Introductory Transport Phenomena

786 Pages·2014·8.101 MB·English

by  R. Byron Bird, Warren E. Stewart, Edwin N. Lightfoot, Daniel J. Klingenberg

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SUMMARY OF FLUX EXPRESSIONS
CONVECTIVEFLUXEXPRESSIONS
Convectivemomentumfluxtensor:
0(c) =ivv or 0(c) =ivv
ij i j
Convectiveenergyfluxvector:
( ) ( )
q(c) = iÛ + 1iv2 v or q(c) = iÛ + 1iv2 v
2 i 2 i
Convectivemassandmolarfluxvectors:
j(c) =ia v or j(c) =ia v
A A Ai A i
J*(c) =cx v* or J*(c) =cx v*
A A Ai A i
MOLECULARFLUXEXPRESSIONS
Molecularmomentumfluxtensor(i=constant,Newtonianfluid):
0=pt−4(∇v+(∇v)†) or 0 =pt −4((𝜕v∕𝜕x)+(𝜕v∕𝜕x))
ij ij j i i j
Conductiveheatfluxvector+workfluxvector(purefluidsonly;seeEqs.19.3-3and
24.1-10):
∑3
q+w=−k∇T+[0⋅v] or q +w =−k(𝜕T∕𝜕x)+ 0 v
i i i ij j
j=1
Diffusivemassandmolarfluxvectors(ordinarydiffusiononly;seeEq.24.1-9):
j =−i𝒟 ∇a or j =−i𝒟 (𝜕a ∕𝜕x)
A AB A Ai AB A i
J* =−c𝒟 ∇x or J* =−c𝒟 (𝜕x ∕𝜕x)
A AB A Ai AB A i
TOTAL(CONVECTIVE+MOLECULAR)FLUXEXPRESSIONS
Totalmomentumfluxtensor:
d=ivv+0 or d=ivv+pt+f
Totalenergyfluxvector(purefluidsonly):
( ) ( )
e= iÛ + 1iv2 v+q+w or e= iĤ + 1iv2 v+q+[f⋅v]
2 2
Totalmassandmolarfluxvectors(ordinarydiffusiononly):
n =ia v+j or n =a (n +n )+j
A A A A A A B A
N =cx v*+J* or N =x (N +N )+J*
A A A A A A B A
Allfluxesdefinedabovehavethesamesignconvention;theyarepositivewhenthe
quantitybeingtransportedismovingfromthenegativesideofasurfacetotheposi-
tiveside.Derivatives𝜕∕𝜕x appearingincomponentformsofequationsareforCarte-
i
siancomponentsonly.
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EQUATIONSOFCHANGEINTERMSOFTHETOTALFLUXES
Theseequationsarevalidonlyforsystemsinwhichgravityisthesoleexternalforce.
Moreinformationcanbefoundin§19.2.
Momentum:
𝜕
iv=−[∇⋅d]+ig
𝜕t
Energy:
( )
𝜕 iÛ + 1iv2 =−(∇⋅e)+i(v⋅g)
𝜕t 2
Mass:
𝜕
ia =−(∇⋅n )+r
𝜕t A A A
EQUATIONSOFCHANGE(RESTRICTEDFORMS)
Momentum(forNewtonianfluidswithconstantiand4):
( )
Dv 𝜕v
i ≡i +[v⋅∇v] =−∇p+4∇2v+ig
Dt 𝜕t
Energy(forNewtonianfluidswithconstantiandk):
( )
iĈ DT ≡iĈ 𝜕T +(v⋅∇T) =k∇2T+4C
p Dt p 𝜕t v
Mass(forbinarymixturesofAandBwithconstanti𝒟 andnocrosseffects):
AB
( )
Da 𝜕a ( )
i A ≡i A + v⋅∇a =i𝒟 ∇2a +r
Dt 𝜕t A AB A A
DIMENSIONLESSGROUPS
(l andv areacharacteristiclengthandvelocity,respectively)
0 0
Re=l v i∕4 Pr=Ĉ 4∕k Sc=4∕i𝒟
0 0 p AB
Ra=GrPr Gr=gvl32T∕l2 Gr =grl32a ∕l2
0 a 0 A
Nu=hl ∕k Pé=RePr Pé =ReSc
0 AB
Sh=k l ∕𝒟 j =Nu∕RePr1∕3 j =Sh∕ReSc1∕3
c 0 AB H D
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Introductory
Transport
Phenomena
R. Byron Bird
Warren E. Stewart
Edwin N. Lightfoot
Daniel J. Klingenberg
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Preface
Transportphenomenaisnowgenerallyrecognizedasakeyscientificsubjectsupporting
studyof several branchesof engineering,agriculture,meteorology, medicine,and envi-
ronmental studies. Historically, the subject had its origins in Europe, and some of the
keyoriginalpapersandbookswerewritteninGermanandFrench,aswellasinEnglish.
