# Numerical simulation and experimental validation of mixing performance of kneading discs in a twin screw extruder.

INTRODUCTION

Simulation and Experimental Validation

Co-rotating intermeshing twin-screw extruders have attracted considerable attention during the last 20 years, mainly in the field of food and polymer processing (1). Among different modules of a co-rotating twin-screw extruder, kneading discs (KDs) arc the dominant ones for mixing efficiency. It remains empirical when it comes to designing KDs and optimizing extrusion conditions for a specific application because of the very large number of geometrical, theological, and operating parameters that may influence the flow. Simplified 1D models are currently the important method used for predicting the temperature, pressure profile, and residence time distribution (RTD) in extrusion processing (2), (3). However, they only provide average values of different parameters along the extruder that may not always be sufficient for describing flow pattern and mixing (4), (5). In recent years, much attention has been paid to full 3D simulation of the flow field in different regions of a co-rotating twin-screw extruder. Manas-Zloczower and coworker investigated the distributive mixing, dispersive mixing and flow field of different regions in co-rotating twin-screw extruders including mixing and conveying elements (6-8). Bravo et al. presented a 3D flow field analysis for KDs (9), (10). Ishikawa et al. performed nonisothermal simulation of KDs, rotors, conveying element, and screw mixing elements (11-13). Lawal and Kalyon discussed mixing mechanisms in co-rotating twin screw extruders (14).

It is not easy to calculate flow patterns in complex geometries such as KDs. As the KDs rotate, time-dependent boundaries make numerical simulations more difficult. Avalosse introduced a so-called mesh superposition technique (MST) to model intermeshing twin-screw extruders without calling upon remeshing (15). They used the Polyflow software (Fluent Inc.) based on the MST to analyze mixing efficiency and flow field in a kneading block section of a twin-screw extruder (16-18). Connelly and Kokini used Polyflow to discuss the effects of shear thinning and viscoelasticity fluid on mixing in a 2D mixer (19), (20). They simulated the flow of a viscous Newtonian fluid in a twin sigma blade mixer using Polyflow code with MST and analyzed the distributive mixing and mixing efficiency of the mixer (21). However, those studies were limited to batch mixers. Breuer et al. simulated the flow pattern in a miniature mixer and compared the performance of the Albert polymer asymmetric minimixer to that of the MiniMAX molder (22). Ficarella et al. investigated the fluid-dynamic behavior of an extrusion cooking process for cereals in a co-rotating twin-screw extruder using Polyflow code (23), (24).

Experimental validation of simulated results is another big challenge. There have been studies on numerical simulations of pressure, temperature, and velocity profiles and their comparison with experiments (9), (11), (25). Potente and Flecke used a 2.5D finite element method to analyze the minimal residence time in conveying elements and compared the minimal residence time with experimental data (26). The modeling was based on the plain unwinding of the intermeshing twin screws. Huneault measured the RTD in co-rotating twin-screw extruders, used the RTD to determine the effect of operating conditions on the degree of fill (27), and compared the experimental data of the degree of fill with the calculated ones. However, to the best of the authors' knowledge, numerical simulations and experimental validations of the local RTD in mixing sections of a twin-screw extruder are still rare because of experimental difficulties, and yet they are crucial.

Kinematic Model of Distributive Mixing

For a viscous fluid, the distributive mixing mechanism is the stretching and reorientation of fluid elements (28). Ottino and coworkers developed a lamellar model to quantify the capacity of a flow to deform matter and to generate interface between two components in a mixture, which gives a kinematic approach to modeling distributive mixing by tracking the amount of deformation experienced by a fluid element (28-32). In the initial fluid domain, when an infinitesimal material area |dA| with normal direction [^.N] is transformed into an area |da| with normal direction [^.n] at time t, the area stretch ratio of material surface [eta] is defined as (29) follows:

[eta] = [|da|/|dA|]. (1)

A good distributive mixing quality requires a high value of [eta] over space and time. An instantaneous stretching efficiency [e.sub.[eta]] based on stretching of material surfaces is defined as (30) follows:

[e.sub.[eta]] = [[[eta]/[eta]]/[(D:D).sup.[1/2]]] = [[d(log [eta])/dt]/[(D:D).sup.[1/2]]] = [[ - D:[^.m][^.m]]/[(D:D).sup.[1/2]]], (2)

where [eta] is a material time derivative of [eta], D is a rate of deformation tensor, and [^.m] is a local normal unit vector of current surface. [e.sub.[eta]] characterizes the fraction of the dissipated energy used for stretching the area and falls in the range [ - 1, 1]. A positive (negative) value indicates that the dissipated energy is used for stretching (shrinking) the surface. A time-averaged efficiency <[e.sub.[eta]]> is defined as (29) follows:

<[e.sub.[eta]]> = [1/t][[integral].sub.0.sup.t][e.sub.[eta]]dt. (3)

The above mixing parameters can be calculated when the velocity field is known. The logarithm of the area stretch ratio log [eta] can be used to compare the distributive mixing performance of different mixers. Distributive mixing efficiency can be characterized by time-average efficiency <[e.sub.[eta]]>. Some researchers defined a distribution function [F.sub.[alpha]] associated with the field x. The quantity [F.sub.[alpha]]([[alpha].sub.p], t) is given as (33), (34) follows:

[F.sub.[alpha]]([[alpha].sub.p], t) = p, (4)

where [[alpha].sub.p] is a critical value, which indicates that p % of marker particles have a value of x lower than or equal to [[alpha].sub.p] at time t. The field [alpha] may be [eta], [e.sub.[eta]], or <[e.sub.[eta]]>.

This work aims at analyzing the mixing of a generalized Newtonian fluid in KDs using numerical simulation and experiments. The velocity field in kneading zones is simulated using a commercial computational fluid dynamics code: POLYFLOW 3.10.0 [R]. The local RTD of the same domains is then calculated based on the velocity field. Simulated results are compared with experimental ones obtained by an in-line measuring instrument (35), (36). The distributive mixing parameters such as the area stretch ratio of material surface [eta], instantaneous stretching efficiency [e.sub.[eta]] and time-averaged efficiency <[e.sub.[eta]]> are calculated using the interface tracking techniques. These parameters are then used to compare the distributive mixing performance and efficiency of different types of KDs.

