burden Polymer 7ownat vokmie 5 Number I (1996)

1017.6020196

A Transient Finite Element Model of Mixing of Rubber

Compounds in a Banbury Mixer

Mir Hamid Reza GhoreishyPolymer Research Center of Iran, P.O.Box 14185/515, Tehran, I .R.Iran

Received: 4 September I995; accepted 13 December 1995

ABSTRACT

The aim of this paper is to develop a mathematical model for the simulationof the flow of elastomeric materials inside a typical Banbury mixer. Themodel combines the non-Newton€an behaviour of rubber compound withnon-isothermal flow regime in a transient condition . The set of the governingequations are solved using the finite element method . The momentumequations are solved using continuous penalty method and the energyequation is solved employing the C3aierkin method. A streamline upwindingtechnique Is used in the solution scheme of the energy equation to provideconvergent and stable results for high Peclet number flow . The solution ofthe transient equations is performed using implicit 8 time stepping method.The overall solution strategy is based on the de-coupling of the flow andenergy equation and use of the Picard iterative scheme. The results of thesimulation inciudothevelocity, pressure and temperature distributions andthe variation of these parameters with time. Comparison of these results withthe available experimental data confirms the general validity of thedevelopment model.

Key Words : mathematical modeling, finite element method, simulation, internal mixers, elastomers

INTRODUCTION

The process of mixing of elastomeric materials withthe compounding ingredients such as carbon black,oil and other additives in internal mixers is consid-ered to be the most important step in determiningthe properties of the final product . Internal mixershave been used as a dominant rubber mixing devi-ce for more than a century . It was, however, onlyin the twentieth century that the widespread use ofrubber compounds resulted in extensive use of int-ernal mixers . It should be pointed out that thereare two types of internal mixers used by rubber

industries namely the Banbury mixer and theFrancis Shaw Intermix[1 ] . These types of mixersdiffer from each other due to the shape of rotorsand speeds at which these rotors rotate.

In this research work we simulate the flowinside the Banbury mixer. The development of thismixer to its current form is attributed to theBirmingham Iron Foundry (later Farrel Corpor-ation) . A Banbury mixer consists of a mixingchamber shaped like the symbol used for digit eightwith a rotor in each part . Each rotor has two orfour wings on its surface. The rotors are operatedat different speeds .The compounding ingredients

30

A Taasieut Finite Eleneat Model or Miring

are fed to the mixing chamber through a verticalchute in which an air or a hydraulic driven ram islocated . The homogenized mixture is dischargedthrough a drop-door at the bottom.

There have been numerous attempts in thepast to develop mathematical models in order tostudy the flow and mixing in internal mixers. This isbecause a mathematical model can predict stress,velocity and temperature distributions in itsdomain . This in turn can be used to optimize thedesign or the operating conditions and thus redu-cing the cost and the energy consumption.

BackgroundThe early flow analysis in an internal mixer can beattributed to Bergen, et aL [21 which dates back tolate 1950s . It is a very simple analytical model forthe shearing flow of a viscous fluid between theblade and the chamber wall . Tadmor[3] alsopresented a more sophisticated but analyticalmodel based on the assumption of isothermal,Newtonian fluid for the flow between two longconcentric cylinders with a short low clearance andnegligible curvature. Following these early models,many investigators tried to use simple numericalmethods to develop more complicated models.

White et al . 141 used the lubrication theoryand modelled the flow in an internal mixer basedon the assumption of an isothermal flow regimeand Newtonian fluid. The (mite difference methodwas used to solve the derived set of the equations.Kim el al. [5] and White et a2. [6] improved thismodel to include the effect of non-Newtonianbehaviour . Kim et aL[7] developed a 3-dimensionalnon-Newtonian non-isothermal model based onthe lubrication theory to predict the longitudinalpumping effect.

Hu et ai. [8—9] introduced a more rigorousapproach than the previous model based on theLubrication theory extended to cylindrical coordina-tes which permits full consideration of curvature ofthe flow domain . This analysis, however, was limit-ed to Newtonian fluids and fully filled chamber.

It is well known that during the mixing ofelastomeric materials in an internal mixer, a verycomplicated 3-dimensional flow field with multiple

free surface is established . Therheologicalbehav-iour of rubber compounds also changes in responseto mixing process and cannot be adequatelydescribed even using apparently complicated equa-tions . Additional difficulties arise from theincorporation of boundary conditions affectingmixer flow field . Consequently there has beenvarious attempts by different researchers who havetried to develop mathematical models of mixingprocess using advanced numerical technique suchas the finite element method.

