Study of Dynamic Flow and Mixing Performances of Tri-Screw Extruders with Finite Element Method

Abstract

There is a special circumfluence in the center region of cross-section for a tri-screw extruder. To study the effect of the dynamic center region on the flow and mixing mechanism of the tri-screw extruder, 2D finite element modeling was used to reduce the axial effects. Based on the particle tracking technology, the nonlinear dynamics of a typical particle motions in the center region was carried out and the mixing process in the tri-screw extruder was analyzed with Poincaré maps. Moreover, mixing evaluations of the tri-screw and twin-screw extruders were analyzed and compared. The results show that there are many multiple and fractional excitation frequencies in the Fast Fourier Transform (FFT) spectrums, and it shows a chaotic motion in the center region. Poincaré section maps in the tri-screw show the better particles diffusive ability than those in the twin-screw extrude due to the excellent diffluent ability. Furthermore, the tri-screw extruder has the better dispersive, distributive mixing abilities and stretching mixing efficiency than the twin-screw extruder in the cross-section.

1. Introduction

The traditional single and twin screw extruders are most popular in the polymer mixing and reaction. With the development of extrusion technology, a new tri-screw extruder is put forward at the beginning of 21st century, which has one dynamic center region and three intermeshing regions [1]. It is different from the traditional single- and twin-screw extruders. Because the tri-screw extruder has more complex geometrical structure than the twin-screw extruder, it has great complex flow and mixing characteristics. So far, the tri-screw extruder has received more attention, and the studies on its flow and mixing mechanisms have grown significantly. Particularly in 2006, the first industrialized triangle arrayed tri-screw extruder in the world has been successfully applied in a Chinese factory [2]. It indicates a new period is coming for the study of the tri-screw extruder. However, the theoretical studies about tri-screw extruder are still in the immature conditions because of the complexity of geometrical structure and mixing mechanism in the tri-screw extruder.

Up to now, a number of numerical simulations and experimental researches have been conducted to investigate the flow and mixing mechanism for the extrusion problem of single-screw and twin-screw extruders [3 –7]. In addition, the chaotic mixing in the twin-screw or single-screw extruder has been carried out to reveal the stretching and folding actions, using Poincaré section, Lyapunov exponent, and so forth [8 –11]. However, the studies of the tri-screw extruder are relatively limited due to its complex geometry and the short development time. Hu and Chen established a 3D isothermal flow model of kneading-disc elements for the tri-screw extruder with FEM method to emphasize particularly on the analysis of flow rules [12]. On this basis, they further studied the mean mixing efficiencies of the tri-screw and twin-screw extruders using CFD package, PolyFlow [13]. Zhu et al. numerically studied the flow rules of polyethylene melts in a tri-screw extruder with a quasi-three-dimensional finite element modeling. The mean mixing evaluations in a tri-screw extruder were analyzed and compared with those in a twin-screw extruder [14]. Moreover, Zhu et al. studied the temperature distribution, energy consumption, and capacity ratio in a tri-screw extruder at different rotational speeds, screw flight numbers, pressure differences, and flow rates. In particular, the comparisons of energy consumption and capacity ratio between the twin-screw extruder, parallel arranged, and triangle arrayed tri-screw extruders were investigated [15]. Jiang and Zhu performed the simulation and experiment to testify the strong conveying capacity and high shear frequency in the tri-screw extruder for processing carbon black master batch [16]. However, the previous studies hardly figured out the special motions of stretching and folding in the center region. In the other hand, no information was presented on the analysis of flowing in the tri-screw extruder by using the dynamic tools such as FFT spectrums. There is a need to understand the complex mixing mechanism in the tri-screw extruders.

It is well known that there is one circumfluence phenomenon in the center region but no material stagnation phenomenon at the axial cross-section for the tri-screw extruder [13, 14]. However, the effects of dynamics center region on the mixing efficiency of the triangularly arranged tri-screw extruder are not understood clearly. In this paper, in order to reduce the effects of axial flow on mixing efficiency in the tri-screw extruder, the 2D and transient finite element (FE) modeling is used to study the dynamics flow and mixing characteristics of tri-screw extruder with particle tracking technology. A typical particle is selected to study the nonlinear dynamic behaviors in the center region. Furthermore, the comparisons of mixing process in the tri-screw and twin-screw extruder are studied with Poincaré maps to learn about the repetitions of stretching and folding in the center region. At last, many parameters describing the transient dispersive and distributive mixing, such as the mixing index, probability of maximal shear rate, segregation scale, logarithm of the length stretch, and mean efficiencies in the tri-screw extruder are calculated and compared with those in the twin-screw extruder. The main purpose of this paper is to further understand the mixing mechanism and the special contributions of center region for the entire tri-screw extruder.

