Engineering Flow and Heat Exchange

Engineering

Flow and

Heat Exchange

Octave Levenspiel

Third Edition

Engineering Flow and Heat Exchange

Octave Levenspiel

Engineering Flow

and Heat Exchange

Third Edition

Octave Levenspiel

Department of Chemical Engineering

Oregon State University

Corvallis, OR, USA

ISBN 978-1-4899-7453-2 ISBN 978-1-4899-7454-9 (eBook)

DOI 10.1007/978-1-4899-7454-9

Springer New York Heidelberg Dordrecht London

Library of Congress Control Number: 2014947869

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Preface

This volume presents an overview of fluid flow and heat exchange.

In the broad sense, fluids are materials that are able to flow under the right

conditions. These include all sorts of things: pipeline gases, coal slurries, tooth￾paste, gases in high-vacuum systems, metallic gold, soups and paints, and, of

course, air and water. These materials are very different types of fluids, and so it

is important to know the different classifications of fluids, how each is to be

analyzed (and these methods can be quite different), and where a particular fluid

fits into this broad picture.

This book treats fluids in this broad sense, including flows in packed beds and

fluidized beds. Naturally, in so small a volume, we do not go deeply into the study of

any particular type of flow; however, we do show how to make a start with each. We

avoid supersonic flow and the complex subject of multiphase flow, where each of

the phases must be treated separately.

The approach here differs from most introductory books on fluids, which focus

on the Newtonian fluid and treat it thoroughly, to the exclusion of all else. I feel that

the student engineer or technologist preparing for the real world should be intro￾duced to these other topics.

Introductory heat transfer books are devoted primarily to the study of the basic

rate phenomena of conduction, convection, and radiation, showing how to evaluate

“h,” “U,” and “k” for this and that geometry and situation. Again, this book’s

approach is different. We rapidly summarize the basic equations of heat transfer,

including the numerous correlations for h. Then we go straight to the problem of

how to get heat from here to there and from one stream to another.

The recuperator (or through-the-wall exchanger), the direct contact exchanger,

the heat-storing accumulator (or regenerator), and the exchanger, which uses a third

go-between stream—these are distinctly different ways of transferring heat from

one stream to another, and this is what we concentrate on. It is surprising how much

creativity may be needed to develop a good design for the transfer of heat from a

stream of hot solid particles to a stream of cold solid particles. The flavor of this

v

presentation of heat exchange is that of Kern’s unique book; certainly simpler, but

at the same time broader in approach.

Wrestling with problems is the key to learning, and each of the chapters has

illustrative examples and a number of practice problems. Teaching and learning

should be interesting, so I have included a wide variety of problems, some whim￾sical, others directly from industrial applications. Usually the information given in

these practice problems has been designed so as to fall on unique points on the

design charts, making it easy for the student and also for the instructor who is

checking the details of a student’s solution.

I think that this book will interest the practicing engineer or technologist who

wants a broad picture of the subject or, on having a particular problem to solve,

wants to know what approach to take.

In the university it could well form the basis for an undergraduate course

in engineering or applied fluids and heat transfer, after the principles have been

introduced in a basic engineering course such as transport phenomena. At present,

such a course is rarely taught; however, I feel it should be an integral part of

the curriculum, at least for the chemical engineer and the food technologist.

My thanks to Richard Turton, who coaxed our idiot computer into drawing

charts for this book, and to Eric Swenson, who so kindly consented to put his skilled

hand to the creation of drawing and sketch to enliven and complement the text.

Finally, many thanks to Bekki and Keith Levien, who without their help this new

revision would never have made it to print.