NUMERICAL SIMULATION

Geometry of KDs

Figure 1 shows the geometry of eight different KDs simulated in this work. They all have the same axial length, i.e., 64 mm, and differ in stagger angle, disc width, and/or disc gap. Table 1 describes the characteristics of KD1 to KD8 and flow channel in Fig. 1. The first three KDs (KD 1 to KD3) are used for experimentally validating the simulated results. KD4 to KD6 have the same disc width and gap, but differ in stagger angle. They are chosen to analyze the effect of the stagger angle on the local RTD and distributive mixing parameters. The effects of the disc width and gap on the axial and distributive mixing are studied with KD6, KD7, and KD8. The KD7 and KD8 do not have disc gap between the two discs. The flow channel includes three parts: inlet section, kneading section, and outlet section. The kneading section accommodates a KD. Sufficiently long inlet and outlet sections are added to the computational domains where fully developed flows are assumed (37). Figure 2 shows the disc gaps of a KD.

[FIGURE 1 OMITTED]

[FIGURE OMITTED]

[FIGURE 2 OMITTED]

Meshing is carried out using the commercial preprocessor software GAMBIT (Fluent Inc.). The meshes of each KD are chosen after a preliminary mesh-discretization study. A convergence analysis compares the velocity and pressure solutions of different mesh numbers along the axial direction. The number of meshes is increased till the solutions are insensitive to it. Table 2 gathers the numbers of 3D elements for KD1 to KD8 and the flow channel used for the simulations.

Numerical Simulation and Material Parameters

Polyflow software incorporates a so-called MST to simulate transient flows with internal moving parts (36). The MST generates finite element meshes for the flow domain and each screw element, respectively, which are then superimposed. Since the position of each screw element is updated at each time step, the flow domain can be calculated by the weak formulation of the Navier-Stokes equations and the screw element domain can be constrained by the screw speed.

The flow is assumed to be a time-dependent, isothermal, incompressible, and generalized Newtonian fluid with no body forces. The governing equations are the momentum and continuity equations and can be written as follows:

[DELTA] * [tau] - [DELTA]P = 0. (5)

[nabla] * u = 0. (6)

where [tau] is the extra stress tensor. P is the pressure, and u is the velocity vector. [tau] is given by

[tau] = 2[eta]D, (7)

where D is the rate of deformation tensor: [eta] is the kinematic viscosity and depends on the shear rate [gamma]. The constitutive equation is represented by the Carreau model:

[eta]([gamma]) = [[eta].sub.0][[1 + [([lambda][gamma]).sup.2]].sup.[(n - 1)/2]], (8)

where [[eta].sub.0] is the zero shear rate viscosity, [lambda] is the natural time, and n is the power law index. The material used in this study is polystyrene (PS) of Yangzi-BASF. China. Its viscosity is measured on a capillary rheometer (Thermo Haake Rheoflixer) for the high shear rate domain and on an advanced rheometric expansion system (ARES: Rheometrics Corp., USA) in a parallel plate mode for the low shear rate domain (see Fig. 3). The parameters for the Carreau model are [[eta].sub.0] = 2814 Pa s. [lambda] = 0.148 s, and n = 0.278.

[FIGURE 3 OMITTED]

The no-slip boundary conditions are imposed to the surfaces of the KDs and barrel bore walls. A volume flow rate is set to the inlet section and a normal stress boundary condition is applied to the outlet section. Table 3 lists the operating conditions for the numerical simulations and experiments. The volume throughput was calculated by dividing the mass throughput by the melt density of the PS (970 kg/[m.sup.3]).

Marker Particle Tracking Analysis

To calculate the local RTD provided by each of the above eight types of KDs, virtual particles are launched at the same time in the flow domain. It is assumed that the marker particles are massless, volumeless, and noninter-acting with each other. Under these assumptions, the particles can be located by integrating the velocity vectors with respect to time. Initially, these particles are randomly distributed in an inlet vertical plane (see Fig. 4) and their trajectories between the inlet and outlet are calculated from the velocity profiles. Along each trajectory, the residence time is computed. The RTD is obtained based on the residence time of each of these particles. The number of marker particles has to be high enough for the simulation. Figure 5 shows that 1000 marker particles are not enough and that 2000 marker particles start yielding results much closer to those of the 3000.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Previous researchers pointed out that the initial location and orientation of interfaces could have an influence on mixing parameters such as the area stretch ratio [eta] and instantaneous stretching efficiency [e.sub.[eta]] (20), (21), (28), (29). According to Chella and Ottino, the stretch rate of the initially vertical interface perpendicular to the streamlines is faster than that of horizontal interface in the rectangular cavity flow (31). They also found that when a large amount of the area is calculated to characterize the mixing performance of different mixers, the influence of initial orientation on the stretch rate is very small. Figure 6 shows the effect of the number of marker particles and that of the initial direction of interface on the arithmetic mean of the area stretch ratio and that of the instantaneous stretching efficiency in KD2. When the number of marker particles is large enough, such as 2000 and 3000 particles, the arithmetic means of the two mixing parameters are insensitive to the initial direction and location nor the number of marker particles. This infers that they can be used to compare the distributive mixing performance and efficiency of different types of KDs. Subsequently, 2000 particles with random direction are used to calculate mixing parameters.

