Numerical study of twin-screw extruders by three-dimensional flow analysis - development of analysis technique and evaluation of mixing performance for full flight screws.
The development of polymer compounds, both blends and alloys, has played a vital role in the polymer industry. The twin-screw extruder is a high-performance device for the mixing of materials. However, the theoretical study of twin-screw extruders has not been sufficient mainly because the screw geometry is very complex, and therefore it is very difficult to analyze the behavior of materials. There are a few papers that report theoretical analyses in the solid conveying and melting zones.
The flow pattern in the melt conveying zone has been analyzed with the assumption that the flow channel was fully filled with material. In an earlier stage, flow analyses have been performed in two-dimensional cross sections perpendicular to the axis and to the rotational direction of the screw. Janssen et al. (1) and Speur et al. (2) analyzed the two-dimensional non-Newtonian flow in a co-rotating intermeshing twin-screw extruder by the finite element method (FEM) and discussed the relationship among the extrusion rate, the shear-thinning effect of viscosity, and the circulation flow. Gotsis and Kalyon (3) and Kalyon et al. (4) reported a nonisothermal non-Newtonian flow analysis in a co-rotating intermeshing extruder and discussed the mixing performance using a simulated spatial distribution of tracer fluid. They also considered the viscous heating in the nip region. Bigio and Zerafati (5) discussed the mixing performance in a counter-rotating nonintermeshing extruder in a similar way. Sastrohartono et al. (6) analyzed the flow in a co-rotating partially-intermeshing extruder and reported the effects of flow rate, pressure gradient, and power-law index on the flow field. However, these reports based on two-dimensional flow analysis could not take into account the real flow pattern because of a disregard of the helix angle and various kinds of leakage flow.
Nguyen and Lindt (7) expanded to a three-dimensional flow analysis in a nonintermeshing twin-screw extruder using a simplified flow model. They simplified the flow channel using rotating zero-helix angle screws and having the barrel move in the opposite direction to the transport direction with apparent forward velocity of the screws. They obtained leakage flows, and their calculated results agreed with the experimental ones as regards the relationship between the flow rate and the pressure gradient. However, the real flow mechanism was not clarified. Three-dimensional flow analysis with real flow geometry is required in order to clarify the flow in detail because the mixing mechanism is strongly related to the complex flow geometry.
On the other hand, several studies using three-dimensional flow analysis were carried out for kneading disc elements. Sebastian and Rakos (8) have developed a three-dimensional analysis method using the flow analysis network (FAN) method. The results obtained agreed with the results by FEM and the reliability of the FAN method was confirmed. Also Gotsis et al. (9) analyzed Newtonian flow by FEM, and Yang and Manas-Zloczower (10) reported on non-Newtonian flow using the FIDAP code.
As for full flight screw elements, the authors have been developing a three-dimensional flow analysis technique in the melt conveying zone with real screw geometry (11). In this paper, we developed a three-dimensional flow analysis technique for the full flight screws with open C-shaped channels and thin flight width in co-rotating and counter-rotating twin-screw extruders. We compare the details of velocity and stress fields in co- and counter-rotating twin-screws with the same screw configuration. Also, we obtained the spatial distribution of tracer and residence time distribution by a numerical tracer experiment and discuss the flow mechanism, the performance of transportation, and mixing, etc.
SCREW GEOMETRY AND OPERATION CONDITION
The illustrations of screw configurations used in this study are shown in Fig. 1. We consider the full flight screws with open C-shaped channels in both co-rotating and counter-rotating twin screw extruders. The definitions of geometric parameters are indicated in Fig. 2. We define the side gap as the axial gap between the flight in one screw and the adjacent flight in the other screw, and the calendar gap as the clearance between the flight in one screw and the screw root in the other screw as shown in Fig. 2. The values of geometric parameters and operation conditions, i.e. rotational speed, used in this study are listed in Table 1.
We use the following assumptions as flow conditions:
(1) The flow channel is fully filled with the polymer melt.
(2) The fluid is incompressible and pure viscous non-Newtonian.
(3) The flow is isothermal and steady.
(4) The inertia and gravitational forces are negligible because of high viscosity fluid.
(5) The fluid sticks to the solid surface.
(6) The flow field far away from the nip region is not influenced by the geometry in the nip region because C-shaped channels are opened. Therefore it is fully developed and is the same as that in a single-screw extruder with the same screw geometry.
The assumptions of isothermal flow and fluid filling are not realistic for actual industrial cases but we believe the obtained results can be used for relative evaluation of mixing performance.
