Study of Mixing Efficiency in Kneading Discs of Co-Rotating Twin-Screw Extruders

HONGFEI CHENG and I. MANAS-ZLOCZOWER*

Department of Macromolecular Science Case Western Reserve University

Cleveland, Ohio 441 06

Co-rotating twin-screw extruders are widely used for compounding and blending in polymer processing. In a modular machine, the dominant elements determining dispersive mixing efficiency are the kneading discs. A fluid dynamics analysis package - FIDAP, using the finite element method was implemented to simulate the 3-D isothermal flow patterns in the kneading disc region of a Werner & Pfleiderer ZSK-53 co-rotating twin-screw extruder. The kneading discs simulated are three- lobe discs. The flow field characteristics relevant for dispersive mixing are shear stresses generated and elongational flow components. We compare these flow characteristics for the three-lobe kneading discs of the ZSK-53 with the two-lobe discs of a ZSK-30 machine. We also discuss the influence of processing parameters on the flow field mixing efficiency.

INTRODUCTION

o-rotating, intermeshing twin-screw extruders are C widely used in polymer compounding and blend- ing. The self-wiping feature of this machine provides some advantages not available in other types of twin- screw machines, such as a complete elimination of any stagnant zone. Among the different modules of the co-rotating twin-screw extruder, the kneading discs are the dominant elements in determining mixing ef- ficiency.

Szydlowski et al. (1, 2), Wang et al. (31, and Gotsis et al. (41 analyzed various models for the flow in a co- rotating twin-screw extruder. Yang and Manas-Zloc- zower (5) used a fluid dynamics analysis package - FIDAP to simulate the 3-D isothermal flow patterns in the kneading block region of a co-rotating Werner & Pfleiderer ZSK-30 machine. In this work, FIDAP was employed to simulate the flow patterns in the knead- ing disc region of a Werner & Pfleiderer ZSK-53 co- rotating twin-screw extruder. The kneading discs sim- ulated are three-lobe discs. We compare the flow characteristics for the ZSK-53 and ZSK-30 machines and discuss the influence of processing conditions on dispersive mixing efficiency.

DESCRIPTION OF METHODS

The kneading blocks in the ZSK-53 twin-screw ex- truder have both forward and reverse configurations.

To whom correspondence should be addressed.

1082

For both forward and reverse configurations the stag- ger angle is 30". Forward kneading discs act like nor- mal screw elements (positive displacement), whereas the reverse kneading discs have a negative pumping effect. Reverse discs are mostly used at the end of the melting zone to create a melt seal and to eliminate any starved sections. They can also be used to generate local high pressure regions as discussed by Booy (6). The two different types of kneading discs will be ana- lyzed in our flow simulations.

Mesh designs for both forward and reverse config- urations are shown in Fig. 1. Also shown in Fig. 1 is the mesh design for the two-lobe discs in the ZSK-30 machine. In the simulations, we tried to maintain the exact dimensions of the machine, including the clear- ances between the rotors and chamber walls and var- ious adjacent discs. The exact dimensions of the kneading discs (tip angle, channel depth) were calcu- lated from the knowledge of the tip number, centerline distance and screw diameter as discussed by Booy (7) and Mielcarek (8). The calculated dimensions were then used to construct the mesh. The geometry spec- ifications of the ZSK-53 are as follows: barrel diame- ter = 53.2 mm; screw tip diameter = 53.0 mm; screw root diameter = 42.0 mm; centerline distance = 48.0 mm; length of kneading disc region = 30 mm; number of discs = 5; stagger angle = 30". The geometry spec- ifications of the ZSK-30 are as follows: barrel diame- ter = 30.85 mm, screw tip diameter = 30.7 mm; screw root diameter = 2 1.55 mm; centerline distance = 26.2 mm; length of kneading disc region = 28 mm; number of discs = 5; stagger angle = 45".

