Discretization of construction

2.1. Discretization of construction

Numerical calculations were conducted using the method of finished elements. As a calculating tool ABAQUS^® programme was used. The construction of discrete

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models was made on the basis of the developed geometrical models of analysed constructions, presented in the previous point.

The discretization of a geometrical model was made using tetragonal solid elements, type C3D4T, which constitute 4-nodal elements with the shape function of the first order and full integration to enable to take into account in a numerical analysis a thermal degree of freedom. In all variants it was accepted that elements of construction – special screws of the extruder - are made of steel marked 40HM.

In strength calculations, a bilinear model of material was accepted, of elastic-plastic characteristics, for which the following material properties were accepted (Table 1):

Table 1. Mechanical properties of steel 40HM Material property – steel 40HM

In thermal calculations, a bilinear model of material was accepted, of elastic-plastic characteristics, for which the following thermal properties were accepted (Table 2):

Table 2. Material properties for thermal calculations Material property – steel 40HM

Density  [kg/m³] 7860

Linear expansion coefficient [1/K ] 1.2 ·10^-5 Heat conduction coefficient

 [W/(m·K)] 58

Specific heat [J/(kg·K) 450

A discrete model of the special screw developed for variant 1 is presented in Figure 24.

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Fig. 24. Discrete model of the special screw – variant 1

A discrete model of the special screw developed for variant 2 is presented in Figure 25.

Fig. 25. Discrete model of the special screw – variant 2

A discrete model of the special screw developed for variant 3 is presented in Figure 26.

Fig. 26. Discrete model of the special screw – variant 3

2.2. Boundary conditions and loading of construction

The numerical analysis of the adopted special extruder screws were carried out for four loading cases:

– case I: the screw loaded with the torque Mc = 200 Nm;

– case II: the screw loaded with the pressure pmax = 25MPa;

– case III: the screw loaded with the torque Mc = 200Nm and the pressure pmax = 25MPa;

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- case IV: the screw heated to the temperature tmax = 190°C paralelly loaded with the torque Mc = 200Nm and the pressure pmax = 25MPa.

a) Boundary conditions and loading of construction – case I

The definition of boundary conditions in a numerical model for case I of calculations was carried out by fixing nodes placed on the surfaces of mounting the special screw (Fig. 27), blocking the possibility of their displacement (translational degrees of freedom of nodes placed on these surfaces) in three directions X, Y and Z.

The torque Mc = 200 Nm was applied to the drive shaft of the screw. Exemplary boundary conditions and the loading of the special screw for selected variants 1, 2 and 3 are presented in Figures 27, 28 and 29.

Fig. 27. Boundary conditions and loading of numerical model of the special screw - variant I

Fig. 28. Boundary conditions and loading of numerical model of the special screw – variant II

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Fig. 29. Boundary conditions and loading of numerical model of the special screw – variant III

b) Boundary conditions and loading of construction – case II

The definition of boundary conditions in a numerical model for case II of calculations was carried out by fixing nodes placed on the surfaces of mounting the special screw (Fig. 30), blocking the possibility of their displacement (translational degrees of freedom of nodes placed on these surfaces) in three directions X, Y and Z. The pressure resulting from the pressures exerted by the transported material was applied by means of mathematical functions reaching the maximum pressure pmax = 25 MPa in the area of the largest diameter of the screw core. Exemplary boundary conditions and the loading of the special screw for selected variants 1, 2 and 3 are presented in Figures 30, 31 and 32.

Fig. 30. Boundary conditions and loading of numerical model of the special screw – variant I

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Fig. 31. Boundary conditions and loading of numerical model of the special screw – variant II

Fig. 32. Boundary conditions and loading of numerical model of the special screw – variant III

c) Boundary conditions and loading of construction – case III

The definition of boundary conditions in a numerical model for case III of calculations was carried out by fixing nodes placed on the surfaces of mounting the special screw (Fig. 33), blocking the possibility of their displacement (translational degrees of freedom of nodes placed on these surfaces) in three directions X, Y and Z.

The torque Mc = 200 Nm was applied to the drive shaft of the screw. The pressure resulting from the pressures exerted by the transported material was applied by means of mathematical functions reaching the maximum pressure pmax = 25 MPa in the area of the largest diameter of the screw core. Exemplary boundary conditions and the loading of the special screw for selected variants 1, 2 and 3 are presented in Figures 33, 34 and 35.

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Fig. 33. Boundary conditions and loading of numerical model of the special screw – variant I

Fig. 34. Boundary conditions and loading of numerical model of the special screw – variant II

Fig. 35. Boundary conditions and loading of numerical model of the special screw – variant III

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d) Boundary conditions and loading of construction – case IV

The definition of boundary conditions in a numerical model for case IV of calculations was carried out by fixing nodes placed on the surfaces of mounting the special screw (Fig. 33), blocking the possibility of their displacement (translational degrees of freedom of nodes placed on these surfaces) in three directions X, Y and Z.

The torque Mc = 200 Nm was applied to the drive shaft of the screw. The pressure resulting from the pressures exerted by the transported material was applied by means of mathematical functions reaching the maximum pressure pmax = 25 MPa in the area of the largest diameter of the screw core. Additionally, the screw was heated by means of a defined mathematical function in order to reflect the physical behaviour of the screw during the polymer extrusion process, the maximum value of the temperature of the screw was accepted T ≈ 190°C (Fig. 36). The initial temperature of the numerical model was accepted as T0 = 22°C. Exemplary boundary conditions and the loading of the special screw for selected variants 1, 2 and 3 are presented in Figures 36, 37 and 38.

Fig. 36. Boundary conditions and loading of numerical model of the special screw – variant 1

Fig. 37. Boundary conditions and loading of numerical model of the special screw – variant 2

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Fig. 38. Boundary conditions and loading of numerical model of the special screw – variant 3