Table Of ContentEntropic Invariants
of Two-Phase Flows
By
Eugene Barsky
Azrieli College of Engineering, Jerusalem
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1.Two-phaseflow.2.Statisticalmechanics.3.Computationalfluiddynamics.I.Title.
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Introduction
Two-phase flows are widely used in industry. In the overwhelming majority of cases, various tech-
nological processes involving such flows are organized in such a way that the carrier fluid flow is
directed bottom-up. Separation regimes of such flows with one part of solid phase moving with the
flow and another part moving counter the flow are rather widespread. For example, processes of
minerals concentration, bulk materials fractionating by particles sizes, processes in boiling bed,
powder materials drying, mostmass-exchange processesare organizedinthisway.
Visual observations and rapid filming of two-phase flows with a poly-fractional solid phase in
a transparent vertical channel show an absolutely chaotic motion of solid particles. They move
upwards and downwards, to the left and to the right, take part in vortical motions of the carrier
medium,collidewitheachotherandwithchannelwalls,formaggregatesthatpermanentlyappearand
decompose.Allthismakesitimpossibletopredictthedirectionandvelocityofevenoneparticle,not
to speak about the behaviourof solidphase on thewhole.
Thiscanexplainwhyscientistshavenotattainedtheoreticalunderstandingofsuchflowsuntilnow.
Itisnotsurprising,becauseuntiltodaythereisnoacceptablegeneraltheoryevenforpurelyone-phase
liquidandgasflows.Itisknownthatmanyprominentspecialistshavebeenworkingformanyyearsin
thisfieldinmanycountriesaroundtheworld.Numeroustheoreticalschemesforpureturbulentflows
have not led to generalizing results, butonly to a grim joke saying that ‘turbulence is a cemetery of
theories’.
However, at the same time, dams, hydroelectric power station, pneumo- and hydro-transport
systems,andmorearebeingbuiltandoperated.Theyallsuccessfullyfunctionandpracticallyalways
reachtheir design characteristics.
Asknown,itisaresultoffragmentary,andnotgeneralregularitiesofflows,whichhavenotbeen
combined into a unified theory yet. Such regularities fix only some ‘patch-like’ particulars of flows.
For one-phase flows, such fragmentary regularities include so-called similarity criteria derived from
rather simple mathematical models. At the same time, these criteria possess an explicit physical
meaning. In hydrodynamics, they are, as a rule, dimensionless, and each of them consists of param-
etersreflectingsomecharacteristicfeaturesofflows.Theseparameterscanacquireaninfinitelylarge
numberof numerical values, while the criterioncan remain constant.
Such relations are usually called invariants. It is a rather broad notion, which can be defined as
follows:
Invariant is a system of values, parameters, or regularities describing a group of phenomena
remaining unchanged at oneor another transformationof parametersinside thissystem.
It is noteworthy that invariants are not obligatorily dimensionless. Even graphical relations can
serveasinvariantsatarespectivechoiceofcoordinates,sothatallexperimentalorestimatedvaluesof
parameters fitted the same curve.
It isveryimportant to study the possibility offinding such invariants for two-phase flows, which
are,forgoodreason,muchmorecomplicatedthanone-phaseflows.Thiswillmakeitpossibletostart
the developmentofa general approach tothe construction oftheir theory.
The initial study of the process started with the study of the behaviour of solitary particles. First
publications on the topic appeared in the second half of the nineteenth century. First, regularities of
settling of solitary spherical particles in an unlimited motionless liquid were studied. In numerous
vii
viii Introduction
subsequent papers devoted to the study of this phenomenon, the effect of various factors (fluid and
materialdensities,finalvelocitiesofparticlessettling,theirdragcoefficients,etc.)ontheseregularities
was revealed. A transition to materials with irregular particles shape has been so complicated that
publicationsdealingwiththeseproblemsarestillappearingnowadays.Anattemptofapplyingmain
regularities obtained when studying the behaviour of solitary particles in moving medium to real
processeshas notgivenpositiveresults.
Todayalltheproblemsoftwo-phaseflowsaresolvedatanempiricallevel.Numerousresearchesare
continuouslyperformedonmodelsofactualindustrialunits.Thetransferoftheobtainedresultsonto
pilot plants is not always successful. A huge volume of experimental studies has allowed the devel-
opmentofnumerousempiricalcomputationmethodsforspecificapparatuses.However,thebestandthe
mostgroundedempiricalformula canbeapplied withsatisfactoryresults onlywithina limitedrange
determined by the conditions of its derivation. Frequently used extrapolation of such dependencies
beyondtheexperimentalrangelimitsleadstobadmistakes.Besides,empiricalrelationsdonotreflect
theregularitiesofthephenomenaunderstudy,butonlyprovidetheirquantitativecharacteristics.Any
carefullyconductedexperimentdoesnotallowtakingintoaccountthediversityofconstantandrandom
factorsdbothquantitativeandqualitativeones.Thelattercanbetakenintoaccountleastofall.
