760352204_ftp_2

The Hybrid FEM/FDM Computer Model for Analysis of the Metering Section of a

Single-Screw Extruder RONG-YEU CHANG and KUEN-JANG LIN

Department of Chemical Engineering National Tsing Hua University

Hsinchu, Taiwan 30043, Republic of China

A hybrid FEM/FDM computer model was employed in this study for simulating the non-Newtonian, nonisothermal polymer melt felt in the metering section of a single-screw extruder. The pressure distribution in the screw surface was obtained by solving the generalized Reynolds equation. Instead of using the energy equation in Eulerian frame, a Lagrangian expression was involved for stabilizing the numerical scheme. The temperature profiles were obtained by finite difference discretization for the energy equation in such element. The screw surface with the screw channels and the flight lands could be modeled as a surface divided into small shell elements. To demonstrate applicability, the results provided by the hybrid FEM/FDM were found to be similar to those of the 2D FDM for the thermally developing flow, through Fenners example. It can also illustrate the leakage flow and the cross-channel effect in the screw pumping problem. The results from the Hybrid FEM/FDM revealed that if the clearance becomes too large, the volumetric flow rate would considerably decrease and the exist melt temperature would increase. In addition, when the clearance is close to the normal design clearance, the leakage flow through the flight lands was found to be small. These computational results were observed to correlate with those of other experimental studies. Finally, the hybrid FEM/FDM approach can in principle be extended to the non-Newtonian, nonisothermal flow in a complex screw surface such as the barrier screw and the Maddock mixing head.

INTRODUCTION channel. Griffith (161, Zamoditis et al (17). and Fen-

xtrusion is one of the most important operations E in the polymer processing industry. A substantial part of every polymer passes through an extruder at least once in its production path from the polymeriza- tion reactor to the finished product (1). The metering section of a single-screw extruder, which controls the throughput and the exit melt temperature of the extruder, has been extensively studied. Understand- ing the transport phenomena under screw pumping has been the focus of several comprehensive investi- gations (1-10). Rowell and Finalyson (1 1, 12) p r e posed the first pumping model for the analysis of Newtonian fluids in a rectangular channel. McKelvey et aL (13,141 used a simple energy balance to derive the analytical solution of Newtonian fluids under isothermal and adiabatic extrusion. Rotem and Shin- nar ( 15) obtained numerical solutions for one-dimensional isothermal power-law fluids in a rectangular

To whom correspondence should be addressed

1748 PO1 .YMER

ner (18) obtained numerical solutions for the 2D, fully developed, nonisothermal, and nowNewtonian flow of melt in an infinitely wide rectangular screw channel. When the thermal convection effects become significant during the extrusion process, the thermally fully developed flow will not be achieved even at the exit of the metering section (5). Generally, the governing equations of the thermally developing flow could not analytically be solved. Yate (19) applied a perturbation expansion in the Brinkman number of the thermally developing model: however, the result- ing Brinkman number was within the range of 0 to 1.5 (21). Fenner (20) made detailed comparisons of the thermally fully developed flow and the thermally developing flow. The polymer melt may possibly possess a pressure backflow, when the extrusion oper- ates at either a high die head pressure or low throughput. Elbirli and Lindt (21) got the stable solution under an appreciable pressure backflow by a p plying the coordinate transformation from an Eule- rian frame toward a Lagrangian frame. Bruker et al

ENGINEERING AND SCIENCE, NOVEMBER 1995, Vol. 35, No. 22

Hybrid FEM/ FDM Computer Model

(22) compared the experimental data with the results of the thermally developing flow, concluding that the thermally developing flow analysis provided an accurate flow description of polymer melt in the metering section. The 2D FDM (21-25) only predicted the flow fields along the screw channel and neglected the flight lands, assuming that the temperature, shear rate, and sear stress remained constant across the cross-channel direction. Pittman et a1 (26) studied the nonisothermal non-Newtonian model with the leakage flow. Rauwendaal et aL (27) found that the effects of the leakage flow on total throughput be- came rather large for small values of the power law index. Increased flight clearance in the melt convey- ing section having a pronounced effect on extruder performance was confirmed by other experimental reports (28-30).

