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Three Dimensional Numerical Simulation of Melt Filling Process in Mold Cavity with Insets
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 126 (2015) 496 – 501
7th International Conference on Fluid Mechanics, ICFM7
Three dimensional numerical simulation of melt filling process in mold cavity with insets Qiang Li School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China
Abstract This paper presents a level set function, named shape function, to describe and treat the irregular surface of mold cavity. With the aid of the shape function, the continuity, momentum and energy equations can be extended for solving two-phase flows with an immersed solid surface. The governing equations are solved by using the finite volume solver with the coupled level set and volume of fluid (CLSVOF) method to track the polymer melt front. As case studies, the melt filling processes are simulated in mold cavities with cylindrical insets. The numerical results show that the mathematical model can successfully depict the phenomenon of race-tracking and fountain, which are very helpful for determining processing conditions and designing mold cavities. ©©2015 Published by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license 2015The TheAuthors. Authors. Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM)
Keywords: Immersed boundary method; level set function; CLSVOF method; finite volume method; injection molding; irregular boundary. 1. Introduction The physical processes of flow and heat transfer in plastic injection molding, especially in complex-shaped molds with insets, are very complicated. The simulation of the filling processes is of most importance to optimize and design the mold cavities, so many researches have been done to simulate the complex flow in injection molding by using the finite element (FE) and finite volume (FV) methods [1-3]. To deal with complex boundaries, in FVM, unstructured grids or body-fitting grids are usually employed to conform to complex molds [3], but the mesh generation is time consuming. During recent years, the immersed boundary method (IBM) has been improved greatly, one of whose advantages is that the governing equations can be solved easily on Cartesian grid with a body force prescribed on boundaries and the boundary can reach to a no-slip condition [4]. In addition, there is another class of methods, usually referred to as “Cartesian grid methods”, which is originally developed for simulating inviscid flows with complex embedded solid boundaries on Cartesian grids [5]. These methods have been extended to simulate unsteady viscous flows and thus have capabilities similar to those of IBM [6]. Another branch of * Corresponding author. Tel.: +86-0391-3987791; fax: +86-0391-3987001. E-mail address: [email protected]
1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM)
doi:10.1016/j.proeng.2015.11.290
497
Qiang Li / Procedia Engineering 126 (2015) 496 – 501
Cartesian grid method, cut-cell method, has succeeded in simulating two-phase flow with embedded solid boundaries, which has been successfully applied to the Euler equations in two and three dimensions, and to flows involving both moving bodies and moving material interfaces [7]. In this review, the term immersed boundary (IB) method is used to encompass all such methods that simulate viscous flows with immersed (or embedded) boundaries on Cartesian grid system. In this paper, the immersed boundary technique is applied to a full three-dimensional complex flow in injection molding process, and a level set function is used to describe and treat the irregular surface of mold cavity. The twophase flow model which is proposed in [2] is adopted to simulate the melt filling, where the Cross-viscosity model is employed to describe the viscous behavior of polymer melt. The governing equations of two-phase flow are solved by using FVM and IBM. Herein two level set functions are employed, one for treating the complex molds, and another for tacking melt front with the aid of volume-of-fluid (VOF) method, which is the so-called CLSVOF (coupled level set and volume of fluid) method [2,8]. The content of this paper is listed as follows. First of all, the mathematical model is proposed in Section Secondly, Section 3 presents the numerical implementation of the FVM and IBM. Last but not least, In Section 4, melt filling processes in two different molds, i.e., ring-shaped mold cavities with two cylindrical insets of the same/different thickness are simulated and analyzed in detail. Some conclusions and future research direction are included in this paper. 2. Mathematical Model 2.1. Governing Equations During the melt filling process, since the air velocity is low, both the air and liquid phases can be regarded as incompressible fluid [2]. The continuity, momentum and energy equations of the incompressible fluids can be written as the unified equations in dimensionless form
wU u 0 wt
(1)
w Uu (2) Uuu p 2KD Ug Pf wt w UCT (3) UCuT NT W : u wt where u is velocity, p pressure, T temperature. U , K , C and N are density, viscosity, thermal capacity and conductivity, respectively.
