ReviewA heuristic method for modeling three-dimensional non-Newtonian flows of polymer melts in single-screw extruders
Introduction
Plasticating single-screw extruders abound in the polymer processing industry. Due to their great versatility, they are employed in many polymer-shaping processes, producing a variety of plastic products such as pipes, sheets, profiles, films, and fibers. Indeed, a significant proportion of polymers passes through a plasticating single-screw extruder at least once after manufacture. The elementary processing steps of the plasticating extrusion process, however, are generally the same: (a) transport and (b) melting of particulate solids, followed by (c) mixing and (d) pumping of the polymer melt. The most critical step involves melt-conveying and pressurization. To force the material through the die at the desired processing rate, the extruder must pump the molten polymer through the melt-conveying zone, also known as metering section, and build up sufficient pressure.
Numerous theoretical studies have examined the complex flow of polymer melts in the metering zone of single-screw extruders. The first scientifically sound theory to model screw viscosity pumps was proposed by Rowell and Finlayson [1], and later refined by Carley et al. [2] and Mohr et al. [3]. These early analyses derived exact analytical solutions for the down-channel and cross-channel flows, assuming the viscosity of the polymer melt to be constant. Tadmor and Klein [4] expanded on their original research work by providing a fundamental review of plasticating single-screw extrusion in general.
The pioneering theories of melt-conveying in single-screw extruders considered Newtonian fluids and provided analytical solutions for the flow rate in the form of two independent terms: (i) a drag flow and (ii) a pressure flow. Including non-Newtonian flow behavior of the polymer melt increases the complexity of the mathematical analysis substantially, since the governing flow equations become non-linear due to the dependency of viscosity on shear rate. To refine the general understanding of melt-conveying in single-screw extruders, efforts have been directed towards numerical analyses of power-law-model based flows. Rotem and Shinnar [5] derived numerical solutions for a one-dimensional non-Newtonian flow between two parallel plates in linear movement. Griffith [6], Zamodits and Pearson [7] and Karwe and Jaluria [8] presented numerical solutions for a two-dimensional non-Newtonian flow in an infinitely wide screw channel. Spalding et al. [9] obtained numerical solutions for a three-dimensional non-Newtonian flow by applying the finite-element method. All these studies applied a power-law model to describe the shear-thinning nature of the polymer melt. To date, however, an exact analytical solution to even the most simplified mathematical model – a one-dimensional non-Newtonian flow under isothermal conditions – remains elusive. Thus, numerical methods are essential for sophisticated analyses of flow in single-screw extruders.
In spite of their high relevance, the practical usefulness of numerical techniques is limited, as they require professional expertise to achieve stable solutions and tend to be time-consuming and cost-intensive. Only a few scientific studies have proposed analytical approximation methods for estimating the effect of non-Newtonian flow behavior on the pumping capability of the metering zone without resorting to numerical techniques. Booy [10] applied an effective viscosity to the original Newtonian theory that considers the non-Newtonian nature of the polymer melt. White and Potente [11] presented approximate equations for predicting the output-pressure gradient behavior of power-law fluids in infinitely wide screw channels. However, their application range is divided into several sections, and thus the equations have both undefined and discontinuous regions. To provide a continuous model, Rauwendaal [12] refined the traditional Newtonian model by introducing correction factors for the non-Newtonian flow behavior. These parameters were determined only for a limited range of helix angles, and – moreover – only positive pressure gradients were applied, leading Pachner et al. [13] to propose a generalized two-dimensional melt-conveying model for power-law fluids. The novelty of their theoretical approach is the construction of a heuristic model from a large number of numerical solutions of scaled flow equations using symbolic regression based on genetic programming. This established type of regression analysis, first investigated in depth by Koca [14], applies evolutionary computation to uncovering intrinsic analytical relations of arbitrary data sets. Thus, it is commonly used for finding mathematical models that best fit sets of scientific data obtained from either theoretical or experimental analysis. Several studies have demonstrated its usefulness in various applications. Bramerdorfer et al. [15] employed symbolic regression based on genetic programming for modeling the nonlinear behavior of permanent-magnet synchronous machines. Lughofer et al. [16] predicted key metrics of tribological systems by applying this type of regression analysis. A detailed review on the fundamentals of the modeling technique was provided by Affenzeller et al. [17], [18].
