Handbook of food engineering practice

CRC PressBoca Raton New York

Copyright 1997 CRC Press, LLC

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Library of Congress Cataloging-in-Publication Data

Handbook of food engineering practice / edited by Enrique Rotstein, R. Paul Singh, and Kenneth J. Valentas.

p. cm.Includes bibliographical references and index.ISBN 0-8493-8694-2 (alk. paper)1. Food industry and trade--Handbooks, manuals, etc.

I. Rotstein, Enrique. II. Singh, R. Paul. III. Valentas, KennethJ., 1938- .TP370.4.H37 1997664--dc21 96-53959

CIP

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The Editors

Enrique Rotstein, Ph.D.,

is Vice President of Process Technology of the Pillsbury Company,Minneapolis, Minnesota. He is responsible for corporate process development, serving allthe different product lines of his company.

Dr. Rotstein received his bachelors degree in Chemical Engineering from Universidaddel Sur, Bahia Blanca, Argentina. He obtained his Ph.D. from Imperial College, Universityof London, London, U.K. He served successively as Assistant, Associate, and Full Professorof Chemical Engineering at Universidad del Sur. In this capacity he founded and directedPLAPIQUI, Planta Piloto de Ingenieria Quimica, one of the leading Chemical Engineeringteaching and research institutes in Latin America. During his academic career he also taughtat the University of Minnesota and at Imperial College, holding visiting professorships. Heworked for DuPont, Argentina, and for Monsanto Chemical Co., Plastics Division. In 1987he joined The Pillsbury Company as Director of Process Analysis and Director of ProcessEngineering. He assumed his present position in 1995.

Dr. Rotstein has been a member of the board of the Argentina National Science Council,a member of the executive editorial committee of the

Latin American Journal of ChemicalEngineering and Applied Chemistry

, a member of the internal advisory board of DryingTechnology, and a member of the editorial advisory boards of

Advances in Drying, PhysicoChemical Hydrodynamics Journal

, and

Journal of Food Process Engineering

. Since 1991 hehas been a member of the Food Engineering Advisory Council, University of California,Davis. He received the Jorge Magnin Prize from the Argentina National Science Council, wasHill Visiting Professor at the University of Minnesota Chemical Engineering and MaterialsScience Department, was keynote lecturer at a number of international technical conferences,and received the Excellence in Drying Award at the 1992 International Drying Symposium.

Dr. Rotstein is the author of nearly 100 papers and has authored or co-authored several books.

R. Paul Singh, Ph.D.,

is a Professor of Food Engineering, Department of Biological andAgricultural Engineering, Department of Food Science and Technology, University of Cali-fornia, Davis.

Dr. Singh graduated in 1970 from Punjab Agricultural University, Ludhiana, India, witha degree in Agricultural Engineering. He obtained an M.S. degree from the University ofWisconsin, Madison, and a Ph.D. degree from Michigan State University in 1974. Followinga year of teaching at Michigan State University, he moved to the University of California,Davis, in 1975 as an Assistant Professor of Food Engineering. He was promoted to AssociateProfessor in 1979 and, again, to Professor in 1983.

Dr. Singh is a member of the Institute of Food Technologists, American Society ofAgricultural Engineers, and Sigma Xi. He received the Samuel Cate Prescott Award forResearch, Institute of Food Technologies, in 1982, and the A. W. Farrall Young EducatorAward, American Society of Agricultural Engineers in 1986. He was a NATO Senior GuestLecturer in Portugal in 1987 and 1993, and received the IFT International Award, Instituteof Food Technologists, 1988, and the Distinguished Alumnus Award from Punjab AgriculturalUniversity in 1989, and the DFISA/FPEI Food Engineering Award in 1997.

Dr. Singh has authored and co-authored nine books and over 160 technical papers. Heis a co-editor of the

Journal of Food Process Engineering.

His current research interests arein studying transport phenomena in foods as influenced by structural changes during processing.


Kenneth J. Valentas, Ph.D.,

is Director of the Bioprocess Technology Institute and AdjunctProfessor of Chemical Engineering at the University of Minnesota. He received his B.S. inChemical Engineering from the University of Illinois and his Ph.D. in Chemical Engineeringfrom the University of Minnesota.

Dr. Valentas career in the Food Processing Industry spans 24 years, with experience inResearch and Development at General Mills and Pillsbury and as Vice President of Engi-neering at Pillsbury-Grand Met. He holds seven patents, is the author of several articles, andis co-author of

Food Processing Operations and Scale-Up.

Dr. Valentas received the Food, Pharmaceutical, and Bioengineering Division Awardfrom AIChE in 1990 for outstanding contributions to research and development in the foodprocessing industry and exemplary leadership in the application of chemical engineeringprinciples to food processing.

His current research interests include the application of biorefining principles to foodprocessing wastes and production of amino acids via fermentation from thermal tolerantmethlyotrophs.


Contributors

Ed Boehmer

StarchTech, Inc.Golden Valley, Minnesota

David Bresnahan

Kraft Foods, Inc.Tarrytown, New York

Chin Shu Chen

Citrus Research and Education CenterUniversity of FloridaLake Alfred, Florida

Julius Chu

The Pillsbury CompanyMinneapolis, Minnesota

J. Peter Clark

Fluor Daniel, Inc.Chicago, Illinois

Donald J. Cleland

Centre for Postharvest and Refrigeration Research

Massey UniversityPalmerston North, New Zealand

Guillermo H. Crapiste

PLAPIQUIUniversidad Nacional del SurCONICETBahia Blanca, Argentina

Brian E. Farkas

Department of Food ScienceNorth Carolina State UniversityRaleigh, North Carolina

Daniel F. Farkas

Department of Food Science and Technology

Oregon State UniversityCorvallis, Oregon

Ernesto Hernandez

Food Protein Research and Development Center

Texas A & M UniversityCollege Station, Texas

Ruben J. Hernandez

School of PackagingMichigan State UniversityEast Lansing, Michigan

Theodore P. Labuza

Department of Food Science and NutritionUniversity of MinnesotaSt. Paul, Minnesota

Leon Levine

Leon Levine & Associates, Inc.Plymouth, Minnesota

Jorge E. Lozano


Jatal D. Mannapperuma

California Institute of Food and Agricultural Research

Department of Food Science and TechnologyUniversity of California, DavisDavis, California

Martha Muehlenkamp

Department of Food Science and NutritionUniversity of MinnesotaSt. Paul, Minnesota

Hosahilli S. Ramaswamy

Department of Food Science and Agricultural Chemistry

MacDonald Campus of McGill UniversitySte. Anne de Bellevue, QuebecCanada


Enrique Rotstein

The Pillsbury CompanyMinneapolis, Minnesota

I. Sam Saguy

Department of Biochemistry, Food Science, and Nutrition

Faculty of AgricultureThe Hebrew University of JerusalemRehovot, Israel

Dale A. Seiberling

Seiberling Associates, Inc.Roscoe, Illinois

R. Paul Singh

Department of Biological and Agricultural Engineering and

Department of Food Science and TechnologyUniversity of California, DavisDavis, California

James F. Steffe

Department of Agricultural Engineering and Department of Food Science and Human Nutrition

Michigan State UniversityEast Lansing, Michigan

Petros S. Taoukis

Department of Chemical EngineeringLaboratory of Food Chemistry

and TechnologyNational Technical University of AthensAthens, Greece

Martin J. Urbicain


Kenneth J. Valentas

University of MinnesotaSt. Paul, Minnesota

Joseph J. Warthesen

Department of Food Science and Nutrition

University of MinnesotaSt. Paul, Minnesota

John Henry Wells

Department of Biological and Agricultural Engineering

Louisiana State University Agricultural Center

Baton Rouge, Louisiana


Preface

The food engineering discipline has been gaining increasing recognition in the food industryover the last three decades. Although food engineers formally graduated as such are relativelyfew, food engineering practitioners are an essential part of the food industrys workforce.The significant contribution of food engineers to the industry is documented in the constantstream of new food products and their manufacturing processes, the capital projects toimplement these processes, and the growing number of patents and publications that spanthis emerging profession.

While a number of important food engineering books have been published over the years,the

Handbook of Food Engineering Practice

will stand alone for its emphasis on practicalprofessional application. This handbook is written for the food engineer and food manufac-turer. The very fact that this is a book for industrial application will make it a useful sourcefor academic teaching and research.

A major segment of this handbook is devoted to some of the most common unit operationsemployed in the food industry. Each chapter is intended to provide terse, to-the-point descrip-tions of fundamentals, applications, example calculations, and, when appropriate, a reviewof economics.

The introductory chapter addresses one of the key needs in any food industrynamely the design of pumping systems. This chapter provides mathematical pro-cedures appropriate to liquid foods with Newtonian and non-Newtonian flow char-acteristics. Following the ubiquitous topic of pumping, several food preservationoperations are considered. The ability to mathematically determine a food steril-ization process has been the foundation of the food canning industry. During thelast two decades, several new approaches have appeared in the literature that provideimproved calculation procedures for determining food sterilization processes.

Chapter 2 provides an in-depth description of several recently developed methodswith solved examples.

Chapter 3 is a comprehensive treatment of food freezing operations. This chapterexamines the phase change problem with appropriate mathematical procedures thathave proven to be most successful in predicting freezing times in food. The dryingprocess has been used for millennia to preserve foods, yet a quantitative descriptionof the drying process remains a challenging exercise.

Chapter 4 presents a detailed background on fundamentals that provide insight intosome of the mechanisms involved in typical drying processes. Simplified mathe-matical approaches to designing food dryers are discussed. In the food industry,concentration of foods is most commonly carried out either with membranes orevaporator systems. During the last two decades, numerous developments havetaken place in designing new types of membranes.

Chapter 5 provides an overview of the most recent advances and key informationuseful in designing membrane systems for separation and concentration purposes.

The design of evaporator systems is the subject of Chapter 6. The procedures givenin this chapter are also useful in analyzing the performance of existing evaporators.

