Expo Ceramicos

Ž .Powder Technology 106 1999 119–131www.elsevier.comrlocaterpowtec

Unsteady state planar divergent flow of extrusion pastes

A.S. Burbidge a,), J. Bridgwater b

a School of Chemical Engineering, UniÕersity of Birmingham, Edgbaston, Birmingham B15 2TT, UKb Department of Chemical Engineering, UniÕersity of Cambridge, Pembroke Street, Cambridge CB2 3RA, UK

Received 6 November 1998; accepted 29 March 1999

Abstract

The divergent downward flow of pastes between parallel plates is examined experimentally and several distinct phases are identified inthe development of the flow. The experimental design allows the distance between the plates to be adjusted. Initially, a bar of paste isformed but forces arising from the sliding contact cause a back pressure on the feed nozzle which leads to bulging of the paste close tothe entry port. This observation allows an alternative method of determining paste properties which potentially gives better estimates ofthe bulk yield stress than those found by a piston extrusion method. In subsequent phases, the flow is radial when close to the feed butbecomes more complex at greater distances. A radial flow model is derived and compared with experimentally determined values of theradial stress component at the inlet. In some cases, the agreement is good. However, the paste does not always flow radially and thesurface contact with the plates is not always good. The flow fields should prove valuable in the development of numerical models; thesealso show how defects can develop in products made by moulding. q 1999 Elsevier Science S.A. All rights reserved.

Keywords: Unsteady state; Divergent flow; Extrusion pastes

1. Introduction

Extrusion is a widely used process of considerablecommercial importance. The products fabricated in thisway range from polymers to ceramics, foods and metals.Despite the importance of extrusion as a process, little isknown about the flow of extrusion pastes. The advent ofinjection moulding for paste systems means that this lackof knowledge needs to be addressed.

A typical extrusion paste is a dense, polydisperse sus-pension of solid phase particles in a lubricating liquidphase. The solid phase may be, say, a ceramic particle of

Ž .small diameter typically -50 mm , although it is com-mon to improve the packing fraction by adding smallerparticles that can partially fill the voids between those oflarger size. The liquid phase is commonly an aqueoussuspension of a rheology modifier, the typical ones beingclays, carbohydrates and water soluble polymers such asmethyl-celluloses. It is important to choose a liquid phase

) Corresponding author. Tel.: q44-121-414-5295; fax: q44-121-414-5377; E-mail: [email protected]

carefully so as to produce a consistent homogeneous prod-uct and to avoid quality control problems.

In a paste flow, there is a complex interaction betweenthe liquid and solid phases. Liquid phase movement withina paste prevents the use of conventional oscillatory coneand plate rheometry, since any sample of paste will soongenerate lubricating layers at the shearing surfaces. Oven-

w xston and Benbow 1 suggested that any changes in cross-section could be approximated by using the pure plasticwork assumption common in metal deformation theory.They combined this with the assumption that in a pipe ofconstant cross-section, there is a plug flow with a thinlayer of lubrication at the wall. Hence, they present asemi-empirical model relating pressure drop to the geome-

w xtry of a ram extruder system. Benbow and Bridgwater 2present this model in detail and describe how it can beadapted for systems other than flow from a barrel into adie in a ram extruder. Although the flow of extrusionpastes can be described semi-empirically for axi-symmet-ric flows, the extension of models to non-symmetric sys-tems is problematical at the present time. The presentpurpose is to develop an understanding of paste injectionbetween parallel plates. Vertical downflow is studied, the

0032-5910r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved.Ž .PII: S0032-5910 99 00069-8

( )A.S. Burbidge, J. BridgwaterrPowder Technology 106 1999 119–131120

Fig. 1. Laboratory piston extruder.

purpose being to gain an understanding of the flow, toexplain it quantitatively and to discover the origin ofdefects in structure.

2. Paste preparation

Three pastes were used in this study: paste A wasŽ .composed of 25.5% wt. each of 3, 17, 37 mm median

Ž .diameter a-alumina particles, 3% wt. each of potatoŽ .starch and bentonite, and 17.5% wt. water. Paste B was a

Ž .boehmite gel composed of 50.5% wt. boehmite, 46.5%Ž . Ž .wt. water and 3% wt. HNO . Paste C was composed of3

Ž .27.3% wt. each of 3, 17, 37 mm median diameter a-Ž .alumina particles, 4% wt. of potato starch and 14.1%

Ž .wt. water. Alumina was obtained from Universal Abra-sives, Bentonite from Steetly Minerals. Potato starch was

Ž .supplied by Sigma and boehmite Plural SB from CondeaChemie. In preparation of A and C, the dry componentswere mixed together for 20 min in a Hobart mixer and

Fig. 2. Parallel plates apparatus.