D 1 2 u uT
and
W K I wui wx j wu j wxi . g
is gravitational
acceleration. f is the body force(virtual force), which can be prescribed on a regular mesh in IBM. solid volume fraction in a computational cell, i.e.
P
P is the
0 for the fluid cell, P 1 for the solid cell, and
0 P 1 for the solid/fluid interface. 2.2. CLSVOF method and shaped level set function In the CLSVOF method, the level set function is adopted to capture the melt front with the aid of VOF function, while the level set and VOF functions are advected by using the following equations, respectively
D) w) (4) u ) 0 Dt wt where ) is level set function I or VOF function F. Please see Ref. [8] for important details about CLSVOF method.
498
Qiang Li / Procedia Engineering 126 (2015) 496 – 501
The melt front is represented by the level set function
I (x, t ) [2]. And the sign of I (x, t )
is that
I !0
in
the melt phase and I 0 in the air phase, respectively. The level set function is employed to treat the discontinuities of density and viscosity near the interface. The density and viscosity are assumed to be constant in each phase, while the fluid properties across the interface are smoothed over a transition band by using the level set function (5) U U I Ua Um Ua HH I
K K I Ka Km Ka HH I C C (I ) Ca Cm Ca HH I N N (I ) N a N m N a HH I
The subscript
(6) (7) (8)
a and m represent air and melt, respectively. The smoothed Heaviside function H H I is if I H 0 ° ° 1 ª I 1 § SI ·º H H I ® «1 sin ¨ ¸» © H ¹¼ °2 ¬ H S °¯1
if I d H
(9)
if I ! H
where H is a parameter related to the interface thickness, herein H 'x . 'x is the grid width along the x direction. The Cross-viscosity model is adopted to describe the rheological property of the polymer melt in this paper [2].
K T , J, p where
W*
K0 T , p
1 K0J W *
1 n
is the model constant that representsthe shear stress rate,
pseudoplastic behavior slope of the melt as
1 n ,
K0
(10)
n is the model constant which symbolizes the
is the melt viscosity under zero-shear-rate conditions.
In this paper ,the melt front and mold shape are described by using two level sets, one for the liquid/air phase, and another for the solid/fluid phase. For example, an irregular solid domain : is embedded in the computational s
domain : , such that :
: \ :s represents the fluid domain where the governing equations are to be
f
discretized. To describe the irregular boundary of the mold, a level set function is employed, such that
If
is
negative in the fluid region : and positive in the solid region : . Meanwhile, the boundary * corresponds to the zero level set of this function, i.e., f
s
dist , x : f ° (11) I x ®0, x* ° dist x , :s ¯ where dist represents the distance between x and the nearest point on the irregular boundary of the mold cavity. 3. Numerical implementation The governing equations are discretized by the finite volume SIMPLE methods on a nonstaggered grid [2]. The level set function belongs to the Hamilton-Jacobi equations, which is discretized in this paper by the fifth-order WENO (weighted essentially non-oscillatory) scheme in space and third-order TVD (total variation diminishing) Runge-Kutta scheme in time, respectively [2]. The VOF function F is employed to conserve the melt mass which would be lost in the level set method. To solve the VOF function F, the flux-splitting algorithm is used. Please refer to Ref. [8] for detailed description.
Qiang Li / Procedia Engineering 126 (2015) 496 – 501
4. Numerical simulation of complex filling process In this section, the injection molding processes of Cross fluid are simulated by using IBM and CLSVOF method, where the shaped level set function is employed to treat complex mold boundaries. 4.1. Filling process in ring-shaped mold cavity with two cylindrical insets of the same thickness Figure 1 demonstrates the melt interfaces in ring-shaped mold cavity with two cylindrical insets of the same thickness at different time. After the melt turns around the circular inserts, two melt branches encounter and the cavitations form among melt and inserts (Fig. 1(c)), and then the seam lines or weld lines begin to form (Fig. 1(b)(c)), which are undesirable in the injection molding.
