9781118532140

Analysis and Modelling of Non-steady Flow in Pipe and Channel Networks

by
  • ISBN13:

    9781118532140

  • ISBN10:

    1118532147

  • Format: Hardcover
  • Copyright: 2013-05-13
  • Publisher: Wiley

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Supplemental Materials

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Summary

Analysis and Modelling of Non-Steady Flow in Pipe and Channel Networks deals with flows in pipes and channel networks from the standpoints of hydraulics and modelling techniques and methods. These engineering problems occur in the course of the design and construction of hydroenergy plants, water-supply and other systems. In this book, the author presents his experience in solving these problems from the early 1970s to the present day. During this period new methods of solving hydraulic problems have evolved, due to the development of computers and numerical methods.

This book is accompanied by a website which hosts the author's software package, Simpip (an abbreviation of simulation of pipe flow) for solving non-steady pipe flow using the finite element method. The program also covers flows in channels. The book presents the numerical core of the SimpipCore program (written in Fortran).

Key features:

  • Presents the theory and practice of modelling different flows in hydraulic networks
  • Takes a systematic approach and addresses the topic from the fundamentals
  • Presents numerical solutions based on finite element analysis
  • Accompanied by a website hosting supporting material including the SimpipCore project as a standalone program

Analysis and Modelling of Non-Steady Flow in Pipe and Channel Networks is an ideal reference book for engineers, practitioners and graduate students across engineering disciplines.