Fig. 3. Elemental force balance for a small radial flow element.

then the water was added before a further 20 min mixing.The paste was then passed twice through the mincerattachment of the mixer to remove agglomerates. To makepaste B, the acid was diluted with the water, this added tothe boehmite powder and the whole lot mixed in a Z-blademixer for 20 min. All pastes were stored in a refrigeratorin PVC bags to prevent loss of liquid phase by evapora-tion. The paste was allowed to return to room temperaturebefore experiments were performed.

3. Experimental method

Experiments were conducted in a modified piston ex-Ž .truder system Fig. 1 constructed around a Dartec com-

puter-controlled 100 kN screw strain frame. The cross-headof the strain frame was connected to the plunger of apiston extruder constructed of stainless steel. A ribbednylon seal ensured that there was no leakage of the sampleat the top surface. The plunger was closely fitting inside a25-mm barrel section that screwed into an angled flangewhich connected self-centering die attachments. The entirepiston arrangement was supported by means of a four-legged stand constructed of mild steel, allowing access tothe outlet of the die attachment. For the experimentsdetailed here, a parallel plate apparatus was attached to the

Ž .piston extruder Fig. 2 . This consists of a pair of 100mm=100 mm=10 mm polycarbonate plates mountedsuch that the separation can be adjusted between 1 and

Fig. 4. Mohr’s circle of stress for small elements of paste at differentradii.

( )A.S. Burbidge, J. BridgwaterrPowder Technology 106 1999 119–131 121

20 mm. The width of the slit in the plane of the plates wasfixed at 20 mm. A Panasonic SVHS video camera could bepositioned normal to the plane of the plates was used toprovide a visual record of the extrusion flows.

The barrel of the extruder was filled with paste and theplunger positioned so that the cross-head of the strainframe was in direct contact with the upper surface of thepaste in the barrel. This was taken as the reference fordisplacement. The extrusion program was then executed,operation being completely automatic until the requiredextrusion had been completed. The cross-head was thenbacked away from the top of the paste, so that it was

possible to remove the extruder assembly from the strainframe.

The experiments were videoed and images processedusing a PC with a frame grabber card. Images could thusbe automatically acquired at a range of time intervals overthe duration of the experiment. The resulting sequences ofimages could then be viewed either individually or as amultimedia movie. Strain frame data was also availabledescribing displacement and extrusion pressure vs. time.

w xBenbow and Bridgwater 2 rheological parameters forthe extrusion pastes were measured in the standard way.This involved extrusions through square entry dies of

Fig. 5. Comparison of extrusions of pastes A, B and C between parallel plates of pitch 10 mm at creeping flow rates.


Fig. 6. Force balance on initial extruded finger of paste at point of yield.

varying lengths at a range of velocities. The parameters arerecovered by means of a set of least squares regressions.

4. Modelling

Consider an element of paste of angle f and thicknessŽ .w Fig. 3 . Define a coordinate frame by the unit vectors

e in the radial direction, e in the tangential direction and1 2

e normal to the plane of the plates. Further, take the sign3

convention that compressive stresses are positive and as-sume a flowfield such that there is uniform velocity in theradial direction of magnitude independent of the angularcoordinate.

Table 1Benbow–Bridgwater extrusion parameters for pastes A, B and C mea-sured by method of finger yielding between parallel plates

Ž . Ž . Ž .Paste Plate pitch s MPa t Mpa b MPa srm0 0Ž .mm

A 4 0.18 0.01 0.00A 10 0.30 0.02 0.03A 15 0.22 0.02 0.11B 4 0.01 0.00 0.00B 10 0.29 0.01 1.26B 15 0.31 0.08 0.00C 4 1.84 0.04 0.00C 10 1.56 0.08 0.00

Consider the geometry of an arc at radius r, the differ-ential annular surface area A normal to the 3 direction,3

i.e., in the plane of the paper is

f 2 2d A s rqd r yr (f rd r 1Ž . Ž .3 2

and the flow area normal to radial direction at radius r is

A sf rw 2Ž .1

Hence, assuming no velocity in the 3 direction, the superfi-cial radial velocity is given by

QÕ s 3Ž .1 wrf

since the material is incompressible. For a paste system

ŽFig. 7. Typical forcerdisplacement data from Dartec strain frame illustrating calculation of net piston force acting on extruded finger of paste paste A,.4-mm gap, creeping flow .


