Patent Publication Number: US-11376777-B1

Title: Molding system for preparing molded article with orientation effect

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims the benefit of U.S. Provisional Patent Application No. 63/177,689 filed Apr. 21, 2021, which is incorporated herein by reference in its entirety. 
    
    
     TECHNICAL FIELD 
     The present disclosure relates to a molding system for preparing a molded article, and more particularly, to an injection-molding system for preparing a molded article using a computer-aided engineering (CAE) simulation. 
     DISCUSSION OF THE BACKGROUND 
     Applications for polymers and plastics can be found in most areas of daily life due to their versatility and economic viability in the manufacturing industry. In plastics manufacturing, the actual flow of polymer melts is transient, non-Newtonian and non-isothermal, with frozen layers building up as the complex mixture flows through the mold cavity. Characteristics of a finished product are determined by many complex factors, such as changes in the direction of flow, inclusion of ribs, and changes in thickness and holes. To control the quality of the products, a deep understanding of complicated flow fields is critical. Nowadays, computer-aided engineering (CAE) software provides realistic simulation and predictive analysis for complex flows of complex fluids. 
     Viscoelasticity is a primary material property of polymer melts that exhibit both viscous and elastic characteristics. The die swell effect is important to understand for viscoelastic fluids. According to results of academic research of fluid mechanics and rheology, the White-Metzner constitutive equation is a nonlinear viscoelastic model of the generalized Newtonian fluids (GNF) available in polymer processing flows. 
     In practice, with the White-Metzner model, it is difficult to simulate that the die swell ratio of polymer melts is increased with the average flow rate or the wall shear rate. This can be attributed to three critical problems with the White-Metzner constitutive equation. In this model, the modulus parameter cannot be determined by experimental data. The extension viscosity can become a divergent risk at high extension rates. Moreover, although the first normal stress difference can be given, the second normal stress difference is zero and not described by the model. To resolve such significant issues, the present disclosure proposes modifications to the White-Metzner constitutive equation to simulate the die swell effect. This is based on extensive research of complex viscoelastic phenomena related to the die swell of viscoelastic fluids. 
     This Discussion of the Background section is provided for background information only. The statements in this Discussion of the Background are not an admission that the subject matter disclosed in this section constitutes prior art to the present disclosure, and no part of this Discussion of the Background section may be used as an admission that any part of this application, including this Discussion of the Background section, constitutes prior art to the present disclosure. 
     SUMMARY 
     The present disclosure provides a molding system for preparing a molded article, comprising a molding machine, including a barrel, a screw mounted for moving within the barrel, a driving motor driving the screw to move a molding material; a mold disposed on the molding machine and connected to the barrel of the molding machine to receive the molding material, and having a mold cavity with a die swell structure for being filled with the molding material; a processing module simulating a filling process of the molding material from the barrel into the molding cavity based on a molding condition including a predetermined screw speed for the molding machine, wherein simulating the filling process of the molding material comprises simulating a die swell effect of the molding material in the die swell structure by taking into consideration of an effective factor, a shear viscosity, an extension viscosity, a shear rate, and a molecular orientation effect of the molding material; and a controller operably communicating with the processing module to receive the molding conditions and with the molding machine to control the driving motor of the molding machine based on the molding conditions to move the screw at the predetermined screw speed to transfer the molding material at a corresponding flow rate to perform an actual molding process for preparing the molded article. 
     In some embodiments, the processing module performs a model to simulate the die swell effect of the molding material, and the model is represented by an expression: 
                     Wi   ⁡     (     γ   .     )         γ   .       ⁢       τ   *     ∇       +   τ     =         2   ⁢       η   W     ⁡     (     γ   .     )       ⁢   D     +     τ   A       =           η   W     ⁡     (     γ   .     )       ⁢   D     +     2   ⁢     η   W     ⁢     N   p     ⁢     D   :     A   4                   
where {dot over (γ)} represents a shear rate of the molding material, Wi({dot over (γ)}) represents an viscoelastic property of the molding material, τ represents a stress distribution, τ ∇ * represents a rate of change of the stress distribution, η W ({dot over (γ)}) represents the weighted viscosity distribution of the molding material, D represents a rate of deformation of the molding material, τ A  represents the molecular orientation effect on the stress distribution, and A 4  represents a molecular orientation distribution of the molding material.
 
     In some embodiments, the molecular orientation effect on the stress distribution is represented by an expression:
 
τ A =2η W   N   p   D:A   4  
 
where N p  represents a dimensionless parameter, D represents a rate-of-deformation distribution, and A 4  represents a molecular orientation distribution of the molding material.
 
     In some embodiments, the rate of change of the stress distribution is represented by an expression: 
                 τ   *     ∇     =         D   ⁢           ⁢   τ     Dt     -       ∇   L     ·   τ     -     τ   ·     L   T                     L   =       ∇   u     -     ξ   ⁢           ⁢   D             
where ∇u represents the standard velocity gradient distribution, L represents an effective velocity gradient distribution and ξ represents the effective factor.
 
     In some embodiments, the effective factor is related to the shear rate of the molding material, and the effective factor is represented by an expression: 
               ξ   ⁡     (     γ   .     )       =       ξ   0         [     1   +       (       γ   .         γ   .     XC       )     2       ]       N   X               
where ξ 0 , {dot over (γ)} XC , and N X  represent parameters determined by using an experimental data.
 
