sionrac-, thiser,

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Y. C. Lame-mail: myclam@ntu.edu.sg

X. Chen

K. C. Tam

Singapore-MIT Alliance Program,Nanyang Technological University,

Singapore 639798,Singapore

S. C. M. YuSchool of Mechanical & Production Engineering,

Nanyang Technological University,Singapore 639798,

Singapore

Simulation of Particle Migrationof Powder-Resin System inInjection MoldingPowder injection molding is an important processing method for producing precimetallic or ceramic parts. Experience, intuition and trial-and-error have been the ptice for the design and process optimization of such molding operations. Howeverpractice is becoming increasingly inefficient and impractical for the molding of largmore complicated and more costly parts. In this investigation, a numerical methosimulating the mold-filling phase of powder injection molding was developed. Thewas modelled using the Hele-Shaw approach coupled with particle diffusion transequation for the calculation of powder concentration distribution. The viscosity offeedstock was evaluated using a power-law type rheological model to account foviscosity dependency on shear rate and powder concentration. A numerical exampresented and discussed to demonstrate the capabilities and limitations of the simualgorithm, which has the potential as an analytical tool for the mold designer. The vation of powder density distribution can be predicted, which is ignored by the exissimulation packages. Preliminary simulation indicated that powder concentration vation could be significant. Non-isothermal analysis indicated that most of the key paeters for filling process would change due to a change in powder concentration disttion. @DOI: 10.1115/1.1580850#

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1 IntroductionPowder injection molding~PIM! is a relatively new process fo

producing metallic or ceramic net shape parts. The main stepthe powder injection molding process are: 1! Mixing a metallic orceramic powder with a polymer or organic binder to form granufeedstock with the desired fraction of powder; 2! Loading thegranular feedstock into a heated barrel and with the shearingtion of a high torque screw, the granular feedstock reaches a smolten state; 3! The semi-molten feed is injected into a coolemold cavity to form the net shape green part; 4! The final part isformed by debinding and sintering of the green part@1#.

The injection molding step is important. Indeed, if the moldpart is significantly heterogeneous, then distortions and defcould appear in the final products. For a successful powder intion molding, an important factor is moldability. The high qualipart depends not only on the viscosity of the feedstock but alsoinjection molding process. Many researchers have focused onappropriate volume fraction of powder and binder system for scessful powder injection molding@2,3#. However, the flow of thepowder-binder mixture in PIM process is difficult to predicFeedstock, usually with 40%–60% powder, is a two-phase mrial that can be viewed as a concentrated suspension. The dedistribution of the powder changes during molding and the moing process exhibits strong nonlinearity that is dependent onpowder density. In the existing numerical algorithms for simuling practical powder injection molding process, the inner strtural changes~i.e., powder density distribution! in the materialduring injection molding are generally ignored. The bulk propties of the concentrated suspension~powder/resin mixture! arerepresented by material constants assuming that the mixtuhomogeneous. This continuum approach@4–8# is sufficient onlyfor describing the macroscopic behavior by using an effectivebulk, viscosity of the mixture. This can provide general useinformation on the overall flow behavior of the feedstock andpredicting the occurrence of short shots. The phenomena of y

Contributed by the Manufacturing Engineering Division for publication in tJOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedMarch 2001; revised October 2002. Associate Editor: R. Smelser.

538 Õ Vol. 125, AUGUST 2003 Copyright ©

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stress and slip had been included also for the description ofPIM filling process@4#. However, this continuum approach neglects the interaction between particle and binder system andnot predict the changes of powder density distribution as a fution of time and location during the filling phase.

Granular media mechanics@9# is an alternative approach thacan be used to predict the motion of individual particles. In gralar media mechanics, the kinematic change of powder densitdirectly taken into account together with explicit evaluationparticle-binder interaction and powder characteristics, such asticle size and distribution. However, such an approach limitsnumber of particles that can be considered before the comptional effort becomes excessive. Thus, its potential for simulatindustrial process is limited.