Table of Contents

1 Chapter Hydraulic Networks 1

1.1 Finite element technique 1

1.1.1 Functional approximations 1

1.1.2 Discretization, finite element mesh 3

1.1.3 Approximate solution of differential equations 6

1.2 Unified hydraulic networks 20

1.3 Equation system 22

1.3.1 Elemental equations 22

1.3.2 Nodal equations 22

1.3.3 Fundamental system 24

1.4 Boundary conditions 27

1.4.1 Natural boundary conditions 27

1.4.2 Essential boundary conditions 28

1.5 Finite element matrix and vector 29

2 Chapter Modelling of incompressible fluid flow 35

2.1 Steady flow of an incompressible fluid 35

2.1.1 Equation of steady flow in pipes 35

2.1.2 Subroutine SteadyPipeMtx 37

2.1.3 Algorithms and procedures 39

2.1.4 Frontal procedure 42

2.1.5 Frontal solution of steady problem 49

2.1.6 Steady test example 54

2.2 Gradually varied flow in time 56

2.2.1 Time-dependent variability 56

2.2.2 Quasy non-steady model 57

2.2.3 Subroutine QuasyUnsteadyPipeMtx 58

2.2.4 Frontal solution of unsteady problem 59

2.2.5 Quasy non-steady test example 61

2.3 Unsteady flow of an incompressible fluid 63

2.3.1 Dynamic equation 63

2.3.2 Subroutine RgdUnsteadyPipeMtx 64

2.3.3 Incompressible fluid acceleration 65

2.3.4 Acceleration test 67

2.3.5 Rigid test example 68

3 Chapter Natural boundary conditions objects 71

3.1 Tank object 71

3.1.1 Tank dimensioning 71

3.1.2 Tank model 73

3.1.3 Tank test examples 76

3.2 Storage 83

3.2.1 Storage equation 83

3.2.2 Fundamental system vector and matrix updating 84

3.3 Surge tank 84

3.3.1 Surge tank role in the hydropower plant 84

3.3.2 Surge tank types 87

3.3.3 Equations of oscillations in the supply system 92

3.3.4 Cylindrical surge tank 93

3.3.5 Model of a simple surge tank with upper and lower chamber 100

3.3.6 Differential surge tank model 103

3.3.7 Example 109

3.4 Vessel 112

3.4.1 Simple vessel 112

3.4.2 Vessel with air valves 115

3.4.3 Vessel model 116

3.4.4 Example 118

3.5 Air valves 120

3.5.1 Air valve positioning 120

3.5.2 Air valve model 123

3.6 Outlets 125

3.6.1 Discharge curves 125

3.6.2 Outlet model 127

4 Chapter Water hammer – classic theory 129

4.1 Description of the phenomenon 129

4.1.1 Surge wave travel following the sudden halt of a locomotive 129

4.1.2 Pressure wave propagation after sudden valve closure 130

4.1.3 Pressure increase due to a sudden flow arrest

 – the Joukowsky water hammer 130

4.2 Water hammer celerity 131

4.2.1 Relative movement of the coordinate system 131

4.2.2 Differential pressure and velocity changes at the water hammer front 133

4.2.3 Water hammer celerity in circular pipes 134

4.3 Water hammer phases 136

4.3.1 Sudden flow stop, velocity change   138

4.3.2 Sudden pipe filling, velocity change   140

4.3.3 Sudden filling of blind pipe, velocity change   141

4.3.4 Sudden valve opening 143

4.3.5 Sudden forced inflow 147

4.4 Underpressure and column separation 148

4.5 Influence of extreme friction 152

4.6 Gradual velocity changes 155

4.6.1 Gradual valve closing 155

4.6.2 Linear flow arrest 157

4.7 Influence of outflow area change 159

4.7.1 Graphic solution 160

4.7.2 Modified graphical procedure 161

4.8 Real closure laws 162

4.9 Water hammer propagation through branches 164

4.10 Complex pipelines 166

4.11 Wave kinematics 166

4.11.1 Wave functions 166

4.11.2 General solution 169

5 Chapter Equations of non-steady flow in pipes 171

5.1 Equation of state 171

5.1.1 p,T phase diagram 172

5.1.2 p,V phase diagram 172

5.2 Flow of an ideal fluid in the streamtube 177

5.2.1 Flow kinematics along the streamtube 177

5.2.2 Flow dynamics along the streamtube 180

5.3 The real flow velocity profile 184

