Deconvolutingboundaryconditions and rheologicalresponsein paste flow

M. J. ADAMS1,B. J. BRISCOE2,M.KAMYAB2and S. S. PANESAR3 1 Unilever Research,PortSunlight Laboratory, QuarryRoadEast,Bebington, Merseyside L633JW,UK2 Department ofChemicalEngineering, Imperial College of Science TechnologyandMedicine,Prince Consort Road,London SW7 2BY,UK3 BP ResearchCentre,Sunbury,MiddlesexTW167N1,UKReceived for APT 14January 1991; accepted17September1991Abstract-Thecomprehensivenumerical simulation of paste processing operations requires aquantita-tiveknowledgeof both the material's evolving rheological response expressed in terms of a constitutive relationshipand the boundaryconditionpertainingat the systemwalls. Plasticines are soft solid materials comprisingconcentratedparticulate suspensions which exhibit visco-elastic-plasticbehaviour with either sliporno-slipboundaryconditions. The paperdescribes an investigationof the viabilityof the uniaxialcompressionofcylindrical specimens forobtainingthegoverning equations associated with the bulk and wall behaviour. 1. INTRODUCTION Theprocessingof concentrated particulate suspensions is common practicein a widerangeof industries. Foods,detergentsandceramicsarematerialswhicharefrequently processed by forming, extrusion and roll milling;thistypeofprocessinghas much in common with hot metal working.The numerical simulation of suchprocessesisconsiderablymore tractable with the advent of readilyavailablecommercial software packagesbased on finite element and finite difference methods.However,inorder to implementaunitrather than asystems analysis it is necessarytoprescribethe constitutive equationfor the feed material and the wall boundarycondition.

Concentratedsuspensions may behave as powerlaw fluids or as visco-elastic-plasticsolids. Here we are concerned with suspensionsthat are representativeof the lattertypeof behaviour. In these cases,elastic contributions can generallybeneglectedforthehighstrain/low rate deformations that are generallyentailed in practice. Here, the term 'visco' refers to plasticsolids that have a rate dependent yield stress.

Theboundaryconditions describe the wayinwhich the material interacts with the equipmentwalls,anddependboth on the material compositionandthecontactconditions,i.e. the wall temperature,the extent of strain and the localdisplacementvelocity.The two limitingcases are wall slipandcompletestick. Wall sliphasbeen studied in some detailformoderatelyconcentratedparticulate suspensions (e.g.Yilmazer and Kalyon [1]). It is thoughttoarisefromparticles migrating away from the wall producinga thin liquid layer, comprising the continuous phases,adjacentto the wall. This thin lubricatinglayerallows the bulk of the suspensiontoslideagainstthe wall so that shearing,in tube flow for example,ispredominantlyconfinedto the layer.

248It is difficult to envisagethat such aprocessoccurs in very highly concentrated

particulatesuspensionswhere the disperse phase volumeapproachesthemaximumpackingvalue.However,it is possiblethatliquiddoes become entrained betweenparticle-wallcontacts. The details of such a boundarycondition are unknownalthough by analogywithhydrodynamiclubrication it mightbeexpectedthat the surface traction increases with the prevailinghydrostaticpressure.Forveryfineparticulate phases andhighlyviscous continuous phases,thetractionalstressesmaybecomesufficiently high toarrest wall slipandthe stick boundaryconditionwilloccur.

What has been said above concerning pastes isequally applicable to metals without the complicationof fluid migration. Metals, like concentrated dispersions,oftenhaveparticulateand continuous phasesand also showelastic-plasticbehaviour.One of the more commontechniquesforcharacterizingthe mechanical propertiesofmetals is the compressionofcylindricalspecimens,aprocesswhich is known as 'upsetting'.Thisconfigurationisoftenpreferredtotensiletestingbecausein the latter,athighstrains,neckingoccurs and strain localization is associated withacomplextriaxial stress state in the neck.Inaddition,compressionactstosuppressfracture of more brittlespecimens. Cylinder upsettinghas been the subjectofexhaustiveexperimentalandanalyticalstudies in themetalworkingfield because of the inherent complexitiesthatmayoccur. One of the majorproblemsis that the specimenmaystick to theplatensor the friction at the platens may behigh.Thisresults in barrellingso that the side of the cylinderdevelopsa convexgeometry.Moresubtly,deformation then occurs byaprocessof'folding'where the side of thecylinderfolds onto the platensurfaces. In order to overcome these problems,metalspecimensarecarefullylubricated to ensure a low platenfriction and hence a homogeneousdeformation where the originalshapeof the cylinderispreserved.The aim of the current work was to investigatethepotentialofcylinderupsettingforcharacterizingconcentratedparticulatesuspensions.Thetechniquealso offers theopportunityforinvestigatingthe wall boundaryconditions. In thispaper,theanalysiswillbe confined to homogeneousdeformation which can be modelled usingsimple equilibrium stress evaluation (seeSection2).Plasticine was used as a model suspensionfor this purpose.2.CYLINDERUPSETTINGThe mean apparent compressive stress in upsetting may beattributed to both the bulkdeformation and the end effects which dependon whether the specimenslipsorsticks at the platens.Inthe case of slip,frictional work is dissipatedat the plateninterface. When the friction is high,or the no-slip boundary conditionprevails,cone-shapedinternal zones of nominallyundeformed material form at theplatensandinternal frictional losses result at these interfaces. The interfaceconstraintimposed by these end effects can thus resultininhomogeneousdeformation. In general,thetotal work done in such deformation is greaterthan for the homo- geneous case; thedifference is known as the redundant work.

