Three-dimensional simulation of primary and secondary penetration in a clip-shaped square tube during a gas-assisted injection molding process.
Injection molding, due to its capability to efficiently produce complex plastic parts for various industrial applications, has become one of the most popular manufacturing techniques for mass production purpose since the first engineering plastic was invented 100 years ago. However, the intricate rheological properties of polymer material generally complicate the adequate choice of operation parameters during molding processes and prevent engineers from optimizing their production processes in a straight-forward way. Because the main complexity of melt flow is due to its rheology, which basically depends on flow characteristics, e.g. temperature, pressure, shear rate etc., the melt flow problem therefore has a highly nonlinear nature, which generally inhibits practical problems from being analytically studied. To meet the need of production effectiveness and geometry precision, the development of simulation methodology based on numerical models has being pursued by numerous researchers over the last three decades. Even with numerical approaches, an extensive analysis of practical melt flow problems with required predictions accuracy and affordable computational efforts becomes possible only after the simplifications of governing equations under reasonable assumptions have been adopted. The most successful model in the past is known as the Hele-Shaw flow formulation, which is capable of describing the polymer melt flow in thin cavities during conventional injection moldings with good prediction accuracy. Many numerical methods and extensions based on this two-and-half dimensional formulation are developed using different discretization schemes, which have been applied to study common injection molding problems, such as melt filling, melt packing, mold cooling, as well as to simulate special molding processes, e.g. co-injection, gas-assisted injection and microchip encapsulation. Despite the success of Hele-Shaw models in some injection molding applications, fully three-dimensional models are further developed to overcome the inherent creeping-flow and thin-wall assumptions of this mid-plane model, which are obviously not valid for fountain flow regions, branching flows within the runner and cavity systems, flows with dominant inertia and gravitational effects and thick/complex geometries with micro features or inserts (1). Several three-dimensional models (2-11) have been developed to simulate a number of melt flow problems with notable successes. These three-dimensional simulations have been conducted to demonstrate their ability to reproduce important three-dimensional features of melt flows with nontrivial progresses. Although a number of published literatures have illustrated the successful applications of three-dimensional models to a number of injection molding problems, such as thick parts, jetting phenomenon, and asymmetric gas penetration, the development of simulation techniques for injection molding problems is still far from complete. For example, the gas and melt compressibility arising in gas-assisted injection molding (GAIM) processes are still not well modeled in aforementioned three-dimensional studies, which results in the well-known primary and secondary penetration phenomenon.
THREE-DIMENSIONAL SIMULATION OF GAIM
The GAIM process has received wide popularity due to several advantages over traditional injection molding processes, such as lower injection pressure required, less material involved, and smaller warpage and shrinkage achieved. Those features are particularly favorable for developing environmentally friendly production techniques, where less energy and materials are consumed. Figure 1 gives a schematic illustration of gas penetration phenomenon in a GAIM process inside tubular parts, where R, S, and W represent the gas core width, skin thickness and tube width, respectively. As elucidated by an early study (12), the gas penetration is divided into the primary and secondary gas penetrations, where the secondary gas penetration is found apparently to have a smaller gas core size than that of the primary one, Fig. 2. During the GAIM processes, the primary gas penetration occurs in the filling phase due to the gas inertia with injected gas pressure in an order of [10.sup.6] Pa (or higher). In this stage, the compressibility of polymer melt is quite limited but the gas compressibility plays a substantial role in the gas penetration process. The secondary gas penetration takes place in the packing and cooling phase of GAIM processes, where the melt compressibility mainly accounts for an additional but nontrivial further penetration. The three-dimensional modeling proposed by Khayat et al. (2) is an early example of attempts devoted to the GAIM simulation conducted on simple geometry, where a Newtonian flow problem neglecting heat transfer effect was solved by a boundary element method. Johnson et al. (4) represented an isothermal and Newtonian flow calculation incorporated with a pseudo-concentration approach to tracking the gas front during GAIM processes for a "dog-bone" shaped tensile specimen. The numerical model given by Haagh et al. (5) demonstrates a typical three-dimensional simulation of GAIM processes for generalized Newtonian fluids using finite element methods, where an upwind Petrov-Galerkin scheme was employed to solve the Stokes-based governing equation set and a pseudo-concentration method was additionally utilized to predict the gas front location. Their numerical predictions for an axis-symmetric cylinder and a plaque with rib were favorably consistent with experimental measurements in the isothermal case, but their nonisothermal calculation only gave a qualitative agreement. The work contributed by Ilinca and Hetu (7) was another example of finite element methods to the simulation of GAIM processes, where a streamline upwind Petrov-Galerkin scheme (SUPG) was adopted. Similar to (5), the incompressible Stokes equation, instead of the complete momentum equation, was solved in the numerical simulation, and again a pseudo-concentration method was used to capture the interfacial fronts. In contrast to (5), two front equations were employed to describe the polymer/air and polymer/air interfaces in their study. After validating their method with isothermal filling of a channel with a Newtonian fluid, they gave a comprehensive numerical study on the penetration characteristics of a thin plate under different operation conditions. They further demonstrated the application of this algorithm to GAIM process in a thick part and coinjection molding process in a rectangular plate (8), Similar to the methodology illustrated in (5), (7), Polynkin et al. (9) implemented another SUPV finite-element code to simulate the three-dimensional GAIM process in a tubular handle, where only one fictitious concentration was solved to track the gas/melt interface. Different from (5), (8), the time-dependent heat coefficient at cavity wall employed in the simulation was determined in a separate analysis of transit heat conduction problem among the melt, the mold tool, and the circulating coolant, where a dynamically updated permeable wall was adopted for the gas but not for the polymer. In their study only qualitative comparison with corresponding measurements was presented. Adopting the same numerical methodology accompanied by corresponding experiments, Polynkin et al. (10) further investigated the primary and secondary gas penetration in a rectangular channel, where a modified continuity equation with source terms was employed. Because of relatively low gas pressures used in their study, the gas density was taken to be constant and the thermal properties and density of melt was assumed to be temperature dependent, while the disclosure of the latter were