Numerical and Experimental Investigation of Three-Dimensional Flow in Extrusion Dies

M E S H GUPTA,* YOGESH JALURIA, VALENTINAS SERNAS, and M O W E D ESSEGHIP"

Department of Mechanical and Aerospace Engineering Rutgers, The State University of New Jersey

Piscataway, New Jersey 08855

and

TAI H. W O N * * *

Department of Mechanical Engineering Pohang Institute of Science & Technology

Pohang, South Korea

The univariant element, QlP0. and the multivariant elements, Q:Po and RgP,, are compared for the numerical simulation of the flow in extrusion dies. The pressure distribution obtained by using the QIPo element was found to be afflicted with the checkerboard pressure mode. On the other hand, the multivari- ant elements, Q:Po and RgP,, gave accurate and physically reasonable velocity and pressure distributions. The computed values of the pressure drop across extrusion dies matched well with the pressure drop determined experimentally.

INTRODUCTION

n the design of extrusion dies, post die changes in I the extrudate profile arising from die swell and cooling shrinkage play an important role. However, one of the main objectives in die design is to establish a flow channel geometry such that, for the specified flow rate of the material, the pressure required at the die entrance does not exceed the pressure available from the extruder. The die must also be robust enough to avoid any significant changes in the flow channel dimensions due to the deformations caused by the internal pressure. Therefore, determining the pressure distribution in the die at the specified flow rate is very important.

In an extrusion die, the cross section of the flow channel changes from the circular entrance cross section to the desired exit profile shape, which may not be symmetric. Moreover, owing to screw rotation, the polymer may have circumferential velocity in the inltial portion of the die, which necessitates a three- dimensional flow analysis.

The univariant element, Q, Po (trilinear velocity and constant pressure), and the multivariant elements?

Q:Po and R; Po, as shown in Fig. 1, were used in the present work. The Q:Po element is obtained by adding one velocity node at the center of each of the six faces of the QIPo element. The nodes at the center of the faces have only one degree of freedom, which is the velocity component normal to the face. Therefore, the velocity components parallel to one of the six faces of the Q: Po element vary bilinearly over the face, whereas the variation of the velocity compe nent normal to a face is biquadratic over the face. The pressure is constant within a Q:Po element. The R$P0 element is obtained from the @;Po element by adding one velocity node at the center of the each edge. Each of the new nodes has only two degrees of freedom, which are the velocity components normal to the edge. Therefore, on a R;P0 element, the veloc- ity components normal to an edge vary quadratically over the edge, whereas the velocity component in the direction of an edge varies linearly over the edge. The pressure is constant within a RgP0 element. A de- tailed analysis of the three types of elements can be found in Refs. 1 and 2.

In this work, the shape functions introduced in Ref. 2 were used to develop finite element programs . -

for simulating viscous, incompressible flows using the QIPo, Q:Po, and RgP0 elements. Using these finite elements codes, a comparative study of the performance of the three types of finite elements, for simulating the flow of incompressible fluids in extru-

*Present address: Mechanical and Aerospace Engineering. Cornell Univer-

'"Previously. assistant professor at Rutgers University.

variant finite element.

sity. Ithaca. NY 14853. **Present address: Polymer Processing Institute. Hoboken. NJ 07030.

' Different velocity components have different variations within a multi-

POLYMER ENGINEERING AND SCIENCE, MID-APRIL 1993, Vol. 33, NO. 7 393

Mahesh Gupta, Yogesh Jaluria, Valentinas Semas, Moharned Esseghir and Tai H. Kwon

BASIC EQUATIONS FOR A VISCOUS, INCOMPRESSIBLE FLOW

For creeping flow of generalized Newtonian fluids, neglecting the inertial and body force terms, the m e mentum equation can be represented as

div(2p.E) - gradp= 0 in R (1)

where p is the viscosity of the fluid, p is the pres- sure, R is the flow domain, and L is the strain rate tensor. The incompressibility constraint can be represented as

divu=O in R (2)

0 Oun

where u is the velocity at a point in R. The boundary conditions to be satisfied are

T = T onr, (3)

u = t i onr , (4)

and

where T is the traction force at the boundary, r, is the part of the boundary where the traction force is given, r,, is the part of the boundary where the veloc- ity is given, T is the prescribed value of traction force on rT, and ti is the prescribed value of velocity on T,,.