2. Numerical Modeling and Conditions

2.1. Physical Modeling

A tri-screw extruder has three intermeshing regions and one center region, and the centerlines between its three screws must constitute an equilateral triangle, as shown in Figure 1(a). The simplified 2D geometric specifications are listed as follows: screw root diameter is 26 mm, screw tip diameter is 34 mm, barrel diameter is 34.8 mm, the centerline distance of three screws is 30.4 mm, the screw clearance is 0.4 mm, and the clearance between screw and barrel is 0.4 mm. It should be noted that the three screws are all counter-clock rotated. The special center region is where the three screws cannot sweep the polymer melt. The barrel and the screw elements are meshed individually with Gambit 2.2 software, and the mesh superposition technique (MST) is used with a commercial CFD analysis package, PolyFlow, without remeshing for the periodically changing geometry [17]. The barrel is meshed with quadrilateral element, and the screws are meshed with the composite quadrilateral and triangular elements. The total number of elements and nodes of the whole tri-screw extruder are 3,474 and 3,841, respectively, as shown in Figure 1(b).

Figure 1:

Geometric and FE models of tri-screw extruder: (a) 2D geometrical model; (b) 2D FE model.

In order to compare the mixing efficiency between the tri-screw and twin-screw extruders, the geometric sizes of the twin-screw in Figure 2(a) are equal to those of the tri-screw extruder. At the same time, the grid density of twin-screw is also equal to that of the tri-screw extruder. The total number of nodes and elements of the whole twin-screw extruder geometry are 2,388 and 2,658, respectively, as shown in Figure 2(b).

Figure 2:

Geometric and FE models of twin-screw extrude in cross-section: (a) 2D geometric model; (b) 2D FE model.

2.2. Mathematical Modeling

In this study, the time-dependent flow conditions are considered. To simulate the behavior of the fluid, some assumptions are used as follows. The non-Newtonian and laminar flow are proposed due to high viscosity of polymers; the flow domain is fully filled with fluid under the conditions of isothermal, incompressible, and nonslip at barrel wall and screw surfaces; the forces of gravity and inertia are neglected. Based on the above assumptions, the governing equations are described as follows [17].

The form of continuity and momentum equations can be expressed by

\begin{matrix} \nabla \cdot \overset{⃑}{v} = 0, \\ - \nabla P + \nabla \cdot \overset{⃑}{T} = 0 . \end{matrix}

(1)

The stress tensor is described as

\overset{⃑}{T} = 2 η (\dot{γ}) \overset{⃑}{D},

(2)

where $\overset{⃑}{v}$ is the velocity vector, P is the pressure, $\overset{⃑}{T}$ is the extra-stress tensor, $\overset{⃑}{D}$ is the rate-of-strain tensor, and $\dot{γ}$ is the shear rate and can be defined as

\dot{γ} = \sqrt{2 \overset{⃑}{D} : \overset{⃑}{D}} .

(3)

The rheology of the polymer melt is described in terms of the Bird-Carreau model

η = η_{\infty} + (η_{0} - η_{\infty}) {[1 + {(λ \dot{γ})}^{2}]}^{(n - 1) / 2},

(4)

where η_∞ is infinite shear viscosity, η₀ is the zero shear viscosity, λ is a model-specific relaxation time, and n is the power-law index.

In this paper, the viscosity of polyethylene (PE) melts only depending on the shear rate that is selected to simulate in the tri-screw extruder. The parameters of this PE melts at 200°C are as follows: ρ = 882 kg/m³, λ = 0.049 s, n = 0.5, η_∞ = 0 Pa·s, and η₀ = 5520 Pa·s. In order to minimize the “High Weissenberg Problem” and the potential 3D effects [4], a little rotational speed of the screws 0.5 rpm is used in this study. The velocity and pressure fields are calculated based on the PolyFlow software.