not explicitly given in the literature. Four different cases were successfully simulated using relatively fine tet-rahedral elements, where the element number was ranging form 0.5 to 1.2 million, where the primary and secondary gas penetrations were qualitatively compared. They also reported several numerical strategies to overcome difficulties encountered in their numerical scheme, such as the handling of source terms occurring in the continuity equation and the improvement of iterative equation solver to reduce the required computational cost. In contrast to the previous study, a fully three-dimensional model based on a finite volume discretization proposed by Chau and Lin (11) was demonstrated to simulate melt filling and gas penetration in a simple and compact way, where a volume-of-fluid method was employed to capture the gas/melt and air/melt interfaces. Without the negligence of the nonlinear convection term in the momentum equation, three-dimensionally nonlinear phenomenon, such as asymmetrical gas penetration and jetting, were successfully reproduced. A quantitative comparison of gas core size in a rectangular plate with half cylinder between numerical results and experimental measurements showed good agreement only for the major region of the primary penetration. Besides, the total gas penetration length was under-predicted mainly due to ignoring the gas and melt compressibility during the GAIM process.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
After surveying published literatures related to three-dimensional modeling of GAIM processes, all except one (10) of these studies focus only on the primary penetration, and none of them has taken the compressibility of injected gas and filling melt during the molding process into account. As indicated by recent studies (10), (11), without adequate modeling of the gas and melt compressibility in the GAIM processes, the overall gas penetration is prone to be substantially underestimated and the predicted penetration behavior of gas in melt could be qualitatively incorrect. Although a previous study (10) has already shown some progresses of three-dimensional models in predicting the secondary penetration in GAIM processes, the full modeling of gas and melt compressibility is still not extensively disclosed. This article proposes a generalized Newtonian model to predict the three-dimensional gas penetration phenomenon in the GAIM process, where the gas and melt compressibility are both taken into account. Thus, both the primary and secondary penetrations in GAIM processes are able to be quantitatively predicted. Additionally, an incompressible model requiring no outflow boundary is also presented to emphasis the influence of gas compressibility on the primary penetration. Based on a finite volume discretization, the proposed numerical model solves the complete momentum equation with two front transport equations, where the nonlinear convection term is not neglected. Two front transport equations are employed to track the gas/melt and air/melt interfaces. The modified Cross-WLF model is adopted to describe the melt rheological behavior. The two-domain modified Tait equation is exploited to represent the melt compressibility, while a polytropic model is used to describe the coupling among pressure, temperature, and density for the injected gas and air. The proposed schemes are first quantitatively validated by the penetration length and gas core distribution of the gas penetration process in a clip-shaped square tube. Then the influences of five major molding parameters, such as the injection pressure, mold temperature, melt temperature, delay time, and melt material on the gas penetration characteristics in the same clip-shaped square tube via the proposed numerical approach are extensively presented and discussed.
FULLY THREE-DIMENSIONAL FLOW MODEL
Basic Governing Equations
First, the basic governing equations for the incompressible model proposed by the author in a recent work (11) are addressed, which give the starting point of the proposed compressible model. The first two governing equations employed in the fully three-dimensional flow model to simulate the gas penetration phenomenon in injection molding processes follows the conservations of mass and momentum, which are expressed in the different form as follows:
[[[partial derivative][rho]]/[[partial derivative]t]] + [nabla]*([rho]u) = 0 (1)
[[[partial derivative]([rho]u)]/[[partial derivative]t]] + [nabla]*([rho]uu) = [nabla]*T + [rho]g (2)
where, [rho], t, u, T, and g denote the density, time, velocity velocity vector, stress tensor, and gravitaional acceleration, respectively. The stress tensor T and rate of strain tensor D are then given as follows:
T = 2[eta]D - [2/3][eta]([nabla]*u)I - pI (3)
D = [1/2]([nabla]u + [nabla][u.sup.T]) (4)
where [eta], I, and [rho] denote the viscosity, unit tensor, and pressure, respectively. The third governing equation adopted in the fully three-dimensional model follows the conservations of energy, which is expressed in the differential form as follows:
[[[partial derivative]([rho]h)]/[[partial derivative]t]] + [nabla]*([rho]uh) = [nabla]*(k[nabla]T) + 2[eta]D:D (5)
where h, k, and T denote the enthalpy, thermal conductivity, and temperature, respectively. The additional two governing used in the fully three-dimensional model follows the conservations of species, assuming all fluids are incompressible, which are expressed in the differential form as follows
[[[partial derivative][c.sub.a]]/[[partial derivative]t]] +][nabla]*([c.sub.a]u) = 0 (6)
[[[partial derivative][c.sub.g]]/[[partial derivative]t]] + [nabla]*([c.sub.g]u) = 0 (7)
Equations 6 and 7 describe the transport phenomenon of the air/melt and the gas/melt interfaces, respectively. In this model, melt is chosen as the background component in both equations, while air and gas are separately employed as the foreground components in these two equations. The value of volume fractions [C.sub.a] and [C.sub.g], defined as the ratio of volume in cell occupied by the foreground species to the total cell volume, comply with the following convection: one for cells fully filled by the foreground component, zero for cells without any fore-ground component, and the value between zero and one for cells partially filled by the foreground component. As a consequence, the volume fraction of melt [C.sub.m] is then implicitly defined by 1-[C.sub.a]-[C.sub.g] to fulfill the conservation of melt in a cell. Because more than one fluid, i.e. air, gas, and melt, are involved in GAIM processes and for the sake of simplicity, they are treated as a single effective fluid in the proposed approach to calculate the velocity and pressure. In this approach, the physical properties of a cell vary in space according to the volume fraction of fluids defined in that cell. Fluid properties [[gamma].sub.eff], such as density and viscosity, are determined by a linear interpolation according to the volume fraction in cell as follows:
[[lambda].sub.eff] = [c.sub.a][[lambda].sub.a] + [c.sub.g][[lambda].sub.g] + [c.sub.m][[lambda].sub.m] (8)
where [[gamma].sub.a], [[gamma].sub.g] and [[gamma].sub.m] denote the fluid property of air, gas and polymer melt, respectively. Different from those schemes treating interfaces explicitly, the melt front is implicitly defined in this approach at the location, where its volume fraction is equal to 0.5. In the order to describe the rheological property of polymer melt, the modified Cross-WLF model is adopted:
[eta](T,[gamma]) = [[[[eta].sub.0]/[1 + [([[[eta].sub.0][gamma]]/[tau*]).sup.1 - n]]]] (9)
[[eta].sub.0](T) = [[eta].sub.g] exp [ - [[C.sub.1.sup.g](T - [T.sub.g])]/[[C.sub.2.sup.g] + T - [T.sub.g]]] (10)
where [gamma] denote the shear rate, [[eta]sub.0] the viscosity of zero shear rate, n, [tau*], [T.sub.g], [[eta].sub.g], [C.sub.1.sup.g], [C.sub.2.sup.g], are parameters of material properties.
