Numerical Simulation and Experimental Verification of Nonisothermal Flow in Counter-Rotating Nonintermeshing Continuous Mixers.
KAZUMORI FUNATSU [*]
We have developed non-Newtonian and nonisothermal flaw simulation codes in twin screw extruders using the finite element method. These codes can simulate the fully filled part of several kinds of screw elements, such as full flight screws, kneading discs, rotors, and their combinations. In this paper, we describe how we applied them to simulate a counter-rotating nonintermeshing continuous mixer, LCM100G, by Kobe Steel, Ltd. The LCM100G is a Farrel-type continuous mixer that has two mixing stages. We focused on the second mixing stage, since the flow domain of this stage is almost filled by polymer melts. Numerical simulations at various flow rates were performed. We also carried out experimental observations to verify the numerical simulations. Pressure and temperature profiles from the simulations were found to be in good agreement with the experimental results.
In the plastics industry, processes that produce tailor-made polymeric products from raw materials using mixing devices are commonly performed. These consist of melting, mixing, degassing, and pelletizing processes, which are sometimes accompanied by a chemical reaction. Through these processes, polymeric materials are blended with polymers, fillers, or other additives and are converted into useful products that have desired properties. All these processes are performed on various types of mixing devices, such as single screw extruders, twin screw extruders (TSEs), batch mixers, and continuous mixers, etc. Of these devices, TSEs and continuous mixers are the most widely used.
TSEs are composed of two kinds of screw elements. One is a conveying element, that is, a full flight screw, and the other is a mixing element, that is, a kneading disc. Continuous mixers consist of rotors designed for batch mixers and conveying elements for a continuous mixing process. The counter-rotating, nonintermeshing continuous mixer treated in this study is one of these continuous mixers.
The characteristics of each device should be understood in order to select the proper machine. For this purpose, theoretical studies of polymer flow in these devices are necessary. In the case of the TSE, there have been many theoretical studies using several numerical techniques. Szydlowski et al [1, 2] and Sebastian et al.  developed the simulation of kneading disc regions in a co-rotating TSE using the Flow Analysis Network (FAN) method. Yang et al  and Kajiwara et al  developed the simulation of isothermal flow in a co-rotating TSE using the finite element method (FEM). In addition to these studies, there have been many others, such as White et al. , Potente et al. , and Kiani et al  for simulations of a TSE.
In the case of counter-rotating nonintermeshing continuous mixers, there have been few theoretical studies. As far as we know, the following are the only two papers concerned with numerical simulations.
Kim et aL  developed the isothermal flow in a Farrel Continuous Mixer (FCM) using the FAN technique. In their study, the flow channel was assumed to be fully filled by polymer melts. Bang et al.  improved their model and took account of the effect of starved flow. However, they did not take account of the heat generation and transfer.
The FAN technique is one of approximation methods and neglects the flow in the radial direction due to the hydrodynamic lubrication approximation. This technique has an advantage of running programs that demand few computer resources. However, it has a disadvantage in accuracy, because the approximation cannot be applied to three-dimensional complicated geometries. The FEM has no limitation of geometry, although it requires a large amount of computer memory and time. Hence the FEM is considered to be more suitable for the simulation of the mixing element treated here due to its three-dimensional complicated geometries.
In this study, we developed three-dimensional non isothermal flow simulations for a continuous mixer using the FEM technique. The simulations were performed for the rotor of a counter-rotating nonintermeshing continuous mixer, LCM100G. The LCM is one of the FCM-type mixers developed by Kobe Steel, Ltd. that has two mixing stages, as illustrated in Fig. 1. Since the flow domain is almost filled by polymer melts, we focused on the second mixing stage. Numerical simulations at various flow rates were carried out. The numerical results, such as velocity vectors, pressure, and temperature profiles were precisely examined. Experimental observations were also carried out in terms of pressure and temperature profiles to verify the accuracy of the numerical simulations.
The flow domain changes with the screw rotation in this process. This moving boundary problem becomes the major difficulty in numerical flow analysis. To solve this, Yang et al.  performed a number of quasi-steady state analyses of the sequential geometries to represent the dynamic motion. Since the Reynolds number of the polymer flow is very small, isothermal flow can be approximated as quasi-steady state. In the case of nonisothermal flow, temperature distribution is dominated by viscous heating and convection because the thermal conductivity of polymer melts is very small. If the temperature distribution is mainly dominated by convection, the quasi-steady state analysis would not be acceptable. However, in this study, we assumed the temperature distribution was mainly dominated by transient viscous heating, and the quasi-steady state analysis was applicable. This assumption was not appropriate at high rotational speed due to the high Peclet number. However, the obtained results can be useful for the re lative evaluation of the flow field by comparison with the following experiments. The fully filled assumption was applied to the simulation because experimental observation confirmed that the flow domain was filled with a polymer melt of more than 90% at any flow rates.