Fig.1.
(a) t=2.2
(b) t=3.85
(c) t=6.05
(d) t=7.15
Melt front in ring-shaped mold cavity with two cylindrical insets of the same thickness at different time.
4.2. Filling process in ring-shaped mold cavity with two cylindrical insets of different thickness Figure 2 shows the filling process in ring-shaped channel with two cylindrical insets of different thickness at different times. When polymer melt flows around the two cylindrical insets with the three-quarter thickness of the other segments in the mold cavity, in a short time, the difference between two melt branches is unobservable. Meanwhile, the fountain phenomenon could be seen clearly, as shown in Fig. 2(a). At t=3.3, the three branches emerge (Fig. 2(b)). Then the difference enlarges gradually due to race-tracking effect. After flowing around the insets, the three branches meet each other (Fig. 2(c)) and the seam lines begin to form which gradually disappears over time (Fig. 2(d)). Figure 2 reveals that the air traps vanish over the thin inset at t=4.4. The above phenomena could only be simulated successfully in 3D case.
499
500
Qiang Li / Procedia Engineering 126 (2015) 496 – 501
(a) t=2.2
(c) t=3.85
(b) t=3.3
(d) t=4.4
Figure 2: Melt front in ring-shaped mold cavity with two cylindrical insets of different thickness at different time.
5. Conclusions In this work, the immersed boundary and CLSVOF methods are proposed to simulate the melt filling processes in complex molds with insets, and the filling processes are simulated in ring-shaped channels with two cylindrical insets of the same/different thickness, respectively. And the conclusions can be drawn as follows. The race-tracking effect is a phenomenon that polymer melt front along the thicker edge moves faster than that in the thinner area, which could cause weld line to form or air to be trapped in the polymer melt. Due to race-tracking effect, the melt flow is unbalanced in the thick and thin segments of the mold, which will influence weightily the quality and performance of the final plastic products. So in injection molding process that the injection port should be added in the thin sections or the each part’s thickness of the mold should be consistent. Acknowledgements This work was partically supported by National Natural Science Foundation of China (No.11301157), NSFC Tianyuan Fund for Mathematics (No.11326232), the Research Fund for the Doctoral Program of Henan Polytechnic University (B2013-057) and Natural Science Foundation of Education Department of Henan Province (No.15A110001). References [1] A.Polynkin, J. F. T.Pittman, J.Sienz, Gas assisted injection molding of a handle: Three-dimensional simulation and experimental verification, Polym. Eng. Sci. 45(2005) 1049–1058.
Qiang Li / Procedia Engineering 126 (2015) 496 – 501
[2] Q.Li, J.Ouyang,G.Wu,X.Xu, Numerical Simulation of Melt Filling and Gas Penetration in Gas Assisted Injection Molding, CMES-Comp. Model. Eng. 82(2011)215–232. [3]B.J.Araujo, J.C.F.Teixeira, A.M.Cunha, C.P.T.Groth, Parallel three-dimensional simulation of the injection molding process, Int.J. Numer. Meth. Fl. 59(2009) 801–815. [4]J.I. Choi, R.C. OPeroi, J.R. Edwards, J.A. Rosati, An Immersed Boundary Method for Complex Incompressible Flows, J.Comput.Phys. 224(2007) 757–784. [5]M.J. Aftosmis, M.J. Berger, J.E. Melton, Robust and efficient Cartesian mesh generation for component-based geometry, AIAA.J.36(1998)952–960. [6]M. Meinke, L. Schneiders, C. Günther, W. Schröder, A Cut-Cell Method for Sharp Moving Boundaries in Cartesian Grids, Comput. Fluids.85(2013) 135–142. [7]C. Günther, M. Meinke, W. Schröder, A flexible level-set approach for tracking multiple interacting interfaces in embedded boundary methods, Comput. Fluids. 102(2014) 182–202. [8]G.Son, Efficient implementation of a coupled level-set and volume-of-fluid method for three-dimensional incompressible two-phase flows,Numer.Heat.Tr. B-Fund. 43(2003) 549–565.
