SimonA.Mathias
Hydraulics,Hydrology andEnvironmental Engineering
SimonA.Mathias DepartmentofEngineering
DurhamUniversity Durham,UK
ISBN978-3-031-41972-0ISBN978-3-031-41973-7(eBook) https://doi.org/10.1007/978-3-031-41973-7
©UniversityofDurham2023
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Preface
ThistextbookhasbeendevelopedfromlecturenotesusedtoteachM.Eng.and M.Sc.engineeringstudentsatDurhamUniversity.Thebookissplitupintothree partscontainingtenchapterseach.PartIisbasedonataughtmoduleconcerningcivilengineeringhydraulics.PartIIisbasedonataughtmoduleconcerning hydrologyandwaterresources.PartIIIisbasedonataughtmoduleconcerning environmentalengineering.
ThefirstfivechaptersofPartIcoverarangeofclassicalhydraulictopics includinghydrostatics,Bernoulliequation,pipeflow,openchannelflowandAiry waves.Theremainingchaptersincludederivationsoftheshallow-waterequations, methodofcharacteristicssolutionsforthekinematicwaveequation,Laplacetransformsolutionsforthediffusionwaveequationandnumericalsolutionsobtainedby thefinitedifferencemethod.Itisalsoshownhowtheadvection-dispersionequation(analogoustothediffusionwaveequation)canbeusedtodescribechemical transportandheattransportinporousmedia.
PartIIincludesacomprehensivesetofhydrologicaltopics,withafocuson mathematicalmodelformulationandderivationofanalyticalsolutions.Thefirst chapterlooksatfloodfrequencyanalysis.Thenextfourchapterslookatgroundwaterflow,coveringtopicsincludingDarcy’slaw,theForchheimerequation,method ofimages,thespecificstoragecoefficient,specificyieldandpumpingtestanalysis. Theremainingchapterslookatunsaturatedflowprocesses,humiditymeasurement, potentialevapotranspiration,solarradiationandrainfallrunoffmodelling.
PartIIIcontainsthreechaptersaboutenvironmentaleconomics,threechapters aboutairpollution,threechaptersaboutwaterqualityandafinalchapterabout environmentalimpactassessmentanduncertaintymanagement.Theobjectiveof theenvironmentaleconomicschaptersistohelpengineeringstudentsbetterascribe valuetoenvironmentalimprovement.Economicstopicscoveredincludetragedy ofthecommons,cashflowanalysis,supplyanddemandmodelling,ecosystem servicesandcostofpollutiononhealth.Airpollutiontopicscoveredincludeair qualitycontroltechnologies,carboncaptureandstorageandatmosphericdispersionmodelling.Waterqualitytopicsincludewaterpollution,watertreatmentand wastewatertreatment.
Readersareinvitedtoactivelyengagewithderivationsofanalyticalsolutions throughavastsetofchallenges,specifiedthroughoutthebook.Eachchapteralso includesaproblemsheetwithworkedsolutions,comprisingrelevantandpracticalapplicationsofthetheoreticalequationsderived.Manyoftheproblemsheets involvewritingsmallMATLABprogrammesaimedtohelpthereadergaindeeper insightintoparametersensitivityandprocessunderstanding.
Thisbookbuildsonalargebodyofreferencedliterature.Nevertheless,itis worthtopointoutthatthewritingstyleandformatofthisbookhasbeenparticularlyinfluencedbyMasseyandWard-Smith[1],StroudandBooth[2],Chapraand Canale[3],CooperandAlley[4],Crittendenetal.[5]andFrazerandJewkes[6]. Thewrittenmaterialalsobuildsheavilyonmyexperienceasastudentunderthe instructionofTonyGrassandJamesCrollatUniversityCollegeLondon,Andrew SpinkattheUniversityofBirminghamaswellasAdrianButler,HowardWheater, NeilMcIntyreandGeoffreyStephensonatImperialCollegeLondon.Iamalso gratefulforimportantadviceandguidanceprovidedbymywife,JinhuaMathias,andcolleagues,includingJeroenvanHunen,AndyAplin,StefanNielsen, NeilGoulty,RiccardoScarpa,StefanoGianiandMohammedSeaidatDurham University,ChrisMacMinnatOxfordUniversity,GrahamSanderatLoughboroughUniversity,AndrewIresonattheUniversityofSaskatchewanandGianni VesuvianoattheCentreforEcologyandHydrology.
Durham,UK April2023
References
SimonA.Mathias
1.B.S.Massey,J.Ward-Smith, MechanicsofFluids.8thedn.(TaylorandFrancis,2006)
2.K.A.Stroud,D.J.Booth, EngineeringMathematics.6thedn.(PalgraveMacmillan,2007)
3.S.C.Chapra,R.P.Canale, NumericalMethodsforEngineers-withProgrammingandSoftware Applications.3rdedn.(McGraw-HillInternationalEditions,1998)
4.C.D.Cooper,F.C.Alley, AirPollutionControl:ADesignApproach.4thedn.(WavelandPress, 2010)
5.J.C.Crittenden,R.R.Trussell,D.W.Hand,K.Howe,G.Tchobanoglous, MWH’swatertreatment:principlesanddesign.3rdedn.(JohnWiley&Sons,2012)
6.N.M.Frazer,E.M.Jewkes, EngineeringEconomics:FinancialDecisionMakingforEngineers. 5thedn.(Pearson,2013)
PartICivilEngineeringHydraulics
1BasicPrinciplesofFluidMechanics
1.1Introduction..
1.2Hydrostatics..
1.2.1HydrostaticPressure..
1.2.2MeasuringPressure...
1.2.3HydrostaticForcesonInclinedPlates
1.2.4Buoyancy ..........................................12
1.3MovingFluids ...............................................18
1.3.1Control-VolumeinaStreamtube
1.3.2TheContinuityEquation..
1.3.3TheMomentumEquation.
1.3.4TheBernoulliEquation...
1.3.5SummaryofKeyResults..
1.3.6MeasuringFluidVelocityUsingaPitotTube
1.3.7TheTorricelliEquation
1.3.8FlowoveraSharpCrestedWeir
2.3.1Viscosity.
2.3.2FlowBetweenTwoParallelPlates..
2.3.3FlowThroughaCylindricalPipe...
2.4SteadyTurbulentFlow.
2.4.1TurbulentFlowThroughaNon-prismatic andNon-cylindricalPipe..
2.4.2LinkingWallShearStresstoFluidVelocity..
2.4.3FrictionFactorforTurbulentFlowSystems..
2.4.4FrictionFactorforLaminarFlowSystems...
2.5EmpiricalLawsforOpenChannelFlow..
2.5.1ChezyEquation...
2.5.2ManningEquation
2.5.3ExpressionsforFrictionFactor.
2.5.4AnExpressionforTotalHead..
2.5.5NormalFlowConditions..
2.6ProblemSheet ...............................................54
2.7WorkedSolutions..
3GraduallyVariedFlowinOpenChannels
3.1Introduction..
3.2GraduallyVariedFlow(GVF)Equation..
3.2.1FluidDepthGradient.. .............................63
3.2.2TheFroudeNumber... .............................64
3.3RectangularCross-Sections. .................................64
3.3.1IncorporatingCriticalDepth ........................65
3.3.2IncorporatingNormalDepth ........................65
3.3.3ApplicationoftheChezyEquation.
3.3.4ApplicationoftheManningEquation
3.4ClassificationofFlowProfiles.. .............................66
3.5SolvingtheGVFEquation..
3.5.1SolutionsforAdverseSlopes...
3.5.2SolutionsforMildSlopes.
3.5.3SolutionsforCriticalSlopes ........................72
3.5.4SolutionsforSteepSlopes.