IntheyearsfollowingWorldWarII,interestintheteachingoftransportphenomenain
U.S.universitiesledtotherapiddevelopmentofcoursesandtextbooksinthisarea,first
forbackgroundinengineeringcurricula,withlatertextbooksinvariousspecializedareas
withinengineeringaswellasoutside.
So what exactly is meant by the term “transport phenomena”? Very simply put, it
includes the transport of momentum (or “fluid mechanics”), the transport of energy (or
“heat transfer”), and the transport of chemical species (or “mass transfer”). Of particular
importancearethenthetransportcoefficients—theviscosity(describingthetransportof
momentum),thethermalconductivity(describingthetransportofenergy),andthediffusiv-
ity(describingthetransportofchemicalspecies).Luckilythereare“conservationlaws”
thatapplytothethreeentitiesbeingtransported,andwewillpayconsiderableattention
totheseconservationlawsthroughoutthistextbook.
Understandably,thebookdividesitselfquitenaturallyintothreeparts,oneforeach
oftheentitiesbeingtransported.Muchemphasiswillbeplacedonthesimilarities—and
differences—betweenthedevelopmentofthethreetypesofconservationequationsand
theirassociatedtransportcoefficients.Eachofthethreeareasjustdescribedmaybestud-
ied at one of three different levels: the molecular level (where we try to understand the
transportcoefficientsintermsofthemolecularinteractions);themicroscopiclevel(where
weregardthematerialsascontinua);andthemacroscopiclevel(whereweexaminelarge
systems—suchaspiecesofequipmentorbiologicalorgans).Itisimportant,wefeel,for
thosefirstmakingacquaintancewiththesubjectoftransportphenomenatounderstand
theconnectionbetweenthethreelevelsofthesubject,aswellasthatbetweenthethree
quantitiesthataretransported.Sincetherearethesevariousconnections,wecanlayout
thegeneralplanofthistextbookasinthetableshowninChapter0—thesubjectmaterial
justorganizesitself!Despitethisapparentlyobviousself-organization,thereareineach
ofthethreemainpartsofthebooksometopicsthatdonotfitintothistidyscheme.Each
ofthethreeparts,therefore,hasan“eighthchapter,”inwhichtheseoddballtopics—but
veryimportanttopics—arediscussed.
Duetothearrangementofthetopicstobediscussed,itispossibleforteachingpur-
posestoconsidertwowaysofproceeding.Inthefirstway,wecanteachthematerial“by
columns”—that is, by teaching the chapters in the order 1, 2, 3, and so on. Clearly this
isthebestcourseofactionforinstructionofundergraduates.Asforgraduatestudents,
teaching“byrows”—thatis,byteachingthechaptersinthesequence1,9,17,2,10,18,
andsoon—maybepreferable.
Thechaptersareprovidedwithasummarysectionaswellasquestionsfordiscus-
sion, and these may be helpful in suggesting ways to organize “problems sessions” or
“quizsections.”Eachchapteralsohas,attheend,anumberofproblems,whichhavebeen
groupedintovariousclasses:ClassA,illustratingdirectnumericalapplicationsofmate-
rialinthetext;ClassB,involvingelementaryanalysisofphysicalsituations;andClassC,
requiringmorematureanalysisormaterialfromseveralchapters.
Dependingonthenumberofcredithoursallottedtothissubject,itmaynotbepossi-
bleforallthematerialinthebooktobecovered.Therefore,inthechapteroutlinesatthe
beginningsofthechapters,asanaidtoinstructors,wedesignateoptionalsectionsby(o).
iii
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iv Preface
Omissionofthetopicssodesignated,willstillleaveenoughmaterialforawell-balanced
introductorycourse.
Thesubjectoftransportphenomenahaslongbeenregardedasarathermathematical
subject.However,itshouldbetaughtasatopicinappliedphysics.Emphasisshouldbe
placedonthevisualizationofphysicalsystems,thephysicalinterpretationoftheresultsof
problemsolving,and,ofcourse,thechoiceofthephysicallawsneeded.Mathematicswill
inevitablyhavetobeusedinordertosolvetheproblems,butthemathematicsissecondary
inimportancetothephysics.Everyeffortismadeinthisintroductorytexttoexplainhow
themathematicsisused,andtoincludesufficientintermediatestepsinderivationssothat
the text will be useful for self-study. Much emphasis should be placed on checking the
mathematicsateachstepinadevelopmentandlearningtoaskwhethertheequationsare
givingresultsinagreementwithphysicalintuition.