Table 2Benbow–Bridgwater extrusion parameters for pastes A, B and C mea-sured by piston extrusion method

Ž . Ž . Ž . Ž .Paste s MPa a MPa srm t MPa b MPa srm0 0

A 0.33 1.13 0.02 0.27B 0.43 0.11 0.05 0.70C 1.86 38.3 0.04 2.94

w xsliding on a surface, Benbow and Bridgwater 2 presentŽ .the following result z)0 :

t Õ0 it s qb Õ 4Ž .i i< <z

where t is the ith component of a lubricated surface sheari

stress vector t due to the slip velocity vector z of thepaste at a point. t is the stress required to initiate slipping0

and b is a velocity factor. This equation amounts toallowing a thin lubricating layer of Bingham fluid at thesurface of a bulk paste flow which is undeformed normalto the plane of slip. Since the thickness of this shear layeris unknown, it is more convenient to work with slipvelocities than shear rates.

The rate of work input at the inner surface normal to theradial direction is

w ss rsr0 wrÕ ss rsr0 Q 5Ž .r 11 1 11

where s rsr0 is the radial normal stress component mea-11

sured on the inner face of the radial flow region. The rate

of work done by surface shear stress on one pasterplatesurface is given by

aw s f rt Õ d r 6Ž .Hs 1 1

r0

the components in directions 2 and 3 being zero sinceÕ sÕ s0. This can be evaluated by substituting from 32 3

into 4 to give

a a 2Q bQw s t d rq d rH Hs 0 2w f rwr r0 0

Qt bQ2 a0w s ayr q ln 7Ž . Ž .s 0 2 ž /w rf w 0

The internal dissipation rate DU within the volume Vassociated with an arbitrary strain rate field ´

U is˙

˙U UD s s :´ dV 8Ž .˙HV

where s is the stress field satisfying the yield criterionover the volume V, which, for a two-dimensional systemalong arbitrary principle directions UU and VV, reduces to

˙UD s s ´ qs ´ dV 9Ž .˙ ˙Ž .H UU UU VV VVV

Ž .For an incompressible material tr ´ s0, i.e., ´ q˙ UUŽ .´ s0. Substituting into Eq. 9 yields˙ VV

˙UD s s ys ´ dV 10Ž . Ž .˙H UU VV UUV

For a perfectly plastic material obeying Tresca’s yield

ŽFig. 8. Typical divergent flow pattern showing two-dimensional nature of the flow pattern in the plane of the plate surfaces paste A, 10 mmrs, 10-mm.plate spacing .


Ž .Fig. 9. Creeping flow summary for paste A 1 mmrs superficial velocity at entry .

criterion s ys ss for which ´ is by definition˙UU VV 0 UU

the maximum principal strain rate further reduces to

˙U U< <D s s ´ dV 11Ž .˙H max0V

which is a well known plasticity result. The components ofthe strain rate tensor in cylindrical polar coordinates are

w xgiven in many standard texts, e.g., Bird et al. 4 . For ourassumed strain field where Õ s0 this results inu

EÕ Qr´ s sy 12Ž .˙11 2E r f r w

1 EÕ Õ Qu r´ s q s 13Ž .˙22 2r Eu r f r w

E Õ 1 EÕu r´ sr q s0 14Ž .˙12 ž /Er r r Eu

where ´ s´ since the strain rate tensor is symmetric.˙ ˙12 21

Hence, directions 1 and 2 are principal axes and thus ´11

and ´ are equal and opposite maximum strain rates. It˙22

should be noted that for this to be true, the shear rate in the1–3 directions needs to be vanishingly small. In the case

Ž .of the experiment, this is approximately satisfied Fig. 8 .Ž . Ž .Thus, from Eqs. 11 and 12

E Q QU< <´ s s 15Ž .˙ max 2ž /E r f rw f r w

so

QUD s s dV 16Ž .H 0 2f r wV

Ž 2 .Noting Vs f r wr2 . Changing the variable from V to rand integrating between outer radius a and inlet radius r0

yields

a s Q a0UD s d rss Qln 17Ž .H 0 ž /r rr 00


An energy balance on the system equating the total workŽ Ž .. Ž Ž ..input Eq. 5 with that output Eq. 6 or dissipated

Ž Ž ..internally Eq. 17 leads to

a 2t 2bQ a0rsr0s ss ln q ayr q lnŽ .11 0 0 2ž / ž /r w rf w0 0

18Ž .