One of the most common computations necessary in designing any evaporator iscalculating the material and energy balance. Several illustrative approaches on howto conduct material and energy balances in food processing systems are presentedin Chapter 7.


After processing, foods must be packaged to minimize any deleterious changes inquality. A thorough understanding of the barrier properties of food packagingmaterials is essential for the proper selection and use of these materials in thedesign of packaging systems. A comprehensive review of commonly availablepackaging materials and their important properties is presented in Chapter 8.

Packaged foods may remain for considerable time in transport and in wholesale andretail storage. Accelerated storage studies can be a useful tool in predicting the shelflife of a given food; procedures to design such studies are presented in Chapter 9.

Among various environmental factors, temperature plays a major role in influencingthe shelf life of foods. The temperature tolerance of foods during distribution mustbe known to minimize changes in quality deterioration. To address this issue,approaches to determine temperature effects on the shelf life of foods are given inChapter 10.

In designing and evaluating food processing operations, a food engineer relies onthe knowledge of physical and rheological properties of foods. The publishedliterature contains numerous studies that provide experimental data on food prop-erties. In Chapter 11, a comprehensive resource is provided on predictive methodsto estimate physical and rheological properties.

The importance of physical and rheological properties in designing a food systemis further illustrated in Chapter 12 for a dough processing system. Dough rheologyis a complex subject; an engineer must rely on experimental, predictive, andmathematical approaches to design processing systems for manufacturing dough,as delineated in this chapter.

The last five chapters in this handbook provide supportive material that is applicable toany of the unit operations presented in the preceding chapters.

For example, estimation of cost and profitability one of the key calculations thatmust be carried out in designing new processing systems. Chapter 13 providesuseful methods for conducting cost/profit analyses along with illustrative examples.

As computers have become more common in the workplace, use of simulationsand optimization procedures are gaining considerable attention in the food industry.Procedures useful in simulation and optimization are presented in Chapter 14.

In food processing, it is imperative that any design of a system adheres to a varietyof sanitary guidelines. Chapter 15 includes a broad description of issues that mustbe considered to satisfy these important guidelines.

The use of process controllers in food processing is becoming more prevalent asimproved sensors appear in the market. Approaches to the design and implementationof process controllers in food processing applications are discussed in Chapter 16.

Food engineers must rely on a number of basic sciences in dealing with problemsat hand. An in-depth knowledge of food chemistry is generally regarded as one ofthe most critical. In Chapter 17, an overview of food chemistry with specificreference to the needs of engineers is provided.

It should be evident that this handbook assimilates many of the key food processingoperations. Topics not covered in the current edition, such as food extrusion, microwaveprocessing, and other emerging technologies, are left for future consideration. While werealize that this book covers new ground, we hope to hear from our readers, to benefit fromtheir experience in future editions.

Enrique RotsteinR. Paul SinghKenneth Valentas


Table of Contents

Chapter 1

Pipeline Design Calculations for Newtonian and Non-Newtonian Fluids

James F. Steffe and R. Paul Singh

Chapter 2

Sterilization Process Engineering

Hosahalli S. Ramaswamy, and R. Paul Singh

Chapter 3

Prediction of Freezing Time and Design of Food Freezers

Donald J. Cleland and Kenneth J. Valentas

Chapter 4

Design and Performance Evaluation of Dryers

Guillermo H. Crapiste and Enrique Rotstein

Chapter 5

Design and Performance Evaluation of Membrane Systems

Jatal D. Mannapperuma

Chapter 6

Design and Performance Evaluation of Evaporation

Chin Shu Chen and Ernesto Hernandez

Chapter 7

Material and Energy Balances

Brian E. Farkas and Daniel F. Farkas

Chapter 8

Food Packaging Materials, Barrier Properties, and Selection

Ruben J. Hernandez

Chapter 9

Kinetics of Food Deterioration and Shelf-Life Prediction

Petros S. Taoukis, Theodore P. Labuza, and I. Sam Saguy

Chapter 10

Temperature Tolerance of Foods during Distribution

John Henry Wells and R. Paul Singh


Chapter 11

Definition, Measurement, and Prediction of Thermophysical and Rheological Properties

Martin J. Urbicain and Jorge E. Lozano

Chapter 12

Dough Processing Systems

Leon Levine and Ed Boehmer

Chapter 13

Cost and Profitability Estimation

J. Peter Clark

Chapter 14

Simulation and Optimization

Enrique Rotstein, Julius Chu, and I. Sam Saguy

Chapter 15

CIP Sanitary Process Design

Dale A. Seiberling

Chapter 16

Process Control

David Bresnahan

Chapter 17

Food Chemistry for Engineers

Joseph J. Warthesen and Martha R. Meuhlenkamp


1

Pipeline Design Calculations for Newtonian and Non-Newtonian Fluids

James F. Steffe and R. Paul Singh

CONTENTS

1.1 Introduction 1.2 Mechanical Energy Balance

1.2.1 Fanning Friction Factor1.2.1.1 Newtonian Fluids 1.2.1.2 Power Law Fluids1.2.1.3 Bingham Plastic Fluids 1.2.1.4 Herschel-Bulkley Fluids1.2.1.5 Generalized Approach to Determine Pressure Drop in a Pipe

1.2.2 Kinetic Energy Evaluation1.2.3 Friction Losses: Contractions, Expansions, Valves, and Fittings

1.3 Example Calculations1.3.1 Case 1: Newtonian Fluid in Laminar Flow1.3.2 Case 2: Newtonian Fluid in Turbulent Flow1.3.3 Case 3: Power Law Fluid in Laminar Flow1.3.4 Case 4: Power Law Fluid in Turbulent Flow 1.3.5 Case 5: Bingham Plastic Fluid in Laminar Flow1.3.6 Case 6: Herschel-Bulkley Fluid in Laminar Flow

1.4 Velocity Profiles in Tube Flow 1.4.1 Laminar Flow1.4.2 Turbulent Flow

1.4.2.1 Newtonian Fluids1.4.2.2 Power Law Fluids

1.5 Selection of Optimum Economic Pipe DiameterNomenclature References


1.1 INTRODUCTION

The purpose of this chapter is to provide the practical information necessary to predictpressure drop for non-time-dependent, homogeneous, non-Newtonian fluids in tube flow. Theintended application of this material is pipeline design and pump selection. More informationregarding pipe flow of time-dependent, viscoelastic, or multi-phase materials may be foundin Grovier and Aziz (1972), and Brown and Heywood (1991). A complete discussion ofpipeline design information for Newtonian fluids is available in Sakiadis (1984). Methodsfor evaluating the rheological properties of fluid foods are given in Steffe (1992) and typicalvalues are provided in Tables 1.1, 1.2, and 1.3. Consult Rao and Steffe (1992) for additionalinformation on advanced rheological techniques.

1.2 MECHANICAL ENERGY BALANCE

A rigorous derivation of the mechanical energy balance is lengthy and beyond the scope of thiswork but may be found in Bird et al. (1960). The equation is a very practical form of theconservation of energy equation (it can also be derived from the principle of conservation ofmomentum (Denn, 1980)) commonly called the engineering Bernouli equation (Denn, 1980;Brodkey and Hershey, 1988). Numerous assumptions are made in developing the equation:constant fluid density; the absence of thermal energy effects; single phase, uniform materialproperties; uniform equivalent pressure (

g h term over the cross-section of the pipe is negligible).The mechanical energy balance for an incompressible fluid in a pipe may be written as

(1.1)

where

F, the summation of all friction losses is

(1.2)

and subscripts 1 and 2 refer to two specific locations in the system. The friction losses includethose from pipes of different diameters and a contribution from each individual valve, fitting,etc. Pressure losses in other types of in-line equipment, such as strainers, should also beincluded in

F.

1.2.1 F

ANNING

F

RICTION

F

ACTOR

In this section, friction factors for time-independent fluids in laminar and turbulent flow arediscussed and criteria for determining the flow regime, laminar or turbulent, are presented.It is important to note that it is impossible to accurately predict transition from laminar toturbulent flow in actual processing systems and the equations given are guidelines to be usedin conjunction with good judgment. Friction factor equations are only presented for smoothpipes, the rule for sanitary piping systems. Also, the discussion related to the turbulent flowof high yield stress materials has been limited for a number of reasons: (a) Friction factorequations and turbulence criteria have limited experimental verification for these materials;(b) It is very difficult (and economically impractical) to get fluids with a significant yieldstress to flow under turbulent conditions; and (c) Rheological data for foods that have a highyield stress are very limited. Yield stress measurement in food materials remains a difficulttask for rheologists and the problem is often complicated by the presence of time-dependentbehavior (Steffe, 1992).

u ug z z P P F W2

2

2

12

12 1

2 1 0( ) ( ) + ( ) +

+ + =

Ff u L

Dk uf

=

( )+

( )22

12 2


The Fanning friction factor () is proportional to the ratio of the wall shear stress in apipe to the kinetic energy per unit volume:

(1.3)

TABLE 1.1 Rheological Properties of Dairy, Fish, and Meat Products

ProductT

(C)n

()K

(Pas

n

)

o

(Pa)

(s

1

)

Cream, 10% fat 40 1.0 .00148 60 1.0 .00107 80 1.0 .00083

Cream, 20% fat 40 1.0 .00238 60 1.0 .00171 80 1.0 .00129

Cream, 30% fat 40 1.0 .00395 60 1.0 .00289 80 1.0 .00220

Cream, 40% fat 40 1.0 .00690 60 1.0 .00510 80 1.0 .00395

Minced fish paste 36 .91 8.55 1600.0 67238Raw, meat batters

15

a

13

b

68.8

c

15 .156 639.3 1.53 30050018.7 12.9 65.9 15 .104 858.0 .28 30050022.5 12.1 63.2 15 .209 429.5 0 30050030.0 10.4 57.5 15 .341 160.2 27.8 30050033.8 9.5 54.5 15 .390 103.3 17.9 30050045.0 6.9 45.9 15 .723 14.0 2.3 30050045.0 6.9 45.9 15 .685 17.9 27.6 30050067.3 28.9 1.8 15 .205 306.8 0 300500

Milk, homogenized 20 1.0 .002000 30 1.0 .001500 40 1.0 .001100 50 1.0 .000950 60 1.0 .000775 70 1.0 .00070 80 1.0 .00060

Milk, raw 0 1.0 .00344 5 1.0 .00305

10 1.0 .00264 20 1.0 .00199 25 1.0 .00170 30 1.0 .00149 35 1.0 .00134 40 1.0 .00123

a

%Fat

b

%Protein

c

%Moisture Content

From Steffe, J. F. 1992.