Corvallis, OR, USA Octave Levenspiel

vi Preface

Contents

Part I Flow of Fluids and Mixtures

1 Basic Equations for Flowing Streams ...................... 3

1.1 Total Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Mechanical Energy Balance . . . . . . . . . . . . . . . . . . . ....... 5

1.3 Pumping Energy and Power: Ideal Case . ................. 6

1.4 Pumping Energy and Power: Real Case Compression . . . ..... 7

1.4.1 Expansion . . ................................ 8

2 Flow of Incompressible Newtonian Fluids in Pipes . . . . . . . . . . . . 21

2.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

References and Recommended Readings . . . . . . . . . . . . . . . . . . . . . 43

3 Compressible Flow of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1 Adiabatic Flow in a Pipe with Friction . . . . . . . . . . . . . . . . . . . 46

3.2 Isothermal Flow in a Pipe with Friction . . . . . . . . . . . . . . . . . . 49

3.3 Working Equations for Flow in Pipes (No Reservoir

or Tank Upstream) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Flow Through an Orifice or Nozzle . . . . . . . . . . . . . . . . . . . . . 52

3.4.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5 Pipe Leading from a Storage Vessel . . . . . . . . . . . . . . . . . . . . . 54

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 Molecular Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.1 Equations for Flow, Conductance, and Pumping Speed . . . . . . . 73

4.1.1 Notation ................................... 73

4.1.2 Laminar Flow in Pipes . ........................ 74

4.1.3 Molecular Flow in Pipes ........................ 75

4.1.4 Intermediate or Slip Flow Regime ................. 76

vii

4.1.5 Orifice, Contraction, or Entrance Effect in the Molecular

Flow Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1.6 Contraction in the Laminar Flow Regime . . . . . . . . . . . 79

4.1.7 Critical Flow Through a Contraction . . . . . . . . . . . . . . . 79

4.1.8 Small Leak in a Vacuum System . . . . . . . . . . . . . . . . . 80

4.1.9 Elbows and Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.1.10 Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Calculation Method for Piping Systems . . . . . . . . . . . . . . . . . . 82

4.3 Pumping Down a Vacuum System . . . . . . . . . . . . . . . . . . . . . . 84

4.4 More Complete Vacuum Systems . . . . . . . . . . . . . . . . . . . . . . 87

4.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

References and Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5 Non-Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.1 Classification of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.1.1 Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.1.2 Non-Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2 Shear Stress and Viscosity of a Flowing Fluid . . . . . . . . . . . . . 102

5.3 Flow in Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.3.1 Bingham Plastics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.3.2 Power Law Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.3.3 General Plastics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.3.4 Comments on Flow in Pipes . . . . . . . . . . . . . . . . . . . . . 110

5.4 Determining Flow Properties of Fluids . . . . . . . . . . . . . . . . . . . 111

5.4.1 Narrow Gap Viscometer . . . . . . . . . . . . . . . . . . . . . . . . 112

5.4.2 Cylinder in an Infinite Medium . . . . . . . . . . . . . . . . . . . 112

5.4.3 Tube Viscometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.5 Discussion on Non-Newtonians . . . . . . . . . . . . . . . . . . . . . . . . 115

5.5.1 Materials Having a Yield Stress,

Such as Bingham Plastics . . . . . . . . . . . . . . . . . . . . . . . 115

5.5.2 Power Law Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.5.3 Thoughts on the Classification of Materials . . . . . . . . . . 117

References and Related Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6 Flow Through Packed Beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.1 Characterization of a Packed Bed . . . . . . . . . . . . . . . . . . . . . . 133

6.1.1 Sphericity ϕ of a Particle . . . . . . . . . . . . . . . . . . . . . . . 133

6.1.2 Particle Size, dp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.1.3 Determination of the Effective

Sphericity ϕeff from Experiment . . . . . . . . . . . . . . . . . . 137

6.1.4 Bed Voidage, ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.2 Frictional Loss for Packed Beds . . . . . . . . . . . . . . . . . . . . . . . . 139

6.3 Mechanical Energy Balance for Packed Beds . . . . . . . . . . . . . . 140

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

viii Contents

7 Flow in Fluidized Beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.1 The Fluidized State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.2 Frictional Loss and Pumping Requirement Needed

to Fluidize a Bed of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.3 Minimum Fluidizing Velocity, umf . . . . . . . . . . . . . . . . . . . . . . 156