Boundary and Initial Conditions
In the filling phase the flow rate, as well as the temperature, is specified at the melt gate, while a zero-gradient assumption is adopted at the prescribed outlet boundaries. The cavity wall is assumed at a given mold temperature, where the no-slip condition is applied for all fluids at the cavity wall. During the gas injection and packing process, including the cooling stage, a prescribed pressure and temperature is specified for the injected gas at the gas inlet, while the melt gate is treated as cavity wall. A zero-gradient boundary condition, instead of the boundary condition specifying a prescribed value, is applied to the calculation of volume fraction at the cavity wall, which ensures a smooth front transport up to solid wall. The initial condition of the air initially residing inside cavity is a zero velocity field with mold temperature.
Incompressible Model Without Outflow Boundary
In the Hele-Shaw flow approximation, the continuity and momentum equations for polymer melt flow is simplified and combined into a single Poisson-type equation in terms of the pressure and fluidity. From this equation transformation, pressure replaces velocity as the principal variable in the merged governing equation, which requires adequate boundary conditions to let the governing equations become well posed. Therefore, only pressure must be specified on boundaries. Outflow boundary specifying mass flux is not required as boundary condition. The mass flux at specific location can be determined in terms of pressure, after the pressure filed has been computed. In the fully three-dimensional models, the nonlinear convection term prevents the direct combination of governing equations, and the velocity and pressure must be decoupled and solved iteratively. With fully three-dimensional approaches, the mass flux at both inflow and outflow is required for Simple-type algorithm (13), as employed in this numerical model, since global mass conservation should be achieved through the mass flux balance on outer boundaries in solving the continuity equation. Otherwise, the Simple-type scheme will fail to give any converged pressure field. For the inflow regions, it is easy to estimate the melt flux through the screw speed and size. However, the outflow boundary condition is quite ambiguous in the traditional injection molding processes. In practices, there is no apparent opening on mold, but small, gaps between mold parts serve as air vents, where air can freely escape from cavity during the molding processes. In the numerical calculation, it is very difficult to really simulate these gaps with their actual sizes. Previous studies (7), (11) favored to specify artificial outflow boundaries in order to simulate the effects of gaps. This approach could provide reasonable approximation, provided that there is a dominant flow direction of melt (7), (11), where a zero-gradient outflow boundary condition is often employed. Nevertheless, for some melt filling problems with more than one or even without any principal flow direction, this simple strategy would fail to function, where the melt could nonphysically escape from the cavity. Another approach to overcome this problem is to use a dynamically updated permeable wall, which allows air to escape but not the melt. As reported by (9), special care should be taken in order to satisfy the global mass conservation, and the work to activate the preamble wall may complicate the numerical algorithm and implementation of code. To overcome these difficulties without bringing new complexity, we propose a simple incompressible model, which could deal with melt filling problems without specifying any outflow boundaries (or flux). This would be very advantageous for problems, where the outflow regimes cannot be easily or explicitly identified or more than one major flow direction occurs, such as the melt filling of parts with several branches. Without specifying outflow flux, we must add an extra source term in the continuity equation, which serve as an artificial sink in the cavity in order to balance the inflow mass flux:
[[[partial derivative][rho]]/[[partial derivative].sub.t]] + [napla]* ([rho]u) = [rho][F.sub.a](x) (11)
where [F.sub.a](x) denotes the additional sink function for the continuity equation, and x represents the space vector. For the purpose of maintaining the global mass conservation, the sink function [F.sub.a](x) must satisfy the following relation:
[[integral].sub.inlet][rho](u*n)dS = [[integral].sub.[OMEGA]][rho][F.sub.a](x)dV (12)
where [OMEGA] denotes the whole volume of cavity to be simulated. Because the choice of the sink function [F.sub.a](x) could affect the front sharpness during the simulation process, the sink function should have a similar nature of the numerical scheme applied to determine the convective flux in the species equation. In this article, for any control volume in space the sink function [F.sub.a](x) is expressed as follows:
[F.sub.a](x) = s(1 - [c.sub.a]) (13)
where s denotes a scaling constant determined by the Eq. 12 in each time step. Similarly, the sink function [F.sub.a](x) should be also applied to the air/melt front equation, because the mass conservation of air must be also satisfied:
[[[partial derivative][c.sub.a]]/[[partial derivative]t]] + u*[nabla][c.sub.a] = [F.sub.a](x) (14)
In this incompressible model, the Eqs. 11 and 14 are solved, instead of the original governing equations, i.e. Eqs. 1 and 6. In this way, no outflow boundary condition is required, since the inflow flux can be balanced by the additional sources added in the corresponding governing equations, where the mass conservation principle is not violated.
