POLYMER MELT FLOW IN CABLE COVERING EQUIPMENT
by
Fusun Nadiri, B.Sc.(Eng.), A.C.G.I.
A thesis submitted for the degree of
Doctor of Philosophy
of the
University of London
and for the
Diploma of Imperial College
August 1978.
Applied Mechanics Group, Department of Mechanical Engineering, Imperial College of Science & Technology, London SW7 2BX.
2
POLYMER MELT FLOW IN CABLE COVERING EQUIPMENT by
Fusun Nadiri
ABSTRACT
The objective of the work is to predict the behaviour of polymer melt
flow encountered in cable covering equipment. In this investigation, the
melt flow is assumed to be non-Newtonian, steady, laminar, viscous and
incompressible.
The finite element technique is used in solving the flow equations.
The complex geometry of the die and the low Reynolds number of the flow
made it more suitable to use the finite element method which is more
versatile and flexible than the finite difference procedure. The analysis
predicts the flow characteristics of polymer melts in various geometries
using a power-law constitutive equation. It solves for two-dimensional
problems of the harmonic type, using constant strain (rate) triangular
elements. In this analysis, flow in cylindrical shallow channels has
been treated using the lubrication approximation.
Experiments were performed on a three-layer head to study melt flow
in cable covering crosshead dies. Measurements included extruder screw
speed, pressures and temperatures. The polymers used in the experiments
were a crosslinking low density polyethylene and an ethylene/vinyl acetate
copolymer. Rheological tests were made with the polymers to study their
physical properties. The experimental results were compared with the FE
predictions.
Temperature development in pressure flows was studied.. The effect
of thermal conduction, convection and viscous dissipation on melt flow was
investigated. Deflector distortion due to the hydrostatic melt pressure
was also studied to see whether or not it had any significance. A method
of analysis was developed by which, given the required output, the die
could be designed for any cable covering application.
Predictions of non-Newtonian melt flow characteristics are useful in
the design and operation of cable'covering equipment. The flow analysis
techniques developed in this work can be used to help in the design of
crosshead dies.
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ACKNOWLEDGEMENTS
I wish to express my deepest gratitude to Dr R.T. Fenner for all the
encouragement, guidance, stimulating discussions and helpful suggestions
he has offered consistently throughout the course of this work.
I offer my sincere thanks to Messrs E.T. Lloyd, L.M. Sloman,
F.T. White and A.E. Williamson of AEI Cables Limited for their practical
advice and many helpful suggestions. The assistance of Messrs G. Easterby,
J.E. Miller and G.R. Williamson in running the experiments is also
gratefully acknowledged. Special appreciation and thanks are offered to
Miss E.A. Quin for carefully typing this thesis.
Finally, I wish to thank AEI Cables Limited both for their generous
financial support and for making available the equipment on which the
experimental trials were carried out.
To my husband, Soyer
4
CONTENTS
Page
Abstract
2
Acknowledgements
3
Contents
4
CHAPTER 1: INTRODUCTION
9
1.1 The Cable Covering Process
9
1.2 Properties of Polymeric Materials
11
1.3 Theoretical Analysis in Relation to Practical Die
Design
13
1.4 Objectives of This Work
14
CHAPTER 2: PHYSICAL PROPERTIES OF POLYMERS
16
2.1 Importance of Polymer Properties in Melt Flow
16
2.2 Measurement of Polymer Viscosity
16
2.2.1 Errors involved in viscosity data
18
2.2.2 Heat effects in capillary rheometers
20
2.2.3 Rabinowitsch shear rate correction
20
2.2.4 The power-law equation
21
2.3 Factors Affecting Polymer Viscosity
22
2.3.1 Effect of temperature
22
2.3.2 Effect of pressure
23
2.3.3 Effect of shear history
24
2.4 Effect of Temperature and Pressure on Other
Properties of Polymers
25
2.5 A Constitutive Equation for Polymer Melts
25
Page
CHAPTER 3: MATHEMATICAL MODEL FOR MELT FLOW IN CROSSHEAD DIES 28
3.1 Introduction 28
3.2 Crosshead Die Geometry
28
3.3 The Lubrication. Approximation 30
3.4 Lubrication Approximation Applied to Conservation
Equations 31.
3.5 Governing Flow Equations for Melt Flow in Crosshead
Dies 35
3.6 Boundary Conditions 37
CHAPTER 4: SOLUTIONS OF THE CROSSHEAD DIE FLOW EQUATIONS
4.1 Review of Existing Solutions
4.2 The Solution Procedure Used in This Work
4.3 The Method of Solution
4.4 Solution Procedure Used for Pressure Distribution
44
44
47
58
61
CHAPTER 5: MULTI-LAYER CABLE COVERING EXPERIMENTS 66.