The power law model was used here to simulate the shear thinning behavior of non-Newtonian fluids. For power law fluids

p = KE 2- ’ (5 )

where z, is the effective shear strain rate, which is defined as

€,= 42(E:Z) (6)

If V and Q are the solution spaces for velocity and pressure, then the variational form of the incom- pressible flow problem described in Eqs 1 to 4 is:

Find (u, p) E V,, x Q such that

2pG(u):E(v)du-! pdivvdx= n

(7)

(d lnqdivudu=O VqEQ (8) FJ& I. Three-dimensional finite elements (a) Q1 Po element (b) Q: Po element (c) R: Po element. where

sion dies, is performed. A rectangular die is analyzed first, followed by a circular die with the conical end of the screw rotating in the initial portion of the die. The flow is assumed to be isothermal. The univariant finite element, Q,Po, does not satisfy the BabuSka- Brezzi condition (3, 4). The flow simulations obtained by using the Q1 Po element exhibit spurious pressure modes. On the other hand, the multivariant finite elements, Q:Po and RgP,,, which do satisfy the BabuSka-Brezzi condition ( 11, give accurate velocity and pressure distributions in the two different extru- sion dies analyzed here.

V , , = { v t v ~ V , v = ~ o n r , }

vo = { v t v ~ v, v = 0 on I-,}

and

FINITE ELEMENT FORMATION

Using the finite element approach, the velocity and the pressure inside an element can be represented as

{U}e= “l{_u)e (9)

Pe = [ Npl{_~}, (10)

394 POLYMER ENGINEERINGAND SCIENCE, MID-APRIL 1993, Vol. 33, No. 7

Investigation of Three-Dimensional Flow in Extrusion Dies

- 2p 0 0 0 0 0 - o 2 p o o o o 0 0 2 p o o o

0 0 0 0 4 p o 0 0 0 0 0 4 p

Lo]= 0 0 0 4p 0 0

- -

where {u}~ is the column matrix of three velocity components at a point inside the element, &}e is the column matrix of velocity components at the nodes of the element, pe is the pressure at a point inside the element and {pie is the column matrix of pressure at the nodes of the element.

Differentiating Eq 9, the six independent strain rate components can be arranged in a column matrix as

(€1 = N'lI_u)e (11)

Using this notation, the discretized linear momentum and continuity equations can be expressed in matrix form as follows:

[ K l l u ) - [Gl lP ) = {f} (12)

[GIT{U1 = (01 (13)

where [ K ] is the global stiffness matrix, [GI is the global matrix for the incompressibility constraint, W is the column matrix for the work equivalent force due to the traction and body forces, {O} is a zero column matrix, {u} and { p } are the column matrices of unknown velocities and pressure at nodes. For each element, matrices [ K] and [GI are:

Neglecting the body forces

tion of a power law fluid flowing in a rectangular channel is not available, a uniform velocity profile is specified at the entrance. For the developed flow at the exit, the component of the traction force along the axis of the die and the two velocity components nor- mal to the axis of the die are zero. The velocity distri- bution in the middle vertical plane of the die, ob- tained by using the QIPo. Q:Po, and the RgPo elements, are shown in Figs. 3(a), (b), and (c), re- spectively. The velocity distributions obtained by us- ing the three different types of finite elements are very close to each other. The pressure variation along the axis of the die is shown in Fig. 4. All three types of elements give almost the same pressure variation along the rectangular die. Flow simulations obtained by using the QIPo elements are often afflicted with the checkerboard pressure mode (2, 6, 7). However,

Flg. 2. Finite-element discretization for a rectangular cort verging die.

Table 1. Material Properties of Low Density Polyethylene (LDPE), From Fenner (5).

~~

Reference viscosity p0 16.0 kPa.s

Reference temperature To 523" K Temperature coefficient of viscosity b, Power law index n 0.3 Densitv P 750 kg/m3

Reference strain rate E, 1 . 0 s - '

0.01" K - '

(d

Fig. 3. Vetocityfield along the rectangular die US^ (d Q1 Po element (b) 9: Po element fc) R,+ Po element.

395 POLYMER ENGINEERING A N D SCIENCE, MID-APRIL 7993, VOl. 33, NO. 7

Mahesh Gupta, Yogesh Jaluria, Valentinas Semas, Mohamed Esseghir and Tai H . Kwon

LEGEND o Using Q,Pn elements 0 Using " Q 'P elements A UsingR'P elements

................. l.-.D . .........................

.-----.---.-.--.-P.-R-------------.

0.00 0.01 0.02 0.03 0.04 0.05 0.06 Flow Direction (m)

Fg. 4. Pressure variation along the rectangular die.

n

in this particular case, even the pressure distribution obtained by using the Q1 Po elements is free from the checkerboard pressure mode. The pressure distribu- tion obtained by using the (&Po elements shows some fluctuations near the entrance, but the fluctua- tions are negligible compared to the absolute value of the pressure.

Figure 5 shows the pressure variation along the die at different flow rates at a processing temperature of 523°K. The pressure drop across the die increases with the flow rate, but the pattern of the pressure variation along the die is the same at all flow rates.