Cookand Larke [2]developedanexperimentalprocedurewhichattemptedtoeliminate the end effects. They compressed cylindrical specimensof the same diameter but differentheightsgivingdifferent initial aspectratios,dolhp,wheredoandhoare the initial diameter and heightof the specimen.Atagiven compressive

249strain, they extrapolated the mean compressive pressures to a zero aspectratioinorder to obtain a 'true' deformation stress,Œo,which,inprinciple,should not be influencedbythe end effects. In this way, they were able to determine a 'true' stress-straincurve.However,currentpracticefor metals usuallyinvolvesusingpolytetrafluorethylene(PTFE)sheets to reduce platenfriction since the extra- polationprocedureis considered unreliable for unlubricated specimens,particularlyathighstrains[3]. Clearly under these circumstances it is not possibletoinvestigatethe influence of theactualboundaryconditionsoperatingbetween the material and the wall.

vanRooyenandBackofen[4]analysedupsettingusinganequilibriumstressevaluation and the results of this analysiswerefullysubstantiatedbythem for metal specimens.Theanalysisassumed that the material deforms homogeneouslyandobeystheyieldcriterion for rigid-plasticsolids,thus

whereŒzandŒ,are the axial and radial principalstresses and Y is the uniaxial yieldstress.Conventionally,inmetalworking,twotypesof frictional boundarycondi-tions are appliedwhich we willincorporateinto a generalfrictioncoefficientIf/.Thefirsttypeinvolves the Tresca friction factor,run(0< m < 1)which is given by

whereT is the wall tractional stress at the platenand k is the shear yieldstress. It is usual to assume that T is a constant. The second possibilityinvolves a Coulombicboundarycondition where

where x (0<f.1<0.57)is the coefficient of friction and p is the wallpressure.In this case both rand p wouldgenerallybe functions of the radial coordinate r which has itsoriginat the axis of the disc. The mean normal pressure pm is thengiven by

where s is the natural uniaxial compressivestrain. The upperlimits on '1/ correspond tono-slipboundaryconditions under whichonly inhomogeneous deformation is possible.Benbow[5]studied the flow ofaqueousbasedaluminasuspensions through acircular die and found a boundaryconditionapplicableto the natural lubricatinglayer.Itmaybewritten in the followingformforupsetting

where v, is the radial wall (platen) velocity andToisthe flow initiation value of the wall tractional stresses. Thevelocitycoefficienta' isgivenbythe ratio t7lx,whereq andx are the viscosityand thickness or effective thickness of the fluid layergeneratedat the interface respectively.Benbow[5]assumed that the wall traction T wasindependentofpressure.Itmaybe shown that for thisboundarycondition,

250the mean normal pressureforcylinder upsetting isgiven by

wherey/=tolk, 13' /(k.J3) and v is the uniaxial compressive velocity. Itcan be seen that each of these boundaryconditions result in a relationshipbetween the mean pressureand a function of the yieldstress which contains the initialaspectratio. The form of this function is such that at a zero aspectratio,themeanpressureisequalto the yieldstress. This is the basis of the Cook and Larke proceduredescribed earlier. Actuallymost materials are not true rigid-plasticandoftendisplaystrainhardening.In such cases the yieldstress can be treated as a flowstress.

3. RESULTS AND DISCUSSION Figure1showsplotsofp. as a function of initial aspectratio at a number of natural strains for Plasticine cylindrical specimens thatwere either unlubricated or lubricated with a silicone oil. The specimenswerecompressed using anInstrontestingmachine(model 6022) at a constant strain rate of 0.1 s-' and a constant ambienttemperatureof 21.5°C. Linear extrapolationto a zero aspectratio was possibleintheregionof lower aspectratios;the deviations from linearitywereconsiderablymorepronouncedfor the unlubricated specimens. Figure 2 shows the true stress-strain curves obtained from each of these data sets usingthe Cook and Larkeprocedure [2] and the results are reasonablyconsistent.These curves correspondto a rigid-plasticbehaviour with some strain hardening.The method is notsufficientlysensitive at low strains to determine unequivocallyifthere is a significantelastic deformation region.