     In some embodiments, the viscoelastic property of the molding material is related to the shear rate of the molding material, and the viscoelastic property of the molding material is represented by an expression: 
               Wi   ⁡     (     γ   .     )       =       Wi   0         [     1   +       (       γ   .         γ   .     WC       )       -   2         ]       N   W               
where Wi 0 , {dot over (γ)} WC , and N W  represent parameters determined by using an experimental data.
 
     In some embodiments, the weighted viscosity distribution of the molding material is represented by an expression: 
               η   W     =         (     1   -   W     )     ⁢     η   S       +     W   ⁢           ⁢     η   E                     W   =         γ   .     E   2         γ   .     T   2                       γ   .     T   2     =         γ   .     S   2     +       γ   .     E   2             
where W represents a weighting function, η S  represents the shear viscosity of the molding material, η E , represents the extension viscosity of the molding material, {dot over (γ)} T  represents a total strain rate of the molding material, {dot over (γ)} S  represents a characteristic shear rate of the molding material, and {dot over (γ)} E  represents a characteristic extension rate of the molding material.
 
     In some embodiments, the viscoelastic property of the molding material is determined by using an experimental data of a first normal stress difference represented by an expression:
 
 N   1 =2 Wiτ   12  
 
where τ 12  represents the shear viscosity of the molding material.
 
     In some embodiments, the effective factor is determined by using an experimental data of a first normal stress difference and a second normal stress difference represented by an expression: 
               N   1     =     2   ⁢   Wi   ⁢           ⁢     τ   12                     N   2     =       -     ξ   2       ⁢     N   1             
where τ 12  represents the shear viscosity of the molding material,
 
               τ   12     =         η   S     ⁢     γ   .         1   +       W   i   2     ⁡     (     1   -     C   N   2       )                 
and the parameter C N ({dot over (γ)}) is given
 
 C   N ({dot over (γ)})=1−ξ({dot over (γ)})
 