In the filling phase, the pressure inside the cavity is not fully developed, and therefore the melt compressibility is significantly smaller than that of air and injected gas. The gas inertia and compressibility plays a dominant role in creating the primary penetration. Hence, the melt compressibility is neglected in the filling phase. The modified continuity and front transport equations, which derive from the conservation of mass and species, employed in the filling phase are expressed as follows:
[nabla]*u = [S.sub.a] + [S.sub.g] (15)
[[[partial derivative][c.sub.a]]/[[partial derivative]t]] + [nabla]*([c.sub.a]u) = [S.sub.a] (16)
[[[partial derivative][c.sub.g]]/[[partial derivative]t]] + [nabla]*([c.sub.g]u) = [S.sub.g] (17)
In contrast to the filling phase, the pressure is fully developed during the packing and cooling phase, where the melt compressibility, mainly accounting for the secondary penetration, becomes the most dominant factor influencing the further penetration. Consequently, the air and gas compressibility is ignored. The modified continuity and front transport equations, which derive from the conservation of mass and species, employed in the packing phase are expressed as follows:
[nabla][miid dot]u = [S.sub.m] (18)
[[[partial derivative][c.sub.a]]/[[partial derivative]t]][nabla]*([c.sub.a]u) = - [S.sub.m] (19)
[[[partial derivative][c.sub.g]]/[[partial derivative]t]][nabla]*([c.sub.g]u) = - [S.sub.m] (20)
The source terms, [S.sub.a], [S.sub.g], and [S.sub.m], appearing on the right-hand side of the modified continuity and front transport equations account for the compressibility of air, gas, and melt, respectively, which are expressed as follows:
[S.sub.i] = - [[c.sub.i]/[[rho].sub.i]]([[partial derivative][[rho].sub.i]]/[[partial derivative]t] + u* [nabla][[rho].sub.i]), i = a,g, and m. (21)
where the subscripts a, g, and m stand for the physical property of air, gas, and melt, respectively. The correlation among pressure, temperature, and density for air and gas is assumed following a polytropic relationship:
P = [[rho].sub.i.sup.m]RT,i = a and g (22)
where R denotes the universal gas constant and m represents the polytropic constant. Because air could partially escape from the cavity but not the injected gas during the GAIM process, they should have different m values to give better agreement with experimental observations. If the melt compressibility is considered, the melt density [[rho].sub.m] is then modeled by the two-domain modified Tait equation as follows:
[[rho].sub.m] = 1/[v.sub.0](T)[1 - [C.sub.0]In(1 + [p/B(T)])] + [v.sub.t](p,T) (23)
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where [T.sub.1] denotes the transition temperature, [C.sub.0], [[beta].sub.1], [[beta].sub.2], [[beta].sub.3], [[beta].sub.4], [[beta].sub.5], [[beta].sub.6], [[beta].sub.7], [[beta].sub.8], and [[beta].sub.9], are parameters of material properties. During the filling calculation, [S.sub.m] is neglected in the modified governing equations due to its magnitude much smaller than [S.sub.a] and [S.sub.g]. In contrast, only [S.sub.m] is considered in the packing calculation, since it becomes the most dominant component among [S.sub.a], [S.sub.g], and [S.sub.m]. This measure could substantially reduce the implementation work and numerical effort in conducting a simulation using the proposed compressible model without losing required prediction accuracy. Following the proposed approach, a full consideration of air, gas, and melt in each phase of injection molding processes is quite straightforward. Similar to the incompressible model without any outflow boundary, only velocity (or pressure) and temperature are required at the melt gate (or gas inlet), where no explicit outflow boundary should be specified.
The differential form of governing equations is further expressed in the integral form over a control volume V as follows:
[[partial derivative]/[[partial derivative]t]][[integral].sub.V][rho][empty set]dV + [[integral].sub.s][rho][empty set](u*n)dS = [[integral].sub.s][GAMMA]([nabla][empty set]*n)ds + [[integral].sub.V][S.sub.[empty set]]dV (24)
where [empty set] [GAMMA], and [S.sub.[empty set]] denote the generic function, the generic diffusion constant and the corresponding source term, respectively, and the control volume V is bounded by a closed surface S with the outward normal vector n. All governing equations, i.e. the continuity equation, momentum equation, energy equation and species equation are all solved using a finite volume discretization (14). All vectors quantities, e.g. position vector, velocity and moment of momentum, are expressed in Cartesian coordinates. Nonstaggered variable arrangement is used to define dependent variables: all physical quantities are stored and computed at cell centers. An interpolation practice of second order accuracy is adopted to calculate the physical quantities at cell-face center [(15) A first-order upwind scheme is used to compute the convection term appearing in the governing equations, except those of the species equations. Following the Picard iteration method, the generalized convective flux [C.sub.f] at cell face f is expressed in terms of the mass flux [[dot]m.sub.f] across the cell] face as follows:
[C.sub.f] = [[integral].sub.s][rho][empty set](u*n)dS[approximately equal to][[m.sub.f][[empty set].sub.f] (25)
Where [empty set] denotes the generalized field variable. The generalized diffusive flux [D.sub.f] at cell face f is expressed as follows:
[D.sub.f] = [[integral].sub.s][GAMMA]([nabla][empty set]*n)dS[approximately equal to][[GAMMA].sub.f][nabla][[empty set].sub.f][S.sub.f] (26)
where [nabla][empty set] needed for approximating the diffusion term follows the second-order approximation demonstrated in (15). The volume, integral [Q.sub.P] accounting for the source term appearing in the governing equations is approximated using the mean value theorem as follows:
[Q.sub.P] = [[integral].sub.v][S.sub.[empty set]dV[approximately equal to]([S.sub.[empty set]).sub.p][V.sub.p] (27)
where ([S.sub.[empty set])P denotes the source term evaluated at node P and [V.sub.P] represents the control volume belonging to node P. The source terms are first expressed in terms of field dependent variables, and then an Euler implicit scheme is employed to approximate these source terms appearing in the governing equations. Correspondingly, implicit Euler scheme is adopted to approximate the time derivative of filed variable [empty set] at node P by integrating the derivative during a time interval [DELTA]t:
[B.sub.P] = [[partial derivative]/[[partial derivative]t][[integral].sub.v][rho][empty set]dV[approximately equal to][[V.sub.P]/[[DELTA].sub.t]]([[([rho][empty set])].sub.P.sup.[n + 1]] - [[([rho][empty set])].sub.P.sup.n]) (28)