Two approaches have been proposed in regard to the entire flow domain in the screw extruder, i.e., the 3D finite element flow analysis and the simplified 2D flow analysis in the screw surface via solving the generalized Reynolds equation. Masberg and Menges (31) proposed the usage of the 3D FEM for the numerical simulation of the 3D flow problems. The 3 D FEM model (32-36) used a solid element to approxi- mate the screw channel: hence, this model could predict the gapwise pressure gradient and the cross- channel flow effect. Rauwendaal (37) and Spalding et aL (38) utilized the 3 D FEM for simulating the polymer melt flow in the screw channel and the flight lands. A comparison of the 3 D FEM results with those of the 2D FDM revealed that these results were similar to each other when the screw channel with a width to depth ratio higher than 10 (7). Fortunately, the aspect ratio of the metering section of a single- screw extruder is almost > 10. Two numerical meth- ods have been proposed to solve the generalized Reynolds equation, i.e., the flow analysis network (FAN) and the hybrid FEM/FDM. The FAN is the powerful method for solving the flow problems in polymer processing, which includes pin barrel screw (39), tangential counter-rotating twin-screw (40-4 1). and intermeshing co-rotating twin screw (42-43). The hybrid FEM/FDM for solving the generalized Reynolds equation has been successfully used to simulate an unsteady filling process, which includes in- jection molding, compression molding, and resin transfer molding (44-48). Matsuoka and Takahashi (49) applied the hybrid FEM/FDM for simulating the isothermal, non-Newtonian flow in the steady profile extrusion coating die process. The purpose of this investigation lies in applying the hybrid FEM/FDM to simulate the flow in the metering section of a single- screw extruder. The generalized Reynolds equation is solved by the FEM, and the energy equation is dis- cretized by the FDM. The leakage flow is also discussed later.

THEORETLCAL ANALYSIS

Most screw channels and flight lands have shell-like geometrical configurations in which its thickness is

much smaller than the other physical dimensions. A screw surface is assumed for the mathematical derivations. The following simplified assumptions are made in order to obtain a solution to the problem (5), i.e.,

1) The flow is steady. 2) The flow is incompressible. 3) The interia force is negligible from the order of

4) The body force is neglected. 5) The polymer melt is purely viscous fluid. 6) The velocity and pressure gradient in a gapwise

7) The lubrication approximation is made. 8) Heat conduction in the direction of flow is negligi-

ble if compared with the conduction in the gap wise direction.

The simplified governing equations for any planar geometry are as follows:

magnitude.

direction are neglected.

d u d u - + - = o (1) d x d y

(2)

(3)

BC1 z = O U = U , U = U , T=T, (5 )

BC2 Z = f f u=ub U = U b T=T, (6) BC3 inlet p = p o T=To (7)

BC4 outlet p = p , (8)

where x, y are the planar coordinates (e.g. x is the axial coordinate and y the circumferential coordinate for the unwound screw surface): z is the gapwise coordinate u,u are the velocity components in x, y directions, respectively: p is the pressure: T is the temperature: v ( j , T ) is the shear viscosity; and + is the shear rate, i.e.,

(9)

In addition, p is the density: C, is the specific heat: k is the thermal conductivity: H is the thickness; (T,, Tb, To) are the screw, barrel and inlet temperature: ( p o , p,) are the inlet and outlet pressure: and (us , us) and (u,, u,) are the screw and barrel velocity components in ( x, y). respectively. The shear viscosity is represented in this work by the power-law model (5).

r ] = 'exp[-b(T-TreJ)l (10)

where mo is the power-law constant: n is the power- law index: b is the temperature sensitivity: and Trcr

POLYMER ENGINEERING AND SCIENCE, NOVEMBER 7995, Vol. 35, No. 22 1749

R.-Y. C h a n g and K . J . Lin

is the reference temperature. The screw temperature in most cases is unknown (5). Usually, two types of the screw thermal boundary conditions, i.e., constant screw temperature (i.e., T, = Tb) and adiabatic thermal condition at the screw surface (i.e. an adiabatic screw), are considered.