3.6DevelopmentofCompositeProfiles.
3.7ProblemSheet ...............................................75
3.8WorkedSolutions..
4RapidlyVariedFlowinOpenChannels
4.1Introduction..
4.2CriticalFlowConditions
4.3HydraulicJumps...
4.3.1ConjugateDepths.
4.3.2TotalHeadLossDuetoaHydraulicJump...
4.4Shallow-WaterWaves..
4.5LocatingHydraulicJumps.. .................................91
4.5.1SupercriticaltoNormalSub-criticalFlow
4.5.2SupercriticaltoNon-normalSub-criticalFlow
4.6ProblemSheet ...............................................94
4.7WorkedSolutions..
5AiryWavesandBasicCoastalDefenceDesign
5.1Introduction.. ...............................................104
5.2AiryWaveTheory. ..........................................105
5.2.1TheTransientBernoulliEquation...
5.2.2DerivationofaMassConservationEquation.
5.2.3TransformationtoaLaplaceEquation...
5.2.4ConceptualModelforAiryWaves..
5.2.5AnalyticalSolution .................................109
5.2.6DisplacementofParticles.
5.2.7TheAiryWaveDispersionEquation
5.2.8Shallow-WaterApproximation..
5.3WaveShoaling ..............................................114
5.3.1Wave-Power. ......................................114
5.3.2WavelengthEquation.. .............................116
5.3.3Wave-HeightEquation
5.3.4Shallow-WaterApproximation..
5.4EmpiricalEquationsforDesign.
5.4.1SignificantWave-PeriodandWave-Height...
5.4.2CriticalSlopeforWave-Breaking...
5.4.3WaveRun-Up ......................................119
5.5ProblemSheet ...............................................120
5.6WorkedSolutions.. ..........................................122 References.. ........................................................125
6Shallow-WaterEquations ..........................................127
6.1Introduction..
6.2TheCauchyMomentumEquation...
6.2.1NetForcesin x , y and z Direction.
6.2.2ApplicationofNewton’sSecondLaw
6.3NavierStokesEquations .....................................134
6.3.1ConstantViscosity
6.3.2ConstantDensityandViscosity.
6.4Shallow-WaterEquations...
6.4.1ContinuityEquation...
6.4.2MomentumEquations.
6.5SaintVenantEquations
6.6DiffusionWaveEquation... .................................143
6.6.1WideChannelwithManning’sEquation.
6.6.2Ogata–BanksSolution. .............................145
6.6.3PecletNumber .....................................145
6.6.4ApplyingthePrincipleofSuperposition.
6.6.5KinematicWaveEquation.
6.7ProblemSheet ...............................................148
6.8WorkedSolutions..
7MethodofCharacteristicsSolutionfortheKinematicWave Equation ............................................................155
7.1Introduction.. ...............................................156
7.2SimilarityTransformSolution...
7.3ConservativeForm. ..........................................157
7.4MethodofCharacteristics... .................................157
7.4.1ConstantWaveSpeedSolution.
7.5ShockWaves. ...............................................158
7.5.1Rankine–HugoniotCondition...
7.5.2UsingtheRankine–HugoniotJumpCondition toLocateaShockFront...
7.6RarefactionWaves. ..........................................161
7.6.1ShowingThataShockWaveDoesNotWork
7.6.2ApplyingaSimilarityTransformSolution... .......162
7.7CombinedShockWaveandRarefactionWave ................164
7.8ProblemSheet ...............................................167
7.9WorkedSolutions.. ..........................................168
8ConvolutionSolutionfortheDiffusionWaveEquation
8.1Introduction..
8.1.1GoverningEquations..
8.2ConstantFlowRateBoundary..
8.2.1ReductiontoaDiffusionEquation..
8.2.2SolutionbyLaplaceTransform.
8.2.3WritinginTermsofErrorFunctions
8.2.4LimitingCases .....................................180
8.3InstantaneousWaterRelease
8.4Time-VaryingFlowRateBoundary..
8.4.1LaplaceTransformSolution
8.4.2WritinginTermsofaConvolutionIntegral..
8.4.3ConsiderationoftheInstantaneousWater ReleaseSolution..
8.5ProblemSheet ...............................................185
8.6WorkedSolutions..
9ChemicalandHeatTransport
9.1Introduction..
9.2UnitsforAqueousConcentrations...
9.3TotalandKinematicPorosity...
9.4Advection ...................................................196
9.4.1PureAdvectionwithInitialCondition...
9.4.2PureAdvectionwithBoundaryCondition
9.5Adsorption...
9.5.1PureAdvectionwithAdsorption
9.5.2LinearEquilibriumAdsorptionPartition Coefficients.. ......................................199
9.6Decay .......................................................201
9.6.1DecayCoefficientsandHalf-Lives.. ................201
9.6.2PureAdvectionwithAdsorptionandDecay. .......202
9.6.3RadioactiveDecayVersusMicrobial Degradation.. ......................................203
9.7MolecularDiffusion...
9.8HydrodynamicDispersion..
9.9Advection,DispersionandDecay...
9.11ProblemSheet ...............................................207
9.12WorkedSolutions.. ..........................................209 References.. ........................................................216
10.2BoundaryValueProblem...
10.3ApplicationofTaylorSeries
10.4FourDifferentSchemes ......................................222
10.4.1ExplicitTime-SteppingwithCentralDifference inSpace(ETCS).. .................................222
10.4.2ExplicitTime-SteppingwithBackward DifferenceinSpace(ETBS) ........................223
10.4.3ImplicitTime-SteppingwithCentralDifference inSpace(ITCS)... .................................224
10.4.4ImplicitTime-SteppingwithBackward DifferenceinSpace(ITBS) .........................225
10.5ExplicitVersusImplicitTime-Stepping...
10.6NumericalStability
10.6.1ExplicitSchemes..
10.6.2ImplicitSchemes.. .................................228
10.6.3Neumann,CourantandPecletNumber.. ...........228
10.7MatrixFormulation ..........................................229
10.7.1ExplicitTime-SteppingSchemes... ................229
10.7.2ImplicitTime-SteppingSchemes... ................230
10.7.3MoreabouttheJacobianMatrix ....................230
10.7.4ImplementationinMATLAB...
10.8NumericalDiffusion... ......................................233
10.8.1NumericalDiffusionfromtheETCSScheme .......233
10.8.2NumericalDiffusionfromtheETBSScheme .......234
10.8.3NumericalDiffusionfromtheITCSScheme. .......234
10.8.4NumericalDiffusionfromtheITBSScheme. .......235
10.8.5CorrectingalltheSchemesforNumerical Diffusion ..........................................236
10.9ImplicitSchemesforNon-linearPDEs...
10.10ProblemSheet ...............................................237
PartIIHydrologyandWaterResources
11HydrologyandFrequencyAnalysis .................................253
11.1Introduction.. ...............................................254
11.2FundamentalHydrology .....................................254
11.2.1TheHydrologicalCycle...
11.2.2CatchmentWaterBalance.
11.2.3RainfallMeasurement. .............................260 11.2.4RiverFlowMeasurement..
11.3FrequencyAnalysis ..........................................262
11.3.1ReturnPeriodandProbability..
11.3.2CumulativeDistributionFunctions..
11.3.3EmpiricalCDF .....................................264
11.3.4ExtrapolationbyMomentMatching ................265
11.3.5TheNormalDistribution..
11.3.6TheGumbelDistribution..
11.3.7AnalysisofAnnualFlowStatistics.
11.4ProblemSheet ...............................................274
12FluidFlowinPorousMedia
12.1Introduction..
12.2Aquifers,AquicludesandAquitards.