where the superscript n and n + 1 denote two consecutive time levels. Further, the fluids sharing a cell are assumed to have the identical velocity, pressure and other field properties. The grid employed to discretize the computation domain extends over all fluids, and it remains fixed during the numerical calculation for the sake of simplicity. The correct prediction of interface sharpness requires an accurate approximation of the convection term in the front transport equations. As discussed in (16). convectional interpolation schemes could bring excessive numerical diffusion, which lead to strong smearing of interface. Enlightened by the work of (17), the two-stage interpolation scheme (18) is employed to compute the convection term in the volume fraction equations, where the nonphysical oscillation is avoided and the sharpness of interface is assured. The convective flux [c.sub.f] at cell face f is expressed as follows, see Fig. 3:
[FIGURE 3 OMITTED]
[C.sub.f] = [[~.[alpha]].sub.f]([c.sub.D] - [c.sub.U]) + [c.sub.U] (29)
[[~.[alpha]].sub.f] = [[~.[alpha]].sub.f][square root of [cos [theta]]] + [[alpha].sub.P](1 - [square root of [cos [theta]]]) (30)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)
[alpha] = [[c - [c.sub.U]]/[[c.sub.D] - [c.sub.U]]] (32)
where [c.sub.f] is the volume fraction defined at the cell-face f, Cu denotes the local Courant number of node P ([equivalent to]u.[S.sub.f] [DELTA]t/[V.sub.p]), [s.sub.f] is the normal vector of cell-face [n.sub.f] and [theta] represents the angle between [s.sub.f] and the normal vector of interface at that cell-face [n sub f]. The constants (a, b) appearing in the adopted interpolation scheme are chosen as (0.3, 0.7) in this study to fulfill the boundness criterion for convection differencing schemes. The typical values of polytropic constant for air and gas are chosen as [m.sub.g]--1.33 and [m.sub.a] = 0.8, respectively, which give the best agreement with corresponding measurements in the studied cases. The typical time step for melt filling is in an order of millisecond, while a smaller time step, e.g. 0.1 ms, is employed for the gas injection phase to obtain required numerical stability.
In an inner iteration of the proposed numerical scheme, the calculation starts from solving the momentum equations to obtain the velocity field. The pressure field is the determined from the continuity equation with a SIMPLE type algorithm (13). The temperature field is computed from the energy equation after the velocity field has been obtained. The front locations are then calculated from the front transport equations. The fluid properties are then updated using the new calculated flow field. Tens of inner iterations are commonly required to obtain a converged solution for each time step. This procedure completes an outer iteration of the proposed numerical scheme and the numerical calculation then advances to the next time step.
MELT FILLING AND GAS PENETRATION IN A SQUARE TUBE
Melt Filling Process
Similar to the gas penetration phenomenon in polymer melt, the filling of polymer melt in an empty cavity also involves the front advancing between two fluids, i.e. the residing air and injected melt. The melt filling and gas penetration in a clip-shaped square tube is chosen to exemplify the applications of proposed incompressible and compressible models in fully three-dimensional melt filling simulations. In addition to the gas penetration process, the proposed numerical models can as well benefit the prediction of melt front during the early filling process. Figure 4 depicts the geometrical characteristics of the studied clip-shaped square tube (12), where / denotes the penetration length defined at the centerline of tube starting from the gas inlet center and L is the length of tube. The melt gate and gas inlet are separately located on different sidewalls close to the inner end of tube. Table 1 summarizes the constants of employed Cross-WLF model for the injected melt. i.e. Polystyrene (PS), while Table 2 gives its related material properties. Simulations employing different numerical models and boundary conditions are conducted to illustrate the advantages of the proposed incompressible and compressible models over traditional incompressible approaches. Table 3 presents the material constants for the modified two-domain Tait equation for PS. All calculated cases are in a melt flow rate of 41.7 g/s with the melt temperature ([T.sub.m]) of 230[degrees]C and the mold temperature ([T.sub.w]) of 60[degrees]C. An identical grid with ~22,000 body-fitted, hexahedral cells is used in four numerical simulations. Fig. 5. Revealed by a grid-dependence test of the adopted grid topology, the primary penetration length of four successive grid levels converges monotonously to a grid-independent solution, where the here employed grid density has a discretization error less than 2%. Case A denotes the numerical result of the traditional incompressible model with outlets specified at both tube ends, and Case B employs the same model and boundary conditions as Case A except the outlet close to melt gate is treated as a solid wall. Case C represents the simulation based on the proposed compressible model, while the proposed incompressible model without outflow boundary is demonstrated in Case D. In contrast to the traditional incompressible flow model, there is no outlet boundary required in the latter two cases. All four cases use the same initial condition, i.e. the cavity is presumed originally occupied by air at room temperature, which is assumed to have constant density except the one following the polytropic law in the compressible model. Figure 6 shows the comparison of melt front prediction on the y = 0 plane close to the melt gate among four cases, where dark region represents the injected melt and the gray region denotes the residing air. The black vectors shown in Fig. 6 represent the velocity field at a specific instance. Because the melt is injected into the cavity from the sidewall of tube, it should flow evenly toward two tube ends, before it reaches the near tube end. After the cavity space neighboring the near tube end has been completely filled, the polymer melt flow switches from two opposite flow directions to only one main flow direction toward the far tube end. After comparing all calculated cases, the proposed compressible and incompressible models (Case C and D) seem to better capture the expected features of the studied melt filling process. Neither melt overflow nor trapped air near the tube end is observed in both cases, which better reflects the actual filling behavior observed in the experiments. Form the vector plots given in Fig. 6, it is easily to find that the aforementioned filling characteristics in an asymmetric melt filling process is easily captured by the proposed compressible and incompressible models. This comparison demonstrates the benefits of adopting the proposed compressible and incompressible models in predicting melt front evolution in the melt filling process, where no outlet boundary is explicitly required. Figure 7 depicts the evolution of melt front at different melt filling percentage before the gas is injected into the cavity, where the proposed compressible model is employed.