In the case of an expanding disc of material, the value of ais then a function of time, and can therefore be substitutedwith a function of Q and elapsed time, t, by means of amass balance.

Ž .When the substitution fs2p is made, Eq. 18 isw xidentical to a result presented by Bates 3 for radial

outflow between a pair of discs. His approach was basedupon integration of a differential force balance.

Consider the stress state at various radii within a disc ofŽ . Ž .paste Fig. 4 . Case a represents the stress state at the

outer edge of the disc of paste. A necessary boundarycondition is that the stress normal to the surface is zeroand this fixes the point s at the origin of Mohr’s circle11

of stress. Assuming that the material is in plastic yield,

Tresca’s criterion fixes the circumferential stress s at a22

value ys . Now consider a small element at some smaller0

radius such that radial stress s is now compressive. It11

can be seen that the hoop stress s becomes less and less22Ž .tensile until finally at a neutral point b it is zero. At radii

less than this neutral point, both stress components areŽ .compressive c .

If the yield criterion is symmetric about the shear stressaxis, i.e., the behaviour of the paste is identical in tensionand compression, then the change from tensile to compres-sive hoop stresses is mechanistically unimportant. How-ever, if the paste has significantly less strength in tensionthan compression, its behaviour is more complex — amatter addressed below.

5. Results and discussion

Fig. 5 shows images captured at various stages duringtypical extrusions of pastes A, B and C. In each frame, thebar that appears along the top of the image is a 100-mmlong brass beam which is an integral part of the frame and

Ž .Fig. 10. Fast flow summary for paste A 10 mmrs superficial velocity at entry .


is external to the plates, and thus has no effect on theobserved flow field. The 10 mm=100 mm square dielandchannel extends to the level of the lowest part of thissupport and hence the view of the divergent region isuninterrupted. The bolts that appear in the lower right andleft corners of the frame hold the plates parallel and at therequired pitch. Each bolt ensures a tight contact betweenthe plates and a spacer which is sandwiched in-between.Rectangular spacers maintain the pitch of the plates at theinjection point at the top of each image and also form theedges of the dieland. The images shown here are for aplate spacing of 10 mm. The width of the injection pointwas thus held constant at 10 mm throughout, so in the caseof these images, the injection point was a square cross-sec-tion of 10 mm=10 mm. In these experiments, the superfi-cial paste velocity through this orifice was maintained at 1mmrs.

In each case, during the initial stages of the flow, thepaste forms a finger of the same cross-sectional area as the

Ž .injector Fig. 5a,b,c . This finger will eventually form the

‘stalk’ of a mushroom shaped extrudate. At this stage,there is no divergence of the flow pattern since the stressstate of the paste at the entry to the parallel plates is lessthan the bulk yield stress. As the outflow continues, thecontact area of the paste and plate becomes larger; there-fore, the stress at the entry due to the surface shear stressincreases. As the yield stress is approached, the fingerbegins to bulge until ultimately there is a change in theobserved flow pattern. In the case of pastes A and B, a

Ž .mushrooming occurs Fig. 5d,e , whereas for paste C theflow splits to one side and the other of the initial fingerwhich is now stationary. Following a mushroom analogyas before, this bulging region forms the ‘cup’ of themushroom. The flow patterns of pastes A and B approachthe expected radial flow pattern predicted by the theory,although cracks appear at the top of the ‘stalk’. As thepaste is undergoing a hoop stretching, there will be atensile hoop stress acting along the free surface of themushroom. It seems likely that the junction of this expand-ing outer surface and the finger originally formed act as

Ž .Fig. 11. Creeping flow summary for paste B 1 mmrs superficial velocity at entry .


stress concentrators causing preferential cracking. Paste CŽ .is behaving differently Fig. 5f and is not consistent with

a radial flow pattern. Rather a sequence of fingers is seen.After some time, the flow patterns of pastes A and B

settle down into a region of radial flow with a collection ofcracked peninsulae at the periphery. It is notable that thecracks that were apparent at small radii at short times haveeither healed or convected with the flow, since they are nolonger visible as the experiment progresses. This behaviouris consistent with the radial flow model since the hoopstress will remain at a maximum at the free outer surfaceand become more compressive at lower radii. Assumingthese are indeed caused by hoop tension, their positionswill therefore be determined by the distance from the outersurface. This would seem to be consistent with convection