Rheological Methods in Food Process Engineering

.

Freeman Press, East Lansing, MI. With permission.

fu

w= ( )

22


can be considered in terms of pressure drop by substituting the definition of the shear stressat the wall:

(1.4)

TABLE 1.2Rheological Properties of Oils and Miscellaneous Products

Product % Total solidsT

(C)n

()K

(Pas

n

)

o

(Pa)

(s

1

)

Chocolate, melted 46.1 .574 .57 1.16Honey

Buckwheat 18.6 24.8 1.0 3.86Golden Rod 19.4 24.3 1.0 2.93Sage 18.6 25.9 1.0 8.88Sweet Clover 17.0 24.7 1.0 7.20White Clover 18.2 25.2 1.0 4.80

Mayonnaise 25 .55 6.4 30130025 .60 4.2 401100

Mustard 25 .39 18.5 30130025 .34 27.0 401100

OilsCastor 10 1.0 2.42

30 1.0 .45140 1.0 .231

100 1.0 .0169Corn 38 1.0 .0317

25 1.0 .0565Cottonseed 20 1.0 .0704

38 1.0 .0306Linseed 50 1.0 .0176

90 1.0 .0071Olive 10 1.0 .1380

40 1.0 .036370 1.0 .0124

Peanut 25.5 1.0 .065638.0 1.0 .025121.1 1.0 .0647 .326437.8 1.0 .0387 .326454.4 1.0 .0268 .3264

Rapeseed 0.0 1.0 2.53020.0 1.0 .16330.0 1.0 .096

Safflower 38.0 1.0 .028625.0 1.0 .0522

Sesame 38.0 1.0 .0324Soybean 30.0 1.0 .0406

50.0 1.0 .020690.0 1.0 .0078

Sunflower 38.0 1.0 .0311


Rheological Methods in Food Process Engineering

. Freeman Press,East Lansing, MI. With permission.

f P RLu

P DLu

=( )

=( )

2 22


TABLE 1.3 Rheological Properties of Fruit and Vegetable Products

ProductTotal solids

(%)T

(C)n

()K

(Pas

n

)

(s

1

)

ApplePulp 25.0 .084 65.03 Sauce 11.6 27 .28 12.7 160340

11.0 30 .30 11.6 55011.0 82.2 .30 9.0 55010.5 26 .45 7.32 .7812609.6 26 .45 5.63 .7812608.5 26 .44 4.18 .781260

ApricotsPuree 17.7 26.6 .29 5.4

23.4 26.6 .35 11.2 41.4 26.6 .35 54.0 44.3 26.6 .37 56.0 .58051.4 26.6 .36 108.0 .58055.2 26.6 .34 152.0 .58059.3 26.6 .32 300.0 .580

Reliable, conc.Green 27.0 4.4 .25 170.0 3.3137

27.0 25 .22 141.0 3.3137Ripe 24.1 4.4 .25 67.0 3.3137

24.1 25 .22 54.0 3.3137Ripened 25.6 4.4 .24 85.0 3.3137

25.6 25 .26 71.0 3.3137Overripe 26.0 4.4 .27 90.0 3.3137

26.0 25 .30 67.0 3.3137Banana

Puree A 23.8 .458 6.5 Puree B 23.8 .333 10.7 Puree (17.7 Brix) 22 .283 107.3 28200

Blueberry, pie filling 20 .426 6.08 3.3530Carrot, Puree 25 .228 24.16 Green Bean, Puree 25 .246 16.91 Guava, Puree (10.3 Brix) 23.4 .494 39.98 15400Mango, Puree (9.3 Brix) 24.2 .334 20.58 151000Orange Juice

ConcentrateHamlin, early 25 .585 4.121 050042.5 Brix 15 .602 5.973 0500

0 .676 9.157 0500 10 .705 14.255 0500

Hamlin, late 25 .725 1.930 050041.1 Brix 15 .560 8.118 0500

0 .620 1.754 0500 10 .708 13.875 0500

Pineapple, early 25 .643 2.613 050040.3 Brix 15 .587 5.887 0500

0 .681 8.938 0500 10 .713 12.184 0500


Pineapple, late 25 .532 8.564 050041.8 Brix 15 .538 13.432 0500

0 .636 18.584 0500 10 .629 36.414 0500

Valencia, early 25 .583 5.059 050043.0 Brix 15 .609 6.714 0500

10 .619 27.16 0500Valencia, late 25 .538 8.417 050041.9 Brix 15 .568 11.802 0500

0 .644 18.751 0500 10 .628 41.412 0500

Naval65.1 Brix 18.5 .71 29.2

14.1 .76 14.6 9.3 .74 10.8 5.0 .72 7.9 0.7 .71 5.9 10.1 .73 2.7 29.9 .72 1.6 29.5 .74 .9

Papaya, puree (7.3 Brix) 26.0 .528 9.09 20450Peach

Pie Filling 20.0 .46 20.22 1140Puree 10.9 26.6 .44 .94

17.0 26.6 .55 1.38 21.9 26.6 .55 2.11 26.0 26.6 .40 13.4 80100029.6 26.6 .40 18.0 80100037.5 26.6 .38 44.0 40.1 26.6 .35 58.5 230049.8 26.6 .34 85.5 230058.4 26.6 .34 440.0 11.7 30.0 .28 7.2 55011.7 82.2 .27 5.8 55010.0 27.0 .34 4.5 1603200

PearPuree 15.2 26.6 .35 4.25

24.3 26.6 .39 5.75 33.4 26.6 .38 38.5 80100037.6 26.6 .38 49.7 39.5 26.6 .38 64.8 230047.6 26.6 .33 120.0 .5100049.3 26.6 .34 170.0 51.3 26.6 .34 205.0 45.8 32.2 .479 35.5 45.8 48.8 .477 26.0 45.8 65.5 .484 20.0 45.8 82.2 .481 16.0 14.0 30.0 .35 5.6 55014.0 82.2 .35 4.6 550

TABLE 1.3 (continued)Rheological Properties of Fruit and Vegetable Products

ProductTotal solids

(%)T

(C)n

()K

(Pas

n

)

(s

1

)


Simplification yields the energy loss per unit mass required in the mechanical energy balance:

(1.5)

There are many mathematical models available to describe the behavior of fluid foods(Ofoli et al., 1987); only those most useful in pressure drop calculations have been includedin the current work. The simplest model, which adequately describes the behavior of the foodshould be used; however, oversimplification can cause significant calculation errors (Steffe,1984).

1.2.1.1 Newtonian Fluids

The volumetric average velocity for a Newtonian fluid (

=

in laminar, tube flow is

(1.6)

PlumPuree 14.0 30.0 .34 2.2 550

14.0 82.2 .34 2.0 550Squash

Puree A 25 .149 20.65 Puree B 25 .281 11.42

TomatoJuice conc. 5.8 32.2 .59 .22 500800

5.8 38.8 .54 .27 5008005.8 65.5 .47 .37 500800

12.8 32.2 .43 2.0 50080012.8 48.8 .43 2.28 50080012.8 65.5 .34 2.28 50080012.8 82.2 .35 2.12 50080016.0 32.2 .45 3.16 50080016.0 48.8 .45 2.77 50080016.0 65.5 .40 3.18 50080016.0 82.2 .38 3.27 50080025.0 32.2 .41 12.9 50080025.0 48.8 .42 10.5 50080025.0 65.5 .43 8.0 50080025.0 82.2 .43 6.1 50080030.0 32.2 .40 18.7 50080030.0 48.8 .42 15.1 50080030.0 65.5 .43 11.7 50080030.0 82.2 .45 7.9 500800


Rheological Methods in Food Process Engineering.

Freeman Press,East Lansing, MI. With permission.

TABLE 1.3 (continued)Rheological Properties of Fruit and Vegetable Products

ProductTotal solids

(%)T

(C)n

()K

(Pas

n

)

(s

1

)

P f Lu

D( )

=2 2

uQR R

P RL

P DL

= =( )

=

( )

2 2

4 218 32


Solving Equation 1.6 for the pressure drop per unit length gives

(1.7)

Inserting Equation 1.7 into the definition of the Fanning friction factor, Equation 1.4, yields

(1.8)

which can be used to predict friction factors in the laminar flow regime, N

Re

< 2100 whereN

Re

=

D u/

. In turbulent flow, N

Re

> 4000, the von Karman correlation is recommended(Brodkey and Hershey, 1988):

(1.9)

The friction factor in the transition range, approximately 2100 < N

Re

< 4000, cannot bepredicted but the laminar and turbulent flow equations can be used to establish appropriatelimits.

1.2.1.2 Power Law Fluids

The power law fluid model (

= K (

)

n

) is one of the most useful in pipeline design work fornon-Newtonian fluids. It has been studied extensively and accurately expresses the behavior ofmany fluid foods which commonly exhibit shear-thinning (0 < n < 1) behavior. The volumetricflow rate of a power law fluid in a tube may be calculated in terms of the average velocity:

(1.10)

meaning

(1.11)

which, when inserted into Equation 1.4, gives an expression analogous to Equation 1.8:

(1.12)

where the power law Reynolds number is defined as

(1.13)

PL

u

D( )

=32

2

f PL

Du

u

DDu N

=( )

=

=

2

322

162 2 2

Re

1 4 0 0 410fN f= ( ) . log .Re

uQR

PLK

n

nR

Rn

n n=

( )

+

+( )

2

1

3 122 3 1

1

PL

u KD

n

n

n

n

n( )=

+ +4 2 61

f PL

Du

u KD

n

n

DLu N

n

n

n

PL

=

=

+

=+ 2

4 2 62

162 1 2

Re,

N D uK

n

n

D uK

n

nPL

n n n n n

n

n

Re, =( )

+ = ( )

+

82 6 8

43 1

2 2

1


Experimental data (Table 1.4) indicate that Equation 1.12 will tend to slightly overpredictthe friction factor for many power law food materials. This may be due to wall slip or time-dependent changes in rheological properties which can develop in suspension and emulsiontype food products.