Circular Die With a Rotating Conical Surface

A three-dimensional flow in a real die system was simulated next to better test the different finite ele- ment types. Figure 6 shows the details of the die assembly used in a laboratory single screw extruder. The die was assumed to consist of all the extrudate passage past the end of the screw, so that the die had the shape of a cone that came together into a small diameter cylindrical hole. The inside surface of the cone consisted of the tapered portion of the end nut that rotated with the screw. Thus the conical die channel had a strong circumferential velocity which made the flow strongly three-dimensional.

The laboratory extruder was fully instrumented with three pressure transducers which allowed the pressure at the end of the screw to be measured by extrapolating the pressure gradient past the last pressure transducer which is shown in Fig. 6. The thermocouple in the die cavity monitored the temper-

L X G W 0 Flow rate = 1.m x lO-lkg/S

____________________------------------ now rate = 1.5 x 10-~k&S--- A

6 mow rate = 225 x 10-~k& n Flow rate = 3.0: lO-'kg/s 0 NOW rate = 3.75 x 10-~kg/s Flow rate = 4.5 x lo-skg/s,,,,~

________________________________________ now rate = 525 x 10-~k&~-

0 Flow rate = 7.5 x 104kds.,,,, .............................................................................

----- - . .............................................................................. 0

I I I I I I W 0.01 0.02 0.03 0.04 0.05 0.06

Axid direction (m) Q. 5. Pressure variation along the rectangular due for dif ferentJow rates obtained by using R;P, elements for LDPE at a processing temperature of 523°K.

Dynisco pressure transducer

End nut

Hex cup 1 The rmocoupl

Die

/- Extruder

-

screw

D!e holder Plug

Q. 6. Die assembly detail of experiment.

ature of the corn syrup extrudate. The material prop erties of the corn syrup, as measured by Esseghir (81, are given in Table 2. The mass flow rate was experi- mentally measured by weighing the extrudate that was collected past the die over a fixed time period.

The finite element mesh used to analyze the flow in this conical die is shown in Fig. 7. At the die en- trance, the outer and the inner diameters of the annulus are 2.171 and 2.021 cm, respectively. The

396 POLYMER ENGINEERING AND SCIENCE, MID-APRIL 1993, Vol. 33, NO. 7

Table 2. Material Properties of Corn Syrup, From Esseghir (8).

Reference viscosity po 105.258 Pas Reference temperature To 293" K Temperature coefficient of viscosity bT Power law index n Density p 1381 kg/m3

0.095" K - 1 .o

Flg. 7 . Finite-element discretization of the circular die with the conical end of the screw rotating in the initial portion of the die.

exit diameter of the die is 2.4 mm. The average veloc- ity for the experimentally determined flow rate is specified at the die entrance. For the developed flow at the exit, the component of the traction force along the axis of the die and the velocity components nor- mal to the axis of the die are specified to be zero.

Figure 8 presents the velocity distribution along the die obtained by using the three types of elements. The velocity distributions obtained by using the three different types of finite elements are very close to each other. Owing to the rotation of the conical end of the screw in the die, the fluid in the annular portion of the die has some velocity in the circumferential direction as well. Figures 9(a), (b) , and (c) show the circumferential velocity distribution across the die at a cross section near the die entrance obtained by using the Q,Po, @Po, and RgP, elements, respec- tively. As expected, the circumferential velocity is maximum at the screw tip. The circumferential veloc- ity decreases with the increasing radius and becomes zero at the die. The pressure distribution obtained by using the QIPo elements was found to be afflicted with the checkerboard pressure mode. The pressure distribution along the circular die obtained after aver- aging the pressure over the adjacent Q, Po elements is shown in Fig. 10. I t is evident from Fig. 10 that even after averaging the pressure over the adjacent Q1 Po elements, the checkerboard pressure mode could not be eliminated completely in this particular case. In general, filtering the checkerboard pressure mode over a distorted b i t e element mesh is a very complicated problem, and the averaging technique is not enough to eliminate the spurious pressure mode (7) over such finite element meshes. The pressure

Investigation of Three-Dimensional Flow in Extrusion Dies

POLYMER ENGINEERING AND SCIENCE, MID-APRIL 1993, Vol. 33, NO. 7 397

Fg. 8. Velocity distributions along the circular die using (d Q1 Po element (b) 9: Po element (c) Rd Po element.

distributions along the die obtained by using the Q;Po and RgP, elements are very close to each other (Fig. 10). The kink in the pressure curves is at the location where the conical portion of the die com- bines with the circular tube.

The flow of corn syrup in the circular die was simulated at seven different rotational speeds of the screw. As a result of the change in the flow rate with the rotational speed of the screw, the velocity and pressure distributions inside the die also change with

Mahesh Gupta, Yogesh Jaluria, Valentinas Semas, Mohamed Esseghir and Tai H. Kwon

the screw speed. The flow rates at different screw speeds were determined experimentally. Figure 11 presents the pressure variation along the die at differ- ent screw speeds. Because of the increase in the flow rate with the screw speed, the pressure drop across the die increases with the rotational speed of the

LEGEND 0 After averaging on Q,P,, elements

Using Q 'P elements

..----.--.-----.-L--~------------------------------- Using R 'P elements A

.................................. 1 ...... 0 ................................................................