Ž .of cracks. In the case of paste C Fig. 5i , there wouldseem also to be radial flow surrounded by a cracked outerregion, although in this case, the radial flow region is verysmall. It seems in this case that the tensile yield stress ofthe material is almost negligible when compared with thatin compression. If this were the case, then radial flowcould only occur at radii small enough that the hoop stress

Ž .was compressive Fig. 4 . This would account for thesmall radial flow region.

Since the initial stages of the flow lead to the extrusionof an undeformed finger of material, from statics, a force

Ž .balance can be attempted Fig. 6 . Assuming that there isgood contact between the plates and the surface of thepaste allows calculation of values for the surface shearstress t , the surface shear stress velocity factor b and the0

bulk yield stress s . The force on the extrusion piston is0Ž .measured as a function time Fig. 7 ; hence, the fraction of

the force acting on the finger of interest can be calculatedby subtraction of a reference pressure noted at the pointthe finger begins. In these experiments, the contribution ofthe self-weight of the paste is negligible being less than1% of the bulk yield stress in all cases, and is thereforeignored in the calculations. If, for the creeping flow exper-iments, it is assumed that the flow rate is small enough thatthe contribution of the b term can be neglected, then

Ž . Ž .s s Frwt and t s Fr2wL where L is the length0 0 1 1

at which the finger first starts to bulge. Consideration of afast flow experiment also at superficial velocity V leading2

to an initial bulging length of L allows calculation of b2

Ž .Fig. 12. Fast flow summary for paste B 10 mmrs superficial velocity at entry .


Ž . Ž .values since b' L t yL t r L V . Values of s , t1 0 2 0 2 2 0 0

and b calculated by this method are shown in Table 1. Insubsequent text, values calculated from these parallel plateexperiments will be referred to as s

U , tU and b

U so as to0 0

distinguish them from values calculated by the standardŽ w xpiston extrusion method Ovenston and Benbow 1 , Table

.2 .For paste A, the values of s

U are consistently lower0

than s values measured by the piston technique. Values0

of tU are in agreement with t for the 10 and 15 mm0 0

spacings although the value calculated from the 4 mmspacing is too low. All values of b

U are far too small, theclosest approach being for a 15-mm spacing giving a valueof 0.11 MPa srm compared with the 0.27 MPa srm fromthe capillary experiment.

For paste B, the value of sU is very low for 4-mm0

plates, although for 10- and 15-mm plates, the estimate isabout 25% lower than that from the standard method.Values of t

U are in fair agreement for 10 and 15 mm0

spacings although the value calculated from the 4-mmspacing is too low. Comments on values of b

U are notsignificant given the degree of scatter. It should be noted

that for paste B the length of the unyielded finger is veryshort leading to a larger error in these measurements thanthat found for pastes A and C.

In the case of paste C, the values of sU are in excellent0

agreement with those values measured by the standardmethod for both spacings tested. t

U estimates are also in0

fair agreement, although values of bU are once more

unreliable. Results are not included for the 15-mm platesince the apparatus was not sufficiently long to cause thefinger to yield before it left the system at the bottom edgeof the plates.

This parallel plate method would seem to provide alower value of the bulk yield stress than the extrusionmethod. This is not surprising, since the extrusion methodassumes that the plastic deformation due to the change offlow area is completely uniform and that the work done isequivalent to the perfect plastic work of the deformation.In fact, the deformation is almost certainly non-uniformleading to extra or ‘redundant’ work being required. Thisredundant work will cause the value of bulk yield stresscalculated by this method to be an overestimate. If there isno extrudate swelling, then the parallel plate method pro-

Ž .Fig. 13. Creeping flow summary for paste C 1 mmrs superficial velocity at entry .


vides a pure compressive stress at the top of the extrudedfinger and would therefore be expected to give a betterestimate of the bulk yield stress. Measurements of theshear stresses t

U and bU are dependent upon the reliabil-0

ity of the surface contact and therefore prone to error whenthere is a low back-pressure. It would thus be expected thatthese parameters measured from the capillary experimentwould provide a better estimate.