Finite Element Approximation for Pressure

Under the boundary conditions 5 and 6, Eqs 2 and 3 are integrated in the gapwise direction so as to obtain the velocity u and u (50, 51). respectively:

1 s, z 1 S O 0 4 dx [ 0 z z 7 ub-u, z l u=u,+- -&+- / -&--/ -& S O L 4 (11)

1 s, z 1 S O 0 4 Jy d p [ o z z n u b - u , z 1 u=u,+- - & + - I-&--/ -& SO L n (12)

where

H 1 so=/ -& 0 4

A second integration of Eqs 11 and 12 with respect to z results in the flow rate per unit width qx and q!,

H

where

The continuity equation averaged across the gapwise direction could be written as

(19)

where q = qxj + q J is the flow rate per unit width. A substitGtion of the flow rate per unit width, Eqs 15 and 16, into the averaged continuity equation yields the pressure equation, which is also called the generalized Reynolds equation (50, 51)

where i;, = u,,! + ud, g , = us! + UJ. The generalized Reynolds equation is a nonlinear elliptic PDE for non-Newtonian fluids. The boundary conditions should be given at all boundaries of the computa-

tional domain. Fortunately, the pressure boundary conditions have little effects on the flow field in the screw pumping problem. They will be discussed later. The generalized Reynolds Eq 20 is solved by the Galerkin weighted residual method, i.e.

/ Q N z . c j d d l = 0 (21)

where Ni is the shape function of the linear triangu- lar element and dl denotes the area of the domain of interest.

Finite Difference Approximation for Temperature

The temperature gradient in the gapwise direction could not be ignored when the viscous dissipation effects become significant as a result of either high viscosity or a high screw rotation speed. After the pressure field is obtained at each vertex node, the finite difference formulation for the energy equation is represented at the gapwise direction at the centroid of each element. The location of temperature nodes is chosen to be the centroid of each element where the velocity, viscosity, and shear rate are most accurate and less averaging is required for the temperature dependent properties (48). Owing to the low thermal conductivity of melt and subsequently high Peclet numbers, an upwind technique is employed for enhancing the numerical stability in the calculation. Therefore, the energy equation in Eulerian frame is transferred here to a Lagrangian expression from the instability point of view (2 1, 52)

where D/Dt denotes the material derivative. The finite difference discretization of the energy equation at the centroid of a triangle element is given as

where denotes the temperature of layer j at the centroid of element: TJ denotes the upstream temperature of T J ; k,+ ,,2 = k((q+, + T,)/2): and A l denotes the length of each triangle element. Figure 1 is a schematic representation of the upstream temperature TJ in each local element. The physical meaning of the Lagrangian expression could be referred to the monotone streamline upwind (MSU) approximation (52). The MSU approximation yields little physical spatial oscillations and possess little numerically false diffusion (52). So far, the governing equations have been derived for a two-dimensional geometry. The system equation for each element defined in the three-dimensional coordinate is derived on the local coordinate. Therefore, additional coordinate transfor-

1750 POLYMER ENGlNEERlNG AND SCIENCE, NOVEMBER 7995, Vol. 35, No. 22


I I I !

I - A I -4

Fg. !. A schematic representation of the upstream temperature 7; in each element.

mations are required to transform the nodal coordinates of an element from a global system to a local system (48). More details can be found from Refs. 48 and 53. The complex screw surface could be modeled by the finite element mesh without using the unwound screw approximation.

RESULTS AND DISCUSSION

To demonstrate the applicability and reliability of the hybrid FEM/FDM, the thermally developing flow in the screw pumping problem is studied. The screw geometrical configuration, material properties of the polystyrene melt, and operating conditions of Fenner's example (5) are listed in Table 1 . In order to see the reliability of this approach, this problem is solved again by the 2D FDM, which follows Fenner's method with the negligible leakage flow assumption. Another reason for solving this problem by the 2D FDM is try to find the boundary conditions for the hybrid FEM/FDM. The computational domain for the unwound screw channel surface used by the 2D FDM

Table 1. Screw Geometrical Configuration and Material Properties.

(a) Extruder Geometry and Operating Conditions Barrell diameter, Db Screw L/D ratio Screw helix, angle, H Screw channel depth, H Screw channel width, W Screw flight width, e Barrell temperature, Tb Screw temperature, Ts Inlet melt temperature, To Screw rotation speed Flow rate, Q

(b) Properties of PS Melt Power-law index, n Power-law constant, m, Reference temperature, T,,, Temperature sensitivity, b Thermal conductivity, k Density, p Specific heat, Cp

