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In recent years, increased attention has been paid to the design of cooling systems in injection molding, as it became clear that cooling affects both productivity and part quality. In order to systematically improve the performance of a cooling system in terms of rapid, uniform, and even cooling, the designer needs a CAE analysis tool. For this, a computer simulation has been developed for three-dimensional mold heat transfer during the cooling stage of an injection molding process. In this simulation, mold heat transfer is considered as cyclic-steady, three-dimensional conduction; heat transfer within the melt region is treated as transient, one-dimensional conduction; heat exchange between the cooling channel surfaces and coolant is treated as steady, as is heat exchange with the ambient air and mold exterior surfaces. Numerical implementation includes the application of a hybrid scheme consisting of a modified three-dimensional, boundary-element method for the mold region and a finite-difference method with a variable mesh for the melt region. These two analyses are iteratively coupled so as to match the temperature and heat flux at the mold-melt interface. Using an example, the usefulness of the simulation developed here in the design of a cooling system for an injection molding process is amply demonstrated.

This work employs a coupled analysis of the fluid flow and heat transfer in the polymer melt during the filling and post-filling stages of the injection-molding process and of mold cooling/heating which occurs during the entire process. Polymer melt analysis (PMA) has been carried out through a unified theoretical model implemented using a hybrid finite-element/finite-difference/control-volume numerical solution of the generalized Hele-Shaw flow of a compressible viscous fluid under non-isothermal conditions. Further, mold-cooling analysis (MCA) has been carried out utilizing a periodic heat conduction model implemented using a modified three-dimensional boundary-element method. To faithfully accommodate the effects of mold cooling on the fluid flow and heat transfer in the polymer melt, PMA and MCA have been coupled for appropriate data exchange and iterations carried out until a convergent solution for mold temperatures and for flow, pressure and temperatures within the polymer melt is obtained. The results obtained from this integrated simulation for different test cases have been compared with experimental data and a favorable agreement has been noticed. Using an illustrative example, the results are discussed in detail.

A saturated porous medium confined between two horizontal cylinders is considered. As a result of a temperature difference between the cylinders, thermal convection is induced in the medium. The flow structure is investigated in a parameter space (R, Ra) where R is the radii ratio and Ra is the Darcy-Rayleigh number. In particular, the cases of R = 2, 2½, 21/4 and 2½ are considered. The fluid motion is described by the two-dimensional Darcy-Oberbeck-Boussinesq's (DOB) equations, which we solve using regular perturbation expansion. Terms up to O(Ra60) are calculated to obtain a series presentation for the Nusselt number Nu in the form \[ Nu(Ra^2) = \sum_{s=0}^{30} N_sRa^{2s}. \] This series has a limited range of utility due to singularities of the function Nu(Ra). The singularities lie both on and off the real axis in the complex Ra plane. For R = 2, the nearest singularity lies off the real axis, has no physical significance, and unnecessarily limits the range of utility of the aforementioned series. For R = 2½, 2¼ and 21/8, the singularity nearest to the origin is real and indicates that the function Nu(Ra) is no longer unique beyond the singular point.Depending on the radii ratio, the loss of uniqueness may occur as a result of either (perfect) bifurcations or the appearance of isolated solutions (imperfect bifurcations). The structure of the multiple solutions is investigated by solving the DOB equations numerically. The nonlinear partial differential equations are converted into a truncated set of ordinary differential equations via projection. The steady-state problem is solved using Newton's technique. At each step the determinant of the Jacobian is evaluated. Bifurcation points are identified with singularities of the Jacobian. Linear stability analysis is used to determine the stability of various solution branches. The results we obtained from solving the DOB equations using perturbation expansion are compared with those we obtained from solving the nonlinear partial differential equations numerically and are found to agree well.

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