a finite piece method for simulating polymer melt flows in extrusion sheet dies

A Finite Piece Method for Simulating Polymer Melt Flowsin Extrusion Sheet Dies

Yong Li, Jianrong Zheng, Chunmin Xia, Wei Zhou

Department of Mechanical Engineering, School of Mechanical and Power Engineering, East China University ofScience and Technology Shanghai, 200237, People’s Republic of China

Received 26 October 2010; accepted 26 May 2011DOI 10.1002/app.34979Published online 2 September 2011 in Wiley Online Library (wileyonlinelibrary.com).

ABSTRACT: A finite piece method is proposed to simu-late three-dimensional slit flows in extrusion sheet dies inthis paper. The simulations concern incompressible fluidsobeying different constitutive equations: generalized New-tonian (Carreau-Yasuda law), and viscoelastic Phan-ThienTanner (PTT) models. Numerical simulations are carriedout for the isothermal and nonisothermal flows of polymermelt through sheet dies. The Picard iteration method is uti-lized to solve nonlinear equations. The results of the finitepiece method are compared with the three-dimensional(3D) finite element method (FEM) simulation and experi-ments. At the die exit, the relative error of the volumetric

flow between the finite piece method and the 3D FEM isbelow 1.2%. The discrepancy of the pressure distributionsdoes not exceed 6%. The Maximum error of the uniformityindex between the simulations and experiments is about2.3%. It shows that the solution accuracy of the finite piecemethod is excellent, and a substantial amount of computingtime and memory requirement can be saved. VC 2011 WileyPeriodicals, Inc. J Appl Polym Sci 123: 3189–3198, 2012

Key words: polymer extrusion; viscoelastic properties;simulations; PTT constitutive equation; the finite elementmethod

INTRODUCTION

The manufacture of plastic films and sheets is fre-quently accomplished by using the so-called fish-tailor coat-hanger dies, in which the polymeric fluidflows through the gap with large ratio of width tothickness. The slit flow is a typical problem for thiskind of dies. The main rheological concern in thedesign of the channels is to obtain as uniform a meltflow distribution as possible. Obviously, knowledgeof how a shear thinning and viscoelastic polymermelts flow inside the various die sections is neces-sary for proper design.

Traditional FEM can handle the 3D flow problemof the polymer melt. In some of the numerical stud-ies, the flows of the polymer melt in sheet dies havebeen simulated using the 3D FEM to solve for thepressures, velocities, and temperatures.1–5 Variousgeneralized Newtonian fluid models have beenincluded in these simulations. But an exact mathe-matical analysis of viscoelastic flows in the dies is avery challenging proposition because of the com-plexities with geometrical and viscoelastic effects.

On the other hand, it can be justified that thepolymeric fluid motion follows the two-dimensional(2D) Hele Shaw model6 between the gaps in the

sheet dies. The 2D simulations provide a computa-tionally efficient alternative to the 3D methods.Vlcek et al.7 used a control volume method to per-formed a 2D simulation on the flow distribution inslit dies. Liu et al.8 combined both the simple one-dimensional lubrication approximation and the 3Dfinite element simulation to design the extrusionsheet die for viscoplastic fluids with a Binghammodel. Yu9 divided the flow regime into 2D and 3Dregions, and proposed a hybrid 3D/2D FEM to sim-ulate motions of Carreau fluid. Moreover, noniso-thermal flows in coat-hanger dies have been investi-gated in some studies.10 Arpin et al.11 developed anew model using a modified flow analysis networkmethod for the calculation of the 2D flow inside acoat-hanger die, coupled with a finite-differencescheme for the calculation of temperature. The diebody was also analyzed coupled to the melt flowanalysis to minimize the die-inlet pressure.12,13

Smith et al.14,15 performed optimization of extrusionsheet dies by using a gradient-based optimizationalgorithm to minimize the total pressure drop acrossthe die and reduce the velocity variation at the dieexit.Among the constitutive models used in the poly-

mer flow simulation, the nonlinear PTT16 constitu-tive equation provides a better fitting to the rheologyof polymer melts than other simpler models such aspower-law, the upper convected Maxwell or Old-royd-B. And it is usually used to test the efficiencyof numerical method. The numerical study using

Correspondence to: Y. Li ([email protected]).