- 120

8 17.66

6 102 12

220 220 220 100 143

0.36 10,800

200 0.022 0.21 990

2000

mm

mm mm mm "C "C "C rPm cm3/sec

Pa. sec " "C "C - 1 W/m-"K kg/m3 j / kg .OK

is shown in Fig. 2. The pressure and temperature fields of the 2D FEM are shown in Figs. 3 and 4. The magnitudes of the pressure in the pushing flight and the trailing flight at the exit of the metering section are 6.2 and 5.0 MPa, respectively. At the entrance, the melt is assumed to have a uniform temperature of 220C. The thermal boundary conditions of the set barrel and screw temperature are also assumed to have an equal temperature profile of 220C (i.e. a cool screw T, = Tb = To in the present case). Based on these assumptions, the averaged temperature elevation of the melt depends only on the vicious dissipation. The metal temperature profiles continuously vary from 220C at the entrance to 235C around the exit: consequently, the flow at the delivery end is still far from being thermally fully developed as shown in

In order to obtain the mesh-independent solution, two types of mesh are designed. The MESH 1 (Rg. 5a)

Flg. 4.

x(cm) Q. 2. Computational domain for the unwound screw channel used via the 20 FDM.

Fig. 3. The pressure profiles predicted by the 20 FDM.

Fig. 4. The averaged melt temperature predicted by the 20 FDM.


R.-Y. Chang a n d K . J . Lin

(b) Fig. 5. (d Finite element mesh MESHl for the 30 screw channels and theflight lands. (b) MESHZ.

and MESH2 (Fig. 5b) contain 1748 and 3836 elements, respectively. The thickness of the flight lands is assumed here to be 0.1 mm. The hybrid FEM/ FDM results are demonstrated by mapping the numerical results from the 3D shell screw surface into the 2D unwound screw surface, as shown in Figs. 6a and 6b. The computational domain of the hybrid FEM/FDM has some discrepancies with that of the 2 D FDM; in addition, the mesh used in Figs. 5a and 5b has a resemblance to the screw surface in the real extrusion process. The inlet and outlet pressure boundary conditions for the hybrid FEM/FDM are assumed to be 0 and 5.0 MPa from the 2 D FDM results. It is easy to find that the solutions of the pressure fields are almost mesh independent from FYg. 7. The discrepancy between the melt temperatures provided by the hybrid FEM/FDM from MESH 1 and MESH2 is within 1C. as shown in Fig. 8. In addition, the discrepancy between the melt temperatures provided by the hybrid FEM/FDM from MESH2 and the 2D FDM is within 4C. Therefore, the results from the MESH1 and MESH2 seem to be acceptable. More precisely, the MESH2 is therefore chosen instead of MESH 1 in the following calculation from the numerical point of view.

- 1 ' I 0 2 4 6 a t o

X/D Fig. 7. The pressure proJles along the screw axis predicted by the hybrid FEM/ FDM from MESHl and MESHZ.

The pressure distribution in the whole domain predicted by the hybrid FEM/FDM from MESH2 is provided in Fig. 9; it is noted that the pressure in the pushing flight is higher than that in the trailing flight. This phenomenon suggests that (a) the melt flows, from the pushing flight toward the trailing flight in the screw channel and (b) the melt leaks from the pushing flight backward the trailing flight in the flight lands. It is discovered that the larger shear rate appears in the thinner element from Fig. 10. There is not much variation in the screw cross-channel direction for the shear rate, as indicated from the comparison with a high shear rate in the flight lands. The averaged shear rate can be approximated by the shear rate caused via the pure drag flow (i.e., Nn-D,/H, where N is the screw rotation speed; and D, is the barrel diameter). The value of the shear rate caused via the pure drag flow is of the order of 105 sec-' in the screw channel, while it reaches a peak of the order of 6283 sec-' in the flight land. Figure 11

(b) Fig. 6. (d Finite element mesh MESHl for the unwound screw surface mapped.frorn Fig. 5a. fb) MESHZ.

" . - L MESH2

220.0 230.0 240.0 250.0

T e m p e r a t u r e ("C) Fig. 8. Comparison between the 2 D FDM and the hybrid F E M / FDM predicted melt temperature across the depth of the channel at L / 2 and L

1752 POLYMER ENGINEERING AND SCIENCE, NOVEMBER 1995, Vol. 35, No. 22


t- d -1 c

0

Fig. 9. The pressure proj?les predicted by the hybrid F E M / F D M under a cool screw.

