centrifugal pumps

Contents XIX List of tables Table 1.3 Equalization of non-uniform velocity distributions 34 Table 2.2 Total dynamic head and net positive suction head NPSH 44 Table 3.7 2 Leakage losses

Centrifugal Pumps

Johann Friedrich Gülich

Centrifugal Pumps

With 372 Figures and 106 Tables

123

Library of Congress Control Number: 2007934043

ISBN 978-3-540-73694-3 Springer Berlin Heidelberg New York

This work is subject to copyright All rights are reserved, whether the whole or part of the material

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Preface

Life is linked to liquid transport, and so are vital segments of economy Pumping devices – be it the human heart, a boiler feeder or the cooling-water pump of a motorcar – are always part of a more or less complex system where pump failure can lead to severe consequences To select, operate or even design a pump, some understanding of the system is helpful, if not essential Depending on the applica-tion, a centrifugal pump can be a simple device which could be built in a garage with a minimum of know-how – or a high-tech machine requiring advanced skills, sophisticated engineering and extensive testing When attempting to describe the state-of-the-art in hydraulic engineering of centrifugal pumps, the focus is neces-sarily on the high-tech side rather than on less-demanding services even though these make up the majority of pump applications

Centrifugal pump technology involves a broad spectrum of flow phenomena which have a profound impact on design and operation through the achieved effi-ciency, the stability of the head-capacity characteristic, vibration, noise, compo-nent failure due to fatigue, as well as material damage caused by cavitation, hy-dro-abrasive wear or erosion corrosion Operation and life cycle costs of pumping equipment depend to a large extent on how well these phenomena and the interac-tion of the pump with the system are understood

This book endeavors to describe pump hydraulic phenomena in their broadest sense in a format highly relevant for the pump engineer involved in pump design and selection, operation and troubleshooting Emphasis is on physical mecha-nisms, practical application and engineering correlations for real flow phenomena, rather than on mathematical treatment and inviscid theories

The first (1999) and second (2004) editions of this book were written in man In the present (third) edition, chapter 10 (vibrations) and chapter 13.2 (two-phase pumping) were considerably extended Additions were also made to chap-ters 2, 3, 5 to 7, 11, 14, 15 and the appendixes, minor additions and corrections in the remaining chapters New methods have been given for NPSH-scaling to lower speeds (chapter 6.2.3) and defining high-energy pumps (chapter 15.4)

Ger-The text has been translated by W Berner, Durban, South Africa, (about 40%

of the text), Mrs R Gülich (about 15%) and the author, who edited the raw lations extensively and is solely responsible for the final version presented

J.F Gülich

Acknowledgements

The English edition owes its existence to the initiative and sponsoring of the agement of Sulzer Pumps For this I am most grateful to Dr A Schachenmann who initiated the project, R Paley and Dr R Gerdes

man-The book benefited much from the help which I got from many colleagues at Sulzer Pumps in the US and the UK to whom my sincere thanks are extended: Most important were the reviews of the English text M Cropper was instrumental

in this activity S Bradshaw, R Davey, Dr J Daly, D Eddy, M Hall,

Dr A Kumar, P Sandford, D Townson, and C Whilde reviewed individual chapters

J.H Timcke meticulously checked most of the chapters for consistency with the 2nd edition and made many suggestions for making the text and figures easier

to understand Mrs H Kirchmeier helped with the figures and computer lems Last but not least, my wife Rosemarie Gülich was a tremendous help in checking and improving the final text

prob-I am grateful to various individuals who provided me with literature and granted permission for using figures: Prof Dr.-Ing F Avellan and Dr M Farhat from the Ecole Polytechnique Lausanne; Prof Dr.-Ing D.H Hellmann and

H Roclawski from the Technical University Kaiserslautern; Prof Dr.-Ing G syna, Prof Dr.-Ing habil U Stark, Mrs Dr.-Ing I Goltz, Mrs P Perez, Dr Ing

Ko-H Saathoff from the Technical University Braunschweig; C.Ko-H van den Berg, MTI Holland; Dr.-Ing H Wurm and F Holz, Wilo Dortmund; and A Nicklas, Sterling Fluid Systems

The 1st and 2nd editions benefited from the reviews of individual chapters vided by: Dr.-Ing G Scheuerer, ANSYS München and Dr P Heimgartner,

pro-W Bolliger, pro-W Schöffler, Dr.-Ing pro-W Wesche, Dr P Dupont, S Berten,

G Caviola, E Leibundgut, T Felix, A Frei, E Kläui (all from Sulzer)

The following organizations or individuals kindly granted permission for using figures:

− Sulzer Pumps Ltd, Winterthur

− Mr T McCloskey, Electric Power Research Institute, Palo Alto, CA, USA

− VDMA, Frankfurt

− VDI-Verlag, Düsseldorf

− Mr J Falcimaigne, Institut Français du Petrole, Paris

− ASME New York

The appropriate references are given in the figure captions

Hints for the reader VII

Hints for the reader

The text is written according to US-English spelling rules

As is customary in English publications, the decimal point is used (I had to

substitute points for commas in all figures, equations and graphs and hope not to have overlooked too many of them) To avoid confusion of readers used to the

decimal comma, large figures are written in the form of 6’150’000.00 (instead of

6,150,000.00)

Nomenclature: Unfortunately there is no commonly accepted nomenclature and

use of technical terms As far as possible I have consulted various standards as to the most accepted terms The reader is referred to the extensive list of symbols given below For easy reference this list defines the chapters, tables or equations where the respective symbols are introduced A number of subscripts from the German original were left unchanged since replacing them by meaningful English abbreviations involved too much of a risk of overlooking some items which are used throughout the text and the equations

Conventions: Equations, tables and figures are numbered by chapter The

geo-metrical dimensions of impellers, diffusers and volutes are defined in Table 0.2 Figures added after printing the first edition are labeled with capital letters after the figure number; e.g Fig 8.2B

To improve the readability, simplified expressions have sometimes been used (for example “volute” instead of “volute casing”) In order to avoid monotonous repetition of technical terms, synonyms are (sparingly) employed

Formulae frequently used in practice were gathered in tables which present the sequence of calculation steps required to solve a specific problem These tables help to find information quickly without looking through a lot of text; they also facilitate programming The equations presented in the tables are labeled by “T” For example Eq (T3.5.8) refers to equation 8 in table 3.5 Most of the tables are labeled as “Table 6.1”, for example Some “data tables” are referred to as “Ta-ble D6.1” for instance; this subterfuge was made necessary by the layout of the German editions which contained “tables” and “plates”