OfimportancetoteacherswillbethedifferencebetweenBSL(TransportPhenomena,
by Bird, Stewart, and Lightfoot) and the present textbook, which we will designate by
BSLK.Readersfamiliarwiththeformerbookwillrecognizethattheoverallorganization
ofthecontentsissimilar,andthatthereisconsiderableoverlapinthecontentsofthetwo
books.Youwillfind,however,thatconsiderablemorespacehasbeendevotedtofilling
inmissingstepsinmathematicalderivationsandtofullerexplanationsofmathematical
developments,includinganenlargementoftheappendixdevotedtomathematicaltopics;
inaddition,muchmaterialhasbeenremovedthatwefeltwasbeyondthelevelofmathe-
maticalpreparationofmostundergraduates.Ourintentionwastomakethisbookreflect
thetopicscoveredinourundergraduatecourse,withtheexceptionofafewadvancedtop-
icsincludedforthebrighteststudents.Finally,weremovedthechaptersdealingwithtwo
independentvariablesandreplacedthemwithchaptersondimensionalanalysis,includ-
ingthemuch-usedBuckinghampitheorem.
Wewishtothankthepeoplewhohaveinfluencedthefinalformofthistextbook,and
atthetopofthelistwemustputOlafAndreasHougen,whoguidedourdepartmentand
emphasizedthescientificbackgroundofchemicalengineering.Also,hefeltverystrongly
thatnotopicshouldbetaughttoengineeringstudentsunlessthereareclearindications
thatitcanbeusedinapplications.Rarelydidaweekgobythatwedidnotgetquestions
fromstudentsorprofessorsatotherinstitutionsaboutunclearportionsofthetextinthe
predecessorsofthisvolume.Respondingtothesequerieshasbeenaveryhelpfulexercise
forthecoauthorsinplanningfutureprintingsofthebookandforbetterunderstanding
thepedagogicalproblemsthatthereadershavehadtocopewith.Everyoneshouldfeel
freetocontactusinconnectionwiththepresentvolume,andweencourageyoutopoint
outerrorsandinconsistenciesinthetext.Nextwewouldliketoacknowledgeourown
studentsandcolleaguesfromwhomwehavelearnedsomuch;conversationswithPro-
fessorsEdwinN.Lightfoot,MichaelD.Graham,EricV.Shusta,A.JeffreyGiacomin,and
RossE.Swaneyhavebeenparticularlyhelpful.Eachandeveryoneofthemhasinsome
waycontributedtoourknowledgeofthesubjectandthesolvingofpedagogicalproblems
encountered.SpecialthanksgotoProfessorCarlosRamirezoftheUniversityofPuerto
Ricoforthetimeandtroublethathetookinsupplyinguswithcopiouscorrigendaforthe
secondeditionofBSL,andtoDr.M.I.HillofColumbiaUniversityforofferingcritiques
ofseveralchaptersforBSLK.
Transportphenomenaisanevolvingsubject.Eachmonth,newapplicationsappear
inthetechnicaljournals,andnewtechniquesforproblemsolvingarepresented.Weare
acutelyawarethatwecannotpossiblycoverallthenewestoftheoreticalandexperimental
developmentsinabeginningtextbook,butwehopethatwehaveprovidedaspringboard
fromwhichthereaderscanlaunchthemselvesintothenewareas.
Andnow,wepointoutthatthechiefcoauthorsofthisBSLKbookareRBBandDJK,
whohavetriedtocarrythetorchforward;DJKhastaughttheundergraduateandgraduate
transportphenomenacoursesforabouttwentyyears,andhisunderstandingofstudents’
challengeshasbeenveryhelpful.RegretfullyENLwasnotabletoparticipatefullyinthis
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Preface v
rewriteoftheTransportPhenomenatextbook,althoughhiscontributionstotheearlierBSL
editions—particularlysomeofthemorechallengingexamplesandproblems—havebeen
retained.TheuntimelypassingofWEShasbeenkeenlyfeltbyour“team,”andtohimwe
alsoowegenerousthanksforhisorganizationalskills,hispassionforaccuracy,andhis
wonderfulsenseofhumor.
RBB
ENL
WES
DJK
Madison,Wisconsin
Summer2013
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