Consequently,we can conclude that the extrapolation procedure forobtainingthetruestress-strain curve is reasonablyreliableprovidedthatonly relatively smallaspectratiospecimensareutilized;itshould be noted that there is a lower practicallimit when double barrellingis observed. The value of the procedureisfurthersupported by thegood agreement with the measured tensile stress-strain curvewhich is also shown inFig.2;the maximum inthis case correspondsto the onset ofnecking.Thelinearityof the plotsof the mean pressureas a function of aspect ratio, inthe lower aspectratiorange,wouldsuggestthat the wall boundaryconditionsare frictional in origin.Itis thus possibletocomputethe friction coefficients from theslopesof the linear regionsinFig.1by employing Eq (4). The variation of y/with natural strain is shown in Fig.3. As mightbeexpected,thecoefficientsare small for the lubricatedspecimens.For the unlubricated specimens,the friction coefficient increases upto a strain of 40% and then decreases with strain. The maximum in the friction coefficient correspondsto the pointatwhichbarrellingandfoldingwere observed to become significant. Consequently, athighstrains,thevaluesof y/ canonlybe regardedas extrapolation parameters due to the inhomogeneousdeformationofthespecimens;the actual boundaryconditions are no-slip.

251

Figure1. The mean compressivestress as a function of initial aspectratio for (a)lubricated and (b)unlubricatedPlasticinecylinders.Each data set correspondsto different natural compressivestrainswhich are denoted on the lines fitted to the data.

On the basis of the current data it is not possibletodistinguishthemechanisticoriginof the slip boundary condition. It is clear, however, from the linearityof the meanpressuredataupto limited values of initial aspectratios that the second order term inEq. (6), which introduces the velocity dependence, isrelativelysmall.Thenon-linearityof these data at greater aspect ratios is mostcertainlydue to the complexnature of the deformation and not an intrinsic characteristic of the boundarycondition.

4.CONCLUSIONSUnlikePlasticine,which was used as a model material in the current work,mostconcentratedsuspensionscannot be convenientlytestedintension to obtain their constitutiverelationship.Inaddition,it is difficult to measure the wall boundary

252

Figure2.(A)and(B):the stress-strain curves derived from Fig.1bya linear extrapolationto a zero initialaspectratioobtained at a strain rate of 0.1 s-'. (C):The tensile stress-strain curve for Plasticine obtained at a strain rate of 0.05 s-'. ¡.

conditions of pastes directly as with manymaterials such as metals and polymers.Uniaxialcompressionsofcylindricalspecimensoffers a useful procedurefor charac- terisingthe bulk and interfacial wall behaviour of pastes.Thetechniqueis not without difficulties because of the complexitiesof the deformation process arising from the interactions at the platens.However,it does appearthatusefuldata can be obtainedby employing specimens withrelativelylow initial aspectratios. The results show that Plasticine is a plasticsolidwhich exhibits a limited degreeof strain hardening.For unlubricated specimens, slip boundaryconditionsoperateat natural strainlessthanabout 0.5. The exact nature of these wall boundaryconditions can notbe elucidated usingtheanalysisdescribedin the present study. Lubricating

CompressiveNatural Strain Figure3. The friction coefficients derived from the linear regionsof the data shown inFig.1.

253theplatenswith a silicone fluid results in a substantial reduction in the wall friction. This leads to nearhomogeneousdeformation as judgedbytheabsence of significantbarrellingwhich was observed under the unlubricated conditions. REFERENCES1. U. Yilmazer and D. M. Kalyon, Slip effects in capillaryandparalleldiscs: torsional flows of highlyfilledsuspensions.J.Rheol., 33, 1197-1212, 1989.2.M. Cook and E. C. Larke,Resistance of copperandcopper alloys tohomogeneousdeformation in

compression.J. Inst. Metals, 71, 371, 1945.3. A. B. Watts and H. Ford,Anexperimental investigation of the yieldingofstripbetween smooth dies.Proc. Inst. Mech. Engrs.,B1, 448-453, 1952.4. G. T. van RooyenandW.A.Backofen,Astudyof the interface friction in plastic compression, Int. J. Mech. Sci., 1, 1-27, 1960.5.J. J. Benbow,Thedependenceofoutputrate on die shape during catalystextrusion. Chem. Engng.Sci.,26, 1467-1473, 1971.