Wissenschaftlicher Zeitschrift Kunststofftechnik Journal ... · into the temperature profile over...

© Carl Hanser Verlag Zeitschrift Kunststofftechnik / Journal of Plastics Technology 5 (2009) 2

eingereicht/handed in: 10.09.2008 angenommen/accepted: 07.01.2009

Dipl.-Ing. Karsten Anger, Prof. Dr.-Ing. Helmut Potente, Prof. Dr.-Ing. Volker Schöppner, Dipl.-Ing. H. Hörmann, Institut für Kunststofftechnik KTP, University of Paderborn

Prof. Dr. Sönke Hansen, Institut für Mathematik, University of Paderborn

Radial Temperature Modelling inside the Screw Channel for Tempered Single Screws To achieve a constant product quality during extrusion and injection moulding processes it is of utmost importance to keep a certain mass temperature inside the screw channel and over the screw length. Local temperature peaks have to be avoided, as they can lead to thermal degradation of the material. For the theoretical characterization of the radial temperature profile inside the screw channel and the resultant temperature profile over the screw length mathematical-physical models are required. With help of these models it is possible to optimize the design of a plasticizing unit for a particular application. Modelling the temperature profiles for screws without tempering is the state of the art. However, the calculation of radial temperature profiles of thermoplastics for tempered screws was impossible on the basis of existing mathematical-physical models until now. So far, the screw was regarded as an adiabat object without heat flux into the screw.

Modellierung des radialen Temperaturprofils im Schneckenkanal von temperierten Einschnecken Im Extrusions- und Spritzgießprozess ist das Erreichen und Einhalten einer bestimmten Masse-temperatur im Schneckenkanal und über der Schneckenlänge in der Plastifiziereinheit von großer Bedeutung für die Qualität des Extrudates. Lokale Überhöhungen der Temperatur müssen vermieden werden, da diese zu einer thermischen Schädigung der Schmelze führen können. Deshalb ist die modelltheoretische Beschreibung des radialen Temperaturprofils im Schneckenkanal und des Tem-peraturprofils über die Schneckenlänge sehr wichtig, um Plastifiziereinheiten optimal für die jeweilige Anwendung auslegen zu können. Zur theoretischen Beschreibung des Temperaturverhaltens sind mathematisch-physikalische Modelle nötig. Die Modellierung dieser Temperaturprofile für Schnecken ohne Innentemperierung stellt den Stand der Technik dar, allerdings ist bisher die Berechnung des radialen Temperaturverlaufs von innentemperierten Thermoplastschnecken auf Basis mathematisch-physikalischer Modelle nicht möglich gewesen, da die Schnecke als adiabat angesehen wurde.

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Anger et al. Radial Temperature Modelling

Journal of Plastics Technology 5 (2009) 2 154

Radial Temperature Modelling inside the Screw Channel for Tempered Single Screws

K. Anger, H. Potente, V. Schöppner, H. Hörmann, S. Hansen

1 INTRODUCTION

The extrusion process is one of the most important value creation processes within polymer processing. It provides the production of endless products in a continuous operation. Within this process the plasticizing unit plays a key role for initialization of the raw material to form the product geometry. Therefore, the theoretical forecast of the process parameters inside the plasticizing unit is of utmost importance.

The temperature profile over the screw length and the initialized mass temperature at the screw tip are influencing the product quality, due to the temperature dependent viscosity and the risk of material degradation.

For a better handling of the temperature profile, tempering systems are used to supply or remove heat in sections. Besides the generally used barrel tempering systems, sometimes, screw cooling systems are beneficial. For example these systems could be used to process thermal sensitive materials or to avoid temperature peaks.

For the theoretical characterization of the radial temperature profile inside the screw channel and the resultant temperature profile over the screw length mathematical-physical models are required. With help of these models it is possible to optimize the design of a plasticizing unit for a particular application. Modelling the temperature profiles for screws without tempering is the state of the art. So far, the screw was regarded as an adiabat object without heat flux into the screw. Therefore, the modelling of radial temperature profiles of thermoplastics for tempered screws is not possible on the basis of existing theoretical basics until now.

Within a state-aided project of the DFG (Deutsche Forschungs Gemeinschaft) a temperature model on basis of an existing approach for the radial temperature profile inside the screw channel was developed. In this case the screw has to be regarded as a non adiabat object. For an analytical calculation of the radial temperature profile and the resultant profile over the screw length, the energy equation has to be reduced and solved. The model of the radial temperature profile for internal tempered screws inside the channel and practical results are presented in this paper.