Journal of Applied Polymer Science, Vol. 123, 3189–3198 (2012)VC 2011 Wiley Periodicals, Inc.

this model was initiated with the simulation of 2Dand 3D steady contraction flow.17–21 More recentstudies consider 3D extrusion flows22 and 3D freesurface flows.23 But it is only used in some simpleflow channels of the 3D problems.

The channel of the sheet dies is wide and thingaps, and the primary focus of most sheet diedesigns is the development of a uniform velocityacross the width of the die exit. In addition geomet-rical complexity and rheological behavior lead to ahuge CPU and pro-process time required for 3Dsimulations of viscoelastic flows in the dies. There-fore a complete 3D analysis is unnecessary. In thisstudy, a finite piece method is established to predictthe isothermal and nonisothermal viscoelastic flowthrough a planar slit channel. It is a finite elementsemianalytical method which can be adopted toreduce the computation memory requirement for 3Dproblems. The mathematical model of the 3D flowsthrough the sheet dies is established with Carreau-Yasuda and PTT constitutive model. The SU technol-ogy24 is employed to improve the computation sta-bility. The flows of LLDPE polymer melts in twokinds of sheet dies are analyzed and the results arecompared with those from 3D FEM and Arpin’sexperiments.11

Governing equations

The flow is governed by the general equations formass and momentum. Considering the characteris-tics of the polymer melts flow in a die channel, foran incompressible fluid under creeping flow condi-tions (Re ¼ 0), the motion equation and the continu-ity equation are given by:

�rPþr � s ¼ 0 (1)

r �V ¼ 0 (2)

where P is the isotropic pressure, s is the extrastresstensor, V is the velocity vector.

Rheological models of two types are adopted forthe simulations. For a non-Newtonian (generalizedNewtonian) fluid, s is expressed by the equation:

s ¼ 2gðIIDÞD ¼ 2gðIIDÞðrVþrVTÞ (3)

where D is the rate of strain tensor, the non-Newto-nian viscosity obeys a Carreau-Yasuada law such that:

gðIIDÞ ¼ g0ð1þ ðk _cÞaÞðn�1Þ=a (4)

where _c ¼ ð2PDÞ1=2 and PD is the second invariantof the rate of strain tensor. a, n, and k are constants.

For a viscoelastic fluid, the polymer extrastresstensor can be splited into a purely viscous part anda polymeric contribution:

s ¼ ss þ S (5)

here S is the extrastress tensor due to viscoelasticityand ss is the stress component of pure Newtonianfluid given by

ss ¼ 2gsD (6)

where gs is the Newtonian-contribution solvent vis-cosity. To complete the description, a constitutiveequation that describes the rheology of the fluid isrequired to determine the polymeric part of the extrastress tensor. The constitutive equation for a PTTfluid may be expressed as:

YðSÞSþ k½ð1� n=2ÞSrþðn=2ÞS

D� ¼ 2gvD (7)

In the above equations, Sr

and SD

denote, respec-tively, the contravariant and the covariant deriva-tives of the stress tensor:

Sr¼ ðV � rÞS�rVT � S� S � rV (8)

SD¼ ðV � rÞSþ S � ðrVÞT þrV � S (9)

The stress function Y(S) follows the exponentialform:

YðSÞ ¼ expððek=gvÞTrS (10)

In the above equations, k is the relaxation time, gv

is the zero shear polymer viscosity. The total viscos-ity g0 ¼ gs þ gv. PTT models have two scalarparameters which can be used to control viscoelastic.The parameter n governs the ratio of the first to thesecond normal stress. The other parameter e imposesan upper limit to the elongational viscosity, whichincreases as this parameter decreases.For the nonisothermal flow, the energy equation is

express as:

qCpDT

Dt¼ kr2T þ s : D (11)

where T is the temperature, q denotes the fluid den-sity, k is the thermal conductivity, Cp is the heatcapacity. The temperature dependence of the viscos-ity and the relaxation time is described by using ashift factor A(T) and B(T). Accordingly, the viscosityg0 and the relaxation time k are expressed in termsof the temperature by means of the followingrelations:

g0ðTÞ ¼ AðTÞg0ðT0Þ (12)

kðTÞ ¼ BðTÞkðT0Þ (13)