Mathematical expressions: Empirical data in the literature are frequently

pre-sented in graphical form In most cases such data are given in this book in the form of approximate equations in order to ease programming and to save space For reasons of simplicity the upper limit of a sum is not specified when there can be no doubt about the variable For example, ∑

i RRi,

represents the sum of the disk friction losses in all stages of a multistage pump

An equation of the form y = a×exp(b) stands for y = a×eb, where “e” is the base

of the natural logarithm

The symbol ~ is used for “proportional to”; for example, PRR ~ d2 stands for

“the disk friction loss is proportional to the 5th power of the impeller diameter”

Frequent reference is made to the specific speed nq which is always calculated with n in rpm, Q in m3/s and H in m For conversion to other units refer to Ta-ble D2.1 or Table 3.4

Many diagrams were calculated with MS-Excel which has limited capabilities for graphic layout For example: 1E+03 stands for 103; curve legends cannot show symbols or subscripts

Equations in the text are written for clarity with multiplier-sign i.e a×b (instead

of a b) This is not done in the numbered equations

The sketches should not be understood as technical drawings; in particular the hatching in sections was not always done in the usual way

Literature: There is a general bibliography quoted as [B.1], [B.2], etc while

standards are quoted as [N.1], [N.2], etc The bulk of the literature is linked to the individual chapters This eases the search for literature on a specific topic The roughly 600 quotations provided represent only 1% (order of magnitude) if not less of the relevant literature This statement applies to all topics treated in this book When selecting the literature, the following criteria were applied: (1) pro-viding the source of specific data or information; (2) for backing up a statement; (3) refer the reader to more details on the particular investigation or topic; (4) to provide reference to literature in neighboring fields In spite of these criteria, the selection of the literature quoted is to some extent coincidental

In order to improve the readability, facts which represent the state of the art are not backed up systematically by quoting literature where they may have been re-ported In many cases it would be difficult to ascertain where such facts were pub-lished for the first time

Patents: Possible patents on any devices or design features are not necessarily

mentioned Omission of such mention should not be construed so as to imply that such devices or features are free for use to everybody

Disclaimer of warranty and exclusion of liabilities: In spite of careful checking

text, equations and figures, neither the Springer book company nor the author:

• make any warranty or representation whatsoever, express or implied, (A) with respect of the use of any information, apparatus, method, process or similar item disclosed in this book including merchantability and fitness for practical purpose, or (B) that such use does not infringe or interfere with privately owned rights, including intellectual property, or (C) that this book is suitable to any particular user’s circumstances; or

• assume responsibility for any damage or other liability whatsoever (including consequential damage) resulting from the use of any information, apparatus, method, process or similar item disclosed in this book

In this context it should well be noted that much of the published information

on pump hydraulic design is empirical in nature The information has been ered from tests on specific pumps Applying such information to new designs har-bors uncertainties which are difficult to asses and to quantify

gath-Finally it should be noted that the technological focus in the various sectors of the pump industry is quite different Low-head pumps produced in vast quantities

Hints for the reader IX

are designed and manufactured to other criteria than engineered high-energy pumps This implies that the recommendations and design rules given in this book cannot be applied indistinctly to all types of pumps Notably, issues of standardi-zation and manufacturing are not addressed in this text

