There is thus a problem with low-conductivity inclusions in foods (such as fats and bone in meat), which will both heat slowly themselves and distort the field so that regions around them also heat slowly. Conventional thermal processes are slow enough that these differences even out through conduction. In an incorrectly designed ohmic formulation, cold spots may occur, and there will be insufficient time for thermal equilibrium to be reached.

8.3. Modeling and Validation 8.3.1. Single-Phase Behavior Most studies of ohmic heating have considered multiphase flow, in which particles can be processed rapidly. Quarini (1995) considered the behavior of single-phase fluids undergoing electrical heating while flowing in a tube, and showed that the variation of velocity across the flow would itself lead to nonuniform temperature, with the hottest regions being at the walls—the APV heater employed wall cooling to minimize this effect. The effect of adding particles is to flatten the flow profile, as discussed below.

8.3.2. Simple 3D Models and Extension to Fluid Flow The sort of effects seen in Section 8.2 led to concerns that high- or low-conductivity inclusions might be possible inside the system, resulting in effects inside the system similar to or the same as those seen in Figure 8.3. Complex “shadow effects” are possible around solids (see, e.g., Fryer et al. 1993; Kemp et al. 1999). In low-viscosity fluids, it might be expected that the temperature differences will even out through mixing, but many food fluids are of high viscosity. De Alwis and Fryer (1992) developed a 2D finite element (FE) model to simulate the behavior of slices of solid immersed in a fluid; that is, the situation of Figure 8.3. Figure 8.4 shows similar 2D simulations, in which the cold regions can be seen. This type of model does not solve Equations 8.8 and 8.9; Equations 8.3 and 8.7 are solved for the field and temperature distributions. When there is little mixing, then the temperature differences between different regions of a food mixture will be more pronounced. A simplistic attempt to model fluid mixing was made by defining an enhanced thermal conductivity, λ* (Fryer et al. 1993), and it was shown that this approach could simulate the different mixing behaviors resulting

Chapter 8

Multiphysics Modeling of Ohmic Heating

161

(b)

(a)

Time = 150 Surface: Temperature (°C) Max: 83.066

Time = 150 Surface: Temperature (°C) Max: 117.341 0.08

115

0.06

110

0.04

105

0.08

0.04

75

0.02

0.02 100

0.00

95

–0.02

65

–0.02

90

–0.04

–0.06

85

–0.06

–0.08

80

–0.08

0

70

0.00

–0.04

–0.06 –0.04 –0.02

80

0.06

0.02 0.04 0.06 Min: 78.311

60 55

–0.06 –0.04 –0.02

0

0.02 0.04 0.06 Min: 52.8

Figure 8.4. Temperature contours for (a) solids having an electrical conductivity of 2 S/m immersed in a fluid of 1 S/m, and (b) solids of electrical conductivity 4 S/m immersed in a fluid with an electrical conductivity of 2 S/m. See color insert.

from changes in viscosity. Both the simulations and the experiments suggest that the coldest spot can arise within the particle (e.g., for a particle of low conductivity) or in the bulk of the fluid (e.g., around the sides of a particle of high conductivity). The problem of modeling the commercial APV heater is that it involves a flowing of solid–liquid mixture. It is not possible to model two-phase solid– liquid flows by full solution of the equation set described above, even without solving for the electric and thermal fields. To develop a model to approximate the behavior of the system, simplifications have to be made. Zhang and Fryer (1993b, 1994) assumed that the flow consisted of a series of spheres evenly distributed in a liquid, so that a 3D “unit cell” could be modeled as representative of the whole, as in Figure 8.5. The heating rates of the two phases (i.e., Qs and Ql) were solved using an FE model to identify the heating rates as a function of solids fraction ϕ and Rκ. Assuming that the two phases travel at constant speed and are of uniform temperature, two heat balances can be written: For the solid:

(a)

(b)

Figure 8.5. Unit cell model: (a) particles on a cubic grid; (b) section of the grid that has to be modeled (from Zhang and Fryer 1993b).

dTS dx

(8.12)

φ dT ⋅ ha(TS − Tl ) + QL = vL (ρC p )L L (1− φ ) dx

(8.13)

−ha(TS − TL ) + QS = vS (ρC p )S while for the liquid:

In this case, the two equations are coupled through the interfacial heat transfer term ha, the product of

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

the heat transfer coefficient, and the interfacial area per unit volume. The changes in the temperature will change the electrical conductivities of the two. The heat generation terms were solved numerically for the unit cell in 3D for a range of Rκ and ϕ, and the data correlated as RQS =

9 Rκ

( Rκ + 1)

2

[1 − f (φ, Rκ )] + f (φ, Rκ ) Rκ

L

d

D

l Solid Mixed Fluid

Rfp

(8.14)

since the heat generation rate will vary between the first term (Eq 8.10, for ϕ = 0) and the second (for ϕ = 1). This model could be used to predict approximate temperatures for the two phases. The 3D solution of the unit cell was computationally expensive (and used a 1 GB hard drive which in 1989 cost more than £20,000!), but the model could then be run relatively efficiency using Equation 8.14 as a function. Clearly, a number of the approximations are not correct; temperatures and velocities are not uniform, nor are the particles spheres. However, it does offer an estimate of the whole system behavior. Benabderrahmane and Pain (2000) developed this type of model further, introducing thermal gradients into the particles and simulating thermal behavior, again for plug flow of the system but with slip between particles and liquids. Studies of how particle–liquid mixtures behave are discussed below.

8.3.3. Identification of Local Worst Case: Recent 3D Models Sastry (1992) and Sastry and Palaniappan (1992) have used an alternative approach to that shown above in which they approximate the 3D system using a circuit analogy and a network of resistors. This approach is probably less accurate than the Zhang and Fryer (1994) approach, as it does not solve for the field distribution, but is much less computationally expensive. The model assumes that the field lines are parallel so that the approximation shown in Figure 8.6 can be made. The problem that has been most studied is that of identification of the worst case, that is the coldest

Rfs1

Rfs2 Rfp

Figure 8.6. Circuit analogy (from Sastry and Salengke 1998). The model simplifies the system into parallel and series resistive elements.

spot in the process, against which the process must be designed. In conduction cooking, this is easy to identify, while for ohmic heating (as for other volumetric heating processes; see Chapters 4–7), it is difficult. Identification of the local worst case is obviously important as it allows the process designer to make changes to the formulation or process. A number of the commercial successes for ohmic heating have been in the processing of very uniform mixtures, such as fruit pieces in syrup, whose conductivity can be controlled and made uniform. Salengke and Sastry (2007) carried out a lengthy series of 3D simulations, both for the FE simulation of particles and the circuit model, and report computational times of several hundred seconds on a CRAY supercomputer, demonstrating the complexity of the equations and their coupling, even for relatively simple systems (linear change in conductivity with temperature was assumed). The simulations of Salengke and Sastry (2007) show, as would be expected, differences between the two models and a range of places in which the coldest spot can arise. In a flow situation, stirring of the fluid will occur, but, as shown by Kemp et al. (1999), particles in the low-field region will underheat even if they are capable of overheating the

Chapter 8

(a)

Multiphysics Modeling of Ohmic Heating

163

(b)

Time = 150 Slice: Temperature (°C)

Max: 95.833

Time = 150 Slice: Temperature (°C)

94

100

92

98

90

96

88

94

0 z

y

0

–0.05

0

86

x

92 z0

y Min: 84.54

Max: 102.253 102

x

–0.05

90 Min: 89.706 Figure 8.7. Three-dimensional solution of the Laplace equation for a fluid with electrical conductivity of 2 S/m and cubes with an edge of 1 cm having electrical conductivities of (a) 1 S/m and (b) 4 S/m. See color insert.

fluid. Further simulations are presented by Marra et al. (2009) based on a solution of the Laplace equation for the field distribution. A 3D solution of the Laplace equation is shown for solid cube particles in a fluid having an electrical conductivity of 2 and 4 S/m are shown in Figure 8.7a and b, respectively. The shadow regions of low temperature around the particle and the temperature differences between the fluid and particles for different conductivity ratios can be seen. A number of the predictions of the models have been validated using thermocouples; however, these are clearly invasive and will cause distortion of the electric field. Elegant visualization of the temperature field around particles was carried out by Ruan et al. (1999) and by Ye et al. (2003a, 2003b), in which magnetic resonance imaging (MRI) methods were used to study the temperatures in particle– liquid mixtures and gels. The method was used both to identify temperatures and to estimate particle– fluid heat transfer coefficients, which are important parameters in developing better models.

8.3.4. Validation of Ohmic Heating, and Measurements of Particle–Fluid Flows The commercial ohmic heating units and models described above involve flow of high-solids fraction mixtures, in which the carrier fluid is often viscous and non-Newtonian, through complicated 3D geometries. The food industry does not believe in models: The need is to validate them. Validating the desktop cell experiments is simple because thermocouples can be used, so long as they are insulated from the electric field. MRI experiments have also shown the effects that can arise. Full-scale monitoring of the commercial system requires measurement of the two-phase temperatures that cannot be done using static thermocouples. Indirect methods can show that particles overheat the fluid, for example, the liquid temperature in the holding tube can increase from inlet to outlet, as a result of heat transfer from the solids to the liquid (Kim et al. 1996a). Measurements to validate the flow field in an ohmic system require measurements of the distribution of

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

times that the two phases spend in the process. This can be done by some means of tagging particles and then measuring the time between entrance and exit. Palaniappan and Sizer (1997) suggest that measurements of 299 particle times is necessary for 99% confidence that the true behavior is being measured— this is a stringent requirement that many measurements have not met. The APV Baker equipment was validated as described by Kim et al. (1996a, 1996b) using particles containing spores of Bacillus stearothermophilus and chemical markers processed in a starch solution with 30–40% solids content. In some cases, higher lethality was observed at the center of the meatballs rather than near the surface, that is, confirming the predictions that particles may overheat the liquid as shown above. The authors considered that a laminar assumption (i.e., that some particles could travel twice as fast as the mean) was a suitable conservative one. More recently, Tucker et al. (2002) described the use of amylase-containing time–temperature indicators (TTIs) to validate an APV ohmic heater being used to pasteurize fruit pieces. Such methods, using microbial or chemical tracers, enable the whole effect of the process to be determined. To determine how particles behave or to model the process, it is, however, necessary to determine particle trajectories in solid–liquid mixtures, but these are difficult to measure. Some data can be obtained optically (such as Lareo et al. 1997a, 1997b), but most real fluids are opaque. Particle flows control both the residence time in the system and the particle–fluid heat transfer coefficient, and are of importance in all of the heating, holding, and cooling sections of a process. A variety of methods have been used to study particle times-of-flight—that is, the time that particles spend in a system. Liu et al. (1993) described a system based on metal detectors to study particle flows, while Tucker and Heydon (1998) described Hall effect-based sensors that can be used—these formed the basis of some of the measurements made by Kim et al. (1996a, 1996b). Marcotte et al. (2000) reported experiments on an ohmic system using an ultrasonic method as well as a colored tracer. Hall effect sensors were also used by Eliot-Godereaux et al. (2001) to find the passage time distribution of

particles in a 10 kW APV ohmic heating system; particles travelled at less than the mean velocity throughout. Recently, Sarang et al. (2009) and Tulsiyan et al. (2009) discussed the measurement of residence time distribution (RTD) using radiofrequency identification (RFID)-tagged particles, and measure flows in the ohmic heater at Ohio State University, which uses stirred vessels between sets of electrodes. Chicken particles were tagged and passed through the process, and more than 299 readings were taken. The results show that the residence time distribution was wide, with some particles passing through significantly faster than the mean residence time (52 s as opposed to 92 s), and some travelling significantly slower. Chen et al. (2010) reported measurements made on a 2,000 L/h ohmic heater made by Rossi and Catelli in Italy, in which experimental temperatures are compared with a model built using the approach of Figure 8.6; agreement is excellent in all cases. The effect of varying various process parameters was also studied and the variation of both exit temperature and output lethality mapped; as expected, fluid and particle electrical conductivity have the greatest effect. Full knowledge of the flow field would be useful to enable the heating patterns of the solids and interphase heat transfer coefficient to be identified. Particle paths are, however, more difficult to identify than residence times, as information on the Lagrangian trajectory would be needed. Fairhurst et al. (2001) and Cox et al. (2003) use positron emission particle tracking (PEPT) to measure the velocity profile and trajectories both of large particles and of small tracers, which are isokinetic with the flow. PEPT enables observation of particle trajectories even through metal pipework. The results of Fairhurst et al. (2001) show that there are cases where particle velocities faster than twice the mean can be found under conditions where plugs of particles pass through the flow rapidly. Such conditions must be avoided in commercial practice. Combination of PEPT and TTIs has been demonstrated by Mehauden et al. (2009), who used PEPT to follow TTIs within a commercial mixer vessel. This type of approach might be valuable for identifying real flow patterns in commercial systems.

Chapter 8

8.4. Further Development of Ohmic Heating and Appropriate Modeling Computational power increases with Moore’s law; what once could be done only on a workstation can now be done on PCs or laptops. Simultaneous simulation of the thermal, electric, and flow fields in the ohmic system remains very difficult, however, but simulations of the single-particle situation are now feasible using relatively inexpensive equipment and codes such as COMSOL Multiphysics™ (COMSOL AB, Stockholm, Sweden). However, some recent developments do suggest how ohmic heating might find use in the industry. The most recent works of Sarang et al. (2009) and of Tulsiyan et al. (2009) have studied chicken chow mein formulations, suggesting that production of ready-to-eat meals is possible. One possible, if exotic, route is suggested by Jun and Sastry (2007), who discuss the use of the process in future space flight for reheating reusable pouches of food and give a computational fluid dynamics (CFD) model for heating and field behavior. The work of Chen et al. (2010) on the sterilization of solid–liquid mixes, from Campbells in the United States, shows that the original design intention may still prove profitable. A novel alternative ohmic process for the processing of single-phase viscous liquids has also been developed and modeled by Ghnimi et al. (2008, 2009). This minimizes any possible problems due to fouling and velocity profile by heating a jet of fluid flowing between two electrodes, as shown in Figure 8.8. Heating will occur only within the jet as the field is concentrated into it. Assuming a uniform radial velocity profile (as there is no shear across the fluid arising from a wall) for a jet of initial velocity u0 and radius R, the authors derive a model first for the velocity and radius of the jet with distance x from the nozzle using Bernoulli’s equation 1/ 2

u( x ) ⎛ 2 g ⎞ r ( x ) ⎛ 2 g ⎞ = ⎜ 1 + 2 x⎟ ; = ⎜ 1 + 2 x⎟ ⎝ ⎝ u0 u0 ⎠ R u0 ⎠

−1 / 4

(8.15)

and then obtain a model for the temperature rise under conditions where the electrical conductivity is linear with temperature by integrating the heat balance along the jet

Multiphysics Modeling of Ohmic Heating

165

Inlet mass flow 5 2 9 1

6 I

4

8

7

v

3

Outlet mass flow Figure 8.8. Liquid jet ohmic heater, showing as follows: 1: fluid jet falling between electrodes; 2: stainless steel electrode connected to the phase; 3: stainless steel electrode connected to the mass; 4: electrical isolation using PEEK bars with nuts and bolts; 5: radar sensor; 6: high-frequency electromagnetic wave generated by the radar sensor; 7: glass tube between electrodes; 8: injection of the nitrogen; 9: electrical insulation; I: alternate current in the jet; V: voltage applied at the boundaries of the electrodes (from Ghnimi et al. 2008).

 p dT = κ 0 (1 + kt )E 2 ⋅ π [r ( x )]2 dx mC

(8.16)

This model is shown to agree well with the experimental results. The potential for this type of process is that it may overcome some of the problems associated with velocity and thermal profiles across pipework. Electric field processing may, however, also be useful in extraction and infusion. That electrical processing can enhance mass transport between solids and liquids was identified by Halden et al. (1990) in experiments on beetroot, in which the purple betanin dye moved much more quickly into the solution under ohmic treatment than under conventional processes. Schreier et al. (1993) quantified this effect by measuring the mass flux of betanin dye between a solid foodstuff (beetroot) and surrounding liquid under conventional and electrical heating at 100°C. Mass transfer of the dye under the influence of a 50 Hz applied electric field increased linearly with

166

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

increasing electric field. This work was repeated (Lima et al. 2001) at uniform temperatures below 100°C, and enhancement was found at 42 and 58°C but not at 72°C. This result was interpreted in terms of differing electrical conductivities due to the different processing methods. Kulshrestha and Sastry (2003) again studied beetroot and found that increasing the field strength increased the efflux of dye. It was proposed that the effect was due to electroporation of the material. Kemp and Fryer (2007) demonstrated that diffusion into solids could also be enhanced using electric fields. Alternating electric fields have been claimed to enhance a number of processes. Sastry and Salengke (1998) outlined a number of potential applications for alternating electric fields, described as “moderate electric field” processing by Sensoy and Sastry (2004). Juice yields of ohmically pre-treated samples were enhanced against conventional treatments, and the drying rates of fruit and vegetable tissue samples were enhanced against either conventional (steam) or microwave heating. Lima and Sastry (1999) studied the effect of electrical treatment on juice yield and suggested that electrical pre-treatment was useful. Zhong and Lima (2003) showed that drying of sweet potato was enhanced by ohmic pre-treatment, and extraction of rice bran oil is also increased (Lakkakula et al. 2004). If the treated material has a higher permeability, then it may be possible to increase extraction rates (Kulshrestha and Sastry 2003, 2010). In terms of modeling of the process, it is now easily possible to simulate complex particle–liquid heating patterns; however, the problem of solving for the flow patterns at the same time is much more complex. Efficient models for two-phase solid– liquid flows are not available as yet; some progress has been made in using discrete element methods in gas–solid flows and in combining CFD and discrete element method (DEM) models (such as Curtis and van Wachem 2004; Tsuji et al. 2008; Guo et al. 2010), and good data for processes such as heat transfer have been developed (Garic´-Grulovic´ et al. 2009). This type of modeling offers hope that the computational and theoretical complexity of modeling solid–liquid behavior can be addressed over the next decade.

8.5. Conclusions Ohmic heating is a method for food processing in which electric current is passed through the food and the material then heats as a result of its inherent electrical resistance. It is possible to see fluids and particles be heated at comparable rates. Preliminary experiments showed that heating is purely resistive, so that the heat generation distribution in the system can be modeled if the distribution of electrical conductivity is known. However, to simulate fluid flow as well as heating involves simultaneous solution of models for the flow, thermal, and electrical fields, with the three being strongly coupled through the temperature dependence of electrical conductivity and viscosity. A series of models have been built for the process that demonstrates various factors that have been shown in practice, such as particle overheating. The difficulty in the process arises when the fluids (or sections of them) have electrical conductivities significantly different from one another: Under those conditions, the models show that local “shadow regions” of low electric field strength can arise, resulting in large differences in temperature between the liquid and the solid. Within the liquid, these changes in temperature can be minimized by improving convective mixing, an effect that is more pronounced for low-viscosity fluids such as water. However, commercial food processes commonly use high-viscosity carrier fluids (such as starch) in which convective processes will be significantly slower. Three-dimensional simulations of the heating patterns that occur have been developed, but only very simple models for the flows—in which phase volumes are constant—have been attempted. The original vision for the APV Baker process to rapidly sterilize heterogeneous mixtures such as ready-to-eat meal formulations has not been realized by ohmic heating; most commercial products processed using this system are homogeneous. However, there are now a number of manufacturers selling ohmic equipment successfully. Alternative uses such as the use of electric fields to enhance mass transfer may also be reaching commercial activity.

Chapter 8

Full simulation of the flows and heating rates found in two-phase systems will require developments both in computational size and in solid–liquid flow modeling.

Notation Cp E g ha I k P P p Q Ql Qs R R RQs RTc RTs Rκ T t Tl Ts u0 V v vl vs κ κ0 κc κl κs λ λ* μ

Heat capacity Electrical field strength Gravity Interfacial heat transfer Current Temperature coefficient Power Pressure Pressure Volumetric heating rate Heating rate for liquid Heating rate for solid Resistance Radius Ratio of heat generation Ratio of heating rates for cylinder Ratio of heating rates for sphere Ratio of electrical conductivities Temperature Time Temperature liquid Temperature solid Initial velocity Voltage Velocity vector Velocity vector for liquid Velocity vector for solid Electrical conductivity Electrical conductivity at reference temperature Electrical conductivity of solid (cylinder) Electrical conductivity of liquid Electrical conductivity of solid (sphere) Thermal conductivity Enhanced thermal conductivity Viscosity

J/kgK V/m m/s2 A W Pa Pa W/m3 W/m3 W/m3 Ω m

K s K K m/s V

S/m S/m S/m S/m S/m W/mK W/mK Pa/s

ρ ϕ x m D L d RfP RfS1 RfS2

Multiphysics Modeling of Ohmic Heating

Density Solids fraction Distance from the nozzle Mass flow rate Diameter Length Diameter of solid Parallel resistive element Series resistive element 1 Series resistive element 2

167

kg/m3 m kg/s m m m Ω Ω Ω

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Chapter 8

Sastry SK. 1992. A model for heating of liquid-particle mixtures in a continuous flow ohmic heater. J Food Process Eng 15:263–278. Sastry SK, Palaniappan S. 1992. Mathematical modeling and experimental studies on ohmic heating of liquid-particle mixtures in a static heater. J Food Process Eng 15:241–261. Sastry SK, Salengke S. 1998. Ohmic heating of solid-liquid mixtures: A comparison of mathematical models under worst-case heating conditions. J Food Process Eng 21:441–458. Schreier PJR, Reid DG, Fryer PJ. 1993. Enhanced diffusion during the electrical heating of foods. Int J Food Sci Technol 28:249–260. Sensoy I, Sastry SK. 2004. Extraction using moderate electric fields. Food Eng Phys Prop 69(1):7–13. Tsuji T, Yabumoto K, Tanaka T. 2008. Spontaneous structures in three-dimensional bubbling gas-fluidized bed by parallel DEM–CFD coupling simulation. Powder Technol 184: 132–140. Tucker GS, Heydon C. 1998. Food particle residence time measurement for the design of commercial tubular heat exchangers suitable for processing suspensions of solids in liquids. Food Bioproducts Process 76:208–216. Tucker GS, Lambourne T, Adams JB, Lach A. 2002. Application of a biochemical time–temperature integrator to estimate pasteurisation values in continuous food processes. Innov Food Sci Emerg Technol 3:165–174. Tulsiyan P, Sarang S, Sastry SK. 2009. Measurement of residence time distribution of a multicomponent system inside an ohmic heater using radio frequency identification. J Food Eng 93:313–317.

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Wang CS, Kuo SZ, Kuo-Huang LL, Wu JSB. 2001. Effect of tissue infrastructure on electric conductance of vegetable stems. J Food Sci 66:284–288. Wang WC, Sastry SK. 1993. Salt diffusion into vegetable tissue as a pretreatment for ohmic heating: Electrical conductivity profiles and vacuum infusion studies. J Food Eng 20:299–309. Ye X, Ruan R, Chen P, Chang K, Ning K, Taub I, Doona C. 2003a. Accurate and fast temperature mapping during ohmic heating using PRF shift MRI thermometry. J Food Eng 59:143–150. Ye X, Ruan R, Chen P, Doona C, Taub I. 2003b. MRI temperature mapping and determination of the liquid–particulate heat transfer coefficient in an ohmically heated food system. J Food Sci 68:1341–1346. Zareifard MR, Ramaswamy HS, Trigui M, Marcotte M. 2003. Ohmic heating behavior and electrical conductivity of twophase food systems. Innov Food Sci Emerg Technol 4:45–55. Zell M, Lyng JG, Cronin DA, Morgan DJ. 2009. Ohmic heating of meats: Electrical conductivities of whole meats and processed meat ingredients. Meat Sci 83:563–570. Zhang L, Fryer PJ. 1993a. Electrical resistance heating of foods. Trends Food Sci Technol 4:364–369. Zhang L, Fryer PJ. 1993b. Models for the electrical heating of solid-liquid food mixtures. Chem Eng Sci 48:633–643. Zhang L, Fryer PJ. 1994. Food processing by electrical heating; the sensitivity of product sterility and quality to process parameters. AIChE J 40:888–898. Zhong T, Lima M. 2003. The effect of ohmic heating on vacuum drying rate of sweet potato tissue. Bioresour Technol 87(3):215–220.

Chapter 9 Basics for Modeling of Pulsed Electric Field Processing of Foods Nicolás Meneses, Henry Jaeger, and Dietrich Knorr

9.1. Introduction The combination of empirical models with numerical simulations has been already implemented and successfully used for the characterization of engineering processes. Based on the basic physical phenomena, such as the conservation of mass, momentum, and energy, several theoretical models have been developed, for example, the Navier– Stokes equations, and have been used to simulate complex systems and processes (Ferziger and Peric 2002). The joule heating, a theoretical model to describe the temperature changes as a function of the electric field strength in conductive media, has been used to simulate pulsed electric field (PEF) processes for food material, among other numerous applications. The resulting temperature increase in the media during PEF processing as a function of the electric field strength and the treatment time was also validated by experimental results showing good prediction accuracy (Fiala et al. 2001; Lindgren et al. 2002; Van den Bosch et al. 2003; Gerlach et al. 2008; Jaeger et al. 2009). Mechanistic models for describing the microbial inactivation phenomena, based on electroporation and electrical breakdown, have been developed

(Zimmermann et al. 1974, 1976; Zimmermann 1986; Tsong 1990), but not yet used for describing the microbial population after a PEF treatment (Lelieveld et al. 2007). On the other hand, only empirical models have been developed to describe the inactivation of microorganisms and enzymes, and the degradation of food compounds in dependency on the PEF process parameters. To date, the most frequently used parameters to describe the PEF processing of foods are the treatment time and the electric field strength. The electric field strength and the treatment time of any given PEF process can be predicted, and, therefore, the combination of empirical with theoretical models becomes important for this purpose. Using experimentally obtained models (i.e., microbial inactivation kinetics) and theoretical models (i.e., mass conservation, joule heating, charge conservation, and so on) and implementing both into a numerical simulation allows the study of the microbial inactivation process not only after the treatment, but also during PEF processing. These concepts can lead to an optimization of the PEF process and assist in the design of an appropriate treatment chamber, with a homogeneous electric field strength distribution, treatment time, temperature, and microbial inactivation.

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

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This chapter is focusing on four key aspects relevant to PEF modeling and its implementation into numerical simulations. The first part is related to the Multiphysics nature of PEF processing, which is described by governing equations such as fluid dynamics, heat transfer, and electrical phenomena. The second part focuses on mathematical equations to describe the PEF process as a continuous system for liquid food pasteurization. Included in the mathematical equations are process parameters such as energy input, pulse energy, pulse shape, and treatment time. An overview of the main treatment chamber designs used for PEF treatment of food together with their relevant electrical properties is also given. Physical properties such as rheological properties, thermal properties, and electrical properties will be discussed in the following section, which points out the interaction of the properties mentioned with fluid flow, heat transfer, and electrical treatment of foods. The last part is focusing on PEF application as pasteurization technology and describes the main kinetic models suitable for further simulation considering microbial and enzyme inactivation.

9.2. Governing Equations for Multiphysics Simulation of PEF Processing PEF processing is a Multiphysics phenomenon. The relevant variables are changing not only with time but also with space; hence, it is necessary to describe the phenomena by partial differential equations (Van Boekel 2009). The three main equations to describe a thermohydrodynamics model are the mass conservation, the conservation of momentum, and the energy conservation equations (Ferziger and Peric 2002; see also other chapters in this book, e.g., Chapters 4, 5, and 11).

∂ρ + ∇ ⋅ ( ρu ) = 0 ∂t

(9.1)

Conservation of momentum:

ρ

∂u T − ∇ ⋅ η ⋅ ∇u + (∇u ) + ρ (u ⋅∇ ) u + ∇p = F ∂t (9.2)

(

)

where ρ denotes the density of the fluid (kg/m3), u represents the average velocity (m/s), η is the dynamic viscosity (kg/m/s), p denotes the pressure (Pa), and F represents the volumetric force vector (m/kg2/s).

9.2.2. Heat Transfer Balance The general equations for an energy balance, which considers heat transfer through convection and conduction is:

ρc p

∂T + ∇ ( −λ ⋅∇T + ρc p Tu ) = Qs ∂t

(9.3)

where cp denotes the specific heat capacity (m2/s2/K), T is temperature (K), λ is the thermal conductivity (kg m/s3/K), ρ is the density (kg m−3), u is the velocity vector (m s−1), and Qs is a sink or source term (kg/m/s−3 or W/m3). If the velocity u is equal to zero, the general heat transfer equation turns into Equation 9.4, which governs pure conductive heat transfer in a solid or liquid:

ρc p

∂T + ∇ ( −λ ⋅∇T ) = Qs ∂t

(9.4)

In addition to the thermo-hydrodynamics equations, the Poisson’s equations are needed to extend the model to an electro-thermo-hydrodynamic problem, in order to consider the potential differential within the treatment chamber.

9.2.1. Conservation of Mass and Momentum

9.2.3. Poisson’s Equations

For incompressible Newtonian liquids, the mass and momentum conservation can be written in a spatiotemporal form as:

Assuming no generation of any electromagnetic forces, the relation between the electric potential and the electric field is given by:

Chapter 9

E = −∇ ⋅ V

Basics for Modeling of Pulsed Electric Field Processing of Foods

(9.5)

where E is the electric field strength (V/m). The equations to solve the electrical potential are based on charge conservation: −∇ ⋅ (σ∇V − J e ) = 0

(9.6)

where σ is the electrical conductivity (S/m), V is the electrical potential (V), and J e (A/m2) is the current density.

9.2.4. Coupling Heat Transfer and Poisson’s Equations During PEF processing, the temperature increases due to resistive (ohmic) heating from the electric current. The generated resistive heat Qs is proportional to the square of the magnitude of the electric current density J. Current density, in turn, is proportional to the electric field, which equals the negative of the gradient of the potential V: Qs ∝ J

2

(9.7)

The coefficient of proportionality is the electric resistance ρe = 1/σ, which is also the reciprocal of the temperature-dependent electrical conductivity σ = σ(T). Combining these facts gives the fully coupled relation: 1 2 2 Qs = J = σ E = σ ∇V σ

2

(9.8)

Thus, the conservation of charge, which describes the electric field strength behavior, can be coupled to the energy Equation 9.4 (Gerlach et al. 2008; see also Chapter 11 of this book). This set of coupled differential equations represents the electric-thermohydrodynamics model, suitable to simulate any continuous PEF processing as well as a batch process (when the velocity component is set to zero). These equations require the thermophysical properties of the food as input parameters. The quality of the process simulation depends on the accuracy of these models in describing the variables as a function of time, temperature, and electric field strength, as well as on given boundary conditions. Models of food properties as a function of temperature and time are

173

available in the literature, but data of food properties as a function of the electric field—in the range of microseconds—do not exist. Since the electric field is applied as pulses of the duration in the range of microseconds, and the time between the pulses is relatively long (milliseconds to seconds) in comparison to the duration of the pulse, any change in viscosity, density, heat capacity, and thermal conductivity of the food affected by the electric field can be neglected. This is not the case for the change of other food compounds or microorganisms during the electrical pulse, since certain structures are irreversibly affected. The effect of the electric field strength, treatment time (in the range of microseconds), and electrical energy on microorganisms has been reported since the early beginning of PEF research (Zimmermann et al. 1974). In order to consider the impact of process parameters, such as temperature, electric field strength, and treatment time on food compounds and/or microorganisms, the use of a new balance approach considering also diffusion of particles is needed. The general balance considering diffusion and reaction rate is: ∂ci + (u∇ ) ci = ∇ ( Di ∇ci ) + Qs ∂t

(9.9)

where ci (mol/m3) is the species or substance concentration, Di (mol/m3/s) denotes the diffusion coefficient, and Qs (W m3) represents a source or sink, typically due to chemical reactions; arbitrary kinetic expressions of the involved species (ci) can be introduced. When assuming ci being the concentration of any food compound and/or microorganism, and expressing it in terms of its relative activity RA, Equation 9.9 turns to: ∂RAi + (u∇ ) RAi = ∇ ( Di ∇RAi ) + Qs (9.10) ∂t In this case, Qs represents a source affecting the relative compounds activity as a function of process parameters, such as electric field strength, treatment time, residence time, and temperature, which can be determined experimentally. With this governing

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equation, any food compound or microbial inactivation kinetic can be coupled with the electro-thermohydrodynamic model and thus can be simulated. Equations 9.1 to 9.10 are a set of coupled partial differential equations, which can be used to simulate any PEF process (batch and continuous processing), independent and dependent on temperature or time. The use of the correct kinetic inactivation model as dependent on PEF process parameters, or the adequate thermophysical property as dependent on temperature is discussed in the next sections.

9.3. PEF System and Modeling of PEF Processing Parameters A basic circuit for the generation of PEFs including the treatment chamber for their application to the food material is shown in Figure 9.1. The food in the treatment chamber can be modeled as a resistor Rc in parallel to a capacitor C. The resistance Rc is inversely proportional to the electrical conductivity of the food, whereas the capacitance C depends on the dielectric permittivity of the food. Both electrical characteristics can be affected by the geometry of the treatment chamber, but due to the short pulse durations, the capacitor properties can be neglected. The electrical conductivity of the media together with the electrical resistance as affected by the electrode configuration determines the total resistance of the treatment chamber (Loeffler 2006). The ratio between the total resistance of the treatment chamber and the rest of the electrical circuit determines the voltage drop and the resulting peak

voltage across the electrodes. The use of a treatment chamber with a high resistance results in a more effective voltage division and higher electric field strength in the treatment chamber. Furthermore, the microbial inactivation is generally enhanced when the medium has a high electric resistivity (Huelsheger et al. 1981; Mizuno and Hori 1988).

9.3.1. Pulse Shape and Pulse Width When the high-voltage switch (Figure 9.1) is set to position A (open), the capacitors will be charged. Then, as the switch is set to position B (closed), the capacitors will be discharged and a current will flow leading to the induction of an electric pulse in the treatment chamber. Two different pulse shapes are commonly used: square wave pulses and exponential decay pulses. Additionally, these pulses can be applied in a monoor bipolar manner (Beveridge et al. 2002). A square wave pulse can be mathematically represented as: V (t , τ ) = V0 ⋅ ( H1 − H 2 ) t<0 ⎧0 ⎪ H1 = ⎨1 2 t = 0 ⎪1 t>0 ⎩

t<0 ⎧0 ⎪ H 2 = ⎨1 2 t = τ ⎪1 t >τ ⎩

(9.12)

The pulse width of a square wave pulse is defined as the period of time in which the maximal voltage is maintained. Square wave pulses can be generated by using a switch with on–off function or an adequate network (Martín et al. 1994). In the case of exponential decay pulses, the voltage decay depends on the capacity of the capacitor(s) and the resistance, in accordance to the following equation (Ho and Mittal 2000): V (t ) = V0 ⋅ e

Figure 9.1. Electrical circuit for the generation of highvoltage pulses.

(9.11)

−

t R⋅C

(9.13)

where V(t) (V) is the voltage across the resistor at a given time t (s), V0 (V) is the initial voltage, R (Ω) is the resistance, and C (F) is the capacitance. The pulse width for exponential decay pulses is defined at the point where the voltage has decreased to 1/e ≈ 36.8% of its initial value, and it depends on

Chapter 9

Basics for Modeling of Pulsed Electric Field Processing of Foods

175

the resistance and capacitance as shown by Equation 9.14 (Barsotti et al. 1999):

τ = R ⋅C

(9.14)

The resistance is normally related to the resistance of the food and the treatment chamber only, since the resistance of the electrical circuit is comparatively low and can therefore be neglected. For a direct current and a treatment chamber with a uniform cross-section, Equation 9.15 is valid to describe the electrical resistance: R=

1 d ⋅ A σ (T )

(9.15)

where d (m) is the distance between the electrodes, A (m2) is the cross section where the electrical current flows and σ (S m−1) is the electrical conductivity of the food as a function of the temperature. Alternatively, Equation 9.16 can be used for any treatment chamber configuration: R=

V I (T , σ )

Figure 9.2. Voltage drop in an exponential decay pulse and pulse width determined at 36.9% of the peak voltage as a function of the temperature (calculation according to Eqs. 9.13 and 9.15).

(9.16)

where V (V) is the applied voltage and I (A) is the electrical current as a function of the temperature and the conductivity. This equation can be preferably used to estimate the treatment chamber resistance in cases where the parameters V and I are directly measured. According to Equation 9.16, the resistance is a function of the temperature and the electrical conductivity of the food. Therefore, different treatment times will result during the PEF treatment with exponential decay pulses, since the temperature is either inhomogeneous in the treatment chamber or not constant during the treatment itself. Considering a pulse generator for exponential decay pulses and a treatment chamber with parallel electrode configuration with an interelectrode gap of 3 mm and an electrode cross section of 0.0015 m2, Equation 9.13 can be used to calculate the voltage decay as a function of time, as shown in Figure 9.2 (C = 19.1 nF and V0 = 8 kV). The resulting resistance as a function of the temperature can be estimated using Equation 9.15 with σ as a linear function of the temperature. As the

pulse width is a function of the resistance, different values for the pulse width are expected to occur at different process temperatures. The higher the increase in the media conductivity with temperature, the higher the pulse width differences will be. It is clear from Figure 9.2 that the temperature is an important factor to be considered during PEF treatments in order to allow the exact characterization of the treatment parameters. To date, the changes of the treatment time as a function of the temperature have not been considered for the performance of PEF experiments using exponential decay pulses. For studies investigating the effect of PEF treatment and temperature, it is not recommendable to use exponential decay pulses. In this case the use of square wave pulses is more appropriate since the pulse width is controlled and constant with small variations in rise and fall time, depending on the media conductivity. There are no general equations relating the pulse width and fall time to media properties and temperature. The exact pulse shape should be obtained by measurements and approximated mathematically (Figure 9.3).

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

field strength intensity can be mainly attributed to differences in the electrode surface area. The electric field strength in a coaxial treatment chamber can be calculated according to Equation 9.18. E=

Figure 9.3. Ideal square wave pulse (solid line) and real pulse shape obtained by voltage measurement (dashed line) in a medium with an electrical conductivity of 4.7 mS/cm at 40°C.

V ⎛r ⎞ r ⋅ ln ⎜ 0 ⎟ ⎝ ri ⎠

This equation shows that the electric field strength is decreasing as a function of the radius of the treatment chamber, where r is the radius at which E is measured, r0 and ri are the radii of the inner and outer electrode surface, respectively (Bushnell et al. 1993). The uniformity of the electric field strength is improved when the difference between the inner and outer radii of the electrode surfaces is negligible compared with the inner radius (Eq. 9.19).

(r0 − ri ) << ri The equations describing the pulse shape are a function of time and can be introduced into the numerical simulation as boundary conditions for the variable voltage (V ) when a time-dependent simulation of PEF processing is desired.

9.3.2. Voltage and Electric Field Strength The electric field strength depends on the voltage, the geometry of the electrodes, and the distance between them (Zhang et al. 1995). In addition, the media conductivity and the temperature have a significant impact on the electric field strength. Considering a fluid-cross treatment chamber of parallel plates, the electric field strength can be mathematically expressed as: E=

V d

(9.17)

where E is the electric field strength (V/m), V is the applied voltage (V), and d is the interelectrode gap (m). This equation is valid when a homogeneous electric field strength distribution is assumed. In the case of a coaxial treatment chamber geometry, the electric field strength is not uniform and depends on the location where it is measured. Differences in the

(9.18)

(9.19)

The electric field strength in a co-field treatment chamber is also not homogeneous, and a different method for the calculation is necessary. According to Gerlach et al. (2008), the calculation of a conversion factor is necessary for an estimation of the electric field strength. This conversion factor relates the applied initial voltage to the resulting electric field strength. A possible way for its calculation is the estimation of an average electric field strength based on the applied voltage. As the electric field strength depends on the treatment chamber dimension and design, a conversion factor should be defined for every treatment chamber configuration. This topic is discussed in the following chapter (Chapter 10 of this book). In addition, the voltage delivered by the pulse generator is always higher than the voltage in the treatment chamber due to losses between the capacitor(s) and the treatment chamber. The voltage should be measured as close as possible to the treatment chamber in order to obtain an accurate estimation of the electric field strength (Pataro 2004).

9.3.3. Treatment Time and Residence Time The treatment time is an important parameter that should be controlled in order to avoid over- or

Chapter 9

Basics for Modeling of Pulsed Electric Field Processing of Foods

underprocessing. The treatment time (tt) is defined as the product of the number of pulses (n) and the pulse duration ( τ ): tt = n ⋅ τ

(9.20)

The pulse number is a function of the applied frequency and residence time of the product in the treatment chamber. n = f ⋅ tres

(9.21)

where f (Hz) is the frequency, which is defined as the number of pulses delivered per second, and tres (s) is the residence time. In a batch treatment, the residence time corresponds to the holding time of the sample in the treatment for the time period of the pulse application. In the case of a continuous treatment, the residence time is a function of the treatment chamber length (L) and the fluid velocity (u), and can be estimated as shown in Equation 9.22: tres =

L u

(9.22)

where L (m) is the length of the treatment chamber and u is the fluid velocity (m/s). According to Holland and Bragg (1995), a basic estimation of the fluid velocity can be calculated according Equation 9.23, considering a laminar flow of a Newtonian fluid in a pipe of circular cross section: ⎛ r2 ⎞ u = 2 ⋅ uave ⎜ 1 − 2 ⎟ ⎝ ri ⎠ uave =

Q (π ⋅ ri2 )

(9.23) (9.24)

where uave (m/s) is the volumetric average velocity, Q (m3/s) is the flow rate, and ri (m) is the internal radius. When the flow becomes turbulent, an approximation of the velocity profile can be given exemplarily by the one-seventh power velocity distribution equation. r⎞ ⎛ u = 2 ⋅ uave ⎜ 1 − ⎟ ⎝ ri ⎠

1/ 7

(9.25)

177

These equations do not consider the thermophysical properties as affected by the temperature, and Equation 9.25 is not valid in the viscous sublayer of the turbulent boundary layer. For a general approximation of the velocity profile and the calculation of an average treatment time, Equations 9.23 and 9.25 can be used. When accurate estimation of the treatment time in continuous systems is desired, more complex equations are necessary. Coupling Equations 9.14 and 9.20–9.24, the treatment time can be calculated according to Equation 9.26 for a continuous PEF treatment with exponential decay pulses in a co-field treatment chamber (Figure 9.4) where the insulator radius equals the electrode radius. t t (r ) =

f ⋅τ ⋅ L Q ⎛ r2 ⎞ 2⋅ ⋅ 1− (π ⋅ ri2 ) ⎜⎝ ri2 ⎟⎠

(9.26)

As the velocity profile depends strongly on the radial coordinate, the treatment time is also not homogeneous. It is evident that by performing continuous treatments, the treatment time should be considered and an average value and the standard deviation should be calculated. A compromise between the maximum treatment time (pipe walls) and the shortest treatment time (center of the pipe) should be found in order to guarantee a minimum treatment time necessary for a desired level of microbial inactivation. When performing inactivation kinetic studies that depend on the treatment time, it has to be taken into account that the high deviation of the residence time occurring in the above described continuous systems will not allow homogeneous conditions and limits the exact characterization of the applied treatment conditions (Meneses et al. 2010).

9.3.4. Specific Energy and Energy per Individual Pulse The energy input is a useful dose parameter for describing a PEF process since a comparison in terms of energy consumption with traditional processes

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Figure 9.4. Treatment time and velocity profile for a laminar fluid flow in a continuous colinear treatment chamber at an average flow rate of 5 L/h.

can be made (Puértolas et al. 2009). The specific energy is related to the pulse energy, the repetition rate of the pulses, and the mass flow rate. A basic definition of the specific energy input during continuous PEF treatments can be performed according to Equation 9.27: Wspec = Wpulse

f m

(9.27)

where Wspec (kJ/kg) is the specific energy input, Wpulse (J/pulse) is the energy per pulse, f is the frequency (Hz), and m is the mass flow rate (kg/s). For batch treatments, the term f m can be easily replaced by n m , where n is the pulse number and m (kg) is the mass of the sample in the treatment chamber. The pulse energy represents the electrical energy delivered by each electrical pulse during the PEF treatment. Depending on the pulse shape, the pulse energy can be estimated by different equations. For exponential decay pulses, it can be calculated for the discharge of the capacitor bank: Wpulse _ exp =

1 ⋅C ⋅V 2 2

(9.28)

where C is the capacity of the capacitor(s) and V is the charging voltage.

For square wave shaped pulses, the pulse energy can be determined by the current intensity, the voltage, and the pulse width according to: Wpulse _ cuad = V ⋅ I ⋅ τ

(9.29)

where I (A) is the current flowing through the media during the pulse width τ (s) as a result of the applied voltage V (V). Based on media conductivity σ and the measured electric field strength E, the specific energy input for exponential decay and rectangular pulses can be calculated by Equation 9.30 according to Heinz et al. (2003): ∞

Wspec = f

1 2 σ (T ) ⋅ E (t ) dt m

∫

(9.30)

0

Since the energy input is a function of the voltage (Eqs. 9.28 and 9.29), it is difficult to study the single effect of the electric field strength on food components or microorganisms because related changes in the energy input may affect the electric field strength impact. Considering a parallel plate treatment chamber with an interelectrode gap of 2 mm and a resistance of 150 Ohm, operating with square wave pulses of 3 μs at a voltage level between 3 and 9 kV, the resulting electric field strength will be 15 and 45 kV/cm,

Chapter 9

Basics for Modeling of Pulsed Electric Field Processing of Foods

respectively. According to Equation 9.16, the resulting electrical current is 20 and 60 Amperes, respectively. Using Equation 9.29, the following pulse energy values can be calculated: 0.18 J/pulse for the case of 15 kV/cm and 1.62 J/pulse for the case of 45 kV/cm, which means a difference of 800%. The same difference can be found by considering Equation 9.28 for exponential decay pulses. Energy differences of about 100% have been reported from microbial inactivation experiments using PEF by Saldaña et al. (2009). It is clear that the characterization of a PEF treatment by the electric field strength and the number of pulses or the treatment time without the consideration of the parameters related to the energy input (pulse energy and total specific energy input) is incomplete. In order to compare results of different research groups, different system configurations, and even different technologies, the total specific energy input can be used as a meaningful dose parameter in combination with the treatment parameters discussed earlier (Heinz et al. 2003; Jaeger et al. 2009; Puértolas et al. 2009). An additional consequence of the applied energy is the temperature increase in the sample during the PEF treatment; the higher the total energy input, the higher the temperature increase is. Not considering the pulse energy as well as the corresponding temperature increase will lead to a misinterpretation of microbial inactivation results at different electric field strength levels. The consequence of the energy input and temperature interrelations is the necessity of determining kinetic parameters depending on electric field strength at constant levels of energy input and temperature. Empirical models for microbial inactivation available in the literature with control of the energy input and the temperature are limited (Sensoy et al. 1997; Schrive et al. 2006; Fox et al. 2008; Jaeger et al. 2010). Two possibilities for controlling the energy input and temperature are: • Implementation of an additional inductance and resistance in the electrical circuit to control the pulse width and the pulse energy.

179

• Application of active cooling of the electrodes to prevent or control the temperature increase and to avoid changes in the electrical conductivity and pulse energy.

9.3.5. Treatment Temperature The treatment temperature is directly related to the electrical energy delivered into the treatment chamber, a phenomenon known as joule heating. The electrical energy (Qs) can be expressed as a function of the electric field strength (E) and the electrical conductivity (σ): Qs = σ (T ) ⋅ E ( x, y, z )

2

(9.31)

Analytical solutions for the temperature estimation have been proposed (Van den Bosch et al. 2003). Based on Equation 9.31 and a simple heat balance, the following expression can be deduced for the average temperature increase: ΔTave =

Wspec cp

(9.32)

This equation is the simplest one, which relates the average temperature increase to the energy input and heat capacity. This equation is valid when the application of a non-pulsating voltage is assumed. The temperature increase in a continuous treatment chamber can be calculated based on the energy delivered per pulse (Wpulse), the pulse frequency ( f ), the mass flow rate ( m ), and the specific heat capacity (cp) according to Equation 9.33: Wpulse ⋅ f = m ⋅ c p ⋅ (Tout − Tin )

(9.33)

The temperature occurring at a certain time t after the liquid has entered the treatment chamber (0 < t ≤ tres with tres referring to the total residence time in the treatment chamber) can be calculated according to Equation 9.34 (Jaeger et al. 2010): Tout = Tin +

Wpulse ⋅ f ⋅ tres m

(9.34)

As the temperature depends on the residence time, and the residence time is a function of the fluid velocity, strong variations can be found along the radial coordinate of the treatment chamber. Equation

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

9.34 is not valid for the thin layer of slow-moving flow close to the walls. An approximation of the temperature close to the wall and on the wall itself can be found in Lelieveld et al. (2007). Basically, the assumption of a pure conductive heat transport in the zone close to the wall is valid.

9.4. Relevant Food Engineering Properties for PEF Simulations Engineering properties of foods are relevant for the design and operation of equipment employed in food unit operations. Besides the application of the electric field itself, the fluid flow and heat transfer are the most important phenomena that are directly connected to the continuous PEF treatment of foods. The pressure or temperature differences as driving forces as well as the application of a difference in the electrical potential across the food material and the formation of an electric field are the basic principles of these unit operations.

9.4.1. Thermophysical Properties of Foods The PEF treatment is considered a nonthermal alternative to traditional pasteurization of liquid foods (Lelieveld et al. 2007) since the inactivation of microorganisms is based on the electromechanical mechanism of electroporation (Crowley 1973; Zimmermann et al. 1974), which allows food processing at moderate temperatures. However, when increasing the temperature an increased fluidity of the cell membrane leads to a reduced membrane stability and facilitates the electroporation process (Stanley 1991; Kanduser et al. 2008). This synergetic effect of temperature during PEF-induced inactivation can be used to improve the inactivation results and/or to reduce the required amount of costintensive electrical energy (Craven et al. 2008; Riener et al. 2008). Besides the combination of the PEF treatment and elevated treatment temperature, a further temperature increase can occur during the PEF treatment as a result of the joule heating (Lindgren et al. 2002). In both cases, temperature plays a major role during PEF processing, as temperature influences

Figure 9.5. Relative changes in thermophysical properties of water as a function of the temperature. Reference temperature of 20°C.

thermophysical properties, mainly the viscosity and, to a lesser extent, the thermal conductivity, the density, and the specific heat capacity (Figure 9.5, see also Chapter 2 of this book). The consideration of the treatment temperature (inlet temperature) and the temperature increase during PEF processing is essential not only for the interpretation of PEF results but also for the numerical simulations of electric field strength distribution and flow characteristics. The electrical conductivity, thermal conductivity, heat capacity, viscosity, density, and some other properties depend on temperature and will affect the temperature increase and temperature distribution by the joule heating during the PEF treatment in turn. Most food products are complex mixtures of many different chemical compounds. Even the water may be bound in one of several different ways which may alter its effect on the thermal properties (Lewicki 2004). Based on the composition of a food, it is possible to estimate the thermal properties such as the specific heat capacity using prediction equations considering different specific heats and the content of proteins (Xp), fats (Xf), carbohydrates (Xc), water (Xw), and ash (Xw) in the food product according to Equation 9.35 (Heldman and Singh 1981):

Chapter 9

Basics for Modeling of Pulsed Electric Field Processing of Foods

c p, food = 1.424 X c + 1.549 X p + 1.675 X f + 0.837 X a + 4.187 X w

(9.35)

For high-moisture foods, one can assume that the change in specific heat mimics the change in specific heat of water with temperature (T in °C). This can be calculated by Equation 9.36 according to (Kessler 2002): c p,water = 4.214 − 2.153 ⋅ 10 −3 ⋅ T + 3.646 ⋅ 10 −5 ⋅ T 2 − 1.4948 ⋅ 10 −7 ⋅ T 3

(9.36)

Thermal conductivity values and their measurement depend on the structure or physical arrangement of the sample (voids, nonhomogeneities, particleto-particle contact, and so on) as well as on the chemical composition. For most liquid foods, an equation based on water, protein, carbohydrate, fat, and ash content appears to be adequate. Choi and Okos (1986) developed Equation 9.37 for the calculation of the thermal conductivity of complex liquid foods depending on their composition:

λ food = 0.61 X w + 0.20 X p + 0.205 X c + 0.175 X f + 0.135 X a

(9.37)

For predicting the thermal conductivity of water depending on temperature, Equation 9.38 can be used as suggested by (Kessler 2002).

λ water = ( 568.96 + 1.88 ⋅ T − 8.2 ⋅ 10 −3 ⋅ T 2 + 6.02 ⋅ 10 −6 ⋅ T 3 ) ⋅ 10 −3

(9.38)

9.4.2. Rheological Properties Information on the rheological properties of the food products to be PEF treated is required for the proper design of the treatment chamber geometry as well as the numerical simulation of heat transfer phenomena and fluid flow. Fluid dynamics is of special importance since the flow velocity distribution in the treatment chamber determines the residence time and therefore the treatment time. Modifications of the treatment chamber geometry can be used to affect the flow behavior and to induce mixing effects that may improve the microbial inactivation as well

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as the retention of heat-sensitive compounds (Jaeger et al. 2009). The viscosity of Newtonian fluids (such as water, milk, and clear fruit juices) is influenced by temperature and composition, but is independent of shear rate and shear history (Borwankar 1992). The rheological behavior of the fluid food plays an important role in continuous PEF treatment since it determines the fluid flow and the velocity profile. Equations describing velocity profiles can be used to examine the influence of different rheological models on the velocity distribution, and to determine the residence time distribution of the fluid particles. Since the viscosity also depends on the temperature, the control or documentation of the temperature is essential for the determination of the viscosity or for the conduction of experiments and calculations based on viscosity data. As an average value, there is about a 2% decrease in viscosity for each degree Celsius change in temperature (Lewis 1987). A product-dependent relation between viscosity and temperature has to be determined experimentally, based on empirical equations or on estimations considering hydrodynamic volumes of the component fractions. According to Bertsch (1983), the kinematic viscosity ν (mPas) of milk depends on temperature (T) and fat content Xf (in the range of 0–12%): ln ν = 3.03 ⋅ 10 −5 ⋅ T 2 − 1.813 ⋅ 10 −2 ⋅ T + 0.609 + X f ( −2.3 ⋅ 10 −6 ⋅ T 2 + 5.49 ⋅ 10 −4 ⋅ T

+ 2.06 ⋅ 10 −3 )

+ X 2f ⋅ ( 2.5 ⋅ 10 −7 ⋅ T 2 + 6.29 ⋅ 10 −5 ⋅ T + 5.42 ⋅ 10 −3 )

(9.39)

The kinematic viscosity ν can be defined by the ratio between the dynamic viscosity η and the density ρ. According to Equation 9.40, adapted from Kessler (2002), the density (kg/m3) of water changes also with temperature:

ρwater = 1000.22 + 1.0205 ⋅ 10 −2 ⋅ T − 5.8149 ⋅ 10 −3 ⋅ T 2 + 1.496 ⋅ 10 −5 ⋅ T 3

(9.40)

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Additional, detailed information on physical properties can be found in Bertsch (1983) for milk or in Constenla et al. (1989) for fruit juice.

9.4.3. Electrical Properties of Food The electric behavior of liquid foods is dependent not only on the chemical composition such as the content of water, fat, protein, carbohydrates, and minerals but also on the effects of dissolved and suspended solids that the liquid may contain. Small particle sizes, as in the case of orange or tomato juice (homogeneous distribution), or products with randomly dispersed solids of relatively large particle size such as vegetable soup (heterogeneous) may occur. Although the overall (volumetric) electrical properties of homogeneous and heterogeneous systems of similar moisture, dissolved salts, and suspended solids content may be nearly identical, the behavior during PEF treatment is different with respect to effects of particle size, homogeneity, and distribution. The electrical properties of foods such as electrical conductivity and ionic strength are of key relevance for PEF treatment since they determine the maximum applicable electric field intensity as well as the current flow. The coupling and distribution of the electric field strength, the energy, and the product’s heating is based on the electrical food properties. The mechanism or mode of energy transfer from the electric field to the product is the energy dissipation caused by joule heating, which couples electrical and thermal food properties. The electrical conductivity of a medium, defined as the ability to conduct electric current, is given in Siemens per meter or the inverse of the resistivity ρe, given in Ωm. The electrical conductivity of liquid food is caused by the presence of ions and shows a linear relation to temperature. The thermal conductivity for different liquid food systems as a function of the temperature is shown in Figure 9.6, and the linear relationships σ(T) are given in Table 9.1. Food conductivity ranges from σ = 0.6 mS/cm (tap water) or even below for pure fats and oils to

Figure 9.6. Relation between electrical conductivity and temperature of different liquid food systems. Adapted from Toepfl et al. (2007).

Table 9.1. Coefficients a and b for the linear relationship of the conductivity σ and the temperature T (σ = aT + b) for different beverages according to Toepfl (2004). Product Apple juice Red grape juice Orange juice Tomato juice Carrot juice Milk (3.5 % fat)

a (mS cm−1°C−1)

b (mS cm−1)

0.037761 0.060964 0.069500 0.316900 0.141250 0.080119

1.38071 1.11286 1.53000 3.31000 2.77500 2.37714

values in excess of σ = 7 mS/cm for milk ultrafiltrates and other high-conductivity products (Barsotti et al. 1999). Treatment chambers with high electrically conductive foods have a poor resistance, and it is necessary to produce higher voltages to achieve the same effect of microbiological inactivation that is achieved during processing of low-conductive foods. It is also more difficult to build sufficient field strength when the conductivity is too high (Wouters et al. 2001). To obtain the same degree of microbiological inactivation in foods with very different conductivity, the treatment conditions, such as the interlectrode gap

Chapter 9

Basics for Modeling of Pulsed Electric Field Processing of Foods

in the treatment chamber, the pulse width, and the voltage, have to be adapted. On the other hand, the presence of ions appears to be necessary to increase the transmembrane potential (Bruhn et al. 1998). The membrane will be weakened and will be more susceptible to an electric pulse in media with higher ionic strength, causing higher permeability and structural changes. However, the bactericidal effect of PEF is inversely proportional to the ionic strength of the suspension; that is, the inactivation is generally enhanced when the medium has a high electric resistivity (Huelsheger et al. 1981; Mizuno and Hori 1988). Liquid or semisolid foods containing suspended phases (i.e., solid particles, air bubbles, or large fat globules) with significantly different properties from those of the continuous phase will be considered heterogeneous. The electric behavior of one phase is essentially different from another phase, resulting in different electric field strength distribution as well as joule heating rates. Figure 9.7 illustrates the impact

of agglomeration of microbial cells and the attachment of microbial cells to fat globules (insulating particles) with completely different electrical properties from the surrounding media on the electric field strength distribution. The numerical simulation conducted by Toepfl et al. (2007) for an external electric field strength of 12.5 kV/cm showed that the peak values for the critical electric field strength were reduced by agglomeration and insulating particles to 84% and 45%, respectively. This fact limits the lethality of the PEF process since the required membrane potential for an electroporation is not reached. Similar effects will occur when the media contains electrically insulating gas bubbles, which can be a result of electrolysis at the electrodes (Gongora-Nieto et al. 2003).

9.4.4. PEF Processing Unit—Relevant Thermophysical Properties Since the thermophysical properties as well as the electrical and rheological properties are dependent on temperature, it is essential to determine the temperature during PEF processing. This includes the temperature during preheating, the temperature increase during PEF treatment due to the joule heating, and the temperature decrease during subsequent cooling steps (Jaeger et al. 2010). The heat transfer coefficient (U) for the heat exchanger in use (either heating or cooling) can be determined experimentally and calculated according to Equation 9.41, where m is the mass flow rate, cp is the specific heat capacity, Tin and Tout are the inlet and outlet temperatures of the product, A is the area for the heat transfer, and ΔTlog is the logarithmic mean temperature difference between the inside and outside (Text) of the heat exchanger. m ⋅ c p ⋅ (Tin − Tout ) = U ⋅ A ⋅ ΔTlog

Figure 9.7. Distribution of the electric field strength and impact on the critical electric field strength for electroporation of microbial cells when attached to each other (case A: agglomeration) or attached to insulating particles (case B: fat globule). Adapted from Toepfl et al. (2007). See color insert.

183

(9.41)

For the implementation of the heat transfer phenomena in the numerical simulation, the outer convective heat transfer coefficient (αo) may be required. The inner convective heat transfer coefficient (αi) can be calculated based on the Nusselt, Reynolds, and Prandtl values (Martin 2002). Together with the overall heat transfer coefficient (U), the inner

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Figure 9.8. Temperature–time profile (line 1 and line 2 with lower specific energy input) for a PEF processing including preheating (A), an intermediate pipe section (B), the PEF treatment (C), an intermediate pipe section (D), and the cooling (E). Lines 3 and 4 represent exemplarily the corresponding thermal inactivation of a heat-sensitive microorganism in the different sections of the equipment. Adapted from Jaeger et al. (2010).

convective heat transfer coefficient (αi), and the conductive heat transfer coefficient (λ), which depends on the material of the heat exchanger with the wall thickness (s), the outer convective heat transfer coefficient (αo) can be calculated based on Equation 9.42: U=

1 1 s 1 + + αi λ αo

(9.42)

Implementation of the known characteristics for the heat transfer in numerical simulation software packages allows the simulation of the temperature during the heating and cooling processes. Equation 9.34 of Section 9.3.5 and the equations obtained for a particular heat exchanger system are combined to calculate the temperature–time profile of a liquid during pre-heating, PEF treatment, and cooling, which is shown in one example in Figure 9.8. The obtained temperature–time profile is the basis for the consideration of temperature-dependent physical properties of foods as well as for the differentiation of thermal inactivation effects occurring during PEF treatment (Jaeger et al. 2010).

9.5. Selected Inactivation Models for Numerical Simulations Since the numerical simulation of PEF processing allows a detailed description of small changes of a variable, such as the electric field strength as a function of the treatment chamber geometry, or even the changes of the fluid velocity as altered by temperature changes, it is important to use an adequate microbial/enzyme inactivation model, capable to represent the dependency of the inactivation on the electric field strength and the treatment time with sufficient accuracy. The main process conditions considered here are the electric field strength and the treatment time. Selected models appropriate for numerical simulations are exclusively obtained under homogeneous treatment conditions. Since the electric field strength is homogeneous only in a treatment chamber with parallel plate electrode configuration, models performed in colinear or coaxial treatment chambers will not be considered in this section. Additionally, only models performed in batch systems will be discussed since the treatment time in a continuous treatment chamber is not homogeneous as discussed in Section 9.3.3.

Chapter 9

Basics for Modeling of Pulsed Electric Field Processing of Foods

9.5.1. Kinetic Models for Microbial Inactivation The main mechanism of PEF inactivation is related to the electroporation of the cell membrane (Zimmermann et al. 1974; Zimmermann 1986). Mechanistic models for the microbial inactivation by PEF have not yet been combined with kinetic models to one single predictive model that is capable to describe the expected microbial inactivation based on the mechanism of inactivation. In contrast to this, several empirical models have been developed; numerous experimental results have been fitted to proposed kinetic models, mainly to describe the evolution of the microbial inactivation as a function of the electric field strength and treatment time. For a detailed description of the mechanism of microbial inactivation by PEF, see Barbosa-Cánovas et al. (1999) and Lelieveld et al. (2007). A detailed description on kinetic models can be found in Álvarez et al. (2007). Empirical models based on Huelsheger et al. (1981) and Peleg (1995) and those that follow a Weibull distribution (Weibull 1951; Van Boekel 2002) have been used to model the microbial inactivation by PEF. An example of different models is given below. Model A. First order exponential decay: S (t ) = e − kt

(9.43)

Model B. Huelsheger et al. (1981): ⎛t⎞ S (t ) = ⎜ ⎟ ⎝ tc ⎠

− ( E − Ec ) k

(9.44)

where S is the survival rate (N/N0), t is the treatment time, tc is the critical treatment time, E is the electric field strength, Ec is the electric field strength at the extrapolated survival fraction of 100%, and k is a constant characteristic of each microorganism. Model C. Peleg (1995). Based on Fermi’s equation: S (t ) =

1 ⎡ ( E − E c ( n )) ⎤ 1 + exp ⎢ k ( n ) ⎥⎦ ⎣

(9.45)

185

Ec(n) is the electric field strength at which the survival level is 50% and k(n) is a kinetic constant that describes the steepness of the sigmoid curve. Ec(n) and k(n) depend on both the number of pulses and the treatment time. Model D. Weibull distribution (Weibull 1951): ⎛ t⎞ S (t ) = exp ⎜ − ⎟ ⎝ δ⎠

α

(9.46)

In addition to these models, alternative equations have been used. The existence of two populations of microorganisms (PEF-sensitive and PEF-resistant subpopulations) can be described by an extension of the exponential model (Pruitt and Kamau 1993; Amiali et al. 2004). Models E and F: S (t ) = β0 + (1 − β0 ) ⋅ e − β2 t

(9.47)

S (t ) = β0 e − β1t + (1 − β0 ) ⋅ e − β2 t

(9.48)

where S(t) is the fraction of total survivors; t is the treatment time (μs); β0 is the fraction of survivors in population 1 (generally related to the PEFsensitive population); β1 is the specific death rate of subpopulation 1; and β2 is the specific death rate of subpopulation 2. Model G. Sigmoidal equations in PEF have been used (Álvarez et al. 2003d) and also applied for thermal inactivation studies (Augustin et al. 1998):

(

S (t ) = 1 + e(tlog − tc ,log )

sd 2

) −1

(9.49)

This model is justified based on a distribution of the PEF sensitivity within the bacterial population. S(t) is the fraction of total survivors; tlog is the log10 of the treatment time (μs); tc,log is the log10 of the time necessary to inactivate 50% of the population (μs); and sd is a parameter proportional to the standard deviation of the PEF resistance (μs0.5). Model H describes upward concave curves. (Peleg and Penchina 2000; Álvarez et al. 2003a): log10 S (t ) = − k ln (1 − k1t )

(9.50)

where S(t) is the survival fraction; t is the treatment time (μs); and k and k1 are characteristic parameters of the equation.

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Model I. A log-log model consisting of a linear regression of the log10 of the survival fraction of microoganisms versus the log10 of the treatment time (Raso et al. 2000). Model G. The log-logistic model (Cole et al. 1993; Raso et al. 2000): log10 S (t ) = k1 + ( k2 − k1 ) / 1

+ exp [ 4 k3 ( k4 − log10 t ) / ( k2 − k1 )] (9.51)

where k1 is the upper asymptote log CFU/mL, k2 is the lower asymptote, k4 is the position of the maximum slope, and k3 is the maximum slope. Table 9.2 summarizes the selected models, which are suitable for numerical simulations. The range of

the treatment parameters for which the model was applied (electric field strength, treatment time, and temperature) is also indicated.

9.5.2. Kinetic Models for Enzyme Inactivation The mechanism of enzyme inactivation is still unclear; many authors assume that PEF affects the native structure of the enzyme, which could result in changes of its activity. Possible theories on enzyme inactivation mechanisms can be found in Martín-Belloso and Elez-Martínez (2005), in which different kinetic models for enzyme inactivation by PEF are shown. A review on enzyme inactivation

Table 9.2. Selected models for the microbial inactivation by PEF. Batch system with parallel plate electrode configuration.

Model

Microorganism

Product

A, I and G

Salmonella senftenberg

Citrate–phosphate McIlvaine buffer (pH 7; 2 mS/cm)

D D

Salmonella serovar Enteritidis and Salmonella serovar Typhimurium Listeria monocytogenes

D

Yersinia enterocolitica

D, F, G and H

Escherichia coli

E and F

Escherichia coli

D

Lactobacillus plantarum

D

Listeria monocytogenes

D

Dekkera anomala, Dekkera bruxellensis, Lactobacillus hilgardii, and Lactobacillus plantarum

Citrate–phosphate McIlvaine buffer (pH 7; 2, 3, 4 mS/cm) Citrate–phosphate McIlvaine buffer (pH 3.8, 5.4; 7, 2 mS/cm) Citrate–phosphate McIlvaine buffer (pH 7; 2 mS/cm) Whole egg (pH 7.8), egg yolk (pH 6.4) and egg white (pH 8.3) Citrate–phosphate McIlvaine buffer (pH 3.5–7.0; 2 mS/cm) Citrate–phosphate McIlvaine buffer (pH 3.5–7.0; 2 mS/cm) Wine (2.02 ± 0.22 mS/cm) and must (1.97 ± 0.23 mS/cm)

Electric field strength, treatment time and temperature 12–28 kV/cm 10–2,000 pulses <35°C 15–28 kV/cm 15–28 kV/cm 0–2,000 μs <32°C 5.5–28 kV/cm 1–15 μs <35°C 15–28 kV/cm 0–2,000 μs <35°C 15 kV/cm 20–100 ms 0°C 12–25 kV/cm 0–1,000 μs <35°C 15–28 kV/cm 10–1,000 μs <35°C 16–31 kV/cm 10–100 pulses <30°C

Author Raso et al. (2000) Álvarez et al. (2003a) Álvarez et al. (2003b) Álvarez et al. (2003c) Álvarez et al. (2003d) Amiali et al. (2004) Gómez et al. (2005a) Gómez et al. (2005b) Puértolas et al. (2009)

Chapter 9

Basics for Modeling of Pulsed Electric Field Processing of Foods

by PEF treatments can be found in Van Loey et al. (2001). The single impact of different PEF parameters, such as electric field strength, treatment time, energy input, and temperature on enzyme inactivation has not been clearly identified and implemented in kinetic equations. In addition, differences in experimental setups, such as media properties and treatment chamber geometries, complicate the understanding of possible mechanisms for the enzyme inactivation by PEF. Nevertheless, several empirical models have been used and successfully validated with experiments. A first approach for differentiating between pure thermal and PEF effects occurring simultaneously in a continuous treatment chamber can be found in Jaeger et al. (2009, 2010). The number of kinetic models for the inactivation of enzymes suitable for numerical simulations is even more limited than for the models describing the microbial inactivation. Table 9.3 summarizes the models, which were obtained under homogeneous

187

distributions of the electric field strength and the treatment time (parallel plate electrode configuration and batch system).

9.6. Conclusion and Outlook The aspects discussed in this chapter pointed out the main basic principles for the numerical simulation of the PEF process as well as for the implementation of numerous additional physical properties of the food material and the treatment chamber. The potential of numerical simulations to combine the kinetic models with simulated characteristics of a distinct PEF treatment in order to predict occurring processing effects was shown. The Multiphysics nature of PEF processing does not only require the consideration of phenomena related to mass transfer, heat transfer, and the electrical field effects, but does also need to take into account the dependency of physical media properties as affected by the above mentioned interrelations. It was pointed out that a complete characterization of a PEF treatment needs to include

Table 9.3. Selected models for the inactivation of enzymes by PEF. Model

Enzyme

Origin/product

First-order Huelsheger based on Fermi

PME

Extract from tomato (0.945 ± 0.01 mS/cm)

First-order

PPO

First-order

PPO

Extracts from apple (2.27 mS/cm) and pear (2.42 mS/cm) Extract from peach

First-order

PG

First-order

LP

First-order Huelsheger based on Fermi Weibull

PE

Enzyme solution in distilled water (13 mS/cm) Pseudomonas fluorescens in SMUFa Aqueous solution (0.4 mS/cm and pH 4.6)

Electric field strength, treatment time, and temperature 5–24 kV/cm 50–400 pulses 4–15°C 5.52–24.6 kV/cm 0–8 ms 4–15°C 2.18–24.30 kV/cm 0–30 ms 4–25°C 5.18–19.39 kV/cm 0–400 pulses 4–25°C 16.4–27.4 kV/cm 10–80 pulses 18–34°C 19–38 kV/cm 0–463 μs 4–37.2°C

Author Giner et al. (2000) Giner et al. (2001) Giner et al. (2002) Giner et al. (2003) Bendicho et al. (2002) Giner et al. (2005)

a SMUF, simulated skim milk ultrafiltrate. PME, pectin metylesterase; PG, polygalacturonase; LP, lipase; PE, pectinesterase; PME, pectin metylesterase; PPO, polyphenoloxidase.

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energy parameters such as pulse energy and total specific energy input apart from treatment time or pulse number in order to distinguish accurately between the impact of these factors on the inactivation of microorganisms or the alteration of food compounds. The implementation of kinetic models obtained under defined treatment conditions into the numerical simulation of PEF treatment systems will be key to further characterizing the underlying mechanisms and the optimization of processing units.

Acknowledgments The authors would like to thank the German Ministry of Economics and Technology (via AiF) and the FEI (Forschungskreis der Ernährungsindustrie e.V., Bonn), Project AiF 15610 N and the Commission of the European Communities, Framework 6, Priority 5 “Food Quality and Safety,” Integrated Project NovelQ FP6-CT-2006-015710. Nicolás Meneses gratefully acknowledges the financial support from DAAD (German Academic Exchange Service) and Universidad Austral de Chile, Valdivia, Chile, for the initial period of the work.