12.3PorosityandVoidRatio ......................................283
12.4Darcy’sLaw.. ...............................................284
12.5BoreholesandPiezometers..
12.6InsightsfromtheDarcy-WeisbachEquation..
12.6.1DarcyFluxandPore-WaterVelocity ................288
12.6.2HydraulicRadius,GrainGeometryandVoid Ratio ..............................................289
12.6.3ATheoreticalHydraulicConductivity Relationship. ......................................290
12.7Permeability.. ...............................................291
12.8Non-DarcianFlow. ..........................................292
12.8.1TheForchheimerEquation .........................292
12.8.2TheErgunEquation...
12.8.3ReynoldsNumberCriterionforNon-Darcian Flow...............................................293
12.9ProblemSheet ...............................................294
12.10WorkedSolutions..
13ConfinedandUnconfinedAquifers
13.1Introduction.. ...............................................300
13.2TwoTypesofAquifer. ......................................300
13.3TemperateandAridEnvironments..
13.4ApplicationsofDarcy’sLaw
13.4.1ConfinedAquifers.
13.4.2UnconfinedAquifers..
13.4.3TheDupuit-ForchheimerAssumption
13.4.4UnconfinedAquiferswithRecharge
13.5ProblemSheet ...............................................308
13.6WorkedSolutions..
14Steady-StateRadialFlowtoWells
14.1Introduction..
14.2TheThiemEquation...
14.3ApplyingthePrincipleofSuperposition..
14.4MethodofImages. ..........................................318
14.4.1FixedHydraulicHeadBoundaries..
14.4.2ImpermeableBoundaries..
14.5Step-DrawdownTests..
14.6Steady-StateNon-darcianFlowtoaWell.
15TransientGroundwaterFlow
15.1Introduction..
15.2FluidCompressibility..
15.4MassConservationinaControl-Volume..
15.4.1WritinginTermsofHydraulicHead ................339
15.4.2WritingasaDiffusionEquation
15.5StorageCoefficients...
15.5.1SpecificStorageCoefficient
15.5.2Storativity...
15.5.3SpecificYield
15.6SolutionbySimilarityTransform...
15.6.1ApplicationofanIndependentVariable Transform ..........................................343
15.6.2ApplicationofaDependentVariableTransform.....345
15.6.3SubstitutingtheComplementaryErrorFunction.....346
15.7ProblemSheet ...............................................347
15.8WorkedSolutions..
16TransientFlowtoWells .............................................351
16.1Introduction.. ...............................................352
16.2MassConservationwithRadialSymmetry...
16.3SolutionbySimilarityTransform...
16.3.1SubstitutingtheExponentialIntegralFunction
16.3.2Large-TimeApproximation
16.3.3TransientRadiusofInfluence...
16.3.4IncorporationofNon-darcyEffects.
16.4PumpingTestAnalysis
16.4.1AnalysisofDrawdownData
16.4.2AnalysisofRecoveryData
16.5ProblemSheet ...............................................364
16.6WorkedSolutions.. ..........................................365 References..
17VadoseZoneProcesses
17.1Introduction..
17.2BasicPrinciples...
17.2.1NegativePressureHeads..
17.2.2SurfaceTensionandWettability
17.2.3Air-EntryPressureofaCapillaryTube..
17.2.4SoilMoistureCharacteristicCurve.
17.2.5UnsaturatedHydraulicConductivity
17.2.6EmpiricalFunctions...
17.3.1PressureHeadFormulation
17.3.2SpecificMoistureCapacity
17.3.3MoistureContentFormulation..
17.4InfiltrationModelling..
17.4.1InfiltrationCapacity...
17.4.2SharpWettingFrontAssumption...
17.4.3ImplicitSolutionfor L
17.4.4ThePhilipEquation...
17.5.1MeasuringMoistureContent...
17.5.2MeasuringNegativePressureHead.
17.5.3ComparisonofFieldMeasurements
17.5.4SoilMoistureDeficitandtheZeroFluxPlane
17.6ProblemSheet
17.7WorkedSolutions..
18Humidity ...........................................................405
18.1Introduction.. ...............................................407
18.2Absolute,SpecificandRelativeHumidity ....................407
18.3IdealGasModelforWaterandAir. .........................408
18.3.1TheIdealGasLaw .................................408
18.3.2IdealGasMixturesandPartialPressures. ...........409
18.3.3Dalton’sLawofAdditivePressures. ................409
18.3.4Amagat’sLawofAdditiveVolumes ................410
18.3.5TheRelationshipBetweenAbsoluteHumidity, SpecificHumidityandVapourPressure.. ...........410
18.3.6LatentHeatofEvaporation .........................413
18.4MeasurementofHumidity.. .................................413
18.4.1EnergyBalanceinaPsychrometer.. ................414
18.4.2DiffusiveApproachtoDefiningFluxes.. ...........414
18.4.3Monin-ObhukovSimilarityTheory. ................415
18.4.4ResistanceFormulation .............................415
18.4.5VapourPressureasaFunctionofDry-Bulb andWet-BulbTemperatures ........................416
18.4.6ThePsychrometricConstant ........................417
18.4.7EstimatingHumidityfromDry-Bulb Temperature. ......................................417
18.5ProblemSheet ...............................................418
18.6WorkedSolutions.. ..........................................419 References.. ........................................................422
19EvaporationandRadiation .........................................423
19.1Introduction.. ...............................................425
19.2EvaporationandEvapotranspiration.
19.2.1PotentialEvapotranspiration
19.2.2Open-WaterEvaporation..
19.2.3NoteAboutPenman...
19.2.4AerodynamicResistance..
19.3Radiation. ...................................................429
19.3.1Black-BodyRadiation. .............................430
19.3.2NetIncomingRFD .................................431
19.3.3MeasuringRFDwithaNetRadiometer..
19.3.4EstimatingSWRFD.. .............................433
19.3.5EstimatingLongwaveRadiation
19.4ProblemSheet ...............................................442
19.5WorkedSolutions..
20RainfallRunoffModelling ..........................................447
20.1Introduction..
20.2WaterBalanceModelling...
20.2.1WaterBalancefortheVegetativeCanopy
20.2.2WaterBalanceforSoil .............................452
20.2.3CombinedWaterBalance..
20.2.4ProbabilityDistributedModel(PDM)
20.2.5NumericalSolution...
20.3RiverFlowRouting ..........................................461
20.3.1MassConservationStatement..
20.3.2SheetFlowApproximation
20.3.3BehaviourDuringRecession...
20.3.4NumericalSolution...
20.3.5ExampleSimulations..
20.4CalibrationandValidation..
20.5ProblemSheet ...............................................468
20.6WorkedSolutions..
PartIIIEnvironmentalEngineering
21CostBenefitAnalysis ...............................................481
21.1Introduction.. ...............................................482
21.2Common-PoolResourcesandPublicGoods..
21.2.1Excludability,Rivalry,GoodsandBads..
21.2.2TheTragedyoftheCommons..
21.2.3MoreDifficultQuestions..
21.3CashFlowAnalysis...
21.3.1Discounting..
21.3.2SequencesofAnnualPayments.
21.3.3ProjectEvaluationMethods
21.3.4EquivalentAnnualCost(EAC).
21.3.5InflationRates .....................................493
21.4ProblemSheet
21.5WorkedSolutions..
22MarketEfficiency
22.1Introduction..
22.2ModellingSupplyandDemand.
22.2.1Producer,ConsumerandSocialSurplus.
22.2.2LinearSupplyandDemandSchedules...
22.2.3MonopolyControlledMarket...
22.2.4CompetitiveMarket...
22.2.5MarketEfficiencyandExternalities.