     In some embodiments, the first normal stress difference is related to the shear viscosity of the molding material. 
     In some embodiments, the second normal stress difference has a non-zero value. 
     In some embodiments, the molding material comprises polystyrene resin. 
     In some embodiments, the molding material comprises a plurality of polymer chains, and the molecular orientation effect of the molding material represents an orientation distribution of the polymer chains. 
     In some embodiments, the molding material comprises a plurality of fibers, and the molecular orientation effect of the molding material represents an orientation distribution of the fibers. 
     The foregoing has outlined rather broadly the features and technical advantages of the present disclosure in order that the detailed description of the disclosure that follows may be better understood. Additional features and advantages of the disclosure will be described hereinafter, and form the subject of the claims of the disclosure. It should be appreciated by those skilled in the art that the conception and specific embodiment disclosed may be readily utilized as a basis for modifying or designing other structures or processes for carrying out the same purposes of the present disclosure. It should also be realized by those skilled in the art that such equivalent constructions do not depart from the spirit and scope of the disclosure as set forth in the appended claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A more complete understanding of the present disclosure may be derived by referring to the detailed description and claims when considered in connection with the Figures, where like reference numbers refer to similar elements throughout the Figures, and: 
         FIG. 1  schematically shows molecular structures of high-density polyethylene (HDPE) and low-density polyethylene (LDPE), respectively. 
         FIG. 2  shows an experimental observation of the die swell with an extrudate cross-section which is greater than a die cross-section. 
         FIG. 3  shows an experimental observation of streamlines in 4:1 planar contraction flow patterns with corner vortex for LDPE and HDPE, respectively. 
         FIG. 4  shows a mixture of polymer matrix, fiber, and coupling agent at the macroscopic level according to the prior art. 
         FIG. 5  shows a mixture of partial aligned polymer chains, fibers, and coupling agents at the microscopic level according to the prior art. 
         FIG. 6  shows normalized transient shear viscosity η + /η e  with respect to shear strains for the unfilled and fiber-filled fluids, respectively wherein η +  and η e  are transient-state and equilibrium-state values of shear viscosity, respectively. 
         FIG. 7  shows the transient shear viscosity for 30 wt % SGF/PBT (polybutylene terephthalate resin containing 30% by weight of short glass fibers) composite at shear rate {dot over (γ)}=1.0 s −1  and 260° C. 
         FIG. 8  is a flowchart showing an injection-molding simulation operation in accordance with some embodiments of the present disclosure. 
         FIG. 9  is a schematic view of an injection-molding apparatus in accordance with some embodiments of the present disclosure. 
         FIG. 10  is a functional block diagram of the computer in  FIG. 9  in accordance with some embodiments of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     Embodiments, or examples, of the disclosure illustrated in the drawings are now described using specific language. It shall be understood that no limitation of the scope of the disclosure is hereby intended. Any alteration or modification of the described embodiments, and any further applications of principles described in this document, are to be considered as normally occurring to one of ordinary skill in the art to which the disclosure relates. Reference numerals may be repeated throughout the embodiments, but this does not necessarily mean that feature(s) of one embodiment apply to another embodiment, even if they share the same reference numeral. 
     It shall be understood that, although the terms first, second, third, etc. may be used herein to describe various elements, components, regions, layers or sections, these elements, components, regions, layers or sections are not limited by these terms. Rather, these terms are merely used to distinguish one element, component, region, layer or section from another element, component, region, layer or section. Thus, a first element, component, region, layer or section discussed below could be termed a second element, component, region, layer or section without departing from the teachings of the present inventive concept. 
     The terminology used herein is for the purpose of describing particular example embodiments only and is not intended to be limited to the present inventive concept. As used herein, the singular forms “a,” “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It shall be further understood that the terms “comprises” and “comprising,” when used in this specification, point out the presence of stated features, integers, steps, operations, elements, or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, or groups thereof. 
     The present disclosure is directed to a molding system for preparing a molded article using a computer-aided engineering (CAE) simulation. In order to make the present disclosure completely comprehensible, detailed steps and structures are provided in the following description. Obviously, implementation of the present disclosure does not limit special details known by persons skilled in the art. In addition, known structures and steps are not described in detail, so as not to limit the present disclosure unnecessarily. Preferred embodiments of the present disclosure will be described below in detail. However, in addition to the detailed description, the present disclosure may also be widely implemented in other embodiments. The scope of the present disclosure is not limited to the detailed description, and is defined by the claims. 
     Injection molding is a technology commonly used for high-volume manufacturing of parts made of synthetic resin, most commonly made of thermoplastic polymers. During a repetitive injection-molding process, a plastic resin, most often in the form of small beads or pellets, is introduced into an injection-molding machine that melts the resin beads under heat, pressure, and shear. The now-molten resin is forcefully injected into a mold cavity having a particular cavity shape. The injected plastic is held under pressure in the mold cavity, cooled, and then removed as a solidified part having a shape that essentially duplicates the cavity shape of the mold. 
     Polymers/Plastics applications can be found in almost all areas of everyday living due to their versatility with an economically attractive choice in the manufacturing industry. Polymer materials can be processed by fast, highly-automated methods, such as injection molding. Thickness-variation or contraction channels are one of familiar geometries for injection-molding products. An obvious vortex is significantly found in the corner of the thicker upstream channel. The vortex size is related to molecular structures of polymer melts. 
       FIG. 1  schematically shows molecular structures of high-density polyethylene (HDPE) and low-density polyethylene (LDPE), respectively. Obviously, the HDPE and LDPE have different molecular structures: the HDPE is consisted of linear chains with “molecular orientation”; in contrast, the LDPE chain is branched in absence of molecular orientation. 
     The die well effect or extrudate swell is a complex viscoelastic phenomenon. As shown in  FIG. 2 , die swell can be experimentally observed as an extrudate with a cross-section D 2  which is greater than a die cross-section D 1 . The die swell ratio, defined as D 2 /D 1 , is increased with the average flow rate or the apparent shear rate (See, for example, Bird, R. B., R. C. Armstrong, and O. Hassager,  Dynamics of Polymeric Liquids: Fluid Mechanics  (Wiley-Interscience, New York, 1987); Morrison, F. A.,  Understanding Rheology  (Oxford University, 2001); the entirety of the above-mentioned publication is hereby incorporated by reference herein and made a part of this specification, which is incorporated herein by reference in its entirety). 