Various investigators had proposed shear-induced particlegration theory to explain a number of flow phenomena for cocentrated suspensions including particle accumulation anblunted velocity profile in simple flow systems. Leighton and Arivos @10,11# suggested phenomenological models for particle mgration in imhomongenous shear flow typically due to migratiand irreversible interactions. Following the earlier studies of Jkins and McTigue@12#, Nott and Brady@13# proposed a suspension balance model for concentrated suspension, in which thecept of hydrodynamic temperature was used as a measure ointensity of the velocity fluctuations of the particles. Using Stoksian dynamics, Nott and Brady@13# had recently carried out dynamic simulations of pressure-driven flow for a suspension itwo-dimensional channel with respect to a monolayer of identspherical non-Brownian particles. Their simulations confirmedLeighton and Acrivos@10# shear induced migration theory anshowed the gradual accumulation of particles toward the centethe channel, leading to a concentration maximum near the celine, and a blunting of velocity profiles. Philipes et al.@14#adapted the scaling arguments of Leighton & Acrivos@10,11#,together with an empirical relationship between the suspenviscosity and particle concentration, to predict particle migratfor imhomogeneous shear flows. There are many accumulatedperimental evidences that show the difference in flow fieldtween concentrated suspensions and an equivalent homogen

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2003 by ASME Transactions of the ASME

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liquid. As reported by Karnis et al.@15#, the velocity profile of aflowing suspension in a circular channel was blunt, comparethe typical parabolic profile. Averbakh et al.@16# and Koh et al.@17# described an experimental method of measuring velocitieslow viscous flows of highly concentrated suspensions with LaDoppler Anemometry technique. Their measurements confirmthat velocity profiles in a concentrated suspension were bluntto the accumulation of particles from high shear area migratelow shear area and it was different from those for a Newtonfluid.

In this investigation, a new numerical approach which haspotential of general applicability, but targeting the simulationpractical powder injection molding process at this instance, is pposed. This approach incorporates the advantages of thetinuum and particulate approaches and can be applied to comindustrial processes. The present analytical and numerical deopment can provide an insight into the mold filling process soto predict the concentration distribution of the powder, a macharacteristic of powder injection molding. The proposed modebased on the generalized Hele-Shaw flow model@18# for thincavities, coupled with a diffusion model which describes theteraction between powder and binder. Flow behavior ofpowder/binder mixture is approximated by non-Newtonian viscity model under nonisothermal conditions. Simulation resultspresented for the prediction of the effects of shear-induced parmigration during powder injection molding. In essence, the prsure, temperature and concentration of the powder in the mixcan be computed, which are used to predict the shape and locof the moving flow front as a function of time. Subsequently, tchanges in powder density distribution, the velocity distributioshear rate and weld line formation can be extracted from thresults. These results of pressure, temperature of the mixtureconcentration of the powder, are processing parameters that athe properties and shape of the green part.

2 Theoretical BasisWe consider the concentrated suspension~mixture of powder

and binder system! occupying a three-dimensional regionV(t) atany timet filling the cavity under the action of high pressure atemperature, as shown in Fig. 1. For clarity of explanation,have restricted the flow to the plane of the figure. The formatof the solid skin layer is neglected in this work. In Cartesicoordinates the boundary is defined by the outer contourC0 in thex, y plane; the gap thickness 2b in thez-direction is much smallerthan the length scale definingC0 . The fluid enters the cavityacross the entry contourCe , and as timet marches, occupies aregion extending to the moving frontCm(t). There may be alsoimpermeable contoursCi within C0 . Our objective is to deter-mine the flow field and the powder distribution within theboundaries occupied by the feedstock.

It must be pointed out that an important aspect common toinjection molding processes, which has been neglected inpresent formulation, is fountain flow of the flow front. This asumption of ignoring fountain flow was employed in establishplastic injection molding software such as Moldflow and goresults can be achieved. The reason why fountain flow can

Fig. 1 Schematic of illustrating the location of the melt insidethe cavity

Journal of Manufacturing Science and Engineering

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ignored through Hele-Shaw approximation and yet good resare obtained is that the effect due to the flow field is cumulatiFountain flow at any location only occurs for a very short duratcompared to the flow at the wake of the flow front, which coforms to Hele-Shaw approximation.

2.1 Flow Governing Equation. The concentrated suspension ~mixture of particle/binder! of PIM is assumed to be a generalized Newtonian fluid. In injection molding the inertial termsthe momentum conservation equation are negligible. The flowbe assumed to be quasi-steady flow and the lubrication apprmation can be used for modelling the global flow behavior inmold cavity. The Hele-Shaw model@18# for thin part can be em-ployed. The resultant sets of equation can be written as:

]~bu!

