Transport Processes Primer

298 Pages·2019·3.605 MB·English

by  Pozrikidis, Constantine

Constantine Pozrikidis
Transport Processes
Primer
Transport Processes Primer
Constantine Pozrikidis
Transport Processes Primer
123
ConstantinePozrikidis
CollegeofEngineering
UniversityofMassachusettsAmherst
Amherst,MA,USA
ISBN978-1-4939-9908-8 ISBN978-1-4939-9909-5 (eBook)
https://doi.org/10.1007/978-1-4939-9909-5
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Preface
Transport phenomena or transport processes is a concept proposed by chemical engineers in an
effort to unify fluid mechanics, heat transfer, and mass transfer in a stationary material or moving
fluid.Thecoreprocedureprescribedintextsandelsewhereinvolvesdefiningafiniteorinfinitesimal
controlvolume,andthenperforminganintegralordifferentialmomentum,heat,mass,orsomeother
typeofbalanceofatransportableentitytoderiveagoverningequation.
Thebalancetypicallystatesthattherateofaccumulationofacertainextensivepropertyofinterest,
such as mass, momentum, or total energy, is determined by the rates of convective and diffusive
transportacrosstheboundariesofthecontrolvolume,aswellasbyappropriateratesofinteriorand
surface loss or production. The control volume itself may be stationary or evolve in an arbitrary
fashion.
One subtlety of the aforementioned approach is that mass, momentum, and energy balances are
consequences of the principle of mass conservation, Newton’s second law of motion, and the first
law of thermodynamics. In their classical form, these natural laws apply to well-defined bodies or
pieces of material (closed systems), as opposed to control volumes that allow matter to cross their
boundaries(opensystems).
Theprincipleofmassconservationispertinenttothemassofasingle-speciesmaterialparcelor
speciesXparcel,
(cid:2)(cid:2)(cid:2) (cid:2)(cid:2)(cid:2)
ρdV, ρ dV,
X
where ρ is the density; Newton’s second law of motion is pertinent to the momentum of a single-
speciesmaterialparcelorspeciesXparcel,
(cid:2)(cid:2)(cid:2) (cid:2)(cid:2)(cid:2)
ρudV, ρ u dV,
X X
where u is the fluid velocity; the first law of thermodynamics is pertinent to the total energy of a
single-speciesmaterialparcelorspeciesXparcel,
(cid:2)(cid:2)(cid:2) (cid:2)(cid:2)(cid:2)
ρ(1 |u|2+v+u)dV, ρ (1 |u |2+v +u )dV,
2 X 2 X X X
v
vi Preface
wherevisthespecificpotentialenergyanduisthespecificinternalenergy.Consequently,introducing
fluid dynamics based on momentum transport over a control volume could be hard to justify in
foresightthoughstraightforwardtoexplaininhindsight.
A way out is to rewrite the natural laws so that they apply to open systems that allow solid
or fluid material to enter or exit a control volume through inlets and outlets. However, the dual
restatement may appear like a band-aid that undermines the omnipotence of the classical approach
andunnecessarilycomplicatesthelogistics.
Afurthercomplicationisthatasystemmaybeclosedwithregardtoonepropertybutopenwith
regard to another.For example, a system may beclosed with respect to mass butopen with respect
toenergy.Asystemthatisclosedwithregardtoeverypossibletransportableentityortransmittable
field iscompletely isolated.Thenotionofopen,closed,and isolated systems has been discussed in
thenaturalsciencesandundertheauspicesofinformationtheory,sociology,biology,anthropology,
linguistics,history,politicalscience,andphilosophy.
Toensureaccuracyandscientificrigor,thegoverningequationsoftransportphenomenaarebest
derivedfromtheclassicalnaturallawsappliedtomaterialparcels.Thetransportapproachmaythen
bevalidatedandemployedasapracticalmethodofformulatingequationsandobtainingsolutionsin
scienceandengineeringapplications.Therecommendedprocedureinvolvesthefollowingsteps:
1. Write an expression for a property of interest attributed to a material parcel, such as mass,
momentum,totalspecificenergy,orspeciesmass.
2. UsetheReynoldstransportequationtoexpresstherateofchangeoftheparcelpropertyinterms
ofaccumulationovertheparcelvolumeandassociatedfluxintegratedovertheparcelsurface.
3. Introduce a physical law for the rate of change of the parcel property. For example, the rate of
changeofmassiszeroandtherateofchangeofmomentumisgivenbyNewton’ssecondlawof
motion.
4. Regardtheparcelasacontrolvolume,orelseconsidertheparceloccupyingacontrolvolumeof
interestataparticularinstanttoobtainanintegraltransportbalance.