Table of contents

1 Fluid dynamic principles 1

1.1 Flow in the absolute and relative reference frame 1

1.2 Conservation equations 2

1.2.1 Conservation of mass 2

1.2.2 Conservation of energy 3

1.2.3 Conservation of momentum 4

1.3 Boundary layers, boundary layer control 7

1.4 Flow on curved streamlines 11

1.4.1 Equilibrium of forces 11

1.4.2 Forced and free vortices 14

1.4.3 Flow in curved channels 16

1.5 Pressure losses 18

1.5.1 Friction losses (skin friction) 18

1.5.2 Influence of roughness on friction losses 21

1.5.3 Losses due to vortex dissipation (form drag) 25

1.6 Diffusers 27

1.7 Submerged jets 31

1.8 Equalization of non-uniform velocity profiles 33

1.9 Flow distribution in parallel channels, piping networks 34

2 Pump types and performance data 39

2.1 Basic principles and components 39

2.2 Performance data 43

2.2.1 Specific work, head 43

2.2.2 Net positive suction head, NPSH 45

2.2.3 Power and efficiency 46

2.2.4 Pump characteristics 46

2.3 Pump types and their applications 47

2.3.1 Overview 47

2.3.2 Classification of pumps and applications 49

2.3.3 Pump types 52

2.3.4 Special pump types 64

3 Pump hydraulics and physical concepts 69

3.1 One-dimensional calculation with velocity triangles 69

Contents XI

3.2 Energy transfer in the impeller, specific work and head 72

3.3 Flow deflection caused by the blades Slip factor 75

3.4 Dimensionless coefficients, similarity laws and specific speed 80

3.5 Power balance and efficiencies 83

3.6 Calculation of secondary losses 85

3.6.1 Disk friction losses 85

3.6.2 Leakage losses through annular seals 90

3.6.3 Power loss caused by the inter-stage seal 98

3.6.4 Leakage loss of radial or diagonal seals 98

3.6.5 Leakage losses in open impellers 99

3.6.6 Mechanical losses 101

3.7 Basic hydraulic calculations of collectors 101

3.8 Hydraulic losses 107

3.9 Statistical data of pressure coefficients, efficiencies and losses 112

3.10 Influence of roughness and Reynolds number 120

3.10.1 Overview 120

3.10.2 Efficiency scaling 121

3.10.3 Calculation of the efficiency from loss analysis 123

3.11 Minimization of losses 129

3.12 Compendium of equations for hydraulic calculations 130

4 Performance characteristics 145

4.1 Head-capacity characteristic and power consumption 145

4.1.1 Theoretical head curve (without losses) 145

4.1.2 Real characteristics with losses 148

4.1.3 Component characteristics 151

4.1.4 Head and power at operation against closed discharge valve 157

4.1.5 Influence of pump size and speed 160

4.1.6 Influence of specific speed on the shape of the characteristics 160

4.2 Best efficiency point 161

4.3 Prediction of pump characteristics 166

4.4 Range charts 167

4.5 Modification of the pump characteristics 169

4.5.1 Impeller trimming 170

4.5.2 Under-filing and over-filing of the blades at the trailing edge 177

4.5.3 Collector modifications 178

4.6 Analysis of performance deviations 179

4.7 Calculation of modifications of the pump characteristics 182

5 Partload operation, impact of 3-D flow phenomena on performance 187

5.1 Basic considerations 187

5.2 The flow through the impeller 190

5.2.1 Overview 190

5.2.2 Physical mechanisms 192

5.2.3 The combined effect of different mechanisms 198

5.2.4 Recirculation at the impeller inlet 200

5.2.5 Flow at the impeller outlet 206

5.2.6 Experimental detection of the onset of recirculation 207

5.3 The flow in the collector 209

5.3.1 Flow separation in the diffuser 209

5.3.2 Pressure recovery in the diffuser 211

5.3.3 Influence of approach flow on pressure recovery and stall 213

5.3.4 Flow in the volute casing 214

5.3.5 Flow in annular casings and vaneless diffusers 215

5.4 The effects of flow recirculation 216

5.4.1 Effects of flow recirculation at the impeller inlet 216

5.4.2 Effect of flow recirculation at the impeller outlet 220

5.4.3 Effect of outlet recirculation on the flow in the impeller sidewall gaps and on axial thrust 226

5.4.4 Damaging effects of partload recirculation 229

5.5 Influence of flow separation and recirculation on the Q-H-curve 230

5.5.1 Types of Q-H-curve instability 230

5.5.2 Saddle-type instabilities 231

5.5.3 Type F instabilities 238

5.6 Means to influence the shape of the Q-H-curve 239

5.6.1 Introduction 239

5.6.2 Influencing the onset of recirculation at the impeller inlet 240

5.6.3 Influencing the onset of recirculation at the impeller outlet 240

5.6.4 Eliminating a type F instability 241

5.6.5 Influencing the saddle-type instability of impellers with nq < 50 242

5.6.6 Influencing the saddle-type instability of impellers with nq > 50 244

5.6.7 Influencing the instability of semi-axial and axial impellers 244

5.6.8 Reduction of head and power at shut-off 248

5.7 Flow phenomena in open axial impellers 249

6 Suction capability and cavitation 257

6.1 Cavitation physics 257

6.1.1 Growth and implosion of vapor bubbles in a flowing liquid 257

6.1.2 Bubble dynamics 259

6.2 Cavitation in impeller or diffuser 262

6.2.1 Pressure distribution and cavity length 262

6.2.2 Required NPSH, extent of cavitation, cavitation criteria 264

6.2.3 Scaling laws for cavitating flows 265

6.2.4 The suction specific speed 269

6.2.5 Experimental determination of the required NPSHR 271

6.2.6 Cavitation in annular seals 281

6.3 Determination of the required NPSH 281

6.3.1 Parameters influencing NPSHR 281

6.3.2 Calculation of the NPSHR 284

6.3.3 Estimation of the NPSH3 as function of the flow rate 288

Contents XIII

6.4 Influence of the fluid properties 291

6.4.1 Thermodynamic effects 292

6.4.2 Non-condensable gases 294

6.4.3 Nuclei content and tensile stresses in the liquid 295

6.5 Cavitation-induced noise and vibrations 298

6.5.1 Excitation mechanisms 298

6.5.2 Cavitation noise measurements for quantifying the hydro-dynamic cavitation intensity 299

6.5.3 Frequency characteristics of cavitation noise 302

6.6 Cavitation erosion 303

6.6.1 Testing methods 304

6.6.2 Cavitation resistance 306

6.6.3 Prediction of cavitation damage based on cavity length 309

6.6.4 Prediction of cavitation damage based on cavitation noise 312

6.6.5 Solid-borne noise measurements for cavitation diagnosis 314

6.6.6 Paint erosion tests to determine the location of bubble implosion 314

6.6.7 Onset of erosion and behavior of material subject to different hydrodynamic cavitation intensities 316

6.6.8 Summarizing assessment 319

6.7 Selection of the inlet pressure in a plant 323

6.8 Cavitation damage: analysis and remedies 326

6.8.1 Record damage and operation parameters 326

6.8.2 Forms of cavitation and typical cavitation damage patterns 327

6.8.3 Reduction or elimination of cavitation damage 332

6.9 Insufficient suction capacity: Analysis and remedies 333

7 Design of the hydraulic components 335

7.1 Methods and boundary conditions 335

7.1.1 Methods for the development of hydraulic components 335

7.1.2 The hydraulic specification 336

7.1.3 Calculation models 337

7.2 Radial impellers 339

7.2.1 Determination of main dimensions 339

7.2.2 Impeller design 348

7.2.3 Criteria for shaping the blades 353

7.2.4 Criteria for suction impeller design 356

