Numerical simulation and experimental study of pressure and residence time distribution of triple-screw extruder.
Triangle arrayed triple-screw extruder (TTSE) is a novel type of polymer processing equipment with special mixing characteristics. It has three meshing zones and a central region which have great advantage in polymer blending. The unique mixing mechanism of TTSE makes it suitable for processing in high viscosity or high filled systems. To promote the industrial compounding and mixing process, numerous simulations and experiments based on twin-screw extruders are carried out to understand the internal mechanism of extruders even with complexity in melting and mixing. For instance, Rios et al.  utilized the simulation based on boundary element method (BEM) to study the internal flow field of a twin-screw extruder for screw optimization. Jaffer et al.  simulated the flow field in the kneading discs of intermeshing twin-screw extruder by a three-dimensional method, and compared with the results of particle image velocimetry (PIV). Takeshi Ishikawa et al.  used finite element analysis to perform the three-dimensional nonisothermal flow simulations in the kneading discs of corotating twin-screw extruders. These studies focus on the pressure and temperature distributions in extruders. Jiang et al.  studied the shearing field of the intermeshing counter-rotating twin screw extruders by the simulation, focusing on the effects of flight thickness and screw channel depth, and the simulated results were compared with experimental results. Zhang et al. [5, 6] evaluated the distribute mixing performance in kneading discs of corotating twin screw extruders through simulation of local residence time distributions (LRTDs), and the simulated results were also verified by an in-line measuring instrument. The overall, partial, and local residence time distributions, residence revolution distributions and residence volume distributions (denoted as RTD, RRD, and RVD, respectively) in a corotating twin screw extruder were also studied to establish the relationships among these parameters. The overall and partial RTDs were measured in-line directly and LRTDs were calculated according to a statistical theory. The effects of the type and geometry of mixing element on RTD were also discussed. LRTDs allowed comparing the performances of mixing elements. It has also been confirmed experimentally and theoretically that specific throughput, defined as the ratio of throughput (Q to screw speed (N), controls all the three types of residence distributions.
Most experiments about mixing equipment concern the screw configuration and mixing technique. Lin et al.  prepared nylon66/clay nano-composites in a corotating twin-screw extruder and the morphological evolution of clay along the screws was examined with different screw configurations and mixing regions. Takamasa  studied the grafting reaction in twin-screw extruders experimentally and theoretically. The authors built the models based on reaction kinetics and constitutive equations, and calculated the flow field with the nonisothermal non-Newtonian flow conditions by flow analysis network (FAN). The reaction conversion and shear viscosity were measured to verify the calculated results.
Numerous literatures show that the simulation results of TTSE are not in high consistence with experiments at present. The accuracy and the guidance to experiments of simulations still cannot be verified comprehensively. Meanwhile, there are no systematic studies on the mixing of triangle arrayed triple-screw extruders. Therefore, in this article, the simulation results of TTSE were compared with experimental results. The direct verification method proves the reliability of simulation by comparing the simulated data with the experimental results which were obtained with the same process conditions and output parameter (axial pressure difference). Indirect verification is used when the experimental conditions for direct verification cannot be reached. Here, we carried out the verification by comparing the RTD function which can be used to characterize the mixing performance of extruder.
The RTD is a probability distribution function that describes the amount of time a fluid could spend inside the extruder. We measured the global residence time distributions (GRTD) and drew the GRTD curves with different inlet flow rates, screw speeds and screw configurations and indirectly verified the reliability of simulation by comparing them with the simulated LRTD and average residence time. The GRTD describes the amount of time that the tracer move from the inlet to the outlet through the extruder, and the LRTD represents the situation that particles experienced through the simulated models in the same theory. We hope to establish higher consistence between theoretical simulation and experimental research, and thereby provide a basis for development of blending modification technology and equipment.
In case of TTSE, many factors would affect the extrusion of polymer melts. Thus the following basic assumptions should be made according to the melting flow characteristics: extrusion in an extruder is steady; the flow field is fully filled with the melts under an incompressible and no-slip boundary; the melt flow is a type of transient isothermal laminar flow; the inertial force, gravity and melt elasticity are ignored.
In a flow system supposed to be isolated with the above surroundings, regardless the type of change or process, the total mass remains constant for the flow system according to the law of mass conservation and the conservation equation is continuous. The differential equation is expressed as:
[partial derivative][rho]/[partial derivative]t + [nabla] x ([rho]u)=0 (1)
where t is time; u is a velocity vector; [rho] is the melt density; [nabla] is Hamilton operator.