]x1

]~bv !

]y50 (1)

]P

]x5

]

]z S h]u

]zD (2)

]P

]y5

]

]z S h]v]zD (3)

where continuity equation is expressed in terms of gapwise aaged velocity componentsu and v, and u and v are velocitycomponents in thex and y directions respectively,b is the halfthickness,h is the apparent shear viscosity andP is the cavitypressure. Following Krieger@19#, the viscosity of concentratedsuspension~mixture of powder/binder! is represented by the inelastic modelh5h(g,T,F), where

g5AF S ]u

]zD 2

1S ]v

]zD 2G (4)

andT is the temperature. The temperature field is described byenergy equation, simplified further by employing the small thicness approximation as follow,

rCvS ]T

]t1u

]T

]x1v

]T

]y D5K]2T

]z2 1hF S ]u

]zD 2

1S ]v]zD 2G (5)

wherer is the feedstock effective density,Cv the specific heat,andK the thermal conductivity. When the no slip boundary codition is employed at the wall and with a given viscosity, Eqs.~1!and ~2! are easily integrated. The result is, for PIM applicatiwhere viscosity is symmetric inz,

u5LxEz

b z

hdz, v5LyE

z

b z

hdz (6)

whereLx5]P/]x, Ly5]P/]y; b5half of thickness of the part.With

S5E2b

b z2

hdz (7)

Integrating equation~1! across the thickness, the followinequation can be obtained:

]

]x S S]P

]x D1]

]y S S]P

]y D50 (8)

It should be noted that the adoption of this symmetricalsumption for this preliminary investigation is for clarity in explanation. This assumption can be easily relaxed.

2.2 Migration of Particles. When a suspension~powder/binder mixture! is subjected to imhomgeneous shear flow in tcavity, diffusion of particles takes place. The particle Peclet nuber Pe5a2g/D, defined in terms of local shear rateg, particleradius a and diffusivity D, is large for PIM, This implies thatBrownian force can be neglected. Phillips et al.@14# proposed a

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phenomenological model for the shear-induced diffusion phenenon for a neutrally buoyant uniform particle size, based onideas suggested earlier by Leighton and Acrivos@10,11#. Theydescribe two mechanisms. The first mechanism concerns theuniform frequency of particle interactions in a shear field. Pticles tend to move from region of high frequency to region of lofrequency. The second mechanism comes from the nonunifviscosity field in which the interaction between two adjacent pticles is asymmetrical. As a results of such an interaction a parmoving towards the low effective viscosity region will movelarger distance than the other particle. Based on these two menisms, Phillips et al.@14# presented a unsteady state diffusiequation which can model particle migration in imhomongeneshear flow.

DF

Dt5Dca

2¹•~F2¹g1Fg¹F!1Dha2¹•S gF21

h

dh

dF¹F D

(9)

whereD/Dt[]/]t1V•¹ is the material time derivative.¹ is thethree-dimensional Laplace operator in thex, y, andz directions,Fis particle concentration by volume,a is the characteristic particleradius, g is the local shear rate,Dc and Dh are the empiricallydetermined diffusion coefficients.

Equation~9! is second order and nonlinear in particle fractivolume F5F(x,y,z,t), where x, y, z are the coordinates in aCartesian coordinate system. For simple system, such as Cflow @11#, this equation can be solved together with the flow goerning equation for a Newtonian fluid with concentratiodependent viscosity to yield both time-dependent and steady-concentration and velocity profiles. However, the solution of E~9! is difficult and time-consuming for PIM without simplificatioas the powder-resin mixture is a non-Newtonian fluid with timdependent flow field and transient concentration and veloprofiles.