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2 SIMPLIFICATION OF THE ENERGY EQUATION

To be able to calculate the temperature profile inside a plasticizing unit, the energy equation has to be used and solved. However, some assumptions and simplifications have to be made to solve the energy equation analytically because of its complexity. After the solving process of this reduced and simplified equation the radial temperature profile can be identified in screw sections of constant characteristics. The boundary conditions and material parameters, except the viscosity, are regarded as constant in the actual section. The screw is divided into a certain amount of separate sections, which is shown in Figure 1.

Figure 1: Sectioning of the screw length with constant characteristics

For each section the radial temperature profile inside the screw channel has to be calculated, which is the basis for the average radial temperature. A radial temperature profile and the appropriate average temperature are shown in Figure 2.

Figure 2: Radial temperature profile inside the screw channel

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The combined calculated average radial temperature for each section results into the temperature profile over the screw length. An axial profile over the screw length is shown in Figure 3.

Figure 3: Axial temperature profile inside the screw channel

In the following the approach to reduce the energy equation for calculating the radial temperature profile in each section is illustrated. The general energy equation without the source term for an open system is [12]:

vTTvt

Tc

2 Equation 1

To be able to solve the energy equation analytically, the following assumptions and simplifications are used to reduce it, this procedure was trusted in the past [1, 2, 3, 9, 10 and 11]:

- kinematic reversal and unwinded screw channel

Figure 4: Kinematic reversal and unwinded screw channel

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- the melt is incompressible, thus ccc pV

- the melt flow is laminar and stationary, resultant 0

t

T

- no velocity components in x - and y - direction, hence 0xv and 0yv

- the channel is completely filled with melt (In this assumption there are no solid parts)

- the melt flow does not slip at wall

- the heat flux in direction of the channel and across the channel is zero, thus 0yq and 0zq

- the height of the channel is much smaller than the width, therefore bh and 0xz

In the following the used material laws are listed:

- the viscosity behaviour of the melt is described by the Power law, thus nTK )(

- all material parameters except the viscosity are temperature invariant in the observed section

- the temperature dependence of the viscosity is described by the reduced

Arrhenius law, thus 0

0)( TTT eKTK

.

With these assumptions, simplifications, material laws and the heat conduction equation of Fourier the energy equation can be reduced to the following [1, 3 and 9]:

0

2

2TT

z ex

T

z

Tvc

Equation 2

In this equation is the dissipated energy across the channels cross-section per volume and zv is the averaged flow velocity in z -direction. For a better handling the energy equation can be written dimensionless as followed ( 10 , 10 ) [1, 3 and 9]:

ZTGz

Br

Gz

exp

12

2

Equation 3

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with

BT

TT 0 ;

h

x ;

z

z

B

nnT

B T

hvK

T

hBr

11

002

0 ; Vzb

hc

z

hvcGz z

2

Equation 4

Nevertheless, Equation 3 is not solvable easily, because of the dependency of the exponential term ZTexp on the temperature T or . Therefore, the exponential term is described by a linearization in sections (Equation 5):

ZZ TccTTT 210 expexp Equation 5

With this linearization the exponential term is subdivided in two terms:

- )(2 ZTc is dependent on temperature T or and

- 1c is independent on temperature T or .

The exponential term is approximated by a secant between the averaged radial temperature of the previous section 0T and the barrel temperature ZT (Figure

5).

Figure 5: Approximation of the exponential term by a secant (Example: =0,008 1/°C; T0=190°C; TZ=230°C)

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Resultant, there are two equations with the two unknowns 1c and 2c .

1.) For 0TT :

)(1exp 002100 TTccTT 11 c Equation 6

2.) For ZTT :

)(exp 0210 TTccTT ZZ

0

02

exp1

TT

TTc

B

B

Equation 7

with

c

tTt

cTtt

aba

aba

1

1log

20

1

20,

20,10 [1, 9]

Equation 8

with

St

St

StR

StR

TT

TT

TT

TT

Tta

0

0

0

6.101

86.8

6.101

86.8

, 10 and

St

St

StR

StR

TT

TT

TT

TT

Tta

206.101

2086.8

6.101

86.8

20,0

0

010 [1, 9]. Equation 9

With this linearization Equation 3 can be written as followed:

BrcBrTcGz B

122

2

Equation 10

This way of simplification and reduction of the energy equation in Equation 1 is a proved approach for its analytical solution [1, 3 and 9].