Notation Latin Variables A C ci cp c p,water c p, food Di d E Ec F f H1 H2 I J

Surface Capacitance Concentration of substance or specie i Specific heat capacity Specific heat capacity of water Specific heat capacity of food Diffusion coefficient of specie i Inter electrode gap Electric field strength Critical electric field strength Volumetric force vector Frequency or pulse repetition rate Heaviside step function 1 Heaviside step function 2 Electrical current Current density

Je k L m m n p r ri ro R RA Rc s S Q Qs ΔTave ΔTlog T Tin Text Tout t tc tc,log tlog tr tt U u uave Wpulse Wpulse_cuad Wpulse_exp Wspecific Xa Xc Xf Xp Xw V V0

Externally generated current density Model kinetic constant Length Mass Mass flow rate Number of pulses Pressure Radius Inner radius Outer radius Resistance Relative activity Treatment chamber resistance Wall thickness Microbial survival rate Flow rate Source term Average temperature difference Logarithmic mean temperature difference Temperature Inlet temperature Exterior temperature Outlet temperature Time Critical treatment time Log10 of the critical treatment time Log10 of the treatment time Residence time Treatment time Overall heat transfer coefficient Velocity vector Average velocity Energy per pulse Energy per pulse for quadratic pulses Energy per pulse for exponential pulses Specific energy input Content of ash Content of carbohydrates Content of fat Content of proteins Content of water Electric potential Initial voltage

Chapter 9

Basics for Modeling of Pulsed Electric Field Processing of Foods

Greek Variables ∇ ∂ ∂t α αi αo β0 β1 β2 λ λ water λ food η π ρ ρe ρwater σ τ ν

Gradient operator Partial derivative Kinetic constant of Weibull model Inner convective heat transfer coefficient Outer convective heat transfer coefficient Fraction of survivors in population 1 Specific death rate of subpopulation 1 Specific death rate of subpopulation 2 Thermal conductivity Thermal conductivity of water Thermal conductivity of food Dynamic viscosity Pi Density Electrical resistivity Density of water Electrical conductivity Pulse width Kinematic viscosity

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Giner J, Gimeno V, Espachs A, Elez P, Barbosa-Cánovas G, Martín O. 2000. Inhibition of tomato (Licopersicon esculentum Mill.) pectin methylesterase by pulsed electric fields. Innov Food Sci Emerg Technol 1(1):57–67. Giner J, Gimeno V, Barbosa-Cánovas G, Martín O. 2001. Effects of pulsed electric field processing on apple and pear polyphenoloxidases. Food Sci Technol Int 7(4):339–345. Giner J, Ortega M, Mesegué M, Gimeno V, Barbosa-Cánovas G, Martín O. 2002. Inactivation of peach polyphenoloxidase by exposure to pulsed electric fields. J Food Sci 67(4):1467–1472. Giner J, Gimeno V, Palomes M, Barbosa-Cánovas GV, Martín O. 2003. Lessening polygalacturonase activity in a commercial enzyme preparation by exposure to pulsed electric fields. Eur Food Res Technol 217:43–48. Giner J, Grouberman P, Gimeno V, Martín O. 2005. Reduction of pectinesterase activity in a commercial enzyme preparation by pulsed electric fields: Comparison of inactivation kinetic models. J Sci Food Agric 85:1613–1621. Gómez N, García D, Álvarez I, Condón S, Raso J. 2005a. Modeling inactivation of Listeria monocytogenes by pulsed electric fields. Int J Food Microbiol 103:199–206. Gómez N, García D, Álvarez I, Raso J, Condón S. 2005b. A model describing the kinetics of inactivation of Lactobacillus plantarum in a buffer system of different pH and in orange and apple juice. J Food Eng 70:7–14. Gongora-Nieto M, Pedrow PD, Swanson BG, Barbosa-Canovas GV. 2003. Impact of air bubbles in a dielectric liquid when subjected to high electric field strengths. Innov Food Sci Emerg Technol 4:57–67. Heinz V, Toepfl S, Knorr D. 2003. Impact of temperature on lethality and energy efficiency of apple juice pasteurization by pulsed electric fields treatment. Innov Food Sci Emerg Technol 4(2):167–175. Heldman DR, Singh RP. 1981. Food Process Engineering. Westport: Avi Publishing. Ho S, Mittal GS. 2000. High voltage pulsed electrical field for liquid food pasteurization. Food Rev Int 16(4):395–434. Holland FA, Bragg R. 1995. Fluid Flow for Chemical Engineers, 2nd ed. London: Elsevier. Huelsheger H, Potel J, Niemann EG. 1981. Killing of bacteria with electric pulses of high field strength. Radiat Environ Biophys 20:53–65. Jaeger H, Meneses N, Knorr D. 2009. Impact of PEF treatment inhomogeneity such as electric field distribution, flow characteristics and temperature effects on the inactivation of E. coli and milk alkaline phosphatase. Innov Food Sci Emerg Technol 10(4):470–480. Jaeger H, Meneses N, Moritz J, Knorr D. 2010. Model for the differentiation of temperature and electric field effects during thermal assisted PEF processing. J Food Eng (in press). Kanduser M, Sentjurc M, Miklavcic D. 2008. The temperature effect during pulse application on cell membrane fluidity and permeabilization. Bioelectrochemistry 74(1):52–57.

Kessler HG. 2002. Food and Bio Process Engineering. München: Verlag A. Kessler. Lelieveld HLM, Notermans S, de Haan SWH, eds. 2007. Food Preservation by Pulsed Electric Fields. Abington: Woodhead Publishing. Lewicki PP. 2004. Water as the determinant of food engineering properties. A review. J Food Eng 61(4):483–495. Lewis MJ. 1987. Physical Properties of Foods and Food Processing Systems. Weinheim: VCH Verlagsgesellschaft. Lindgren M, Aronsson K, Galt S, Ohlsson T. 2002. Simulation of the temperature increase in pulsed electric field (PEF) continuous flow treatment chambers. Innov Food Sci Emerg Technol 3:233–245. Loeffler M. 2006. Generation and application of high intensity pulsed electric fields. In: J Raso, V Heinz, eds., Pulsed Electric Fields Technology for the Food Industry, 27–72. Heidelberg: Springer-Verlag. Martin H. 2002. Dimensionslose Kenngrößen. In: VD Ingenieure, ed., VDI Wärmeatlas. Heidelberg: Springer-Verlag. Martín O, Zhang Q, Castro AJ, Barbosa-Cánovas GV, Swanson BG. 1994. Pulsed electric fields of high voltage to preserve foods. Microbiological and engineering aspects of the process. Span J Food Sci Technol 34:1–3. Martín-Belloso O, Elez-Martínez P. 2005. Enzymatic inactivation by pulsed electric fields. In: D-W Sun, ed., Emerging Technologies for Food Processing, 155–181. London: Academic Press. Meneses N, Jaeger H, Moritz J, Knorr D. 2010. Impact of insulator shape and flow characteristics on inactivation of E. coli using a continuous co-linear PEF system. J Food Eng (in press). Mizuno A, Hori Y. 1988. Destruction of living cells by pulsed high-voltage application. IEEE Trans Ind Appl 24(3): 387–394. Pataro G. 2004. The utilization of pulsed electric fields in food stabilization. PhD dissertation, University of Salermo, Salermo. Peleg M. 1995. A model of microbial survival after exposure to pulsed electric fields. J Sci Food Agric 67:93–99. Peleg M, Penchina CM. 2000. Modelling microbial survival during exposure to lethal agent varying intensity. Crit Rev Food Sci Nutr 40(2):159–172. Pruitt K, Kamau DN. 1993. Mathematical models of bacteria growth, inhibition and death under combined stress conditions. J Ind Microbiol 12:221–231. Puértolas E, López N, Condón S, Raso J, Álvarez I. 2009. Pulsed electric fields inactivation of wine spoilage yeast and bacteria. Int J Food Microbiol 130:49–55. Raso J, Álvarez I, Condón S, Sala FJ. 2000. Predicting inactivation of Salmonella senftenberg by pulsed electric fields. Innov Food Sci Emerg Technol 1:21–30. Riener J, Noci F, Cronin DA, Morgan DJ, Lyng JG. 2008. Combined effect of temperature and pulsed electric fields on apple juice peroxidase and polyphenoloxidase inactivation. Food Chem 109(2):402–407.

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Basics for Modeling of Pulsed Electric Field Processing of Foods

Saldaña G, Puértolas E, López N, García D, Álvarez I, Raso J. 2009. Comparing the PEF resistance and occurrence of sublethal injury on different strains of Escherichia coli, Salmonella Typhimurium, Listeria monocytogenes and Staphylococcus aureus in media of pH 4 and 7. Innov Food Sci Emerg Technol 10:160–165. Schrive L, Grasmick A, Moussiere S, Sarrade S. 2006. Pulsed electric field treatment of Saccharomyces cerevisiae suspensions: A mechanistic approach coupling energy transfer, mass transfer and hydrodynamics. Biochem Eng J 27:212–224. Sensoy I, Zhang Q, Sastry S. 1997. Inactivation kinetics of Salmonella Dublin by pulsed electric field. J Food Process Engineering 20(5):367–381. Stanley DW. 1991. Biological membrane deterioration and associated quality losses in food tissues. In: FM Clydesdale, ed., Critical Reviews in Food Science and Nutrition, New York: CRC Press. Toepfl S. 2004. Erhöhung des letalen Effektes gepulster elektrischer Felder bei der Inaktivierung ausgwählter Mikroorganismen durch Variation der Impulscharakteristik. Diploma thesis, Berlin University of Technology, Berlin. Toepfl S, Heinz V, Knorr D. 2007. High intensity pulsed electric fields applied for food preservation. Chem Eng Process 46:537–546. Tsong TY. 1990. Voltage modulation of membrane permeability and energy utilization in cells. Biosci Rep 3:487–505. Van Boekel MAJS. 2002. On the use of the Weibull model to

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describe thermal inactivation of microbial vegetative cells. Int J Food Microbiol 74:139–159. Van Boekel M. 2009. Kinetic Modeling of Reactions in Foods. Boca Raton, FL: CRC Press/Taylor & Francis. Van den Bosch H, Morshuis P, Smit J. 2003. Temperature distribution in continuous fluid flows treated by PEF. Paper read at XIII International Symposium on High Voltage (ISH), Delft, the Netherlands. Van Loey A, Verachtert B, Hendrickx M. 2001. Effects of high electric field pulses on enzymes. Trends Food Sci Technol 12:94–102. Weibull W. 1951. A statistical distribution function of wide applicability. J Appl Mech 51:293–297. Wouters PC, Álvarez I, Raso J. 2001. Critical factors determining inactivation kinetics by pulsed electric field food processing. Trends Food Sci Technol 12(3–4):112–121. Zhang Q, Barbosa-Cánovas GV, Swanson BG. 1995. Engineering aspects of pulsed electric field pasteurization. J Food Eng 25:261–281. Zimmermann U. 1986. Electric breakdown, electropermeabilization and electrofusion. Rev Physiol Biochem Pharmacol 105:175–256. Zimmermann U, Pilwat G, Riemann F. 1974. Dielectric breakdown in cell membranes. Biophys J 14:881–899. Zimmermann U, Pilwat G, Beckers F, Riemann F. 1976. Effects of external electrical fields on cell membranes. Bioelectrochem Bioenerg 3:58–83.

Chapter 10 Computational Fluid Dynamics Applied in Pulsed Electric Field Preservation of Liquid Foods Nicolás Meneses, Henry Jaeger, and Dietrich Knorr

10.1. Introduction Pulsed electric field (PEF) treatment can be considered a nonthermal food preservation technology since the microbial inactivation mechanism is based on electroporation (Knorr et al. 1994). Nevertheless, due to the electrical conductivity of the food material, a substantial current will flow as an unwanted side effect during the application of the high voltage (Raso and Heinz 2006). To reduce the energy consumption and to limit the product heating, a treatment chamber with a high electrical resistance is desired. This parameter is of outstanding importance in the case of treating products with high electrical conductivity, where a high electrical current flow will lead to increased joule (ohmic) heating effects and an unwanted temperature increase. Many authors (Fiala et al. 2001; Lindgren et al. 2002; Van den Bosch et al. 2002; Gerlach et al. 2008) have described the temperature distribution in a PEF treatment chamber and reported the occurrence of high local temperatures due to the inhomogeneous distribution of the electrical field, limited flow velocity, and recirculation of the liquid. These temperatures are only detectable by numerical simulations since temperature measurement in the small

volume of the treatment chamber is hardly feasible without interference of the measuring device with the flow and the electric field. Treatment homogeneity in terms of homogeneous electric field distribution as well as residence and treatment time of the product and uniform temperature distribution presents the most challenging aspect during the design of a treatment chamber (Lindgren et al. 2002; see also Chapter 9 of this book). In this chapter the application of computational fluid dynamics using commercial software packages for simulating PEF processing is discussed. The first section gives an overview on different treatment chamber configurations and validation methods of numerical simulations used by different research groups. The second section covers the topic of transient simulation of PEF processing in batch systems. The third section deals with the steady-state simulation of PEF and the comparison of two different treatment chamber setups focusing on electric field strength, temperature, and velocity profiles. The numerical background on how to carry out a simulation using a commercial software package is an important aspect and therefore included in each

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

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section of this chapter. Governing equations, already shown in detail in the previous chapter (Chapter 9 of this book), the implementation of boundary conditions, and thermophysical properties of liquid food, the numerical grid, and solution methods will be explained, followed by a presentation of methods for the experimental validation. The insertion of static mixing devices in the treatment chamber is presented in order to provide a possibility of altering the undesirable laminar flow profiles occurring in most of the experiments performed in small-scale continuous systems. The impact of static mixing devices on treatment homogeneity, microbial inactivation, and product quality will be shown using milk alkaline phosphatase (ALP) and Escherichia coli as indicators for the treatment effectiveness and temperature distribution. The final part of this chapter focuses on the application of numerical simulations for the characterization of the PEF process conditions.

10.2. PEF Treatment Chamber Design The treatment chamber design should consider the uniformity of the electric field distribution in combination with the flow characteristics in continuous applications, as well as the extent of the temperature increase. An adequate shape of the electrodes and the insulator will reduce dielectric breakdown effects since local high electric field strength levels can be avoided, for example, by providing a round-edged insulator geometry (Dunn and Pearlman 1987). Dielectric breakdowns are undesirable as they can cause arcing, which leads to the destruction of the food, damage on the electrode and insulator surface, or even an explosion of the treatment chamber due to the pressure increase. Electric field homogeneity and the avoidance of low field intensities are desirable not only from the microbial inactivation point of view, but also from the fact that energy dissipation and power consumption will take place in low field regions without contributing to the microbial inactivation. Since PEF processing at an industrial scale will be applied for a continuous product flow, the efficiency of the treatment strongly depends on the

design of the flow-through treatment chamber, which is basically composed of two electrodes and an insulating body. Various treatment chamber designs such as parallel plates, coaxial cylinders, or colinear configurations have been used for PEF processing, and some modifications of these basic designs have been proposed (Alkhafaji and Farid 2007; Huang and Wang 2009; Jaeger et al. 2009). An overview of different treatment chamber configurations can be found in Barbosa-Cánovas et al. (1999), Lelieveld et al. (2007), and Huang and Wang (2009). Parallel plates are the simplest in design and produce the most uniform distribution of the electric field (Jeyamkondan et al. 1999), but their lower electrical resistance limits the application possibilities. The geometry consists of a rectangular duct of insulating material with two limited electrodes on opposite sides (see Section 10.4). Only the electrode length and distance determine the electric field distribution. In a co-axial treatment chamber, the food is placed between two cylinders, one internal cylinder used as high-voltage electrode and one external cylinder used as ground electrode. The electric field within this treatment chamber is not uniform because it is distributed in descendant order, from the central cylinder toward the external cylinder. One of the major advantages of this type of chambers is that peak values of local electric field strength are minimized or eliminated. The liquid food is flowing through the thin gap between the two cylinders so that the application of this chamber is restricted to liquid foods with only small particles. Furthermore, the effective area of the electrodes is very large, causing a high current flow and a low resistance of the treatment chamber. The co-field (or colinear) treatment chamber is one of the most commonly utilized to operate in continuous systems. A flow pattern suitable for food processing can be achieved with the colinear design. A hollow high voltage and grounded electrode with a circular inner hole are kept on a defined distance by an insulating spacer (Figure 10.1). The product is pumped through the drilling forming the electrical load of the high-voltage discharge circuit.

Chapter 10

Figure 10.1. Treatment chamber and treatment zone for a colinear configuration. Ri denotes the insulator radius, Di is the treatment zone radius, and De is the electrode (highvoltage and ground electrodes) radius.

The electric field strength is not homogeneous and depends strongly on the insulator geometry that is placed between the two electrodes (Meneses et al. 2010). By modification of the insulator geometry, it is possible to obtain different electric field strength distributions, which can be simulated with numerical software (Gerlach et al. 2008; see also Chapter 11 of this book). To achieve sufficient treatment intensity for all volume elements as well as to prevent overprocessing or arcing, the electric field should be free of local peak values. Relative to the electrodes, the inner diameter of the insulator should be slightly smaller than the inner diameter of the electrodes in order to produce a more homogeneous electrical field (Toepfl et al. 2007). Co-field chambers with a small electrode surface area and a large electrode gap will provide a high load resistance. Another important aspect for the electrode design and the numerical simulation of occurring electric field and thermal effects is the possibility of cooling the electrodes in order to control the temperature. Examples of batch treatment chambers with temperature control used for kinetic inactivation studies are given by Qin et al. (1994) and Bazhal et al. (2006).

10.3. Experimental Validation of PEF Simulations: A Review Steady-state simulations of PEF have been performed with commercial code packages in colinear

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treatment chambers with thermophysical properties implemented into the numerical code and considered dependent or independent on temperature. Due to the small dimensions of the treatment chamber in comparison to the dimension of the temperature probe, it is difficult to perform adequate measurements at different locations in the treatment chamber without perturbation of the flow and alteration of the experimental temperature measurement by the measuring device itself. Nonetheless, a good correlation between the simulated temperature and experimental measurements has been reported (Fiala et al. 2001; Lindgren et al. 2002; Van den Bosch et al. 2002; Gerlach et al. 2008; Jaeger et al. 2009). Fiala et al. (2001) studied the effect of the pulse frequency and the applied voltage on the product temperature using ANSYS, a finite element software package that includes several modules which can be used either individually or for a variety of coupled-field phenomena. The electro-hydrodynamic model was used, and solved stationary (timeindependent) and thermophysical properties were implemented into the numerical code and considered to be temperature-dependent. A co-field treatment chamber geometry with Di = 1.5 mm and De = 1.5 mm (Figure 10.1) was used in that study. The temperature was measured 6.6 cm upstream of the mid-chamber position along the axis as a function of the pulse frequency and electrode voltage. No information on the position of the temperature probe regarding the radial coordinate was given. The product temperature was a linear function of f × τ (pulse frequency and pulse width) and was in very good agreement with the experimental results. The authors concluded that the differences between the predicted and measured temperature at high values of f × τ are due to physical phenomena such as partial discharges or bubble formation that occurred in the treatment chamber but were not included in the model. Furthermore, the presence of probes that are interfering with the fluid flow has a considerable impact on the temperature distribution. Van den Bosch et al. (2002) applied an analytical method to estimate the temperature at the chamber

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walls without considering the inhomogeneities of the electric field inside the treatment chamber. In the same study, a commercial finite element method (FEM) package was used to solve the set of differential equations described in the previous chapter. The standard k-ε turbulence model was used for the fluid dynamics model, and the thermophysical properties used in the numerical simulation were considered to be dependent on temperature. In that study, a co-field treatment chamber with Di = 6.4 mm, De = 6.4 mm, Ri = 4 mm, and Li = 17.4 mm was used (Figure 10.1). No information was given on the exact coordinate for the temperature measurement. It was concluded that because of the slow flow velocity near the wall, the temperature of the highvoltage electrode is much higher than the average fluid temperature. Lindgren et al. (2002) investigated the modeled temperature increase in the outflow for different colinear flow-through PEF treatment chamber designs with Di varying from 0.635 to 1.15 mm, De = 1.15 mm, and Li varying from 2.3 to 4.6 mm (Figure 10.1) The governing equations presented in the previous chapter were solved independent from time using FEMLAB™, a FEM software also named COMSOL Multiphysics™ (COMSOL AB, Stockholm, Sweden). Thermophysical properties were set up in the numerical code to be independent of temperature. The average simulated temperature obtained from the FEM model was compared with an average estimated temperature according to the following equations: ΔTestimated = Wspec ( ρ ⋅ c p )

(10.1)

ΔTcalc = Qs ⋅ tt ( ρ ⋅ c p )

(10.2)

Qs = σ ⋅ E 2

(10.3)

Li ⋅ τ ⋅ f u

(10.4)

tt =

where Wspec is the specific energy input (J/kg); ρ is the density (kg/m3); cp is the specific heat capacity (m2/s2/K); Qs is the absorbed power density (W/m3); σ is the conductivity of the liquid food (S/m); E is the electric field strength (V/m); and tt is the treatment time, which is proportional to the length Li of

the treatment zone (m), the pulse width τ (s), the frequency f (s−1), and the velocity u (m/s). The estimated and calculated temperatures were compared with the measured temperature on the outside of the pipe (no exact information about the position of the temperature element was given) for a wide range of frequency values between 250 and 444 Hz and an electric field strength between 23 and 31 kV/cm. The authors found a direct correlation between energy input and temperature increase, which was confirmed by measurements and simulations. It was also shown that the temperature at the treatment chamber walls is higher because of the laminar flow conditions. It was concluded that a decrease in the insulator radius (Figure 10.1) not only makes it possible to avoid local zones of high electric field strength but also gives rise to a possible development of recirculation regions immediately downstream of the insulators. Gerlach et al. (2008) used the commercial FEM software FIDAP (ANSYS Inc., Canonsburg, PA) to solve the set of differential equations presented in the previous chapter (Chapter 9 of this book). In their work, the temperature distribution was simulated in a colinear treatment chamber (Figure 10.1) with Di = 2 mm, De = 3 mm, Ri = 1 mm, and Li = 2 mm. The product temperature was experimentally measured 7 cm downstream of the second insulator. The experiments and simulations were performed at different pulse repetition rates. The exact position of the temperature sensor was located 1.5 mm from the wall. Experimental results were also in good agreement with simulations. It was found that the predicted temperature differs strongly along the radial coordinate of the pipe, even 7 cm downstream of the insulator zone. The last study on this topic reported in the literature is from Jaeger et al. (2009). It focused on the simulation of electric field strength, fluid velocity, and temperature distributions in a colinear treatment chamber (Figure 10.1) with Di, De, Ri, and Li as in Gerlach et al. (2008). The standard k-ε model was used to solve the governing equations, as done by Van den Bosch et al. (2002). The simulated temperature was compared with experimental measurements at a distance of 3.5 cm from the second insulator and

Chapter 10

1.5 mm from the wall for different treatment conditions. Additional experimental measurements of the temperature were also conducted in a larger colinear treatment chamber with a treatment zone radius (Di) of 10 mm and an electrode radius (De) of 15 mm, allowing the temperature measurement at different locations on the radial coordinate and at different distances from the second insulator (see Figure 10.2). As shown in Figure 10.2, using a large treatment chamber, it was possible to detect temperature differences between the wall and the center of the treatment chamber, which cannot be measured in a smaller treatment chamber geometry. Local areas of high temperature in zones downstream of the second insulator and close to the walls can be identified and have to be taken into consideration when evaluating thermal effects during the PEF treatment.

Figure 10.2. Temperature obtained by simulation and experimental measurement in a colinear treatment chamber with an inner electrode diameter (De) of 30 mm and a treatment zone length (Li) of 30 mm. Temperature was measured at two different positions in the radial coordinate (wall and center of the treatment chamber) for each distance. The experiment was performed at a flow rate of 60 L/h and an initial temperature of 18.6°C. The voltage was set to 18 kV, pulse rate to 100 Hz, and pulse width to 6 μs; conductivity of the salt NaCl solution was adjusted to σ (T) 4.6 mS/cm (at 20°C); and total specific energy input was 87 kJ/kg (adapted from Jaeger et al. 2009). RMS, root mean square.

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10.4. Unsteady Simulation of PEF Batch Processing The basic model used for the time-dependent simulation consists of two coupled models: a thermal and an electrostatic model in a two-dimensional geometry. In this example, a treatment chamber with parallel plate electrode configuration is used. The partial differential equations are solved using a time-dependent solver in COMSOL Multiphysics™. Lagrange-quadratic elements are chosen as the basis functions with triangular-shaped elements. The maximum mesh size is 2 × 10−4 and expands over the whole geometric domain. Over 30,000 elements were used and the total number of degrees of freedom was around 124,000 (Figure 10.3). The governing equations for the thermoelectrostatic model consist of the thermal energy

Figure 10.3. Geometric domain and numerical mesh for an unsteady-state simulation in a treatment chamber of parallel plate electrode configuration. Boundary conditions (A, B, C, and D) as shown in Table 10.1. Electrode length (Le) of 20 mm, electrode thickness (s) of 1 mm, and interelectrode gap (d) of 2.5 mm.

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Table 10.1. Boundary conditions for Figure 10.3 assuming thermal insulated walls. Boundary (location)

Figure 10.4. Electric field strength and temperature increase during one single pulse. Thermal-insulated electrodes are assumed. Simulation was conducted for a parallel plate treatment chamber geometry with salt solution of a conductivity according to Equation 10.6.

equation and Poisson’s equations, as described in Chapter 9. The unsteady-state simulation is performed by modifying the source of heat in the thermal energy equation. The modification consists of coupling the source of heat with the pulsating joule heating: Qs = σ E = σ ∇V 2

2

(10.5)

where V is the applied voltage as a function of time (rectangular function assuming a square wave pulse of the pulse width τ according to Figure 10.4). Since the pulse width is 1.45 μs in the considered case, the minimal time step for a transient simulation should be lower than 1.45 μs; in this case, the minimal time step was set to 0.1 μs. For the interval between the pulses (1 Hz) the time step was set to 0.1 second. The boundary conditions are summarized in Table 10.1 and correspond to those shown in Figure 10.3. Density, viscosity, and heat capacity of a salt solution were assumed to be similar to water. The correlation between the electrical conductivity (σ in S/m) and the temperature (T in K) was determined experimentally and can be expressed according to Equation 10.6 according to a media conductivity of σ = 0.23 S/m at 25°C.

σ = 6 ⋅ 10 −4 × T + 0.058

(10.6)

Value

Electrostatic model • Ground electrode (A) • High voltage electrode (B) • Insulator (C and D)

V = V0 V =0 n ⋅ σ ⋅ ∇V = 0

Thermal model • Insulated walls (A, B, C, D)

n ⋅ ( λ∇T ) = 0

The thermal properties are set up in the numerical code as dependent on the temperature. The square wave pulse and the temperature as function of the process time are shown in Figure 10.4 for a treatment chamber with a parallel plate electrode geometry. In the case of a heat flux condition, where the solid walls exchange energy with the surrounding environment through convective heat transfer, it is necessary to set the heat transfer coefficient αo and the external bulk temperature Tinf. Taking into account the boundary conditions of Table 10.1, the heat transfer model can be expressed as shown in Equation 10.7: n ⋅ (λ∇T ) = α o (Tinf − T )

(10.7)

The value of αo depends on the geometry and the ambient flow conditions. In the conducted simulations, forced convection between the walls and the surrounding media was assumed. For a laminar external forced convection, Equation 10.8 can be used (COMSOL 2008).

λ 0.928 Pr 0.33 Re 0.5 α o,aveg = ⎛⎜ ⎞⎟ 0.67 0.25 ⎝ L⎠ ⎛ 0.0207 ⎞ ⎞ ⎛ ⎜⎝1 + ⎜⎝ Pr ⎟⎠ ⎟⎠

(10.8)

where λ is the thermal conductivity (W/m/K), L is the characteristic length (m), Pr is the Prandtl number, and Re is the Reynolds number. The impact of the heat transfer coefficient and the pulse number on the temperature increase is shown in Figure 10.5.

Chapter 10

10.5. Steady-State Simulation of Continuous PEF Processing The overall model consists of the following coupled set of models: a thermofluiddynamic model and an electric model in two-dimensional cylindrical coordinates. The equation system is solved using a steadystate solver in COMSOL Multiphysics™. Lagrangequadratic elements are chosen as the basis functions with triangular-shaped elements. The simulation was performed for a colinear treatment chamber consisting of two treatment zones (Figure 10.6). The energy dissipation in the first treatment zone is important to be considered

Figure 10.5. Impact of the heat transfer coefficient on the simulated temperature. The calculation is based on a square wave pulse width of 1.4 μs.

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since it affects the temperature and the corresponding properties of the fluid; therefore, the whole treatment chamber was simulated. The maximum mesh size is 0.075 mm and expands over the whole geometric domain by an element growth rate of 1.075. Close to 117,000 elements were used and the total number of degrees of freedom was around 1,000,000 (see Figure 10.6). The governing equations for the thermofluiddynamic model consist of the total continuity equation, the momentum equation (incompressible Navier– Stokes equation), and the thermal energy equation. The electrostatic model is based on Poisson’s equations, assuming no generation of an electromagnetic force. The boundary conditions for the fluid dynamics model assume a fully developed laminar flow profile at the inlet of the treatment chamber and no-slip conditions at the walls. For the thermal model, a temperature T0 is assumed at the inlet of the treatment chamber. Since the treatment chamber used for the numerical simulation was surrounded by an insulating material, thermal insulation of the walls was considered as boundary condition. The boundary conditions for the steady-state simulation are summarized in Table 10.2. The heat source of the thermal energy equation is coupled to the maximum electrical energy continuously delivered; thus, it has to be multiplied by f × τ (pulse frequency and pulse width) to obtain values for averaged electrical energy input (steady source term; see also Chapter 11 of this book).

Figure 10.6. Geometric model (A) and numerical mesh (B) for a colinear treatment chamber consisting of two treatment zones (numbers 7 and 8). Boundary conditions (1–8) as shown in Table 10.2. Due to symmetry conditions, only one half of the treatment chamber is shown.

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Table 10.2. Boundary conditions for a steady-state simulation of the PEF process (see Figure 10.6 for location of the boundaries). Boundary and number

Value

Electrostatic model • HV electrode (6) • Ground electrode (4,5) • Insulator (1,2,7,8) • Axial symmetry (3)

V = V0 V =0 n ⋅ σ ⋅ ∇V = 0 r=0

Thermal model • Inflow (1) • Outflow (2) • Walls (4,5,6,7,8) • Axial symmetry (3)

T = T0 n ⋅ q = 0 , q = − λ∇T n ⋅ λ∇T = 0 r=0

Flow model • Inflow (1) • Outflow (2) • Wall (4,5,6,7,8) • Axial symmetry (3)

u = u0 u=0 n⋅u = 0 r=0

Figure 10.7. Contour plots of the electric field strength (E), the velocity profile (u), and the temperature (T) distribution as simulated for the second treatment zone of the colinear treatment chamber. See color insert.

This calculation is based on the assumption that the residence time of the sample in the treatment zone is large compared with the time period in between subsequent pulses; that is, the product receives many pulses while passing the treatment zone (Fiala et al. 2001). The thermophysical properties are implemented in the numerical simulation as temperature dependent. The density, viscosity, and heat capacity of a salt solution were assumed to be similar to water (Gerlach et al. 2008). The correlation between the electrical conductivity (σ in S/m) and the temperature (T in K) can be expressed according to Equation 10.9 for a medium with a conductivity of σ = 0.47 S/m at 25°C

σ = 1 ⋅ 10 −3 × T + 0.132

(10.9)

The simulated electric field strength, the velocity profile, and the temperature distribution in the colinear treatment chamber are shown in Figure 10.7. The inhomogeneity of the electric field strength in the colinear treatment chamber illustrated in Figure 10.7 has to be considered when studying inactivation kinetics, especially when the kinetics

depend on the electric field strength and the treatment time (Jaeger et al. 2009). The use of a treatment chamber with homogeneous treatment conditions should be preferred in that case. The maximum increase in temperature is reached at the wall due to the laminar flow conditions causing the residence time of the liquid sample in the electric field to become longer, thus allowing the liquid to receive a greater treatment extent. An alteration of the flow behavior in order to minimize an increase in temperature is an important task to be solved to allow a nonthermal process. Therefore, the next section refers to a possible improvement of a continuous colinear treatment chamber.

10.6. Improvement of the PEF Treatment Homogeneity Homogeneity of the PEF treatment in a continuous treatment chamber is linked to the electric field strength, the flow velocity profile, and the distribution of the temperature. As described in Section 10.5, the temperature increases due to the dissipation of the electrical energy, depending on the electric field

Chapter 10

strength, frequency, pulse width, and electrical conductivity of the media. The homogeneity of the temperature can be controlled if the electric field strength and the flow velocity, and thus the residence time of the liquid product, are homogeneous. In the treatment chamber, the exposure time of the liquid sample to the electric field and the corresponding joule (ohmic) heating are proportional to the residence time between the electrodes. In turn, the residence time in the treatment chamber is proportional to the inverse of the flow velocity (Lindgren et al. 2002). Thus, a laminar flow profile of the liquid should be avoided in order to increase the homogeneity of the residence time in the treatment zone and consequently the distribution of temperature. A possible system to avoid a laminar profile, without the necessity to increase the fluid velocity and producing turbulence, is the utilization of grids as mixing devices. According to Castro et al. (2003), a grid promotes an improved homogeneity of a velocity profile and produces higher turbulence intensity. A modification of the colinear treatment chamber shown in Figure 10.6 was performed by inserting a polypropylene mesh or a stainless-steel woven wire cloth. Both grids had a mesh size of 350 μm and a wire diameter of 180 μm. The grids were located at different positions within the treatment chamber, according to Figure 10.8 and the following designs: Design A: Treatment chamber without insertion of grids (Figure 10.6). Design B: Two grids made of polypropylene, located upstream of each treatment zone of Figure 10.6. Design C: Four grids made of stainless steel, located upstream and downstream of each treatment zone of Figure 10.6. A three-dimensional (3D) simulation was performed for the electric field strength calculations, whereas a two-dimensional simulation was applied for the joule (ohmic) heating and the calculation of velocity profiles because of the limited available computer capacity. The insertion of the stainless-steel grids upstream and downstream of the treatment zone alters the

CFD Applied in PEF Preservation of Liquid Foods

201

Figure 10.8. Location of grids in the treatment chamber (only one treatment zone is shown) and comparison of electric field strength for the designs B and C. Two slices of the simulated electric field strength in a 3D geometry with V0 = 15 kV are shown. The mean electric field strengths and standard deviations are 23.33 ± 2.83, 22.50 ± 3.08, and 30.83 ± 1.58 kV/cm for the design A, B, and C, respectively. Calculation based on the total volume of the treatment zone of 4 mm length (Li), according to Figure 10.1. See color insert.

electrode configuration. The distance between the ground and high-voltage electrode is shorter, which increases the electric field strength. Furthermore, the electrode surface is also increased, which leads to a higher flow of electrical current, thus producing an increase in the dissipated heat. Nonetheless, the insertion of grids improves the fluid mixture and alters the common parabolic profile of laminar flows, which results in lower standard deviations of the flow velocity considering a radial coordinate in the treatment zone (Figure 10.9). Figure 10.10 shows the simulated temperature after the insertion of grids and reveals a more homogeneous temperature distribution within the treatment chamber, which is a result of the reduction in temperature peaks. The maximum occurring temperature can be reduced from 80 to 55°C due to

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Figure 10.9. Comparison of the velocity profile for the designs A, B, and C. The mean velocity and standard deviation are based on the radial coordinate at 1, 2, and 3 mm downstream of the beginning of the second treatment zone of 4 mm length (Li).

mixing effects downstream of the grids. The temperature in the center of the treatment chamber can be increased by increasing the electrical field intensity between the two metal grids.

10.6.1. Impact of the Insertion of Grids on the PEF Effect on ALP and the Inactivation of Escherichia coli The insertion of grids in the treatment chamber aims to modify the flow characteristics, electric field strength, and temperature distribution. An increased field strength from 23.33 to 30.83 (see Figure caption 10.8), the alteration of the flow characteristics (Figure 10.9), and the avoidance of peaks of high temperature (Figure 10.10) could be shown in the conducted investigations. The reduction in temperature hot spots during the PEF treatment will contribute to the retention of heat-sensitive compounds. The effect of the insertion of grids on ALP inactivation during PEF treatment was analyzed. ALP is a heat-sensitive enzyme used as a heat indicator during conventional pasteurization of milk. Considering the low thermal stability of ALP, which already shows detectable inactivation

Figure 10.10. Temperature distribution in the colinear treatment chamber for the designs A, B, and C and residual activity of milk alkaline phosphatase (ALP) after a PEF treatment at a voltage of 18 kV, a frequency of 96 Hz, a flow rate of 10 L/h, and a specific energy input of 130 kJ/kg in a colinear treatment chamber with insertion of different grids. See color insert.

at 60°C for holding times below 5 s (heat inactivation data not shown), a higher retention of ALP activity can be interpreted as a result of the improved temperature homogeneity in the treatment chamber.

Chapter 10

CFD Applied in PEF Preservation of Liquid Foods

Winput =

Figure 10.11. Inactivation results of E. coli after the insertion of grids. Inlet temperature was 30°C, flow rate was 10 L/h, and treatment time was in the range of 12–19 μs for the different energy input levels.

The relative residual activity of ALP in raw milk after a PEF treatment at a voltage of 18 kV, a frequency of 96 Hz, a flow rate of 10 L/h, and a specific energy input of 130 kJ/kg is shown in Figure 10.10. It can be observed that the retention of ALP activity is increased with the design B and is even higher in the model C. The ALP activity can be retained because of the elimination of peaks of high temperature as shown by the numerical simulations (Figure 10.10), and the ALP activity has proven to be a capable tool for the validation of the simulations. The PEF inactivation of E. coli was performed in Ringer solution (conductivity 3 mS/cm) and was used as an indirect indicator of increased average electric field strength due to the treatment chamber modifications. The inactivation of E. coli by the PEF treatment for the treatment chamber designs A and C is shown in Figure 10.11. The microbial inactivation was enhanced by an average of 0.6 log-cycles for the considered treatment intensity (specific energy input) range of 80– 120 kJ/kg. Due to varying current flows for the different treatment chamber configurations, the energy input was aimed to be constant and adapted by changing the pulse repetition rate and was calculated according to Equation 10.10:

I ⋅V ⋅τ ⋅ f m

203

(10.10)

where I is the electrical current, V is the applied voltage, τ is the pulse width, f is the pulse repetition rate, and m is the mass flow. The improved inactivation of E. coli observed in the design C is mainly due to the increased electric field strength obtained when using stainless-steel grids. The higher homogeneity of the electric field strength in the design C (37.6 ± 1.9 kV/cm) in comparison to the design A (28.6 ± 3.4 kV/cm) allows for a more homogeneous treatment and avoids the exposure of the microorganisms to zones with low electric field strength. Furthermore, the slightly increased overall temperature in the treatment zone (Figure 10.10) can improve the microbial inactivation due to synergistic effects between temperature and electric field strength (Toepfl 2006).

10.7. Turbulence Simulation in PEF Processing The effects produced by turbulence in the treatment chamber may or may not be desirable, depending on the application. Turbulent flow leads to higher skin friction and improved heat transfer in comparison to laminar flows. An intense mixing is desirable during PEF treatments since the electric field strength, temperature, and velocity profiles are not homogeneous. The exposure of particles (e.g., microbial cells) to different locations of the treatment zone with different electric field strengths is only possible by increasing the mixing effects. Temperature increases considerably near the walls in laminar flow conditions because of the low fluid velocity in this zone. The treatment time, as discussed in Sections 10.5 and 10.6, is not uniform because it depends on the velocity profile. Therefore, avoiding the laminar profile and generating a turbulent one will lead to a more homogeneous treatment. In a fully developed turbulent flow through a pipe, a turbulent boundary layer will exist with a thin viscous sublayer immediately adjacent to the wall (Holland and Bragg 1995). The strong chaotic

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

motion causes the various layers of the fluid to mix intensely. An extra-fine mesh on the walls should be used for simulation in order to obtain an accurate prediction of the turbulent fluctuations that exist in this zone. PEF processing at high flow rates requires accurate models of fluid dynamics, and the introduction of a turbulent model is necessary. Several models for the description of turbulence exist and their application depends on the specific requirements of the particular case. A detailed explanation about turbulence models can be found in Ferziger and Peric (2002). In the present study, the standard k-ε turbulence model was used. The momentum and heat transfer equations presented in the previous chapter have to be modified and the introduction of two new variables, the turbulent viscosity in the momentum equation and the turbulent thermal conductivity in the heat transfer equation, is required. The equations for the momentum balances and continuity considering a Newtonian fluid are the following:

ρ

C μ kT2 ⎞ ∂u ⎡⎛ T ⎤ − ∇ ⋅ ⎢⎜ η + ρ ⋅ ∇u + ( ∇u ) ⎥ ⎟ σk ε ⎠ ∂t ⎣⎝ ⎦ + ρu ⋅ ∇u + ∇p = 0

(

)

(10.11)

where ρ denotes the density of the fluid (kg/m3), u represents the average velocity (m/s), η is the dynamic viscosity (kg/m/s), p is the pressure (Pa), kT is the turbulent energy (m2/s2), and ε is the dissipation rate of the turbulence energy (m2/s3). The turbulence energy is given by Equation 10.12:

ρ

Cμ k ⎞ ∂kT ⎡⎛ ⎤ − ∇ ⋅ ⎢⎜ η + ρ ⋅ ∇kT ⎥ + ρu ⋅ ∇k σ k ε ⎟⎠ ∂t ⎣⎝ ⎦ 2 T

= ρCμ

(

kT2 T ∇u + ( ∇u ) ε

)

2

− ρε

(10.12)

And the dissipation by Equation 10.13:

ρ

C μ kT2 ⎞ ∂ε ⎡⎛ ⎤ − ∇ ⋅ ⎢⎜ η + ρ ⋅ ∇ε ⎥ + ρu ⋅ ∇ε ⎟ ⎝ ⎠ σε ε ∂t ⎣ ⎦

(

)

T 2

= ρCε 1Cμ kT ∇u + ( ∇u )

− ρCε 2

ε2 kT

(10.13)

The model constants used in the turbulence energy and dissipation equations were set to the following values: Cμ: 0.09, Cε1: 1.44, Cε2: 1.92, σk:0.9, and σε: 1.62 (Wilcox 1998). The general equation for heat transfer, which considers convection and conduction, is:

ρc p

∂T + ∇ ( − ( λ + λT ) ⋅ ∇T + ρc pTu ) = Qs ∂t

λT = c pυT

(10.14) (10.15)

where cp denotes the specific heat capacity (m2/s2/K), T is the temperature (K), λ is the thermal conductivity (kg m/s3/K), λT is the turbulent thermal conductivity, υT is the turbulent kinematic viscosity (kg/m/s), ρ is the density (kg/m3), u is the velocity vector (m/s), and Qs is a sink or source term (kg/m/s3). The criterion for using the k-ε model instead of a laminar model is based on the consideration of turbulence according to the Reynolds number. Turbulent flow in pipes is considered at Reynolds number above 3,000 (McComb 1990; Martin 2002). A turbulent model for PEF simulation might also be interesting from an industrial point of view since PEF processing at a high mass flow is desirable, which can be related to turbulences inside the treatment zone.

10.8. Characterization of PEF Processing Conditions Assisted by Numerical Simulations A continuous PEF treatment in a colinear treatment chamber shows an inhomogeneous electric field strength distribution. The treatment time is also inhomogeneous, since the velocity profile differs considerably from the wall to the center of the treatment zone. These inhomogeneities can lead to differences in the treatment intensity and produce local heating of the sample. Numerical simulations allow the characterization of the process conditions to a certain extent, since the obtained results can be analyzed by a statistical software. The electric field strength is related to the applied voltage, the geometry of the treatment chamber, and

Chapter 10

the insulator. This relation can be approximated by a constant. The constant is a defined conversion factor (g), which relates the applied voltage to the resulting electric field strength in a given treatment chamber and can be implemented according to Equation 10.16: E = V ⋅g

(10.16)

where E is the electric field strength (V/m), V is the applied voltage (V), and g is the conversion factor (m−1). In the case of a treatment chamber with parallel electrode configuration, the conversion factor can be defined as the reciprocal of the interelectrode gap d (m): g =1 d

(10.17)

Subsequently, in this case, Equation 10.16 is valid for each point of the treatment chamber and results in the same field strength value independent of the location in the treatment chamber. For other electrode configurations, the most suitable method to estimate the conversion factor is to consider the entire treatment zone, where an electric field is present. A calculation method according to Equations 10.18 and 10.19 can be applied (Gerlach et al. 2008): Eaveg =

1 Vtreat

N

⋅

∑ E ⋅ δV i

i

(10.18)

i =1

N

Vtreat =

∑ δV

i

(10.19)

i =1

where Eaveg is the average electric field strength simulated over all N elements of the numerical grid, Ei is the electric field strength occurring in a volume element δVi, and Vtreat is the total volume inside the treatment chamber, where the electric field strength is considered to reach significant values for affecting the treatment media, for example, >10 kV/cm for most bacteria and >4.7 kV/cm for yeasts (Grahl and Märkl 1996). The treatment time is another important parameter necessary to describe a PEF process. The treatment time depends on the number of pulses, the pulse length, and the residence time of the sample in the treatment zone (Eq. 10.4).

CFD Applied in PEF Preservation of Liquid Foods

205

Table 10.3. Flow velocity, treatment time, and electric field strength for a colinear treatment chamber (design A).

Coordinate

0.1 mm from wall

1 mm from wall

2 mm from wall (pipe center)

v (m/s) ttreat (μs) Eaveg (kV/cm)

0.11 8.42 37.1 ± 1.9

0.27 3.42 31.8 ± 1.7

0.28 3.22 30.2 ± 1.9

Salt solution (conductivity 3 mS/cm) at a flow rate of 10 L/h, PEF treatment at a pulse frequency of 75 Hz, pulse duration of 3 μs, an initial voltage of 22 kV, and an initial temperature of 30°C.

An inhomogeneous flow velocity, as occurring in laminar flow conditions, will result in an inhomogeneous residence time of the sample, and thus an inhomogeneous treatment time. The fluid velocity closest to the walls is expected to be lower than in the center of the treatment chamber, leading to a lower treatment time in this central part. Considering three different sections in the treatment chamber design A (Figure 10.8), 0.1 mm from the wall, 1 mm from the wall, and in the center of the treatment chamber, the resulting fluid velocity is shown in Table 10.3. The treatment time (tt) was calculated according to Equation 10.4 considering the residence time of the liquid in the insulator zone (length L = 4 mm, flow velocity u), the pulse frequency ( f ), and the pulse width (τ). A large variation of the treatment time was observed depending on the location within the treatment chamber. The treatment time between the location close to the wall and the center of the treatment zone varies between 8.42 and 3.22 μs, respectively. Due to the lower flow velocity near the wall, residence and therefore treatment times are significantly higher (153%) than in the center.

10.9. Conclusion and Outlook One key aspect of maximum PEF effectiveness is the performance of the treatment chamber. The treatment chamber design plays a major role, especially

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

for continuous treatment of liquid foods in terms of assuring food safety and stability. Maximum microbial inactivation with minimal negative impact on other valuable food constituents and bioactive components requires an accurately defined treatment intensity including homogeneous treatment conditions. This uniform treatment can be achieved by homogeneous electric field distribution, flow velocity distribution and residence time resulting in a uniform temperature distribution in the treatment chamber. However, most treatment chamber designs do not provide a uniform distribution of the electric field, and laminar fluid flow characteristics lead to residence time deviations. The examples discussed in this chapter show how the numerical simulation of PEF processes with computational tools allows the characterization of PEF treatment parameters such as the temperature in the treatment chamber in addition to the measurement of an average outlet temperature as performed during experimental studies. Simulation of electric field distribution and fluid flow characteristics is the basis for the treatment chamber optimization in order to improve the treatment homogeneity as a key parameter for kinetic inactivation studies as well as the industrial implementation of the PEF technology.

Acknowledgments The authors would like to thank the German Ministry of Economics and Technology (via AiF) and the FEI (Forschungskreis der Ernährungsindustrie e.V., Bonn), Project AiF 15610 N, and the Commission of the European Communities, Framework 6, Priority 5 “Food Quality and Safety,” Integrated Project NovelQ FP6-CT-2006-015710. Nicolás Meneses gratefully acknowledges the financial support from DAAD (German Academic Exchange Service).

Notation Latin Variables A A0 cp

Enzyme activity Initial enzyme activity Specific heat capacity

Di De d E Eaveg Ei f g I kT L Le Li m n N p r Ri q s Qs ΔT T T0 Tinf t tt u Wspec V V0 Vi Vtreat

Treatment zone radius Electrode radius Inter electrode gap Electric field strength Average electric field strength Electric field strength occurring in a volume element Frequency or pulse repetition rate Conversion factor Electrical current Turbulent energy Characteristic length Electrode length Length of treatment zone Mass flow rate Normal vector Element of the numerical grid Pressure Radius Insulator radius Heat flux Wall or electrode thickness Source term Temperature difference Temperature Initial temperature External bulk temperature Time Treatment time Velocity vector Specific energy input Electric potential Initial voltage Volume element Total volume of treatment zone

Abbreviations ALP HV FEM PEF RA

Milk alkaline phosphatase High voltage Finite element method Pulsed electric field Relative activity

Chapter 10

Greek Variables ∇ ∂ ∂t αo α o,aveg Cε1 Cε2 Cμ λ η ρ σ σk σε τ υT

Gradient operator Partial derivative Outer convective heat transfer coefficient Average outer convective heat transfer coefficient Dissipation rate equation production coefficient Dissipation rate equation dissipation coefficient Eddy viscosity coefficient Thermal conductivity Dynamic viscosity Density Electrical conductivity Turbulent Prandtl number for kinetic energy Turbulent Prandtl number for dissipation rate Pulse width Turbulent kinematic viscosity

References Alkhafaji S, Farid M. 2007. An investigation on pulsed electric fields technology using new treatment chamber design. Innovat Food Sci Emerg Technol 8(2):205–212. Barbosa-Cánovas GV, Góngora-Nieto MM, Pothakamury UR, Swanson BG. 1999. Preservation of Foods with Pulsed Electric Fields. San Diego: Academic Press. Bazhal M, Ngadi M, Raghavan GSV, Smith JP. 2006. Inactivation of Escherichia coli O157:H7 in liquid whole egg using combined pulsed electric field and thermal treatments. LWT 39:419–425. Castro H, Marighetti J, De Bortoli M, Natalini M. 2003. Modificación de la intensidad de la turbulencia en el canal de aire de la UNNE. Chaco, Argentina: Universidad Nacional del Nordeste. COMSOL. 2008. COMSOL User ’s Guide. Version 3.5a. Burlington, VT: COMSOL. Dunn JE, Pearlman JS. 1987. Methods and apparatus for extending the shelf life of fluid food products. United States: patent. Ferziger JH, Peric M. 2002. Computational Methods for Fluid Dynamics. Berlin: Springer. Fiala A, Wouters P, Bosch E, Creyghton Y. 2001. Coupled electrical-fluid model of pulsed electric field treatment in a model food system. Innovat Food Sci Emerg Technol 2:229–238.

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Gerlach D, Alleborn N, Baars A, Delgado A, Moritz J, Knorr D. 2008. Numerical simulations of pulsed electric fields for food preservation: A review. Innovat Food Sci Emerg Technol 9(4):408–417. Grahl T, Märkl H. 1996. Killing of microorganisms by pulsed electric fields. Appl Microbiol Biotechnol 45:148–157. Holland FA, Bragg R. 1995. Fluid Flow for Chemical Engineers, 2nd ed. London: Elsevier. Huang K, Wang J. 2009. Designs of pulsed electric fields treatment chambers for liquid foods pasteurization process: A review. J Food Eng 95(2):227–239. Jaeger H, Meneses N, Knorr D. 2009. Impact of PEF treatment inhomogeneity such as electric field distribution, flow characteristics and temperature effects on the inactivation of E. coli and milk alkaline phosphatase. Innovat Food Sci Emerg Technol 10:470–480. Jeyamkondan S, Jayas DS, Holley RA. 1999. Pulsed electric field processing of foods: A review. J Food Prot 62(9):1088–1096. Knorr D, Geulen W, Grahl T, Sitzmann W. 1994. Food application of high electric field pulses. Trends Food Sci Technol 5:71–75. Lelieveld HLM, Notermans S, de Haan SWH. 2007. Food Preservation by Pulsed Electric Fields. Abington, MA: Woodhead Publishing. Lindgren M, Aronsson K, Galt S, Ohlsson T. 2002. Simulation of the temperature increase in pulsed electric field (PEF) continuous flow treatment chambers. Innovat Food Sci Emerg Technol 3:233–245. Martin H. 2002. Dimensionslose Kenngrößen. In: VD Ingenieure, ed., VDI Wärmeatlas. Heidelberg: Springer-Verlag. Meneses N, Jaeger H, Moritz J, Knorr D. 2010. Impact of insulator shape, flow rate and electrical parameters on inactivation of E. coli using a continuous co-linear PEF system. DOI: 10.1016/ j.ifset.2010.11.007. Qin B-L, Zhang Q, Barbosa-Cánovas GV, Swanson BG, Pedrow PD. 1994. Inactivation of microorganisms by pulsed electric fields of different voltage waveforms. IEEE Trans Dielectr Electr Insulat 1(6):1047–1057. Raso J, Heinz V. 2006. Pulsed Electric Fields Technology for the Food Industry. Food Engineering Series. Heidelberg: Springer-Verlag. Toepfl S. 2006. Pulsed electric fields (PEF) for permeabilization of cell membranes in foodand bioprocessing—Applications, process and equipment design and cost analysis. PhD dissertation, Berlin University of Technology, Berlin. Toepfl S, Heinz V, Knorr D. 2007. High intensity pulsed electric fields applied for food preservation. Chem Eng Process 46:537–546. Van den Bosch H, Morshuis P, Smit J. 2002. Temperature distribution in continuous fluid flows treated by PEF. Paper read at XIII International Symposium on High Voltage (ISH), Delft, the Netherlands. Wilcox DC. 1998. Turbulence Modeling for CFD. La Cañada, CA: DCW Industries.

Tad 1.00 0.95 0.89 0.84 0.79 0.74 0.68 0.63 0.58 0.53 0.47 0.42 0.37 0.32 0.26 0.21 0.16 0.11 0.05 0.00

u/(L/tprocess)

T

18 16 14 12 10 8 6 4 2 0

W=0 W = 0.25 W = 0.5

0

0.2

0.4

0.6

0.8

1

t/tprocess

Figure 4.2. Development of temperature field and vertical velocity near the cylindrical wall at mid-height of the autoclave during a high-pressure process with aqueous sucrose solution containing W = 0, 0.25, 0.5 kg/kg sucrose (1, 5: forced convection; 2: development of thermal/ hydrodynamic layer; 3: fully developed thermal/hydrodynamic layer; 4: horizontal temperature stratification). (a) A* 1.00 0.95 0.89 0.84 0.79 0.74 0.68 0.63 0.58 0.53 0.47 0.42 0.37 0.32 0.26 0.21 0.16 0.11 0.05 0.00 (b) A* 1.00 0.95 0.89 0.84 0.79 0.74 0.68 0.63 0.58 0.53 0.47 0.42 0.37 0.32 0.26 0.21 0.16 0.11 0.05 0.00

PPO

LOX

BSA

BGLU

PPO

LOX

BSA

BGLU

Figure 4.3. Dimensionless enzyme activity A* (enzymes suspended in water) at the end of pressure-holding time of a highpressure process (autoclave 2.3 L; pressure ramp 400 MPa/s; pressure-holding time 120 s at 700 MPa; initial temperature of process [a] 323 K and [b] 293 K). PPO, polyphenoloxidase; LOX, lipoxygenase; BSA, bacillus subtilis α-amylase; BGLU, β-glucanase (Rauh et al. 2009). Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

Dimensionless temperature increase

1.0 Particle 1 Particle 2

0.8

0.6

0.4

0.2

0.0

–0.2

0.0

0.2

0.4

0.6

0.8

1.0

Dimensionless pressure application time tt* Figure 4.4. Different treatment histories of two enzymes due to nonuniform temperature distributions and convective and diffusive transport phenomena during the process. Left: particle tracks of two different enzymes (Particle 1, Particle 2) during a high-pressure process; right: temperatures faced by the two enzymes (Particle 1, Particle 2) at different locations and times.

(a)

(b)

log S 0

T (°C) –2

Vessel wall

110 –4

Vessel wall

3-L unit

35-L unit

100 –6

90 80

–8

70

–10

60

–12 –14

Carrier walls Figure 5.7. Comparison of the temperature distribution predicted by CFD models of two high-pressure sterilization vessels including a PTFE carrier at the end of 315 s holding time (initial temperature 90°C and 600 MPa): (a) a 35-L vessel; (b) a 3-L vessel (not to scale).

(a) without

(b) metal

(c) PTFE

carrier

Figure 5.9. Predicted distribution of extent of C. botulinum log reduction extent inside a 35-L pilot-scale high-pressure vessel in three scenarios: (a) vessel without carrier, (b) vessel including a metal composite carrier, and (c) vessel including a Teflon (PTFE) carrier (from Knoerzer et al. 2007).

(a)

Z

(b) r

Time=350 Surface:Temperature [K] Max: 394 Arrow: Velocity field [m/s] 1.4

390

1.35

metal lid

1.3 1.25

top water entrance

380

1.2 1.15

air layer

1

1.1

370

1.05 1

2

0.95

360

0.9 stainless steel

0.85 3

0.8

350

0.75 4

0.7 0.65

340

0.6 water

5

0.55 0.5

330

0.45 6

0.4 0.35

PTFE carrier

7

320

0.3 0.25 0.2

metal valve

310

0.15 0.1 0.05

300

0 –0.05 –0.34 –0.26 –0.18 –0.1 –0.02 0.06

water inlet

Min: 293

Figure 5.10. CFD model of a 35-L vessel including carrier, packages, steel walls, and metal lid: (a) computational domains of the model structure and (b) thermal and flow profile in the vessel at the end of holding time (315 s) at 600 MPa (from Juliano et al. 2009). Conditions simulated include starting temperature and pressurization rate to a final pressure of 600 MPa and a holding time of 315 s. Arrows proportional to the maximum velocity at specific time.

(a)

(b)

(c)

(d)

log S 0

–2

–4

–6

–8

–10

–12

–14

–16

Figure 5.11. Predicted distribution of C. botulinum log reduction extent according to four selected kinetic inactivation models using a CFD model platform for a high-pressure 35-L sterilization system: (a) traditional log-linear kinetic model, (b) Weibull distribution model, (c) nth-order kinetic model, and (d) combined discrete log-linear and nth-order kinetic model (adapted from Juliano et al. 2009).

× 10–3 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5

18 16 14 12 10 8 6 4 2 2

4 2

2

4

6

8

10

12

14

16

18

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 × 10–3 Side view (y-z) 4 2

2

4

6

8

10

12

14

16

18

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 × 10–3

4 6 8 10 12 14 16 18 Top view of central plane (x-y)

Front view (x-z)

(a) Electric field distribution (density) inside mashed potato with 0% salt content. × 10–3

18

4 2

3.5

16

12

14

2.5

3

3.5 × 10–3

12

14

16

0.5 1 1.5 2 2.5 4 6 8 10 12 14 16 18 Top view of central plane (x-y) Front view (x-z) (b) Electric field distribution inside mashed potato with 0.5% salt content.

3

14

3

12

2.5

10

2

8

2 0.5

4 2

1

4

6 1

8 1.5

10 2

2

4

6

8

10

0.5

2

2

× 10–3

18

4 2

3.5

16 14

3

12

2.5

10

2

8 4

1

2

0.5 2

4

6

8

10

12

14

2

4

0.5

6

1

8

1.5

10 2

12

14

2.5

3

Side view (y-z) 4 2

1.5

6

16

18

Side view (y-z)

1.5

6

4

2 0.5

16

Top view of central plane (x-y)

4 1

6 1.5

8

10 2

12

14

2.5

3

18

18

3.5 × 10–3

16

18

3.5 × 10–3

16

18

3.5 × 10–3

Front view (x-z)

(c) Electric field distribution (density) inside mashed potato with 1.0% salt content. Figure 6.14. Electric field distribution (top, front, and side views) inside a food package containing mashed potato with different salt contents. (a) Electric field distribution (density) inside mashed potato with 0% salt content. (b) Electric field distribution inside mashed potato with 0.5% salt content. (c) Electric field distribution (density) inside mashed potato with 1.0% salt content.

120 115

120

110

110

105

100

100 95

90

90 85

80 70 20 15 10 y

Temperature (°C)

Temperature (°C)

130

80 5

2

0 1

3

z

5

4

6

75

120

125 120 115 110 105 100 95 90 85 80 20

115 110 105 100 95 90 15 10 y

5

120 115 110 105

10

y

5

0 0

2

z

4

6

122 120 118 116 114 112 110 108 106 104

85

124 122

125 Temperature (°C)

Temperature (°C)

125

15

z

6

4

t = 150 s

t = 100 s

100 20

2

0 0

120

120

118

115

116 114

110 105 20

112 110 15

t = 270 s

10 y

5

0 1

2

3

4

5

6

108

z

t = 320 s

Figure 6.16. Temperature distribution at a vertical plane in mashed potato for different heating times. Mashed potato was immersed in 125oC water while heated with 915 MHz energy.

Phase image at Tref

Δ f (Hz) 60

Phase image at T

ΔT (K) 325 320

40

315 20 (a)

(b)

(c)

310 (d)

0

305

Figure 7.5. (a) MRI phase image of a model food at known initial temperature (Tref = const); (b) phase image of heated sample; (c) phase difference distribution (c = a − b); and (d) temperature distribution (calculated from phase difference with knowledge of a chemical shift of Δf/ΔT = 0.01 ppm/°C).

T (K) 310

14

30 simulation

25 20

12

measurement

308

10 306

z (mm)

8 slices

15 10 5 0 60 40 x (mm

304

4 2 0 50 30 x (mm 20 )

40 30 20 y (mm)

)

6

302 60 40 y (mm)

300

Figure 7.11. Visual comparison between the simulated (left) and the measured (right) heating of a model food cylinder (at a discrete time): PMW = 19 W; t = 250 s; Text = 298 K.

T (K) 308

306

304

302

simulation

measurement

Figure 7.12. Visual comparison between the simulated (left) and the measured (right) heating of the center cross-section (h = 16 mm) of the microwave-heated model food cylinder (at a discrete time).

z-dimension (mm)

T (K) 350 60

340

50

330

40

320

30 20

310

10 t = 800 s 0 60 40 x-dimen s

60 20 40 m) 20 0 0 (m n io s n ion (mm y-dime )

300

Figure 7.17. 3D temperature distribution in a chicken wing after microwave treatment of 600 s (power was switched on after 200 s) as calculated using the new simulation procedure (flowchart in Figure 7.10).

(b)

(a)

Time = 150 Surface: Temperature (°C) Max: 83.066

Time = 150 Surface: Temperature (°C) Max: 117.341 0.08

0.08

115

0.06

110

0.04

80

0.06 0.04

105

75

0.02

0.02 100

0.00

95

–0.02

70

0.00

65

–0.02

–0.04

90

–0.04

–0.06

85

–0.06

60

55 –0.08 80 0.02 0.04 0.06 Min: 78.311 –0.06 –0.04 –0.02 0 0.02 0.04 0.06 Min: 52.8 Figure 8.4. Temperature contours for (a) solids having an electrical conductivity of 2 S/m immersed in a fluid of 1 S/m, and (b) solids of electrical conductivity 4 S/m immersed in a fluid with an electrical conductivity of 2 S/m. –0.08

–0.06 –0.04 –0.02

0

(b)

(a) Time = 150 Slice: Temperature (°C)

Max: 95.833

Time = 150 Slice: Temperature (°C)

94

100

92

98

90

96

88

94

0 z

y

0

–0.05

0

86

x

92 z0

y Min: 84.54

Max: 102.253 102

x

–0.05

90 Min: 89.706

Figure 8.7. Three-dimensional solution of the Laplace equation for a fluid with electrical conductivity of 2 S/m and cubes with an edge of 1 cm having electrical conductivities of (a) 1 S/m and (b) 4 S/m.

Figure 9.7. Distribution of the electric field strength and impact on the critical electric field strength for electroporation of microbial cells when attached to each other (case A: agglomeration) or attached to insulating particles (case B: fat globule). Adapted from Toepfl et al. (2007).

Figure 10.7. Contour plots of the electric field strength (E), the velocity profile (u), and the temperature (T) distribution as simulated for the second treatment zone of the colinear treatment chamber.

Figure 10.8. Location of grids in the treatment chamber (only one treatment zone is shown) and comparison of electric field strength for the designs B and C. Two slices of the simulated electric field strength in a 3D geometry with V0 = 15 kV are shown. The mean electric field strengths and standard deviations are 23.33 ± 2.83, 22.50 ± 3.08, and 30.83 ± 1.58 kV/cm for the design A, B, and C, respectively. Calculation based on the total volume of the treatment zone of 4 mm length (Li), according to Figure 10.1.

Figure 10.10. Temperature distribution in the colinear treatment chamber for the designs A, B, and C and residual activity of milk alkaline phosphatase (ALP) after a PEF treatment at a voltage of 18 kV, a frequency of 96 Hz, a flow rate of 10 L/h, and a specific energy input of 130 kJ/kg in a colinear treatment chamber with insertion of different grids.

(a)

(b)

(c)

Figure 11.4. Electro-hydrodynamic fields in the standard PEF chamber for f = 20 Hz, τ = 10 μs, f = 15 kV, RI = 2 mm, and m· = 5.2 kg/h. (a) Velocity field and streamline (flow is from left to right); (b) electric field strength and electric potential; (c) temperature field.

Figure 11.5. Electric field strength computed with temperature-independent material properties (f = 20 Hz, τ = 10 μs, f = 15 kV, RI = 2 mm, and m· = 5.2 kg/h).

Figure 11.8. Residual activity field in the PEF chamber for f = 20 Hz, τ = 10 μs, f = 15 kV, RI = 2 mm, m· = 5.2 kg/h, and n = 25, resulting in a treatment time of 192 μs.

Figure 11.15. Insulator geometry defined by splines.

Pressure [Pa]

Max: 4.722e5 5 –0.05 ×10 0 0.05 4 0.1

0.05

Max: 7.534e ×104 –0.05 7 0 0.05 6 0.1

Int [W/m2]

3

5

2

4 0.05

z x

y –0.05

0

0

0 0.05

2

z

y

x

–0.05

0

0 0.05

Min: –8.939e4

(A) Acoustic pressure

–0.05 0 0.05 0.1

1

4 3.5 3 2.5

0.05

3

1

Max: 4.4e ×103

Intensity, norm [W/m2]

2 1.5

z x

1 y –0.05

0 Min: 0

(B) Intensity (plane wave approximation)

0

0.5

0 0.05

0 Min: 0

(C) Intensity

Figure 12.7. Acoustic pressure and sound intensity simulation for R (radius of the reactor) = H (distance from reactor top to horn tip) = D (distance from horn tip to reactor base) = 50 mm; r (horn radius) = 6.5 mm and λ = 75 mm. (A) Acoustic pressure. (B) Klima’s calculation of the intensity assuming plane wave (Eq. 12.47). (C) Corrected calculation at CFNS via Equation 12.39.

Pressure [Pa]

Int [W/m2]

Max: 9.716e6 ×106 0 8 0.1 6 4

0.05

0 –2

z x

–4 y 0

0

–6

Min: –7.999e6

(A) Acoustic pressure

0

Max: 254 ×107 4

0.1

2.5

1.5

0.05

x

0.5

y 0

0

0 Min: 0

(B) Intensity (plane wave approximation)

2 1.5

1 z

3.5 3

2

2 0.05

Intensity, norm [W/m2]

Max: 3.189 ×107 0 3 0.1 2.5

z x

1 y 0

0

0.5 0 Min: 0

(C) Intensity

Figure 12.8. Acoustic pressure and sound intensity simulation for optimized geometrical configuration: R (radius of the reactor) = 45 mm; H (distance from reactor top to horn tip) = 25 mm; D (distance from horn tip to reactor base) = 77 mm; r (horn radius) = 6.5 mm and λ = 75 mm. (A) Acoustic pressure. (B) Klima’s calculation of the intensity assuming plane wave (Eq. 12.47). (C) Corrected calculation at CFNS via Equation 12.39.

a m–1 0.01

Acoustic pressure Pressure [Pa] 0.1

Intensity Max: 4.722e5 ×105

Max: 1.00e5 ×105 1

Intensity, norm [W/m2] 0.1

4 0.08

0.9 0.7

2

z

1 0.04

0.06

0.6 0.5

z

0.06

0.8

0.08

3

0

0.04

0.4

–1

0.02

–2 0 –0.02

0

1

0.02 r

–3

0.04

0.06 0.08 Min: –3.337e5

Pressure [Pa]

Max: 4.722e5 ×105

0.1

2

z

0.06

1

0.04

0 0.02

–1 –0.02

0

10

0.02 r

0.04

0.06 0.08 Min: –2.492e5

Pressure [Pa]

Max: 4.722e5 ×105

0.1

1 0.5

0

0 –0.02

0

0.02 r

0.04

0.06 0.08 Min: –3.064e4

0.6 0.5

0.04

0.4 0.3

0.02

0.2 0.1

0 –0.02

0

0.02 r

0.04

0.06

Intensity, norm [W/m2] 0.1

0.08 0 Min: 0 Max: 1.00e5 ×105 1 0.9

0.08

0.8 0.7

0.06

0.6 0.5

z

z 0.02

Max: 1.00e5 ×105 1

0.7

2.5 1.5

0.08 0 Min: 0

0.8

0.06

3 2

0.06

0.9

3.5

0.04

0.04

0.08

4

0.06

0.02 r

0.1

4.5

0.08

0

Intensity, norm [W/m2]

–2

0

0.1 –0.02

z

3

0.2

0

4 0.08

0.3

0.02

0.04

0.4 0.3

0.02

0.2 0.1

0 –0.02

0

0.02 r

0.04

0.06

0.08 0 Min: 0

Figure 12.9. Optimized geometry from Figure 12.8 modeling the effect of the absorption coefficient on acoustic pressure and intensity.

Max: 1.953 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 Figure 12.15. Axial–symmetrical CFD velocity distribution predicted from approach 2 for P/V = 35 kW/m3 and S = 0.00281 (Trujillo and Knoerzer 2009).

Min: 3.544e-7

1 NODAL SOLUTION OCT 16 2009 12:42:56

STEP =1 SUB =8 FREQ =22045 /EXPANDED UZ (AVG) RSYS =0 DMX =7.707 SMN =–7.521 SMX =3.987

Y Z

–7.521

–6.242

–4.963

–3.685

–2.406

–1.128

.150918

1.43

2.708

Figure 13.12. Vibrational displacements in the cylindrical tube of the CRT system along the Z-axis.

3.987

ANSYS 6.1 JAN 21 2003 10:43:32 AVG ELEMENT SOLUTION STEP =1 SUB =1 FREQ=22269 /EXPANDED SPL (AVG) DMX =.537E–05 SMN =110.313 SMX =169.424 XV =1 YV =1 ZV =1 DIST =.172893 YF =.155 =–.025 ZF Z-BUFFER 110.313 116.881 123.449 130.017 136.585 143.152 149.72 156.288 162.856 169.424

Drying chamber: acoustic distribution Figure 13.14. 2D standing wave field pattern of the acoustic pressure inside the radiating tube obtained in ANSYS code.

Feq_acax(1)=21625 Surface: Sound pressure level

Max:184.58

0.3

180

0.25

170

0.2

160

0.15

150

0.1 140

0.05 130

0 –0.15

–0.1

–0.05

0

0.05

0.1

0.15

0.2

0.25

120 Min:119.856

Figure 13.15. 2D standing wave field pattern of the acoustic pressure inside the radiating tube obtained in COMSOL code.

(a)

160 165 160

155

SPL (dB)

155 150

150

145 140

145

135 130

140

125 100 80

135

400 60

300 40

200 20 0 0

Length (mm) (b)

130

100 Radius (mm)

SPL (dB) 0 160 50 155

LENGTH (mm)

100 150 150 145

200

140

250

135

300

130 0

10

20

30

40

50

60

70

80

90

RADIUS (mm) Figure 13.17. (a) 3D-sound pressure level distribution pattern (in decibels) inside the cylindrical radiator: electrical power applied, 90 W. (b) 2D-sound pressure level distribution pattern (in decibels) inside the cylindrical radiator: electrical power applied, 90 W (Gallego-Juárez et al. 2010).

0.1600 0.1565 0.1530 0.1495 0.1460 0.1425 0.1390 0.1355 0.1320 0.1285 0.1250 0.1215 0.1180 0.1145 0.1110 0.1075 0.1040 0.1005 0.0970 0.0935 X 0.0900

Z = 0.155

Z = 0.125

Z = 0.095 Z Pathlines Colored by Z-Coordinate (m)

Y

Figure 14.8. Path lines of fluid released from the annular gaps with Reaxial = 5 and Ta = 200.

39 37 35 33 31 29 27 25 23 21 20 18 16 14 12 10 8 6 4 2 0

Zoom 5

1 2 43

12345

Y X

Z Path Lines Colored by Particle ID

Figure 14.9. Path lines of fluid released from the annular gaps displayed in the X-Y-plane.

6.00e+00 5.70e+00 5.40e+00 5.10e+00 4.80e+00 4.50e+00 4.20e+00 3.90e+00 3.60e+00 3.30e+00 3.00e+00 2.70e+00 2.40e+00 2.10e+00 1.80e+00 1.50e+00 1.20e+00 9.00e–01 Y 6.00e–01 X 3.00e–01 Z Contours of long outlet 0.00e+00 Figure 14.13. Spore survival distribution at the exit of a multiple-lamp UV reactor.

z x

Reactor B, Q=125 m3/h, SAK=3 Figure 15.2. Fluence distribution at the outlet of the reactor B.

3417 2813 2210 1606 1003 400 Fluence, J/m2

912 780 648 516 384 252 Fluence, J/m2 Figure 15.4. Fluence distribution at the outlet of the reactor C.

Z X

Y

Reactor F, Q=125 m3/h, SAK=3 Figure 15.6. Fluence distribution at the outlet of the reactor F.

994 895 795 696 596 497 Fluence, J/m2

768 682 595 509 422 335 Fluence, J/m2 Figure 15.8. Fluence distribution at the outlet of the reactor G.

