(161y) Model of Nonisothermal Blown Film Extrusion Using the Perturbation Expansion Technique

Blown film extrusion is a technology for the manufacturing of thin plastic films, with products including heat-resistant medical films, food wrap, and transport packaging (Cantor, 2006). The mathematical modeling of blown film extrusion is complicated by the rheology of the polymer melt, which is non-Newtonian and heavily influenced by highly nonlinear crystallization kinetics. In addition, the film geometry is unknown ahead of time and must be calculated as part of the numerical solution.

The effects of model parameters on the film properties and on operational stability have been investigated using several models including the thin-shell (Pearson and Petrie, 1970a,b), quasi-cylindrical (Doufas and McHugh, 2001), and perturbation models (Housiadas et al., 2007). Some numerical models considered only steady-state operation (Pearson and Petrie, 1970a,b), whereas other studies have analyzed dynamic behavior (Yeow, 1976; Cain and Denn, 1988; Pirkle and Braatz, 2003a). Blown film extrusion can exhibit a variety of interesting instabilities, which can be axisymmetric (Yeow, 1976; Cain and Denn, 1988) or non-axisymmetric (Housiadas et al., 2007). The effects of heat transfer and flow-induced crystallization on stability have been investigated (Pirkle and Braatz, 2011).

Due to the strong nonlinearities associated with blown film extrusion, the simulation results are a strong function of the rheological constitutive equation used to calculate the viscous stress tensor. Constitutive equations that have been explored in blown film extrusion are the quasi-Newtonian (Pearson and Petrie, 1970a,b), upper convected Maxwell (Luo and Tanner, 1985), Marucci (Cain and Denn, 1988), Phan-Thien-Tanner (Housiadas et al., 2007), and two-phase microstructural (Doufas and McHugh, 2001; Pirkle and Braatz, 2003b) models. Polymer crystallization is induced both by cooling and flow (Doufas and McHugh, 2001). While several researchers have shown that the best of these models can qualitatively and semi-quantitatively describe many of the film product properties, state variables during operation, and flow instabilities, the quantitative agreement with some of the states has been quite poor, at least for some operating conditions (Liu, 1991, 1994; Liu et al., 1995; Pirkle et al., 2010).

This work extends the isothermal perturbation model described by Housiadas et al. (2007) to nonisothermal operations by incorporating an energy balance equation and crystallization kinetic expression. This extension is important in practical applications, in which the film undergoes significant cooling, and the polymer rheology is highly sensitive to temperature.

The nonisothermal model is able to predict the freeze line height, which is a parameter which must be specified a priori in isothermal simulations. The perturbation model has the capability of resolving spatial gradients in the states across the thin dimension of the film. Variations in the radial direction, which have been reported (Housiadas et al., 2007) to have a significant effect on the final product film properties, have been assumed negligible in most blown film extrusion models, including the thin-shell model of Pearson and Petrie (1970a,b).

Differential index analysis of the nonisothermal perturbation model is also presented. The perturbation model includes an interfacial tension balance as one of its governing equations; this vector equation is shown to introduce implicit constraints with respect to the axial direction, limiting the number of boundary conditions that may be independently specified. As a result, the selection of boundary conditions is nontrivial compared to that of the thinfilm model, which contains no interfacial tension balances or associated implicit constraints (Schiesser, 1996). Index analysis informs the selection of the boundary conditions. The minimum-order reduction boundary condition is used as the outflow boundary condition (Pirkle and Braatz, 2003a), as this approach has been shown to produce more physically meaningful results than alternatives (Schiesser, 1996).

Details are provided for the numerical algorithm for carrying out these simulations. To reduce the high condition number associated with large matrix equations with many states, the full nonlinear dynamical system is separated into more easily solved subsystems, which are then solved iteratively to converge to the solution of the full set of equations. The splitting approach is informed by the properties of the physical system, analysis of the governing equations, and singular value decomposition of the discretized and linearized system.