       FIG. 3  shows an experimental observation of streamlines in 4:1 planar contraction flow patterns with corner vortex for LDPE and HDPE, respectively. As shown in  FIG. 3 , LDPE&#39;s vortex size is usually large than HDPE&#39;s. 
     In addition, understanding fibers filled in a polymer melt is an important challenge in fundamental rheology with viscoelastic properties strongly depending on the fiber orientation. During the startup shear flows with small shear rates, the change of fiber orientation induces a viscosity overshoot for fiber-filled fluids, whereas no such finding occurs in unfilled fluids (see, Eberle, A. P. R., G. M. Vélez-Garcia, D. G. Baird, and P. Wapperom, “Fiber Orientation Kinetics of a Concentrated Short Glass Fiber Suspension in Startup of Simple Shear Flow,” J Non-Newtonian Fluid Mech 165 110-119 (2010), which is incorporated herein by reference in its entirety). 
     In a macroscopic view, a “fiber-reinforced composite” refers to as a material containing fibers embedded in a matrix with well-defined interfaces between two constitutive phases. At a macroscopic level, as shown in  FIG. 4 , the coupling agents  45  are small difunctional molecules acting as a bridge between the fibers  43  and the fluid (matrix)  41 . If necessary, these additional coupling agents can be added to strengthen the cohesion between the inorganic fiber and organic polymer. At a microscopic level,  FIG. 5  represents a mixture that exists in the practical suspension where the fibers  43  are immersed in a matrix of partial aligned polymer chains  41  by adding a small amount of the coupling agent  45 . The coupling agent  45  is an interface medium between the fibers  43  and the polymer chains  41 . The long polymer chains are an entanglement. Very probably, the fibers  43  are also entwined by the chains or adsorbed into the chains  41  through the coupling agent  45 . 
       FIG. 6  shows normalized transient shear viscosity η + /η e  with respect to shear strains for the unfilled and fiber-filled fluids, respectively wherein η +  and η e  are transient-state and equilibrium-state values of shear viscosity, respectively. The waveforms in  FIG. 6  imply that the rheological nonlinearity of viscosity overshoot is relatively increased, when fibers are filled to a neat polymer. 
     Viscoelasticity (VE) is a primary property of polymer materials that exhibit both viscous and elastic characteristics. The corner vertex is a viscoelastic fluid behavior. Recently, Tseng et al. have proposed a modified White-Metzner (WM) viscoelastic fluid model (see, U.S. Pat. No. 11,027,470, which is incorporated herein by reference in its entirety). In the present innovation, such a viscoelastic model is further extended with the effect of molecular orientation on stress tensor for polymer melts, which is incorporated herein by reference in its entirety). 
     A typical injection-molding procedure comprises four basic operations: (1) heating the plastic resin in the injection-molding machine to allow it to flow under pressure; (2) injecting the melted plastic resin into a mold cavity or cavities defined between two mold halves that have been closed; (3) allowing the plastic resin to cool and harden in the cavity or cavities while under pressure; and (4) opening the mold halves to cause the cooled part to be ejected from the mold. In the conventional injection molding of synthetic resin by an injection-molding machine, the weight of the injected synthetic resin varies with the molten resin pressure, the molten resin specific volume, the molten resin temperature and other molten resin conditions. Therefore, it is difficult to form products of a consistent quality. 
     In general, the setting of molding conditions of the injection-molding machine requires a large number of trial molding operations and a lengthy setting time because the setting work largely depends on the know-how and experience of an operator of the injection-molding machine, and various physical values affect one another as well. 
     To control the quality of the molded products, a deep understanding of complicated flow fields is critical. Computer-aided engineering (CAE) software provides realistic simulation and predictive analysis for complex flows of complex fluids. In the academic research of fluid mechanics and rheology, nonlinear viscous models of the generalized Newtonian fluids (GNF) have been available in polymer processing flows. However, with these techniques it is difficult to simulate the viscoelastic behaviors encountered in the die swell. Rheological researchers have developed viscoelastic constitutive equations of the stress tensor for polymer liquids. The first and second normal stress differences, N 1  and N 2 , are important material functions of viscoelastic fluids under simple shear flow (See, for example, Bird, R. B., R. C. Armstrong, and O. Hassager,  Dynamics of Polymeric Liquids: Fluid Mechanics  (Wiley-Interscience, New York, 1987); Morrison, F. A.,  Understanding Rheology  (Oxford University, 2001); the entirety of the above-mentioned publication is hereby incorporated by reference herein and made a part of this specification). The White-Metzner constitutive equation is a nonlinear viscoelastic model with the relaxation time function, which is related to the GNF shear viscosity and the modulus (See, for example, Morrison, F. A.,  Understanding Rheology  (Oxford University, 2001); White, J. L. and A. B. Metzner, “Development of Constitutive Equations for Polymeric Melts and Solutions,” J Appl Polym Sci 7 867-1889 (1963); the entirety of the above-mentioned publication is hereby incorporated by reference herein and made a part of this specification). In practice, however, this model has difficulties simulating the die swell. There is a numerical divergent issue due to the strain hardening of extension viscosity with at high extension rates. In addition, the modulus parameter is not experimentally determined since the relaxation time is not easy to be measured. The first normal stress difference can be given, but the second one is not described. Therefore, the present disclosure proposes modifications to the White-Metzner constitutive equation to simulate the die swell effect. 
     The actual flow of polymer melts is transient, non-Newtonian and non-isothermal, with frozen layers building up as the complex mixture flows through the mold cavity. The governing equations of the fluid mechanics include the equation of continuity, the equation of motion, and the equation of energy to describe the transient and non-isothermal flow motion as follows: 
                         ∂   ρ       ∂   t       +       ∇     ·   ρ       ⁢           ⁢   u       =   0           (   1   )                       ∂               ∂   t       ⁢     (     ρ   ⁢           ⁢   u     )       +     ∇     ·     (     ρ   ⁢           ⁢   uu     )           =       -     ∇   P       +     ∇     ·   τ       +     ρ   ⁢           ⁢   g               (   2   )                 ρ   ⁢           ⁢       C   P     ⁡     (         ∂   T       ∂   t       +     u   ·     ∇   T         )         =       ∇     ·     (     k   ⁢     ∇   T       )         +     τ   :   D               (   3   )               
where ρ represents the density; u represents the velocity vector; t represents the time; τ represents the extra stress tensor (stress distribution); ∇u represents the velocity gradient tensor (velocity gradient distribution); D represents the rate-of-deformation tensor (i.e., symmetric tensor of ∇u); g represents the acceleration vector of gravity; p represents the pressure; C p  represents the specific heat; T represents the temperature; and k represents the thermal conductivity.
 