[FIGURE 6 OMITTED]

EXPERIMENTAL

RTDs are measured under the same operating conditions as the numerical simulations. Figure 7 shows the screw configuration used in this study. There are two test points marked as Probes 1 and 2. The screw elements beneath Probes 1 and 2 were cylinders. The cylinders ensured full fill of the screw elements beneath Probes 1 and 2 and a constant depth between the probes and the screw element surfaces. Meanwhile, using the cylinder was to make sure that the flow was fully developed in the inlet and outlet sections as it is assumed for the numerical simulations. Briefly, the two probes were bifurcated optical fibers and the detection principles were based on fluorescent emission. Details about their characteristics and working principles can be found in the literature (35), (36). There were four identical KDs between Probes 1 and 2, called the test zone. It was the object of the numerical simulation. However, the test zone has four pairs of identical KDs instead of two for the numerical simulation. The reason was that computing four pairs of KDs was computationally costly. To overcome this problem, four pairs of KDs were divided into two "two pairs." It was assumed that the statistical theory for RTDs is applicable for calculating the RTD of the four pairs from those of the two "two pairs." Based on the statistical theory (38-40), the delay time of the four pairs was twice that of the two "two pairs." [E.sub.1] (t) and [E.sub.2] (t) were measured by Probes 1 and 2, respectively. Meanwhile, [E.sub.2] (t) was calculated by the convolution of [E.sub.1] (t) and [E.sub.12] (t) obtained by numerical simulations and was used to compare with the experimentally measured [E.sub.2] (t).

[FIGURE 7 OMITTED]

RESULTS AND DISCUSSION

Experimental Validation

To gain better understanding of the flow occurring in the KDs of the co-rotating twin-screw extruder, Fig. 8 shows the streamlines of two particles initially located at different places of the inlet. They followed different streamlines to the exit, implying that they had different residence times. Particle 1 proceeded around the screws in a "figure-eight" mode, which is in agreement with the experimental results of previous researchers (41), (42). For particle 2, its motion first followed a "figure-eight" mode and then circled around one screw. It finally returned a "figure-eight" mode. The above two different motion modes show that as time evolved, the fluid particles underwent different degrees of shear and stress. The complex flow in the KDs is indicative of good mixing.

[FIGURE 8 OMITTED]

Figure 9 compares the numerical local RTD results with the experimental ones for KD2. The overall trend was in good agreement. However, the former were wider than the latter. Reasons for those differences could be as follows. The simulation assumed that the flow channel be fully filled and that the flow be isothermal. Moreover, the melt density of the polymer was assumed to be constant and did not vary with pressure and temperature, and the polymer melt was assumed to be a generalized Newtonian fluid. Those assumptions were not necessarily fully met in practice. It was reported that subtle differences in rheology could impart significant differences to mixing (20). Figure 10 compares the simulated and experimental RTD for different screw configurations. Again the 90[degrees] KDs has the highest axial mixing performance while the 45[degrees] and 60[degrees] KDs have similar axial mixing performance.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

Effect of Geometrical Parameters on Local RTD

The KD4, KD5, and KD6 have the same disc width and gap but different stagger angles. Figure 11 shows the effect of the stagger angle on the local RTD. The delay time and average residence time increased with an increase in the stagger angle. Also the 90[degrees] KD provides the highest axial mixing quality.

[FIGURE 11 OMITTED]

Figure 12 shows the effects of the disc gap and width on the local RTD. The three local RTD curves are similar in delay time and shape, indicating that the KD6, KD7, KD8 have similar axial mixing quality.

[FIGURE 12 OMITTED]

Distributive Mixing Performance and Efficiency

The area stretch was used as a measure for comparing the distributive mixing performance of different types of KDs. An effective KD requires high values of the logarithm of the area stretch ratio over the axial distance. Distributive mixing efficiency was characterized by time-average efficiency. Figure 13a shows the arithmetic mean of log [eta] from the entry to the exit. For KD4, KD5, and KD6, it increased more or less linearly with increasing axial distance Z, indicating that [eta] increased exponentially with Z. Among the three screw configurations, for a given axial distance the arithmetic mean of log [eta] followed the order: KD6 > KD5 > KD4, indicating that the distributive mixing performance increased with increasing stagger angle of the KD. Figure 13b shows the critical value of the arithmetic mean of log [eta] along Z for 10, 50, and 90% marker particles for KD4, KD5, and KD6, respectively. Again it followed the order: KD6 > KD5 > KD4, regardless of the percentile of marker particles.

[FIGURE 13 OMITTED]

From Fig. 13a, the arithmetic mean of log [eta] vs. Z curves fluctuated, especially at a higher axial distance. This means that not all particle surfaces are stretched along Z in a steady manner. At the initial stage of mixing, it is easy to stretch the surface area and the arithmetic mean of log [eta] increases linearly with Z. As the surface area is further stretched, area stretching becomes more difficult. Meanwhile, Ishikawa et al, pointed out that the disc gap plays an important role for mixing because of the large amount of forward and backward flow (11). The latter may bring about stretch and shrink of surfaces. Figure 13b shows that stretch and shrink of surfaces occur primarily at high area stretch ratio.

Figure 14a shows the effects of the disc gap and disc width on the arithmetic mean of log [eta] as a function of Z for 90[degrees]KDs. The arithmetic mean of log [eta] of a KD with a gap (KD6) is higher than that of a corresponding KD without disc gap (KD7). On the other hand, A KD with a smaller disc width a smaller disc width (KD7) has a higher arithmetic mean of log [eta] than a corresponding one (KD8), indicating that the distributive mixing performance decreases with increasing disc width. This result is in agreement with that of Andersen (43). It is also noted that unlike KD7 and KD8 whose arithmetic mean of log [eta] increases linearly and smoothly with Z, that of KD6 also increases linearly but in a fluctuating manner. In fact, each fluctuation of the arithmetic mean of log [eta] corresponds to a passage through a gap. Figure 14b shows the critical values for 10, 50, 90% marker particles. There are no fluctuations at low percentiles such as 10 and 50%. Fluctuations are visible only at high percentiles of 90%.