Consequently, Equation 10 is the reduced energy equation. It can be solved analytically to describe the radial temperature profile in an unwinded screw channel. In the past the Equation 10 was solved with the assumptions of a constant temperature at the cylinder and an adiabat screw [1, 3 and 9]. This solution is not allowable for tempered screw, because there is a not neglectable heat flux between the melt and the screw. This heat flux has to be included into the examinations.

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The heat flux at the screw ground is assumed as a constant temperature at the screw, as is shown in Figure 6.

Figure 6: Constant temperatures at the screw and at the barrel wall

The following equation show the boundary conditions of constant temperatures at the barrel wall 1 and the screw ground 0 :

B , if 1 and S , if 0 Equation 11

Furthermore, the solution of the reduced and simplified energy equation is made for the following starting condition (third-order polynomial). This third-order polynomial is used because of the degrees of freedom:

432

23

10 uuuu Equation 12

Afterwards, the Laplace-Transformation is used to solve Equation 10, which is a 2. order differential equation. The subsidiary equation in the Laplace area is:

0ˆ)(ˆ1

012

2

2

pGz

Brc

Gz

BrTcp

d

d

GzZ

with 10

Equation 13

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The assumption that in the subfield the temperature is only depending on one space coordinate (in this case ) is used to simplify the calculation. Furthermore, the showed boundary conditions have to be transformed into the Laplace area:

pZ

at the barrel 1 and

pS

at the screw 0 .

Equation 14

The solution of Equation 13 with Equation 12 and Equation 14 in the Laplace area is:

)sinh(

)sinh(1

)(

6

)sinh(

)sinh(

)(

)sinh(

)sinh(

)(

)sinh(

)sinh(

)(

)5.0cosh(

))5.0(cosh(1

)(

2

)5.0cosh(

))5.0(cosh(1

)5.0cosh(

))5.0(cosh(1

)(

)sinh(

))1(sinh(

)sinh(

)sinh(ˆ

22

1

2

3

2

2

2

3

2

1

22

2

2

4

2

1

Gz

BrTcpGz

u

Gz

BrTcp

u

Gz

BrTcp

u

Gz

BrTcp

u

Gz

BrTcpGz

u

Gz

BrTcp

u

Gz

BrTcppGz

Brc

pp

Z

Z

Z

Z

Z

Z

Z

Sz

Equation 15

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with

GzGz

BrTcp Z

2

Equation 16

Then, the Inversion Theorem is applied to each of the nine expressions on the right hand of Equation 15 [4, 5 and 6]. Hence, the dimensionless solution is:

222

2221

4

0

12/

331321

222

1

/

2

2

2

1

2

2

2

2

)12(

2

)12(

)12(sin)12(

4

)cos(

6

)cos(

cos

sin2

50cosh

)50(cosh1

sinh

)1(sinh

sinh

sinh

222

222

nπ

u

BrTcπn

Brcu

πnπn

e

nπnπ

u

nπnπ

uuu

BrTcπn

nπΘn)(π

Θ

πne

BrTc.

.BrTc

Tc

c

BrTc

BrTcΘ

BrTc

BrTcΘΘ

B

n

BrTcπ)n(Gz

B

SB

n

BrTcπnGz

B

B

B

B

BS

B

BB

B

B

Equation 17

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3 SPECIFICATION OF THE INFRARED THERMOMETER

To measure the radial temperature profile inside the screw channel for the experimental comparisons, a special developed thermometer, which is able to measure the temperature profile over the channel height, is needed. Therefore, the thermometer must have a radial adjustability. For this kind of investigations FOS Messtechnik GmbH in cooperation with the Institut für Kunststofftechnik at the University of Paderborn have developed a special IR-thermometer with an axial shiftable measuring tip. The front end of the thermometer can be seen in Figure 7.

Figure 7: The front end of the IR-thermometer with radial adjustment of the sensor to measure the radial temperature profile in the channel

To shift the front end of the IR-thermometer into the screw channel and to use the radial adjustability of the cone point, a gap has to be made through the screw flight, as it can be seen in Figure 8.

Figure 8: Gap for the IR-thermometer

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It is a standard size melt-thermometer with ½ inch-20 thread, 45° cone sealing and a front diameter of Ø7.80 mm. In the centre of the measuring tip a small sapphire window can be seen and also a thermocouple (“TIP 3”) is place next to the infrared thermometer. Thus, the radial temperature profile can be measured by using the infrared thermometer and also the thermocouple. In these investigations the thermocouple was used for measuring the radial profile.

