MODELLING AND OPTIMISATION OF SINGLE SCREW EXTRUSION ...€¦ · Therefore, it is the objective of this work to investigate the influence of the viscous behaviour of polymer melts

Covas and Gaspar-Cunha, e-rheo.pt, 1 (2001) 41-62

A COMPUTATIONAL INVESTIGATION ON THE EFFECT OF POLYMER RHEOLOGY ON THE PERFORMANCE OF A

SINGLE SCREW EXTRUDER

J. A. Covas* and A. Gaspar-Cunha

Dept. of Polymer Eng., University of Minho, Campus de Azurém, 4800-058 Guimarães, Portugal. Phone: +351 253510245; Fax: +351 253510249.

e-mails: [email protected], [email protected].

* - Corresponding author

Key words: extrusion modelling, single screw extruder, plasticating extrusion, polymer extrusion

Abstract: This work investigates the influence of the rheological characteristics of a polymer on the

(predicted) performance of a typical single screw extruder. A global modelling package is developed

in order to yield important process responses such as axial pressure, melting rate, melt temperature,

power consumption and degree of mixing. The effect of the power law constants is studied both in

terms of the general process behaviour and of the sensitivity of the extruder to small changes of the

input conditions.

1. INTRODUCTION

Plasticating single screw extrusion can probably be considered as the most important unit

operation in polymer processing technology. Extruders are a fundamental part of any

extrusion line (for producing pipes, profiles, blown or flat film, filaments, coated wire, etc),

are often used in compounding (e.g., incorporation of additives) and blow moulding (for

producing bottles and other hollow containers) and, in modified form, are used as plasticating

units of injection moulding machines.

Therefore, it is not surprising that extrusion has been the subject of many studies, focusing on

the physical understanding and on the mathematical modelling of the process, on innovative

technological developments, on new applications, and on monitoring and optimisation. In

particular, the physical phenomena developing along the screw during the practical operation

of a single screw extruder have been extensively studied in the last four decades [e.g. see the

reviews of 1, 2, 3]. Consequently, a number of computational tools has been made available,

the level of sophistication varying from simple 1D analytical models to complex 3D

approaches using finite element methods [2,4,5]. Unfortunately, some of these programmes

41

mailto:[email protected]

mailto:[email protected]


are black boxes, i.e., little is known about the underlying physics, while others only cover the

final stages of the process, when melt pressure and/or drag flows develop.

Generally, the models encompassing the flow of a polymer from the hopper to the die exit

predict the value of important process parameters such as output, degree of mixing, melt

temperature, melting rate, residence time, power consumption, for a particular

extruder/polymer/die combination. The completeness of the predictions and the corresponding

degree of accuracy depend, obviously, on the sophistication of the mathematical models

involved and of the constitutive equations adopted. The predictions are valid for a specific set

of operating conditions, screw/die geometry and polymer properties. The rheological

behaviour of the polymer is of paramount importance in defining the type of response of this

complex system. Surprisingly, although the general trends are well known [4,6], a quantitative

evaluation of the effects of rheology on the performance of single screw extrusion is

apparently not yet available.

Therefore, it is the objective of this work to investigate the influence of the viscous behaviour

of polymer melts on the performance of single screw extruders. A simple power law model

will be adopted, as it evidences clearly the effects of viscosity levels, pseudoplasticity

character and temperature dependence. Computational experiments encompassing the entire

screw will be carried out for this purpose.

2. THEORETICAL FRAMEWORK

In a conventional extruder an Archimedes-type screw rotates inside a heated barrel of

diameter D (Figure 1). Generally, the screw has three distinct geometrical sections (Figure 1-

A), denoted as feed (constant channel depth H1), compression (varying channel depth along

the axis) and metering (constant smaller channel depth H2), with lengths L1, L2 and L3,

respectively.

For a specific system geometry and polymer, the main operating parameters are the screw

speed and the barrel temperature profile. The (solid) polymer is fed through the hopper.

Gravity-induced flow guarantees material transfer to the barrel. The polymer progresses along

the screw due to a balance of friction forces and, as a result of a combination of pressure, heat

transfer and dissipation of mechanical energy, it melts. In the remaining of the screw

distributive mixing and pressure generation occur, so that the polymer flows through the die

42


located downstream at a given rate, where it is shaped into the desired cross-section. These

phenomena developing sequentially inside the machine (Figure 1-B) have been analysed

thoroughly and are usually referred to as [1,3,4]:

i) - solids conveying of material in the hopper;

ii) - drag solids conveying in the initial turns of the screw;

iii) - delay in melting, due to the development of a thin film of melted material

separating the solids from the surrounding metallic wall(s);

iv) - melting, where a specific melting mechanism develops, depending on the local

pressure and temperature gradients;

v) - pumping involving the complex but regular helical flow pattern of the fluid

elements towards the die;

vi) - flow through the die.

