3 Pipe Forming Tools - Smithers · PDF file64 Design of Extrusion Forming Tools Spider leg...

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63 Jean-Marc Haudin, Michel Vincent, Bruno Vergnes 3.1 Introduction The production of pipes is usually performed according to the scheme presented in Figure 3.1: Circular saw Haull-off system Cooling bath Pipe Die Extruder Calibrator Figure 3.1 Extrusion line for pipe production An extruder continuously feeds an axisymmetric die, by means of which the geometry of the pipe is obtained. At the die exit, the pipe passes through a calibrator, in order to cool down its surface and to freeze the outside diameter. Then, the pipe is pulled through different cooling baths, before being eventually cut and stored at the end of the extrusion line. In the present chapter, we will focus only on the die and the calibrator. Different types of die may be used [1, 2], but the most common is the spider leg die, shown in Figure 3.2. 3 Pipe Forming Tools

Transcript of 3 Pipe Forming Tools - Smithers · PDF file64 Design of Extrusion Forming Tools Spider leg...

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Jean-Marc Haudin, Michel Vincent, Bruno Vergnes

3.1 Introduction

The production of pipes is usually performed according to the scheme presented in Figure 3.1:

Circular saw

Haull-offsystem Cooling bath

Pipe

Die Extruder

Calibrator

Figure 3.1 Extrusion line for pipe production

An extruder continuously feeds an axisymmetric die, by means of which the geometry of the pipe is obtained. At the die exit, the pipe passes through a calibrator, in order to cool down its surface and to freeze the outside diameter. Then, the pipe is pulled through different cooling baths, before being eventually cut and stored at the end of the extrusion line. In the present chapter, we will focus only on the die and the calibrator.

Different types of die may be used [1, 2], but the most common is the spider leg die, shown in Figure 3.2.

3 Pipe Forming Tools

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Design of Extrusion Forming Tools

Spider leg

Mandrel

z

θ

r

Figure 3.2 Example of geometry of a spider leg die

The axisymmetric die consists of a mandrel connected to the carter by streamlined parts, called spider legs, which are regularly distributed around the periphery. The molten polymer flows around these spider legs and joins again downstream, forming welding lines. The final part of the die, called the land, has a constant cross section and defines the final dimensions of the pipe. This type of die allows one to produce a wide variety of pipes, from medical catheters of millimetre size to water pipes up to 2 m in diameter.

3.2 Flow Through Pipe Dies

3.2.1 The Different Approaches from One-dimensional to Three-dimensional

Even though the geometry of a pipe die is three-dimensional (3D), different assumptions may be used to simplify the calculation of the flow.

First of all, inside the die, the flows are confined, without free surface and without abrupt changes of cross-section (see Figure 3.2). They are primarily controlled by shear, with elongation playing a limited role. Accordingly, as explained in Section 2.2.2, elastic effects can be neglected and a viscous behaviour (power or Carreau type laws) is generally sufficient to obtain a good approximation of the flow conditions. However, thermal effects can be important and a coupling between mechanical and thermal equations is necessary.

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As pipes are usually extruded horizontally (see Figure 3.1), mass and inertia forces can be neglected (see Section 2.1). Moreover, due to the progressive change of the cross-section area in the flow direction, lubrication approximations can be systematically applied in order to simplify the flow equations [3]. As a consequence, three approaches can be developed to characterise the flow conditions in a pipe die:

(a) a very simple one-dimensional (1D) approach, by means of which the orders of magnitude of pressure and temperature along the die may be obtained (see Section 3.2.2);

(b) a two-dimensional (2D) approach, using lubrication approximations and based on finite volume or finite element calculations (see section 3.2.4); and

(c) a full 3D approach, for example for calculating the flow around the spider legs.

3.2.2 One-dimensional Calculation

Let us consider the die presented in Figure 3.2, used to produce polyvinyl chloride (PVC) pipes at a flow rate of 300 kg/h. The pipe has an internal diameter of 10 mm and a thickness of 5 mm. If we neglect the zone around the spider legs, the flow can be considered as axisymmetric. Applying the lubrication approximations (the velocity in the radial direction is negligible compared to the velocity in the flow direction) and considering a power law behaviour, Stokes equations reduce to the following expression, relating the pressure gradient dp/dz to the volume flow rate Q:

2 2

1 1

1 12 2

2 ( 2 )n

nn nR r R R

R R R r

dp K Qdz R Rr u dudr r u dudr

u u

π∗

/ /∗ ∗

− /= − + − ∫ ∫ ∫ ∫

(3.1)

where K is the polymer consistency (see Section 2.2.1), n the power law index, R1 and R2 the radius of the mandrel and carter, respectively, and R* the location where the velocity is at its maximum (Figure 3.3). This expression is written locally, meaning that R1, R2 and R* vary with the axial coordinate z.