The model was based on pure viscous non-Newtonian and incompressible flow with stress tensor [tau], pressure p, and velocity vector v. The continuity and momentum equations are
[nabla] . v = 0 (1)
- [nabla]p + [nabla] * [tau] = 0 (2)
The energy equation with density [rho], temperature T, heat capacity [C.sub.p], and thermal conductivity k is
[rho][C.sub.p]v * [nabla]T = k[[nabla].sup.2] + [tau]:[nabla]v. (3)
The constitutive equation and its temperature dependency are assumed to be represented by the Carreau model and Arrhenius' law,
[tau] = 2[eta]D (4)
[eta] = H(T)F([gamma]H(T)), [gamma] = [square root of]2[[pi].sub.D] (5)
F = [[eta].sub.0][[1 + [[[lambda].sup.2].sub.c](2[[pi].sub.D]).sup.(n-1)/2] (6)
H(T) = exp[-[beta](T - [T.sub.[alpha]])] (7)
where D is the rate deformation tensor, [[eta].sub.0] the zero shear rate viscosity, [[pi].sub.D] the second invariant of the rate deformation tensor, [[lambda].sub.c] and n the Carreau model's parameters, [beta] the Arrhenius' law's parameter, and [T.sub.[alpha]] the reference temperature, respectively.
The standard FEM Galerkin method was used in deriving the flow field, and the Streamline upwind/Petrov-Galerkin [11, 12] scheme was applied to the temperature field to obtain oscillation-free solutions. The flow field and temperature field were solved separately and the convergence was obtained by an iterative scheme. The flow diagram of this simulation is shown in Fig. 2. 2.
Kobe Steel, Ltd.'s LCM100G is a kind of FCM with a 100 mm barrel diameter. It is composed of conventional screw elements and rotors. Each rotor has three twisted wings and each wing consists of two sections: a forward pumping section and a backward pumping section (see Fig. 1). 1). In this study, only the rotor region of the second mixing stage was analyzed. The finite element mesh of the LCM100G and its barrel and rotor surface are shown in Fig 3. The process conditions are listed in Table 1, and the boundary conditions of the flow and temperature analysis are listed in Tables 2 and 3, respectively.
In order to represent the dynamic motion of rotors, we chose eight geometries in a sequential 15[degrees] increment due to the symmetry of the rotors. The geometries representing a one-third cycle are illustrated in Fig. 4 and are labeled by the angle [alpha] between the X-axis and the right rotor wing tip.
The material used in this study was high density polyethylene produced by Mitsubishi Chemical Corp. and its viscosity was measured by a capillary rheometer. The curve fit results by the Carreau model as shown in Fig. 5, and its parameters are listed in Table 4. The other parameters in Table 4 are data typical of high density polyethylene.
The online measurements of pressure and temperature in the second mixing stage of the LCM100G were carried out. Figure 6 shows the barrel section equipped with several ports for pressure and temperature measurements. A port for pressure measurements was located at Z = 80 mm and five ports for temperature were placed at Z = 40, 80, 120, 180, and 210 mm, respectively. The pressure gauge and thermocouples were placed in the port and the values of the polymer melts in the translation region (away from the nip) were directly measured. The barrel temperature was not controlled; hence the barrel was only cooled by ambient air.
In Fig. 6, the locations of the cross sections that were used for contour plots of the numerical results are also shown. The cross section A-A' is located in the orward pumping section and the cross section C-C' in the backward pumping section. The cross section B-B' is in the middle of the Z position between A-A' and C-C' and shows neutral pumping.
RESULTS AND DISCUSSION
Figure 7 shows temperature profiles along the axial distance Z of numerical results and experimental results at different flow rates. The flow rate means the feed rate in the practical process. In this figure, the temperature of numerical results is the average value of each rotational degree [alpha]. In this figure, we assumed that all the inlet temperatures were equal to 200[degrees]C in spite of different flow rates. This figure shows that the temperature increases with the decreasing flow rate. Since the viscous dissipation is almost the same at various flow rates due to the same rotational speed. This is because of the large thermal convection in the axial direction with the increasing flow rate. The tendency of the numerical and experimental results is similar. However, the absolute values are somewhat different. This is due to the boundary conditions on the inlet cross section. The result of imposing different temperatures with flow rates on the inlet boundary is shown in Fig. 8. The inlet temperature s are 215[degrees]C, 203[degrees]C, and 200[degrees]C at Q = 375, 455, and 500 kg/h, respectively. The results show good agreement with the experiments. Hence, we adopted the same boundary condition as in Fig. 8 in the subsequent numerical studies. Figure 9 shows the temperature distributions on the rotor and barrel surface at each flow rate. The upper left region of each flow domain was removed to show the rotor surface. Figure 10 shows the temperature distributions on the B-B' cross section at a different rotational degree [alpha] at Q = 500 kg/h. The position of the cross sections is shown in Fig. 6. The temperature on the barrel surface is lower than that on the rotor surface in all cases. This distribution is caused by the cooling effect from the barrel surface and the adiabatic boundary condition on the screw surface.