A

D

��

��

��

Heater band

��

L1 L2 L3

Barrel Screw Die

FEED COMPRESSION METERING

D

��

��

��

Heater band

��

L1 L2 L3

Barrel Screw Die

FEED COMPRESSION METERING

B

i)

��

��

��

iv) v)iii)ii) vi)

Transversalcuts

i)

��

��

��

iv) v)iii)ii) vi)

Transversalcuts

Figure 1- Plasticating single screw extruder. A) typical geometry; B) Physical phenomena.

The hopper can be considered as a sequence of vertical and/or convergent columns subjected

to static loading, given the difference between their flow capacity and the effective discharge

rate [6]. The vertical pressure profile, hence the pressure at the bottom of this device, can be

determined by performing a force balance on an elemental slice of the bulk solids [7]. This

corresponds to establishing the inlet condition of the extruder.

43


Modelling of the solids conveying zone of the screw (Figure 2-ii) considers sliding of a non-

isothermal elastic solid plug with heat dissipation at all (screw and barrel) surfaces. The solids

temperature increases due to the contribution of conduction from the hot barrel and of friction

near the polymer/metal interfaces, heat convection along the channel taking place due to the

polymer motion [8,9]. The pressure generated can be determined from force and torque

balances made on differential down-channel elements [8].

The model developed considers that the delay zone is sub-divided into two sequential steps,

as demonstrated by previous experimental observations [1,3]. The local higher temperatures

and friction forces favour the formation of a melt film, C, near to the inner barrel wall (Figure

2-iii-A). Eventually, depending on the operating conditions (output and screw temperature),

the material in contact with the screw surfaces may melt in this zone (films B, D and E) by the

same mechanism (Figure 2-iii-B). The first step was described mathematically using the

approach of Kacir and Tadmor [10] to compute the pressure and temperature profiles in the

solid and the power consumption. The film thickness and temperature can be computed from

the momentum and energy equations assuming heat convection in the down-channel and

radial directions and heat conduction in the radial direction [5]:

∂∂

∂∂

=∂∂

yV

yxP xη

(1)

0=∂∂

yP

(2)

∂∂

∂∂

=∂∂

yV

yzP zη

(3)

22

2

)( γηρ &+∂∂

=∂∂

yTk

zTyVc mzpm

(4)

where ρm , cp and km denote melt density, specific heat, and thermal conductivity,

respectively, and η is the melt viscosity, which is calculated using a temperature dependent

power law:

( )[ ] 10exp −−−= nTTak γη & (5)

where k0 , a, T0 and n are the usual constants and γ is the shear rate, given by: &

21

22

∂∂

+

∂∂

=y

Vy

V zxγ& (6)

44


ii)

Barrel

Screw root

Screw flightScrewflight

qb

qs

qf qf

y

x

∆z

H

W

Barrel

Screw root

Screw flightScrewflight

qb

qs

qf qf

y

x

∆z

H

WW

iii)-A

Vbz

Vbx Vsz

H

T(y)

Tb

Tm

Tso

Ts

A

Vb

CδC

Vbz

Vbx Vsz

H

T(y)

Tb

Tm

Tso

Ts

A

Vb

CδC

iii)-B

xT(y)

B

Vbz

VbxVsz

H

Tb

Tm

Tso

Ts

DATm

C

Vb

δDE

WB

δC

D

Ex

T(y)

B

Vbz

VbxVsz

H

Tb

Tm

Tso

Ts

DATm

C

Vb

δDE

WB

δC

D

E

iv)

Vbz

VbxVsz

H

T(y)

Tb

Tm

Tso

Ts

B

E

DA

Tm

D

C

C

Vb

δDE

WB

δC

Vbz

VbxVsz

H

T(y)

Tb

Tm

Tso

Ts

B

E

DA

Tm

D

CC

CC

Vb

δDE

WB

δC

v)

Vbz

Vbx

H

T(y)

Tb

Tb

Vb Vbz

Vbx

H

T(y)

Tb

Tb

Vb

Cartesian Coordinates zy

x

Cartesian Coordinates zy

x

zy

x Figure 2- Physical models of the various functional zones. The identification of ii) to v) is given in

Figure1b.

45


Neglecting leakage flow, the melt must recirculate in the x-direction:

∫ =C dyyVx

δ

00)( (7)

where δC is the melt film thickness. The relevant boundary conditions are:

====

====

−====

b

m

bzz

szz

bxx

x

T)y(TT)y(T

V)y(VV)y(V

V)y(V)y(V

δδδ0000

(8)

where Tm is the melting temperature (in the case of semi-crystalline polymers, or a

temperature slightly above the glass transition point for amorphous polymers). Solving

equations (1), (3), (4) and (7), coupled to the boundary conditions (8), yields the melt film

velocity and temperature fields. However, since the viscosity depends on these variables the

equations are non-linear, thus involving the use of a specific finite difference discretisation

scheme [11,12]. For example, the solution of equations 1 and 3 can be obtained using the

implicit Crank-Nicholson scheme, as shown by the algorithm of Figure 3.