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R1(z)

R1(z)

R*(z)

R2(z)

R2(z)

z

r

Figure 3.3 Local geometry of a pipe die

The average temperature T across the flow thickness can be calculated by solving:

(3.2)

where α is the thermal diffusivity, Tw the temperature of the carter, assumed to be constant, and rcp the heat capacity. The mandrel is supposed to be adiabatic and Nu is a Nusselt number, qualifying the heat transfer between polymer and carter [4]. Equations 3.1 and 3.2 are coupled through the value of the consistency K, which depends on temperature, for example according to the Arrhenius law (see Section 2.2.3):

0

0

1 1exp aEK KR T T

= −

(3.3)

where T is the temperature, K0 the consistency at the reference temperature T0, Ea the activation energy and R the gas constant.

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The resolution of Equations 3.1 and 3.2 step by step all along the pipe die enables the evolution of the pressure and temperature to be calculated, as shown in Figure 3.4. We observe that the pressure drop is 21 MPa, mainly concentrated along the final land. The average temperature increases also principally along the die land, from 186 to 192 °C (the fixed carter temperature Tw is 185 °C).

PRE

SSU

RE

p (

bar)

DIE LENGTH z (cm)10 30 50

193

191

189

187

185100

200

ME

AN

TE

MPE

RA

TU

RE

Spider legs

P

T

Figure 3.4 Pressure and average temperature evolution along the pipe die. Adapted from B. Vergnes and J.F. Agassant, Advances in Polymer Technology, 1986, 6,

441. ©1986, Wiley [3]

These first results show that the die land is the ‘sensitive’ zone of the die where the thickness is the weakest and, consequently, the shear rates the highest. To better understand the flow conditions in this section, we can now develop a 2D approach for the temperature.

3.2.3 Temperature Computations

In this section, we keep the same mechanical 1D approach, but we consider a temperature profile across the flow thickness. For that, we have to solve the following equation using, for example, a finite difference method:

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1n

pT k T dwc w r Kz r r r dr

r+∂ ∂ ∂ = + ∂ ∂ ∂

(3.4)

where w is the velocity component in z direction and k the thermal conductivity.

It is possible to show that the adiabatic condition on the mandrel, previously used in the 1D model, is no more realistic [5]. We assume now that the mandrel is in thermal equilibrium: it receives a heat flux from the flowing polymer and releases calories to the external carter through the spider legs. After a certain time, it will reach equilibrium, with a temperature resulting from the flow conditions.

The calculations have been performed in the conditions mentioned previously (300 kg/h, T0 = 183 °C, Tw = 185 °C). Figure 3.5 shows the temperature profiles at different locations between the spider legs and the land end (from points A to C, see Figure 3.4).

Temperature (°C)

Dim

ensi

onle

ss r

adiu

s

190 2001800

0,2

0,6

1A B C

Figure 3.5 Temperature profiles at different locations along the pipe die. The dimensionless radius is defined as (r-R1)/(R2-R1)

We can see that the temperature of the mandrel is 198 °C, i.e., much higher than the imposed temperature on the carter (185 °C). It induces a large temperature increase close to the mandrel wall, which could possibly lead to polymer degradation (especially in the case of PVC extrusion). The temperature profile is heterogeneous and presents two hot spots in the sheared zones, close to the walls. Overheating can reach 17 °C at the land exit, which could lead to possible troubles in the post-extrusion operations (calibration, cooling, residual stresses, etc). A possible solution would be to implement

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a thermal regulation of the mandrel, by circulation of a cooling fluid, in order to impose a controlled temperature.

3.2.4 An Example of Nonaxisymmetric Flow

In the previous sections, the flow in the pipe die was considered as axisymmetric. It can sometimes be different. To illustrate this point, we refer to the problem of thickness homogeneity encountered in the production of pipes, in the conditions previously presented [6]. Instead of having a constant thickness of 5 mm, the pipe presented a thinner value in the vertical plane (4.7 mm) and a thicker one in the horizontal plane (5.3 mm) (Figure 3.6).

emin = 4.7 mm

emax = 5.3 mm

e = 5 mm

(a) (b)

Figure 3.6 Example of correct pipe (a); and defect of symmetric thickness heterogeneities (b); e is the pipe thickness

Because of the symmetry, this defect could not be due to a geometrical problem (mandrel off-centring would lead to an asymmetric defect). In fact, it was related to temperature heterogeneities created by the flow in the counter-rotating twin screw extruder used to plasticise the PVC and to feed the die. Measurements in the channel between extruder and die have effectively shown that, in the cross-section, two hot zones were present in a horizontal plane (Figure 3.7).

