(181bq) Mathematical Modeling of Blown Film Extrusion Using the Discrete Slip-Link Rheological Model

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 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 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 modelings 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 under some operating conditions (Liu, 1991, 1994; Liu et al., 1995; Pirkle et al., 2010).

This work explores the dynamics of blown film extrusion for a more modern rheological approach that is known as the discrete slip-link model (DSM) (Schieber et al., 2003; Khaliullin and Schieber, 2009; Andreev et al., 2013). The DSM was chosen in this work for describing the dynamics of flexible polymer melts due to its ability to predict the viscoelasticity of polydisperse linear and branched entangled polymer systems. The DSM models the entanglement of polymer chains within a melt more accurately than tube models, but does not incur the full computational cost of molecular dynamics simulations (Baig et al., 2006; Kim et al., 2007; Tzoumanekas and Theodorou, 2006; Stephanou et al., 2010; Nicholson and Rutledge, 2016).

The fully three-dimensional axisymmetric model of blown film extrusion is solved numerically, without using any thin-shell or similar approximations of the underlying momentum and energy conservation equations. The variations in the radial direction are spatially resolved, which has been reported (Housiadas et al., 2007) to have a significant effect on the final product film properties in spite of having been assumed negligible in most blown film extrusion models. 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, which employs strategies to retain high numerical stability and accuracy, while having a manageable computational cost. The spatial derivatives are handled using the same numerical-method-of-lines strategy (Schiesser, 1991) used to simulate blown film extrusion for the two-phase microstructural model (Pirkle and Braatz, 2003a). Unlike Pirkle and Braatz (2003a), this work employs a different method for dynamically coupling the rheological constitutive equation with momentum and energy conservation equations, and does not use the thin-shell approximation. The different numerical coupling method is required because the DSM is not describable in the same type of simple functional form than for the rheological constitutive equations used in past blown film extrusion studies.

The numerical simulation results obtained from the proposed blown film extrusion model are compared to the results obtained by other models reported in the literature, in order to investigate the inaccuracies created by different approximations. The comparisons include the product film properties and the spatiotemporal behavior of operational states such as the crystallinity, film geometry, temperature, and the total stress tensor.

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