2.3. Evaluation of Mixing Efficiency

Distributive mixing and dispersive mixing are very important characteristics to evaluate the mixing equipments. The scale of segregation, S(t), is an important parameter to evaluate the distributive mixing and can be expressed by

S (t) = \int_{0}^{ζ} R (r, t) d r,

(5)

where R(r, t) is a correlation coefficient for the concentration, and it gives the probability of finding a pair of random points with a relative distance r and with the same concentration [17]. The segregation scale is a measure of the size of the regions of homogenous concentration and decreases when mixing improves.

After running of the screws, the initial material particles disperse and go to even distributions. The difference between the actual distribution and the ideal distribution of the material particles is called the cluster distribution index, ∊, and is defined as

∊ = \frac{\int_{0}^{\infty} {[C (r) - C {(r)}_{ideal}]}^{2} d r}{\int_{0}^{\infty} {[C {(r)}_{ideal}]}^{2} d r},

(6)

where C(r) is the coefficient of the probability density function and the value of ∊ varies from 0 with ideal distribution to 1 with no distribution [18].

Dispersive mixing index, λ_MZ, is used to evaluate the dispersive mixing efficiency and can be described as

λ_{MZ} = \frac{| \overset{⃑}{D} |}{| \overset{⃑}{D} | + | \vec{W} |},

(7)

where $| \overset{⃑}{D} |$ is the magnitude of rate-of-strain tensor and $| \vec{W} |$ is the magnitude of vorticity tensor. λ_MZ ranges from 0 for implying pure rotation to 1.0 for implying pure elongation. When λ_MZ is 0.5, it means that the fluid is in the simple shear condition for screw extruders.

The instantaneous efficiency of stretching flow is defined as a local instantaneous efficiency of mixing given by

e_{λ} = \frac{\dot{λ} / λ}{{(\overset{⃑}{D} : \overset{⃑}{D})}^{1 / 2}} .

(8)

The instantaneous efficiency, e_λ, can be thought of as the fraction of the energy dissipated locally that is used to stretch a fluid element at a given instant in a purely viscous fluid [17].

Based on the instantaneous efficiency, the time-averaged mixing efficiency of stretching flow is used to describe the distributive mixing and can be defined as [17]

〈 e_{λ} 〉 = \frac{1}{t} \int_{0}^{t} e_{λ} d t .

(9)

3. Results and Discussions

3.1. Flow Visualizations

The velocity vector distributions in the cross-section and magnified vector in the center region with different angles are visualized at a rotational speed of 0.5 rpm, as shown in Figure 3. As can be seen in Figure 3(a), even if the area of the center region is very small, there is circumfluence phenomenon in the center region [14]. It implies the poor flow and mixing efficiency for the tri-screw extruder. Moreover, with the increase of the area for the center region in Figure 3(b), the area of triangular poor flow zone in center region also increases, in which there are three regions with small velocities. The poor flow zone always exists in the center of the center region for the tri-screw extruder. It is different from twin-screw extruders and must have great effects on the mixing characteristics of the tri-screw extruder.

Figure 3:

Velocity vector distributions in the tri-screw extruder with different angles: (a) 0°; (b) 180°.

3.2. Particle Motions and Nonlinear Dynamic Behaviors

To learn about the special flow rules in the tri-screw extruder, three typical particle trajectories are selected to describe the motions from the center region to screw channel regions, as shown in Figure 4. It is found that there are not only the “∞” motional orbits, which is similar to the twin-screw extruder, but also the flowing motion only through one screw, which is similar to single screw extruder. Moreover, there is a common particle moving through the three screws after flowing out the center region. So the flow rules and mixing mechanisms of polymer melt in the tri-screw extruders are more complex than those in single-screw and twin-screw extruders. In Figure 4(b), the typical particle moves 19 cycles in the center region referring to zero degree. Based on the transient displacement and velocity of this particle in Figure 4(b), many parameters such as velocity, pressure, and logarithm of length stretch were calculated to study the dynamic behaviors of the particle, as illustrated in Figures 5 and 6, respectively. The main emphasis is to study the nonlinear dynamic behaviors of the particle motions in the center region.

Figure 4:

Orbits of three particles in the tri-screw extruder: (a) three particles; (b) typical motions in the tri-screw extruder.

