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

538 Pages·2013·3.841 MB·English

ANALYSIS AND
MODELLING OF
NON-STEADY FLOW IN
PIPE AND CHANNEL
NETWORKS
ANALYSIS AND
MODELLING OF
NON-STEADY FLOW IN
PIPE AND CHANNEL
NETWORKS
VinkoJovic´
UniversityofSplit,Croatia
A John Wiley & Sons, Ltd., Publication
Thiseditionfirstpublished2013
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LibraryofCongressCataloging-in-PublicationData
Jovic,Vinko.
Analysisandmodellingofnon-steadyflowinpipeandchannelnetworks/VinkoJovic.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-1-118-53214-0(hardback:alk.paper)–ISBN978-1-118-53686-5(mobi)–ISBN978-1-118-53687-2
(ebook/epdf)–ISBN978-1-118-53688-9(epub)–ISBN978-1-118-53689-6(wileyonlinelibrary)
1. Pipe–Hydrodynamics. 2. Hydrodynamics. I. Title.
TC174.J692013
621.8(cid:3)672–dc23
2012039412
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ISBN:978-1-118-53214-0
Typesetin9/11ptTimesbyAptaraInc.,NewDelhi,India
Contents
Preface xiii
1 HydraulicNetworks 1
1.1 Finiteelementtechnique 1
1.1.1 Functionalapproximations 1
1.1.2 Discretization,finiteelementmesh 3
1.1.3 Approximatesolutionofdifferentialequations 6
1.2 Unifiedhydraulicnetworks 21
1.3 Equationsystem 23
1.3.1 Elementalequations 23
1.3.2 Nodalequations 24
1.3.3 Fundamentalsystem 25
1.4 Boundaryconditions 28
1.4.1 Naturalboundaryconditions 28
1.4.2 Essentialboundaryconditions 30
1.5 Finiteelementmatrixandvector 30
Reference 36
Furtherreading 36
2 ModellingofIncompressibleFluidFlow 37
2.1 Steadyflowofanincompressiblefluid 37
2.1.1 Equationofsteadyflowinpipes 37
2.1.2 SubroutineSteadyPipeMtx 40
2.1.3 Algorithmsandprocedures 42
2.1.4 Frontalprocedure 45
2.1.5 Frontalsolutionofsteadyproblem 51
2.1.6 Steadytestexample 57
2.2 Graduallyvariedflowintime 59
2.2.1 Time-dependentvariability 59
2.2.2 Quasinon-steadymodel 60
2.2.3 SubroutineQuasiUnsteadyPipeMtx 61
2.2.4 Frontalsolutionofunsteadyproblem 63
2.2.5 Quasi-unsteadytestexample 65
2.3 Unsteadyflowofanincompressiblefluid 65
2.3.1 Dynamicequation 65
2.3.2 SubroutineRgdUnsteadyPipeMtx 68
2.3.3 Incompressiblefluidacceleration 69
vi Contents
2.3.4 Accelerationtest 72
2.3.5 Rigidtestexample 72
References 75
FurtherReading 75
3 NaturalBoundaryConditionObjects 77
3.1 Tankobject 77
3.1.1 Tankdimensioning 77
3.1.2 Tankmodel 79
3.1.3 Tanktestexamples 83
3.2 Storage 90
3.2.1 Storageequation 90
3.2.2 Fundamentalsystemvectorandmatrixupdating 91
3.3 Surgetank 91
3.3.1 Surgetankroleinthehydropowerplant 91
3.3.2 Surgetanktypes 94
3.3.3 Equationsofoscillationsinthesupplysystem 99
3.3.4 Cylindricalsurgetank 101
3.3.5 Modelofasimplesurgetankwithupperandlowerchamber 108
3.3.6 Differentialsurgetankmodel 112
3.3.7 Example 117
3.4 Vessel 121
3.4.1 Simplevessel 121
3.4.2 Vesselwithairvalves 124
3.4.3 Vesselmodel 126
3.4.4 Example 127
3.5 Airvalves 128
3.5.1 Airvalvepositioning 128
3.5.2 Airvalvemodel 133
3.6 Outlets 135
3.6.1 Dischargecurves 135
3.6.2 Outletmodel 137
Reference 138