The significance of the various term in Eq.~9! could be ob-tained by examining the values of the parameters. Estimatesthe values of the characteristic parameters of PIM are containeTable 1. We use the following scales:

x5Lx* 5~b/d!x* , y5Lx* 5~b/d!y* ,

z5bz* , t5~L/V!t* 5~b/dV!t* , (10)

u5~L/t !u* 5Vu* , v5~L/t !v* 5Vv* ,

w5~b/t !w* 5dVw* , h5h0h* , F5F0F*

When these scalings and values are introduced into diffusion~9!, the transient term, convection term and particle migratterm normal to the shear face~z direction! are at least one order omagnitude larger than the other terms in Eq.~9!. Thus, we obtainthe simplified diffusion equation as follow:

]F

]t1u

]F

]x1v

]F

]y5

]

]z S D1

]F

]z D1]

]z S D2

]g

]z D (11)

Table 1 Values of characteristic parameters of PIM

Characteristic parameters Characteristic values

Cavity thickness~b! b51023 mCavity length~L! L5b/d m whered5b/L!1

Velocity of feedstock (V) V51021 m/sViscosity of feedstock (h0) h05104 Pa•s

Thermal conductivity of feedstock~k! k513100 W/m•KDensity of feedstock~r! r553103 kg/m3

Initial concentration (F0) F05531021

Particle diameter (a0) a051025 m

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In the simplified diffusion Eq.~11!, the diffusion coefficients,D1and D2 are functions of the particle size and density, shear rand viscosity:

D15S DcF1DhF2ga21

h

]h

]F D ga2 (12a)

D25DcF2a2 (12b)

where g and a are the shear rate and the particle radius resptively. For polydisperse concentrated suspension with varyingticle size, a mean particle size could be employed in Eqs.~12a!and ~12b!. Shear-induced diffusion is assumed to be independof interparticle interactions, which include frictional and colloidforces. These effects are incorporated in the constantsDc andDhof the above equation, which could be determined by conducthe appropriate experiments@11,14,20#. If a nonhomogeneousshear flow initiated in powder-resin system with uniform particconcentration, the second term of the right hand side of Eq.~11!produces a flux, which in turn causes a concentration gradienthence induces a second flux by the first term. Equation~11! mustbe supplemented with appropriate condition at the boundwhich is simply that the particle could not migrate through tphysical boundary.

2.3 Material Models. In this investigation, the viscosity othe bulk material is taken to be a function of the neat bindviscosity and the volume fraction of the powder. We have adopthe following Krieger rheological model@19#.

h5hbS 12F

FcD 2m

(13)

where hb is the viscosity of binder.F and Fc are the powderconcentration and the critical powder loading respectively. Trheological model describes the effect of the volume fractionthe powder on the flow behavior of powder/binder mixture. In thpreliminary investigation, monodisperse particle size~uniform par-ticle size! was used. For polydisperse concentrated suspenwith varying particle size, empirical correlation method@21# canbe employed to determine the constants of the Krieger rheologmodel.

As the powder distributionF is calculated, the feedstock effective densityr and specific heatCv can be determined respectiveusing the linear rule of mixture as follows:

r5r1F1r2~12F! (14)

Cv5Cv1F1Cv2~12F! (15)

wherer1 andr2 are the densities of the binder system and powparticle respectively.Cv1 and Cv2 are the specific heats of thbinder system and powder particle respectively.

To estimate the thermal conductivityK, Jeffrey’s equation isadopted, which should give a better approximation than the linrule of mixture. The Jeffrey’s equation can be represented as@6#:

K5H 113zF1F2S 3z213z3

41

9z3

16

l12

2l131

3z4

64 D J •Km

(16)

wherel5K f /Km andz5(l21)/(l12), K, K f , andKm are thethermal conductivity of feedstock, particles and binder matrixspectively.F is the particle volume fraction.

2.4 Numerical Procedures. It should be pointed out thathe Krieger rheological model, Eq.~13!, diffusion Eq. ~11! andflow governing Eqs.~1!–~3!, are employed for this preliminaryinvestigation. We believe that with these simple relationships, tare sufficient to capture the effect of powder migration duringfilling stage in PIM. A numerical scheme of finite element/finidifference method has been implemented for solving the presstemperature and powder distribution fields. In the present metha single finite element mesh representing the mid-surface of

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Fig. 2 Schematic of control volume and finite difference grid used for pressure,temperature, and particle concentration computations respectively

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cavity, is used to solve for moving flow front as reported by Hber and Shen@18#. Thin shell triangular elements are used to dscribe a three-dimensional thin part. The triangular elementsfurther divided into three sub-volumes, which are considered toa net work of control volume, as shown in Fig. 2. The nodcontrol-volume formulation is derived from the Galerkin aproach, as described by Hieber and Wang@18,22–23#. The nu-merical equation for the net flowq( l ) of elementl can be writtenas:

qi~ l !5S~ l !(

k51

3

Rik~ l !PN , i 51,2 or 3 (17)

whereS( l ) andRik( l ) are the flow conductance and the coefficient

the nodal pressure to the net flow in elementl respectively. Massconservation for the control volume at nodeN can be written as:

(l

S~ l !(k51

3

Rik~ l !PN50 (18)

Nodal Eq.~18! can be solved for the nodal pressure.A finite difference grid, as shown in Fig. 2, is mapped onto ea

element centroid for solving the unsteady energy and diffusequations and to capture the through thickness variation of tperature, and through thickness particle migration. The nodalergy and diffusion equation with averaged temperature andticle concentration evaluated at the node of each element in mform can be written as:

@K#N$Mk11%N5$F%N (19)

where@K#N is the global stiffness matrix,$Mk11%N is the pending

matrix vector such as nodal vector of temperature or powder ccentration at time stepk11. $F%N is the force vector including theconvection, viscous heating and the initial temperature or partconcentration terms. Equation~19! can be solved for temperaturor powder distribution.

A FAN approach@23# of which the rectangular cells are replaced by the control volumes is employed to track the flow fr

ring Science and Engineering

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advancement as a function of time. To distinguish the filled nofrom melt front nodes, a fill factorf i j , having a value between 0and 1, which is associated with each element to indicate thecentage of filling, is defined by:

f i j 5DVi j /Vi (20)

whereVi and DVi j denote control volume and filled volume respectively ofi th vertex node at thej time step. Numerical calcu-lation of the pressure field is based on mass conservation incontrol volume which can be either empty (f i j 50), partiallyfilled (0, f i j ,1) or completely filled (f i j 51) with feedstock. Ineach time step, the pressure field is calculated to obtain the veity distribution in the flow domain. It is important to note that foF(z) in the newly-filled volume, its initial value may be taken athe average value of its neighbor upper-stream element.

The main numerical step employed for flow analysis of powdinjection molding can be summarized as:

a. The viscosity of molten powder-mixture is calculated usiEq. ~13! for each element.

Table 2 Composition of the binder †7‡

Materials

Binder

Densityr~kg/m3!

WeightPercent of binder

Polypropylene 0.903 20Carrnauba Wax 0.970 10

Paraffin 0.900 69Stearic Acid 0.850 1

Table 3 Physical characteristics of the powder †7‡

MaterialDensityr~kg/m3!

Specific heatCp(J/kg•K)

ConductivityCoefficientl~W/m•K!

Critical particleloading

Fc

Iron 7.87 328.3 75.8 0.6

Table 4 Physical and rheological characteristics of the materials †7‡

Material

Constantn

Eq. ~21a!

ConstantTa ~K!

Eq. ~21b!

Constantm0(Pa•sn)Eq. ~21b!

Specificheat

Cp(J/kg•K)

Conductivitycoefficientl~W/m•K!

Constantm

Eq. ~13!

Binder 0.99 2138 2.86431025 2790 0.024 -Feedstock 0.20 6660 6.531025 1313 45.5 21.82

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Fig. 3 Mesh of the rectangular plate cavity

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b. Global stiffness matrix equation from flow conductance otained using Eq.~7!.

c. At each instant we have a boundary value problem wnormal gradients specified alongCe , C0 , Ci andCm . The pres-sure field Eq.~8!, which is quasi-linear in pressureP if the tem-perature is known, is solved iteratively as flow conductanceS isdependent onu, v and also the temperatureT. T should be deter-mined from Eq.~5! simultaneously. The convergence of the sotion is achieved using a relaxation scheme at each time step bfinite element method.

d. Temperature field Eq.~5! is solved using finite differencemethod. Its variation at entry and along the wall must be pscribed. An implicit method is used for the conduction term. Tviscous dissipation and convection terms are evaluated usingsolution from the previous time step. This method allows the cculation of the temperature as a function of time and positionthrough thickness direction from node to node.

e. Concentration variation, which is a transient thredimensional field, is solved using Eqs.~11!, and~12! at the nodalpoints of the finite element mesh, with the flow determined frothe previous time step by the finite difference procedure inconservative form for every filling steps.

f. Subsequent to the convergence of pressure, temperatureconcentration, the melt front is updated by calculating the fluxeach frontal control volume.

This procedure is carried out until the cavity is completefilled.