[FIGURE 14 OMITTED]

In a closed system such as a batch mixer, the time-average efficiency exhibits three characteristic features: <[e.sub.[eta]]> decays as [t.sup.-1] for flow with no reorientation; <[e.sub.[eta]]> decays as [t.sup.-1] but oscillates for flow with moderate reorientation; <[e.sub.[eta]]> oscillates without decay for flow with strong reorientation (20), (31), (32). For an open system such as an extruder or a Kenics mixer, [Z.sup.-1] is equivalent to [t.sup.-1] Figure 15a shows the evolution of the arithmetic mean of the time-average efficiency as a function of Z for KD4, KD5, and KD6. The three curves have periodic oscillations without decay as [Z.sup.-1], indicating that all the three KDs bring about strong reorientation. Among the three KDs, the arithmetic mean of the time-average efficiency is the highest for KD6 is highest, indicating that KD6 provides strongest reorientation. Figure 15b shows the time-average efficiency as a function of Z for different percentiles of marker particles. At low percentiles such as 10 and 50%, the critical values of the time-average efficiency are similar for the three types of KDs. At a high percentile such as 90%, KD6 yields significantly higher time-average efficiency. This indicates that at low percentiles the three types of KDs are similar in time-average efficiency. However, at high percentiles, KD6 provides the highest time-average efficiency. It is also noted that when marker particles start moving, most of them are stretched. Nevertheless, some of them have shrunk. Therefore, they exhibit negative time-average efficiency at the initial stage. They are all stretched eventually.

[FIGURE 15 OMITTED]

As for the three 90[degrees] KDs (KD6, KD7, and KD8), they have similar RTD curves (see Fig. 12) but differ in log [eta] (see Fig. 14). KD6 provides the highest log [eta], thus the highest distributive mixing performance. Figure 16 compares these three types of KDs in terms of <[e.sub.[eta]]>. Clearly KD8 provides the highest mixing efficiency and not KD6. The reason is that <[e.sub.[eta]]> is not only a function of d(log [eta])/dt but also of D:D and RTD. Figure 17 shows the evolution of d(log [eta])/dt and [(D:D).sup.[1/2]] as a function of Z for KD6, KD7, and KD8, respectively. The arithmetic mean of d(log [eta])/dt is very close for the three types of KDs. On the other hand, these three types of KDs differ more or less in [(D:D).sup.[1/2]], the energy dissipated. The value of [(D:D).sup.[1/2]] is the lowest for KD8, indicating that the fraction of the energy dissipated for the area stretch is the highest for KD8. In sum, KD8 provides the lowest distributive mixing performance (characterized by log [eta]) but highest mixing efficiency (characterized by <[e.sub.[eta]]>).

[FIGURE 16 OMITTED]

[FIGURE 17 OMITTED]

CONCLUSIONS

This article has reported on three-dimensional numerical simulations of flow and mixing in KDs in a co-rotating twin-screw extruder. The simulations are validated against experimentally measured local RTDs for different types of KDs and experimental conditions.

In addition to operating conditions, the geometry of the KD may have a significant effect on the local RTD. The delay time and mean residence time increase with increasing stagger angle of the KD: 45[degrees] < 60[degrees] < 90[degrees]. The axial mixing characterized by the spread of the local RTD follows the opposite trend.

The area stretch ratio [eta] and the time-average efficiency <[e.sub.[eta]]> characterize the distributive mixing performance and efficiency, respectively. The former increases with increasing axial distance Z in an exponential manner for all types of KDs simulated in this work, indicating that they all have good distributive mixing performance. The magnitude of increase in [eta] and <[e.sub.[eta]]> follows the order: 45[degrees] < 60[degrees] < 90[degrees].

The disc gap and disc width play an important role in distributive mixing performance and efficiency. A KD with a disc gap provides higher area stretch ratio and time-average efficiency than a corresponding one without disc gap. A KD with a larger disc width has a lower distributive mixing performance than a corresponding one with a smaller disc width. However, its distributive mixing efficiency could be higher because of lower energy dissipation, implying that for thermally sensitive polymers, it would be better to use a KD with a larger disc width than a KD with a smaller disc width. In general, KDs with disc gaps, small disc widths, and large stagger angles provide good distributive mixing performance.

ACKNOWLEDGMENTS

The authors express their sincere thanks to the reviewers for their thoughtful and constructive comments on this manuscript.

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Xian-Ming Zhang, (1) Lian-Fang Feng, (2) Wen-Xing Chen, (1) Guo-Hua Hu (3), (4)

(1) Key Laboratory of Advanced Textile Materials and Manufacturing Technology, Ministry of Education, Zhejiang Sci-Tech University, Hangzhou 310018, China

(2) State Key Laboratory of Polymer Reaction Engineering, College of Materials Science and Chemical Engineering, Zhejiang University, Hangzhou 310027, China

(3) Laboratory of Chemical Engineering Sciences, CNRS-ENSIC-INPL, 54001 Nancy, France

(4) Institut Universitaire de France, Maison des Universites, 75005 Paris, France

Correspondence to: Lian-Fang Feng: e-mail: fenglf@zju.edu.en or GuoHua Hu: e-mail: hu@ensic.inpl-nancy.fr

Contract grant sponsor: National Natural Science Foundation of China; contract grant number: 50390097; contract grant sponsor: Key Laboratory of Advanced Textile Materials and Manufacturing Technology (Zhejiang Sci-Tech University), Ministry of Education; contract grant number: 2008QN05.

DOI 10.1002/pen.21404

Simulation and Experimental Validation

Co-rotating intermeshing twin-screw extruders have attracted considerable attention during the last 20 years, mainly in the field of food and polymer processing (1). Among different modules of a co-rotating twin-screw extruder, kneading discs (KDs) arc the dominant ones for mixing efficiency. It remains empirical when it comes to designing KDs and optimizing extrusion conditions for a specific application because of the very large number of geometrical, theological, and operating parameters that may influence the flow. Simplified 1D models are currently the important method used for predicting the temperature, pressure profile, and residence time distribution (RTD) in extrusion processing (2), (3). However, they only provide average values of different parameters along the extruder that may not always be sufficient for describing flow pattern and mixing (4), (5). In recent years, much attention has been paid to full 3D simulation of the flow field in different regions of a co-rotating twin-screw extruder. Manas-Zloczower and coworker investigated the distributive mixing, dispersive mixing and flow field of different regions in co-rotating twin-screw extruders including mixing and conveying elements (6-8). Bravo et al. presented a 3D flow field analysis for KDs (9), (10). Ishikawa et al. performed nonisothermal simulation of KDs, rotors, conveying element, and screw mixing elements (11-13). Lawal and Kalyon discussed mixing mechanisms in co-rotating twin screw extruders (14).