Figure 5:

Transient velocity and FFT curves of the typical particle: (a) velocity; (b) FFT.

Figure 6:

Transient pressure and logarithm of length stretch curves of the typical particle: (a) pressure; (b) amplified pressure; (c) FFT spectrum; (d) logarithm of length stretch.

As can be seen in Figure 5(a), the velocities of the particle in the center region are relatively smaller than those in the screw channel, but their frequencies in the center region are more frequent in comparison with the screw channel. The Fast Fourier Transform (FFT) results of the velocity are shown in Figure 5(b). It shows a chaotic motion in the center region with continuous spectrum in a wide span of frequencies, while quasiperiodic motion appears in a more regular way [19]. Moreover, the FFT spectrum in the center region shows that many multiple and fractional excitation frequencies appear in the velocity spectrum.

The pressure changes of the particle with mixing time in Figure 6 show that the pressures in the center region are relatively small. It results in a little flow velocities of polymer melt in the center region. The FFT results of the pressures in the center region show that many multiple excitation frequencies appear in the pressure spectrum, as depicted in Figure 6(c). It also implies the chaotic dynamics behaviors of the particles. As can be seen in Figure 6(d), the logarithm of length stretch distributions of the particle in Figure 6(b) shows that the logarithm of length stretch increases exponentially for the particle with the increasing of the mixing time. It can be useful in enhancing the mixing efficiency for the tri-screw extruder. Moreover, the logarithm of length stretch is less than zero from 0 s to 70 s in the center region, indicating that the particle is in the compress conditions.

3.3. Mixing Process with Poincaré Map

Initially, 5000 particles were set along the midline of the tri-screw extruder. Based on the 4th-order-Runge-Kutta method, Poincaré sections of the tri-screw extruder with different periods are depicted in Figure 7, where each dot represents the location of each fluid particle at different periods. It can be seen from Figure 7 with an increase of mixing time, the midline of particles is stretched and its length increases rapidly. And then the martial line is folded and the adjacent fluid particles are detached rapidly. It changes the orderly arrangement of fluid particles into chaotic distributions. After 5 periods in Figure 7(c), the Poincaré section exhibits a relatively uniform distribution in the tri-screw extruder except for the center region, suggesting the achievement of better mixing. In addition, the amplified Poincaré sections in the center region are shown in Figure 7(d). It can be seen from Figure 7(d) that the repetitions of stretching and folding appear in the center region, which is useful in increasing mixing efficiency.

Figure 7:

Poincaré sections of initial positions in the midline for the tri-screw extruders: (a) 0.5T; (b) T; (c) 5T; (d) in the center region with 5T.

To further understand and compare the mixing performance in the tri-screw and twin-screw extruders, three square zones (2 × 2 mm) in the flow region are selected in the tri-screw and twin-screw extruders, as shown in Figures 8(a) and 8(b), respectively. In every zone, 5000 particles are set free in the two kinds of extruders. Furthermore, the Poincaré sections of the three zones in the tri-screw and twin-screw extruders are depicted from Figures 9 to 12.

Figure 8:

Initial positions of the 5000 particles in the different zones of two extruders: (a) tri-screw extruder; (b) twin-screw extruder.

Figure 9:

Poincaré sections in the initial P₁ zone for the tri-screw and twin-screw extruders: (a) tri-screw extruder; (b) twin-screw extruder.

Figure 9 depicts Poincaré sections of the initial particles in P₁ zone for the tri-screw and twin-screw extruder. When T = 0.5, the stretch length of initial particles in the tri-screw extruder is greater than that in the twin-screw extruder. With the increase of periods such as T = 3 and T = 5 in Figures 9(a) and 9(b), respectively, the particles are diffusive in the two extruders, but the relative uniformity of particles in the tri-screw extruder is always better than that in the twin-screw extruder. This is mainly because the tri-screw extruder has a better dispersive ability of particles in the center region. The amplified Poincaré map in Figure 9(a) shows that there are blank regions in the center region, in which the particles flow out because of the big flow velocities. However, there are still many concentrated particles in the middle of the center region after 5 periods due to the circumfluence phenomenon in the tri-screw extruder.