It is possible to measure the temperature profile from the barrel wall up to 5 mm down to the ground of the screw channel. Figure 9 shows all the measuring points being integrated into this device, whereas the output of these signals is an amplified output-signal for experimental process investigations:

IR (1): Sapphire window for the IR-radiation input,

Sensor-Body (2): thermocouple being used for internal compensation purposes,

TIP(3): Thermocouple to detect the front end temperature of the tip,

Front (4): Thermocouple to detect the internal surface temperature at the interface of the barrel and the melt and

Sealing (5): Thermocouple to detect the temperature of the barrel in the sealing area.

Figure 9: Measuring points being integrated in the infrared thermometer [8]

4 SPECIFICATION OF THE MATERIAL PARAMETERS

Two independent series of experimental measurements were operated. For the first series of measurements a Polyethylene, DOWLEX SC 2108 G, and a Polypropylene, DOW R338-02N Resin, developed by DOW Chemical Company, were used. For the second series of measurements an Acrylonitrile Butadiene Styrene (ABS), DOW MAGNUM 3904, and a Polypropylene, Borealis FG 5190, were used.

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The Polyethylene DOWLEX SC 2108 G is suitable for the production of cast films with high stiffness, good tear strength and adequate toughness. Its Melt Index (190°C/2.16kg) is 2.5g/10min and its density 935kg/m³. The Vicat Softening Temperature is 118°C.

The natural Polypropylene Copolymer DOW R338-02N is a random copolymer developed for extrusion and blow moulding applications. It is a high molecular weight copolymer with medium ethylene weight distribution and it features high clarity. It is suitable for blow moulded containers, film, shrink film, thermoformed containers, stretch blow moulded bottles and calendaring. Its Melt Flow rate (230°C/2.16kg) is 2.0g/10min and its density is 900 kg/m³. The Vicat-Softening-Temperature is 129°C.

The Acrylonitrile Butadiene Styrene DOW MAGNUM 3904 is a resin with super-high impact. It is used in a wide range of technical applications. It can be processed by injection moulding, extrusion or thermoforming with excellent processability and with a stable light base colour that is ideal for self-colouring. It is suitable for the fabrication of luggage, protective covers, automotive interior trim and transportation as well as leisure applications. Its Melt Flow rate (220°C/10kg) is 4.7g/10min and its density is 1050 kg/m³.

The Borealis FG 5190 is a Butylene Linear Low Density Polyethylene for film extrusion. It includes antioxidants. The grade is developed for medium and heavy duty bags, bin liners, dual stretch films and film applications demanding high puncture resistance and strength. Its Melt Flow rate (190°C/2.16kg) is 1.2g/10min and its density is 919 kg/m³.

5 EXPERIMENTAL RESULTS

A 45 mm diameter extruder with a 31.22 length-to-diameter ratio (L/D) was used to collect extrusion process data. This extruder had seven pressure sensors along the axis of the barrel to measure pressures along the screw. Furthermore, it had seven temperature sensors along the axis of the barrel, whereof six were thermocouples and one was a radial adjustable infrared thermometer with several thermocouples, which was positionioned at Pos.2, Pos.3, Pos.4 and Pos.5.

The extruder was equipped with six barrel temperature control zones. The measurement technique was connected to Dasylab, 16-bit data acquisition system, which allows pressure and temperature data to be collected at rates up to 100 KHz.

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This report describes results from some sets of experiments conducted on a conventional screw with a feeding, melting and melt conveying section. The screw used was single-flighted and had a channel depth in the feeding section of 8.95 mm for 8.22 L/D, a transition length of 7 L/D, and a depth in the melt conveying section of 4 mm for 16 L/D. The length of the melt conveying section is of that length due to the specified measuring points. These measuring points are shown in Figure 10.

Figure 10: Measuring points along the screw length its distances

Oil was used as cooling medium for tempering the screw and its regulated temperature at the inlet of the tempering unit was 80°C. The regulated volume flow of the cooling medium was 4.2 l/min. Thus, the flow velocity of the oil inside the tube was about 1.4 m/s and the flow velocity of the oil around the tube was about 5.5 m/s. The cooling medium flows inside the siphon tube to the screw tip and around the siphon tube out of the screw (Figure 11).