Initial values for:Vx0,j(y) (e.g., a linear profile between 0 and Vbx)Vz0,j(y) (e.g., linear variation between Vsz and Vbz)T0,j(y) (e.g., linear variation between Tm and Tb)

do {do {

solve equation 1 (to obtain Vxi,j and ∂P/∂x)solve equation 3 (to obtain Vzi,j and Pi)

} while (Vxi,j, ∂P/∂x, Vzi,j and Pi have not converged)solve equation 4 (to obtain Ti,j)

} while (Ti,j has not converged)

Initial values for:Vx0,j(y) (e.g., a linear profile between 0 and Vbx)Vz0,j(y) (e.g., linear variation between Vsz and Vbz)T0,j(y) (e.g., linear variation between Tm and Tb)

do {do {

solve equation 1 (to obtain Vxi,j and ∂P/∂x)solve equation 3 (to obtain Vzi,j and Pi)

} while (Vxi,j, ∂P/∂x, Vzi,j and Pi have not converged)solve equation 4 (to obtain Ti,j)

} while (Ti,j has not converged)

Figure 3- Algorithm for solving the system of non-linear equations for the delay zone.

The second step of this stage (Figure 2-iii-B) can be considered as a particular case of the

melting zone. It develops while the width of the melt film B remains smaller than the channel

height [13].

Melting involves the simultaneous development of 5 regions (identified as A to E in Figure 2-

iv). The model implemented was initially proposed by Lindt et al [13,14] and considers that

46


the solid bed velocity is constant and that cross-channel flow exists. Each of the five

individual regions require different forms of the momentum and energy equations, coupled to

the relevant boundary conditions and force, heat and mass balances. The flow of the melt

films C, D and E can be described by equations (1) to (6), together with specific boundary

conditions (equation 8) and a condition of cross-channel flow (equation 7). Flow in the melt

pool is taken as two-dimensional, hence equations (2), (3) and (6) are replaced, respectively,

by:

∂∂

∂∂

+

∂∂

∂∂

=∂∂

yV

yxV

yzP zz ηη

(9)

22

2

2

2

)( γηρ &+

∂∂

+∂∂

=∂∂

yT

xTk

zTyVc mzpm

(10)

21

222

∂∂

+

∂∂

∂∂

=y

Vy

Vy

V xzxγ&

(11)

These equations are solved using finite differences and the following boundary conditions [5]:

========

========

−====

b

s

m

s

bzz

z

szz

z

bxx

x

THyTTyTTWxT

TxT

VHyVyV

VWxVxV

VHyVyV

)()0()(

)0(

)(0)0(

)(0)0(

)(0)0(

(12)

The non-isothermal two-dimensional flow of the same power-law fluid in the pumping zone

occurs, i.e., equations (1), (9), (10) and (11) for the melting step remain valid, if coupled to

the following conditions [5]:

========

========

−====

b

s

s

s

bzz

z

z

z

bxx

x

THyTTyTTWxT

TxT

VHyVyV

WxVxV

VHyVyV

)()0()(

)0(

)(0)0(0)(

0)0()(

0)0(

(13)

Melt distributive mixing depends on the growth of the interfacial area between two adjacent

fluid components and on their average residence time for flow. Since the variation of

interfacial area is proportional to the shear strain of the melted polymer, the average strain can

be used as a simple criterion to estimate the degree of mixing [15]. In this work, an weighted-

average total strain function, WATS, was computed, which requires the integration of the

47


strain experienced by a particle, γ , with respect to the residence time distribution, f(t)

[5,15,16].

)(y

∫∞

=0

)()( dttfyγγ (14)

Function f(t) is defined as:

( ) ( )**

**

C

C

QQQQd

dttf++

= (15)

and represents the fraction of melt whose residence time lies in the range t to t+dt, Q*+Q*c

being the total volumetric flow rate (i.e., Q), and dQ* and dQ*c the differential flow associated

with the neighbourhood of planes y and yc, respectively, so that:

( ) ( ) ( )[ ] WdyyVyVQQd czzC +=+ ** (16)

These planes contain the same fluid element in the upper (y) and lower (yc) portions of the

channel, as the result of polymer re-circulation in the x direction. Velocity ( )yzV is the

average of the velocity Vz(x,y) determined from the resolution of the momentum equations.

Finally, pressure flow in the die was computed assuming the existence of successive channels

with uniform cross-section in the downstream direction and the non-isothermal two-

dimensional flow of a non-Newtonian fluid, where equations (1), (9) and (10) are solved by

finite differences.

The global plasticating extrusion model links sequentially the above stages through the

appropriate boundary conditions. Coherence of the physical phenomena between any two

adjacent zones must obviously be ensured. After estimating an initial output from volumetric

considerations, calculations are carried out for small screw increments along its helical

channel (Figure 4). If the predicted pressure drop at the die exit is not sufficiently small

(theoretically it should be nil), the output is changed and a new iteration is carried out.

