University of Minho School of Engineering <IPC/I3N – Institute for Polymers and Composites>

Uma Escola a Reinventar o Futuro – Semana da Escola de Engenharia - 24 a 27 de Outubro de 2011

AbstractThe inherent design freedom promoted by the employment of thermoplastic profiles is one of the major reasons for their attractiveness. Theoretically, thermoplastic profiles can be produced with any cross section suited for a specific application. The design of the corresponding extrusion dies usually employ a methodology based on experimental trial-and-error approaches, being highly dependent on the experience of the designer and highly demanding in terms of resources. These difficulties are obviously more evident when the plastic profile has a complex geometry.

This research team is involved since the mid-nineties on the development of computational tools to aid the design of thermoplastic profile extrusion dies. Initially, the numerical code employed was based on structured meshes that limited its use to simple geometries.

In this work, a numerical modelling code developed to work with unstructured meshes is described and employed in a case study involving the design of a extrusion die for the production of complex cross section profile.

The results obtained show that the developed code can be a useful tool to aid the design of complex profile extrusion dies.

Numerical Modelling CodeFor modelling purposes the Navier Stokes equations for an incompressible generalized Newtonian fluid, under isothermal conditions, were considered. This set of equations, in tensorial notation, comprises the mass:

and linear momentum:

conservation equations, were u is the velocity, the specific mass, p the pressure and the deviatoric stress tensor.For the case of a generalized Newtonian fluid the deviatoric stress tensor is given by:

where the shear viscosity is a function of the second invariant of the rate of deformation tensor

NELSON D. GONÇALVES* Supervisor: João M. Nóbrega, Co-supervisor: Olga S. Carneiro

* nelson.goncalves@dep.uminho.pt

COMPUTER AIDED DESIGN OF EXTRUSION DIES FOR COMPLEX GEOMETRY PROFILES

with

For this work, a Bird-Carreau constitutive equation was employed for the shear viscosity function:

Being the zero shear-rate viscosity, the shear viscosity at very high shear rates, the characteristic time and n the power-law index.

The governing equations were discretized following the Finite Volume Method for unstructured meshes.

Case StudyThe numerical modelling code described in the above was used to optimise the flow distribution in a profile extrusion die required to produce a complex profile whose cross section is illustrated in Fig. 1a. Due to symmetry, just half of the geometry was modelled, as shown both in the cross section geometry, Fig. 1b, and in the flow channel geometry used for the first trial, Fig. 1c.

where Qi and Qobj,i are the current and required (objective) volumetric flow rates of a specific ES, respectively, Ai and A are the area of each ES and of the total cross section, respectively, and NES the number of ES.

The polymer melt employed in the numerical runs was modelled using a generalized Newtonian constitutive equation, Bird-Carreau, with the following parameters

, , , .

From the results shown in Figures 2 and 3 it is evident that the initial trial geometry promote a highly unbalanced flow distribution, but a significant improvement was achieved with the final solution.

Figure 2 – Outlet cross section of the flow channel for the tested trials and contours of the outlet flow distribution obtained.

Figure 3 – Results obtained for the trial geometries: (a) objective function and (b) ratio between actual and required flow rate per ES.

ConclusionIn this work the development of a 3D numerical modelling code, based on the Finite Volume Method, able to model the flow of Generalized Newtonian Fluid with unstructured meshes, was described.The results obtained allow to conclude that the numerical code developed is able to deal with complex geometrical problems, being an essential tool to aid the design of profile extrusion dies.

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uu ux x x

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ES3

ES4

(a) (b) (c)

Figure 1 – Case study geometry: (a) cross section of the profile (dimensions in mm); (b) half of the cross section considered for the computational model, showing the division into Elemental Sections;

(c) typical flow channel geometry.

The flow distribution was monitored by the division of the profile cross section into 4 Elemental Sections (ES), as depicted in Fig. 1b.The performance of a specific solution was evaluated using the following Objective Function:

40 5.58 10 Pa s 0Pa s 3.21 0.3014n

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i iobj

i obj i

Q AFQ A

0 8velocity (m/s)

T1 T2 T3 T4 T50

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Trial

Fobj

T1 T2 T3 T4 T50

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0.4

0.6

0.8

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1.2

1.4

1.6

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Trial

Qi / Qobj,i

(a) (b)