Since it is unrealistic at present to discuss the possibility of developing a general theory of two-
phase flows, the efforts should be concentrated on the search for physically grounded invariants for
them.Thechaoticdisorderinthemotionofsolidparticlesintwo-phaseflowsbynomeanspointstothe
absenceofgeneralregularities.Onthecontrary,rigidinternalregularitiescanmanifestthemselvesin
such systems only through a general chaotization, as it was established, for example, during the
development of kinetic theory of gases. This theory was developed within the framework of ther-
modynamics. To understand the logics and the essence of the approach to the problem that led to
the development of the kinetic theory of gases, we will try to understand how the development of
thermodynamics led tothis theory.
Initially it was known that fire releases heat that can lead to an increase in the volume of any
bodiesdsolid, liquid, and gases. As a result, work is performed. Fire led people to a steam engine,
which became the basis of the industrial society development in the beginning of the nineteenth
century.Tocreatesteamengines,itwasnecessarytoestablishtheconnectionbetweenheatandwork.
Insearchofasolutiontothisproblem,energyconservationlaw(thefirstlawofthermodynamics)was
formulated.Itimpliesthatheatisofthesamenatureasenergy.Inaheat-engine,heatisconvertedinto
work,buttheamountofenergyisconserved.Thelawoftotalpotentialandkineticenergyconservation
wasestablishedmuchearlier,inclassicalmechanicsdevelopedbyNewton,Lagrange,andHamilton.It
wasoutstandinglysuccessfulandisstilldevelopingatpresent.Oneofitsdistinctivefeaturesisastrict
determinism.
Classicalmechanicscanpredictthebehaviourofasystemwitheitherasmallnumberofelements
orwithmanysymmetricallyarrangedelements.Iftheinitialconditionsforasystemarespecified,and
forcesactingonitaredetermined,onecanfollowthechangesinthissystembothinthefutureandin
the past using classical mechanics.
An especially important conclusion of this mechanics was that due to Newton’s works it has
becomeclearthatnatureobeyssimpleuniversallawsthatarecognizableandcanbeexpressedusing
exactmathematicallanguage.Sincethen,experiments,quantitativestudyofvariousphysicalvaluesin
theirinterconnection,mathematicalrelationsbetweenthemhaveformedthebasisfortheinsightinto
the secrets ofnature.
Introduction ix
Usingclassicalmechanics,ithasbecomepossibletocarrytheproblemofcelestialbodiesmotion
toitsconclusion,butitprovedtobeabsolutelyhelplessfacingtheproblemofthreebodiesmotionin
a general case, nottospeak about mass motion ofmanybodies.
Intheseventeenthandeighteenthcenturies,Newton’sworkshadsuchahighprestigethatinsistent
attemptsweremadetoreducealllawsofnature,discoveredbythattime,toclassicalmechanics.Inthe
long run, they were unsuccessful. However, at least in one field of science these attempts led to
excellentresults,namely,inthetheoryofheatphenomena,butithappenedonlyinthemiddleofthe
nineteenth century. By that time, confidence in the molecular structure of substances started
strengthening. Therefore, efforts were made to find out how macroscopic properties of a substance
dependonthebehaviourofimaginarymoleculesitiscomposedof.Thefirstsuccessesinthisdirection
concerned gases in which molecules interact with a negligible force. The obtained results allowed
expressingpressure,temperature,andothermacroscopicparametersofgasthroughsuchanaveraged
characteristicofmoleculesastheirkineticenergy.Therefore,thistheorywascalledakinetictheoryor
statistical mechanics. It is based on fundamental Boltzmann’s works, where speculative analysis
allowed establishing statistical relations between the molecular structure of substance and its
microscopic properties at amicroscopic level.
Statistical mechanics was based on classical mechanics, but it established new relations and
introduced new concepts. A starting point for Boltzmann was atomistic theory stating that matter
consists ofimmensenumberofsmall moving balls.