5.3.1 Reynolds number, flow regimes 184

5.3.2 Velocity profile in the developed boundary layer 184

5.3.3 Calculations at the cross-section 186

5.4 Control volume 187

5.5 Mass conservation, equation of continuity 187

5.5.1 Integral form 187

5.5.2 Differential form 188

5.5.3 Elastic liquid 188

5.5.4 Compressible liquid 190

5.6 Energy conservation law, dynamic equation 190

5.6.1 Total energy of control volume 190

5.6.2 Rate of change of internal energy 191

5.6.3 Rate of change of potential energy 191

5.6.4 Rate of change of kinetic energy 191

5.6.5 Power of normal forces 192

5.6.6 Power of resistance forces 193

5.6.7 Dynamic equation 193

5.6.8 Flow resistances, the dynamic equation discussion 194

5.7 Flow models 196

5.7.1 Steady flow 196

5.7.2 Non-steady flow 197

5.8 Characteristic equations 200

5.8.1 Elastic liquid 200

5.8.2 Compressible fluid 203

5.9 Analytical solutions 206

5.9.1 Linearization of equations – wave equations 206

5.9.2 Riemann general solution 206

5.9.3 Some analytical solutions of water hammer 207

6 Chapter Modelling of non-steady flow of compressible liquid in pipes 211

6.1 Solution by the method of characteristics 211

6.1.1 Characteristic equations 211

6.1.2 Integration of characteristic equations, wave functions 212

6.1.3 Integration of characteristic equations, variables h, v 213

6.1.4 Water hammer is the pipe with no resistance 215

6.1.5 Water hammer in pipe with friction 222

6.2 Subroutine UnsteadyPipeMtx 229

6.2.1 Subroutine FemUnsteadyPipeMtx 230

6.2.2 Subroutine ChtxUnsteadyPipeMtx 233

6.2.3 Comparison tests 238

 7 Chapter Valves and Joints 243

7.1 Valves 243

7.1.1 Local losses of energy head at valves 243

7.1.2 Valve status 245

7.1.3 Steady flow modelling 245

7.1.4 Non-steady flow modelling 247

7.2 Joints 256

7.2.1 Energy head losses at joints 256

7.2.2 Steady flow modelling 258

7.2.3 Non-steady flow modelling 260

7.3 Test example 265

8 Chapter Pumping units 269

8.1 Introduction 269

8.2 Euler's equations of turbo engines 270

8.3 Normal characteristics of the pump 273

8.4 Dimensionless pump characteristics 277

8.5 Pump specific speed 280

8.6 Complete characteristics of turbo engine 281

8.6.1 Normal and abnormal operation 281

8.6.2 Presentation of turbo engine characteristics depending on the direction of rotation 281

8.6.3 Knapp circle diagram 282

8.6.4 Suter curves 284

8.7 Drive engines 286

8.7.1 Asynchronous or induction motor 286

8.7.2 Adjustment of rotational speed by frequency variation 287

8.7.3 Pumping unit operation 288

8.8 Numerical model of pumping units 290

8.8.1 Normal pump operation 290

8.8.2 Reconstruction of complete characteristics from normal characteristics 294

8.8.3 Reconstruction of a hypothetic pumping unit 297

8.8.4 Reconstruction of the electric motor torque curve 298

8.9 Pumping element matrices 299

8.9.1 Steady flow modelling 299

8.9.2 Unsteady flow modelling 303

8.10 Examples of transient operation stages modelling 308

8.10.1 Test example A) 309

8.10.2 Test example B) 312

8.10.3 Test example C) 314

8.10.4 Test example D) 315

8.11 Analysis of operation and types of protection against pressure excesses 319

8.11.1 Normal and accidental operation 319

8.11.2 Layout 319

8.11.3 Supply pipeline, suction basin 320

8.11.4 Pressure pipeline and pumping station 322

8.11.5 Booster station 324

8.12 Something about protection of sewage pressure pipelines 326

8.13 Pumping units in pressurized system with no tank 328

8.13.1 Introduction 328

8.13.2 Pumping unit regulation by pressure switches 329

8.13.3 Hydrophor regulation 331

8.13.4 Pumping unit regulation by variable rotational speed 333

 