Figure 11: Principle sketch of the tempered screw with a siphon tube

The applied rotation speeds were 25, 75 and 125 rpm to measure identifiable profiles for PE (SC 2108) and PP. The feeding temperature of the resin was between 20 and 25 °C. The pressure at the screw tip was for PE with 25 rpm approximately 48 bar, for 75 rpm 88 bar and for 125 rpm 119 bar. For PP 25 rpm the measured pressure at the screw tip was approximately 106 bar, for 75 rpm 146 bar and for 125 rpm 203 bar.

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For the theoretical calculation of the temperature profiles along the axis of the screw and inside the screw channel the temperatures of the boundary layer of the melt at the barrel wall are needed. Therefore, it is not expedient to use the adjusted temperatures of the heating tapes, because the measured temperatures of the heating tape control are measured inside the barrel material. Hence, these adjusted heating tape temperatures are not the real needed temperatures for the melt temperature calculation. Thus, the melt temperatures of the boundary layer of the melt at the barrel wall were measured during these experimental investigations.

The adjusted temperature profiles of the heating tapes were regulated for all sets of experiments (PE SC2108, PP, ABS and PE FG5190) for n=25 1/min without tempering the screw. The measured temperatures of the boundary layer at the barrel wall along the axis of the barrel for PE (SC 2108) and PP are shown in Figure 12 und Figure 13.

Figure 12: Measured temperature profile over the screw length (SC2108)

left: Polyethylene, without tempering, n=25, 75 and 125 rpm right: Polyethylene, with tempering, n=25, 75 and 125 rpm

Figure 13: Measured temperature profile over the screw length

left: Polypropylene, without tempering, n=25, 75 and 125 rpm right: Polypropylene, with tempering, n=25, 75 and 125 rpm

rpm

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The boundary layer temperature at the barrel in the feeding section at Pos. 7 is rising for all sets of experiments by heightening the rotation speed. Rising shearing caused by increased rotation speeds are the reason for this effect.

Between the measuring point in the feeding section (Pos. 7) and the measuring point in the melting section (Pos. 6) the boundary layer temperature of the melt at the barrel decreases for some sets of experiments. The reason is that the adjusted temperature profile of the heating tapes was fixed for 25 rpm and a non tempered screw. For higher rotation speeds this fixed temperature of the heating tape in this section is lower than the measured temperature at the boundary layer of the melt at Pos. 7, which was heightened by shearing effects. This is because at Pos. 7 the solid bed is much bigger, whereby, the melt film thickness is less than at Pos. 6. At Pos. 7 the thin melt film between the barrel wall and the solid bed can be heated up more effective by shearing effects than at Pos. 6. At Pos. 6 the solid bed is essentially smaller and hence, the shear rate is not as high as at Pos. 7. Therefore, in some experiments the melt temperature is cooled down by the colder solid bed and by the barrel tempering.

The applied rotation speeds for ABS and PE (FG5190) were 25, 125 and 175 rpm to measure identifiable profiles. The feeding temperature of the resin was also between 20 and 25 °C. The pressure at the screw tip was for ABS with 25 rpm approximately 139 bar, for 125 rpm 208 bar and for 175 rpm 243 bar. For PE 25 rpm the measured pressure was approximately 192 bar, for 125 rpm 333 bar and for 175 rpm 364 bar. The measured temperature profiles of the boundary layer at the barrel wall for the second series of measurements for ABS and PE (FG 5190) are shown in Figure 14 and Figure 15.

Figure 14: Measured temperature profile over the screw length

left: Acrylonitrile Butadiene Styrene, without tempering, n=25, 125 and 175 rpm

right: Acrylonitrile Butadiene Styrene, with tempering, n=25, 125 and 175 rpm

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Figure 15: Measured temperature profile over the screw length (FG5190)

left: Polyethylene, without tempering, n=25, 125 and 175 rpm right: Polyethylene, with tempering, n=25, 125 and 175 rpm

These figures show nearly the same effects as observed before. However, the temperature differences between the different rotation speeds are more visible in the melt conveying section.

For all sets of experiments the tempering of the screw influences the temperature profile in the melt conveying section. Tempered screws are able to influence the homogeneity of the axial temperature profile in this section.