9 0- N

-0 &- el 2 E :-

0

0

1- 10.015 -0.blO -0.b05 0.600 0.605 0.010

Axial direction (m) 115

Q. 10. Pressure variation along the circular die at screw frequency= 2.333Hz andflow rate= 4.21667X kg/s.

LEGEND 0 Screw frequency = 0.333 Hz Flow rate = 3.778xlO-' kg/s

......................................... Screw frequency ........................................................................................... = 0.667 Hz, Flow rate = 8.3611xlO-' kg/s,,,

0 Screw freguency = 1.333 Hz, Flow rate = 2.14187~10-~ kg/s, * Screw fnquency = 1.667 HG Flow rate = 2.87222~10-' kg/s 0 Screw frequency = 20 Hz, Flow rate = 3.47222x10-' kg/s

............................................................................................................................................ * Screw frequency = 2.333 HZ Flow rate = 421667x10-' kg/s

."... ~ ~ . ~ ~ ~ ~ ~ ~ r . ~ ~ ~ ~ ~ , ~ ~ ~ ~ ~ ~ ~ ~ = ~ ~ ~ ~ ~ ~ ~ ~ . ~ ~ ......

(C )

Fig. 9. Velocity distributions across the circular die with the screw tip rotating in the computational domain using (d 9, Po element (b) Q,? Po element (c) R; Po element.

15

Q. 1 1 . Pressure variation along the circular die.

398 POLYMER ENGINEERING AND SCIENCE, MID-APRIL 1993, VOl. 33, NO. 7

Investigation of Three-Dimensional Flow in Extrusion Dies

screw. But the pattern of the pressure variation along the die is the same at all flow rates.

Figure 12 compares the numerical and the experi- mentally obtained values of the pressure at the die entrance. The experimental and the numerical re- sults for the die entrance pressure match very well for the given range of the flow rate.

CONCLUSIONS

The flows in two different extruder dies were simu- lated by using the univariant finite element, Q,Po, and the multivariant finite elements, Q:Po and Ripo.

LEGEND 0 Numerical results using Ql+PO elements

0 Experimental results ._ ... , .. . . . . _.... . . . . ... ..... , .. . . , . ...., ... . .. ... ... . . . .. .. ..... . _. . . . . ..... .. . . , . . .. . _. . . . . . . .. . .. . . . . .. . . . ... . . . . . .. . . . . . . I b o 2 3 - 3 1

Fig. 12. Numerical and experimental values of the pressure at the entrance of the circular die.

Very similar velocity distributions were obtained from the three types of finite elements. The pressure distri- butions along a circular converging die, obtained by using the univariant element, Q, Po, was found to be aMicted with spurious pressure modes. Over the complicated finite element mesh required for simulat- ing the flow in the circular converging die, the checkerboard pressure mode could not be eliminated completely by averaging the pressure over the neigh- boring Q1 Po elements. The pressure distributions ob- tained by using the Q:Po and Ripo elements, which satisfy the BabuSka-Brezzi condition, are free from spurious pressure modes. The numerical values of the pressure drop across the circular die matched very well with the pressure drop determined experi- mentally.

ACKNOWLEDGMENT

We wish to thank San Diego Supercomputer Cen- ter for making their computing resources available to us. The authors are also thankful to Dr. M. Karwe, Dr. S. Gopalakrishna, and Mr. T. Sastrohartono for discussions throughout this work. This work was partially supported by a grant from N.J. Center for Advanced Food Technology (CAFT) at Rutgers Univer- sity.

REFERENCES

1. M. Fortin, Int. J. Numer. MethodsFluids, 1, 347(1981). 2. M. Gupta, T. H. Kwon, and Y. Jaluria, Int. J. Numer.

3. I. BabuSka, Numer. Math. 16, 322(1971). 4. F. Brezzi, RAIRO. A n d Num. 8,R2, 129 (1974). 5. R. T. Fenner. Principles of Polymer Processing, Chemical

6. M. Gupta and T. H. Kwon, Polyrn Eng. Sci., 30, 1420

7. R. Sani. P. M. Gresho, R. L. Lee, D. F. Griffiths. and M. Engleman, Int. J. Numer. Methods Ffuids, 1, 17-43 and

8. M. Esseghir, PhD thesis, Rutgers University, New

Methods Fluids, 14. 557 (19921..

Publishing, New York (1979).

(1990).

171-204 (1981).

Brunswick, N.J . (1990).

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