FUNDAMENTALSFUNDAMENTALS OFOF FLUIDFLUID MECHANICSMECHANICS ChapterChapter 77 DimensionalDimensional AnalysisAnalysis Modeling,Modeling, andand SimilitudeSimilitude JyhJyh--CherngCherng ShiehShieh Department of Bio-Industrial Mechatronics Engineering National Taiwan University 12/24/2007 1 MAINMAIN TOPICSTOPICS DimensionalDimensional AnalysisAnalysis BuckinghamBuckingham PiPi TheoremTheorem DeterminationDetermination ofof PiPi TermsTerms CommentsComments aboutabout DimensionalDimensional AnalysisAnalysis CommonCommon DimensionlessDimensionless GroupsGroups inin FluidFluid MechanicsMechanics CorrelationCorrelation ofof ExperimentalExperimental DataData ModelingModeling andand SimilitudeSimilitude TypicalTypical ModelModel StudiesStudies SimilitudeSimilitude BasedBased onon GoverningGoverning DifferentialDifferential EquationEquation 2 DimensionalDimensional AnalysisAnalysis 1/41/4 AA typicaltypical fluidfluid mechanicsmechanics problemproblem inin whichwhich experimentationexperimentation isis required,required, considerconsider thethe steadysteady flowflow ofof anan incompressibleincompressible NewtonianNewtonian fluidfluid throughthrough aa long,long, smoothsmooth-- walled,walled, horizontal,horizontal, circularcircular pipe.pipe. AnAn importantimportant characteristiccharacteristic ofof thisthis system,system, whichwhich wouldwould bebe interestinterest toto anan engineerengineer designingdesigning aa pipeline,pipeline, isis thethe pressurepressure dropdrop perper unitunit lengthlength thatthat developsdevelops alongalong thethe pipepipe asas aa resultresult ofof friction.friction. 3 DimensionalDimensional AnalysisAnalysis 2/42/4 TheThe firstfirst stepstep inin thethe planningplanning ofof anan experimentexperiment toto studystudy thisthis problemproblem wouldwould bebe toto decidedecide onon thethe factors,factors, oror variables,variables, thatthat willwill havehave anan effecteffect onon thethe pressurepressure drop.drop. PressurePressure dropdrop perper unitunit lengthlength ∆p = f (D,ρ,µ, V) l 9 Pressure drop per unit length depends on FOUR variables: sphere size (D); speed (V); fluid density (ρ); fluid viscosity (m) 4 DimensionalDimensional AnalysisAnalysis 3/43/4 ToTo performperform thethe experimentsexperiments inin aa meaningfulmeaningful andand systematicsystematic manner,manner, itit wouldwould bebe necessarynecessary toto changechange onon ofof thethe variable,variable, suchsuch asas thethe velocity,velocity, whichwhich holdingholding allall otherother constant,constant, andand measuremeasure thethe correspondingcorresponding pressurepressure drop.drop. DifficultyDifficulty toto determinedetermine thethe functionalfunctional relationshiprelationship betweenbetween thethe pressurepressure dropdrop andand thethe variousvarious factsfacts thatthat influenceinfluence it.it. 5 SeriesSeries ofof TestsTests 6 DimensionalDimensional AnalysisAnalysis 4/44/4 Fortunately,Fortunately, therethere isis aa muchmuch simplersimpler approachapproach toto thethe problemproblem thatthat willwill eliminateeliminate thethe difficultiesdifficulties describeddescribed above.above. DDCollectingCollecting thesethese variablesvariables intointo twotwo nondimensionalnondimensional combinationscombinations ofof thethe variablesvariables (called(called dimensionlessdimensionless productproduct oror dimensionlessdimensionless groups)groups) 9 Only one dependent and one D∆p ⎛ ρVD ⎞ independent variable l = φ⎜ ⎟ 2 ⎜ µ ⎟ 9 Easy to set up experiments to ρV ⎝ ⎠ determine dependency 9 Easy to present results (one graph) 7 PlotPlot ofof PressurePressure DropDrop DataData UsingUsing …… 3 D∆p L(F/ L ) 0 0 0 l = =& F L T ρV2 (FL−4T2 )(FT−1)2 −4 2 −1 ρVD (FL T )(LT )(L) 0 0 0 =& = F L T µ (FL−2T) dimensionless product or dimensionless groups 8 BuckinghamBuckingham PiPi TheoremTheorem 1/51/5 A fundamental question we must answer is how many dimensionless products are required to replace the original list of variables ? The answer to this question is supplied by the basic theorem of dimensional analysis that states If an equation involving k variables is dimensionally homogeneous, it can be reduced to a relationship among k-r independent dimensionless products, where r is the minimum number of reference dimensions required to describe the variables. Buckingham Pi Theorem Pi terms 9 BuckinghamBuckingham PiPi TheoremTheorem 2/52/5 GivenGiven aa physicalphysical problemproblem inin whichwhich thethe dependentdependent variablevariable isis aa functionfunction ofof kk--11 independentindependent variables.variables. u1 = f (u 2 , u 3 ,....., u k ) DDMathematically,Mathematically, wewe cancan expressexpress thethe functionalfunctional relationshiprelationship inin thethe equivalentequivalent formform g(u1, u 2 , u 3 ,....., u k ) = 0 Where g is