Governing Equations and Material Constants
From assumptions (2), (3), and (4), the equations of continuity and motion can be reduced to
[Nabla] [multiplied by] v = 0 (1)
- [Nabla]p + [Nabla] [multiplied by] [Tau] = 0 (2)
where v is the velocity vector, p is the hydrostatic pressure and, [Tau] is the extra stress tensor. We used the Carreau model as the constitutive model of pure viscous non-Newtonian fluid described as
[Tau] = 2[Eta]D (3)
[Mathematical Expression Omitted] (4)
where D is the rate of deformation tensor, [Eta] is the shear viscosity, [[Eta].sub.0] is the zero-shear-rate viscosity, and [[Lambda].sub.c] and n are the material constants. We determined the values of the material constants from the experimental data of the shear viscosity curve for a low-density polyethylene melt. The experimental data and the fitting curve by the Carreau model are shown in Fig. 3. The values of [[Eta].sub.0], n, and [[Lambda].sub.c] are 19,500 Pa [center dot] s, 0.52, and 5.5 s, respectively.
Table 1. Values of Geometric Parameters and Operation Conditions Used in This Study. Pitch L (mm) 40 Flight width B (mm) 10 Barrel radius Rb (mm) 20 Screw radius Rs (mm) 15 Side gap [Delta]s (mm) 10 Calender gap [Delta]c (mm) 0.5 Helix angle [Theta] (deg) 17.65 Rotational speed N (rpm) 60
Analysis Domain and Moving Coordinate System
The analysis domain can be reduced to a one-pitched segment from the steady condition and the system's periodic nature. Also, it can be reduced to the mid-region containing the intermeshing zone based on assumption (6), as shown in Fig. 4. In other words, the channel flow region far from the nip region is omitted from the analysis domain.
If we use a fixed coordinate system, the geometry of analysis domain changes with time as shown in Fig. 5 a. However, we can fix the geometry of the analysis domain if we use a new coordinate system moving in the axial direction with the apparent forward velocity of the screws. In this coordinate system, a given point can be observed to move in a direction parallel to the flight.
Figure 6 shows the three-dimensional representation of the analysis domain. The boundary conditions are as follows:
(1) Nonslip condition on the screw and barrel surfaces.
(2) The flows in the boundary cross sections perpendicular to the rotational direction, i.e. the cross sections A and B, are fully developed.
(3) The flows in the boundary cross sections parallel to the flight, i.e. the cross sections C and D, correspond to that in the central cross section of the C-shaped channel in the opposite-side screw, i.e. the cross section E, because of the periodic steady condition.
As for boundary condition (2), the flow field can be obtained in the same way as for the single-screw extruder. As regards boundary condition (3), we could not find the correct flow fields in cross sections C and D at first. So we calculated the whole flow field using an initial guess for these cross sections, and the obtained flow field in cross section E is used as are those in cross sections C and D in the second iterative step. This procedure is repeated until converging solutions can be obtained.
Finite Element Mesh
The finite elements used were brick elements with eight nodal points. The velocity components and pressure were interpolated by means of triquadratic and trilinear polynomials. The meshing pattern used in the calculation for the counter-rotating twin-screw is shown in Fig. 7. The number of elements is 248, and the total number of degrees of freedom is 9490. Also, we performed the calculation using finer meshes and confirmed that the maximum errors of velocity components for both meshes was less than 3%.
(1) The values of the geometric parameters of the twin-screw extruder and the rotational speed are given.
(2) The finite element meshes are generated using geometric parameters.
(3) The velocity fields in cross sections A and B are calculated in the same way as for the single-screw extruder.
(4) The velocity fields in cross sections C and D are assumed as the initial condition.
(5) The velocity components and the pressure are simultaneously solved from Eqs 1 through 4.
(6) The velocity field in cross section E is obtained and is substituted for the velocity fields in cross sections C and D.
(7) Steps (5) and (6) are continued until the velocity fields in cross sections C and D coincide with that in cross section E.
(8) The stress fields in the whole region are calculated from Eqs 3 and 4.
(9) The spatial distribution of tracer, the residence time distributions, the history of tracer motion, etc. can be obtained by the numerical tracer experiment in which the tracer motion is pursued by means of the Euler method using the obtained velocity fields.