     For polymer melts, the extra stress tensor T is defined by the generalized Newtonian fluid (GNF) viscous model as follows:
 
τ=2η S ( T,P ,{dot over (γ)}) D   (4)
 
In general, the Cross-William-Landel-Ferry (Cross-WLF) flow curve model has been used in the present disclosure to describe the shear viscosity η S  as a function of the temperature T, pressure P, and strain rate {dot over (γ)} (see, Cross, M. M., “Relation between Viscoelasticity and Shear-Thinning Behaviour in Liquids,” Rheol Acta 18 609-614 (1979), which is incorporated herein by reference in its entirety).
 
     Furthermore, the flow curves of shear viscosity dominate the flow behaviors of a variety of materials. Commonly, the well-known Cross-WLF model used in polymer rheology and processing simulations can describe complex viscosity behaviors, including viscosity varying with shear rate for the Cross model and the zero-shear-rate viscosity, depending on temperature and pressure for the WLF model as follows: 
                       η   S     ⁡     (       γ   .     ,   T   ,   P     )       =         η   0     ⁡     (     T   ,   P     )         1   +       (         η   0     ⁢     γ   .         τ   *       )       1   -   n                   (   5   )                   η   0     ⁡     (     T   ,   P     )       =       D   1     ⁢     exp   ⁡     (       -       A   1     ⁡     (     T   -     T   c       )             A   2     +     (     T   -     T   c       )         )                 (   6   )                 T   c     =       D   2     +       D   3     ⁢   P               (   7   )                 A   2     =         A   ~     2     +       D   3     ⁢   P               (   8   )               
where seven parameters are fit by related experimental data, including n, τ*, A 1 , Ã 2 , D 1 , D 2  and D 3 .
 
     The extensional viscosity is always non-dimensionalized with the shear viscosity. This ratio Tr is called the Trouton ratio: 
     
       
         
           
             
               
                 
                   Tr 
                   = 
                   
                     
                       η 
                       E 
                     
                     
                       η 
                       S 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     For isotropic Newtonian viscosity, the Trouton ratio ideally equals 3, 4, and 6 in the uniaxial, planar, and biaxial extension flows, respectively. In practice, it is difficult to directly model the extensional viscosity via a mathematical relationship (see, Sarkar, D. and M. Gupta, “Further Investigation of the Effect of Elongational Viscosity on Entrance Flow,” J Reinf Plast Compos 20 1473-1484 (2001), which is incorporated herein by reference in its entirety). Therefore, Sarkar and Gupta provided a strain-rate dependence of the Trouton ratio function to determine the extensional viscosity as follows: 
                     Tr   ⁡     (     γ   .     )       =     3   +     δ   ⁡     [     1   -     1       1   +       (     λ   ⁢           ⁢     γ   .       )     2             ]                 (   10   )                   η   E     ⁡     (     γ   .     )       =         η   S     ⁡     (     γ   .     )       ⁢     Tr   ⁡     (     γ   .     )                 (   11   )               
where the parameters δ and λ are fitted by experimental extension viscosity data. This model can complete the extension viscosity curve with low-strain-rate extension thickening.
 
     Based on the previous work of Sarkar and Gupta, Tseng (see, Tseng, H.-C., “A Revisitation of Generalized Newtonian Fluids,” J Rheol 64 493-504 (2020), which is incorporated herein by reference in its entirety) recently proposed a new Trouton ratio expression: 
                     Tr   ⁡     (     γ   .     )       =     3   +       T   0         [     1   +       (       λ   T     ⁢     γ   .       )       -   2         ]       n   T                   (   12   )               
where three parameters of T 0 , λ T , and n T  are fitted by experimental extension viscosity data to describe the significant extension thinning and extension thickening characteristics. Moreover, Tseng also derived the weighted shear/extensional viscosity η W ({dot over (γ)}), called the eXtended GNF (GNF-X) viscous model, as below:
 
                     η   W     =         (     1   -   W     )     ⁢     η   S       +     W   ⁢           ⁢     η   E                 (   13   )                 1   -   W     =         γ   .     S   2         γ   .     T   2               (   14   )               W   =         γ   .     E   2         γ   .     T   2               (   15   )                   γ   .     T   2     =         γ   .     S   2     +       γ   .     E   2               (   16   )               
where W is the weighting function, and also called the extension fraction; η S  and η E  are the GNF shear viscosity and extensional viscosity with respect to strain rates, respectively. {dot over (γ)} S  and {dot over (γ)} E  are the principal shear rate and principal extensional rate, respectively; and {dot over (γ)} T  is the total strain rate. Note that 1−W and W represent the shear-rate and extension-rate percentages, respectively. When W=0, the GNF-X weighted viscosity returns to the GNF shear viscosity. The details of the GNF-X weighted viscosity are available elsewhere (see, Tseng, H.-C., “A Revisitation of Generalized Newtonian Fluids,” J Rheol 64 493-504 (2020), which is incorporated herein by reference in its entirety). Note that the generalized extensional viscosity η E  is respectively given as
 
                 η   E     =       η   UE     3       ,       η   E     =       η   PE     4       ,         
and
 
               η   E     =       η   BE     6           
for uniaxial, planar, and biaxial extensional flows, wherein η UE  is the uniaxial extensional (UE) viscosity; the planar extensional (PE) viscosity η PE ; the biaxial extensional (BE) viscosity η BE .
 
     Based on the GNF-X weighted viscosity, Tseng et al. (see, U.S. Pat. No. 11,027,470, which is incorporated herein by reference in its entirety) have the invention to effectively modify the White-Metzner viscoelastic model below, which can successfully simulate the die swell phenomenon of polymer melts. 
                             W   i     ⁡     (     γ   .     )         γ   .       ⁢       τ   *     ∇       +   τ     =     2   ⁢       η   W     ⁡     (     γ   .     )       ⁢   D             (   17   )                 Wi   ⁡     (     γ   .     )       =       Wi   0         [     1   +       (       γ   .         γ   .     WC       )       -   2         ]       N   W                 (   18   )                   τ   *     ∇     =         D   ⁢           ⁢   τ     Dt     -       ∇   L     ·   τ     -     τ   ·     L   T                 (   19   )                   D   ⁢           ⁢   τ     Dt     =         ∂   τ       ∂   t       +     u   ·     ∇   τ                 (   20   )               L   =       ∇   u     -     ξ   ⁢           ⁢   D               (   21   )                 ξ   ⁡     (     γ   .     )       =       ξ   0         [     1   +       (       γ   .         γ   .     XC       )     2       ]       N   X                 (   22   )               
where Wi({dot over (γ)}) and ξ are the Weissenberg number and slip factor and depend on the strain rate {dot over (γ)}; All of model parameters ξ 0 , {dot over (γ)} XC , N X , Wi 0 , {dot over (γ)} WC , and N W  are fitted by experimental data.
 