3190 LI ET AL.

Journal of Applied Polymer Science DOI 10.1002/app

In these equations, g0 (T0) and k (T0) denote therespective viscosity and relaxation time at a refer-ence temperature T0. The shift factor are given by11

AðTÞ ¼ exp C11

T� 1

T0

� �� (14)

BðTÞ ¼ exp C21

T� 1

T0

� �� (15)

where C1 and C2 are the WLF parameters.For the slit flow channel, the size of thickness is

much less than others. We postulate the pressuredoes not change along the thickness direction andcreate Cartesian coordinate system which the Z axisis along the thickness direction of the slit channel, asshown in Figure 1, so the eqs. (1) and (2) can berewritten as the component type:

� @p

@xþ @sxx

@xþ @syx

@yþ @szx

@z¼ 0 (16)

� @p

@yþ @sxy

@xþ @syy

@yþ @sxz

@z¼ 0 (17)

� @p

@z¼ 0 (18)

@vx@x

þ @vy

@yþ @vz

@z¼ 0 (19)

Postulating the lower surface of the channel is h1(x,y), the upper surfaces is h2 (x,y), the integral of eq.(19) along the thickness isZ h2

h1

@vx@x

dzþZ h2

h1

@vy

@ydzþ

Z h2

h1

@vz@z

dz ¼ 0 (20)

and defining the volumetric flow as follow:

qx ¼Z h2

h1

vxdz (21)

qy ¼Z h2

h1

vydz (22)

we obtain:

@qx@x

þ @qy

@y¼ vxðh2Þ @h2

@xþ vyðh2Þ @h2

@y� vzðh2Þ

� vxðh1Þ @h1@x

þ vyðh1Þ @h1@y

� vzðh1Þ� �

ð23Þ

It can be found that the right end item is the nor-mal velocity of the upper and lower surface which isobviously zero, so the continuity equation related tothe volumetric flow is:

@qx@x

þ @qy

@y¼ 0 (24)

Numerical scheme

The length and width of sheet dies in the X, Y direc-tion is much longer than the thickness in the Z direc-tion, and a well-designed die shall provide smoothand uniform flow in the X-Y plane. Thus the key issueof the simulation is to acquire the flow distribution inthe X, Y direction. Since the flow without secondaryflow (e.g., vortex) is creeping, the flow channel withvarying thickness can be divided into several smallparallel regions (elements) of different height. Theflow inside the element can be assumed to be thesame as those inside the parallel flow channel; alsothe velocity vz and extrastress Szz can be ignored.Because the velocity and extrastress are still the func-tion of the coordinate Z, it still remains to be a 3Dproblem. The finite piece method which is one of thefinite element semianalysis methods with volumetricflow as unknown variables will be utilized to calculatethe flow distribution of the X, Y plane. In this way, thekey issue mentioned above can be simplified to a 2Dsolving procedure to obtain final solution.For the finite piece method, the distribution func-

tions of the velocity vx, vy and the extrastress S ten-sor related to the Z direction is assumed to beknown so that the finite element interpolating poly-nomial can be constructed without the Z coordinate.As the Fourier series can fit any continuous functionwithin the given region, it is easy to approximatethe curve by using the Fourier series. In otherwords, the finite piece method is about to constructan approximation function of the velocity and theextrastress by using the finite element interpolatingpolynomial in some directions and by continuoussmooth Fourier series fitting the boundary condi-tions in other directions. Assuming that:

S¼XRk¼1

Skðx;yÞFkðzÞ¼XRk¼1

XMi¼1

wiðx;yÞSki FkðzÞ¼XRk¼1

XMi¼1

Wki S

ki

(25)

Figure 1 Schematic diagram of the fish-tail die channeland the coordinate system.