RESULTS AND DISCUSSION
Figure 8 shows the calculated results of velocity fields in the counter-rotating twin-screw. Cross sections I and II are the cross sections perpendicular to the screw axes across the mid-point of the side gap and across the mid-point of the calendar gap as shown in Fig. 8 a, respectively. The contours of the axial velocity component in cross sections I and II are shown in Figs. 8 b and d, while the velocity vectors are shown in Figs. 8 c and e, respectively. As regards the velocity fields in the intermeshing zone, the velocity components in the x-y plane become larger in the side gap but smaller in the calender gap, as shown in Figs. 8 c and e. This is because the fluid flows more easily in the wide side gap than in the narrow calender gap. In Figs. 8 b and d, the negative values of the axial velocity show the velocity in the transport direction. It is found that the fluid flows totally in the transport direction.
The corresponding results in the co-rotating twin-screws are shown in Fig. 9. Fig. 9 c shows that the fluid flowing in the barrel side moves from one screw to the other in the intermeshing zone. The most interesting result is that the fluid has a large axial velocity in the transport direction, i.e. a larger negative value, in comparison with that in the counter-rotating twin-screw. This suggests that the co-rotating twin-screw has a higher transport performance than the counter-rotating type in the case of the screw configuration used in this study.
However the results obtained are those for the full flight screw having the geometrical conditions of thin flight width, larger side gap, and open C-shaped channels. These results are not general because many kinds of screw types are used in industry. We should perform the simulation for the various kinds of screw configurations and compare the results with experiment.
Spatial Distribution of Tracers
Numerical determination of the spatial distribution of tracers is useful for evaluating mixing performance. Figures 10 and 11 show examples of the spatial distribution of tracers in the rotational direction in the counter-rotating and co-rotating twin-screws, respectively. We set the initial positions of the tracers in one screw channel far from the intermeshing zone, as shown in Figs. 10 and 11. The tracer particles are aligned in 15 lines over the channel width direction, and each line involves 80 particles of tracer from the barrel to the screw root. Figures 10 and 11 show the spatial distributions of tracer particles at 0.4 and 0.8 s after the injection of tracer. All particles are projected on the cross section perpendicular to the screw axis in these Figures and are also distributed in the axial direction. We can find that most of the tracer particles move in the channel of one screw in the counter-rotating twin-screw, while considerable amounts move to the other screw side in the co-rotating twin-screws. These results suggest that the co-rotating twin-screw is expected to achieve effective distributive mixing in comparison with the counter-rotating screw from the viewpoint of distribution in the rotational direction.
Flow Rates of Axial Transportation and Leakage Flows
We calculated several kinds of flow rates in order to quantitatively evaluate the performance of transportation and mixing described in the previous sections.
Table 2 shows the transport flow rate and the average velocity in the axial direction, which were obtained by the integration of axial flow fields over the cross-sectional plane perpendicular to the axis. The transport flow rate in the co-rotating twin-screw is 1.4 times as much as that in the counter-rotating. In the co-rotating screw extruder with thin flight width, when the fluids flow through the intermeshing zone from one screw to the other, it is easier to flow in the forward channel than in the backward channel because the flow channel to the forward direction is wider than to the backward direction as shown in Fig. 1 b. Therefore a large velocity in the transport direction is generated in the intermeshing zone as shown in Fig. 9. (The velocity in the transport direction is shown as a negative value owing to the definition of coordinates in this study.) Most leakage from one screw to the other is forward flow in the axial direction. On the other hand, a large amount of fluid passes through the calender gap and side gap, and flows in the channel in the same screw for the counter-rotating extruder having thin flight width and open C-shaped channel. This fluid does not contribute to fluid transport.
The flow rates of various kinds of leakage flows are listed in Table 3. [Q.sub.C] and [Q.sub.S] represent the flow rates of fluids that pass through the side cross section and the calender cross section, as shown in Fig. 12. [Q.sub.AB] is the net flow rate of fluid that moves from screw A to B passing through the intermeshing zone. These values are divided by the flow rate of fluid flowing in the tangential direction in the screw channel, Q. In both counter-rotating and co-rotating twin-screws most fluid flows in the wide side gap avoiding the narrow calender gap, and therefore [Q.sub.c] is very small. This fluid flowing in the side gap passes through the side cross section and moves in the same screw channel in the counter-rotating twin-screw, while most fluid flowing in the side gap changes flow direction on this side of the side cross section and moves in the other screw channel in the co-rotating twin-screw. Therefore [Q.sub.s] is considerably smaller and [Q.sub.AB] is much larger in the co-rotating twin-screw in comparison with those in counter-rotating screws. The co-rotating twin-screw is expected to homogenize the materials by rearrangement of fluid.