3 Results and Discussions

3.1 Material Data. The material data employed for the numerical examples in this paper are tabulated in Tables 2, 3, arespectively.

The viscosity relationship for the binder is approximated usthe following power law viscosity model@24#:

h5g~T!gn21 (21a)

g~T!5m0 exp~Ta /T! (21b)

wheren, m, m0, and Ta are material constants. The power laindex n and the temperature dependent material constant forbinder and feedstock are contained in Table 4.

3.2 Rectangular Plate Cavity. A rectangular plate cavityhas been chosen to demonstrate the effects of powder dedistribution on flow and heat transfer during the mold filling stagThe dimensions of the rectangular plate are 10 cm by 20 cm wa thickness of 0.2 cm. The finite element mesh used in this si

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lation is shown in Fig. 3~384 elements, 211 nodes!. NodeG is thegate location. The molten feedstock temperature is 200°C, filtime is 2 s~flow rate is 20 cm3/s!. The mold temperature is 80°C

To investigate the effects of powder density on the filling stapowder concentration distribution and effective viscosity weresumed to be uniform at the gate. Non-isothermal simulations wthe assumption of no particle migration and with particle migtion were carried out. Without particle migration, uniform partic

Fig. 4 „a… Predicted bulk powder concentration „%… distribu-tion with Dc ÕDhÄ0.66 and FsÄ45% „b… Predicted bulk powderconcentration „%… distribution with Dc ÕDhÄ0.90 and FsÄ45%

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Fig. 5 „a… Bulk temperature distribution without particle migration at the end of filling „tÄ2 s… „b… Bulk temperature distribution with particle migration at the end of filling „tÄ2 s…

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density distribution would always be predicted, which is not reistic. With particle migration, a final inhomogeneous green pwould be predicted.

3.3 Effect of Diffusion ConstantsDc and Dh. There areno experimental data on the diffusion coefficients for iron powmixtures. Thus, three sets of diffusion constantsDc andDh werechosen:~0. and 0.!, ~0.43 and 0.65!, ~0.55 and 0.65!. This is toobserve the effect of the diffusion constants on particle migratIn addition, mono-size iron powder particles of 50mm in radiuswere assumed for each simulation. Non-isothermal simulatiwere carried out. The nonhomogeneous shear flow field dumolding is existed in the part. As expected, from shear-indudiffusion model, Eq.~11!, powder migrated toward low shear regions from high shear regions resulting in powder accumulaas shown in Fig. 4~a!. For Dc /Dh50.66 (Dc50.43 andDh50.65) and initial concentrationFs545%, the maximum bulkpowder concentration is approximately 47.1%. The minimum bpowder concentration is approximately 43.1%.

According to Eqs.~12a! and~12b!, the diffusion coefficientsD1and D2 are proportional to the diffusion constantsDc and Dh .Figure 4~b! shows the through thickness average~bulk! distribu-

acturing Science and Engineering

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tion powder concentration of the part withDc /Dh50.90 (Dc50.55 andDh50.65) at the end of filling with the same initiaconcentration as in Fig. 4~a!. The maximum bulk powder concentration is approximately 47.8%. The minimum bulk powder cocentration is approximately 42.6%. As expected, with largDc /Dh , there was greater particle density variation.

3.4 Effect of Particle Migration on Temperature andPressure. Particle density has an effect on thermal conductiviand therefore on temperature distribution. In addition, it hapronounced effect on the through thickness viscosity distributiand thus work done through shear heating. Figures 5~a! and ~b!respectively show the temperature distribution~through thicknessaverage! without and with particle migration for the whole domain. It indicates that for the same input volume flow rate, teperature without particle migration was higher than with partimigration. A better appreciation can be obtained by considerthe global picture: without particle migration, the reductiondissipation rate was lower. Thus, the network done had tohigher and this would result in higher overall increase in tempeture. Obviously, the temperature distribution depends on the loparticle concentration. A specific node is chosen to show