Cheng et aL[10—14] used FIDAP (a fluiddynamic package based on the finite elementmethod) to analyze the flow and to study thedispersive mixing in a Banbury mixer (type B). Theanalyses were performed based on 2-dimensional,steady state and transient along with the isothermaland non-isothermal flow regimes, as well as thenon-Newtonian (power-law) fluid conditions.

Yang et aL [15] developed a 3-dimensionalmodel for the flow of rubber inside a BB-2 typeBanbury mixer in order to study the materialmotion in the direction normal to the rotor wingsusing FIDAP package. Wang et aL[16], in a recentpaper, started the study of the distributive mixingby using their previously mentioned model basedon FIDAP.

Nassehi, et aL[17] performed a wide series offinite element analysis for rubber mixing andcarried out finite element analysis of steady state,non-isothermal power-law fluids within a2-dimensional domain representing real mixergeometry . A fully filled chamber was selected . Inorder to study the effect of viscoelastic behaviourof rubbery materials, Nassehi et aL[18] used a newfinite element algorithm to simulate a 2-dimen-sional steady viscoelastic flow inside an internalmixer . The main aim of this study was to find thelimiting stress at which the .material is preventedfrom entering the narrow gap between the bladeand chamber wall . Nassehi el aL [19] attempted toapply a viscometric constitutive equation known as(CEF) model (a simple viscoelastic model) to thesimulation of predominantly circumferential flow ina 2-dimensional representation of an internal mixerusing polar coordinate system.

31

Iranian Polymer Journal Volume 5 Number I [1995}

Gboreiehy Ni K R

Present StudyThe present research work is devoted to themathematical modelling q f flow of an elastomericmaterial inside the Banbury mixer . The set of thegoverning equations are solved using finite elementtechnique via an in-house developed computercode. This computer program is used to simulatethe flow inside a typical mixer.

MATHEMATICAL MODEL

The governing equations of the transient, non-iso-thermal flow of an incompressible non-Newtonianfluid in a two dimensional Cartesian coordinatesystem in the absence of body force are given asthe following equation:

-continuity equation:

ax +=U

(1)

-x component of the momentum equation:

+ u au + au)= — a~ + az- + aT1Y

ax

ay

ax

ax

ay

(2)

where t1 is shear dependent non-Newtonian visco-sity of the fluid. The rate-of-deformation tensor isdefined as:

` au

&v + cru

ax

ax

ay

(6)ay +

ax

l ay

Viscosity it in the present study is given by apower-law type equation known as Carreau model

[20]:(n-1)n

t] = 7~o [1 - Ao2( 2—I2)]

e'bgTO)

(7)

where sio is the consistency of the fluid, n is thepower-law index, To is a reference temperature, bis the temperature sensitivity coefficient, .o is therelaxation time reflecting the elastic behaviour ofthe fluid and I2 is the second invariant of rate-of-deformation tensor defined as:

I2=(24) 2 + 2(--+ a)2 +(2a )2

(8)

The source term Q in the energy equation(4)represents the viscous heat dissipation effect and isexpressed by:

p( at

-y component of the momentum equation:

p(a +u—+va)–

as + '9Q = 2''712

(9)

(3)-energy equation:

pCK— aT aT

=—ks ) + a (k~T }+Qat + u ax + v--)

ay ax ax ay ay

(4)In these equations u and v are the x and y

components of the velocity vector, p is thepressure, T is the temperature, p is the materialdensity, Cp is the heat capacity of the rubber, k isthe thermal conductivity of the rubber, T,s, Tv, Ty?,

are the components of viscous stress tensor r whichare given for a generalized Newtonian model interms of rate-of-deformation tensor y by:

(5)

FINITE ELEMENT FORMULATION

Flow EquationsThe finite element formulation of flow equationscan be based on either a pressure-velocity (mixedor u-v-p) scheme or the use of the penalty meth-ods . In the present study we have selected thepenalty technique because it produces a morecompact set of working equations, thus, reducingthe required computer storage and computationalcost . Furthermore, there is some evidence whichshows that for highly viscous fluid the penaltymethod gives more accurate solution than themixed method . The pressure field is found by asecondary calculation based on the variationalrecovery method[21].