Figures 10 and 11 show Poincaré section maps of the initial particles in P₂ and P₃ zones for the tri-screw and twin-screw extruders, respectively. In the same periods, the tri-screw always has better particles diffusive ability than the twin-screw extruder, because the special center region of the tri-screw extruder has excellent diffluent ability. In addition, the screw channel regions in the twin-screw and tri-screw extruders always appear in the blank zones, where the particles cannot reach, because there is plug flow in the screw channel regions with lack of stretching mechanism. Moreover, the stretching, folding, and compressing phenomena can be observed in Figures 11(a) and 11(b). It shows the complex mixing mechanisms in the tri-screw and twin-screw extruders.

Figure 10:

Poincaré sections of initial positions in P₂ zone: (a) tri-screw extruder; (b) twin-screw extruder.

Figure 11:

Poincaré sections of initial positions in P₃ zone: (a) tri-screw extruder; (b) twin-screw extruder.

Figure 12:

Cluster distribution index in the two kinds of screw extruders.

The mixing uniformity after diffusing of particles from the initial positions is evaluated by the cluster distribution index [18], as plotted in Figure 12. The cluster distribution index in P₁ zone is obviously smaller in the tri-screw extruder than in the twin-screw extruder, due to the disruption effect of the center region that allows the cluster of particles to diffuse out in the tri-screw extruder. The rule of cluster distribution index in P₂ zone is similar to that in P₁ zone for the two kinds of screw extruders. Moreover, the values of the cluster distribution index in the center of the screw channel (in P₃ zone) of the tri-screw extruder are similar to those in the twin-screw extruder, due to the similar mixing mechanism in this region. In addition, the oscillation of cluster distribution index of the tri-screw extruder in P₁ zone is obviously smaller than that in other zones because of the strong stretching and folding effects in the center region.

3.4. Mixing Evaluations

3.4.1. Dispersive Mixing

The distributions of dispersive mixing index with different angles for the tri-screw and twin-screw extruders are calculated as shown in Figure 13. From Figure 13(a), the dispersive mixing indices in the intermeshing regions and the middle of the screw channel are higher than those in other regions for the tri-screw extruder due to great shear rates. In particular, the values of dispersive mixing indices in the center region are always between 0.1 and 0.4, corresponding to the simple shear flow mixing. The dispersive mixing index distributions of the twin-screw extruder are described in Figure 13(b). It is clear that the dispersive mixing index distributions in the twin-screw extruder are similar to those in the tri-screw extruder except for the center region. But the areas with large values of dispersive mixing index in the tri-screw extruder are bigger than those in the twin-screw extruder, because the tri-screw extruder has three intermeshing regions, and the twin-screw extruder has only one intermeshing region.

Figure 13:

Dispersive mixing index distributions of the twin-screw and tri-screw extruders with angle 0°: (a) tri-screw extruders; (b) twin-screw extruder.

The comparisons of the probability of mixing index for 10%, 50%, and 90% of particles in the twin-screw and tri-screw extruders are plotted in Figure 14. With the increase of mixing time, the dispersive mixing indices of 10%, 50%, and 90% of particles in the tri-screw extruder are bigger than those in the twin-screw extruder, respectively. For example, when the mixing time is 400 s, the probabilities of obtaining maximal mixing index for 90% and 50% of particles in the twin-screw extruder are 99.05% and 95.72%, respectively, while increase to 99.73% and 98.11% in the tri-screw extruder. It further reveals that the tri-screw extruder has better dispersive mixing efficiency than the twin-screw extruder in the cross-section due to one more intermeshing region, where the dispersive mixing index is relatively higher than the other regions.

Figure 14:

Comparisons of the probability of dispersive mixing index for the two extruders.

The shear rate is an important parameter to evaluate the dispersive mixing for the screw extruders. Figure 15 describes the comparison of the probability of maximal shear rate for the twin-screw and tri-screw extruders. It can be seen that with the increasing of mixing time, the ratio of particles with great maximal shear rate increases gradually. When the periodic center region has periodically maximal area, corresponding to 100 s, 200 s, and 300 s, the tri-screw extruder has higher probability to obtain great maximal shear rate than the twin-screw extruder. So the tri-screw extruder has better dispersive mixing ability induced by the shear rates than the twin-screw extruder.

Figure 15:

Comparison of the probability of maximal shear rate of the two-screw extruders.