an unspecified function, different from f. 10 BuckinghamBuckingham PiPi TheoremTheorem 3/53/5 TheThe BuckinghamBuckingham PiPi theoremtheorem stastatestes that:that: GivenGiven aa relationrelation amongamong kk variablesvariables ofof thethe formform g(u1, u 2 , u 3 ,....., u k ) = 0 TheThe kk variablesvariables maymay bebe groupedgrouped intointo kk--rr independentindependent dimensionlessdimensionless products,products, oror ΠΠ terms,terms, expressibleexpressible inin functionalfunctional formform byby Π1 = φ(Π 2 , Π 3 ,,,, Π k−r ) rr ???? ΠΠ???? or φ(Π1, Π 2 , Π 3 ,,,, Π k−r ) = 0 11 BuckinghamBuckingham PiPi TheoremTheorem 4/54/5 TheThe numbernumber rr isis usually,usually, butbut notnot always,always, equalequal toto thethe minimumminimum numbernumber ofof independindependentent dimensionsdimensions requiredrequired toto specifyspecify thethe dimensionsdimensions ofof allall thethe parameters.parameters. UsuallyUsually thethe referencereference dimensionsdimensions requiredrequired toto describedescribe thethe variablesvariables willwill bebe thethe basicbasic dimensionsdimensions M,M, L,L, andand TT oror F,F, L,L, andand T.T. TheThe theoremtheorem doesdoes notnot predicpredictt thethe functionalfunctional formform ofof φφ oror ϕϕ .. TheThe functionalfunctional relationrelation amongamong thethe independentindependent dimensionlessdimensionless productsproducts ΠΠ mustmust bebe determineddetermined experimentally.experimentally. TheThe kk--rr dimensionlessdimensionless productsproducts ΠΠ termsterms obtainedobtained fromfrom thethe procedureprocedure areare independent.independent. 12 BuckinghamBuckingham PiPi TheoremTheorem 5/55/5 AA ΠΠ termterm isis notnot independentindependent ifif itit cancan bebe obtainedobtained fromfrom aa productproduct oror quotientquotient ofof thethe ototherher dimensionlessdimensionless productsproducts ofof thethe problem.problem. ForFor example,example, ifif 2Π Π 3 / 4 Π = 1 or Π = 1 5 6 2 Π 2 Π 3 Π 3 thenthen neitherneither ΠΠ5 nornor ΠΠ6 isis independentindependent ofof thethe otherother dimensionlessdimensionless productsproducts oror dimensionlessdimensionless groupsgroups.. 13 DeterminationDetermination ofof PiPi TermsTerms 1/121/12 SeveralSeveral methodsmethods cancan bebe usedused toto formform thethe dimensionlessdimensionless products,products, oror pipi term,term, thatthat arisearise inin aa dimensionaldimensional analysis.analysis. TheThe methodmethod wewe willwill describedescribe inin detaildetail isis calledcalled thethe METHODMETHOD ofof repeatingrepeating variables.variables. RegardlessRegardless ofof thethe methodmethod toto bebe usedused toto determinedetermine thethe dimensionlessdimensionless products,products, oneone beginsbegins byby listinglisting allall dimensionaldimensional variablesvariables thatthat areare knownknown (or(or believed)believed) toto affectaffect thethe givengiven flowflow phenomenon.phenomenon. EightEight stepssteps listedlisted belowbelow outlineoutline aa recommendedrecommended procedureprocedure forfor determiningdetermining thethe ΠΠ terms.terms. 14 DeterminationDetermination ofof PiPi TermsTerms 2/122/12 StepStep 11 ListList allall thethe variables.variables. 1 >>ListList allall thethe dimensionaldimensional variablesvariables involved.involved. >>KeepKeep thethe numbernumber ofof variablesvariables toto aa minimum,minimum, soso thatthat wewe cancan minimizeminimize thethe amountamount ofof laboratorylaboratory work.work. >>AllAll variablesvariables mustmust bebe independent.independent. ForFor example,example, ifif thethe crosscross--sectionalsectional areaarea ofof aa pipepipe isis anan importantimportant variable,variable, eithereither thethe areaarea oror thethe pipepipe diameterdiameter couldcould bebe used,used, butbut notnot both,both, sincesince theythey areare obviouslyobviously notnot independent.independent. γ=ρ×g, that is, γ,ρ, and g are not independent. 15 DeterminationDetermination ofof PiPi TermsTerms 3/123/12 StepStep 11 ListList allall thethe variables.variables. 2 >>LetLet kk bebe thethe numbernumber ofof variables.variables. >>Example:Example: ForFor pressurepressure dropdrop perper unitunit length,length, k=5.k=5. (All(All variablesvariables areare ∆∆pp ,, D,D,µµ,,ρρ,, andand VV )) l ∆p = f (D,ρ,µ,V) l 16 DeterminationDetermination ofof PiPi TermsTerms 4/124/12 StepStep 22 ExpressExpress eacheach ofof thethe variablesvariables inin termsterms ofof basicbasic dimensions.dimensions. FindFind thethe numbernumber ofof referencereference dimensions.dimensions. >>SelectSelect aa setset ofof fundamentalfundamental (primary)(primary) dimensions.dimensions. >>ForFor example:example: MLT,MLT, oror FLT.FLT. >>Example:Example: ForFor pressurepressure dropdrop perper unitunit lengthlength ,, wewe choosechoose FLT.FLT. ∆p