ed042p611
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C. E. Gall and R. R. Hudgins I Teaching Dimensionless Groups in . University of Waterloo Waterloo, Ontario, Canada I Chemical Engineering The traditional approach to the deriva- tion of dimensionless groups involves two steps- "guessing" appropriate variables describing the system, and employing the Buckingham Pi Theorem1 to collect them into groups. By contrast, the approach dis- cussed in this article is from a more rigorous engineering science viewpoint. A paradigm is presented to show visually the st,ructure of chemical engineering dimen- sionless groups.l To the student approaching this subject for the first time, the traditional approach tends to obscure a number of features of the structure of dimensionless groups. In the first place, the central importance of convective fluid motion may not he obvious. In addi- tion, it is possible to overlook the fact that there are a number of dimensionless groups that relate the same transport processes as a result of diierent mechanisms. Such a case is exemplified by the Reynolds, Grashof, Galileo, and Taylor numbers (defined in the table), which all relate convective and molecular momentum transport under the influence of various predominant mechanisms. As a background to further discussion, it must first of all be assumed that the mechanisms of transport are identical on the molecular level; i.e., molecular motion is the basis of transport of heat, mass, and momentum. Neglected will be any discussion of radiant heat transfer and electronic conduction of heat, as in liquid metals. A general diffusivity, E, may he defined in the follow- ing manner: Transfer rste of s quantity of concentration € Area across which transfer occurs E = Driving force expressed as the change in concentration € with distance in the direction of transfer As an exam~le. consider Fourier's law of heat transfer The molecular transport flux equations have counter- parts written for convective and total transport. In the case of total transport of mass, heat, and momen- tum, two forms of the flux equations may be written- either in terms of the point gradient of the transported quantity or in terms of the overall gradient existing in the system. In an analogous fashion, convective transfer can be considered as a product of a diiusivity and concentra- tion gradient. The diiusivity in this case is apparent from the following example for convected heat transfer. The heat transfer rate per unit area cross section by the convection of a fluid of velocity v, from point 1 to point 2, is by a simple heat balance equal to: " where D is the distance from point 1 to point 2. The diffusivity in this expression is recognized as Dv. The same diffusivity, Dv, is to be recognized as that for transport by convection of heat, mass, or momentum. In the case of heat transfer to the walls of a cylinder by fluid flowing with velocity v in the direction of the axis, the convective transfer is proportional to a radial velocity of v~ of the eddies within the fluid and a heat concentration difference & ' , (Tb - T,) between the hulk and the wall. The turbulent eddy velocity in the radial direction is proportional in some manner to the average axial velocity; therefore, the heat flux by the turbulent Vorious Forms of the Reynolds Number by conducti&' The diffusivity for heat transfer is d e Symbol v for identified Comments fined at constant density and heat capacity as: r.mlln no/" II$ on v k QIA - q__ Reynolds Nn. u Average velocity far chan- PC, -pCP(dTldZ) -pCn(dTldz) nel or pipe flow Reynolds Nna 1 - ) Average velocity of stream where a = thermal diffusivity, k = thermal conduc- inside a packed bed tivity, p = density, C, = heat capacity at constant ~ ~ l i l ~ ~ No. D2g~ - Velocity resulting from pressure, Q = rate of heat transport, A = area, T = gravitational forces in "i"rona li"llirl~ . .... -. ..-. ~-. temperature, q = heat flux, and z = axial coordinate. Grashof Nor DaB(AT)g~ Velocity resulting fmm This eauation is more commonly seen as buoyancy forces in vis- which is the molecular transport flux equation for heat. SILBEBBERG, I. H.. AND MCKEWA, J. J., JR., Petrol. Refiner, 32, No. 4, 179; No. 5, 147; No. 6, 101 (1953). A mare comprehensive summary of the structure of dimen- sionless groups has been made from a different point of view by ELINKENBERG, A,, AND MOOY, H. H., Chern. Eng. Progr., 44,17 (1948). Taylor Dean 1 Power cous liquid Velocity for flow in curved channels Velocity for power input to pmpeller with grhvita- taonal and drag forces present v = kinematic viscosity, r = orosity of bed, g~ = gravita- tional constant, 0 = thermal CoefRcient of expansion, L = length, P = power input to propeller, g, = conveeion factor, and n = rotational speed of propeller. Volume 42, Number 1 I, ~ovember 1965 / 61 1