9 Chapter Open channel flow 337

9.1 Introduction 337

9.2 Steady flow in a mildly sloping channel 337

9.3 Uniform flow in a mildly sloping channel 339

9.3.1 Uniform flow velocity in open channel 339

9.3.2 Conveyance, discharge curve 342

9.3.3 Specific energy in a cross-section. Froude number 345

9.3.4 Uniform flow programming solution 349

9.4 Non-uniform gradually varied flow 351

9.4.1 Non-uniform flow characteristics 351

9.4.2 Water level differential equation 352

9.4.3 Water level shapes in prismatic channels 354

9.4.4 Transitions between supercritical and subcritical flow, hydraulic jump 355

9.4.5 Water level shapes in non-prismatic channel 362

9.4.6 Gradually varied flow programming solutions 365

9.5 Sudden changes in cross-sections 368

9.6 Steady flow modelling 372

9.6.1 Channel stretch discretization 372

9.6.2 Initialization of channel stretches 372

9.6.3 Subroutine SubCriticalSteadyChannelMtx 375

9.6.4 Subroutine SuperCriticalSteadyChannelMtx 376

9.7 Wave kinematics in channels 377

9.7.1 Propagation of positive and negative waves 377

9.7.2 Velocity of the wave of finite amplitude 377

9.7.3 Elementary wave celerity 379

9.7.4 Shape of positive and negative waves 381

9.7.5 Standing wave – hydraulic jump 381

9.7.6 Wave propagation through transitional stretches 382

9.8 Equations of non-steady flow in open channels 384

9.8.1 Continuity equation 384

9.8.2 Dynamic equation 385

9.8.3 Law of momentum conservation 387

9.9 Equation of characteristics 391

9.9.1 Transformation of non-steady flow equations 391

9.9.2 Procedure of transformation into characteristics 392

9.10 Initial and boundary conditions 392

9.11 Non-steady flow modelling 394

9.11.1 Integration along characteristics 394

9.11.2 Matrix and vector of the channel finite element 396

9.11.3 Test examples 400

10 Chapter Numerical modelling in karst 405

10.1 Underground karst flows 405

10.1.1 Introduction 405

10.1.2 Investigation works in karst catchment 405

10.1.3 The main development forms of karst phenomena in the Dinaric area 406

10.1.4 The size of the catchment 410

10.2 Conveyance of the karst channel system 413

10.2.1 Transformation of rainfall into spring hydrographs 413

10.2.2 Linear filltration law 414

10.2.3 Turbulent filtration law 416

10.2.4 Complex flow, channel flow and filtration 418

10.3 Modelling of karst channel flows 420

10.3.1 Karst channel finite elements 420

10.3.2 Subroutine SteadyKanalMtx 421

10.3.3 Subroutine UnsteadyKanalMtx 423

10.3.4 Tests 425

10.4 Method of catchment discretization 428

10.4.1 Discretization of karst catchment channel system without diffuse flow 428

10.4.2 Equation of the underground accumulation of karst sub-catchment 431

10.5 Rainfall transformation 433

10.5.1 Uniform input hydrograph 433

10.5.2 Rainfall at the catchment 436

10.6 Discretization of karst catchment with diffuse and channel flow 437

11 Chapter Convective-dispersive flows 441

11.1 Introduction 441

11.2 A reminder of continuum mechanics. 442

11.3 Hydrodynamic dispersion 445

11.4 Equations of convective-dispersive heat transfer 446

11.5 Exact solutions of convective-dispersive equation 448

11.5.1 Convective equation 448

11.5.2 Convective-dispersive equation 450

11.5.3 Transformation of convective - dispersive equation 450

11.6 Numerical modelling in hydraulic network 451

11.6.1 The selection of solution basis, shape functions 451

11.6.2 Elemental equations. Equations integration on finite element 453

11.6.3 Nodal equations 456

11.6.4 Boundary conditions 456

11.6.5 Matrix and vector of finite element 457

11.6.6 Numeric solution test 458

11.6.7 Heat exchange of water table 460

11.6.8 Equilibrium temperature and linearization 461

11.6.9 Temperature disturbance caused by artificial sources 461

12 Chapter Hydraulic vibrations in networks 465

12.1 Introduction 465

12.2 Vibration equations of a pipe element 466

12.3 Harmonic solution for the pipe element 467

12.4 Harmonic solutions in the network 469

12.5 The vibrations source modelling 471

12.6 Hints to implementation in SimpipCore 472

12.7 Illustrative examples 475

A. Appendix Program solutions 479

A.1 SimpleSteady – a textbook example of FEM program 479

A.1.1 File SimpleSteady.f90 – main program 479

A.1.2 Memory module - file SsglobalVars.f90 479

A.1.3 File SSIosubs.f90 480

A.1.4 File SSmatrix.f90 482

A.1.5 File SSsolve.f90 483

A.1.6 File SShydraulics.f90 486

A.1.7 Test example 487

A.2 Numerical solution of problems with initial conditions 488

A.2.1 Equation system 488

A.2.2 Explicit method 489

A.2.3 Implicit method 490

A.2.4 Mixed method 490

A.2.5 Runge-Kutta 4th order method 490

A.2.6 Runge-Kutta-Fehlberg method 491

A.2.7 Program solution - module ODE 493

A.2.8 Program source - module ODE 496

A.3 Program solution for oscillations in surge tank 502

A.4 Program Vessel 508

B. Appendix SimpipCore program 513

B.1 Main program 513

B.2 SimpipCore project 513

B.3 Input / Output syntax 515

B.3.1 Parser 516

B.3.2 SimpipCore user manual 518

B.4 Execution of SimpipCore program 518

Index

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