handbook of food engineering practice

Steffe Department of Agricultural Engineering and Department of Food Science and Human Nutrition Michigan State University East Lansing, Michigan Petros S.. Paul Singh CONTENTS 1.1 Intro

Acquiring Editor: Harvey M Kane

Project Editor: Albert W Starkweather, Jr.

Cover Designer: Dawn Boyd

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, Kenneth J., 1938-

TP370.4.H37 1997

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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 bachelor’s degree in Chemical Engineering from Universidaddel Sur, Bahia Blanca, Argentina He obtained his Ph.D from Imperial College, University

of London, London, U.K He served successively as Assistant, Associate, and Full Professor

of 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 taught

at the University of Minnesota and at Imperial College, holding visiting professorships Heworked for DuPont, Argentina, and for Monsanto Chemical Co., Plastics Division In 1987

he 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 Chemical Engineering and Applied Chemistry, a member of the internal advisory board of DryingTechnology, and a member of the editorial advisory boards of Advances in Drying, Physico Chemical 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, with

a 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 Following

a 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, Institute

of 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 He

is a co-editor of the Journal of Food Process Engineering His current research interests are

in 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, and

is co-author of Food Processing Operations and Scale-Up.

Dr Valentas received the “Food, Pharmaceutical, and Bioengineering Division Award”from 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

Kraft Foods, Inc.

Tarrytown, New York

Chin Shu Chen

Citrus Research and Education Center

Centre for Postharvest

and Refrigeration Research

Massey University

Palmerston North, New Zealand

Guillermo H Crapiste

PLAPIQUI

Universidad Nacional del Sur–CONICET

Bahia Blanca, Argentina

Brian E Farkas

Department of Food Science

North Carolina State University

Raleigh, North Carolina

Ruben J Hernandez

School of PackagingMichigan State UniversityEast Lansing, Michigan

Canada

and Agricultural Engineering and

Department of Food Science and Technology

University of California, Davis

Davis, California

James F Steffe

Department of Agricultural Engineering

and Department of Food Science

and Human Nutrition

Michigan State University

East Lansing, Michigan

Petros S Taoukis

Department of Chemical EngineeringLaboratory of Food Chemistry and Technology

National Technical University of AthensAthens, Greece

Martin J Urbicain

PLAPIQUIUniversidad Nacional del Sur–CONICETBahia Blanca, Argentina

Baton Rouge, Louisiana

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 industry’s 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 review

of 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 description

of 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 given

in 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 how

to conduct material and energy balances in food processing systems are presented

• 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 must

be 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 methods

to estimate physical and rheological properties

• The importance of physical and rheological properties in designing a food system

is further illustrated in Chapter 12 for a dough processing system Dough rheology

is 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 variety

of sanitary guidelines Chapter 15 includes a broad description of issues that must

be 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 implementation

of process controllers in food processing applications are discussed in Chapter 16

• Food engineers must rely on a number of basic sciences in dealing with problems

at 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 Rotstein

R Paul Singh Kenneth 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

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

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 Introduction1.2 Mechanical Energy Balance1.2.1 Fanning Friction Factor1.2.1.1 Newtonian Fluids1.2.1.2 Power Law Fluids1.2.1.3 Bingham Plastic Fluids1.2.1.4 Herschel-Bulkley Fluids1.2.1.5 Generalized Approach to Determine Pressure Drop in a Pipe1.2.2 Kinetic Energy Evaluation

1.2.3 Friction Losses: Contractions, Expansions, Valves, and Fittings1.3 Example Calculations

1.3.1 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 Flow1.3.5 Case 5: Bingham Plastic Fluid in Laminar Flow1.3.6 Case 6: Herschel-Bulkley Fluid in Laminar Flow1.4 Velocity Profiles in Tube Flow

1.4.1 Laminar Flow1.4.2 Turbulent Flow1.4.2.1 Newtonian Fluids1.4.2.2 Power Law Fluids1.5 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 found

in 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 (ρ 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

1.2.1 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 used

in 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 flow

of 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)

2 2

2 1 2

The Fanning friction factor (ƒ) 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

Product

T (°C)

n (–)

K (Pa·s n )

( )2σ

ƒ can be considered in terms of pressure drop by substituting the definition of the shear stress

at the wall:

(1.4)

TABLE 1.2 Rheological Properties of Oils and Miscellaneous Products

T (°C)

n (–)

K (Pa·s n )

=( )δ = ( )