     Early, the pioneers of Lipscomb et al. (see, Lipscomb II, G. G., M. M. Denn, D. U. Hur, and D. V. Boger, “The Flow of Fiber Suspensions in Complex Geometries,” J Non-Newtonian Fluid Mech 26 297-325 (1988) and Wang, J., J. F. O&#39;Gara, and C. L. Tucker III, “An Objective Model for Slow Orientation Kinetics in Concentrated Fiber Suspensions: Theory and Rheological Evidence,” J Rheol 52 1179-1200 (2008), which is incorporated herein by reference in its entirety) developed a constitutive equation of dilute fiber suspensions in which the effect of fiber orientation on flow stress can be modeled:
 
τ=2η m   D+ 2η m   ϕN   p   D:A   4   (23)
 
where τ is the extra stress tensor; D is the rate-of-deformation tensor; η m  is the matrix viscosity; ϕ is the fiber volume fraction; N p  is a dimensionless parameter. The fourth-order orientation tensor of fibers A4 is determined by using a higher order polynomial closure approximation in terms of the second-order orientation tensor A.
 
From the above equation, the first term is the viscous contribution of a polymer matrix, and the second term is the effect of fiber orientation on the stress tensor (stress distribution) of the fluid. Therefore, the present disclosure can assume the effect of molecular orientation on stress tensor for neat polymer matrix as follows:
 
τ A =2η W   N   p   D:A   4   (24)
 
     Furthermore, the modified WM viscoelastic model with molecular orientation can be extended as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
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     This can be known as the extended WM viscoelastic model. Finally, the extended WM viscoelastic model is integrated into the Moldex3D, as shown in  FIG. 8 . 
     In the present disclosure, a planar contraction flow simulation was performed via the Moldex3D, incorporating the extended WM model with the molecular orientation effect. Two materials of interest are the branched LDPE and the linear HDPE. The contraction flow is given at the isothermal temperature 150° C. and the constant flow rate of 150 mm 3 /s (or the apparent shear rate=122 s −1 ). As a result, it is obvious that LDPE&#39;s vortex size is large than HDPE&#39;s. This is in good agreement with the experimental observation. Therefore, the present invention is verified to be an effective method for the effect of molecular orientation on viscoelastic fluid behaviors. 
     In addition,  FIG. 7  shows the transient shear viscosity for 30 wt % SGF/PBT (polybutylene terephthalate resin containing 30% by weight of short glass fibers) composite at shear rate {dot over (γ)}=1.0 s −1  and 260° C. The suspended WMT-X model predictions are in good agreement with experimental data, especially for the viscosity overshoot (see, Eberle, A. P. R., G. M. Vélez-Garcia, D. G. Baird, and P. Wapperom, “Fiber Orientation Kinetics of a Concentrated Short Glass Fiber Suspension in Startup of Simple Shear Flow,” J Non-Newtonian Fluid Mech 165 110-119 (2010), which is incorporated herein by reference in its entirety). 
       FIG. 8  is a flowchart showing an injection-molding simulation operation in accordance with some embodiments of the present disclosure. As shown in  FIG. 8 , the CAE simulation software can calculate the stress tensor τ by applying the modified White-Metzner constitutive equation for the subsequent die swell effect analysis. In addition, after calculating the stress tensor τ, if the numerical calculation results are not convergent, the fiber parameters or the molding conditions may be adjusted, and another simulation can be performed to obtain updated calculation results. By applying the modified White-Metzner constitutive equation, the die swell effect of the molding resin can be effectively simulated. 
     In some embodiments, the simulation operation may be repeated with different boundary conditions to obtain a feasible molding condition (e.g., a screw speed with a flow rate for transferring desired amount of molding material into the mold cavity during the very short injection phase); then, in the actual molding step, the controller operates the molding machine (controlling the driving motor  230  to move the screw  220  at a predetermined speed) with the feasible molding condition for transferring the fluid molding material into the mold cavity to perform an actual molding process for preparing the molding article. If the quality of the prepared molding article is different from the designed, the simulation operation may be further repeated with different boundary conditions to obtain another feasible molding condition (e.g., an updated screw speed with an updated flow rate for transferring desired amount of molding material into the mold cavity) in the subsequent actual molding step. 
     With continuing reference to  FIG. 8 , in injection-molding simulation operations, the governing equations of fluid mechanics that describe the transient flow behaviors are as follows: 
                         ∂   ρ       ∂   t       +       ∇     ·   ρ       ⁢           ⁢   u       =   0           (   28   )                     ∂     ∂   t       ⁢     (     ρ   ⁢           ⁢   u     )       +     ∇     ·     (     ρ   ⁢           ⁢   uu     )           =       -     ∇   P       +     ∇     ·   τ       +     ρ   ⁢           ⁢   g               (   29   )                   pC   p     ⁡     (         ∂   T       ∂   t       +     u   ·     ∇   T         )       =       ∇     ·     (     k   ⁢     ∇   T       )         +     τ   ⁢     :      ⁢   D               (   30   )                       Wi   ⁡     (     γ   .     )         γ   .       ⁢       τ   *     ∇       +   τ     =         2   ⁢       η   w     ⁡     (     γ   .     )       ⁢   D     +     τ   A       =           η   W     ⁡     (     γ   .     )       ⁢   D     +     2   ⁢     η   W     ⁢     N   p     ⁢   D   ⁢     :      ⁢     A   4                   (   31   )               
where ρ represents the density; u represents the velocity vector; t represents the time; τ represents the extra stress tensor; ∇u represents the velocity gradient tensor (velocity gradient distribution); g represents the acceleration vector of gravity; p represents the pressure; C p  represents the specific heat; T represents the temperature; k represents the thermal conductivity; τ ∇ * represents the effective upper convected time derivative of the extra stress tensor; Wi({dot over (γ)}) represents the Weissenberg number; η W ({dot over (γ)}) represents the weighted viscosity of the molding resin; D represents a rate of deformation of the molding resin; and {dot over (γ)} represents the shear rate of the molding resin; τ A  represents the molecular orientation effect on the stress distribution, N p  represents a dimensionless parameter, and A 4  represents a molecular orientation distribution of the molding material.
 