7.2.5 Exploiting three-dimensional effects in design 358

7.3 Radial impellers for specific speeds below nq ≈ 18 359

7.3.1 Two-dimensional blades 359

7.3.2 Pumping disks with channels of circular section 361

7.3.3 Impellers with straight radial blades 363

7.3.4 Double-acting impeller with straight radial blades 364

7.4 Radial impellers for non-clogging pumps 366

7.5 Semi-axial impellers 368

7.6 Axial impellers and diffusers 373

7.6.1 Features 373

7.6.2 Calculation and selection of main dimensions 374

7.6.3 Basic properties of airfoils 379

7.6.4 Blade design 383

7.6.5 Profile selection 388

7.6.6 Design of axial diffusers 390

7.7 Inducers 392

7.7.1 Calculation of inducer parameters 393

7.7.2 Design and shaping of an inducer 398

7.7.3 Matching the inducer to the impeller 399

7.7.4 Recommendations for inducer application 400

7.8 Volute casings 402

7.8.1 Calculation and selection of main dimensions 402

7.8.2 Design and shaping of volute casings 406

7.8.3 Influence of the volute shape on hydraulic performance 410

7.9 Radial diffusers with or without return channels 412

7.9.1 Calculation and selection of main dimensions 412

7.9.2 Design and shaping of radial diffusers 418

7.10 Semi-axial diffusers 421

7.11 Volutes combined with a diffuser or stay vanes 422

7.12 Annular casings and vaneless diffusers 423

7.13 Inlet casings for between-bearing pumps 424

8 Numerical flow calculations 429

8.1 Overview 429

8.2 Quasi-3D-procedures and 3D-Euler-calculations 431

8.2.1 Quasi-3D- procedures 431

8.2.2 Three-dimensional Euler-procedures 432

8.3 Basics of Navier-Stokes calculations 433

8.3.1 The Navier-Stokes equations 433

8.3.2 Turbulence models 434

8.3.3 Treatment of near-wall flows 439

8.3.4 Grid generation 441

8.3.5 Numerical procedures and control parameters 444

8.3.6 Boundary conditions 446

8.3.7 Initial conditions 448

8.3.8 Possibilities of 3D-Navier-Stokes-calculations 449

8.4 Averaging and post-processing 452

8.5 Impeller calculations 459

8.5.1 Global performance at best efficiency flow rate 459

8.5.2 Velocity profiles 462

8.5.3 Influence parameters 463

8.5.4 Sample calculation 463

8.6 Calculation of collectors and stages 466

8.6.1 Separate calculation of the collector 466

Contents XV

8.6.2 Steady calculations of stages or complete pumps 467

8.6.3 Unsteady calculations 469

8.7 Two-phase and cavitating flows 470

8.8 Calculation strategy, uncertainties, quality issues 473

8.8.1 Uncertainties, sources and reduction of errors 473

8.8.2 CFD quality assurance 475

8.8.3 Comparison between calculation and experiment 486

8.9 Criteria for assessment of numerical calculations 488

8.9.1 General remarks 488

8.9.2 Consistence and plausibility of the calculation 488

8.9.3 Will the specified performance be reached? 489

8.9.4 Maximization of the hydraulic efficiency 489

8.9.5 Stability of the head-capacity curve 492

8.9.6 Unsteady forces 492

8.10 Fundamental considerations on CFD-calculations 493

9 Hydraulic forces 495

9.1 Flow phenomena in the impeller sidewall gaps 495

9.2 Axial thrust 508

9.2.1 General procedure for calculating axial thrust 508

9.2.2 Single-stage pumps with single-entry overhung impeller 511

9.2.3 Multistage pumps 515

9.2.4 Double-entry impellers 519

9.2.5 Semi-axial impellers 520

9.2.6 Axial pumps 520

9.2.7 Expeller vanes 520

9.2.8 Semi-open and open impellers 522

9.2.9 Unsteady axial thrust 523

9.3 Radial thrust 524

9.3.1 Definition and scope 524

9.3.2 Measurement of radial forces 525

9.3.3 Pumps with single volutes 526

9.3.4 Pumps with double volutes 531

9.3.5 Pumps with annular casings 532

9.3.6 Diffuser pumps 533

9.3.7 Radial thrust created by non-uniform approach flows 533

9.3.8 Axial pumps 535

9.3.9 Radial thrust balancing 535

9.3.10 Radial thrust prediction 536

10 Noise and Vibrations 539

10.1 Unsteady flow at the impeller outlet 539

10.2 Pressure pulsations 542

10.2.1 Generation of pressure pulsations 542

10.2.2 Noise generation in a fluid 543

10.2.3 Influence parameters of the pump 544

10.2.4 Influence of the system 545

10.2.5 Scaling laws 546

10.2.6 Measurement and evaluation of pressure pulsations 547

10.2.7 Pressure pulsations of pumps in operation 549

10.2.8 Damaging effects of pressure pulsations 552

10.2.9 Design guidelines 552

10.3 Component loading by transient flow conditions 553

10.4 Radiation of noise 555

10.4.1 Solid-borne noise 555

10.4.2 Air-borne noise 556

10.5 Overview of mechanical vibrations of centrifugal pumps 559

10.6 Rotor dynamics 561

10.6.1 Overview 561

10.6.2 Forces in annular seals 562

10.6.3 Hydraulic impeller interaction 569

10.6.4 Bearing reaction forces 570

10.6.5 Eigen values and critical speeds 571

10.6.6 Rotor instabilities 574

10.7 Hydraulic excitation of vibrations 577

10.7.1 Interactions between impeller and diffuser blades 577

10.7.2 Rotating stall 581

10.7.3 Other hydraulic excitation mechanisms 582

10.8 Guidelines for the design of pumps with low sensitivity to vibrations 586

10.9 Allowable vibrations 589

10.10 General vibration diagnostics 592

10.10.1 Overview 592

10.10.2 Vibration measurements 593

10.10.3 Vibration diagnostics 595

10.11 Bearing housing vibrations: mechanism, diagnostics, remedies 601

10.11.1 Hydraulic excitation mechanisms 602

10.11.2 Mechanical reaction to hydraulic excitation 606

10.11.3 Hydraulic versus mechanical remedies 608

10.11.4 Bearing housing vibration diagnostics 610

10.12 Hydraulic and acoustic excitation of pipe vibrations 621

10.12.1 Excitation of pipe vibrations by pumps 622

10.12.2 Excitation of pipe vibrations by components 624

10.12.3 Acoustic resonances in pipelines 624

10.12.4 Hydraulic excitation by vortex streets 629

10.12.5 Coupling of flow phenomena with acoustics 631

10.12.6 Pipe vibration mechanisms 635

11 Operation of centrifugal pumps 639

11.1 System characteristics, operation in parallel or in series 639

11.2 Pump control 644

Contents XVII

11.3 Static and dynamic stability 651

11.4 Start-up and shut-down 653

11.5 Power failure, water hammer 657

11.6 Allowable operation range 658

11.7 The approach flow to the pump 661

11.7.1 Suction piping layout 662

11.7.2 Transient suction pressure decay 664

11.7.3 Pump intakes and suction from tanks with free liquid level 670

11.7.4 Can pumps 685

11.8 Discharge piping 685

12 Turbine operation, general characteristics 689

12.1 Reverse running centrifugal pumps used as turbines 689

12.1.1 Theoretical and actual characteristics 689

12.1.2 Runaway and resiatnce characteristics 695

12.1.3 Estimation of turbine characteristics from empirical correlations 696

12.1.4 Behavior of turbines in plants 701

12.2 General characteristics 704

13 Influence of the medium on performance 711

13.1 Pumping highly viscous fluids 711

13.1.1 Effect of viscosity on losses and performance characteristics 711

13.1.2 Estimation of viscous performance from the characteristics measured with water 718

13.1.3 Influence of viscosity on the suction capacity 724

13.1.4 Start-up of pumps in viscous service 725