Because the polymer melt is assumed to be incompressible ([partial derivative][rho]/ [partial derivative]t=0), Eq. 1 is simplified as:
[nabla] x u=0 (2)
The law of momentum conservation based on an incompressible steady flow can be applied in the melt flow of polymers and thus the momentum equation is derived as follows
[nabla]p = [nabla][tau] (3)
where p is pressure of the fluid element, and [tau] is the stress acting on the surface of the system.
When the law of energy conservation is applied to the polymer melt which exchanges heat with the surrounding, the energy equation for the discussed system can be derived as follows:
[rho][C.sub.v] [partial derivative]T/[partial derivative]t = [nabla] x (k x [nabla]T[([partial derivative]p/[partial derivative]T).sub.p] x ([nabla] x u) + ([tau]: u) + [rho]s (4)
where [C.sub.v] is specific heat capacity, T is the fluid temperature, k is the heat transfer coefficient of fluid, and [rho]s is the item of external heat source.
According to the above assumptions, the item of energy equation is ignored.
The Carreau--Yasuda (CY) model reflects the shear thinning in moderate shear rate, as well as the Newtonian flow characteristics in high and low shear rates separately. Therefore, this model was utilized to mathematically express the rheological properties of polymers as the constitutive equation:
[eta]([??]) - [[eta].sub.[infinity]] = ([[eta].sub.0] - [[eta].sub.[infinity]])[1 + [[([[lambda].sub.c] x [??]).sup.a]].sup.(n-1)/a] (5)
where [??] is the shear rate, [[eta].sub.[infinity]] is the viscosity at infinite shear rate, [[eta].sub.0] is the viscosity at zero shear rate, [[lambda].sub.c] is relaxation time, n is the non-Newtonian index, and a is the viscosity transition rate constant.
Equations 2,3, and 5 can be summarized as the mathematical models to constitute the numerical model.
A torque rheometer was used to obtain the rheological properties of homo-polypropylene and to deliver the parameters for establishment of the CY model (supposing a = 2 and [[eta].sub.[infinity]] = 0). The shear rate range of rheometer was set to 10-1000 [s.sup.-1] which was close to the range of extruder. The characteristics of homo-polypropylene F280Z for simulation and experiment is shown in Table 1.
POLYFLOW is a widely used simulation tool for polymer processing. The Gambit program of the software was used to build up the models of triple-screw elements and barrel elements, which were divided into tetrahedral and hexahedral meshes. The elements were superposed so they could be inputted into the main program for simulation. The barrel-screw gap should be divided into two mesh layers. The model is shown in Fig. 1.
The modeling is based on the TTSE for the experiment, with a lead of the screw element and kneading disc of 36 mm, a screw diameter of 35.2 mm, a screw root diameter of 28 mm, a barrel diameter of 35.6 mm, a screw center distance of 32 mm, a flight clearance of 0.4 mm, and a barrel-screw clearance of 0.2 mm. The five kneading discs have a 1-mm gap between each other, and the thickness of each disc is 6 mm.
The 3D How fields were simulated in different inlet flow rates, screw speeds and screw configurations. The flow fields of five screw configurations are shown in Table 2 with their finite element models and number of meshes. The configurations are represented as, A: SE36/36+SE36/36+SE36/36, B: SE36/ 36+SE36/36+KB30/6/36, C: SE36/36+KB30/6/36+KB60/6/36, D: KB30/6/36+SE36/36+KB60/6/36, E: KB30/6/36+KB60/6/ 36+ KB30/6/36L.
The 3D flow field of TTSE was simulated in 13 process conditions. The inlet flow rates and outlet pressures were set as boundary conditions, and the values of different tasks are shown in Table 3.
The pressure differences in different conditions were calculated by the post-processing software FIELDVIEW, after the simulation by POLYFLOW. With the three screw elements in Fig. 2 as example, the right side of a screw is the outlet of the model which is set to be z = 108 mm, so the entrance is z = 0 mm. The distance between sections I and II is equal to that between the pressure transducers in experiment. The positive direction of the z-axis is the extrusion direction, and each screw rotates counterclockwise. The weighted average pressures of cross-section at z = 30 mm ([p.sub.s1]) and z = 108 mm ([p.sub.s2]) are calculated to obtain a pressure difference [DELTA][p.sub.s] = [p.sub.s2] - [p.sub.s1].
Particle tracking method can be used to simulate the particles traveling in the channel and obtain the system's local average residence time (LART) which indicated the average time the particles flow through the model. We placed 1500 particles randomly at the system's inlet, and then calculated their path line by using the mixing task in POLYFLOW. After data processing LRTD and LART can be obtained.
Homopolypropylene F280Z was provided by Sinopec Zhenhai Refining and Chemical Company with a melt flow rate of 2.8 g/10 min.