Equation 1.12 is appropriate for laminar flow which occurs when the following inequalityis satisfied (Grovier and Aziz, 1972):

(1.14)

The critical Reynolds number varies significantly with n (Figure 1.1) and reaches a maximumvalue around n = 0.4.

When Equation 1.14 is not satisfied, can be predicted for turbulent flow conditionsusing the equation proposed by Dodge and Metzner (1959):

(1.15)

This equation is simple and gives good results in comparison to other prediction equations(Garcia and Steffe, 1987). The graphical solution (Figure 1.2) to Equation 1.15 illustrates thestrong influence of the flow-behavior index on the friction factor.

1.2.1.3 Bingham Plastic Fluids

Taking an approach similar to that used for pseudoplastic fluids, the pressure drop per unitlength of a Bingham plastic fluid (

=

pl

=

o

) can be calculated from the volumetric flowrate equation:

(1.16)

TABLE 1.4Fanning Friction Factor Correlations for the Laminar Flow of Power-Law Food Products Using the Following Equation: = a (N

Re,PL

)

b

Product(s) a* b* Source

Ideal power law 16.0 1.00 Theoretical predictionPineapple pulp 13.6 1.00 Rozema and Beverloo (1974)Apricot puree 12.4 1.00 Rozema and Beverloo (1974)Orange concentrate 14.2 1.00 Rozema and Beverloo (1974)Applesauce 11.7 1.00 Rozema and Beverloo (1974)Mustard 12.3 1.00 Rozema and Beverloo (1974)Mayonnaise 15.4 1.00 Rozema and Beverloo (1974)Applejuice concentrate 18.4 1.00 Rozema and Beverloo (1974)Combined data of tomato concentrate and apple puree 29.1 .992 Lewicki and Skierkowski (1988)Applesauce 14.14 1.05 Steffe et al. (1984)* a and b are dimensionless numbers.

N nn n

NPL n n PLRe, Re, 2394. Friction loss coefficients are the sameas those found for Case 2: k,entrance = 0.5 ; k,valve = 9 ; k,elbow = 0.45. The friction factor isfound by iteration of Equation 1.15:

yielding = 0.0051. Then

k

k

k

f,entrance

f,valve

f,elbow

=

( )( )=

=( )( )

=

=( )( )

=

. .

.

.

.

.

.

.

.

55 2 1 2 500323 9

1 42

9 500323 9

13 89

45 500323 9

0 69

f = =16323 9

0 0494.

.

F J kg= ( )( ) ( ) + + + ( )( ) ( ) + =2 0494 1 66 10 50348 1 42 13 89 3 691 66

280 0 189 1

2 2. . .

.

. . .

.

. .

= ( ) + ( ) + =

( ) = ( ) ( ) =W J kg

P kPap

9 81 2 5 1 661 2

189 1 215 9

215 9 1250 270

2

. .

.

.

. .

.

N PLRe, . ..

.

.

,( ) = ( )+ ( )( )

+

=+( ) +critical6464 45

1 3 45 12 45

2 3942

2 45 1 45

1 445

6736 6 0 4450 75 10

1 45 21 2f

f= ( )

( ) ( )

( )( )( ).

log . ..

.

.

.

F J kg= ( )( ) ( ) + + + ( )( ) ( ) + =2 0051 1 66 10 50 348 55 9 3 451 66

280 0 103 5

2 2. . .

.

. .

.

. .


and

1.3.5 CASE 5: BINGHAM PLASTIC FLUID IN LAMINAR FLOW

Assume, pl = 0.34 Pa s and o = 50 Pa making NRe,B = 212.4 and NHe = 654.8. To checkthe flow regime, cc is calculated from Equation 1.24:

giving cc = 0.035. The critical value of NRe,B is determined from Equation 1.23:

meaning the flow is laminar because 212.4 < 2229. Friction loss coefficients may be deter-mined from Table 1.5, Equations 1.39, 1.41, and 1.42; however, in this particular problem,NRe,B = NRe,PL = 212.4, so the friction loss coefficients in this example are the same as thosefound in Case 1: kf,entrance = 2.59; kf,valve = 21.18; kf,elbow = 1.06. , a function of c (Figure1.16), is taken as 1 (the worst case value) for the calculations. The friction factor is foundby iteration of Equation 1.20:

resulting in = 0.114. Then,

and

1.3.6 CASE 6: HERSCHEL-BULKLEY FLUID IN LAMINAR FLOW

Assume, K = 5.2, o = 50 Pa and n = 0.45 giving NRe,PL = 323.9 and NHe,M = 707.7. Flow islaminar (Figure 1.11) and the friction loss coefficients are the same as those found for Case3 because the Reynolds numbers are equal in each instance: k,entrance = 0.83; k,valve = 13.89;k,elbow = 0.69. Also, = 1.2 can be taken as the worst case (Figure 1.16). The friction factoris calculated by averaging the values found on Figures 1.11 and 1.12:

= ( ) + ( ) + =

( ) = ( ) ( ) =W J kg

P kPap

9 81 2 5 1 662

103 5 129 4

129 4 1250 162

2

. .

.

. .

.

c

c

c

c1

654 816 8003

( ) =.

,

N BRe,.

.

. . ,( ) = ( ) ( ) + ( ) critical 654 88 035 1 43 035 13 035 2 2294

1212 4 16

654 86 212 4

654 83 212 42

4

3 8.

.

.

.

.

= ( ) +( )( )

ff

F J kg= ( )( ) ( ) + + + ( )( ) ( ) + =2 114 1 66 10 50 348 2 59 21 18 3 1 061 66

280 0 306 7

2 2. . .

.

. . .

.

. .

= ( ) + ( ) + =( ) = ( ) ( ) =

W J kg

P kPap

9 81 2 5 1 66 306 7 334 0

334 0 1250 418

2. . . . .

.


Then

and

1.4 VELOCITY PROFILES IN TUBE FLOW

1.4.1 LAMINAR FLOW

It is important to know the velocity profiles present in pipes for various reasons such ascalculating the appropriate length of a hold tube for a thermal processing system. Expressionsgiving the velocity profiles in laminar flow are easily determined from the fundamentalequations of motion. With a Newtonian fluid the result is

(1.48)

and, for the case of a power law material,

(1.49)

By considering the volumetric flow rate, the relationship between the mean velocity (u =Q/R2)) and maximum velocity (located at the center line where r = 0) may also be calculated:

(1.50)

In the case of a Bingham plastic fluid, the velocity profile equation is

(1.51)

The velocity in the plug, at the center of the pipe, where o for r ro is

(1.52)

f = + =0 071 0 0812

0 076. . .

F J kg= ( )( ) ( ) + + + ( )( ) ( ) + =2 076 1 66 10 50348 1 42 13 89 3 691 66

280 0 230 3

2 2. . .

.

. . .

.

. .

= ( ) + ( ) + =( ) = ( ) ( ) =

W J kg

P kPap

9 81 2 5 1 66 230 3 257 1

257 1 1250 321

2. . . . .

.

u f r PL

R r= ( ) = ( ) ( )

42 2

u f r PLK

n

nR r

n n

n

n

n= ( ) = ( )

+

+ +2 1

1 1 1

u

u

n

nmax

=+

+

11 3

u f r P RKL

r

Rr

Rr

Ro

= ( ) = ( )

2 24

1 2 1

u f r P RKL

r

Ro

= ( ) = ( )

2 2

41


where the value of the critical radius, ro, is calculated from the yield stress:

(1.53)

The velocity profile for a Herschel-Bulkley fluid may also be determined:

(1.54)

1.4.2 TURBULENT FLOW

It is difficult to predict velocity profiles for fluids in turbulent flow. Relationships for New-tonian fluids are reliable. Those for power-law fluids are available but they have not receivedadequate experimental verification for fluid foods.

1.4.2.1 Newtonian Fluids

Semi-theoretical prediction equations for the velocity profile of Newtonian fluids in turbulentflow, in smooth tubes, are well established and discussed in numerous textbooks (e.g., Brodkeyand Hershey, 1988; Denn, 1980; Grovier and Aziz, 1972). The equations are presented interms of three distinct regions of the pipe:For the viscous sublayer

(1.55)

For the transition zone where turbulent fluctuations are generated

(1.56)

For the turbulent core

(1.57)

where

(1.58)

(1.59)

(1.60)

and, y, the distance from the pipe wall is

rL

Poo

= ( )

2

u f r L

Pn

K

P rL

n

w on

o

n

= ( ) =( ) +

( ) ( )

++

21 1 2

1

1 1

1 1

u y+ += y+ 5

u y+ += + ( )3 05 11 513 10. . log y+<

(1.61)

The origin of the coordinate system is located at the wall, r = R; therefore, the velocity iszero at r = R where y = 0 and a maximum at the center of the pipe where r = 0 and y = R.Combined, the above equations constitute the universal velocity profile.

A common problem facing the engineer in the food industry is to predict the maximumvelocity found in a pipe. To illustrate an approach to this problem, consider the followingexample.

EXAMPLE 1.1Assume: = 0.010 Pa s; D = 0.0348 m; u = 1.75 m/s; = 1225 kg/m3; NRe = 7460; =0.0084. The velocity is maximum at the center line where y = R and the friction velocity is

The calculations proceed as

The maximum velocity may be calculated from the definition of the turbulent velocity, u+ =u/u*, as

Once the maximum velocity has been determined, the power-law equation may beused to approximate u/umax at other locations:

(1.62)

For example, the velocity halfway between the center line and the wall (r = .5R) would becalculated from

making

The power-law equation does a reasonable job of predicting velocity profiles in spite ofthe fact that it is independent of NRe. Grovier and Aziz (1972) note that Equation 1.62 ismost appropriate for 0.1 < y/R < 1.0 and 3000 < NRe < 100,000. Also, the exponent mayvary from at NRe = 4000 to at NRe = 3,200,000.

y R r=

u uf

m s* ..