Table 2. Flow Rate and Average Velocity in the Transport Direction. Average Flow Rate Velocity ([cm.sup.3]/s) (cm/s) Counter-rotating 10.84 1.408 Co-rotating 15.14 1.670 Table 3. Flow Rates of Various Kinds of Leakage Flows. [Q.sub.C]/Q [Q.sub.S]/Q [Q.sub.AB]/Q Counter-rotating 0.058 0.910 0.016 Co-rotating 0.015 0.194 0.831 [Q.sub.C]: Flow rate of fluid passing through the calender cross-section C indicated in Fig. 12. [Q.sub.S]: Flow rate of fluid passing through the side cross section S indicated in Fig. 12. [Q.sub.AB]: Net flow rate of fluid moving from the screw A to B. Q: Flow rate of fluid flowing in the tangential direction in the screw channel.
Residence Time Distribution
The residence time distribution is one of the important indices for the evaluation of extruder performance, for example, mixing, thermal degradation, polymerization in reactive extrusion, etc. We obtained the time traveled through one pitch of the screw for each tracer particle in the numerical tracer experiment and calculated the flow rate of the fluid with a given residence time. Then we obtained the residence time distributions for both counter-rotating and co-rotating twin-screws, as seen in Fig. 13, where we show normalized cumulative flow rates.
The counter-rotating twin-screw has a wider distribution and a longer mean value of residence time than the co-rotating twin-screw. The reason for this is the same as that which explains why the flow rate in the transport direction in the co-rotating extruder is larger than that in the counter-rotating one, as mentioned above. In Fig. 13, we also represented the theoretical values of average residence time obtained from the channel volume and throughput. In the numerical tracer experiments, some particles were difficult to pursue and disappeared. The particles that are set near the wall initially are difficult to pursue, and these particles seemed to have larger residence times. Therefore, the average residence time obtained from the simulation results is shorter than theoretical values.
From the viewpoint of mixing, a wider distribution of residence time means a wider distribution of tracer particles in the axial direction. Although it is suggested from the spatial distribution of the tracers in the rotational direction that the co-rotating twin-screw has higher performance for distributive mixing, as described in the previous sections, the distribution of tracers in the axial direction is found to be more suitable in the counter-rotating twin-screw. A quantitative evaluation method for such performance using the simulation results should be accomplished in the future.
The spatial distribution of tracers is strongly related to distributive mixing. We had to evaluate the other aspects of mixing performance, i.e., dispersive mixing, which means the break up of agglomerates (for example, filling particles in the polymer matrix). This dispersive mixing is thought to be strongly related to the magnitude of stress acting on the agglomerates. Therefore, we calculated the characteristic shear stress as one evaluation index, which is invariant and is defined as
[Mathematical Expression Omitted] (5)
where [([[Tau].sub.max]).sub.xy] denotes the maximum shear stress in the x-y plane and can be obtained as
[Mathematical Expression Omitted] (6)
[Mathematical Expression Omitted] (7)
Figures 14 and 15 show the contours of the characteristic shear stress in the typical two planes perpendicular to the axis for the counter-rotating and co-rotating twin-screws, respectively. These planes are the same as those in Figs. 8 and 9, and were previously explained. The larger values occur on the flight in the calender gap region, but there is little fluid passing through this region. On the whole, the values of the characteristic shear stress for both rotating types are almost the same.
We have developed a technique for predicting the three-dimensional flow field in the melt conveying zone of a twin-screw extruder. We obtained information about velocity and stress fields for full flight screws with thin flight width and open C-shaped channels in both co-rotating and counter-rotating twin-screw extruders, and calculated the flow rates of transportation and various kinds of leakage flows. We also obtained the spatial distribution of tracers and residence time distribution using numerical tracer experiments.
The results suggest that the co-rotating twin-screw is suitable for strong transportation and tangential distribution of fluid, while the co-rotating twin-screw provides greater axial distribution in distributive mixing. There is no difference in dispersive mixing for both rotating types. All results were thought to be reasonable from the phenomenological and empirical viewpoints. However, the results obtained are those for a full flight screw having the geometrical conditions of thin flight width, larger side gap, and open C-shaped channels. These results is not general because many kinds of screw types are used in industry.
After this we should confirm the applicability to actual problems by comparing with experimental results and then establish evaluation methods for several kinds of equipment performance using the simulation results. We will also modify this simulation technique in order to apply it to the intermeshing type twin-screw extruder or other mixing elements, for example, the kneading disk and rotor.
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|Author:||Kajiwara, Toshihisa; Nagashima, Yuki; Nakano, Yoshio; Funatsu, Kazumori|
|Publication:||Polymer Engineering and Science|
|Date:||Aug 1, 1996|
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