POLYMER ENGINEERING AND SCIENCE, JUNE 1997, Vol. 37, No. 6

Study of Mixing Efliiency in Kneading Discs

manner (20' increment for each time step). There is a total of 18 geometries for one complete cycle, but be- cause of the symmetry of the rotors it is sufficient to analyze only 6. The geometries in Fig. 2 are labeled by the angle a! between the left rotor tip and the X-axis for the first disc. The elements used were brick elements with 8 nodal points in each element. The total number of nodal points for each geometry is 18,525 and the total number of elements is 15.900.

The field equations for the steady state, isothermal flow of an incompressible fluid were solved for each ge- ometry.

The rheology of the fluid was described in terms of a power law model with a Newtonian plateau region:

The model fluid used was HDPE (Alathon 7040) with a consistency index, rn = 4680 N a power law index, n = 0.59, a plateau viscosity, qo = 1160 Pa * s, and a maximum shear rate for Newtonian behavior yo = 30 s-'.

Non-slip boundary conditions for the chamber wall and rotor surfaces were employed. Fluid elements are stationary on the chamber wall and move with an angular velocity equivalent to the specified rpm on the rotor surface. We also specified the axial pressure difference between the entrance and exit planes.

In the simulations, a fully filled condition was as- sumed. Although most co-rotating twin-screw extrud- ers are starved fed, because of the reverse pumping effect of the left-handed screw elements and/or kneading discs, some sections are fully filled. In this light, the assumption of a filling factor of one in our flow simulations may still be considered realistic. All the simulations were carried out on the Cray Y-MP supercomputer. The cpu time required to perform the flow simulations varied between 8000 and 20,000 s depending on the flow conditions and different types of kneading discs employed.

Dispersive mixing efficiency was gauged in terms of shear stress distributions and elongational flow compo- nents. Yang and Manas-Zloczower (5) have used similar criteria to discriminate among various processing pa- rameters with respect to dispersive mixing emciency.

Larson (9) proposed a frame invariant parameter, S,, as a measure of the flow strength. The flow strength parameter is defined as:

2(trD2I2

\

Fig. 1. Mesh desgn (a) ZSK-53 forward configuration; (b) ZSK-53 reverse configuration; (c) ZSK-30 forward confisura- tion; rotors rotating anticlockwise.

s, =

Because of the time-dependent flow boundaries, a t r ( 6 ) . number of sequential geometries were chosen to rep- resent a complete mixing cycle. In FQ. 2 we show six different geometries for the ZSK-53 in a sequential

0

where p is the rate of deformation tensor and D is the

(4)

POLYMER ENGlNEERfNG AND SCIENCE, JUNE 7997, Vol. 37, No. 6 1083

Hongfei Cheng and 1. Manas-Zloczower

Fig. 2. Mesh designs representing one complete cycle.

Jaumann time derivative of D 0.e. the time derivative of D with respect to a frame that rotates with the angular velocity of the fluid element). The parameter Sf varies from 0 for pure rotational flow to infinite for pure elongational flow (Sf = 1 for simple shear flow).

For graphical purposes, we normalized the flow strength parameter according to:

1084

( 5 )

where N , ranges from 0 to 1. Calculation of the flow strength parameter requires

computation of the second derivatives for the velocity. In order to minimize the numerical error, high density mesh designs are required. In the case of the kneading disc region of a co-rotating Nn-screw extruder, owing to the complex geometry, this requirement was im- peded by computer limitations.

Another way to quantify the elongational flow com- ponents is by considering the relative magnitudes of

POLYMER ENGINEERING AND SCIENCE, JUNE 1997, Vol. 37, No. 6

Study of Mixing Emiency in Kneading Discs

the rate of deformation and vorticity tensors. In our previous work, we have defined a parameter A to re- flect flow characteristics important for dispersive mix- ing, namely its elongational components. The param- eter A is given by

M X . Y E C PL.OT'I> ,, ,*891.00

(6) A=- ID1

ID1 + I W I

Fig. 3. X-Y plane projections of the velocity vectors for the geometry with a = 30" at dtfferent axial distances; rpm = 100; AP = lo6 Pa (a) Z = 0.9 cm; (b) Z = 2.1 cm.

