dimensional analasys and scaling laws ppt
TRANSCRIPT
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Sharath Chandra. J (2012H148035H)
Sharabh Kochar (2012H148037H)
K. Venkatesh (2012H148038H)
M.E (Thermal Engineering)Department of Mechanical Engineering
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2
Dimensional Analysis Buckingham Pi Theorem.
Determination of Pi Terms.
Comments about Dimensional Analysis.
Common Dimensionless Groups in Fluid Mechanics Correlation of Experimental Data.
Modeling, Similitude
Similitude Based on Governing Differential Equation.
Scaling Laws
Contents
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A typical fluid mechanics problem in which experimentation is required, consider the
steady flow of an incompressible Newtonian fluid through a long, smooth- walled,
horizontal, circular pipe.
An important characteristic of this system, which would be interest to an engineer
designing a pipeline, is the pressure drop per unit length that develops along the pipe as
a result of friction.
The first step in the planning of an experiment to study this problem would be to
decide on the factors, or variables, that will have an effect on the parameter under
consideration.
For e.g. Let us consider Pressure drop per unit length
pl =f(D,,,V) PressuredropperunitlengthdependsonFOURvariables:
spheresize(D);speed(V);fluiddensity();fluidviscosity
Contd
DimensionalAnalysis Dimensional Analysis
Buckingham Pi
Theorem
Determination of Pi
Terms
Comments about
Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics
Modeling, Similitude
Scaling Laws.
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Toperformtheexperimentsinameaningfuland systematicmanner,itwouldbe
necessarytochangeoneof thevariable,suchasthevelocity,while holdingallother
constant,andmeasurethecorrespondingpressuredrop.
Difficultytodeterminethefunctionalrelationshipbetween thepressuredropand
thevariousfactsthatinfluenceit.
Fortunately,thereisamuchsimplerapproachtothe problemthatwilleliminatethe
difficultiesdescribed above. Collectingthesevariablesintotwonon-dimensional
combinationsofthevariables(calleddimensionless productordimensionlessgroups).
Onlyonedependentandone independentvariable Easytosetupexperimentsto determinedependency
Easytopresentresults(onegraph)
Dimensional Analysis
Buckingham Pi
Theorem
Determination of Pi
Terms
Comments about
Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics
Modeling, Similitude
Scaling Laws.
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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
I fanequationinvolvingkvariablesisdimensional ly homogeneous,i tcanbereduced
toarelationshipamong k-rindependentdimensionlessproducts,whereristhe
minimumnumberofreferencedimensionsrequiredto describethevariables.
The above theorem is call edBucki ngham Pi Theorem, and the terms are calledNon-
Dimensional Parametersafter being rearr anged as functions.
Givenaphysicalprobleminwhichthedependentvariable isafunctionof (k-1)
independentvariables.
u1=f(u2,u3,.....,uk)
Mathematically,wecanexpressthefunctionalrelationship inthe
equivalentform
g(u1,u2,u3,.....,uk)=0
Wheregisanunspecifiedfunction,differentfromf . Contd
BuckinghamPiTheorem Dimensional Analysis
Buckingham Pi
Theorem
Determination of Pi
Terms
Comments about
Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics
Modeling, Similitude
Scaling Laws.
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TheBuckinghamPitheoremstatesthat:
Givenarelation amongk variablesoftheform
g(u1,u2,u3,.....,uk)=0
Thekvariablesmaybegroupedintok-rindependent dimensionless
products,orterms,expressiblein functionalformby
Dimensional Analysis
Buckingham Pi
Theorem
Determination of Pi
Terms
Comments about
Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics
Modeling, Similitude
Scaling Laws.
Thenumber r isusually,butnotalways, equal to the minimum numberof
independent dimensions required to specify the dimensions of all the
parameters. Usually the reference dimensions required to describe the
variables willbethebasicdimensionsM,L,andTorF,L,andT.
The theoremdoesnotpredict the functional formof .The functional
relation among the independent dimensionless products must be
determined experimentally.
Thek-rdimensionlessproductstermsobtained
fromtheprocedureareindependent.
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Themethodwe most commonly use to determine the Pi terms iscalledthe METHOD of
repeatingvariables.Eightstepslistedbelowoutlinearecommended procedurefordeterminingtheterms.
Dimensional Analysis
Buckingham Pi
Theorem
Determination of Pi
Terms
Comments about
Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics
Modeling, Similitude
Scaling Laws.
Determinationof PiTerms
Step1Listallthevariables.
Step2Expresseachofthevariablesintermsof basicdimensions.Findthenumberof
reference dimensions.
Step3Determinetherequirednumberofpiterms.
Step4SelectaK repeatingvariables, whereK = number ofreferencedimensions.
Step5Formapitermbymultiplyingoneofthe non-repeatingvariablesbytheproduct
ofthe repeatingvariables,eachraisedtoanexponentthat willmakethecombination
dimensionless.