Iranian Framer Journal Volume 5 Number 1 11996)

32

A Tnviext F;oilr Banco[ Model of Arians

The basic step in the penalty formulation isthe elimination of the pressure term in momentumequations using:

p= –A(

+ -)

(10)

where 2 is a penalty parameter . It can be shownthat if we choose 1 to be a relatively large number,the continuity equation will be satisfied. It is re-commended that the value of the . A be consideredas a function of viscosity to ensure uniform con-tinuity enforcement in non-Newtonian problems[22] . Therefore equation (10) is written as:

p=-%(a + -)

(11)

where:

A. = ?IA

(12)

In this equation tl is the local viscosity and). is alarge positive number (say, 10 19) . There are twoways of using the relation (11) in order to eli-minate the pressure in the momentum equations.These are known as the continuous and the dis-crete penalty methods[22] . In the continuouspenalty method the relation (11) is directlyincorporated into the momentum equations whilein the discrete penalty method the modified con-tinuity equation (11) is treated in the usualGalerkin finite element method to give the discret-ized form . In the present research work, thederived set of working equations are based on theuse of continuous penalty method.

Following the basic procedures of theweighted residuals finite element techniques, themomentum equations are multiplied by weightfunctions and the integral of the obtained residualsover the solution domain (Qe) are set to be zero,thus we obtain:

foilix au+u_

4321 +v au)+

2 (2,Tau )at

ax

ay

ax ax

axa au

[rJ(

+ a)}}dQ

(13)r3y

ay

fnw2{p( aV +u ' + Vav)+~– a (297-)at

ax

ay

ay ay

ay

a [tl(sy a]]}d5a

(14)

Substituting the equation (11) into aboveequations and applying the divergence theorem, wewill have the weak variational forms of thegoverning equations as:

fp{[p(a +uat + au

au)]wt+[271 +Aax +ay

ax

)] 1 +[li(aau av

y + ax )] as

yyay}dS~-awl{[2i ax +

ay ax

A'(- +ay—An. + (- + a-x )ny}ds = 0 (15)

fa{[F► ( at + u + v8—y )]

w2 + [2a,- +A'( +

av am

auyy—)]—+ [~(

a a }dsa—~kw2{[2tj aiay

ay

A'(ax + aY )iny + o (a, + av ),.)ds = 0 (16)

In the finite element context the solutiondomain is discretized into a mesh consisting offinite elements (Q.) and over each element weapproximate u and v by interpolation functions as:

11

v = vi(t)Wi(x,Y)

( 17 )1 =3

where'i(x,y) are the interpolation functions andu;(t) , v,(t) are the nodal velocities . Using Galerkinmethod in which the weight functions are the sameas the interpolation functions, the finite elementmethod working equations associated with penaltyformulation are derived as

: [Kul Pcli

[Ill01] [M' l i] [ [ 1 ]+ [ Kms ] K9 J[ [v] J

(Mll)y = (M = ff"% dxdy

(19)

(kll)i, = ffp ( t J1 ' + q•

) dxdy +.ax

ay

f f (moi ai ate; + a?il, a!i1;) dxdy + f fA. a lit;

ax ax

ay ay

ax

(18)[Fl ][F2]

where:

33

Irwiimt Polymer Journal Volume 5 Number I (1996)

Gkoreii y M. a R

~aW.

7Idxdy

( 2o)

(kt2)ii = ff a$i ate;dxdy + ff'{

.a i a ► ; dxdyay ax

ax ay

(21)

(k2i ) ii = ff

j' dxdy + f fA'4 ` alFJdxdyair

(22)

(k9ri=ffP(> r

+

' )dxdy+ff(2:1

ax

ay

ayate; + ami ate; )dxdy + f

f). a-i a7ii dxdyay

ax ax

ay ay

(23)

(F')i= f'Pi{jai

au

aunx+i7( + a)ny — pnx}dFax

ay

(24)

(F2)i = f'Pi{2i ay ny + rf( ay +

)n, — pny} dT(25)

Equation (18) can be written in the compact formas:

[M]IX} + [K]{X} = {F}

(26)

where [M] is the mass matrix, [K] is the stiffnessmatrix, {F} is the right hand side (load vector),{X} is the first derivative of vector of unknownand {X) is the solution vector . In order to obtain anon-trivial solution the penalty terms in thestiffness matrix coefficients must be evaluatedusing reduced integration . For example, whenusing 9-noded element, 2-point Gauss quadraturemust be used to evaluate the terms includingpenalty parameters while the 3-point Gaussquadrature is used to calculate the non-penaltyterms[ 23].

