(289d) Non-Isothermal Blown Film Extrusion Including Crystallization: Instabilities, Multiplicities, and Mapping of Stable Operating Regions

Blown film extrusion is a widespread commercial process for the manufacture of plastic tubes or sheets such as for medical packaging, container liners, shrink wrap and laminated films for food transportation and storage, and agricultural films used to trap heat and moisture in greenhouses [1,2]. Simulation of this process was facilitated by the development of the thin shell model [3,4]. Blown film extrusion processes can exhibit a variety of instabilities that can limit machine productivity and product quality. The region of stable operations depends on the effect of crystallization on the rheological properties of the polymer. Stability operating regions including such effects have been mapped for a simplified version of the thin shell model [5] that does not include two essential pairs of constraints (constant bubble inflation pressure and machine tension, or constant bubble air mass and take-up ratio). In real operation, the air bubble inflation pressure and modified machine tension are not fixed and are not easy to tightly control [6-10]. This is remedied by fixing the more controllable bubble air mass, which remains approximately constant once the air inlet valve is shut, and the take-up ratio, which is controlled by the speed of the nip rollers. This presentation describes the simulation of the dynamic thin-shell model with non-isothermal crystallization kinetics and a quasi-Newtonian constitutive relation for material properties typical of linear low density polyethylene (LLDPE) and laboratory operating conditions [11,12]. Regions of stable operations are mapped and under certain conditions, spontaneously oscillating solutions and multiple steady states are generated. The nature of the various steady states is examined using a local stability analysis based on the linearized system. To facilitate the solution of the generalized eigenvalue problem resulting from the spatial discretization and linearization steps, the dynamic model is rearranged so that only one time derivative is present per equation. This diagonalized model leads to matrices in the generalized eigenvalue problem that are easily handled by sparse matrix solvers [13], which enables higher spatial resolution.

For a given bubble air mass, the take-up ratio is used as the continuation parameter for mapping steady-state solutions [14]. The take-up ratio correlates smoothly, but not necessarily monotonically, with the machine tension. Curves of either blow-up ratio or thickness reduction versus take-up ratio reveal that there are take-up ratios where no, one, or multiple solutions exist. The heat transfer coefficient from the polymer film to the external air and surroundings has a marked influence on the qualitative and quantitative features of the blow-up ratio versus thickness reduction curves. The stability zone is mapped by different assemblages of stable blow-up ratio versus thickness reduction plots. Generalized eigenvalue analysis of the linearized blown film equations indicates that increasing the heat transfer rate increases the stability of operations. There is a corresponding decline, however, in the thickness reduction of the blown film for a given blow-up ratio. Spontaneous oscillations representing either helical or draw-resonance instabilities [15,16] are also revealed for conditions lying outside the stability zones.

Also, by comparing stability zones corresponding to curves of constant inflation pressure force and those corresponding to constant air bubble mass, respectively, it was discovered that the former approach can lose track of stable regions due to hysteresis phenomena. Using the latter approach, all stable states are readily revealed. This indicates that the preferred method of mapping out the stability region is to hold the bubble air mass constant and vary the take-up ratio.

References

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[12] Pirkle, Jr., J.C., M. Fujiwara, and R.D. Braatz, ?Maximum-likelihood parameter estimation for the thin-shell quasi-newtonian model for a laboratory blown film extruder,? Ind. Eng. Chem. Res., in press (2010).

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