Step6RepeatStep5foreachoftheremaining nonrepeatingvariables.
Step7Checkalltheresultingpitermstomake suretheyaredimensionless.
Step8Expressthefinalformasarelationship amongthepiterms
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SelectionofVariables
Oneofthemostimportant,anddifficult,stepsinapplying dimensionalanalysistoanygiven
problemistheselection ofthevariablesthatareinvolved.
Thereisnosimpleprocedurewherebythevariablecanbe easilyidentified.Generally,one
mustrelyonagood understandingofthephenomenoninvolvedandthe governingphysical
laws.
Ifextraneousvariablesareincluded,thentoomanypi termsappearinthefinalsolution,and
itmaybedifficult, timeconsuming,andexpensivetoeliminatethese experimentally.Ifimportantvariablesareomitted,thenanincorrectresult willbeobtained;andagain,this
mayprovetobecostly anddifficulttoascertain.
Mostengineeringproblemsinvolvecertainsimplifying assumptionsthathaveaninfluence
onthevariablestobe considered.
Usuallywewishtokeeptheproblemsassimpleas possible,perhapsevenifsomeaccuracy
issacrificed.
Contd
Dimensional Analysis
Buckingham Pi
Theorem
Determination of Pi
Terms
Comments about
Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics
Modeling, Similitude
Scaling Laws.
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Asuitablebalancebetweensimplicityandaccuracyisan desirablegoal.
Variablescanbeclassifiedintothreegeneralgroup:
Geometry:lengthsandangles.
MaterialProperties:relatetheexternaleffectsandthe responses.
ExternalEffects:produce,ortendtoproduce,achange inthesystem.Suchas
force,pressure,velocity,or gravity.
Pointsshouldbeconsideredintheselectionofvariables:
Considerothervariablesthatmaynotfallintoonethe threecategories.
Forexample,timeandtimedependent variables.
Besuretoincludeallquantitiesthatmaybeheld constant(e.g.,g).
Makesurethatallvariablesareindependent.
Dimensional Analysis
Buckingham Pi
Theorem
Determination of Pi
Terms
Comments about
Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics
Modeling, Similitude
Scaling Laws.
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A list of variables that commonly arise in fluid mechanics and heat transfer
problems. Possibletoprovidea physicalinterpretationto thedimensionlessgroups
whichcanbehelpfulin assessingtheirinfluence inaparticularapplication.
CommonDimensionlessGroups Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics and Heat
Transfer
Correlation of
Experimental Data.
Modeling, Similitude
Scaling Laws.
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Dimensional analysis only provides the dimensionless groups describing the
phenomenon,andnotthespecificrelationship betweenthegroups. Todeterminethis
relationship,suitableexperimentaldatamustbe obtained. Thedegreeofdifficulty
dependsonthenumberofpiterms.
OnePiTerm
ThefunctionalrelationshipforonePiterm.
1=C ; whereCisaconstant. The value of the constant is to
be determined by an experimental procedure
Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics and Heat
Transfer
Correlation of
Experimental Data.
Modeling, Similitude
Scaling Laws.
CorrelationofExperimentalData
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Assume that thedrag,D,actingona sphericalparticle that falls very slowly througha
viscousfluidisafunctionoftheparticlediameter, d,theparticlevelocity,V,andtheflui
viscosity,.Determine, withtheaidthedimensionalanalysis,howthedragdependson
the particlevelocity.
FlowwithOnlyOnePiTerm Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics and Heat
Transfer
Correlation of
Experimental Data.
Modeling, Similitude
Scaling Laws.
Thedrag
D=f(d,V,)
d=L =FL-2T
V=LT-1
D=F
=ML-3
D V
D=CVd=CD
Vd
1=
Foragivenparticleandfluids,thedragvaries
directlywiththevelocity
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ProblemswithTwoPiTerms
1 = (2)thefunctionalrelationship amongthevariablescanthe bedeterminedbyvarying2and
measuringthe correspondingvalueof1.
Theempiricalequation relating2and1byusing astandardcurve-fitting technique.
Anempiricalrelationshipis validovertherangeof2.
Dangerousto extrapolate
beyondvalidrange
65
Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics and Heat
Transfer
Correlation of
Experimental Data.
Modeling, Similitude
Scaling Laws.
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1 = (2, 3)
Familiescurveofcurves
Todetermineasuitableempiricalequation
relatingthethreepiterms.
Toshowdatacorrelationsonsimplegraphs.
Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics and Heat
Transfer
Correlation of
Experimental Data.
Modeling, Similitude
Scaling Laws.
ProblemswithThreePiTerms
M d l P
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Modelvs.PrototypeWhat is a Model?
Amodelisarepresentationofaphysicalsystemthatmay beusedtopredictthebehavior
ofthesysteminsomedesiredrespect. Mathematicalorcomputermodelsmayalsoconform
tothis definition,ourinterestwillbeinphysicalmodel.