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

8 Solid Particles Falling Through Fluids . . . . . . . . . . . . . . . . . . . . . 167

8.1 Drag Coefficient of Falling Particles . . . . . . . . . . . . . . . . . . . . 167

8.1.1 The Small Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

8.1.2 Nonspherical Particles . . . . . . . . . . . . . . . . . . . . . . . . . 168

8.1.3 Terminal Velocity of Any Shape

of Irregular Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 169

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Part II Heat Exchange

9 The Three Mechanisms of Heat Transfer: Conduction,

Convection, and Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

9.1 Heat Transfer by Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . 179

9.1.1 Flat Plate, Constant k . . . . . . . . . . . . . . . . . . . . . . . . . . 181

9.1.2 Flat Plate, k ¼ k0 (1 + βT) . . . . . . . . . . . . . . . . . . . . . . . 181

9.1.3 Hollow Cylinders, Constant k . . . . . . . . . . . . . . . . . . . . 181

9.1.4 Hollow Sphere, Constant k . . . . . . . . . . . . . . . . . . . . . . 181

9.1.5 Series of Plane Walls . . . . . . . . . . . . . . . . . . . . . . . . . . 182

9.1.6 Concentric Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . 182

9.1.7 Concentric Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

9.1.8 Other Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

9.1.9 Contact Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

9.2 Heat Transfer by Convection . . . . . . . . . . . . . . . . . . . . . . . . . . 183

9.2.1 Turbulent Flow in Pipes . . . . . . . . . . . . . . . . . . . . . . . . 184

9.2.2 Turbulent Flow in Noncircular Conduits . . . . . . . . . . . . 185

9.2.3 Transition Regime in Flow in Pipes . . . . . . . . . . . . . . . 186

9.2.4 Laminar Flow in Pipes (Perry and Chilton,

pg. 168 (1984)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

9.2.5 Laminar Flow in Pipes, Constant Heat Input

Rate at the Wall (Kays and Crawford 1980) . . . . . . . . . 186

9.2.6 Laminar Flow in Pipes, Constant Wall

Temperature (Kays and Crawford 1980) . . . . . . . . . . . . 187

9.2.7 Flow of Gases Normal to a Single Cylinder . . . . . . . . . . 188

9.2.8 Flow of Liquids Normal to a Single Cylinder . . . . . . . . 189

9.2.9 Flow of Gases Past a Sphere . . . . . . . . . . . . . . . . . . . . . 189

9.2.10 Flow of Liquids Past a Sphere . . . . . . . . . . . . . . . . . . . 189

9.2.11 Other Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Contents ix

9.2.12 Condensation on Vertical Tubes . . . . . . . . . . . . . . . . . 190

9.2.13 Agitated Vessels to Jacketed Walls . . . . . . . . . . . . . . . 190

9.2.14 Single Particles Falling Through Gases

and Liquids (Ranz and Marshall 1952) . . . . . . . . . . . . 191

9.2.15 Fluid to Particles in Fixed Beds

(Kunii and Levenspiel, 1991) . . . . . . . . . . . . . . . . . . . 191

9.2.16 Gas to Fluidized Particles . . . . . . . . . . . . . . . . . . . . . . 192

9.2.17 Fluidized Beds to Immersed Tubes . . . . . . . . . . . . . . . 192

9.2.18 Fixed and Fluidized Particles to Bed Surfaces . . . . . . . 192

9.2.19 Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . . . 192

9.2.20 Natural Convection: Vertical Plates

and Cylinders, L > 1 m . . . . . . . . . . . . . . . . . . . . . . . . 193

9.2.21 Natural Convection: Spheres and Horizontal

Cylinders, d < 0.2 m . . . . . . . . . . . . . . . . . . . . . . . . . 194

9.2.22 Natural Convection for Fluids in Laminar

Flow Inside Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

9.2.23 Natural Convection: Horizontal Plates . . . . . . . . . . . . . 195

9.2.24 Other Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

9.3 Heat Transfer by Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

9.3.1 Radiation from a Body . . . . . . . . . . . . . . . . . . . . . . . . 196

9.3.2 Radiation onto a Body . . . . . . . . . . . . . . . . . . . . . . . . 197

9.3.3 Energy Interchange Between a Body

and Its Enveloping Surroundings . . . . . . . . . . . . . . . . 197

9.3.4 Absorptivity and Emissivity . . . . . . . . . . . . . . . . . . . . 197

9.3.5 Greybodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

9.3.6 Radiation Between Two Adjacent Surfaces . . . . . . . . . 199

9.3.7 Radiation Between Nearby Surfaces

with Intercepting Shields . . . . . . . . . . . . . . . . . . . . . . 199

9.3.8 View Factors for Blackbodies . . . . . . . . . . . . . . . . . . . 200

9.3.9 View Factor for Two Blackbodies

(or GreyBodies) Plus Reradiating Surfaces . . . . . . . . . 202

9.3.10 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

9.3.11 Estimating the Magnitude of hr . . . . . . . . . . . . . . . . . . 208

References and Related Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

10 Combination of Heat Transfer Resistances . . . . . . . . . . . . . . . . . . . 211

10.1 Fluid–Fluid Heat Transfer Through a Wall . . . . . . . . . . . . . . . . 212

10.2 Fluid–Fluid Transfer Through a Cylindrical Pipe Wall . . . . . . . 214

10.3 Conduction Across a Wall Followed

by Convection and Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 216

10.4 Convection and Radiation to Two Different

Temperature Sinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

10.5 Determination of Gas Temperature . . . . . . . . . . . . . . . . . . . . . 218

10.6 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

x Contents

11 Unsteady-State Heating and Cooling of Solid Objects . . . . . . . . . . . 223

11.1 The Cooling of an Object When All the Resistance

Is at Its Surface (Bi ¼ hL/ks!0) . . . . . . . . . . . . . . . . . . . . . . . 225

11.2 The Cooling of an Object Having Negligible Surface

Resistance (Bi ¼ hL/ks!1) . . . . . . . . . . . . . . . . . . . . . . . . . . 226