In principle, it is possible to describe the motion of a system of independent particles from the
standpoint of classical mechanics. Writing a differential equation for each particle, one can obtain
comprehensiveinformation on thesystemmotion.However, forasystemwithan immensenumber
of particles N, it is necessary to solve in a general form 2N differential equations, which is prac-
tically unrealisable due to the impossibility to specify initial conditions for each particle. At first
sight, we could conclude that with increasing number of particles, properties of the system under
study should become incredibly more complicated, which excludes the possibility of finding some
regularitytherein.However,itisnottrue,sinceatalargenumberofparticles,newpeculiarrelations
arise, which by no means can be reduced to purely mechanical phenomena. Here qualitatively new
regularitiesarisehavingcharacteristicofstatisticalmechanicsonly.Theirmaindistinctivefeatureis
the use of probabilistic notions based on the understanding of ambiguity, randomness, and uncer-
tainty of mass events. Before the works of Boltzmann, physicists believed that probability and
uncertainty lead to a total randomness and incognizability, i.e. to chaos. In fact, it has turned out
that a strict scientific study of uncertainty of events is possible. It had started long before
Boltzmann’s works with the analysis of roulette wheel results carried out by Bernoulli and Pascal.
This analysis required calculation of probabilities, which had not been considered serious
mathematics for a long time.
The main idea of probability determination is usually illustrated by the fact that it is totally
undeterminablewhetherweobtainheadsortailswhentossingacoin.However,ifwetossitverymany
times, overall occurrence of heads or tails approaches 50%. A similar result is observed at simulta-
neoustossingofalargenumberofidenticalcoins.Thistransitionfromacompleteindefinitenesstoan
almostexactdefinitenessincaseofalongsequenceofeventsorlargesystemsisthemainideainthe
studyof randomness.
Principallawsofclassicalmechanicsdescribereversibleprocessesanddonotimplythepossibility
ofirreversibility.Time(t)appearsintheselawsintheseconddegree.Therefore,substitutionof(cid:1)tfor
x Introduction
+tgivesirreversibility.Thisallowsustoestimatefutureinarespectiveproblemwiththesamedegree
ofcertainty asthe past.
Theirreversibilityrevealedinthermodynamicswassubstantiatedbystatisticalmechanics.Thefirst
stepinirreversibilityformalizationwasmadewhendeterminingentropy.Thereexistsarulecalledthe
second law of thermodynamics. It reads that in any physical process entropy remains constant or
increases,andifitincreases,theprocessisirreversible.Allthisseemedrathermysteriousandnotvery
satisfactoryatthattime.Whatisthesenseofentropy?Whydoesitalwaysincreaseandnotdecrease?
Boltzmann tried to solve these problems on the basis of ‘atomistic’ hypothesis and obtained
amazing results. They were as important for physics of the twentieth century as the discovery of
relativitytheoryandquantummechanics.During morethanhundredrecentyears,hisconceptshave
demonstratedtheirscientificpowerandarenowappliedtovarioussituationsfarbeyondthelimitsof
physical problems they were meant for. To the first approximation, it is very popular to explain the
notion of entropy by the following reasoning. We imagine gas atoms contained under normal
conditions in a closed vessel with the volume of, say, one cubic meter. Atoms in this vessel are in
acertainconfigurationwithrespecttoeachotherandvesselwalls,buttheypermanentlymove,andthe
system configuration permanently changes. It means that a system consisting of a multitude of
particlescanhaveagreatnumberofconfigurations.Foranexternalobserveralltheseconfigurations
lookthesamedasacubicmeterofgas.Foraninternalobserverwhoisabletodiscernseparateatoms,
thisvolumeofgasisratherambiguous.Boltzmanndefinedthemeasureofthisambiguityasentropy
parameter.
Ourinterestineverythingstatedaboveisduetoacertainphysicallysubstantiatedanalogyobserved
between the kinetic theory of gases and two-phase flows. These flows can be considered as a mass
process with a simultaneous participation of a large number of particles of various sizes. Although
these two phenomena are essentially different, the analogy between them (at least in particulars)
allowsclarifyingthegeneralpatternofatwo-phaseflow.Abroadrangeofvariousrandomfactors(the
flow turbulence, non-uniform pressure in the flow and solid phase concentration, shape, mass and
dynamicdragofparticles,theirinteractionwitheachotherandwithapparatuswalls)affectsthisflow.
Allthispredeterminesmarkedlyirreversiblecharacterofthetwo-phaseflow.Toclarifyatleastitsmost
generalregularities,itisnecessarytousemainideasofstatisticalmechanicsapplyingthem,naturally,
tospecific conditions ofthe problem under study.
Afundamentaldistinctionofthisapproachisthatitisbasedonthenecessitytodeterminethestate
oftheentiresystematonce,irrespectiveofitssize.Herethemethodsofmassprocessesanalysisare
understood as naturally statistical. Practice of mathematical modelling of technological processes
showsthattoconstructasuccessfulmodel,onedoesnothavetoaimattheirthoroughdescription.Itis
enough to find a simple scheme, if it reflects, to a sufficient degree, the essence of the modelled
phenomenon.