22.3EnvironmentalPolicyOptions...
22.3.1PrivateOwnership. .................................514
22.3.2CommandandControlRegulation..
22.3.3PollutionPermitTrading..
22.4ProblemSheet ...............................................518
22.5WorkedSolutions..
23EvaluatingtheEnvironment
23.1Introduction..
23.2EvaluationMethods
23.2.1CostBasedMethods..
23.2.2RevealedPreferenceMethods..
23.2.3StatedPreferenceMethods
23.3EcosystemServices ..........................................529
23.3.1CategoriesofServicesandBiomes. ................529
23.3.2EvaluatingEcosystemServices.
23.3.3LostValuebyEcosystemConversion
23.4PollutionandHealth... ......................................536
23.4.1HealthCostsofPollution..
23.4.2ValueofStatisticalLife(VSL).
23.4.3Disability-AdjustedLifeYear(DALY)...
23.4.4WTPtoEliminateaDALY
23.4.5DiscountingYLLandYLD
23.5ProblemSheet
23.6WorkedSolutions..
24AirQualityControl
24.1Introduction..
24.2AirQualityGuidelinesandStandards
24.3.1SettlingChambers.
24.3.2CycloneSeparators .................................559
24.3.3ElectrostaticPrecipitators.. .........................559
24.3.4FabricFilters. ......................................561
24.3.5WetDustScrubbers...
24.4Ozone
24.5NitrogenOxides...
24.5.1Fuel,ThermalandPromptNO x
24.5.2FuelSwitching .....................................564
24.5.3CombustionControl... .............................564
24.5.4FlueGasTreatment... .............................565
24.6SulphurOxides(SO x ).......................................566
24.6.1RecoveringSulphurfromSourGas.
24.6.2RecoveringSulphuricAcidfromSulphideOres.....566
24.6.3FormingGypsumFromSulphurDioxide inFlueGas........................................567
24.7ParticleSettlingTheory ......................................567
24.7.1CollectionEfficiency.. .............................568
24.7.2SettlingVelocity.. .................................569
24.7.3CyclonicSeparation...
24.8ProblemSheet ...............................................576
24.9WorkedSolutions..
25CarbonCaptureandStorage
25.1Introduction..
25.2CarbonCapture ..............................................583
25.2.1SeparationProcesses..
25.2.2IndustrialSources. .................................587
25.2.3AlternativeCombustionOptions
25.3CarbonStorage ..............................................589
25.3.1StorageOptions... .................................589
25.3.2TrappingMechanisms. .............................592
25.3.3EnvironmentalImpacts
25.3.4StorageCapacityEstimation
25.4AbsorptionTheory. ..........................................597
25.4.1TwoFilmModel.. .................................597
25.4.2CountercurrentAbsorptionModel..
25.5ProblemSheet ...............................................603
25.6WorkedSolutions..
26AtmosphericDispersion ............................................611
26.1Introduction.. ...............................................613
26.2TheAdvectionDispersionEquation.
26.2.1MassConcentrationofWasteGases ................613
26.2.2MassConservationinaControl-Volume. ...........614
26.2.3ReynoldsDecomposition.. .........................615
26.2.4GaussianPlumeModelling .........................617
26.2.5ReflectionFromtheLandSurface.. ................620
26.3AtmosphericStability.. ......................................621
26.3.1DryAdiabaticLapseRate. .........................621
26.3.2ImpactonPlumeBehaviour ........................623
26.3.3ImpactonDispersion. .............................625
26.4EffectiveStackHeightEstimation... .........................626
26.4.1ConceptualModel. .................................626
26.4.2Briggs’PlumeRiseEquations.. ....................628
26.4.3AnalyticalSolutiontoProvideTheoreticalBasis....628
26.5ProblemSheet ...............................................635
26.6WorkedSolutions..
27WaterPollution ....................................................643
27.1Introduction..
27.2DrinkingWater ..............................................645
27.2.1PathogenicOrganisms. .............................645
27.2.2TotalDissolvedSolids(TDS)...
27.2.3OtherToxicSubstances...
27.3Wastewater... ...............................................649
27.3.1DissolvedOxygen(DO)...
27.3.2BiochemicalOxygenDemand(BOD)
27.3.3ChemicalOxygenDemand(COD).
27.3.4SuspendedSolids(SS)
27.3.5TotalPandN
27.4ModellingBODandDO
27.4.1BODDecay..
27.4.2MassBalanceforaRiverandDischarger
27.4.3DOConsumptioninRivers
27.5ProblemSheet ...............................................660
27.6WorkedSolutions..
28.1Introduction..
28.2WaterTreatmentProcesses..
28.2.1Screening
28.2.2DissolvedAirFloatation..
28.2.3Clarification..
28.2.4CoagulationandFlocculation...
28.2.5Filtration. ..........................................669
28.2.6Adsorption...
28.2.7Disinfection..
28.2.8Aeration.
28.2.9Desalination.
28.2.10IonExchange
28.2.11Remineralisation..
28.3GranularFiltrationTheory..
28.3.1HydraulicHeadLoss..
28.3.2FiltrationEfficiency... .............................676
28.3.3RemovalRateCoefficientCorrelations...
28.3.4BackwashHydraulics.
28.4ReverseOsmosisTheory
28.4.1ReverseOsmosisExplained
28.4.2ConservationEquations...
28.4.3ConcentrationPolarisation.
28.4.4OsmoticPressure. .................................685
28.4.5SoluteDiffusionThroughtheMembrane
28.4.6FeedRateEstimation.
28.5ProblemSheet
29WastewaterTreatment
29.1Introduction..
29.2WastewaterTreatmentProcesses
29.2.1AttachedGrowthReactors.
29.2.2SuspendedGrowthReactors
29.2.3NitrogenRemoval. .................................699
29.2.4WasteStabilizationPonds.
29.2.5SludgeDisposalandReuse
29.3ActivatedSludgeTheory
29.3.1MassConservationStatements..
29.3.2MonodEquation..
29.3.3CompletelyMixedConditions..
29.3.4IncompletelyMixedConditions
29.3.5LimitingCases .....................................710
29.3.6DispersionCoefficientEstimation..
29.4ProblemSheet
29.5WorkedSolutions..
30EIAandUncertaintyManagement
30.1EnvironmentalImpactAssessment..
30.1.1OutlineoftheEIAProcess
30.1.2ProjectScreening.
30.1.3Scoping..
30.1.4ImpactSignificanceAssessment
30.1.5RiskRanking
30.2HandlingUncertainty..
30.2.1UncertaintyinModelPredictions...
30.2.2CumulativeDistributionFunctions..
30.2.3ParameterDistributions
30.2.4MonteCarloSimulation...
30.2.5TornadoPlots ......................................738
30.2.6SimpleExample..
30.3ProblemSheet ...............................................742
30.4WorkedSolutions..
BasicPrinciplesofFluidMechanics
Abstract
Thischapterintroducessomebasicprinciplesoffluidmechanics.Equations arederivedtodescribehydrostaticforcesoninclinedplates.Theconceptof Archimedes’principleisexplainedandananalyticalsolutionforassessingthestabilityoffloatingofobjectsisdeveloped.Thecontinuity,momentumandBernoulli equationsarederivedusingacontrol-volumewithinastreamtube,locatedwithin amovingincompressibleandinviscidfluid.TheBernoulliequationisusedto derivetheTorricelliequationandequationsdescribingflowoversharpcrested weirs.Practicalapplicationsarepresentedthroughaproblemsheetwithworked solutions.
Notation
a Acceleration[LT 2 ].
a1 Depthoflowdensityfluidinau-tubemanometer[L].
a2 Elevationdifferencebetweenfluidinterfacesinau-tubemanometer[L].