5. Applythedivergencetheoremtoconvertallboundaryintegralsintovolumeintegrals,anddiscard
the integral signs to derive governing differential balances in the form of differential equations
writteninconservativeornonconservativeform.
My main goal in this book is to review the basic concepts and notions of transport processes
and illustrate the origin of the governing equations by deriving and summarizing the equations of
mass, momentum, energy, enthalpy, entropy, and other related transport for homogeneous fluids
and mixtures of fluids in the context of mechanical, chemical, biological, biomedical, and other
mainstreamscienceandengineering.
Noteworthy features ofthe discourse with regard to mass transportincludesthe interpretationof
diffusionintermsofspeciesparcelkinematics,thediscussionofFick’sandfractionaldiffusionlaws,
the introduction of partial stresses and associated equations of motion for individual species in a
mixture,andthestudyofspeciesandmixtureenergetics.
A summary of transport equations in differential and integral forms are presented and unified
in Appendix A. All necessary relations from thermodynamics employed in the text are derived in
AppendixBforaself-containeddiscourse.
Preface vii
Matlab1 programsperformingnumericalsimulationsofrandomwalksthatillustratethenatureof
ordinaryandfractionaldiffusionarelistedinthetext.
Amherst,MA,USA ConstantinePozrikidis
1Matlab(cid:2)R is a proprietary computing environment for numerical computation and data visualization. Matlab and
Simulink are registered trademarks of The MathWorks, Inc. For product information, please contact: The Math-
Works, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, USA, Tel: [phone], Fax: [phone], E-mail:
[email protected],Web:www.mathworks.com.
Contents
1 HomogeneousFluids .......................................................... 1
1.1 TheConceptofFluidVelocity ............................................. 1
1.1.1 ContinuumApproximation......................................... 1
1.1.2 VelocityField.................................................... 2
1.1.3 Differentiability .................................................. 2
1.1.4 MaterialPointParticles............................................ 2
1.1.5 MaterialParcel................................................... 3
1.1.6 Exercise ........................................................ 3
1.2 DensityandOtherIntensiveFields.......................................... 3
1.2.1 DensityasaFieldFunction ........................................ 3
1.2.2 ExtensiveandIntensiveProperties .................................. 4
1.2.3 TemperatureField ................................................ 4
1.2.4 Exercise ........................................................ 4
1.3 FundamentalDecompositionofKinematics .................................. 4
1.3.1 Velocity-GradientTensor .......................................... 5
1.3.2 DivergenceoftheVelocity ......................................... 5
1.3.3 FundamentalDecompositionofKinematics........................... 6
1.3.4 VorticityVector .................................................. 6
1.3.5 TheVorticityIstheCurloftheVelocity .............................. 6
1.3.6 FourTypesofLocalMotion........................................ 7
1.3.7 Translation ...................................................... 7
1.3.8 Rotation ........................................................ 7
1.3.9 Deformation..................................................... 8
1.3.10 ContractionandExpansion......................................... 8
1.3.11 Exercises........................................................ 8
1.4 LagrangianMappingofPointParticles ...................................... 9
1.4.1 RGB ........................................................... 9
1.4.2 Point-ParticlePosition............................................. 9
1.4.3 Point-ParticleVelocity ............................................ 9
1.4.4 LabelingandPhysicalSpaces ...................................... 10
1.4.5 JacobianMatrix .................................................. 10
1.4.6 MaterialVectors.................................................. 10
1.4.7 VolumetricCoefficient ............................................ 11
1.4.8 Exercise ........................................................ 11
1.5 TheMaterialDerivative................................................... 11
1.5.1 RateofChangeFollowingaPointParticle............................ 11
1.5.2 RelationtoEulerianDerivatives .................................... 12
ix
x Contents
1.5.3 DirectionalDerivative............................................. 12
1.5.4 Point-ParticleVelocity ............................................ 13
1.5.5 Point-ParticleAcceleration......................................... 13
1.5.6 PropertiesoftheMaterialDerivative................................. 13
1.5.7 FundamentalLawofKinematics.................................... 13
1.5.8 VolumetricEvolutionEquation ..................................... 14
1.5.9 ExpansionorContractionofaMaterialParcel......................... 14
1.5.10 Exercises........................................................ 14
1.6 MaterialParcelsandNaturalLaws.......................................... 14
1.6.1 ParcelProperties ................................................. 15
1.6.2 IntegrationinLabelingSpace....................................... 15
1.6.3 EvolutionofParcelProperties ...................................... 15
1.6.4 NaturalLaws .................................................... 16
1.6.5 SurfaceFluxandVolumeProduction ................................ 16
1.6.6 ConductiveFluxandSurfaceTraction ............................... 16
1.6.7 NarrativeForm................................................... 17