3. MATERIAL AND EQUIPMENT

The characteristics of a High Density Polyethylene (NCPE 0928, from BOREALIS) were

used in the computational experiments. They are presented in Table 1 and were either

obtained from the manufacturer or measured experimentally [17].

48


The geometry of an existing Leistritz LSM 36 laboratorial single screw extruder fitted with a

conventional polyethylene-type three-zone screw and a simple rectangular die, as shown in

Figure 5, was considered. The typical operating conditions are also indicated in Figure 5.

Start

Define:Q1 andQ2

•Hopper•Solids Conveying•Delay•Melting•Melt Conveying

End

Yes

No

Die

Pexit<ε

New Q1 andQ2

For each ∆Z

Start

Define:Q1 andQ2

•Hopper•Solids Conveying•Delay•Melting•Melt Conveying

End

Yes

No

Die

Pexit<ε

New Q1 andQ2

For each ∆Z

Figure 4- Flowchart of the global modelling package.

D =

36

mm ��

��

��

Tb = 190 ºC

L = 936 mm

N= 50 rpm

2.7D

6 x

2m

m

350 mm

��

12D 7D 7D

D =

36

mm ��

��

��

Tb = 190 ºC

L = 936 mm

N= 50 rpm

2.7D

6 x

2m

m

350 mm

��

12D 7D 7D

D =

36

mm ��

��

��

Tb = 190 ºC

L = 936 mm

N= 50 rpm

2.7D

6 x

2m

m

350 mm

��

12D 7D 7D

D =

36

mm ��

��

��

Tb = 190 ºC

L = 936 mm

N= 50 rpm

2.7D

6 x

2m

m

350 mm

��

12D 7D 7D

Figure 5- Geometry and operating conditions of the extruder.

49


Table 1- Polymer properties. Property Law Value

Solids density [18]

( ) PF0 e∞∞ ρ−ρ+ρ=ρ

with

TTb

TbTbbFg

32210 −

+++=

ρ∞ = 948 kg/m3

ρ0= 560 kg/m3 Tg= -125 °C b0= -1.276e-9 1/Pa b1= 8.668e-9 1/°C Pa b2= -5.351e-11 1/°C2 Pa b3 = -1.505e-4 °C/Pa

Melt density [1]

PTgPgTgg 3210m +++=ρ g0 = 854.4 kg/m3

g1 = -0.03236 kg/m3 °C g2 = 2.182e-7 kg/m3 Pa g3 = 3.937e-12 kg/m3 °C Pa

Friction coefficients

--- polymer-barrel = 0.45 polymer-screw = 0.25

Solids thermal conductivity

--- 0.186 W/m °C

Melt thermal conductivity

--- 0.097 W/m °C

Heat of fusion --- 196802 J/kg Solids specific heat --- 1317 J/kg Melt specific heat [19]

2210m TCTCCC ++= C0 = -1289 J/kg

C1 = 86.01 J/kg °C C2 = -0.3208 J/kg Pa

Melting temperature

--- 119.6 °C

Viscosity

( )[ ] 10exp −−−= nTTak γη & n = 0.345

k = 29.94 kPa sn a = 0.00681 1/°C T0 = 190 °C

4. COMPUTATIONAL EXPERIMENTS

This work will investigate the influence of the power law constants (n, k and a) on the

performance of an extruder in terms of the axial pressure, presence of solids, average melt

temperature and power consumption profiles along the screw length, of the temperature field

development along the pumping zone and of the mass output, channel length required for

complete melting, average melt temperature at die exit, total mechanical power consumption

and degree of mixing. A Central Composite experimental design approach was adopted, as it

yields quantitative information on the effect of the individual parameters and on their

interactions. As shown in Table 2, the three independent rheological variables require that 15

50


runs (reom1 to reom15) must be carried out. Usually, the central point (reom9) is replicated in

order to tackle the experimental variation, but this is obviously not required in the present

situation. The lower (-1), medium (0) and higher level (1) of each variable was coded as

shown. The corresponding values are 5000, 27500 and 50000 Pa.sn for k, 0.25, 0.45 and 0.65

for n, 0.0001, 0.001 and 0.01 ºC-1 for a. The direct comparison between runs reom13, reom9

and reom12, runs reom14, reom9 and reom4 and runs reom11, reom9 and reom2 demonstrate

the effect of n, k and a, respectively.

Table 2- Central composite design of the computational experiments

Run n k (Pa.sn) a (ºC-1) n k A reom1 0.25 5000 0.01 -1 -1 1 reom2 0.45 27500 0.01 0 0 1 reom3 0.65 50000 0.01 1 1 1 reom4 0.45 50000 0.001 0 1 0 reom5 0.25 38750 0.0001 -1 0.5 -1 reom6 0.65 5000 0.01 1 -1 1 reom7 0.55 38750 0.0001 0.5 0.5 -1 reom8 0.25 5000 0.0001 -1 -1 -1 reom9 0.45 27500 0.001 0 0 0 reom10 0.65 5000 0.0001 1 -1 -1 reom11 0.45 27500 0.0001 0 0 -1 reom12 0.65 27500 0.001 1 0 0 reom13 0.25 27500 0.001 -1 0 0 reom14 0.45 5000 0.001 0 -1 0 reom15 0.25 50000 0.01 -1 1 1

The sensitivity of the extruder to small changes in the rheological constants was estimated

from the results of the runs presented in Table 3. This study reproduces the unavoidable

practical oscillations of the material characteristics and operating conditions around an

average value.