It is not easy to calculate flow patterns in complex geometries such as KDs. As the KDs rotate, time-dependent boundaries make numerical simulations more difficult. Avalosse introduced a so-called mesh superposition technique (MST) to model intermeshing twin-screw extruders without calling upon remeshing (15). They used the Polyflow software (Fluent Inc.) based on the MST to analyze mixing efficiency and flow field in a kneading block section of a twin-screw extruder (16-18). Connelly and Kokini used Polyflow to discuss the effects of shear thinning and viscoelasticity fluid on mixing in a 2D mixer (19), (20). They simulated the flow of a viscous Newtonian fluid in a twin sigma blade mixer using Polyflow code with MST and analyzed the distributive mixing and mixing efficiency of the mixer (21). However, those studies were limited to batch mixers. Breuer et al. simulated the flow pattern in a miniature mixer and compared the performance of the Albert polymer asymmetric minimixer to that of the MiniMAX molder (22). Ficarella et al. investigated the fluid-dynamic behavior of an extrusion cooking process for cereals in a co-rotating twin-screw extruder using Polyflow code (23), (24).

Experimental validation of simulated results is another big challenge. There have been studies on numerical simulations of pressure, temperature, and velocity profiles and their comparison with experiments (9), (11), (25). Potente and Flecke used a 2.5D finite element method to analyze the minimal residence time in conveying elements and compared the minimal residence time with experimental data (26). The modeling was based on the plain unwinding of the intermeshing twin screws. Huneault measured the RTD in co-rotating twin-screw extruders, used the RTD to determine the effect of operating conditions on the degree of fill (27), and compared the experimental data of the degree of fill with the calculated ones. However, to the best of the authors' knowledge, numerical simulations and experimental validations of the local RTD in mixing sections of a twin-screw extruder are still rare because of experimental difficulties, and yet they are crucial.

Kinematic Model of Distributive Mixing

For a viscous fluid, the distributive mixing mechanism is the stretching and reorientation of fluid elements (28). Ottino and coworkers developed a lamellar model to quantify the capacity of a flow to deform matter and to generate interface between two components in a mixture, which gives a kinematic approach to modeling distributive mixing by tracking the amount of deformation experienced by a fluid element (28-32). In the initial fluid domain, when an infinitesimal material area |dA| with normal direction [^.N] is transformed into an area |da| with normal direction [^.n] at time t, the area stretch ratio of material surface [eta] is defined as (29) follows:

[eta] = [|da|/|dA|]. (1)

A good distributive mixing quality requires a high value of [eta] over space and time. An instantaneous stretching efficiency [e.sub.[eta]] based on stretching of material surfaces is defined as (30) follows:

[e.sub.[eta]] = [[[eta]/[eta]]/[(D:D).sup.[1/2]]] = [[d(log [eta])/dt]/[(D:D).sup.[1/2]]] = [[ - D:[^.m][^.m]]/[(D:D).sup.[1/2]]], (2)

where [eta] is a material time derivative of [eta], D is a rate of deformation tensor, and [^.m] is a local normal unit vector of current surface. [e.sub.[eta]] characterizes the fraction of the dissipated energy used for stretching the area and falls in the range [ - 1, 1]. A positive (negative) value indicates that the dissipated energy is used for stretching (shrinking) the surface. A time-averaged efficiency <[e.sub.[eta]]> is defined as (29) follows:

<[e.sub.[eta]]> = [1/t][[integral].sub.0.sup.t][e.sub.[eta]]dt. (3)

The above mixing parameters can be calculated when the velocity field is known. The logarithm of the area stretch ratio log [eta] can be used to compare the distributive mixing performance of different mixers. Distributive mixing efficiency can be characterized by time-average efficiency <[e.sub.[eta]]>. Some researchers defined a distribution function [F.sub.[alpha]] associated with the field x. The quantity [F.sub.[alpha]]([[alpha].sub.p], t) is given as (33), (34) follows:

[F.sub.[alpha]]([[alpha].sub.p], t) = p, (4)

where [[alpha].sub.p] is a critical value, which indicates that p % of marker particles have a value of x lower than or equal to [[alpha].sub.p] at time t. The field [alpha] may be [eta], [e.sub.[eta]], or <[e.sub.[eta]]>.

This work aims at analyzing the mixing of a generalized Newtonian fluid in KDs using numerical simulation and experiments. The velocity field in kneading zones is simulated using a commercial computational fluid dynamics code: POLYFLOW 3.10.0 [R]. The local RTD of the same domains is then calculated based on the velocity field. Simulated results are compared with experimental ones obtained by an in-line measuring instrument (35), (36). The distributive mixing parameters such as the area stretch ratio of material surface [eta], instantaneous stretching efficiency [e.sub.[eta]] and time-averaged efficiency <[e.sub.[eta]]> are calculated using the interface tracking techniques. These parameters are then used to compare the distributive mixing performance and efficiency of different types of KDs.

NUMERICAL SIMULATION

Geometry of KDs

Figure 1 shows the geometry of eight different KDs simulated in this work. They all have the same axial length, i.e., 64 mm, and differ in stagger angle, disc width, and/or disc gap. Table 1 describes the characteristics of KD1 to KD8 and flow channel in Fig. 1. The first three KDs (KD 1 to KD3) are used for experimentally validating the simulated results. KD4 to KD6 have the same disc width and gap, but differ in stagger angle. They are chosen to analyze the effect of the stagger angle on the local RTD and distributive mixing parameters. The effects of the disc width and gap on the axial and distributive mixing are studied with KD6, KD7, and KD8. The KD7 and KD8 do not have disc gap between the two discs. The flow channel includes three parts: inlet section, kneading section, and outlet section. The kneading section accommodates a KD. Sufficiently long inlet and outlet sections are added to the computational domains where fully developed flows are assumed (37). Figure 2 shows the disc gaps of a KD.