3.4.2. Distributive Mixing

Distributive mixing can be effectively evaluated by means of segregation scale. In order to statistically calculate the segregation scale, 5000 particles were evenly distributed in the flow domain in the initial time. After 5 revolutions, the segregation scales for the two kinds of extruders are shown in Figure 16. During 0∼100 s, the values of segregation scales in all two extruders show a rapid drop because of the largely initial segregated areas in the twin-screw and tri-screw extruders. After 100 s, the segregation scale of the tri-screw extruder fluctuates globally smaller than that of twin-screw extruder. At the same time, the values of segregation scale in the tri-screw extruder are less than in the twin-screw extruder, because the center region of tri-screw extruder is continually varied from big to small and disturbs the streamline. It implies that the tri-screw extruder has better distributive mixing than the twin-screw extruder.

Figure 16:

Segregation scale for the two kinds of extruders.

3.4.3. Stretching Mixing Efficiency

It is well known that the stretching flow is more effective than the shear flow for the polymer processing [20]. The mean logarithm of the length stretch of different screw extruders after 5 revolutions is depicted in Figure 17. It can be seen that with the increase of operating time, the logarithm of length stretch increases exponentially in the twin-screw and tri-screw extruders, and it is helpful to the efficiency mixing. But the values of the logarithm of length stretch in the tri-screw extruder are obviously larger than those in the twin-screw extruder because the particles in the center region are in the conditions of folding and stretching. It implies that the tri-screw extruder has better stretching mixing efficiency than the twin-screw extruder.

Figure 17:

Logarithm of length stretch of different screw extruders.

The larger the value of mean time-averaged efficiency is, the higher the mixing efficiency the screw extruder gets. The predicted mean time-averaged efficiencies of the tri-screw and twin-screw extruder are described in Figure 18. It shows the rapid rise in the time-averaged efficiency at the initial mixing time, because the initial martial is prone to stretch due to its very small area. As the time increases, the time-averaged efficiencies of the two screw extruders decrease and level off at a value above zero, because the stretching area of material increases. It is necessary for an effective mixer with positive value of time-averaged efficiency, indicating the strong reorientation effect. By comparing, it is found that the values of the tri-screw extruder are bigger than those of the twin-screw extruder. For example, the time average efficiencies are 0.0345 and 0.0301 in the twin-screw at 300 s and 500 s, respectively, but increase to 0.0537 and 0.0469 in the tri-screw extruder. It indicates that the tri-screw extruder has more efficient stretching mixing than the twin-screw extruder.

Figure 18:

Mean time-averaged efficiencies of the two kinds of extruders.

4. Conclusions

In this study, a 2D finite element modeling method is employed to solve the flow in the new tri-screw extruder. The nonlinear motions of typical particles in the center region are studied with particle tracking and FFT technology. The dispersive mixing processes in the whole tri-screw and twin-screw extruders are carried out by using Poincaré maps. Moreover, the comparisons of mixing efficiencies in the tri-screw and twin-screw extruders can give us a great insight to the mixing mechanism in the tri-screw extruder. The following main conclusions can be drawn.

For the local region, the circumfluence phenomena of polymer melts always exist in the center region and mainly depend on the shear flow mixing. Under the excitations of the three screws, the polymer particle in the center region has greatly changing frequencies in the velocity and pressure profiles, and their FFT spectrums present abundant frequency components, such as subharmonics and superharmonics. It shows a nonlinear and chaotic motion of the particle. Moreover, it is interesting to note that the repetitious actions of stretching and folding appear in the center region.

Overall, it is noted that the center region plays an important role in increasing the overall mixing efficiency for the tri-screw extruder. Due to the excellent diffluent ability and dynamic changes of area, the tri-screw extruder has better particles dispersive ability than the twin-screw extruder in the Poincaré sections. At the same time, the tri-screw extruder has better distributive mixing and stretching efficiencies than the twin-screw extruder with the evaluations of the segregation scale, mean logarithm of the length stretch, and mean time-averaged efficiency.

Footnotes

Acknowledgments

The authors are thankful for the financial support for this work by the Natural Science Foundation of China (Grant no. 50903042) and the Science Foundation of Liaoning Educational Committee of China (Grant nos. L2010249, L2010247, 2009A431).

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