Fig. 10. The auei raged shear rate predicted by the hybrid F E M / F D M under a cool screw.

shows that the averaged melt temperature profiles under a cool screw gradually increase from 220C at the entrance to 233C at the end of the metering section. The convection in the energy equation has been considered both in the down-channel and the cross-channel direction when using the hybrid FEM/FDM; whereas only the down-channel convection is considered by the 2 D FDM. However, the discrepancies between the melt temperatures produced by the hybrid FEM/FDM and the 2 D FDM are appar- ently small in light of the fact that the down-channel velocities are typically three times larger than the

cross-channel velocities (37). The volumetric flow rate predicted by the hybrid FEM/FDM with 0.1 mm clearance is 142.1 cm3/sec, as listed in Table 2. From the above results, it can be conchded that the pressure and melt temperature profiles obtained by the hybrid FEM/FDM are similar to those of the 2 D FDM in this case.

The results described above are based on isothermal screw temperature. Another special case, an adiabatic screw, is considered (2). Figure 1 2 shows that the averaged melt temperatures under an adiabatic condition gradually vary from 220C at the entrance


R:Y. Chang and K . J . Lin

\

-.. 1.. ..

F g . 1 I . The averaged melt temperature predicted b y the hybrid FEM/ FDM under a cool screw.

Table 2. Comparison of the Flow Rate (cm3/sec) at 100 rpm.

FEM d=0.6 d=0.3 d=0.15 d=0.1 d=O.O6

FDM mm mm mm mm mm

143.0 116.4 135.5 142.0 143.4 144.3

to 239C at the exist of the metering section. The melt temperature elevation under such a process is higher than that under a isothermal condition (i.e., a cool screw) because the viscous dissipation is conducted away to the barrel and screw under a cool screw. In

order to compare the different thermal situation in the screw channel and the flight clearance, the exit melt temperature profiles across the depth of the screw channel and the flight clearance are plotted in fig. 13. The melt temperature profiles in the flight clearance are lower than those in the screw channel no matter what a cool screw or an adiabatic screw is considered, as shown in Fig. 13. Similar numerical results have been reported by Rauwendaal(37).

The throughput versus the screw rotation speed with various clearances produced by the hybrid FEM/FDM under a cool screw is provided in Fig.

Q. 12. The averaged melt temperature predicted b y the hybrid FEM/ FDM under an adiabatic screw.

1754 POLYMER ENGINEERING AND SCIENCE, NOVEMBER 1995, Vol. 35, No. 22

llybrid FEM/ FDM Computer Model

1.0

0 8

0.6

0.4

0.2

0.0 220 230 240 250 260

T e m p e r a t u r e ("C)

Fig. 13. The exit melt temperature profrles across the depth oj-lhe screw channels andflight lands.

160 1-1

I20 - d=0.06 mn h I L) 3

E c: v

L--

0 20 40 60 80 100 120

240

h

Lj ", 235

i

2 230 3 i CI a E 225 e: b

RPM (a)

7-

- d= 0.6 mm - d = 0.3 mm - d = 0.1 mm - d=O.06 mm

1 220 LA- I I 1 1

0 20 40 60 60 100 120

R P M (b)

Fig. 14. (4, The throughput us. rpm with vanomflight clearances provided by the hybrid F E M / F D M under a cool screw.lb) The averaged exit melt temperature us. rpm with various flight clearances provided by the hybrid FEM/ FDM under a cool screw.

14a It is shown that the relationship between the throughput and the screw rotation speed is almost linear. and the throughput, as expected, decreases as the clearance increases. The leakage flow through the flight lands is quite small under any screw rotation speed if the clearance is less than 0.1 mm. I t implies that this screw with a 0.1 mm clearance in the present case could provide efficient pumping. This value is very close to the normal design clearance (27). i.e., 0.001 Db. If the clearance is 0.6 mm and the screw rotation speed is 100 rpm, much reduction of throughput can be achieved (say, 18%). In addition, the reduction of throughput with increasing clearance becomes larger for higher screw rotation speed. The same qualitative result was also reported in other studies (54-58).