Consecutively, the results of the experimental measurements concerning the radial temperature profile inside the screw channel for the two series of measurements are presented. The results are measured with the thermocouple TIP (3). The radial temperatures are measured up to 3.5 mm from the barrel wall into the screw channel. In Figure 16 to Figure 19 the radial temperature profiles at different measuring points over the screw length (Figure 10), applied rotation speeds of 25, 75/125 and 125/175 rpm as well as with and without tempering of the screw are shown (x-axis: ][0 mm is in accordance with 0 screw ground and ][4 mm is in accordance with 1 barrel wall; y-axis: temperature [°C]). Furthermore, the first series of measurements (PE SC2108 and PP) are containing a variation of the oil temperature at the inlet of the siphon tube between 80°C and 120°C.

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Figure 16: Measured radial temperature profiles while processing PE (SC 2108)

Figure 17: Measured radial temperature profiles while processing PP

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Figure 18: Measured radial temperature profiles while processing ABS

Figure 19: Measured radial temperature profiles while processing PE (FG5190)

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The radial temperature profiles for a rotation speed of 25 rpm shows that the tempering of the screw has no influence on the temperature level and profile until the middle of the melt conveying section. At the end of the melt conveying section there is a clearly decrease of the temperature level for tempered screws. Furthermore, no clear effects can be seen by varying the oil temperatures at the inlet of the siphon tube. Therefore, in the second series of measurements the inlet temperatures were not varied.

The effects of the screw tempering are visible for all applied rotation speeds at the end of the melt conveying section. In this area the cooling medium is leaving the siphon tube, while the heat is transported from the melt into the cooling medium. Therefore, the temperature level for tempered screws is always lower than for non-tempered. All the described effects are nearly material independent.

While observing the temperature development for higher speeds like 75, 125 and 175 rpm and in direction from the screw tip to the melting section a decrease of the temperature from the barrel wall to the screw ground can be observed. A characteristic S-shaped profile occurs. The causing for this S-shaped profile is the shear rate. The shear rate is increasing by escalating the rotation speed. This S-shaped profile is developed more strongly in the melting section than in the melt conveying section as it is shown in Figure 17 to Figure 19. The more the material is conveyed to the screw tip, the merrier these S-shaped profile characteristics can be balanced by mixing effects. But especially for high rotation speeds these S-shaped profile keep visible, even at the screw tip.

6 COMPARISON OF THE CALCULATED AND MEASURED RADIAL TEMPERATURE PROFILES

To be able to calculate the radial temperature profile with Equation 17, the starting and boundary conditions are needed. Therefore, four boundary conditions have to be fixed to solve the starting condition in Equation 12. This is necessary because the starting condition is a third-order polynomial. Due to the experimental investigations the starting condition has to proceed like it is displayed in the Figure 20.

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Figure 20: Measured values to display the starting condition

The following four boundary conditions Equation 18, Equation 19, Equation 20 and Equation 21 are used for the solving operation of Equation 12. Equation 20 and Equation 21 are simplifications to display the Equation 17 easily. The simplification in Equation 21 is acceptable as it can be seen in Figure 20.

Z 0,10 Equation 18

S 0,00 Equation 19

Z 0,9.00 Equation 20

00,5.020

2

Equation 21

Consequently, Equation 17 can be solved with the starting and boundary conditions and an analytical radial temperature profiles can be calculated. For the calculations the measured boundary layer temperatures of the melt in the viewed section are used. In Figure 21 to Figure 22 the theoretical and experimental investigated temperature profiles at position 5 (blue points) and position 2 (red points) of the PE SC2108 are compared. The difference between these two measuring points was divided into ten parts for the calculation.

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Te

mp

era

ture

[°C

]

Te

mp

era

ture

[°C

]

Te

mp

era

ture

[°C

]

Te

mp

era

ture

[°C

]

Figure 21: Calculated temperature development (PE SC2108)

left: Polyethylene, without tempering, n=125 rpm right: Polyethylene, with tempering (80°C), n=125 rpm

Figure 22: Calculated temperature development (PE SC2108)

left: Polyethylene, without tempering, n=125 rpm right: Polyethylene, with tempering (80°C), n=125 rpm

In Figure 21 the calculation in sections can be seen. In this case, the viewed melt conveying section was divided into 10 smaller sections, were the radial temperature profile was calculated. This is because of the boundary conditions and material parameters, except the viscosity, which are regarded as constant in the actual section. At the end of every section a medial radial temperature was calculated and transferred as the starting temperature 0T into the next

section.