Inner concentration profiles

1.9 g/L

1.9 g/L

1.5 g/L

1.5 g/L

0.75 g/L

Raffinate

Withdrawal concentration profiles

1.1 g/L 0.38 g/L 0 g/L 0 min

1.1 g/L

1.9 g/L

0.76 g/L

42 min

84 min

1.3E02 min

1.7E02 min

2.1 E02

42 min

84 min

1.3E02 min

1.7E02 min

2.1 E02

Extract

1.5 g/L 1.1 g/L

0.38 g/L

0.75 g/L 0.38 g/L

0 g/L Desorbent

Extract

Feed

Raffinate

0 g/L 0 min

Figure 16.13. Internal concentration profiles and transient extract and raffinate concentrations.

Chapter 11 Novel, Multi-Objective Optimization of Pulsed Electric Field Processing for Liquid Food Treatment Jens Krauss, Özgür Ertunç, Cornelia Rauh, and Antonio Delgado

11.1. Introduction In recent decades, product innovations and innovative food technologies have become of crucial importance for ensuring competitive advantages in the saturated food market of industrialized countries. In addition, recent developments in food innovations have attempted increasingly to meet the requirements imposed by the developing countries and thereby to contribute particularly to solutions of the world nutrition problem. In this context, the most promising advantages are assigned to emerging technologies that can replace classical thermal processes and, at the same time, offer benefits regarding the quality, ecology, resource management, and economics of products and processes. Socalled nonthermal technologies are mostly based on microbiological or physical effects. Regarding the latter, high pressure (Knorr 1996; Hendrickx et al. 1998; Cheftel et al. 2000; Sanz and Otero 2000; Le Bail et al. 2003; Delgado et al. 2008), ultrasonics (Knorr et al. 2004; Patist and Bates 2008) and pulsed electric fields (PEFs) (Zimmermann et al. 1974; Grahl and Märkl 1996; Wouters et al. 1999; Ho and Mittal 2000; Mañas and Pagán 2005; Teopfl et al.

2006; Gerlach et al. 2008) represent the most promising physical (nonthermal) effects induced for food treatment. This chapter deals with the optimization of PEFs as a method for treating biological matter. This method relies on electric pulses that affect biological matter in such a way as to change the material or structural conformation and has the potential to be used in a wide variety of applications (Teopfl et al. 2006). With respect to food processing, current research activities are focused on the preservation and extraction of valuable components. One of the best known effects of PEF is the permeabilization of biological cells. In addition, it has also been reported that biological cells respond to exposed PEFs by an electric breakdown of the cell membrane, which leads to electromechanical membrane perforation (electroporation) (Jeyamkondan et al. 1999). A (typical) example of this type of application is the permeabilization of plant cells by means of PEF, which facilitates access to valuable cell contents, for example, for their extraction. Furthermore, irreversible perforation of cell membranes of microorganisms can be employed for increasing food safety by inactivating pathogens. As

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

209

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

a result of these effects, the PEF method extends shelf life by inactivation of microorganisms that induce spoilage. PEF has also been shown to be an appropriate method for preventing enzymatically driven spoilage. However, the mechanism of inactivation of enzymes obviously differs from those acting on microorganisms. Additionally, the mechanisms affecting detrimental enzymes behave in a strongly enzyme-dependent manner and are not yet fully understood (Mañas and Vercet 2006). Regardless of the inactivation mechanism involved, the experimental investigations presented in the literature revealed that the electric field strength and treatment time are the major process parameters affecting the perforation efficiency and the inactivation rate of microorganisms and enzymes. The secondary influencing parameters are (1) the characteristics of the PEF treatment chamber, (2) the amplitude and frequency of the electric pulse, (3) the material consistency, structural conformation, and viability of the biomatter, and (4) the electric and thermo-fluid-dynamic properties of the liquid food (Jeyamkondan et al. 1999; Barbosa-Cánovas and Altunakar 2006). Owing to the complex interaction of the PEF process with the medium and spoiling agents, numerous experiments were necessary to quantify the inactivation performance of the PEF process for different foods, microorganisms, and enzymes. Álvarez et al. (2006) and Mañas and Vercet (2006) summarized the inactivation measurements available in the literature for different microorganisms and enzymes, respectively. In contrast to the availability of a substantial body of publications based on specific experiments, the scientific research of PEF processes for modeling and simulation seems to be far from complete. This is clearly revealed by the large gap in knowledge even when basic effects induced by PEF are considered. Basically, the performance of PEF relies on applying electric pulses of short duration (1−100 μs) and high intensity (10−50 kV/cm) to a food specimen placed between electrodes. For example, Figure 11.1 depicts graphically the basic construction of a colinear PEF treatment chamber as defined by Gerlach et al. (2008) as standard configuration for their investigations. The chamber comprises a high-voltage

electrode

insulator

ground

(R-RI) LI

R

r

RI z

Figure 11.1. Half cross-section of a colinear PEF chamber with LI = 4 mm − 2 ∗ (R − RI ) and R = 3 mm.

electrode, a ground electrode, and an insulator. For making use of the benefits of continuous treatment technologies, food capable of flowing is forced through the treatment chamber. In general, flow is driven by a pressure gradient generated by a pump. As a consequence, overlapping electric and forced convection fields causes electro-fluid-dynamic effects to appear. This results in field heterogeneities, which in turn may have a substantial impact on the quality of the product and the process. From these considerations, it is concluded that dealing with the description and systematic optimization of PEF processes demands adequate treatment of this process with field theories, but the need for such a treatment has been recognized only very recently. As a consequence, only few studies that elaborate on PEF using somewhat simplified formulations are available in the literature (Misaki et al. 1982; Qin et al. 1995; Góngora-Nieto et al. 2003; Toepfl et al. 2007; Gerlach et al. 2008). The urgent necessity to take the theory of various fields into consideration can be deduced directly from further inhomogeneities inherently linked to the basic effects of PEF. The term “nonthermal processes” as used in the literature often leads to confusing interpretations. In general, no sufficient explicit discrimination of nonthermal and isothermal processes is made. Instead of the availability of a strict isothermal field, that is, of a homogeneous temperature distribution, nonthermal technologies aim at processing food at a temperature level that is sufficiently low to preserve the sensory, nutritional, and functional properties of food. This is also the case for PEF. The energy of the electric pulses required for the permeabilization of the biological cells alters the level of the thermal energy in the treatment chamber due to ohmic

Chapter 11

Novel, Multi-Objective Optimization of PEF Processing for Liquid Food Treatment

heating. Thus, PEF can fulfill isothermal conditions only approximately for vanishing electric energy and very high values of thermal diffusivity. However, this case is only academically relevant: Permeabilization takes place only if a certain level of the electric energy is exceeded. Zimmermann et al. (1974) showed that the critical transmembrane electric potential required for the breakdown of the cell is nearly 1 V. Furthermore, isothermal processing leads to long treatment times and to dissipation of energy to a large extent. Clearly, both consequences of isothermal conditions are in crucial contradiction to the goals of modern food processing. Under practical conditions, PEF always generates thermal fields that interfere with the electromagnetic field used for cell permeabilization and the flow field that serves as a basis of the continuous transport of food capable of flowing. Thereby, any thermal heterogeneity produces density differences that can induce a natural flow field in foods consisting of pure liquids or dispersed fluid systems. Thus, natural convection occurs in the presence of gradients perpendicular to, and in the direction of, gravity. This is also the case with other “nonthermal” processes such as high-pressure treatment of food (Delgado et al. 2008; Rauh et al. 2009), in which natural convection phenomena are also available (see also Chapters 4 and 5 of this book). However, other analogies have also recently appeared: Delgado and Hartmann (2003) and Pehl and Delgado (1999) have demonstrated highpressure treatment acting instantaneously. This is due to the fact that the pressure waves propagate rapidly, that is, with the speed of the sound. Thus, the propagation time within the pressurized chamber takes values that are substantially smaller than those of the characteristic times of the biological and the biochemical reactions, with the exception of very fast bioreactions. In agreement with this, the propagation of electric impulses occurs with the velocity of light; hence, the instantaneity of the impact of the electric field must be expected. In summary, PEF acts instantaneously but does not operate completely uniform, even when assuming homogeneous food materials. Hence, investigation and optimization of inhomogeneous fields

211

demand adequate experimental, theoretical, and numerical methods. Experimental procedures are considered being out of the frame of this contribution. The mathematical complexity of the overlapping effects of the transport of momentum and thermal energy in the fluid flow and also the electric field makes pure mathematical analytical treatment almost impossible. However, numerical methods as used in computational fluid dynamics (CFD) deliver the tools capable of simulating the mentioned fields involved in the PEF treatment of food capable of flowing (see also Chapters 9 and 10 of this book). Numerical methods also represent an adequate and, at the same time, powerful basis for optimizing and for scaling up PEF processes as required in industrial applications. In this context, the optimization of the processes in the strong interrelated electrofluid-dynamic and thermal fields requires particularly the optimization of the initial and boundary conditions realized in the treatment chamber (Gerlach et al. 2008). The most influential boundary condition is the geometric shape of the chamber as it influences both electro-fluid-dynamic and thermal effects. Owing to the inhomogeneous field strength distribution, local overheating of the liquid food might occur, which destroys valuable constituents of the food. Hence, the impact of PEF on the inactivation of food-spoiling microorganisms and enzymes in the liquid food can best be understood when the PEF process is designed in accordance with simulation results. This provides an appropriate base for analyzing the spatiotemporal distribution of electric field strength, velocity, and temperature fields, and also of inactivation in the treatment chamber for ensuring the efficient application of PEF. Hence, it appears convenient to address two different thematic approaches in this chapter. In the first approach, modeling and numerical simulation of the PEF process are discussed at a level that provides a systematic understanding of the most relevant features regarding the goal of this contribution. This discussion of the first approach includes, in particular, (1) a presentation and explanation of the model equations, (2) a brief overview of the simplifications employed in the literature and their consequences, and (3) some exemplary interpretations of numerical

212

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

results presented at examples from the literature. Further details on the numerical treatment of PEF processes can be found in Gerlach et al. (2008). The second thematic approach deals with an optimization procedure that meets optimization objectives and restrictions systematically. The approach is based on specific numerical procedures that are aimed at reducing field heterogeneities by optimizing the shape of the treatment chamber. To ensure convergence, the basic field equations and object function and constraints must be treated before starting with the optimization. Results of optimization are presented at examples and discussed.

11.2. Modeling and Simulation of Fields Induced by PEF in Food Capable of Flowing Modeling of fields represents a common task in applied mathematics, natural sciences, and engineering. Classically, mathematical models are deduced by balancing the quantity in question. This is especially the case for conservative quantities such as mass, momentum, energy, and electric charges that also play a decisive role in PEF processes. The temporal change of the quantity of interest for a given fixed balance volume corresponds to (1) diffusive and convective transport through the surface of the volume and (2) generation within the volume. In addition, the constitutive equations, appropriate formulations of the kinetics of molecular and cellular systems in question, and the initial and boundary conditions must be available for ensuring completeness of the mathematical model and thus its solubility. As a rule, balancing leads to a system of integral or differential equations. Generally, CFD aims at solving them by using numerical finite approaches such as finite differences, volumes, elements, or boundary elements. Finite methods translate the basic equations into a system of algebraic equations that must be solved in such a way that the residue, that is, the differences between the exact solution of the basic equations and the approximate solution of the discrete equations, are minimized (Brebbia and Ferrante 1978). For ensuring numerical consistency, stability, and convergence, the spatiotemporal

discretization must meet resolution requirements in space and time as imposed by the physicochemical effects under consideration (Ferziger and Peric´ 2002). In general, this results in very large systems of algebraic equations that, in turn, must be solved as efficiently as possible. As a matter of fact, the calculation costs, that is, the CPU time and the storage volume required, represent often decisive hurdles. This statement is valid regardless of the enormous progress that has been observed in highperformance computing in recent decades. Even though there have been drastic increases in floating point operations up to petaflops, vectorized and parallelized numerical algorithms, local refinement of numerical mesh, and novel computer architectures, the simulation of processes that can cover several orders of magnitude in space must be considered as being beyond of the computational possibilities accessible at present and, most likely, in the near future. Fortunately, such limits of feasibility are expected for PEF only if fluid dynamic transport occurs in the turbulent regime and, additionally, resolution of turbulence in the different spatial and temporal scales is to be resolved. The analysis of the impact of PEF on biomatter occurs at the molecular or mesoscale level. However, such considerations have not yet been addressed in the literature. Hence, this chapter focuses on treatment chambers and fluid dynamic velocities that are sufficiently small to ensure laminar transport. The next section deals with the basic equations necessary for modeling electro-fluid-dynamic and (unwanted but also unavoidable) thermal effects and inactivation during the PEF process.

11.2.1. Governing Model Equations for the Electro-Fluid-Dynamic and Thermal Fields Within the continuum framework assumed here, the electro-fluid-dynamic and thermal processes active during the application of PEF can be modeled by using the conservation equations of mass, momentum, energy, and electric charge, as generally done in CFD (Bird et al. 2007; Gerlach et al. 2008; see also Chapters 9 and 10 of this book). In most studies

Chapter 11

Novel, Multi-Objective Optimization of PEF Processing for Liquid Food Treatment

presented in the literature, the liquid food in question is assumed to have properties similar to those of water. The density of water ρ is a weak function of temperature T within the temperature range of interest for PEF that lies, as a rule, in the range of 283.15–373.15 K. Therefore, the constitutive models used postulate the validity of incompressibility and Newtonian behavior. With respect to the impact of the electric field on the transport processes, the electric pulses are additionally considered to alter the energy level and the flow behavior in the chamber owing to the temperature dependence of material properties, such as viscosity, thermal, and electric conductivity. Hence, the model equation for the conservation of mass reads: ∇⋅u = 0

(11.1)

where u denotes the velocity vector and ∇ is the nabla operator. With respect to the remarks regarding balancing models given earlier, Equation 11.1 permits the simple interpretation that a change in density is impossible within a fixed fluid volume due to incompressibility; that is, any mass transport crossing the surface of the volume must compensate to zero. The conservation of momentum is governed by the following model equation: ∂u ρ ⎡⎢ + (u ⋅ ∇)u ⎤⎥ = −∇p + μ (T )∇2 u ⎣ ∂t ⎦

(11.2)

where p and μ represent the pressure and the dynamic viscosity of the fluid, respectively. In contrast to the density ρ , the viscosity of liquids μ exhibits a strong dependence on temperature T . Equation 11.2 indicates that for a given balance volume, the convective momentum transport in the liquid food as described by the left-hand side is driven by pressure forces (first term on the righthand side) that must also compensate the viscous momentum transport. Owing to the low-velocity gradients, viscous dissipation can be neglected when balancing energy that is expressed here by the temperature for convenience. The correct formulation of energy conservation must take into consideration the effect of the

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electric field strength E on the temperature distribution as expressed by the following balance model: ∂T ρc p (T ) ⎡⎢ + u ⋅ ∇T ⎤⎥ = k (T )∇2T + σ (T ) E 2 ⎣ ∂t ⎦

(11.3)

where c p , k , and σ denote the specific isobaric heat capacity, thermal conductivity, and electric conductivity, respectively. In accordance with Equation 11.3, the temporal change of the thermal energy in the fixed balance volume of fluid (first term on the left-hand side) is due to the convective (second term on the left-hand side) and diffusive (first term on the right-hand side) energy flows through the surface. Furthermore, the change of energy in the balance volume is elevated by the transformation of the electric energy to thermal energy (joule heating), as taken into account by the last term in Equation 11.3. The PEF processes employ very short pulse widths τ with frequency f , so the unsteady joule heating can be modeled as a steady source term. Consequently, τ and f appear as a coefficient in front of the joule heating term in the energy equation (Gerlach et al. 2008; see also Chapter 10 of this book): ∂T ρc p (T ) ⎡⎢ + u⋅∇T ⎤⎥ = k (T )∇ 2T + τ fσ (T )E 2 (11.4) ⎣ ∂t ⎦ In the literature, it is generally assumed that the pulsating electric field does not induce an unsteady magnetic field, that is, ∇ × E = 0 . Hence, the electric field strength vector becomes the (negative) gradient of the electric potential φ (Gerlach et al. 2008): E = −∇φ

(11.5)

As a result of charge conservation, the electric potential can be derived from ∇ ⋅ J = ∇ ⋅ [σ (T )∇φ ] = 0

(11.6)

where J represents the current density. In summary, modeling results in the system of nonlinear partial differential Equations 11.1, 11.2, and 11.4–11.6 that describes the electro-fluiddynamic and thermal fields taking place in a PEF chamber. It is worth mentioning that these equations

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are strongly interrelated. In principle, the conservation equations for mass (Eq. 11.1) and momentum (Eq. 11.2) provide a sufficient base for determining both the velocity field and the pressure. However, the temperature dependence of the material properties requires the simultaneous solution of the energy equation. In turn, the energy equation is linked to the model for the conservation of electric charge, Equation 11.6, via Equation 11.4. An additional link to the other conservation laws is the appearance of the temperature-dependent electric conductivity in the charge balance, Equation 11.6. The strong interrelations of the governing equations through the constitutive parameters, in connection with their lack of availability for most fluid foods (Heinz et al. 2001), impose substantial restriction on the numerical simulation of PEF processes. These aspects, in addition to the strong nonlinear characteristics of the model equations, which result in high computational costs, make the numerical simulation of PEF processes a tedious task. In fact, the number of publications using numerical tools in this field has been rather limited to date. In particular, this explains the lack of publications treating the complete model equations (Gerlach et al. 2008). In most studies, only the electric field is simulated as described by Equations 11.4 and 11.5, and an improvement of its uniformity is attempted. Nonuniform distributions of the field strength can be caused under practical conditions by the geometry of the chamber composed of insulators and electrodes (Misaki et al. 1982; Qin et al. 1995) or by impurities in the treatment medium such as air bubbles or other dielectric materials such as fat globules (Góngora-Nieto et al. 2003; Toepfl et al. 2007). In principle, dealing with Equations 11.4 and 11.6 can contribute, at least to a certain level, to the design of treatment chambers that offer improved homogeneity of the electric field strength. More precisely, the transmembrane potential must be achieved in any point of the chamber to ensure adequate homogeneous treatment. On the other hand, field intensity peaks must be avoided as they might cause overtreatment or even dielectric breakdowns in the medium (Qin et al. 1995; Góngora-Nieto et al. 2003; Toepfl et al. 2007).

However, any optimization of PEF by considering the model Equations 11.4 and 11.6 must exhibit solely systematic errors. First, this is due to the fact that optimization of PEF does not represent a single objective, but is always a multi-objective task. Hence, focusing on the goal of electric field uniformity only can lead to results that are even contradictory to those required for example to ensure a uniform dwell time or temperature. Furthermore, solving Equations 11.4 and 11.6 apart from the other governing equation corresponds to the case of a constant electric conductivity σ . This restriction not only implies “frozen values” of σ but, even more crucially, it also leads to a solution that does not depend on σ . As a consequence, optimization based purely on the electric field cannot claim adaptability either to different foods or to the required forced convection and to the unwanted but practically unavoidable thermal effects. In contrast, some groups (Fiala et al. 2001; Lindgren et al. 2002; Gerlach et al. 2008) studied in more detail the local interplay of the heating of the fluid due to the electric field and the heat transfer due to heat convection and conduction. In these studies, the flow field, electric field, and the temperature field were solved in a coupled manner. This offers unique possibilities of exploiting the advantages of PEF as a nonthermal technology by monitoring and optimizing the whole “treatment history” (Rauh et al. 2009) at every location in the treatment chamber. In this context, modeling and simulation must include, of course, the main objective of PEF, namely the perforation of biological cells or the inactivation of enzymes or microorganisms. As no conclusive model for describing perforation is directly available in literature, the discussion in the next section concentrates on inactivation.

11.2.2. Models for PEF-Induced Inactivation In literature, two different modeling approaches for managing PEF-induced inactivation can be found. The most classical one treats inactivation of microorganisms or enzymes within the pure framework of kinetics. This corresponds to an inactivation depend-

Chapter 11

Novel, Multi-Objective Optimization of PEF Processing for Liquid Food Treatment

ing purely on the time for a given treatment procedure and the food. In contrast, a novel modeling approach for inactivation (Delgado and Hartmann 2003; Delgado et al. 2008) is being increasingly accepted, which balances the residual activity ( RA ) or survival rate (for convenience, inactivation of microorganism is not discussed here explicitly, as it can be modeled in a similar way) as other conservative quantities already discussed. As shown by Delgado et al. (2008) for the first time, this is possible as RA represents a conversion of mass which, therefore, obeys strictly the partial continuum equation. This permits dealing numerically with inactivation in a similar manner to an arbitrary scalar quantity. However, almost all investigations reported in the literature are based on the pure kinetic approach. The inactivation performance of the PEF process for a certain spoiling agent is measured after a specific food has been exposed to a field strength E for a treatment time t *. Hence RA can be calculated by: RA( E, t * ) = 100 ×

A( E, t * ) A0

(11.7)

where A and A0 are the enzyme activity in the sample after and before the treatment, respectively. RA is commonly modeled in two steps (Álvarez et al. 2006). The primary model depicts the effect of the dominant process parameters, E and t * , and the secondary model formulates the influence of the characteristics of the medium and the enzyme. Based on the trends of the measurements, the following primary model of Hülsheger et al. (1981) has found most acceptance: ⎛ t* ⎞ RA = ⎜ * ⎟ ⎝ tc ⎠

[ − ( E − Ec ) / κ ]

(11.8)

where Ec and tc* represent the critical electric field strength and critical treatment time, respectively, above which the inactivation starts and κ is a constant factor characteristic for each microorganism or enzyme. Accordingly, it is clear from this equation that electric field strength is the most dominant factor in an inactivation process that is followed by the treatment time. The secondary models formulate

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the relations for Ec , tc* , and κ as functions of properties of the medium (e.g., temperature, pH, conductivity), pulse characteristics, and type of microorganism or enzyme (Álvarez et al. 2006). It is worth mentioning that Equation 11.8 reflects only a pure time dependency in accordance with the introductory remarks in this section. However, as liquid food volumes with different values of RA are transported in the chamber by forced and natural convection through regions of different electric field strength and temperature, the novel transport model mentioned earlier must be employed to describe adequately the “treatment history,” as follows: ⎡ ∂RA ⎤ * ⎢⎣ ∂t + u ⋅ ∇RA⎥⎦ = ∇( D∇RA) + S (T , E, t )

(11.9)

where D is the diffusion coefficient. Similarly to the interpretations given earlier, the balance Equation 11.8 expresses that the temporal change in the residual activity depends on the convective and diffusive flow through the surface of the balance volume. However, additionally, a source term S (T , E , t * ) is available, which in the case of an inactivation represents a negative source term. Generally, a source term depends on the temperature T , the electric field strength E and the treatment time t * , and, if necessary, other influencing factors. Enzymes and microorganisms have different resistances to temperature. It is known that some enzymes are temperature resistant (Vora et al. 1999; Van Den Broeck et al. 2000), and especially for such enzymes, PEF treatment at low temperatures becomes an alternative process of preservation, such that the source term becomes a function of electric field strength and treatment time, S ( E, t * ) . The diffusion term on the right-hand side of Equation 11.9 can be neglected because of its relatively small influence in comparison with the other terms present in the equation (Delgado et al. 2008). In the literature, different empirical models for the source term in the transport equation for RA have been suggested. As an example, Giner et al. (2005) conducted inactivation experiments between two parallel plates, which yield a highly homogeneous electric field, in a batch mode without any mean flow of the processed liquid. They maintained

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the temperature of the chamber at less than 313.15 K, so that the thermal inactivation was kept as low as possible (Vora et al. 1999; Van Den Broeck et al. 2000). Based on the experimental data, they proposed the following first-order fractional model of RA based on the measurements conducted for the enzyme PE: RA = RA0 ⋅ exp (( 4.2 ⋅ 10 −3 − 266 ⋅ 10 −6 E ) ⋅ t *)

(11.10)

This model is a function of the treatment time t * and also the electric field strength E . Because of the highly homogeneous field and controlled temperature in the experiments, this model is implemented into a coupled field solver (see Section 11.2.8) for the simulations of inactivation. The relation between the real time and the treatment time in the pulsating chamber is: t * = tτ f .

(11.11)

Hence the source term is simply the time derivative of the empirical RA model (10): dRA = S ( E ) = RA0 ⋅ ( 4.2 ⋅ 10 −3 − 266 ⋅ 10 −6 E ) ⋅ τ f ⋅ RA. dt (11.12) It should be noted that in order to maintain the validity of this model, the temperature at the exit of a PEF chamber should not exceed 313.15 K, as was the case in the experiments of Giner et al. (2005). The transport equation for RA becomes: ⎡ ∂RA ⎤ ⎢⎣ ∂t + u ⋅ ∇RA⎥⎦ = S ( E )

(11.13)

Equations 11.1, 11.2, 11.4–11.6, 11.12, and 11.13 form the coupled system of partial differential equations whose solutions with defined initial and boundary conditions and the constitutive parameters yield the description of the flow field, temperature field, electric field, and the field of relative residual activity.

11.2.3. Numerical Method For the simulation of coupled fields, which are modeled by the coupled equation system (Eqs. 11.1, 11.2, 11.4–11.6, 11.12, and 11.13), common numerical

methods can be employed to discretize the mathematical model, such as finite difference, finite element, and finite volume methods (Hoffmann and Chiang 2000; Ferziger and Peric´ 2002). The number of investigations employing numerical simulations of PEF chambers is limited and the available studies comprise different levels of equation coupling and different numerical discretization methods. Qin et al. (1995) simulated the electric field in a colinear PEF chamber using a finite element method. Lindgren et al. (2002) and Gerlach et al. (2008) simulated the coupled flow, and thermal and electric fields also using a finite element method. Recently, Rauh et al. (2010) coupled the inactivation field of PE to the electro-hydrodynamic field and simulated the complete PEF process by using a finite volume method implemented in the open source field simulation package OpenFOAM (2007a, 2007b) environment. The electro-hydrodynamic fields were solved as a steady-state problem; only the RA field was integrated in time by using the steady-state solution of the velocity and the electric field strength. As the Reynolds number Re = ρUb D / μ of the flow within the PEF chamber, which is based on the bulk velocity Ub and the diameter of the inlet d = 2 R , is approximately 380 at the inlet, no turbulence model is used. Using a turbulence model in closed conduits when the local Reynolds number is larger than 2,000 is recommended.

11.2.4. Fluid Properties and Boundary Conditions The material properties in the governing equations depend on temperature. Temperature-dependent properties of the liquid foods are discussed by Heinz et al. (2001) for selected liquid foods. For most of the liquid foods, the properties of water are good approximations for simulations, given that the viscosity of the liquid food can be accepted to have Newtonian character. In order to avoid icing and seeding, the temperature field in the simulations should be limited to a range of between 283.15 and 373.15 K . The changes in the density ρ and specific heat capacity c p with temperature are very small, so these can be assumed to be constant in the

Chapter 11

Novel, Multi-Objective Optimization of PEF Processing for Liquid Food Treatment

Table 11.1. Standard boundary and initial conditions (part 1). Parameter Inlet u

p T RA

φ

Electrode

Insulator

0 m/s 0 m/s Hagen– Poiseuille profile for a mass rate of 5.2 kg/h Zero Zero Zero gradient gradient gradient Zero Zero 283.15 K gradient gradient Zero Zero 100% gradient gradient Zero Zero 15 kV gradient gradient

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Table 11.2. Standard boundary and initial conditions (part 2).

Ground

Parameter

Outlet

Symmetry axis

Initial field

0 m/s

u p

Zero gradient 1.013E05 Pa Zero gradient Zero gradient Zero gradient

Symmetry axis Symmetry axis Symmetry axis Symmetry axis Symmetry axis

0 m/s 1.013E05 Pa 283.15 K 100% 0V

T RA φ Zero gradient Zero gradient Zero gradient 0 kV

reduce the number of grid points. To achieve this, “symmetry axis boundary” conditions can be applied on the axis of the PEF chamber.

11.2.5. Computational Grid and Grid Dependence Study simulations; for example, they can take values at the mean operating temperature. However, the dynamic viscosity μ (T ) , the thermal conductivity k (T ), and the electric conductivity σ (T ) exhibit strong dependence on temperature (Schmidt 1969; Gerlach et al. 2008). Foremost, σ (T ) influences the locally dissipated electric power. The viscosity and thermal conductivity of water can be taken from the standard literature, but not the electric conductivity. The electric conductivity of some liquid foods can be approximated by salt solutions of water. For example, Gerlach et al. (2008) used 2.44% salt solution for the electric conductivity of liquid foods. In order to incorporate the temperature effects in the simulations, at each iteration step the material properties should be modified by using the local temperatures simulated in the previous iteration step. Specific initial and boundary conditions should be defined for all the fields to be simulated. An example set of conditions, which were employed in the simulations of the PEF process in the colinear PEF chamber (see Figure 11.1) (Gerlach et al. 2008; Rauh et al. 2010), are given in Tables 11.1 and 11.2. Since the colinear PEF chamber is axisymmetric, solving one azimuthal slice of the flow domain is a good approximation and a common practice to

The domain in which various fields are simulated should be discretized into grid points on which the variables will be calculated. The grid can be structured, block structured, or unstructured depending on the complexity of the geometry and the capability of the solver. In general, with structured grids the number of grid points can be kept lower than that in the other methods. However, only very simple geometries can be modeled with that type of grid because of the rapid decrease in the quality of the grid with increase in the complexity of the grid structure. Therefore, block-structured grids are a good compromise when the geometry of interest becomes complex. Nowadays, commercial solvers can deal with all of the grid types mentioned. Gerlach et al. (2008) employed block-structured grids. In contrast, Rauh et al. (2010) used unstructured hexahedra and split-hexahedra cells, which are known to have better numerical accuracy. An example picture of the computational domain used by Rauh (2010) is shown in Figure 11.2. This computational grid comprises 36,954 cells. The azimuthal direction is discretized with one cell corresponding to a 1° slice. It is expected that the results should converge to a value with increase in the number of grid points. Hence, it is a common practice to perform a grid independence study; that is, one has to check the

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Figure 11.2. Computational grid of a colinear PEF chamber, RI = 2 mm , generated by the snappyHexMesh tool of OpenFOAM.

Figure 11.3. Comparison of the number of cells with the average temperature in the insulator volume and the maximum of  = 5.2 kg/h. the temperature in the treatment chamber for f = 20 Hz, τ = 10 μ s, φ = 15 kV, n = 25, and m

dependence of the results on the number of grid points and find the minimum number of grid points with converged results. As a result, the accuracy of the simulations can be monitored and the computational resources can be used economically if many simulations have to be performed. It was found that the most sensible variable is the maximum local temperature. For example, a grid independence study (see Figure 11.3) using the type of grid in Figure 11.2 shows that around 37,000 cells are sufficient to proceed with the analysis.

11.2.6. Electro-Hydrodynamic Field in a Colinear PEF Treatment Chamber Results of a simulation for the electro-hydrodynamic field are presented in Figure 11.4. A long, stable separation region behind the insulator is a noticeable feature of the (laminar) flow in the colinear PEF chamber. This separation region is the main reason for the accumulation of heated fluid behind the insulator. Similar phenomena should be expected for any scalar variable such as RA.

Chapter 11

Novel, Multi-Objective Optimization of PEF Processing for Liquid Food Treatment

219

(a)

(b)

(c)

Figure 11.4. Electro-hydrodynamic fields in the standard PEF chamber for f = 20 Hz, τ = 10 μ s, φ = 15 kV, RI = 2 mm,  = 5.2 kg/h. (a) Velocity field and streamline (flow is from left to right); (b) electric field strength and electric potential; and m (c) temperature field. See color insert.

Figure 11.5. Electric field strength computed with temperature-independent material properties ( f = 20 Hz, τ = 10 μ s,  = 5.2 kg/h). See color insert. φ = 15 kV, RI = 2 mm, and m

It is interesting to observe that the electric field strength E in Figure 11.4b is not symmetric on the insulator. This is caused by the temperaturedependent material properties, mainly σ (T ) , which leads to changes in the electric field strength at constant voltages but changing resistivity and therefore current flow. As viscosity and electric conductivity of the liquid are strong functions of temperature, it is desirable to keep them variable in the simulations. In Figure 11.5, the electric field computed with constant material properties is shown. There is a clear difference between the results in Figures 11.4b and 11.5, especially close to the downstream edge of the insulator, where the temperature varies strongly. Therefore, unrealistic inactivation will be computed when temperature-independent material properties are used.

Figure 11.4c shows a higher temperature (313.15 K) over the right corner of the insulator than appeared in the paper by Giner et al. (2005). One might think that in this region thermal inactivation might take place in reality. However, the influence of temperature on the residual activity can be ignored because the critical volume and the associated velocity field result in a relatively short treatment time for this enzyme to become inactive by thermal means.

11.2.7. Proper Selection of Process Parameters In order to avoid the destruction of valuable food components and boiling of the fluid, the average temperature at the exit of the PEF chamber should

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

 = 5.2 kg/h Figure 11.6. Temperature in the standard PEF chamber as a function of τ f for (a) RI = 2 mm, Tin = 283.15 K, and m  = 5.2 kg/h. and (b) φ = 15 kV, Tin = 283.15 K, and m

be kept below 373.15 K and the maximum temperature should not exceed 373.15 K at any point within the PEF chamber, respectively. With the help of simulations, it is possible to determine the average and maximum temperature development as a function of τ f in a PEF chamber. The temperature at the inlet of the PEF chamber in the simulations

was 283.15 K . As can be seen in Figure 11.6, the average temperature in the insulator volume and the maximum temperature in whole PEF chamber are strong functions of the insulator ’s radius and the electric potential applied. Figure 11.6 shows that the maximum temperature can exceed 373.15 K for τ f ≈ 300, φ = 15 kV, RI = 2 mm, and m = 5.2 kg/h,

Chapter 11

Novel, Multi-Objective Optimization of PEF Processing for Liquid Food Treatment

above which simulations cannot be performed with the selected numerical methods. These observations reveal that optimization of the insulator geometry would be a complex trade-off between the minimization of the RA and minimization of the excessive temperatures.

11.2.8. Implementation of Experimental Inactivation Models The treatment time for a fluid particle exposed to a electric field strength E as it moves through the PEF chamber can be approximated by using the time of travel through the insulator region ( LI + 2( R − RI )) / Ub as t * = τ f ( LI + 2( R − RI )) / Ub , where Ub is the bulk velocity of the fluid in the insulator region. The maximum allowed temperature can be selected to be 353.15 K and, for the preservation of the valuable food components, the average temperature at the outlet of PEF chamber can be chosen to be a maximum of 313.15 K . The choices of τ = 10 μs , f = 20 Hz , RI = 2 mm and φ = 15 kV satisfy the temperature limits. As a result, the treatment time becomes 7.67 μs. Simulation with

221

these process parameters showed a volume-averaged field strength of 27.2 kV/cm in the insulator. Therefore, according to the inactivation kinetics, Equation 11.10, the average RA at the exit of the PEF chamber cannot be less than 95% . In other words, many of the PEF chambers should be serially connected to achieve a reasonable reduction in RA and the fluid should be cooled to prevent excessive fluid temperatures between each PEF chamber (Bendicho et al. 2003). Hence, in addition to the energy required for the generation of an electric field, energy is required for the cooling of fluid to 283.15 K between each treatment chamber. Following this, the cost of the PEF process increases linearly with the number of treatment chambers. The cost of the process, however, can be minimized with an optimized PEF chamber and process. A reduction in RA along serially connected chambers can be simulated by using periodic boundary conditions only for the RA field; that is, the outlet values of RA in one PEF chamber can be used as the inlet boundary condition in the next simulation of the PEF chamber. These types of simulations made by Rauh et al. (2010) (Figure 11.7) reveal that

Figure 11.7. Change in residual activity as a function of the number of treatment chambers and the comparison of simulations with prolonged treatment time and successive treatment chambers.

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

only after 29 chambers does the RA drop to below 50% with the selected flow rate, chamber geometry, and τ f . The number of treatment chambers necessary can be reduced by increasing τ f , but at the cost of increasing detrimental effects of excessive temperature. The simulation of RA with periodic boundary conditions is a good approximation of the PEF treatments in experiments. Nevertheless, this method is still computationally very expensive for the optimization of the insulator shape in a PEF chamber. Therefore, in order to reflect increased treatment time with increased number of PEF chambers, the treatment time in the inactivation kinetics, Equation 11.10, is modified to involve the number of chambers;

that is, the treatment time, Equation 11.11, is redefined as: t* = τ f t n

(11.14)

where n is the number of treatment chambers. Accordingly, S ( E ) in Equation 11.12 becomes: dRA = S (E ) dt = RA0 ⋅ ( 4.2 ⋅10 −3 − 266 ⋅ 10 −6 E ) ⋅ τ f n ⋅ RA. (11.15) The simulations conducted with this prolonged treatment time are compared with successive PEF chamber simulations in Figure 11.7. As the difference is not more than 6% in RA, the assumptions

 = 5.2 kg/h, and Figure 11.8. Residual activity field in the PEF chamber for f = 20 Hz, τ = 10 μ s, φ = 15 kV, RI = 2 mm, m n = 25, resulting in a treatment time of 192 μ s. See color insert.

Figure 11.9. Comparison of the simulated RA in a colinear PEF chamber and the measured RA in a parallel plate chamber  = 5.2 kg/h, and n = 25). (Giner et al. 2005) (simulation constants: τ = 10 μ s, f = 20 Hz, RI = 2 mm, m

Chapter 11

Novel, Multi-Objective Optimization of PEF Processing for Liquid Food Treatment

made for a prolonged treatment time can be accepted as being realistic. Moreover, the simulation approach with a prolonged treatment time decreased the simulation time inversely proportional to the selected number of serially connected PEF chambers, and therefore it can be used to optimize systematically the insulator geometry and the process parameters. Hence, simulations with the prolonged treatment time, Equation 11.14, are a fast method for analyzing and optimizing a PEF process. The simulated RA field corresponding to the electro-hydrodynamic fields in Figure 11.4 is shown in Figure 11.8. As can be seen from the E field and RA field, the highest decay of RA occurs as the fluid approaches the insulator and is exposed to E for the first time. Comparison of the simulated RA and the measured RA by Giner et al. (2005) in Figure 11.9 shows that the implementation is successful even if the PEF chambers were different in the simulations and the experiments. A detailed validation of this implementation is documented by Rauh et al. (2010).

11.3. Numerical Optimization of Treatment Chamber and PEF Process The geometry of the insulator in the PEF chamber and the process parameters, such as flow rate, frequency, pulse width, and electric potential, determine the level of inactivation and the homogeneity of various fields. The developed simulation tool can deliver the impact of process and shape parameters on RA and the other field variables, and therefore optimization of the PEF chamber and process are possible in principle in order to achieve safe and high-quality food with reduced process costs. Owing to the large number of control variables, optimization should be performed by using automatic optimization algorithms. These algorithms can be classified in two main groups: deterministic and stochastic methods. The deterministic approaches are fast but prone to finding a local optimum for a single objective function, whereas the latter, such as evolutionary algorithms, can be used to find global optima for multi-objective functions. A brief and

223

incomplete classification of optimization methods is provided in Figure 11.10. However, a hybrid of these methods can boost the convergence toward a global optimum. Hence, optimization algorithms are nowadays indispensable when processes with a complex physical nature are considered, as is the case for the PEF process. The target of an optimization task is defined via object (cost) functions and additional constraints for different variables. A simple optimization task is defined as: min ci ( x ), 1, … , k s.t. h j ( x ) = 0, j = 1, … , q h j ( x ) ≤ 0, a≤x≤b

j = q, … , m (11.16)

where ci is the object function(s), x is the vector of process and shape parameters, h j are the constraints, and a, b are the limitation of x . Hence, the optimization can be achieved by coupling the numerical simulation of the equation system (Eqs. 11.1, 11.2, 11.4–11.6, 11.12, and 11.13) and a deterministic or a stochastic optimizer as shown in Figure 11.11. Numerical optimization of the PEF chamber geometry and PEF process might have multiple objectives and constraints. Optimization of the insulator geometry for minimum RA of spoiling enzymes or bacteria is one objective. The simulations can be extended with additional RA models of spoiling and useful enzymes, which would allow optimization of the PEF processing for multicomponent enzyme mixtures existing in liquid foods. Temperature constraints can also be implemented in the optimization procedure to eliminate designs generating excessive local temperatures, which can be detrimental to valuable components of a specific food. In the following, examples for PEF chamber optimization and PEF process are provided.

11.3.1. Speeding Up the Convergence of Optimization The convergence speed of optimization can be enhanced by taking certain measures in the simulation

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

OPTIMIZATION

Probabilistic (Stochastic search)

Deterministic

Gradient based

Direct search

Evolutionary compt. (algorithms)

Simulated annealing

Based on f and — f

Exhaustive search

Genetic algorithms

Based on f, — f and — 2f

Nelder–Mead Simplex algorithm

Evolutionary strategies

Particle swarm optimization

Hooke–Jewees algorithm

Multidirectional search method Figure 11.10. Classification of optimization algorithms (Ertunç and Krauss 2009).

Figure 11.11. Coupling of numerical simulation with numerical optimizers.

phase and in the selection of the optimization method. Owing to the large number of control variables, it is still desirable to reduce the time for a single simulation. An axisymmetric assumption along the axis of the PEF chamber and simulating only a slice of the flow represent one method to reduce the simulation time. Further, the grid independence study reveals the minimum number of cells (grid points) from which simulations deliver results with acceptable accuracy. In order to reduce the numerical instability, Rauh et al. (2010) decoupled the simulation of the RA field from the simulation of the electro-hydrodynamic field, and they obtained an additional saving by the steady simulation of the electro-hydrodynamic field. As discussed in the previous section, the implementation of the inactivation kinetics can be modified to involve serially connected PEF chambers by

Chapter 11

Novel, Multi-Objective Optimization of PEF Processing for Liquid Food Treatment

prolonging the treatment time, so that not only can a reasonable inactivation be achieved, but also the simulation time for serially connected chambers can be reduced inversely proportional to the necessary number of serially connected PEF chambers. In the simulations of RA , the circulation region at the wake of the insulator (see Figure 11.4a) together with the zero gradient boundary outlet condition hinders the convergence of RA , in particular, when the recirculation region cannot be fully captured within the computational domain. Rauh et al. (2010) showed that an artificial diffusion term added to the transport equation for RA, Equation 11.13, which was activated after the insulator, decreased the time for convergence, with little influence on the area-averaged RA at the outlet. As a result of these measures taken without sacrificing the accuracy of the results, a large number of simulations required for the numerical optimization of PEF process and the chamber can be afforded. When selecting the optimization method, a tradeoff should be made between the number of control variables, the number of object functions, and the computational resources needed for the simulation. Deterministic approaches need fewer iterations and, therefore, they can deal with a large number of control variables, but they are prone to obtaining local optima. Therefore, these methods are used when the simulation necessitates a very long time and a priori knowledge exists about the location of the optimum. In contrast, stochastic methods are suited to obtaining global optima with multiple object functions. However, they need an excessive number of iterations; that is, when these methods are selected, one simulation should not demand a long computation time. In the following, we provide examples of PEF chamber optimization and PEF process optimization, both of which were conducted using a genetic algorithm, MOGA II (multi-objective genetic algorithm), of modeFrontier (Rauh et al. 2010). In order to speed up convergence, the elitism option in the algorithm was activated, which opts to keep the best individual of one generation in the next generation (see, for example, Weise 2007).

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11.3.2. Treatment Chamber Optimization The residual activity depends on t * and E , Equation 11.10. Only within the volume in the insulator (see Figure 11.4b) is E high enough for the inactivation of PE. Therefore, the optimum design of the insulator geometry is of interest for minimum residual activity at the outlet of the PEF chamber with temperature constraints. Here, only the internal radius of the insulator ( RI ) is used as a control variable by keeping the length of the insulator LI and all other process parameters constant. Consequently, a shape optimization problem can be defined as: min c( RI ) = RAavg s.t. Tavg ≤ 313.15K Tmax ≤ 353.15K 1.2 mm ≤ RI ≤ 2.5mm

(11.17)

where RAavg is the average residual activity at the outlet, Tavg is the average temperature in the volume of the insulator, and Tmax is the maximum temperature in the whole treatment chamber. The optimizer iterated 12 generations, each comprising 10 individuals; that is, 120 individual designs were simulated. The dependences of RAavg , Tavg , and Tmax on RI are shown in Figure 11.12. It should be noted that each point in this figure corresponds to one shape iteration. It can be seen in Figure 11.12a that the optimizer automatically increased the internal radius, so that the treatment time increased and, therefore, RAavg decreased. Convergence was obtained at RI = 2.23 mm as a further decrease in RI was limited by the temperature constraints. In the work of Gerlach et al. (2008), the dependence of the homogeneity of the electric field strength on the PEF chamber geometry was analyzed as the homogeneity was thought to be an indicator of better inactivation. Therefore, in addition to the object function, the standard deviation of electric field strength ( ErSD ) was monitored to see whether there is a relation between the average residual activity and the homogeneity of the electric field strength. The results in Figure 11.13 indicate that the minima of ErSD and RAavg approach each other, which means

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Figure 11.12. Dependence of (a) RAavg and (b) Tavg and Tavg on RI (Rauh et al. 2010).

that minimization of ErSD can be used as an object function. However, a strong gradient also exists in the vicinity of the minimum RAavg . Therefore, minimization of RAavg should be considered a more direct and accurate object function, especially when more complicated shapes and inactivation of multiple food components are considered.

11.3.3. Process Optimization Here, optimization of process parameters is demonstrated for a selected PEF geometry. The objective is to minimize residual activity and increase the flow rate while satisfying temperature constraints. The residual activity depends on t * and E, Equation 11.10.

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Novel, Multi-Objective Optimization of PEF Processing for Liquid Food Treatment

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Figure 11.13. Relation between RAavg and ErSD (Rauh et al. 2010).

Hence t * and E are the basic process variables. t * is proportional to f , τ , n, and t, Equation 11.14. In the optimization problem, the number of serially connected chambers and RI are kept constant at 25 and 2.23 mm, respectively. τ f and E are selected to be the control variables. Consequently, a constraint multi-objective process optimization problem is defined as: min c1 (τ f , m , φ ) = RAavg

11.4. Conclusions and Outlook

max c2 ( m ) = m s.t. Tavg ≤ 313.15 K Tmax ≤ 353.15 K 3 ≤ m ≤ 7 kg/h 1 ≤ fτ ≤ 500 Hz μ s 10 ≤ φ ≤ 15 kV

RAavg by increasing φ and τ f (Figure 11.14b,c). From these results, it is clear that there is still potential to have a mass flow rate higher than 7 kg/h within the selected limits for τ f and φ ; however, RAavg would remain at the same level as the radius is not allowed to change. Therefore, a combined shape and process optimization would help to exploit the capacity of the PEF process.

(11.18)

In the optimization, 23 generations, each comprising 10 individuals; that is, 230 individual designs were simulated. Only the results of the valid designs are shown in Figure 11.14. In general, the stochastic nature of the optimization process can be observed from the scatter of the data. The plot of two object functions (Figure 11.14a) indicates that the mass flow rate was rapidly increased up to the given limit of 7 kg/h and further iterations were made to reduce

Modeling, numerical treatment, and numerical optimization of the PEF process were addressed. We demonstrated the simulation of coupled thermofluid-dynamic fields in a PEF process of liquid food. The method of implementing the residual rest activity model of PE to the electro-hydrodynamic field simulations was rigorously shown as an example. Temperature considerations indicated that the process parameters should be properly selected so that valuable food contents are not damaged. Furthermore, it was shown that to reach a reasonable level of inactivation, a large number of PEF chambers should be serially connected and the outgoing liquid from each chamber should be cooled. After taking measures to speed up the simulations, examples of the optimization of the PEF

 , and the relation between RAavg (b) and φ (c) and τ f (Rauh et al. 2010). Figure 11.14. (a) Object functions RAavg versus m

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229

Figure 11.15. Insulator geometry defined by splines. See color insert.

chamber and process were shown to be successful for finding the optimum geometry and process parameters for minimum residual rest activity, within the given temperature limits. The results presented here clearly show that the simulations and numerical optimization can be extended to conduct optimization with more complicated insulator geometries, for example, as shown in Figure 11.15, and multiple residual activity models, which might be applicable simultaneously for both spoiling agents and valuable food components.

Acknowledgments This work was carried out with financial support from the Commission of the European Communities, Framework 6, Priority 5 “Food Quality and Safety,” Integrated Project NovelQ FP6-CT-2006-015710. The authors gratefully acknowledge the support received. The authors thank Mr. Cagatay Köksoy and Dr. Ing. Daniel Gerlach for their valuable suggestions and Mr. Balkan Genç for proofreading of the text.

Notation ∇ κ

μ ρ σ τ φ

Nabla operator Constant factor characteristic for each microorganism or enzyme Dynamic viscosity Density Electric conductivity Pulse width Electric potential

a A A0 b ci cp d D Dart E Ec E rSD f hj J k LI n p r R RI RA RA0 Re S t t* tc*

Lower limit of x Enzyme activity in the sample after treatment Enzyme activity in the sample before treatment Lower limit of x Object (cost) functions Specific isobaric thermal capacity Diameter Diffusion coefficient of RA Artificial diffusion coefficient of RA Electric field strength Critical electric field strength Relative standard deviation of electric field strength Pulse frequency Constraints of the optimization problem Current density Thermal conductivity Insulator length Number of treatment chambers Local pressure Radial coordinate Inner electrode radius Internal insulator radius of the conduit Relative residual activity (shortened to residual activity) Relative residual activity before treatment Reynolds number Source term of the inactivation Time Treatment time Critical treatment time

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T u U1 Ub x z

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Temperature Velocity Velocity component in the z-direction Bulk velocity of the fluid in the insulator region Vector of the optimization parameters Axial coordinate

Abbreviations CFD PE PEF

Computational fluid dynamics Pectin esterase Pulsed electric field

References Álvarez I, Condón S, Raso J. 2006. Microbial inactivation by pulsed electric fields (Ch. 4.). In: J Raso, V Heinz, eds., Pulsed Electric Fields Technology for the Food Industry: Fundamentals and Applications, 97–130. New York: Springer. Barbosa-Cánovas GV, Altunakar B. 2006. Pulsed electric fields processing of foods: An overview (Ch. 1.). In: J Raso, V Heinz, eds., Pulsed Electric Fields Technology for the Food Industry: Fundamentals and Applications, 3–26. New York: Springer. Bendicho S, Barrosa-Cánovas GV, Martín O. 2003. Reduction of protease activity in simulated milk ultrafiltrate by continuous flow high intensity pulsed electric field treatments. J Food Sci 68(3):952–957. Bird RB, Stewart WE, Lightfoot EN. 2007. Transport Phenomena, 2nd ed. New York: John Wiley & Sons. Brebbia CA, Ferrante AJ. 1978. Computational Methods for the Solution of Engineering Problems. London: Pentech Press. Cheftel JC, Lévy J, Dumay E. 2000. Pressure-assisted freezing and thawing: Principles and potential applications. Food Rev Int 16(4):453–483. Delgado AC, Hartmann C. 2003. Pressure treatment of food: Instantaneous but not homogeneous effect. In: R Winter, ed., Proceedings of the 2nd International Conference on High Pressure Bioscience and Biotechnolgy. Advances in High Pressure Bioscience and Biotechnolgy, 459–464. Dortmund, September 16–19, 2002. Berlin: Springer. Delgado AC, Rauh C, Kowalczyk W, Baars A. 2008. Review of modelling and simulation of high pressure treatment of materials of biological origin. Trends Food Sci Technol 19(6): 329–336. Ertunç Ö, Krauss J. 2009. Optimization Methods and Their Applications in Fluid Mechanics. Lecture Notes of Applied Fluid Mechanics. Erlangen-Nuremberg: FriedrichAlexander-University. Ferziger JH, Peric´ M. 2002. Computational Methods for Fluid Dynamics, 3rd ed. Berlin: Springer.

Fiala A, Wouters PC, Van den Bosch E, Creyghton YLM 2001. Coupled electric-fluid model of pulsed electric field treatment in a model food system. Innovat Food Sci Emerg Technol 2(4):229–238. Gerlach D, Alleborn N, Baars A, Delgado AC, Moritz J, Knorr D. 2008. Numerical simulations of pulsed electric fields for food preservation: A review. Innovat Food Sci Emerg Technol 9(4):408–417. Giner J, Grouberman P, Gimeno V, Martín O. 2005. Reduction of pectinesterase activity in a commercial enzyme preparation by pulsed electric fields: Comparison of inactivation kinetic models. J Sci Food Agric 85(10):1613–1621. Góngora-Nieto MM, Pedrow PD, Swanson BG, BarbosaCánovas GV. 2003. Impact of air bubbles in a dielectric liquid when subject to high field strengths. Innovat Food Sci Emerg Technol 4(1):57–67. Grahl T, Märkl H. 1996. Killing of microorganisms by pulsed electric fields. Appl Microbiol Biotechnol 45(1–2):148–157. Heinz V, Álvarez I, Angersbach A, Knorr D. 2001. Preservation of liquid foods by high intensity pulsed electric fields—Basic concepts for process design. Trends Food Sci Technol 12(3–4):103–111. Hendrickx MEG, Ludikhuyze LR, Van den Broeck I, Weemaes CA. 1998. Effects of high pressure on enzymes related to food quality. Trends Food Sci Technol 9(5):197–203. Ho SY, Mittal GS. 2000. High voltage pulsed electric field for liquid food pasteurization. Food Rev Int 16(4):395–434. Hoffmann KA, Chiang ST. 2000. Computational Fluid Dynamics, 4th ed. Wichita: Engineering Education System. Hülsheger H, Potel J, Niemann EG. 1981. Killing of bacteria with electric pulses of high field strength. Radiat Environ Biophys 20(1):53–65. Jeyamkondan S, Jayas DS, Holley RA. 1999. Pulsed electric field processing of foods: A review. J Food Prot 62(9):1088–1096. Knorr D. 1996. Advantages, opportunities and challenges of high hydrostatic pressure application to food systems. In: R Hayashi, C Balny, eds., High Pressure Bioscience and Biotechnology, Proceedings of the International Conference on High Pressure Bioscience and Biotechnology, Kyoto, 279–287. Amsterdam: Elsevier. Knorr D, Zenker M, Heinz V, Lee DU. 2004. Applications and potential of ultrasonics in food processing. Trends Food Sci Technol 15(5):261–266. Le Bail ALE, Boillereaux L, Davenel A, Hayert M, Lucas T, Monteau JY. 2003. Phase transition in foods: Effect of pressure and methods to assess or control phase transition. Innovat Food Sci Emerg Technol 4:15–24. Lindgren M, Aronsson K, Galt S, Ohlsson T. 2002. Simulation of the temperature increase in pulsed electric field (PEF) continuous flow treatment chambers. Innovat Food Sci Emerg Technol 3(3):233–245. Mañas P, Pagán R. 2005. A review: Microbial inactivation by new technologies of food preservation. J Appl Microbiol 98(6): 1387–1399.

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Mañas P, Vercet A. 2006. Effect of pulsed electric fields on enzymes and food constituents (Ch. 5.). In: J Raso, V Heinz, eds., Pulsed Electric Fields Technology for the Food Industry: Fundamentals and Applications, 131–152. New York: Springer. Misaki T, Tsuboi H, Itaka K, Hara T. 1982. Computation of threedimensional electric field problems by a surface charge method and its application to optimum insulator design. IEEE Trans Power Apparatus Syst 101(3):627–634. OpenFOAM. 2007a. Programmer ’s guide—Version 1.4.1. Available at: http://www.opencfd.co.uk/openfoam. OpenFOAM. 2007b. User guide—Version 1.4.1. Available at: http://www.opencfd.co.uk/openfoam. Patist A, Bates D. 2008. Ultrasonic innovations in the food industry: From the laboratory to commercial production. Innovat Food Sci Emerg Technol 9(2):147–154. Pehl M, Delgado AC. 1999. An in-situ technique to visualize temperature and velocity fields in liquid biotechnical substances at high pressure. In: H Ludwig, ed., Advances in High Pressure Bioscience and Biotechnology, 519–522. Heidelberg: Springer. Qin BL, Zhang YQ, Barbosa-Cánovas GV, Swanson BG, Pedrow PD. 1995. Pulsed electric field treatment chamber design for liquid food pasteurization using a finite element method. Trans ASAE 38(2):557–565. Rauh C, Baars A, Delgado AC. 2009. Uniformity of enzyme inactivation in short-time high-pressure process. J Food Eng 91(1):154–163. Rauh C, Krauss J, Ertunc Ö, Delgado A. 2010. Numerical simulation of nonthermal food preservation. In: Proceedings

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ICNAAM 2010, 19–25. September 2010, Rhodos, Greece, American Institute of Physics. Sanz PD, Otero L. 2000. High-pressure shift freezing. Part 2. Modelling of freezing times for a finite cylindrical model. Biotechnol Progress 16(6):1037–1043. Schmidt E. 1969. Properties of Water and Steam in SI-Units. Berlin: Springer. Teopfl S, Mathys A, Heinz V, Knoor D. 2006. Review: Potential of high hydrostatic pressure and pulsed electric fields for energy efficient and environmentally friendly food processing. Food Rev Int 22:405–423. Toepfl S, Heinz V, Knorr D. 2007. High intensity pulsed electric fields applied for food preservation. Chem Eng Process Process Intensificat 46(6):537–546. Van Den Broeck I, Ludikhuyze LR, Van Loey AML, Hendrickx MEG. 2000. Inactivation of orange pectinesterase by combined high-pressure and -temperature treatments: A kinetic study. J Agric Food Chem 48(5):1960–1970. Vora HM, Kyle WSA, Small DM. 1999. Activity, localisation and thermal inactivation of deteriorative enzymes in Australian carrot (Daucus carota L) varieties. J Sci Food Agric 79(8):1129–1135. Weise T. 2007. Global optimization algorithms—Theory and application. Available online as e-book at http://www.itweise.de/. Wouters PC, Dutreux N, Smelt JPPM, Lelieveld HLM. 1999. Effects of pulsed electric fields on inactivation kinetics of Listeria innocula. Appl Environ Microbiol 65(12):5364–5371. Zimmermann US, Pilwat G, Riemann F. 1974. Dielectric breakdown of cell membranes. Biophys J 14(11):881–899.

Chapter 12 Modeling the Acoustic Field and Streaming Induced by an Ultrasonic Horn Reactor Francisco Javier Trujillo and Kai Knoerzer

12.1. Introduction Low-frequency high-power ultrasound has become a very active research area in food technology and food processing engineering over the last two decades. One of the well-developed applications of power ultrasound is for cleaning and surface decontamination (Mason et al. 1996). Power ultrasound, given that it can damage living cells, has also been suggested for inactivation of vegetative organisms. Although this technology does not achieve complete microbial inactivation, let alone total inactivation of bacterial spores, ultrasound-assisted microbial inactivation has found applications in the pasteurization of raw whole milk at lower temperatures (BermúdezAguirre et al. 2009). Thermo-sonicated milk shows better color retention than conventionally pasteurized milk due to shorter processing times and lower temperatures. One of the original uses of power ultrasound in biochemistry was to break down biological cells to liberate the cell contents (Mason et al. 1996) and to enhance extraction from plant materials (Vilkhu et al. 2008). For example, high amounts of phenolics can be extracted from coconut shells by applying an ultrasound-assisted extraction

technology (Rodrigues and Pinto 2007). Power ultrasound can inactivate and denaturize enzymes such as pectin methylesterase in orange juice, which improves the cloud stability of the juice. Interestingly, ultrasound can also have a positive effect on enzyme activity, although only at intensities below a certain threshold (Mason et al. 1996). Another important application of power ultrasound is in emulsification. In this case, bubbles collapsing near the interface of two immiscible liquids cause a very efficient mixing of the layers and the disruption of the droplets of the dispersed phase due to the formation of micro jets. Very fine oil-in-water emulsions can be produced this way, which has found use in the beverage industry (Kentish et al. 2008). Other applications that have recently gained interest in the science community include the modification of foaming properties of, for example, protein suspensions, and the modulation of textures of various food materials by high-power ultrasound (Jambrak et al. 2008, 2009). The principle of the above mentioned ultrasonic technologies is based mainly on the mechanical and sonochemical effects that can be observed when ultrasonic waves propagate in a liquid medium

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

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(Dahlem et al. 1999). Bubbles are generated by pressure changes during the propagation of highintensity ultrasound waves in liquids. The bubbles grow and collapse during the compression passage of waves, resulting in local temperatures of up to 5,000 K and pressures of up to 50 Mpa (Suslick 1988). This phenomenon is called cavitation, and it generates extreme conditions that induce molecular fragmentation, sonoluminescense, hydroxyl radicals, streaming, and enormous shear forces. Potential applications, making use of the various effects in high-power ultrasound processing, are diverse. As in many other innovative food processing technologies (at their time), the development of specific ultrasound applications was, and still is, mostly based on time- and labor-intensive trial-and-error approaches. Furthermore, scaling up of ultrasound processes has proven to be difficult and not straightforward, as opposed to many conventional food processing technologies, due to the complexity of the interactions between the sound field, cavitation, and the induced flow and temperature distributions. Much work has been devoted to simulating bubble dynamics, bubble generation, and the bubbles’ interaction with sound fields (Commander and Prosperetti 1989; Horst et al. 1996; Servant et al. 2001; Horst 2007) during the last two decades. However, there is a great need for applying Multiphysics modeling for the characterization of the flow phenomena, and the levels of turbulence induced by acoustic fields in ultrasound reactors. This information can, for example, be useful for the better design and optimization of ultrasound tank reactors as well as flowthrough cells. The level of induced turbulence and mixing will allow conclusions to be drawn on whether the entire (liquid) food product is likely to undergo the same or similar treatment history, regardless of whether the sound field is uniform or not. The theoretical basis of acoustic streaming was first described by Rayleigh at the end of the 19th century (Rayleigh 1896), but it is still an active area of research for physicists, given that the interesting interactions between nonlinear acoustic and fluid dynamics have not been completely unveiled. As mentioned by Lighthill (1978a), acoustic streaming

is “quite an old topic, which at present is somehow rejected.” Perhaps the reason for this statement is the fact that acoustic streaming publications appear dispersed during this century. It started with Rayleigh in 1896 (Rayleigh 1896). Then, significant work was done in this area in the 1940s and 1950s, where the pioneering work of Nyborg (1953, 1958), Eckart (1948) and Westervelt (1953) laid out the fundamental mathematics of low-power streaming. In 1978, Lighthill (1978a) formulated an acoustic theory for high-power streaming which he called “Stuart streaming.” In the 1990s, Tjotta (1999, 2000) worked in streaming based on nonlinear acoustics. Still in this decade, physicists such as Loh et al. (2006) published acoustic streaming articles in physics journals. This is because acoustic streaming is a complex nonlinear effect that has not yet been completely formulated, especially for high Reynolds numbers and acoustic sources of high power. The dispersion of publications in acoustic streaming through the century, which has been approached by different authors at different times, makes the understanding of these phenomena difficult to new modelers interested in this area of research. For instance, Nyborg (1953, 1958) developed his acoustic streaming theory based on the method of successive approximations, while Lighthill (1978a) explained acoustic streaming theory in terms of Reynolds stress. Authors like Piercy and Lamb (1954) and Nowicki et al. (1997) quantify the streaming driving force in terms of “acoustic radiation pressure,” while Lighthill quantifies it in terms of the spatial variation of Reynolds stress. The aim of this review is to summarize and compile the basic knowledge of acoustic streaming in a critical way to clarify and explain the fundamentals of this theory. The review is oriented toward Multiphysics modeling of ultrasonic horn reactors, which usually produce high acoustic power. Therefore, the Reynolds low power theory, developed by Rayleigh (1896), Nyborg (1953, 1958) and Westervelt (1953) (RNW theory), is critically compared with the high power “Stuart streaming” theory proposed by Lighthill (1978a). This chapter is divided into four main sections. Section 12.2 explains how to model the acoustic field based on linear acoustics. It also reviews basic

Chapter 12

Acoustic Field and Streaming in an Ultrasonic Horn Reactor

concepts that are necessary to understanding acoustic modeling. Section 12.3 presents a brief introduction to nonlinear acoustics, which is essential to understanding the nonlinear effect of streaming. Section 12.4 explains the RNW theory, which applies only to low Reynolds numbers, and sources of low power. It also compares the successive approximation method, which is used to develop the RNW theory, followed by the Reynolds stress approach developed by Lighthill (1978a). Finally, Section 12.5 presents the “Stuart streaming” approach to model streaming for sources of high power, which is the most frequently encountered case in ultrasonic horn reactors used in the food industry.

12.2. Modeling the Acoustic Field—Linear Acoustics Acoustics is mainly concerned with the generation, transmission, and reception of energy propagated as vibrational waves in matter. This is a field that normally is not taught in schools of food, processing, or chemical engineering, which specialize in understanding and modeling transformation processes that are based on thermodynamics and the laws of conservation of mass, momentum, and energy. However, acoustic modeling is based on the same laws of conservation. This section will review linear acoustics, which is a simplification valid only for low-power sound waves in lossless media, emphasizing the derivation of the wave equation from the laws of conservation. The aim of this review is to facilitate the understanding of nonlinear effects such as streaming, which cannot be explained with linear acoustics, but that are based on the same conservation principles without incorporating linear simplifications. Basic concepts such as complex notation of harmonic waves, plane waves, acoustic energy, intensity, impedance, and rays of sound will be briefly reviewed so they can be taken from this chapter as a quick reference. The interested reader can find more detailed information in the books of Kinsler et al. (2000), Morse and Ingard (1986), Filippi et al. (1999), Lighthill (1978b), Leighton (1997), and Raichel (2006), which are the main

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sources of this section of the chapter. The section concludes with an example of computational fluid dynamics (CFD) modeling of the acoustic intensity distribution in a horn reactor developed by Klíma et al. (2007).

12.2.1. Derivation of the Linear Wave Equation Acoustic waves are generated by the balance between inertia and elastic restoring forces that arise when molecules of fluids or solids are displaced from their normal configuration (Kinsler et al. 2000). It is the balance between the compressibility and the inertia of the fluid that govern the propagation of sound waves (Lighthill 1978b). The acoustic field is generally modeled by solving the linear wave equation, which is a simplification obtained from the equations of conservation of momentum and energy, continuity, and the state of the fluid (Raichel 2006). The following assumptions allow derivation of the acoustic wave equation: 1. The unperturbed fluid has definite values of pressure p0 and temperature T0 , which are time independent and denoted by the subscript 0 (Raichel 2006). 2. Viscous stresses are neglected; the fluid is assumed to be inviscid. 3. The unperturbed fluid does not undergo macroscopic motion. Hence, the unperturbed velocity  v0 = 0 of the fluid is set to zero and convection can be neglected.1 4. The fluctuation of density ( ρ − ρ0 ) is assumed to be very small ( ρ ≈ ρ0 ). 5. The compression of the fluid is assumed to be thermodynamically reversible and adiabatic. Perturbations, which are the fluctuations from equilibrium of the unperturbed fluid, can be noted    as p1 = ( p − p0 ) , ρ1 = ( ρ − ρ0 ) , v 1 = ( v − v0 ) , and T 1 = (T − T0 ) for pressure, density, velocity, and 1

This simplification eliminates the possibility of modeling streaming via linear acoustic given that streaming is the macroscopic motion generated by sound absorption.

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temperature, respectively. The equations of state and conservation of energy, mass, and momentum will be consequently applied, in conjunction with the simplifications above, to derive the linear wave equation. 12.2.1.1. Equation of State and Conservation of Energy The equation of state, p = ρ RT

(12.1)

relates the thermodynamic variables pressure p , density ρ , and temperature T for an ideal gas. During passage of acoustic waves, the fluid compresses and expands, meaning that work − ∫pdV is exerted on a fluid element by the surrounding fluid pressure. Under the assumption that this compression–expansion is adiabatic (assumption 5), the law of conservation of energy requires that the exerted work must be balanced by the internal energy of the fluid element (Raichel 2006) dU = − pdV

(12.2)

Hence, during the reversible adiabatic compression– expansion of an ideal gas, the pressure and density change together in a way that: pρ −γ = constant

(12.3)

where γ = c p cv . Derivation of Equation 12.3 can be found in the thermodynamics book of Smith and Ness (Smith and Ness 1987). By differentiating Equation 12.3 over time it follows that ∂p ∂ −γ ( ρ ) + ρ − γ ∂t ∂t ∂ρ ∂p −γ pρ −γ −1 + ρ −γ ∂t ∂t ∂ ρ ∂ p −γ pρ −1 + ∂t ∂t ∂p ∂t p

=0 =0 ∂ρ ∂t

(12.4)

By using assumptions 1 and 4 it follows that: ∂p γ p0 ∂ρ ∂ρ = = c2 ∂t ρ0 ∂ t ∂t where

γ p0 = γ RT ρ0

(12.6)

By rearranging Equation 12.5: ∂ρ ∂p = c2 ∂t ∂t ∂p ∂t = c2 ∂t ∂ ρ the speed of sound can be expressed thermodynamically as: ⎛ ∂p ⎞ c2 = ⎜ ⎟ ⎝ ∂ρ ⎠ adiabatic

(12.7)

12.2.1.2. Continuity Equation The continuity equation is a mathematical expression of the conservation of mass of the fluid: ∂ρ  = −∇ ⋅ ( ρv ) ∂t

(12.8)

By applying assumption 4 to the right-hand side of Equation 12.8, the continuity equation can be expressed as: ∂ρ  = − ρ0 ∇ ⋅ ( v ) ∂t

(12.9)

Combining Equations 12.9 and 12.5 results in: 1 ∂p  = − ρ0 ∇ ⋅ ( v ) 2 c ∂t

(12.10)

This equation represents the compressibility of the  fluid. It shows that changes in the velocity ∇ ⋅ ( v ) will cause the fluid to compress or expand. For  instance, a negative divergence −∇ ⋅ ( v ) indicates mass entering a fluid element. Hence, according to Equations 12.9 and 12.10, density and pressure will increase with time.

=0

= γ pρ −1

c2 =

(12.5)

12.2.1.3. Momentum Equation The inertia of the fluid is expressed by the Navier–Stokes equation, which arises by applying Newton’s second law of motion to the fluid:  ∂v     ρ ⎛⎜ + v ⋅∇v ⎞⎟ = −∇p + μ∇ 2 v + F (12.11) ⎝ ∂t ⎠

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Acoustic Field and Streaming in an Ultrasonic Horn Reactor

 By applying assumptions 2 ( μ∇ 2 v = 0 ) and 3   ( v ⋅∇v = 0 ), and neglecting external forces ( F ), the Navier–Stokes equation can be reduced to the Euler equation:  ⎛ ∂v ⎞ (12.12) ρ0 ⎜ ⎟ = −∇p ⎝ ∂t ⎠ Assumption 4 allows expressing the density in the left-hand term of the equation as equal to ρ0 . Equation 12.12 represents the inertia of the fluid showing that changes in pressure causes the fluid to  accelerate ∂v ∂t . 12.2.1.4. Linear Wave Equation As mentioned earlier, sound waves are governed by the balance between the compressibility (see Eq. 12.10) and the inertia of the fluid (see Eq. 12.12) (Lighthill 1978b). Thus, the linear wave equation can be obtained by first taking the divergence of Equation 12.12:  ⎛ ∂v ⎞ (12.13) ∇ ⋅ ⎜ ρ0 ⎟ = −∇ 2 p ⎝ ∂t ⎠ Then, taking the time derivate of Equation 12.10 and using the facts that space and time are independent and ρ0 is a weak function of time (Kinsler et al. 2000), we get:  1 ∂2 p ⎛ ∂v ⎞ (12.14) ρ = −∇ ⋅ 0 ⎜⎝ ⎟ c 2 ∂t 2 ∂t ⎠ Finally, elimination of the divergence term between Equations 12.13 and 12.14 gives: ∇2 p =

1 ∂2 p c 2 ∂t 2

(12.15)

Alternatively, the wave equation can be expressed in terms of the pressure perturbation p1 = ( p − p0 ) ∇ 2 p1 =

1 ∂ 2 p1 c 2 ∂t 2

(12.16)

12.2.1.5. Irrotational Velocity Field in Linear Acoustics This subsection demonstrates that the linear wave equation can be expressed in terms of the velocity potential ( Φ ). The concept of velocity potential is used in the derivation of Equation 12.31

237

later in this chapter, which is an important expression for plane waves. The derivation starts by considering that in vector calculus the curl of a gradient is always zero. Thus, taking the curl of Equation 12.12 is equal to zero:  ∂  ⎡ ⎛ ∂v ⎞ ⎤ ∇ × ⎢ ρ0 ⎜ ⎟ ⎥ = ρ0 [∇ × v ] = −∇ × (∇p ) = 0 ∂t ⎣ ⎝ ∂t ⎠ ⎦ (12.17) since the curl of ∇p must be zero. This implies that  the particle velocity2 is irrotational ∇ × v = 0 . The fundamental theorem of vector calculus says that irrotational fields can be expressed as the gradient of a scalar. Hence, the particle velocity can be expressed as:  (12.18) v = ∇Φ where Φ is a velocity potential. Expressing Equation 12.12 in terms of the excess pressure p1 and substituting Equation 12.18 into Equation 12.12 gives: ⎤ ⎡ ∂Φ ∇ ⎢ ρ0 + p1 ⎥ = 0 ⎣ ∂t ⎦

(12.19)

For gradual changes, both terms inside the brackets must vanish identically; therefore: p1 = p − p0 = − ρ0

∂Φ ∂t

(12.20)

Thus, Φ (velocity potential) also satisfies the wave equation: 1 ∂2Φ (12.21) ∇2Φ = 2 2 c ∂t

12.2.2. Harmonic Waves and Complex Notation Sound waves produced in ultrasonic horn reactors are sinusoidal or harmonic (see Figure 12.1). Mathematically they are solutions of the wave Equation 12.15 that are represented as sinusoidal functions: p (r , t ) = P cos (ω t − kr + φ ) 2

(12.22)

The velocity of the fluid element during compression or expansion caused by the passage of a wave is usually referred to as “particle velocity.”