501
ScienceDirect Procedia Engineering 126 (2015) 496 – 501
7th International Conference on Fluid Mechanics, ICFM7
Three dimensional numerical simulation of melt filling process in mold cavity with insets Qiang Li School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China
Abstract This paper presents a level set function, named shape function, to describe and treat the irregular surface of mold cavity. With the aid of the shape function, the continuity, momentum and energy equations can be extended for solving two-phase flows with an immersed solid surface. The governing equations are solved by using the finite volume solver with the coupled level set and volume of fluid (CLSVOF) method to track the polymer melt front. As case studies, the melt filling processes are simulated in mold cavities with cylindrical insets. The numerical results show that the mathematical model can successfully depict the phenomenon of race-tracking and fountain, which are very helpful for determining processing conditions and designing mold cavities. ©©2015 Published by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license 2015The TheAuthors. Authors. Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM)
Keywords: Immersed boundary method; level set function; CLSVOF method; finite volume method; injection molding; irregular boundary. 1. Introduction The physical processes of flow and heat transfer in plastic injection molding, especially in complex-shaped molds with insets, are very complicated. The simulation of the filling processes is of most importance to optimize and design the mold cavities, so many researches have been done to simulate the complex flow in injection molding by using the finite element (FE) and finite volume (FV) methods [1-3]. To deal with complex boundaries, in FVM, unstructured grids or body-fitting grids are usually employed to conform to complex molds [3], but the mesh generation is time consuming. During recent years, the immersed boundary method (IBM) has been improved greatly, one of whose advantages is that the governing equations can be solved easily on Cartesian grid with a body force prescribed on boundaries and the boundary can reach to a no-slip condition [4]. In addition, there is another class of methods, usually referred to as “Cartesian grid methods”, which is originally developed for simulating inviscid flows with complex embedded solid boundaries on Cartesian grids [5]. These methods have been extended to simulate unsteady viscous flows and thus have capabilities similar to those of IBM [6]. Another branch of * Corresponding author. Tel.: +86-0391-3987791; fax: +86-0391-3987001. E-mail address: [email protected]
1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM)
doi:10.1016/j.proeng.2015.11.290
497
Qiang Li / Procedia Engineering 126 (2015) 496 – 501
Cartesian grid method, cut-cell method, has succeeded in simulating two-phase flow with embedded solid boundaries, which has been successfully applied to the Euler equations in two and three dimensions, and to flows involving both moving bodies and moving material interfaces [7]. In this review, the term immersed boundary (IB) method is used to encompass all such methods that simulate viscous flows with immersed (or embedded) boundaries on Cartesian grid system. In this paper, the immersed boundary technique is applied to a full three-dimensional complex flow in injection molding process, and a level set function is used to describe and treat the irregular surface of mold cavity. The twophase flow model which is proposed in [2] is adopted to simulate the melt filling, where the Cross-viscosity model is employed to describe the viscous behavior of polymer melt. The governing equations of two-phase flow are solved by using FVM and IBM. Herein two level set functions are employed, one for treating the complex molds, and another for tacking melt front with the aid of volume-of-fluid (VOF) method, which is the so-called CLSVOF (coupled level set and volume of fluid) method [2,8]. The content of this paper is listed as follows. First of all, the mathematical model is proposed in Section Secondly, Section 3 presents the numerical implementation of the FVM and IBM. Last but not least, In Section 4, melt filling processes in two different molds, i.e., ring-shaped mold cavities with two cylindrical insets of the same/different thickness are simulated and analyzed in detail. Some conclusions and future research direction are included in this paper. 2. Mathematical Model 2.1. Governing Equations During the melt filling process, since the air velocity is low, both the air and liquid phases can be regarded as incompressible fluid [2]. The continuity, momentum and energy equations of the incompressible fluids can be written as the unified equations in dimensionless form
wU u 0 wt
(1)
w Uu (2) Uuu p 2KD Ug Pf wt w UCT (3) UCuT NT W : u wt where u is velocity, p pressure, T temperature. U , K , C and N are density, viscosity, thermal capacity and conductivity, respectively.