Furtherreading 138
4 WaterHammer–ClassicTheory 141
4.1 Descriptionofthephenomenon 141
4.1.1 Travelofasurgewavefollowingthesuddenhaltofalocomotive 141
4.1.2 Pressurewavepropagationaftersuddenvalveclosure 141
4.1.3 Pressureincreaseduetoasuddenflowarrest–theJoukowskywaterhammer 143
4.2 Waterhammercelerity 143
4.2.1 Relativemovementofthecoordinatesystem 143
4.2.2 Differentialpressureandvelocitychangesatthewaterhammerfront 145
4.2.3 Waterhammercelerityincircularpipes 147
4.3 Waterhammerphases 149
4.3.1 Suddenflowstop,velocitychangev →0 151
0
4.3.2 Suddenpipefilling,velocitychange0→v 154
0
4.3.3 Suddenfillingofblindpipe,velocitychange0→v 156
0
4.3.4 Suddenvalveopening 159
4.3.5 Suddenforcedinflow 161
Contents vii
4.4 Under-pressureandcolumnseparation 164
4.5 Influenceofextremefriction 167
4.6 Gradualvelocitychanges 171
4.6.1 Gradualvalveclosing 171
4.6.2 Linearflowarrest 174
4.7 Influenceofoutflowareachange 176
4.7.1 Graphicsolution 178
4.7.2 Modifiedgraphicalprocedure 179
4.8 Realclosurelaws 180
4.9 Waterhammerpropagationthroughbranches 181
4.10 Complexpipelines 183
4.11 Wavekinematics 183
4.11.1 Wavefunctions 183
4.11.2 Generalsolution 187
Reference 187
Furtherreading 187
5 EquationsofNon-steadyFlowinPipes 189
5.1 Equationofstate 189
5.1.1 p,Tphasediagram 189
5.1.2 p,Vphasediagram 190
5.2 Flowofanidealfluidinastreamtube 195
5.2.1 Flowkinematicsalongastreamtube 195
5.2.2 Flowdynamicsalongastreamtube 198
5.3 Therealflowvelocityprofile 202
5.3.1 Reynoldsnumber,flowregimes 202
5.3.2 Velocityprofileinthedevelopedboundarylayer 203
5.3.3 Calculationsatthecross-section 204
5.4 Controlvolume 205
5.5 Massconservation,equationofcontinuity 206
5.5.1 Integralform 206
5.5.2 Differentialform 207
5.5.3 Elasticliquid 207
5.5.4 Compressibleliquid 209
5.6 Energyconservationlaw,thedynamicequation 209
5.6.1 Totalenergyofthecontrolvolume 209
5.6.2 Rateofchangeofinternalenergy 210
5.6.3 Rateofchangeofpotentialenergy 210
5.6.4 Rateofchangeofkineticenergy 210
5.6.5 Powerofnormalforces 211
5.6.6 Powerofresistanceforces 212
5.6.7 Dynamicequation 212
5.6.8 Flowresistances,thedynamicequationdiscussion 213
5.7 Flowmodels 215
5.7.1 Steadyflow 215
5.7.2 Non-steadyflow 217
5.8 Characteristicequations 220
5.8.1 Elasticliquid 220
5.8.2 Compressiblefluid 223
viii Contents
5.9 Analyticalsolutions 225
5.9.1 Linearizationofequations–waveequations 225
5.9.2 Riemanngeneralsolution 226
5.9.3 Someanalyticalsolutionsofwaterhammer 227
Reference 229
Furtherreading 229
6 ModellingofNon-steadyFlowofCompressibleLiquidinPipes 231
6.1 Solutionbythemethodofcharacteristics 231
6.1.1 Characteristicequations 231
6.1.2 Integrationofcharacteristicequations,wavefunctions 232
6.1.3 Integrationofcharacteristicequations,variablesh,v 234
6.1.4 Thewaterhammeristhepipewithnoresistance 235
6.1.5 Waterhammersinpipeswithfriction 243
6.2 SubroutineUnsteadyPipeMtx 251
6.2.1 SubroutineFemUnsteadyPipeMtx 252
6.2.2 SubroutineChtxUnsteadyPipeMtx 255
6.3 Comparisontests 261