ρ

δρ

2

TABLE 1.3 Rheological Properties of Fruit and Vegetable Products

Product

Total solids (%)

T (°C)

n (–)

K (Pa·s n )

Guava, Puree (10.3 Brix) — 23.4 494 39.98 15–400 Mango, Puree (9.3 Brix) — 24.2 334 20.58 15–1000 Orange Juice

Total solids (%)

T (°C)

n (–)

K (Pa·s n )

·

γγγγ

(s –1 )

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

Product

Total solids (%)

T (°C)

n (–)

K (Pa·s n )

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, NRe < 2100 where

NRe = ρD u/µ In turbulent flow, NRe > 4000, the von Karman correlation is recommended(Brodkey and Hershey, 1988):

(1.9)

The friction factor in the transition range, approximately 2100 < NRe < 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 (σ = K (γ·)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:

uD

uD

u KD

nn

n n

u KD

nn

D

n n

D uK

nn

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 inequality

is satisfied (Grovier and Aziz, 1972):

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 (σ = µplγ·= σo) can be calculated from the volumetric flowrate equation:

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)

* a and b are dimensionless numbers.

Written in terms of the average velocity, Equation 1.16 becomes

=σσ = = ( )

σδ

σρ

2

Equations 1.18 or 1.20 may be used for estimating ƒ in steady-state laminar flow which

occurs when the following inequality is satisfied (Hanks, 1963):

(1.23)

where cc is the critical value of c defined as

(1.24)

FIGURE 1.2 Fanning friction factor (ƒ) for power-law fluids from the relationship of Dodge and

Metzner (1959) (From Garcia, E J and Steffe, J F 1986, Special Report, Department of Agricultural

Engineering, Michigan State University, East Lansing, MI.)

He B

( )

2 2

Re, ≤ 8 1−4 +  =( )Re,

3

13

cc varies (Figure 1.3) from 0 to 1 and the critical value of the Bingham Reynolds number

increases with increasing values of the Hedstrom number (Figure 1.4)

The friction factor for the turbulent flow of a Bingham plastic fluid can be considered aspecial case of the Herschel-Bulkley fluid using the relationship presented by Torrance (1963):

FIGURE 1.3. Variation of cc with the Hedstrom number (NHe) for the laminar flow of Bingham plastic

fluids (From Steffe, J F 1992, Rheological Methods in Food Process Engineering, Freeman Press,

East Lansing, MI With permission.)

FIGURE 1.4. Variation of the critical Bingham Reynolds number (NRe,B) with the Hedstrom number

Lansing, MI With permission.)

Increasing values of the yield stress will significantly increase the friction factor (Figure 1.5)

In turbulent flow with very high pressure drops, c may be small simplifying Equation 1.25 to

(1.26)

1.2.1.4 Herschel-Bulkley Fluids

The Fanning friction factor for the laminar flow of a Herschel-Bulkley fluid (σ = K (γ·n +

σo) can be calculated from the equations provided by Hanks (1978) and summarized byGarcia and Steffe (1987):

(1.27)

where

(1.28)

c can be expressed as an implicit function of NRe,PL and a modified form of the Hedstrom

FIGURE 1.5 Fanning friction factor (ƒ) for Bingham plastic fluids (NRe,PL) from the relationship of Torrance (1963) (From Garcia, E J and Steffe, J F 1986, Special Report, Department of Agricultural Engineering, Michigan State University, East Lansing, MI.)

n

1.2.1.5 Generalized Approach to Determine Pressure Drop in a Pipe

Metzner (1956) discusses a generalized approach to relate flow rate and pressure drop fortime-independent fluids in laminar flow The overall equation is written as

(1.31)

FIGURE 1.6 Fanning friction factor (ƒ) for a Herschel-Bulkley fluid with n = 1.0, based on the

relationship of Hanks (1978) (From Garcia, E J and Steffe, J F 1986, Special Report, Department

of Agricultural Engineering, Michigan State University, East Lansing, MI.)

n n

n( )

FIGURE 1.7 Fanning friction factor (ƒ) for a Herschel-Bulkley fluid with n = 0.9, based on the relationship of Hanks (1978) (From Garcia, E J and Steffe, J F 1986, Special Report, Department

of Agricultural Engineering, Michigan State University, East Lansing, MI.)

FIGURE 1.8 Fanning friction factor (ƒ) for a Herschel-Bulkley fluid with n = 0.8, based on the relationship of Hanks (1978) (From Garcia, E J and Steffe, J F 1986, Special Report, Department

of Agricultural Engineering, Michigan State University, East Lansing, MI.)