Figure 22 shows the comparison between experimental results which were displayed as crosses and the calculated radial temperature profiles in each section. The blue crosses show the measured radial starting temperature and the red crosses show the measured radial end temperature of the viewed screw

Measured values

Calculated values

Measured values

Calculated values

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Te

mp

era

ture

[°C

]

Te

mp

era

ture

[°C

]

Te

mp

era

ture

[°C

]

Te

mp

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ture

[°C

]

area. The red lines show the calculated radial temperature profiles. In Figure 23 and Figure 24 the theoretical and experimental investigated temperature profiles at the positions 5 (blue points) and 2 (red points) of the ABS are compared.

Figure 23: Calculated temperature development

left: Acrylonitrile Butadiene Styrene, without tempering, n=125 rpm right: Acrylonitrile Butadiene Styrene, with tempering (80°C), n=125

rpm

Figure 24: Calculated temperature development

left: Acrylonitrile Butadiene Styrene, without tempering, n=125 rpm right: Acrylonitrile Butadiene Styrene, with tempering (80°C), n=125

rpm

These comparisons show that the experimental investigations and the calculated results of the radial temperature profiles are qualitative consistent. Furthermore, the quantitative evaluation is also acceptable. The correlations are good for the case without tempered screws and for the case of tempered screws. Thus, the temperature model with the boundary conditions of constant temperatures (Figure 6) is adequate for the theoretical describing of the temperature effects of tempered screws.

Measured values

Calculated values

Measured values

Calculated values

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7 CONCLUSIONS

The experimental investigations show a homogenization of the melt temperature over the screw length by using a tempered screw. A reduction about 15 to 20°C of the melt temperature at the screw tip can be achieved.

Furthermore, S-shaped radial temperature profiles were measured inside the screw channel. The S-shaped characteristics are more visible the more the measuring position is placed from the screw tip to the melting section and the more the rotation speeds are increased.

The comparison of the experimental investigation and the calculated radial temperature profiles show that the theoretically developed equation offers acceptable results to describe the radial and the resultant axial temperature profile in a plasticizing unit.

However, for the calculation the measured temperature at the screw ground was used. This is only possible because of the experimental investigation. If there are no measured values of the screw temperatures they must be calculated. The methods for the calculation of the temperature at the screw ground and the temperatures of the cooling medium will be published soon.

8 REFERENCES

The results presented were obtained in the context of research project PO171/89-1 which was sponsored by the Deutsche Forschungsgemeinschaft (DFG). We would like to thank the DFG for their support.

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9 TABLE OF SYMBOLS

Roman Symbols

ta [-] temperature shifting factor

20, Tta [-] temperature shifting factor for 20T

b [m] channel width

b [s] reciprocal transition shear rate

Br [-] Brinkmann number

c [-] slope (Carreau-law)

c [KJ/(kg•K)] heat capacity

Vc [KJ/(kg•K)] heat capacity for constant volume

Pc [KJ/(kg•K)] heat capacity for constant pressure

21,cc [-] constants

D [m] screw diameter

e [m] flight width

g [-] actual screw section

Gz [-] Graetz number

h [m] channel height

i [-] total number of screw sections

)(TK [Pa•sn] consistency factor in dependence on T

TK0 [Pa•sn] coefficient of the power law in dependence on T

and on the shear rate s

11

L [m] screw length

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n [-] power law exponent

p [-] Laplace-Parameter

P [m] pitch

q [W/m²] heat flux density

xq [W/m²] heat flux density in x-direction

zq [W/m²] heat flux density in z-direction

t [s] time

T [°C] actual temperature

BT [°C] barrel temperature

RT [°C] reference temperature (Carreau-law)

StT [°C] standard temperature (Carreau-law)

0T [°C] averaged temperature at the end of the section

before

4321 ,,, uuuu [-] constants for starting condition (polynomial of the

third order)

V [m³/s] melt flow rate

x [-] coordinate in the channel depth direction

y [-] coordinate transverse to channel direction

z [-] coordinate in the channel direction

z [m] length of one / the actual channel section in

channel direction

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Greek Symbols

[1/°C] temperature coefficient of the Arrhenius law

[1/s] shear rate

[m] screw clearance

[-] dimensionless channel length in the regarded

screw section

[-] dimensionless temperature

B [-] dimensionless temperature at the barrel wall

S [-] dimensionless temperature at the screw ground

0 [-] dimensionless temperature at the end of the

section before

[-] dimensionless temperature in the Laplace area

[W/(m•K)] heat conductivity

v

[m/s] velocity vector

zv [m/s] averaged velocity in z-direction

0v [m/s] circumferential velocity

xx vv 0 [m/s] x-component of the barrel velocity

zz vv 0 [m/s] z-component of the barrel velocity

[-] dimensionless channel heigth

[kg/m³] density

[N/m²] shear stress

0 [-] dissipated energy across the channels cross-

section per volume and averaged flow speed in z -

direction zv

[°] angle of the channel pitch

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Literatur

[1] Potente, H. Auslegen von Schneckenmaschinen-Baureihen - Modellgesetze und ihre Anwendung