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change of physical variables during the filling stage. Figures 6~a!and ~b! show the temperature variation at internal node 148~asshown in Fig. 3! and pressure variation at the gate~point G!respectively as the cavity was being filled. The results weretained with Dc /Dh50.66 (Dc50.43 and Dh50.65) and Fs545%. In addition, the simulation was carried out with the asumption of no particle migration and particle migration underconditions of non-isothermal flow. Figure 6~a! shows that the tem-perature for the assumption of no particle migration was contently higher than with particle migration. The reason being twith particle migration, the thermal properties, such as therconductivity and specific heat, had changed for the blend. Lotemperature with particle migration would translate into highviscosity and thus higher resistance to flow. With the same flrate, higher pressure at the gate would be expected as showFig. 6~b!. This means that the driving force, such as pressgradient, of the feedstock was also affected by the migrationthe particles. Different particle volume fraction, distribution avolume flow rate would cause different pressure gradient. Tshows clearly that changes in particle concentration and effecviscosity distributions could result in a change of power requito drive the flow. This prediction is consistent with experimenobservation of Seifu@25# for viscous flows of concentrated supensions exhibiting shear-induced particles migration.

Fig. 6 „a… Temperature profile at node 148 as the cavity wasfilled — j— Without migration, aÄ50 mm; —m— Dc ÕDhÄ0.66,aÄ50 mm „b… Pressure profile at the gate G as the cavity wasfilled — j— Dc ÕDhÄ0.66, aÄ50 mm; —m— Without migration,aÄ50 mm

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3.5 Through Thickness Powder Concentration and Veloc-ity Distribution. To further illustrate the effect of powder migration, 2D contour plots of particle density concentration ataken at three different cross-sections of the part. Figures 7~a!, ~b!,and ~c! show the contour plots of cross-sections A-A, B-B, C-respectively, their locations are shown in Fig. 3. Note that wsymmetry, only a quarter of the cross-sections are shown.evident that powder concentration at the midplane increasedsiderably with a corresponding significant decrease in powconcentration at the outer surface~i.e., along they-axes,z50).This was caused by the relatively high shear rate at the osurface~next to the wall! and negligible shear rate at the midplanof the part. With an increase in distance in the axial direction, i

Fig. 7 „a… Contour plots of particle concentration „%… at „a…Section A-A, „b… Section B-B, „c… Section C-C; The value of thecontour of particle volume concentration „%… are: „1… 52.0; „2…50.2; „3… 48.4; „4… 46.7; „5… 44.9 „6… 43.1; „7… 41.3; „8… 39.5; „9…37.8; „10… 36.0

Fig. 8 Through thickness velocity profiles at end of filling „tÄ2 s… at node 148 in Fig. 3 — l— fÄ0.45; —j— fÄ0.35;—m— fÄ0.25; —× fÄ0.0

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from section A-A to C-C, particle migration became more prnounced. With a decrease in concentration at the region ofwall, there was an increase in concentration considerably atstagnation area such as the corner~nodes 2 and 3 in Fig. 3!, seeFigs. 4~a! and ~b!.

The changes in the through thickness direction for concention and effective viscosity were further reflected in the velocprofiles as shown in Fig. 8. The migration of particles to tmidplane resulted in an increase in effective viscosity. Thisturns blunted the velocity profiles at the midplane. With ancrease in the initial particle concentration, the velocity profibecame more blunted. This prediction is in agreement withcomputational results of Brady@13#, who employed Stokesian dynamics method for viscous flows of concentrated suspensionshibiting shear-induced particles migration.

Fig. 9 Gate pressure evolution for different mesh sizes — l—146 meshes; — j— 384 meshes; — m— 680 meshes

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3.6 Effect of the Mesh Size. In order to evaluate the senstivity of the results with mesh size, three mesh sizes have btested:~i! 146 elements,~ii ! 384 elements as shown in Fig. 3 an~iii ! 680 elements. The computational times were 96 s, 112 s,200 s on a PC586 Pentinum III~RAM 284 Mb! for cases~i!, ~ii !,and ~iii !, respectively.

The evolution of pressure versus filling time at the gate hbeen plotted for the three cases in Fig. 9. The pressure at thewas slightly higher for the coarse mesh~cases~i!! but there waslittle difference between cases~ii !, and~iii !.