.= = =

21 75 0084

20 1134

y yu+ =

=( )( )

=* . / .

.

.

0348 2 1134 1225010

241 71

u y+ += + ( ) = + ( ) =5 5 5 756 5 5 5 756 241 71 19 2210 10. . log . . log . .

u u u m smax

* . . .= = ( ) =+ 19 22 1134 2 18

u

u

yR

R rR

max

= =

1 7 1 7

u

u

R rR

max

. .= = ( ) =

1 71 75 0 9067

u = ( )( ) =. . .9067 2 18 1 98


1.4.2.2 Power-Law Fluids

Dodge and Metzner (1959) derived equations to describe the velocity profile of power-lawfluids in tube flow. Small errors were corrected by Skelland (1967) and the final equationswere presented as

(1.63)

for the laminar sublayer and

(1.64)

for the turbulent core, where y+ incorporates the flow-behavior index needed for the consid-eration of power-law fluids:

(1.65)

Constants were obtained from friction-factor measurements so the thickness of the laminarsublayer was not obtained. The above equation can be used to predict the maximum velocityin a pipe. Consider the following example problem.

EXAMPLE 1.2Assume: K = 0.31 Pa sn; D = 0.0348 m; u = 1.75 m/s; = 1225 kg/m3; n = 0.40; NRe, PL =7741; = 0.0045. The velocity is maximum at the center line where y = R. The friction velocity is

and

The maximum velocity may be calculated from the definition of the turbulent velocity, u+ =u/u*, as

An alternative equation for predicting velocity in the turbulent core for power-law fluids waspresented by Clapp (1961)

(1.66)

u yn+ +

= ( )1

un

yn n

n nn

+ += ( ) + + +

5 66 0 566 3 475 1 960 0 815 1 628 3 175 10 1 2 75 10. log . . . . . log. . .

y y uK

n n+

=( )* 2

u uf

m s* ..

.= = =

21 75 0045

20 083

y y uK

u

n n

+

+

=( )

=( ) ( ) ( )

=

= ( ) ( ) ( ) + ( ) + ( ) ( ) +

* . / ..

.

.

.

log . ..

.

.

. . . . . log.

. .

. . .

2 4 2 4

75 10 1 2 75 10

0348 2 0838 122531

14 6

5 664

14 6 0 5664

3 4754

1 960 0 815 4 1 628 4 3 14

= 23 86.

u u u m smax

* . . .= = ( ) =+ 23 86 083 1 98

un

yn

+ += ( ) +2 78 2 303 3 8010. . log .


This equation correlated well with experimental data for 0.698 < n < 0.813 and 5480 < NRe,PL< 42,800.

1.5 SELECTION OF OPTIMUM ECONOMIC PIPE DIAMETER

The selection of pipe diameter for food processing systems is usually based on the require-ments of the processing equipment such as inlet port size for a pump; however, optimumsolutions can be used if sufficient economic data is available. Denn (1980) has discussed thesolution for Newtonian fluids. In addition, Darby and Melson (1982) used dimensionalanalysis to develop graphs from which the optimum pipe diameter could be obtained directlyfor Newtonian, Bingham plastic, and power-law fluids. The problem has been solved forpumping Herschel-Bulkley fluids by Garcia and Steffe (1986a).

NOMENCLATURE

A1 Upstream cross-sectional area, m2A2 Downstream cross-sectional area, m2c Yield stress/shear stress at the wall, o/w, dimensionlesscc Critical value of c, dimensionlessD Pipe diameter, m Fanning friction factor, dimensionlessg Acceleration due to gravity, 9.81 m/s2k Friction-loss coefficient, dimensionlessK Consistency coefficient, Pa snKE Kinetic energy per unit mass, J/kgL Length of pipe, mN Any dimensionless numberNHe Hedstrom number for a Bingham plastic fluid, dimensionlessNHe,M Modified Hedstrom number for a Herschel-Bulkley fluid, dimensionlessNRe Reynolds number for a Newtonian fluid, dimensionlessNRe,B Reynolds number for a Bingham plastic fluid, dimensionless(NRe,B)critical Critical Reynolds number for Bingham plastic fluid, dimensionlessNRe,PL Reynolds number for a power-law fluid, dimensionless(NRe,PL)critical Critical Reynolds number for a power-law fluid, dimensionlessn Flow-behavior index, dimensionlessro Critical radius, mr Radial coordinate, mR Pipe radius, mQ Volumetric flow rate in a pipe, m3/su Velocity, m/su+ Turbulent velocity [u/u*], dimensionlessu* Friction velocity , m/s

u Volumetric average velocity [Q/(R2)], m/sumax Maximum velocity in the tube, m/sW Work output per unit mass, J/kgy Distance from pipe wall into fluid [R r], my+ Distance from the tube wall [u*y/K], dimensionlessz Height above a reference plane, m Kinetic energy correction coefficient, dimensionless Constant defined by Equation 1.42, dimensionlessP Pressure drop over a pipe of length, L, Pa(P)p Pressure drop across a pump, Pa Shear rate, 1/s Newtonian viscosity, Pa s

w

u f=[ ]2


pl Plastic viscosity of a Bingham fluid, Pa s Density, kg/m3 Shear stress, Pao Yield stress, Paw Shear stress at the wall of a pipe [(PR)/(2L)], Pa

REFERENCES

Bird, R. B., Stewart, W. E., and Lightfoot, E. N., 1960, Transport Phenomena, John Wiley and Sons,New York.

Brodkey, R. S. and Hershey, H. C., 1988, Transport Phenomena, McGraw-Hill, New York.Brown, N. P. and Heywood, N. I., Eds., 1991, Slurry Handling: Design of Solid-Liquid Systems, Elsevier,

New York.Clapp, R. M., 1961, Turbulent heat transfer in pseudoplastic non-Newtonian fluids, III.A. Int. Dev. Heat

Transfer, ASME, Part III, Sec. A., 652661.Crane Co., 1982, Flow of fluids through valves, fittings and pipe, Technical Paper No. 410M, 21st

printing, Crane Co., 300 Park Ave., New York.Darby, R. and Melson, J. D., 1982, Direct determination of optimum economic pipe diameter for non-

Newtonian fluids, J. Pipelines (2):1121.Denn, M. M., 1980, Process Fluid Mechanics, Prentice-Hall, Englewood Cliffs, NJ.Dodge, D. W. and Metzner, A. B., 1959, Turbulent flow of non-Newtonian systems, AIChE J

5(7):189204.Edwards, M. F., Jadallah, M. S. M., and Smith, R., 1985, Head losses in pipe fittings at low Reynolds

numbers, Chem. Eng. Res. Des. 63:4350.Garcia, E. J. and Steffe, J. F., 1986, Review of friction factor equations for non-Newtonian fluids in

pipe flow, Special Report, Department of Agricultural Engineering, Michigan State University,East Lansing, MI.

Garcia, E. J. and Steffe, J. F., 1986a, Optimum economic pipe diameter for pumping Herschel-Bulkleyfluids in Raminar flow, J. Food Proc. Eng. 8(2):117136.

Garcia, E. J. and Steffe, J. F., 1987, Comparison of factor equations for non-Newtonian fluids in tubeflow, J. Food Proc. Eng. 9(2):93120.

Griskey, R. G. and Green, R. G., 1971, Flow of dilatant (shear-thickening) fluids, IAIChE J17(3):725728.

Grovier, G. W. and Aziz, K., 1972, The Flow of Complex Mixtures in Pipes, R. E. Krieger, Malabar, FL.Hanks, R. W., 1963, Laminar-turbulent transition of fluids with a yield stress, AIChE J 9(3):306309.Hanks, R. W., 1978, Low Reynolds turbulent pipe flow of pseudohomogenous slurries, Paper C-2 in

Proc. 5th Int. Conf. on Hydraulic Transport of Solids in Pipes (Hydrotransport 5), Hanover, FederalRepublic of Germany, May 811, BHRA Fluid Engineering, Cranfield, Bedford, England.

Houska, M., Sesrk, J., Jeschke, J., Adam, M., and Prida, J., 1988, in Progress and Trends in Rheology,II, Giesekus, H., Ed., Proceedings of the Second Conference of European Rheologists, Prague,June 1720, 1986, pp 460463, Springer-Verlag, New York.

Kittredge, C. P. and Rowley, D. S., 1957, Resistance coefficients for laminar and turbulent flow throughone-half inch valves and fittings, Trans. ASME 79:17591766.

Lewicki, P. P. and Skierkowski, K., 1988, Flow of fruit and vegetable purees through pipelines, inProgress and Trends in Rheology, II, Giesekus, H., Ed., Proceedings of the Second Conference ofEuropean Rheologists, Prague, June 1720, 1986, pp 443445, Springer-Verlag, New York.

Lord. D. L., Hulsey, B. W., and Melton, L. L., 1967, General turbulent pipe flow scale-up correlationfor rheologically complex fluids, Soc. Petrol. Engrs. J. 7(3):252258.

Metzner, A. B., 1956, Non-Newtonian technology: fluid mechanics, mixing, heat transfer, in Advancesin Chemical Engineering, Vol. 1, Drew, T. B. and Hoopes, J. W., Eds., Academic Press, New York.

Ofoli, R. Y., Morgan, R. G., and Steffe, J. F., 1987, A generalized rheological model for inelastic fluidfoods, J. Texture Stud. 18(3):213230.

Osorio, F. A. and Steffe, J. F., 1984, Kinetic energy calculations for non-Newtonian fluids in circulartubes, J. Food Sci. 49(5):12951296, 1315.

Rao, M. A. and Steffe, J. F., Eds., 1992, Viscoelastic Properties of Foods, Chapman and Hall, New York.