Fig. 4 . V, contours for the geome- try with a = 30" at 100 rpm and under an axial pressure dtfference of 10'Pa ( a ) Z = 0.3 cm: ( b ) Z = 0.9 cm; (c) Z = 1.5 cm: (d) Z = 2.1 cm; (e) Z = 2.7 cm.

where I@( and 101 are the magnitudes of the rate of strain and vorticity tensors, respectively. A varies be- tween 0, for a pure rotational flow, to 1, for a pure elongational flow.

RESULTS AND DISCUSSION

Velocity profiles for the ZSK-53 at two cross sec- tions in the region are shown in Fig. 3. The boundary conditions were a rotational speed of 100 rpm and an axial pressure rise of lo6 Pa. The fluid undergoes a circular motion following the rotation of each rotor. In the bridge region there is a material exchange between the two screws. Figure 4 plots V, contours from the velocity profiles at different cross sections in the re- gion. Areas of back flow can be observed primarily near the rotor tips.

The pumping capacity of the kneading discs in the ZSK-53 machine is illustrated in Rg. 5. As expected,

L.. (a)

i. (b)

POLYMER ENGINEERING AND SCIENCE, JUNE 1997, Vol. 37, No. 6

Honsfei Cheng and 1. Manas-Zoczower

+ ZSK-53 forward; loorpm

+& ZSK-53 reverse; 100rpm

150.00 ZSK-53 forward; 200rpm

100.00

50.00

0.00

-50.00

-100.00

-2.00E+6 -1.00E+6 O.OOE+O 1.00E+6 2.00€+6 A P (Pa)

Fig. 5. Correlation between instantaneousflow rate and awial pressure difference for the kneading discs in the ZSK-53 machine.

ZSK-53

+ n =0.4 +,=I0 2s"

+..- n=0.5 ~,=16.3s"

+ n=0.59 3,=30.0s'

-4- n=0.7 ~,=104.5s~' * n=0.8 f,=i068 9s.'

200.00

150.00

100.00

h 2 50.00

-6 5 0.00

-50.00

-100.00

-150.00

-2.00E+6 -1.00E+6 O.OOE+O 1.00E+6 2.00E+6 A P (Pa)

Fig. 6. The influence of melt pseudoplasticity on the pumping capacity of kneading discs in the ZSK-53 machine.

the output increases with rpm and decreases with pressure rise. The reverse configuration has a nega- tive pumping effect.

The pumping capacity of extruders is strongly af- fected by the rheology of the fluid. In Fig. 6 we show the influence of power law index on the correlation output-axial pressure difference. The model fluids used in the simulations have the same consistency index, rn = 4680 N * s"/rn2, but different power law indexes and different lengths for the Newtonian pla- teau region. The slope of the screw characteristic curves (in the kneading disc region) changes dramat- ically with the power law index of the pseudoplastic polymeric melts.

-+% ZSK-53; iOOrpm

-++ ZSK-30, loorpm

+ ZSK-53,200rpm

4 ZSK-30.2OOrpm

150.00 1 h m

-50.00 ~, -2.00E+6 -1 .OOE+6 O.OOE+O 1.00€+6 2.00E+6

A P (Pa)

FQ. 7. Comparison of pumping characteristics for kneading discs in ZSK-53 and ZSK-30 machines.

Kneading discs are primarily used as dispersive mixing elements. In order to study the influence of disc design and processing variables on dispersive mixing performance, we took the example of a block of 5 forward discs in the ZSK-53 machine (three-lobe discs with a stagger angle of 30") and compared it with the performance of a similar block in the ZSK-30 ma- chine (two-lobe discs with a stagger angle of 45").