     Solving the governing equations (28)-(31) requires a transient state analysis, which can be performed numerically using a computer (See, for example, Rong-Yeu Chang and Wen-hsien Yang, “Numerical simulation of mold filling in injection molding using a three-dimensional finite volume approach,” International Journal for Numerical Methods in Fluids Volume 37, Issue 2, pages 125-148, Sep. 30, 2001; the entirety of the above-mentioned publication is hereby incorporated by reference herein and made a part of this specification, which is incorporated herein by reference in its entirety). During the transient state analysis, the process variables that change with time are not zero; i.e., the partial derivatives ∂/∂t in the governing equations (28)-(31) are not considered zero. 
     The true 3D Finite Volume Method (FVM) is employed due to its robustness and efficiency to solve the transient flow fields in a complex 3D geometrical article. In some embodiments of the present disclosure, the simulation flow in  FIG. 8  can be implemented using a commercial injection-molding simulation software, Moldex3D (CoreTech System Co. of Taiwan), to facilitate the orientation predictions of the fiberless molding resin. 
     In some embodiments, the simulating process may be repeated with different boundary conditions to obtain a feasible molding condition (e.g., a screw speed with a flow rate for transferring desired amount of molding material into the mold cavity); then, in the actual molding step, the controller operates the molding machine (control the driving motor  230  to move the screw  220  at a predetermined screw speed) with the feasible molding condition for transferring the fluid molding material into the mold cavity to perform an actual molding process for preparing the molding article. If the size of the prepared molding article is different from the simulating result, the simulating process may be further repeated with different boundary conditions to obtain another feasible molding condition (e.g., an updated screw speed with an updated flow rate for transferring desired amount of molding material into the mold cavity) in the subsequent actual molding step. 
       FIG. 9  is a schematic view of an injection-molding apparatus  10  in accordance with some embodiments of the present disclosure. Referring to  FIG. 9 , the injection-molding apparatus  10  that can be used to carry out molding includes a molding machine  20 , a mold  30 , a clamping assembly  40  and a computer  50 . The molding machine  20  includes a barrel  210  having a downstream end  212  connected to the mold  30 . The mold  30  includes mold halves  310  and  320  to define a mold cavity  330  and a runner  340  in communication with the mold cavity  330 . 
     The clamping assembly  40  is in operative connection with the mold  30  for clamping the mold halves  310  and  320 . In some embodiments, the clamping assembly  40  includes a fixed plate  410 , a plurality of tie bars  420  mounted on the fixed plate  410 , and a moving plate  430  slidably engaged with the tie bars  420  and guided by a is driving cylinder  440 . The mold half  310  proximal to the barrel  210  is secured on the fixed plate  410 , and the mold half  320  distal to the barrel  210  is secured on the moving plate  430  in any suitable manner, wherein the driving cylinder  440  drives the moving plate  430  to open or close the mold  30 . In some embodiments, the barrel  210  includes a nozzle  2102  adapted to engage a sprue  450  in the fixed plate  410 . In some embodiments, the sprue  450  is in communication with the runner  340  as the mold half  310  is assembled with the fixed plate  410 . In some embodiments, the fixed plate  410  may be equipped with a sprue bush  452  including the sprue  450  and receiving the nozzle  2102  during an injection time. A molding material  100  under pressure is delivered to the sprue bush  452  from the nozzle  2102  pressed tightly against the sprue bush  452  in order to deliver the molding material  100  to the sprue  450  during a filling stage of the injection time. In some embodiments, the molding material  100  may be a molding resin such as polystyrene resin, for example. 
     In some embodiments, the clamping assembly  40  further includes an ejector plate  460  mounted with at least one ejector pin (not shown), wherein the moving plate  430  is disposed between the fixed plate  410  and the ejector plate  460 . In some embodiments, the ejector plate  460  is fixed on the tie bar  420 . In some embodiments, the driving cylinder  440  penetrates the ejector plate  460  and directly connects to the moving plate  430  to open or close the mold  30 . After the mold halves  310  and  320  are separated (i.e., the mold  30  is opened), a distance between the moving plate  430  and the ejector plate  460  is reduced, so the ejector pin can penetrate through the ejector plate  460  to push a molded product out of the mold  30 . 
     A screw  220  is mounted for moving within the barrel and is operably connected, at an upstream end  214  opposite to the downstream end  212  of the barrel  210 , to a driving motor  230 . The molding machine  20  processes material, such as plastic granules  102 , by feeding the material through a hopper  240  to the barrel  210  in order to make the material soft and force the molding material  100  into the mold  30  by the use of the screw  220 , wherein the plastic granules  102  change phase from solid to liquid by at least one heater band  250  surrounding the barrel  210 . In some embodiments, the molding machine  20  further includes a check valve  260  mounted on the screw  220 , wherein the check valve  260  is in tight contact with the barrel  210  during the filling stage, and the check valve  260  is open for allowing the liquid material to flow to the downstream end  212  of the barrel  210  during a packing stage. In some embodiments, if the mold cavity  330  is almost filled with the molding material  100 , a packing process proceeds. In some embodiments, the screw  220  rotates and moves toward the upstream end  214  of the barrel  210  during the packing stage. 