The velocity vectors at Q = 500 kg/h on cross section B-B' are shown in Fig. 11. Large velocity vectors are shown at the wing tip and in the intermeshing regions. Figure 12 shows the Z component of velocity Vz on a different cross section and (A) shows the forward, (B) the neutral, and C) the backward pumping section. In Fig. 12a, the forward flow is observed in the screw root and intermeshing regions. This flow is dominant on this cross section. However, backward flows are also observed at the wing tip. The Vz value of cross section (B) is very small in comparison with (A) and (C) because Vz is caused by only pressure flow in the Z direction on this cross section. In Fig. 12c the reverse distributions of (A) are observed; namely, backward flows are observed in the regions of the rotor root and intermeshing. These distributions are caused by the forward and backward pumping mechanism by helically twisted wings.
The pressure distributions at Q = 500 kg/h on cross section B-B' at different rotational degrees are shown in Fig. 13. In all cases, the pressure in front of the wing tip in the rotational direction is shown higher than that behind the tip due to screw rotation. The comparison between the calculated and experimental results of pressure variation with screw rotation is shown in Fig. 14. The calculated result has a peak at 0[degrees] (120[degrees]), but the experimental results show three peaks in pressure. Since the wing tip passes in front of the pressure gauge just one time in one-third of a rotation cycle, this phenomenon is caused by a very small amount of unfilled part. The numerical results show good agreement with experimental observation except for these unexpected pressure peaks.
SUMMARY AND CONCLUSIONS
In this study, we developed three-dimensional non-Newtonian and nonisothermal flow simulations using FEM and applied this to a counter-rotating, nonintermeshing continuous mixer, LCM100G, produced by Kobe Steel, Ltd. We also carried out experimental verification, and the following conclusions were obtained.
(1) The temperature values become low with the increasing flow rate because of the large thermal convection in the axial direction, since the viscous dissipation is almost same at various flow rates due to the same rotational speed.
(2) The rotor surface shows a higher temperature value than the barrel surface because of the cooling effect from the barrel and the adiabatic boundary of the rotor surface.
(3) In the rotor root and the intermeshing region, the forward flow is observed in the forward pumping section and the backward flow in the backward pumping section. However, the reverse distributions are seen at the top of the wingtip.
(4) The pressure in front of the wing tip in the rotational direction is shown higher than that behind the wing tip.
(5) The numerical results of temperature profiles show good agreement with experimental observations. In the case of pressure, the numerical results also show good agreement with the experimental results except for unexpected peaks in the experimental results.
The authors would like to express their gratitude to Mr. Yoshinori Kuroda of Kobe Steel, Ltd. for experimental support and useful discussions.
(1.) On leave from Mitsubishi Chemical Corporation. Yokkaichi Plant.
(*.) To whom correspondence should be addressed.
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Process Conditions of LCM100G. Barrel temperature No temperature control Rotational speed N[rpm] 420 Flow rate Q [kg/h] 375, 455, 500 Flow Boundary Conditions. Inlet cross section Constant flow rate (375, 455, 500 kg/h) Barrel inner surface No slip Rotor surface Tangential velocity by screw rotation Outlet cross section Outflow Temperature Boundary Conditions. Inlet cross section Constant temperature Barrel inner surface Heat flux: q = h(T-Ta) h = 1200 [W/([m.sup.2].K)] Ta = Ambient temp. (30[degrees]C) Rotor surface Adiabatic Outlet cross section Adiabatic (dT/dz = 0) Material Data of High Density Polyethylene. Carreau model parameter [lambda] 1.360 Carreau model parameter n 0.296 Zero shear rate viscosity [[eta].sub.0] [Pa.s] 52930.0 Arrhenius' law parameter [beta] [1/K] 0.01 Reference temperature [T.sub.[alpha]] [K] 503.0 Density [rho] [kg/[m.sup.3]] 752.0 Specific heat [C.sub.p] [J/(kg.K)] 1900.0 Thermal conductivity k [W/(m.K)] 0.225
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|Author:||ISHIKAWA, TAKESHI; KIHARA, SHIN-ICHI; FUNATSU, KAZUMORI; AMAIWA, TERUO; YANO, KAZUNORI|
|Publication:||Polymer Engineering and Science|
|Article Type:||Brief Article|
|Date:||Feb 1, 2000|
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