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

(A) Harmonic sound field

Compression

Expansion sound speed c

(B) Movement of particles

Unperturbed or equilibrium

(C) Acoustic pressure

Maximum Compression

Acoustic perturbation l

Maximum Expansion

l Zero displacement

(D) Displacement

Sinusoidal waves

Zero displacement

(E) Particle velocity

Figure 12.1. Harmonic longitudinal sound waves.

where r is the distance from the source r = f ( x, y, z ), P is the amplitude of the wave, φ is the phase angle, and k is the wave number, k=

2π λ

(12.23)

λ is the wavelength, and ω is the angular frequency: ω = 2πν =

2π T

(12.24)

Equation 12.22 can be simplified by converting it into a complex expression via Euler ’s formula. The advantage of this notation is that it facilitates operating and combining sinusoidal waves, avoiding the need for complex algebra and trigonometric identities. It was probably for this reason that Richard Feynman called Euler ’s formula “our jewel” and “one of the most remarkable, almost outstanding, formulas in all of mathematics” (Feynman 1977). Complex notation of sinusoidal functions is the

Chapter 12

Acoustic Field and Streaming in an Ultrasonic Horn Reactor

Imaginary

∇2 P − j

e jq = cos q + j sin q

sin q q cos q

l Real

ω2 P=0 ρ0 c 2

239

(12.28)

where P is the amplitude of pressure. This equation results from applying the technique of the separation of variables to reduce the wave equation by assuming that the wave function p (r , t ) is in fact separable as a function of position P (r ) and time T (t ), p (r , t ) = P (r ) T (t ) . Solving Equation 12.28 allows mapping the amplitude of the pressure in ultrasonic reactors, which can be used to determine the acoustic intensity distribution. Hence, this equation is “more important” for transformation process modelers, than the linear wave equation.

Figure 12.2. Geometrical representation of Euler’s formula.

12.2.4. Unattenuated Plane Sound Waves

standard in physics and engineering to represent harmonic oscillations, sound waves, electromagnetic waves, and variable current circuit, among others. The complex simplification of Equation 12.22 is achieved by adding the imaginary term jP sin (ω t − kr ) : p (r , t ) = P cos (ω t − kr ) + jP sin (ω t − kr )

(12.25)

In Equation 12.25 the phase angle was assumed φ = 0. The added imaginary term does not affect the real pressure expression, but, by using Euler ’s relation, e jx = cos x + j sin x

(12.26)

Equation 12.25 can be expressed as: p (r , t ) = Pe j (ωt − kr )

Plane waves have been widely studied in acoustics. Any acoustic wave far from the source, where λ is smaller than the distance to the source in a space of interest, can be considered a plane wave. Plane waves are incorporated in this review because there are many important formulas, often encountered in streaming articles, that hold true for plane waves. However, it is important to take into account that in horn reactors the distance to the source, the horn tip, cannot be considered far (e.g., negligible curvature of wavefront). An unattenuated plane wave is a disturbance of the medium that propagates with the same “waveform” at constant velocity. Hence, a solution of Equation 12.21, for a plane wave traveling in the positive x direction with velocity c can be expressed mathematically as: Φ = f ( x − ct )

(12.27)

Graphically (see Figure 12.2), the pressure can be represented as a vector rotating counter clockwise with an angular frequency ω in the complex plane (recall that ω = dθ dt ). The length of the vector represents the maximum value of the pressure P . The actual value of the pressure can determined as the real part of Equation 12.27.

(12.29)

where f ( x ) is the waveform at t = 0, while the waveform at a later time t has identical shape but is shifted by a distance ct in the positive x direction. The wave is plane and longitudinal and consequently it satisfies (Lighthill 1978b): v = f ′ ( x − ct )

(12.30)

For such a traveling plane wave, Equation 12.20 gives:

12.2.3. Helmholtz Equation The Helmholtz equation represents the timeindependent form of the wave Equation 12.15:

∂Φ = − ρ0 f ′ ( x − ct ) ( −c ) ∂t (12.31) p1 = p − p0 = ρ0 cv

p1 = p − p0 = − ρ0

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

By applying Equation 12.7 ( dp dρ ) = c 2 :

12.2.5. Acoustic Energy and Intensity A characteristic property of waves is that they transport energy without the need for any net transport of material (Lighthill 1978b). “Acoustic energy” is defined as the variation of energy produced by the acoustic perturbation (Filippi et al. 1999), while “acoustic intensity” is the rate of transport of acoustic energy. Both concepts, acoustic energy and intensity, can be used to characterize acoustic fields in sonoreactors. Indeed, the acoustic intensity distribution is crucial for the prediction of possible sonochemical effects and cavitation zones (Klíma et al. 2007). In this subsection, both concepts are presented, differentiating between values that are time dependent, or instantaneous, and values that are time-averaged over the period. Time-averaged values are derived from instantaneous values. The derivation of those terms is presented with the aim of clarifying their physical meaning. The end of the subsection shows expressions of energy and intensity of plane harmonic waves, which are widely used in the acoustics and streaming literature. 12.2.5.1. Acoustic Energy The energy transported by acoustic waves through a fluid medium is of two forms: (1) the kinetic energy of the moving elements and (2) the potential energy of the compressed fluid (Kinsler et al. 2000). The instantaneous kinetic energy density (energy per unit of volume) is associated with the fluctuation of velocity of the fluid element Ek =

1 1 2 ρ0 v 2

(12.32)

 where v 1 is the norm of the velocity fluctuation vector. The instantaneous potential energy is associated with the work conducted on the fluid by action of only the excess pressure p1 = p − p0 in a fluid element by compression to density ρ from the undisturbed density ρ0 . By definition, dW = pdV , where W is total work and V is volume. From conservation of mass ρV = ρ0V0 = constant, and after differentiation ρdV + Vdρ = 0 , the following expression is obtained: dV = − (V ρ ) dρ

(12.33)

dV = − (V ρc 2 ) dp

(12.34)

The work is obtained by integration, using the excess pressure p1 = p − p0 (instead of just the pressure p) and Equation 12.34: V

W=

∫ ( p − p ) dV 0

V0

p− p0

=−

∫

( p − p0 ) (V ρ c 2 ) d ( p − p0 ) (12.35)

p0 − p0

=

1 ( p − p0 )2 (V0 ρ0 c 2 ) 2

The instantaneous potential energy is obtained by dividing Equation 12.35 by V0 to express it in the unit of J m−3: 1 ( p − p0 ) 1 ( p1 ) Ep = = 2 2 c ρ0 2 c 2 ρ0

2

2

(12.36)

The instantaneous energy density Ei is the sum of the kinetic and potential energy densities: 1  2 1 ( p1 ) ρ0 v + 2 2 c 2 ρ0

2

Ei = Ek + E p =

(12.37)

The time average of Ei gives the energy density at any point in the fluid. For a harmonic wave, the time average can be obtained by integration over the period T of the wave: T

1 E = Ei = Ei dt T 0

∫

(12.38)

 12.2.5.2. Acoustic Intensity The intensity I i of a sound wave is the instantaneous rate per unit area at which work is done by one element of fluid on an adjacent element (Kinsler et al. 2000). Such rate of work is the product of (1) the force acting across a plane perpendicular to the direction of displacement of the wave, which per unit area is equal to the pressure, with (2) the velocity component in that direction. Although the true rate of working per unit

Chapter 12

Acoustic Field and Streaming in an Ultrasonic Horn Reactor

 area is pv , the instantaneous acoustic intensity is defined as:    (12.39) I i = ( p − p0 ) v 1 = p1v 1 This means that work is accounted for only when there is an excess pressure p1 = p − p0 (Lighthill 1978b). The intensity I is the time averaged ( )  of the instantaneous intensity I i :    (12.40) I = ( p − p0 ) v 1 = p1v 1 Equation 12.40 is the general three-dimensional  case, where the acoustic intensity is a vector I representing the rate of transport of acoustic energy. Unattenuated waves satisfy the equation of conservation of acoustic energy:  (12.41) ∇⋅I = 0  which means that the acoustic energy ∫ in I ⋅ dA entering a volume  element must be equal to the acoustic energy ∫ out I ⋅ dA leaving the volume element. 12.2.5.3. Acoustic Energy and Intensity for Plane Waves For a plane wave, the instantaneous acoustic energy density can be found by combining Equations 12.37 and 12.31:

( p1 ) 1 1 ρ0 v 2 + ρ0 v 2 = ρ 0 v 2 = ρ0 c 2 2 2 (12.42) 2

Ei = Ek + E p =

The averaged acoustic energy can be found from Equation 12.38: E = Ek + E p + =

1 1 ρ0 v 2 + ρ0 v 2 = ρ 0 v 2 3 2 2

(12.43)

Showing that for traveling plane waves the averaged values of kinetic and potential energy are equal. The instantaneous intensity of the wave can be found by combining Equations 12.31 and 12.39: I i = ρ0 cv 2 =

3

( p1 )2 ρ0 c

(12.44)

In this chapter the time averaged over the period of a variable a can be represented as a or a .

241

These values show that, for a plane wave, the rate of energy transported per unit area by the sound wave I i is c times the energy per unit volume Ei : I i = c ⋅ Ei

(12.45)

The intensity can be obtained by integration of Equation 12.44 over the period of the harmonic plane wave: T

I = Ii

T

1 1 = ρ0 cv 2 dt = ρ0 cv 2 = cρ0VA2 T 0 2

∫

(12.46)

where v 2 is the average over the period of the square of the particle velocity. Given that the particle velocity is a sinusoidal function, v 2 = (1 2 )VA2 , where VA is the amplitude of the particle velocity. Alternatively, the intensity can be obtained by integration of the right-hand side term of Equation 12.44: I = Ii

T

1 = T

T

∫ 0

( p1 )2 dt = ρ0 c

P02 2 ρ0 c

(12.47)

where P0 is the amplitude of the sinusoidal pressure wave. This result comes from the fact that the integral over a period of the square of a sinusoidal function is equal to half the amplitude of the wave.

12.2.6. Acoustic Radiation Pressure The acoustic radiation pressure is customarily interpreted as the time-averaged pressure acting on an object in a sound field. The concept of “radiation pressure” is borrowed from the theory of electromagnetism, and was initially proposed by James Clerk Maxwell after he realized that electromagnetic waves can transport momentum, showing that it is possible to exert a pressure (a radiation pressure) on an object by shining light on it. Thirty years after Maxwell’s theoretical prediction, Nichols and Hull used a torsion balance technique to measure radiation pressure by measuring the torsion angle caused by the force acting on one of the mirrors of the equipment just by irradiating light on it (Halliday and Resnick 1988). In 1902, Rayleigh started inquiring if other kinds of vibration, such as acoustic waves, can also exert pressure, finally arriving to the

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

I

Wall

c Δt Figure 12.3. Plane wave carrying an acoustic intensity I and moving at speed c. The plane wave impinges on the wall on the right-hand side.

concept of “acoustic radiation pressure” or “Rayleigh radiation pressure” (Beyer 1978). If a plane wave impinges on an obstacle, by definition, the acoustic radiation pressure must be numerically equal to the time-averaged momentum flux (or mean rate of change of momentum d (mv) / dt ) per unit area of the obstacle’s surface (Gol’dberg 1971). If the obstacle is a wall that absorbs acoustic energy, the absorption of that energy will generate a force on the wall. Consider Figure 12.3 where a plane acoustic wave travels in the positive x direction, approaching a wall in the yz-plane of an area A. If the wave has the intensity I , and the wave energy is absorbed by the wall in a time interval Δt, then the energy absorbed is equal to IAΔt . From Newton’s first law of movement, the wall must apply a force Fw in the negative x direction to stop the wave motion, which in a time Δt acts over a distance cΔt . Therefore, the work done by the wall is equal to Fw cΔt . This work equals the absorbed energy Fw cΔt = IAΔt . Therefore, the radiation pressure is defined as the force per unit area Prad =

Fw I = A c

(12.48)

Thus, from Equation 12.46 follows: Prad =

I 1 = ρ0 v 2 = ρ0V02 c 2

(12.49)

Comparing with Equations 12.45 and 12.49, it can be seen that the radiation pressure is numerically

equal to the acoustic energy density E. However, they are different; the acoustic energy is a scalar while the radiation pressure is a vector (Gol’dberg 1971). Radiation pressure is often expressed in terms of the acoustic energy density, but Equations 12.48 and 12.49 are true for an ideal fluid (Gol’dberg 1971). When the wave is just moving throughout the fluid, instead of impinging on obstacles, the radiation pressure is equal to the rate of acoustic momentum through a unit surface area. In a lossless fluid, it is impossible to see the effects of radiation pressure. But when the medium is attenuating, it produces a spatial gradient of radiation pressure, which, similarly to a gradient of “hydrostatic pressure,” causes movement of the fluid, or streaming.

12.2.7. Acoustic Impedance The concept of impedance is widely used in physics and refers to the ratio of a general driving force to the velocity response (Leighton 1997). A common case of using impedance to model oscillating driving force is the implementation of Ohm’s law to AC circuits where the impedance is the ratio of the driving force (voltage) to the velocity response (current). In acoustics, the driving force is the pressure perturbation p1 and the velocity response is the  particle velocity v 1. Hence, the specific acoustic impedance is defined as: p1 z = 1 v

(12.50)

For plane waves this ratio is: z = ± ρ0 c

(12.51)

The choice of sign depends on whether propagation is in the positive or negative x direction. The term ρ0 c is called the specific acoustic impedance of a material. Although the specific acoustic impedance of the medium is a real number for progressive plane waves, this is not true for standing plane waves or for diverging waves. Similar to the impedance in electric circuits, z is generalized as a complex number: z = r + jx

Chapter 12

Acoustic Field and Streaming in an Ultrasonic Horn Reactor

where r is the specific acoustic resistance ρ0 c and x is the specific acoustic reactance of the medium for the considered wave.

12.2.8. Boundary Conditions As with any waveforms, acoustic waves can be reflected and transmitted at interfaces between two different media (see Figure 12.4). The ratios of the pressure amplitudes and intensities of the reflected and transmitted waves to those of the incident wave depend on the characteristic acoustic impedances, the speed of sound of the two media, and the angle of incidence (Kinsler et al. 2000). The pressure transmission and reflection coefficients are defined as:

243

2. Continuity of normal component of velocity: the normal components of the particle velocities on both sides of the boundary must be equal. This indicates that the fluids remain in contact at the boundary.       (12.55) n ⋅ vi + n ⋅ vr = n ⋅ vt From Equations 12.54 and 12.55, other interface boundary conditions can be derived, depending on the angle of incidence and whether the material is another fluid or a rigid wall. Some of these boundaries will be explained below. 12.2.8.1. Plane Wave, Normally Incident on a Planar Boundary (see Figure 12.4) In this case, Equations 12.52–12.54 imply that:

T = pt pi

(12.52)

1+ R = T

R = pr pi

(12.53)

Division of Equation 12.54 by Equation 12.55 yields:

where subscripts i , r , and t stand for incident, reflected, and transmitted waves, respectively. There are two boundary conditions to be satisfied for all times and at all points on the interface boundary: 1. Continuity of pressure: the acoustic pressures on both sides of the boundary must be equal. This boundary condition indicates that there cannot be a net force acting on the interface separating the fluids. pi + pr = pt

Medium 1

(12.54)

Medium 2

pi pt pr

pi + pr pt = = z2 vi + vr vt

(12.56)

(12.57)

which is a statement of the continuity of the normal specific acoustic impedance across the boundary. For a plane wave p v = ±r = ± ρc , the sign depending on the direction of propagation. Hence, after some algebraic manipulations: R=

r2 − r1 r2 + r1

(12.58)

T=

2r2 r2 + r1

(12.59)

and

12.2.8.2. Plane Wave, Oblique Incident on a Planar Boundary (see Figure 12.5) This case is similar to the reflection and transmission of light trough materials of different refractive indices. The angle of reflection and transmission can be found via the law of reflection: sin θ i = sin θ r

(12.60)

sin θ i sin θ t = c1 c2

(12.61)

and Snell’s law: Interface Figure 12.4. Reflection and transmission of a plane wave normally incident on a planar boundary.

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Medium 1

where ∂ ∂n denotes the gradient in the direction of the normal to the wall boundary following the nomenclature used by COMSOL Multiphysics™.

Medium 2

qi qt

qr

Interface Figure 12.5. Reflection and transmission of a plane wave (oblique incident on a planar boundary).

The following subsection explains some standard boundary conditions in modeling packages, such as COMSOL Multiphysics™ (COMSOL AB, Stockholm, Sweden). These are “sound hard boundary wall,” “sound soft boundary,” and “pressure source.” 12.2.8.3. Sound Hard Boundary Wall This is used to model rigid surfaces, which are walls made of particles that do not move. In order to satisfy the continuity of normal component of velocity,       n ⋅ vi + n ⋅ vr = n ⋅ vt (Eq. 12.55), the normal component of the particle velocity on the fluid side must vanish because the “velocity” of the particles on the   solid side n ⋅ vt is zero. Because there is no normal velocity at the interface, this boundary condition is   equivalent to normal acceleration n ⋅ ( ∂v ∂t ) being equal to zero. Hence, multiplying each side of  Equation 12.12 per the normal unit vector n   ⎛ ∂v ⎞  ⎡ 1 ⎤ (12.62) n ⋅ ⎜ ⎟ = n ⋅ ⎢ − ∇p ⎥ ⎝ ∂t ⎠ ρ ⎣ 0 ⎦ allows expression of the sound hard boundary condition as:  ⎡1 ⎤ n ⋅ ⎢ ∇p ⎥ = 0 ⎣ ρ0 ⎦

(12.63)

This is also equivalent to the normal derivate of the pressure at the boundary equal to zero ∂p =0 ∂n

(12.64)

12.2.8.4. Sound Soft Boundary When r1 = r2 the wave will be completely transmitted as seen with reflection and transmission coefficients calculated with Equations 12.58 and 12.59 ( R = 0 , T = 1 ). In this case the two media are said to be “impedance matched.” However, the opposite occurs in liquid– gas boundaries such as the air–water interface. The impedance of air is 4 × 102 kg/m2/s, while that of water is 1.5 × 106 kg/m2/s. Hence, a wave in air impinging on water has a pressure reflection coefficient of R = 0.999 . When the wave moves from water to air, R = −0.999 . Therefore, in both cases, the wave is almost entirely reflected, and in the second case, the wave is also inverted (Leighton 1997). This large difference in impedance between the two media causes the wave to reflect so the two media are said to be “impedance mismatched.” Assuming R = 1 implies that the amplitude of the reflected wave is equal to the incident wave Pr = Pi (see Eq. 12.53). Hence, in order to satisfy continuity of pressure at the boundary pi + pr = pt (see Eq. 12.54), the acoustic pressure pt = pi = pr must vanish at the interface: p=0

(12.65)

This boundary condition is called sound soft boundary and is appropriate to model liquid–gas interfaces that are characterized by a high “impedance mismatch.” 12.2.8.5. Pressure Source For harmonic waves, this boundary condition is applied to specify the amplitude of the harmonic pressure P at the boundary. p=P

(12.66)

Radiation boundaries should be applied for acoustics sources, but they will not be explained in this chapter for the sake of simplicity. A complete explanation of this type of boundaries can be found in chapter 7 of Fundamentals of Acoustics (Kinsler et al. 2000).

Chapter 12

Acoustic Field and Streaming in an Ultrasonic Horn Reactor

12.2.9. Rays of Sound In some cases the sound propagation can be expressed in terms of rays rather than in terms of waves (see Figure 12.6). This concept will be introduced in this section for unattenuated waves, but it will be further developed in Section 12.5.1, when modeling “Stuart streaming,” which is the highpower jet-like streaming caused by the attenuation of focused sound sources that are modeled as rays of sound. Rays of sound are reminiscent of the rays of light studied in geometrical optics (Lighthill 1978b). For instance, the speed of sound (and consequently the impedance r = ρc ) is often a function of space so the direction of the propagation of the wave changes as it moves through the media (Kinsler et al. 2000). Certain sound radiation patterns, such as highfrequency sound radiated from planar circular pistons (e.g., of ultrasonic horns), can also be considered rays of sound (or Gaussian beams) (Kamakura et al. 1995; Perov and Rinkevich 2008). These cases can be studied under the assumption that the acoustic energy is transmitted along a tube or beam of sound, whose transversal area is perpendicular to the direction of the sound propagation. The direction of the sound propagation is equal to the direction of the sound intensity vector I (see Figure 12.6). If there is no absorption or production of sound within the beam, the conservation of acoustic energy (see Eq. 12.41) says that the energy entering the beam must be equal to the energy leaving the beam. Given that the intensity is the rate of transport of acoustic energy per unit of area, the energy entering or leaving the beam is equal to the

Area A1

Ray of sound

245

product of the intensity I and the transversal area of the beam A. In other words, the energy conservation of the beam can be expressed as: IA = constant

(12.67)

12.2.10. Numerical Simulations of Ultrasonic Horn Reactor Klíma et al. (2007) modeled the intensity distribution in an ultrasonic reactor, which is necessary for the determination of cavitation zones in the reactor. The acoustic pressure amplitude was obtained by solving the Helmholtz equation (Eq. 12.28) while neglecting sound absorption. The following boundary conditions were applied to the model: 1. p = P0 4 at the horn tip and ∂p ∂n = 0 (sound hard boundary) at the side walls of the horn. 2. p = 0 (sound soft boundary) at all other walls. Sound soft boundaries around the reactor allow total reflection at the liquid–air boundary.5 Hence, by adjusting the geometry of the reactor, it is possible to find an optimal configuration where standing waves and maximum ultrasound intensities zones are formed. Figures 12.7 and 12.8 show a replication of Klima’s work that was conducted at CSIRO Food and Nutritional Sciences (CFNS).6 As seen in the figures, changes in reactor geometry have a significant impact on the sound intensity distribution, and configurations that produce zones of very highpressure amplitude and intensity where cavitation can occur can be identified. This type of optimization is empirically applied for designing ultrasonic baths, working at low acoustic power levels in order to avoid cavitation damage of the tank walls (Mason 1998). However, horn reactors usually operate at very high power, and under these conditions, sound absorption cannot be neglected. In fact, it is widely

Area A2 4

Intensity I1 Intensity I2 A1I1 = A2I2 Figure 12.6. Unattenuated ray of sound.

This is the amplitude of the sinusoidal function of pressure p (r , t ) = Pei(ωt −kr ) . 5 In this case the effect of the reactor wall is neglected. 6 Klima calculated the intensity via Equation 12.47, which applies for plane waves. However, this approximation does not hold true for horn reactors; the calculation of the intensity can be corrected via Equation 12.39.

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Pressure [Pa]

Max: 4.722e5 5 –0.05 ×10 0 0.05 4 0.1

Int [W/m2]

Max: 7.534e ×104 –0.05 7 0 0.05 6 0.1

3 0.05

y x –0.05

0

0 0.05

0.05

z x

y –0.05

0

0 0.05

Min: –8.939e4

(A) Acoustic pressure

1 0 Min: 0

3.5 2.5

0.05

3 2

4 3

4

2

0

–0.05 0 0.05 0.1

5

1 z

Max: 4.4e ×103

Intensity, norm [W/m2]

2 1.5

z x

1 y –0.05

(B) Intensity (plane wave approximation)

0

0.5

0 0.05

0 Min: 0

(C) Intensity

Figure 12.7. Acoustic pressure and sound intensity simulation for R (radius of the reactor) = H (distance from reactor top to horn tip) = D (distance from horn tip to reactor base) = 50 mm; r (horn radius) = 6.5 mm and λ = 75 mm. (A) Acoustic pressure. (B) Klima’s calculation of the intensity assuming plane wave (Eq. 12.47). (C) Corrected calculation at CFNS via Equation 12.39. See color insert.

Pressure [Pa]

Int [W/m2]

Max: 9.716e6 ×106 0 8 0.1 6 4

0.05

0 –2

z x

–4 y 0

0

–6

Max: 254 ×107 4

0.1

z

2.5

1.5

x

0.5

y 0

0

0 Min: 0

(B) Intensity (plane wave approximation)

3.5 3

0.05

2 1.5

1

Min: –7.999e6

(A) Acoustic pressure

0

2

2 0.05

Intensity, norm [W/m2]

Max: 3.189 ×107 0 3 0.1 2.5

z x

1 y 0

0

0.5 0 Min: 0

(C) Intensity

Figure 12.8. Acoustic pressure and sound intensity simulation for optimized geometrical configuration: R (radius of the reactor) = 45 mm; H (distance from reactor top to horn tip) = 25 mm; D (distance from horn tip to reactor base) = 77 mm; r (horn radius) = 6.5 mm and λ = 75 mm. (A) Acoustic pressure. (B) Klima’s calculation of the intensity assuming plane wave (Eq. 12.47). (C) Corrected calculation at CFNS via Equation 12.39. See color insert.

reported that in horn reactors, all the sound power is absorbed within a very narrow region from the horn tip (Gogate et al. 2002).

far from the source, and λ much smaller than the distance to the source (Kinsler et al. 2000). The sources of dissipation caused by the medium can be divided into three categories:

12.3. Nonlinear Acoustics, Sound Absorption, and Acoustic Streaming

1. Viscous losses: The linear wave equation neglects viscosity (assumption 2). However, viscous losses occur when there is a relative motion between adjacent portions of the fluid such as during compression and expansion on passage of sound waves. Taking into account viscous losses requires incorporating the viscous shear term in the Navier–Stokes equation (Eq. 12.11).

The linear wave equation neglects any losses of acoustic energy in spite of the fact that ultimately all acoustic energy is converted into thermal energy. Assumptions 1–5 in Section 12.2.1 apply to many situations for low acoustic power levels, distances

Chapter 12

Acoustic Field and Streaming in an Ultrasonic Horn Reactor

2. Heat conduction losses: The linear wave equation assumes that the compression–expansion is thermodynamically reversible and adiabatic (assumption 5). Hence, as seen in previous chapters of this book on high-pressure processing (e.g., Chapters 4 and 5), the temperature of the fluid must increase during compression and decrease during expansion on passage of the acoustic wave. However, those temperature changes do not necessarily follow a reversible adiabatic process, and some heat may flow from the compressed and “hot” parts of the medium to the adjacent parts that are expanded and “cold.” In this case, irreversible thermodynamics offers an appropriate framework to take into account heat conduction losses. This will be explained in Section 12.3.1.4). 3. Molecular relaxation losses: These types of losses are caused by processes on the molecular level, such as conversion of the kinetic energy of the molecules into stored potential energy formed by structural rearrangement of clusters, and rotational and vibrational energies of polyatomic molecules, among others. The nonlinear theory of sound allows considering absorption and attenuation of sound waves in thermo-viscous fluids (fluids that may undergo viscous and heat conduction losses). This theory is based on the equation of state and the laws of conservation of mass, momentum, and energy, which is the same framework from where the linear wave equation was derived but without incorporating linearization assumptions 1–5. The nonlinear theory is a wider generalization that permits analysis not only of absorption and attenuation but also of streaming and heat generation (Tjotta 1999, 2000).

12.3.1. Nonlinear Acoustics The nonlinear theory of sound propagation in homogeneous thermo-viscous fluids is based on the following fundamental equations: the equation of state, the continuity equation (or law of conservation of mass), the conservation of energy (or entropy equation in irreversible thermodynamics), and the momentum equation (Tjotta 1999).

247

12.3.1.1. State Equation This is a thermodynamic equation that describes the state of a material being considered. It correlates state variables such as density, temperature, volume, internal specific energy, and specific entropy, among others, to describe the momentary condition of a thermodynamic state:

ρ = ρ ( p, T ); s = s ( ρ, T )

(12.68)

Equation 12.1, for example, is the state equation of ideal gases. 12.3.1.2. Continuity Equation This is the law of conservation of mass and is given by: ∂ρ  = −∇ ⋅ ( ρv ) ∂t

(12.69)

12.3.1.3. Momentum Equation The law of conservation of momentum of a compressible fluid, in the absence of external body forces, is expressed by the nonlinear Navier–Stokes equation bellow7:  ∂v   ρ ⎛⎜ + v ⋅∇v ⎞⎟ ⎝ ∂t ⎠ (12.70)  ⎛1  ⎞ = −∇p + μ∇ 2 v + ⎜ μ + μ B ⎟ ∇ (∇ ⋅ v ) ⎝3 ⎠ where μ B is the volume viscosity or bulk viscosity. The last term on the right-hand side appears when taking into account the fluid compressibility. This term disappears for incompressible fluids, where the  divergence (∇ ⋅ v ) equals to zero as a consequence of incompressibility ( ρ = constant, ∂ρ ∂t = 0 ) from the continuity Equation 12.69. In that case, Equation 12.70 transforms into Equation 12.11 when neglecting external forces. 12.3.1.4. Energy Conservation Equation Irreversible thermodynamics allow rewriting of the energy conservation equation in terms of entropy. This is done by applying the thermodynamic relation of internal energy du = Tds − pdv 7

Equation 12.11 applies to compressible fluids.

(12.71)

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

to the energy conservation equation. Equation 12.71 indicates that the internal energy ( u) can be expressed in terms of entropy ( s) and specific volume ( v ): ⎛ ∂u ⎞ ⎛ ∂u ⎞ du = ⎜ ⎟ ds − ⎜ ⎟ dv ⎝ ∂s ⎠ v ⎝ ∂v ⎠ s

(12.72)

Thus, Equations 12.71 and 12.72 thermodynamically define temperature as T = ( ∂u ∂s )v and pressure as p = ( ∂u ∂v )s , allowing expression of the conservation of energy as (Filippi et al. 1999):

ρT

ds  = τ : D − ∇⋅q + r dt

(12.73)

is the viscosity stress tensor, where τ    D = 1 2 (∇v + ∇v T ) is the strain rate tensor, q is the heat flux vector, and r is a heat source per unit volume and unit time. Equation 12.73 indicates how entropy moves under perturbations from the equilibrium and accounts for heat conduction losses. The linear wave equation neglected heat conduction and assumed that the compression and expansion is reversible and adiabatic (assumption 5). In that case, (ds dt ) = 0 or Δs = ΔQrev T = 0. By applying the first law of thermodynamics dU = − pdV ⇒ Cv dT = − pdV and the ideal gas equation p = ρ RT , the reversible adiabatic compression–expansion Equation 12.3 is found ( pρ −γ = constant ), which was used to derive the linear wave equation. Equation 12.73 is a general expression of heat exchange processes that deviate from ideal conditions (adiabatic irreversibility), and is used to account for heat conduction losses during sound absorption and attenuation.

12.3.2. Absorption from Viscous Losses Losses caused by viscosity can be estimated via the nonlinear Navier–Stokes Equation 12.70 when the  viscous term ( μ∇ 2 v ) is not neglected. By assuming an adiabatic process and linearizing the continuity equation (see Eq. 12.9), the lossy wave equation for a viscous fluid can be obtained: ∂⎞ 2 1 ∂2 p ⎛ ⎜⎝ 1 + τ s ⎟⎠ ∇ p = 2 ∂t c ∂t

(12.74)

where τ s is a relaxation time. Under the assumption of monofrequency motion, this wave equation is

reduced to a lossy Helmholtz equation (Kinsler et al. 2000): ∇2 p + κ 2 p = 0

(12.75)

where κ is a complex wave number:

κ = k − jα s

(12.76)

and α s is the absorption coefficient associated to viscosity. The solution of Equation 12.75 is a damped traveling wave solution: p = P0 e −α s x e j (wt − kx )

(12.77)

The pressure amplitude of the wave at a location x is given by P0 e −α s x , where e −α s x is a damping factor caused by viscous losses. For plane waves, the damped amplitude can be replaced into Equation 12.47 to obtain the intensity as a function of x: I ( x ) = ( P0 e −α s x ) 2 ρ0 c = I 0 e −2α s x 2

(12.78)

where I 0 = P0 2 ρ0 c is the intensity at x = 0. A similar analysis can be conducted for heat conduction losses to determine the absorption coefficient associated to thermal conduction. For small losses, it is possible to calculate a total absorption coefficient α as the sum of absorption coefficients of individual loss mechanisms:

α=

∑α

i

(12.79)

i

Hence, Equations 12.75–12.78 can be used by replacing α s = α . Figure 12.9 shows the predictions of acoustic pressure and intensity using a model that solves Equation 12.75 at different values of the absorption coefficient. The figure shows that at high α the sound can be completely attenuated in the close vicinity of the horn tip as reported in the literature (Gogate et al. 2002).

12.3.3. Acoustic Streaming Acoustic streaming is a term that describes the timeaveraged velocity induced in a flow near a vibrating element, or by the absorption of acoustic waves by the medium. Two main factors are known to induce acoustic streaming, friction between a medium and a vibrating object, and the spatial attenuation

Chapter 12

a m–1 0.01

Acoustic Field and Streaming in an Ultrasonic Horn Reactor

Acoustic pressure Pressure [Pa] 0.1

Intensity Max: 4.722e5 ×105

Max: 1.00e5 ×105 1

Intensity, norm [W/m2] 0.1

4 0.08

0.9 0.7

2

0.04

0.06

0.6 0.5

z

z

1

0.8

0.08

3

0.06

0

0.04

0.4

–1

0.02

–0.02

0

1

0.02 r

–3

0.04

0.06 0.08 Min: –3.337e5

Pressure [Pa]

Max: 4.722e5 ×105

0.1

2

z

0.06

1

0.04

0 0.02

–1 –0.02

0

10

0.02 r

0.04

0.06 0.08 Min: –2.492e5

Pressure [Pa]

Max: 4.722e5 ×105

0.1

1 0.5

0

0 –0.02

0

0.02 r

0.04

0.06 0.08 Min: –3.064e4

0.6 0.5

0.04

0.4 0.3

0.02

0.2 0.1

0 –0.02

0

0.02 r

0.04

0.06

Intensity, norm [W/m2] 0.1

0.08 0 Min: 0 Max: 1.00e5 ×105 1 0.9

0.08

0.8 0.7

0.06

0.6 0.5

z

z 0.02

Max: 1.00e5 ×105 1

0.7

2.5 1.5

0.08 0 Min: 0

0.8

0.06

3 2

0.06

0.9

3.5

0.04

0.04

0.08

4

0.06

0.02 r

0.1

4.5

0.08

0

Intensity, norm [W/m2]

–2

0

0.1 –0.02

z

3

0.2

0

4 0.08

0.3

0.02

–2 0

249

0.04

0.4 0.3

0.02

0.2 0.1

0 –0.02

0

0.02 r

0.04

0.06

0.08 0 Min: 0

Figure 12.9. Optimized geometry from Figure 12.8 modeling the effect of the absorption coefficient on acoustic pressure and intensity. See color insert.

of waves in the medium (Kwon et al. 2007). The former is called Rayleigh streaming, associated with boundary layers at solid surfaces (Riley 1998). Rayleigh streaming has two components: inner and outer streaming. The inner streaming is created within the boundary layer due to friction between the medium and the wall. Subsequently, inner

streaming induces larger scale steady streaming outside the boundary layer (Loh et al. 2002). Work in this type of streaming has been conducted by Rayleigh (1896), Nyborg (1958), Jackson (1960), and more recently by Kwon, Loh, Lee, and coworkers (Loh et al. 2002, 2006; Kwon et al. 2007), and Alassar (2008).

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

This chapter focuses on the streaming caused by the absorption or attenuation of sound waves. This is also known as “Eckart streaming” (Eckart 1948) or “quartz wind” (Riley 1998) because it became more popular when piezoelectric ultrasonics came into use. The analysis of this streaming is divided in two sections. The first one (Section 12.4) explains the classical treatment of Rayleigh (1896), Nyborg (1953), and Westervelt (1953), which Lighthill (1978a) called RNW streaming, and has been shown to be valid only for acoustic sources of very low power. The second one (Section 12.5) explains streaming at high acoustic powers, which generates turbulent jets as a consequence of sound absorption. So far in this review, the concept that streaming is caused by sound absorption (or attenuation) has been given without explanation, but this will be demonstrated in Section 12.4.3 after the RNW streaming theory and the Reynolds stress concept are explained.

12.4. Low-Power Acoustic Streaming Rayleigh, Nyborg, and Westervelt (RNW) Streaming Theory 12.4.1. Successive Approximations Versus Reynolds Stress Equations 12.68–12.70 and 12.73 give the mathematical framework for analyzing nonlinear acoustic effects such as streaming and heat generation. As mentioned before, “quartz winds” or “Eckart streaming” is the time-averaged flow induced by the spatial absorption of sound waves. To analyze this phenomenon, Nyborg (1953) started from the continuity and momentum Equations 12.69 and 12.70. Then, the solution of those equations was approached by using the method of successive approximations, where each variable was expanded as series of terms representing the excess pressure p1 = ( p − p0 ), excess density ρ1 = ( ρ − ρ0 ), and excess velocity    v 1 = ( v − v0 ) at any point by: p1 = p1 + p2 + …

ρ1 = ρ1 + ρ2 + …    v 1 = v1 + v2 + …

(12.80)

(A) successive approximations method n1

n2

(B) Reynolds approach, for average f and time-dependent f(t) values. f(t) f f¢ = f(t) – f Figure 12.10. Comparison of the successive approximation method versus Reynolds stress approach.

 where the quantities p1 , ρ1 , and v1 are the firstorder approximation and are usually the solution of the wave equation; they vary sinusoidally in time with frequency ω and thus represent the sound field (Nyborg 1953). The second order approximation p2 ,  ρ2 v2 are time independent or “time-averaged” terms that yield corrections to be added to the first  order quantities. The second order velocity v2 is the   time-averaged streaming velocity. v1 and v2 can  be orders of magnitude different. v1 represents the velocity of contraction and expansion of particles or elements of fluid during the passage of sound waves. That velocity is of the same order of magnitude of the speed of sound in the medium (approximately 1,500 m/s for water). The streaming velocity  v2 reported in literature ranges from only a few millimeters up to 1–2 meters per second. Figure 12.10 shows the superposition effect of adding the first and second approximation of velocity. As seen in the   figure, the time average of v1 + v2 is equal to the  time-independent term v2 . This mathematical approach of modeling streaming by expanding variables as first- and second-order quantities was initiated by the work of Nyborg (1953), Eckart (1948), and Westervelt (1953), and it is still followed by physicists and acousticians today (Nowicki et al. 1997, 1998; Loh et al. 2006; Kwon et al. 2007). However, the same results can be obtained by transforming the instantaneous8 continuity (see Eq. 8

Steady state equation at the time scale of the perturbation.

Chapter 12

Acoustic Field and Streaming in an Ultrasonic Horn Reactor

12.69) and momentum (see Eq. 12.70) equations into the Reynolds equations, which are used to study the influence of turbulent fluctuations on the mean flow. Reynolds showed that turbulent variations of velocity around the mean value act as an effective stress on the flow. Then, Lighthill (1978b) demonstrated that wave propagating through a flow acts similarly to Reynolds stress. Reynolds equations are time-averaged versions of Equations 12.69 and 12.70 used to model turbulence, and can be applied to analyze the time-averaged effect of acoustic fluctuations. Reynolds stress analysis facilitates the point of view of researchers working on fluid dynamics and also process modelers, who are familiar with turbulent models such as k − ε and k − ω . Reynolds introduced the mean value of random variations into the theory of turbulence. This approach is the base of common turbulent models such as the mixing length, k − ε and k − ω . Reynolds equations are obtained by defining time-dependent  flow properties φ ( p , ρ , v , etc.) as the sum of an average mean component φ and a time-varying fluctuating component φ ′ :

φ (t ) = φ + φ ′

(12.81)

τ ijR = ρvi v j

251

(12.84)

As with any stress system, the spatial variation of the Reynolds stress ∇ ⋅ τ R can cause a net force per unit volume F to act on the fluid:  (12.85) F = −∇ ⋅ τ ijR So Equation 12.83 becomes:

 ρ ( v ⋅∇v ) = −∇p + μ∇ 2 v + F

(12.86)

Dimensional inspection of Equation 12.83 may help to realize that the term ∇ ⋅ τ R is a force per unit volume as each term of the equation has units of N m−3. Following the Einstein notation used by Lighthill, the j component of force per unit volume can be written as:

(

 ∂ ρvi v j Fj = − ∂xi

)

(12.87)

where j can be 1, 2, or 3 and the repeated suffix i is summed from 1 to 3. Thus, for j = 2:

) ( ) (

(

∂ ρv22  ∂ ρv1v2 ∂ ρv3 v2 F2 = − − − ∂x1 ∂x2 ∂x3

)

(12.88)

Equation 12.87 can be written as the sum of two terms:

By definition, the average of the fluctuating component must be zero ( φ ′ = 0 ). Hence, averaging both sides of Equation 12.81 gives φ (t ) = φ . This means that v (t ) = v , T (t ) = T , and so on. However, when averaging a term that contains a product of two terms such as:

ρvi (t ) v j (t ) = ρvi v j + ρvi′v ′j ρvi′v ′j is not equal to zero. Thus, after taking the time average of the components of the incompressible continuity and Navier–Stokes equations, the terms whose average does not vanish form a tensor called the Reynolds stress: ∇ ⋅ (v ) = 0

(12.82)

ρ ( v ⋅∇v ) = −∇p + μ∇ 2 v − ∇ ⋅ τ R

(12.83)

where τ R is the Reynolds stress tensor. The terms of τ R are given by:

(

)

 ∂ ρvi v j Fj = = − v j ∂ ( ρvi ) ∂xi + − ρvi ∂ ( v j ) ∂xi ∂xi (12.89)

(

) (

)

12.4.2. RNW Streaming Theory Equation 12.89 is equivalent to the force obtained by following the successive approximations method of Nyborg (1953), Eckart (1948), and Westervelt (1953):      (12.90) F = − ρ0 ( v1 ⋅∇ ) v1 + v1 (∇ ⋅ v1 ) where the square brackets indicate the time average  and v1 is the first-order time-dependent velocity coming from the successive approximation method (see Eq. 12.80). Rayleigh, Nyborg, and Westervelt approached streaming via the continuity and momentum equations, but neglecting the left-hand side of Equation 12.83:

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

 0 = −∇p + μ∇ 2 v + F

(12.91)

The argument for doing this is that acoustic streaming is a second-order effect. However, this approximation is known to be valid only for very slow flows (called “creeping motion”), with the condition of Reynolds number Re < 1. Lighthill (1978a) proved that by neglecting the convective term v ⋅∇v the RNW streaming theory is applicable only to low sources of acoustic powers of microwatts or less (Lighthill 1978a). The majority of publications in acoustic streaming expresses F via Equation 12.90, probably for historical reasons, following the pioneering work of Nyborg and Eckart, even though both approaches, successive approximations and Reynolds stress, produce the same results. However, the Reynolds stress approach offers a framework that links turbulence with acoustic perturbations. Indeed, acoustic streaming shows a symmetry with the sound generated by turbulent fluid flows flow noise: Reynolds stress appears by averaging the effect of turbulent fluctuations. The mean value of Reynolds stress is the turbulent momentum flux, whose fluctuations in a flow field generates noise (sound). In the acoustic case, sound is a perturbation moving through a fluid. The averaging of that perturbation is a Reynolds stress, whose spatial variation generates flow (streaming). For instance, the turbulent flow in jet engines produces sound. But not only a jet generates sound, also sound can generate a jet (as explained in Section 12.5 for high sources of acoustic power). Lighthill, who is the founder of aeroacoustics, developed the theory of sound generation by fluid flows, linking turbulence and sound via the Reynolds stress (Lighthill 1952, 1954). Reynolds stress analysis also allows the study of the propagation of sound through moving fluids, which is capable of stimulating the fluid motion to generate additional acoustic energy (Lighthill 1972). Hence, analyzing nonlinear acoustic effects via Reynolds stress facilitates study of not only streaming but also other important effects observed in horn reactors such as the generation of heat and turbulence at high acoustic powers. Therefore, this approach rather than the successive approximation method will be used in this review.

12.4.3. Sound Attenuation or Absorption as the Cause of Streaming As explained in Section 12.4.2, the force that causes streaming is the spatial variation of the Reynolds stress ∇ ⋅ τ R . This is similar to saying that the spatial variation in pressure ∇p is a driving force in the momentum equation (Eq. 12.70). Pressure is a force per unit area, but if the pressure does not change between two spatial points, there is no driving force between those points regardless of the actual pressure magnitude. Similarly, there must be spatial variation of the Reynolds stress (not just the presence of Reynolds stress) for streaming to occur. The question is: what causes spatial variations of Reynolds stress? The answer is attenuation or sound absorption, and this will be shown for a plane wave. If the plane wave is moving in positive x direction, the driving force calculated with Equation 12.87 or 12.88 is: F=−

( )

∂ ρv 2 ∂x

(12.92)

From Equation 12.46 the intensity is defined as: I = ρ0 cv 2

(12.93)

Hence, ρ v 2 = ( I c ) and Equation 12.92 can be expressed in terms of the acoustic intensity: F=−

1 dI 1 = − ∇I c dx c

(12.94)

If the wave is unattenuated ∇I = 0 (see Eq. 12.41), the force F vanishes. But if the plane wave is attenuated, the intensity of the sound changes spatially according to Equation 12.78 ( I = I 0 e −2α x ). Therefore, for a plane wave, a spatial variation of the intensity causes streaming. In that case, Equation 12.94 can be expressed in terms of the absorption coefficient α and the intensity I : F=

2α −2α x 2α I0e = I c c

(12.95)

Again, in the absence of absorption α = 0, the force vanishes and streaming does not occur. Figures 12.7B and 12.8B show spatial variation of the acoustic intensity. Does this mean that streaming

Chapter 12

Acoustic Field and Streaming in an Ultrasonic Horn Reactor

occurs in that reactor? The answer is “No” because Equations 12.94 and 12.95, which are frequently encountered in the streaming literature (Nowicki et al. 1997; Tjotta 1999), are true only for plane waves. The modeling exercise in Section 12.2.10 is a theoretical example of modeling the acoustic field in a sonoreactor based on linear acoustics. If linear acoustics hold true, streaming is “prohibited” by the linear assumption 3. The intensity distribution displayed in those figures follows the law of conservation of acoustic energy (see Eq. 12.41), which says that the divergence of the intensity vector9 is zero ( ∇ ⋅ I = 0). That means that a lossless acoustic wave may spread out from the source, changing its intensity pattern, but still being divergence-less. This will be clarified further in Section 12.5.1 for attenuation of rays of sound. Other publications (Piercy and Lamb 1954) quantify the streaming driving force as the negative spatial variation of radiation pressure, which is equivalent to spatial variation of Reynolds stress. This will also be demonstrated for plane waves. In this case the radiation pressure is given by Equation 12.49 Prad =

I c

(12.96)

Defining the streaming driving force, for a plane wave moving in the positive x direction, as the negative of the spatial variation of the acoustic radiation pressure gives: F=−

dPrad I dI I =− = − ∇I dx c dx c

(12.97)

Again, for unattenuated waves, ∇I = 0 (see Eq. 12.41), and the force F vanishes. For attenuated plane waves, Equation 12.78 applies ( I = I 0 e −2α x ). Hence, the force is given by: F=

2α −2α x 2α I0e = I c c

9

Section 12.4.4 expresses the gradient of the acoustic radiation pressure as the streaming driving force.

12.4.4. Experimental Method to Determine Sound Absorption Coefficients from Measurements of Acoustic Streaming The knowledge of absorption coefficients is essential for appropriately modeling an ultrasound process with associated wave attenuation and the generation of heat and motion. This section will present and discuss an approach based on the RNW theory for the determination of absorption coefficients. The experiment was designed by Piercy and Lamb (1954) based on the streaming generated by sound absorption. Figure 12.11 shows a schematic representation of the apparatus, which is basically an ultrasonic radiator on one side of the main tube, with a sensor on the opposite side. The absorption of the acoustic energy produces streaming in the main tube. The bleeder (smaller) tube allows the fluid to return. For a damped or attenuated plane wave, the intensity is given by Equation 12.78: I ( x ) = I 0 e −2α x Piercy and Lamb used the radiation pressure concept to analyze streaming. For a plane wave, the radiation pressure at any point in the medium is a force per unit area acting in the direction of propagation of the sound wave with a magnitude equal to the energy density (see left-hand side terms of Eq. 12.49): Prad =

Intensity is a vector in 2D and 3D, while in 1D is a scalar.

I I 0 e −2α x = c c

(12.99)

Then, the acoustic driving force, F , will be equal to the negative gradient of radiation pressure dPrad dx (see Eqs. 12.97–12.98).

(12.98)

As can be seen, Equations 12.97 and 12.98 are identical to Equations 12.94 and 12.95. The example on determining absorption coefficients described in

253

F=−

dPrad 2α I 0 e −2αl = dx c

(12.100)

where x has been replaced by the length of the tube l . The large tube exhibits streaming due to the force F , while the smaller tube allows the mass flow to return (see Figure 12.11). The mathematical analysis

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

absorber

main tube

acoustic source

acoustic sensor bleeder or smaller tube

Figure 12.11. Experimental equipment design by Piercy and Lamb (1954).

conducted by Piercy and Lamb did not result in an expression of the velocity distribution in the main tube. Instead, from the law of momentum conservation, Piercy and Lamb concluded that the acoustic driving force in the main tube must be balanced with the pressure drop in the bleeder tube, with a velocity profile following Poisseuille’s law: vb =

Δp ( Rb2 − rb2 ) 4 μl

(12.101)

where vb is the velocity in the bleeder tube, rb is the radial distance from the bleeder tube axis to a given point, Rb is the radius of the bleeder tube, Δp is the pressure drop in the bleeder tube, and l is the length of the bleeder. The pressure difference Δp arises from the change in the radiation pressure due to the attenuation of the sound wave over a path length l in the main tube. Then, the driving force in the bleeder tube, which is the pressure gradient: dp dx ≈ Δp Δ x = Δp l , must be equal to the driving force that causes streaming in the main tube Δp l = F = 2α I 0 e −2αl c . Hence, substituting Δp l into Equation 12.101 gives: 2α I 0 e −2αl ( Rb2 − rb2 ) vb = 4 μc

(12.102)

This equation allows calculation of the velocity profile in the bleeder tube and can be used to determine the absorption coefficient α from experimental measurements of (1) velocities in the bleeder

tube vb , measured by microscope observation of the motion of small aluminum particles suspended in the liquid, and (2) the intensity at the right-hand side of the main tube ( I 0 e −2αl ), which is measured with the sensor (see Figure 12.11). The flow equation in the main tube will be derived based on Equation 12.91, as an exercise of the application of the RNW theory. Piercy and Lamb’s mathematical analysis will be completed in this chapter by understanding that streaming is only caused by the radiation pressure generated after sound attenuation. The equation of momentum (Eq. 12.91) in radial coordinates is: 0=−

∂p μ ∂ ⎡ ∂v ⎤ + r +F dz r ∂r ⎢⎣ ∂r ⎥⎦

(12.103)

Here, the term ∂p ∂z is neglected given that any observed “pressure difference” in the main tube is really a “radiation pressure” whose driving force is F. According to the continuity equation: ∂v =0 dz

(12.104)

Hence, v is only dependent of r ; therefore from Equation 12.103 follows: d ⎡ dv ⎤ r r =− F dr ⎢⎣ dr ⎥⎦ μ

(12.105)

and after integration: r

dv r2 =− F+A dr 2μ

(12.106)

where the integration constant A = 0 is evaluated with the boundary condition dv dr = 0 at r = 0 . After a second integration follows: v=−

r2 F+B 4μ

(12.107)

The constant B is evaluated with the boundary condition v = 0 at r = R . Hence, after replacing Equation 12.100 and rearranging follows: vm =

2α I 0 e −2αl 2 ( Rm − rm2 ) 4 μc

(12.108)

where the subscript m stands for main tube.

Chapter 12

Acoustic Field and Streaming in an Ultrasonic Horn Reactor

12.5. High-Power Ultrasonic Streaming—Jet Flow Behavior The RNW streaming theory is applicable only for acoustic sources at low power levels, where weak streaming motion is generated at very low speeds. However, this theory is inadequate to explain any substantial acoustic streaming motion such as that generated in horn reactors normally used in food processing. Stuart (1963) pioneered the idea that for higher Reynolds numbers, which are observed at higher acoustic power levels, the full equation of motion (see Eq. 12.86) must be used including the important inertia terms. Zarembo (1971) also confirmed that the assumptions in the RNW10 theory only applies to Reynolds numbers lower than 1, and developed a “fast streaming” theory. Alternatively, Lighthill (1978a) introduced the term “Stuart streaming” to describe acoustic streaming at higher Reynolds numbers, resulting from the application of a concentrated high-power acoustic beam, which is likely to be the most important type of streaming in food process engineering. As described by Lighthill (1978a), “it is hardly an exaggeration to say that all really noticeable acoustic streaming motions are Stuart streaming.” Lighthill (1978a) proved that at powers above 4 × 10−4 W, the acoustic streaming takes the form of an inertially dominated turbulent jet. In 1999 and based on Zarembo’s (1971) “fast streaming” theory, Dahlem et al. (1999) developed the first CFD simulation of a horn sonoreactors, linking the acoustic and hydrodynamic fields at higher Reynolds numbers. In 2009, Trujillo and Knoerzer (2009) developed a CFD model to predict the streaming induced in a high-power ultrasonic horn reactor based on the Stuart streaming acoustic concept proposed by Lighthill. The model assumed that the horn tip is an inlet where all acoustic energy absorbed by the liquid is converted into turbulent motion, the jet. Using this assumption, the Navier– Stokes and k-ε turbulent equations were solved using COMSOL Multiphysics™ to determine the hydrodynamic field in the reactor; the results were compared with the experimental data obtained by 10

Zarembo calls that theory “slow streaming,” but it corresponds to what Lighthill calls RNW theory.

255

Kumar et al. (2006). The geometry of the CFD model developed by Trujillo and Knoerzer was based on the configuration used in Kumar ’s experiments. This will be explained in more detail in Section 12.5.3.

12.5.1. Attenuation of a Sound Beam As mentioned in Section 12.2.9, high Reynolds jet streaming is caused by the attenuation of concentrated high-power ultrasound waves that can also be considered as rays of sound. Figure 12.12 shows a narrow beam of sound with an acoustic power W = IA at its source. In the absence of attenuation, the acoustic power in the beam is conserved as expressed by Equation 12.67. It is important to remember that even if the power is constant, the intensity can change along the beam if the transversal area of the beam changes. If the sound beam is attenuated, the acoustic power will be reduced according to: W = W0 e − β X

(12.109)

where W0 = I 0 A0 is the acoustic power at the source, X is the distance from the source, and e − β X is a damping term that accounts for the spatial attenuation of the acoustic beam. Lighthill (1978a) used the attenuation coefficient β , defined as the proportional loss of acoustic energy per unit distance covered by a traveling wave. Other authors such as Nyborg (1953), Piercy and Lamb (1954), and Tjotta (1999) expressed attenuation with the absorption coefficient α that comes from the lossy Helmholtz equation (Eq. 12.75), where the pressure solution contains a damped term e −α x as seen in Equation 12.77:

Area A0

Ray of sound

Area A1

Intensity I0 W0 = A0I0

Intensity I1 W1 = A1I1 = W0e–bx X Absorption W0 (1–e–bx)

Figure 12.12. Attenuated ray of sound.

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

p = P0 e −α x e j (wt − kx )

(12.110)

Therefore, the difference between β and α is that β accounts for the damping of the acoustic energy or power, while α accounts for the damping of the pressure. For plane waves (see Eq. 12.78) I = I 0 e −2α X , then I = I0e

−2α X

W = W0 e −2α X Comparing this with Equation 12.109, it can be seen that for plane waves β = 2α . Most of the publications on acoustic streaming express the attenuation in terms of 2α , probably due to the roots from the Helmholtz equation, while Lighthill’s work reports it in terms of β . Alongside the acoustic energy flow (see Eq. 12.109) there is known to be an acoustic momentum flow rate: W W0 − β X = e c c

(12.111)

Fa is the acoustic momentum flow rate or force (by definition, force is the rate of change of momentum F = d (mv) / dt ). Equation 12.111 gives the acoustic momentum flow rate after attenuation as a function of the distance X from the source. Fa decreases from W0 c at the source to (W0 c ) e − β X at the distance X . By applying the law of conservation of momentum it can be concluded that any reduction of “acoustic momentum” must appear as “hydrodynamic momentum,” that is, streaming: Fh =

W0 W0 − β X W0 − e = (1 − e− β X ) c c c

(12.112)

where Fh is the hydrodynamic momentum flow rate. The spatial rate of decay of this momentum flow acts as a net force per unit length generating motion: FL = −

dFa β β = W0 e − β X = W dX c c

F=

β β (W0 A0 ) e− β X = I 0 e− β X c c

(12.114)

which is equal to Equations 12.95 and 12.98 when β = 2α .

W = IA = I 0 A0 e −2α X

Fa =

as explained in Section 12.4.3, the net force would vanish. Hence, spatial variations of FL are needed for streaming to occur. Since for a plane wave the intensity is c times the energy or power transmitted, the force per unit volume can be expressed as:

(12.113)

where FL is the force per unit length. If there were no spatial differences of the momentum flow rate,

12.5.2. Stuart Streaming Stuart streaming is the flow acoustically induced at higher Reynolds numbers resulting from the application of a concentrated high-power acoustic beam. In this case, the inertia term in Equation 12.86 must be included as initially proposed by Stuart (1963):  ρ ( v ⋅∇v ) = −∇p + μ∇ 2 v + F (12.115) Applying the solution from Squire (1951) for the motion equation above, when a concentrated force  F = W c is applied by an acoustic beam of power W , Lighthill (1978a) demonstrated that the streaming motion becomes a jet. He assumed that the acoustic source releases its power as a narrow beam, where the net force (or rate of momentum) at a distance X along the sound beam is given by Equation 12.112. This equation represents the force that causes streaming (jet flow) after attenuation of the sound intensity, and is equal to the rate of momentum delivered at the source W0 c minus the acoustic momentum flow remaining, where the beam has been attenuated by a factor e − β X . The equation below represents the kinematic momentum, which also increases with distance X along the beam as: K = ρ0 Fh = ρ0

W0 (1 − e− β X ) c

(12.116)

If the attenuation coefficient is very high, the streaming motion generated by the acoustic beam is a turbulent jet, delivering momentum at a rate W0 c . Schlichting (1979) showed that the mean flow of a turbulent jet is similar to the laminar jet solution by taking a constant eddy viscosity equal to:

Chapter 12

μt = 0.016 ( K )

12

Acoustic Field and Streaming in an Ultrasonic Horn Reactor

(12.117)

where K = ρ W c . For low absorption coefficients, the eddy viscosity increases along the beam as the kinematic momentum increases according to Equation 12.116. In that case, Equation 12.117 must be used in conjunction with Equation 12.116 to describe the increase of turbulent viscosity along the sound beam. Lighthill (1978a) assumed that the velocity profile of the acoustically generated jet flow follows a Gaussian distribution: ⎛ 2K ⎞ v=⎜ 2 2⎟ ⎝ ρ πS ⎠

12

e[−(r S )]

(12.118)

where r is the distance from the beam axis and S = S ( X ) is a measure of the width of the jet. Equation 12.118 can be justified if the intensity of the projected sound beam also has a Gaussian form. This equation also satisfies the conservation of momentum rate of the generated jet (Schlichting 1979): ∞

∫

ρ v 2 2π r dr = Fh

(12.119)

0

257

12.5.3. CFD Modeling of High-Power Ultrasonic Streaming As mentioned earlier, Kumar et al. (2006) studied the effect of ultrasonic power on the velocity profile and turbulence (turbulent energy) distribution in a horn reactor filled with water by using laser doppler anemometry (LDA). Trujillo and Knoerzer (2009) developed and validated a CFD model of that reactor by using the LDA data obtained by Pandit’s group. The ultrasonic reactor (Figure 12.13) is a cylindrical vessel with a diameter of 0.135 m and a volume of 2,000 mL with a horn tip submerged 0.02 m into the liquid. The diameter of the horn tip is 0.013 m. Three levels of power density were studied, 15, 25, and 35 kW/m3, which corresponds to a total power of 30, 50, and 70 W, respectively. The model assumed that the horn tip is an inlet where all acoustic energy is released as a turbulent jet. The hydrodynamic rate of momentum of the incoming jet was assumed to be equal to the total acoustic momentum rate W0 c emitted by the acoustic power source. The side of the horn was assumed to be an outlet to fulfill the conservation of mass. The Navier–Stokes Equation 12.115 was solved in

0.05 Wall slip

0 –0.05

Outlet pressure = 0

Horn

0.05

Velocity inlet

Horn tip Axial symmetry 0 0.05

Wall Iogarithm wall function

0 Reactor

–0.05 Figure 12.13. Schematic diagram of an ultrasonic horn reactor. 3D geometry on the left and 2D axial symmetric geometry with boundary conditions on the right (Trujillo and Knoerzer 2009).

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

COMSOL Multiphysics™, along with the k-ε turbulent model, with a two-dimensional axis-symmetric representation of the geometrical configuration. Figure 12.13 shows the geometry and the boundary conditions of the model. For the inlet boundary condition, the turbulent viscosity is assumed constant and is calculated with Equation 12.117. Given that the k-ε turbulent model requires two boundary conditions, the turbulent length scale ( ι ) was calculated as:

ι = 0.07rH

(12.120)

where rH is the horn radius. The velocity profile at the inlet was calculated via the following two approaches: turbulent circular jet solution (TCJS) and Gaussian jet velocity distribution (GJVD) 12.5.3.1. Approach 1: TCJS (Schlichting 1979) The velocity profile at the inlet is estimated with equations taken from Schlichting (1979) for a turbulent jet releasing its kinematic momentum from an orifice: v=

3 K 1 8π ρμt x ⎛ 1 2 ⎞ 2 ⎜⎝ 1 + η ⎟⎠ 4 1 3 K r η= 4 π μt x

(12.121)

imposed horn. In the case of the acoustically generated jet flow, the “orifice” is fictitious. Thus, x can be considered a fitting parameter that modulates the velocity profile at the inlet releasing the same kinematic momentum (see Eq. 12.119). 12.5.3.2. Approach 2: GJVD (Lighthill 1978a) In this case the inlet velocity profile is calculated with Equation 12.118, where the velocity depends on the kinematic momentum K , the radial distance r, and S = S ( X ) , which is a measure of the width of the streaming jet. Similar to the previous approach, the conservation of momentum rate of the jet applies, and S can be considered a fitting parameter to obtain a particular velocity profile. 12.5.3.3. Results Figure 12.15 shows a velocity distribution inside the ultrasonic reactor, predicted

Max: 1.953 1.8

(12.122)

where x is the distance from the orifice. Figure 12.14 shows the velocity profile of such a jet flow, which must satisfy the conservation of momentum rate (see Eq. 12.119) at any distance x , and a super-

1.6 1.4 1.2 1 0.8

Horn

∞ r n 22pr dr = Fh 0

0.6 0.4

rH

b 0.2

Orifice X

Figure 12.14. Schematic diagram of jet flow velocity profile superimposed in the horn (Trujillo and Knoerzer 2009).

Min: 3.544e-7 Figure 12.15. Axial–symmetrical CFD velocity distribution predicted from approach 2 for P V = 35 kW/m3 and S = 0.00281 (Trujillo and Knoerzer 2009). See color insert.

Chapter 12

Acoustic Field and Streaming in an Ultrasonic Horn Reactor

following approach 2. The velocity pattern is in agreement with the experimental data. Figure 12.16 shows the axial velocity below the horn tip at three acoustic powers following both approaches 1 and 2. Optimum values of the fitting parameters x and S were obtained by minimizing the root-mean-squareerror. As can be seen in the figure, both approaches produce identical results. The CFD predictions show excellent agreement with the experimental data at all power densities via both approaches. The advantage of the model is that it links the acoustic and hydrodynamic fields via only two inputs: the absorbed acoustic power and a fitting parameter (x or S). For acoustic powers W ≥ 50 W, the total acoustic momentum is equal to the total momentum rate of the jet generated at the horn inlet, which is calculated with Equation 12.119. The agreement of the velocity profiles, and the almost perfect match between the total acoustic and hydrodynamic momentum, confirms the validity of assuming a high attenuation coefficient, and that the horn tip is an inlet where all the acoustic momentum is absorbed and converted into hydrodynamic momentum. This also helps to explain a finding reported by Kumar et al. (2006) that most of the

259

turbulent kinetic energy (85%) is dissipated in the 2% of the volume near the horn tip and that the cavitational activity is restricted to a small zone around the transducer tip (Gogate et al. 2002; Kumar et al. 2007). For acoustic powers less than W < 50 W, there is also an excellent agreement with the velocity profile, but the total hydrodynamic momentum at the inlet is 85% of the total acoustic momentum when approach 2 is followed. This suggests that at this acoustic power level ( W < 50 W), the “overall” absorption coefficient is still high enough to absorb the totality of the acoustic energy within a few millimeters from the horn tip. As explained in Chapter 2 (Section 2.2.8.2), systems containing bubbles may exhibit strong multiple scattering. The overall absorption coefficient accounts for the combined effect of the attenuation due to absorption alone and strong scattering due to bubbles formed during cavitation. Overall attenuation coefficients that allow total absorption of the acoustic energy in the close vicinity of the source must have values several orders of magnitude higher than attenuation coefficients due to absorption alone, which are reported in the literature (Piercy and Lamb 1954). This is not surprising given the strong sound attenuation caused P/V = 15 kW/m3, x = 0.060 P/V = 25 kW/m3, x = 0.025 P/V = 35 kW/m3, x = 0.025

1.8

P/V = 15 kW/m3

Vz (m/s)

1.6

P/V = 25 kW/m3

1.4

P/V = 35 kW/m3

1.2

P/V = 15 kW/m3, S = 0.00673

1

P/V = 25 kW/m3, S = 0.00281 P/V = 35 kW/m3, S = 0.00281

0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

z (Dimensionless) Figure 12.16. Axial velocity distribution below the horn tip using optimum values of x and S for approaches 1 and 2, respectively (Trujillo and Knoerzer 2009). See color insert.

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

by scattering and reflection of sound waves by bubbles in a cavitating system. A more precise calculation should account for the distance of a few millimeters taken by the sound beams to be completely absorbed, converting the acoustic energy into hydrodynamic momentum. The most important acoustic and fluid flow patterns are occurring in this narrow region near the horn tip. The near jet vortex formation, detachment, and consequent entrainment of fresh liquid is important for characterizing cavitation performance. The accurate representation of the flow (at a millimeter scale) in this region would require incorporating cavitation models to simulate bubble generation, bubble dynamics, and interaction with sound fields (Commander and Prosperetti 1989; Horst et al. 1996; Servant et al. 2001; Horst 2007). This, however, is beyond the scope of the discussed work, and as mentioned earlier, the proposed method of “patching” the Gaussian-like velocity profile allows for accurate prediction of the flow field in all areas of the reactor—except the narrow absorbing region near the horn tip—enabling the characterization of the overall hydrodynamics of the system. These reactors exhibit highly inhomogeneous sound and flow fields, and, therefore, the information obtained by the proposed method can be used to develop improved reactor designs exhibiting a better mixing of the fluid in the vessel. Also, continuous flowthrough systems, operating at low frequency and high power can be optimized with respect to their flow and turbulence energy distribution to ensure that the entire liquid food product undergoes a similar treatment by thorough mixing, despite the fact that the acoustic field is unevenly distributed.

12.6. Final Remarks and Outlook High-power ultrasound is an innovative technology that has been suggested for several new potential applications in the food processing industry. Although some applications, such as ultrasound-assisted cleaning, have been established, there is an urgent need to utilize Multiphysics models for the characterization, design, scale up, and optimization of ultrasound reactors. These numerical models will facilitate the

development of specific applications for the food industry instead of utilizing conventional time- and labor-intensive trial-and-error approaches. The lack of these models to date can be explained by the high complexity of the interaction between sound, cavitation, induced flow, and heat generation. This chapter offers a critical review of acoustic (i.e., ultrasound induced) streaming, which is one of the important effects in high-power ultrasonication. Acoustic power in such applications is generally high; thus, the Reynolds low power RNW theory was critically compared with Lighthill’s high power “Stuart streaming” theory. The chapter aimed to explain the basics of acoustics and also introduced nonlinear acoustics to enable the understanding of streaming. It discussed the derivation of the wave equation from the laws of conservation, to facilitate a basic understanding of streaming to food processing and chemical engineering modelers who are familiar with thermodynamics and the laws of conservation of mass, momentum, and energy. Particular emphasis was put on Lighthill’s Reynolds stress approach, as it greatly assists in understanding the turbulent jet-like behavior exhibited by high-power horn reactors. The chapter was concluded with the work on modeling the jet flow induced by sound absorption at high sources of acoustic power conducted at CFNS. This example is, to our best knowledge, the first of its kind to date. Although only the ultrasound-induced jet streaming in water was discussed, the same approach can be utilized for any ultrasound horn reactor application (operating at high power and low frequency) for the treatment of any homogeneous liquid food (at any viscosity). The governing equations for acoustics were, in this case, not coupled with the equations for mass, momentum, and energy conservation. For describing the ultrasound-induced flow pattern in a high-power horn reactor, simultaneously solving these differential equations is not needed. The model was successfully validated by comparing the predicted flow distribution with the distribution measured by means of LDA. It can be utilized for optimizing fluid flow, turbulence, and induced mixing in high-power low-frequency ultrasound chambers and flow-through cells. Furthermore, the model

Chapter 12

Acoustic Field and Streaming in an Ultrasonic Horn Reactor

can greatly assist in equipment scale-up, from labover pilot- to industrial scale in larger tanks or cells, also incorporating more ultrasound transducers. The fitting constants x and/or S may have to be adjusted, however, for different liquid foods and horn characteristics. More work is needed on the treatment of heterogeneous liquid food systems, such as emulsions or suspensions, where the two phases (continuous and dispersed phase) significantly increase the degree of complexity, not only from the CFD viewpoint but more importantly from the acoustic perspective; the small dispersed particles scatter the ultrasound waves and a force on the particles (caused by differences in ultrasound absorption) causes a motion of the dispersed phase relative to the continuous phase. In standing wave applications, utilized for enhanced separation of particles with similar densities to their continuous phase, this movement directs the secondary phase toward the pressure nodes (Woodside et al. 1999) of the standing wave, forming bands of high concentration at half-wavelength intervals (Miles et al. 1995). In such cases, simultaneously solving the governing equations of acoustics and energy, mass, and momentum conservation for each of the phases is essential to describing and predicting the discussed phenomena. Although these types of multiphase complexities have not yet been fully described in Multiphysics models, the increase in computational power, together with improvements in commercial numerical software packages, will, without doubt, make this possible in the near future. Such applications can then be optimized with respect to reactor design and process conditions and advanced to a stage where they will find immediate industrial applications for separation of phases (e.g., enhancing the separation of fat globules from whole milk).

Latin Letters

b

Transversal area; or integration constant Characteristic width of the jet approach 1

cv D E Ei Ek Ep  F Fj Fa Fh FL F w I Ii j k K  n p P  q r

rH R

Re s S

Notation A

B c cp

m2 m

t T

Integration constant Speed of sound Specific heat capacity at constant pressure Specific at constant volume Strain rate tensor Time averaged of Ei ( E = Ei ) Instantaneous energy density ( = Ek + E p ) Instantaneous kinetic energy density Instantaneous potential energy Force vector per unit of volume  j component of F Acoustic force or momentum rate Hydrodynamic force or momentum rate Force per unit length Force at the wall Sound intensity vector Instantaneous acoustic intensity vector Imaginary unit number Wave number Kinematic momentum Normal unit vector to a surface Pressure Pressure amplitude Heat flux vector Radial distance; or specific acoustic resistance; or heat source per unit volume and unit time Horn radius Ideal gas constant; or reflection coefficient; or radius Reynolds number Specific entropy Measure of the width of the jet approach 2 Time Temperature; or period; or transmission coefficient

261

m/s J/Kg/K J/Kg/K s−1 J/m3 J/m3 J/m3 J/m3 N/m3 N N N/m N W/m2 W/m2

Radian/m Kg2/m2/s2 Pa Pa J/s/m2 m Kg/m2/s J/s/m3 0.0065 m

m J/K/Kg m s K s

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

U  v

Total internal energy Velocity vector (can be streaming or particle velocity) Velocity (can be streaming or particle velocity) Total volume Amplitude of the sinusoidal oscillation of particle velocity Volume of the unperturbed fluid Distance; or distance from the orifice: or specific acoustic reactance Distance from acoustic source i component of position vector Acoustic impedance; or axial distance Acoustic power, or work Acoustic power emitted by the source

v V VA V0 x

X xi z W W0

J m/s

αs β Cμ ε φ Φ γ η

ι κ λ μ μt τ τR τs ν

Absorption coefficient—damping of the acoustic pressure Absorption coefficient associated to viscosity—damping of the acoustic pressure Absorption coefficient—damping of the acoustic energy or power Constant (0.09) Energy dissipation rate Flow property Velocity potential = c p cv Value defined by Equation 12.122 Length scale Turbulent kinetic energy Wavelength Dynamic viscosity Turbulent viscosity Viscosity stress tensor Reynolds stress tensor Relaxation time Frequency

Angle Density Angular frequency

Radian kg/m3 rad s−1

m/s m3 m/s

Subscripts Equilibrium value ( v0 is Equilibrium value of v ); or initial value ( P0 is initial value of P ); Bleeder tube in equipment Figure 12.11 Incident ( Pi is the incident pressure) Main tube in equipment Figure 12.11 Reflection ( Pr is the reflected pressure) Reversible Transmission ( Pt is the transmitted pressure)

0

3

m m m Kg/m2/s m Kg/m2/s m W J W

b i m r rev t

Superscripts 1

Greek Letters α

θ ρ ω

′

Perturbation ( a1 is perturbation of a; a1 = ( a − a0 ) ) Perturbation ( ϕ ′ is perturbation of ϕ )

−1

m

m−1 m−1

m2/s3

m m2/s2 m Pa s Pa s Kg/m/s2 Kg/m/s2 s Hz

Other Symbols a , a  a

Time average of a Vector a

References Alassar RS. 2008. Acoustic streaming on spheres. Int J NonLinear Mech 43(19):892–897. Bermúdez-Aguirre D, et al. 2009. Modeling the inactivation of Listeria innocua in raw whole milk treated under thermosonication. Innovat Food Sci Emerg Technol 10(2):172–178. Beyer RT. 1978. Radiation pressure—The history of a mislabeled tensor. J Acoust Soc Am 63:1025–1030. Commander KW, Prosperetti A. 1989. Linear pressure waves in bubbly liquids: Comparison between theory and experiments. J Acoust Soc Am 85:732–746. Dahlem O, Reisse J, Halloin V. 1999. The radially vibrating horn: A scaling-up possibility for sonochemical reactions. Chem Eng Sci 54:2829–2838. Eckart C. 1948. Vortices and streams caused by sound waves. Phys Rev 73:68–76. Feynman RP. 1977. The Feynman Lectures on Physics, Vol. 1. Wokingham: Addison-Wesley.