POLYMER MELT FLOWS IN EXTRUSION SHEET DIES 3191


V ¼XRk¼1

Vkðx; yÞFkðzÞ ¼XRk¼1

XMi¼1

/iðx; yÞVki FkðzÞ

¼ PRk¼1

PMi¼1

UkiV

ki ð26Þ

The pressure approximation function still is:

p ¼XNi¼1

uiðx; yÞpi (27)

In the equations, the velocity V include the com-ponents vx and vy; the extrastress S include the com-ponents Sxx,Syy,Sxy,Sxz,Syz; Fk(z) is Fourier serieswhich meet the boundary conditions in the thicknessdirection; Ski , V

ki are the nodal extrastress and veloc-

ity corresponding to the kth item of the series,respectively; pi is the node pressure; wi (x,y), /i (x,y),ui (x,y) are respectively, the shape function of extra-stress, velocity, and pressure; Wk

i ¼ wi (x,y)Fk (z),Uk

i ¼ /i (x,y)Fk (z); M is total number of the nodewith stress and velocity; N is total number of thenode with pressure, R is the total number of theterm taken in the series.

In this article, we use quadrilateral elements witha quadratic interpolation for the velocity and extra-stress, a linear interpolation for the pressure field.According to above ideas, the element can only bedivided in the X-Y plane, but the ‘‘element’’ is thepieces 110220330440 with certain thickness of chan-nels, as shown in Figure 2, thus it is called thefinite piece method. For the finite elements method,the elements are connected by nodes but for thefinite piece method they are connected by lines, forexample 220, 330, and 440 in Figure 2. After theapproximation function of X-Y plane is constructed,according to eqs. (25) and (26), the velocity andextrastress distributions of the entire region can bedetermined.

For a viscoelastic fluid (PTT constitutive equa-tion), to overcome the effects of convection term theinconsistent SU method24 is adopted here and anadditional term of the weighted function isintroduced:

�Wki ¼

�kV

ðV �VÞ � rWki (28)

the coefficient �k is defined by the velocity compo-nents vn and vg at the element center:

�k ¼ ðv2n þ v2gÞ1=2.2 (29)

The Galerkin weighted residual method isadopted for the discretization of eqs. (1), (7), and

(24) where let residual value equal to Wki , U

ki , and ui.

The additional term only affects the purely advectiveterm of the constitutive equations and the followingelemental equations are obtained:

ZX

ðWki ðYðSÞSþ kðð1� n=2ÞS

rþðn=2ÞS

DÞ

�2gvDÞdXþZC

�Wki kV � rSdX ¼ 0 ð30Þ

ZX

ðrUki ÞT � ð�pIþ sÞdX ¼

ZC

tUki dC (31)

ZX

uir � qdX ¼ 0 (32)

In the equations, t is the known surface force, q isthe volumetric flow.The Picard iteration method is applied to solution

of above nonlinear equations. The algorithm of thenumerical method is described by a flow chart inFigure 3. Indeed, let S(n), V(n), q(n), and p(n) denoterespectively, the values of polymeric stress tensor,velocity, volumetric flow and pressure after last iter-ation, we use below relation:

sðnþ1Þ ¼ SðnÞ þ gsðrVðnþ1Þ þ rVðnþ1ÞTÞ (33)

Substituting it into eqs. (31) and (32):

ZX

� Pðnþ1Þ þ 2g

@vðnþ1ÞÞx

@x

!@Uk

i

@xdX

þZX

g

@v

ðnþ1Þx

@yþ @v

ðnþ1Þy

@x

!@Uk

i

@ydXþ

ZX

g@dvðnþ1Þ

x

@z

@Uki

@zdX

¼ �ZX

� Pþ @S

ðnÞxx

@x

!@Uk

i

@xdX�

ZX

@SðnÞyx

@y

@Uki

@ydX

�ZX

@SðnÞzx

@z

@Uki

@zdXþ

ZC

txUki dC ð34Þ

Figure 2 The finite piece mesh with thickness H.