Figure 7 compares the pumping characteristics of the two devices a t different rotational speeds. Clearly, the bigger ZSK-53 machine shows in general higher output values a t comparable axial pressure differ- ences. Also, the slope of the characteristics curves is larger for the three-lobe discs configuration in the ZSK-53 machine by comparison with the two-lobe discs in the ZSK-30 device.

The flow field characteristics relevant for dispersive mixing are shear stresses generated and elongational flow components. Table 1 shows the average values of the parameter A, shear rate, shear stress and flow rate for a complete cycle for the right handed kneading discs of the ZSK-53 machine at 100 rpm and an axial pressure rise of lo6 Pa. The average values were ob- tained by weighing the corresponding parameter for each element in the flow field by the volume of the element itself for the entire flow domain. Note that all the average flow characteristics remain almost un- changed when going from one geometry to another. This result indicates that for process analysis it is sufficient to analyze the data for one single geometry.

Figure 8 compares the average flow field character- istics for the forward and reverse configurations in the ZSK-53. The shear stresses generated in the forward configuration are higher for pressure drop and lower for pressure rise by comparison with the reverse con- figuration. Elongational flow components are not af- fected when changing from the forward to the reverse configuration.

1086 POLYMER ENGINEERING AND SCIENCE, JUNE 1997, Vol. 37, No. 6

Study of Miring Eflwiency in Kneading Discs

Table 1. Average Values for Various Flow Characteristics for Different Geometries in a Complete Cycle. Material HDPE, rpm = 100, Pressure Difference = 10' Pa.

2

0.00

Geometry, a A +, s-l r, Pa Q, cm3/s

I I I 1 ' 1

30" 50" 70" 90" 110" 130"

0.5702 344.44 0.571 1 346.54 0.5722 347.01 0.5715 343.08 0.5709 345.34 0.571 8 342.36

9.3477 x 104

9.3475 x 104 9.3481 X lo4

9.3468 X lo4 9.3483 X lo4 9.3465 X lo4

7.228 7.221 7.335 7.344 7.196 7.232

0.80 "0°

l.lOE+5 '2 a,

1.00E+5 Vl

0.20

0.00 -- 8.00€+4

-2.00E+6 -1.00E+6 O.OOE+O 1.00€+6 2.00E+6 A P(Pa)

FYg. 8. Comparison of theflow fwld characteristks for the forward and reverse configuration in the Z S K - 5 3 device.

-& J. , ZSK-53; 200rpm

+ 1 : ZSK-30: lwrpm

+ ; ZSK-30; 200rpm

+ 7 , ZSK-53; 1OOrpm

7 , ZSK-53; 200rpm

-ff+ , ZSK-30; 100rpm

7 , ZSK-30.200rpm

OXO 1 r 1.60E+5 0.58 1.40E+5

m a $! z

0.56 1.20E+5 u) u)

d

0.54 1.00€+5 W S v )

0.52 8.00E+4

0.50 6.00E+4

-2.00€+6 -1.00€+6 O.OOE+O 1.00E+6 2.00€+6 A P (Pa)

Fig. 9. Influence of processing conditions on the average val- ues of shear stress and parameter A in the two devices.

Figure 9 shows the influence of axial pressure dif- ference and rotational speed on the flow field charac- teristics in the ZSK-53 and ZSK-30 machines. The shear stress increases with rotational speed and with the magnitude of the axial pressure difference. The shear stresses generated are consistently higher in the ZSK-53 machine by comparison with the ZSK-30 device. Elongational flow components show little vari- ation with rpm and generally decrease with increasing pressure difference over the region. The values of A are slightly higher for the ZSK-53 machine.