Energy EquationApplying the Galerkin method to equation (4) andcarrying out the integration by parts (divergencetheorem), the weak variational form of the energyequations is obtained as:

Iranian Polymer J,,wnni Volume 5 Number I (FM)

fp i pC a-Td(2 + f.wt pCp(

+ v -)dQ +

fpk(OW' al

i~t-

aWi

aT+

ay )dQ= frWi(k)dF+ax ax

ay

an

f AwiodQ (27)

The temperature field over a typical element isapproximated in terms of interpolation functionsas:

i=1

T =lT;(t)Wi(x,y)

(28)i .i

Thus the finite element working equations ofthe energy equation will be given as:

[M] {T) + {[Kv] + [Kc]} {T} _ {F}

(29)

where [M] is the mass matrix, [KY] and [Kg] are theconvection and conduction stiffness matrices, ( T}and {T} are the solution vector of the temperaturefield and its first order derivative, respectively, and{F} is the load vector. These matrices are given bythe following relations:

(M)ii = f fPepciWidxdy (30)

alP'= f fPCp [wi

+

a' (31)(K`)ii

(t

v

)]may

(K = ffk(a

i a ►F; + ami ate; )dxdy (32)ax

ax

ay

ay

(F)i = frMi(kan

)dT + ,[',f 1F Qdxdy (33)

The weight functions wi have been used for theconvection term, (KY) to distinguish it from the Tiused in other terms. This is due to the very lowthermal conductivity of elastomeric materials whichresults the heat transfer to be essentially byconvection mechanism. Mathematically this meansthat the first order derivatives in equation (4) aredominant . To overcome any numerical instabilityassociated with the treatment of first order deriva-tives a streamline upwinding approach[24] speci-ally developed for biquadratic element is used. Theweight function for convective terms is written as:

wi = Ti + di(u 5— + v ')

(34)x

ay

34

A Tr .n.innr Finite Elruoas Model of

g

where 4' is a multiplier known as upwindingparameter . The value of d) determines the amountof upwinding applied into the solution scheme.The optimum value of 4) for two-dimensionalbiquadratic elements is given as:

= I & al vt2+IA,l v,,2

1M 2I v l

where v is the velocity vector, vt and v, are thevelocity components in directions of e t and e,, (unitvectors directed along the mapped local coordin-ates at integration points) finally At and A, are twoparameters defined in terms of the mesh Pecletnumber[25].

Solution of Transient EquationsThe previously developed finite element workingequations are first order set of differentialequations with respect to time . q f those availabletime integration methods for the solution of thesetype of equations, we use a well known single timestep procedure belonging to 0 family of approxim-ation methods . In this technique direct numericalintegration is applied and a recurrence formulawhich relates the values of solution vector at time(t) to the values of the solution vector at a latertime (t + At) is obtained . To illustrate this methodbriefly, let (to) denote a typical time between (to)

and (tn+i) so that to = to + Oat where 0 S 0 � 1.We write the final form of finite element

formulation at time to as:

[M]{X}a + [K]{X}, = {F}e (36)

By introducing the following approximations :

(37)At

{X} = (1—B){X}n + B{X}n+ t (38)

{F}e = (I–0){F}n + 9{F} .+i (39)

and substituting the above equations into equation(36), we obtain:

QM] + OAt[K]) {X} .+, _ ([M] -- (1–9)s t[K])

{X}n + ((1–O){F} 1 + B{F},+ t )At

(40)

where {X},+t on the left hand side is unknown andall of the terms on the right hand side are known.For a given value of 0, equation (40) represents arecurrence formula in which the nodal values{X}o+t are calculated from the known values of{X}n at the beginning of the time step . It thepresent study, we selected the B = 1 (backward orfull implicit method), thus, provided an uncondi-tionally stable scheme.

