4 Buckingham Pi theorem

Buckingham Pi is a procedure for determining dimensionless groups from the variables in the problem. 4.1 The Rules. Let us assume that there are n = 3 ... [next][prev][prev-tail][tail][up] 4BuckinghamPitheorem Assuggestedinthelastsection,iftherearemorethan4variablesintheproblem,andonly3dimensional quantities(M,L,T),thenwecannotfindauniquerelationbetweenthevariables.Thebestwecanhopeforis tofinddimensionlessgroupsofvariables,usuallyjustreferredtoasdimensionlessgroups,onwhichthe problemdepends.Infact,thisisquiteagoodapproach,forreasonsthatwillbediscussedinthenext section. BuckinghamPiisaprocedurefordeterminingdimensionlessgroupsfromthevariablesinthe problem. 4.1TheRules Letusassumethattherearen=3dimensionalquantitiestoconsider-mass,lengthandtime.Theproblem involvesm=6variables,denotedA...F.Ingeneralwecanderivem-ndimensionlessgroups,oftendenoted 1,2...,usingthefollowingprocedure WritedownthedimensionsforallvariablesA...F Selectnofthevariables-sayA,B,C.Thesearecalledtherepeatingvariables,andwillappearinallthe terms.Notethattherearecertainrestrictionsonourchoice : Noneoftherepeatingvariablescanbedimensionless Notworepeatingvariablescanhavethesameoveralldimension.Forinstance,D,thepipe diameter,andr,theroughnessheight,bothhavedimensionofL,andsocannotbothbeused asrepeatingvariables. Selectoneothervariable-sayD.SomecombinationofA,B,C,Disdimensionless,andformsthefirst termordimensionlessgroup.Wecanfindthecombinationbydimensionalanalysis,bywritingthe groupintheform Equatingcoefficientsgives3equationsfor4unknowns,sowecanexpressallthecoefficientsintermsof justone. Repeatthisprocedurewiththerepeatingvariablesandthenextvariable,souseA,B,C,E.Continue untilnovariablesareleft. Havingworkedoutallthedimensionlessgroups,therelationshipbetweenthevariablescanbeexpressed asarelationshipbetweenthevariousgroups.Typicallywecanwritethisasonegroup(forexample1as afunctionoftheothers 4.2Workedexample Thisisprobablybestillustratedbyaworkedexample.Theheadlossinahorizontalpipeinturbulent flowisrelatedtothepressuredropp,andisameasureoftheresistancetoflowinthepipe.It dependsonthediameterofthepipeD,theviscosityanddensity,thelengthofthepipel,the velocityoftheflowvandthesurfaceroughness.Westartbylistingthedimensionsofthese parameters D[L] v[LT-1] [ML-3] p[ML-1T-2] [ML-1T-1] l[L] [] WewillchooseD,vandasrepeatingvariables.Ourfirstdimensionlessgroupinvolvesp,inthe form Thisgivestherelations 0=c+d 0=-b-2d 0=a+b-3c-d equatingcoefficientsofM,TandLrespectively.Writingthemallintermsofd, c=-d b=-2d 0=a-2d+3d-da=0 Resubstituting Wecanrepeattheprocesswithtogettheseconddimensionlessgroup : Equatingcoefficients 0=c+d 0=-b-d 0=a+b-3c-d Solvingagainintermsofdgivesc=-d,b=-danda=-d,thus Infact,werecognisethisgroupastheReynoldsnumber,writtenupsidedown,so ThenextdimensionlessgroupwillinvolvelwithdimensionL.Howeveroneoftherepeatingvariablesisthe diameterD,andsotheratioofthetwoisalreadydimensionless.Sothenextdimensionlessgroup is Finally,isalreadydimensionlessanyway,beingtheratiooftheheightoftheroughnesstothepipediameter D.so Fromthisanalysiswehavesuccessfullydeterminedthattheturbulentflowinaroughenedpipedependson aheadlossparameter togetherwiththeReynoldsnumber,theratioofthepipelengthtodiameter,andthesurfaceroughness coefficient.Wecanwritethisrelationasarelationbetweenthegroups orinfull Thisisasfaraswecangousingdimensionalanalysis.Experimenthowevershowsthatthepressuredrop dependslinearlyonthelengthofthepipe,sowecanmakethisrelationexplicit : Wecanrearangethisalittlefurther,rewritingitas whichisofcoursetheDarcyequationforheadlossinapipe. 4.3Helpfulhints Thereareanumberofhelpfulshortcutsthatcansimplifymatters : Ifaquantityisdimensionless,itisatermalready Ifanytwovariableshavethesamedimensions,theirratiowillbeaterm Inthefinalexpression,thedimensionlessgroupscanappearinanyfunctionalform.Inparticular, theexpressionmaydependon-1aseasillyason,soanydimensionlessgroupcanbeinverted ifthatismoreconvenient. Anytermmaybeexpressedintermsoftheothers. Alloftheseshortcutswereusedinthepreviousexample. [next][prev][prev-tail][front][up]