1.6.8 TransportBalances ............................................... 17
1.6.9 GeneralizedLaws ................................................ 17
1.6.10 Exercises........................................................ 17
1.7 TheReynoldsTransportEquation .......................................... 17
1.7.1 FormalDerivation ................................................ 18
1.7.2 PowersoftheMetricCoefficient .................................... 18
1.7.3 ReynoldsTransportEquation....................................... 18
1.7.4 ConvectionbyNormalMotion...................................... 19
1.7.5 OrientationoftheUnitNormalVector ............................... 19
1.7.6 EvolutionofaParcel’sVolume ..................................... 19
1.7.7 EssentialStep.................................................... 20
1.7.8 EvolutionEquations .............................................. 20
1.7.9 Exercise ........................................................ 20
1.8 IntegralBalanceOveraControlVolume..................................... 20
1.8.1 ControlVolumeOccupiedbyaParcel................................ 21
1.8.2 EvolvingControlVolume.......................................... 22
1.8.3 Accumulation.................................................... 22
1.8.4 ConvectiveTransport ............................................. 22
1.8.5 Control-VolumeBalance........................................... 23
1.8.6 BalanceOveranOpenSystem...................................... 23
1.8.7 Exercise ........................................................ 23
1.9 MassConservationandMaterialContinuity .................................. 23
1.9.1 MassConservationforaClosedSystem.............................. 23
1.9.2 InstanceofaGeneralLaw ......................................... 24
1.9.3 ContinuityEquation .............................................. 24
1.9.4 MaterialContinuity............................................... 25
1.9.5 EulerianForm ................................................... 25
1.9.6 IncompressibleFluids............................................. 25
1.9.7 EvolutionofaSpecificParcelProperty............................... 26
1.9.8 Traveler’sTimeDerivative ......................................... 26
1.9.9 Exercises........................................................ 27
Contents xi
1.10 Rankin–HugoniotDensityCondition........................................ 27
1.10.1 EnsuringRegularity............................................... 27
1.10.2 One-DimensionalFlow............................................ 28
1.10.3 Exercise ........................................................ 29
1.11 IntegralMassBalance .................................................... 29
1.11.1 MassAccumulation............................................... 29
1.11.2 MassAccumulationDuetoConvection .............................. 29
1.11.3 StationaryorRecirculatingFluid.................................... 30
1.11.4 EngineeringAnalysis ............................................. 30
1.11.5 MassConservationforanOpenSystem .............................. 30
1.11.6 Exercise ........................................................ 30
1.12 TracersandPackets ...................................................... 30
1.12.1 LabelingTracers ................................................. 30
1.12.2 ReynoldsTransportEquation....................................... 31
1.12.3 RateofChangeofaPacketProperty................................. 32
1.12.4 FluxandProduction .............................................. 33
1.12.5 PacketPropertyConservation ...................................... 33
1.12.6 Accumulation.................................................... 34
1.12.7 ControlVolumeOccupiedbyaPacket ............................... 34
1.12.8 Mixtures ........................................................ 35
1.12.9 Exercise ........................................................ 35
2 MomentumandForces ........................................................ 37
2.1 TractionandtheStressTensor ............................................. 37
2.1.1 ForceinTermsoftheTraction...................................... 38
2.1.2 TractioninTermsoftheStressTensor ............................... 38
2.1.3 Hydrostatics ..................................................... 39
2.1.4 PressureinHydrodynamics ........................................ 39
2.1.5 DeviatoricStressTensor........................................... 39
2.1.6 IncompressibleNewtonianFluids ................................... 39
2.1.7 Exercise ........................................................ 40
2.2 ParcelMomentum ....................................................... 40
2.2.1 MomentuminTermsofthePoint-ParticleAcceleration................. 40
2.2.2 AlternativeDerivations............................................ 41
2.2.3 Exercise ........................................................ 41
2.3 Cauchy’sEquationofMotion .............................................. 41
2.3.1 InstanceofaGeneralForm ........................................ 41
2.3.2 Point-ParticleAcceleration......................................... 42
2.3.3 Euler,Bernoulli,andNavier–StokesEquations ........................ 42
2.3.4 EulerianForm ................................................... 43
2.3.5 ConservativeForm ............................................... 43
2.3.6 Stress–MomentumTensor ......................................... 43
2.3.7 Exercise ........................................................ 44
2.4 Rankin–HugoniotMomentumCondition .................................... 44
2.4.1 One-DimensionalFlow............................................ 45
2.4.2 Exercise ........................................................ 46