Rozema, H. and Beverloo, W. A., 1974, Laminar isothermal flow of non-Newtonian fluids in circularpipe, Lebensmitt. Wissenschaft und Technologie 7:223228.

Sakiadis, B. C., 1984, Fluid and particle mechanics, in Perrys Chemical Engineers Handbook, 6thed., Sec. 5. Perry, R. H., Green, D. W., and Maloney, J. D., Eds., McGraw-Hill, New York.

Skelland, A. P. H., 1967, Non-Newtonian Flow and Heat Transfer, John Wiley and Sons, New York.Steffe, J. F., 1984, Problems in using apparent viscosity to select pumps for pseudoplastic fluids, Trans.

ASAE 27(2):629634.Steffe, J. F., 1992, Rheological Methods in Food Process Engineering, Freeman Press, East Lansing, MI.Steffe, J. F., Mohamed, I. O., and Ford, E. W., 1984, Pressure drop across valves and fittings for

pseudoplastic fluids in laminar flow, Trans. ASAE 27(2):616619.Steffe, J. F. and Morgan, R. G., 1986, Pipeline design and pump selection for non-Newtonian fluid

foods, Food Technol. 40(12):7885. [Addendum: Food Technol. 41(7):32].Torrance, B. McK., 1963, Friction factors for turbulent non-Newtonian fluid flow in circular pipes,

South African Mech. Engr. 13:8991.


2

Sterilization Process Engineering

Hosahalli S. Ramaswamy and R. Paul Singh

CONTENTS

2.1 Introduction2.2 Principles of Thermal Processing 2.3 Thermal Resistance of Microorganisms

2.3.1 Survivor Curve and D Value2.3.2 Thermal Death Time (TDT) and D Value2.3.3 Temperature Dependence and z Value2.3.4 Reaction Rate Constant (k) and Activation Energy (E

a

)2.3.5 Lethality Concept

2.4 Heat Transfer Related to Thermal Processing2.4.1 Conduction Heat Transfer

2.4.1.1 Steady-State Conduction2.4.1.2 Unsteady-State Conduction2.4.1.3 Solution to Unsteady-State Heat Transfer Problem

Using a Spherical Object as an Example2.4.2 Convection Heat Transfer

2.4.2.1 Steady-State Convection Heat Transfer2.4.2.2 Unsteady-State Convection Heat Transfer

2.4.3 Characterization of Heat Penetration Data2.4.4 Heat Penetration Parameters2.4.5 The Retort Come-Up Time

2.5 Thermal Process Calculations2.5.1 The Original General Method2.5.2 The Improved General Method2.5.3 The Ball-Formula Method

2.5.3.1 Come-Up Time Correction and the Ball-Process Time2.5.4 The Stumbo-Formula Method2.5.5 The Pham-Formula Method

References

2.1 INTRODUCTION

Conventional thermal processing generally involves heating of foods packaged in hermeticallysealed containers for a predetermined time at a preselected temperature to eliminate thepathogenic microorganisms that endanger the public health as well as those microorganismsand enzymes that deteriorate the food during storage. The original concept of in-container


sterilization of foods has come a long way since Nicholas Appert first introduced the art ofcanning in 1810 (Lopez, 1987). Today, the consumer demands more than the production ofsafe and shelf-stable foods and insist on high quality foods with convenient end use. Hightemperature-short time (HTST) and ultra-high temperature (UHT) techniques have beendeveloped to minimize the severity of heat treatment and promote product quality. Asepticprocessing and packaging further minimize the heat severity by quick heating and coolingof the food under aseptic conditions prior to packaging. Thin profile, thermostable, micro-wavable packages have been developed for promoting faster heat-transfer rates which mini-mizes the heat damage to product quality while adding the convenience of package micro-wavability. Whatever the specific procedure employed, it is essential to design a processwhich will deliver the required heat treatment to the food. In this chapter, the principles ofthermal processing are detailed emphasizing the use of process calculation methods forestablishing thermal processes.

2.2 PRINCIPLES OF THERMAL PROCESSING

Generally, thermal processing is not designed to destroy all microorganisms in a packagedproduct. Such a process would result in low product quality due to the long heating required.Instead, the pathogenic microorganisms in a hermetically sealed container are destroyed andan environment is created inside the package which does not support the growth of spoilagetype microorganisms. In order to determine the extent of heat treatment, several factors mustbe known (Fellows, 1988): (1) type and heat resistance of the target microorganism, spore,or enzyme present in the food; (2) pH of the food; (3) heating conditions; (4) thermo-physicalproperties of the food and the container shape and size; and (5) storage conditions followingthe process.

Foods have different microorganisms and/or enzymes that the thermal process is designedto destroy. In order to determine the type of microorganism on which the process should bebased, several factors must be considered. In foods that are vacuum packaged in hermeticallysealed containers, low oxygen levels are intentionally achieved. Therefore, the prevailingconditions are not conducive to the growth of microorganisms that require oxygen (obligateaerobes) to create food spoilage or public-health problems. Further, the spores of obligateaerobes are less heat resistant than the microbial spores that grow under anaerobic conditions(facultative or obligate anaerobes). The growth and activity of these anaerobic microorganismsare largely pH dependent. From a thermal-processing standpoint, foods are divided into threepH groups: (1) high-acid foods (pH < 3.7; e.g., apple, apple juice, apple cider, apple sauce,berries, cherry (red sour), cranberry juice, cranberry sauce, fruit jellies, grapefruit juice,grapefruit pulp, lemon juice, lime juice, orange juice, plum, pineapple juice, sour pickles,sauerkraut, vinegar); (2) acid or medium-acid foods (3.7 < pH < 4.5; e.g., fruit jams, fruitcocktail, grapes, tomato, tomato juice, peaches, pimento, pineapple slices, potato salad, prunejuice, vegetable juice); and (3) low-acid foods (pH > 4.5; e.g., all meats, fish, vegetables,mixed entries, and most soups).

With reference to thermal processing, the most important distinction in the pH classifi-cation is the dividing line between acid and low acid foods. Most laboratories dealing withthermal processing devote special attention to

Clostridium botulinum

which is a highly heat-resistant, rod-shaped, spore-forming, anaerobic pathogen that produces the

botulism

toxin. Ithas been generally accepted that

C. botulinum

does not grow and produce toxins below a pHof 4.6. Hence, pH 4.5 is taken as the dividing line between the low acid and acid groupssuch that, with reference to processing of acid foods (pH < 4.5), one need not be concernedwith

C. botulinum

. On the other hand, in the low acid foods (pH > 4.5), the most heat-resistantspore former that is likely to be present and survive the process is

C. botulinum

which canthrive comfortably under the anaerobic conditions that prevail inside a sealed container to


produce the potent exotoxin. There are other microorganisms, for example

Bacillus stearo-thermophilus

,

B. thermoacidurans,

and

C. thermosaccolyaticum

, which are more heat resis-tant than

C. botulinum

. These are generally

thermophilic

in nature (optimal growth temper-ature ~ 5055C) and hence are not of much concern if the processed cans are stored attemperatures below 30C.

The phrase minimal thermal process was introduced by the US Food and Drug Admin-istration in 1977 and defined as the application of heat to food, either before or after sealingin a hermetically sealed container, for a period of time and at a temperature scientificallydetermined to be adequate to ensure the destruction of microorganisms of public healthconcern (Lopez, 1987).

C. botulinum

is the microorganism of public health concern in thelow-acid foods and due to its high-heat resistance, temperatures of 115125C are commonlyemployed for processing these foods. With reference to the acid and medium-acid foods, theprocess is usually based on the heat-resistant spoilage-type vegetative bacteria or enzymewhich are easily destroyed even at temperatures below 100C. The thermal processes forsuch foods are therefore normally carried out in boiling water.

2.3 THERMAL RESISTANCE OF MICROORGANISMS

The first step prior to establishing thermal processes is identification or designation of themost heat-resistant or target microorganism/enzyme on which the process should be based.This requires the microbiological history of the product and conditions under which it issubsequently stored rendering it somewhat product specific. The next step is evaluation ofthermal resistance of the test microorganism which must be determined under the conditionsthat normally prevail in the container. In order to use thermal destruction data in processcalculation, they must be characterized using an appropriate model. Further, since packagedfoods cannot be heated to process temperatures instantaneously, data on the temperaturedependence of microbial destruction rate is also needed to integrate the destruction effectthrough the temperature profile under processing conditions. The various proceduresemployed for experimental evaluation of thermal destruction kinetics of microorganisms aresummarized in Stumbo (1973) and Pflug (1987).

2.3.1 S

URVIVOR

C

URVE

AND

D V

ALUE

Published results on thermal destruction of microorganisms generally show that they followa first-order reaction indicating a logarithmic order of death. In other words, the logarithmof the number of microorganisms surviving a given heat treatment at a particular temperatureplotted against heating time (survivor curve) will give a straight line (Figure 2.1). Themicrobial destruction rate is generally defined in terms of a decimal reduction time (D value)which is the heating time in minutes at a given temperature required to result in one decimalreduction in the surviving microbial population. In other words, D value represents a heatingtime that results in 90% destruction of the existing microbial population. Graphically, thisrepresents the time between which the survival curve passes through one logarithmic cycle(Figure 2.1).

Mathematically

(2.1)

where a and b represent the survivor counts following heating for t

1

and t

2

min, respectively.The logarithmic nature of the survivor or destruction curve indicates that complete destructionof the microbial population is not a theoretical possibility, since a decimal fraction of thepopulation should remain even after an infinite number of D values. In practice, calculated

D t t a b= ( ) ( ) ( )[ ]2 1 log log


fractional survivors are treated by a probability approach; for example, a surviving populationof 10

8

/unit would indicate one survivor in 10

8

units subjected to the heat treatment.Traditionally, in thermal processing applications, survivor curves are plotted on specially

constructed semilog papers for easy handling and interpretation of results. The survivor countsof microorganisms are plotted directly on the logarithmic ordinate against time on the linearabscissa. The time interval between which the straight line portion of the curve passes througha logarithmic cycle is taken as the D value. In engineering approaches, one can prepare a logN vs. t computer graph on spreadsheet and run a linear regression of log N on t in the rangein which the points represent a reasonable straight line. The negative reciprocal slope of sucha regression equation for the straight line gives the D value. Visual observation of data pointsprior to regression is desirable for proper selection of the regression range.