13.1.5 Viscous pumping applications - recommendations and comments 725 13.2 Pumping of gas-liquid mixtures 727

13.2.1 Two-phase flow patterns in straight pipe flow 727

13.2.2 Two-phase flow in pumps Physical mechanisms 730

13.2.3 Calculation of two-phase pump performance 740

13.2.4 Radial pumps operating with two-phase flow 746

13.2.5 Helico-axial multiphase pumps 752

13.2.6 System curves 756

13.2.7 Slugs and gas pockets 757

13.2.8 Free gas, dissolved gas and NPSH 759

13.3 Expansion of two-phase mixtures in turbines 760

13.3.1 Calculation of the work transfer 760

13.3.2 Prediction of turbine characteristics for two-phase flow 762

13.4 Hydraulic transport of solids 765

13.5 Non-Newtonian liquids 773

14 Selection of materials exposed to high flow velocities 777

14.1 Impeller or diffuser fatigue fractures 778

14.2 Corrosion 790

14.2.1 Corrosion fundamentals 790

14.2.2 Corrosion mechanisms 791

14.2.3 Corrosion in fresh water, cooling water, sewage 795

14.2.4 Corrosion in sea water and produced water 798

14.3 Erosion corrosion in demineralized water 803

14.4 Material selection and allowable flow velocities 812

14.4.1 Definition of frequently encountered fluids 812

14.4.2 Metallic pump materials 814

14.4.3 Impellers, diffusers and casings 820

14.4.4 Wear ring materials 831

14.4.5 Shaft materials 834

14.4.6 Materials for feedwater and condensate pumps 835

14.4.7 Materials for FGD-pumps 836

14.4.8 Composite materials 837

14.5 Hydro-abrasive wear 839

14.5.1 Influence parameters 839

14.5.2 Quantitative estimation of hydro-abrasive wear 842

14.5.3 Material behavior and influence of solids properties 848

14.5.4 Material selection 852

14.5.5 Abrasive wear in slurry pumps 853

15 Pump selection and quality considerations 857

15.1 The pump specification 858

15.2 Determination of pump type and size 860

15.3 Technical quality criteria 866

15.3.1 Hydraulic criteria 866

15.3.2 Manufacturing quality 870

15.4 High-energy pumps 875

Appendices 881

A1 Units and unit conversion 881

A2 Properties of saturated water 883

A3 Solution of gases in water 886

A4 Physical constants 889

A4.1 Atmospheric pressure 889

A4.2 Acceleration due to gravity 889

A5 Sound velocity in liquids 890

Literature 891

Index 917 List of tables XIX Symbols and abbreviations XXIII

Contents XIX

List of tables

Table 1.3 Equalization of non-uniform velocity distributions 34

Table 2.2 Total dynamic head and net positive suction head (NPSH) 44

Table 3.7 (2) Leakage losses II: Radial or diagonal gaps, open impellers 138

Table 3.7 (3) Screw-type pumping grooves in turbulent flow 139

Table 3.9 Statistical efficiency data and efficiency scaling 142

Table 3.10 Influence of roughness and Reynolds number on efficiency 143

Table 4.1 Diffuser/volute characteristic and best efficiency point 162

Table 5.1 Interpretation and modification of pump characteristics 246-247

Table 6.1 Assessment of the risk of cavitation damage 320-321

Table 6.2 Selection of available NPSHA to ensure safe operation 324

Table 7.4 Airfoils 386

Table 9.1 Rotation of the fluid in the impeller sidewall gaps 512-513

Table 10.1 Hydraulically induced damage to pump components 550

Table 10.4 Noise emission of pumps, A-weighted levels in dB A 557

Table 10.6 Assessment of shaft vibrations from bearing clearance 590 Table 10.7 Assessment of shaft vibrations, ISO 7919-3 and API 610 590 Table 10.8 Assessment of bearing housing vibrations, ISO and API 591

Table 10.10 Impact of hydraulic design on bearing housing vibrations 611 Table 10.11A Fundamental frequencies and data treatment 614-615

Table 10.11G Data relevant for interaction with rotor dynamics 620

Table 11.1 Calculation of suction pressure decay transients 669

Table 11.3 Empirical data for air-drawing surface vortices 677

Table 13.1 Estimation of characteristics for pumping viscous fluids 721 Table 13.2 (1) Estimation of pump characteristics for viscous fluids, [B5] 723 Table 13.2 (2) Estimation of pump characteristics for viscous fluids, [N.12] 724

Table 13.3 (2) Polytropic compression of Gas-Liquid-Mixtures 745

Contents XXI

Table 14.4 Electrochemical potential in flowing sea water 793

Table 14.7 Metal loss due to erosion corrosion in demineralized water 808

Table 14.10 Determination of the allowable head per stage 822

Table 14.14 Materials for shafts exposed to the fluid pumped 835

Table 14.16 Estimation of metal loss due to hydro-abrasive wear 846

Table 15.2 Tolerances for impellers, diffusers and volutes 872

Table 15.3 Quality requirements for impellers and diffusers 873

Table A2-2 Approximate formulae for water and vapor properties 883

List of data tables

Table D6.3 Coefficients λc and λw at shockless flow 284

Table D7.1 Suction impeller design 358

Table D9.1 Flow and pressure around the impeller circumference 529

Table D10.3 Allowable pipe vibration velocities, ISO 10816-1 638

Symbols and abbreviations

Unless otherwise noted all equations are written in consistent units (SI-System)

Most symbols are defined in the following As appropriate, the equation or

chap-ter is quoted where the symbol has been defined or introduced Vectors in the text

and in equations are printed as bold characters Symbols with local significance

only are defined in the text

The following tables may help the understanding of the physical meaning of

various parameters of prime importance:

• Table 0.1 and 0.2: Geometric dimensions of the flow channels, flow angles

and velocities

• Table 2.2: Head and net positive suction head (NPSH)

• Tables 3.1 and 3.2: Velocity triangles

• Table 3.4: Model laws and dimensionless parameters

Chapter or Equation

A2q area between vanes at impeller outlet (trapezoidal: A2q = a2 ×b2)

A3q diffuser/volute inlet throat area (trapezoidal: A3q = a3 ×b3)

BEP best efficiency point

b acceleration

b width of channel in the meridional section

b2 impeller outlet width; if double-entry, per impeller side

b2tot (b2ges) impeller outlet width including shrouds Eq (9.6)

CV solid-borne noise as RMS of acceleration

CV* dimensionless solid-borne noise acceleration CV* = CV×d1/u1

c3q average velocity in diffuser throat c3q = QLe/(zLe ×A3q)

ceq roughness equivalence factor Eq (1.36b)

cp pressure recovery coefficient Eqs (1.11), (1.40), (T9.1.5)

db arithmetic average of diameters at impeller or diffuser

e.g d1b = 0.5 (d1 + d1i); defined such that: A1 = π×d1b ×b1 Table 0.2

dd inner diameter of discharge nozzle

dm geometric average of diameters at impeller or diffuser, e.g

) d d ( 5 0

ds inner diameter of suction nozzle

ER maximum erosion rate (at location of highest metal loss) Table 6.1

F force

Fax axial force (“axial thrust”)

FDsp radial thrust correction factor for double volutes, Fig 9.18 Table 9.4

FMat material factor for cavitation: Table 6.1, for abrasion: Table 14.16

f frequency

fL influence of leakage flow on disk friction Eq (T3.6.7), Table 3.6

fn frequency of rotation fn = n/60

fq impeller eyes per impeller: single-entry fq = 1; double-entry fq = 2

fH correction factor for head (roughness, viscosity) Eq (3.32)

fη correction factor for efficiency (roughness, viscosity) Eq (3.31)