What is a Prototype?Thephysicalsystemforwhichthepredictionaretobe made.
Modelsthatresembletheprototypebutaregenerallyofadifferent size,mayinvolve
differentfluid,andoftenoperateunderdifferent conditions.
Usuallyamodelissmallerthantheprototype. Occasionally,iftheprototypeisverysmall,itmaybeadvantageous tohaveamodelthatislargerthantheprototypesothatitcanb
moreeasilystudied.Forexample,largemodelshavebeenusedto studythemotionofRBCs.
Withthesuccessfuldevelopmentofavalidmodel,itispossibleto predictthe
behavioroftheprototypeunderacertainsetof conditions.
Thereisaninherentdangerintheuseofmodelsinthatpredictions canbemade
thatareinerrorandtheerrornotdetecteduntilthe prototypeisfoundnottoperformas
predicted.
Itisimperativethatthemodelbeproperlydesignedandtestedand thatthe
resultsbeinterpretedcorrectly.
Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics and Heat
Transfer
Modeling, Similitude
Scaling Laws.
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SimilarityofModelandPrototypeWhatconditionsmustbemettoensurethesimilarityofmodeland prototype?
GeometricSimilarityModelandprototypehavesameshape.
Lineardimensionsonmodelandprototypecorrespondwithin constantscale
factor.
KinematicSimilarity
Velocitiesatcorrespondingpointsonmodelandprototypediffer onlybya
constantscalefactor.DynamicSimilarity
Forcesonmodelandprototypedifferonlybyaconstantscale factor.
Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics and Heat
Transfer
Modeling, Similitude
Scaling Laws. `
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Theratioofamodelvariabletothecorresponding prototypevariableiscalledthe
scaleforthatvariable.
LengthScale
VelocityScale
DensityScale
ViscosityScale
ModelScales
Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics and Heat
Transfer
Modeling, Similitude
Scaling Laws.
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Di i l A l i
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Consider a problem from the field of conduction heat transfer.
Plate plunged at t=0 into a highly conducting fluid such that at surface
Suppose that we are interested in estimating the time needed by the thermal front to
penetrate the plate, the time until the center of the plate feels the heating imposed
on the outer surface.
Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics and Heat
Transfer
Modeling, Similitude
Scaling Laws.
Dimensional Anal sis
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Estimate the order of magnitude of each of the term
appearing on LHS
On RHS
Equating the two orders of magnitude
Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics and Heat
Transfer
Modeling, Similitude
Scaling Laws.
Dimensional Analysis
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Rule 1 . Always define the spatial extent of the region in which you
perform the scale analysis. Size of the region of interest is D/2
Rule 2. one equation constitutes an equivalence between the scales of two
dominant terms appearing in the equation .
Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics and Heat
Transfer
Modeling, Similitude
Scaling Laws.
Dimensional Analysis
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Dimensional Analysis
Common Dimensionless Groups
in Fluid Mechanics and Heat
Transfer
Modeling, Similitude
Scaling Laws.
Dimensional Analysis
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e s o a a ys s
Common Dimensionless Groups
in Fluid Mechanics and Heat
Transfer
Modeling, Similitude
Scaling Laws.
Dimensional Analysis
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Nature offers us clear sign that the phenomenon of transition is
associated with a fundamental property of fluid flow.
Laminar flow is characterized by a critical number that serveas a landmark for laminarturbulent transition.
SCALING LAWS OF TRANSITIONy
Common Dimensionless Groups
in Fluid Mechanics and Heat
Transfer
Modeling, Similitude
Scaling Laws.
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Ob i Dimensional Analysis
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1. Cigarette smoke plume is in one plane.
2. The meander is most visible from the special viewing direction that happens
to be perpendicular to the plane of meander.
3. This observation is important because it contradicts the belief that the
transitional shape of the buoyant jet is spiral. Batchelor and Gill postulated the
existence of helical, not plane sinusoidal disturbances.
Observations Common Dimensionless Groups
in Fluid Mechanics and Heat
Transfer
Modeling, Similitude
Scaling Laws.
The flow appears to have the natural property to meander with a characteristic
wave length during transition, regardless of the nature of the disturbing agent.
This observation is important because it it illustrates the conflict between
hydrodynamic stability thinking, to which the postulate of disturbances is a
necessity
Dimensional Analysis
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Common Dimensionless Groups
in Fluid Mechanics and Heat
Transfer
Modeling, Similitude
Scaling Laws.
Dimensional Analysis
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"L'ignoranza accecante ci inganna.
O! Miseri mortali, aprite gli occhi!
its a Leonardo da Vinci quote but i am curious as to what it would
be in correct Italian
Again, the quote is "Blinding ignorance does mislead us. O!
Wretched mortals, open your eyes!`
Common Dimensionless Groups
in Fluid Mechanics and Heat
Transfer
Modeling, Similitude
Scaling Laws.
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