Boundary and Initial ConditionsFor momentum equations, it is assumed that theshear stress at walls is less than a critical value,thus, the rubber compound completely sticks to thewall (no-slip) . In this case all of the prescribedboundary conditions are either of the first type(essential or Dirichlet boundary conditions) or ofthe second type (natural or von-Neumann bound-ary conditions). The flow domain which we study inthis research work consists of a fully filled chamberwith a Banbury type rotor blade (Figure 1) . Weassume that the pressure is zero along the inlet andoutlet surfaces of the flow domain : Therefore theline integral in the right hand side of the workingequations vanishes and the only boundary condi-tions which need to be imposed are Dirichlet (firsttype) conditions on the domain walls . The boun-dary conditions for the energy equation are only ofthe first type i.e . prescription of temperaturevalues at chamber wall, blade surface and inlet ofthe flow domain (Figure 1) . Zero flux condition is

Figure 1 . Flow domain with boundary conditions.

(35)

35

Iranian Polymer Journal Volume 5 Nvmbcr 1 {1996,1.

Q1arrisby M. H . R.

given at the domain exit . For the initial conditions,it is assumed that all of the materials and boundar-ies have a zero velocity and uniform temperatureequal to 298 K (25 'C).

GLOBAL SOLUTION STRATEGY

Using the isoparametric mapping[21] the workingequations of the present scheme are cast into alocal (natural) coordinate system. The members ofthe coefficient matrices are then computed foreach element by a Gauss quadrature method . Theresulting algebraic equations are assembled into aglobal matrix and after imposing the appropriateset of boundary conditions are solved by a frontalsolution algorithm[2F] . The presence of the con-vective terms in the momentum and energy equa-tions as well as the dependency of the localviscosity gradients, make this set of equationsnonlinear . Consequently a de-coupled iterativeprocedure based on the successive substitutionmethod (Picard iteration method) [22] was adopt-ed. In this procedure, at the beginning of the firstiteration, the velocity field is set to zero and thetemperature field is set to be equal to the initialconditions, and the coefficient matrices are compu-ted and assembled . The global equations are solvedto obtain the velocity field . The obtained velocityfield in turn is used in the calculation of viscosityand the solution of the energy equation. Using thecomputed velocity and temperature fields at theend of the first iteration as initial estimates, theprocedure is repeated until the velocity and thetemperature fields are converged . The converg-ence criterion used in this work is given by:

N

LI x-r+1 — Xi' I 2

al

bs (5

(41)

E I xr+b l 2i = l

where X i' denotes the flow variables (velocity ortemperature) at degree of freedom i at iterationcycle of r, and d is the convergence tolerance.After the solution of the governing equations, thesolution procedure is forwarded for a new time

step and this is repeated until the end of thenumber of desired time steps has been reached.

RESULTS AND DISCUSSION

A mixing chamber of 0.lm radius and gap width(distance between the blade tip and the chamberwall) of 0 .006m is considered . The chamber wallrotates at 0.2m/sec and the blade is assumed to bestationary. The temperature at both walls (chamberand blade surface) is maintained at 323 K and thetemperature at the inlet of the domain is 360 K(Figure 1) . The physical and rheological propertiesof the rubber compound are as:

r]o = 2 X 10' Pa .s'n = 0.2b = 0.014K-1

To = 373 KRo = 1 .15 sp = 1055 kglm3Cp = 1255 J/kg.Kk = 0 .13 W/m I{.

The finite element mesh for this analysisconsists of 210 9—noded Lagrangian elements withtotal number of nodes equal to 923 (Figure 2).

The problem was analyzed under both steadystate and transient conditions for a non-isothermalflow regime. For the steady state case, the results

Figure 2 . Finite element mesh model.

Iranian Pdymer Journal Volume 5 Number 1 (19%)

36

A Transient Finite Pianism Model of Mixing

are shown in Figures 3-6 . Figure 3(a) shows thevelocity vector field in the flow domain. The detailsof the velocity field at the gap region is drawn inFigure 3(b) . The flow streamlines are alsopresented in Figure 4 . As it can be seen, the flowdomain is divided into two parts . The first partinvolves the region between the blade tip and thechamber wall and its associated trailing zone . Theflow pattern in this region is similar to drag orCouette flow . In the other region, the fluidundergoes a circular motion . These streamlines arein total accordance with the velocity fieldpresented in Figure 3(a,b) . The pressure distribu-tion is shown in Figure 5 . The highest value for thepressure is about 0 .3 x10 7 Pa and it occurs asexpected at the vicinity of the blade tip . Theexperimentally measured pressure at the proximityof the blade is between 0 .25 X 107 and 0.35X107[27] and as it can be seen it is in good agreement

with calculated results . The temperature distribu-tion is presented in Figure 6. The maximum com-puted temperature is around 371 K which belongsto the region near the inlet where the thickness of-rubber compound is high . It is expected that, thelow thermal conductivity of the material shouldresult in the rise of temperature at this region.