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Acoustic Field and Streaming in an Ultrasonic Horn Reactor

Filippi P, et al. 1999. Acoustics: Basic Physics, Theory, and Methods. London: Academic Press. Gogate PR, et al. 2002. Mapping of sonochemical reactors: Review, analysis, and experimental verification. AIChE J 48(7):1542–1560. Gol’dberg ZA. 1971. Part II Acoustic radiation pressure. In: LD Rozenberg, ed., High Intensity Ultrasonic Fields, pp. 75–133. New York, London: Plenum Press. Halliday D, Resnick R. 1988. Fundamentals of Physics. New York: John Wiley & Sons. Horst C. 2007. Ultrasound reactors. In: FJ Keil, ed., Modeling of Process Intensification, pp. 193–226. Weinheim: Wiley-VCH. Horst C, et al. 1996. Design, modeling and performance of a novel sonochemical reactor for heterogeneous reactions. Chem Eng Sci 51:1837–1846. Jackson FJ. 1960. Sonically induced microstreaming near a plane boundary. II. Acoustic streaming field. J Acoust Soc Am 32:1387–1395. Jambrak AR, et al. 2008. Effect of ultrasound treatment on solubility and foaming properties of whey protein suspensions. J Food Eng 86(2):281–287. Jambrak AR, et al. 2009. Physical properties of ultrasound treated soy proteins. J Food Eng 93(4):386–393. Kamakura T, et al. 1995. Acoustic streaming induced in focused Gaussian beams. J Acoust Soc Am 97:2740–2746. Kentish S, et al. 2008. The use of ultrasonics for nanoemulsion preparation. Innovat Food Sci Emerg Technol 9(2):170–175. Kinsler LE, et al. 2000. Fundamentals of Acoustics. New York: John Wiley & Sons. Klíma J, et al. 2007. Optimization of 20 kHz sonoreactor geometry on the basis of numerical simulation of local ultrasonic intensity and qualitative comparison with experimental results. Ultrason Sonochem 14:19–28. Kumar A, et al. 2006. Characterization of flow phenomena induced by ultrasonic horn. Chem Eng Sci 61:7410–7420. Kumar A, Gogate PR, Pandit AB. 2007. Mapping the efficacy of new designs for large scale sonochemical reactors. Ultrason Sonochem 14(5):538–544. Kwon K, Loh B-G, Lee D-R. 2007. Experimental and analytical investigation of acoustic streaming generated by standing ultrasonic waves in an open boundaries. Eur Phys J Appl Phys 40:343–449. Leighton TG. 1997. The Acoustic Bubble. San Diego, CA: Academic Press. Lighthill J. 1952. On sound generated aerodynamically. I. General theory. Proc R Soc Lond A Math Phys Sci 211(1107):564–587. Lighthill J. 1954. On sound generated aerodynamically. II. Turbulence as a source of sound. Proc R Soc Lond A Math Phys Sci 222(1148):1–32. Lighthill J. 1972. The fourth annual fairey lecture: The propagation of sound through moving fluids. J Sound Vib 24:471–492. Lighthill J. 1978a. Acoustic streaming. J Sound Vib 61(3): 391–418. Lighthill J. 1978b. Waves in Fluids. Cambridge: Cambridge University Press.

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Vilkhu K, et al. 2008. Applications and opportunities for ultrasound assisted extraction in the food industry—A review. Innovat Food Sci Emerg Technol 9(2):161–169. Westervelt PJ. 1953. The theory of steady rotational flow generated by a sound field. J Acoust Soc Am 25:60–67. Woodside SM, et al. 1999. Acoustic force distribution in resonators for ultrasonic particle separation. AIChE J 44: 1976–1984. Zarembo LK. 1971. Acoustic streaming. In: LD Rozenberg, ed., High Intensity Ultrasonic Fields, pp. 137–199. New York, London: Plenum Press.

Chapter 13 Computational Study of Ultrasound-Assisted Drying of Food Materials Enrique Riera, José Vicente García-Pérez, Juan Andrés Cárcel, Victor M. Acosta, and Juan A. Gallego-Juárez

13.1. Introduction The driving force behind the improvement of drying technologies is the need to both produce better quality products (Chou and Chua 2001) and save energy. Despite the fact that convective drying processes have been widely addressed before, there is still room for improvement (Vega-Mercado et al. 2001). The slow drying rates during the falling rate period as well as the loss of quality provoked by the heating constitute a challenge for the research community. Several strategies have been applied to improve the air drying process, and of these, the introduction of alternative energy sources to hot air drying should be emphasized. Microwave, radio frequency, infrared radiation, and high-power (intensity) ultrasound have been used to improve the drying process (García-Pérez et al. 2007). The improvement of the drying rate is mainly due to the heating effect of microwave, radio frequency, and infrared radiation. Heating is not the main effect associated with the application of power ultrasound in convective drying processes, which avoids quality degradation due to high temperatures (GallegoJuárez 1998). It is widely recognized that power ultrasound produces mechanical effects both on the

gas–solid interfaces and in the material being dried, which may facilitate water removal without introducing a high amount of thermal energy during the process (Cárcel et al. 2010). Therefore, application of power ultrasound to either dry heat-sensitive materials (Gallego-Juárez et al. 1999; Mulet et al. 2003b) or low-temperature drying processes has great potential. Despite the promising effects linked to applying ultrasound in drying processes, some technical drawbacks have made the full development of ultrasonic drying very difficult. The high energy attenuation in gas systems as well as the high impedance mismatch between ultrasonic radiators and air makes the acoustic energy transfer during drying more complicated than in liquid media. This is well demonstrated by the fact that very little has been published on ultrasound-assisted drying (GallegoJuárez et al. 1999) compared with ultrasonic applications in liquid media, such as the extraction of natural products (Riera et al. 2004a; Cravotto et al. 2008), osmotic dehydration (Cárcel et al. 2007b), and meat (Cárcel et al. 2007c) and cheese brining (Sánchez et al. 1999) and pretreatment (Jambrak et al. 2007). Therefore, the development of new,

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

265

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1 NODAL SOLUTION JAN 18 2010 16:07:34

STEP =1 SUB =25 FREQ =22045 REAL ONLY /EXPANDED UX (AVG) RSYS =0 DMX =.005018 SMN =–.001202 SMX =.001132

X

–.001202

Y

–.001202

0

.100E–07

.001132

Figure 13.1. Design of the power ultrasonic system by FEM.

highly efficient ultrasonic transducers to be applied in drying processes, and a rigorous analysis of the influence of ultrasound on mass transfer processes may be considered matters of relevant research. This work deals not only with the design, development, characterization, and validation of an airborne power ultrasonic system, but also with the analysis of ultrasonic application on mass transfer phenomena that take place during drying. The main objective in the design was to generate an intense airborne ultrasonic field inside a convective drying chamber of cylindrical shape. To this end, the drying chamber was replaced by a cylindrical cavity vibrating in one of its flexural modes with 12 nodal lines

(NL) (see Figure 13.1). To drive such a cavity at this vibration mode, a power piezoelectric vibrator was developed and tuned to the corresponding resonant frequency of the cylindrical cavity. As a result, a new ultrasonic transducer with a cylindrical radiator was developed. In order to design and develop the new transducer correctly, a numerical study was carried out by using finite element methods (FEM), and a model was implemented (by following the protocol described in Figure 13.2) and experimentally validated. The main elements of the ultrasonically assisted drier where the new transducer was installed are described in detail in the following sections. Using

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Computational Study of Ultrasound-Assisted Drying of Food Materials

267

PRELIMINARY STUDY ULTRASONIC TRANSDUCER RESONANCE MODE MATERIAL CHARACTERIZATION

ANALYTICAL PRE-DESIGN

MODELING AND ANALYSIS BY FEM - STATIC ANALYSIS - MODAL ANALYSIS

ANALYSIS OF THE RESULTS - NODAL POSITION - IMPEDANCE -POWER CAPACITY NO

YES

OK

ANALYTICAL REDESIGN

ANALYSIS OF THE ACOUSTIC FIELD BY FEM NO

YES

OK

FINAL DESIGN AND DEVELOPMENT LOW- AND HIGH-POWER ELECTRICAL CHARACTERIZATION

NO

OK

STUDY OF THE CAUSE

YES

CORRECT MODELING EXPERIMENTAL TRIALS

Figure 13.2. Protocol for the design and modeling of the new transducer.

this prototype, drying experiments were carried out over a wide range of operating conditions, and different products were tested. The drying kinetics were thoroughly analyzed by using different computational strategies, all of which are described in this work. Thereby, the influence of power ultrasound on mass transfer phenomena that take place during drying is shown.

13.2. Computational Design of the Power Ultrasonic System: Numerical Study by FEM of the Constituent Elements This section presents the computational design carried out using FEM of the new power ultrasonic system that is designated as the cylindrical radiator

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

transducer (CRT) and used as a drying chamber. Such a transducer is essentially made up of a cylindrical radiating chamber driven at its surface center by a piezoelectric vibrator formed by an element of transduction in a sandwich configuration and a solid mechanical transformer that acts as a vibration amplifier. The finite element analysis was performed on the transducer structure using the commercial software ANSYS® (ANSYS Mechanical, Canonsburg, PA). Modal and harmonic analyses of the CRT were performed in accordance with a procedure summarized as follows:

of common ground, and the other poles were connected to 1 V. Harmonic analysis including pre-stress to study the frequency response of the transducer.

Measurement of the orthotropic mechanical properties of the different constituent materials. Introduction of the dielectric relative permittivity matrix, piezoelectric matrix, and compliance matrix for the piezoelectric ceramics in the model. No internal losses (damping) were assumed. Modeling of the ultrasonic vibrator (sandwich transducer and the mechanical amplifier) and the radiating tube. Application of the symmetry boundary conditions for the three-dimensional (3D) Quart model. Application of pre-stress to the piezoelectric ceramics by the central bolt. Application of electric open-circuit condition. Only one of the negative poles or positive poles of the ceramic rings were connected to zero voltage of common ground, and the other poles were left free without any connection. Static analysis for transmitting pre-stress to the ceramics through the bolt of the sandwich transducer. Application of electric short-circuit or “resonance” condition (a constant voltage of zero was applied at all electrical contacts of ceramic rings or all voltage potentials are connected to common ground). Modal analysis of the CRT to determine the working frequency. Application of electric load for harmonic analysis. Only one of the negative or positive poles of the piezoceramic rings was connected to zero voltage

13.2.1.1. Piezoelectric Sandwich Transducer The most widely used piezoelectric transducer configuration for high-power applications is the wellknown sandwich, also called the Langevin-type, transducer. The sandwich transducer is a half-wave resonant length-expander structure, which, in its simpler version, consists of a disk (or paired discs) of piezoelectric ceramics sandwiched between two identical metal blocks. When used in pairs, the ceramics are mounted and connected in parallel with their polarization axis in opposite directions and separated by electrodes connected to the highvoltage load. Coupling between the piezoelectric elements and the metal end-sections as well as an increase in the tensile strength is achieved by mechanically pre-stressing the assembly in the axial direction by means of a bolt (Van Randeraat and Stterington 1974). The dimensions of the sandwich transducer can be calculated by using the expression (Neppiras 1973; Gallego-Juárez 1997; Radmanovic and Mancic 2004):

13.2.1. Ultrasonic Vibrator The ultrasonic vibrator has two components, a piezoelectric sandwich transducer and a mechanical transformer that acts as mechanical amplifier. In the next sections both components are described in detail.

⎛ ωc ⎞ ⎛ ω a ⎞ Z c ρc vc Ac tan ⎜ ⎟ tan ⎜ = = ⎝ v1 ⎟⎠ Z1 ρ1 v1 A1 ⎝ vc ⎠

(13.1)

After choosing the materials, the resonant frequency, and two of the three geometrical quantities, the preceding equation can be used to fix the unknown dimension. In the present study, the following design procedure has been used: (1) analytical calculation of the approximate dimensions of the sandwich

Chapter 13

Computational Study of Ultrasound-Assisted Drying of Food Materials

transducer from Equation 13.1 and (2) numerical modeling of the transducer by FEM. Thus, the data calculated by the analytical expression are subsequently used as the first input data for the numerical modeling. In this case, the piezoelectric transducer was made up of four PZT-802 (MTC Electroceramics, Southampton, U.K.) piezoceramic rings, two stainless-steel cylinders (AISI-420) constituting the back and front mass, and a brass flange that acts as a central electrode. A central bolt is required to apply pre-stress on the PZT ceramics. The bolt was selected to ensure life under fatigue loading by considering both static load (pre-stress) and dynamic load. The bolt was made of steel following the specifications DIN912 high-tensile grade 12.9. One important point here is that the bolt should be fine-threaded to prevent it from coming loose under operation. Another point in the design and development of this type of piezoelectric transducer is to minimize energy losses. For this purpose, a practical procedure was developed and used to select the piezoelectric ceramics that constitute the stacks of high-power sandwich transducers (Chacón et al. 2006). As the values of density, Young modulus, shear modulus, and Poisson ratios of the materials must be used in the design process, all these properties were accurately determined by measuring the longitudinal and shear acoustic velocities. Two different techniques were used, depending on the dimensions and attenuation of the materials: transmission and echo-pulse configurations using tone burst excitation. Normal incidence was applied to the surface of the material to propagate both longitudinal and shear waves generated by Panametrics (Olympus NDT Inc., Waltham, MA) contact-type wideband transducers. Once the density, the longitudinal and transverse velocities, and the thicknesses of the material samples are measured, the mechanical properties are calculated along their three orthogonal directions (X, Y, Z). The measured orthotropic mechanical properties of each material are shown in Tables 13.1–13.3. For the transducer construction, PZT-802 piezoelectric ceramics (Morgan Electro Ceramics, Piezoelectric Ceramics 2006) were selected because

269

Table 13.1. Material properties of stainless-steel backing and front cylinders. Magnitude Measured density ρ (Kg/m3) Young’s modulus EX (N/m2) Young’s modulus EY (N/m2) Young’s modulus EZ (N/m2) Poisson’s ratio μXZ Poisson’s ratio μYZ Poisson’s ratio μXY Rigidity GXZ (N/m2) Rigidity GYZ (N/m2) Rigidity GXY (N/m2)

7,715 2.1874 × 1011 2.1826 × 1011 2.1881 × 1011 0.279 0.278 0.280 8.55 × 1010 8.56 × 1010 8.52 × 1010

Table 13.2. Material properties of brass flange. Magnitude Measured density ρ (Kg/m3) Young’s modulus EX (N/m2) Young’s modulus EY (N/m2) Young’s modulus EZ (N/m2) Poisson’s ratio μXZ Poisson’s ratio μYZ Poisson’s ratio μXY Rigidity GXZ (N/m2) Rigidity GYZ (N/m2) Rigidity GXY (N/m2)

8,350 1.0141 × 1011 9.2404 × 1010 1.0049 × 1011 0.348 0.345 0.365 3.73 × 1010 3.77 × 1010 3.38 × 1010

Table 13.3. Material properties of copper–beryllium electrodes. Magnitude Measured density ρ (Kg/m3) Young’s modulus EX (N/m2) Poisson’s ratio μXY

8,250 1.250 × 1011 0.300

they can efficiently generate high-amplitude vibrations at ultrasonic frequencies. The material properties of such piezoceramics, as listed by the manufacturer, are the following: Compliance matrix [sE] for PZT-802 under constant electric field (polarization axis along the Z-axis)

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Y Z YZ XZ XY ⎤ ⎡ X ⎢11.5 −3.7 − 4.8 0 0 0 ⎥ ⎢ ⎥ 11.5 − 4.8 0 0 0 ⎥ ⎢ ⎥ 13.5 0 0 0 ⎥ [ s E ] = ⎢⎢ ⎢ 31.9 0 0 ⎥ ⎢ ⎥ 31.9 0 ⎥ ⎢ ⎢⎣ 30.4 ⎥⎦ −12 2 x10 ( m / N ) Piezoelectric stress matrix (stress developed/electric field applied at constant strain), [e] (polarization axis along the Z-axis): − 4.1⎤ 0 ⎡ 0 ⎢ 0 − 4.1⎥ 0 ⎢ ⎥ 0 14 ⎥ ⎢ 0 [ e] = ⎢ ⎥ (C / m) 0 ⎥ ⎢ 0 10.3 ⎢10.3 0 0 ⎥ ⎢ ⎥ 0 0 ⎦ ⎣ 0 Dielectric relative permittivity matrix at constant strain, [ εTS ] (polarization axis along the Z-axis): 0 0⎤ ⎡900 ⎢ [ε ] = 0 900 0⎥ ⎢ ⎥ 0 600 ⎦⎥ ⎢⎣ 0 S T

Publications on piezoelectric materials use the following criteria to define the piezoelectric matrix [ s E ] ⎡X ⎢s E ⎢ 11 ⎢ E [ s ] = ⎢⎢ ⎢ ⎢ ⎢ ⎢⎣

Y E s12 E s22

Z E s13 E s23 E s33

YZ XZ XY ⎤ 0 0 0 ⎥ ⎥ 0 0 0 ⎥ ⎥ 0 0 0 ⎥ sE44 0 0 ⎥ ⎥ E s55 0 ⎥ E ⎥ s66 ⎦

This criterion is based on the ANSI/IEEE Standard on Piezoelectricity 176-1987. However, the ANSYS code requires a transformation of the matrix into the form

⎡X Y ⎢s E s E ⎢ 11 12 E s22 ⎢ E [ sANSYS ] = ⎢⎢ ⎢ ⎢ ⎢ ⎢⎣

Z E s13 E s23 E s33

XY YZ XZ ⎤ 0 0 0 ⎥ ⎥ 0 0 0 ⎥ ⎥ 0 0 0 ⎥ E s66 0 0 ⎥ ⎥ E s55 0 ⎥ sE44 ⎥⎦

As already mentioned, in sandwich transducers a central bolt is required. A static analysis was carried out under open-circuit electrical condition to apply 250 kg/cm2 of pre-stress on the ceramics. For the sandwich transducer, modal analysis by FEM is used to determine the natural frequencies, mode shapes, and the location of the nodal plane, which are important parameters in the design of a transducer. The modal analysis was performed under resonance conditions: A constant voltage of zero was applied at all electrical contacts of the ceramic disks. This is a condition of short-circuit where all voltage potentials are connected to common ground. 3D Quart modeling, meshing, and analysis was employed to study the vibration behavior of the transducer at the working frequency as well as the presence of other nearby vibration modes. In this case, a one-quarter symmetry sector with symmetry boundary conditions was meshed by appropriate selected shape elements. SOLID5 elements were used for piezoelectric rings and SOLID185 for other components (see Figure 13.3). SOLID5 has a hexahedral geometry with 3D magnetic, thermal electric, electric, piezoelectric, and structural field capability. The element has eight nodes with up to six degrees of freedom at each node. SOLID185 also has a hexahedral geometry with eight nodes with up to three degrees of freedom per node and has structural and thermal field capabilities. PRETS179 elements were used for the central part of the bolt. It has one translation degree of freedom UX (where UX represents the defined displacement and X pre-tension direction, respectively). ANSYS transforms the geometry of the problem so that, internally, the pre-tension force is applied in the specified pre-tension load direction,

Chapter 13

1

Computational Study of Ultrasound-Assisted Drying of Food Materials

ELEMENTS /EXPANDED MAT NUM

271

SEP 29 2009 09:02:44

Z Y

Figure 13.3. 3D meshing modeling of the piezoelectric transducer element with sandwich configuration.

regardless of how the model is defined. The selected element size was 2 mm and the numerical model was carried out with a total number of about 5,000 nodes and 3,600 elements. Initially, a static analysis of the sandwich transducer was performed, applying a pre-stress of 250 kg/cm2 on the surface area of the ceramic rings (Arnold and Mühlen 2001). Next, the modal analysis was carried out to find the resonance under shortcircuit (resonance) and pre-stress conditions. The value obtained was 22,065 Hz (see Figure 13.4). The harmonic analysis was performed to determine the structural response to a harmonic load by applying 1 V to the electrical contacts of the ceramic rings over a frequency span about which the resonance frequency was obtained using modal analysis. The results showed that, although the model is more difficult to be constructed and more timeconsuming to run than in two-dimensional (2D) axisymmetric modeling, 3D modeling could take all the

modes of vibration into consideration (Iula et al. 2002; Abdullah and Pak 2008). 13.2.1.2. Assembly of Sandwich Transducer and Mechanical Amplifier In processing applications, the sandwich transducer is used, but it is generally connected to a specially shaped metallic transmission line, which produces displacement amplification at the working end. Such a transmission line, called mechanical amplifiers or horns, is formed by a half-wavelength resonant bar of variable crosssection that vibrates extensionally (Merkulov 1957; Merkulov and Kharitonov 1959–60; Rozenberg 1969). The horn must be designed to resonate at the same frequency as the sandwich transducer that is driving it. The vibration amplitude at the radiating surface depends on the geometry of the mechanical amplifier. In the present case, the mechanical amplifier was a stepped horn made of titanium alloy. This element consists of a bar with cylindrical sections, each λ /4

272

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

NODAL SOLUTION SEP 29 2009 10:49:32

STEP =1 SUB =3 FREQ =21065 /EXPANDED UZ (AVG) RSYS =0 DMX =10.108 SMN =–7.273 SMX =2.84

–7.273

Y Z

–6.15

–5.026

–3.902

–2.779

–1.655

–.53119

.592496

1.716

X

2.84

Figure 13.4. Mode shape of the sandwich transducer from modal analysis in 3D modeling.

long and of different diameters, D1 and D2. The displacement amplification produced by the horn can be obtained from the following equation: M=

S1 ⎛ D1 ⎞ =⎜ ⎟ S2 ⎝ D2 ⎠

2

(13.2)

which indicates the ratio of the areas of the two sections or the ratio of the squared diameters (Belford 1960; Einsner 1963; Frederick 1965; Ensminger 1973; Gallego-Juárez 1997; Abramov 1998). The transition between the two steps of the stepped horn is rounded to avoid a fatigue failure in that region when the ultrasonic vibrator is working under highpower conditions. The amplification of the vibration

motion produced by the stepped horn is needed for the power excitation of the cylindrical tube radiator to generate a high-intensity ultrasonic field in its interior. The assembly formed by the piezoelectric sandwich and the stepped horn constitutes the vibrator that drives the cylindrical radiator. Such a radiator was designed following the same procedure as described earlier for the single piezoelectric sandwich. In this way, the mechanical properties of the titanium alloy bar (Table 13.4) used for the fabrication of the mechanical amplifier were also measured by acoustic methods to be included in the FEM model. Next, 3D Quart modeling, meshing, and analysis was employed to simulate the vibrator. The

Chapter 13

Computational Study of Ultrasound-Assisted Drying of Food Materials

different components of the vibrator were calculated to be resonant at the working frequency of the cylindrical radiator. In Figure 13.5 the 3D meshing modeling is shown with SOLID5 elements for piezoelectric rings, PRETS179 elements for the central part of the bolt and SOLID185 elements for other components. The selected element size was

Table 13.4. Material properties of titanium alloy. Magnitude Measured density ρ (Kg/m3) Young’s modulus EX (N/m2) Young’s modulus EY (N/m2) Young’s modulus EZ (N/m2) Poisson’s ratio μXZ Poisson’s ratio μYZ Poisson’s ratio μXY Rigidity GXZ (N/m2) Rigidity GYZ (N/m2) Rigidity GXY (N/m2)

1

ELEMENTS /EXPANDED MAT NUM

4,414 1.28 × 1011 1.12 × 1011 1.28 × 1011 0.326 0.307 0.307 4.91 × 1010 4.91 × 1010 4.21 × 1010

273

2 mm, and the numerical model was carried out with a total number of 9,000 nodes. Mode shape and node location are shown for the 2D axi-symmetric modeling in Figures 13.6 and 13.7. The ultrasonic vibrator has a total length equal to one wavelength and vibrates with two vibrational nodes, one located in the sandwich transducer and the other placed in the stepped horn. The resonance frequency computed by modal analysis was found to be 22,031 Hz. The resonance frequency obtained by harmonic analysis in the 22,000–22,080 Hz frequency range with an electric load coincides with that obtained by modal analysis, as shown in Figure 13.8. This frequency value was used to design the resonant mode of the cylindrical radiator, which is presented in the following section.

13.2.2. Cylindrical Radiator The last component of the power ultrasonic system consists of a cylindrical aluminum radiator that acts

OCT 7 2009 10:17:43

Y

Z

X

Figure 13.5. 3D meshing modeling of the ultrasonic vibrator (sandwich transducer mounted with mechanical amplifier).

274

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

NODAL SOLUTION STEP =1 SUB =3 FREQ =22031 /EXPANDED UZ (AVG) RSYS =0 DMX =10.852 SMN =–10.852 SMX =1.887

–10.852

SEP 29 2009 13:50:37

Y Z

X

–9.437

–8.021

–6.606

–5.19

–3.775

–2.359

–.94386

.47165

1.887

Figure 13.6. Mode shape of the ultrasonic vibrator from modal analysis in 3D modeling.

as a drying chamber. For the correct design of this cavity, we first gauged a single cylindrical tube (length, diameter, and wall thickness) to resonate in one of its natural flexural modes at the same resonant frequency as the ultrasonic vibrator. At this frequency the tube has a vibration mode with 12 NL (Gallego-Juárez et al. 2010). The isotropic mechanical properties of the aluminum are shown in Table 13.5. The dimensions of the tube were about 300 mm in length, 100 mm in diameter, and 10 mm in thickness. Full 3D modeling, meshing, and analysis were employed to study the vibration pattern of the tube and to determine its natural resonant frequency. For the volume meshing, SOLID186 elements of 4 mm size were used. Although this element is similar in

Table 13.5. Material properties of aluminum. Magnitude Measured density ρ (Kg/m3) Young’s modulus EX (N/m2) Poisson’s ratio μXY

2,807 7.09 × 1010 0.340

shape and size to SOLID185, it has 20 nodes per element. The model was carried out with a number of about 30,000 nodes. Figure 13.9 shows the phase map of the flexural mode of the symmetrical tube at 21,654 Hz with 12 NL. It can also be seen that there is a phase change in the thickness due to a bending of the tube wall.

1

PATH= PATH1 VALUE= UY1

MAY 5 2009 12:39:57

X Y

–6.041

Z

–5.283

–4.525

–3.766

–3.008

–2.25

–1.492

–.7336

.024583

.782765

Figure 13.7. Location of nodes of the ultrasonic vibrator from modal analysis in 3D Quart modeling without cylindrical radiator. 1

POST26 AMPLITUDE MIPS

SEP 29 2009 14:08:57 .4 .36 .32 .28 .14

VALU .2 .16 .12 .08 .04 0

(×10**1) 2200

2201.6 2203.2 2204.8 2206.4 2206 2200.8 2202.4 2204 2205.6 2207.2 FREQ

Figure 13.8. Amplitude versus frequency of the ultrasonic vibrator from harmonic analysis in 3D Quart modeling without a cylindrical radiator.

275

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

1

NODAL SOLUTION FEB 19 2010 12:23:33

SUB =1 FREQ =21654 USUM (AVG) RSYS =0 DMX =1.733 SMN =.242466 SMX =1.733

.242466

.408089

.573711

.739334

.904957

1.071

1.236

1.402

1.567

1.733

Figure 13.9. Phase map (along the Z-axis) of the radiating tube with 12 NL at 21,654 Hz from modal analysis in full 3D modeling.

For the practical realization of the cylindrical radiator, a central ring is added to the simple cylinder as a driving region where the vibrator is connected. Such a modification required a redesign of the single cylindrical tube, for which 3D meshing modeling was used. SOLID186 elements were also used in the modeling with two element sizes: 2 mm for the central ring and 4 mm for the other part of the tube. The model was constructed with a larger number of nodes. A modal analysis was carried out to determine the resonant frequency of the radiator as well as the presence of other nearby modes. If other natural modes close to the working frequency were present, it would be necessary to modify the geometry and the dimensions of the cylindrical cavity to change the distribution of the vibration modes in order to mitigate modal interaction. The resonant frequency of the new radiator for the mode with 12 NL was found to be 22,109 Hz,

which is closer than the resonance frequency of the single cylindrical tube to the working frequency of the ultrasonic vibrator. However, changes in the vibration mode are appreciated in comparison with the symmetrical tube (see Figure 13.10). There is a lack of symmetry in the vibration, which is likely caused by the presence of the central ring and a hole in the center of the cylinder to connect the vibrator. Due not only to the axi-symmetric and punctual excitation, but, in all likelihood, also to the influence of the ultrasonic vibrator, the distribution of the displacements on the opposite the excitation area is altered, leading to the tube losing part of its axial symmetry.

13.2.3. CRT The objective of this section is to study the acoustic behavior of the whole transducer (CRT) made up of

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Computational Study of Ultrasound-Assisted Drying of Food Materials

277

1 NODAL SOLUTION OCT 9 2009 08:43:29

STEP =1 SUB =21 FREQ =22109 /EXPANDED UY (AVG) RSYS =0 DMX =5.332 SMN =–2.766 SMX =2.726

Z X

–2.766

–2.156

–1.546

–.93573

Y

–3.25524 .894886 .284681 1.505

2.115

2.726

Figure 13.10. Mode shape of the new tube vibrating at its flexural mode with 12 NL from modal analysis in full 3D modeling.

the piezoelectric sandwich, the stepped horn, and the cylindrical radiator. The CRT was studied by using 3D Quart modeling and meshing. Symmetry boundary conditions and a volume meshing generated by filling the volume of the constituent components of the CRT by adequately selected shape elements was used. For this modeling, the selected element sizes were 1 and 4 mm, depending on the components. A total number of about 72,000 nodes were used in the modal analysis. Figure 13.11 shows the meshing of the CRT system, while Figure 13.12 shows the vibrational displacements. The harmonic analysis of the model was carried out by applying a voltage of 1 V to the electrodes of the piezoceramics. To calculate the distribution of mechanical stresses in the hollow cylinder (see Figure 13.13), it was assumed that no electric charges were present and that a vibrational displacement of 10 microns was applied

in the center of the tube by the vibrator. In this way, the maximum value of the mechanical stress in the tube was determined and the limiting value to avoid fatigue problems in the behavior of the CRT system was estimated. Harmonic analysis of the cylinder was carried out to calculate the radiated power in the air medium by applying a harmonic load of 10 microns on the excitation point. To calculate the power capacity of the CRT, the following numerical equation was applied: Acoustic Power = (2π ) ρ0 c0 f 2 R1 3

L

∑ Δl ⋅U

2 r

(13.3)

0

Taking into account that the limiting stress for the material of the cylindrical radiator (aluminum alloy) is of about 100 MPa and the power capacity of the CRT was about 90 W.

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

1

ELEMENTS /EXPANDED MAT NUM

OCT 9 2009 Z

Y X

Figure 13.11. Meshing of the CTR for modal analysis.

13.2.4. Acoustic Field inside the CRT The ANSYS code was also used to determine sound pressure levels (SPL) inside the vibrating tube. Modal and harmonic analyses were used to study the acoustic pressure of the standing wave field established inside the resonant cylindrical chamber and the fluid–structure interaction. The wave equation is derived from fundamental equations of state, continuity, and motion. It was assumed that the air behaves as an ideal acoustic medium. In this case, for harmonic analysis, the linear wave equation is given by (ANSYS 1992, 2007; Khan et al. 2002): ∇2 p =

1 ∂2 p c 2 ∂t 2

ω2 (13.5) po c2 Since FEM is used, the preceding equation is converted to matrix form: ∇ 2 po = −

(13.4)

where, c 2 = κ / ρ . The acoustic pressure p is assumed to be harmonic, so p = po e jω , which inserted in the preceding equation, gives:

[ M ]{p} + [ K ]{p} = {F}

(13.6)

In this way the finite element matrix equation for the fluid mesh is:

[ M f ]{p} + [ K f ]{p} = {Ff }

(13.7)

In parallel, the equivalent form of the finite element matrix equation for the structural mesh is:

[ M s ]{U} + [ K s ]{U} = {Fs}

(13.8)

where {p} is a vector of unknown nodal pressures, {Ff} is a vector of applied fluid loads, {U} is a vector of unknown nodal displacements, and {Ff} is a vector of applied structural loads. The interaction of the fluid and the structure at a mesh interface is such that the acoustic pressure

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Computational Study of Ultrasound-Assisted Drying of Food Materials

279

1 NODAL SOLUTION OCT 16 2009 12:42:56

STEP =1 SUB =8 FREQ =22045 /EXPANDED UZ (AVG) RSYS =0 DMX =7.707 SMN =–7.521 SMX =3.987

Y Z

–7.521

–6.242

–4.963

–3.685

–2.406

–1.128

.150918

1.43

2.708

3.987

Figure 13.12. Vibrational displacements in the cylindrical tube of the CRT system along the Z-axis. See color insert.

exerts a force on the structure and the structural motions produce an effective fluid load. The governing finite element matrix equation is obtained by combining Equations 13.4 and 13.5, which then become: ⎡ Ms ⎢ρ RT ⎣ o

0 ⎤ ⎛ U ⎞ ⎡ K s + M f ⎥⎦ ⎜⎝  p ⎟⎠ ⎢⎣0

− R ⎤ ⎛ U ⎞ ⎛ Fs ⎞ = K f ⎥⎦ ⎜⎝ p ⎟⎠ ⎜⎝ Ff ⎟⎠

(13.9)

where R is a coupling matrix that represents the effective surface area associated with each node on the fluid–structure interface, Equation 13.8 implies that nodes on the fluid–structure interface have both displacement and pressure degrees of freedom. Equation 13.9 is unsymmetrical in terms of both the stiffness and the mass. In the present study, the acoustic field inside the cylinder was calculated following the above FEM procedure with 17,000 mesh elements. A 2D axi-

symmetric model of the hollow cylindrical radiator was developed for both modal and harmonic analyses. The model was defined on the plane passing over the symmetrical axis of the cylinder in position X = 0. The cartesian Z-axis was assumed to be the axis of symmetry. The boundary conditions were described in terms of the acoustic and structural elements. The hollow cylinder was modeled with PLANE42 elements used for structural components and FLUID29 acoustic elements for fluid-air components. ANSYS has FLUID29 as a specific acoustic element used for acoustic analysis. The element accepts fluid density and speed of sound as material input. The element is defined by four nodes having three degrees of freedom at each node. PLANE42 is used for the 2D modeling of solid structures. The element can be used as a plane or as an axi-symmetric element. The element is defined by four nodes

280

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

1

NODAL SOLUTION OCT 21 2009 12:48:19

SUB =1 FREQ =22045 IMAGINARY /EXPANDED SINT (AVG) DMX =.493E–04 SMN =.126e+07 SMX =.155E+09

.126E+07 .355E+08 .697E+08 .104E+09 .138E+09 .184E+08 .526E+08 .867E+08 .121E+09 .155E+09 Figure 13.13. Distribution of mechanical stresses in the cylindrical tube.

having two degrees of freedom at each node (translation in the nodal z- and x-directions). For this modeling, the selected element sizes were 1 and 2 mm in the structure and in the fluid, respectively. In the numerical procedure that follows the calculation of the average SPL inside the tube, harmonic analysis was used. In this way, both maximum stress and an SPL were obtained for a vibration displacement of 10 microns applied in the center of the hollow cylinder. The maximum value of the mechanical stress that can support the material of the tube, and the average value of the acoustic pressure were obtained. The SPL was calculated and a value of about 156 dB was obtained. Figure 13.14 shows the computed standing wave field structure inside the tube. To gain more knowledge of the acoustic pressure distribution inside the

tube, a second simulation was carried out with COMSOL® Multiphysics (Burlington, MA) modeling and simulation code by following a procedure similar to the one described above with ANSYS code. In this case, 2D axial symmetry elements were used for meshing and modeling. The computed acoustic pressure pattern distribution inside the cylinder is shown in Figure 13.15. It is clear from the comparison between Figures 13.14 and 13.15 that a similar pressure-filled structure was obtained in both cases.

13.3. Development of the Power Ultrasonic Transducer Prototype The CRT system was built for a resonance frequency of about 21,897 Hz with 12 NL. The electroacoustic characteristics of such a system were measured

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Computational Study of Ultrasound-Assisted Drying of Food Materials

281

ANSYS 6.1 JAN 21 2003 10:43:32 AVG ELEMENT SOLUTION STEP =1 SUB =1 FREQ=22269 /EXPANDED SPL (AVG) DMX =.537E–05 SMN =110.313 SMX =169.424 XV =1 YV =1 ZV =1 DIST =.172893 YF =.155 =–.025 ZF Z-BUFFER 110.313 116.881 123.449 130.017 136.585 143.152 149.72 156.288 162.856 169.424

Drying chamber: acoustic distribution Figure 13.14. 2D standing wave field pattern of the acoustic pressure inside the radiating tube obtained in ANSYS code. See color insert.

giving the following values: electrical impedance Z = 365 Ω, bandwidth Bw = 17 Hz, quality factor Q = 1,288, and a power capacity in the order of 90 W (Riera et al. 2004b). The CRT developed by the Power Ultrasonics Group of the Instituto de Acústica de Madrid (CSIC) is shown in Figure 13.16. This group has also developed other types of high-power ultrasonic transducers for use in food processing based on vibrating plates (GallegoJuárez et al. 1999, 2000, 2002, 2007; Mason et al. 2005; Zheng and Sun 2005).

13.3.1. Acoustic Field Measurements inside the CRT The experimental validation of the CRT was carried out by measuring the acoustic field inside the cylin-

drical cavity. It was measured in a semi-anechoic chamber. A PC-controlled 3D measuring system was used. The X, Y, Z coordinates related to the microphone position are governed by means of numerical control equipment (LabView code, National Instruments, Austin, TX), and they are sent to an analogue treatment stage. 2D Raster scans, which measure the SPL (planes of symmetry X–Z), were made with a 1/8” GRAS microphone (GRAS Sound and Vibration, Holte, Denmark) parallel to the axis of symmetry (Z-axis) of the cavity and perpendicular to the radial distance (X-axis) to the walls. The microphone was stepped in 0.75 mm increments in both Z- and X-directions. The electrical output signal of the microphone was captured and stored on a PC. The output voltage level from the microphone was converted to the equivalent acoustic pressure (Pa)

282

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Feq_acax(1)=21625 Surface: Sound pressure level

Max:184.58

0.3

180

0.25

170

0.2

160

0.15

150

0.1 140

0.05 130

0 –0.15

–0.1

–0.05

0

0.05

0.1

0.15

0.2

0.25

120 Min:119.856

Figure 13.15. 2D standing wave field pattern of the acoustic pressure inside the radiating tube obtained in COMSOL code. See color insert.

by using the known sensitivity of the microphone. In this way, the ultrasonic pressure distribution pattern in the tube was obtained. From these measurements, the average SPL obtained inside the cylindrical radiator was about 154.1 dB with an electric power applied to the CRT of about 90 W (Figure 13.17a,b). Therefore, a high-level ultrasonic field inside the cavity was reached using relatively low applied power. Such a result confirms the FEM predictions for the acoustic field and, as a consequence, the feasibility of using such a system to assist ultrasonically conventional convective drying. The CRT system was used as the drying chamber of a standard convective drier in the facilities of the Polytechnic University of Valencia without heating the samples significantly.

13.4. Ultrasonically Assisted Convective Drier The high-power ultrasonic application system described in the previous section was assembled in the laboratory-scale convective drier of the ASPA Research Group (Food Technology Department, Polytechnic University of Valencia) (Sanjuán et al. 2004). A scheme of the ultrasonically assisted convective drier is depicted in Figure 13.18. In the new prototype, the conventional drying chamber was replaced with the CRT. This element not only contains the samples during drying but also radiates the acoustic energy into the air medium and the samples. The main elements of this prototype are described in detail in the following sections.

Chapter 13

Computational Study of Ultrasound-Assisted Drying of Food Materials

Figure 13.16. Picture of the developed cylindrical radiator transducer (CRT).

13.4.1. Automatic Control: Air Velocity, Temperature, and Sample Weight The drier operates automatically; the air velocity and temperature are controlled and samples are weighed at preset times. The air circulation system is made up of a centrifugal fan (Figure 13.18:1) of medium pressure (COT-100, Soler and Palau, Spain), which forces the air through the heating elements toward the drying chamber. A frequency inverter acts over the fan in order to control the turn speed (Inverter DV-551, Panasonic, Secaucus, NJ) providing an air velocity ranging between 0.1 and 14 m/s. Air velocity is measured (± 0.1 m/s) by using a vane anemometer (Figure 13.18:3) (Wilh. Lambrecht GmbH, Göttingen, Germany). Both the anemometer and the frequency inverter are wired to a process controller (E5CK, OMRON, Kyoto, Japan), thereby keeping the air velocity at the desired target by using a proportional integral derivative (PID) control mode. The electric heating elements (maximum power 3,300 W, 220 V) are placed in the air duct between the fan and the anemometer. A voltage inverter

283

(80 A, Nixa, Barcelona, Spain) actuates over the heating elements (Figure 13.18:2) in order to control the energy radiated into the air. The air drying temperature is measured by using a Pt-100 placed near the drying chamber. In the same way as for the air velocity control, the temperature sensor (Figure 13.18:5) and voltage inverter are wired to a process controller (E5CK, OMRON) for a PID control. The sample load in the drying chamber may be carried out by using two different setups. One of them consists of a tray load device with 10 trays (diameter 84 mm), 34 mm apart, and made of a square wire mesh (side 3 mm) (Figure 13.18:11). The other loading system is made up of a metallic central axis (diameter 3 mm) with 20 randomly placed metallic hooks (diameter 1 mm). By combining these systems, the samples may be arranged in order or at random in the drying chamber. The dimensions of both devices were shortened (in comparison with the ultrasonic wavelength in air) to avoid introducing any perturbation in the acoustic field. The devices are wired by their central axis to a chamber made of methacrylate (Figure 13.18:6) (diameter 100 mm and height 100 mm). A sponge was used as coupling material (Figure 13.18:7) and placed between the vibrating cylinder and the chamber in order to achieve an optimal vibration mode and avoid air losses. A balance (Figure 13.18:12) (PM4000, Mettler Toledo, Columbus, OH) wired to the methacrylate chamber allowed the samples to be weighed at preset times by using two pneumatic moving arms (Figure 13.18:8). A programmable logic controller (PLC) (CQM41, OMRON) arranged the weighing system in sequence. The balance and the two process controllers are connected to a PC by RS-232 interfaces. An application was developed in Visual Basic™ (Microsoft) to supervise the whole process from the PC.

13.4.2. Additional Elements in the Ultrasonic System The inclusion of the CRT in the drier required the use of additional elements to provide an adequate excitation of the ultrasonic device. An ultrasonic power generator (Figure 13.18:15) (maximum power

(a)

160 165 160

155

SPL (dB)

155 150

150

145 140

145

135 130

140

125 100 80

135

400 60

300 40

200 20 0 0

Length (mm) (b)

130

100 Radius (mm)

SPL (dB) 0 160 50 155

LENGTH (mm)

100 150 150 145

200

140

250

135

300

130 0

10

20

30

40

50

60

70

80

90

RADIUS (mm) Figure 13.17. (a) 3D-sound pressure level distribution pattern (in decibels) inside the cylindrical radiator: electrical power applied, 90 W. (b) 2D-sound pressure level distribution pattern (in decibels) inside the cylindrical radiator: electrical power applied, 90 W (Gallego-Juárez et al. 2010). See color insert.

284

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17

8

285

16

T,HR 12

14 13

9 11 10 8 7

15

6 5

3

2

Hot air

Room air

1

4

1. Fan, 2. Heating unit, 3. Anemometer, 4. 3-Way valve, 5. Pt-100, 6.Methacrylate chamber, 7. Coupling material, 8. Pneumatic moving arms, 9. Ultrasonic transducer, 10. Vibrating cylinder, 11. Loading device, 12. Balance, 13. Impedance matching unit, 14. Digital power meter, 15. High-power generator, 16. PC, 17. Room conditions sensor. Figure 13.18. Diagram of the ultrasonically assisted convective drier (Cárcel et al. 2010).

500 W, LC-9601, Institute of Acoustics, CSIC, Madrid, Spain) supplies and amplifies the electric signal at high frequencies (greater than 20 kHz). An impedance matching unit (Figure 13.18:13) (Institute of Acoustics, CSIC) was included in the ultrasonic circuit in order to fit the impedance output of the generator to the input impedance of the ultrasonic application system. This adapter consists of an impedance transformer (from 50 Ω to 200–500 Ω) and a variable inductance (between 5 and 9 mH) to

compensate the interelectrodic capacity (Co) of the ultrasonic prototype. The main goal of the impedance matching unit is to optimize the acoustic energy transfer from the high-power generator to the piezoelectric transducer. The electric properties of the output signal of the impedance matching unit were measured by using a digital power meter (Figure 13.18:14) (WT210, Yokogawa, Japan). Among others, the frequency (Hz), power (W), intensity (A), voltage (V), and

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

30

21620

2

55

Voltage

Frequency Phase

21610

21590

Voltage (V)

20

1.9

Phase (º)

Frequency (Hz)

21600

Intensity

54

25

53 1.8 52

15 21580

Intensity (A)

286

1.7 51

21570

10 0

10

20

30

40

50

Measurements

50

1.6 0

10

20

30

40

50

Measurements

Figure 13.19. Electric properties of the ultrasonic signal during drying of lemon peel slabs (thickness 10 mm) (40°C, 1 m/s). Measurement time interval: 2 minutes.

phase (°) were monitored during the ultrasonically assisted drying experiments. The power meter permits the optimal excitation of the ultrasonic power system, fitting the frequency in order to attain the 12 NL mode identified in the design. The excitation of other modes would lead to a significant reduction in the energy irradiated by the vibrating cylinder, and applying high power could produce the breakdown of the material provoked by intense stress. The power meter is connected to the PC (Figure 13.18:16) via the RS-232 interface. An application was developed in LabVIEW (National Instruments, Austin, TX) to log the electric properties during the experiments. Thereby, the behavior of the power ultrasonic system during drying may be monitored and compared with that previously observed during characterization. In addition, anomalous behavior may be identified.

13.4.3. Behavior of the Ultrasonic System during Drying Figure 13.19 shows some of the electric properties of the ultrasonic signal measured by means of the digital power meter during drying of lemon peel slabs. The values of the electric parameters were similar to those previously obtained during characterization (Section 13.3). Thus, the frequency

achieved was close to the previously measured frequency (21.8–21.9 kHz). As can be observed in Figure 13.19, this parameter was reduced during drying due to the heating of the ultrasonic power system. Although the phase was slightly higher, this fact does not lead to a significant reduction in the effective power supplied to the ultrasonic transducer. The voltage was slightly lower compared with the characterization (60 V) and, as a consequence, higher electric intensities were measured (1.8–1.9 A). This, in fact, has a positive effect on the energy irradiated from the walls of the vibrating cylinder since the vibration amplitude of the piezoelectric ceramics is proportional to the electric intensity. This improvement could be explained by the cooling effect caused by the air flow in the vibrating cylinder, which avoids overheating and, therefore, provides optimal working conditions for the ultrasonic system. Unfortunately, as the cooling effect of the air is linked to the air temperature applied, this effect may disappear in experiments carried out at high temperatures. In order to identify the influence of the air velocity, the SPL generated by the cylindrical chamber was measured using the preset pathway already described in Section B.1 of García-Pérez et al. (2006a). The sound pressure in the drying chamber decreased as the air velocity increased, remaining

Chapter 13

Computational Study of Ultrasound-Assisted Drying of Food Materials

158

SPL (dB)

156

154

152 0

4

8

12

16

Air velocity (m/s) Figure 13.20. Influence of air velocity on average sound pressure level.

287

identifying the effects involved. Modeling is essentially a way of representing processes or phenomena to explain the observed data and to predict behavior under different conditions (Mulet 1994). Two types of models can be used to describe water migration during the falling rate period: theoretical and empirical models (Blasco et al. 2006). The most popular theoretical models are those based on diffusion theory; they attempt to explain the problem using physical laws. Empirical models do not explain the process; their only aim is to determine the relevant variables and quantify the kinetic. Academic research should avoid these approaches whenever possible, but empirical models may indeed be very useful for industrial applications or for simplifying complex phenomena, for which their usefulness is beyond doubt.

13.5.1. Empirical Models almost constant above 8 m/s (Figure 13.20). That means that there was a reduction in the acoustic energy in the medium at high air flow rates. The effect of turbulence on the acoustic field disruption has also been addressed in a liquid medium: The acoustic energy produced by an ultrasonic bath (20 kHz) decreased when a mechanical agitator was used, as compared with those found without agitation (Mulet et al., 2003a, 2003b). Therefore, air flow also affects the behavior of the ultrasonic application system. Low air velocities (linked to low air temperatures) may produce a beneficial effect due to a cooling effect, which avoids overheating. However, high air velocities cause a disturbance in the acoustic field generated inside the drying chamber, reducing the energy available for the drying process.

13.5. Computational Analysis of Ultrasonically Assisted Convective Drying It is essential to know the influence of power ultrasound on the moisture removal mechanism during the drying process to fully develop this technology. In order to attain this objective, modeling is a fundamental tool, not only for quantifying but also for

Weibull’s and Peleg’s models are two of the most popular empirical models employed to describe drying kinetics. Weibull’s model, also named Page, is a probabilistic function used to describe the behavior of changeable complex systems (Cunha et al. 1998). Initially, it was applied in materials resistance to describe the fatigue-induced material failure. Drying may be considered a failure of the food system when subjected to stress by hot air; therefore, the application of power ultrasound could increase the stress produced in the product during drying. Page (1949) developed an equation to describe the thin-layer drying of shelled corn using the Weibull’s function (Eq. 13.10). It has been used to model the drying kinetics of foods such as figs, currants, and plums (Karathanos and Belessiotis 1997), onions (Wang 2002), grapes (Azzouz et al. 2002), rough rice (Basunia and Abe 2001), lettuce and cauliflower (López et al. 2000), dates (Hassan and Hobani 2000; Cunha et al. 2001), curcuma (Blasco et al. 2006), kiwi (Simal et al. 2005), carrot, and lemon peel (García-Pérez et al. 2006a). ⎛ ⎛ t ⎞α ⎞ W (t ) = We + (Wc − We ) exp ⎜ − ⎜ ⎟ ⎟ ⎝ ⎝ β⎠ ⎠

(13.10)

288

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Weibull

4.5

Peleg 1.5

3 2.5

kl–1

b –1 (10–4 L/s)

3.5

2 1.5

external resistance

1

internal resistance

(103 kg water/s kg dm)

4 1.25 1 0.75 0.5

external resistance

0.25

0.5 0

internal resistance

0

0

1

2

3

4

5

v (m/s)

0

1

2

3

4

5

v (m/s)

Figure 13.21. Use of empirical model to establish the velocity threshold that separates the mass transfer control into external and internal resistance in the drying of Curcuma longa rhizomes (Blasco et al. 2006).

where α is a shape parameter that is a behavioral index for the process, and β is a scale (kinetic) parameter. Actually, the β parameter is inversely associated with the drying rate and includes all the effects on the mass transfer. Peleg (1998) proposed a two parameter, nonexponential empirical equation to describe moisture sorption curves. The model has been very useful to describe the rehydration kinetics of several foodstuffs (Pan and Tangratanavalee 2003), although it has also been applied to model drying kinetics (Sopade and Kaimur 1999; Blasco et al. 2006). W = Wc −

t k1 + k2 t

(13.11)

where k1 is a kinetic parameter inversely proportional to the drying rate, while k2 is a parameter that is strongly linked to final product characteristics; in a drying process, it is related to equilibrium moisture content. Actually, empirical models provide a good description of drying kinetics, and interesting conclusions about the mass transfer process may also be extracted from the analysis of the kinetic parameters. Figure 13.21 shows the evolution of the kinetic parameters of Weibull’s and Peleg’s models as

affected by air velocity (Blasco et al. 2006). Figure 13.21 illustrates that, for drying samples of Curcuma longa, an air velocity threshold close to 2 m/s was identified separating the mass transfer control into external (low air velocities) and internal (high air velocities) resistance. According to these results, the use of air velocities of over 2 m/s is not justified on an industrial scale because, although it will not provide higher drying rates, it will require higher energy costs. Despite the usefulness of empirical models to quantify the influence of process variables during drying and to provide a good description of drying kinetics, they present important drawbacks preventing their application. The results obtained using an empirical model cannot be extrapolated to different experimental conditions. Therefore, its application to optimization problems is very limited.

13.5.2. Diffusion Models Mechanistic models are built by following some assumptions, of which the most common is to consider mass transport inside the food as being solely driven by moisture concentration differences, that

Chapter 13

Computational Study of Ultrasound-Assisted Drying of Food Materials

is, a Fickian mechanism. It is known that other mechanisms coexist, such as capillarity, and also that the food structure could affect mass transport. Nevertheless, if properly defined, Fickian models usually describe the process well. The diffusion model is based on the Fick’s second law; it was formulated by Lewis (1921) and further developed by Sherwood (1929). These models have been widely used mostly for regular shaped geometries (Simal et al. 1998). The differential equation of diffusion can be obtained by combining Fick’s law and the microscopic mass balance. This equation for isotropic solids, which assumes effective moisture diffusivity to be constant, is written for slab (Eq. 13.12), infinite length cylinder (Eq. 13.13), and finite length cylinder (Eq. 13.14) geometries as follows: ∂τ ( x , t ) ⎛ ∂ 2 ( x, t ) ⎞ = De ⎜ ⎝ ∂x 2 ⎟⎠ ∂t

(13.12)

∂τ (r , t ) ⎛ ∂ 2 τ (r , t ) 1 ∂τ (r , t ) ⎞ (13.13) = De ⎜ + ⎝ ∂r 2 ∂t r ∂r ⎟⎠ ∂τ (r , x, t ) ∂t ⎛ ∂ 2 τ (r , x, t ) ∂ 2 τ (r , x, t ) 1 ∂τ (r , x, t ) ⎞ = De ⎜ + + ⎝ ∂x 2 ∂r 2 r ∂r ⎟⎠ (13.14) In order to solve the diffusion equations, initial and boundary conditions are needed. The assumptions considered determine the difficulty of solving the mass transfer equation. When evaluating models with different degree of complexity, it is advisable both to take the effort needed to obtain closer fits into consideration and to manage the complexity more efficiently (Mulet 1994). Indeed, the recommended complexity of a model depends on the use it is intended for. Some of the most common assumptions considered are (Simal et al. 1996; Cárcel et al. 2007a): The initial moisture content and temperature are uniform inside the sample. Heat transfer is neglected, assuming sample temperature is close to air drying temperature.

289

The solid shape is assumed to be constant during the drying period; no shrinkage is considered. The solid symmetry will be considered in order to solve the model. Considering these assumptions, there are two types of models of different complexities, one where only diffusion controls the process and the other where, additionally, external convection should be considered. The first type is simpler and is the one most commonly utilized. 13.5.2.1. Neglecting External Resistance to Mass Transfer (NER Models)—Analytical Solutions Diffusion-controlled drying refers to a drying process in which the diffusion of internal moisture to the solid surface is the slowest and, therefore, rate-limiting step when compared with the rate at which the surface moisture is removed by the carrier gas flow (Simal et al. 2005). This aspect is included in the model through a boundary condition considering moisture content on the solid surface to be at equilibrium with the air during drying (Eq. 13.15 for slab geometry):

τ ( L, t ) = We

{t > 0; x = L}

(13.15)

Crank (1975) presented a compilation of analytical solutions to the diffusion problem for different regular geometries using the separation of variables method. Once the solution is reached, it should be integrated to obtain the average moisture content (García-Pérez et al. 2009b). The solutions for slab (thickness 2L) and cylinder (radius R) infinite geometries in terms of the average moisture content are presented in Equations 13.16 and 13.17, respectively: W (t ) = We + (Wc − We ) ×

∑

⎛ ⎛ − De (2 n + 1)2 π 2 t ⎞⎞ 8 × exp ⎜⎝ ⎟⎠⎟⎠ n=0 ⎜ 4 L2 ⎝ (2 n + 1)2 π 2

∞

(13.16) W (t ) = We + (Wc − We )

∑

∞ n =1

8 ⎛ D ⎞ exp ⎜ − 2e α n2 t ⎟ ⎝ R ⎠ α n2 (13.17)

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

The diffusion equations for finite length geometries may be adequately obtained by combining the solution for infinite geometries. Thereby, the solution for finite cylinder (radius R, half-length L) is written (Eq. 13.18) in terms of the solution for infinite cylinder and slab geometries. In the case of a cube geometry (Eq. 13.19), the equation is assembled by combining the solution for three infinite slabs. ⎡ ⎢ ⎢ ⎢ W (t ) = We + (Wc − We ) ⎢ ⎢ ⎢ ⎢ ⎢⎣

∑ ∑

8 ⎞ ⎤ ⎛ × 2 2 ⎜ ⎟ ⎥ (2n + 1) π ∞ ⎜ ⎟ ×⎥ n=0 ⎜ ⎛ − De (2 n + 1)2 π 2 t ⎞ ⎟ ⎥ ⎜ exp ⎜ ⎟⎠ ⎟⎠ ⎥ 4 L2 ⎝ ⎝ ⎥ ⎥ ∞ 8 ⎛ D ⎞ ⎥ exp ⎜ − 2e α n2 t ⎟ 2 n =1 α ⎝ R ⎠ ⎦⎥ n (13.18) 3

8 ⎛ ⎞⎞ ⎛ × ⎜ ∞ ⎜ (2 n + 1)2 π 2 ⎟⎟ ⎟⎟ ⎜ W (t ) = We + (Wc − We ) ⎜ 2 2 ⎜ n=0 ⎜ ⎛ − De (2 n + 1) π t ⎞ ⎟ ⎟ ⎜ ⎜ exp ⎜ ⎟⎠ ⎟⎠ ⎟⎠ 4 L2 ⎝ ⎝ ⎝ (13.19)

∑

The effective moisture diffusivity parameter considered in these models includes all the kinetic effects, not only those associated with the diffusion mechanism, but also those linked to unknown mechanisms, as well as deviation due to simplifications carried out in the modeling (Mulet 1994). If the approach of negligible external resistance deviates from real behavior during drying, effective moisture diffusivity would include this effect. This fact may be observed in Figure 13.22, where effective diffusivity is depicted at different air velocities for the drying of C. longa rhizomes. As air velocity has no physical influence on the effective moisture diffusivity, the increase in diffusivity at low air velocities suggests a significant external resistance. A similar conclusion was obtained from the analysis of empirical models (Figure 13.21). 13.5.2.2. Considering Significant External Resistance to Mass Transfer (ER Models). Numerical Solutions In some cases, moisture transport during the drying process could be affected

3.5 3.0

De (10–9 m2/s)

290

2.5 2.0 1.5 external resistance

1.0

internal resistance

0.5 0.0 0

1

2

3

4

5

Velocity (m/s) Figure 13.22. Use of diffusion models neglecting external resistance to mass transfer to establish the velocity threshold that separates the mass transfer control into external and internal resistance in the drying of Curcuma longa rhizomes (Blasco et al. 2006).

by both the internal resistance of water transfer inside the samples and the external resistance between the solid surface and the surrounding air. This fact can be considered in the model by replacing the boundary conditions assumed in Equation 13.15 with those in Equation 13.20 (slab geometry), which consider the equality of the diffusion and convective mass fluxes on the solid–air interface. As consequence, mass transfer is jointly controlled by effective diffusivity and mass transfer coefficient. − De ρds

∂τ ( L , t ) = k (ϕ e ( L, t ) − ϕ air ) ∂x

{t > 0; x = L} (13.20)

In Equation 13.20, the relative humidity of the drying air (φair) may be calculated by measuring the relative humidity of the environment (φe) and considering the psicrometric expressions. This model, which assumes a diffusion mechanism and considers external resistance to mass transfer to be significant, can be used to identify and quantify the influence of power ultrasound on the kinetics of a process. In addition, the ER model makes it easier to separate the ultrasonic effect on external and

Chapter 13

SYMMETRY PLANE

Computational Study of Ultrasound-Assisted Drying of Food Materials

SOLID SURFACE

Δx =

Δx

–L 0

1

2 · · · i · · · n–1

L n

x j =i Δx; i∈[0,n] L n

x

Figure 13.23. Location of nodes for infinite slab geometry.

internal resistance to mass transfer. However, the boundary condition stated in Equation 13.20 makes it difficult to find the analytical solution to the model, in which case it is common to use numerical methods to solve the diffusion equation as is reported in the literature. In addition, the usefulness of numerical methods is beyond any doubt when diffusion models consider other phenomena such as shrinkage or a nonconstant effective diffusivity value. The main alternatives used in the literature to solve the diffusion equation are finite difference and FEM. The finite difference method provides good results for regular geometry bodies (Simal et al. 1996; Wang and Sun 2003; Mulet et al. 2005) and needs fewer computational requirements than finite element modeling. The symmetry of regular geometries allows their volume to be divided into equal parts delimited by symmetry planes, and to be solved by addressing only one section of the volume (Figure 13.23). For the application of the finite difference method, the original control volume is divided into a constant number of elements (subvolumes), which constitute the characteristic network of the solid. The node is the characteristic point of the sub-volume, and the location of the nodes selected in the solid section for an infinite slab geometry is shown in Figure 13.23. Equation 13.21 shows the relationship of the local moisture content for a node, being a function of the moisture content

291

both of the neighbor nodes and of the same node at a previous time. The particular expression at each kind of node must be obtained by adequately combining the boundary conditions (Simal et al. 2003). ⎡ ⎛ ⎛ Δx 2 ⎞ ⎞ ⎤ De Δt ⎢τ (i, t ) ⎜ ⎜ ⎟⎠ + 2⎟⎠ − ⎥ (13.21) ⎝ ⎝ Δ D t τ (i, t − Δt ) = e ⎥ Δx 2 ⎢ ⎢⎣ τ (i + 1, t ) − τ (i − 1,, t ) ⎥⎦ For the infinite slab, the position of the node in the axial direction is characterized by the x-coordinate, L the separation between nodes by Δx = , the n −1 number of nodes in the x-direction by n and, finally, the time interval considered by Δt. When applying the finite difference method, such as reported in Equation 13.21, a set of implicit equations must be solved to determine the moisture content in each node of the network. Thereby, the moisture profile in the solid is determined as well as its evolution during drying. The average moisture content of a sample is calculated by integrating the local moisture content for the whole solid volume. The use of computational tools is necessary to solve the diffusion equation, following the aforementioned procedure. In this work, a series of functions were programmed using the MATLAB (The MathWorks Inc., Natick, MA) code to solve the diffusion equation for different geometries.

13.5.3. Ultrasonic Effect on Mass Transport Phenomena The main aim of this section is to gain insight into the influence of power ultrasound on mass transfer mechanisms that occur during convective drying. To do this, power ultrasound-assisted convective drying experiments (ultrasound [US] experiments) of different products (carrots, lemon and orange peel, eggplant, potato, and persimmon) were carried out under different experimental conditions using the CRT described in the previous sections. To identify the effect of ultrasound on the drying process, US experiments were compared with those carried out

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3.50 US 3.00 AIR

W (d.b.)

2.50 2.00 1.50 1.00 0.50 0.00 0

10000

20000

30000

40000

Time (s) Figure 13.24. Drying kinetics of orange peel slabs (thickness 5.9 ± 0.4 mm) at 40°C and 1 m/s with (US, 37 kW/m3) and without (AIR) power ultrasound application.

under the same experimental conditions but without applying power ultrasound (AIR experiments). The results obtained show a significant improvement of drying kinetics when ultrasound was applied. Convective drying kinetics of orange peel slabs carried out at 40°C and 1 m/s (AIR) and ultrasonically assisted by applying an acoustic power of 37 kW/m3 (US) are depicted in Figure 13.24 for comparison. Ultrasonic application reduced drying time by 50%, which confirms a significant improvement of drying kinetics and, as a consequence, a significant effect on mass transfer phenomena. Nevertheless, the improvement of the drying rate was dependent on the experimental conditions established during drying: The influence of drying air velocity (García-Pérez et al. 2006a, 2007; Cárcel et al. 2007a), temperature (García-Pérez et al. 2006b), and applied acoustic power (García-Pérez et al. 2009a), among others, on the ultrasonic application was identified. In addition, when experiments were carried out with different products but under the same experimental conditions, there was a different effect of ultrasound on the drying rate, which also confirms the influence of raw matter properties on the ultrasonic application.

A simple analysis of drying kinetics does not provide relevant information about the ultrasonic influence on external and/or internal mass transfer resistance. In addition, the ultrasonic influence needs to be quantified. Modeling may be considered not only a useful tool with which to clarify these issues, but also one that may be used to predict the behavior of the system under different operational conditions (Mulet 1994), which is very useful in drier design and optimization. The influence of power ultrasound on mass transfer phenomena can be seen at the example of AIR and US (31 kW/m3) drying experiments of persimmon cylinders carried out at 40°C at several air velocities ranging from 0.5 to 14 m/s (Cárcel et al. 2007a). In the same way as in Figure 13.22, the use of an NER model allowed the mass transfer to be separated into the internal and external resistance, thereby establishing an air velocity threshold between 4 and 6 m/s (Figure 13.26). The fit of the NER model to experimental data was better at high air velocities due to the fact that drying conditions were closer to the assumed boundary condition of negligible external resistance to mass transfer (Figure 13.25). As a consequence, there is no need to consider a more complex model, including external resistance, for experiments using air velocities of over 6 m/s. Nevertheless, the NER model does not adequately describe the drying kinetics when low air velocity experiments are applied. Thus, an ER model was fitted to drying kinetics carried out at less than 6 m/s, providing in this case good agreement with the experimental data. Figure 13.26 depicts the effective diffusivities identified using the NER model for high air velocities (internal resistance control) and the ER model for low air velocities (shared control by internal and external resistance). The effective diffusivities identified for US experiments at low air velocities were significantly higher than the values obtained for AIR experiments. This result confirms a significant effect of power ultrasound on internal diffusion, which means the water moves easily in the product toward the surface. The explanation of improvement of water diffusion may be supported by the alternate

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Sample moisture (kg W/kg dry matter)

5

4

3

2

1

0 0

5,000

10,000

15,000

20,000

Time (s)

Figure 13.25. Experimental AIR drying kinetics of persimmon cylinders (height 30 mm and diameter 13 mm) (50°C) at 0.5 m/s (䊊) and 8 m/s (䉭) and calculated using an NER model (Cárcel et al. 2007a).

8.00 7.00

De (10–10 m 2/s)

6.00 5.00 4.00 3.00

External + Internal Resistance

Internal Resistence

2.00 AIR

1.00

US 0.00 0

2

4

6 8 Velocity (m/s)

10

12

14

Figure 13.26. Influence of air velocity on effective moisture diffusivities identified fitting NER (air velocities > 6 m/s) and ER models (air velocities < 6 m/s) to AIR and US (31 kW/m3) drying kinetics (50°C) of persimmon cylinders (height 30 mm and diameter 13 mm) (Cárcel et al. 2007a).

293

expansions and contractions produced by power ultrasound in the materials (Gallego-Juárez et al. 1999). In the literature, this fact is referred to as a “sponge effect” due to its similarity to a sponge being squeezed and released. The influence of ultrasound on the effective diffusivity disappeared at high velocities (Figure 13.26), which is explained by the fact that acoustic energy decreases with an increase in air velocity (see Section 13.4.3). At high air velocities, the acoustic field generated in the drying chamber is disturbed by air flow, reducing the energy available to the drying process. These results agree with those found by Gallego-Juárez et al. (1999) on the ultrasonic drying of carrots using an airborne ultrasonic system: Gallego-Juárez et al. found that the recorded weight loss during drying decreased when air velocity changed from 1 to 3 m/s. Additional drying experiments were carried out with lemon peel slabs under the same experimental conditions (40°C, 1 m/s). The results showed an effect of ultrasound on the effective diffusivity over the whole range of air velocities tested. Despite reducing acoustic energy by increasing the air velocity, the acoustic level was able to affect the mass transfer through a more sensitive product like lemon peel (García-Pérez et al. 2007). In all likelihood, product structure has an influence on the ultrasonic effects, which is addressed at the end of this section in more detail. The use of the ER model to describe the drying kinetics of persimmon cylinders at under 6 m/s also contributed to identification of a significant influence of ultrasonic application on the mass transfer coefficient. The mass transfer coefficient identified by fitting the ER model to experimental data was higher for US experiments over the entire range tested (Figure 13.27). Obviously, the higher the applied air velocity, the higher the mass transfer coefficient, due to the control of mass transfer by external resistance. The literature reports significant mechanical effects of ultrasound on the interfaces (Gallego-Juárez 1998). Among others, ultrasound introduces pressure variations, oscillating velocities, and micro-streaming, which may reduce the diffusion boundary layer thickness, thus improving

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the transfer rate from the solid surface to the air medium. The microstructural effects induced by power ultrasound application in drying of orange peel were studied by cryo-scanning electron microscopy (Ortuño et al. 2010). The analysis reported a high degradation of albedo cells (internal issue) (Figure 13.28) as well as a very intense effect on the surface of the orange peel (Figure 13.29). The albedo is

–3.50 AIR

–3.90

US y = 0.46x – 4.63 R2 = 0.99

US

ln k

–4.30

–4.70 AIR y = 0.50x – 4.81 2 R = 0.93

–5.10

–5.50 –1.0

–0.5

0.0

0.5

1.0

1.5

ln v Figure 13.27. Influence of air velocity on mass transfer coefficients identified fitting an ER model to AIR and US (31 kW/ m3) drying kinetics (50°C) of persimmon cylinders (height 30 mm and diameter 13 mm) (Cárcel et al. 2007a).

A

100 μm

B

characterized in fresh samples by long tubular cells (Figure 13.28A). Air drying provokes in the cells the loss of the tubular shape due to water removal (Figure 13.28B), although a more intense effect on the structure was observed in the US samples (Figure 13.28C) due to a larger compression in the albedo cells (Figure 13.28C). The acoustic wave acts repeatedly (sponge effect) over the cell, facilitating water removal and inducing a structural change. In the case of the orange surface (Figure 13.29A), air drying scattered the waxy components, closing the pores and creating a waterproof barrier (Figure 13.29B). Despite scattering, the ring-shaped accumulations around the pores continued to be well defined in AIR dried samples (Figure 13.29B). However, in the case of US experiments, results show the ring-shaped structure around the pore to be less distinguishable (Figure 13.29C). The aforementioned effects of power ultrasound on the interfaces are also responsible for a more intense scattering of the waxy compounds on the cuticle surface. The effects observed in the albedo cells and in the orange surface reveal a very intense effect of ultrasound on the internal issue and the interfaces, respectively, and support the results about the influence of ultrasound on internal and external resistance to mass transfer. The literature also reports significant heating effects of power ultrasound on liquid media (Cárcel et al. 2007b, 2007c). These effects are mainly provoked by cavitation, which could produce an effect less intense in air than in liquid media. Moreover,

C

100 μm

100 μm

Figure 13.28. Orange peel surface: (A) fresh orange; (B) AIR dried (40°C, 1 m/s); (C) US dried (37 kW/m3).

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A

Computational Study of Ultrasound-Assisted Drying of Food Materials

B

40 μm

295

C

30 μm

30 μm

Figure 13.29. Albedo cells (internal tissue of orange peel): (A) fresh orange; (B) AIR dried (40°C, 1 m/s); (C) US dried (37 kW/m3).

–6.8

–7.2

ln k

a buffer effect on the temperature could be produced by the air flow. Therefore, an improvement in mass transfer during drying caused by the heating effect provoked by power ultrasound application may be considered as almost negligible. Taking this fact into consideration, drying experiments of carrot cubes were carried out at 1 m/s and at different air temperatures (30, 40, 50, 60, and 70°C) with (US, 31 kW/m3) and without (AIR) power ultrasound application (García-Pérez et al. 2006b). An ER model was fitted to drying kinetics, showing the aforementioned effect of power ultrasound on the external and internal resistance to mass transfer. The effect of power ultrasound on the mass transfer coefficient and the effective moisture diffusivity decreased as the temperature rose, and it was negligible at 70°C (Figure 13.30). This indicates that ultrasound application introduces a given amount of energy into the drying process, thus affecting water mobility. A rise in temperature increases the water mobility linked to higher air temperatures and, as a consequence, reduces the influence of ultrasonic energy in the mass transfer process. This fact supports the negligible effect of power ultrasound on kinetic parameters observed in Figure 13.30. However, more experimentation is needed to confirm this hypothesis. The high efficiency of the ultrasonic application at low temperatures supports a potential use in the drying of heat sensitive materials. In these cases, the use of low temperatures is linked to low

–7.6 AIR

–8

–8.4 0.0028

US

0.003

0.0032

0.0034

1/T (K–1) Figure 13.30. Influence of air temperature on mass transfer coefficients identified fitting an ER model to AIR and US (31 kW/m3) drying kinetics (1 m/s) of carrot cubes (side 8.5 mm) (García-Pérez et al. 2006b).

drying rates; high drying rates could not be attained by increasing the temperature due to the fact that products are prone to heating. Therefore, due to its mechanical effects, the use of power ultrasound may increase the drying rate without significantly heating the material and, as a consequence, without reducing its quality attributes. This issue is discussed in Section 13.6 in more detail.