A Area[L2 ]. ˆ
A Meancross-sectionalarea[L2 ].
B Breadth[L].
E Totalhead[L].
F Force[MLT 2 ].
Fb Buoyantforce[MLT 2 ].
Fg Gravitationalforce[MLT 2 ].
g Gravitationalacceleration[LT 2 ].
h Hydraulichead[L].
h v Velocityhead[L].
H Depth[L].
©UniversityofDurham2023
S.A.Mathias, Hydraulics,HydrologyandEnvironmentalEngineering, https://doi.org/10.1007/978-3-031-41973-7_1
H F Depthofahydrostaticforce[L].
I yy Secondmomentofareaaboutacentroidinthe y -direction[L4 ].
L Length[L].
L BG Distancebetweenthecentreofbuoyancyandthecentreofgravity[L].
L BM Distancebetweenthecentreofbuoyancyandthemetacentre[L].
M Mass[M].
p Fluidpressure[ML 1 T 2 ].
p0 Atmosphericpressure[ML 1 T 2 ].
pa Absolutepressure[ML 1 T 2 ].
Q Flowrate[L3 T 1 ].
s Distancealongastreamtube[L].
t Time[T].
U Moment[ML2 T 2 ].
Ub Momentduetoabuoyancyforce[ML2 T 2 ].
Ub Momentduetoabuoyancyforcefollowingarotation[ML2 T 2 ].
V I Immersedvolume[L3 ].
v Fluidvelocity[LT 1 ].
y Distancefromaliquidsurfaceintheplaneofasubmergedandinclined plate[L].
y F Distancefromaliquidsurfacetothelocationwhereahydrostaticforce applies,intheplaneofasubmergedandinclinedplate[L].
yc Distancetoacentroidfromadatum[L].
yc Distancetoacentroidfromadatumfollowingarotation[L].
z Elevation[L].
θ Inclinationangle[ ].
ρ Massdensityofafluid[ML 3 ].
ρs Massdensityofasolidobject[ML 3 ].
ψ Pressurehead[L].
CoBCentreofbuoyancy.
CoGCentreofgravity.
1.1Introduction
Thischapterprovidesabriefintroductiontosomebasicprinciplesoffluidmechanics. Thefirstsectionfocusesonhydrostatics.Thesecondsectionfocusesonmovingfluids intheabsenceoffrictionalforces.Alltheresultsderivedinthischapterareclassic. Similarresultswithdifferentderivationscanbefoundinmanyexistingtextbooks concerningthemechanicsoffluids.Thereaderisdirectedto[1]foralternativeand sometimesmoredetailedexplanations.
1.2Hydrostatics
Thestudyofforcesinstaticfluidisreferredtoashydrostatics.Inthissectionwe willdeveloptheconceptofhydrostaticpressureanduseittoderivemethodsfor measuringpressure,determininghydrostaticforcesoninclinedplatesandassessing thestabilityoffloatingobjects.
1.2.1HydrostaticPressure
Consideratankofstaticliquidwithafreeuppersurfaceopentotheatmosphere (Fig. 1.1).Let A [L2 ]betheplanareaofthetank, H [L]bethedepthoftheliquid withinthetank, p0 [ML 1 T 2 ]betheatmosphericpressure, ρ [ML 3 ]bethemass densityoftheliquidand g [LT 2 ]begravitationalacceleration.Thedownward forceappliedtothebaseofthetankwillbe A ( p0 + ρ gH ).Theabsolutepressure, pa [ML 1 T 2 ],appliedtothebaseofthetankistherefore p0 + ρ gH
Inhydraulics,wearemoreconcernedwitharelativepressure,whichrepresents thedifferencebetweenabsolutepressureandatmosphericpressure.Hereafter,the termfluidpressure, p [ML 1 T 2 ],istakentomeantherelativepressureofafluid, p ≡ pa p0 [ML 1 T 2 ].
ReferringbacktothetankofliquidinFig. 1.1, p = 0,attheliquidsurfaceand p = ρ gH atthebaseofthetank.Furthermore,if z [L]representselevationabove thebaseofthetank
Equation(1.1)isoftenreferredtoasahydrostaticpressureprofile.Interestingly,it willapplyirrespectiveofthegeometryofthetankofconcern.Furthermore,pressure isisotropic,whichmeansthatitsmagnitudeisindependentofdirection.Thisimplies thattheliquidappliesauniformforcedistributiontothebaseofthetankandalinear forcedistributiontothesidewallsofthetank(asindicatedbythearrowsinthetank showninFig. 1.1).
Fig.1.1 Hydrostaticfluid pressureinatankofliquid
Fluidpressure, p
1.2.2MeasuringPressure
Notethathydrostaticpressureisindependentofthequantityoffluidpresent.Instead, itisdependentontheelevationofthehydraulicallyconnectedfreesurfaceexposed totheatmosphere.Forexample,considerFig. 1.2.Thefluidpressureatthebaseof eachofthefirstthreedevices(Fig. 1.2 a,bandc)shouldbethesamebecausethe elevationoftheexposedfreesurface,ineachdevice,isthesamedistanceabovethe baseoftheunderlyingtanks.
ThedeviceshowninFig. 1.2bisinfactameasuringdevicereferredtoasa piezometer.Thefluidpressureatthetopofthetankcanbemeasuredbymultiplyingtheelevationdifferencebetweenthetopofthetankandthefreesurfaceinthe piezometerby ρ g .
ThedeviceshowninFig. 1.2cisanalternativemeasuringdevicereferredtoas au-tubemanometer.Thefluidpressureatthetopofthetankcanbemeasuredby multiplyingtheelevationdifferencebetweenthetopofthetankandthefreesurfacein themanometerby ρ g .Theadvantageoftheu-tubemanometer,overthepiezometer, isthatfluidswithdifferentdensitiestothatbeingmeasuredcanbeusedtoeither providemoreorlesssensitivity.Furthermore,itispossibletomeasureabsolute pressures,whicharelessthanatmosphericpressure.
Figure 1.2dshowsatankwiththesamepressureasthatinFig. 1.2cbutwitha u-tubemanometercontainingalessdensefluid(thelightershade).Consequently, thefreesurfaceexposedtotheatmosphereishigherinFig. 1.2dascomparedtoin Fig. 1.2c.Smallchangesinpressurewillleadtolargerchangesinfluidlevelwhen usingalessdensefluid,henceitismoresensitive(ascomparedtowhenusinga commonfluid).
Figure 1.2eshowsatankwiththesamepressureasthatinFig. 1.2cbutwitha u-tubemanometercontainingamoredensefluid(thedarkershade).Consequently, thefreesurfaceexposedtotheatmosphereislowerinFig. 1.2eascomparedtoin Fig. 1.2c.Largepressuresleadtosmallerfluidlevelswhenusingamoredensefluid, henceasmalleru-tubecanbeused(ascomparedtowhenusingacommonfluid).
Fig.1.2 Examplesofdifferentvesselsthathavethesamefluidpressureatthebase. a Inverted conicalvase. b Tankwithapiezometer. c Tankwithau-tubemanometercontainingcommonfluid. d Tankwithau-tubemanometercontaininglessdensefluid(thelightershade). e Tankwithau-tube manometercontainingmoredensefluid(thedarkershade)
(a)(b)(c)(e) (d)
Fig.1.3 Examplesofdifferentfluidpressuremeasuringdevicesinacommontankofliquid. a Invertedconicalvase. b Piezometer. c U-tubemanometercontainingcommonfluid. d U-tube manometercontaininglessdensefluid(thelightershade). e U-tubemanometercontainingmore densefluid(thedarkershade)
Fig.1.4 Highdensityfluidu-tubemanometersconnectedtoapipe.Thecirclesrepresentacrosssectionthroughthepipe. a Measurementofanabsolutefluidpressuregreaterthanatmospheric pressure. b Measurementofanabsolutefluidpressuregreaterthanatmospheric
NotethatgivenallfivevesselsinFig. 1.2 arehydrostatic,exactlythesamefluid levelswouldbeobservediftheywereeachlinkedtothesametank,suchasshown inFig. 1.3.