A feature of the flow is its extreme two-dimensionalityŽ .in the plane of the plate surfaces Fig. 8 . Note particularly

the extremely sharp square edge at each of the plate pasteinterfaces and the straightness of the free surface joiningthe plates. In the case of a typical Newtonian fluid, thiswould be a parabolic surface, but the surface observed isconsistent with a lubrication assumption as was used in thederivation of the radial flow model. Defects in the pasterun from one plate to the other. There is negligible dieswell in these experiments.

A crude test of the radial flow theory can be made bymeasuring the ‘pressure’ at the injection point as a func-tion of time. From a knowledge of the flow rate theapparent radius a can be calculated if it is assumed that the

material flows radially at all times. We know that notablyfor pastes A and B the paste does not flow radially at earlytimes, although as the experiment progresses the flowbecomes increasingly radial in nature and a then becomesa better characteristic length scale. The extrusion force onthe piston can be measured as previously detailed, allow-ing the calculation of a ‘pressure’ at the injection point ofthe plates. There is a degree of uncertainty in associatingthis pressure with the stress components predicted by themodel, since the edge of the injection slot does not alignconveniently with the cylindrical polar coordinate frameemployed in the model. That is, we are trying to relate themeasurement of an axial stress to a predicted radial stress.This issue can be addressed by carrying out a forcebalance on a control volume chosen to totally enclose thepiston and have a bottom boundary that coincident with a5-mm radius semicircle that just touches either side of theinjection port. Hence, this radial normal stress componentis lower than the axial normal stress calculated along thephysical edge of the inlet by a factor of pr2. Experimentalradial stresses were all calculated in this manner, and theserepresent the normal stresses at the 5-mm semicircle.

Ž .Fig. 14. Fast flow summary for paste C 10 mmrs superficial velocity at entry .


Figs. 9–14 and Fig. 15 show comparisons betweenexperimental radial stress components as determined bythis method and predictions from the radial flow theory.The rheological parameters used in the theory were those

Ždetermined by the standard piston extrusion method Table.2 . In all cases, the stress data has been normalised so that

when as0.005 m, the radial normal stresses are zero, butno other adjustments are made. This can be tricky ininterpreting some experiments, the results of this beingparticularly prevalent in the case of the larger plate gaps.For example, see Fig. 9 with a 15-mm gap where at latertimes the forms of the theoretical and experimental curvesare very similar, although the absolute values are displaceddue to anomalous behaviour at the start of the experiment.

For paste A, experimental results for creeping flowŽ .seem to show good agreement with theory Fig. 9 , al-

though the theory is an overestimate for larger gaps. Thiscan probably be attributed to the normalisation procedurediscussed in the previous paragraph and also to a reductionin the contact at the interface of the plates and paste. For

Ž .fast flow Fig. 10 , the 4-mm gap data again show thecorrect functional behaviour although the absolute value ofthe theory is too low. The theoretical gradients for thelarger gaps are overestimated. It is unclear why this effectis seen at higher superficial flow rates.

Ž .Experiments with paste B Figs. 11 and 12 show verypoor agreement with theory beyond radii of 0.02 m. In allcases, the experimentally measured stresses become con-stant or start to fall, while the theoretical predictionscontinue to rise. Again, these effects can almost certainlybe attributed to contact problems at the sliding pastesurface. Paste B is a gel and hence has considerableelasticity and therefore one could anticipate a Poissoncontraction normal to the plane of straining, this contrac-tion reducing the contact of paste and plate and thereforelowering the observed stress gradient. Such effects are notincluded in the current model since perfect plasticity isassumed with elastic strains being zero. This mechanism ofPoisson contraction had been suggested previously by

w xBates 3 .For completeness, we consider finally the experimentr

Ž .theory comparisons for paste C Figs. 13 and 14 . Gener-ally, there is little sign of agreement except in the case of

Ž .fast flow with a 10-mm gap Fig. 14 . This is probablynothing more than coincidence since in all four cases pasteclearly does not flow radially and hence the radial flowtheory is not applicable.

6. Conclusions

Experiments have been performed on pastes betweenparallel flat plates of varying gaps with three differentpastes. The development of the flow patterns and injectionpressures with time have been recorded. In each case, theflow patterns are found to be controlled mainly by the

formulation of the pastes: one exhibiting radial flow; asecond mushrooming; and a third fingering. This fingeringis thought be due to a significantly reduced yield stress intension. Paste properties such as tensile yield stress appearto be extremely sensitive to formulation. In order for apaste to flow in a divergent manner, it must be able tosupport a hoop tension; therefore, formulation of pastes formoulding is a very important consideration.