HAAKE PolySoft OS torque rheometer was produced by Thermo Fisher Scientific. A triangle-arranged TTSE connected with a melt pump system was self-made. It has a screw diameter of 35.2 mm and a length-diameter ratio (L/D) of 28. The temperatures of six zones from the feed inlet to the head are 170, 180, 190, 200, 210, and 210[degrees]C. The positions of the pressure transducers along the axial direction are showed in Fig. 3 and Table 4.
An automatic screw feeder was used to make the feed rates the same operating conditions as the inlet flow rates of simulations in Table 3.
Measurement of Pressure Difference
Pressure differences were measured by the pressure transducers on the screw extruder, and the test points were set the same as in the simulation. The pressures in the experiment were named as [p.sub.e1] and [p.sub.e2], so their difference was [DELTA][p.sub.e]. Noticeably, a melt pump between the extruder head and TTSE was used to keep [p.sub.e2] equal to [p.sub.s2].
Measurement of GRTD
To get the experimental values, the calcined tracer method was used to measure GRTD off-line. After the TTSE was operating normally, 3 g of CaC'03 as a tracer was added into the feed inlet. Then we sampled at the outlet at a constant time interval for 3 min. The mass fraction of CaC03 can be measured through high-temperature calcination. With the CaC03 concentrations at different time points, the GRTD and the experimental LART can be obtained.
All the measurements above were carried out under the same operating conditions as the numerical simulations.
RESULTS AND DISCUSSION
The reliability of simulation is proved by comparing the simulated axial pressure difference with the experimental results. The simulation objects are three screw elements at the end of the screw, and the lead of each screw elements is 36 mm. Therefore, the pressure difference is obtained from the outlet to the upstream of screw till 78 mm. The simulated and experimental pressure differences at different inlet flow rates ([Q.sub.v]), screw speeds (N) and screw configurations are shown in Figs. 4-6, respectively. The simulated and experimental pressure differences are basically consistent. Table 5 shows the absolute values and relative error between the simulated and experimental differences. Through correlation analysis between the simulated and experimental values, the correlation coefficients of [Q.sub.v], N and screw configuration are 0.985, 0.996, and 0.999, respectively. In summary, the simulation is highly reliable.
With the indirect verification method, the experimental GRTDs and global average residence time (GART) curves were compared with the simulated LRTDs and LARTs at varying [Q.sub.v], N and screw configurations. The reliability of simulation can be proved through comparison at the same condition.
The simulated and experimental LRTD density function curves at varying [Q.sub.v] are shown in Fig. 7. Obviously, the curves move to the left and the peak values increase when [Q.sub.v] increases. Therefore, the variations of simulated LRTD curves and experimental GRTD curves are consistent at varying [Q.sub.v].
All of the simulated LRTD curves have a long trail which cannot be seen in the figure. The simulated RTD values are <0.001 after 40 s and the minimum of experimental RTD in 3 min is also 0.001. To make the minimums of the simulated and experimental RTD densities equal, we only counted the average residence time of particles in the first 40 s of simulation. The simulated LART and the experimental GART are shown in Fig. 8 respectively. Obviously, the average residence time decreases with the increasing [Q.sub.v]. It means that a larger [Q.sub.v] results in a shorter residence time in the channel, which is unfavorable for mixing. The figures show that the trends of the simulated LART and the experimental GART are similar.
The density function curves of the simulated LRTD and experimental GRTD at varying screw speed (N) are shown in Fig. 9. Obviously, the RTDs move to the left as N increases, and the peak values increase when N decreases. Therefore, the variations of the simulated LRTD curve and experimental GRTD curve at varying N are similar. The figures also show little effect of screw speed on both RTD curves, it means that the residence time of melt does not change when the other process parameters are steady.
The simulated LART in early 40 s and the experimental GART at varying screw speed (TV) are shown in Fig. 10. Both of them show that the average residence time decreases with the increasing N. It means that a larger N results in a shorter residence time in the channel, which is partly unfavorable for distributive mixing. Moreover, the trends of simulated LART and experimental GART are similar at varying N.
Figure 11 shows the density function curves of the simulated LRTD and experimental GRTD with different screw configurations. Obviously, as the number of kneading discs increases, the curves move to the right and the peak values decrease (except for groups D and E). In consideration of the off-line measurement, there must be some errors. But the errors do not affect the final conclusion, the variations of simulated LRTD curve and experimental GRTD curves at different screw configurations are similar. Moreover, the figures show that screw configuration has a weak effect on RTD. It means that the residence time of melt in extruder is not significantly affected by the screw design.
The simulated LART in early 40 s and the experimental GART with different screw configurations are shown in Fig. 12. The trends of group A-C and E show that the increase of kneading elements extends the residence time of particles, especially for the group with inverse kneading discs. Because the simulated data contain some errors, the simulated average residence time of group C is lower than group D and the experimental results are reversed. In summary, the trends of simulated LART and experimental GART are similar.
