Calculation of the Temperature Field

3.5.1.1 General Presentation

The temperature field is calculated by solving the energy equation with appropriate boundary conditions [20-22]. The process is broken down into different cooling steps, characterised by their length and the boundary conditions on the pipe surfaces. During the first step (path in air between die exit and calibrator entrance), the polymer can be stretched.

Pipe extrusion is considered as a steady-state process and temperature is assumed to depend only on r and z. Heat conduction in the flow direction (z) is neglected with respect to heat convection in that direction. Conversely, convection along r is neglected. Viscous heat dissipation is supposed to be small compared to other terms.

With such assumptions, the energy equation is reduced to:

2

This equation is solved by an explicit finite difference method, with a 2D mesh along z (i) and r (j).

3.5.1.2 Boundary Conditions

In air or water, convective conditions are considered at the pipe surfaces:

(

_{surface fluid}

)

where h is the heat transfer coefficient, T_surface is the surface temperature of the pipe and T_fluid is the temperature of the cooling fluid. When temperature measurements are available, h is usually adjusted to fit the experimental data.

For thermal exchanges in the calibrator, the heat transfer coefficient h_cal can be calculated by a simple model (Figure 3.30):

1 1

cal film of the water film between the pipe and the calibrator (~ 100 mm) and k_water is the conductivity of water (0.63 W.m^-1. °C^-1). Hence 1/h_cal ~ 10^-3 + 10^-6 + 10^-4, which shows that the major role in the heat transfer is played by convection between the calibrator and the water bath. Therefore, h_cal can be taken to be identical to the heat transfer coefficient for direct contact between the pipe and the cooling water.

Water of the cooling tank

Calibrator Water film PA12 tube q

Figure 3.30 A simple model for thermal exchanges in the calibrator. Reproduced with permission from J.M. Haudin, A. Carin, O. Parant, A. Guyomard, M. Vincent, C. Peiti and F. Montezin, International Polymer Processing, 2008, 23,

55. ©2008, Hanser [22]

3.5.1.3 Crystallisation

Crystallisation has been treated in two different ways. In a first approach, the energy equation was modified to take into account the heat liberated by crystallisation:

z

Pipe Forming Tools

where DH is the actual enthalpy of crystallisation per unit mass and χ(r,z) is the transformed volume fraction (solidified fraction) at point (r, z). The kinetic law χ(t) was described by the extension of Ozawa’s equation to nonconstant cooling rate (see Section 2.4.1).

In a second simplified approach, the numerical code directly used an experimental crystallisation curve recorded at an appropriate cooling rate. The gradual release of the latent heat between the onset (T_onset) and the end (T_end) of crystallisation gives rise to an exothermal peak, which is approximated by an isosceles triangle with an area equal to the enthalpy of crystallisation per unit mass DH. At each node (i, j), the temperature is first calculated by solving Equation 3.9, i.e., without taking crystallisation into account. Then, the temperature value is modified to take into account the heat released by the mass element at node (i, j). The transformed fraction χ is defined as the ratio of the heat already released to the total heat of crystallisation.

In both approaches, the crystallisation temperature is usually defined as the temperature corresponding to χ = 0.5.

3.5.1.4 Validation of the Model and Typical Results

In the case of PE, the temperature of the internal surface of the pipe could be measured, and the two treatments of crystallisation presented above could be compared. It appeared that the simplified approach gave the better results. Indeed, a good agreement between calculations and experiments is observed both for different line speeds (Figure 3.17) and different pipe thicknesses (Figure 3.31). Therefore, this simplified method was used in subsequent stress calculations, since such calculations require an accurate description of the temperature field.

For PA12 pipes, the experimental analysis presented above has shown the importance of drawing in the calibrator on structure development. Therefore, the model should consider the influence of flow on crystallisation kinetics. Unfortunately, no data are available for crystallisation under elongation, and this is the reason why the simplified approach has been retained. This approach works here since it allows, in a practical way, a balance between cooling effects (which tend to decrease the crystallisation temperature), and drawing ones (which tend to increase it). The temperature and transformed fraction profiles obtained by using the model are presented in Figure 3.32 for three locations in the pipe thickness: external surface, centre, and internal surface.

250

Figure 3.31 Evolution of inner temperature of LDPE pipes for different thicknesses. Comparison between calculations by the simplified crystallisation

model (solid lines) and experiments (symbols). Pipe thickness: 0.45 mm (1), 0.65 mm (2), and 0.90 mm (3)

The temperature evolution is calculated from the die exit to the end of the process (8 m). It gives access to the mean cooling rates in the calibrator: 400 °C.s^-1 (external surface), 60 °C.s^-1 (middle) and 30 °C.s^-1 (internal surface). This shows a strong cooling of the pipe external surface during calibration. With the scale used in Figure 3.32a, crystallisation appears as a very small accident on the temperature evolution, which is visible only at the internal surface. Therefore, we also use the transformed fraction curves (Figure 3.32b) and an enlargement of the temperature evolution at the external surface to analyse the crystallisation phenomena in more depth (Figure 3.33). In the external zone, the crystallisation exothermal peak is observed at about 0.11 m from the die exit, at a temperature of 125 °C. Crystallisation ends (transformed fraction equal to 95%) at 0.15 m from the die exit, compared to 1.5 and 1.8 m for the central and internal zones, respectively. Taking into account the distance between the die and the calibrator entrance (0.065 m), this means that most of the crystallisation of the external surface occurs inside the calibrator. The solidified layer at the calibrator exit is about 20-30 mm thick.

Pipe Forming Tools

Figure 3.32 Evolution of: (a) temperature, and (b) transformed fraction for three locations in the pipe thickness

200

100 150

50

0 0.05 0.15

Distance from extrusion die (m)

Temperature (ºC)

0.25 0.1

Crystallisation Calibrator entrance

Calibrator exit

0.2 0.3

Figure 3.33 Enlargement of the temperature evolution at the external surface of the tube. Location of the crystallisation zone

In document Design of Extrusion Forming Tools (Page 108-113)