6.3.1 Testexample 261
6.3.2 Conclusion 263
Furtherreading 264
7 ValvesandJoints 265
7.1 Valves 265
7.1.1 Localenergyheadlossesatvalves 265
7.1.2 Valvestatus 267
7.1.3 Steadyflowmodelling 267
7.1.4 Non-steadyflowmodelling 269
7.2 Joints 279
7.2.1 Energyheadlossesatjoints 279
7.2.2 Steadyflowmodelling 279
7.2.3 Non-steadyflowmodelling 282
7.3 Testexample 288
Reference 290
Furtherreading 290
8 PumpingUnits 291
8.1 Introduction 291
8.2 Euler’sequationsofturboengines 291
8.3 Normalcharacteristicsofthepump 295
8.4 Dimensionlesspumpcharacteristics 301
8.5 Pumpspecificspeed 303
8.6 Completecharacteristicsofturboengine 305
8.6.1 Normalandabnormaloperation 305
8.6.2 Presentationofturboenginecharacteristicsdependingonthedirection
ofrotation 305
8.6.3 Knappcirclediagram 305
8.6.4 Sutercurves 308
8.7 Driveengines 310
8.7.1 Asynchronousorinductionmotor 310
Contents ix
8.7.2 Adjustmentofrotationalspeedbyfrequencyvariation 311
8.7.3 Pumpingunitoperation 312
8.8 Numericalmodelofpumpingunits 314
8.8.1 Normalpumpoperation 314
8.8.2 Reconstructionofcompletecharacteristicsfromnormalcharacteristics 318
8.8.3 Reconstructionofahypotheticpumpingunit 321
8.8.4 Reconstructionoftheelectricmotortorquecurve 322
8.9 Pumpingelementmatrices 323
8.9.1 Steadyflowmodelling 323
8.9.2 Unsteadyflowmodelling 327
8.10 Examplesoftransientoperationstagemodelling 333
8.10.1 Testexample(A) 334
8.10.2 Testexample(B) 336
8.10.3 Testexample(C) 339
8.10.4 Testexample(D) 341
8.11 Analysisofoperationandtypesofprotectionagainstpressureexcesses 345
8.11.1 Normalandaccidentaloperation 345
8.11.2 Layout 345
8.11.3 Supplypipeline,suctionbasin 346
8.11.4 Pressurepipelineandpumpingstation 348
8.11.5 Boosterstation 350
8.12 Somethingaboutprotectionofsewagepressurepipelines 353
8.13 Pumpingunitsinapressurizedsystemwithnotank 355
8.13.1 Introduction 355
8.13.2 Pumpingunitregulationbypressureswitches 355
8.13.3 Hydrophorregulation 358
8.13.4 Pumpingunitregulationbyvariablerotationalspeed 360
Reference 362
Furtherreading 362
9 OpenChannelFlow 363
9.1 Introduction 363
9.2 Steadyflowinamildlyslopingchannel 363
9.3 Uniformflowinamildlyslopingchannel 365
9.3.1 Uniformflowvelocityinopenchannel 365
9.3.2 Conveyance,dischargecurve 368
9.3.3 Specificenergyinacross-section:Froudenumber 372
9.3.4 Uniformflowprogrammingsolution 377
9.4 Non-uniformgraduallyvariedflow 378
9.4.1 Non-uniformflowcharacteristics 378
9.4.2 Waterleveldifferentialequation 380
9.4.3 Waterlevelshapesinprismaticchannels 382
9.4.4 Transitionsbetweensupercriticalandsubcriticalflow,hydraulicjump 383
9.4.5 Waterlevelshapesinanon-prismaticchannel 391
9.4.6 Graduallyvariedflowprogrammingsolutions 395
9.5 Suddenchangesincross-sections 398
9.6 Steadyflowmodelling 401
9.6.1 Channelstretchdiscretization 401
9.6.2 Initializationofchannelstretches 402
x Contents
9.6.3 SubroutineSubCriticalSteadyChannelMtx 404
9.6.4 SubroutineSuperCriticalSteadyChannelMtx 406
9.7 Wavekinematicsinchannels 407