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15.00

1.8 y = 0.203x + 6.52 R2 = 0.98

y = 0.017x + 1.02 R2 = 0.92

1.7 De (10 –10 m2/s)

De (10–10 m2/s)

13.00 11.00 9.00 7.00

1.6 1.5 1.4 1.3 1.2

5.00 0

20

40 60 P (kW/m3)

80

100

Figure 13.31. Influence of acoustic power on effective diffusivities identified fitting an ER model to US (31 kW/m3) drying kinetics (40°C, 1 m/s) of lemon peel slabs (thickness 10 mm).

The effect of power ultrasound on mass transfer phenomena was dependent on the acoustic power applied in the drying chamber, as shown by the ultrasonically assisted drying experiments of lemon peel slabs (40°C and 1 m/s) (Figure 13.31) (GarcíaPérez et al. 2009a). Acoustic power was defined as the electric power applied to the ultrasonic transducer divided by the volume of the drying chamber (vibrating cylinder). Effective diffusivities and mass transfer coefficients were identified by fitting an ER model to drying kinetics. The fact that effective diffusivity increased in line with an increase in the applied acoustic power is shown by the linear relationship in Figure 13.31 (García-Pérez et al. 2009a). The same behavior was found in the case of the mass transfer coefficient. Therefore, the higher the acoustic power, the lower the external and internal resistance to mass transfer. Similar results concerning the applied acoustic power have been found in applications in liquid media (Cárcel et al. 2007b, 2007c). Actually, the literature shows that the effects of power ultrasound are reduced at high acoustic powers, so that the proportionality shown in Figure 13.31 is lost and the effect disappears above a certain acoustic energy level. However, acoustic levels of

1.1 0

10

20 P (kW/m3)

30

40

Figure 13.32. Influence of acoustic power on mass transfer coefficients identified fitting an NER model to AIR and US drying kinetics (40°C, 1 m/s) of carrot cubes (side 8.5 mm) (García-Pérez et al. 2009a).

over 37 kW/m3 could not be tested in the ultrasonic application system used in this work. Additionally, drying experiments were carried out on carrot cubes under the same experimental conditions (40°C and 1 m/s) and at the same applied acoustic power. In this case, the influence of power ultrasound on the kinetic parameters of carrot drying did not appear until the acoustic power exceeded a minimum value. That means that a minimum acoustic energy level in the drying chamber was needed to affect effective diffusivity and mass transfer coefficient in the drying of carrot samples. This minimum level was found to be between 8 and 12 kW/m3 (Figure 13.32). Above this threshold, the values of the kinetic parameters were also proportional to the applied acoustic power (GarcíaPérez et al. 2009a). The existence of an acoustic energy threshold has already been stated in different applications of power ultrasound on the mass transfer process, such as osmotic dehydration (Cárcel et al. 2007b), meat brining (Cárcel et al. 2007c), and the convective drying of surimi (Nakagawa et al. 1996). The threshold was not observed for the effective moisture diffusivity values identified in the drying experiments

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Computational Study of Ultrasound-Assisted Drying of Food Materials

of lemon peel (Figure 13.31), as this value is probably lower than the minimum applied ultrasonic power. On the other hand, the slope of the linear relationship between effective diffusivity and acoustic power was much higher in lemon peel (Figure 13.31) experiments than in those of carrots (Figure 13.32) (0.084 vs. 0.017). This fact reveals a greater influence of the application of power ultrasound on the water mass transfer in lemon peel than in carrot drying. The fact that lemon peel and carrots behave in a different way when acoustic energy is applied can be explained by taking the structure into consideration. Indeed, lemon peel (porosity 0.4) is considered a more porous product than carrot (porosity 0.04) (Boukouvalas et al. 2006; García-Pérez et al. 2009a). Therefore, porosity may be considered one of the most important structural variables for determining the acoustic effectiveness in foodstuffs. Large intercellular spaces are found in high-porosity products, making the product more prone to alternating compression and expansion cycles produced by ultrasonic waves (sponge effect) (Gallego-Juárez 1998) and facilitating the water transfer through the solid. Small intercellular spaces, characteristic of low porosity products, produce higher resistance to internal water transfer. As a consequence, high acoustic energy levels are required to affect mass transfer in carrot, which is considered a low-porosity product. By comparing AIR (40°C, 1 m/s) and US experiments (40°C, 1 m/s, 37 kW/m3) of products of differing porosities, potato (0.037), lemon peel (0.303), and eggplant (0.504), the porosity effect was also observed in the reduction of the drying time. Ultrasonic application reduced the drying time of these products by 35, 50, and 75%, respectively. Therefore, the higher the product porosity, the greater the effect of power ultrasound on the drying kinetic. The effect of porosity could also be explained by considering a greater acoustic energy absorption in high-porosity products due to a larger gas volume, which may produce an increase in the internal energy available in the particles. This fact would lead to more intense compressions and expansions (sponge effect). Moreover, the acoustic effects on

297

Table 13.6. Influence of the acoustic power on drying time reduction, total energy usage, and energy saving for drying experiments of orange peel slabs (thickness 5.9 ± 0.4 mm) at 40°C and 1 m/s. Acoustic power (kW/m3) 0 6 12 18 25 30 37

Drying time (h)

Total energy (kWh)

Drying time reduction (%)

9.3 7.7 7.6 6.5 6.4 5.3 5.0

1.7 1.5 1.6 1.5 1.5 1.4 1.3

17.1 18.0 29.7 30.9 42.3 45.9

Energy saving (%) 10.2 4.3 12.2 7.9 18.3 18.9

the boundary layer of intercellular spaces could be more intense in high-porosity products due to larger pores in the food matrix. Indeed, these phenomena may also contribute to reducing internal resistance to mass transfer. The structure of the product to be dried may be one of the most important variables to be considered when addressing the ultrasonically assisted convective drying of foodstuffs. A previous analysis of product structure may contribute to establishing its response to acoustic energy and predicting the potential use of power ultrasound to improve mass transfer process and, therefore, reduce the energy cost involved in the drying process. In order to fully develop ultrasonic-assisted drying, it is essential to evaluate if the reduction of drying time due to ultrasound also leads to energy saving. This issue was investigated by measuring the total energy consumption using a digital power meter (FLUKE 430, Fluke Ibérica, Madrid, Spain). The main elements to be considered when quantifying the energy consumption were the heating elements, the ventilation system (fan), and the power ultrasonic generator. Table 13.6 reports the results obtained in ultrasonic-assisted drying experiments of orange peel carried out by applying different acoustic powers. As already mentioned, the reduction in drying time was dependent on the acoustic power: the higher the acoustic power, the faster the drying kinetic. The application of ultrasound led to an energy saving, which was also dependent on the

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acoustic power applied. Thus, the maximum energy saving was obtained for experiments carried out by applying 37 kW/m3. The ultrasonic influence on energy aspects was also evaluated from modeling by computing the second law of thermal efficiency (García-Pérez et al. 2008). Thus, in drying experiments of grape stalk, a significant increase in the thermal efficiency was obtained for experiments carried out at 40°C (0.0181 and 0.0289 in AIR and US experiments, respectively). The improvement in the thermal efficiency by the ultrasonic application was reduced when experiments were carried out at 60°C. Therefore, the effect of power ultrasound lessened at high temperatures, such as already mentioned for drying experiments of carrot (García-Pérez et al. 2006b). Therefore, these results suggest a potential use of power ultrasound on drying process carried out at low temperatures.

13.6. Future Trends This section states the future trends of the research in which the Power Ultrasonics (CSIC) and ASPA Groups (Polytechnic University of Valencia) are currently involved addressing ultrasonic-assisted drying. Two main topics will be addressed: the potential application of power ultrasound on drying processes at low temperatures, and specifically the atmospheric freeze drying, and the scaling up for industrial applications. To date, hot air drying is the most common method for dehydrating products at the industrial level due to high process rates, relatively low energy use, and low investment; however, it results in lowquality products. Vacuum freeze drying is an alternative to hot air drying in order to obtain high-quality products. At freezing conditions, degradation reactions slow down considerably and product shape is well-maintained. However, vacuum freeze drying requires batch operation and high investment costs because it operates at low pressures. For that reason, vacuum freeze drying is used mainly in products with high economical interest. Atmospheric freeze drying is considered an adequate alternative to vacuum freeze drying, as it facilitates the operation at atmospheric pressures. The operation consists mainly of circulating an air flow through the product

at temperatures below the thawing point; as a consequence, it allows continuous processing. As reported in the literature, atmospheric freeze drying provides dried products with similar quality attributes to vacuum freeze drying. Despite the strong interest in atmospheric freeze drying, the full development of this technology to date is limited, especially due to energy consumptions at long drying times and working below the freezing point at atmospheric pressures. Drying rates could be improved by adequately combining other energy sources during drying, such as microwave, infrared radiation, and radio frequency, which improve the drying rate by gentle heating. As discussed earlier, in comparison to these technologies, the effects associated with power ultrasound are not based on heating but on mechanical effects; therefore, its use minimizes quality loss otherwise caused during drying by heating. For that purpose, the Power Ultrasonics Group of the Instituto de Acústica de Madrid is developing a new kind of ultrasonic transducer at pilot plant scale. Their design will incorporate strategies to eliminate or mitigate modal interactions frequently produced at high-power operation as a consequence of the nonlinear behavior of the piezoelectric ceramics and materials. In parallel, new high-quality materials such as titanium and special aluminum alloys with high elastic performance are considered for use in the construction of more powerful cylindrical radiators. In addition, cylindrical radiators of higher dimensions (length and diameter) are also considered for development. However, cylindrical radiators will imply more complex studies due to the fact that such radiators will operate at higher modes, which increase the potential interferences between these modes and the working mode of the new transducers. To achieve these challenging goals, including the design of appropriate systems and the optimization of process conditions as well as the equipment, Multiphysics modeling is an essential tool to avoid costly and time-consuming trial-and-error experiments. The numerical simulations will allow the development of such systems at low cost and with superior performance, similar to what was shown in Section 13.2.

Chapter 13

Computational Study of Ultrasound-Assisted Drying of Food Materials

Acknowledgments

v1

The authors acknowledge the financial support from the projects DPI2009-14549-C04-01 and DPI2009-14549-C04-04.

R R1 r R S1

Notation a AC A1 c

c0 D1 D2 d.b. De F i K k k1, k2 L l, L M

p P Ur vC

Metal section thickness Sectional area of the ceramics Sectional area of the metal pieces Ceramic section thickness; acoustic wave speed Speed of sound in air Larger diameter of the stepped horn Smaller diameter of the stepped horn Dry basis Effective moisture diffusivity Applied fluid vector; structural loads Node position Stiffness matrix Mass transfer coefficient Peleg’s model parameters Half thickness Length of the tube Magnification factor of the amplitude vibration; mass matrix Acoustic pressure Acoustic power Vibration amplitude in radial direction Extensional-mode velocity of sound in the ceramic

m S2 m2 m2

SPL t W

m; m/s Wc m/s m

We x

Extensional-mode velocity of sound in the metal pieces Coupling matrix Inner radius of the tube Radial direction Radius Sectional area of the larger diameter Sectional area of the smaller diameter Sound pressure level Time Average moisture content Critical moisture content Equilibrium moisture content Axial direction

299

m/s

m2 m

m m2 m2 dB s kg water/kg dry matter kg water/kg dry matter kg water/kg dry matter

m kg water/kg dry matter m2/s

N/m kg water/m2/s

m m dimensionless; kg

Pa (kW/m3) m/s

Greek Symbols α αn β φair φe λ κ ρ ρ0 ρc ρ1 ρds τ

Weibull’s shape parameter Eigen values Weibull’s kinetic parameter Air relative humidity Equilibrium relative humidity Wavelength in the material Fluid bulk modulus Density of the fluid Density of the air Density of the ceramics Density of the metal pieces Dry solid density Local moisture content

ω

Angular resonant frequency

s

m Pa kg/m3 kg/m3 kg/m3 kg/m3 kg/m3 kg water/kg dry matter radian/s

References m/s

Abdullah A, Pak A. 2008. Correct prediction of the vibrational behavior of a high power ultrasonic transducer by FEM simulation. Int J Adv Manuf Technol 39:21–28.

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Abramov OV. 1998. High-Intensity Ultrasonics. Theory and Industrial Applications. The Netherlands: Gordon and Breach Science Publishers. ANSYS. 1992. Acoustic and fluid—Structure interaction, a revision 5.0 tutorial. ANSYS. Inc., June. ANSYS. 2007. Theory reference for ANSYS and ANSYS workbench, release 11.0. ANSYS, Inc., January. Arnold FJ, Mühlen SS. 2001. The resonance frequencies on mechanically pre-stressed ultrasonic piezotransducers. Ultrasonics 39:1–5. Azzouz S, Guizani A, Jomaa W, Belghith A. 2002. Moisture diffusivity and drying kinetic equation of convective drying of grapes. J Food Eng 55:323–330. Basunia MA, Abe T. 2001. Thin-layer solar drying characteristics of rough rice under natural convection. J Food Eng 47:295–301. Belford JF. 1960. The stepped horn. In: Proceedings National Electronics Conference, Chicago, 814–822. Blasco M, García-Pérez JV, Bon J, Carreres JE, Mulet A. 2006. Effect of blanching and air flow rate on turmeric drying. Food Sci Tecnol Int 12:315–323. Boukouvalas CJ, Krokida MK, Maroulis ZB, Marinos-Kouris D. 2006. Density and porosity: Literature data compilation for foodstuffs. Int J Food Prop 9:715–746. Cárcel JA, García-Pérez JV, Riera E, Mulet A. 2007a. Influence of high intensity ultrasound on drying kinetics of persimmon. Dry Technol 25:185–193. Cárcel J, Benedito J, Rosselló C, Mulet A. 2007b. Influence of ultrasound intensity on mass transfer in apple immersed in a sucrose solution. J Food Eng 78:472–479. Cárcel J, Benedito J, Bon J, Mulet A. 2007c. High intensity ultrasound effects on meat brining. Meat Sci 76:611–619. Cárcel JA, García-Pérez JV, Riera E, Mulet A. 2010. Improvement of convective drying of carrot by applying power ultrasound. Influence of mass load density. Dry Technol. DOI 10.1080/ 07373937.2010.483032. Chacón D, Rodríguez G, Gaete L, Riera E, Gallego-Juárez JA. 2006. A procedure for the efficient selection of piezoelectric ceramic constituting stacks in high-power ultrasonic sandwich transducers. Ultrasonics 44:517–521. Chou SK, Chua KJ. 2001. New hybrid drying technologies for heat sensitive foodstuffs. Trends Food Sci Technol 12:359–369. Crank J. 1975. The Mathematics of Diffusion, 2nd ed. Oxford, UK: Clarendon Press. Cravotto G, Boffa L, Mantegna S, Perego P, Avogadro M, Cintas P. 2008. Improved extraction of vegetable oils under highintensity ultrasound and/or microwaves. Ultrason Sonochem 15:898–902. Cunha LM, Oliveira FAR, Oliveira JC. 1998. Optimal experimental design for estimating the kinetic parameters of processes described by the Weibull probability distribution function. J Food Eng 37:175–191. Cunha LM, Oliveira FAR, Aboim AP, Frías JM, Pinheiro-Torres A. 2001. Stochastic approach to the modeling of water losses during osmotic and improved parameter estimation. Int J Food Sci Technol 36:253–262.

Einsner E. 1963. Design of sonic amplitude transformers for high magnification. J Acoust Soc Am 35(9):1367–1377. Ensminger D. 1973. Ultrasonics: The Low- and High-Intensity Applications. New York: Marcel Dekker Inc. Frederick R. 1965. Ultrasonic Engineering. New York: Wiley. Gallego-Juárez JA. 1997. High-power ultrasonic transducers. In: LA Crum, TJ Mason, JL Reisse, KS Suslick, eds., Sonochemistry and Sonoluminiscence, NATO ASI Series, Vol. 524, 259–270. Dordrecht: Kluwer Academic Publishers. Gallego-Juárez JA. 1998. Some applications of power ultrasound to food processing. In: MJW Povey, TJ Mason, eds., Ultrasound in Food Processing, pp. 127–143. Glasgow: Blackie Academic & Professional. Gallego-Juárez JA, Rodríguez-Corral G, Gálvez-Moraleda JC, Yang TS. 1999. A new high intensity ultrasonic technology for food dehydration. Dry Technol 17:597–608. Gallego-Juárez JA, Rodriguez G, Riera E, Campos C, Vázquez F, Acosta VM. 2000. A macrosonic system for industrial processing. Ultrasonics 38 N°1-8:331–336. Gallego-Juárez JA, Rodríguez G, Riera E, Vázquez F, Campos C, Acosta VM. 2002. Recent development in vibrating-plate macrosonic transducers. Ultrasonics 40:889–893. Gallego-Juárez JA, Riera E, De la Fuente S, Rodríguez G, Acosta VM, Blanco A. 2007. Application of high-power ultrasound for dehydration of vegetables: Processes and devices. Dry Technol 25(11):1893–1901. Gallego JA, Rodríguez G, Acosta VM, Riera E. 2010. Power ultrasonic transducers with extensive radiators for industrial processing. Ultrason Sonochem 17:953–964. García-Pérez JV, Cárcel JA, De la Fuente S, Riera E. 2006a. Ultrasonic drying of foodstuff in a fluidized bed. Parametric study. Ultrasonics 44:e539–e543. García-Pérez JV, Rosselló C, Cárcel JA, De la Fuente S, Mulet A. 2006b. Effect of air temperature on convective drying assisted by high power ultrasound. Defect Diffus Forum 258–260:563–574. García-Pérez JV, Cárcel JA, Benedito J, Mulet A. 2007. Power ultrasound mass transfer enhancement in food drying. Food Bioprod Process 85:247–254. García-Pérez JV, Cárcel JA, García-Alvarado M, Riera E, Mulet A. 2008. Effect of power ultrasound on second law thermal efficiency of food convective drying. In: 16th International Drying Symposium (IDS2008), Hyderabad, India. García-Pérez JV, Cárcel JA, Riera E, Mulet A. 2009a. Influence of the applied acoustic energy on the drying of carrots and lemon peel. Dry Technol 27:281–287. García-Pérez JV, Cárcel JA, García-Alvarado MA, Mulet A. 2009b. Simulation of grape stalk deep bed drying. J Food Eng 90:308–314. Hassan BH, Hobani AI. 2000. Thin-layer drying of dates. J Food Process Eng 23:177–189. Iula A, Vazquez F, Pappalardo M, Gallego JA. 2002. Finite element three-dimensional análisis of the vibrational behavior of the Langevin-type transducer. Ultrasonics 40: 513–517.

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Computational Study of Ultrasound-Assisted Drying of Food Materials

Jambrak AR, Mason TJ, Paniwnyk L, Lelas V. 2007. Accelerated drying of button mushrooms, Brussels sprouts and cauliflower by applying power ultrasound and its rehydration properties. J Food Eng 81:88–97. Karathanos VT, Belessiotis VG. 1997. Sun and artificial air drying kinetics of some agricultural products. J Food Eng 31:35–46. Khan MS, Cai C, Hung KC. 2002. Acoustics field and active structural acoustic control modeling in ANSYS. In: 2002 International ANSYS Conference Proceedings. Available at: http://www.ansys.com/events/proceedings/2002.asp. Lewis WK. 1921. The rate of drying of solids materials. Ind Eng Chem 13(5):427–432. López A, Iguaz A, Esnoz A, Virseda P. 2000. Thin-layer drying behavior of vegetable wastes from wholesale market. Dry Technol 18:995–1006. Mason TJ, Riera E, Vercet A, López-Bues P. 2005. Application of ultrasound. In: D-W Sun, ed., Emerging Technologies for Food Processing, 323–351. Amsterdam: Elsevier Academic Press. Merkulov LG. 1957. Design of ultrasonic concentrations. Soviet Phys Acoust 3:246–255. Merkulov LG, Kharitonov AV. 1959–60)Theory and analysis of sectional concentrators. Soviet Phys Acoust 5:183– 190. Morgan Electro Ceramics, Piezoelectric Ceramics. 2006. Properties and applications. Chapters 2: Physical basis; 6: Transducers, and 8: Testing PZT discs and plates. Available at: http://www.morganelectroceramics.com/pzbook.html. Mulet A. 1994. Drying modeling and water diffusivity in carrots and potatoes. J Food Eng 22:329–348. Mulet A, Cárcel J, Benedito J, Rosselló C, Simal S. 2003a. Ultrasonic mass transfer enhancement in food processing. In: J Welti-Chanes, J Vélez-Ruiz, G Barbosa-Canova, eds., Transport Phenomena in Food Processing, 265–278. EEUU. New York: CRC Press. Mulet A, Cárcel JA, Sanjuán N, Bon J. 2003b. New food drying technologies—Use of ultrasound. Food Sci Technol Int 9:215–221. Mulet A, Blasco M, García-Reverter J, García-Pérez JV. 2005. Drying kinetics of Curcuma longa rhizomes. J Food Sci 7:E318–E323. Nakagawa S, Yamashita T, Miura H. 1996. Ultrasonic drying of walleye Pollack surimi. Nipon Shokutin Kagaku Kagaku Kaishi 43:388–394. Neppiras EA. 1973. The pre-stressed piezoelectric sandwich transducer. In: Ultrasonics International Conference Proceedings, 295–302. Ortuño C, Pérez-Munuera I, Puig A, Riera E, García-Pérez JV. 2010. Influence of power ultrasound application on mass transport and microstructure of orange peel during hot air drying. Phys Procedia 3:153–159. Page GE. 1949. Factors influencing the maximum rates of air drying shelled corn in thin layers. MSc thesis, Purdue University, Lafayette, USA.

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Pan Z, Tangratanavalee W. 2003. Characteristics of soybeans as affected by soaking conditions. Lebensmittel Wissenschaft und Technologie 36:143–151. Peleg M. 1998. An empirical model for the description of moisture sorption curves. J Food Sci 53:1216–1219. Radmanovic MD, Mancic DD. 2004. Design and Modeling of the Power Ultrasonic Transducer. Le Locle, Switzerland: MP Interconsulting. Riera E, Golas Y, Blanco A, Gallego-Juarez JA, Blasco M, Mulet A. 2004a. Mass transfer enhancement in supercritical fluids extraction by means of power ultrasound. Ultrason Sonochem 11:241–244. Riera E, Rodríguez G, Vázquez F, De la Fuente S, Campos C, Gallego-Juárez JA, Mulet A. 2004b. Abstract Book of the 9th Meeting of the European Society of Sonochemistry (ESS9), April 25–30, Badajoz, Spain, 79–80. Rozenberg LD. 1969. Sources of High-Intensity Ultrasound, Vol. 2. New York: Plenum Press. Sánchez ES, Simal S, Femenia A, Benedito J, Rosselló C. 1999. Influence of ultrasound on mass transport during cheese brining. Eur Food Res Technol 209:215–219. Sanjuán N, Lozano M, García-Pascua P, Mulet A. 2004. Dehydration kinetics of red pepper (Capsicum annuum L var Jaranda). J Sci Food Agric 83:697–701. Sherwood TK. 1929. The drying of solids. Ind Eng Chem 21(1):12–16. Simal S, Rosselló C, Mulet A. 1998. Modeling of air drying in regular shaped bodies. Trends Chem Eng 4:171–179. Simal S, Mulet A, Catala PJ, Cañellas J, Rosselló C. 1996. Moving boundary model for simulating moisture movement in grapes. J Food Sci 61:157–160. Simal S, Femenia A, García-Pascual P, Rosselló C. 2003. Simulation of the drying curves of a meat-based product: Effect of the external resistance to mass transfer. J Food Eng 58:193–199. Simal S, Femenía A, Garau MC, Rosselló C. 2005. Use of exponential, Page’s and diffusional models to simulate the drying kinetics of kiwi fruit. J Food Eng 66:323–328. Sopade PA, Kaimur K. 1999. Application of Peleg’s equation in desorption studies of food systems: A case study with sago (Metroxylon sagu rottb.) starch. Dry Technol 17:975– 989. Van Randeraat S, Stterington RE. 1974. Piezoceramic Ceramics, 2nd ed. London: Mullard Ltd. Vega-Mercado H, Góngora-Nieto MM, Barbosa-Cánovas GV. 2001. Advances in dehydration of foods. J Food Eng 49:271–289. Wang J. 2002. A single-layer model far-infrared radiation drying of onion slices. Dry Technol 20:1941–1953. Wang L, Sun WD. 2003. Recent developments in numerical modeling of heating and cooling processes in the food industry—A review. Trends Food Sci Technol 14:408–423. Zheng L, Sun D-W. 2005. Ultrasonic assistance of food freezing. In: D-W Sun, ed., Emerging Technologies for Food Processing, 603–626. Amsterdam: Elsevier.

Chapter 14 Characterization and Simulation of Ultraviolet Processing of Liquid Foods Using Computational Fluid Dynamics Larry Forney, Tatiana Koutchma, and Zhengcai Ye

14.1. Introduction: UV Light Processing of Liquid Foods and Beverages U.S. Food and Drug Administration (FDA) approval of ultraviolet light (UV) as an alternative treatment to thermal pasteurization of fresh juice products (US FDA 2000) led to the growing research and interest in UV technology in the food industry as a low-cost nonthermal method. UV light inactivates microorganisms by damaging their nucleic acid, thereby preventing microorganisms from replicating. The use of UV light is well established for air and water treatment and surface decontamination. Mercury lamps, the dominant sources for UV treatment, are highly developed and provide good efficiency, long life, and compact size for various applications. Low-pressure mercury lamps (LPML) are monochromatic sources and easy to install and operate. They are readily available with a wellestablished and quantified emission spectrum at comparatively low cost. Medium-pressure lamps (MPM) have higher emission intensity in the UV-C range; however, the source is polychromatic. The lamp source operates at high temperatures and at higher electrical potential. Lamp material and enclo-

sures of MPM lamps age faster than LPM (Masschelein and Rice 2002). In addition to mercury lamps, special technologies lamps are promising due to instant start, robust packaging with no mercury in the lamp. Light sources such as pulsed lamps or excimer lamps hold promise for future food applications. Although recent advances in the science and engineering of UV light irradiation have demonstrated that this technology holds considerable promise as a nonthermal preservation method for liquid foods (egg whites or whole eggs), soft drinks and beverages (fresh fruit and vegetable juices, teas, milk), and ingredients (liquid sweeteners, whey, or cheese proteins), its commercial use is still limited (Altic et al. 2007; Singh and Ghaly 2007; Geveke 2008; Koutchma 2009). The challenges of UV processing of liquid foods and beverages are that, compared with water, they have a broad range of optical and physical properties, diverse chemical compositions that influence UV light transmittance (UVT), dose delivery, momentum transfer, and consequently microbial inactivation. Low and ultra-low UVT results in low penetration depth in the germicidal UV-C range and specifically at 253.7 nm. In

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

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addition, some food components are strong UV light absorbers. Other key factors that influence efficacy of UV treatment of low UVT liquids include UV reactor design and fluid dynamics parameters. The flow pattern strongly influences total delivered UV dose since the position and the residence time of the microorganisms in certain regions of the irradiance field can vary significantly. The UV reactors have to be designed in such a way as to ensure delivery to all volume of fluid close to the UV source. Thus, optimal design of UV reactors can reduce the interference of high UV absorbance and viscosity associated with some liquid products and therefore improve inactivation efficiency. Lastly, UV sources should be selected for each specific application to match the absorption spectra of the treated product and to increase UV light penetration. The implementation of computational fluid dynamics (CFD) codes allows development of numerical results for new industrial processes in food engineering and optimizing their efficiency and cost-effectiveness. The use of CFD in food processing has grown substantially in the past 10 years. Thus, it is no surprise that CFD can be used as a powerful tool to address some specific issues and concerns about UV processing of liquid foods based on history and experiences of applying CFD modeling in water treatment. CFD processing solves the governing equations across each cell by means of an iterative procedure to predict profiles of velocity and concentration. The use of CFD for the evaluation of UV reactor performance involves the calculation of flow, coupling of flow to UV light irradiation phenomena, and inactivation kinetics models. A few recent applications of CFD for low UVT liquids are reported in the literature and include evaluation of UV inactivation based on transport and irradiation phenomena (Ye et al. 2007); computation of fluence distribution and identification of the location of the least treated liquid or so-called dead spot in the reactor (Chiu et al. 1999; Unluturk et al. 2004); evaluation and comparison of the inactivation performance of various existing UV reactors; calculation of dimensions and determination of geometry; and particularly determination of the optimal gap width of annular reactors (Ye et al. 2008), as well as

biodosimetry validation studies (Koutchma and Parisi 2004). The key to successful application of CFD is to accurately characterize the input parameters to CFD such as liquid physical and optical parameters, microbial inactivation kinetics, and lamp UV output value. This chapter will review (1) critical characteristics of liquid foods in relation to UV treatment and CFD modeling, (2) features of microbial inactivation and dose delivery in annular and coiled UV reactors, (3) summary of CFD applications for modeling of flow patterns, and (4) optimization of microbial performance of the Taylor–Couette reactor. General guidelines for validation of CFD models will be proposed.

14.1.1. Characterization of Liquid Foods in Relation to UV Treatments Liquid food products and beverages have a wide range of physical, chemical, and optical properties. Each group of properties needs to be properly assessed in order to design the preservation process and optimize the inactivation performance of the UV reactor. Optical properties are the major factors impacting on UV light transmission and consequently microbial inactivation in liquid foods. Chemical composition, pH, dissolved solids (degree Brix), and water activity are considered hurdles that can modify UV inactivation efficacy. Physical properties (viscosity, density) influence the effectiveness of fluid momentum transfer and flow pattern. 14.1.1.1. UV Light Absorption in Liquid Foods Absorbance, absorption coefficient, penetration depth, and absorption spectra are critical optical characteristics of any liquid food that will undergo UV treatment (see also Chapter 2 of this book). The Lambert–Beers law (Equation 14.1) (Jagger 1967) is the linear relationship between absorbance (A), concentration of an absorber of electromagnetic radiation (c, mol/L), and extinction coefficient (ε, L/mol/cm) or molar absorptivity of the absorbing species, which is a measure of the amount of light absorbed per unit concentration

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absorbance or optical density, and path length of light (d, cm) A = ε ×c×d

(14.1)

The absorption coefficient (α), base e (αe) called Naperian absorption coefficient, or base 10 (α10), called the logarithmic coefficient, is also used in the calculations and is defined by Equation 14.2 as the absorbance divided by the path length (m−1) or (cm−1). The absorption coefficient is a function of the wavelength of the electromagnetic wave.

α e = 2.303 A / d

(14.2)

Penetration depth (λ) is the depth (cm) where the initial flux I0 drops by a specified percentage of its value at the quartz sleeve, for example, 95% or 99%. The penetration depth is defined by Equation 14.3

λ = 1/ α e

(14.3)

Scattering is the phenomenon that includes any process that deflects electromagnetic radiation from a straight path through an absorber when light waves interact with a particle. UV light scattered from particles is capable of killing microbes. Much of the scattered light is in the forward direction and is a significant portion of the transmitted UV light. The scattering phenomenon plays an important role in processing liquid foods with particles. The liquid itself and the concentration of the suspended units can be transparent if A << 1, opaque if A >> 2 or semitransparent for anything in between these extremes; that is, 1 < A < 2. In a majority of cases, liquid foods will absorb UV radiation. Experimental measurements are usually made in terms of transmittance of a substance (T, %), which is defined as the ratio of the transmitted to the incident light irradiance. As opposed to the absorption coefficient, which is a characteristic of the material only, the transmittance depends on the geometric dimensions of the material. A convenient way of presenting information about UVT of materials is to give the values of their absorption coefficient at various wavelengths. The transmittance of UV light through fresh juices and other liquid foods is low compared with water due to their high optical density. Often UVT is less than 1% and results from

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high UV light absorption and scattering. Accordingly, other terms used to characterize liquid foods in relation to UV treatment are “ultra-low and low UV transmittance liquids.” The variations of UV absorption coefficients and physicochemical properties among seven varieties of juices were reported by Koutchma et al. (2007). Apple juice with α10 = 25.9 cm−1 was characterized as semitransparent liquid, while carrot and orange juices were almost opaque liquids with α10 > 60 cm−1. The variety of fresh juices tested represented different pH groups, Brix levels, and viscosities. Apple and orange juices belonged to the high acid food group (pH < 3.7). Pineapple juice (pH 3.96) was within the group of acid or medium acid foods (3.7 < pH < 4.5), whereas carrot (pH 5.75) and watermelon (pH 5.19) juices were in the group of low acid foods (pH > 4.5). In terms of viscosity, apple and watermelon juice were less viscous, Newtonian fluid products, whereas carrot, orange, and pineapple juices were characterized by higher viscosity and non-Newtonian behavior. These differences showed not only the importance of a full characterization of liquid foods to design a proper preservation process but also the choice of the correct UV reactor to deliver a scheduled process. Due to very limited data on absorptive properties of liquid foods, beverages, and fresh juices, it is highly desirable that sufficient UV absorbance data will be properly measured, characterized, and reported. For validation of CFD models, it is important to understand that complicated optical phenomena can occur in liquid foods that will not follow the Lambert–Beer law. The absorbance measurement techniques should account for the presence of the suspended particles and their effect on the estimation of the absorbed UV dose.

14.1.2. Designs of UV Reactors and Dose Delivery A number of continuous flow devices were developed and validated for a variety of drinks and beverages ranging from raw milk, whey protein, exotic tropical juices, and nectars to the more common apple cider and apple juice. These UV reactors are

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commercially available or currently under development for use in the study of low-UVT treatment of liquids. The reactor designs include traditional annular, thin-film, static and dynamic mixers, and coiled tube devices. In annular UV reactors (with singular or multiple lamps) liquid is pumped through the gap formed by two concentric stationary cylinders. For example “ultradynamics” UV reactor (Ye et al. 2007) uses a single large quartz sleeve as the inner reactor wall with a surrounding metal cylinder as the outer reactor wall. Reports are available where annular type laminar reactors were used for treatment of apple juice and cider (Worobo 1999), as well as mango nectar and cheese whey (Guerrero-Beltran and Barbosa-Canovas 2006; Singh and Ghaly 2007). The length and gap size can vary depending on the type of treated liquids or flow rate. Thin-film reactors form a thin fluid layer (less than 1 mm), either between fixed boundaries or by flowing fluid on a fixed surface or by creating a “sheet” of fluid in air to decrease the path length and thus avoid problems associated with lack of penetration. Thin-film reactors are characterized by laminar flow with a parabolic velocity profile. Extensive research on the application of UV light for fresh apple cider by Worobo (1999) yielded a design and production model of a thin film with a 0.8-mm gap CiderSure UV reactor. UV treatment of orange juice was reported by Tran and Farid (2004) using a vertical single LPM UV lamp thin-film reactor. The thickness of the film was approximately 0.21–0.48 mm. Another commercial thin-film reactor is the PureUV/SurePure reactor that was used for treatment of apple juice, guava-and-pineapple juice, mango nectar, strawberry nectar, and two different orange and tropical juices (Keyser et al. 2008). This reactor is a single-lamp system with a thin fluid film formed between the lamp surface and a surrounding rippled or undulating outer wall. The reactor consisted of inlet, outlet chambers, and a corrugated spiral tube between the chambers. The tangential inlet created both a high velocity and turbulence (Re > 7,500) even at a minimum flow rate of 3,800 L/h in the inlet chamber, and liquid product

was brought into contact with the UV radiation while flowing in a gap between the sleeve and the spiral tube. Latros of Dundee, Scotland, is the best known example of the use of static mixers in UV treatment of low UVT fluids. In this design, 10 or more “half twist” static mixers are encased in a section of Teflon sleeve, with radiation applied by surrounding UV lamps. The static mixer consists of a series of alternating right- and left-hand helical elements with 180° rotations, each juxtaposed at 90° to the element preceding it. The static mixers achieve flow division, flow reversal, radial mixing, and axial differentiation of the fluid stream, which constantly changes the thin film at the inner wall of the pipe, thereby exposing more bacterial cells to the UV light during treatment. Tests of this reactor have been published in the academic literature. This unit reactor is repeated in series to achieve higher doses, and a pilot unit is available with three of these sections “folded” within a single treatment zone. Altic et al. (2007) reported UV light inactivation of Mycobacterium avium subsp. paratuberculosis in Middlebrook 7H9 broth and whole and semi-skim milk using a laboratory-scale UV device. Another type of static mixers is in so-called coiled tube UV reactors used to increase liquid delivery by enhanced mixing due to the Dean effect. Salcor Inc. (Fallbrook, CA) has promoted a UV reactor in which Teflon tubes are coiled in a helix, with 12 lamps inside and 12 lamps outside the helix. Fluid is pumped through the coiled tubing. The curved flow path can result in a pair of counterrotating vortices with their axis along the length of the coil. Koutchma et al. (2007) validated the performance of a coiled UV module 420 model (Salcor Inc., Fallbrook, CA) for fresh tropical juices pasteurization. Geveke (2005, 2008) processed apple cider and liquid egg white with a single lamp UV reactor surrounded by a coil of UV transparent Chemfluor tubing. A UV coil reactor was designed and used for the online sterilization of cheese whey (Singh and Ghaly 2007). Its microbial destruction efficiency was compared to the conventional annular UV reactor. Both reactors had the same geometry (840 mL volume

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and 17 mm gap size) and were tested at 11 flow rates. The flow was laminar in both reactors. The phenomenon of Dean flow was observed in the coil reactor, and Dean vortices resulted in higher microbial destruction efficiency in the coil reactor in a shorter retention time. The results showed that despite the high turbidity of cheese whey, it could be sterilized using UV radiation if the proper reactor design and flow rate are used. Ye (2007) investigated the inactivation efficiency of a Taylor–Couette flow reactor as an example of a dynamic mixer. Details of optimum performance for this reactor are discussed in Forney et al. (2008). This reactor has a stationary outer cylinder with a rotating inner cylinder. The rotation of the inner cylinder causes the formation of counter-rotating vortices with the axis of the vortex along the circumference of the cylinders and brings fluid from the dark outer cylinder to the high-intensity inner wall region. The microbial inactivation results are reported in terms of logarithmic reduction (log10) of the number of target pathogenic organism or total count while attempting to compare efficiency in terms of dose (mJ/cm2) or energy (mJ/L). However, more work is needed in the design of UV reactors capable of providing sufficient and uniform UV doses to all parts of the treated liquid. Quantitave tools and indicators need to be introduced to evaluate the efficacy of UV treatments in food matrices.

14.2. Mathematical Modeling of UV Processing of Liquid Foods Using CFD 14.2.1. CFD Modeling Approaches for the Simulation of UV Processing of Liquid Foods Most CFD codes contain three main elements: (1) a pre-processor, (2) a solver, and (3) a post-processor. 14.2.1.1. Pre-Processor The pre-processor defines the geometry of interest or computational domain. The latter is divided into a number of subdomains called cells or control volumes with fluid properties expressed at the center of each cell. Boundary conditions for the cells that touch the domain boundaries are specified as either solid walls, inlets,

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or outlets. Finally, the physical phenomena to be modeled in the computational domain such as laminar or turbulent flow are selected and the fluid properties such as density or viscosity are specified. The accuracy of a CFD solution is dependent on the fineness of the grid. Optimal meshes are often nonuniform and smaller in cell size in regions where large variations occur from cell to cell, for example, the fluid velocity near a solid wall. 14.2.1.2. Solver Most commercially available CFD codes use the finite volume method for solving the equations of fluid flow over all volume elements of the solution domain. This requires substitution of the partial differential equations by algebraic approximations representing flow processes such as convection, diffusion, and sources for each cell. The resulting system of coupled nonlinear algebraic equations that represent the conservation of mass, momentum, and microbes for each cell are then solved with an iterative method (see also other chapters of this book for description of various numerical methods). For example, the conservation of a general flow variable such as a velocity or microbe concentration within a finite control volume can be represented as a balance between various processes tending to either increase or decrease the magnitude of the variable. Thus, within the control volume (Versteeg and Malalasekera 1995) [ Rate of change of ϕ ] = [ Net flux of ϕ due to convection ] + [ Net flux of ϕ due to diffusion ] + [ Net rate of creation of ϕ ].

(14.4)

Commercial codes provide a number of solution procedures such as the SIMPLE algorithm to provide an iterative solution to the nonlinear equations. Since the accuracy of the solutions may depend on the grid size, commercial solvers will provide additional solutions for comparison by halving cell dimensions. 14.2.1.3. Post-Processor CFD software often also provides capabilities for data visualization tools

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such as domain geometry and grid display, shaded contour plots, and particle tracking, as well as functions to evaluate and integrate parameters over domains or boundaries in one, two, and three dimensions.

Quartz sleeve

Outlet

Stainless steel

UV lamp

14.2.2. Modeling of Flow Fields Since liquid foods are considered incompressible fluids, the flow field can be predicted by solving the governing equations for mass and momentum conservation (see also, e.g., Chapters 4, 5, 9, 11, and 12 for governing equations in CFD). In this chapter the discussion refers to Newtonian fluids where the viscous stresses are proportional to the rates of fluid deformation. In this case the fluid viscosity μ is a fluid property and is constant throughout the flow field independent of the local value of stress.

Fluid gap

14.2.2.1. Laminar If the fluid hydraulic Reynolds number Reh < 2,100, the flow is laminar in the reactor (Bird et al. 2002). Here, Re h = Dh u / ν

(14.5)

where u is the average fluid velocity through the reactor, ν = μ / ρ is the kinematic viscosity, μ is the fluid viscosity, ρ is the fluid density, and Dh is the hydraulic diameter. The hydraulic diameter is useful since many flow-through reactors have noncircular cross-sections. In all cases Dh = 4 A f / P , where Af is the cross-sectional area of the flow and P is the perimeter of Af. For example, in the case of flow through, a circular pipe Dh is equal to the pipe diameter D. However, if the geometry is defined by flow between concentric cylinders of diameter D2 > D1, then Dh = D2 − D1. Figure 14.1 represents a schematic of a laminar thin-film reactor where fluid passes from the inlet to the outlet through a fluid gap d = R2 − R1 (Ye et al. 2007, 2008; Forney et al. 2008). The inner radius of the quartz sleeve is R1 and the outer diameter of the stainless steel tube is R2. When a fluid containing microbes passes through the fluid gap, the local microbe concentration balance for N in units of viable microbes per unit volume can be determined for each cell in the numerical grid. The

Inlet Figure 14.1. Schematic of a thin-film annular reactor.

concentration distribution of viable microbes N(r,z) within the fluid gap can be determined numerically by solving the steady-state equation of the form ur

∂N ∂N ⎛ 1 ∂ ⎛ ∂N ⎞ ∂ 2 N ⎞ + uz = Dm ⎜ r + − kIN ⎝ r ∂r ⎜⎝ ∂r ⎟⎠ ∂z 2 ⎟⎠ ∂r ∂z (14.6)

where ur and uz are radial and axial velocity components, I(r) is the radiation intensity in units of mJ/cm2, and k is the microbe rate constant in units of cm2/mJ-s. Dm in Equation 14.6 represents the microbe diffusion coefficient in units of cm2/s. The terms on the left of Equation 14.6 represent the concentration terms, while the first term on the right is the diffusion term followed by the rate expression or source for the inactivation of the microbe. Numerical codes approximate the differential terms in Equation 14.6 by expressing all derivatives in algebraic form for each cell or control volume

Chapter 14

UV Processing of Liquid Foods Using Computational Fluid Dynamics

in the grid. This procedure provides a nonlinear algebraic approximation to Equation 14.6 that conserves N within each control volume as discussed in reference to Equation 14.4. For example, the term ∂N/∂r could be approximated as N (r , z ) − N (r − Δr , z ) Δr

(14.7)

on the inside boundary of the control volume where N(r,z) is the microbe concentration at the center of the control volume or cell located at the position r,z and Δr is the radial distance between adjacent cells. Numerical approximations of the velocities ur and uz in Equation 14.6 are also obtained by solving the laminar Navier–Stokes equations (see other chapters in this book for the incompressible Navier– Stokes equations) using a commercial software code. User defined functions are also necessary to process the radiation source terms for I(r) in Equation 14.3 with the same commercial software. 14.2.2.2. Turbulent If the hydraulic Reynolds number Reh > 2,100 for the reactor shown in Figure 14.1, the flow is turbulent (Bird et al. 2002). The conservation equation for the microbe in axisymmetrical coordinates can be written for the steadystate case in the form ur

∂N ∂N μ ⎛ 1 ∂ ⎛ ∂N ⎞ ∂ 2 N ⎞ + uz = t ⎜ − kIN ⎜r ⎟+ ∂r ∂z ρSct ⎝ r ∂r ⎝ ∂r ⎠ ∂z 2 ⎟⎠ (14.8)

where ur and uz are time-averaged radial and axial velocity components, and μt is the turbulent viscosity. These three variables, ur, uz, and μt, can be obtained by solving the time-averaged form of the Navier–Stokes equations with the same numerical procedure outlined for laminar flow. For example, one must define the grid of cells over the solution domain as done in the laminar case. The numerical code then defines algebraic expressions for conservation of both mass and momentum along with the conservation of microbes or Equation 14.8 for each cell in the domain.

309

In the case of turbulent flows, there are more variables than independent conservation expressions so that two additional approximate expressions are implemented in the numerical code. These two expressions conserve the turbulent kinetic energy κ and dissipation rate ε defined for each cell in the grid. Commercial numerical software commonly provides solutions with either the standard κ-ε model (Launder and Spalding 1972), the RNG κ-ε model (Yakhot and Orszag 1986), or the realizable κ-ε model (Shih et al. 1995). Similar to the solutions in laminar flow, user-defined functions must be developed for the radiation source term on the right of Equation 14.8. Moreover, the turbulent Schmidt number Sct, which relates the turbulent transfer of momentum to the turbulent transport of the microbe, is set at a constant such that Sct = 0.8 (Fox 2003). 14.2.2.3. Dean Flow Coiled tube geometries promote a secondary flow within the tube consisting of a pair of counter rotating vortices called the Dean effect. (Dean 1927, 1928; Fellouah et al. 2006). A schematic of a Dean reactor is shown in Figure 14.2. It was demonstrated by Dean (1927) that a secondary flow field develops in the fluid through a coiled tube because of the centrifugal forces acting on the fluid within the tube. The Dean number De is the similarity parameter governing fluid motion in such a flow configuration. The Dean number is calculated as De = Re( D / Dc )1 / 2 , where D is the tube diameter, Dc is the coil diameter, and Re is the tube Reynolds number Re = uD / ν ,

Outlet UV lamp

Teflon tube wound in helix pattern

Inlet

Figure 14.2. Schematic of Dean flow reactor.

310

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

where u is the average fluid velocity within the tube. Such secondary flows are promoted within the range of 0.03 < D/Dc < 0.1 and can be either laminar or turbulent for Re either less than or greater than 2,100, respectively. Numerical simulation of steady flow in a coiled tube will introduce centrifugal forces created by the circumferential motion. An additional source term on the order of

ρuz2 Rc

(14.9)

must be included in the conservation of radial momentum where Rc = Dc / 2. For turbulent Dean flow, one should consider using an advanced numerical model such as the RNG κ-ε or the realizable κ-ε models, which are better suited to simulate the effects of swirl. Numerical simulation of Dean flow requires calculation of the velocity components uz, ur, and uθ, where the last reflects the unsymmetrical effects of swirl. Microbe conservation expressions must also include uθ in both the convection terms and either the laminar or turbulent diffusion terms. The coupling between swirl and radial pressure gradients may lead to instabilities in the solution process. These instabilities can be reduced by introducing a sufficiently refined grid to resolve large gradients in pressure and swirl velocity. It may also be useful to calculate the flow field at low volume flow rates. These initial results can be used as input to additional calculations at larger flow rates. Additional problems occur with Dean flow with respect to the calculation of UV intensity. UV lamps can be placed at the center of the coiled tube or external to the coil as shown in Figure 14.2. In either case the UV radiation intensity is not symmetrical along the tube axis. Laboratory experiments should be performed with a small section of tubing with the same wall thickness and material as the prototype and at the same distance from the UV lamp to determine the radiation intensity over the tube crosssection. A mathematical model for the radiation intensity I(r,θ) must then be developed for use with the solution grid.

14.2.3. Modeling of UV Radiation Accurate predictions of microbial inactivation with UV reactors require detailed knowledge of the UV radiation levels within the reactor. This task is much easier if the UV radiation is applied in a symmetrical geometry. One useful example is the placement of the UV lamp at the center of the thin-film reactor in Figure 14.1. The geometry of Figure 14.1 ensures that the radiation exposure to the fluid is uniform along the axis of flow. Such a geometry simplifies the application of the kinetic rate term kI(r)N on the right of the microbe conservation expressions represented by Equations 14.10 and 14.11. For the geometry of Figure 14.1 radiation levels can be accurately expressed for all cells in the computational grid. The radiation intensity I(r), also called the fluence rate in units of mJ/cm2, is I (r ) = I o

R1 exp(−α (r − R1 )) r

(14.10)

for R1 ≤ r ≤ R2 where Io is the incident intensity at the inner wall r = R1 bounding the fluid gap, and R2 is the radius of the outer wall. The fluid absorbance α = 2.3 A where A is the base 10 absorption coefficient for the liquid. Therefore, the liquid absorption coefficients are defined by I = e −α x = 10 − Ax Io

(14.11)

It is useful to define the radiation penetration depth x = λ as the radiation path length at which the fluence rate is 10% of the incident fluence rate or I = Io (0.1). As shown in Equation 14.8, the radiation path length occurs when λA = 1 or λ = 1/A (Forney et al. 2008).

14.2.4. Modeling of UV Fluence The fluence distribution for microbes passing through a UV reactor is computed numerically by following a large number of microbes through the reactor while summing the number of photons incident on each microbe. These fluence distributions are useful since they can be used, for example, to

Chapter 14

UV Processing of Liquid Foods Using Computational Fluid Dynamics

compare the effects of different flow patterns within a reactor caused by laminar or turbulent conditions. To illustrate the computation of the UV fluence distribution, one can numerically track a large number of microbes through the reactor where the microbes are evenly distributed initially across the reactor inlet. Examples of fluence distributions are presented in Figure 14.3 for the laminar reactor shown in Figure 14.1. Here, the fluence distribution function E (log It ) is defined by the expression (Forney et al. 2004) ⎛ 1 ⎞ ⎛ dN ⎞ E (log It ) = ⎜ ⎝ N T ⎟⎠ ⎜⎝ d (log It ) ⎟⎠

(14.12)

In Equation 14.12, it is often practical to use the dimensionless fluence of It / I av , where the average fluence Iav of an ideal plug flow reactor (PFR) assumes a uniform velocity v such that R2

2π I av =

∫ I (r )rdr

R1

(14.13)

π (R − R ) 2 2

2 1

and t = L / u is the fluid residence time where L is the reactor length and u is the average fluid velocity or

A=0/cm A=3/cm A=6/cm A=12/cm

E(log10It)

8

u = Q / π ( R22 − R12 )

14.2.5. Modeling of Microbial Inactivation In this section the simple first-order inactivation model is compared with a useful series event model. Source terms are presented for each model. 14.2.5.1. First-Order Model The first-order inactivation model is the simplest model used to describe UV inactivation kinetics since the model depends on a single value of the microbe concentration N that can be related to the microbe concentration in neighboring cells. The model assumes that the inactivation rate changes with respect to pathogen concentration and fluence rate such that dN = − k1 IN dt

(14.14)

where k1 is first-order inactivation constant in units of cm2/mJ. The right side of Equation 14.14 would now appear in the microbe concentration Equations 14.6 and 14.8 written earlier representing the loss term for each cell in the numerical grid. 14.2.5.2. Series-Event Model The series event inactivation model (Severin et al. 1983) was proposed to account for the lag at low fluence by introducing a constant n, called the threshold level. It assumes that the inactivation of microorganism elements takes place in a stepwise fashion and the rate at each step is first-order with respect to fluence rate I,

12 10

311

6

dN i = k SE I ( N i −1 − N i ) dt

4 2 0 –5

–4

–3

–2

–1

0

1

2

3

log10It Figure 14.3. Dimensionless function.

UV

fluence

distribution

(14.15)

where kSE is the inactivation constant in the series event inactivation model and subscript i is event level. Here, kSE is assumed to be the same for different event levels. When n elements of the microorganisms (a threshold) have been inactivated, the microorganisms will become nonviable (Severin et al., 1983, 1984).

312

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

For example, if the threshold level is n = 4, then the number of viable microbes N at the reactor outlet is determined by summing the nonviable microorganisms N4 corresponding to n = 4 on each cell at the reactor outlet and setting ⎤ ⎛ 1⎞⎡ N = ⎜ ⎟ ⎢QN 0 − q cell N 4 ⎥ ⎝ Q⎠ ⎣ ⎦ cells

∑

(14.16)

Here, qcell is the volume flow rate of fluid for each cell at the reactor outlet and Q is the total flow rate through the reactor. A comparison of numerical computations for the laminar reactor in Figure 14.1 is shown in Figure 14.4.

14.2.6. Calculations of Optimum Geometry (Gap Width) To illustrate the calculation of the optimum geometry for a given reactor design, inactivation levels N/N0 are computed numerically for three possible flow patterns in the concentric cylinder geometry of Figure 14.1 (Ye et al. 2008). The three flow patterns are laminar, turbulent, and Taylor–Couette flows. Figure 14.5 is the comparison of log reductions among the three flow patterns where the radius of

the inner cylinder is 1.225 cm and series event inactivation model of Escherichia coli is used. The radius of the outer cylinder is changed from 1.235 to 1.74 cm in order to create different gap widths. According to Figure 14.5, laminar Taylor–Couette flow achieves a higher log reduction than either laminar or turbulent flow with the same dosage Ioτ and absorption coefficient α. For example, when α = 40 cm−1 and λ/d = 0.417, the inactivation levels N/N0 are 8.2 × 10−10, 8.1 × 10−3, and 0.27 for Taylor– Couette flow (Ta = 200), turbulent flow, and laminar flow, respectively. Both turbulent and laminar flows achieve poor inactivation levels when the absorption coefficients of juices are high and λ/d is small. Moreover, the optimum λ/d for Taylor–Couette flow is reduced to λ/d = 0.5 compared with an optimum value of λ/d = 1.0 for turbulent and λ/d = 1.5 for laminar flow. Thus, for a given juice, Taylor–Couette flow is suitable for disinfecting juices with high absorption coefficients and with larger gap widths d as shown in Figure 14.5. It should be noted that government regulations require an inactivation level of N/N0 = 10−5 to meet current standards. This value is achieved for the Taylor–Couette flow pattern at small λ/d ∼ 0.1 corresponding to d ∼ 2.5 mm. This

1 1.E+00

laminar turbulent Ta = 100 Ta = 200

0.1 1.E-01 1.E-02

0.01

1.E-03

N/N0 0.001

1.E-04

N/N0 1.E-05 0.0001

1.E-06 1.E-07

0.00001 0

20

40

60

Average Residence Time, s experiments, A = 3.45/cm 1st order, A = 3.45/cm

experiments, A = 6.02/cm 1st order, A = 6.02/cm

series-event, A = 3.45/cm

series-event, A = 6.02/cm

Figure 14.4. Comparison of E. coli microbial reduction experiments and numerical predictions.

1.E-08 1.E-09 1.E-10 0

0.5

l /d

1

1.5

Figure 14.5. Comparison of inactivation for three flow patterns. Dosage is I0τ = 2870 mJ/cm2 and absorption coefficient is α = 40 cm−1.

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UV Processing of Liquid Foods Using Computational Fluid Dynamics

313

Figure 14.6. Schematic of a Taylor–Couette reactor.

gap d would increase with smaller fluid absorption coefficients α < 40.

14.3. Application of CFD Modeling to Predict Microbial Performance of the Taylor–Couette Reactor The schematic of a Taylor–Couette reactor is shown as Figure 14.6. The stator of the Taylor–Couette reactor was constructed using PVC with a 15.42-cm

inner diameter and has a quartz rotor with a 14.76cm outer diameter corresponding to a gap width of 3.3 mm. The radiation source was a low-pressure mercury UVC lamp with 0.78 W/in.

14.3.1. Inner Cylinder Rotates While Outer Cylinder Is Stationary When fluid passes the annular gap between two concentric cylinders, flow patterns can be laminar

314

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

or turbulent Taylor–Couette flow (Taylor 1923; Lueptow et al. 1992) when the outer cylinder is fixed, the inner cylinder is rotating, and the rotating speed of the inner cylinder exceeds a certain value. Figure 14.7a,b is Lueptow’s results where mainly seven flow regimes are shown, including: 1. Taylor vortices. 2. Wavy vortices. 3. Random wavy vortices.

4. 5. 6. 7.

Modulated wavy vortices. Turbulent modulated wavy vortices. Turbulent wavy vortices. Turbulent vortices.

In Figure 14.7, the axial Reynolds number is defined as: Re axial =

U av dρ μ

(14.17)

(a) 40 HV

CP 30

TRA

HWV

Re

TWV RWV

20

TV

WV

10 LV

TMW 0

SHV 0

500

1000 MWV

2000

1500 Ta

2500

3000

(b) 40 Acronyms for flow regimes 30

HWV HV

Re 20

CP

RWV

10 LV

SHV

WV

CP HV HWV LV MWV RWV SHV TMW TRA TV TWV WV

Couette-Poiseuille flow Laminar helical vortices Laminar helical wavy vortices Laminar vortices Laminar modulated wavy vortices Random laminar wavy vortices Stationary helical vortices Turbulent modulated wavy vortices Transitional flow Turbulent vortices Turbulent wavy vortices Laminar wavy vortices

0 0

50 100 150 200 250 300 350 Ta

Figure 14.7. (a) Flow regimes for Taylor–Couette flow with an imposed axial flow. (b) Details of flow regimes for low Taylor numbers (Reprinted with permission from Lueptow et al., Physics of Fluids, Vol. 4, Issue 11, Page 2446–2455, 1992. Copyright 1992, American Institute of Physics).

Chapter 14

UV Processing of Liquid Foods Using Computational Fluid Dynamics

where the gap width d between two cylinders was chosen as the characteristic length. Taylor number is defined as: Ta =

R1Ω1 d υ

(14.18)

Because Equation 14.18 is similar to the Reynolds number, it is also called the rotating Reynolds number. However, since the effect of the radius ratio is not considered in Equation 14.18, the following form including the effect of the radius ratio can be defined as Ta =

R1Ω1 d ⎛ d ⎞ ⎜ ⎟ υ ⎝ R1 ⎠

12

(14.19)

which has been used more extensively (Coeuret and Legrand 1981; Sczechowski et al. 1995; Xue et al. 2002; Forney et al. 2003). The form of Equation 14.19 will be used in this chapter unless otherwise noted.

0.1600 0.1565 0.1530 0.1495 0.1460 0.1425 0.1390 0.1355 0.1320 0.1285 0.1250 0.1215 0.1180 0.1145 0.1110 0.1075 0.1040 0.1005 0.0970 0.0935 X 0.0900

When Reaxial = 5 and Ta = 200 (which is equal to Ta = 488 in Figure 14.7), the flow pattern is that of laminar wavy vortices. The path lines of fluid released from the annular gaps with Z = 0.095 m, Z = 0.125 m, and Z = 0.155 m (Z = 0 m is the base of the Taylor–Couette reactor) are shown in Figure 14.8. They show that laminar wavy vortices lose their axial symmetry and are three-dimensional flows. The path lines of fluid released from the annular gaps with Z = 0.115 m are displayed in the X-Y-plane in Figure 14.9. They are colored by particle ID and show that the fluid starting near the inner cylinder has chances to approach the outer cylinder, while the fluid starting near the outer cylinder has chances to approach the inner cylinder. This explains that Taylor–Couette flow has a more narrow dosage distribution than laminar and turbulent flows. By combining the path lines of Figures 14.8 and 14.9, the flow pattern of laminar wavy vortices in figure 7 (a) of Lueptow et al. 1992 can be inferred.

Z = 0.155

Z = 0.125

Z = 0.095 Z Y

315

Path Lines Colored by Z-Coordinate (m)

Figure 14.8. Path lines of fluid released from the annular gaps with Reaxial = 5 and Ta = 200. See color insert.

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

39 37 35 33 31 29 27 25 23 21 20 18 16 14 12 10 8 6 4 2 0

Zoom 5

1 2 43

12345

Y X

Z Path Lines Colored by Particle ID

Figure 14.9. Path lines of fluid released from the annular gaps displayed in the X-Y-plane. See color insert.

1

0.1

Re = 40 Re = 20 Re = 10 Re = 5, Turbulent Re = 5

N/N0 0.01

0.001

0.0001 0

100

200

300

400

Ta

Figure 14.10. Log reductions with different Ta and Reaxial.

Figure 14.10 shows calculated microbial log reductions with different Taylor number and Reaxial (residence time). For the curve of Re = 5, turbulent, the real Taylor number is 10 times Ta displayed as axis X in the figure, namely, the real Ta = 2,000,

3,000, and 4,000. The series event model with k = 0.67474 cm2/mJ and n = 4 is used for theoretical predictions. Figure 14.10 shows that log reductions mostly increase with Taylor number. Path lines of fluids in Taylor–Couette flow can be divided into two parts (Wereley and Lueptow 1999). One part is within the vortices and advances downstream with a uniform velocity; it will be called the vortex part. Another part is winding around the vortices and alternately flowing near the inner and outer cylinders; it will be called the winding part. The winding part should be as small as possible in order to achieve a narrow dosage distribution and better disinfection efficiency. The winding part mostly decreases with the increase in Taylor number. The latter results in the increase in log reductions. However, increasing Taylor number infinitely cannot always increase log reductions. The survival curve of Re = 5 in Figure 14.10 shows the optimum of microbial inactivation when Ta is between 200 and 300, whereas the curve of Re = 10 shows the log reductions approach a maximum value when Ta > 300. Furthermore, the flow pattern in the TaylorCouette reactor will change from laminar TaylorCouette flow to turbulent Taylor-Couette flow if the Taylor number exceeds a certain value. The survival curve at Re = 5, turbulent, in Figure 14.10 shows that log reductions in turbulent Taylor–Couette flow do not increase with increasing Taylor number. On the contrary, the log reductions in turbulent Taylor– Couette flow decrease with an increase in Taylor number. The latter is caused by axial turbulent mixing. Turbulent flow is a three-dimensional flow and intensifies mixing along all directions. Radial mixing is absolutely necessary to improve the performance of Taylor–Couette reactors. However, turbulent Taylor–Couette flow increases both the radial and axial mixing where the latter should be avoided. The axial mixing at turbulent Taylor–Couette flow is a type of back mixing and will reduce disinfection results. With the increase in Taylor number in turbulent Taylor–Couette flow, the flow pattern is approaching that of a continuous stirred tank reactor (Resende et al. 2004). By contrast, the axial mixing is negligible in laminar Taylor–Couette flow, which

Chapter 14

UV Processing of Liquid Foods Using Computational Fluid Dynamics

can be approximated by a PFR (Kataoka et al. 1975). Therefore, the Taylor-Couette reactors should be operated in laminar Taylor–Couette flow to achieve optimal inactivation performance.

14.3.2. Both Cylinders Are Stationary When both cylinders are stationary, the flow pattern in the annual gap can be laminar Poiseuille flow or turbulent flow depending on flow rates. Since the residence time in the current UV reactor configuration is too short for turbulent flow to achieve significant log reductions, only laminar flow will be addressed here. UV disinfection with turbulent flow can be found in Ye (2007) and Ye et al. (2008). Figure 14.11 shows log reductions with different residence time and λ/d, where λ is the penetration depth at which the fluence rate is 10% of the incident radiation fluence rate, or Iλ = 10% I0

(14.20)

The penetration depth will be the reciprocal of the absorption coefficient α (10 base) or λ = 1/A. The series event model with k = 0.67474 cm2/mJ and n = 4 is used for the theoretical predictions. Previous research (Ye 2007) already showed that the optimum gap width for laminar flow is λ/d = 1.5. When λ/d is near the optimum gap width, for

1 0.1 0.01

N/N0

l /d = 0.25 l /d = 0.50 l /d = 0.76 l /d = 1.01

0.001 0.0001 0.00001

0.000001 20

60

100

140

Average Residence Time, s

Figure 14.11. E. coli log reductions with different residence time and λ/d.

317

example, λ/d = 0.76 and 1.01 in Figure 14.11, the log reductions can easily reach 5 (required inactivation performance) by increasing residence time. On the other hand, when λ/d is far away from the optimum gap width, for example, λ/d = 0.25 and 0.5 in Figure 14.11, the log reductions increase very slowly by increasing the residence time. Therefore, the gap width of UV reactors operating in laminar Poiseuille flow has to be designed near its optimum gap width.

14.4. Validation of CFD Models CFD modeling for evaluation of UV reactor performance involves modeling of flow, coupling of flow to UV light irradiation, and microbial inactivation. The key to success of the application of CFD is biodosimetrical validation. Biodosimetry validation is the final check of the accuracy of sub-models such as fluids and light irradiation, inactivation kinetics model, model input parameters, and their mathematical processing. Validated CFD models need to demonstrate consistent agreement with experimentally measured biodosimetry over the entire range of flow rates and UV lamp power levels. According to water treatment validation protocols, ±0.2 log of variation in microbial reduction is normally expected. Therefore, validated CFD models do not match experimental results by biodosimetry perfectly. It is important to note that experimentally measured microbial reduction is associated with uncertainties and errors like any other experimental method. According to protocols (Trojan Technologies Inc and Fluent Inc. 2006) the CFD results should be in agreement with biodosimetry to within 20%. Quality control of CFD modeling should be managed by ensuring that physical variables are accurate and modeling variables are robust. The following CFD checklist is recommended to ensure the use of proper model inputs. Physical variables include dimensions of reactor, flow rate, physical properties of fluid such as density and viscosity, UVT, action spectrum of organisms, lamp power, spectrum, efficiency, sleeves UV transmission, inactivation kinetics, boundary conditions, and CFD model setup.

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

The following examples show how CFD modeling was validated to evaluate effect of UV lamps output power on inactivation performance of the annular UV reactor and how the validated CFD model was used to access the delivered dose in commercial scale UV reactor.

14.4.1. Efficacy of High-Intensity Low-Pressure Amalgam Lamps (LPAL) for Inactivation of Food Pathogens Gorlitt (2007) studied the effects of the output power of the high-intensity LPAL using a bench scale single tube annular UV reactor. The reactor was assembled with two lamps with nominal output power of either 320 W or 450 W. According to the manufacturer (American Ultraviolet Co, Lebanon, MI) 90% of UV output was within the range 233.7 to 273.7 nm. The lamp arc length is 149 cm with an outer diameter (OD) of 19 mm. The lamp’s sleeve is clear fused quartz circular tubing with a transmission rate of 94%. After the UV lamps were turned on for 10 minutes, an IL 1700 Research Radiometer equipped with an SUD240 detector (International Light, Inc., Newburyport, MA) was used to measure the incident irradiation fluence rate. The fluence rate was measured along the entire length of the quartz sleeve and averaged. The obtained value of the average irradiation fluence rate on the surface of the quartz sleeves was 49.7 mW/cm2 and 60.4 mW/cm2 for 320 W and 450 W LPAL, respectively. The measured values were used in the calculations of UV fluence. A water–caramel–syrup solution with absorption coefficients α ranging from 1.58 to 4.42 cm−1 was used to mimic the low UVT fluids. The model solutions were inoculated with E. coli K12 and treated by UV light at the flow rates of 15, 38, and 49 mL/s, respectively. The calculated Re numbers were between 104 and 341, indicating a laminar flow regime in the reactor. The measured inactivation of E. coli was compared with the results calculated using the first-order model and the series-event inactivation model. The velocity, residence time, UV fluence rate, and UV fluence distribution along the

reactor radius were computed for each flow rate, absorption coefficient, and UV lamp intensity. The effect of the variation of absorption coefficient of the model solution on reduction of E. coli K12 was measured first when the 320-W and 420-W output power LPAL were installed. The difference in microbial inactivation between 320 and 450 W LPALs was less than 0.56 logs for all tested model solutions. The largest difference in the reduction of E. coli K12 equal to 0.9 logs was found in the model solution with an absorption coefficient of 3.52 cm−1. A one-way ANOVA statistical analysis confirmed that there was no significant difference (α = 0.05) in inactivation performance between lamps in the range of absorbancies tested except for α = 3.524 cm−1. Next, the first-order model and the series-event model were evaluated for predictions of reduction of E. coli K12 in the UV reactor. The inactivation rate constant k = 0.325 cm2/ mJ reported by Ye et al. (2007) was used in calculations. After comparison of predicted data with experimental results, it was evident that both the models underestimated microbial inactivation. However, the first-order model was found to be more accurate for the prediction of LPAL at 320-W output power in the range of absorption coefficients of 2.6 and 3.5 cm−1. Figure 14.12 shows the performance of the firstorder inactivation model in predicting log reduction of E. coli K12 for the range of absorptions coefficients when a 320-W UV source was used. The

320 W UV LPAL Calculated log reduction

318

4 3.5 3 2.5 2 1.5 1 0.5 0 –0.5

y = 0.9586x – 0.267 R2 = 0.9522

0

1

2 3 Measured log reduction

4

Figure 14.12. Comparison of experimental and calculated reduction of E. coli using first-order reaction model.

Chapter 14

UV Processing of Liquid Foods Using Computational Fluid Dynamics

difference between measured and calculated microbial reduction did not exceed 20% for both models. It was concluded that both models can be used to calculate microbial inactivation for a range of low UVT of model solutions tested. However, the models have to be validated to predict inactivation in food matrices, taking into account physical and chemical properties (pH, water activity) of real foods.

14.4.2. Application of CFD Modeling to Evaluate a Delivered UV Dose by Biodosimetry Biodosimetry or bioassay is a practical method commonly used for measuring the UV dose delivery in a UV reactor. Biodosimetry is a biological approach since inoculated liquid product is passed through the reactor. Inactivation of microorganisms is determined by comparing the concentration of viable microorganisms in the samples taken from the reactor ’s influent (N0) and effluent (NF). The relationship between UV dose and specific log survival of the target bacteria is first developed using a collimated beam (CB) LPML in a static UV system. The CB test procedure is not appropriate for low UVT fluids due to a nonuniform fluence in the sample. A correct dose measurement would result from biodosimetry only in the case that all microorganisms receive the same dose in an ideal reactor. In addition, the CB is not perfectly parallel. Ye et al. (2007) proposed a novel method to overcome the disadvantages of the traditional CB approach for liquid foods to measure UV inactivation using a single lamp annular UV reactor. This approach and CFD modeling was used to determine a delivered dose in a multiple lamp annular UV reactor designed to pasteurize liquid food sweeteners. In order to develop the relationship between UV dose and specific log survival, a laboratory-scale single lamp annular reactor was designed and built. It was operated with a lowpressure amalgam arc lamp (LPAL) light (90% of the lamp’s output is from 233 to 273 nm) that was similar to the lamps used in a multiple-lamp prototype commercial-scale reactor and a similar flow regime. Tests were conducted to determine the level

319

of inactivation of Bacillus subtilis ATCC 6633 spores after one pass of the liquid sweetener through the annular reactor. Flow rates ranged from 151.2 to 308.7 mL/s to construct a dose–response curve. The inactivation constant of B. subtilis was determined by CFD modeling and fitting experimental data using first-order and series-event reaction models as was reported by Ye et al. (2007). When eight experimental points of B. subtilis inactivation data were fitted by the series-event inactivation model with threshold n = 4, the obtained value of the inactivation constant was kSE = 0.094 cm2/mJ with R2 = 0.97. The resulting values of the decimal reduction dose of B. subtilis spores were in the range of 23.86 to 24.47 mJ/cm2. The UV dose for a one log reduction of B. subtilis ATCC 6633 reported by Chang et al. (1985) was 36 mJ/cm2. Sommer et al. (1999, 2000) reported the dose of 20–22 mJ/cm2 for a one log reduction of this strain. CB procedure was used in both studies. The measured D10 dose using an annular UV reactor in food model system was comparable with earlier reported data (Cairns 2006). After proving that B. subtilis could be inoculated into a sweetener solution, held, and recovered under all processing conditions excluding the UV treatment, liquid inoculated at 105 CFU/mL solutions were UV treated in a prototype commercial UV reactor containing seven low-pressure, highintensity amalgam lamps. Microbial reduction was accomplished by utilizing UV at various flow rates of up to a maximum average laminar flow velocity of 0.22 m/s and a minimum residence time of 3.70 s. CFD modeling was used to predict spore survival distribution as shown in Figure 14.13. The reactor obtained a 5.1-log10 reduction of B. subtilis spores. The UV dosage delivered to the solution was determined as 121.7 mJ/cm2.

14.4.3. Validation of CFD Modeling for the Fluid Flow Field CFD simulation of UV processing of liquid foods as discussed earlier requires both an accurate representation of the fluid flow field and a detailed summary of the UV dosage intercepted by all microbes. A useful procedure to validate the CFD simulation of

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6.00e+00 5.70e+00 5.40e+00 5.10e+00 4.80e+00 4.50e+00 4.20e+00 3.90e+00 3.60e+00 3.30e+00 3.00e+00 2.70e+00 2.40e+00 2.10e+00 1.80e+00 1.50e+00 1.20e+00 9.00e–01 Y 6.00e–01 X 3.00e–01 Z Contours of long outlet 0.00e+00 Figure 14.13. Spore survival distribution at the exit of a multiple-lamp UV reactor. See color insert.

the fluid flow field is to compute the residence time distribution (RTD) for the flow-through device. Most commercial CFD codes provide instruction to determine the RTD by following a large number of point particles distributed evenly across the entrance to the UV reactor. The CFD code then produces a distribution of resident times for the collection of particles for any reactor design. The CFD-simulated RTD can be compared with the experimental RTD for the UV reactor. Here, the experimental RTD is determined by injecting a pulse of a UV-sensitive dye at the reactor inlet and recording the distribution of dye at the reactor outlet. Comparison of the simulated RTD and experimental values would be useful for validating the CFD simulation. The simulated RTD can also be used directly in place of experimental values in the segregation model of UV reactor performance as illustrated recently (Koutchma et al. 2009) for the Dean flow reactor.