3192 LI ET AL.


ZX

g

@v

ðnþ1Þy

@xþ @v

ðnþ1Þx

@y

!@Uk

i

@xdX

þZX

� Pðnþ1Þþ 2g

@vðnþ1Þy

@y

!@Uk

i

@ydXþ

ZX

g@v

ðnþ1Þy

@z

@Uki

@zdX

¼ �ZX

@SðnÞxy

@x

@Uki

@xdX�

ZX

� Pþ @S

ðnÞyy

@y

!@Uk

i

@ydX

�ZX

@SðnÞzy

@z

@Uki

@zdXþ

ZC

tyUki dC ð35Þ

ZX

@q

ðnþ1Þx

@xþ @q

ðnþ1Þy

@y

!/idX ¼ 0 (36)

Moreover, eqs. (34) and (35) are tenable onlywhen they are used inside the element. The velocityvalues of the same nodes in different elements with

different thickness are not equal; however, the volu-metric flow values are equal. Thus eqs. (34) and (35)are converted into equations with volumetric flow asthe unknown variables to combine the element equa-tions into a global equation.In this article, we only use the first item of series,

so the velocity is given by following form:

vxðx; y; zÞ ¼ vxðx; yÞ sin� pHz�

(37)

vyðx; y; zÞ ¼ vyðx; yÞ sin� pHz�

(38)

H is the thickness of the channel, integrating alongthe thickness, one obtains:

qxðx; yÞ ¼ZH0

vxðx; y; zÞdz ¼ 2H

pvxðx; yÞ (39)

qyðx; yÞ ¼ZH0

vyðx; y; zÞdz ¼ 2H

pvyðx; yÞ (40)

Substitute for vx,vy in eqs. (34) and (35) and changeUk

i to Ui ¼ /i sinpH z� �

, we can obtain following FEMequations with the variable of volumetric flow.

Aijqxj þ Cijqyj �DinPn ¼ Xi (41)

Cjiqxj þ Bijqyj � EinPn ¼ Yi (42)

Djnqxj þ Ejnqyj ¼ 0 (43)

Xi and Yi are the right-side items of eqs. (34), (35) iand j indicate the number of the velocity node, n isthe number of the pressure node, and:

Aij ¼ p2H

ZX

2g@Uj

@x

@Ui

@xdX

0@

þZX

g@Uj

@y

@Ui

@ydXþ

ZX

g@Uj

@z

@Ui

@zdX

1A

Bij ¼ p2H

ZX

g@Uj

@x

@Ui

@xdX

0@

þZX

2g@Uj

@y

@Ui

@ydXþ

ZX

g@Uj

@z

@Ui

@zdX

1A

Cklij ¼

p2H

ZX

g@Uj

@x

@Ui

@ydX;

Din ¼ZX

un

@Ui

@xdX; Ein ¼

ZX

un

@Ui

@ydX:

The unknown variable of eqs. (41), (42), and (43)is the volumetric flow q. Because the volumetric

Figure 3 Flow chart of the computational procedure.



flow is continuous at nodes, the global stiffness ma-trix equation for all the elements can be assembledafter the elemental calculation.

By the finite piece method, we divide elementwith thickness only in the XY plane. Compared with3D FEM, the number of the elements becomes muchless, and the preprocessing becomes more simple.

The extrastress tensor S in Xi and Yi of eqs. (41)and (42) is calculated by solving linear eq. (30). Theiterative algorithm is given by following form:

ZX

ðWki ðYðSðnÞÞSþ kðð1� n=2ÞSðnþ1Þ

rþðn=2ÞSðnþ1Þ

D

Þ

� 2gvDðnÞÞdXþ

ZC

�Wki kV

ðnÞ � rSðnþ1ÞdX ¼ 0 ð44Þ

In the equations, S(n), V(n), and D(n) denote respec-tively, the values of polymeric stress tensor, velocity,and rate of strain tensor after last iteration. Theextrastress tensor is assumed to be following formaccording to the boundary conditions:

Sijðx; y; zÞ ¼ Sijðx; yÞ sin� pHz�; i; j ¼ x; y (45)

Sijðx; y; zÞ ¼ Sijðx; yÞ cos� pHz�; i ¼ x; y; j ¼ z (46)

If the flow is nonisothermal, after the flow andstress fields are solved, a second iteration loop is nec-essary to calculate the temperature field by using a fi-nite-difference method given by Arpin.11 The temper-ature field is considered 3D and the thermal balance

includes convection in the X and Y direction, viscousdissipation and conduction through the walls. Con-duction in X and Y can be neglected. For the steadyflow, the energy eq. (11) can be written as:

qCp

�vx

@T

@xþ vy

@T

@y

�¼ k

@2T

@z2þ s : D (47)

The global algorithm is illustrated in Figure 3.