The axial pressure difference and RTD of TTSE were measured by numerical simulation and experiments at different inlet flow rates, screw speeds and screw configurations. The reliability of simulation was proved by comparing the axial pressure differences directly. With the indirect verification method, the simulated LRTD and the experimental GRTD, and the simulated LART and the experimental GART were compared to prove the reliability.
Through the verification, we conclude that the simulated and the experimental pressure differences are basically consistent. The simulated and the experimental pressure difference curves at different process conditions are highly similar which directly verified the reliability of simulation.
The trends of experimental GRTD and simulated LRTD were similar, and based on this, the GART and LART at different process conditions are in similar trends, which indirectly verified the reliability of simulation. The comparisons also show that the screw speed and screw configuration have a weak effect on RTD, however, they influence the shear rate and extension rate of the flow field in TTSE which play an important role in mixing. In general, the conclusions will be helpful for numerical simulation to improve the modification blending technology and equipment design.
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Kunxiao Yang, (1) Chunling Xin, (1) Dongquan Yu, (1) Baorui Yan, (1) Junjian Pang, (1) Yadong He (1,2)
(1) College of Mechanical and Electrical Engineering, Beijing University of Chemical Technology, Beijing 100029, People's Republic of China
(2) The Engineering Research Center of Polymer Processing Equipment at Ministry of Education, Beijing 100029, People's Republic of China
Correspondence lo: Yadong He; e-mail: firstname.lastname@example.org
Contract grant sponsor: Natural Science Foundation of China; contract grant number: 51273019.
TABLE 1. Characteristics of homo-polypropylene F280Z. Viscosity at zero Relaxation Non shear rate [[eta].sub.O] time Newton (Pa x [s.sup.-1]) [[lambda].sub.c] (s) index n 16,851 3.376 0.369 Viscosity at Viscosity at zero Viscosity infinite shear rate shear rate [[eta].sub.O] transition [[eta].sub.[infinity] (Pa x [s.sup.-1]) rate constant a (Pa [s.sup.-1]) 16,851 2 0 TABLE 3. Boundary condition. Inlet flow rates Outlet Qv (x [10.sup.-6] pressures Screw [m.sup.3] [P.sub.s2] speeds Screw Task [s.sup.-1]) (MPa) N (rpm) configuration 1# 1.60 12 120 A 2# 2.24 12 120 A 3# 2.88 12 120 A 4# 3.52 12 120 A 5# 4.16 12 120 A 6# 2.88 12 60 A 7# 2.88 12 90 A 8# 2.88 12 150 A 9# 2.88 12 180 A 10# 2.88 12 120 B 11# 2.88 12 120 C 12# 2.88 12 120 D 13# 2.88 12 120 E TABLE 4. Axial locations of pressure transducers. Pressure Pressure transducer 1 (Pel) transducer 2 (Pe2) Axial location (mm) 900 978 TABLE 5. Absolute values and relative error of pressure difference between simulation and experiment. Task 1# 2# 3# 4# Absolute value; [absolute value of 0.04 0.11 0.04 0.12 [DELTA][p.sub.s]-[DELTA][p.sub.e]] Relative error [absolute value of 0.7 2.1 0.8 2.6 [DELTA][p.sub.s]-[DELTA][p.sub.e]]/ [DELTA][p.sub.s] x 100% Task 5# 6# 7# Absolute value; [absolute value of 0.19 0.57 0.02 [DELTA][p.sub.s]-[DELTA][p.sub.e]] Relative error [absolute value of 4.3 20.8 0.5 [DELTA][p.sub.s]-[DELTA][p.sub.e]]/ [DELTA][p.sub.s] x 100% Task 8# 9# 10# Absolute value; [absolute value of 0.17 0.31 0.03 [DELTA][p.sub.s]-[DELTA][p.sub.e]] Relative error [absolute value of 3.0 4.9 1.0 [DELTA][p.sub.s]-[DELTA][p.sub.e]]/ [DELTA][p.sub.s] x 100% Task 11# 12# 13# Absolute value; [absolute value of 0.02 0.18 0.43 [DELTA][p.sub.s]-[DELTA][p.sub.e]] Relative error [absolute value of 1.7 16.7 21.6 [DELTA][p.sub.s]-[DELTA][p.sub.e]]/ [DELTA][p.sub.s] x 100%
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|Author:||Yang, Kunxiao; Xin, Chunling; Yu, Dongquan; Yan, Baorui; Pang, Junjian; He, Yadong|
|Publication:||Polymer Engineering and Science|
|Date:||Jan 1, 2015|
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