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C.E.Gall andR.R.HudginsITeachingDimensionlessGroupsin . UniversityofWaterloo Waterloo, Ontario, CanadaIChemicalEngineering The traditionalapproachtothederiva- tionofdimensionlessgroupsinvolvestwosteps- "guessing"appropriate variables describing the system, and employing the Buckingham Pi Theorem1 to collect themintogroups.Bycontrast,theapproachdis- cussed in this article is from a more rigorous engineering science viewpoint.Aparadigmispresentedtoshow visuallythest,ructure ofchemicalengineeringdimen- sionless groups.l To the studentapproachingthis subject for the first time,thetraditionalapproachtendstoobscurea numberoffeatures ofthestructureofdimensionless groups.I n thefirstplace,thecentralimportanceof convective fluid motionmay not he obvious.In addi- tion, it ispossible to overlook the fact that there are a numberofdimensionless groupsthatrelatethesame transport processes as a result ofdiierent mechanisms. SuchacaseisexemplifiedbytheReynolds,Grashof, Galileo,andTaylornumbers(definedinthetable), whichallrelate convectiveandmolecularmomentum transportundertheinfluence ofvariouspredominant mechanisms. Asabackgroundto further discussion, it mustfirst ofall be assumed that the mechanisms oftransportare identical on the molecular level;i.e.,molecular motion isthe basis oftransport of heat, mass, and momentum. Neglected will be any discussion ofradiant heat transfer and electronic conduction ofheat, as in liquid metals. Ageneral diffusivity, E, may he defined in the follow- ing manner: Transfer rst e ofs quantityofconcentration Area across whichtransfer occurs E= Drivingforce expressedasthechangeinconcentration with distance in the direction of transfer Asan exam~le.consider Fourier'slaw ofheattransfer Themoleculartransportflux equationshavecounter- partswrittenforconvectiveandtotaltransport.I n thecaseoftotal transportofmass, heat, and momen- tum, two forms ofthe flux equations may be written- either in terms ofthe point gradient ofthe transported quantity or in terms ofthe overall gradientexisting in the system. I n an analogous fashion,convectivetransfercanbe consideredas a productofa diiusivity and concentra- tiongradient.The diiusivity in thiscase is apparent from the following example for convected heat transfer. The heat transferrate perunitarea cross section by theconvection ofa fluidofvelocityv, frompoint1 to point 2, is by a simple heat balanceequal to: " where D is the distance from point1 to point2.The diffusivity in this expression is recognizedas Dv. The samediffusivity,Dv, istoberecognizedasthatfor transport by convection ofheat, mass, or momentum. I n the case ofheat transfer to the walls of a cylinder byfluidflowing withvelocityvinthedirectionofthe axis, the convective transferisproportional to a radial velocityofv~ofthe eddies withinthe fluid and a heat concentrationdifference&' , (Tb - T,)betweenthe hulk and the wall. Theturbulenteddyvelocityintheradialdirection isproportionalinsomemannertotheaverageaxial velocity;therefore,theheatfluxbytheturbulent VoriousFormsoftheReynolds Number by conducti&'The diffusivity for heat transfer is d eSymbolv foridentifiedComments fined atconstantdensity and heat capacityas:r.mllnn o / " II$on v kQI A-q__ReynoldsNn.uAveragevelocityfarchan- PC,-pCP(dTldZ)-pCn(dTldz)nelor pipe flow ReynoldsNna1- )Averagevelocityof stream wherea= thermaldiffusivity,k= thermalconduc-inside a packedbed tivity,p=density,C, =heatcapacityat constant~ ~ l i l ~ ~ No.D 2 g ~ -Velocityresultingfrom pressure,Q= rate ofheattransport,A= area,T= gravitational forces in "i"ronal i " l l i r l ~. ....-. ..-.