Dimensional Analasys and Scaling Laws Ppt

7/28/2019 Dimensional Analasys and Scaling Laws Ppt

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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


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)


Buckingham Pi

Theorem

Determination of Pi

Terms

Comments about



in Fluid Mechanics


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



in Fluid Mechanics


Scaling Laws.


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TheBuckinghamPitheoremstatesthat:

Givenarelation amongk variablesoftheform

g(u1,u2,u3,.....,uk)=0

Thekvariablesmaybegroupedintok-rindependent dimensionless

products,orterms,expressiblein functionalformby


Buckingham Pi

Theorem

Determination of Pi

Terms

Comments about



in Fluid Mechanics


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.


Buckingham Pi

Theorem

Determination of Pi

Terms

Comments about



in Fluid Mechanics


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


Buckingham Pi

Theorem

Determination of Pi

Terms

Comments about



in Fluid Mechanics


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.


Buckingham Pi

Theorem

Determination of Pi

Terms

Comments about



in Fluid Mechanics


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


in Fluid Mechanics and Heat

Transfer

Correlation of

Experimental Data.


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




Transfer

Correlation of

Experimental Data.


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



Transfer

Correlation of

Experimental Data.


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

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Transfer

Correlation of

Experimental Data.


Scaling Laws.


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1 = (2, 3)

Familiescurveofcurves

Todetermineasuitableempiricalequation

relatingthethreepiterms.

Toshowdatacorrelationsonsimplegraphs.




Transfer

Correlation of

Experimental Data.


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.




Transfer


Scaling Laws.


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SimilarityofModelandPrototypeWhatconditionsmustbemettoensurethesimilarityofmodeland prototype?

GeometricSimilarityModelandprototypehavesameshape.

Lineardimensionsonmodelandprototypecorrespondwithin constantscale

factor.

KinematicSimilarity

Velocitiesatcorrespondingpointsonmodelandprototypediffer onlybya

constantscalefactor.DynamicSimilarity

Forcesonmodelandprototypedifferonlybyaconstantscale factor.




Transfer


Scaling Laws. `


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Theratioofamodelvariabletothecorresponding prototypevariableiscalledthe

scaleforthatvariable.

LengthScale

VelocityScale

DensityScale

ViscosityScale

ModelScales




Transfer


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.




Transfer


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




Transfer


Scaling Laws.



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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 .




Transfer


Scaling Laws.



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Transfer


Scaling Laws.



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e s o a a ys s



Transfer


Scaling Laws.



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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



Transfer


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


Transfer


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



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Transfer


Scaling Laws.



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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!`



Transfer


Scaling Laws.


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Dimensional Analasys and Scaling Laws Ppt