[FIGURE 1 OMITTED]

[FIGURE OMITTED]

[FIGURE 2 OMITTED]

TABLE 1. The description of kneading discs KD1 to KD8 and flow channel in Fig. 1. Description KD1 Two pairs of intermeshing 45/5/32 kneading discs KD2 Two pairs of intermeshing 60/4/32 kneading discs KD3 Two pairs of intermeshing 90/5/32 kneading discs KD4 One pair of intermeshing 45/10/64 kneading discs KD5 One pair of intermeshing 60/10/64 kneading discs KD6 One pair of intermeshing 90/10/64 kneading discs KD7 One pair of intermeshing 90/10/64 kneading discs KD8 One pair of intermeshing 90/5/64 kneading discs Flow channel Inlet section (2 mm) + kneading section (64 mm) outlet section (4 mm)

Meshing is carried out using the commercial preprocessor software GAMBIT (Fluent Inc.). The meshes of each KD are chosen after a preliminary mesh-discretization study. A convergence analysis compares the velocity and pressure solutions of different mesh numbers along the axial direction. The number of meshes is increased till the solutions are insensitive to it. Table 2 gathers the numbers of 3D elements for KD1 to KD8 and the flow channel used for the simulations.

TABLE 2. The number of 3D elements for KDl to KD8 and the flow channel. Number of 3D elements KD1 8960 KD2 8122 KD3 8512 KD4 9880 KD5 10,100 KD6 9976 KD7 9260 KD8 9260 Flow channel 21,000

Numerical Simulation and Material Parameters

Polyflow software incorporates a so-called MST to simulate transient flows with internal moving parts (36). The MST generates finite element meshes for the flow domain and each screw element, respectively, which are then superimposed. Since the position of each screw element is updated at each time step, the flow domain can be calculated by the weak formulation of the Navier-Stokes equations and the screw element domain can be constrained by the screw speed.

The flow is assumed to be a time-dependent, isothermal, incompressible, and generalized Newtonian fluid with no body forces. The governing equations are the momentum and continuity equations and can be written as follows:

[DELTA] * [tau] - [DELTA]P = 0. (5)

[nabla] * u = 0. (6)

where [tau] is the extra stress tensor. P is the pressure, and u is the velocity vector. [tau] is given by

[tau] = 2[eta]D, (7)

where D is the rate of deformation tensor: [eta] is the kinematic viscosity and depends on the shear rate [gamma]. The constitutive equation is represented by the Carreau model:

[eta]([gamma]) = [[eta].sub.0][[1 + [([lambda][gamma]).sup.2]].sup.[(n - 1)/2]], (8)

where [[eta].sub.0] is the zero shear rate viscosity, [lambda] is the natural time, and n is the power law index. The material used in this study is polystyrene (PS) of Yangzi-BASF. China. Its viscosity is measured on a capillary rheometer (Thermo Haake Rheoflixer) for the high shear rate domain and on an advanced rheometric expansion system (ARES: Rheometrics Corp., USA) in a parallel plate mode for the low shear rate domain (see Fig. 3). The parameters for the Carreau model are [[eta].sub.0] = 2814 Pa s. [lambda] = 0.148 s, and n = 0.278.

[FIGURE 3 OMITTED]

The no-slip boundary conditions are imposed to the surfaces of the KDs and barrel bore walls. A volume flow rate is set to the inlet section and a normal stress boundary condition is applied to the outlet section. Table 3 lists the operating conditions for the numerical simulations and experiments. The volume throughput was calculated by dividing the mass throughput by the melt density of the PS (970 kg/[m.sup.3]).

TABLE 3. Operating conditions for the numerical simulations and experiments carried out in this work. (a) Exp. 1 Exp. 2 Exp. 3 Exp. 4 Screw speed (rpm) 120 150 150 150 Mass throughput (kg/h) 10.7 10.7 13.4 17.8 Volume throughput Q (X [10.sup.-6] 3.06 3.06 3.84 5.10 [m.sup.3]/s) (a) Barrel temperature: 220[degrees]C.

Marker Particle Tracking Analysis

To calculate the local RTD provided by each of the above eight types of KDs, virtual particles are launched at the same time in the flow domain. It is assumed that the marker particles are massless, volumeless, and noninter-acting with each other. Under these assumptions, the particles can be located by integrating the velocity vectors with respect to time. Initially, these particles are randomly distributed in an inlet vertical plane (see Fig. 4) and their trajectories between the inlet and outlet are calculated from the velocity profiles. Along each trajectory, the residence time is computed. The RTD is obtained based on the residence time of each of these particles. The number of marker particles has to be high enough for the simulation. Figure 5 shows that 1000 marker particles are not enough and that 2000 marker particles start yielding results much closer to those of the 3000.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Previous researchers pointed out that the initial location and orientation of interfaces could have an influence on mixing parameters such as the area stretch ratio [eta] and instantaneous stretching efficiency [e.sub.[eta]] (20), (21), (28), (29). According to Chella and Ottino, the stretch rate of the initially vertical interface perpendicular to the streamlines is faster than that of horizontal interface in the rectangular cavity flow (31). They also found that when a large amount of the area is calculated to characterize the mixing performance of different mixers, the influence of initial orientation on the stretch rate is very small. Figure 6 shows the effect of the number of marker particles and that of the initial direction of interface on the arithmetic mean of the area stretch ratio and that of the instantaneous stretching efficiency in KD2. When the number of marker particles is large enough, such as 2000 and 3000 particles, the arithmetic means of the two mixing parameters are insensitive to the initial direction and location nor the number of marker particles. This infers that they can be used to compare the distributive mixing performance and efficiency of different types of KDs. Subsequently, 2000 particles with random direction are used to calculate mixing parameters.