2.3.2 T

HERMAL

D

EATH

T

IME

(TDT)

AND

D V

ALUE

In food microbiology, another term, namely thermal death time (TDT), is commonlyemployed which somewhat contradicts the logarithmic-destruction approach. TDT is theheating time required to cause complete destruction of a microbial population. Such data areobtained by subjecting a microbial population to a series of heat treatments at a giventemperature and testing for survivors. TDT then represents a time below the shortest destruc-tion and the longest survival times. The difference between the two are sequentially reducedand/or geometrically averaged to get an estimate of TDT. The death in this instancegenerally indicates the failure of a given microbial population, after the heat treatment, toshow a positive growth in the subculture media. Comparing TDT approach with the decimalreduction approach, one can easily recognize that the TDT value depends on the initialmicrobial load (while D value does not). Further, if TDT is always measured with referenceto a standard initial load or load reduction, it simply represents a certain multiple of D value.For example, if TDT represents the time to reduce the population from 10

9

to 10

3

, then TDTis a measure of 12 D values. In other words

FIGURE 2.1

Typical survivor curve.


(2.2)

where n is the number of decimal reductions.It should be noted that there are several causes for deviations in the logarithmic behavior

of the survivor curve. Generally, there is a lag at the start of the heating period when first-order behavior doesnt fit the observed data and frequently, deviation from first order alsooccurs at the tail end of the survivor curve referred to as tailing. In his book

Thermobac-teriology in Food Processing

, Stumbo (1973) has detailed several factors causing apparentdeviations of the logarithmic order of microbial death showing typical survivor curves foreach situation: (1) heat activation for spore germination; (2) mixed flora; (3) clumped cells;(4) flocculation during heating; (5) deflocculation during heating; (6) nature of the subculturemedium; and (7) anaerobiosis. Stumbo (1973) has also summarized the various factors thatinfluence the thermal resistance of bacteria: conditions present during sporulation (tempera-ture, ionic environment, organic compounds, lipids, age, or phase of growth) and conditionspresent during heat treatment (pH and buffer components, ionic environment, water activity,composition of the medium).

2.3.3 T

EMPERATURE

D

EPENDENCE

AND

Z

V

ALUE

The D value depends strongly on the temperature with higher temperatures resulting in smallerD values. The temperature sensitivity of D values at various temperatures is normallyexpressed as a thermal resistance curve with log D values plotted against temperature(Figure 2.2). The temperature sensitivity indicator is defined as z, a value which representsa temperature range which results in a ten-fold change in D values or, on a semilog graph,it represents the temperature range between which the D value curve passes through onelogarithmic cycle. Using regression techniques, z value can be obtained as the negativereciprocal slope of the thermal resistance curve (regression of log D values vs. temperature).

FIGURE 2.2

A typical thermal resistance curve.

TDT n D=


Mathematically

(2.3)

where D

1

and D

2

are D values at T

1

and T

2

, respectively.The D value at any given temperature can be obtained from a modified form of the above

equation using a reference D value (D

o

at a reference temperature, T

o

, usually 250F forthermal sterilization)

(2.4)

Equation 2.3 also can be written with reference to TDT values and the z value can be obtainedfrom

(2.5)

where TDT

1

and TDT

2

are TDT values at T

1

and T

2

, respectively. Graphically, as with the Dvalue approach, the z value can be obtained as the negative reciprocal slope of log TDT vs.temperature curve (Figure 2.3; TDT curve). When using this approach, it is advisable to plotthe longest survivor times and shortest destruction times (on logarithmic scale) vs. temperature(linear scale). The regression line could be based on the evaluated TDT as described earlier.As is noted in Pflug (1987) it will be necessary to make sure that the TDT curve is aboveall survivor data points (higher in temperature or longer in time). The TDT curve should beparallel to the general trend of the survival and destruction points.

FIGURE 2.3

A typical TDT curve.

z T T D D= ( ) ( ) ( )[ ]2 1 1 2log log

D Do

T T zo=

( )10

z T T TDT TDT= ( ) ( ) ( )[ ]2 1 1 2log log


2.3.4 R

EACTION

R

ATE

C

ONSTANT

(

K

)

AND

A

CTIVATION

E

NERGY

(E

A

)

The first-order rate thermal destruction of microorganisms is also sometimes expressed interms of a reaction rate constant, k, obtained from the first-order model:

(2.6)

which simplifies to

(2.7)

Several theories have suggested different models that relate the effect of temperature toreaction rates. The most well known and perhaps most frequently used theory in the area ofbiological engineering is that proposed by Arrhenius which is applicable to reactions insolutions and heterogeneous processes. Using a thermodynamic approach, Arrhenius sug-gested that in every system, at any instant of time there is a distribution of energy level amongthe molecules (Figure 2.4). For a molecule to enter into a reaction, it must possess a certainminimal amount of energy, which is called the activation energy (E

a

). The mean of thefrequency distribution of molecule energy levels is a function of temperature (Figure 2.4).The probability that a molecule will possess energy in excess of an amount, E

a

, per mole attemperature T (absolute) is e

E

a

/RT

(where R = universal gas constant) (Figure 2.5). For areaction to occur, molecules that are capable of reacting and have energy at or greater thanE

a

must encounter each other. Thus, according to Arrhenius, if the collision frequency of themolecule is given by k

o

, the dependence of the reaction rate (k) on the absolute temperature(T) is

(2.8)

An Arrhenius plot is constructed by plotting the natural logarithm of reaction rate constantvs. the reciprocal of absolute temperature. A typical Arrhenius plot is shown in Figure 2.5and the E

a

can be calculated from such a plot as (slope)

R. A frequent argument is madethat the Arrhenius model is superior to the z-value model (Bigelow model) because the z valueis temperature dependent where E

a

is not. Comparisons probably cannot be made that simply,but it should be noted that while E

a

is usually assumed to be constant, from absolute-rateand collision theory, we learn that the plot of lnk vs. 1/T will not always give a straight line.Several researchers have shown that either of the two methods can be used in thermalprocessing applications. Generally the z value is related to E

a

using the equation

(2.9)

where E

a

is the activation energy, R is the universal gas constant, and T is the absolutetemperature. However, caution should be exercised when converting z value to E

a

or viceversa because it has been shown that errors associated with interconversion of E

a

and z arefunctions of the selected reference temperature and the temperature range used (Ramaswamyet al. 1989). This discrepancy can be minimized by replacing T

2

with the product of theminimum (T

min

) and maximum temperature (T

max

) of the temperature range between whichkinetic data were obtained (i.e., T

2

= T

min

. T

max

).

loge o

N N kt( ) =

k D= 2 303.

k k eo

E RTa=

E RT za= 2 303 2.


FIGURE 2.4

Energy distribution in a population of molecules.

FIGURE 2.5

Typical Arrhenius plot: ln(k) vs. reciprocal absolute temperature.


2.3.5 L

ETHALITY

C

ONCEPT

Lethality (F value) is a measure of the heat treatment or sterilization processes. In order tocompare the relative sterilizing capacities of heat processes, a unit of lethality needs to beestablished. For convenience, this is defined as an equivalent heating of 1 min at a referencetemperature, which is usually 250F (121.1C) for the sterilization processes. Thus, the Fvalue would represent a certain multiple or fraction of D value depending on the type ofmicroorganism; therefore, a relationship like Equation 2.5 also holds with reference to F value

(2.10)

The F

o

in this case will be the F value at the reference temperature (T

o

). A reference (orphantom) TDT curve is defined as a curve parallel to the real TDT or thermal resistancecurve (i.e., having the same z value) and having a TDT (F value) of 1 min at 250F. With aphantom TDT curve so defined, it will be possible to express the lethal effects of any time-temperature combination in terms of equivalent minutes at 250F or lethality.

(2.11)

Thus, an F value of 10 min at 240F is equivalent to an F

o

of 2.78 min while the F value of10 min at 260F is equivalent to an F

o

of 35.9 min when z = 18F. In these situations, it isassumed that the heating to the appropriate temperatures and the subsequent cooling areinstantaneous. For real processes where the food passes through a time-temperature profile,it should be possible to use this concept to integrate the lethal effects through the varioustime-temperature combinations. The combined lethality so obtained for a process is calledthe process lethality and is also represented by the symbol F

o

. Further, with reference to theprocessing situation, the lethality can be expressed as related to a specific location (normallythermal center) or any other arbitrarily chosen location or a sum of lethality at all pointsinside a container. In terms of microbiological safety, the assurance of a minimal lethality atthe thermal center is of utmost importance, while from a quality standpoint it is desirable tominimize the overall destruction throughout the container.

The criterion for the adequacy of a process must be based on two microbiologicalconsiderations: (1) destruction of the microbial population of public health significance; (2)reduction in the number of spoilage-causing bacteria. For low-acid foods, the microorganismof public health significance is

C. botulinum

and hence destruction of the spores of thisorganism is used as the minimal criterion for processing. Once again, it has been arbitrarilyestablished that the minimum process should be at least as severe to reduce the populationof

C. botulinum

through 12 decimal reductions (

bot cook

). Based on published information,a decimal reduction time of 0.21 min at 250F (Stumbo, 1973) is normally assumed for

C.botulinum

. A twelve-decimal reduction would thus be equivalent to an F

o

value of 12

0.21 =2.52 min. The minimal process lethality (F

o

) required is therefore 2.52 min. Several low-acidfoods are processed beyond this minimum value. An F

o value of 5 min is perhaps morecommon for these foods. The reason for this is the occurrence of more heat-resistant spoilage-type microorganisms which are not of public health concern. The average Do for these spoilagemicroorganisms may be as high as 1 min. An Fo value of 5 min would then be adequate onlyto achieve a 5 D process with reference to these spoilage microorganisms. It is thereforeessential to control the raw-material quality to keep the initial count of these organisms below100 per container on an average, if the spoilage rate were to be kept below one can in athousand (102 to 103 = 5D).

