WANG Lei( )，JI Jun-yang()，LIU Ting-ting()，GAN Xue-hui()*

1 Engineering Research Center of Advanced Textile Machinery，Ministry of Education，Donghua University，Shanghai 201620，China

2 College of Mechanical Engineering，Donghua University，Shanghai 201620，China

Introduction

The extrusion swelling of polymer is always a hot spot in the polymer molding process.Many scholars have done a lot of work for studying extrusion swelling.Zhaoetal.[1- 2]deduced the extrusion swelling equation for polymer melt in conical short die theoretically and quantitatively analyzed the influence factors of the extrusion swelling for the polymer melt in conical short die.Alokuetal.[3]simulated the polymer foaming processes in extrusion flow，and quantified the effect of polymer viscoelasticity on the die-swell by comparison with the simulation results of LDPE foaming in extrusion flow using the Bird-Carreau model and the SVM.Mitsoulis[4]undertook the numerical simulation about the flow of three Boger fluids，which reflected that the extrudate swell and excess pressure losses，as well as the shape and extent of the free surface increased rapidly and monotonically with increasing elasticity level.Russoetal.[5]analyzed the effect of the viscosity ratio on the swelling ratio，which showed that the overall elastic response of the fluid increased as the polymeric viscosity increases.Xuetal.[6]simulated the viscoelastic extrudate swell through elliptical ring die，and obtained that the extrudate swell ratio increased markedly with the increase of volumetric flow rate and relax time.However，the difference of the material properties for two polymers makes theoretical research and numerical simulation more complex.Ganviretal.[7- 8]，Martynetal.[9],and Arda[10]etal.studied the polymer extrusion swelling based on numerical simulation and experimented to dynamically analyze the key factors of extrusion swelling.But there are two kinds of free boundary problems in the co-extrusion process of the multi-component polymers，the unknown interface of the two fluids in the co-extrusion process and the extrusion swell surface of the polymer melt，which make the setting boundary condition and the numerical solving more difficult and complex.There are many researches on co-extrusion for the multi-component polymer[11-16]，which mainly aim at the large size of the die.Huangetal.[11-12]applied PTT constitutive equation and finite element method to simulate the co-extrusion process for polypropylene/polystyrene two melts through rectangular die，and also verified the results by the experiment.Zhangetal.[13]simulated the forming process of the co-extrusion interface for non-Newtonian fluid.Mitsoulisetal.[14]researched the shape and stability for co-extrusion interface by the experiment and simulation.Paulietal.[17]studied the effect of the transition region’s structure between outflow and inflow of the die on the velocity distribution at the outflow and the swell behavior by numerical simulation，which are the most prominent die-related factors influencing the profile quality.However，the micro hole’s size is small and the change of the micro hole size is big，which gives the theoretical research and numerical simulation great difficulty for multi-component composite melt on extrusion swelling in circular micro hole.

The PET/PA6 composite melt in the spinneret will be simulated by using PTT constitutive equations through Polyflow based on the theory of the rheology and the spinning process and the structure of the spinning components，in addition the flow’s influence law for the velocity distribution and the micro hole’s structure’s influence law for extrusion swelling will be analyzed，which has a good guiding role for looking for the best process route and control conditions for the composite spinning and spinning the high performance fiber.

1 Numerical Simulation

1.1 Numerical model

The geometric model is adopted as shown in Fig.1 (a)，which has flow convergence region，straight flow region，and extrusion swell region.The convergence angleαis 30°/45°/60°/75° and the length of the straight flow regionLis 0.2 mm/0.4 mm/0.6 mm/0.8 mm respectively.The meshing model is shown in Fig.1 (b)，in order to get high-quality meshes，using the regular hexahedron unit and zoning the geometric model to mesh，and refine the meshes in the region of the export.The number of the elements is 7 371，the number of the nodes is 9 500，and the skewness is 0.426.

(a) Geometric model

(b) Meshing model

(c) Extrusion swelling model

1.2 Material model

PTT model is used as the constitutive equation to represent the rheological behavior of the polymer melt.

Choosing the materials PA6 and PET as the two component experimental materials respectively，parameters of the two experimental materials are shown in Table 1.

Table 1 Material parameters

1.3 Mathematical model

In order to reduce the amount of the calculation，do some assumptions for the stability flow of the polymer[15-16，18]: ① the polymer melt is incompressible non-isothermal steady flow; ② the flow on the wall is no slip flow，namely the different component of the velocity is zero; ③because of the high viscosity，ignore the influence of the inertia force and mass force; ④the flow in the die is fully developed.In these conditions，the melt satisfies the mass conservation equation，momentum conservation equation，and energy conservation equation in the flow field.

1.4 Boundary conditions

The wall of the micro hole:Vn=Vs= 0.The inflow:QAis the flow of the PA6，QBis the flow of the PET，andQA=QB=2×10-8(m3/s).The interface of two melts: ① kinematics conditions，namely the velocity field in the interface is continuous; ② dynamic conditions，namely the shear stress and normal stress is continuous on both sides of the interface.The outflow:fn=fs= 0.The iteration method: in order to reduce the amount of the calculation，the linear iteration is used for pressure and stress，Picard iteration is used for viscosity,and the mini-element iteration is used for velocity.

2 Effect of Flow Ratio on Velocity Distribution and Extrusion Swelling

Figure 1(c) shows the extrusion swelling for the composite melt at the exit of the micro hole.The fiber’s cross section has an obvious swelling，then tends to be stable.Figure 2 shows the velocity contour maps of the spinning hole in different places.