Figure 1.4ashowsau-tubemanometermeasuringtheabsolutepressure, pa ,of afluidflowinginapipe,wheretheabsolutepressureisgreaterthanatmospheric pressure, p0 .Themassdensityofthelowdensityfluidinthepipeis ρ1 [ML 3 ]and themassdensityofthehighdensityfluidintheu-tubeis ρ2 [ML 3 ].
Challenge1.1 UsetheconceptofhydrostaticpressuretodetermineanexpressionfortheabsolutepressureinthepipeshowninFig. 1.4a.
Theabsolutepressureinthepipe,showninFig. 1.4a,canbeobtainedbyequating twoexpressionsfortheabsolutepressureattheinterfacebetweenthelowandhigh densityfluidsintheu-tube.
Thehydrostaticpressurecontributionduetothelowdensityfluidis ρ1 a1 g .There isalsotheabsolutepressureinthepipe, pa .Theabsolutepressureattheinterface betweenlowdensityandhighdensityfluidsintheu-tubemanometeristherefore pa + ρ1 a1 g .
Ontheotherhand,thehydrostaticpressurecontributionduetothehighdensity fluidis ρ2 a2 g .Thereisalsotheatmosphericpressure, p0 .Itcanthereforealsobe saidthattheabsolutepressureattheinterfacebetweenlowdensityandhighdensity fluidsintheu-tubemanometeris p0 + ρ2 a2 g .
Equatingthetwoexpressionsaboveandsolvingfor pa givesus
Figure 1.4bshowsau-tubemanometermeasuringtheabsolutepressure, pa ,ofa fluidflowinginapipeweretheabsolutepressureislessthanatmosphericpressure, p0 .Again, ρ1 [ML 3 ]isthemassdensityofthelowdensityfluidinthepipeand ρ2 [ML 3 ]isthemassdensityofthehighdensityfluidintheu-tube.
Challenge1.2 UsetheconceptofhydrostaticpressuretodetermineanexpressionfortheabsolutepressureinthepipeshowninFig. 1.4b.
Theabsolutepressureinthepipe,showninFig. 1.4b,canbeobtainedbydetermininganexpressionfortheabsolutepressureatthehighdensityfluidfreesurface andequatingthiswiththeatmosphericpressure.
Thehydrostaticpressurecontributionduetothelowdensityfluidis ρ1 a1 g .The hydrostaticpressurecontributionduetothehighdensityfluidis ρ2 a2 g .Thereisalso theabsolutepressureinthepipe, pa .Theabsolutepressureatthehighdensityfluid freesurfaceistherefore pa + ρ1 a1 g + ρ2 a2 g .Equatingthiswiththeatmospheric pressure, p0 ,andsolvingfor pa givesus
1.2.3HydrostaticForcesonInclinedPlates
Herewewillderiveexpressionsforthetotalforceappliedtoasubmergedinclined plateduetohydrostaticpressure.Wewillthenusemomentmatchingtodeterminethe depthatwhichtheforceapplies.Tobeginwith,wewillfocusonarectangularplate. Thesametheorywillthenbeextendedtoaccountforplatesofarbitrarygeometries.
1.2.3.1RectangularPlate
Consideranimmersedrectangularplateofbreadth, B [L],andlength, L [L], inclinedatanangle, θ [ ],tothehorizontalaxis(seeFig. 1.5).Thedepthof
Fig.1.5 Hydrostatic pressureonaninclined rectangularplate
Sideviewof inclinedplate
Frontviewof inclinedplate
fluidabovethetopandbottomoftheplatearedenoted H0 [L]and H1 [L], respectively.Let y [L]beadistancefromtheliquidsurfaceintheplaneof theplate.Theoriginofthe y -axisattheliquidsurfaceisdenoted O .Thedistance,alongthe y -axis,from O tothetopandbottomoftheplatearedenoted y0 [L]and y1 [L],respectively.Notethat L = y1 y0 , H0 = y0 sin θ and H1 = y1 sin θ.
Challenge1.3 Determineanexpressionfortheforce, δ F [MLT 2 ],applied toathinstripofarea, δ A [L2 ],locatedatadistance, y ,alongthe y -axisdue tohydrostaticpressure.Yourfinalexpressionshouldbeintermsof ρ , g , θ, y and δ A .
Considerathinstripofarea, δ A [L2 ],locatedatadistance, y ,alongthe y -axis. Thefluiddepthatthislocationis y sin θ.Thereforethehydrostaticpressureatthis pointis ρ gy sin θ.Itfollowsthattheforceappliedtothestrip, δ F [MLT 2 ],dueto hydrostaticpressureisfoundfrom
Challenge1.4 Determineanexpressionforthetotalforce, F [MLT 2 ],applied totheinclinedplateduetohydrostaticpressure.Yourfinalexpressionshould beintermsof ρ , g , B , L , H0 and H1 .
Thetotalforce, F [MLT 2 ],appliedtotheinclinedplateisfoundbyintegrating Eq.(1.4)
Notingthattheplateexistsfor y ∈[ y0 , y1 ] and dA dy = B ,itfollowsthat
Giventhat L = y1 y0 , H
θ and H1 = y1 sin θ,itcanbefurtherstated that
Challenge1.5 Determineanexpressionforthemomentabout O , δ U [ML2 T 2 ],appliedtoathinstripofarea, δ A [L2 ],locatedatadistance, y , alongthe y -axis,duetohydrostaticpressure.Yourfinalexpressionshouldbe intermsof ρ , g , θ, y and δ A .
Thehydrostaticforceappliedtoathinstrip,givesrisetoamomentabout O , δ U = δ Fy ,where δ F isgivenbyEq.(1.4).Itfollowsthat
Challenge1.6 Determineanexpressionforthetotalmomentabout O , U [ML2 T 2 ],appliedtotheinclinedplateduetohydrostaticpressure.
Thetotalmomentabout O , U [ML2 T 2 ],appliedtotheinclinedplatedueto hydrostaticpressureisfoundbyintegratingEq.(1.7)
Notingagainthattheplateexistsfor y ∈[ y0 , y1 ] and dA dy = B ,itfollowsthat
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saavat maata riittävästi, kun siitä kuitenkaan ei ole puutetta. Jos siihen mennessä tulen itse jo vanhaksi, niin onhan poika… Sillä tuntuukin olevan halu maahan.
— Niinkö Joel luulee?
— Niin. Muuten ei tule hyvää tästä yhteiskunnasta.
Miehet seisahtivat, Ilmassa tuntui väkevää tervassavua. Aapo katsahti kysyvästi Joeliin, ja tämä naurahti, tunnustellen, mistäpäin savu tuli tielle.
— Mennäänpä katsomaan, joko tuolla olisi valmista, sanoi Joel ja lähti menemään metsään.
— Ettäkö olisi juhannusviinat siellä tulossa?
— Ka, tulehan perässä, mutta hiljaa.
Eikös ollutkin siellä pata kuusen juurella, hiljainen tervastuli alla. Laitos oli äsken pantu käyntiin, mutta valmistaja oli livahtanut tiehensä. Täysinäinen rankkiamme oli toisen puun juurella, ja Joel hämmenteli sitä muhoillen.