Table 3- Runs for studying the sensitivity of the extruder to small variations of input values

Run Aim n k (Pa.sn) a (ºC-1) reos1 0.3 reos2 0.35 reos3

Influence of n 0.4

30000 0.01

reos4 25000 reos5 30000 reos6

Influence of k 0.35 35000

0.01

51


5. RESULTS AND DISCUSSION

Figures 6 to 8 show the influence of n, k and a, respectively, on the axial development of

melting (reduced solids width), pressure, power consumption and average melt temperature.

Viscous dissipation, pressure generation and mechanical power consumption increase with

increasing n. In turn, the increase in polymer temperature due to viscous dissipation reduces

slightly the length of screw required to complete melting. Obviously, an increasing n

corresponds to higher viscosity levels at moderate to high shear rates.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Red

uced

solid

s wid

th

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Red

uced

solid

s wid

th

0

2040

6080

100120

140

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Pres

sure

(MPa

)

0

2040

6080

100120

140

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Pres

sure

(MPa

)

0

2000

4000

6000

8000

10000

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Pow

er C

onsu

mpt

ion

(W)

0

2000

4000

6000

8000

10000

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Pow

er C

onsu

mpt

ion

(W)

120

140

160

180200

220

240

260

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Mel

t tem

pera

ture

(ºC

)

120

140

160

180200

220

240

260

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Mel

t tem

pera

ture

(ºC

)

n = 0.25 n = 0.45 n = 0.65n = 0.25n = 0.25 n = 0.45n = 0.45 n = 0.65n = 0.65

Figure 6- Effect of n on the evolution along the screw length (runs reom13, reom9 and reom12).

As expected, the effect of the consistency index is qualitatively similar (Figure 7). For k

values above 27500 Pa.sn the average melt temperature grows above the set value during most

of the pumping zone. Differences of more than 40ºC may develop. The practical operation

under such conditions should be avoided, as the machine’s control system is ineffective and

polymer degradation may occur.

Variations of a of two orders of magnitude do not seem to affect substantially the

performance of the extruder (Figure 8).

52


0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Red

uced

solid

s wid

th

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Red

uced

solid

s wid

th

0

2040

6080

100120

140

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Pres

sure

(MPa

)

0

2040

6080

100120

140

0.0 0.2 0.4 0.6 0.80

2040

6080

100120

140

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Pres

sure

(MPa

)

0

2000

4000

6000

8000

10000

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Pow

er C

onsu

mpt

ion

(W)

0

2000

4000

6000

8000

10000

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Pow

er C

onsu

mpt

ion

(W)

120

140

160

180200

220

240

260

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Mel

t tem

pera

ture

(ºC

)

120

140

160

180200

220

240

260

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Mel

t tem

pera

ture

(ºC

)

k = 5000 Pa.s k = 27500 Pa.s k = 50000 Pa.sk = 5000 Pa.s k = 27500 Pa.s k = 50000 Pa.s

Figure 7- Effect of k on the evolution along the screw length (runs reom14, reom9 and reom4).

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Red

uced

solid

s wid

th

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Red

uced

solid

s wid

th

0

2040

6080

100120

140

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Pres

sure

(MPa

)

0

2040

6080

100120

140

0.0 0.2 0.4 0.6 0.80

2040

6080

100120

140

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Pres

sure

(MPa

)

0

2000

4000

6000

8000

10000

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Pow

er C

onsu

mpt

ion

(W)

0

2000

4000

6000

8000

10000

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Pow

er C

onsu

mpt

ion

(W)

120

140

160

180200

220

240

260

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Mel

t tem

pera

ture

(ºC

)

120

140

160

180200

220

240

260

0.0 0.2 0.4 0.6 0.8 1.0L (m)

Mel

t tem

pera

ture

(ºC

)

a = 0.0001 ºC-1 a = 0.001 ºC-1 a = 0.01 ºC-1a = 0.0001 ºC-1 a = 0.001 ºC-1 a = 0.01 ºC-1

Figure 8- Effect of a on the evolution along the screw length (runs reom11, reom9 and reom2).

The two-dimensional cross-channel melt temperature profiles at the beginning and at the end

of the pumping zone are presented in Figures 9 and 10. As seen above, the importance of

53


viscous dissipation increases as the rheological behaviour becomes more Newtonian-like

(Figure 9). At the beginning of the melt conveying zone it is possible to detect regions with

lower temperature, which correspond to the presence of freshly molten polymer. The average

melt temperature is smaller than the set value (190ºC), a progressively more complex

temperature profile developing as the material advances along the helical channel.