(a) At the orifice of the jet hole

(b) At the exit of the micro hole

(c) Spinning micro hole

As shown in Fig.2 (a)，the inlet flow of two melts is the same，but the velocity distributions in the spinning hole are changed greatly.When two melts are composited to go into the spinning hole，the velocity distributions of two melts are the same，and the maximum velocity occurs at the center of the spinning hole.After the composite spinning melt goes into the spinning hole，the maximum velocity of it occurs on the side of the PET，as shown in Fig.2 (b).After compositing two melts，because the viscosity of the PET is low，its anelasticity is small，which makes its velocity fast in Fig.2 (c).

Figure 3 (a) shows the velocity distribution curves in co-extrusion process in different places in the spinning hole.

(a) PET∶PA6=1∶1 in different places

(b) Different flow ratios at the exit

As shown in Fig.3 (a)，the maximum velocity of the composite melt occurs atz=-0.05 mm or so，namely on the side of the PET，and the maximum velocity is 1.4m/s.Decreasing the distance to the exit of the spinning micro hole，the maximum velocity becomes slow.Because the PET melt surrounds the PA6 melt，and the PET’s viscosity is lower than the PA6’s viscosity，which makes the velocity of the composite melt more uniform and the maximum velocity reduce.In the condition of guaranteeing the total flow constant，when PET∶PA6= 1∶3 and PET∶PA6= 3∶1，the velocity distribution curves are shown in Fig.3 (b).

When PET∶PA6= 3∶1，the velocity distribution of two melts becomes more non-uniform，and the maximum velocity becomes 1.5 m/s，then leads to the extrusion swelling more serious，which impact the fiber’s mechanical properties.When PET∶PA6= 1∶3，the velocity distribution of two melts becomes more uniform，and the maximum velocity decreases to under 1.2 m/s.The uniform velocity distribution can decrease the extrusion swelling ratio effectively，improving the fiber’s mechanical properties as shown in Fig.4.

Fig.4 The influence of flow ratio on extrusion swelling ratio (L=0.6 mm)

3 Effect of Viscosity on Extrusion Swelling

Another less viscosity material PP than PA is chosen to constitute another two components with PET.Figure 5 shows the extrusion swelling ratio under different melt viscosity.The viscosity of PET/PA6 is close to PP’s and its extrusion swelling ratio is declined.The overall elastic response of the fluid increases as the polymeric viscosity increases，then the extrusion swelling ratio increases with the elastic response increasing.Numerical results show good quantitative agreement with previous simulation results[5].

Fig.5 The influence of viscosity on extrusion swelling ratio (L=0.6 mm)

4 Effect of Die Exit Geometry on Extrusion Swelling

Figures 6 and 7 show the extrusion swelling ratio under different lengths of the straight flow region and convergence angle respectively.Along with rising length of the straight flow region，the extrusion swelling ratio decreases.However，along with decreasing convergence angle，the extrusion swelling ratio decreases.When the length changes from 0.6 mm to 0.8 mm or the convergence angle changes from 45° to 30°，the extrusion swelling ratio change is not obvious.Namely，when the length of the straight flow region increases to a certain value or the convergence angle decreases to a certain value，the extrusion swelling ratio approximation tends to a constant value.

Fig.6 The influence of the straight flow region on extrusion swelling ratio (α=60°)

Fig.7 The influence of the convergence angle on extrusion swelling ratio (L=0.6 mm)

The numerical results agree well with previous relevant experiments，particularly the experiments of Liang[19]and Dangtungeeetal.[20]measured the swell ratio by means of a melt flow rate instrument to investigate the effect of die geometry on the die swell behavior of the melt.It showed that the swell ratio increased nonlinearly with increase of the diameter，whereas it reduced nonlinearly with increase of the die length/diameter ratio.Also Mullneretal.[21]studied the effect of the die length/diameter ratio on the swell value，and the result showed that the value of the swell ratio decreased rapidly at low values of length/diameter and then leveled off as length/diameter was increased further，which indicated that the elastic behavior of the melt depended on the residence time of the melt in the die.In 2013，Musiletal.[22]experimentally investigated the effects of flared length and die exit angle on the extrusion swelling，and obtained the optimum values for the die exit angle and dimensionless flared length.

The elastic deformation of the viscoelastic fluid caused by elongational flow in the convergent duct leads to the extrusion swelling phenomenon.So the melt’s tensile elastic deformation will be more relaxed in the long straight flow region，and the extrusion swelling ratio becomes smaller.When the length of the straight flow region is long enough to make the elastic deformation fully recover，the extrusion swell ratio approximation tends to a constant value.Similarly，when the melt goes through the small convergence angle，the less the tensile elastic deformation is，the smaller the extrusion swelling ratio is.

5 Conclusions

(1)The different viscosity of components has certain influence on the extrusion swelling.The smaller the difference of the melt viscosity is，the smaller the extrusion swelling ratio is，and the better the fiber’s performance is.

(2)The length of the straight flow region has certain influence on the extrusion swelling，appropriately increasing the length can reduce the extrusion swelling ratio and then improve the fiber’s performance.

(3)The convergence angle has certain influence on the extrusion swelling，appropriately decreasing the convergence angle can reduce the extrusion swelling ratio and then improve the fiber’s performance.

(4) The flow of the composite melt has a leading role on extrusion swelling，appropriately increasing the PA6’s flow and decreasing the PET’s flow can reduce the velocity’s difference，in order to improve the fiber’s performance.

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