TABLE 1. Cross-WLF model constants for different polymers. PS ABS PP [eta] 0.2749 0.1815 0.274 [tau*] (Pa) 20015 136290 29374 [[eta].sub.g] (Pa s) 2.68 x 10 (11) 7.29 x 10 (9) 1.48 x 10 (14) [C.sub.g.sup.1] 25.878 20.462 29.968 [C.sub.2] 51.6 51.6 51.6 [T.sub.g] (K) 373.15 373.15 TABLE 2. Material properties for different polymers. PS ABS PP Apparent density [rho] (kg/[m.sup.3]) 948.15 949.33 752.52 Specific heat [c.sub.p] (J/kg K) 2100 2400 3055 Thermal conductivity k (W/m K) 0.18 0.18 0.239 Viscosity (Pa s) at T = 230 [degrees]C and 1124.5 2430.6 1445.7 dot[lambda] = 10 [s.sup.-1] TABLE 3. Material coefficients for the two-domain modified Tait equation. PS ABS PP [[beta].sub.1m] 1.000 x 9.692 x 1.231 x ([m.sup.3]/kg) [10.sup.-3] [10.sup.-4] [10.sup.-3] [[beta].sub.1s] 1.000 x 9.692 x 1.150 x ([m.sup.3]/kg) [10.sup.-3] [10.sup.-4] [10.sup.-3] [[beta].sub.2m] 6.800 x 6.139 x 1.150 x ([m.sup.3]/kg K) [10.sup.-7] [10.sup.-7] [10.sup.-3] [[beta].sub.2s] 2.481 x 3.021 x 4.990 x ([m.sup.3]/kg K) [10.sup.-7] [10.sup.-7] [10.sup.-7] [[beta].sub.3m] (Pa) 1.637 x 2.032 x 1.040 x [10.sup.8] [10.sup.8] [10.sup.8] [[beta].sub.3s] (Pa) 2.215 x 2.545 x 1.560 x [10.sup.8] [10.sup.8] [10.sup.8] [[beta].sub.4m] (1/K) 4.879 x 5.269 x 4.338 x [10.sup.-3] [10.sup.-3] [10.sup.-3] [[beta].sub.4s] 2.877 x 4.331 x 3.263 x (1/K) [10.sup.-3] [10.sup.-3] [10.sup.-3] [[beta].sub.5] (K) 376.51 366.03 395.15 [[beta].sub.6] (K/Pa) 3.106 x 2.550x 1.730 x [10.sup.-7] [10.sup.-7] [10.sup.-7] [[beta].sub.7] 0 0 8.110 x ([m.sup.3]/kg) [10.sup.-5] [[beta].sub.8] (1/K) 0 0 1.909 x [10.sup.-3] [[beta].sub.9] (1/K) 0 0 3.510 x [10.sup.-8]
[FIGURE 4 OMITTED]
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Validation of Gas penetration Characteristics
After demonstrating the application of the proposed compressible and incompressible models to the melt filling process, this section focuses on the validation of the proposed numerical models by comparing gas penetration length and gas core size between numerical prediction and available experimental measurements (12). Figure 8 depicts the predicted growth of gas penetration in PS by the proposed compressible model for a injection pressure ([p.sub.in]) of 60 bar and melt filling of 82% without any delay time ([t.sub.d]), where the room-temperature air is employed as the injected gas, the melt temperature ([T.sub.m]) is regulated at 220[degrees]C and the mold temperature ([T.sub.w]) is set to be 60[degrees]C. The duration of gas injection is 1 s. In the packing phase, the packing pressure is chosen as the employed gas injection pressure. The total packing and cooling time is 3 s. Different from the melt filling behavior, the gas penetration develops only toward the far-end side along the length direction, which agrees well with the corresponding experimental observations. Fig. 2. This is easily understood by the fact that the only available free space left in cavity for possible gas expansion solely locates at the far end. The numerical simulations also show obvious corner effects, where the center of gas core is located off-centered. Additionally, the gas core thickness gradually decreases along the penetration direction in the primary penetration regime. Besides, the proposed compressible model successfully reproduces the secondary gas penetration behavior observed during the packing and cooling phase, which clearly reflects the nontrivial influence of melt compressibility in the packing and cooling phase resulting in a further gas penetration.
[FIGURE 8 OMITTED]
Beside the qualitative comparison between the numerical predictions and experimental results, a quantitative evaluation of numerical predictions is also conducted. Figure 9 compares the primary and total gas penetration length subjected to different injected gas pressure between the predictions given by different numerical models and available experimental measurements, where lines represent the numerical predictions and symbols denote the experimental measurements reported in (12). The numerical result of Hele-Shaw model is obtained from a Moldflow simulation with about 80,000 tetrahedral cells, which gives the worst prediction accuracy despite of employing about four times of cells as in the compressible model. The Hele-Shaw model has under-predicted the primary penetration length at [p.sub.in] = 60 bar for about 40%, while the proposed incompressible model apparently has a better prediction accuracy and hence prevails over the traditional Hele-Shaw-type model. In contrast to these two models showing a linear dependency of gas penetration length on gas pressure, the proposed compressible model suggests that the gas penetration length increases with the reciprocal of the imposed gas pressure in a nonlinear manner, which emphasizes the necessity of modeling gas compressibility in GAIM processes. Through the modeling of melt compressibility, the compressible result further gives the expected secondary gas penetration phenomenon occurring in the packing and cooling phase during a GAIM process. The primary penetration, as well as the secondary penetration, predicted by the compressible model agrees not only qualitatively but also quantitatively well with the experimental measurements.