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Mandrel

10 °C

a

aa

a

Figure 3.7 Temperature field in a cross-section of the channel between the twin screw extruder and the pipe die. The grey zones are the hottest. The temperature

profile along the a-a line is also shown

These hot zones were around 10 °C hotter than the polymer in the periphery. We can then imagine that, when arriving on the mandrel, these zones were separated and flowed on each side of the mandrel, leading to a local increase of flow rate due to the lower viscosity. To validate this assumption and to calculate the flow conditions, it is no longer possible to consider an axisymmetric situation. We have to use a 2D model, applying lubrication approximations. Moreover, to simplify the problem, we focus on the final die land (we have seen previously that it was the ‘sensitive’ part) and we ‘unroll’ the annular space between mandrel and carter as shown in Figure 3.8.

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Figure 3.8 Unrolling of the die land zone

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The calculation of the flow and temperature fields can be achieved by solving the following set of equations [3]:

22 2 2 2

2 2

22

2 1

21 0

n p h p h h p K p Kh pn x x y y Kn x x y y

p p p p p p px x x y x y y ynh

n p px y

+ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∆ + + − + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ ∂ + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂− + = ∂ ∂ + ∂ ∂

(3.5)

(1 ) 221 (2 1) 2

(1 )/221 (2 1) 2

1 2 1( )2 1 2

1 2 1( )2 1 2

n nn n n

n nn n n

n h p p pu x yh n K x y x

n h p p pv x yh n K x y y

− // + /

−/ + /

∂ ∂ ∂ , = − + + ∂ ∂ ∂

∂ ∂ ∂ , = − + + ∂ ∂ ∂

(3.6)

(3.7)

where p(x, y) is the pressure, h(x, y) the local thickness, u and v the two components of the average velocity, and T (x, y) the local average temperature. The consistency K is a function of the average temperature through an Arrhenius law (Equation 3.3).

Equations 3.5 to 3.7 are solved using a finite difference iterative model. The results are shown in Figure 3.9. We have selected initial conditions corresponding to the experimental measurements, i.e., initial temperatures varying from 180 °C (cold zones) to 192 °C (hot zones).

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Tr(°C)

200185180

193

192 192 192192

180

181

184

186

188

184

187

181

180

194

0,9

0,96

1,04

1,081,1

1 1

194

y

x

y

y

Tr(°C)

200

180

193

191 191 191

191

(b)(a)

Positionalong y

Positionalong y

191190

190 190

x

y

qx/Q

qx/Q

Figure 3.9 Regulation temperature rT , temperature field in the die land and final flow rate distribution (from top to bottom); a) homogeneous thermal regulation,

and b) heterogeneous thermal regulation; xq is the local flow rate and Q the average one

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If we consider a homogeneous wall temperature Tw of 185 °C (Figure 3.9a), we can see that, during the flow along the die land, the temperature increases slightly, due to viscous dissipation and heat transfer. Even though the temperature difference between cold and hot zones is reduced (from 12 to 7 °C), we observe that the exit flow rate is heterogeneous. We have reported the ratio of the local flow rate to the average flow rate in Figure 3.9. A homogeneous distribution would provide a constant value of 1. In the present case, we observe higher values (+ 10%) at positions corresponding to the hot zones and lower values (– 10%) at positions corresponding to the cold zones. This confirms that the initial heterogeneity in temperature is at the origin of the thickness defect. Moreover, modelling may also provide a solution to overcome this problem. Let us assume now that we impose around the die a heterogeneous thermal regulation, with higher values of Tw (200 °C) in the cold zones and lower values (180 °C) in the hot zones (Figure 3.9b). We can see that, in this case, the temperature difference can be suppressed before the land exit, leading to a much better homogeneity of the flow rate, and hence of the pipe thickness. It is now common practice to use such heterogeneous regulation systems in pipe extrusion.

3.3 Pipe Calibration – Experimental

3.3.1 Technological Review

After the die exit, the pipes travel on a short distance in the air before entering a calibration device, which is placed in a tank where a cold water bath or jets ensure cooling and polymer solidification (Figure 3.10a). The objective of calibration is to control and adjust the outer circular section of the pipe before solidification allows it to resist gravity.

Even if internal calibration exists, where the internal radius is ensured by a calibrating device, the simpler outer calibration is more widely used. A difference of pressure between the inside and outside of the pipe is imposed, so that the outer surface sticks to the calibrating device. The difference of pressure is usually imposed by an adjustable pressure lower than the atmospheric one in the cooling tank, although a higher pressure inside the tube may also be used.

The calibration device may be simply rings with decreasing diameter, but the preferred system is a tubular sizing device (Figure 3.10). The calibrator is generally made of brass, copper or bronze. Holes are drilled in the sizing device, so that the pressure difference sticks the tube against the calibrator, ensuring the correct tube dimension. The calibrator diameter is usually several per cent larger than the final required tube diameter to counterbalance the effect of thermal shrinkage.