SIMULATIONS OF MELT FLOWS IN SHEETDIES

Isothermal results

Because the simulations of fluids obeying PTT con-stitutive equations are usually used to investigatethe accuracy and efficiency of the numerical tech-nique, the isothermal flows of LLDPE in both thefishtail and coat-hanger dies are analyzed respec-tively, using PTT model and the results are com-pared with the 3D FEM.The Material parameters of LLDPE are given in

Table I.25 Boundary conditions are imposed accord-ing to the extrudate flow characteristics in the chan-nel. Average velocity for the required flow rate isimposed at the entry section. The volumetric flowrate Q ¼ 1 cm3/s. As for fully developed flow at theexit section, velocity components perpendicular tothe die axis X and the traction force along the axis Xare specified to be zero. Nonslip condition isimposed on the stationary solid wall. The symmetrycondition is imposed on the X-Z symmetry plane.The major features of the flow for the fishtail die

have been given in Figure 1. A typical coat-hangerdie consists of a manifold and a slit section asshown in Figure 4. The geometric parameters usedfor the simulation are given in Table II and III. Tosimplify the mesh of the coat-hanger die, the sectionof manifold is treated as square. The simulating ofthe 3D finite element is carried out using the methodgiven by Crochet.26 The finite element meshes forthe 3D approach are displayed in Figure 5 and 6.Because of symmetry the flow in a quarter of thechannel is analyzed. The flow region is divided into

TABLE IMaterial Parameters of LLDPE for PTT Model

Parameter gv (Pa s) gs (Pa s) k n e

Value 11830 1479 0.46 0.01 0.1

Figure 4 Schematic diagram of the coat-hanger die chan-nel and the coordinate system.

TABLE IIGeometric Parameters of the Fishtail Die (mm)

W1 W2 L1 L2 T H1 H2

65 500 80 253 50 4 2

TABLE IIIGeometric Parameters of the Coat-Hanger Die (mm)

D W L1 L2 T R H h

10 204 10 82 26 3.1 2.5 1

3194 LI ET AL.


3D elements by using mapping mesh generationtechnology. The number of element divisions alongthickness is 4. To compare the results, the mesh ofthe finite piece method is the same as that of the 3DFEM in X-Y plane.

Comparisons of simulation results by the finitepiece method to the 3D FEM are shown. Figure 7and 8 show the volumetric flow distributions qx/q0in the lateral scan on the exit. Where qx is the flowrate of unit width which is calculated according toeq. (21), q0 is the average flow rate at the entry. Bycomparison, we find that the results of the finitepiece method are in good agreement with the pre-dictions of the 3D FEM. The discrepancy at allpoints is less than 1.2%. Moreover, it can be seenthat the velocity profile of the coat-hanger die getsflatter than the fishtail die. Figure 9 and 10 show thevolumetric flow distributions on the X-Z symmetryplane (y ¼ 0). The predictions based on the finite pi-ece method are quite close to those based on thecomplete 3D simulation except at the contractionmouth. And the discrepancies are restricted to asmall region. The discrepancies can be easily under-stood because the flow channel with varying thick-ness was divided into parallel regions for the finitepiece method.