-I 80.00

i -

60.00 K 0 0 .- L

E 40.00

0

(u

.- L c

- 5 8 20.00

i 60.00

40.00

20.00

0.00 I L 0.00 0.20 0.40 0.60 0.80 1.00

h

[bl

Fig. 10. Volumetric distributions of the parameter A: material: HDPE: rpm = 100; pressure dnerence = 1 O6 P a fa) ZSK-30 and (b) ZSK-53 .

POLYMER ENGINEERING AND SCIENCE, JUNE 1997, Vol. 37, No. 6 1087

Honsfef Cheng and I . Manas-Zoczower

v

c .- ... e U

40.00

20.00

0.00

Shear Stress ( Pa )

(a)

7 5E3 1 OE4 2 5E4 5 OE4 7.5E4 1 OE5 2 5E5 5 OE5 7 5E5 1 OE6 2 5E6

Shear Stress ( Pa )

(bl

Ftg. 1 1 . Volumetric distributions of shear stress: material: HDPE: rpm = 100; pressure dtfference = 1 O6 P a (a) Z S K - 3 0 and [bj ZSK-53 .

In order to compare the mixing performance of the kneading discs in the two devices, we take the exam- ple of similar operating conditions for both machines, namely a rotational speed of 100 rpm and a pressure rise over the entire region of lo6 Pa. The output in the two machines is similar, although not identical (e.g. 5.99 cm3/s for the ZSK-30 and 7 . 2 3 cm3/s for the ZSK-53). Figures 10 and 11 compare the volumetric

Fig. 12. N, distribution plot on an X-Yplane for the two-lobe ZSK-30: conditbns as in Rg. 10.

distributions of the parameter A and shear stress, respectively, for the two devices. The three-lobe discs of the ZSK-53 showed improved dispersive mixing per- formance in terms of both higher shear stresses gen- erated and more elongational flow components.

For a more accurate description of the flow strength, we also calculated the frame invariant flow parameter N,, for only one disc in the kneading block. The calculations were made increasing the mesh den- sity by a factor of 5 and maintaining the same oper- ating conditions. As mentioned previously, we could not increase the mesh density for the entire region of kneading discs because of computer limitations.

Distribution plots of the parameter N , at a cross section in the mid-plane of the disc are shown in Figs. 12 and 13 for the two-lobe and three-lobe configura- tions, respectively. In both devices the strong elonga- tional flows appear mostly in the converging regions. Volumetric distributions of the parameter N, and av- erage values (for one disc) are presented in o. 14.

Calculations have shown that the parameter N , is not affected by the operating conditions, i.e. rotational speed or pressure gradient. The results presented in Figs. 12-14 also confirm the better mixing perfor- mance of the three-lobe disc configuration in the ZSK-53 by comparison with the two-lobe configura- tion in the ZSK-30 machine.

We have also computed the torque required for op- erating the two devices under different processing conditions. The torque, T, was calculated from the cross product of the force E, resulting from the stress exerted by the fluid on the screw surfaces, and the displacement vector _r :

where is the stress tensor, a is the unit normal vector to the surface and S is the screw surface (we added the torque required to operate the two co-rotat- ing screws).

1088 POLYMER ENGINEERING AND SCIENCE, JUNE 1997, Vol. 37, No. 6

Study of Mixing E m i e n c y in Kneading Discs

Frg. 13. N, distribution plot on an X-Y plane for the three-lobe ZSK- 53: conditions as in Fig. 10.

Figure 15 plots the values of the torque required to operate the two devices for different flow rates and rotational speeds. As expected, the torque values are consistently higher for the bigger ZSK-53 machine.

1 80.00

v 60.00 4 Average = 0.5277

r

40.00

20.00

0.00 I 0.00 0.20 0.40 0.60 0.80 1.00

80.00 --l

h

$ 60.00 c 0

0

LL 40.00 0

a,

v

.- - E .- L - - 5 3 20.00

0.00

Average = 0.5452

0.00 0.20 0.40 0.60 0.80 1.00 N S

(b)

Fig. 14. Volumetric distributions of N, (a) ZSK-30 and (b) ZSK- 53; conditions as in Fig. 10.