— Tulehan pois kokki sieltä! hoilasi Joel metsään, jossa tiesi keittäjän varmasti piilottelevan. — Annat hieman tihuntia meillekin, niin ei puhuta mitään.
— Kenen luulet olevan tämän laitoksen? kysyi Aapo melkein hämillään. Olihan se hänen maallaan. Kuka vietävä oli ollut niin rohkea, että uskalsi tulla hänen alueelleen?
— Tunnen minä tämän tiinun, ei se kaukaisia ole, naurahteli Joel.
— Tulehan katsomaan.
Joel osoitteli kirveensä varrella nimimerkkiä tiinun laidassa. Mitä? Eikö hän ollut tuota samaa astiaa nähnyt joskus Savuniemessä?
— Hentun laitoksia tämä on, vahvisti Joel.
— Ja että se piru uskalsi tulla!
Aapo löi vanteet poikki kirveellään, ja tahmea neste levisi maahan.
— Älä hitossa! Sehän olisi pitänyt antaa lehmille… jumalan viljaa… voi turkanen!
Vihan vimmassa silpoi Aapo laudatkin kappaleiksi ja potkaisi padan kumoon. Kirkas neste oli jo alkanut valua torvesta.
— Ka… nyt teit tyhmyyden. Olisi viety pata ja sammio Hentulle itselleen, sanoi Joel. — Se olisi ollut sille kova paikka.
— Siitä se vähät olisi välittänyt.
Aapo kolhi vielä padankin pieniksi kappaleiksi kirvespohjalla.
Miehet poistuivat, Aapolla kainalossaan lauta, jossa oli Hentun nimimerkki. Siinä se oli. Irtolaisia ja yhteiskunnan hylkiöitä syytettiin tästäkin paheesta, ja kuitenkin olivat viljojen kasvattajat ja parempiosaiset pääsyyllisiä. Aapo tuli tästä sanoneeksi Joelillekin.
— Kukapa heistä, hylkymiehistä, tehtävään rupeaisi, jos sillä ei olisi isoisten kannatusta, virkkoi hänkin. — Irtolaisten syynä se on saanut mennä tähän asti, vaikka tietensä ovat viljat antaneet ja
tuotteet käyttäneet. Siinäkin on yksi vikasolmu tässä yhteiskunnassa.
Kunhan tulee vielä toinen aika…
Joel keskeytti ja jäi miettimään.
— Niin mitä? kysyi Aapo.
— Sitä vain tässä, että kyllä se tuokin konsti niiltä loppuu, kun maat tulevat tiheämmin asutuiksi. Kukapa sitä sitten enää antaa maallaan valmistaa.
— Taidat olla oikeassa, myönsi Aapokin. — Mutta siihen on vielä aikaa.
Miehet kävelivät metsässä ääneti. Äsken näkemänsä oli vienyt Aapon mielen tasapainosta, johon hän oli jo tuntenut pääsevänsä.
Vasta kun hän pääsi kotipihalleen koivukuormineen ja näki vaimonsa iloisena häärivän tuvan ja aittojen välillä, kykeni hän karistamaan pois painavat mietteensä.
Mitäpä auttoi mietiskellä epäkohtia. Kaikesta huolimatta oli kuitenkin yhteiskunta kulkemassa valoisampaa tulevaisuutta kohti.
— Missä pihlajankukat? kysäisi Liina heidän pihaan tultuaan.
— Eihän niitä täällä… jos mentäisiin illallisen jälkeen Harjamaasta noutamaan?
Aapo katsoi kysyvästi vaimoonsa, joka näytti miettivän.
— Niin, en tiedä, mutta voimmehan mennäkin, sanoi Liina hieman epäröiden ja pyörähti tupaan illallista valmistamaan. Miehet kävivät saunaan.
Takaliston uudistalossa on juhannusaaton ehtoo erilainen kuin talossa, jossa ympäristö ja viljelykset ovat vuosikymmenien, jopa satojenkin ajalla rehevöittyneet. Aapo tunsi sen, istuessaan saunan jälkeen puhtoisissaan pihamaan kivellä. Talon ympärillä metsissä ja pellon pientareilla olivat metsän villit kukat auenneet ja täyttivät huumaavalla tuoksullaan pihamaan. Korvesta tuli monenlaista ääntä, jota turhaan sai odottaa kuulevansa vanhan talon pihamaalle. Siellä soitteli satakieli, ja rastaan laulu tuli monista pienistä suista yhtenä sävelhurmina. — Kahdella eri haaralla kukahteli käki. Silloin tällöin vihelteli kuovi rämeessä.
Aapo istuu yhä paikallaan kukkasmeren ja luonnonäänien ympäröimänä, hengittäen voimakkaita tuoksuja. Joel kähmii aittansa ovella ja menee sitten pirttiin ja sanoo jotakin Aapolle mennessään. Liina menee saunaan äsken taittamansa vasta kainalossa.
— Miks'et mene syömään? Vai odotatko minua… minäkään en ole vielä syönyt, sanoo hän Aapolle mennessään.
— Minä odotan tässä sinua.
Liina alkaa riisuutua saunan edessä sileällä nurmella. Aapo on hyvillään, että saunan edessä on niin sileä kaunis nurmi, jossa on nuoren naisen hyvä riisua vaatteensa. Koivu on vielä saunan nurkalla, ja sen riippuvat lehvät melkein hipovat Liinan hiuksia.
Liina heittää hameensa reippaalla liikkeellä menemään. Paidan olkanapit aukenevat, ja pyöreät olkapäät paljastuvat. Niitä muistaa Aapo joskus suudelleensa pyhän riemun vallassa. Paita putoaa kokonaan nurmelle, ja Liina seisoo siinä nuoressa kauneudessaan. Kuulakka valo väreilee hänen punertavalla ihollaan, ja Aapo tuntee, ettei sillä hetkellä voisi häneen kädelläänkään koskea. Täällä
korvessa on kaikki samaa luontoa. Riisutaan pihamaalla ja juostaan saunaan alasti. Kun Liina tekee sen, on siinä kuin jotain pyhää, sykäyttävää riemua, ja tuntee miten suuri voima kohottaa rintaa.
— Siinähän sinä vielä istut. Tule nyt pian aterialle, sitten mennään Harjamaahan.
Liinan posket ja täyteläiset käsivarret ovat tulleet saunassa vielä kauniimmin punertaviksi, ja Aapo sivelee kädellään hänen kimmoista ihoaan.
— Sinä olet… niin…
Aapo koettaa etsiä sanoja, mutta ei löydä. Liina vain hymyilee kaunista hymyään.
* * * * *
Pienissä tuvissa tien varrella jo nukuttiin, kun Aapo vaimonsa kanssa vaelteli Harjamaahan.
Siinä oli Tiensivu, jossa oli kuusihenkisen perheen isänä vanha päivätyöläinen kylästä. Hän oli ottanut mäkituvan palstan, kivikkorinnettä, johon oli koetettu kuokkia peltoa pieniä kaistaleita. Talo ei kuitenkaan antanut puumetsää eikä laidunta, ja nyt katui mies niin, että arveltiin hänen päänsä pettävän. Pojat varastelivat puita toisten talojen mailta ja omansakin, ja eukko kantoi lehmälle heinää, jota riipi talojen niityistä.
Ja tuossa asusti Sumpun Eero. Tuvan vieressä oli kamarin salvos, joka joskus oli tullut siihen rakennetuksi isännän hyvän tahdon puuskan kannustamana, mutta jäänyt keskeneräiseksi. Eerollakin oli suuri perhe, ja uloskäsky odotti häntä, kun ei suostunut maapalaa
ottamaan. Miestä sanottiin polsevikiksi, ja yhteiskunta teki hänenkin suhteensa uutta rakennustyötä. Hänenkin varalleen saataisiin varata kunnan köyhäinhoitolaan yksi koppi.