~-. temperature, q= heat flux,and z= axialcoordinate. GrashofNorDaB( AT) g~ Velocityresulting fmm This eauation is more commonly seen asbuoyancyforcesinvis- which is the molecular transport flux equation for heat. SILBEBBERG,I. H..ANDMCKEWA,J. J., JR., Petrol.Refiner, 32,No.4,179;No.5,147;No.6,101(1953). Amarecomprehensivesummaryofthe structureofdimen- sionlessgroupshas beenmade from a differentpoint ofview by ELINKENBERG,A,, ANDMOOY,H. H.,Chern. Eng. Progr.,44,17 (1948). TaylorDean1 Power cous liquid Velocityfor flowincurved channels Velocityfor power input to pmpellerwithgrhvita- taonalanddragforces present v= kinematic viscosity,r=orosityofbed,g~=gravita- tional constant, 0= thermal CoefRcient ofexpansion, L= length, P= powerinput t o propeller,g,= conveeionfactor, andn = rotational speed ofpropeller. Volume42,Number1 I, ~ovember1965/61 1 mechanisms varies as Thus, the ratio of heat flux by turbulent mechanism to heat flux by molecular mechanism is whereristheradialcoordinate.If fpCp(dT/dr)dris replaced by an average gradient, i.e. where Risthe radiusofthecylinder, wecanimagine thatitispossibletocharacterizethedimensionless ratio Dv/ m as the ratioofthe average heat flux bythe turbulenttransporttotheaveragefluxbymolecular transport.Thedimensionlessgroups,therefore,are seentoariseasratiosofthediffusivities ofvarious quantitiesunderdifferenttransportmechanisms,and can be regarded as ameasure ofthe ratioofmigration rates of quantities by their respective mechanisms. The figure shows, within the circles, the characteristic diffusivity relatedtothe flux ofthe transportedquan- titygivenbythecoordinatesofthediagram.The dimensionless groups are formed by taking ratios of the various pairs of diffusivities, as shown on the appropri- ate connecting lines.Thecentral importanceofcon- vectivemomentumtransportisshownbythelarge TRANSPORT Molecular Heat PCJ Momentum DV ConvectiveTotal "Point"Overall Gradient"Gradient" Mass C Thestructureofdimensionlessgroups.Solidlinesconnectgroupscommonly I E= concentration .a= eddy diffurivily ofheat Ea= total diffurivity forheat NN,,= Nusselt number h = heat transfercoefficient Np., = Pesletnumberforheattmnrfer,commonlycalled Groeh number Np,= Prondtl number Nsa= Schmidt number B = diffvrivity formolecular transport ofmoss 9)K = Knvdren diffurivity Nxn= Knvdsen number Npmm= Peclet number formoss tronrfor ~sed; thedashedlinerconnect less common gmupr Nsb= Shewood number em= eddy diffusivity ofmoss e r = eddy diffurivityofmomentum Em= total diffurivily for mas,k,= overall mar,transfercoef6sient N s ~ = Stanton number formass transfer f= Fanning friction factor Et= total diffwivily formomentum N q = Stanton number for heot transfer A= mean freepoth P = vis~osity G = average speed ofmolecules d = capillarydiameter 612/ JournalofChemicalEducation numberofdimensionlessgroupswhicharederived from it.Whythis occurs is also implied inthe figure. The eddy diffusivity ofheatand mass transport are de- fined only by analogy with the corresponding molecular diffusivities.Since they are dependent on fluid motion for their existence, any meaningful dimensionless group relatingmolecular andconvective heatormasstrans- portwillhavetoberelatedtotheconvectivemo- mentum transport. Theanalogous relationshipof heatandmasstrans- port variables,and ofthe dimensionless groups relating them, maybeseen from the symmetry of the diagram inthefigureaboutthemomentumaxis.Fromthis diagram,itisalsoclearhowsuchquantitiesasthe modifiedPecletandStanton numbersrelatefluxes of thesametransportedquantitiesbasedondifferent gradients. For everysituation in whichconvective momentum transportoccurs,the diffusioncoefficient Dv willhave to beidentified specifically.D will be acharacteristic lengthofthesystem,suchasalengthofpipe,vessel diameter,particlediameter,etc.Likewisevwill re- quireidentification.FortheReynoldsnumber,this velocityisidentifiedforanumberofspecialcases, giveninthetable.Thus,itcanbeseenfromthe figurethateachofthefundamentaldimensionless groupscovers anumberofother dimensionless groups arising out of specific situations. Characterizing the various transportedquantities by meansofdiffusivities inthemannershownherehas certainadvantages ofpresentation.Fromthe overall symmetryof thefigure,astudentshouldbeableto appreciatequicklythe analogybetweenvarioustypes ofheat,mass,andmomentumtransport.He should also be able to grasp the central importance of the con- vectivemomentumtransportinformulatingdimen- sionless groups.Fi al l y,he should he able to see from this scheme thesimilarityof manygroupsofdifferent names whichrelateidenticaltransportedquantitiesin different physical situations. Volume42,Number1 1 , November1965/ 613