[FIGURE 6 OMITTED]

EXPERIMENTAL

RTDs are measured under the same operating conditions as the numerical simulations. Figure 7 shows the screw configuration used in this study. There are two test points marked as Probes 1 and 2. The screw elements beneath Probes 1 and 2 were cylinders. The cylinders ensured full fill of the screw elements beneath Probes 1 and 2 and a constant depth between the probes and the screw element surfaces. Meanwhile, using the cylinder was to make sure that the flow was fully developed in the inlet and outlet sections as it is assumed for the numerical simulations. Briefly, the two probes were bifurcated optical fibers and the detection principles were based on fluorescent emission. Details about their characteristics and working principles can be found in the literature (35), (36). There were four identical KDs between Probes 1 and 2, called the test zone. It was the object of the numerical simulation. However, the test zone has four pairs of identical KDs instead of two for the numerical simulation. The reason was that computing four pairs of KDs was computationally costly. To overcome this problem, four pairs of KDs were divided into two "two pairs." It was assumed that the statistical theory for RTDs is applicable for calculating the RTD of the four pairs from those of the two "two pairs." Based on the statistical theory (38-40), the delay time of the four pairs was twice that of the two "two pairs." [E.sub.1] (t) and [E.sub.2] (t) were measured by Probes 1 and 2, respectively. Meanwhile, [E.sub.2] (t) was calculated by the convolution of [E.sub.1] (t) and [E.sub.12] (t) obtained by numerical simulations and was used to compare with the experimentally measured [E.sub.2] (t).

[FIGURE 7 OMITTED]

RESULTS AND DISCUSSION

Experimental Validation

To gain better understanding of the flow occurring in the KDs of the co-rotating twin-screw extruder, Fig. 8 shows the streamlines of two particles initially located at different places of the inlet. They followed different streamlines to the exit, implying that they had different residence times. Particle 1 proceeded around the screws in a "figure-eight" mode, which is in agreement with the experimental results of previous researchers (41), (42). For particle 2, its motion first followed a "figure-eight" mode and then circled around one screw. It finally returned a "figure-eight" mode. The above two different motion modes show that as time evolved, the fluid particles underwent different degrees of shear and stress. The complex flow in the KDs is indicative of good mixing.

[FIGURE 8 OMITTED]

Figure 9 compares the numerical local RTD results with the experimental ones for KD2. The overall trend was in good agreement. However, the former were wider than the latter. Reasons for those differences could be as follows. The simulation assumed that the flow channel be fully filled and that the flow be isothermal. Moreover, the melt density of the polymer was assumed to be constant and did not vary with pressure and temperature, and the polymer melt was assumed to be a generalized Newtonian fluid. Those assumptions were not necessarily fully met in practice. It was reported that subtle differences in rheology could impart significant differences to mixing (20). Figure 10 compares the simulated and experimental RTD for different screw configurations. Again the 90[degrees] KDs has the highest axial mixing performance while the 45[degrees] and 60[degrees] KDs have similar axial mixing performance.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

Effect of Geometrical Parameters on Local RTD

The KD4, KD5, and KD6 have the same disc width and gap but different stagger angles. Figure 11 shows the effect of the stagger angle on the local RTD. The delay time and average residence time increased with an increase in the stagger angle. Also the 90[degrees] KD provides the highest axial mixing quality.

[FIGURE 11 OMITTED]

Figure 12 shows the effects of the disc gap and width on the local RTD. The three local RTD curves are similar in delay time and shape, indicating that the KD6, KD7, KD8 have similar axial mixing quality.

[FIGURE 12 OMITTED]

Distributive Mixing Performance and Efficiency

The area stretch was used as a measure for comparing the distributive mixing performance of different types of KDs. An effective KD requires high values of the logarithm of the area stretch ratio over the axial distance. Distributive mixing efficiency was characterized by time-average efficiency. Figure 13a shows the arithmetic mean of log [eta] from the entry to the exit. For KD4, KD5, and KD6, it increased more or less linearly with increasing axial distance Z, indicating that [eta] increased exponentially with Z. Among the three screw configurations, for a given axial distance the arithmetic mean of log [eta] followed the order: KD6 > KD5 > KD4, indicating that the distributive mixing performance increased with increasing stagger angle of the KD. Figure 13b shows the critical value of the arithmetic mean of log [eta] along Z for 10, 50, and 90% marker particles for KD4, KD5, and KD6, respectively. Again it followed the order: KD6 > KD5 > KD4, regardless of the percentile of marker particles.

[FIGURE 13 OMITTED]

From Fig. 13a, the arithmetic mean of log [eta] vs. Z curves fluctuated, especially at a higher axial distance. This means that not all particle surfaces are stretched along Z in a steady manner. At the initial stage of mixing, it is easy to stretch the surface area and the arithmetic mean of log [eta] increases linearly with Z. As the surface area is further stretched, area stretching becomes more difficult. Meanwhile, Ishikawa et al, pointed out that the disc gap plays an important role for mixing because of the large amount of forward and backward flow (11). The latter may bring about stretch and shrink of surfaces. Figure 13b shows that stretch and shrink of surfaces occur primarily at high area stretch ratio.

Figure 14a shows the effects of the disc gap and disc width on the arithmetic mean of log [eta] as a function of Z for 90[degrees]KDs. The arithmetic mean of log [eta] of a KD with a gap (KD6) is higher than that of a corresponding KD without disc gap (KD7). On the other hand, A KD with a smaller disc width a smaller disc width (KD7) has a higher arithmetic mean of log [eta] than a corresponding one (KD8), indicating that the distributive mixing performance decreases with increasing disc width. This result is in agreement with that of Andersen (43). It is also noted that unlike KD7 and KD8 whose arithmetic mean of log [eta] increases linearly and smoothly with Z, that of KD6 also increases linearly but in a fluctuating manner. In fact, each fluctuation of the arithmetic mean of log [eta] corresponds to a passage through a gap. Figure 14b shows the critical values for 10, 50, 90% marker particles. There are no fluctuations at low percentiles such as 10 and 50%. Fluctuations are visible only at high percentiles of 90%.