Figure 11 and 12 show the contours of pressuregiven by the finite piece method. The pressure distri-butions on the X-Z symmetry plane based on thetwo numerical approaches are plotted in Figure 13and 14. The results of 3D FEM are the average pres-sure along the thickness. By analyzing the results of3D FEM, we found the discrepancies of pressure

along the thickness direction are less than 2%. It canbe concluded that it is reasonable to assume that thepressure does not change along the thickness. Weobserve that there is an abrupt variation in the trendof pressure distribution near the contraction mouth.Because of thinner channel, the pressure drop ofcoat-hanger dies is greater than the fishtail die.Overall, for two kinds of dies the pressure obtainedby the finite piece method shows satisfactory agree-ment with the results of 3D FEM. It can be seen thepressure predicted by the finite piece method isslight smaller than 3D FEM but not exceeding 6%.And the large discrepancies are restricted to a smallregion. This can be explained by the fact that thesurface force components of wall in X,Y directionare ignored, and only the first item of Fourier seriesis utilized to approach the velocity distributionsalong the Z direction. However, the pressure dropcaused by the shear viscosity in the slit region ismore significant than the normal stresses of the con-verging section, and the surface force of X, Y direc-tion has little influence on the results of the pressure

Figure 5 The 3D finite elements mesh of the fishtail die.

Figure 6 The 3D finite elements mesh of the coat-hangerdie.

Figure 7 Axial flow rate profile predicted by the finite pi-ece method and the 3D FEM in the exit section of the fish-tail die.

Figure 8 Axial flow rate profile predicted by the finite pi-ece method and the 3D FEM in the exit section of the coat-hanger die.



for sheet dies. Accordingly there is a minor discrep-ancy between the results obtained from the finitepiece method and the 3D FEM.

The most significant contribution of the finitepiece method is that a substantial amount of CPUtime and memory requirement can be saved. In thisarticle convergent criterion is set to 10�4 and thecomputational data are shown in Table IV. It can beseen that the number of the unknowns in the finitepiece method is far less than that of 3D FEM. This isbecause the number of its elements is quarter for the3D mesh and the unknowns of nodal velocity arereduced from three to two (vz ¼ 0). Accordingly, thememory requirement is significantly reduced, andCPU time per iteration can be saved by more than80%. We also found the number of iterations is veryclose to that of the 3D FEM. This shows that it doesnot decrease the rate of convergence to predict ve-locity distributions by Fourier series.

Nonisothermal results

The simulations of LLDPE nonisothermal flows inthe coat-hanger die were carried out using Carreau-Yasuda model also. The uniformity index27 and thepressure drop are compared with the results fromArpin’s experiment. The rheological coefficients ofLLDPE have been given in Ref. 11. The temperaturefield is calculated by using a finite-differencemethod. The boundary conditions are applied inaccordance with experiment. The die wall isassumed to be adiabatic: @T

@z ¼ 0.The uniformity index was created to analyze the

performance of flat dies which has been used up tonow. Computing the uniformity index is a goodqualitative comparison tool. Separating the die into

Figure 9 Axial flow rate profile predicted by the finite pi-ece method and the 3D FEM in the X-Z symmetrical planeof the fishtail die.

Figure 11 The contours of pressure in the fishtail die pre-dicted by the finite piece method.

Figure 10 Axial flow rate profile predicted by the finitepiece method and the 3D FEM in the X-Z plane symmetri-cal of the coat-hanger die.

Figure 12 The contours of pressure in the coat-hangerdie predicted by the finite piece method.

3196 LI ET AL.


N sections across the width, the uniformity index isdefined as:

U ¼ 1� 1

N

XNi¼1

NQi

Q� 1

!2

1=2

(48)

where Qi is volumetric flow through i-th section, Qis total volumetric flow.

To investigate the sensitivity of the simulationresults with respect to the mesh size, we computed thenumerical solution on three meshes. The mesh data inTable V comprise the total number of element andnode, degrees of freedom (DOF) and the minimummesh spacing normalized with the widthW (Rmin).

Figure 15 shows the numerical prediction for theuniformity index as a function of the volumetricflow rate in comparison with the experimental data.The overall agreement of the uniformity indexbetween the finite piece method and the experimentis good except for the individual point. The maxi-mum error is about 2.3%.