14.5. Future Research Needs for Understanding of Performance of UV Reactors through CFD Modeling for Food Applications UV light processing can be a viable nonthermal alternative for pasteurizing liquid foods, ingredients,

and beverages. There is sufficient evidence of the success of UV treatment for eliminating or reducing the levels of most types of undesirable microorganisms reported in the literature. Compared with other nonthermal processing methods, UV treatment has received less attention. Little is known about the interaction of UV light with a complex food matrix that can often be referred to as radiative transfer in a semi-transparent or opaque medium. Within a food product, several factors such as low UVT, pH, water activity, and food matrix can influence the delivery of the UV dose. Changes and variations in UV absorbance need to be considered in the design of the preservation process and reactor performance. The standard approach for measuring absorbance and absorption coefficients of low UVT liquids should be determined. The method for measuring and verifying UV dose in food matrices is not yet established. A number of UV light continuous flow systems that included annular laminar and turbulent flow reactors, thin-film devices, and static and dynamic mixers were developed and validated for pasteurization of milk and a variety of beverages ranging from exotic tropical juices to the more common apple cider and apple juice. However, more CFD modeling work is needed in the design of UV reactors capable of providing sufficient UV doses to all parts of the treated liquid and inactivating microorganisms to a sufficient extent. The optimal UV reactor design can reduce the interference of high UV absorbance and viscosity associated with some liquid products and can thereby improve the inactivation efficacy. CFD is a powerful tool for addressing issues related to the application of UV in pasteurization of liquid foods and beverages. Mathematical modeling can be used to improve the efficacy of UV light pasteurization by simulating and optimizing process performance and result in the design of UV reactors with matching irradiation sources. However, CFD input parameters must be measured properly and CFD models must be validated by biodosimetry. The results of CFD modeling of UV inactivation in laminar Poiseuille, turbulent, and laminar Taylor–Couette flow for single- and multiple-lamp

Chapter 14

UV Processing of Liquid Foods Using Computational Fluid Dynamics

concentric reactors demonstrated that laminar Taylor–Couette flow and Dean flow reactors resulted in superior inactivation performance. The optimum ratios of the UV penetration depth to gap widths (λ/d), which are critical design parameters, were obtained and were 1.5, 1, and 0.5 for laminar, turbulent, and laminar Taylor flow reactors, respectively. The optimum value of gap width can be found for each value of absorption coefficient and reactor. Additionally, CFD can be used for biodosimetry validation and address regulatory issues within operating parameters.

Notation Latin Symbols A Af C D Dm Dc De d E (log It ) I0 I(r) Iav i N(r,z) k kSE k1 L N0 Ni

Absorbance Cross-sectional area of the flow (m2) Concentration of absorber (mol/L) Hydraulic diameter (m) Microbe diffusion coefficient (cm2/s) Coil diameter (m) Dean number Gap width (m) Fluence distribution function (mJ/cm2) Incident intensity at the inner wall (mW/cm2) Radiation intensity in units (mW/cm2) Average fluence (mJ/cm2)N microbial concentration (CFU/mL) Event level Microbial concentration at the position r,z (CFU/mL) Microbial rate constant in units (cm2/mJ-s) Microbial rate constant in the series-event inactivation(cm2/mJs) Microbial rate constant in the first order inactivation (cm2/mJ-s) Reactor length (m) Initial microbial concentration (CFU/mL) Concentration of viable microbes at the event i

N(r,z) n P Q qcell Δr R1 R2 Rc Re Sct Ta t = L/v ur uz uθ v x z

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Concentration distribution of viable microbes (CFU/mL) Threshold level Perimeter of Af (m) Fluid volume flow rate (m3/s) Volume flow rate of fluid for each cell at the reactor outlet (m3/s) Radial distance between adjacent cells (m) Inner radius of the quartz sleeve (m) Outer diameter of the tube (m) Coil radius (m) Reynolds number Turbulent Schmidt number Taylor number Fluid residence time (second) Radial velocity component (m/s) Axial velocity component (m/s) Unsymmetrical swirl velocity component (m/s) Average fluid velocity (m/s) Penetration depth (m) Axial coordinate (m)

Greek Symbols α ε ρ ν μ μt λ Ω1

E base absorption coefficient (cm−1) Molar extinction coefficient (L/mol/cm) Fluid density (kg/m3) Kinematic fluid viscosity (m/s2) Dynamic fluid viscosity (Pa s) Turbulent viscosity (Pa s) Radiation path length (m) Angular velocity of inner cylinder (1/s)

References Altic LC, Rowe MT, Grant IR. 2007. UV light inactivation of Mycobacterium avium subsp. Paratuberculosis in milk as assessed by FASTPlaque TB phage assay and culture. Appl Environ Microbiol 2007:3728–3733. Bird RB, Stewart WE, Lightfoot EN. 2002. Transport Phenomena, 2nd ed. New York: Wiley. Cairns B. 2006. UV dose required to achieve incremental log inactivation of bacteria, protozoa and viruses. IUVA News, 8(1):38–45.

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Chang JCH, Ossoff SF, Lobe DC, Dorfman MH, Dumais CM, Qualls RG, Johnson JD. 1985. UV inactivation of pathogenic and indicator microorganisms. Appl Environ Microbiol 49(1):361–1365. Chiu K, Lyn DA, Savoye P, Blatchley ER. 1999. Effect of UV system modification on disinfection performance. J Envir Eng 125:7–16. Coeuret F, Legrand J. 1981. Mass transfer at the electrodes of concentric cylindrical reactors combining axial flow and rotation of the inner cylinder. Electrochim Acta 26:865–872. Dean WR. 1927. Note on the motion of fluid in a curved pipe. Philos Mag 4:208–223 Sp. Iss. 7th Series. Dean WR. 1928. The stream-line motion of fluid in a curved pipe. Philos Mag 5:673–695 Sp. Iss. 7th Series. Fellouah H, Castelain C, Moctarand OE, Peerhossaini H. 2006. A criterion for detection of the onset of Dean instability in Newtonian fluids. Eur J Mech B Fluids 25:505–531. Forney LJ, Goodridge CF, Pierson JA. 2003. Ultraviolet disinfection: Similitude in Taylor-Couette and channel flow. Envir Sci Technol 37:5015–5020. Forney LJ, Pierson JA, Ye Z. 2004. Juice irradiation with TaylorCouette flow: UV inactivation of Escherichia coli. J Food Protect 67:2410–2415. Forney LJ, Ye Z, Koutchma T. 2008. UV disinfection of E. coli between concentric cylinders: Effects of the boundary layer and a wavy wall. Ozone Sci Technol 30:405–412. Fox RO. 2003. Computational Models for Turbulent Reacting Flows. Cambridge: Cambridge University Press. Geveke D. 2005. UV inactivation of bacteria in apple cider. J Food Protect 68(8):1739–1742. Geveke D. 2008. UV inactivation of E. coli in liquid egg white. Food Bioproc Technol 1(2):201–206. Gorlitt M. 2007. Experimental and theoretical evaluation of performance of the single tube annular UV reactor for high absorptive liquid models. Master of Science in Food Process Engineering, Illinois Institute of Technology. Guerrero-Beltran J-A, Barbosa-Canovas G-V. 2006. Reduction of Saccharomyces cerevisiae, Escherichia coli and Listeria innocua in apple juice by ultraviolet light. J Food Process Eng 28:437–452. Jagger J. 1967. Introduction to Research in UV Photobiology. Englewood Cliffs, NJ: Prentice-Hall, Inc. Kataoka K, Doi H, Hongo T, Futagawa M. 1975. Ideal plugflow properties of Taylor vortex flow. J Chem Eng Jpn 8: 472–476. Keyser M, Mullera I, Cilliersb F-P, Nelb W, Gouwsa P-A. 2008. UV radiation as a nonthermal treatment for the inactivation of microorganisms in fruit juices. Innovat Food Sci Emerg Technol 9(3):348–354. Koutchma T. 2009. Advances in UV light technology for nonthermal processing of liquid foods. Food Bioproc Technol 2(2):138–155. Koutchma T, Parisi B. 2004. Biodosimetry of Escherichia coli UV inactivation in model juices with regard to dose distribution in annular UV reactors. J Food Sci 69(1):E14–E22.

Koutchma T, Parisi B, Patazca E. 2007. Validation of UV coiled tube reactor for fresh fruit juices. J Envir Eng 6:319–328. Koutchma T, Forney LJ, Moraru C. 2009. Ultraviolet Light in Food Technology: Principles and Applications. Boca Raton, FL: Taylor & Francis/CRC Press. Launder BE, Spalding DB. 1972. Lectures in Mathematical Models of Turbulence. London, England: Academic Press. Lueptow RM, Dotter A, Min K. 1992. Stability of axial flow in an annulus with a rotating inner cylinder. Phys Fluids 4:2446–2455. Masschelein WJ, Rice RG. 2002. Ultraviolet Light in Water and Wastewater Sanitation. Lewis Publication. Resende MM, Vieira PG, Sousa R, Giordano RLC, Giordano RC. 2004. Estimation of mass transfer parameters in a TaylorCouette-Poiseuille heterogeneous reactor. Braz J Chem Eng 21:175–184. Sczechowski JG, Koval CA, Noble RD. 1995. A Taylor vortex reactor for heterogeneous photocatalysis. Chem Eng Sci 50:3163–3173. Severin BF, Suidan MT, Engelbrecht RS. 1983. Kinetic modeling of UV disinfection of water. Water Res 17:1669–1678. Severin BF, Suidan MT, Rittmann BE, Engelbrecht RS. 1984. Inactivation kinetics in a flow-through UV reactor. J WPCF 56:164–169. Shih T-H, Liou WW, Shabbir A, Yang Z, Zhu J. 1995. A new k-ε eddy-viscosity model for high Reynolds number turbulent flows—Model development and validation. Computers Fluids 24:227–238. Singh JP, Ghaly AE. 2007. Effect of flow characteristics on sterilization of cheese whey in UV reactors. Appl Biochem Biotechnol 142(1):1–16. Sommer R, Cabaj A, Sandu T, Lhotsky M. 1999. Measurement of UV radiation using suspensions of microorganisms. J Photochem Photobiol 53(1–3):1–5. Sommer R, Lhotsky M, Haider T, Cabaj A. 2000. UV inactivation, liquid-holding recovery, and photoreactivation of Escherichia coli O 157 and other pathogenic Escherichia coli strains in water. J Food Protect 63:1015–1020. Taylor GI. 1923. Stability of a viscous liquid contained between two rotating cylinders. Proc R Soc London Ser A Math Phys Sci 223:289–343. Tran M-T, Farid M. 2004. Ultraviolet treatment of orange juice. Innovative Food Sci Emerg Technol 5(4):495–502. Trojan Technologies Inc, Fluent Inc. 2006. Using computational fluid dynamics in the validation of site-specific installations of UV disinfection systems. White paper. U. S. Food and Drug Administration. 2000. 21 CFR Part 179. Irradiation in the production, processing and handling of food. Fed Regist 65:71056–71058. Unluturk S, Koutchma T, Arastoopour H. 2004. Modeling of UV dose distribution in a thin film UV reactor for processing of apple cider. J Food Process 65(1):125–136. Versteeg HK, Malalasekera W. 1995. An Introduction to Computational Fluid Dynamics—The Fluid Volumes Method. England: Longman Scientific & Technical.

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Wereley ST, Lueptow RM. 1999. Velocity field for Taylor– Couette flow with an axial flow. Phys Fluids 11:3637–3649. Worobo R. 1999. Efficacy of the CiderSure 3500. Ultraviolet light unit in apple cider. CFSAN Apple cider food safety control workshop. Xue W, Yoshikawa K, Oshima A, Nomura M. 2002. Continuous emulsion polymerization of vinyl acetate. II Operation in a single Couette-Taylor vortex flow reactor using sodium lauryl sulfate as emulsifier. J Appl Polym Sci 86:2755–2762. Yakhot V, Orszag SA. 1986. Renormalization group analysis of turbulence: I. Basic theory. J Sci Comput 1:1–51.

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Ye Z. 2007. UV disinfection between concentric cylinders. PhD thesis, Georgia Institute of Technology, Department of Chemical and Biomolecular Engineering. Ye Z, Koutchma T, Parisi B, Larkin J, Forney LJ. 2007. Ultraviolet Inactivation Kinetics of E. coli and Y. pseudotuberculosis in Annular Reactors. J Food Sci 72:E271–E278. Ye Z, Forney LJ, Koutchma T, Giorges AT, Pierson JA. 2008. Optimum disinfection between concentric cylinders. Ind Eng Chem Res 47:3444–3452.

Chapter 15 Multiphysics Modeling of Ultraviolet Disinfection of Liquid Food—Performance Evaluation Using a Concept of Disinfection Efficiency Huachen Pan

15.1. Introduction Ultraviolet (UV) disinfection reactors are used for many disinfection applications such as for drinking water, waste water, and food processing. They are devices that use UV radiation to inactivate the microorganisms in the fluid flowing through the reactor. To design a reactor with high disinfection performance and low energy consumption is the goal of mechanical design optimization. UV disinfection is used in the food industry in several ways. It can be used for disinfection of liquid and solid (powder) foods (Adhikari et al. 2005; Keyser et al. 2008; Lszl et al. 2008), for workshop air cleaning (Wirtanen et al. 2002), and for waste water purification (Mavrov et al. 1997) after food processing. In this chapter the focus is on UV reactors for liquid food disinfection. The application of UV disinfection in the food industry is summarized by Koutchma et al. (2009). In the food industry, the main problem in applying UV disinfection technique is the low transmittance of the liquid fluid (Guerrero-Beltran and BarbosaCanovas 2004; Koutchma 2009a). Several strategies can be applied to overcome this difficulty. One is to use thin-film flow passage to make UV light penetration easier (Tran and Farid 2004; Unluturk et al.

2004; Koutchma et al. 2006). This thin-film concept can also be strengthened by producing Taylor vortices inside the thin film flow in a Taylor-vortex reactor, which is formed by two concentric cylinders (Ye 2007). Another solution is to use strong turbulence to let fluid particles have equal chance to be exposed to UV light (Koutchma et al. 2006; Keyser et al. 2008). Recently, a new concept (Franz 2009) was proposed, using a helical tube wrapped around UV lamps. In this reactor, low transmittance liquid food particles, though in laminar flow form going through the tube, have equal chance to be exposed to UV light due to the secondary flow phenomenon inside the helical tube. In this chapter, a definition of disinfection efficiency, which can be obtained using computational fluid dynamics (CFD) modeling results is suggested. To demonstrate the concept of disinfection efficiency, a traditional UV reactor design is used. Such reactors are not typically used in the food processing industry. It is proposed that the concept of disinfection efficiency is universal and not dependent on a particular reactor shape. For a UV reactor, given the UV lamp properties, the key in the design is to arrange optimal lamp locations and flow patterns to ensure that microorganisms in the fluid receive even exposure to the UV

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

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radiation. To do that, one has to define quantitative performance indicators such as device efficiency to guide the design. The overall “efficiency” of a reactor is influenced by several factors such as the UV lamp efficiency, the spectrum of the lamps (the disinfection of microorganisms is spectrumdependent), the transmittance of the fluid, and the mechanical design of the reactor shape, which determines the flow and the radiation pattern. Some factors such as UV lamp efficiency and the spectrum of the lamp are factors that a reactor designer can hardly control and optimize. However, with available lamp types a mechanical designer can maximize the reactor performance, provided that the mechanism of disinfection is well understood. CFD technology has been widely used for understanding and predicting the performance of the UV disinfection devices. For example, Janex et al. (1998) used CFD to optimize a reactor by modifying its geometry, aiming at reducing zones of low UV dose or UV fluence. Taghipour and Sozzi (2005) also found that the disinfection effect was strongly influenced by the reactor hydrodynamics, where a more uniform flow pattern and radiation distribution provided improved reactor performance under a similar flow rate and UV lamp power. Sozzi and Taghipour (2006) and Unluturk et al. (2004) intended to use CFD to accurately predict the UV fluence, and compared CFD with a biodosimetric test. The attempt to predict the UV radiation dose or UV fluence accurately by CFD is difficult due to a number of factors. For example, accurate information on lamp efficiency, which also changes with time when the lamps are getting older, is difficult to obtain. Prediction of radiation intensity is also strongly influenced by the radiation model itself. For example, Bolton (2000a) found that the reactor reflection can have an influence of greater than 20% on the UV intensity level. In addition, the transmittance level of the lamp quartz sleeve may change when fouling or scaling occurs over time. However, even if the “electric efficiency,” which indicates what portion of electric energy is converted into usable UV radiation energy, is uncertain, the “mechanical efficiency,” indicating the percentage of radiation energy being converted into disinfection

efficiency, can still be accurately predicted by CFD simulations. In that sense, CFD may not be an accurate tool for predicting the reactor fluence level; however, it can be a powerful tool for optimizing the reactor ’s mechanical design. Recently, Elyasi and Taghipour (2006) suggested a definition of reactor efficiency, which is defined by a calculated fluence divided by a theoretical maximum fluence. However, the maximum fluence defined in Elyasi and Taghipour (2006) is based on a volume-weighted fluence rate and an average speed in the reactor. It is a rough estimation only of the maximum fluence. Therefore, the definition of reactor efficiency itself may not be accurate. In this study, a refined concept of reactor disinfection efficiency is suggested, based on a more reasonable definition of the average fluence that a reactor can achieve at its exit. Some CFD simulation examples will be given to show the disinfection efficiency of several reactor products.

15.2. Modeling UV Processing 15.2.1. The CFD Method For flow modeling, the Navier–Stokes equations (see, e.g., Chapter 5 of this book for more information on Navier–Stokes equations) were solved. The CFD code used was based on the finite volume method, using multi-block structured grids to handle complicated geometries and multigrid acceleration to ensure efficient solution on larger grids. Turbulent viscosity is modeled by a standard k-ε model for core flow together with an algebraic turbulence model for boundary-layer flow. The nearwall treatment is carefully handled so that the skin friction accuracy is less dependent on the grid density near the wall. For describing the steady viscous flow over a three-dimensional (3D) domain, the continuity equation is ∂ ∂ ∂ ( ρ u) + ( ρ v ) + ( ρ w) = 0 ∂x ∂y ∂z and momentum equations

(15.1)

Chapter 15

∂ ∂ ∂ (ρ uu) + (ρ vu) + (ρ wu) ∂x ∂y ∂z ∂p ∂ ∂ ∂ = − + (τ xx ) + (τ yx ) + (τ zx ) ∂x ∂x ∂y ∂z ∂ ∂ ∂ (ρ uv) + (ρ vv) + (ρ wv) ∂x ∂y ∂z ∂p ∂ ∂ ∂ = − + (τ xy ) + (τ yy ) + (τ zy ) ∂y ∂x ∂y ∂z ∂ ∂ ∂ (ρ uw) + (ρ vw) + (ρ ww) ∂x ∂y ∂z ∂p ∂ ∂ ∂ = − + (τ xz ) + (τ yz ) + (τ zz ) ∂z ∂x ∂y ∂z

Multiphysics Modeling of UV Disinfection of Liquid Food

(15.2)

τ yy

(15.3)

(15.4)

∂u ∂x

15.3.1. The Fluence Field If a microorganism moves with the liquid flow along an infinitively short streamline ds for a time fraction dt, it receives the fraction of the UV disinfection fluence dF, which can be expressed as

∂w ∂z

⎛ ∂v ∂u ⎞ τ xy = τ yx = μ ⎜ + ⎟ ⎝ ∂x ∂y ⎠

dF = Idt

⎛ ∂w ∂u ⎞ + τ xz = τ zx = μ ⎜ ⎝ ∂x ∂z ⎟⎠ ⎛ ∂v ∂w ⎞ τ yz = τ zy = μ ⎜ + ⎝ ∂z ∂y ⎟⎠

The UV intensity calculation is based on the multiple point source summation (MPSS) method, which has been described by Bolton (2000a). The model takes quartz sleeve reflection and refraction into account. During modeling each lamp is divided into 100 sections and the UV intensity produced by each lamp piece is calculated in each grid cell. In UV intensity modeling, it is essential that light absorption in air, quartz, and water is accounted for. Wall shadowing is taken into consideration when the path of light is calculated. In addition, the reflection rate at reactor walls is also considered. Wide-band modeling for lamp power, germicidal effect, and absorption is used in the study of medium-pressure lamp reactor performance.

15.3. Fluence

∂v = 2μ ∂y

τ zz = 2 μ

of the velocity distributions there have minimum effects on the flows in the areas of interest. The length of the inlet pipe is seven times the pipe diameter, to achieve fully developed turbulent pipe flow.

15.2.2. The UV Intensity Field Model

where x, y, and z are Cartesian coordinates, ρ is density, and u, v, and w are velocities. p is pressure and τ is shear stress. For Newtonian fluids, under the assumption of incompressible flow, the shear force is expressed as:

τ xx = 2 μ

327

(15.6)

where I is local UV intensity, or dF = I

(15.5)

where μ is viscosity. Further details of the flow solution method can be found in Pan (2000). For the boundary conditions, velocity is given at all solid and inlet boundaries. At the exit boundary, velocities are extrapolated but with adjustment according to the inlet flow rate to meet the mass flow balance requirement. The grid is extended far enough at the exit boundaries so that assumptions

dt ds ds

(15.7)

or dF =

I →

ds

(15.8)

V →

where V is velocity vector defined as →

V=

ds dt

(15.9)

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dF is a different vector, so Equation ds 15.8 can be expressed as Clearly,

→

V⋅

dF =I ds

(15.10)

∂ ∂ ∂ (uF ) + (vF ) + (wF ) ∂x ∂y ∂z

or u

∂ ∂ ∂ (F ) + v (F ) + w (F ) = I ∂x ∂y ∂z

(15.11)

where F is fluence, and u, v, and w are velocities. Introducing the continuity equation, assuming density ρ is a constant, the continuity equation becomes: ∂ ∂ ∂ (u) + (v) + (w) = 0 ∂x ∂y ∂z

(15.12)

Then Equation 15.11 becomes: ∂ ∂ ∂ (uF ) + (vF ) + (wF ) = I . ∂x ∂y ∂z

(15.13)

From Equation 15.11 it can be seen that the fluence is just like any other physical scalar quantity transported by the flow with a source term I. Up to this point, any viscosity diffusion effect on this scalar quantity has not yet been considered. To add viscous diffusive effect, we can examine the convection–diffusion equation of a general scalar in a viscous flow field. For a scalar φ transported in a viscous flow, we have (Versteeg and Malalasekera 1995): ∂ ∂ ∂ (uφ ) + (vφ ) + (wφ ) ∂x ∂y ∂z =

∂ ⎛ ∂φ ⎞ ∂ ⎛ ∂φ ⎞ ∂ ⎛ ∂φ ⎞ λ + ⎜ λ ⎟ + S ( x, y, z ) ⎜λ ⎟ + ∂x ⎝ ∂x ⎠ ∂y ⎜⎝ ∂y ⎟⎠ ∂z ⎝ ∂z ⎠ (15.14)

∂ ∂ ∂ (uφ ) + (vφ ) + (wφ ) are convection ∂x ∂y ∂z ∂ ⎛ ∂φ ⎞ ∂ ⎛ ∂φ ⎞ ∂ ⎛ ∂φ ⎞ terms, λ + ⎜ λ ⎟ are ⎜λ ⎟ + ∂x ⎝ ∂x ⎠ ∂y ⎜⎝ ∂y ⎟⎠ ∂z ⎝ ∂z ⎠ diffusion terms, and S ( x, y, z ) is a source term; λ is the diffusion coefficient, usually about the same value as the kinetic viscosity. where

Comparing Equations 15.13 and 15.14, if the fluence as a scalar quantity behaves the same way as any other scalar in the flow, the diffusion terms used for any scalar in Equation 15.14 can be added. Then Equation 15.13 becomes:

=

∂ ⎛ ∂F ⎞ ∂ ⎛ ∂F ⎞ ∂ ⎛ ∂F ⎞ λ + ⎜λ ⎜λ ⎟ + I ( x, y, z ) ⎟+ ∂x ⎝ ∂x ⎠ ∂y ⎜⎝ ∂y ⎟⎠ ∂z ⎝ ∂z ⎠ (15.15)

Here it is assumed that the diffusion coefficient is the same as the kinetic viscosity. In turbulent flow, the diffusion coefficient is identical to the effective turbulent kinetic viscosity, usually determined by a turbulence model in the practical computations. After obtaining the UV intensity and the flow field, Equation 15.15 is solved to obtain a full 3D fluence distribution (J/m2): F = F ( x, y, z )

(15.16)

Since only the fluence at the exit of the reactor is important, the fluence at a cross-section close to the intersection of the reactor and exit pipe is collected and post-processed. This is done to reduce distortion of results as a consequence of the mixing phenomenon along the exit pipe.

15.3.2. Fluence Value Handling at the Exit of the Reactor To measure the mechanical efficiency of the reactor, the calculated reduction equivalent fluence (CEF) at a chosen exit was defined. The CEF is defined the same way as the reduction equivalent fluence (REF) measured by biodosimetry. The CEF calculation is based on the microbial survival function shown in Equation 15.17, the fluence distribution at an exit cross-section, and the local flow rate distribution on the same cross-section. The relation between fluence F and microbial survival rate N/N0 can be expressed as: F = − A × log10 ( N / N 0 ) + B

(15.17)

Chapter 15

Multiphysics Modeling of UV Disinfection of Liquid Food

where A and B are constants. N is the number of microorganisms after irradiation and N0 is the number of microorganisms before irradiation. The total microbial survival rate at the exit can be expressed as the flow-averaged value of the local microbial survival rate, assuming N0 is equal at all cells at the reactor inlet: ⎛ N⎞ = ⎜⎝ ⎟⎠ N 0 total

∑ 10

− ( Flocal − B) / A

Qlocal

(15.18)

Q

where (N/N0)total is the total survival rate of bacteria for a reactor, Flocal is local fluence at a local calculation cell at the reactor exit, Qlocal is the flow at the local exit cell face, and Q is the total flow. Then the CEF can be expressed as: CEF = − A × log10 (( N / N 0 )total ) + B

(15.19)

In addition to fluence distribution, the model calculates minimum fluence Fmin, maximum fluence Fmax, and average fluence Fave in water exiting the reactor. The average fluence is calculated as the mass flow average across the exit cross-section.

15.3.3. Disinfection Efficiency At the reactor exit, if the fluence distribution is absolutely even and has a constant value, then CEF equals Fave, so that the maximum efficiency is 100%. However, for a practical reactor, the fluence at the exit is not evenly distributed and CEF is always lower than Fave. Therefore, a parameter

η = CEF / Fave

(15.20)

can be defined as the disinfection efficiency of the reactor. It is called disinfection efficiency because it is an indicator for the reactor to achieve the maximum utilization of the given UV radiation resources.

15.4. Simulation Examples Eight reactors from four manufacturers were evaluated (Orava 2003). The reactors are presented in Table 15.1. Four reactors use medium-pressure

329

Table 15.1. Reactors modeled in the study and the number of mesh cells used in computational models. Reactor type A B C D E F G H

Lamp type

Nominal flow, m3/h

Number of CFD mesh cells

Medium-pressure Medium-pressure Medium-pressure Medium-pressure Low-pressure Low-pressure Low-pressure Low-pressure

450 150 150 700 400 150 150 80

153,552 130,824 157,264 184,296 301,600 184,912 170,640 147,240

lamps, which have broadband UV spectra. The other four lamps use low-pressure UV-C lamps. Calculation was done using two different flow rates and three SAK values for each reactor. The SAK value is the water spectral absorption coefficient. SAK-1 means a water transmittance of 97.7% at a 10-mm penetration length for the UV-C with a wavelength of 254 nm. SAK-3 and SAK-5 mean water transmittances of 93.3% and 89.1%, respectively. Data on geometries and UV spectra and lamp power were collected from the manufacturers to obtain values as close to the values provided by the manufacturer as possible to get results that are comparable to the manufacturer table values. The varying power of medium-pressure lamps was taken into account by using the germicidal correction factor as defined by von Sonntag (Bolton 2000b). The lamp power was calculated at 5-nm intervals, which were multiplied with the appropriate germicidal correction factors. The wavelength dependency of the absorption of water was measured using nine samples from Finnish water utilities. The absorbance curves of surface water and ground water were used for modeling mediumpressure lamps. Also, the absorbance of quartz glass varies, so the absorbance values were taken from Bolton (2000a), except in one case where manufacturer values were used. Reflection from the reactor walls was studied from 0 to 30%, and 10% was selected for the modeling, partly based on a study by Sommer et al. (1996).

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An “ideal” microorganism (survival curve coefficients A = 100 and B = 0, Equation 15.19) was used to calculate the equivalent fluence, whereupon fluence needed for 1 log inactivation is 100 J/m2. The survival curve coefficients of a real Bacillus subtilis strain were used in one calculation to compare the difference between ideal and real bacteria. The difference in CEF was only 0.6%, suggesting that B. subtilis can be modeled using an ideal microorganism’s survival curve coefficients around a fluence of 400 J/m2. If higher or lower fluences are used, the difference may be greater.

15.4.1. Results and Discussion To check the accuracy of the model, one computation was done using identical conditions such as the measured lamp power and water quality used in a biodosimetric test. The test result was provided by the manufacturer. The survival curve of B. subtilis was used. The reactor volume was 0.04 m3. The flow rate was 400 m3/s. The calculated equivalent fluence of 468 J/m2 was only 5% higher as the biodosimetric fluence. However, in most of the simulations, there were many conditions, for example, lamp efficiency, quartz sleeve transmittance, and reactor reflection rate, which cannot be accurately obtained. The CEF is highly dependent on those given conditions. Therefore, an accurate prediction of the fluence level in general is difficult, if essential conditions such as lamp efficiency have to be estimated. Looking at Equation 15.20, the uncertainty in the computational factors, such as the lamp power have the same influence on CEF and on Fave by increasing or decreasing their values by a constant factor. For example, if we have an overestimation of the lamp power by 20%, we may predict CEF and Fave 20% more than they should be. However, the disinfection efficiency η remains the same whether or not CEF and Fave have increased by 20%. The disinfection efficiency is solely determined by the unevenness of fluence distribution at the exit. Such uneven distributions are mainly caused by the developed flow pattern and lamp locations and are therefore mainly determined by the mechanical design of the reactor. These “mechanical factors” such as flow patterns

and radiation patterns are highly predictable by the computational models. Therefore, in this study the focus is on the disinfection efficiency. Figure 15.1 shows the disinfection efficiency for all eight products with different flow rates and water transmittances. It can be seen that the individual designs have pronounced differences in their disinfection efficiencies, ranging from around 50% to around 90%. It is interesting to notice that all reactors equipped with medium-pressure lamps have lower efficiencies (51–73%, plotted using unfilled symbols) than those of the reactors equipped with low-pressure lamps (79–93%, plotted using filled symbols). This is likely due to the larger number of lamps in the low-pressure-lamp reactors, resulting in smaller differences between luminous and dark areas. The disinfection efficiency also changes with changing flow rate or water transmittance. However, those changes are smaller compared with the differences among different reactors. The disinfection efficiency, with one exception, increases as the flow rate increases. The disinfection efficiency does not seem to have a clear correlation with the water transmittance, as indicated in Figure 15.1. Of eight reactor products, four reactors, B, C, F, and G, are for the same design flow rate, 150 m3/h. They exhibit pronounced differences in efficiency. Figures 15.2–15.9 show the exit fluence distributions of the four reactors, and the flow rate shares of their fluence levels using a water transmittance of SAK-3 and a flow rate of 125 m3/h. Figures 15.2 and 15.3 show the results of the reactor B, which has the lowest efficiency. It can be seen that the reason for a low efficiency is an extremely uneven exit fluence distribution, which produces a large flow share of the low fluence level. Reactors C, F, and G, which have a much more even fluence distribution at the exit and less flow shares of the low fluence level, have significantly higher efficiencies. Their fluence distributions and the flow shares at the reactor exit are shown in Figures 15.4–15.9, respectively.

15.5. Conclusions CFD and fluence modeling is a useful tool for describing the physical phenomenon inside the UV

100

Disinfection Efficiency, %

90 80 70 60

Reactor-A, SAK1 Reactor-A, SAK3 Reactor-A, SAK5 Reactor-B, SAK1 Reactor-B, SAK3 Reactor-B, SAK5 Reactor-C, SAK1 Reactor-C, SAK3 Reactor-C, SAK5 Reactor-D, SAK1 Reactor-D, SAK3 Reactor-D, SAK5 Reactor-E, SAK1 Reactor-E, SAK3 Reactor-E, SAK5 Reactor-F, SAK1 Reactor-F, SAK3 Reactor-F, SAK5 Reactor-G, SAK1 Reactor-G, SAK3 Reactor-G, SAK5 Reactor-H, SAK1 Reactor-H, SAK3 Reactor-H, SAK5

50 40 30 20 10 0

0

200

400

600

800

3

Flow, m /h Figure 15.1. Computed disinfection efficiencies of the eight reactors with different flow rates and water transmittances.

40

Disinfection efficiency = 52.2%

Flow share, %

30

z x

Reactor B, Q=125 m3/h, SAK=3

3417 2813 2210 1606 1003 400 Fluence, J/m2

Figure 15.2. Fluence distribution at the outlet of the reactor B. See color insert.

20

10

0 0

500

1,000

1,500

2,000

2,500

3,000

Fluence, J/m2

Figure 15.3. The flow share of the fluence levels at the outlet of the reactor B.

331

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

994 895 795 696 596 497

Z 912 780 648 516 384 252 Fluence, J/m2 Figure 15.4. Fluence distribution at the outlet of the reactor C. See color insert.

500

750

Disinfection efficiency = 83.0%

30

10

1,000

2

Fluence, J/m

Fluence, J/m2

Figure 15.6. Fluence distribution at the outlet of the reactor F. See color insert.

Disinfection efficiency = 69.7%

20

0

Y

Reactor F, Q=125 m3/h, SAK=3

Flow share, %

Flow share, %

30

X

20

10

0

500

600

700

800

900

1,000

Fluence, J/m2

Figure 15.5. The flow share of the fluence levels at the outlet of the reactor C.

Figure 15.7. The flow share of the fluence levels at the outlet of the reactor F.

disinfection reactors. The computational tools’ accurate prediction of the fluence level may be difficult due to the difficulty of obtaining the accurate values for lamp efficiency, the quartz sleeve transmittance, and the reactor wall reflection rate. However, those

parameter uncertainties have much less influence on predicting the disinfection efficiency. CFD and fluence simulations on eight reactors from four manufacturers show pronounced differences in disinfection efficiencies among those

Chapter 15

Multiphysics Modeling of UV Disinfection of Liquid Food

768 682 595 509 422 335 Fluence, J/m2 Figure 15.8. Fluence distribution at the outlet of the reactor G. See color insert.

Flow share, %

30

Disinfection efficiency = 79.5%

20

Whether the fluence has an even or uneven distribution on the exit is highly determined by the flow pattern inside the reactor and by the lamp numbers and locations inside the reactor. Those factors are mainly mechanical design issues, which can be optimized using the developed CFD and fluence modeling program. The disinfection efficiency defined in this study is a useful concept for manufacturers to optimize their reactors’ design. The disinfection efficiency is also a useful concept for the reactor end users to select their water supply systems.

Acknowledgments The funding for the CFD and fluence modeling was provided by Rambol Finland Oy. The funded work was mainly done at CFD-Finland Oy. Special thanks are given to Mr. Timo Laitinen, whose vision made this work possible, and Ms. Marika Orava, who helped with reactor data, results analysis, and part of the CFD meshing. Also thanked is Mr. Tapio AlaPeijari for many useful discussions.

Notation A

10

0 300

333

400

500

600

700

800

Fluence, J/m2

Figure 15.9. The flow share of the fluence levels at the outlet of the reactor G.

reactors. The simulation results show that the nonuniformity of the fluence distribution at the reactor exit is the reason for a low efficiency, especially when a low fluence area in the reactor exit has large flow share.

Constant in logarithm microbial survival rate function B Constant in logarithm microbial survival rate function Calculated reduction equivalent CEF fluence, J/m2 Reduction equivalent fluence, J/m2 REF UV fluence, J/m2 F UV radiation intensity or fluence rate, I w/m2 Number of survived microorganisms N N0 Number of microorganisms before UV irradiation Pressure, N/m2 p Flow rate, m3/h Q General source term S Water spectral absorption coefficient SAK u, v, and w Velocity components, m/s x, y, and z Cartesian coordinates, m

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φ η λ μ ρ τ

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General scalar quantity in a flow field Disinfection efficiency Diffusion coefficient, m2/s Viscosity, kg/m · s Density, kg/m3 Shear stress, N/m2

References Adhikari C, Koutchmar T, Beecham-Bowden T. 2005. Evaluation of HHEVC (4, 40,400-tris-di-B-hydroxyethyl aminotriphenylacetonitrile) dye as a chemical actinometer in model buffers for UV treatment of apple juice and cider. LWT 38(7):717–725. Bolton J. 2000a. Calculation of ultraviolet fluence rate distributions in an annular reactor: Significance of refraction and reflection. Water Res 34(13):3315–3324. Bolton J. 2000b. Terms and definitions in ultraviolet disinfection. Proceedings of Disinfection 2000: Disinfection of Wastes in the New Millennium, New Orleans, USA. Elyasi S, Taghipour F. 2006. Simulation of UV photoreactor for water disinfection in Eulerian frame work. Chem Eng Sci 61(14):4741–4749. Franz C. 2009. UV-C-inactivation of microorganisms in naturally cloudy apple juice using novel inactivation equipment based on Dean vortex technology. Food Control 20(12): 1103–1107. Guerrero-Beltran J, Barbosa-Canovas G. 2004. Advantages and limitations on processing foods by UV light. Food Sci Technol Int 10(3):137–147. Janex ML, et al. 1998.Impact of water quality and reactor hydrodynamics on wastewater disinfection by UV, use of CFD modeling for performance optimisation. Water Sci Technol 38(6):71–78. Keyser M, et al. 2008. Ultraviolet radiation as a non-thermal treatment for the inactivation of microorganisms in fruit juice. Innovat Food Sci Emerg Technol 9(3):348–354. Koutchma T, et al. 2009. Ultraviolet Light in Food Technology: Principles and Applications (a volume in the Contemporary

Food Engineering series). London: CRC Press/Taylor & Francis. Koutchma T. 2009a. Advances in ultraviolet light technology for non-thermal processing of liquid foods. Food Bioprocess Technol 2(2):138–155. Koutchma T, Parisi B, Unluturk S. 2006. Evaluation of UV dose in flow-through reactors for fresh apple juice and cider. Chem Eng Commun 193(6):715–728. Lszl Z, et al. 2008. Comparison of the effects of ozone, UV and combined ozone/UV treatment on the color and microbial counts of wheat flour. Ozone Sci Eng 30(6):413–417. Mavrov V, Fahnrich A, Chmiel H. 1997. Treatment of lowcontaminated waste water from the food industry to produce water of drinking quality for reuse. Workshop Membr Drink Water Product Technical Innovat Health Aspects, June at L’Aquila, Italy. 113(2–3):197–203. Orava M. 2003. Ultraviolet disinfection of potable water. Master ’s thesis, Tampere University of Technology, Tampere Finland: Department of Environmental Engineering (in Finnish). Pan H. 2000. Flow simulations for turbomachineries. Proceedings of the 4th Asian Computational Fluid Dynamics Conference, Mianyang, China, 290–295. Sommer R, Cabaj A, Haider T. 1996. Microbicidal effect of reflected UV radiation in devices for water disinfection. Water Sci Technol 34(7–8):173–177. Sozzi A, Taghipour F. 2006. UV reactor performance modelling by Eulerian and Lagrangian methods. Envir Sci Technol 40(5):1609–1615. Taghipour F, Sozzi A. 2005. Medeling and design of ultraviolet reactors for disinfection by-product precursor removal. Desalination 176(1–3):71–80. Tran M, Farid M. 2004. Ultraviolet treatment of orange juice. Innovat Food Sci Emerg Technol 5(4):495–502. Unluturk SK, et al. 2004. Modeling of UV dose distribution in a thin-film UV reactor for processing of apple cider. J Food Eng 65(1):125–136. Versteeg HK, Malalasekera W. 1995. An Introduction to Computational Fluid Dynamics. Essex, UK: Longman Scientific & Technical. Wirtanen G, et al. 2002. Clean air solutions in food processing. VTT Publications No. 482. ISSN: 12350621. Ye Z, 2007. UV disinfection between concentric cylinders. Doctoral dissertation, Georgia Institute of Technology.

Chapter 16 Continuous Chromatographic Separation Technology— Modeling and Simulation Filip Janakievski

16.1. Introduction Simulated moving bed (SMB) chromatography is a continuous chromatographic technique that allows countercurrent separation of a feed mixture into two product streams. The SMB technology is particularly useful for applications in the food and pharmaceutical industries, where thermal separation technologies such as distillation cannot be used due to product degradation and the similarity of the physical properties of the components, such as boiling point. Furthermore, in recent years, SMB chromatography has been recognized by the food industry as a key separation technology for the costeffective isolation of novel ingredients with high functional and nutritional value (De Silva et al. 2003). The successful design and operation of an SMB process depends on the correct selection of the operating conditions such as flow rates, pressure drops, choice of stationary phase, composition of mobile phase, temperature, pH, switching times, column configurations, and geometries. Due to the complex dynamics of the SMB process, the choice of operating conditions as identified earlier is a challenging and arduous process. The trial-and-error approach often requires many

experiments, and ultimately high product requirements cannot be achieved, and therefore a mathematical modeling approach is required.

16.2. True Moving Bed (TMB) Chromatography In order to understand SMB, it is important to understand the concept of the TMB process. TMB is a theoretical concept where the adsorbent bed moves in the opposite direction to the fluid. It is mainly used for process design and optimization. The TMB unit described in Figure 16.1 is divided into four zones, with each zone having a specific function. Two liquid streams enter the TMB and two liquids are withdrawn from the unit. The liquid streams entering the unit are the feed stream containing the mixture to be separated and the desorbent/ eluent stream. The extract and the raffinate streams are withdrawn from the unit. When a binary mixture is introduced into the unit via the feed, the strongly adsorbed component will migrate toward Zone 2 and will be collected as the extract. The weakly adsorbed component will migrate toward Zone 4 with the liquid phase and will be withdrawn as the

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

335

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Figure 16.1. Schematic representation of a TMB process. Liquid and stationary phases are flowing in opposite directions between the four zones, which are divided by two inlet and two outlet streams.

raffinate. The desorbent/eluent is introduced to Zone 1 to regenerate the adsorbent. The bulk of the separation of the two feed components takes place in Zones 2 and 3, where the components are moving in opposite directions. The liquid and solid flows are continuously recycled between Zones 1 and 4. In practical terms, the operation of a TMB process is difficult to achieve as mechanical damage caused by pumping the resin countercurrently can destroy the resolution and cause severe back mixing of the components. Only very low flow rates can be used to avoid the solid being pushed back by the liquid.

Following several cycles, steady state is achieved and constant concentration profiles are reached at the outlet columns. It has been shown (Hidajat et al. 1986) that for an infinite number of columns, SMB converges to the TMB. While the traditional SMB process utilizes port switching to mimic the countercurrent separation, other methods are also possible where the simulation is achieved by column movement in a carousel or by valve switching.

16.2.1. SMB Chromatography

In the last two decades, there has been significant research and development in the SMB chromatography field, with research leading to various novel SMB systems and new industrial applications. Industrial SMB installations can be classified into two major groups based on their valve design and operating mode. The first group is based on a multiport switching valve and includes the continuous ionic separation (ISEP), continuous chromatographic separation (CSEP), and the continuous reaction separation (RSEP) systems developed by Calgon Carbon (Calgon Carbon Corp., Pittsburg, PA), the rotary valve developed by Universal Oil Products (Universal Oil Products, UOP, Des Plaines, IL) as well as pilot and laboratory units developed by Knauer (Knauer Wissenschaftliche Geraetebau Dr. Ing. Herbert Knauer GmbH, Berlin, Germany). The second group utilizes on–off valves to switch the ports in an SMB system and includes the system

The SMB process was developed to overcome the limitations of TMB while retaining the advantages of continuous countercurrent separation. The original SMB process is based on the simulation of a true countercurrent separation between the solid and liquid phases. The SMB process shown in Figure 16.2 is achieved by connecting several single chromatographic columns in series (Dunnebier and Klatt 2000). The solid phase is fixed within the columns. The simulation is achieved by rotating the position of the two inlet and outlet streams in the direction of the fluid stream at a defined switching time. The time between the switching steps is known as an index. When the total number of columns is equal to the number of indexes, the SMB process has completed a full cycle.

16.2.2. Different Variations of the SMB Technology

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Continuous Chromatographic Separation Technology

337

Figure 16.2. Schematic representation of a simulated moving bed showing the four sections with two columns in each zone.

developed by NovaSep (Groupe Novasep, Nancy, France). The main features of the CSEP/ISEP (Figure 16.3) system is a multiport distribution valve that has a stationary and a rotating head. The stationary head is connected to the inlet and outlet streams, while the rotating head is connected to the columns. The carousel can carry up to 30 columns, and the system can be configured to perform different column arrangements. During the complete cycle, each column is subjected to a loading, washing, elution, and regeneration step. The CSEP system is applied to the separation of amino acids, decolorization of sugars, and in water treatment applications. The rotary valve designed by UOP is commonly known under its trade name Sorbex®. The Sorbex® process is illustrated in Figure 16.4, which shows

how the rotary valve is used to periodically change the position of the eluent, extract, feed, and raffinate streams along the adsorbent chamber. The design of this system results in unused lines between the inlet and outlet streams of the adsorbent chamber. As a consequence, this can lead to increased admixing and cross-contamination of the extract stream with the feed stream. The original UOP valve design has since been modified and improved to perform advanced SMB operations, improve configuration flexibility, and reduce valve complexity and cost. UOP has over 100 large-scale installations around the world (Chin and Wang 2004), including Parex® units for the separation of p-xylene, Molex® for the separation of n-parafins, and Sarex® process for the separation of fructose from corn syrup (Gomes et al. 2006).

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Figure 16.3. Commercial CSEP installation showing the columns located on a carousel and the fluid connections to the lower part of the valve.

The limitation of the CSEP and UOP valves is that they are designed primarily to perform synchronous switching. Asynchronous switching is possible but requires additional multi-port valves that connect the inlet and outlet streams to a number of ports on the main rotary valve. Asynchronous switching of the columns leads to better utilization of the solid bed. Recent developments have led to new variants of SMB chromatography that are capable of improving the performance of the classical SMB systems. More sophisticated systems have been proposed such as Powerfeed, Modicon, Varicol (SeidelMorgenstern et al. 2008), and outlet swing stream (OSS) SMB. The Powerfeed process is superior to that of the classical SMB process as the fluid flow rates can be changed during the switching period (Zhang et al. 2003). The Varicol process is characterized by asynchronous shifting of the inlet and outlet ports. This results in a more efficient use of the adsorbent (Toumi et al. 2002). With the Modicon process the feed concentration is varied within each switching cycle. This can lead to increased product

D

es

or

Ex

be

nt

Rotary Valve

Extract Extract Column

tra

ct

Concentrated Extract

Desorbent

Ra tti na c

e

d Fee Concentrated Raffinate Raffinate Adsorbent Chamber

Raffinate Column

Feed Pumparound Pump Figure 16.4. The Sorbex® process showing the stationary adsorbent chamber and rotary valve. (Reprinted with permission of John Wiley & Sons, Inc.)

Chapter 16

concentrations and reduced solvent consumption compared with the traditional SMB process(Schramm et al. 2003). Outlet swing stream is based on the concept of dynamic collection fronts in the equivalent TMB model leading to enrichment of the extract and raffinate streams(Gomes and Rodrigues 2007). The other SMB approach makes use of two way valve systems to replicate the switching mechanism; as mentioned earlier NovaSep has used this design in their commercial systems. The two-way valve design has been used for sugar separation (Beste et al. 2000), the JO process of Japan Organo Co. (Masuda et al. 1993), and the separation of enantiomers using supercritical fluid SMB chromatography (Denet et al. 2001). These systems are extremely versatile and can be used to set up complex configurations. However, they require a large number of valves compared with a rotary valve design system. Generally, these systems can have four or even six two-way valves per column. As a consequence, an SMB system with eight columns would require up to 48 two-way valves. Each valve is connected between the column and one of the inlet streams (feed/desorbent) or outlet streams (extract/raffinate).

16.2.3. SMB Reactors Perhaps the most complex system is the simulated moving bed reactor (SMBR), which is a continuous and countercurrent operation combining chemical reaction and adsorptive separation within one single unit. This system is particularly important for processes where enzymatic hydrolysis is reversible and equilibrium limited. Higher conversions can be achieved by removing one or more of the products, thereby shifting the equilibrium and preventing accumulation of products that can reversibly or irreversibly inhibit the reaction. The equilibrium dispersive model (discussed in Section 16.4.1) has commonly been used to design and optimize SMBRs. However, reactor setup can also influence the modeling approach. Reactors can be set up to contain individual reaction and separation zones (Hashimoto et al. 1983) or perform the reaction and separation in situ (Pilgrim et al. 2006).

Continuous Chromatographic Separation Technology

339

16.3. Modeling Chromatographic Separations The modeling of moving bed processes may follow two approaches: the TMB and the SMB models. Both the TMB and SMB models are based on the following assumptions: (1) the flow in the liquid phase is described by an axially dispersed plug flow model, and (2) the countercurrent solid flow is represented by a plug flow model in the TMB approach; and isothermal operating conditions exist. The TMB unit will achieve a steady state with every process variable remaining constant and assumes an equivalent countercurrent movement of the solid phase. With SMB, each column is analyzed individually and the system is treated as an arrangement of static columns. The SMB system achieves a cyclic steady state; therefore the boundary conditions for each column change after the end of each switch time interval. In order to model the stationary regime of an SMB, a time-dependent model is required. The TMB model has the advantage of reduced computational complexity compared with the SMB. Furthermore, it can be used to predict the steadystate performance of SMB units and is particularly useful for first analysis in design and optimization experiments. The SMB approaches the behavior of the TMB for an infinite number of columns.

16.3.1. Different Modeling Approaches The mechanisms involved in the modeling of a chromatographic column are depicted in Figure 16.5. The column is filled with porous micron-sized adsorbent particles that form the solid phase. While this is the most typical way of producing columns, other approaches such as monolithic beds have been successfully used. The liquid containing one or more of the components to be separated is introduced into the column and forms the liquid phase. The driving force for mass transfer between the two phases is due to the concentration difference of the components in the liquid phase and the solid phase. Many different modeling approaches with varied complexities have been proposed in the literature.

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Figure 16.5. Schematic representation of a chromatographic column showing the mass transfer phenomena which take place during the adsorption of a molecule.

The different model approaches can be classified by the physical phenomena they take into account, and by the number of effects considered in solving the differential mass balance. Most models consider at least two of the following physical phenomena: • • • • •

Convection. Dispersion. Film diffusion. Pore diffusion. Adsorption.

The simplest model is the ideal model, which assumes convection and adsorption equilibrium, while the most complex model is the general rate model, which assumes convection, dispersion, diffusion, and adsorption. These and other models are summarized in Figure 16.6.

16.4. Column Models The cyclic steady state of the SMB process is calculated on the basis of the single column models described in this section.

16.4.1. The Equilibrium Model (Ideal Model) The equilibrium model is the simplest model and takes into account convective transport and thermodynamics. This model assumes equilibrium between the liquid and solid phases and neglects the influence of axial dispersion (Dünnebier et al. 1998). The material balance is commonly expressed by the following equation: ∂ci 1 − ε ∂qi ∂c + = −u i ∂t ε ∂t ∂x

(16.1)

where ci is the concentration of component i in the liquid phase, qi is the concentration of component i in the solid phase, ε is the bed porosity, and u is the velocity of the fluid. The equilibrium model can provide information on retention times of elution peaks. However, if mass transfer effects are significant, peak shapes cannot be predicted accurately. Equation 16.1 can be rearranged to obtain an expression that predicts the propagation velocity ( wi ) of the different components inside the column.

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Continuous Chromatographic Separation Technology

341

GENERAL RATE MODEL Convection Axial Dispersion Film Mass Transfer Particle Diffusion Adsorption Dynamics

REACTION DISPERSIVE MODEL Adsorption Kinetics Convection Axial Dispersion

THOMAS MODEL Adsorption Kinetics Convection

LUMPED RATE MODELS

EQUILIBRIUM TRANSPORT DISPERSIVE MODEL Mass Transfer Resistance Convection Axial Dispersion

EQUILIBRIUM DISPERSIVE MODEL

EQUILIBRIUM TRANSPORT MODEL

Axial Dispersion Convection

Mass Transfer Resistance Convection

IDEAL MODEL Convection

Figure 16.6. Classification of different column models showing the physical phenomena each model represents. General rate model (most complex) and ideal model (simplest) at opposite ends (Engell and Toumi 2005).

wi =

u 1 − ε ∂qi 1+ ε ∂ci

(16.2)

16.4.2. The Equilibrium Dispersive Model The equilibrium dispersive model is the most commonly used model and has been frequently used to

quantify chromatographic processes under overload conditions (Seidel-Morgenstern 2004). In order to overcome some of the shortfalls of the equilibrium model, a term describing axial dispersion and mass transfer has been added to the mass balance of the mobile phase. This model assumes that the effects of mass transfer and axial dispersion can be lumped into the single parameter Dap known

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Innovative Food Processing Technologies: Advances in Multiphysics Simulation

as the apparent dispersion coefficient. This assumption holds true in the case of linear isotherms as described by van Deemter et al. (1956). The dispersion coefficient accounts for any band broadening effects, particularly when the mass transfer in the column is controlled by molecular diffusion across the mobile phase and the exchange between the solid and liquid phases is very fast (Guiochon et al. 1994). The mass balance for each solute component between planes at distances x + dx from the inlet of the column over a time period t to t + dt results in a partial differential equation. The mass balance can be expressed by the following equation: ∂ci ∂c 1 − ε ∂qi (c) ∂ ci (16.3) +u i + = Dap ∂t ∂x ε ∂t ∂x 2 2

i = 1, N with c = (c1,c2,…,CN) The equilibrium dispersive model provides a very accurate and computationally efficient column model (Klatt et al. 2000). For highly efficient columns the relationship between the apparent dispersion coefficient Dap and the number of theoretical plates of the column can be obtained by: N = uL / 2 Dap or

(16.4)

Dap = uL /2 N

(16.5)

where L is the column length, Dap is the apparent dispersion coefficient, u is the velocity of the fluid, and N is the number of theoretical plates of the column.

16.4.3. The General Rate Model The general rate model is the most comprehensive and rigorous of all the models. Different modeling approaches using the general rate model can be found in the literature. Each model takes into account the effects of axial dispersion, mass transfer between the liquid and solid phase, pore diffusion, and adsorption kinetics within the pore. The mass balance for the fluid phase can be expressed by the following equation:

∂ci ∂ci ∂ 2 c (1 − ε ) 3k film,i +u − Dax 2 + (ci − c p,i ) (16.6) ∂t ∂x ∂x ε rp The mass balance for the solid phase and inside the pores is described by:

εp

∂c p,i (1 − ε ) ∂qi ∂ 2 c 1 ∂ ⎛ ∂c p , i ⎞ + = ε p D p ,i 2 2 ⎜ r2 ⎟ ∂t ∂t ∂x r ∂r ⎝ ∂r ⎠ ε (16.7)

where ε p is the particle porosity, Dp is the diffusion coefficient inside the pores, c p is the concentration in the fluid phase inside the particle pores, r is the particle radius, and k film is the mass transfer coefficient.

16.5. Modeling of the SMB process The two modeling strategies commonly applied to the SMB process are discussed in this section. The first one uses the equivalent TMB to represent the SMB process. The second considers the system as an arrangement of static columns and takes into account the movement of the solid.

16.5.1. The TMB Model The models described in the section on Column Models can be used to represent the TMB model. As the TMB model assumes countercurrent movement of the solid phase, a term describing the movement of the solid has to be added to the mass balance. For the equilibrium model, the mass balance for component i in the liquid phase in zone j can be expressed as: ∂ci , j 1 − ε ∂qi , j ∂ci , j 1 − ε ∂qi , j + = uf ,j + us ε ∂t ε ∂x j ∂t ∂x j

(16.8)

j = 1, 2, 3, 4 In Equation 16.8, u f , j represents the velocity of the liquid in zone j in the TMB model; ci,j is the concentration of component i in the liquid phase in zone j; qi,j is the corresponding concentration in the solid phase; ε is the bed porosity; t denotes the time and xj is the axial coordinate in zone j; us is the equivalent solid phase velocity and is given by us = L / Δt with L the length of the column; and Δt is the switching period.

Chapter 16

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343

16.5.2. The SMB Model

QE = Q1 − Q2

(16.13)

With the SMB model, the solid phase is stationary and countercurrent movement is achieved by switching the valve of the interconnected columns. The dynamic model of the SMB process is achieved by connecting the column models of each of the four sections in the system through a mass balance equation at every node of the unit. The columns are connected by the node model equations 16.9 and 16.10, which are integrated numerically. For the equilibrium dispersive model, the mass balance in the fluid phase can be expressed by:

in cE ,i = c1out ,i = c2 ,i

(16.14)

The Desorbent (D) Node QD = Q1 − Q4

(16.15)

cD,i QD = c Q1 − c Q4 in 1,i

out 4 ,i

(16.16)

The Raffinate (R) Node QR = Q3 − Q4

(16.17)

in cR ,i = c3out ,i = c4 ,i

(16.18)

where Δt is the switching period and p is the period number.

where QF , QE , QD , and QR are the volumetric flow rates in the feed, extract, desorbent and raffinate sections, respectively, while cF , cE , cD , and cR are the concentrations of component i in the four sections. The superscripts in and out are the concentrations of component i at the inlet and outlet of the column. The liquid phase velocity in the column is determined from u = Q Aε , where A is the cross-sectional area of the column. The transient behavior of the SMB unit can be predicted by solving the above equations as well as the appropriate initial and boundary conditions of each column model.

16.5.3. The Node Model

16.5.4. Review of Modeling Strategies

The dynamic model of the SMB process can be attained by connecting the dynamic models of the single chromatographic columns while considering the cyclic port switching. The inlet concentrations and the fluid velocities in the four sections of the SMB unit can be calculated by mass balances around the inlet and outlet nodes. This is known as the node model. The node model relates the internal flow rates, and concentrations to the external flow rates and concentrations. The Feed (F) Node

Dunnebier (Dunnebier and Klatt 2000) compared three different modeling approaches for chromatographic separation processes with nonlinear adsorption thermodynamics. The general rate model required more computation but provided qualitative information on the dynamics of the process during start-up. However, the accuracy of the ideal model was not sufficient for design, control, or optimization purposes because of the unrealistic assumptions of ideality. The kinetic model proved to be effective for columns with low efficiency, but in practical terms this also was unrealistic. Zhong and Guiochon (1996) derived an analytical solution for linear ideal model SMB systems. Algebraic expressions were obtained for the concentration profiles of the raffinate and extract in the columns; also, the operation of the SMB under

∂ci ∂c 1 − ε ∂qi (c) ∂ ci +u i + = Dap ∂t ∂x ε ∂t ∂x 2 2

(16.9)

At the end of the switching period, the position of the inlet and outlet ports are rotated by one column. Therefore, the concentration profiles in column j at the beginning of the switching period are equal to the profiles obtained in column j + 1 at the end of the previous period (Garcia et al. 2006). ci , j = (t p = 0, x j ) = ci , j +1 (t p −1 = Δt , x j +1 )

QF = Q3 − Q2 cF ,i QF = c3in,i Q3 − c2out,i Q2 The Extract (E) Node

(16.10)

(16.11) (16.12)

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steady-state conditions as well as during transition from start-up to steady state were determined. A comparison of SMB and TMB chromatography showed the main difference between the two systems to be the time taken to reach steady state. Seidel-Morgenstern and Guiochon (1993) used the equilibrium dispersive model to calculate the elution profiles of racemic mixtures of two enantiomers. Excellent agreement was obtained between the experimental and calculated elution profiles of the racemic mixture. Therefore, it was concluded that the equilibrium dispersive model can be applied even when the column efficiency is as low as 100 theoretical plates. Chan et al. (2008) successfully applied the equilibrium dispersive and general rate models for modeling protein purification. Overall, the general rate model was shown to give better predictions, but it also presented some difficulties in obtaining its model parameters. The equilibrium dispersive model showed similar model behavior and was preferable due to the reduced complexity. This approach can be applied to different bioseparation isotherm models and can be used in the estimation of isotherm parameter values. The key feature of this approach was the development of a procedure for approximating the feed concentration for mixtures where this is unknown. Although the SMB process has been applied to large-scale separations in the sugar and hydrocarbon industries, only recently has it attracted more attention from the biotechnology and food industries. Some of the examples include the separation of fructose and glucose from cashew apple juice (Azevedo and Rodrigues 2005); the treatment of potato juice from starch production to obtain potato proteins (Andersson Jonatan et al. 2008); and fractionation of bioactive components capable of angiotensinconverting enzyme inhibition from casein hydrolysate for use in functional foods (Ottens et al. 2006). Houwing et al. (2003) demonstrated the chromatographic separation of bovine serum albumin (BSA) and myoglobin by size exclusion chromatography in an SMB. Viard and Lameloise (1992) developed a methodology for modeling glucose–fructose separation

on an ion exchange resin. Henke et al. (2008) showed how modeling and simulation can be used to isolate betaine from molasses.

16.6. Adsorption Isotherms Adsorption isotherms describe the equilibrium between the adsorbed concentration and the concentration in the fluid phase at constant temperature. The equilibrium adsorption isotherms can be described using various models including linear (Dünnebier et al. 1998), Langmuir (Dunnebier and Klatt 2000), Bi-Langmuir (Gentilini et al. 1998), and modified Langmuir (Mazzotti et al. 1997). The simplest relationship between the solid phase q and the fluid phase c concentration is the linear isotherm: qi = Hi ci

(16.19)

The Henry coefficient Hi is the slope of the single component adsorption isotherm at the infinite dilution. Therefore, the linear model is applicable mainly at low concentrations, and it does not represent the competition between the molecules.

q (g/L)

c (g/L) Figure 16.7. Example of a two component Langmuir isotherm showing the equilibrium between the adsorbed concentration (q, g/L) and the concentration in the liquid phase (c, g/L) at constant temperature.

Chapter 16

The most popular of all the nonlinear isotherms is the Langmuir isotherm (Figure 16.7). The Langmuir isotherm can be expressed by: qi =

qs K i ci 1 + K i ci

(16.20)

where K i is the equilibrium constant, and qs is the saturation capacity of the stationary phase. The same applies for all the following variations of the Langmuir equation (there are two values for different adsorption sites in the Bi-Langmuir equation). The Langmuir isotherm assumes that there is single adsorption per binding site, that adsorption sites are of equal energy (homogeneous), and that there is negligible interaction between the adsorbed molecules. The Langmuir isotherm can be extended by adding a term that covers the nonspecific adsorption of the molecule, and gives the multi-Langmuir equation. qi =

qs K i ci 1 + ∑ K i ci

i = 1, 2 …

(16.21)

qs1 K1i ci qs2 K 2 i ci + 1 + ∑ K1i ci 1 + ∑ K 2 i ci

(16.22)

Another useful isotherm is the Freundlich isotherm, which can be applied to heterogeneous adsorbents over a wide range of concentrations. The Toth isotherm has also been applied to heterogeneous adsorbents and has the advantage of bringing together the Langmuir and Freundlich equations.

16.7. SMB Design Research activities in the area of SMB design, optimization, and control have accounted for a significant amount of the literature on modeling. Two important design methods are presented in this section.

345

16.7.1. The Triangle Theory The triangle theory developed by Storti et al. (1989) can be used to determine the optimal operating conditions of the SMB unit in order to achieve the desired separation. This approach is particularly important during the design stage and is used to determine the flow rates in each zone and the switching time. SMB units are often operated under overload conditions, which lead to nonlinear competitive adsorption (Migliorini et al. 1998). In this situation the triangle rule can be successfully applied to achieve complete separation. The operating conditions can be expressed in terms of the net flow rate ratios (mj, j = 1,2,3,4) in each section by the following equation: mj =

Q j Δt − Vε* V (1 − ε*)

(16.23)

where Δt is the switching time, V is the column volume, Q j is the volumetric flow rate, and ε* is the overall porosity of the column determined from:

ε* = ε + ε p (1 − ε )

The multi-Langmuir isotherm can be further modified to incorporate two Langmuir terms, and assumes two different adsorption sites. This isotherm is known as the multicomponent Bi-Langmuir isotherm: qi =

Continuous Chromatographic Separation Technology

(16.24)

where ε is the bed porosity, and ε p is the particle porosity. The triangle theory describes the operating points of Sections 16.2 and 16.3 by plotting the flow rate ratios m2 and m3 in a triangle diagram (Figure 16.8). For nonlinear isotherms the feed concentration has a strong influence on the shape of the separation region. The triangle becomes distorted and the separation region shrinks. As the feed concentration increases and the triangle region decreases, it becomes increasingly difficult to achieve complete separation. However, the productivity increases at higher feed concentrations. Therefore, selecting the appropriate feed concentration is a delicate balance between maximum productivity and maximum purity. Absolute purity in both extract and raffinate is often not desirable and achieving complete separation would require an extremely large inventory. This theory is not very robust for systems where mass transfer resistance inside particles is of importance (Rodrigues and Minceva 2005). As a result, the concept of separation volume was developed.

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0.45 0.4 0.35 0.3 m1 0.25 0.2 0.15 0.58 0.56 0.54 0.52 0.5 0.48 0.46 m3

0.45

0.5 m2

0.55

Figure 16.9. Separation volume concept (Rodrigues and Minceva 2005) showing the relationship between the flow rates in Zones 1, 2, and 3. Figure 16.8. The triangle diagram for a linear isotherm. Region A represents the point of no pure products (i.e., both raffinate and extract are impure). Region B represents the pure raffinate only, region C is pure extract only, and region D is pure raffinate and pure extract. In region D, complete separation is achieved. Points E and F are obtained from the Henry coefficients of the two components.

16.7.2. Separation Volume Theory Another important method used in the design and optimization of an SMB process is the separation volume method. Azevedo and Rodrigues (1999) investigated the influence of flow rate constraints on mass transfer effects in Zone 1 and Zone 4 of the separation region. Instead of the traditional twodimensional plane (m2 × m3), a three-dimensional separation volume is used to define the separation region, as depicted in Figure 16.9. The flow rate ratios of either Zone 1 or Zone 4 can be included in defining the region of complete separation. In the case where Zone 1 is included, the flow rate in Zone 4 is fixed and vice versa. Azevedo and Rodrigues also found that the flow rate in Zone 4 was less influential on the size of the separation region. The advantage of this model is that it takes into account mass transfer resistances, and it considers the effects of the fluid–solid velocity ratios in the regeneration sections.

16.8. SMB Model Parameter Determination The successful design and optimization of an SMB process involves the selection of the correct operating conditions, especially the flow rates (feed, desorbent, raffinate, and extract) in each section and the switching times (Toumi et al. 2007). In order to select the appropriate operating conditions, first the adsorption isotherms (thermodynamics) must be determined. The adsorption isotherms are determined experimentally, and several experimental methods have been developed for that purpose. Those methods can be used to determine single as well as multicomponent isotherms, and they differ in terms of experimental complexity and accuracy. The experimental methods can be generally classified into two categories: static methods and dynamic methods. While static methods are relatively easy to set up, they can often be more timeconsuming and inaccurate. They are prone to errors in determining the adsorbent required and the uncertainty concerning the time needed to reach equilibrium (Seidel-Morgenstern 2004).

16.8.1. Static Methods The static methods use the overall mass balance to measure the adsorption isotherms. With the first static method, known as the batch method, a known amount of adsorbent is contacted with a solution of a known concentration for a sufficient time to reach equilibrium. The final concentration of the solution is subtracted from that of the initial solution to give the adsorbed concentration. For every concentration a single point is obtained on the isotherm, and therefore, several concentrations or adsorbent quantities need to be investigated. The second method is the adsorption-desorption method. With this method, a solution of known concentration is pumped through the column until equilibrium is reached. The column is then desorbed with an eluant and the eluant is usually analyzed by HPLC. The concentration of the adsorbed component can be calculated by the mass balance. Both methods can be used to determine single solute adsorption as well as adsorption of mixtures.

16.8.2. Dynamic Methods Dynamic methods are based on the examination of concentration profiles corresponding to different disturbances of an equilibrated column. There are a number of dynamic methods in the literature, and three of those will be discussed in this section. 16.8.2.1. Elution by Characteristic Point (ECP) Method The ECP method can be used to determine dispersed fronts of a single peak in order to determine the adsorption isotherm. The isotherm is derived from the rear part of the overloaded elution profile. The ECP method should only be used with highly efficient columns where contributions to band broadening are negligible. Evaluation of the isotherm is based on the ideal model. ECP can be applied only to single-solute adsorption. If the isotherm does not show a convex shape (Langmuir-like behavior) but a concave shape, a dispersive tail instead of a dispersive front appears. In many cases, the isotherm cannot be calculated from a single peak, as most detectors do not

Continuous Chromatographic Separation Technology

0.9 C2 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

347

Concentration

Chapter 16

Time

t1

Figure 16.10. Example of a breakthrough curve for a single component. At a given time (t1) the curve reaches a plateau and the outlet concentration is equal to c2.

provide a signal that can directly be converted to concentration units. 16.8.2.2. Frontal Analysis Method The frontal analysis method is based on a step response disturbance. The step response is obtained by making a series of step changes at the column inlet. The result is a breakthrough curve as shown in Figure 16.10. The solid phase concentration can be estimated using one of three procedures: from the retention time of the inflection point of the curve, from the half-height of the plateau (the middle point), or with the equal area method (Andrzejewska et al. 2009). The characteristics of the breakthrough curve are analyzed and they provide the first point in the adsorption isotherm. This procedure is repeated with different feed concentrations until a sufficient number of isotherm points are obtained. The frontal analysis method is the most popular of the dynamic methods and is also regarded as the most accurate. 16.8.2.3. Minor Disturbance (Perturbation) Method The minor disturbance (perturbation) method can be used to determine the adsorption isotherms from the retention time of a small pulse injected into the column. The column must initially be equilibrated at a given concentration, and then a small sample is injected to disturb the equilibrium state. The slope of the isotherm can be calculated from the small peak (Figure 16.11).

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Concentration

348

Time Figure 16.11. The principle of the perturbation method showing a stepwise increase in concentration for a single component. The arrows indicate the time small samples are injected to perturb the equilibrium state (Heuer et al. 1998).

2.94 2.85 2.77 2.68 2.6 2.51 2.43 2.34 2.26 2.17 2.09 2 1.92

Compl. sep. region Heory coeff. weak Heory coeff. strong Operating point 2.73 < ml m lV < 1.76

m II

1.66 1.77 1.88 1.98 2.09 2.2 2.31 2.42 2.53 2.64 2.75 2.85 2.96

Figure 16.12. Triangle theory showing the complete separation zone and the operating point.

1.9 g/L

Inner concentration profiles

m III

Bi-Langmuir model delivers only a rough design estimation

1.9 g/L

1.5 g/L

1.5 g/L

0.75 g/L

Raffinate

Withdrawal concentration profiles

1.1 g/L 0.38 g/L 0 g/L 0 min

1.1 g/L

1.9 g/L

0.76 g/L

42 min

84 min

1.3E02 min

1.7E02 min

2.1 E02

42 min

84 min

1.3E02 min

1.7E02 min

2.1 E02

Extract

1.5 g/L 1.1 g/L

0.38 g/L

0.75 g/L 0.38 g/L

0 g/L Desorbent

Extract

Feed

Raffinate

0 g/L 0 min

Figure 16.13. Internal concentration profiles and transient extract and raffinate concentrations. See color insert.

Like the ECP method, this method is based on the ideal model of chromatography. As only small perturbations from equilibrium occur, this method is less dependent on high plate counts. Furthermore, mixtures can be analyzed directly.

16.9. Simulation and Validation There are several software packages such as Knauer SMB Guide (Knauer Wissenschaftliche Geraetebau Dr. Ing. Herbert Knauer GmbH, Berlin, Germany) and Aspen Chromatography (Aspen Technology, Inc., Burlington, MA), which can be adopted for the

simulation of SMB processes. These simulation packages are used to predict the effect of operating variables on the performance of the SMB process and to define the regions for optimum separation. Figure 16.13 shows the concentration profiles at the different zones within the system. This simulation is based on the triangle theory discussed in Section 16.7.1. Simulation is done with Knauer SMB-Guide simulation software. The triangle theory is used to define an operating point (Figure 16.12). Dynamic calculation is then performed by means of the equilibrium dispersive model. Internal

Chapter 16

concentration profiles are shown for the beginning (narrow lines) and the end of a tact (wide lines) in Figure 16.13. In order to perform the separation, the general parameters (number of columns in each section, column dimensions, theoretical plate numbers, and column porosity) and the adsorption isotherms must be provided. The adsorption isotherms can be determined by the methods described in Section 16.8. Once the values have been specified, the simulation can be performed. The flow rates, switching times, and the purity of the extract and raffinate streams are calculated. These values can be used to perform the experimental validation. If good agreement is attained between the simulated and the experimental results, the model can be used to consider different configurations, optimize the separation, and for scale-up.