FIGURE 1.9 Fanning friction factor (ƒ) for a Herschel-Bulkley fluid with n = 0.7, based on the relationship of Hanks (1978) (From Garcia, E J and Steffe, J F 1986, Special Report, Department

of Agricultural Engineering, Michigan State University, East Lansing, MI.)

FIGURE 1.10 Fanning friction factor (ƒ) for a Herschel-Bulkley fluid with n = 0.6, based on the relationship of Hanks (1978) (From Garcia, E J and Steffe, J F 1986, Special Report, Department

of Agricultural Engineering, Michigan State University, East Lansing, MI.)

FIGURE 1.11 Fanning friction factor (ƒ) for a Herschel-Bulkley fluid with n = 0.5, based on the relationship of Hanks (1978) (From Garcia, E J and Steffe, J F 1986, Special Report, Department

of Agricultural Engineering, Michigan State University, East Lansing, MI.)

FIGURE 1.12 Fanning friction factor (ƒ) for a Herschel-Bulkley fluid with n = 0.4, based on the relationship of Hanks (1978) (From Garcia, E J and Steffe, J F 1986, Special Report, Department

of Agricultural Engineering, Michigan State University, East Lansing, MI.)

FIGURE 1.13 Fanning friction factor (ƒ) for a Herschel-Bulkley fluid with n = 0.3, based on the relationship of Hanks (1978) (From Garcia, E J and Steffe, J F 1986, Special Report, Department

of Agricultural Engineering, Michigan State University, East Lansing, MI.)

FIGURE 1.14 Fanning friction factor (ƒ) for a Herschel-Bulkley fluid with n = 0.2, based on the relationship of Hanks (1978) (From Garcia, E J and Steffe, J F 1986, Special Report, Department

of Agricultural Engineering, Michigan State University, East Lansing, MI.)

In the general solution, n′ may vary with the shear stress at the wall and must be evaluated

at each value of σw Equation 1.31 has great practical value when considering direct

scale-up from data taken with a small diameter tube viscometer or for cases where a well-definedrheological model (power law, Bingham plastic or Herschel-Bulkley) is not applicable Lord

et al (1967) presented a similar method for scale-up problems involving the turbulent flow

of time-independent fluids

Time-dependent behavior and slip may also be involved in predicting pressure losses inpipes One method of attacking this problem is to include these effects into the consistencycoefficient Houska et al (1988) give an example of this technique for pumping minced meatwhere K incorporated property changes due to the aging of the meat and wall slip

FIGURE 1.15 Fanning friction factor (ƒ) for a Herschel-Bulkley fluid with n = 0.1, based on the relationship of Hanks (1978) (From Garcia, E J and Steffe, J F 1986, Special Report, Department

of Agricultural Engineering, Michigan State University, East Lansing, MI.)

δπ

1.2.2 K INETIC E NERGY E VALUATION

Kinetic energy (KE) is the energy present because of the translational rotational motion ofthe mass KE, defined in the mechanical energy balance equation (Equation 1.1) as u2/α, isthe average KE per unit mass It must be evaluated by integrating over the radius becausevelocity is not constant over the tube

The KE of the unit mass of any fluid passing a given cross-section of a tube is determined

by integrating the velocity over the radius of the tube (OSorio and Steffe, 1984):

(1.34)

The solution to Equation 1.34 for the turbulent flow of any time-independent fluid is

(1.35)

meaning α = 2 for these cases With Newtonian fluids in laminar flow, KE = (u)2 with α =

1 In the case of the laminar flow of power law fluids, α is a function of n:

(1.36)where

1.2.3 F RICTION L OSSES : C ONTRACTIONS , E XPANSIONS , V ALVES ,

AND F ITTINGS

Experimental data are required to determine friction loss coefficients (kƒ) Most publishedvalues are for the turbulent flow of water taken from Crane Co (1982) These numbers aresummarized in various engineering handbooks such as Sakiadis (1984) Laminar flow dataare only available for a few limited geometries and specific fluids: Newtonian (Kittredge andRowley, 1957), shear-thinning (Edwards et al., 1985; Lewicki and Skierkowski, 1988; Steffe

et al., 1984), and shear-thickening (Griskey and Green, 1971) In general, the quantity of

engineering data required to predict pressure losses in valves and fittings for fluids, particularlynon-Newtonian fluids, in laminar flow is insufficient.