The overall pressure drop versus volumetric flowrate is shown in Figure 16. It can be seen that the

predicted pressure levels off at the highest flow ratesafter an initial rapid increase, similar to the experi-mental data. However, the pressure drop predictionis smaller than experimental. With the increasing ofthe flow rate, the discrepancy becomes larger, andultimately reaches 5.6% (M1).We can observe in Figure 15 and 16 that the nu-

merical solutions obtained using the three meshesare in good agreement. Using a mesh size (Rmin) of0.0056 instead of 0.00167, which is one-third the size,gives a uniformity index that is only 0.5% lower,and the pressure drop increases by 2.8%. Moreover,we can see that as the mesh is refined the numericalsolutions converge to the experimental data. The

TABLE IVComparison of Computational Data Between the Finite

Piece Method and the 3D FEM

Method

Fishtail die Coat-hanger die

3DFEM

finitepiece

method3DFEM

finitepiece

method

Degrees of node 10 8 10 8No. of node 3939 615 12621 1981No. of element 736 184 2496 624No. of unknowns 36531 4521 116984 14546No. of iteration 22 23 25 27CPU time(s) 946 207 3650 486CPU time(s)/iteration 43 9 146 18

TABLE VMajor Characteristics of the Computational Meshes

Mesh M1 M2 M3

elements 624 1448 2472nodes 1981 4503 7625DOF(V, p) 4641 10534 17827Rmin 0.0167 0.0084 0.0056

Figure 13 Pressure predicted by the finite piece methodand the 3D FEM in the X-Z symmetrical plane of the fish-tail die.

Figure 14 Pressure predicted by the finite piece methodand the 3D FEM in the X-Z symmetrical plane of the coat-hanger die.

Figure 15 Prediction of the uniformity index for the coat-hanger die as a function of the flow rate in comparisonwith experimental data.



coarse mesh M1 leads to a larger uniformity indexand a smaller pressure drop. For this die, consider-ing the sufficient accuracy and computationally cost-effective, mesh M2 is a good comprise which corre-sponds to 40 elements across the half-width.

CONCLUSIONS

In this study, a new finite element technique namedthe finite piece method is proposed to simulate theviscoelastic polymer flow in the sheet dies. In thismethod, the distributions of velocities areapproached using the approximating polynomial inthe X-Y plane and Fourier series in Z direction. Themesh is divided in X-Y plane, so the 3D flow prob-lem is reduced to the 2D case.

Analyzing the behavior of the polymer melt flowthrough the fishtail and coat-hanger dies has beencarried out by both the finite piece method and the3D FEM. The results of the finite piece element arecompared with those of the 3D FEM simulationsand Arpin’s experiments. The solution accuracy isexcellent for most of the cases presented althoughsome assumptions were made to simplify the prob-lem. The volumetric flow distributions obtained bythe finite piece method show satisfactory agreementwith that of 3D FEM. At the die exit the relativeerror between the two methods is below 1.2%. Thereis a minor discrepancy between the pressure distri-butions obtained by the finite piece method and the3D FEM. For the whole die, the discrepancies causedby varying thickness of channel are restricted to asmall region, and it does not exceed 6%. For the fi-nite piece method, the memory requirement is sig-nificantly reduced, and CPU time per iteration canbe saved by more than 80%. The overall agreementof the uniformity index between the finite piecemethod and Arpin’s experiment is satisfactoryexcept for the individual point in which the error isabout 2.3%. The pressure drop prediction of the

finite piece method on the whole is not as close tothe experiment as to the volumetric flow distribu-tions. The maximum discrepancy reaches 5.6% (M1).It may be attributed to the following reasons: (1) thematerial parameters in the constitutive equation arenot accurate enough. (2) Only the first item of Fou-rier series is utilized to approach the velocity distri-butions along the Z direction at high shear rates.(3) Nonslip condition is imposed on the solid wall.It is concluded that the finite piece method is

effective in simulating the slit flow in the sheet dies.Owing to little demand on computational memory,the finite piece method can be used to solve large-scale 3D problems.It can be inferred that the space slit flow can be

analyzed by the finite piece method too. One of fur-ther work is to give the way to solve space problemby this method. Furthermore, the finite piece methodcombined with the 3D FEM to simulate the flow inthe complex 3D channel is another future issue.

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Figure 16 Prediction of the pressure drop for the coat-hanger die as a function of the flow rate in comparisonwith experimental data.

3198 LI ET AL.


a finite piece method for simulating polymer melt flows in extrusion sheet dies

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