11.3 The Cooling of an Object Where Both Surface

and Internal Resistances to Heat Flow Are Important

(0.1 < Bi < 40) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

11.4 The Cooling of a Semi-infinite Solid for Negligible

Surface Resistance (Bi ¼ hL/ks! 1) . . . . . . . . . . . . . . . . . . . . 230

11.5 The Cooling of a Semi-infinite Body Including

a Surface Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

11.6 Heat Loss in Objects of Size L for Short Cooling Times . . . . . . 239

11.7 The Cooling of Finite Objects Such as Cubes, Short

Cylinders, Rectangular Parallelepipeds, and So On . . . . . . . . . . 239

11.8 Intrusion of Radiation Effects . . . . . . . . . . . . . . . . . . . . . . . . . 239

11.9 Note on the Use of the Biot and Fourier Numbers . . . . . . . . . . 240

11.9.1 Assumption A. Particle Conduction

Controls: Bi !1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

11.9.2 Assumption B. Film Resistance

Controls: Bi !0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

11.9.3 Assumption C. Accounting for Both Resistances . . . . . 243

References and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

12 Introduction to Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . 253

12.1 Recuperators (Through-the-Wall Nonstoring Exchangers) . . . . . 253

12.2 Direct-Contact Nonstoring Exchangers . . . . . . . . . . . . . . . . . . 254

12.3 Regenerators (Direct-Contact Heat Storing Exchangers) . . . . . . 256

12.4 Exchangers Using a Go-Between Stream . . . . . . . . . . . . . . . . . 257

12.4.1 The Heat Pipe for Heat Exchange at a Distance . . . . . . 257

12.4.2 Solid–Solid Heat Transfer . . . . . . . . . . . . . . . . . . . . . 258

12.4.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

13 Recuperators: Through-the-Wall Nonstoring Exchangers . . . . . . . 261

13.1 Countercurrent and Cocurrent Plug Flow . . . . . . . . . . . . . . . . . 263

13.1.1 No Phase Change, Cp Independent

of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

13.1.2 Exchangers with a Phase Change . . . . . . . . . . . . . . . . 266

13.2 Shell and Tube Exchangers ........................... 267

13.3 Crossflow and Compact Exchangers . . . . . . . . . . . . . . . . . . . . 273

13.4 Cold Fingers or Bayonet Exchangers . . . . . . . . . . . . . . . . . . . . 278

13.5 Mixed Flow L/Plug Flow G Exchangers . . . . . . . . . . . . . . . . . . 281

13.6 Mixed Flow L/Mixed Flow G Exchangers ................ 282

13.7 Heating a Batch of Fluid ............................. 282

Contents xi

13.8 Uniformly Mixed Batch L/Mixed Flow G Exchangers . . . . . . . 283

13.9 Uniformly Mixed Batch L/Isothermal, Mixed Flow G

(Condensation or Boiling) Exchangers . . . . . . . . . . . . . . . . . . . 285

13.10 Uniformly Mixed Batch L/Plug Flow G Exchangers . . . . . . . . . 286

13.11 Uniformly Mixed Batch L/External Exchanger

with Isothermal G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

13.12 Uniformly Mixed Batch L/External Shell

and Tube Exchanger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

13.13 Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

References and Related Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