Friction loss coefficients for many valves and fittings are summarized in Tables 1.5 and1.6 The kƒ value for flow through a sudden contraction may be calculated at

(1.39)

where A1 equals the upstream cross-sectional and A2 equals the downstream cross-sectionalarea Losses for a sudden enlargement, or an exit, are found with the Borda-Carrot equation

(1.40)

Equations 1.39 and 1.40 are for Newtonian fluids in turbulent flow They are derived using

a momentum balance and the mechanical-energy balance equations It is assumed that lossesare due to eddy currents in the control volume In some cases (like Herschel-Bulkley fluidswhere α is a function of c), each section in the contraction, or expansion, will have a differentvalue of α; however, they differ by little and it is not practical to determine them separately.The smallest α (yielding the larger kƒ value) found for the upstream or downstream section

is recommended

After studying the available data for friction loss coefficients in laminar and turbulentflow, the following guidelines — conservative for shear thinning fluids — are proposed(Steffe, 1992) for estimating kƒ values:

FIGURE 1.16 Kinetic energy correction factors (α) for Herschel-Bulkley fluids in laminar flow.

(From Osorio, F A and Steffe, J F 1984, J Food Science, 49(5):1295–1296, 1315 With permission.)

α

TABLE 1.5 Friction Loss Coefficients for the Turbulent Flow

of Newtonian Fluids through Valves and Fittings

Tee, standard, along run, branch blanked off 0.4

1 For Newtonian fluids in turbulent or laminar flow use the data of Sakiadis (1984)

or Kittredge and Rowley (1957), respectively (Tables 1.5 and 1.6)

2 For non-Newtonian fluids above a Reynolds number of 500 (NRe, NRe,PL, or NRe,B),use data for Newtonian fluids in turbulent flow (Table 1.5)

3 For non-Newtonian fluids in a Reynolds number of 20 to 500 use the followingequation

a This is pressure drop (including friction loss) between run and branch, based on velocity in the main stream before branching Actual value depends on the flow split, ranging from 0.5 to 1.3 if main stream enters run and 0.7 to 1.5 if main stream enters branch.

b The fraction open is directly proportional to steam travel or turns of hand wheel Flow direction through some types of valves has a small effect on pressure drop For practical purposes this effect may be neglected.

c Values apply only when check valve is fully open, which is generally the case for velocities more than 3 ft/s for water.

Data from Sakiadis, B C 1984 Fluid and particle mechanics In: Perry,

R H., Green, D W., and Maloney, J O (ed.) Perry’s Chemical neers’ Handbook, 6th ed., Sect 5 McGraw-Hill, New York.

Engi-TABLE 1.6 Friction Loss Coefficients (k ƒ Values) for the Laminar Flow of Newtonian Fluids through Valves and Fittings

N Re =

Data from Sakiadis, B C 1984 Fluid and particle mechanics In: Perry,

R H., Green, D W., and Maloney, J O (ed.) Perry’s Chemical neers’ Handbook, 6th ed., Sect 5 McGraw-Hill, New York.

Engi-TABLE 1.5 (continued) Friction Loss Coefficients for the Turbulent Flow

of Newtonian Fluids through Valves and Fittings

where N is NRe, NRe,PL, or NRe,B depending on the type of fluid in question and β

is found for a particular valve or fitting (or any related item such as a contraction)

by multiplying the turbulent flow friction loss coefficient by 500:

(1.42)

Values of A for many standard items may be calculated from the kƒ values provided

in Table 1.5 Some A values can be determined (Table 1.7) from the work ofEdwards et al (1985) where experimental data were collected for elbows, valves,contractions, expansions, and orifice plates The Edwards study considered fivefluids: water, lubrication oil, glycerol-water mixtures, CMC-water mixtures (0.48

< n < 0.72, 0.45 < K < 11.8), and china clay-water mixtures (0.18 < n < 0.27, 3.25

< K < 29.8) Equations 1.41 and 1.42 are also acceptable for Newtonian fluidswhen 20 < NRe < 500

The above guidelines are offered with caution and should only be used in the absence ofactual experimental data Many factors, such as high extensional viscosity, may significantlyinfluence kƒ values