D 1 2 u uT
and
W K I wui wx j wu j wxi . g
is gravitational
acceleration. f is the body force(virtual force), which can be prescribed on a regular mesh in IBM. solid volume fraction in a computational cell, i.e.
P
P is the
0 for the fluid cell, P 1 for the solid cell, and
0 P 1 for the solid/fluid interface. 2.2. CLSVOF method and shaped level set function In the CLSVOF method, the level set function is adopted to capture the melt front with the aid of VOF function, while the level set and VOF functions are advected by using the following equations, respectively
D) w) (4) u ) 0 Dt wt where ) is level set function I or VOF function F. Please see Ref. [8] for important details about CLSVOF method.
498
Qiang Li / Procedia Engineering 126 (2015) 496 – 501
The melt front is represented by the level set function
I (x, t ) [2]. And the sign of I (x, t )
is that
I !0
in
the melt phase and I 0 in the air phase, respectively. The level set function is employed to treat the discontinuities of density and viscosity near the interface. The density and viscosity are assumed to be constant in each phase, while the fluid properties across the interface are smoothed over a transition band by using the level set function (5) U U I Ua Um Ua HH I
K K I Ka Km Ka HH I C C (I ) Ca Cm Ca HH I N N (I ) N a N m N a HH I
The subscript
(6) (7) (8)
a and m represent air and melt, respectively. The smoothed Heaviside function H H I is if I H 0 ° ° 1 ª I 1 § SI ·º H H I ® «1 sin ¨ ¸» © H ¹¼ °2 ¬ H S °¯1
if I d H
(9)
if I ! H
where H is a parameter related to the interface thickness, herein H 'x . 'x is the grid width along the x direction. The Cross-viscosity model is adopted to describe the rheological property of the polymer melt in this paper [2].
K T , J, p where
W*
K0 T , p
1 K0J W *
1 n
is the model constant that representsthe shear stress rate,
pseudoplastic behavior slope of the melt as
1 n ,
K0
(10)
n is the model constant which symbolizes the
is the melt viscosity under zero-shear-rate conditions.
In this paper ,the melt front and mold shape are described by using two level sets, one for the liquid/air phase, and another for the solid/fluid phase. For example, an irregular solid domain : is embedded in the computational s
domain : , such that :
: \ :s represents the fluid domain where the governing equations are to be
f
discretized. To describe the irregular boundary of the mold, a level set function is employed, such that
If
is
negative in the fluid region : and positive in the solid region : . Meanwhile, the boundary * corresponds to the zero level set of this function, i.e., f
s
dist , x : f ° (11) I x ®0, x* ° dist x , :s ¯ where dist represents the distance between x and the nearest point on the irregular boundary of the mold cavity. 3. Numerical implementation The governing equations are discretized by the finite volume SIMPLE methods on a nonstaggered grid [2]. The level set function belongs to the Hamilton-Jacobi equations, which is discretized in this paper by the fifth-order WENO (weighted essentially non-oscillatory) scheme in space and third-order TVD (total variation diminishing) Runge-Kutta scheme in time, respectively [2]. The VOF function F is employed to conserve the melt mass which would be lost in the level set method. To solve the VOF function F, the flux-splitting algorithm is used. Please refer to Ref. [8] for detailed description.
Qiang Li / Procedia Engineering 126 (2015) 496 – 501
4. Numerical simulation of complex filling process In this section, the injection molding processes of Cross fluid are simulated by using IBM and CLSVOF method, where the shaped level set function is employed to treat complex mold boundaries. 4.1. Filling process in ring-shaped mold cavity with two cylindrical insets of the same thickness Figure 1 demonstrates the melt interfaces in ring-shaped mold cavity with two cylindrical insets of the same thickness at different time. After the melt turns around the circular inserts, two melt branches encounter and the cavitations form among melt and inserts (Fig. 1(c)), and then the seam lines or weld lines begin to form (Fig. 1(b)(c)), which are undesirable in the injection molding.
Fig.1.