16.10. Conclusion The separation of biological molecules or components with similar adsorption properties can be achieved efficiently by continuous countercurrent processes such as SMB chromatography. The ever increasing demand for health promoting foods has intensified research activities in this field and its potential application in the food industry. The requirement for high-purity fractions and lowering of operating costs has led to the evolution of modeling and simulation strategies for predicting and controlling the behavior of SMB systems. Modeling of SMB processes are widely documented in the literature, and researchers have demonstrated good agreement between the predicted and experimental results. Most of the SMB models essentially differ in the description of the kinetic and dispersive effects, which is directly associated with the complexity of the models. Modeling and simulation have become increasingly important, especially when considering the recent introduction of numerous advanced operating modes that have expanded the number of manipulated variables and the overall complexity of the SMB process. In addition to these innovative operating modes, there is potential for wide-ranging applications in food processing by

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349

bringing together the SMB technology with new adsorbent materials such as molecular imprinted polymers, which have predetermined molecular recognition sites for specific molecules. Moving forward, SMB modeling will play a significant role in designing the right process configurations for the separation of value-adding ingredients in the food industry and support the continuing expansion of the SMB technology.

Notation Abbreviation CSEP ECP ISEP OSS RSEP SMB SMBR TMB UOP

Continuous chromatographic separation Elution by characteristic point Continuous ionic separation Outlet swing stream Continuous reaction and separation Simulated moving bed Simulated moving bed reactor True moving bed Universal Oil Products

Latin Letters A ci cp Dap Dp Hi j k film Ki L N p Q QD QE

Cross-sectional area Concentration of component i in the liquid phase Concentration in the fluid phase inside the particle pore Apparent dispersion coefficient Diffusion coefficient Henry coefficient Zone Mass transfer coefficient Equilibrium constant of component i Column length The number of theoretical plates of the column Period number Volumetric flow rate Volumetric flow rate in the desorbent Volumetric flow rate in the extract

350

QF QR qi qs r u us V wi x

Innovative Food Processing Technologies: Advances in Multiphysics Simulation

Volumetric flow rate in the feed Volumetric flow rate in the raffinate Concentration of component i in the liquid phase Saturation capacity of the solid phase Particle radius Velocity of the liquid phase Velocity of the solid phase Column volume Propagation velocity of component i Axial coordinate

Greek Letters Δt ε εp ε*

Switching time Bed porosity Particle porosity Overall porosity of the column

References Andersson J, Sahoo D, Mattiasson B. 2008 . Isolation of potato proteins using simulated moving bed technology. Biotechnol Bioeng 101(6):1256–1263. Andrzejewska A, Kaczmarski K, et al. 2009 . Theoretical study of the accuracy of the pulse method, frontal analysis, and frontal analysis by characteristic points for the determination of single component adsorption isotherms. J Chromatogr A 1216(7):1067–1083. Azevedo DCS, Rodrigues AE. 1999. Design of a simulated moving bed in the presence of mass-transfer resistances. AIChE J 45(5):956–966. Azevedo DCS, Rodrigues AE. 2005. Separation of fructose and glucose from cashew apple juice by SMB chromatography. Separat Sci Technol 40(9):1761–1780. Beste YA, Lisso M, et al. 2000 . Optimization of simulated moving bed plants with low efficient stationary phases: Separation of fructose and glucose. J Chromatogr A 868(2): 169–188. Chan S, Titchener-Hooker N, et al. 2008 . A systematic approach for modeling chromatographic processes—Application to protein purification. AIChE J 54(4):965–977. Chin CY, Wang NHL. 2004. Simulated moving bed equipment designs. Separat Purif Rev 33(2):77–155. De Silva K, Stockmann R, et al. 2003 . Isolation procedures for functional dairy components: Novel approaches to meeting the challenges. Aust J Dairy Technol vol:58(2):148–152. van Deemter JJ, Zuiderweg FJ, et al. 1956 . Longitudinal diffusion and resistance to mass transfer as causes of nonideality in chromatography. Chem Eng Sci 5(6):271–289.

Denet F, Hauck W, et al. 2001. Enantioseparation through Supercritical Fluid Simulated Moving Bed (SF-SMB) chromatography. Ind Eng Chem Res 40(21):4603–4609. Dünnebier G, Engell S, et al. 1998. Modeling of simulated moving bed chromatographic processes with regard to process control design. Comput Chem Eng 22:S855–S858. Dunnebier G, Klatt KU. 2000. Modeling and simulation of nonlinear chromatographic separation processes: A comparison of different modeling approaches. Chem Eng Sci 55(2):373–380. Engell S, Toumi A. 2005. Optimisation and control of chromatography. Comput Chem Eng 29(6):1243–1252. Garcia M-SG, Balsa-Canto E, et al. 2006. Dynamic optimization of a Simulated Moving Bed (SMB) chromatographic separation process. Ind Eng Chem Res 45(26):9033–9041. Gentilini A, Migliorini C, et al. 1998. Optimal operation of simulated moving-bed units for non-linear chromatographic separations: II. Bi-Langmuir isotherm. J Chromatogr A 805(1–2):37–44. Gomes PS, Minceva M, et al. 2006. Simulated moving bed technology: Old and new. Adsorption 12(5–6):375–392. Gomes PS, Rodrigues AE. 2007. Outlet Streams Swing (OSS) and multifeed operation of simulated moving beds. Separat Sci Technol 42(2):223–252. Guiochon G, Golshan-Shirazi S, et al. 1994. Fundamentals of preparative and nonlinear chromatography. Boston: Academic Press. Hashimoto K, et al. 1983. A new process combining adsorption and enzyme reaction for producing higher-fructose syrup. Biotechnol Bioeng 25(10):2371–2393. Henke S, et al. 2008. The new simulated moving bed pilot plantmodeling, simulation and application. J Food Eng 87(1):26–33. Heuer C, Küsters E, et al. 1998. Design of the simulated moving bed process based on adsorption isotherm measurements using a perturbation method. J Chromatogr A 827(2):175–191. Hidajat K, Ching CB, et al. 1986. Simulated countercurrent adsorption processes—A theoretical-analysis of the effect of subdividing the adsorbent bed. Chem Eng Sci 41(11):2953–2956. Houwing J, Billiet HAH, et al. 2003. Mass-transfer effects during separation of proteins in SMB by size exclusion. AIChE J 49(5):1158–1167. Klatt K-U, Hanisch F, et al. 2000. Model-based optimization and control of chromatographic processes. Comput Chem Eng 24(2–7):1119–1126. Masuda T, Sonobe T, et al. 1993. Process for fractional separation of multi-component fluid mixture. US Patent 5, 198, 120. Mazzotti M, Storti G, et al. 1997. Optimal operation of simulated moving bed units for nonlinear chromatographic separations. J Chromatogr A 769(1):3–24. Migliorini C, Mazzotti M, et al. 1998. Continuous chromatographic separation through simulated moving beds under linear and nonlinear conditions. J Chromatogr A 827(2):161–173. Ottens M, Houwing J, et al. 2006. Multi-component fractionation in SMB chromatography for the purification of active fractions from protein hydrolysates. Food Bioproducts Process 84(C1): 59–71.

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Pilgrim A, Kawase M, et al. 2006. Modeling of the simulated moving-bed reactor for the enzyme-catalyzed production of lactosucrose. Chem Eng Sci 61(2):353–362. Rodrigues AE, Minceva M. 2005. Modeling and simulation in chemical engineering: Tools for process innovation. Comput Chem Eng 29(6):1167–1183. Schramm H, Kaspereit M, et al. 2003. Simulated moving bed process with cyclic modulation of the feed concentration. J Chromatogr A 1006(1–2):77–86. Seidel-Morgenstern A. 2004. Experimental determination of single solute and competitive adsorption isotherms. J Chromatogr A 1037(1–2):255–272. Seidel-Morgenstern A, Guiochon G. 1993. Modeling of the competitive isotherms and the chromatographic separation of two enantiomers. Chem Eng Sci 48(15):2787–2797. Seidel-Morgenstern A, Kessler LC, et al. 2008. New developments in simulated moving bed chromatography. Chem Eng Technol 31(6):826–837.

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Storti G, Masi M, et al. 1989. Optimal design of multicomponent countercurrent adsorption separation processes involving nonlinear equilibria. Chem Eng Sci 44(6):1329–1345. Toumi A, Engell S, et al. 2007. Efficient optimization of simulated moving bed processes. Chem Eng Process Process Intensific 46(11):1067–1084. Toumi A, Hanisch F, et al. 2002. Optimal operation of continuous chromatographic processes: Mathematical optimization of the VARICOL process. Ind Eng Chem Res 41(17):4328–4337. Viard V, Lameloise ML. 1992. Modeling glucose fructose separation by adsorption chromatography on ion-exchange resins. J Food Eng 17(1):29–48. Zhang Z, Mazzotti M, et al. 2003. PowerFeed operation of simulated moving bed units: Changing flow-rates during the switching interval. J Chromatogr A 1006(1–2):87–99. Zhong GM, Guiochon G. 1996. Analytical solution for the linear ideal model of simulated moving bed chromatography. Chem Eng Sci 51(18):4307–4319.

Chapter 17 The Future of Multiphysics Modeling of Innovative Food Processing Technologies Peter J. Fryer, Kai Knoerzer, and Pablo Juliano

17.1. Introduction This book contains a number of chapters detailing the Multiphysics modeling of innovative food processing technologies. In modeling these processes, the first stages are about conceptual understanding, that is, understanding the basic physics and the chemistry of a process. By understanding the process variables and the determination of process parameters, conceptual models can be defined. Conceptual model development is followed by collating the fundamental equations describing the physics of the process, and solving them by means of numerical approximations. In food-related applications solving the equations completely is rarely possible; hence, the food industry has tended to focus on less accurate conceptual models and approximations alone. However, as is shown in this book, it is possible to represent complete food processes using mathematical models. The depth of complexity of the developed models is usually a function of the computational resources. Following Moore’s law, computing power doubles every two years. For example, when models were developed for predicting the ohmic heating patterns around parti-

cles (De Alwis and Fryer 1990) it was feasible to have a two-dimensional solution, which took several hours to run on the University of Cambridge’s mainframe; when producing the simulations for this book (Chapter 8), more complex three-dimensional solutions (albeit axis-symmetric) took only a couple of minutes to run on a laptop computer. For the models reported in Zhang and Fryer (1993), a 1 GB drive was required, which cost more than £20,000 (US$30,000) in 1989; now drives of more than 1 GB are commonly given away as adverts. The pace of improvement in computing power and data handling shows no sign of slowing; and this leads to a corresponding continual increase in the complexity of the models that can be built and operated with affordable resources within a sensible time. Solving the complex Multiphysics problems discussed in this book will benefit from further increases in computational power. Currently, the largest scale computer modeling problems being studied using supercomputers include climate predictions and nuclear simulations (IBM 2010). Therefore, new types of modeling will become increasingly accessible as hardware becomes cheaper and faster. Complexity of models can be defined by the level of discretization required to accurately depict the

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

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geometry of the food processing system and/or the food product. For example, the model could include both microstructural details (including its composition) of the food product and its arrangement in the processing system at the macroscale level (number of packages or volume of product contained or flowing through the system). The model could also show its behavior at several scales of the process (laboratory, pilot, and commercial scale) in batch or continuous form. This type of model would require multiscale meshing, connecting the mesoscale of the microstructure to the meter-scale, but would be able to predict commercial-scale behavior from lab- or pilot-scale data. The innovative processes covered in this book contain an extra Multiphysics dimension added to conventional processing (see also Chapter 1); that means that in addition to thermofluiddynamic phenomena, electromagnetic, electric energy, and acoustic energy, and high-pressure phenomena, must be included in the models. Therefore, the models are complex in that they integrate equations involving different interacting physical phenomena. The modeling systems require the addition of property changes that are mostly nonlinear, adding a further degree of complexity. For example, viscosity may change by the interactions of pulsed electric fields (PEF) in liquid egg, or the texture of plant material may be affected by the disruption of cellulosic material by means of low-frequency ultrasound. The material interactions within foods are largely controlled by soft matter physics and colloidal interaction for which useful mesoscale models are beginning to be available (e.g., Ubbink et al. 2008; Fujita and Yamaguchi 2010). Currently, most of the models based on computational fluid dynamics (CFD) used in the food industry cannot yet incorporate food microstructural variations into the model. The authors expect that, within only a few years, the gap from current academic research in this area to its integration into CFD models will be bridged. In some cases, these models might have immediate applications in the food industry. Based on the above-defined complexities, the questions that need to be addressed to consider into the future are then:

• What can usefully be modeled? It is now possible to simulate whole processes from an engineering perspective, that is, looking at the process variables and not taking into account the soft physics of the food itself. This book contains chapters that cover modeling case studies with their validation based on process variables; for example, high-pressure low-temperature and high-pressure high-temperature processing (Chapters 4 and 5), microwave heating (Chapters 6 and 7), continuous ohmic heating (Chapter 8), and PEF processing (Chapters 9–11). A few of these chapters refer to models validated at industrial scale, which shows that the processes that are taking place are sufficiently well understood. • What additional data are needed? Several chapters in this book stated the need for data in a number of areas. First, thermophysical properties have to be measured and implemented as a function of process variables to achieve accurate predictions. Second, predictive models representing microbial and/or enzyme inactivation, or chemical reactions as a consequence of the application of Multiphysics technologies, need to be further explored. Nonetheless, little has been reported on the design of accurate Multiphysics models that depict the most important overall quality outcomes of a related product. Without accurate data, the predictions of process models cannot be trusted. • How much more detail is needed? Multiphysics models require a number of assumptions in order to reach a practical solution. It is important to make a point on how much level of detail is really required to satisfy the needs of application of the models in prediction, control, and process and equipment design. For example, the assumptions made in the models such as non-deforming solids in high-pressure processing (HPP; Chapter 5) or the implementation of simple food geometries such as spheres for ohmic heating (Chapter 8) may already provide the level of information required. However, in other cases, for example, where the Multiphysics process variables are influencing the structure of the food materials (e.g., ultrasonic detexturization or high-pressure protein gelation) or causing a certain physical phenomenon such as

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coalescence, agglomeration, or emulsification (e.g., ultrasonic emulsification at low frequencies), a high level of detail will make the models much more useful. This chapter is not going to provide any definite answers but will aim to suggest some future research directions and places where research ought to arrive.

17.2. What Can Usefully Be Modeled? 17.2.1. The Way Forward on Current Innovative Process Modeling This book contains many discussions on innovative processes and how they have been modeled. The main reasons for developing these models are to (1) aid in the design of processing units utilizing specific process physics that can be coupled to thermofluiddynamic models, (2) provide understanding on how these units are to be adjusted to the process line (to make the process more effective and efficient), (3) prove that processes can consistently deliver safe food products, and (4) provide safety, efficiency, and sustainability in the processing lines. Several chapters addressed optimization of innovative processes, predominantly from an engineering perspective. Models take advantage of the capabilities that Multiphysics software provide in coupling other specific process physics to CFD, as well as kinetic equations such as for microbial or enzyme inactivation for process optimization, equipment design, and process scale-up. For example, Chapter 5 shows the application of a CFD model to determine the optimum volume of an insulating food package carrier for a high-pressure thermal processing chamber by evaluating processing scenarios with carriers of various wall thicknesses. Chapter 7 describes the use of a feedback-controlled Multiphysics simulation of a MW process for the determination of a MW power pulse program enabling efficient and uniform MW heating applications. Chapter 10 includes a Multiphysics model used to predict the electric field uniformity and associated temperature peaks for the re-design of a PEF treatment chamber so that better uniformity of both temperature and electric field is achieved. Other

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authors in the PEF modeling area utilized a multiobjective optimization algorithm not only to improve field uniformities, but also to ensure the uniformity of safety and quality (uniform microbial and enzyme inactivation; Chapter 11). The change in physical state has also been considered in previous publications at the mesoscale level. For example, Otero et al. (2007) have utilized CFD to map the process of high-hydrostatic-pressure thawing. Based on this study, models can be developed further that predict changes in fat crystallization in different products at low temperatures. Many of the processes described in this book can now be modeled effectively at the full-process scale. The combination of flow, temperature, and field modeling has been performed in various cases (e.g., Chapters 4 and 5, high pressure processing; Chapters 6 and 7, microwave heating; Chapter 8, ohmic heating; Chapters 9–11, pulsed electric field processing; Chapters 12 and 13, ultrasound processing; Chapters 14 and 15, UV processing), but the next stage must be to extend this to other systems (i.e., new process concepts or modified versions of the processes covered in the book). As computing power increases, so can model complexity. For example, the ohmic model of Chen et al. (2010) assumes plug flow of the particle–liquid mixture, as did the model of Zhang and Fryer (1993), while in reality new computer capacities will allow the modeling of both slip between particles and liquids and a velocity profile across the column, as shown in the experiments of Fairhurst et al. (2001). A further practical stage is to facilitate the processes to become more commercially accepted. This requires processes to produce food products that are clearly better for the consumer—the food industry is about products, not processes (however fascinating the engineers find the processes to be). Models have been built about the need to show safety, but to make a process commercially profitable, the products must be of a quality and at a price that the consumer will buy. There is little in the literature about modeling quality, in part because quality is much more difficult to define than safety. There is some discussion of quality in the modeling of conventional processes, but the information is very

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limited to date. For example, Sendin et al. (2010) developed a multi-objective optimization algorithm for quality improvements in thermal processing (e.g., focusing on color changes and nutrient retention). The next generation of models must involve identification of both the economics (in the form of a cost–benefit model) of the processes and the quality attributes of the products. Furthermore, models must be able to represent the process in its full scale. By optimizing the process in terms of energy and water consumption, models can provide the tool to maximize the sustainability benefit of the process as well. The need is to substantiate that new processes can routinely result in genuinely better products than those that can be made by conventional processing, and at lower (or at least acceptable) costs and in a sustainable manner. In this way, the cost and risk of installing new plants can be mitigated and justified by the models. Furthermore, process plants will need to make a range of products, both because many product launches fail because consumers are not sufficiently interested in them, and because retailers continually demand new products in return for continuing to provide shelf space allocated to specific brands. Hence, the need is to build models that can be used to predict how different formulations will behave, and how to optimize operating conditions to maximize the commercial return, and provide minimum water consumption and greenhouse gas emissions. Current commercial applications of the processes described in this book are limited, but it is likely that the economics will become more favorable in the near future. For example, in the United Kingdom at present the supermarkets have supply chains that can move pasteurized chilled food around the whole country. There is limited impetus from an economics perspective to use innovative processes as long as this supply chain works. As the cost of energy increases, the cost of both refrigerated transport and storage will increase; there will come a point in time where this business model is no longer acceptable. At that point, the production of, for example, ambient stable meals of equivalent quality to cookchilled meals may well become interesting to both

consumers and manufacturers. In the United States and Australia, with much longer supply chains, production and transportation tend to be more regionalized. Nonetheless, often that is not possible and the costs of transport are high. It would be useful to develop models that assist in designing innovative technologies that can provide lower or competitive energy requirements (cost) as well as shelf stability so that supply chain costs are minimized.

17.2.2. Extension of Conventional Processes Modeling to Innovative Processes This book has discussed innovative processes (some of these have been studied for many years). Modeling most conventional food processes already shows a significant level of difficulty due to their degree of complexity; adding a Multiphysics dimension provides an additional challenge. For example, the baking of bread involves • a fluid mixing problem in which the viscosity of the mix changes by several orders of magnitude, creating a highly non-Newtonian fluid in which air is entrained, followed by • a heating process in which a combination of radiative and convective heat transfer to the baked surface, and heat and mass transfer within the baked material, results in expansion of the air matrix and development of the rheological properties of the dough, creating • a viscoelastic solid with a structure involving a hard outer crust and a soft core, both containing air cells of different sizes, which has developed flavors and aromas as a result of sets of reactions that are strongly dependent on both moisture content and temperature (Deshlahra et al. 2009). Any mathematical model accurately describing this process will have to replicate behavior on length scales covering the specific aspects of the chemical reactions taking place (Deshlahra et al. 2009) through to the meter-scale of the baking oven (Wong et al. 2007). In this case, the theory is immensely more complicated than the experiment if nutrient or quality attribute/component variation needs to be considered in the model. In practice to date, models have

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only been developed to optimize sections or aspects of the process, for example, to ensure that heating is even in the oven or to predict crust formation. If the baking of bread is assisted by, for example, radio frequency or microwave heating, the electromagnetic fields have to be simultaneously coupled to the energy, mass, and momentum conservation equations. The model will provide electromagnetic field and temperature distributions according to the interaction of the electromagnetic waves with the oven and in the dough domains. If the resolution of the model is increased to focus on the dough system during processing, the bubble formation can be predicted in a similar way to the conventional model. However, the prediction will be different due to the electromagnetic wave dissipation, which will affect the resulting bread matrix. If the material’s chemistry (e.g., nutrient content; color and flavor development) is also affected by the presence of electromagnetic waves, then kinetic models considering these effects will need to be coupled as well to reach an understanding of the rheological changes in the nanoscale. The case for mathematical modeling of processes does not rest on the need to make the processes work; bakers have existed for thousands of years without finite element, volume, or difference models. In many cases a conceptual model is found to be most useful, such as to know the factors that give rise to good quality, and the role that every ingredient has in the mixture. Such understanding can be used, for example, to reduce incrementally the amount of salt that bread contains. However, as will be discussed later, to deliver the desired step-change in quality, nutrition or process efficiency, numerically based understanding is needed. In the case of bread, understanding is required of how the fields associated with the selected technology affect the properties of the dough, which in turn affect the properties of the baked product. Other examples where conventional processes can be enhanced by an innovative process intervention include: the enhancement of extraction processes by applying ultrasonic waves (possible at various frequencies) at the pre-screw press; ultrasound at low frequencies applied in heat exchangers

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to reduce fouling; or a PEF component to enhance pasteurization processes at lower temperatures. Again, a Multiphysics approach considering the fields associated with the innovative process coupled to equations for mass, momentum, and energy conservation will enable differentiation of the beneficial, or not so beneficial, effects of the innovative intervention. The models will also be useful in providing a picture in terms of the extent of inactivation (both for microbes and enzymes), as well as preservation of color, nutrients (vitamins, proteins, antioxidants), and other quality attributes, and any desirable (or undesirable) material modification (gelation, gelatinization, structure breakage, etc.).

17.2.3. Models for Nutrition and Health of Foods Transformed by Innovative Processing The issue of obesity in the developed world, with major consequences in terms of public health, and the undersupply of food elsewhere, together with the major environmental consequences of food production, require much greater through-chain understanding of food (and feed) production. This will assist not only in increasing farming efficiency and optimizing the supply chain to the consumer and intermediate and final processing, but also in understanding and quantifying how food processing affects digestion. Many food products are made of highly structurally complex components, and breakdown of this structure and nutrient absorption in the body are critical: • breakdown in the mouth determines the taste, texture, and eating pleasure of each product (e.g., Norton et al. 2006; Chen 2009), whilst • breakdown in the stomach and gastro intestinal tract determines what nutrients are transferred to the body and at what rate (e.g., Kong and Singh 2008a, 2008b; Thakaran et al. 2010). The breakdown and nutrient absorption may occur differently at different stages of the digestive tract if the food is transformed by innovative processing after exposure to electric/electromagnetic fields, pressure waves, or high-pressure conditions. To

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control breakdown, so that the food is optimal in the mouth and in the digestive tract, requires a combination of multiscale understanding of food chemistry and material science, together with the information provided by the models discussed in Sections 17.2.1 and 17.2.2, which show how processing affects food structure, chemistry, and quality. However, to do this requires understanding from molecular to macroscales. The gut itself is multiscale, from its meterscale length to the nanoscale of the receptors in the gut wall which receive the nutrients. To explain food uptake, and thus to design foods that deliver targeted benefits to mouth and stomach, more sophisticated modeling is needed. As discussed earlier, quality or sensory attributes can be part of the model prediction or expression according to the process target. Hence, in the same way, parameters that determine the response in the body can be interpreted as a model output. Conceptual understanding of the processes can be used to make new food structures (e.g., by means of ultrasound or HPP), for example materials structured to control the release of macronutrients and slow stomach emptying, thus limiting the amount of food that people effectively consume and retain (Norton et al. 2006, 2007). Computational understanding here is in its infancy, but a full understanding of the effect of a food should involve the physics, chemistry, and biochemistry of its creation in-process, and the ways in which it is broken down, tasted, perceived, and digested in-body.

17.3. What Extra Data Are Needed? A number of modeling challenges have been posed for novel and conventional foods and their processes. A critical issue in any model is the accuracy of the data. The chemical industry has built process models of great complexity but has the advantage of their material properties being thermodynamical state functions, and of a large amount of rate data being available. For example, Rojnuckarin et al. (1996) optimized methane conversion to ethylene and acetylene, using 338 kinetic processes all of which have rate constant data, and more than 50 chemical species for which thermodynamic data are available. In the

food industry, little such data is available, and the problem is much more complicated; even something as simple as the thermal conductivity of apple will depend on the variety, any variation within the variety, and the history (growing conditions, handling, and processing) of the material (apple puree will have different properties from raw apple). The identification of adequate methods to determine the properties just mentioned, for example, at highpressure conditions or under electromagnetic or acoustic fields, will enable better modeling of different innovative processes.

17.3.1. Thermophysical Properties As discussed in Chapter 2, as well as in other chapters of this book, reliable models for innovative processing depend on the accurate determination and implementation of thermophysical properties as a function of the process variables. However, only very limited data on thermophysical properties of foods, as a function of innovative processing variables, is available. Many mathematical models have been developed assuming thermophysical properties of food to be constant during processing, for example, the electrical conductivities used in some models mentioned in Chapter 8. Other examples of thermophysical property equations as a function of process variables can be found throughout the book in the various case studies on modeling innovative process technologies. In particular, Table 2.1 (Chapter 2) presents a summary of the thermophysical properties needed for each processing technology, and processing variables affecting them are described through the chapter. For accurate simulations, accurate data are needed, but it is critical to know the required accuracy level. In the absence of “better” data, the properties of water are often assumed, such as in models for HPP (Knoerzer et al. 2007; Juliano et al. 2009; also Chapters 4 and 5 of this book).

17.3.2. Microbial, Enzyme, Quality, and Nutrient Kinetics As discussed before, Multiphysics models can be used to predict process variables (e.g., temperature, pressure, electric/electromagnetic field, and sound

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intensity) at high temporal and spatial resolution. The predicted process variable distribution can be coupled to other kinetic functions (differential equations) to show the distribution according to the desired target (e.g., microbial inactivation, enzyme degradation, and chemical reactions). For example, the effect of several types of models representing the inactivation of enzymes (Chapter 4) or Clostridium botulinum (Chapter 5) in HPP or enzymes in PEF processing (Chapter 11) has been illustrated. However, so far there is no such model that can predict as an output the overall quality, nutrition, and safety of the product. For instance, there are very few kinetic models that can express C. botulinum inactivation during high-pressure high-temperature processing as function of pressure, temperature, and time in different food matrices (Chapter 5). Likewise, there is little information on the inactivation kinetics during ultrasound and PEF processing for different microorganisms or enzymes as a function of the respective process variables. In some cases the link of pre-processing to microbial inactivation and quality outputs after processing through innovative technologies has been unexplored. For example, kinetic models utilized for high-pressure thermal sterilization processes ignore the fact that the food package requires a preheating step at temperatures between 70 and 90°C, where spore germination can be triggered. The main issue here is that different preheating rates will depend on various parameters (the preheating system, packaging material, package size, and product thermal properties) and therefore impact on the kinetics of germination. Therefore, kinetics will be needed to describe the inactivation of spores as function of preheating rate, preheating temperature, holding time during preheating, and the actual process variables in the high-pressure process (refer to Chapter 5). This is extremely important when establishing the final processing conditions based on microbial outputs. The same might apply to the evaluation of the product quality and nutrient retention. More kinetic models representing the texture and color change (e.g., Leadley et al. 2008), the loss or gain of nutrients during processing, and the change in structural

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and chemical parameters that may control the shelf life of the material are needed as functions of Multiphysics process variables. Once the kinetics of different specified attributes making a final product have been determined, Multiphysics models will be able to provide the full range of attributes describing the product as an outcome of the process. This then will enable the selection of the most appropriate process parameters for the material involved and the desired final outcome or intended use for the product.

17.3.3. Food Structure (Geometry Discretization) Until now, models have been designed by discretizing the equipment parts to map the process variables and performance outputs. When processing semisolid/ solid or structured food materials, some published studies have included the food product in the Multiphysics model. However, this has been mostly done by assuming that the food product is uniform in terms of its composition, structure, and shape (geometrical properties). For example, Chapter 5 discusses cylindrical food packages in high-pressure thermal processing; Chapter 8 mentions the use of spheres in modeling ohmic heating (and other models do not even include semisolid or solid food products!). Better representative models will require the establishment of a baseline structure that includes the compositional distribution, porosity, shape, and corresponding thermophysical properties of each component. One example where such data are included in the model is discussed in the microwave heating scenario of a chicken wing in Chapter 7. However, nutrient distribution is largely unknown in living materials as well as processed foods (e.g., dried, thermally processed, and extruded). Once models are established to describe these distributions and their possible degradation in time, further understanding of these systems will be achieved. Multiphysics modeling will provide the platform for comparing nutrient degradation effects after both conventional and innovative processing. Another example is that Multiphysics modeling can help provide visualization of mass transfer phenomena occurring in ultrasonics processing, which

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is currently not understood. One application is the ultrasonic modification of food structures by enzyme modulation due to a mixture of different mass transfer effects (e.g., enhanced microstreaming and enzyme availability resulting from ultrasoundinduced cell breakage in plant materials). In such cases, the structure of the plant material should be included in the model geometry at the microscale level. Then a Multiphysics model will be able to show the extent of cell breakage and map reaction rates across the structure.

17.4. Where Shall We Stop? Although on a professional level, modelers never believe that the model is finished, in practice, the decision to stop modeling in industrial research is often made when the law of diminishing returns sets in. All of the processes discussed here are currently actual or potential manufacturing processes, and the appropriate use of models will be to enable real products, which have controlled physical and microbiological properties, to be developed and manufactured efficiently. No model can be completely accurate. Often, it is the failure of the model to fit the experimental data that makes the next part of the model possible. At this stage, the role of the modeler is to examine how well the model agrees with the measurements, and then adjust the model or explain the difference— this is not yet an automated process, despite some attempts at automation (King et al. 2009). The need to build better models will reflect continuing uncertainties in how a food product will behave. The computational power will continue to increase to a point where much more detail in the model is possible. Much of that detail is going to be of practical use and value; the ability to incorporate different scales into the model, perhaps combining models for texture or microbiology at the microor mesoscale and their variation with temperature, electric field, and pressure, among other process variables. Continuum mechanics approaches (i.e., considering the various domains of the modeled scenario at the mesoscale level as homogeneous) are the easiest

for modeling the systems studied here; however, approaches such as • lattice-Boltzmann simulations of complex geometries, which is being applied to systems such as extrusion (Buick and Cosgrove 2006) and to the analysis of packed beds (Vidal et al. 2009); • particle dynamics simulations, which are now widespread in particle flow and mixing and in particle–fluid simulations (Lemieux et al. 2008; Van Liedekerke et al. 2009); • combined models, such as the mixture of particle dynamics and CFD of Guo et al. (2010), or the combination of atomic and finite element models used to model crack propagation of Lee et al. (2009) may well be of value. These methods have not been applied significantly in the food industry, but could be applied, if, for example, there are sufficient data available as discussed in Section 17.3. Examples discussed in the book, where more detailed models could provide additional, beneficial information, are included in the section below.

17.4.1. High-Pressure Processing The solid food packages described during highpressure processing (HPP) in Chapter 5 are assumed not to undergo deformation. However, it is well known that there is 15–20% compression of liquids and, therefore, a certain deformation of solids under those conditions, which depends on the food materials contained in the packages. The hydrostatic deformation processes occurring during HPP have not yet been fully quantified in both food structures and other materials contained in the vessel during the process. Models that are able to represent the rate of deformation of packages and contained foods, as well as carriers, inside the vessel will provide further understanding and accuracy on the following: • Heat transfer phenomena at the package level and for different food materials, as well as situations where the vessel is operated at full capacity. • Prediction of the capacity of the vessel, which will then enable maximization of the usable volume and provide information for improving process economics.

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In order to obtain this information, structural deformation kinetics at the mesoscale (millimeter scale) level would suffice. In this case, compositional variations should be coupled with continuum mechanics. However, we believe that not much more useful information for characterizing the industrial process can be gathered if the structural changes at the microscale level are determined. Furthermore, an assumption previously made is that chemical changes will not affect heat transfer. We believe that this assumption is valid in most situations and, therefore, there is no need to detail the model further, at least from the thermofluiddynamic point of view. Most or all other phenomena occurring during HPP can be better explained from CFD modeling at the microscale level. For example, insights into the formation of solid structures due to enhanced gelation processes during HPP or the shrinkage of porous materials can be obtained and incorporated into the models.

17.4.2. PEF Processing One of the uses of PEF is the extraction enhancement of plant cellular material (e.g., beta-carotene from carrot pulp) by means of electroporation. In this type of process, changes provoked at the microstructural level need to be further understood. However, Multiphysics models can assist in providing information on extraction efficiency according to the process conditions selected (electric field strength, flow rate profile, and temperature). If the kinetics of extraction is determined as a function of electric field and temperature, it is already possible to come up with predictive models on efficiency distribution in the chamber without having to implement further microstructural detail. The same applies to the application of PEF for cold pasteurization, where microbial cells are inactivated by means of electroporation (Chapters 9–11). In this case, a microbial inactivation kinetic model will be required as function of electric field, temperature, and other potentially relevant conditions, such as pH.

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17.4.3. Microwave, Radio Frequency, and Ohmic Heating The main challenge often identified about microwave, radio frequency, and ohmic heating is the nonuniformity of the electric and electromagnetic field and the associated uneven temperature distributions in the food material. Currently, Multiphysics models already allow performing prediction in terms of electromagnetic field, temperature, and inactivation/degradation distributions. For this purpose, having the food material defined at the mesoscale level, considering each component as homogeneous, would suffice; for example, a heterogeneous meat material with bones and tissue can be considered as consisting of homogeneous components (Chapter 7). A highly detailed microstructure model is not likely to provide further relevant information on the distribution of the associated field and temperature distributions. The models of microwave and ohmic heating (e.g., Chapters 7 and 8) have also tended to consider only simple solids such as spheres and cylinders, rather than the true shape of the food product including its microstructure. Even though these simplifying assumptions were made, these models have been validated successfully, which shows that they provide a close representation of their application.

17.4.4. Ultrasound Processing Examples of applications for ultrasound processing include low-frequency high-power emulsification; texture modification induced by stress responses in living tissue; as well as particle separations under high-frequency standing waves. Some of these applications may benefit from some degree of visualization at the microscale level to be able to properly characterize the process. In ultrasonic emulsification at low-frequency and high-power conditions, once the field distributions are predicted in an ultrasonic chamber (e.g., pipe or tank), the kinetics of particle size reduction can be applied to map the rate of particle formation, as well as their location. However, bubble cavitation (bubble growth and high-energy implosion) provides a further significant degree of complexity to the

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system. The ultrasound field is scattered and attenuated by the cavitation bubble cloud, which can only be captured adequately if the models can be resolved at the scale of the bubbles. Likewise, liquid or solid particles present in ultrasonic separation applications (by means of agglomeration or coalescence) at high frequencies are likely to alter the ultrasonic field distribution. The “true” impact on the field can only be captured when modeling is conducted at the scale of the suspended particles. However, kinetic models representing the rate of agglomeration of particles under ultrasound standing waves may adequately be able to depict the occurrence of this phenomenon throughout the processing chamber when the model is solved at the mesoscale level. Another approximation made in Chapter 12 is that all acoustic intensity is converted into fluid motion in the form of a jet at the tip of the sonotrode caused by cavitation at low-frequency high-power ultrasound without solving the equation governing the acoustic phenomena. If the Multiphysics model needs to include the ultrasound field and the jets to represent reality, that will require simultaneously solving the governing equations for acoustics and thermofluiddynamics with a higher resolution. This will enable the prediction of cavitation, which leads to the jet formation (fluid motion).

17.4.5. Remaining Questions As discussed above, some of the innovative technologies explored in this book require further understanding of their effects at the microstructural level to provide a more complete picture of their processing outcomes. It is questionable, however, if the influence of all Multiphysics phenomena on food behavior at various scales will ever be completely known. Once these technologies are adopted into food processing lines, it is unlikely that the food industry will need Multiphysics models that exactly predict the behavior of all possible variables inside the food. The level of detail required by the models should be determined by the need of the industry to answer questions, which might be posed by process and product designers as well as by regulators.

Nonetheless, accurate and detailed models may reveal phenomena and effects at the microscale level that are hidden or obscured by effects and observations at the macroscale level. This, in turn, can provide clues for new paradigms in the design of equipment and processes.

17.5. Conclusions The achievable complexity of models is a function of the power of the computers on which they run, and computing power doubles every two years. With the development of new computing models and better hardware, it will become possible to increase the complexity of the models. However, much more basic data are needed on how food structure, food chemistry, and microbiology are affected by pressure, temperature, electromagnetic and electric fields, as well as data that describe the distribution of those fields. Better understanding of the breakdown of foods in the mouth and body, which requires combination of physics and biology, will also aid in the design of products. • What can usefully be modeled? Many of the processes described in this book can now be modeled on an industrial scale, at least in an engineering sense. Models are needed that incorporate new strategies and approaches being developed in other parts of science, such as soft matter physics models, which potentially allow connections to be made between the mesoscale of food structure and the macroscale of process equipment. • What additional data are needed? The limiting factor in accuracy is often the availability of data; simple data can be used to show how processes work; real data is needed for real products and processes. More thermophysical data are needed as well as data that could extend the models to predict quality. Without accurate data, the predictions of process models cannot be relied on. • How much more detail is needed? The models have to make simplifications to work. In cases where either the Multiphysics process variables influence the structure of the food (e.g., ultrasonic detexturization or high-pressure protein gelation)

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The Future of Multiphysics Modeling of Innovative Food Processing Technologies

or result in physical phenomena such as coalescence, agglomeration, or emulsification (e.g., ultrasonic emulsification at low frequencies), more detail will be needed. To work effectively, the models need to be developed in the light of knowledge of how foods behave in the mouth and body, so that consideration of nutrition and health aspects can be added. In conclusion, for innovative processes, Multiphysics models that • allow full processes to be simulated; • incorporate ways of predicting quality as well as safety; • consider the economics of manufacture and supply chain design, addressing issues such as the energy consumption; and • make processes environmentally friendly by reducing water consumption, greenhouse gas emissions, and undesirable by-products should be developed. Any “true” multiscale model of these processes may have to replicate behavior on length scales from the nanometer detail of the chemistry through to the meterscales of manufacture. The availability and ease of use of robust software is critical for wider acceptance and implementation of models. Some of the models still require complex programs that take time and effort to operate; as computing power increases, the programs should develop to the point where the food industry can routinely use and adapt Multiphysics models for equipment and process design, modification, and characterization.

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a centrifugal fertilizer spreader. Powder Technol 190(3): 348–360. Vidal D, Ridgway C, Pianet G, Schoelkopf J, Roy R, Bertrand F. 2009. Effect of particle size distribution and packing compression on fluid permeability as predicted by lattice-Boltzmann simulations. Comput Chem Eng 33(1):256–266. Wong SY, Zhou WB, Hua JS. 2007. CFD modeling of an industrial continuous bread-baking process involving U-movement. J Food Eng 78(3):888–896. Zhang L, Fryer PJ. 1993. Models for the electrical heating of solid liquid food mixtures. Chem Eng Sci 48(4):633–642.

Index

Note: Page numbers in italics refer to figures, those in bold to tables. ABCs, use in computer simulation of microwave heating, 115–116 absorption coefficient, 305 acoustic energy, 240, 241 acoustic impedance, 242–243, 244 acoustic intensity, 240–241 acoustic momentum flow rate, 256 acoustic properties, 31, 35 sound attenuation coefficient, 33–34 speed of sound, 32–33 acoustic radiation pressure, 241– 242, 242 acoustic streaming, 234, 248–250 and acoustic momentum flow rate, 256 Eckart streaming, 250 high power, 255–260, 255, 257, 258, 259 low power, 250–254, 250, 254 Rayleigh streaming, 249 Reynolds stress approach, 250– 251, 250, 252 RNW streaming theory, 251–252 sound absorption, 252–254, 254 sound attenuation, 252–253, 255– 256, 255

Stuart streaming, 245, 256–257 successive approximations method, 250, 250 acoustics, linear, 235–236 acoustic energy, 240, 241 acoustic impedance, 242–243, 244 acoustic intensity, 240–241 acoustic radiation pressure, 241– 242, 242 boundary conditions, 243–244, 243 continuity equation, 236 harmonic waves, 237–239, 238 Helmholtz equation, 239 linear wave equation, 235–237, 246–247 plane sound waves, 239, 241, 243, 243, 244 rays of sound, 245, 245 sinusoidal waves, 237–239, 238 state equation, 236 velocity potential, 237 acoustics, nonlinear, 247–248 adsorption isotherms, 344–345, 344, 346, 347–348, 347, 348 analytical solutions to Maxwell’s equations, 103–104

cylindrical coordinate system, 104, 104 rectangular coordinate system, 103–104, 104 spherical coordinate system, 104– 105, 105 ANNs (artificial neural networks). See neural networks annular UV reactors, 306, 307 artificial neural networks. See neural networks atmospheric freeze drying, 298 balancing equations, 62–64 biodosimetry, 317, 319, 320 boundary conditions in high-pressure high-temperature processing, 81–82, 81, 133 in microwave processing, 102– 103, 113, 141–142 in pulsed electric field processing, 211, 217, 217 in ultrasound processing, 243– 244, 243 bubbles, effect on acoustic properties, 32, 34, 35, 234, 259–260

Innovative Food Processing Technologies: Advances in Multiphysics Simulation, First Edition. Edited by Kai Knoerzer, Pablo Juliano, Peter Roupas, and Cornelis Versteeg. © 2011 by John Wiley & Sons, Ltd. and Institute of Food Technologists. Published by John Wiley & Sons, Ltd. ISBN: 978-0-813-81754-5

365

366

Index

cell membrane rupture, in microwave processing, 8 cellular reactions, influence of pressure, 60–61, 63 CFD. See computational fluid dynamics chemical marker methods for determining microwave heating patterns, 113 chromatographic separation technologies, 335 adsorption isotherms, 344–345, 344, 346, 347–348, 347, 348 equilibrium dispersive model, 341–342, 344 equilibrium (ideal) model, 340, 343 general rate model, 342, 343, 344 ideal (equilibrium) model, 340, 343 modeling approaches, 339–340, 340, 341 See also simulated moving bed chromatography; true moving bed chromatography circuit analogy, in ohmic heating, 159, 160 for pulsed electric field processing, 174, 174 coaxial cylinders, in pulsed electric field processing chamber, 194 co-field design for pulsed electric field processing chamber, 194– 195, 195 coiled tube ultraviolet reactors, 306–307 colinear treatment chamber, in pulsed electric field processing chamber, 194–195, 195, 218– 219, 219 compressibility, and high-pressure processing, 60 compression heating coefficient, 28–30 in high-pressure high-temperature processing, 78–80, 79, 84 computational fluid dynamics, 4, 16, 69–70, 70 dimensionless parameters to express process performance, 93–96, 94

and disinfection of liquid food using ultraviolet light processing, 326–327, 330, 332–333 and high-pressure hightemperature processing, 76, 80–97, 81, 83, 87, 88, 89, 90, 91, 92, 93, 94, 95 and high-pressure processing, 57–59 and inlet velocity, 85–86, 87 investigations of impact of process inhomogeneities on molecular/cellular systems, 67–69, 68, 68, 69 and process sterility, distribution of, 88, 90–93, 90, 91, 92, 93 temperature uniformity/flow, prediction of, 84–88, 87, 88, 89 and ultrasonic streaming, 257– 260, 257, 258, 259 and ultraviolet fluence, 310–311, 311, 318 and ultraviolet light processing, 304, 307–321, 308, 309, 311, 312, 313, 314, 315, 316, 317, 318, 320 validation tools for, 96–97 computational grids, used in pulsed electric field processing simulation, 217–218, 218 computational power, and Multiphysics modeling, 353– 354, 362 computer simulation for microwave processing, 101–102, 125–126 heating, 113–120, 114, 118, 119, 120 sterilization, 121–123, 121, 122, 122, 123, 124, 125, 126 conductivity electrical, 31 thermal, 27–28 constitutive relations, 132–133 continuity equation in linear acoustics, 236 in nonlinear acoustics, 247 continuous chromatographic separation. See chromatographic separation technologies

convective drier, in ultrasonic system for drying of food materials, 282–283, 284, 285, 285–287 cylindrical coordinate system, use in Maxwell’s equations, 104, 104 cylindrical radiator, in ultrasonic system for drying of food materials, 273–274, 274, 276, 276, 277 Dean flow, 309–310, 309 density, 24–25 dielectric breakdowns, in pulsed electric field processing 194 dielectric materials, dissipated power in, 112 dielectric properties, 30–31 diffusion effects, and high-pressure processing, 63 dimensionless parameters to express process performance in highpressure high-temperature processing, 93–96, 94 direct measurements, in model validation, 17–18, 17 disinfection efficiency, 329, 330, 331, 333 disinfection of liquid food using ultraviolet light processing, 325–326, 330, 332–333 computational fluid dynamics modeling, used in, 326–327, 330, 332–333 disinfection efficiency of, 329, 330, 331, 333 reactors for, 325–326 simulation of, 329–330, 329, 331, 332 ultraviolet fluence, 327–329, 330, 331, 332, 332–333 ultraviolet intensity field model, 327 dynamic mixers, used in ultraviolet light processing, 306 Eckart streaming, 250 ECP (elution by characteristic point) method of determining adsorption isotherms, 347

Index

Eigenvalue method, and numerical stability, 109–110 electric field processing, and ohmic heating, 165–166 electric field strength, in pulsed electric field processing, 176 electrical conductivity, 31 and ohmic heating, 158, 159–160, 166 in pulsed electric field processing, 182, 182, 183, 193 electrical properties of foods, and pulsed electric field processing, 182–183, 182, 182, 183 electro-hydrodynamic field in colinear treatment chamber, 218–219, 219 electromagnetic heating. See microwave processing electromagnetic wave equations, 102–103 electroporation in microwave processing, 8 in pulsed electric field processing, 180 in ultrasound processing, 233 elution by characteristic point method of determining adsorption isotherms, 347 EM (electromagnetic wave) equations, 102–103 emulsification, use of ultrasound processing in, 233, 261, 361–362 energy conservation of, 81, 213, 236, 247–248 saving, and ultrasonic system for drying of food materials, 297– 298, 297 transport of, 59–60 enzyme inactivation in pulsed electric field processing, 186–187, 187, 188, 210, 214– 216, 223 in ultrasound processing, 233 equilibrium dispersive model, in chromatographic separation technologies, 341–342, 344

equilibrium (ideal) model, in chromatographic separation technologies, 340, 343 ER (external resistance to mass transfer) models, 290–291, 291, 292, 293, 293, 294 Euler’s formula, 238–239, 239 exponential decay pulses, 174–175, 175, 178, 179 external resistance to mass transfer models, 290–291, 291, 292, 293, 293, 294 extraction, use of ultrasound processing, 233 FDM (finite difference method). See finite difference time domain method FDTD method. See finite difference time domain method FEM method. See finite element method fiber optic probes, use in temperature measurement (mapping) in microwave fields, 135 finite difference method. See finite difference time domain method finite difference time domain method, use in microwave processing, 101–102, 105–110, 107, 113–114 ABCs, 115–116 communication algorithm between electromagnetic and thermal fields, 117–118 electromagnetic energy related to microwave heating, 116 heat transfer equations, 116–117 model formulation, 114–115, 114 source specification, 116 sterilization, simulation model for, 121–123, 121, 122, 122, 123, 124, 125, 126 validation of model, 118–120, 118, 119, 120 finite element method use in microwave processing, 101, 110–111, 110 use in pulsed electric field processing, 196

367

use in ultrasonic systems for drying of food materials, 267– 274, 269, 271, 272, 273, 273, 274, 274, 275, 276, 276–280, 277, 278, 279, 280, 281, 282 first-order microbial inactivation model, 311, 312 flow fields, modeling of, 308, 319–320 Dean flow, 309–310, 309 laminar, 308–309, 308, 312, 312, 316–317 Taylor-Couette flow, 312, 312 turbulent, 309, 312, 312, 316, 317 fluence. See ultraviolet fluence fluid properties, and pulsed electric field processing, 216–217 frontal analysis method of determining adsorption isotherms, 347, 347 Gaussian jet velocity distribution approach, used in modeling ultrasonic streaming, 258–259, 258, 259 general rate model, in chromatographic separation technologies, 342, 343, 344 GJVD (Gaussian jet velocity distribution) approach, used in modeling ultrasonic streaming, 258–259, 258, 259 grids, used in pulsed electric field processing, 200–203, 201, 202, 203 harmonic waves, in linear acoustics, 237–239, 238 health of foods, 357–358 heat conduction losses, 247, 248 heat loss in high-pressure hightemperature processing, 77, 78–80 heat transfer balance, 80, 172 coefficient, 28, 83, 94, 183–184 equations, 80, 112–113, 116–117, 172, 173–174 heating equations, in microwave processing, 112–113, 133

368

Index

heating models, in microwave processing, 140 boundary conditions, 141–142 computational methods, 143, 144, 145 geometry, 140, 141, 142 initial conditions, 141 material properties, 142–143 model food cylinder, controlled heating of, 148–149, 148, 149 process conditions, 140–141 simulated heating of chicken wing, 147–148, 148 validation of simulation, 145, 145, 146, 147, 147, 149–150 Helmholtz equation, in linear acoustics, 239 HHP systems. See high hydrostatic pressure systems high hydrostatic pressure systems macroscopic model for thermal exchange, 50–51, 52 neural networks, use in, 47–49 simple representation/modeling of thermal processes, 49, 49, 50, 51, 52 high-power ultrasonic streaming computational fluid dynamics modeling of, 257–260, 257, 258, 259 and sound attenuation, 255–256, 255 Stuart streaming, 256–257 high-pressure high-temperature processing, 75–80, 77 boundary conditions, 81–82, 81, 133 and compression heating, 78–80, 79, 84 and computational fluid dynamics models, 76, 80–97, 81, 83, 87, 88, 89, 90, 91, 92, 93, 94, 95 dimensionless parameters to express process performance, 93–96, 94 heat loss, 77, 78–80 heat transfer balance, 80 integrated temperature distributor, 94–96, 95

physical properties of foods/other materials, 82–83, 83 preheating methods, 78 process uniformity, 94, 94 and spore inactivation, 75, 88, 89–96, 90, 91, 92, 93, 95, 97 and thermal conductivity, 82 vessel, 76–77, 83–84, 86–88, 88, 89 and viscosity, 82–83 high-pressure processing, 5–7, 5, 57–59, 69–70, 360–361 balancing equations, 62–64 and compressibility, 60 and computational fluid dynamics, 57–59 control of process impact, 64–69, 65, 66 diffusion effects, 63 limitations of, 6–7 macroscopic model for thermal exchange, 50–51, 52 and mass, momentum, and energy, 59–60 mathematical modeling, 61–64 molecular/cellular reactions, influence on, 60–61, 63 neural networks, use in, 46–49 numerical simulation, 61–64 phases of, 66–67, 66 scaling up, 64–67, 65, 66 thermofluiddynamic phenomena under high-pressure conditions, 59–61 See also high hydrostatic pressure systems; high-pressure hightemperature processing high-pressure thermal sterilization, 6–7 high-temperature short-time processing, 155, 156 homogeneity of pulsed electric field processing treatment, 200–203, 201, 202, 203 HPHT processing. See high-pressure high-temperature processing HPP. See high-pressure processing HPTS (high-pressure thermal sterilization), 6–7 HTST (high-temperature short-time processing), 155, 156

ideal (equilibrium) model, in chromatographic separation technologies, 340, 343 inactivation and determining distribution of process sterility, 88, 90–93, 90, 91, 92, 93 enzyme, 186–187, 187, 188, 210, 214–216, 223, 233 microbial, 185–186, 186, 188, 209–210, 214–216, 233, 303, 307, 311–312, 312, 317, 318, 319 and numerical simulations in pulsed electric field processing, 184–187, 186, 187, 188, 221– 223, 221, 222 indirect measurements, in model validation, 17, 18 inflow velocity boundary, in highpressure high-temperature systems, 81–82 infrared imaging, 113 infrared thermography, 135–136 inhomogeneities process, 60, 67–69, 68, 68, 69, 156 temperature, 60 inlet velocity and computational fluid dynamics, 85–86, 87 at laminar conditions, 85 at turbulent conditions, 85–86, 87 innovative food processing technologies and conventional processes modeling, 356–357 and nutrition/health of foods, 357–358 process modeling of, 355–356 and thermophysical properties of food, 23–36, 24, 358 integrated temperature distributor, in high-pressure hightemperature processing, 94–96, 95 IR (infrared) imaging, 113 IR (infrared) thermography, 135–136

Index

kinetic models for enzyme inactivation, 186–187, 187, 188, 214, 215 for microbial inactivation, 185– 186, 186, 188, 214, 215 Kohonen’s self-organization map, 41–43, 42

and sound attenuation, 252–253 successive approximations method vs. Reynolds stress approach, 250–251, 250 low-pressure amalgam lamps, used in ultraviolet light processing, 318–319, 318 low-pressure mercury lamps, used in ultraviolet light processing, 303 LPAL (low-pressure amalgam lamps), used in ultraviolet light processing, 318–319, 318 LPM (low-pressure mercury) lamps, used in ultraviolet light processing, 303

laminar flow fields, modeling of, 308–309, 308, 312, 312, 316–317 lamps used in ultraviolet light processing, 303, 318–319, 318 linear acoustics, 235–236 acoustic energy, 240, 241 acoustic impedance, 242–243, 244 acoustic intensity, 240–241 acoustic radiation pressure, 241– 242, 242 boundary conditions, 243–244, 243 continuity equation, 236 harmonic waves, 237–239, 238 Helmholtz equation, 239 linear wave equation, 235–237, 246–247 plane sound waves, 239, 241, 243, 243, 244 rays of sound, 245, 245 sinusoidal waves, 237–239, 238 state equation, 236 velocity potential, 237 linear wave equation, 235–237, 246–247 liquid crystal foils, use in temperature measurement (mapping) in microwave fields, 136, 136 low power acoustic streaming, 250– 254, 250, 254 RNW streaming theory, 251–252 and sound absorption, 252–254, 254

macroscopic modeling, 49–51, 49, 50, 51, 52, 53 magnetic field coupling, in microwave processing, 8 magnetic resonance imaging, use in temperature measurement (mapping) in microwave fields, 137–140, 137, 138, 139, 140 mass conservation of, 80–81, 172, 213, 236, 247 transport of, 59–60, 291–298, 292, 293, 294, 295, 296 mathematical modeling, and highpressure processing, 61–64 Maxwell’s equations, 102, 132 analytical solutions, 103–105, 104, 105 numerical solutions, 105–112, 107, 110 medium-pressure mercury lamps, used in ultraviolet light processing, 303 mercury lamps, used in ultraviolet light processing, 303 method of moment (MoM), use in microwave processing, 111–112 microbial inactivation in pulsed electric field processing, 185–186, 186, 188, 209–210, 214–216 in ultrasound processing, 233 in ultraviolet processing, 303, 307, 311–312, 312, 317, 318, 319

isothermal pressurization, 60 ITD (integrated temperature distributor), in high-pressure high-temperature processing, 94–96, 95 joule heating, 171, 179, 180

369

microwave processing, 7–9, 131– 132, 361 boundary conditions, 102–103, 113, 133, 141–142 computer simulations for, 101– 102, 113–123, 114, 118, 119, 120, 121, 122, 122, 123, 124, 125, 125–126, 126 constitutive relations, 132–133 electromagnetic wave equations, 102–103 and electroporation, 8 finite difference time domain method, use in, 101–102, 105– 110, 107 finite element method, use in, 101, 110–111, 110 heating equations, 112–113, 133 heating models, 140–143, 141, 142, 144, 145, 145, 146, 147, 147–149, 148, 149 limitations of, 8–9 magnetic field coupling in, 8 Maxwell’s equations, 102, 103– 112, 104, 105, 107, 110, 132 method of moment (MoM), use in, 111–112 nonthermal effects, 8 nonuniformity in, 8–9 selective heating in, 8 sterilization, simulation model for, 121–123, 121, 122, 122, 123, 124, 125, 126 temperature distributions, 131– 132, 149–150 temperature measurement (mapping), in microwave fields, 134–140, 136, 137, 138, 139, 140 temperature responses, measurement of, 113 thermal effects, 8 thermal modeling, 132–134 microwave radiometry, use in temperature measurement (mapping) in microwave fields, 136 minor disturbance (perturbation) method of determining adsorption isotherms, 347–348, 348

370

Index

model food cylinder, controlled heating of, in microwave processing, 148–149, 148, 149 model substances, use in temperature measurement (mapping) in microwave fields, 135 molar absorptivity, 34 molecular reactions, influence of pressure, 60–61, 63 molecular relaxation losses, in acoustic energy, 247 MoM (method of moment), use in microwave processing, 111–112 momentum conservation of, 81, 172, 213, 236–237, 247 transport of, 59–60 MPM (medium-pressure mercury) lamps, used in ultraviolet light processing, 303 MRI (magnetic resonance imaging), use in temperature measurement (mapping) in microwave fields, 137–140, 137, 138, 139, 140 Multiphysics modeling, x, 15–18, 353–363 complexity of, 354, 362 and computational power, 353– 354, 362 definition, 4 and economics, 356 and food structure, 359–360 and kinetics (microbial, enzyme, quality, nutrient), 358–359 and nutrition/health of foods, 357–358, 359 and product quality, 355–356 and sustainability, 356 and thermophysical properties of food, 23–36, 24, 358 validation, 16–18, 17 neglecting external resistance to mass transfer models, 289–290, 290, 292, 293 NER (neglecting external resistance to mass transfer) models, 289– 290, 290, 292, 293

neural networks, 51, 53 basis of, 40–43, 40 BP algorithm, 40–41, 42 in chemical industry, 43–45 in food industry, 45–51 in high-pressure processes, 46–49 history of, 39 supervised learning process, 40, 41 unsupervised learning process, 40, 41–43, 42 weights, 40, 41 Newtonian fluids, 26 NNs. See neural networks nonhomogeneity in processes, 156 nonlinear acoustics, 247–248 non-Newtonian fluids, 26 nonthermal effects of microwave processing, 8 nonuniformity, 3–4, 6–7 in microwave processing, 8–9 in ohmic processing, 10 in pulsed electric field processing, 12 numerical dispersion, and Maxwell’s equations, 108–109 numerical simulations and high-pressure processing, 61–64 of pulsed electric field processing, 204–205, 205, 206, 211, 212– 223, 217, 218, 219, 220, 221, 222 numerical solutions to Maxwell’s equations, 105 finite difference time domain method, 105–110, 107 finite element method, 110–111, 110 method of moment (MoM), 111–112 numerical dispersion, 108–109 numerical stability, 109–110 Yee algorithm, 106–108, 107 numerical stability, and Maxwell’s equations, 109–110 nutrition of foods, 357–358 ohmic heating, 9–10, 155–156, 157, 166–167, 361 circuit analogy, 159, 160

and electric field processing, 165–166 and electrical conductivity, 158, 159–160, 166 equations in, 157–158 limitations of, 10 nonuniformity in, 10 particle-fluid flows, measurements of, 163–164 potential model, 159, 159 and shadow regions, 160, 163, 166 simple models, 159–160, 159, 160 single-phase behavior, 160 3D models, 161–163, 161, 162, 163 2D models, 160, 160–161, 161 use in processing single-phase viscous liquids, 165, 165 validation of, 163–164 worst case, identification of, 162–163 optimization of pulsed electric field processing, 211, 212, 214, 223–227, 224, 226, 227, 228, 229 parallel plates, as design for chamber, 194 particle-fluid flows, in ohmic heating, 163–164 PATP (pressure-assisted thermal processing). See high-pressure high-temperature processing PATS (pressure-assisted thermal sterilization). See high-pressure high-temperature processing PEF processing. See pulsed electric field processing penetration depth, and ultraviolet light absorption in liquid foods, 305 permeabilization of cells, and pulsed electric field processing, 209, 210, 211 perturbation (minor disturbance) method of determining adsorption isotherms, 347–348, 348

Index

physical properties of foods/other materials in high-pressure hightemperature processing, 82–83, 83 piezoelectric sandwich transducer, 268–271, 269, 271, 272 plane sound waves, 239, 241, 243, 243, 244 Poisson’s equations, 172–174 porosity, 25, 297 preheating methods in high-pressure high-temperature processing, 78 pressure boundary, in high-pressure high-temperature systems, 82 pressure-assisted thermal processing. See high-pressure hightemperature processing pressure-assisted thermal sterilization. See high-pressure high-temperature processing process impact, control of, in highpressure processing, 64–69, 65, 66 process inhomogeneities, 60, 67–69, 68, 68, 69, 156 process parameters, in pulsed electric field processing, 219– 221, 220 process sterility, distribution of, 88, 90–93, 90, 91, 92, 93 process uniformity in high-pressure high-temperature processing, 94, 94 product quality, and Multiphysics modeling, 355–356 pulse energy, 178–179 pulse shape, in pulsed electric field processing, 174–175, 176, 176 pulse width, in pulsed electric field processing, 175 pulsed electric field processing, 10–12, 171–172, 187–188, 193–194, 209–212, 210, 361 boundary conditions, 211, 217, 217 circuit for, 174, 174 computational grids, 217–218, 218 dielectric breakdowns in, 194

and electric conductivity, 182, 182, 183, 193 electric field strength, 176 and electrical properties of foods, 182–183, 182, 182, 183 electro-hydrodynamic field in colinear treatment chamber, 218–219, 219 and electroporation, 180 and enzyme inactivation, 186– 187, 187, 188, 210, 214–216, 223 equations in, 172–174, 212–214 finite element method, use in, 196 and fluid properties, 216–217 and heat transfer coefficient, 183–184 homogeneity of treatment, improvement in, 200–203, 201, 202, 203 inactivation models for numerical simulations, 184–187, 186, 187, 188, 221–223, 221, 222 joule heating, 171, 179, 180 limitations of, 11–12 and microbial inactivation, 185– 186, 186, 188, 209–210, 214–216 nonuniformity in, 12 numerical simulations, 204–205, 205, 206, 211, 212–223, 217, 218, 219, 220, 221, 222 optimization of, 211, 212, 214, 223–227, 224, 226, 227, 228, 229 and permeabilization of cells, 209, 210, 211 process parameters, 219–221, 220 pulse energy, 178–179 pulse shape, 174–175, 176, 176 pulse width, 175 residence time, 177, 201 and rheological properties of foods, 181–182 specific energy, 177–179 steady-state simulation of continuous processing, 199– 200, 199, 200, 200 temperature, 179–180, 219–221, 220

371

temperature-time profile, 184, 184 and thermal conductivity, 181 and thermophysical properties of foods, 180–181, 180 treatment chamber, 194–195, 195, 205–206 treatment temperature, 179–180 treatment time, 176–177, 178, 221–223 turbulence simulation, 203–204 unsteady-state simulation of batch processing, 197–198, 197, 198, 198, 199 and viscosity, 181 validation of simulations, 195– 197, 195, 197 voltage, 176 “quartz wind” (Eckart streaming), 250 Rayleigh streaming, 249 rays of sound, 245, 245 reactors in simulated moving bed chromatography, 339 for ultraviolet light processing, 305–307, 309, 309, 313–317, 313, 314, 315, 316, 317, 320, 325–326 rectangular coordinate system, use in Maxwell’s equations, 103– 104, 104 residence time, in pulsed electric field processing, 177, 201 Reynolds stress approach, 250–251, 250, 252 rheological properties of foods, and pulsed electric field processing, 181–182 RNW streaming theory, 251–252 sandwich transducer, 268–271, 269, 271, 272 scaling up, in high-pressure systems, 64–67, 65, 66 scattering, and ultraviolet light absorption in liquid foods, 305 selective heating, in microwave processing, 8

372

Index

self-organizing maps, 41–43, 42 series-event microbial inactivation model, 311–312, 312 shadow regions, in ohmic heating, 160, 163, 166 shrinkage, 25 simulated moving bed chromatography, 335, 336, 337, 339, 349 dynamic methods of determining adsorption isotherms, 347–348 ECP (elution by characteristic point) method of determining adsorption isotherms, 347 elution by characteristic point method of determining adsorption isotherms, 347 frontal analysis method of determining adsorption isotherms, 347, 347 minor disturbance (perturbation) method of determining adsorption isotherms, 347–348, 348 modeling of process, 342–344 node model, 343 parameters, determination of, 346–348 perturbation (minor disturbance) method of determining adsorption isotherms, 347–348, 348 reactors, 339 separation volume theory of design, 346, 346 simulation of, 348–349, 348 static methods of determining adsorption isotherms, 347 triangle theory of design, 345, 346 and true moving bed chromatography model, 342 validation of, 349 variations of, 336–339, 338 single-phase viscous liquids, ohmic processing of, 165, 165 sinusoidal waves, in linear acoustics, 237–239, 238 SMB. See simulated moving bed chromatography

SMBRs (simulated moving bed reactors), 339 SOMs (self-organizing maps), 41–43, 42 sound absorption, 252–254, 254 sound attenuation, 33–34, 252–253, 255–256, 255 sound, rays of, 245, 245 sound, speed of, 32–33 sound waves, plane, 239, 241, 243, 243, 244 specific energy, in pulsed electric field processing, 177–179 specific heat capacity, 27 speed of sound, 32–33 spherical coordinate system, use in Maxwell’s equations, 104–105, 105 spore inactivation in high-pressure high-temperature processing, 75, 88, 89–96, 90, 91, 92, 93, 95, 97 square wave pulses, 174, 175, 178 state equation in linear acoustics, 236 in nonlinear acoustics, 247 static mixers, used in ultraviolet light processing, 306 steady-state simulation of continuous pulsed electric field processing, 199–200, 199, 200, 200 sterilization, simulation model for, 121–123, 121, 122, 122, 123, 124, 125, 126 streaming, acoustic. See acoustic streaming Stuart streaming, 245, 256–257 subdomains, boundaries between, in high-pressure high-temperature systems, 82 successive approximations method, 250, 250 surface heat transfer coefficient, 28 sustainability, and Multiphysics modeling, 356 symmetry boundary, in highpressure high-temperature systems, 81

Taylor/Couette flow modeling of, 312, 312 reactor, used in ultraviolet light processing, 307, 313–317, 313, 314, 315, 316, 317 TCJS (turbulent circular jet solution) approach, used in modeling ultrasonic streaming, 258–259, 258, 259 temperature distributions, in microwave processing, 131– 132, 149–150 temperature, in pulsed electric field processing, 179–180, 184, 184, 219–221, 220 temperature inhomogeneities, 60 temperature measurement (mapping), in microwave fields, 134 fiber optic probes, 135 infrared thermography, 135–136 liquid crystal foils, 136, 136 magnetic resonance imaging, 137–140, 137, 138, 139, 140 microwave radiometry, 136 model substances, 135 thermo paper, 136 thermocouples, 135 temperature responses, measurement of, in microwave processing, 113 temperature uniformity/flow, prediction of, 84–88, 87, 88, 89 temperature-time profile, in pulsed electric field processing, 184, 184 thermal conductivity, 27–28 and high-pressure hightemperature processing, 82 in pulsed electric field processing, 181 thermal diffusivity, 27–28, 82 thermal effects of microwave processing, 8 thermal expansion coefficient, 29, 30, 82 thermal modeling, in microwave processing, 132–134

Index

thermal/pressure processing application of neural networks, 47–49 macroscopic model for thermal exchange, 49–51, 49, 50, 51, 52 thermo paper, use in temperature measurement (mapping) in microwave fields, 136 thermocouples, use in temperature measurement (mapping) in microwave fields, 135 thermofluiddynamic phenomena under high-pressure conditions, 59 influence of pressure on molecular/cellular reactions, 60–61 transport of mass, momentum, and energy, 59–60 thermophysical properties, 23–24, 24, 34–36, 358 acoustic properties, 31–34, 35 and balancing equations, 62–64 compression heating coefficient, 28–30 density, 24–25 dielectric properties, 30–31 electrical conductivity, 31 of foods, and pulsed electric field processing, 180–181, 180 molar absorptivity, 34 porosity, 25 shrinkage, 25 specific heat capacity, 27 thermal conductivity, 27–28 thermal diffusivity, 27–28 thermal expansion coefficient, 29, 30 viscosity, 25–27 thin-film reactors, used in ultraviolet light processing, 306, 308, 308 TMB. See true moving bed chromatography transmittance, and ultraviolet light absorption in liquid foods, 305 treatment chamber, in pulsed electric field processing, 194–195, 195, 205–206, 223–226, 224, 226, 227

treatment temperature, in pulsed electric field processing, 179–180 treatment time, in pulsed electric field processing, 176–177, 178, 221–223 true moving bed chromatography, 335–336, 336, 339, 342 turbulence simulation of continuous pulsed electric field processing, 203–204 turbulent circular jet solution approach, used in modeling ultrasonic streaming, 258–259, 258, 259 turbulent flow fields, modeling of, 309, 312, 312, 316, 317 ultrasonic effect on mass transport phenomena, 291–298, 292, 293, 294, 295, 296 ultrasonic horn reactor, simulations/ modeling of, 245–246, 246, 257–260, 257, 258, 259 ultrasonic systems for drying of food materials, 265–267, 266, 267 acoustic behavior, 276–282, 278, 279, 280, 281, 282, 283, 284, 287, 293, 296 behavior of system during drying, 286–287, 286, 287 computational analysis, 287–298, 288, 290, 291, 292, 293, 294, 295, 296, 297 convective drier, 282–283, 284, 285, 285–287 cylindrical radiator, 273–274, 274, 276, 276, 277 diffusion models for analysis of, 288–291, 290, 291 digital power meter, 285–286 empirical models for analysis of, 287–288, 288 and energy saving, 297–298, 297 external resistance to mass transfer models, 290–291, 291, 292, 293, 293, 294 and freeze drying, 298 future trends in, 298

373

impedance matching unit, 285 neglecting external resistance to mass transfer models, 289–290, 290, 292, 293 numerical study of constituent elements using finite element method, 267–274, 269, 271, 272, 273, 273, 274, 274, 275, 276, 276–280, 277, 278, 279, 280, 281, 282 and porosity of food product, 297 prototype, 280–283, 283, 284, 285 ultrasonic effect on mass transport phenomena, 291–298, 292, 293, 294, 295, 296 ultrasonic power generator, 283, 285 ultrasonic vibrator, 268–273, 269, 271, 272, 273, 273, 274, 275 vibration behavior, 270–271 ultrasonic vibrator, in system for drying of food materials, 268, 272–273, 273, 274, 275 mechanical amplifier, 271–273, 273, 273 piezoelectric sandwich transducer, 268–271, 269, 271, 272 ultrasound processing, 12–14, 233– 234, 260–261, 361–362 acoustic momentum flow rate, 256 acoustic streaming, 234, 245, 248–260, 250, 254, 255, 257, 258, 259 boundary conditions, 243–244, 243 and electroporation, 233 emulsification, use in, 233, 261, 361–362 enzyme inactivation, use in, 233 extraction, use in, 233 and heat conduction losses, 247, 248 limitations of, 13–14 linear acoustics, 235–246, 238, 239, 242, 243, 244, 245, 246 microbial inactivation, use in, 233 and molecular relaxation losses, 247

374

Index

ultrasound processing (continued) nonlinear acoustics, 247 Rayleigh streaming, 249 simulations/modeling of ultrasonic horn reactor, 245– 246, 246, 257–260, 257, 258, 259 sound absorption, 252–254, 254 sound attenuation, 252–253, 255– 256, 255 Stuart streaming, 245, 256–257 and turbulence, 234, 251, 252, 255 and viscous losses, 246, 248, 249 See also ultrasonic systems for drying of food materials ultraviolet fluence, 330 computational fluid dynamics modeling of, 310–311, 311, 318 and disinfection of liquid food, 327–329, 330, 331, 332, 332–333 field, 327–328 value handling at reactor exit, 328–329 ultraviolet intensity field model, 327 ultraviolet light processing, 14–15, 303–304, 320 computational fluid dynamics modeling, used in, 304, 307– 321, 308, 309, 311, 312, 313, 314, 315, 316, 317, 318, 320 and Dean flow, 309–310, 309 and disinfection of liquid food, 325–330, 329, 331, 332, 332– 333, 333

efficacy, factors affecting, 303–304 future research needs, 320–321 limitations of, 15 lamps, 303, 318–319, 318 microbial inactivation in, 303, 307, 311–312, 312, 317, 318, 319 optimum geometry (gap width), calculation of, 312–313, 313, 317 and properties of liquid foods, 304–305 reactors, 305–307, 309, 309, 320 ultraviolet light absorption in liquid foods, 304–305 ultraviolet radiation, modeling of, 310 unsteady-state simulation of batch pulsed electric field processing, 197–198, 197, 198, 198, 199 UV processing. See ultraviolet light processing vacuum freeze drying, 298 validation of computational fluid dynamics models, in ultraviolet light processing, 317–320, 318, 320 direct measurements, in model validation, 17–18, 17 of finite difference time domain method, 118–120, 118, 119, 120 indirect measurements, in model validation, 17, 18 of microwave heating models, 145, 145, 146, 147, 147, 149–150

and Multiphysics modeling, 16–18, 17 of ohmic heating models, 163–164 of pulsed electric field processing models, 195–197, 195, 197 of simulated moving bed chromatography, 349 tools, for computational fluid dynamics, 96–97 velocity potential, in linear acoustics, 237 vessels for high-pressure hightemperature processing, 76–77, 83–84, 86–88, 88, 89 viscosity, 25–27 and high-pressure hightemperature processing, 82–83 and pulsed electric field processing, 181 viscous losses, in acoustic energy, 246, 248, 249 voltage, in pulsed electric field processing, 176 water content effect on specific heat of foods, 27 effect on thermal conductivity of foods, 28 worst case, identification of, and modeling ohmic heating, 162–163 Yee algorithm, and Maxwell’s equations, 106–108, 107