1.3 EXAMPLE CALCULATIONS

Consider the typical flow problem illustrated in Figure 1.17 The system has a 0.0348 mdiameter pipe with a volumetric flow rate of 1.57 × 10–3 m3/s (1.97 kg/s) or an averagevelocity of 1.66 m/s The density of the fluid is constant (ρ = 1250 kg/m3) and the pressuredrop across the strainer is 100 kPa Additional friction losses occur in the entrance, the plugvalve, and in the three long radius elbows Solving the mechanical energy balance, Equation1.1, for work output yields

TABLE 1.7 Values of ββββ, for Equation 1.41

90° Short curvature elbow, 1 and 2 inch 842 1–1000 Fully open gate valve, 1 and 2 inch 273 1–100 Fully open square plug globe valve, 1 inch 1460 1–10 Fully open circular plug globe valve, 1 inch 384 1–10 Contraction, A2/A1 = 0.445 110 1–100 Contraction, A 2 /A 1 = 0.660 59 1–100 Expansion, A2/A1 = 1.52 88 1–100 Expansion, A 2 /A 1 = 1.97 139 1–100

Note: Values are determined from the data of Edwards, M F., Jadallah,

M S M., and Smith, R 1985 Chem Eng Res Des 63:43–50.

kN

f = β

β =( )kf turbulent( )500

Subscripts 1 and 2 refer to the level fluid in the tank and the exit point of the system,respectively Assuming a near empty tank (as a worst case for pumping), P2 = P1 and u1 = 0,simplifies Equation 1.43 to

(1.47)

In the following example problems, only the rheological properties of the fluids will bechanged All other elements of the problem, including the fluid density, remain constant

1.3.1 C ASE 1: N EWTONIAN F LUID IN L AMINAR F LOW

Assume, µ = 0.34 Pa · s giving NRe = 212.4 which is well within the laminar range of NRe

< 2100 Then, from Table 1.5, Equations 1.39, 1.41, and 1.42

FIGURE 1.17 Typical pipeline system (From Steffe, J F and Morgan, R E 1986, Food Technol.,

40(12):78–85 With permission.)

− =W ( )u −( )u + ( − )+ − +

2 2

2 1 2

2

ΣF fu LD

32

The friction factor is calculated from Equation 1.8:

Then, the total friction losses are

and

1.3.2 C ASE 2: N EWTONIAN F LUID IN T URBULENT F LOW

Assume, µ = 0.012 Pa · s giving NRe = 6018, a turbulent flow value of NRe Friction losscoefficients may be determined from Equation 1.39, and Table 1.2: kƒ,entrance = 0.55; kƒ,valve =

9 ; kƒ,elbow = 0.45 The friction factor is determined by iteration of Equation 1.9:

giving a solution of ƒ = 0.0089 Continuing,

and

1.3.3 C ASE 3: P OWER L AW F LUID IN L AMINAR F LOW

Assume, K = 5.2 Pa · sn and n = 0.45 giving NRe,PL = 323.9, a laminar flow value of NRe,PL.Then, from Table 1.5, Equations 1.37, 1.41, and 1.42

The friction factor is calculated from Equation 1.12:

Then

and, using Equations 1.37 to calculate α,

1.3.4 C ASE 4: P OWER L AW F LUID IN T URBULENT F LOW

Assume, K = 0.25 Pa · sn and n = 0.45 giving NRe,PL = 6736.6 The critical value of NRe,PL

may be calculated as

meaning the flow is turbulent because 6736.6 > 2394 Friction loss coefficients are the same

as those found for Case 2: kƒ,entrance = 0.5 ; kƒ,valve = 9 ; kƒ,elbow = 0.45 The friction factor isfound by iteration of Equation 1.15:

1.3.5 C ASE 5: B INGHAM P LASTIC F LUID IN L AMINAR F LOW

Assume, µpl = 0.34 Pa · s and σo = 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 mined from Table 1.5, Equations 1.39, 1.41, and 1.42; however, in this particular problem,

deter-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 α, a function of c (Figure1.16), is taken as 1 (the worst case value) for the calculations The friction factor is found

by iteration of Equation 1.20:

resulting in ƒ = 0.114 Then,

and

1.3.6 C ASE 6: H ERSCHEL -B ULKLEY F LUID IN L AMINAR F LOW

Assume, K = 5.2, σo = 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 Case

3 because the Reynolds numbers are equal in each instance: kƒ,entrance = 0.83; kƒ,valve = 13.89;

kƒ,elbow = 0.69 Also, α = 1.2 can be taken as the worst case (Figure 1.16) The friction factor

is calculated by averaging the values found on Figures 1.11 and 1.12:

cc

c c

max

= ++

rR

rR

KL

rR