9.7.1 Propagationofpositiveandnegativewaves 407
9.7.2 Velocityofthewaveoffiniteamplitude 407
9.7.3 Elementarywavecelerity 409
9.7.4 Shapeofpositiveandnegativewaves 411
9.7.5 Standingwave–hydraulicjump 412
9.7.6 Wavepropagationthroughtransitionalstretches 413
9.8 Equationsofnon-steadyflowinopenchannels 414
9.8.1 Continuityequation 414
9.8.2 Dynamicequation 416
9.8.3 Lawofmomentumconservation 417
9.9 Equationofcharacteristics 422
9.9.1 Transformationofnon-steadyflowequations 422
9.9.2 Procedureoftransformationintocharacteristics 423
9.10 Initialandboundaryconditions 424
9.11 Non-steadyflowmodelling 425
9.11.1 Integrationalongcharacteristics 425
9.11.2 Matrixandvectorofthechannelfiniteelement 427
9.11.3 Testexamples 431
References 434
Furtherreading 435
10 NumericalModellinginKarst 437
10.1 Undergroundkarstflows 437
10.1.1 Introduction 437
10.1.2 Investigationworksinkarstcatchment 437
10.1.3 ThemaindevelopmentformsofkarstphenomenaintheDinaricarea 438
10.1.4 Thesizeofthecatchment 443
10.2 Conveyanceofthekarstchannelsystem 446
10.2.1 Transformationofrainfallintospringhydrographs 446
10.2.2 Linearfiltrationlaw 447
10.2.3 Turbulentfiltrationlaw 449
10.2.4 Complexflow,channelflow,andfiltration 451
10.3 Modellingofkarstchannelflows 453
10.3.1 Karstchannelfiniteelements 453
10.3.2 SubroutineSteadyKanalMtx 454
10.3.3 SubroutineUnsteadyKanalMtx 456
10.3.4 Tests 458
10.4 Methodofcatchmentdiscretization 463
10.4.1 Discretizationofkarstcatchmentchannelsystemwithoutdiffuseflow 463
10.4.2 Equationoftheundergroundaccumulationofakarstsub-catchment 466
10.5 Rainfalltransformation 468
10.5.1 Uniforminputhydrograph 468
10.5.2 Rainfallatthecatchment 473
10.6 Discretizationofkarstcatchmentwithdiffuseandchannelflow 474
References 477
Furtherreading 477
Contents xi
11 Convective-dispersiveFlows 479
11.1 Introduction 479
11.2 Areminderofcontinuummechanics 479
11.3 Hydrodynamicdispersion 483
11.4 Equationsofconvective-dispersiveheattransfer 485
11.5 Exactsolutionsofconvective-dispersiveequation 487
11.5.1 Convectiveequation 487
11.5.2 Convective-dispersiveequation 488
11.5.3 Transformationoftheconvective-dispersiveequation 490
11.6 Numericalmodellinginahydraulicnetwork 490
11.6.1 Theselectionofsolutionbasis,shapefunctions 490
11.6.2 Elementalequations:equationintegrationonthefiniteelement 492
11.6.3 Nodalequations 495
11.6.4 Boundaryconditions 495
11.6.5 Matrixandvectoroffiniteelement 496
11.6.6 Numericsolutiontest 497
11.6.7 Heatexchangeofwatertable 499
11.6.8 Equilibriumtemperatureandlinearization 500
11.6.9 Temperaturedisturbancecausedbyartificialsources 501
References 503
Furtherreading 503
12 HydraulicVibrationsinNetworks 505
12.1 Introduction 505
12.2 Vibrationequationsofapipeelement 506
12.3 Harmonicsolutionforthepipeelement 508
12.4 Harmonicsolutionsinthenetwork 509
12.5 Vibrationsourcemodelling 512
12.6 HintstoimplementationinSimpipCore 512
12.7 Illustrativeexamples 515
Reference 518
Furtherreading 518
Index 519