[FIGURE 14 OMITTED]

In a closed system such as a batch mixer, the time-average efficiency exhibits three characteristic features: <[e.sub.[eta]]> decays as [t.sup.-1] for flow with no reorientation; <[e.sub.[eta]]> decays as [t.sup.-1] but oscillates for flow with moderate reorientation; <[e.sub.[eta]]> oscillates without decay for flow with strong reorientation (20), (31), (32). For an open system such as an extruder or a Kenics mixer, [Z.sup.-1] is equivalent to [t.sup.-1] Figure 15a shows the evolution of the arithmetic mean of the time-average efficiency as a function of Z for KD4, KD5, and KD6. The three curves have periodic oscillations without decay as [Z.sup.-1], indicating that all the three KDs bring about strong reorientation. Among the three KDs, the arithmetic mean of the time-average efficiency is the highest for KD6 is highest, indicating that KD6 provides strongest reorientation. Figure 15b shows the time-average efficiency as a function of Z for different percentiles of marker particles. At low percentiles such as 10 and 50%, the critical values of the time-average efficiency are similar for the three types of KDs. At a high percentile such as 90%, KD6 yields significantly higher time-average efficiency. This indicates that at low percentiles the three types of KDs are similar in time-average efficiency. However, at high percentiles, KD6 provides the highest time-average efficiency. It is also noted that when marker particles start moving, most of them are stretched. Nevertheless, some of them have shrunk. Therefore, they exhibit negative time-average efficiency at the initial stage. They are all stretched eventually.

[FIGURE 15 OMITTED]

As for the three 90[degrees] KDs (KD6, KD7, and KD8), they have similar RTD curves (see Fig. 12) but differ in log [eta] (see Fig. 14). KD6 provides the highest log [eta], thus the highest distributive mixing performance. Figure 16 compares these three types of KDs in terms of <[e.sub.[eta]]>. Clearly KD8 provides the highest mixing efficiency and not KD6. The reason is that <[e.sub.[eta]]> is not only a function of d(log [eta])/dt but also of D:D and RTD. Figure 17 shows the evolution of d(log [eta])/dt and [(D:D).sup.[1/2]] as a function of Z for KD6, KD7, and KD8, respectively. The arithmetic mean of d(log [eta])/dt is very close for the three types of KDs. On the other hand, these three types of KDs differ more or less in [(D:D).sup.[1/2]], the energy dissipated. The value of [(D:D).sup.[1/2]] is the lowest for KD8, indicating that the fraction of the energy dissipated for the area stretch is the highest for KD8. In sum, KD8 provides the lowest distributive mixing performance (characterized by log [eta]) but highest mixing efficiency (characterized by <[e.sub.[eta]]>).

[FIGURE 16 OMITTED]

[FIGURE 17 OMITTED]

CONCLUSIONS

This article has reported on three-dimensional numerical simulations of flow and mixing in KDs in a co-rotating twin-screw extruder. The simulations are validated against experimentally measured local RTDs for different types of KDs and experimental conditions.

In addition to operating conditions, the geometry of the KD may have a significant effect on the local RTD. The delay time and mean residence time increase with increasing stagger angle of the KD: 45[degrees] < 60[degrees] < 90[degrees]. The axial mixing characterized by the spread of the local RTD follows the opposite trend.

The area stretch ratio [eta] and the time-average efficiency <[e.sub.[eta]]> characterize the distributive mixing performance and efficiency, respectively. The former increases with increasing axial distance Z in an exponential manner for all types of KDs simulated in this work, indicating that they all have good distributive mixing performance. The magnitude of increase in [eta] and <[e.sub.[eta]]> follows the order: 45[degrees] < 60[degrees] < 90[degrees].

The disc gap and disc width play an important role in distributive mixing performance and efficiency. A KD with a disc gap provides higher area stretch ratio and time-average efficiency than a corresponding one without disc gap. A KD with a larger disc width has a lower distributive mixing performance than a corresponding one with a smaller disc width. However, its distributive mixing efficiency could be higher because of lower energy dissipation, implying that for thermally sensitive polymers, it would be better to use a KD with a larger disc width than a KD with a smaller disc width. In general, KDs with disc gaps, small disc widths, and large stagger angles provide good distributive mixing performance.

ACKNOWLEDGMENTS

The authors express their sincere thanks to the reviewers for their thoughtful and constructive comments on this manuscript.

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Xian-Ming Zhang, (1) Lian-Fang Feng, (2) Wen-Xing Chen, (1) Guo-Hua Hu (3), (4)

(1) Key Laboratory of Advanced Textile Materials and Manufacturing Technology, Ministry of Education, Zhejiang Sci-Tech University, Hangzhou 310018, China

(2) State Key Laboratory of Polymer Reaction Engineering, College of Materials Science and Chemical Engineering, Zhejiang University, Hangzhou 310027, China

(3) Laboratory of Chemical Engineering Sciences, CNRS-ENSIC-INPL, 54001 Nancy, France

(4) Institut Universitaire de France, Maison des Universites, 75005 Paris, France

Correspondence to: Lian-Fang Feng: e-mail: fenglf@zju.edu.en or GuoHua Hu: e-mail: hu@ensic.inpl-nancy.fr

Contract grant sponsor: National Natural Science Foundation of China; contract grant number: 50390097; contract grant sponsor: Key Laboratory of Advanced Textile Materials and Manufacturing Technology (Zhejiang Sci-Tech University), Ministry of Education; contract grant number: 2008QN05.

DOI 10.1002/pen.21404

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Author: | Zhang, Xian-Ming; Feng, Lian-Fang; Chen, Wen-Xing; Hu, Guo-Hua |
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Publication: | Polymer Engineering and Science |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Sep 1, 2009 |

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