As for the influence of the consistency index, Figure 10 demonstrates that for k values lower

or equal to 10000Pa.sn the viscous dissipation is low, while for k values of the order of

50000Pa.sn the temperature develops steadily downstream, local self-heating reaching values

up to 70ºC. Again, the influence of a is insignificant, and therefore it will not be illustrated.

The contour plots shown in Figures 11 to 13 show the influence of the power law constants on

the values of mass output and degree of mixing, WATS (Figure 11), length of screw required

for melting and average melt temperature at die exit (Figure 12) and mechanical power

consumption (Figure 13), respectively. Each graph plots the correlation between k and n for a

constant value of a. The red points refer to the “experimental” points used by the design to

generate the regression surface. Although this is not the aim of the present work, the careful

analysis of these plots would allow the optimisation of the process for a specific set of

prescribed objectives (e.g. maximise the output, minimise the power consumption, ensure a

good level of mixing, etc).

Keeping the operating conditions constant (see Figure 5), an increase of both n and k (Figure

11) will augment linearly the output from 7.6kg/hr to 9kg/hr, i.e., almost 20%. This is

obviously related to the pressure generation capacity under these conditions discussed above.

The effect of a is again smaller, but noticeable.

The influence on the degree of mixing, WATS, is quite different. A minimum performance

around n=0.45 and k=10000Pa.sn, i.e., at intermediate levels, can be perceived, while the best

mixing happens for high values of the rheological constants. As demonstrated by equation

(12), mixing depends on the residence time, i.e., on both the effective length of the melt

conveying zone (which, in turn, is dictated by the melting efficiency) and on the mass output,

as well as on the level of shear rates - and strains - developed (output again). High viscosity

levels favour output, hence shear rate, but compromise the residence time, because although

the melting efficiency is increased, the flow is faster. Fluidity affects output, and consequently

shear rate, but the residence time is favoured since the effect of lower melting efficiency is

54


probably off-set by that of a slower flow. The temperature dependence of viscosity seems to

affect mainly the minimum level of mixing, but not the highest values of WATS.

As observed previously (Figure 8), the behaviour of the melting efficiency (length of screw

required for melting) depends on the level of a (Figure 12). Anyway, melting is always faster

for the highest values of the rheological constants. This is easily explained by the increasing

contribution of viscous dissipation within the melt films (see Figure 2), which will induce a

more effective heat transfer with the solid bed. The same type of reasoning can be applied to

the drag and pressure flow of melt in the screw channel and die. Thus, one would anticipate

higher average melt temperatures at the die exit for a more Newtonian and/or more viscous

fluid, which is in fact observed (Figure 12).

Finally, and not surprisingly, the mechanical power consumption follows the behaviour of

mass output (Figure 13), i.e., the higher the fluidity the easier the flow, therefore the lower the

pressure generation (Figures 6 to 8) and the lower the power consumption. Keeping the

operating conditions but changing the polymer rheology (in practice, the polymer) can

increase the power consumption more than tenfold.

The results discussed so far illustrate (and quantify) the effect of changing significantly the

values of the rheological parameters on the behaviour of a conventional single screw extruder.

In practice, these changes can be originated by changing the polymer, or the grade of the

polymer being processed. In practice, small oscillations of the operating conditions, which are

inherent to any machine, or small variations in the properties of the polymer, which occur

often from batch to batch, will happen. Hence, it interesting to estimate the resulting changes

in the response of the machine. Table 4 presents the same type of results discussed throughout

this section, for the conditions indicated in Table 3. The maximum pressure and maximum

temperature refer to the maximum values attained somewhere in the screw channel. The

sensitivity in terms of percentage is characterised in Table 5. Limited changes in the

rheological parameters may affect significantly the mechanical power consumption and the

maximum pressure generated, but variations on the degree of distributive mixing, melting

efficiency, and viscous dissipation (melt temperature at the die exit and maximum

temperature) are also noticeable. Conversely, the output is relatively stable (as a variation of

the power law constants affects both the flow along the extruder and die), and so is the

average shear rate at the die exit. In turn, the latter determines the stability of the cross-

section, since it correlates with the extrudate-swell.

55


n=0.25 n=0.45 n=0.65n=0.25 n=0.45 n=0.65

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

Beginning of the pumping zone

End of the pumping zone

290 274 258 241 225 209 193 176 160

Melt temperature (ºC)k = 27500 Pa.sn; a=0.001

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax



290 274 258 241 225 209 193 176 160290 274 258 241 225 209 193 176 160

Melt temperature (ºC)k = 27500 Pa.sn; a=0.001

Figure 9- Cross-channel temperature profiles in the pumping zone (runs reom13, reom9 and reom12).