Previous approximate models on non-Newtonian flows in single-screw extruders were based on a two-dimensional modeling framework, analyzing the conveying characteristics of power-law fluids in screw channels of infinite width. Thus, the rate-limiting influence of the flight flanks was generally omitted. In a recent paper [19], we presented a heuristic relationship for predicting the three-dimensional output-pressure gradient behavior of power-law fluids. This work, which applied symbolic regression based on genetic programming, focused exclusively on pressure-generating metering sections, which are typically found in smooth-bore single-screw extruders. In industrial applications, however, pressure development may already take place early in the solids-conveying zone (e.g., in grooved-feed extruders), which causes the downstream functional zones to be overridden, as indicated by a negative pressure gradient. Hence, we present a generalized heuristic relationship that takes into account both the pressure-generating and the pressure-reducing capabilities of single-screw extruders.
The main issue of our analysis is essentially a flow of a generalized Newtonian fluid in an annular duct. Typically, these flows are not only related to melt-conveying in single-screw extruders, but also arise in many other processes. There is a considerable amount of scientific literature on modeling flows in annular gaps, considering both stationary and moving walls. Carrasco-Teja and Frigaard [20], [21] analyzed Newtonian and non-Newtonian displacement flows along narrow eccentric annuli with a moving inner cylinder during primary cementing of horizontal oil and gas wells. The non-Newtonian flow behavior was mathematically described by means of the Herschel-Bulkley model. Fitt and Please [22] studied three-dimensional flows of power-law fluids in scraped surface heat exchangers widely used in food industry. Unlike screw extruders, these devices are equipped with a smooth rotating inner cylinder and thus, the flow is not affected by the presence of flights. Instead, the flow is mainly governed by an axial pressure gradient provided by a pump. Chapman et al. [23] investigated flows of power-law fluids in extrusion, placing special emphasis to materials where the power-law index tends to zero (e.g. clay suspensions). In this limit, the flow characteristics are highly sensitive to the value of the power-law exponent.
Section snippets
Newtonian flow analysis
The most frequently used model for calculating isothermal flows in shallow screw channels, widely known as the classic pumping model, is based on the flat-plate approximation. The helical screw channel is unwound from the screw and located on an infinite flat plate that represents the barrel surface, as shown in Fig. 1. Physically, this means that the curvature of the screw is omitted, which is a reasonable simplification if the channel depth is considerably smaller than the outer diameter of
Problem definition
Applying the flat plate assumption, we consider a straight rectangular screw channel of constant cross-section and infinite length, as shown in Fig. 1. The system is described by means of a Cartesian coordinate system and thus, the curvature of the screw channel is ignored. The clearance between the screw flights and the barrel surface is assumed to be very small, and therefore the effect of leakage flow on the flow rate is ignored. In accordance with the flat-plate approximation with moving
Simulation
To calculate numerical solutions for each of the 87,840 design points derived by applying the theory of similarity, we used the software package ANSYS Fluent 16.2 that is based on the finite-volume method. To allow a fast computational solving process, the simulation set-up was parameterized in terms of geometry, material properties, and processing conditions (including the pressure gradient in the down-channel direction and the velocity boundary conditions). The front and side views of the
Heuristic modeling
The results analyzed in the previous section represent numerical solutions for each design point derived by application of the theory of similarity. However, a mathematical relationship between the characteristic input quantities (h/w, t/Db, n, and Πp,z) and the target variable (Πv) is not yet achieved. To avoid numerical calculations, we sought an approximate analytical relationship that predicts the numerical results without the need for simulations:
We used the open-source
Conclusion
We have developed an algebraic relationship for calculating the pumping capability of power-law fluids in three-dimensional screw channels under isothermal conditions. Our equation approximates accurately the solutions of an extensive parametric design study in which 87,840 physically independent set-ups derived according to the theory of similarity were solved numerically. By applying dimensional analysis, the three-dimensional extruder flow was shown to be defined by four dimensionless input
Acknowledgments
This work was supported by the Austrian COMET K2 program of the Linz Center of Mechatronics (LCM), and was funded by the Austrian Federal Government and the Federal State of Upper Austria.
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