[FIGURE 9 OMITTED]
Except the global penetration characteristic, such as the penetration length, the local penetration feature, the gas core size distribution is also investigated for a injection pressure of 100 bar and melt filling of 82% without any delay time, where the room-temperature air is employed as the injected gas, the melt temperature is regulated at 220[degrees]C and the mold temperature is set to be 60[degrees]C. Figure 10 compares the skin thickness ratio (S/W) between the predictions given by different numerical models and available experimental measurements, where lines represent the numerical predictions and symbols denote the experimental measurements reported in (12). The experimental results suggest that the skin thickness ratio is approximate equal to 0.34 for the first 50% of gas penetration, while the gas core quickly reduces its size at IIL = 0.5 and finally vanishes at about IIL = 0.68. The Hele-Shaw result is the worst prediction among numerical results, which gives a smaller skin thickness and hence a larger gas core. The incompressible model without outflow boundary delivers a favorable agreement with experimental measurements before IIL = 0.5, but it fails to reflect the gas core characteristic after IIL = 0.5 due to the negligence of gas and melt compressibility. The proposed compressible model agrees quite well with the corresponding measurements, especially the steep region between IIL = 0.5 and 0.75, which bridges the transition zone from the primary penetration to the second penetration. From the validation of both global and local gas penetration features in a square tube, the proposed compressible model is proven to have sufficient prediction accuracy in simulating the gas penetration phenomena in GAIM processes.
[FIGURE 10 OMITTED]
INFLUENCES OF MOLDING PARAMETERS ON GAS PENETRATION CHARACTERISTICS
Gas Injection Pressure
One of the most important molding parameters influencing the gas penetration characteristics is the gas injection pressure. As indicated by related experimental evidences (10), (12), the increase of gas injection pressure in tubular parts is usually accompanied by the decrease of gas penetration length. This important gas penetration characteristic seems first to contradict the common expectation that higher gas pressure delivers more gas inertia, which would result in deeper penetration in melt. Because gas pressure acts on no predominant direction in cavity, the injected gas penetrates the melt in all possible directions. Therefore, the increase of gas pressure enhances the penetration along not only the longitudinal but also the radial direction in a tubular cavity. Thus, high gas injection pressure naturally gives large gas core during penetration. As a consequence, the melt excluded by gas penetration in the radial direction will automatically accumulate in front of the advancing gas front, which obviously impedes the further longitudinal development of gas penetrating into the melt. This explains why a short primary penetration with large gas core is resulted by high injection pressure, while the primary penetration length might have a nonlinear dependence on the gas injection pressure. Fig. 9. This argument is clearly supported by numerical simulations of the proposed compressible model, where Fig. 11 depicts the distribution of gas core ratio for four different gas injection pressures (82% melt filling of PS, [T.sub.m] - 220[degrees]C, [T.sub.2][degrees]C, and [t.sub.d] = 0 s) and the numerical prediction has good agreement with the available experimental result at [p.sub.in] = 100 bar. Two curves of gas core size with low gas injection pressure show a convex-type distribution, whereas two high gas injection pressures result in a concave-type one. This should be related to the steep increase of primary gas penetration for the gas pressure between 50 and 80 bar due to the nonlinear behavior of gas compressibility. Despite the negative dependence of the primary penetration on the gas injected pressure, the secondary penetration seems to grow at high gas injection pressure, which is owing to the limited melt compressibility at low gas injection pressure, as reflected by both experimental and numerical results given in Fig. 11.
[FIGURE 11 OMITTED]
Three mold temperatures, i.e. [T.sub.w] = 40[degrees]C, 60[degrees]C, and 80[degrees]C, are studied numerically to identify their influences on the gas penetration characteristics (82% melt filling of PS, [p.sub.in] = 100 bar, [T.sub.m] = 220[degrees]C, and [t.sub.d] = 0 s). The experimental measurements suggest that a lower mold temperature is favorable to increase the primary gas penetration length in a tubular cavity. This is mainly due to that the solidified skin near cavity wall becomes thicker at a lower mold temperature, which apparently hinders the injected gas from penetrating in the radial direction and forces the injected gas to flow more concentrated along the longitudinal direction. Accordingly, high mold temperature tends to give a short but thick primary gas core, because melt is easier to be excluded in the radial direction at high mold temperature. The variation of primary and secondary penetration due to the change of mold temperature is shown in Fig. 12, where the comparison of gas penetration length among different mold temperatures is presented with good experimental agreement. The proposed compressible model, giving discrepancy between measurements and predictions less than 3% for the case of [T.sub.w] = 40[degrees]C, suggests a linear dependence of the gas penetration length on mold temperature, where the primary and secondary penetration length both decreases with the increase of mold temperature. The temperature difference between the mold and melt mainly accounts for the penetration behavior of secondary penetration, where melt experiences large density variation when subjected to high temperature difference. Figure 13 depicts the comparison of gas core ratio among different mold temperature, which evidently confirms the aforementioned tendency of gas core size due to mold temperature variation. And all the numerical results at different mold temperatures exhibit similar tendency of gas core distribution, where the gas core has minor change in size.
[FIGURE 12 OMITTED]
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The gas penetrations at three melt temperatures, i.e. [T.sub.m] = 210, 230, and 250[degrees]C, are predicted to show the difference in the gas penetration behaviors (82% melt filling of PS, [P.sub.in] = 100 bar, [T.sub.w] = 60[degrees]C, and [t.sub.d] = 0 s). The experimental observations show that a lower melt temperature is favorable to increase the primary gas penetration length in a tubular cavity, which is similar to the effect of the mold temperature. High melt temperature results in thinner solidified melt layer near cavity wall when compared with low melt temperature, which clearly facilitate the injected gas to penetrate in the radial direction. Hence, high melt temperature, similar to high mold temperature, tends to give a thick but short primary gas core. This characteristic is clearly indicated in Fig. 14, where the numerical predictions agree well with the measurements except the data point at [T.sub.m] =210[degrees]C with error smaller than 5%. In contrast to the influence of mold temperature where the secondary penetration obviously decreases with the temperature, the secondary penetration raises its length with the increasing melt temperature. This is just understood by the fact that low melt temperature yields smaller temperature variation of melt before solidification and hence a smaller density variation of melt. The numerical results of proposed compressible model, giving a prediction error less than 5%, propose a quasi-linear dependence of the gas penetration length on melt temperature, where the penetration length decreases with the increase of melt temperature. Figure 15 compares the gas core ratio among different melt temperatures. The numerical prediction at [T.sub.m] = 210[degrees]C, apparently cause a much smaller gas core size than those of other two melt temperatures. This suggests that the temperature difference between mold and melt, as well as the melt temperature, can bring nontrivial effects on the penetration length and gas core size.