56


y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax



260 245 230 215 200 185 170 155 140

Melt temperature (ºC)

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

n = 0.45 ; a=0.001

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax



260 245 230 215 200 185 170 155 140

Melt temperature (ºC)

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

y/ymax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

0.000.00

0.50

0.50

1.00

1.000.25 0.75x/xmax

n = 0.45 ; a=0.001

k=5000Pa.sn k=27500Pa.sn k=50000Pa.snk=5000Pa.sn k=27500Pa.sn k=50000Pa.sn

Figure 10- Cross-channel temperature profiles in the pumping zone (runs reom14, reom9 and reom4).

57


Output (kg/hr) – a=0.0001ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

8.01477

8.19358

8.37239

8.5512

7.83767

7.65143

Output – a = 0.001ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

8.01477

8.19358

8.37239

8.5512

8.73002


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

8.19358

8.37239

8.5512

8.73002

8.93514

WATS – a = 0.0001ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

184.887

184.887

201.896

218.904

235.913

167.997

161.772

WATS – a = 0.001ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

184.887

201.896

201.896

218.904235.913

252.922

172.438168.216

WATS – a = 0.01ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

206.022

218.904

235.913

235.913252.922

200.009


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

8.01477

8.19358

8.37239

8.5512

7.83767

7.65143


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

8.01477

8.19358

8.37239

8.5512

7.83767

7.65143


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

8.01477

8.19358

8.37239

8.5512

8.73002


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

8.01477

8.19358

8.37239

8.5512

8.73002


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

8.19358

8.37239

8.5512

8.73002

8.93514


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

8.19358

8.37239

8.5512

8.73002

8.93514

WATS – a = 0.0001ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

184.887

184.887

201.896

218.904

235.913

167.997

161.772

WATS – a = 0.0001ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

184.887

184.887

201.896

218.904

235.913

167.997

161.772

WATS – a = 0.001ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

184.887

201.896

201.896

218.904235.913

252.922

172.438168.216

WATS – a = 0.001ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

184.887

201.896

201.896

218.904235.913

252.922

172.438168.216

WATS – a = 0.01ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

206.022

218.904

235.913

235.913252.922

200.009

WATS – a = 0.01ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

206.022

218.904

235.913

235.913252.922

200.009

Figure 11- Contour plots of output and WATS.

58


Length for Melting – a = 0.001ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.70.324657

0.370405

0.416153

0.4619010.507649

0.53

1117

0.519428


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

0.370405

0.416153

0.416153

0.461901

0.461901

0.445656

0.445656

0.438514

0.438514

0.438514


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

0.324657

0.370405

0.4161530.507649

0.614429

0.561852

Melt Temperature – a = 0.0001ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

206.35

222.179

238.008

253.836

269.665


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

206.35

222.179

238.008

253.836

269.665


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

206.35

222.179

238.008

253.836


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.70.324657

0.370405

0.416153

0.4619010.507649

0.53

1117

0.519428


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.70.324657

0.370405

0.416153

0.4619010.507649

0.53

1117

0.519428


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

0.370405

0.416153

0.416153

0.461901

0.461901

0.445656

0.445656

0.438514

0.438514

0.438514


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

0.370405

0.416153

0.416153

0.461901

0.461901

0.445656

0.445656

0.438514

0.438514

0.438514


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

0.324657

0.370405

0.4161530.507649

0.614429

0.561852


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

0.324657

0.370405

0.4161530.507649

0.614429

0.561852


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

206.35

222.179

238.008

253.836

269.665


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

206.35

222.179

238.008

253.836

269.665


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

206.35

222.179

238.008

253.836

269.665


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

206.35

222.179

238.008

253.836

269.665


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

206.35

222.179

238.008

253.836


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

206.35

222.179

238.008

253.836

Figure 12- Contour plots of length of melting and melt temperature.

59


mmmPower Consumption – a = 0.0001ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

576.6868

1096.503

2084.827

3963.967

7536.852

576.6868

1096.503

2084.827

3963.967

7536.852

Power Consu ption – a = 0.001ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7Power Consumption – a = 0.01ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

576.6868

1096.503

2084.827

3963.967

7536.852

Power Consumption – a = 0.0001ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

576.6868

1096.503

2084.827

3963.967

7536.852


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

576.6868

1096.503

2084.827

3963.967

7536.852

576.6868

1096.503

2084.827

3963.967

7536.852


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

576.6868

1096.503

2084.827

3963.967

7536.852


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7Power Consumption – a = 0.01ºC-1

n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

576.6868

1096.503

2084.827

3963.967

7536.852


n

log

(k)

0.25 0.35 0.45 0.55 0.653.7

3.9

4.2

4.4

4.7

576.6868

1096.503

2084.827

3963.967

7536.852

Figure 13- Contour plots of power consumption.

60


Table 4 – Sensitivity to small changes in rheology.