F Fo

T T To=

( ) (10

Lethality or Fo=

( )F T T zo10


2.4 HEAT TRANSFER RELATED TO THERMAL PROCESSING

The rate of heating of an object or a product in a container is a function of the geometry ofthe object or container, its physical properties, and the heat-transfer characteristics of theobject or the container. There are three possible mechanisms of heat transfer to a thermallyprocessed food product: (a) conduction, (b) convection, or (c) broken heating (which is asequential combination of conduction and convection or convection and conduction). Thenature or consistency of a food or pharmaceutical product, the presence of particles, and theuse of thickening agents and sugars in the covering liquid are some of the factors thatdetermine whether the product heats by convection or by conduction.

When we heat or cool objects or products in containers, we are dealing with unsteady-state heat-transfer processes. Unsteady-steady state heat transfer is mathematically quitecomplex; however, once the equations have been derived or developed they can be simplifiedso their use is not difficult. Such simplification procedures that are extremely useful in dealingwith heat-transfer processes in the food sterilization area have been developed and welldocumented (Ball and Olson, 1957). The following are brief mathematical analyses of bothconduction and convection unsteady-state heat transfer conditions.

2.4.1 CONDUCTION HEAT TRANSFER

2.4.1.1 Steady-State Conduction

Fouriers law is the fundamental differential equation for heat transfer by conduction:

(2.12)

where dQ/dt is the rate of flow of heat, A is the area of cross section perpendicular to thedirection of heat flow, and dT/dx is the rate of change of temperature with distance in thedirection of the flow of heat, i.e., the temperature gradient. The factor, k, is the thermalconductivity and is a property of the material through which the heat is flowing.

For the steady flow of heat, the term dQ/dt is constant and may be replaced by q (rateof heat flow, BTU/h or W). If k and A are independent of temperature (T) and distance (x),the above equation becomes:

(2.13)

2.4.1.2 Unsteady-State Conduction

In unsteady-state conduction, temperature changes with time and solutions-to-heat transferequations get more complicated. The governing partial-differential equation for unsteady-state conduction heating involving a three-dimensional (3D) body is given by

(2.14)

where T is the product temperature, t is the time, x, y, and z are the distances in the x, y, andz directions, and is the thermal diffusivity. The assumptions associated with the aboveequation are as follows: (1) the product temperature is uniform at the start of heating; (2) thesurface temperature of the 3D body is constant after the start of heating; (3) the productthermal diffusivity is constant with time, temperature, and position in the body. Examplesof conduction-heated foods are tightly packaged solid products or highly viscous liquid/semi-solid foods such as vegetable puree, meat ball in gravy, etc.

dQ dt k A dt dx=

q k A T x=

[ ] = ( ) + ( ) + ( )[ ]T t T x T y T z 2 2 2 2 2 2


2.4.1.3 Solution to Unsteady-State Heat Transfer Problem Using a Spherical Object as an Example

The partial differential equation involving a sphere in spherical coordinates is

(2.15)

where r represents the radius of the sphere. The solution for the spacial temperature in theform of an infinite summation series is given by

(2.16)

where x represents the radial location and i is the ith root of the equation

(2.17)

It has been generally recognized that after a short heating time [i.e., Fourier number (Fo),t/r2 0.2], the above series will rapidly converge to just the first term. Thus, the first termapproximation can be written as

(2.18)

and

(2.19)

In most thermal processing applications, the heating behavior is characterized by a heatingrate index, fh, and a lag factor, jch (explained in detail in a later section). Representing fh andjch by the following expressions (Equations 2.20 and 2.21), the equation for the temperaturedistribution in the sphere can be written as shown in Equation 2.22.

(2.20)

(2.21)

(2.22)

Details of the equations related to the other geometries can be found in Ball and Olson (1957).

[ ] = ( ) + ( ) ( )[ ]T t T r r T r 2 2 2

T T T T

x r x r t r

i

i

n

i i i i i i i i i

1 1

1

2 22

( ) ( )[ ] =( ) ( )[ ] ( ){ } ( )[ ] [ ]

=

sin cos sin cos sin exp

Bi i i= 1 cot

T T T T

x r x r t r

i1 1

1 1 1 1 1 1 1 1 12 22

( ) ( )[ ] =( ) ( )[ ] ( ){ } ( )[ ] [ ]sin cos sin cos sin exp

Bi = 1 1 1 cot

T T T T ji ch t fh1 1 10( ) ( )[ ] =

j x r x rch = ( ) [ ] ( ){ } ( )[ ]2 1 1 1 1 1 1 1 1sin cos sin cos sin

f r= [ ] 12 22 303.


2.4.2 CONVECTION HEAT TRANSFER

Convection heat transfer involves the transfer of heat from one location to the other throughthe actual movement or flow of a fluid. When the fluid flow involved is produced wholly bydifferences in fluid density as a result of changes in temperature, the heat transfer is callednatural convection and if the fluid flow is aided by pumping/fan or some other type ofmechanical device, the heat transfer is called forced convection. Since heat transfer byconvection involves fluid flow and heat energy changes, to a great extent, it defies rigorousanalysis. However, since convection heat transfer is extremely important in most processes,procedures for the engineering analysis and models have been developed by empirical methods.

2.4.2.1 Steady-State Convection Heat Transfer

There are many examples in the natural environment where heat transfer by convection takesplace on a steady-state basis. In these systems, the rate of heat flow is constant and or quantityof heat flow is straight forward and simple compared to unsteady-state heat transfer.

2.4.2.2 Unsteady-State Convection Heat Transfer

In unsteady-state convection heat transfer in enclosed areas or confined volumes of fluid, thetemperature at all locations changes with time. The rate of fluid flow inside the enclosedvolume is determined by product characteristics including viscosity; however, the rate of heattransfer from or to an external source through the walls of the container will have a majoreffect on the rate of heating or cooling of the fluid inside the container. The thickness of theboundary layer between the flowing fluid and the wall is a critical factor in the heat transferrate. This stagnant layer or transition velocity zone offers significant resistance to heat transfer.Therefore, in convection heat-transfer processes the wall-to-fluid film coefficient must beevaluated. Empirical expressions have been used for calculating the film coefficient fordifferent fluids under different physical conditions. The equation that follows describes therate of heat flow across the wall into or out of the container.

(2.23)

where U is the overall heat transfer coefficient. In the heating of fluids in enclosures orconfined volumes, the change in the quantity of heat in the fluid per unit of time is a functionof the mass of the fluid (density, , times volume, V), the specific heat (Cp), and the meantemperature, T. If we assume ideal thermal convection inside the container of product, thesevariables can be related using the following equation

(2.24)

The solution to the unsteady-state convection heat-transfer problem is usually obtained fromNewtons law: based on the assumption that the heat flowing into a container is absorbed bythe contents, a temperature change of the product results. Combining Equations 2.23 and2.24, we can write

(2.25)

Separating the variables and integrating over temperature and time, we get

dQ dt UA T T= ( )1

dQ dt C V dT dtp=

U A T T C V dT dtp1 ( ) =


(2.26)

or

(2.27)

or, in terms of temperature and time

(2.28)

Replacing the constant in the parentheses with (1/fh), we get the shortened form of the equation

(2.29)

which is similar to the one we obtained for conduction with jch = 1.0. It should be noted thatEquation 2.29 was derived assuming that the system complies with Newtons Law of heat-ing/cooling. Convection heating under these ideal conditions implies at time-zero a stepchange in the heating-medium temperature and an instantaneous surface response to the newtemperature. Under ideal conditions, there will be no temperature gradients inside the body.These conditions occur only in a unit of infinitely small volume. The associated Biot numberwill be zero.

The heating characteristics in a container may be very different in agitated systems (forcedconvection) compared to nonagitating systems (natural convection). There is only slightdeviation in heating characteristics of agitating systems compared to an idealized system.For practical purposes, in the agitated container, a flow pattern already exits at the timeheating begins. All the resistance to heat transfer and the resulting temperature gradient is inthe stagnant layer of the product adjacent to the wall. There is essentially no temperaturegradient in the bulk contents. On the other hand, in the still (nonagitated) container, a finitetime period is required for the establishment of the temperature-induced flow pattern. Tem-perature gradients continue to exist in the bulk of the fluid even after the flow pattern hasbeen established. For these natural-convection systems the driving force for fluid flowdecreases as the temperature gradient decreases and the system approaches heating-mediumtemperature. This phenomenon tends to cause the fh-value to increase with heating time. This,in turn, may affect the intercept value of the heating curve, especially if data points collectedat longer heating times are heavily weighed.

It has been generally recognized that it is convenient to treat experimental-heating or -cooling data using the f and j concept advocated by Ball (1923) and use it as a data fittingtool for both conduction and convection heating foods. This concept is explained in greaterdetail in the next section.

2.4.3 CHARACTERIZATION OF HEAT PENETRATION DATA

In order to establish thermal process schedules, information on the temperature history ofthe product going through the process is needed in addition to thermal resistance character-istics of the test microorganism (z and Fo). The temperature history of the product undergoingthe process depends on several factors: (1) the heating process (sill vs. agitated cook; in-package vs. aseptic processing); (2) the heating medium (steam, water (immersion or spray)with or without air over pressure, steam/air mixtures); (3) the heating conditions (retort

T T T T eiU A t Cp V

1 1( ) ( ) = ( )

T T T TiU A t Cp V

1 12 30310( ) ( ) = ( ).

T T T TiU A Cp V t

1 12 30310( ) ( ) = ( )[ ].

T T T Tit fh

1 1 10( ) ( ) =


temperature, initial temperature, loading pattern); (4) the product type (solid, semisolid, liquid,particulate liquid; thermophysical properties of the product); and (5) the container type, shape,and size.

Thermal processing may be applied to packaged foods as in the conventional way (canningfor example) or foods may be heated and cooled, filled

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