[FIGURE 14 OMITTED]
[FIGURE 15 OMITTED]
Gas Delay Time
Gas delay time is an important means to manipulate the gas penetration characteristics without requiring any change in pressure and temperature setup of system. Cases of four delay times, i.e. [t.sub.d] = 0 s, 0.5 s, 1.0 s, and 1.5 s, are simulated to understand their influences on the gas penetration phenomenon (82% melt filling of PS, [P.sub.in] = 100 bar, [T.sub.w] = 60[degrees]C, and [T.sub.m] = 220[degrees]C). The experimental observations indicate that large gas delay time is favorable to increase the primary gas penetration length in a tubular cavity, because the thickness of solidified skin near cavity wall increases with the gas delay time, which can be regarded as an additional cooling time before gas injection. So the increase of gas delay time is prone to constrain the gas core penetrating along the longitudinal direction and leads to a longer primary gas penetration, which is clearly supported by Fig. 16, where the numerical predictions agree well with experimental measurements (prediction errors less than 1%). Similar conclusion was also drawn in a recently related study (10). The secondary penetration seems to be less affected by the change of gas delay time, since the two major parameters (variation of pressure and temperature) governing the melt compressibility (represented by the difference of melt density) are almost independent of gas delay time. Figure 17 further compares the gas core ratio among different gas delay time. It is interesting to note that the introduction of gas delay time not only reduces the gas core size but also brings a more uniform distribution of gas core ratio when compared with those created by the change of other molding parameters, where this influence appears to become saturated for the gas delay time greater than 1 s. In this investigated case, the numerical predictions suggest that mean gas core size can be quite uniformly reduced by about 28%. The gas delay time seems to be the most effective molding parameter, which could deliver a small and uniform distribution in gas core size.
[FIGURE 16 OMITTED]
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The last molding parameter to be examined is the nature of polymer material employed in the molding process. Three different polymers, two amorphous (PS and ABS), and one crystalline (PP) materials, are adopted to study the influences of material property on the gas penetration characteristics (82% melt filling, [p.sub.in] = 100 bar, [T.sub.w]60[degrees]C, [T.sub.m] 220[degrees]C, and [t.sub.d] = 0s) As indicated by Fig. 18, ABS melt delivers larger primary penetration but accompanied by smaller secondary penetration, and it gives the largest total penetration among three studied polymers. Since PS and ABS melts have similar physical properties, such as density, specific heats and thermal conductivity, the remaining factor dominating the primary penetration length is the melt viscosity. The noticeable viscosity difference between ABS and PS melts (see Table 2) accounts for a smaller radial penetration depth, i.e., smaller gas core, for the more viscous ABS under the same gas injection pressure (see Fig. 19), which clearly yields the possibility to obtain longer primary penetration length. Because of having a lower density (PS and ABS are about 20% denser than PP), PP melt has the least inertia to withstand the gas penetration. As a result, it results in the thickest but shortest primary gas core. As discussed earlier, the p-[rho]-T relation of melt principally dominates the behavior of secondary penetration, where the melt subjected to large density variation obviously leads to longer secondary penetration. After examining Eq. 23, PS melt experiences the most significant density variation during the GAIM process, which clarifies why the secondary penetration of ABS (and PP) melt is only about one third of that found in PS melt.
[FIGURE 18 OMITTED]
[FIGURE 19 OMITTED]
This article successfully proposes a generalized Newtonian model to predict the three-dimensional gas penetration phenomenon in the GAIM process, where the gas and melt compressibility can be both taken into account. The proposed compressible scheme is quantitatively validated by the global and local characteristics of the gas penetration process in a clip-shaped square tube, while several incompressible models, including a Hele-Shaw calculation, are also presented for the purpose of comparison. In addition to being capable of delivering the expected secondary penetration successfully, the proposed compressible model is superior to other models in giving high prediction accuracy, as well as the prediction of nonlinear penetration growth with respect to pressure, which can not be predicted by incompressible models neglecting the gas compressibility. Then the influences of five major molding parameters, such as the injection pressure, mold temperature, melt temperature, delay time, and melt material on the gas penetration characteristics in the same clip-shaped square tube via the proposed numerical approach are also presented. Because the increase of gas pressure enhances the penetration along not only the longitudinal but also the radial direction in a tubular cavity, high gas injection pressure naturally gives large gas core during primary penetration. This explains why a short primary penetration length with large gas core is resulted by high injection pressure. The secondary penetration grows at high gas injection pressure, which is due to the limited melt compressibility at low gas injection pressure. Lower mold temperature is favorable to increase the primary gas penetration length in a tubular cavity. This is mainly owing to that the solidified skin near cavity wall becomes thicker at a lower mold temperature, which apparently hinders the injected gas from penetrating in the radial direction and forces the injected gas to flow more concentrated along the longitudinal direction. The temperature difference between the mold and melt mainly accounts for the penetration behavior of secondary penetration, where melt experiences large density variation when subjected to high temperature difference. Lower melt temperature is helpable to increase the primary gas penetration length in a tubular cavity, whereas the secondary penetration raises its length with the increasing melt temperature, which is due to that low melt temperature yields smaller density variation of melt before solidification. Large gas delay time is beneficial to increase the primary gas penetration length in a tubular cavity, because the thickness of solidified melt layer near cavity wall increases with the gas delay time. The secondary penetration is less affected by the change of gas delay time, since the two major parameters governing the melt compressibility are almost independent of gas delay time. ABS melt delivers larger primary penetration but accompanied by smaller secondary penetration and it gives the largest total penetration among three studied polymers.
The author thanks his student, Mr. Y.J. Juang for his technical assistance in conducting this research.
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Department of Mechanical Engineering, Chung Yuan University, Jhongli, Republic of China
Correspondence to: S.-W. Chau; e-mail:firstname.lastname@example.org
Contract grant sponsor: National Science Council,, R.O.C.; contract grant number: 93-2212-E-033-006.
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|Publication:||Polymer Engineering and Science|
|Article Type:||Technical report|
|Date:||Sep 1, 2008|
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