The problem is also analyzed for transientcase. The time increment for this model is selectedto be equal to 0 .05 s . The variations of pressureand temperature for three nodes located at near tothe entrance to the gap, inside and outside of thegap (see Figure 7) are given in Figures 8 and 9.Figure 8 shows the variation of the pressure as afunction of time. As can be seen, the pressuredecreases from a maximum value at the initial timeto its steady state value at the final time. A similartrend was experimentally measured[27].

The variation of the temperature for three

Figuer 3a . Velocity field, (the velocity vectors and the flow domain are plotted using different scales).

37

Iranian Pu6rner Journal Yolwne 5 Number I {1996)

Ghoreidy M . H. a

Figure 3b. Enlarged view of the velocity field for the gap region.

mentioned nodes are also shown in Figure 9. Agradual increment is obtained from the initial value(298 K) to the final values which are equal towardthe steady state results. Comparison of the obtain-ed results at the final time with those from thesteady state ones, show the expected gradual trendof the values towards a steady state plateau. Thusthe program predicts the quantitative transient be-haviour of the model accurately.

CONCLUSION

Figure 4 . Streamline contours of the flow.

We have developed a computer model based onthe combination of a 2-dimensional mathematicalmodel, a finite element solution scheme and acomputer program. This computer model is used to

a typical Banbury mixer under fully filled condition.simulate the flow of an elastomeric material inside

Existing experimental data are in good agreement

Imnime Polymer Journal Volume 5 Number I (1996)

38

A hveient Finite Element Model of Mixing

Figure 5. Distribution of pressure in flow domain.

with model predictions . It should be pointed outthat, however, the lack of a laboratory mixer whichis specially equipped with highly sophisticatedinstrumentations to measure the model predictedparameters accurately, prevents us to check thevalidity of the model more exceedingly . On theother hand, there are various parameters whichshould be included into the described model toconsider the flow and the mixing process in a real

mixer . The most important of them can be express-ed as the tracking of the established free surfaces,viscoelastic rheological behaviour of the rubbercompounds changing in response to mixing, andslipstick of rubber on mixer walls.

We plan to continue this research project toinclude the mentioned parameters into our model.This will enable us to tackle the mixing problem ina more realistic condition, thus promoting the

Figure 6 . Distribution of temperature in flow domain.

39

1rmwmx Polymer Journal Vohme 5 Number 1 {1996)

OLouae ' M. IL R.

Figure 7. Location of the nodes used or the describing the

variations of pressure and temperature with time.

agreement of results of the computer simulation tothe obtained data from experiments more greatly.

ACKNOWLEDGEMENT

The author wishes to thank the Chemical Engi-neering Department of Loughborough Universityof Technology, UK for their guidance and Dr.V.Nassehi in the same department for his veryhelpful suggestions to carry this research work_

NOMENCLATURE

b

Temperature sensitivity.Cr. Heat capacity.e6 e, Unit vectors directed along the mapped local

coordinates.PiLoad vector.I2

Second invariant of rate-of-deformation ten-sor.

k

Thermal conductivity.Stiffness matrix.

Mo Mass matrix.n

Power-law index.ni,ny Components of the unit vector normal to the

boundary.pPressure.

5 .00E+05 -.

4.00E+05a

3.00E+05 .

1 2.00E+05 .

1 .00E+05 .

'3 4 5 6 7 8 9 10

Time (s)

Figure 0. Variation of pressure with time.

Q

Generated heat due to the viscosity.t

Time.T Temperature.To Reference temperature.u,v Components of the velocity vector.vf,v1 Components of the velocity vector in direc-

tion of ef,e,,.wt,w2 Weight functions.Xi Solution vector.y

Rate-of-deformation tensor.d

A small pre-defined value used in conver-gence criterion.Viscosity.