3.7 Box Cavity. We have performed many numerical simlations for various three dimensional cavity geometries. Asexample of showing its potential to new industrial PIM producthe nonisothermal numerical simulations of the filling stage foriron box show in Fig. 10 will be presented in this investigatioFigures 11, 12, and 13 show melt front positions as a functiontime, bulk temperature and powder concentration contours at

Fig. 10 Mesh of the box cavity

Fig. 11 Predicted melt front during the filling stage

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Fig. 12 Predicted bulk temperature distribution at the end of filling„tÄ5 s…

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end of a cavity filling, respectively, when one gate, which isG,was employed. The filling time was 5 seconds, the maximum btemperature near the gate is the same as the injection temperof 510 K, the minimum temperature was 438 K, and the mamum concentration in the corner of the box was found to be 0With a uniform initial concentration of 0.45, and the minimuconcentration was found to be 0.43.

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Modifying the design by having two gates but keeping the vume flow rate the same as in the case for one gate. Theconcentration of powder changed significantly as shown in F14. This example demonstrated the potential of evaluating dechanges using the present procedure such that a satisfactorsign and processing conditions can be achieved.

Fig. 13 Predicted bulk powder concentration „%… distribution withDc ÕDhÄ0.66 and FsÄ45% with one gate

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Fig. 14 Predicted bulk powder concentration „%… distribution withDc ÕDhÄ0.66 and FsÄ45% with two gates

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4 ConclusionsIn this investigation, a numerical model based on a FEM/FD

hybrid method has been developed to simulate the powder intion molding process with particle migration. Powder distributican be predicted by shear-induced particle migration throughintroduction of diffusion equation at the nodal control volumlevel. Thus, the model can predict the variation of powder dendistribution which is ignored by the existing simulation packagParticle migration reduced the pressure required for the proceviscous dissipation of energy was reduced. Preliminary simulaindicated that powder concentration variation could be significaNon-isothermal analysis indicated that most of the key paramefor filling process would change due to a change in powder ccentration distribution. These numerical results elucidated theportance of particle migration on non-isothermal flow.

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tries Federation, Princeton, N.J.@2# Hens, K. F., and Lee, D., 1991, ‘‘Process Analysis of Injection Molding w

Powder Products,’’ Advances in Powder Metallurgy, MPIF, APMI,2, p. 127.@3# Moller, James C., and Deeyong, L., 1994, ‘‘Constitutive Behavior of

Powder/Binder System: Molding and Thermal Debinding,’’ Int. J. PowdMetall., 30~1!, p. 103.

@4# Kwon, T. H., and Park, J. B., 1995, ‘‘Finite Element Analysis ModellingPowder Injection Molding Filling Process Including Yield Stress and Slip Pnomena,’’ Polym. Eng. Sci.,35, p. 741.

@5# Wang, C. M., Leonard, R. L., and McCabe, T. J., 1993, ‘‘ComputatioAnalysis of Fluid Flow Injection Molding,’’Advances in Powder Metallurgy &Particulate Materials,3, p. 31.

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@16# Averbakh, A., Shauly, A., Nir, A., and Semiat, R., 1997, ‘‘Slow Viscous Flowof Highly Concentrated Suspensions-Part I: Laser-Doppler VelocimetryRectangular Ducts,’’ Int. J. Multiphase Flow,23, p. 409.

@17# Koh, C., Leal, J. L. G., and Hookham, P. A., 1994, ‘‘An Experiment Invesgation of Concentrated Suspension Flow in a Rectangular Channel,’’ J. FMech.,256~1!, p. 1.

@18# Hieber, C. A., and Shen, S. F., 1980, ‘‘A Finite-Element/Finite-DifferenSimulation of the Injection-Molding Filling Process,’’ J. Non-Newtonian FluMech.,7~1!, p. 1.

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@20# Lam, Y. C., Chen, X., Tan, K. W., Ma, J., Tam, K. C., and Yu, S. C. M., 200‘‘Experimental Measurements and Simulation of Particle MigrationPressure-Driven Tube Flow: Application to Flow in Runners,’’ Journal of Ijection Molding Technology,6~1!, p. 45.

@21# Kordon, A. K., Pitchumani, R., Beris, A. N., Karbhari, V. M., and Dhurjati,S., 1993, ‘‘A Rheological Model for Particulate Ceramic Slurries at Low Teperatures,’’ Scr. Metall. Mater.,29, p. 1095.

@22# Wang, V. W., Hieber, C. A., and Wang, K. K., 1985, ‘‘An Interactive Simulation of Cavity in Injection Molding with Color Graphics,’’Proc. 43rd SPEANTEC, pp. 826–829.

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