Symbols and abbreviations XXV

h casing wall thickness (at location of accelerometer) Table 6.1

Jsp integral over diffuser or volute throat area Eq (3.15); (4.13)

k rotation of fluid in impeller sidewall gap k = β/ω Eq (9.1), Table 9.1

kE, kz rotation of fluid at inlet to impeller sidewall gap Fig 9.1

kn blockage caused by hub: kn = 1 – dn/d1

kR,D radial thrust coefficient referred to d2 (steady) Table 9.4

kR,dyn dynamic (unsteady) radial thrust coefficient Table 9.4

kR,tot total radial thrust coefficient (steady and unsteady) Table 9.4

kR,o radial thrust coefficient (steady) for operation at Q = 0 Table 9.4

L length

M torque

m difference of impeller and diffuser periodicity Chap 10.7.1

m& mass flow rate

NPSH net positive suction head

NPSHA net positive suction head available Table 2.2, Table 6.2

NPSHi net positive suction head required for cavitation inception

NPSHR net positive suction head required according to a specific cavitation

NPSHx net positive suction head required for operation with x-per cent head

NL fluid-borne sound pressure as RMS value; NL* = 2NL/(ρ×u1 )

n rotational speed (revolutions per minute)

n(s) rotational speed (revolutions per second)

nq specific speed [rpm, m3/s, m] Table D2.1, Chap 3.4, Table 3.4

nss suction specific speed [rpm, m3/s, m] Chap 6.2.4, Table 3.4

P power; without subscript: power at coupling

Pm mechanical power losses Table 3.5

Pu useful power transferred to fluid Pu = ρ×g×Htot ×Q Table 3.5

Ps3 power loss dissipated in inter-stage seal Tables 3.5, 3.7(1)

pamb ambient pressure at location of pump installation (usually

atmos-pheric pressure)

pe pressure above liquid level in suction reservoir Table 2.2

QLa flow rate through impeller: QLa = Q + Qsp + QE + Qh = Q/ηv

QLe flow rate through diffuser: QLe = Q + Qs3 + QE

QE flow rate through axial thrust balancing device

Qh flow rate through auxiliaries (mostly zero)

Qsp leakage flow rate through seal at impeller inlet Tables 3.5, 3.7(1)

Qs3 leakage flow rate through inter-stage seal Tables 3.5, 3.7(1)

q* flow rate referred to flow rate at best efficiency point: q* ≡ Q/Qopt

Re Reynolds number, channel: Re = c×Dh/ν; plate or blade: Re = w×L/ν

SG specific gravity; SG ≡ ρ/ρRef with ρRef = 1000 kg/m3

sax axial distance between impeller shrouds and casing Fig 9.1

T temperature

t time

U wetted perimeter (of a pipe or channel)

V volume

Symbols and abbreviations XXVII

w1q average velocity in impeller throat area w1q = QLa/(zLa ×A1q)

x dimensionless radius x = r/r2 Table 9.1

x gas (or vapor) mass content; mass concentration of solids Chap 13

Ysch≡ Yth specific work done by the impeller blades: Yth = g×Hth Table 3.3

Yth ∞ specific work done by the impeller blades with vane congruent flow

Zh hydraulic losses (impeller: ZLa diffuser: ZLe)

zVLe number of vanes of pre-rotation control device

zLe number of diffuser vanes (volute: number of cutwaters)

zpp number of pumps operating in parallel

α ≡ GVF gas content, gas volume fraction, void fraction Table 13.2

α angle between direction of circumferential and absolute velocity

β angle between relative velocity vector and the negative direction of

circumferential velocity

β angular velocity of fluid between impeller and casing Chap 9.1

∆pp-p pressure pulsations measured peak-to-peak Chap 10.2.6

εsp wrap angle of the inner volute (for double volutes) Table 0.2

ηst stage efficiency Eq (T3.5.7)

λ angle between vanes and side disks (impeller or diffuser) Table 0.1

µ dynamic viscosity: µ = ρ×ν

ν kinematic viscosity: ν = µ/ρ

ν1, ν2 vibration orders, natural numbers (1, 2, 3, ….)

ρ" density of gas or saturated vapor

ρmat density of material

ρp density of casing material

ρs density of solids suspended in the fluids Chap 13.4, 14.5

ψp pressure coefficient of static pressure rise in impeller Table 3.3

Ωlimit orbit frequency of stability limit Eq (10.9)

Subscripts, superscripts, abbreviations

Sequence of calculations: in the pumping mode the fluid flows from station 1 to 6,

in the turbine mode from 6 to 1:

1 impeller blade leading edge (low pressure)

2 impeller blade trailing edge (high pressure)

3 diffuser vane leading edge or volute cutwater

Symbols and abbreviations XXIX

A plant

al allowable

ax axial

a,m,i outer, mean, inner streamline

B blade angle (impeller, diffuser, volute cutwater)

NDE non-drive end (of a pump shaft)

opt operation at maximum (best) efficiency (BEP)

PS (DS) pressure surface (pressure side)

q average velocity calculated from continuity (to be distinguished

from velocity vector)

Rec (Rez) recirculation

r radial

Sp volute

SS suction surface (suction side)

Ts (RS) rear shroud or hub

th theoretical flow conditions (flow without losses)

tot total (total pressure = static pressure + stagnation pressure)

v loss

w resistance curve in turbine mode locked rotor (n = 0) Chap 12

zul (al) allowable

The following are superscripts:

* dimensionless quantity: all dimensions are referred to d2 e.g

b2* = b2/d2,velocities are referred to u2, e.g w1* = w1/u2

Symbols and abbreviations XXXI

Table 0.1 Dimensions and flow parameters

Location, main

Vane angles

sin sin d e z 1

1

λ β π

sin sin d e z 1

1

λ β π

sin sin d e z 1

1

λ α π

Inlet:

d5, b5, a5, e5

5B 5 5 R 5

sin d e z 1 1

α π

sin d e z 1 1 α π

Without special subscript: u1≡ u1a and d1≡ d1a as well as d2≡ d2a, if d2a = d2i

The meridional velocity components are equal in relative and absolute system: wm = cmCircumferential velocity components c u and w u are not influenced by blade blockage

Impeller shroud Blade

Influence of blade inclination on blade blockage (twisted blades)

Table 0.2 (1) Geometric dimensions

Outer volute

Inner volute

Outer channel

Symbols and abbreviations XXXIII

Table 0.2 (2) Geometric dimensions

Im peller sidew all gap

Im peller sidewall gap Cham fer

Further pump styles are defined on page 880

OHH – Overhung Single Stage

ISO 13709 (API 610) Type OH2

OHC – Overhung Single Stage

with Canned Motor

API 685

OHM – Overhung Singe Stage

ISO 13709 (API 610) Type OH3

BBS – Between Bearings Single Stage ISO 13709 (API 610) Type BB2

GSG – Inline or Opposed Impeller Diffuser Barrel Type

ISO 13709 (API 610) Type BB5

1 Fluid dynamic principles

The nearly inexhaustible variety of flow phenomena – from the flow through blood vessels, the flow in centrifugal pumps to global weather events − is based

on a few basic physical laws only In this chapter these will be briefly reviewed and their general nature illuminated Emphasis will be on the phenomena which are of special interest and significance to the pump engineer Basic knowledge of the terms of fluid dynamics is taken for granted There is a wealth of textbooks and handbooks on fluid dynamics, only a few can be quoted: [1.1 to 1.7, 1.14]