380.00 -

370.00 -

360.00 -

A :N857. s.eol=986.53

B:N527, a .sol =385 .76

C:N761, s.so1=98242

290.00 , ,0 1 2 3 4 5 6 7 8 9 10

Time (s)

Figure 9. Variation of temperature with time.

0.00E+00 .

-1 .00E+050 1 2

Node-527, steady eolutlall=0.224e+5

Node-e57. steady sol Ion=0.1120+5

tfada, 81, steady sol n=-0.64e+5

350.00 -

340.00

E. 330.00 -

320.00 -

Iranian Polymer Journal Volume 5 Number I (1995)

40

A Tramauiu Haire Menem Model al Minas

t']o

Fluid consistency.B

Parameter in time approximation.A

Penalty parameter.Ao

Relaxation time.ArViscosity dependent penalty parameter.Ae , A,'Iwo parameter used in upwinding formul-

ation.p

Density.3 Viscous stress tensor.• Upwinding parameter.~l Shape function.Q Solution domain.

REFERENCES

1. White J . L., Rub. Chem. Technol., 65, 527, 1992.2. Bergen J . T ., Mixing and Dispersing Processes. Processing

of Thermoplastic Materials, Bernhardt E. C . Ed., Chapter

7, Reinhold, New York, I960.

3. Tadmor Z . and Gogos C_G ., Principles of PolymerProcessing, Chapter 11, Wiley, New York, 1979.

4. White J . L. et al., Polym. Composites, 9, 368, 1988.

5. Kim J . K., White J . L ., Min K. and Szylowski W ., Int.

Polym. Proc ., 4, 9, 1989.

6. White J . L . and Kim J. K., !. AppL Polym. Sci . Appl.

Poly'''. Symp., 44, 59, 1989.

7. Kim J. K. and White J. L, Ins. Polym Proc ., 6, 103, 1991.

& Ho B . and White J . L., Int. Pdym Atm ., 8, 18, 1993.9. Hu B . and White J . L., Rubber Chem. Technol ., 66, 257,

1993.

10. Cheng J. J. and Manas-Zloczower I ., 7. AppL Polym. Sci:AppL Polym Syenp ., 44, 35, 1989.

11. Cheng J . J . and Manas-Zloczower I_, Polym. Eng Sci., 29,1059, 1989,

12. Cheng J . J . and Manaa-Zloczower I ., ANTEC 89, 1870,

1989.13. Cheng J. J. and Manaa-Zlocanwer I ., In[ Polym . Proc ., 5,

178, 1990.

14. Cheng J . L and Manas-Zloczower I., Polyvn. Eng Sci., 29,701, 1989.

15. Yang H. H. and Manaa-Zloczower I ., Ins. Polym. Proc ., 7,195, 1992.

16. Wong T. H. and Manas-Zioczower L, Ins. Polym Proc ., 9,3, 1994.

17. Nassehi V . and Freakley P . K., Int. Polym . Proc., 6, 91,

1991.

18. Nassehi V. and Ghavami Naar G ., International Congresson Computational Methods in Engineering, Shiraz 2—6

May 1993.

19. Nassehi V. and Salemi R., Int. Polym. Proc ., 9, 199, 1994.

20. Bird R. B ., Armstrong R . C . and Hassager q ., Dynamics

of Polymeric Liquids, 2nd ed., Vol . 1 : Fluid Mechanics,

Wiley, New York, 1987.

21. Zienkiewicz 0 . C ., Taylor R . L., The Finite Element

Method, McGraw-Hill, London, 1989.22. Pittman J . F. T., P sndwnsnsals of Computer Modeling for

Polymer Processing, Tucker Ed ., Chapter 6, Hawser,

Munich, 1990.

23. Reddy J. N ., An Introduction to the Finite Element

Method, McGraw-HE, New York, 1984.

24. Petera J ., Nassehi V ., and Pittman J. F . T., Int. I. Num.

Meth. Heat Fluid Flow, 3, 205, 1993.

25. Donea J. et al., Computer Methods in Applied Mechanics

and Engineering, 48, 25, 1985.

26. Hood P., Lis. J. Nuns. Meth. Eng., 10, 379, 1976.

27. Freakley P . K. and Patel S . R ., Rubber Chem . Technol .,

58, 751, 1985.

41

Lrmian Polymer Journal Volume 5 Number 1 (1996)