Atthepresentleveloftheknowledgeoftwo-phaseflows,substantiationofinvariants connecting
principal parameters characteristic of such flows may be considered a great success.
It is noteworthy that by now some invariants of separating flow regimes have been empirically
established. Their checking in numerous experiments has shown that they are valid for two-phase
flowswith developedturbulence. Determiningthemechanismofinvariantsformationfor suchflows
will make it possible to deepen our insight and, possibly, provide a key to finding invariants for
laminar and transient regimes. It is very important to understand how invariants of one-phase flows
are formed.
Introduction xi
This problem and modelling methods are examined in detail in this book. Since the statistical
definitionofentropywasformulatedin1872,ithasbeenconsiderednotonlyasasignofirreversibility,
butalso asa measureof disorder or uncertainty of the respectivesystems.
One can imagine that a complicated dynamic system with a large number of objects, whose
characteristics allow their interpretation through probabilities, should generate entropy. This idea is
widelyusedinmodernscienceinmanyfieldsfarfromthermodynamics,whosemainobjectsaremass
systems. As a rule, entropy plays the role of the main numerical invariant of transformations in the
theory of these systems. Therefore, it is necessary to know the ways of this parameter formation in
thermodynamics,to understandits properties and clarify somedetails.
CHAPTER
1
Modern Conceptions of
Thermodynamic Entropy
CHAPTER OUTLINE
1. Entropysubstantiation........................................................................................................................1
2. Entropy,ensembleofstates,probability...............................................................................................7
3. Problematicaspectsofentropy.........................................................................................................10
1. Entropy substantiation
AfunctionintroducedbyClausius,in1864,whichhenamedentropy,obtainedunexpectedresonance
in science. It appeared inthe course of his attempts to understand the laws of heat transfer and con-
versioninto other kinds of energyand hada rather simpleform:
DQ
H ¼ (1.1)
T
where DQis heat consumption andTis the absolutetemperature ofthe consumptionprocess.
ClausiusintroducedthisfunctiontocharacteriseimperfectionsoftheCarnotcycle.Whilethinking
abouttheprinciplesformingthebasisofsteamengineoperation,Carnotconcludedaboutabasiclimit
ofworkperformedbythisengineattheexpenseofacertainamountofheat.Hefoundadependenceon
temperatures causing the heatfluxand independence ofthe enginedesign oruniversalityandon the
wayofperformingwork.ThemainideadevelopedbyCarnotisthenotionofacycle;inthecourseof
its operation, heat engine passes through a cycle of states. An engine performing mechanical work
owingto aheat fluxreturns toits initial state each time.
Itwasnecessarytofindawayofexpressingthedifferencebetweenusefulenergyexchangesinthis
cycle and irreversibly lost dissipative energy. Examining entropy increment dH during a short time
interval dt, heseparated itinto two parts:
dH ¼deHþdiH:
ThevaluedeHisduetoanexternalcause–massexchangewiththeenvironment.Thispartofentropy
is reversible, which meansthat it can be both positiveandnegative.
In real heat engines, irreversible processes of heat loss and friction take place, side by sidewith
reversibleheatexchange.Theyleadtoentropy increase, andthispartofentropydiHisinternalwith
respecttothesystemandneverchangessign.Inthelongrun,itreflectsirreversiblechangesinsidethe
system, which corresponds to a spontaneous evolution of the system. This refers to such processes
1
EntropicInvariantsofTwo-PhaseFlows.http://dx.doi.org/10.1016/B978-0-12-801458-5.00001-0
Copyright©2015ElsevierInc.Allrightsreserved.
2 CHAPTER 1 Modern Conceptions of Thermodynamic Entropy
insidethesystemasheatconductivityordiffusion.Undertheinfluenceoftheseprocesses,thesystem
passesintoanequilibriumstatecorrespondingtothemaximalentropy.Itisacceptedthatentropyisthe
energylossrequiredforareversiblereturnofasystemintoitsinitialstateafteranadiabaticprocess
completed at the temperature corresponding to said initial state. This point of view is retained until
nowandisreflectedinmostcoursesofthermodynamics.Anotherpointofviewstatesthatentropydoes
nothave the dimensionofenergyandreflects otherfeaturesofa complicated system.
Themain featureofthissummarised functionisthatitchangesinonedirectiononlyandalways
grows:
dH ¼deHþdiH >0:
In contrast to other physical quantities, such as temperature or pressure, entropy cannot be directly
perceivedorreadilymeasured.Itisdeterminedbyamathematicalexpressionandlooksabstractand
evenartificial.Clausiusattributeduniversalfeaturestothisparameter.Heassertedthattheentropyof
the universe tends tothe maximum, which gave riseto the “heat death” theory.