14 Direct-Contact Gas–Solid Nonstoring Exchangers . . . . . . . . . . . . . 305

14.1 Fluidized Bed Heat Exchangers: Preliminary

Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

14.2 Mixed Flow G/Mixed Flow S or Single-Stage

Fluidized Bed Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

14.3 Counterflow Stagewise Fluidized Bed Exchangers . . . . . . . . . . 308

14.4 Crossflow Stagewise Fluidized Bed Exchangers . . . . . . . . . . . . 310

14.5 Countercurrent Plug Flow Exchangers . . . . . . . . . . . . . . . . . . . 311

14.6 Crossflow of Gas and Solids . . . . . . . . . . . . . . . . . . . . . . . . . . 313

14.6.1 Well-Mixed Solids/Plug Flow Gas . . . . . . . . . . . . . . . 313

14.6.2 Solids Mixed Laterally but Unmixed

Along Flow Path/Plug Flow Gas . . . . . . . . . . . . . . . . . 314

14.6.3 Solids Unmixed/Plug Flow Gas . . . . . . . . . . . . . . . . . 315

14.7 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

14.8 Related Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

15 Heat Regenerators: Direct-Contact Heat Storing Exchangers

Using a Batch of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

15.1 Packed Bed Regenerators: Preliminary . . . . . . . . . . . . . . . . . . . 326

15.1.1 Spreading of a Temperature Front . . . . . . . . . . . . . . . . 326

15.1.2 Models for the Temperature Spread . . . . . . . . . . . . . . 327

15.1.3 Measure of Thermal Recovery Efficiency . . . . . . . . . . 329

15.1.4 Periodic Cocurrent and Countercurrent Operations . . . . 330

15.2 Packed Bed Regenerators: Flat Front Model . . . . . . . . . . . . . . . 331

15.2.1 Cocurrent Operations with ^t h¼^t c¼^t . . . . . . . . . . . . . . 331

15.2.2 Countercurrent Operations with ^t h¼^t c¼^t . . . . . . . . . . 332

15.2.3 Comments on the Flat Front Model . . . . . . . . . . . . . . . 332

15.3 Packed Bed Regenerators: Dispersion Model . . . . . . . . . . . . . . 333

15.3.1 Evaluation of σ2

, the Quantity Which Represents

the Spreading of the Temperature Front . . . . . . . . . . . 333

15.3.2 One-Pass Operations; Dispersion Model . . . . . . . . . . . 335

15.3.3 Periodic Cocurrent Operations with Equal Flow

Rates of Hot and Cold Fluids or ^t h¼^t c¼tsw

Dispersion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

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15.3.4 Periodic Countercurrent Operations with Equal Flow

Rates of Hot and Cold Fluids or ^t h¼^t c

Dispersion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

15.3.5 Comments on the Dispersion Model . . . . . . . . . . . . . . 340

15.4 Fluidized Bed Regenerators . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

15.4.1 Efficiency of One-Pass Operations . . . . . . . . . . . . . . . 341

15.4.2 Efficiency of Periodic Operations . . . . . . . . . . . . . . . . 343

15.4.3 Comments on Fluidized Bed Regenerators . . . . . . . . . 344

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

16 Potpourri of Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

Appendix: Dimensions, Units, Conversions, Physical Data,

and Other Useful Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

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