(a) t=2.2
(b) t=3.85
(c) t=6.05
(d) t=7.15
Melt front in ring-shaped mold cavity with two cylindrical insets of the same thickness at different time.
4.2. Filling process in ring-shaped mold cavity with two cylindrical insets of different thickness Figure 2 shows the filling process in ring-shaped channel with two cylindrical insets of different thickness at different times. When polymer melt flows around the two cylindrical insets with the three-quarter thickness of the other segments in the mold cavity, in a short time, the difference between two melt branches is unobservable. Meanwhile, the fountain phenomenon could be seen clearly, as shown in Fig. 2(a). At t=3.3, the three branches emerge (Fig. 2(b)). Then the difference enlarges gradually due to race-tracking effect. After flowing around the insets, the three branches meet each other (Fig. 2(c)) and the seam lines begin to form which gradually disappears over time (Fig. 2(d)). Figure 2 reveals that the air traps vanish over the thin inset at t=4.4. The above phenomena could only be simulated successfully in 3D case.
499
500
Qiang Li / Procedia Engineering 126 (2015) 496 – 501
(a) t=2.2
(c) t=3.85
(b) t=3.3
(d) t=4.4
Figure 2: Melt front in ring-shaped mold cavity with two cylindrical insets of different thickness at different time.
5. Conclusions In this work, the immersed boundary and CLSVOF methods are proposed to simulate the melt filling processes in complex molds with insets, and the filling processes are simulated in ring-shaped channels with two cylindrical insets of the same/different thickness, respectively. And the conclusions can be drawn as follows. The race-tracking effect is a phenomenon that polymer melt front along the thicker edge moves faster than that in the thinner area, which could cause weld line to form or air to be trapped in the polymer melt. Due to race-tracking effect, the melt flow is unbalanced in the thick and thin segments of the mold, which will influence weightily the quality and performance of the final plastic products. So in injection molding process that the injection port should be added in the thin sections or the each part’s thickness of the mold should be consistent. Acknowledgements This work was partically supported by National Natural Science Foundation of China (No.11301157), NSFC Tianyuan Fund for Mathematics (No.11326232), the Research Fund for the Doctoral Program of Henan Polytechnic University (B2013-057) and Natural Science Foundation of Education Department of Henan Province (No.15A110001). References [1] A.Polynkin, J. F. T.Pittman, J.Sienz, Gas assisted injection molding of a handle: Three-dimensional simulation and experimental verification, Polym. Eng. Sci. 45(2005) 1049–1058.
Qiang Li / Procedia Engineering 126 (2015) 496 – 501
[2] Q.Li, J.Ouyang,G.Wu,X.Xu, Numerical Simulation of Melt Filling and Gas Penetration in Gas Assisted Injection Molding, CMES-Comp. Model. Eng. 82(2011)215–232. [3]B.J.Araujo, J.C.F.Teixeira, A.M.Cunha, C.P.T.Groth, Parallel three-dimensional simulation of the injection molding process, Int.J. Numer. Meth. Fl. 59(2009) 801–815. [4]J.I. Choi, R.C. OPeroi, J.R. Edwards, J.A. Rosati, An Immersed Boundary Method for Complex Incompressible Flows, J.Comput.Phys. 224(2007) 757–784. [5]M.J. Aftosmis, M.J. Berger, J.E. Melton, Robust and efficient Cartesian mesh generation for component-based geometry, AIAA.J.36(1998)952–960. [6]M. Meinke, L. Schneiders, C. Günther, W. Schröder, A Cut-Cell Method for Sharp Moving Boundaries in Cartesian Grids, Comput. Fluids.85(2013) 135–142. [7]C. Günther, M. Meinke, W. Schröder, A flexible level-set approach for tracking multiple interacting interfaces in embedded boundary methods, Comput. Fluids. 102(2014) 182–202. [8]G.Son, Efficient implementation of a coupled level-set and volume-of-fluid method for three-dimensional incompressible two-phase flows,Numer.Heat.Tr. B-Fund. 43(2003) 549–565.
501