— Ja tuossa, osoitti Aapo sormellaan. — Kyllähän sinä Liina tiedät tuonkin perheen historian. Sekin on liian surullinen ajateltavaksi juhannusyönä.
— Kyllä. Vein sinne monta kertaa leipää, kun ei heillä riittänyt.
— Ja katsohan tuota laajaa ahoa tuossa. Siinä on maata yhteensä kolme hehtaaria. Mitattiin kerta huvin vuoksi se pala mökin miehen kanssa. Mies on himoinnut sitä ikänsä peltomaakseen, mutta se on siinä vain. Ei kasva puuta eikä heinää, kun karja tallaa sitä joka päivä. Multa siinä on kuin kahvijauhoa, ja se elättäisi suuremmankin perheen.
— Poiketaan tuolle metsätielle, pyysi Liina. — Minä en jaksa enää katsella tällä kertaa noiden mökkitönöjen surkeutta.
— Mennäänpä Jänkän kautta. Nähdään, mitä Simo on siellä puuhannut.
Kuljettiin ahojen ja lepikkölehtojen poikki. Karjan kellot kalahtelivat, ja jostakin kaukaa kuului harmonikan säveliä. Siellä tanssittiin. Järveltä, joka oli metsän takana, kuului airojen loisketta.
— Katsohan sinä näitä ahoja ja väkevämultaisia lepikoita, sanoi Aapo Liinalle. — Näihin voisi vielä entisten tupien lisäksi rakentaa uusia, ja kaikille olisi maata riittävästi. Nyt ne ovat muka karjan laitumena, ja kuitenkaan ei niistä karja mitään hyödy.
— Mutta minäpä luulen, että nämä ahot saavat vielä kerran asukkaansa, sanoi Liina.
— Se on varma se. Silloin uhoo viljavuutta tämä karu maa, ja metsät ovat kuin hyvin hoidettuja puistoja.
Jänkän uudistalon aita tuli näkyviin. Aitaan nojaten katselivat he viljelyksiä, jotka olivat kuin hyvin hoidettua puutarhaa. Ruislaiho, joka oli lähinnä aitaa, kohotti jo tähkiään aidan tasalle, ja suvitouko oli niin täyteläistä, ettei yhtään aukkoa huomannut saroilla.
— Kas kun Simo on ehtinyt jo rakentaa uuden aittarivinkin muiden töittensä ohella. Sanoppa, Liina, mistä luulet hänen sellaista työintoa saaneen, että aivan ihmeitä saa aikaan?
— Oma maa kai sitä on hänelle antanut.
— Niin, ja kun se on saatu kohtuullisella hinnalla, ettei velka vie omistajan tarmoa.
Kun he lähenivät Harjamaan pellon aitaa, hävisi Aapon reipas tuuli.
Mitä? Eikö hänen kätensä vapissut porttia avatessa?
Siinä se oli siis entinen koti raunioina. Pihapuut olivat lehdittyneet tuuheiksi niinkuin aina ennenkin, ja koko tienoo tuoksui pihlajankukilta.
Aapo istui ääneti pihakivelle. Häneltä ei nyt riittänyt huomiota edes rakastamalleen vaimolle. Liina vaistosi tulleensa syrjäytetyksi ja antoi Aapon olla rauhassa. Kadutti, että oli luvannut lähteä tänne, jossa Aapolla oli kaikki Huoruutensa muistot.
Aapo käveli hitaasti rantatielle ja katseli rikkaruohon peittämiä peltosarkoja. Hän ei ollut tahtonut niitä enää kylvää, kun kerran ne oli häneltä riistetty, ja Hentu ei ollut myöskään niihin kajonnut. Heinäpelto näytti lupaavalta. Jättäisikö hän kauniin kukkivan apilan siihen korjaamatta? Ehkäpä Hentu jo katseli sitä ahnain silmin.
Sydäntä katkeroi niin sanomattomasti kaikkea katsellessa.
Jääköön siihen heinäpeltokin, kun on jäänyt kaikki muukin. Hän ei kajoa enää sormellaankaan. Korjatkoon korkea oikeus Hentun kanssa heinät ja muut. Hän kyllä voi elää korvessaan, johon on ajettu. Tälle kiihtyneelle mielelleen hän tiesi saavansa Liinalta kannatusta. Liina tulikin häntä vastaan rantatiellä ja katsoi arasti silmiin.
— Et usko, miltä minusta tuntuu, kun katselen kaikkea tätä häviötä, sanoi Aapo tarttuen vaimonsa käteen, kuin voimaa hakien.
— Kyllä minä sen hyvin ymmärrän… niin kaunista kuin täällä aina oli.
— Ja on vieläkin. Katsohan noita pihapuita ja pihlajia. Näyttää niinkuin nekin surisivat. Katso tuota haapaa, jonka lehtiä yön henki liikuttaa.
Liinakin näytti kärsivän miehensä kärsimyksistä.
— Eikö mennä jo kotiin? virkkoi hän.
— Kotonahan minä olenkin, sanoi Aapo kolkosti. — Voi mennä pitkä aika, ennenkuin tunnen sinne korpeen kotiutuvani. Niin, minunhan piti kysyä sinulta, mitä tehdään näille Harjamaan
heinävainioille. Annetaanko Hentulle ja hänen korkeille apulaisineen vaivojensa palkkioksi?
Hän tunsi herjaavansa, mutta se teki hyvää hänen kipeälle mielelleen.
Liinan silmiin syttyi uhman ilme, ja pieni pää nousi jäykästi pystyyn.
— Antaa heidän pitää vain kaikki, sanoi hän.— Kyllä me tulemme omillamme toimeen. Mitä emme saa vielä uudismaasta, sitä on kyllä meille riittävästi Puromäessä. Luulen isänkin pitävän siitä, ettei oteta korttakaan.
Vaikealta tuntui Aaposta lähtö paluumatkalle, mutta Liina veti häntä kädestä, kainalossaan kimppu pihlajan kukkivia oksia.
— Minunkin pitäisi ottaa niitä.
Hellävaroen taitteli Aapo muutamia oksia pihlajista ja painoi sitten päättävästi hatun silmilleen.
— Tule, mennään. Kohta nousee aurinko ja minä ajattelin olla sen nousua katsomassa sen korpiasunnon rannalla sinun kanssasi.
— Älä sano asunnon, vaan kodin, sillä se siitä täytyy tulla meille varmasti, sanoi Liina.
— Niin, sinun avullasi, ei muuten.
Kun he pääsivät kotipihaansa, nousi aurinko.
— Nyt emme ennättäneetkään rannalle, sanoi Aapo.
— Kaunistahan on tässäkin, katso vain ympärillesi.
Tuhansien lintujen viserrys helisi heidän ympärillään. Aapo tunsi povestaan häviävän kaiken jäykän ja jäätävän. Liinan avulla hän
todellakin tahtoi tehdä tästä kukoistavan kodin. Olihan se vielä kerran niin monen muun tehtävä koskemattomaan metsään, kun maat vapautuisivat viljelykselle.
Liina hymyili jo siinä pihlajankukkiensa ympäröimänä ja kurotti huulensa suudeltaviksi. Aapo kiersi kätensä hänen ympärilleen ja tunsi surujensa haihtuvan nuoren onnensa täyteläisyyteen. * * * * *
Suvituuli heräsi lehdossa ja lähti kulkemaan yli soiden ja asumattomain ahojen, jotka odottivat vapauttajiaan.
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