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2.2 Filling simulation of the melt
2.2.1 Mathematical modeling
Computer simulation techniques have had success in predicting filling behavior in extremely complicated geometries. However, most of the current numerical implementation is based on a hybrid finite-element/finite-difference solution with the middle plane model. The application process of simulation packages based on this model is illustrated in Fig. 2-1. However, unlike the surface/solid model in mold-design CAD systems, the so-called middle-plane (as shown in Fig. 2-1b) is an imaginary arbitrary planar geometry at the middle of the cavity in the gap-wise direction, which should bring about great inconvenience in applications. For example, surface models are commonly used in current RP systems (generally STL file format), so secondary modeling is unavoidable when using simulation packages because the models in the RP and simulation systems are different. Considering these defects, the surface model of the cavity is introduced as datum planes in the simulation, instead of the middle-plane.
According to the previous investigations [4–6], filling governing equations for the flow and temperature field can be written as:
where x, y are the planar coordinates in the middle-plane, and z is the gap-wise coordinate; u, v,w are the velocity components in the x, y, z directions; u, v are the average whole-gap thicknesses; and η, ρ,CP (T), K(T) represent viscosity, density, specific heat and thermal conductivity of polymer melt, respectively.
Fig.2-1 a–d. Schematic procedure of the simulation with middle-plane model. a The 3-D surface model b The middle-plane model c The meshed middle-plane model d The display of the simulation result
In addition, boundary conditions in the gap-wise direction can be defined as:
where TW is the constant wall temperature (shown in Fig. 2a).
Combining Eqs. 1–4 with Eqs. 5–6, it follows that the distributions of the u, v, T, P at z coordinates should be symmetrical, with the mirror axis being z = 0, and consequently the u, v averaged in half-gap thickness is equal to that averaged in whole gap thickness. Based on this characteristic, we can pide the whole cavity into two equal parts in the gap-wise direction, as described by Part I and Part II in Fig. 2b. At the same time, triangular finite elements are generated in the surface(s) of the cavity (at z = 0 in Fig. 2b), instead of the middle-plane (at z = 0 in Fig. 2a). Accordingly, finite-difference increments in the gap wise direction are employed only in the inside of the surface(s) (wall to middle/center-line), which, in Fig. 2b, means from z = 0 to z = b. This is single-sided instead of two-sided with respect to the middle-plane (i.e. from the middle-line to two walls). In addition, the coordinate system is changed from Fig. 2a to Fig. 2b to alter the finite-element/finite-difference scheme, as shown in Fig. 2b. With the above adjustment, governing equations are still Eqs. 1–4. However, the original boundary conditions in the gap wise direction are rewritten as:
Meanwhile, additional boundary conditions must be employed at z = b in order to keep the flows at the juncture of the two parts at the same section coordinate [7]:
where subscripts I, II represent the parameters of Part I and Part II, respectively, and Cm-I and Cm-II indicate the moving free melt-fronts of the surfaces of the pided two parts in the filling stage.
It should be noted that, unlike conditions Eqs. 7 and 8, ensuring conditions Eqs. 9 and 10 are upheld in numerical implementations becomes more difficult due to the following reasons:
1. The surfaces at the same section have been meshed respectively, which leads to a distinctive pattern of finite elements at the same section. Thus, an interpolation operation should be employed for u, v, T, P during the comparison between the two parts at the juncture.
2. Because the two parts have respective flow fields with respect to the nodes at point A and point C (as shown in Fig. 2b) at the same section, it is possible to have either both filled or one filled (and one empty). These two cases should be handled separately, averaging the operation for the former, whereas assigning operation for the latter.