Carl Hanser Publishers, Munich, 1981

[2] Rauwendaal, C. Polymer Extrusion

Carl Hanser Publishers, Munich, 2001

[3] Mitterfellner, H. Vergleich verschiedener Lösungen zur Analyse der Schmelzetemperaturentwicklung im Schneckenkanal von Einschneckenextrudern

Seminar Paper, Institut für Kunststofftechnik (KTP), University of Paderborn, 1991

[4] Carslaw, H.S.

Jaeger, J.C.

Conduction of Heat Solids

Oxford University Press, London, 1959

[5] Carslaw, H.S.

Jaeger, J.C.

Operational Methods in Applied Mathematics

Dover Publications Inc., New York, 1963

[6] Bronstein, I.N.

Semendjajew, K.A.

Taschenbuch der Mathematik

Harri Deutsch Publishers, Frankfurt/Main, 1989

[7] Tautz, H. Wärmeleitung und Temperaturausgleich

Chemie Publishers, Weinheim, 1971

[8] Giese, E. Handout, FOS-Messtechnik GmbH,

www.fos-messtechnik.de

[9] Potente, H. Kunststofftechnisches Seminar: rechnergestützte Extruderauslegung, Universität Paderborn, 1992

[10] Pohl, T. Entwicklung schnelldrehender Einschneckensysteme für die Kunststoffverarbeitung auf Basis theoretischer Grundlagenuntersuchungen

Dissertation, Institut für Kunststofftechnik, Universität Paderborn, 2003

[11] Schramm, D. Möglichkeiten und Grenzen einer verbesserten Prozessbeschreibung für Kautschukextruder

Dissertation, Institut für Kunststofftechnik, Universität Paderborn, 2003

[12] Baehr, H. D.

Stephan K.

Wärme- und Stoffübertragung, Springer-Verlag, Berlin, (2004)

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Stichworte: Einschneckenextruder, Schneckenkanal, radiales Temperaturprofil, Temperaturmessung

Keywords: Single Screw Extrusion, Screw Channel, Radial Temperature Profile, Temperature Measurement

Autor/author: Dipl.-Ing. Karsten Anger Prof. Dr.-Ing. Helmut Potente Prof. Dr.-Ing. Volker Schöppner Institut für Kunststofftechnik Universität Paderborn Warburger Str. 100 33098 Paderborn Prof. Dr. Sönke Hansen Institut für Mathematik Universität Paderborn Warburger Str. 100 33098 Paderborn

E-Mail-Adresse: [email protected] [email protected] Webseite: www.ktpweb.de Tel.: +49(0)5251/60-2451 Fax: +49(0)5251/60-3821

Herausgeber/Editor: Europa/Europe Prof. Dr.-Ing. Dr. h.c. Gottfried W. Ehrenstein, verantwortlich Lehrstuhl für Kunststofftechnik Universität Erlangen-Nürnberg Am Weichselgarten 9 91058 Erlangen Deutschland Phone: +49/(0)9131/85 - 29703 Fax.: +49/(0)9131/85 - 29709 E-Mail-Adresse: [email protected]

Amerika/The Americas Prof. Prof. h.c Dr. Tim A. Osswald, responsible Polymer Engineering Center, Director University of Wisconsin-Madison 1513 University Avenue Madison, WI 53706 USA Phone: +1/608 263 9538 Fax.: +1/608 265 2316 E-Mail-Adresse: [email protected]

Verlag/Publisher: Carl-Hanser-Verlag Jürgen Harth Ltg. Online-Services & E-Commerce, Fachbuchanzeigen und Elektronische Lizenzen Kolbergerstrasse 22 81679 Muenchen Tel.: 089/99 830 - 300 Fax: 089/99 830 - 156 E-mail-Adresse: [email protected]

Beirat/Editorial Board: Professoren des Wissenschaftlichen Arbeitskreises Kunststofftechnik/ Professors of the Scientific Alliance of Polymer Technology

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