1.1 Flow in the absolute and relative reference frame

In turbomachine design a flow described in fixed coordinates is called an lute” movement, while a flow in the rotating reference frame is termed “relative” The flow in the relative reference frame corresponds to the movement that would

“abso-be seen by a co-rotating observer A point on a rotating disk is stationary in the relative reference frame, while it describes a circle in the absolute system If a

mass moves radially outward in a guide on a rotating disk, it follows a straight path in the relative reference frame, while it describes a spiral-shaped movement

in the absolute reference frame When transforming a movement from the absolute

to the relative system, the centrifugal and Coriolis forces must be introduced The absolute acceleration babs is then obtained as a vectorial sum from relative, cen-trifugal and Coriolis acceleration as:1

The velocity conditions are described by three vectors: the peripheral velocity

u = ω×r, the relative velocity w and the absolute velocity c which is obtained

through the vectorial addition of u and w: c = u + w In turbomachine design this

addition is usually presented graphically by “velocity triangles” according to Fig 1.1 (see Chap 3.1)

1.2 Conservation equations

The conservation laws for mass, energy and momentum form the actual basis of fluid dynamics These describe the (not strictly derivable) observation that neither mass nor energy nor momentum can be destroyed or created in a closed system or control volume Accordingly, the following balance equation applies to each of the above quantities X:

0 Z t

X X

∆

∆ +

In Eq (1.1) X1 stands for the input, X2 isthe output, ∆X/∆t is the change in the control volume over time step ∆t, and Z is an additional quantity (supplied or re-moved mass, heat or work) In this general form, the conservation laws apply to steady and unsteady processes of any complexity with or without losses Only steady processes with ∆X/∆t = 0 will be discussed in the following If no mass,

work or heat is supplied or removed (i.e Z = 0), Input = Output applies

The consistent application of these seemingly trivial balance equations is often the only means of quantitatively treating complex problems without resorting to experiments By applying the conservation laws to an infinitesimal volume ele-ment of a flowing fluid, partial differential equations are obtained These com-pletely describe the three-dimensional flow field (continuity and Navier-Stokes equations) which, in general, cannot be solved analytically but only numerically

1.2.1 Conservation of mass

To formulate the conservation laws, consider any control volume according to Fig 1.2 which can be a streamline, a pipe or a machine At the inlet cross section with the control surface A1, fluid enters with the velocity c1 and the density ρ1 The corresponding quantities at the outlet control surface are described by the sub-script 2 The conservation of mass according to Eq (1.1) yields with Z = 0:

1 1 1 2 2 2

m & = ρ A c = A c = constant ρ (1.2) and for incompressible flow (constant density): A1 ×c1 = A2 ×c2 This is the continu-ity equation It states that the incoming and outgoing mass flows are identical in magnitude for a given control volume under steady conditions

= P + P h m h

In Eq (1.3) hTot is the total enthalpy [B.3]; it is equal to the sum of the internal ergy per unit mass U, the static pressure energy p/ρ, the kinetic energy ½c2 and the potential energy g×z:

en-z g + 2

c + p + U

=

Pi is the internal power; it is the sum of the mechanical energy transmitted to the fluid and all the losses which heat up the fluid, see Chap 3.5 For exact calcula-tions on high-pressure pumps, e.g the thermometric determination of efficiency,

as well as for all turbomachines with compressible flows, the generally applicable form of Eq (1.5) must be utilized The enthalpy difference is obtained from Eqs (1.4) and (1.5) as follows:

) z (z g + 2 c c + p p + U

- U

= m

Since the exchange of heat with the environment has been neglected, the change

of the internal energy U is obtained solely from the heating due to losses within the control volume or the machine This is true for incompressible flow only; for liquids it is thus possible to set (U2 – U1) = ∆pv/ρ

Equation (1.6) follows from the first law of thermodynamics; it describes the energy on a streamline without external work transmission (∆hTot = 0):

∫ ∂∂

ρ +

∆ ρ ρ ρ

1

s s v 2 2 2 1 2

t

c p

+ z g + c 2 + p

= z g + c 2

+

This is Bernoulli’s equation for incompressible flows Since all flow processes are inevitably affected by losses, it includes the loss element ∆pv Bernoulli’s equation must only be employed along streamlines or closed channels, because the ex-change of mass and energy with the environment or adjacent streamlines is as-sumed to be zero To be able to use Eq (1.7) for unsteady processes, it has been expanded by the term on the very right, which is obtained from Eq (1.22) With

∂c/∂t ≠ 0, integration follows the path from position s1 to s2

In the above form Eqs (1.5) and (1.6) comprise all losses which lead to the heating of the fluid If in Eq (1.6) only the hydraulic losses (generated by the flow

through the machine) are substituted for (U2 - U1) = ∆pv/ρ, the total enthalpy ference ∆hTot corresponds to the theoretical work Yth of the pump:

dif-) z (z g + 2 c c + p p + p

=

ρ

− ρ

The quantity Yth thus constitutes the work transmitted to the pumped medium per unit mass Usually Yth is largely converted into useful work, while the losses (∆pv/ρ) lead to a − generally negligible − heating of the fluid

If we set U2 = U1, Eq (1.6) yields the specific useful (isentropic) work

∆his = Y = g×H of a pump which constitutes the increase of the total pressure ated by the pump, see Chap 2.2

cre-1.2.3 Conservation of momentum

According to Newton’s 2nd law of mechanics the change of the momentum (ρ×Q×c) of a mass over time is equal to the vectorial sum of all volume and sur-face forces acting on the mass Such are: (1) pressure forces p1 ×A1 and p2 ×A2 at the boundaries of the control volume, (2) external forces on fixed walls Fw, (3) gravity or other accelerations acting on a mass (“body forces”) and (4) friction forces due to wall shear stresses Fτ Considering a control volume as per Fig 1.2, the momentum conservation for steady incompressible flows can be described by:

(p1 + ρ c1) A1 n1 + (p2 + ρ c2) A2 n2 = Fvol + Fw + Fτ (1.9)

n1 and n2 are unit vectors directed outward normal to the areas A1 and A2 If the volume flow rate is explicitly introduced into Eq (1.9), we get:

1.2 Conservation equations 5

p1 A1 n1 + ρ Q c1 n1 + p2 A2 n2 + ρ Q c2 n2 = Fvol + Fw + Fτ (1.10)

When applying the conservation of momentum in the form of Eqs (1.9) or (1.10),

the following must be observed: a) The equations are valid for steady

incom-pressible flow with uniform velocity and pressure distributions on the areas A1 and

A2 b) A1 and A2 must be perpendicular to the velocity vectors c) The signs of the

outward-directed unit vectors must be treated carefully: c1 = - c1 ×n1 and c2 = c2 ×n2

d) All terms are vectors and must be added according to the rules of vector

calcu-lation e) The successful application of momentum conservation equations often

hinges on the appropriate choice of the control volume which must allow the

quantification of pressures and velocities at the control surfaces f) Avoid

generat-ing indefinable forces, when a control surface is placed through structures

Consider a sudden channel expansion according to Fig 1.3 (“Carnot shock”)

as an example for the application of the conservation of momentum and

Ber-noulli’s equation Control surface 1 is positioned immediately downstream of the

step; here pressure p1 is acting over the entire cross section A2, since the same

pressure prevails in the jet and in the wake Control surface 2 is selected

down-stream far enough for the flow to be uniform again No external forces are acting

normal to the channel walls: Fw = 0 Neglecting the gravity (Fvol = 0) and the force

Fτ due to wall shear stresses (which are not transmitted in the wake), the pressure

recovery according to Eq (1.10) is obtained from:

=

−

2

1 2

1 p 2

p 1

A 1 A

A 2 c where c 2 c p

From Eq (1.7) it is then possible to calculate the pressure loss using Eq (1.11):

2 2

1 1

2 1 2 2 1

A 1 e wher c

2 ) c - c ( 2

According to [1.3] the pressure recovery measured reaches approximately 95% of

the theoretical values calculated from Eq (1.11)

Fig 1.3 Sudden expansion a to both sides; b orifice; c to one side

To get an essentially uniform flow velocity through exchange of momentum, a

certain length of channel downstream of the expansion is required It is given by:

L

For an expansion to both sides according to Figs 1.3a and b the factor b = 10

ap-plies, while b = 20 must be used for a one-sided expansion according to Fig 1.3c

Both values of b correspond to an angle of expansion of just under 3° which

“lim-its” the wake (these relationships were derived from data in [1.3]) Equations

(1.11) and (1.12) can also be employed for orifices or other cases with a jet

con-traction, if A1 is substituted for the contracted cross section Ae (also termed “vena

contracta”) as indicated in Fig 1.3b The contracted cross section is calculated

from Ae = µ×A1; the contraction coefficient for sharp-edged orifices is

approxi-mately µ = 0.61, if A1 << A2

Conservation of angular momentum: Another consequence of Newton’s 2nd law

of mechanics is the conservation of the angular momentum (or the “moment of

momentum”) which is of fundamental significance to all turbomachines

Accord-ingly, the change of the angular momentum is equal to the sum of the external

moments Angular momentums ρ×Q×r×cu at inlet and outlet, an external torque M

and friction moments due to shear stresses Mτ are acting on an impeller or a

dif-fuser Since no pressure forces are created on cylindrical surfaces in the

circum-ferential direction, it is possible to write Eq (1.10) as:

ρ Q (c2u r2 – c1u r1) = M + Mτ (1.13) This is Euler’s turbine equation where c2u is the circumferential component of the

flow velocity at the outlet of the control volume, r2 is the outer radius of the

impel-ler and c1u and r1 are the corresponding quantities at the inlet As will be explained

in Chap 3.2, the specific work done by an impeller is obtained from Eq (1.13)

with M×ω = P and neglecting Mτ:

Substituting Eq (1.14) in Eq (1.8), Bernoulli’s equation in the relative reference

frame is obtained (with z1 = z2) In the derivation, c2 = w2 - u2 + 2×u×cu was used

which follows from the velocity triangles, Chap 3.1:

v 2 2 2 2 2

2

w 2 + p

= u 2

w 2 +

This consideration shows that, with an incompressible flow, the conservation of

energy is always satisfied simultaneously with the conservation of momentum

If no external moments are active – i.e M and Mτ in Eq (1.13) are zero –

con-serving the angular momentum requires c u×r = constant for a flow free of forces

This relationship is of fundamental significance to all flows moving with

tangen-tial velocity components in radial direction Examples are the flow exiting from an

impeller into a collector, the flow in radial inlet chambers, in pump sumps, and, in

general terms, the creation and movement of vortices (e.g a tornado)

1.3 Boundary layers, boundary layer control 7

1.3 Boundary layers, boundary layer control

In technically relevant fluid processes, the flow occurs relative to stationary walls (e.g pipelines or channels) or moving structures (such as air foils or impellers) Although the fluid near the wall comprises only a fraction of the flow field, it largely determines losses and velocity distributions in or around a component In classical fluid dynamics the flow field is described by an “inviscid” main flow (which is considered as being free of friction) and a boundary layer flow capturing the processes near solid walls In this concept it is assumed that the relative veloc-ity is zero immediately at the wall (“no-slip condition”) and that no gradients of the static pressure occur perpendicular to the wall within the boundary layer (∂p/∂y = 0) Consequently, the core flow imparts the pressure to the boundary layer and determines the pressure distribution in the flow field under considera-tion All streamlines have the same pressure gradient ∂p/∂x in the flow direction but possess different amounts of kinetic energy

Owing to the no-slip condition, the fluid adhering to a stationary wall (e.g a

pipe or a diffuser) has an absolute velocity of zero In contrast, on the walls of a rotating impeller it has a relative velocity of zero Accordingly, a fluid particle ad-

hering to the walls or blades of an impeller moves with the absolute velocity

cu = u = ω×r

Apart from molecular diffusion no exchange between the individual streamlines

takes place in a laminar flow In contrast, mixing movements perpendicular to the flow occur in turbulent flows in which eddies of finite magnitude ensure the lat-

eral transport This exchange of momentum caused by turbulence largely mines the thickness of the turbulent boundary layer and the velocity distribution over the channel cross section In this way, the fluid layers near the wall are sup-plied with energy through increased turbulence, resulting in a reduced boundary layer thickness and a fuller velocity profile

deter-The velocity distribution in a pipe of radius R with developed turbulent flow is approximately given by the following formula:

ef-Figure 1.4 depicts schematically the development of a two-dimensional ary layer in a channel Suppose the channel has a constant cross section in domain

bound-A, thus w = constant and ∂wx/∂x = 0 At the inlet the boundary layer thickness tends to zero and grows with increasing flow path length The boundary layer is laminar at the inlet Provided that the main flow is turbulent, the boundary layer,