Atfirst,thisparameterwasmetwithdistrust,and,untilnow,heateddebatesaboutthephysicsofthe
notion of entropy arise repeatedly. Such a subtle and peculiar notion attracted many scientists and
intellectuals, whose interests were sometimes far from thermodynamics, who tended to attribute a
physical and evenphilosophical (not always righteous) meaningto entropy.
Today,thenotionofentropyisreferrednotonlytothermodynamic,butalsotophysicochemicalor
biologicalprocesses,crystallography,cosmology,blackholestheory,aswellastoinformationtheory.
Clausiusformulatedthenotionofentropyusingmacroscopicparametersonly.Hisworkcompleted
thedevelopmentofthetheoryofthermodynamicsallowingtheestimationofanyprocessesconnected
to heat exchange. In addition, he contributed to the change in general ideas of the nature of heat by
demonstrating that heat does not differ, in principle, from mechanical energy. He also suggested the
idea thatheat is realised inthe formof kinetic energyof particlesconstitutinga body under study.
Afteratime,thisideawasdevelopedbyBoltzmannandallowedtherevelationofthemechanismof
thethermal laws ofoperationandpeculiarities ofentropyformation.Beforediscussing Boltzmann’s
ideas,itseemsexpedienttoconsider,inbrief,thetheoryofheattransferingaseoussystems.Fromthe
standpoint of the analysis of the revealed invariants, physics of these processes is very close to our
problems.
OneofthefirstquantitativelawsdescribingthebehaviourofgaseswasBoyle’slaw.Itreadsthatat
afixedtemperature T,gas volumeVis inverselyproportionalto the pressure,as follows:
fðTÞ
V ¼ :
p
Much later, Charles formulated the law of gas volume dependence on temperature at a constant
pressure, as follows:
V
¼4ðpÞ:
T
Avogadrosuggestedahypothesisthatmolarvolumesofallgasescontainequalnumbersofmolecules,
asfollows:
1
NA ¼6:022$1023 :
mol
1. Entropy substantiation 3
Hence, gas volume at T¼const and p¼const is proportionalto the number ofmoles n.
This relation for agas can bewrittenas follows:
pV
¼R (1.2)
T
whereRisanabsolutegasconstant.Thus,theabsolutegasconstantacquiresthesenseofaninvariant
for the three main parameters characterising the macrostate of a gaseous system. Not all the three
parameters in Eq. (1.2) can be independent.Only two of them can be specified arbitrarily, while the
magnitude of the third is connectedto the absolutegasconstant.
Gay-Lussac revealed that at the gas outflow into emptiness, the temperature remains unchanged
because,inthiscase,workisnotperformed,andgasenergyremainsunchanged.Itfollowsfromthe
factthatthetemperatureremainsunchanged,whilethevolumeandpressurevary,thattheenergyofan
idealgas depends ontemperature andis independentofthevolumeandpressure.
Hence,theenergyofanidealgasdependsonitstemperatureandthenumberofmoles,asfollows:
U ¼nUmðTÞ
where Umis self-energy per moleand n is the number ofmoles.
Theenergyofamixtureofkcomponentsofanidealgasequalsasumofthecomponents’energies,
as follows:
X
U ¼ nkUmðTÞ:
k
Speaking about an ideal gas, we imply a gas of point particles having weight, but ideally having
no size. Note that even in Boltzmann’s works, non-zero particle size was accepted for the ideal
gas, too.
Foramixtureofidealgases,Dalton’slawisvalid,whichreadsthatthepressuregeneratedbyeach
componentofthegasmixtureisindependentofothercomponentsandisdescribedbythesamelaw:
pkU ¼nkRT (1.3)
wherepkisthepartialpressureofthek-thcomponent,andnkisthenumberofmolesofthiscomponent.
It is another invariant, whose analogy with Dalton’s law is examinedlater.
Gay-Lussacexperimentallyprovedthatthethermalexpansioncoefficientofallgasesataconstant
pressure is approximatelythe sameand equals 1= per centigrade.
273
Van der Waals suggested an ideal gas law taking into account molecule sizes and interactions
between them.
According to the modern theory, mean kinetic energy per one particle is proportional to the
temperature:
1 3
mv2 ¼ kT (1.4)
2 2
wherevisthemeanmoleculevelocity,misthemoleculemass,andkistheBoltzmannconstant.The
gas law can bewrittenin asimpleform:
p¼nkT: (1.5)