     The molding machine  20  further includes a controller  270  for controlling and monitoring the real-time functions of the molding machine  20 , and a display  280  for displaying data related to the performance and operation of the molding machine  20  to on-site technicians. In some embodiments, the display  280  is further configured to accept input data from the on-site technicians. In other words, the display  280  is provided with a communications link directly with the controller  270  to provide real-time control of the molding to machine  20  by the on-site technicians particularly where the on-site technicians&#39; intervention is required. 
     In some embodiments, the molding machine  20  can further include operation interface communication links among the controller  270 , the display  280  and peripheral devices, and a program sequence of operation which allows the operation interface to monitor diagnostic functions of the controller  270  and the molding machine  20 , trigger sound and/or light alarms regarding conditions of the molding machine  20 , receive performance data from the molding machine  20 , and receive input data from the display  280 . 
     The computer  50  is associated with the molding machine  20  and is configured to execute CAE simulation software and transmit at least one simulation result to the controller  270  through a connection such as a hard wire connection or a wireless coupling. In some embodiments, the computer  50  includes a standardized operating system capable of running general-purpose application software for assisting with the analysis and process of the performance data and for communicating with the controller  270  and the display  280  via communication ports of each. 
       FIG. 10  is a functional block diagram of the computer  50  in  FIG. 9 . Referring to  FIG. 10 , the computer  50  includes a processing module  510  such as a processor adapted to perform a computer-implemented simulation method for use in injection molding, an input/output (I/O) interface  520  electrically coupled to the processing module  510 , and memories, which may include a read-only memory (ROM)  530 , a random access memory (RAM)  540  and a storage device  550 . The ROM  530 , the RAM  540  and the storage device  550  are to operably communicate with the processing module  510 . 
     The computer  50  further includes a communication port  560  associated with the controller  270  of the molding machine  20 . The computer  50  may further include one or more accompanying input/output devices including a display  570 , a keyboard  580  and one or more other input devices  590 . The input devices  590  may include a card reader, an optical disk drive or any other device that allows the computer  50  to receive input from the on-site technicians. In some embodiments, the input devices  590  are configured to input computer instructions (software algorithms) stored in a non-transitory computer-readable medium  500 , and the processing module  510  is configured to execute operations for performing a computer-implemented injection-molding simulation method according to the computer instructions. In some embodiments, the processing module  510  reads software algorithms from the other input device  590  or the storage device  550 , executes the calculation steps, and stores the calculated result in the RAM  540 . 
     The present disclosure provides a molding system for preparing a molded article, comprising a molding machine, including a barrel, a screw mounted for moving within the barrel, a driving motor driving the screw to move a molding resin; a mold disposed on the molding machine and connected to the barrel of the molding machine to receive the molding resin, and having a mold cavity with a die swell structure for being filled with the molding resin; a processing module simulating a filling process of the molding resin from the barrel into the molding cavity based on a molding condition including a predetermined screw speed for the molding machine, wherein simulating the filling process of the molding resin is performed taking into consideration of a die swell effect of the molding resin; and a controller operably communicating with the processing module to receive the molding conditions and with the molding machine to control the driving motor of the molding machine based on the molding conditions to move the screw at the predetermined screw speed to transfer the molding resin at a corresponding flow rate to perform an actual molding process for preparing the molded article. 
     Although the present disclosure and its advantages have been described in detail, it should be understood that various changes, substitutions and alterations can be made herein without departing from the spirit and scope of the disclosure as defined by the appended claims. For example, many of the processes discussed above can be implemented in different methodologies and replaced by other processes, or a combination thereof. 
     Moreover, the scope of the present application is not intended to be limited to the particular embodiments of the process, machine, manufacture, composition of matter, means, methods and steps described in the specification. As one of ordinary skill in the art will readily appreciate from the present disclosure, processes, machines, manufacture, compositions of matter, means, methods, or steps, presently existing or later to be developed, that perform substantially the same function or achieve substantially the same result as the corresponding embodiments described herein may be utilized according to the present disclosure. Accordingly, the appended claims are intended to include within their scope such processes, machines, manufacture, compositions of matter, means, methods, and steps.