A consideration of the early stages of flow allows anestimation of the compressive bulk yield stress, the surfaceshear yield stress and the surface shear stress velocityfactor of the flowing material. Comparisons with measure-ments made by the conventional piston extruder method

w xdue to Ovenston and Benbow 1 and Benbow and Bridg-w xwater 2 lead to the conclusion that this plate method

provides a potential for an improved measurement of s ,0

but is less reliable for measuring the surface characteristicst and b.0

A radial flow model has been derived for a segment oflubricated, perfectly plastic paste in radial outflow. Theresult has been shown to be identical to that previously

w xpresented by Bates 3 , although it is derived by a consid-eration of work and not a force balance. Comparisonsbetween the predicted and experimentally determined val-ues of radial stress at the inlet have been made for threedifferent pastes, three different plate spacings at bothcreeping and fast flow rates. The radial flow theory seemsto generally produce an overestimate of the measuredstress values, the degree of overestimate depending on howclose the experiment approaches radial flow. Fast flowvalues are also less good estimates than those for creepingflow, suggesting that the surface shear values are notcorrect. This may be due to a Poisson’s ratio effect reduc-ing the contact between the plates and the paste.

The work gives rise to the need for further studies inseveral areas. The next step will be to vary the angle ofsegment by further constraining the flow and comparetheory and experiment, this being the steady-state experi-ment in contrast to those of unsteady state detailed above.Ultimately, this idea will be extended to a consideration ofsystems with parallel restraining walls and very narrowseparations as found in honeycomb die systems. Furtherinto the future, flow tracking techniques will be developedin order to better understand the flows in this twin platesystem. Additionally, the effects of injection velocity andplate spacing will be further investigated. Attempts arealso underway to measure independently the tensile yieldproperties of extrusion pastes, a necessary corollary ofestablished methods employed in compression.

7. Nomenclature

Ž .a Outer radius of paste segment mA Surface area of paste segment in plane normalx

Ž 2 .to x direction m


˙U Ž .D Rate of internal work dissipation Jrse Unit vector in i directioni

Ž .F Net force acting on paste finger MNL Length at which extruded finger begins to1

Ž .bulge during creeping flow mL Length at which extruded finger begins to2

Ž .bulge during fast flow mŽ 3 .Q Volumetric flow rate of paste m rs

Ž .r Radius mŽ .r Inner radius of paste segment injector radius0

Ž .mŽ .t Width of injector m

Ž 3.V Volume of paste segment mŽ .V Velocity of paste in x direction mrsx

Ž .w Thickness of paste sample mŽ .w Work done by injection stress Jr

Ž .w Surface shear work JsŽ .b Surface shear stress velocity factor MParm

bU Surface shear stress velocity factor estimated

Ž .from plates experiment MParmU Ž y1 .´ Assumed strain rate field tensor s˙

Ž .f Angle of paste segment radŽ .s Stress tensor MPa

Ž .s Bulk yield stress MPa0

sU bulk yield stress estimated from plates experi-0

Ž .ment MPaŽ .s i, j component of stress tensor MPai j

s rsr0 Radial normal stress component at injector11Ž .MPa

t Component of surface shear stress vector in iiŽ .direction MPa

Ž .t Shear yield stress of surface layer MPa0

tU Shear yield stress of surface layer estimated0

Ž .from plates experiment MPa

Acknowledgements

Thanks are due to both A. Bates and D. Horrobin fordiscussions regarding the modelling and to Z. Saracevicfor assistance with experimental work. F. Kolenda at theInstitut Francais du Petrole, Solaize kindly supplied us´with some of the experimental materials. This work wascarried out under an EPSRC Soft Solids grant.

References

w x Ž .1 A. Ovenston, J.J. Benbow, Trans. Br. Ceram. Soc. 67 1968 543.w x2 J.J. Benbow, J. Bridgwater, Paste Flow and Extrusion, Oxford Univ.

Press, Oxford, UK, 1993.w x3 A.J.D. Bates, PhD Thesis, Univ. of Birmingham, Birmingham, UK,

1995.w x4 R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena,

Wiley, New York, 1960.

Documents

Expo Ceramicos