Table 2 summarizes the average flow field charac- teristics obtained with the two-lobe and three-lobe discs, under similar processing conditions. The ZSK-53 device shows overall better dispersive mixing performance but requires a higher power input.

SUMMARY AND CONCLUSIONS

In this paper we presented a comparative study of dispersive mixing efficiency in the kneading disc re- gion of two co-rotating twin-screw extruders, the Werner & Pfleiderer ZSK-53 and ZSK-30. The study was based on the results of isothermal 3-D flow sim- ulations using a fluid dynamics analysis package em- ploying the finite element method. We analyzed the pumping characteristics as well as the flow field char- acteristics relevant for dispersive mixing in the two devices. The bigger machine ZSK-53 shows higher output values at comparable axial pressure differ- ences. However, the change in output with pressure

4 ZSK-53,lWrpm

+ ZSK-30;lOOrprn

++ ZSK-53; ZWrprn

+ ZSK-302Wrprn 70.00

60.00

30.00

20.00

10.00

0.00 40.00 80.00 120.00 160.00 Q ( c d / s )

Fig. 15. Torque required for ZSK-30 and ZSK-53 at daerent operating conditions.

POLYMER ENGINEERING AND SCIENCE, JUNE 1997, Vol. 37, No. 6 1089

Hongfei Cheng and I . Manas-Zloczower

Table 2. Average Values for Various Flow Characteristics of ZSK-30 and ZSK-53.

Device Q, cm3/s rPm AP, Pa A N: T, Pa T , N . m

ZSK-30 5.986 100 1 06 0.5609 0.5277 6.7613 X lo4 18.59 ZSK-53 7.228 100 1 06 0.5702 0.5452 9.3477 x 104 38.63

Calculated for one disc.

ACKNOWLEDGMENT The authors would like to acknowledge financial

support from the DuPont Company and the use of computing service from the Ohio Supercomputer Center.

gradient is more pronounced for the ZSK-53 by com- parison with the ZSK-30 device. The shear stresses generated are higher in the ZSK-53 machine by com- parison with the ZSK-30 device. The magnitude of stresses generated is affected by rotational speed and axial pressure gradient. The elongational flow compo- nents, quantified in terms of a parameter A, show little variation with processing conditions, and are slightly higher in the three-lobe discs of the ZSK-53 by com- parison with the two-lobe discs of the ZSK-30. Similar conclusions were obtained when using the frame in- variant flow strength parameter, N,, or when using the more conventional parameter A. Overall, the three- lobe kneading discs of the ZSK-53 machine seem to provide better flow field characteristics for dispersive mixing by comparison with the two-lobe discs of the ZSK-30 device. However, the torque /power input to operate the ZSK-53 machine is higher than the one required for the smaller ZSK-30 device.

1.

2.

3.

4.

5.

6. 7. 8. 9.

REFERENCES W. Szydlowski, R. Brzoskowskt and J. L. White, Inter. Polyrn. Roc . . 11. 42 (1988). W. Szydlowski and J. L. White, J. Non-Newt. Fluid Mech., 28, 29 (1988). Y. Wang, J. L. White, and W. Szydlowski, Inter. Polyrn. Proc., N, 262 (1989). A. D. Gotsis. Z . Ji, and D. M. Kalyon, SPEAJVTEC Tech. Papers, 38, 139 (1990). H.-H. Yang and I. Manas-Zloczower. Polyrn. Eng. Sct., 32. 1411 (1992). M. L. Booy, Polyrn. Eng. Sci., 18, 973 (1978). M. L. Booy, Polyrn. Eng. Scl., 20, 1220 (1980). D. F. Mielcarek. Chern. Eng. Prog.. 83, 59 (1987). R. G. Larson. Rheol. Acta., 24, 443 (1985).

Recelued Nou. 12, 1996 Reuised February 1997

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