Output (kg/hr)

WATS

Length for

melting (m)

Melt temperature at die exit (ºC)

Power consumpti

on (W)

Maximum

pressure (MPa)

Maximum

temperature (ºC)

Average shear rate at die exit

(s-1) 0.30 8.8 260 0.629 199 1832 35.4 199 13.41 0.35 8.9 236 0.595 214 2285 44.0 220 13.48 n 0.40 8.9 217 0.551 219 2861 52.0 225 13.52

25000 8.8 241 0.603 211 1915 36.3 216 13.32 30000 8.9 236 0.595 214 2285 44.0 220 13.48 k

(Pa.sn) 35000 8.9 229 0.580 217 2664 50.0 225 13.48

Table 5- Sensitivity to small changes in rheology (in percentage).

Output (%)

WATS (%)

Length for

melting (%)

Melt temperature at die exit

(%)

Power consumpti

on (%)

Maximum

pressure (%)

Maximum

temperature (%)

Average shear rate at die exit

(%) n (33%) 1 17 12 10 56 47 13 1 k (40%) 1 5 4 3 39 38 4 1

6. CONCLUSIONS

The availability of a relatively sophisticated plasticating extrusion modelling routine enabled

the study of the influence of the rheological behaviour of a polymer (described by a

temperature dependent power law) on the performance of an extruder. It was shown that the

effect of n and k on major extrusion parameters is significant, whereas the influence of a is

much smaller. Moreover, even limited changes of these parameters may cause important

process instabilities.

The Central Composite analysis carried out can be used as a preliminary optimisation step,

enabling the identification of the level of the rheological constants that produce the best

performance.

7. REFERENCES

[1] Z. Tadmor, I. Klein, Engineering Principles of Plasticating Extrusion, Van Nostrand Reinhold, New York (1970).

[2] K. O´Brian, Computer Modelling for Extrusion and Other Continuous Polymer Processes, Carl Hanser Verlag, Munich (1992).

61


[3] J.F. Agassant, P. Avenas, J. Sergent, La Mise en Forme des Matiéres Plastiques,

Lavoisier, 3rd edition, Paris (1996).

[4] C. Rauwendaal, Polymer Extrusion, Hanser Publishers, Munich (1986).

[5] A. Gaspar-Cunha, Modelling and Optimisation of Single Screw Extrusion, PhD Thesis, University of Minho, Guimarães (2000).

[6] M.J. Stevens, J.A. Covas, Extruder Principles and Operation, 2nd ed., Chapman & Hall, London (1995).

[7] D.M. Walker, An Approximate Theory for Pressures and Arching in Hoppers, Chem. Eng. Sci., 21, pp. 975-997 (1966).

[8] E. Broyer , Z. Tadmor, Solids Conveying in Screw Extruders – Part I: A modified Isothermal Model, Polym. Eng. Sci., 12, pp. 12-24 (1972).

[9] Z. Tadmor, E. Broyer, Solids Conveying in Screw Extruders – Part II: Non Isothermal Model, Polym. Eng. Sci., 12, pp. 378-386 (1972).

[10] L. Kacir, Z. Tadmor, Solids Conveying in Screw Extruders – Part III: The Delay Zone, Polym. Eng. Sci., 12, pp. 387-395 (1972).

[11] A.R. Mitchell, D.F. Griffiths, The Finite Difference Method in Partial Differential Equations, John Wiley & Sons, Chichester (1980).

[12] O.C. Zienkiewicz, K. Morgan, Finite Elements and Approximation, John Wiley & Sons, New York (1983).

[13] J.T. Lindt, B. Elbirli, Effect of the Cross-Channel Flow on the Melting Performance of a Single-Screw Extruder, Polym. Eng. Sci, 25, pp. 412-418 (1985).

[14] B. Elbirli, J.T. Lindt, S.R. Gottgetreu, S.M. Baba, Mathematical Modelling of Melting of Polymers in a Single-Screw Extruder, Polym. Eng. Sci., 24, pp. 988- 999 (1984).

[15] G. Pinto, Z. Tadmor, Mixing and Residence Time Distribution in Melt Screw Extruders, Polym. Eng. Sci., 10, pp. 279-288 (1970).

[16] D.M. Bigg, Mixing in a Single Screw Extruder, Ph. D. Thesis, University of Massachusetts (1973).

[17] J. Covas, A. Gaspar-Cunha and P. Oliveira, Optimisation of Single Screw Extrusion – Experimental Assessment of Theoretical Predictions, International Journal of Forming Processes, 3, pp. 323-343 (1998).

[18] K.S. Hyun, M.A. Spalding, Bulk Density of Solid Polymer Resins as a Function of temperature and Pressure, Polym. Eng. Sci, 30, pp. 571- 576 (1990).

[19] M.A. Spalding, K.S. Hyun, Coefficients of Dynamic Friction as a Function of Temperature, Pressure, and Velocity for Several Polyethylene Resins, SPE-ANTEC Tech. Papers, pp. 2542-2545 (1992).

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MODELLING AND OPTIMISATION OF SINGLE SCREW EXTRUSION ...€¦ · Therefore, it is the objective of this work to investigate the influence of the viscous behaviour of polymer melts