The averaged melt temperature a t the exit of the metering section versus the screw rotation speed obtained via the hybrid FEM/FDM under a cool screw is illustrated in Fig. 14b. For constant flight clearance, the melt temperature strongly increases with the screw rotation speed. I t is not surprising that the melt temperature rise with the screw rotation speed is not a linear function since the viscous dissipation of power-law fluids is a nonlinear form. For constant screw rotation speed, the melt temperature rise has two opposite effects as the clearance is increased. On the one hand, the shear rate in the clearance is reduced, which subsequently leads to lower rates of viscous dissipation. On the other hand, the length of the radial conduction path in the clearance is increased, which consequently leads to a lower heat transfer between the melt in the screw and barrel. The former contribution to the melt temperature is represented in volume weighting and the latter is represented in area weighting when dealing with the energy equation. I t could be confirmed that the latter is the key role in the present case from Fig. 14b.

The same trend that the melt temperature increases by increasing the clearance has been reported by Rauwendall (29) and Pittman e t al (26). If the clearance becomes too large, however, the leakage flow increases and the exit melt temperature rises. The same remarks are also available to the throughput and the averaged exit melt temperature under an adiabatic screw, which are shown in Figs. 15a and 15b. A conclusion can be made that the screw with a normal clearance can provide efficient pumping from Figs. 14a and 15a The above results provided by the hybrid FEM/FDM are based on constant inlet and outlet pressure boundary conditions (BCs); however, the pressures a t the screw channel and the flight lands would not be the same. There- fore, the influence of the pressure BCs is examined by either interpolating or extrapolating the pressure profiles of the 2D FDM as the inlet and outlet pressure BCs of the hybrid FEM/FDM. Although the constant pressure BCs are replaced by linear-like pressure BCs at the inlet and outlet, the pressure profile also gradually increases as a zigzag pattern, which is

POLYMER ENGINEERING AND SCIENCE, NOVEMBER 1995, VoI. 35, No. 22 1755

R:Y. Chang and K . J . Lin

--- 160

- d= 0 6 mrn -- d= 0 3 mrn - d= 0 I mm - d=O06 mrn

0 20 40 60 80 100 120

RPM (a)

- d = 0 8 rnm - d= 0 3 m r n h

a,

4 K!

230 I

l a t: ra 225 j 3

I

220 - 2-1- I I 0 20 40 60 80 100 120

RPM (W

Fig. 15. fd The throughput us. rpm with uariousJight clearances provided by the hybrid FEM/ FDM under an adiabatic screw. fb) The averaged exit melt temperature us. rprn with various flight clearances provided by the hybrid FEM/ FDM under an adiabatic screw.

shown in Fig. 16. The pressure BCs do not appar- ently disturb the pressure profiles in the inner re- gions, as indicated in Figs. 9 and 16. The throughput predicted by the hybrid FEM/FDM with linear-like pressures BCs is 146.1 cm3/sec, and the averaged exit melt temperature is 233C. The results are similar to those using constant pressure BCs. Therefore, those results will not be addressed again.

CONCLUSIONS

The hybrid FEM/FDM computer model was employed in this investigation for simulating the polymer melt flow in the metering section of a single-screw extruder. With fewer assumptions, the hybrid FEM/FDM can provide similar results to those of a conventional 2 D FDM modeling for the thermally fully developed flow. The hybrid FEM/FDM can predict the flow over the flight lands. Additionally, the results suggest that the flights are capable of providing efficient pumping as long as the clearance is close to the normal design clearance, i.e., 0.001 Db. If the clearance becomes too large, the leakage flow would increase and the exist melt temperature would rise. The hybrid FEM/FDM uses the 3 D shell elements to r e p resent the screw surface. Therefore, the hybrid FEM/ FDM approach can be easily extended towards the non-Newtonian, nonisothermal flow in a complex screw surface such as the barrier screw and the Maddock mixing head. In addition, this work can be extended to other continuous polymer processes, e.g. extrusion dies.

ACKNOWLEDGMENTS

The authors would like to thank the National Sci- ence Council of the Republic of China for its financial support under Contract No. NSC 8 1 -0405-E-007-586.

1. ... ..

. 1.

/-/----

1. 0 -. I

, --. .... . - . ..

-\ ..,, I ..&

_- ---> 40 30

I 20

.---.-I-. -2 L:. 2sL-\. 50 .,\. -..

x(cm) 75 -------.I,->/--

Y (cm) loo 0

Rg. J 6. The pressure projles predicted by the hybrid FEM/ FDM with linear-like pressure BCs.

1756 POLYMER ENGlNEERlNG AND SCIENCE, NOVEMBER 1995, Vol. 35, No. 22


1.

2.

3.

4.

5.

6.

7.

8.

9.

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