5.1 Introduction 66
5.2 The Cable Covering Process in General 68
5.3 The Cable Covering Process Employed in the Trials 71
5.3.1 Crosslinking and semi-conducting materials 72
5.3.2 The extrusion crosshead 73
5.3.3 Experimental measurements and
instrumentation 74
5.3.4 Experimental procedure 76
5.4 Experimental Results 78
5.4.1 Polymer properties and data processing 82
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Page
5.4.2 Analysis of the experimental results 87
5.5 Finite Element Formulation of the Problem 89
5.5.1 Mesh generation 89
5.5.2 Convergence and accuracy 91
5.5.3 Computation of stream function values and
pressure distribution 91
5.5.4 Computation of residence time distribution 93
5.6 Comparison of Theory With the Experimental Results 94
5.6.1 Comparison of theoretical and experimental
thickness distributions 95
5.6.1.1 Effects of gravitational forces 96,
5.6.1.2 Effects of geometric imperfections 97
5.6.1.3 Thickness tolerances for high
voltage cables insulated with
crosslinked polyethylene 98
5.6.2 Comparison of theoretical and experimental
pressure distributions 100
5.6.2.1 Deflector distortion 103
5.7 Conclusions 103
CHAPTER 6: POSSIBLE CAUSES OF CABLE ECCENTRICITY 151
6.1 Introduction 151
6.2 Head Misalignment in Relation to the Catenary 152
6.3 Misalignment of the Crosshead Components 156
6.4 Theoretical Analysis of Deflector Distortion 159
6.5 Cable Eccentricity Due. to Gravitational Forces 166
6.5.1 Finite element formulation of a two-
dimensional biharmonic equation 169
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Page
6.6 Melt Elasticity Effects in Polymer Flow 177
6.6.1 Die swell 177
6.6.2 Melt fracture 179
CHAPTER 7: TEMPERATURE DEVELOPMENT IN PRESSURE FLOWS
192
7.1 Introduction 192
7.2 Basic Equations Defining the Problem 194
7.3 Dimensionless Parameters Describing Melt Flow and
Heat Transfer Characteristics 196
7.4 Thermal Boundary Conditions 200
7.5 Velocity Analysis 200
7.6 Temperature Analysis 203
7.7 Solution Procedure Used 207
7.7.1 Stability 208
7.8 Energy Balance 208
7.9 Results and Discussion 212
CHAPTER 8: USE OF THE METHOD OF ANALYSIS IN DESIGN 226
8.1 Introduction 226
8.2 Case Study 228
8.3 Deflector Design by Inversion of the Analysis 231
8.4 Discussion of Results 238
CHAPTER 9: GENERAL DISCUSSION AND CONCLUSIONS
9.1 Discussion
9.2 Conclusions
259
259
262
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Page
APPENDICES
AI Representation of Cl, C2 and C3 in Terms of the
Nodal Point Values of Stream Function 266
A2 h2-Extrapolation Technique 269
A3 Pressure Drop Between Transducer and Melt Inlet
to the Deflector 271
A4 Derivation of the Equilibrium Equations for a Thin
Cylindrical Shell 275
A5 Derivation of the Differential Equations for the
Displacements of a Thin Cylindrical Shell 282
A6 Generation of a Circular Mesh of Triangular
Elements 285
Notation
287
References
301
Work Accepted for Publication:
"Finite Element Analysis of Polymer Melt Flow in Cable
Covering Crossheads"
9
CHAPTER 1
INTRODUCTION
1.1 THE CABLE COVERING PROCESS
The polymer processing field is concerned with the conversion of
polymers into useful products. The most important polymer processing
technique today is the extrusion process which is one of shaping a
material by forcing it through a die. The cable covering process is used
to coat continuous lengths of wire, cable, tube and a variety of products
with a layer of extruded thermoplastic material. The extruders used are
usually of the ram or screw type, the latter being the more widely used.
The screw extruder efficiently and continuously converts solid polymer into
melt and pumps the very high viscosity melt through a die at high pressure.
The principle of operation of an extruder is that of a screw pump,
the action of which depends on the material in the screw channel being
dragged along by the surface of the barrel in which the screw rotates.
Although extruders are sometimes fed with melt, the polymer is normally
supplied in the form of pellets, chips, beads or powder. Such
plasticating extruders perform the function of melting in addition to
mixing and pumping. These machines extrude a tremendous variety of
products at high rates, including cables, wires, pipes, films, sheets,
paper coatings, monofilaments and various contoured profiles. On leaving
the screw, the melt is usually forced through a perforated breaker plate
or other mixing and filtering devices before reaching the die.
The function of an extrusion die is to form the molten material
delivered by the screw into a required cross-section. The die is a
channel whose profile changes from that of the extruder bore to an orifice
which produces the required form. The quality of the extruded product
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depends not only on the extruder and the material used, but also on the
efficiency of the die.
Extrusion dies may be attached to the extruder in three different
ways according to the requirements of the extrusion process. Straight-
through dies are those dies whose axes are in line with the direction of
supply of melt. They are commonly used for the extrusion of pipe, rod,
profiles and sheet. In some cases, a straight-through die is attached to
a curved feed channel to change the direction of take-off. This is used
for the extrusion of tubular and flat film.
Crosshead dies are arranged with their axes at an angle to their feed
supply, usually 90°, but 45° and 30° are also used. Dies of this form
are generally used for the production of insulated wires and cables, or in
other processes where it is necessary to introduce a continuous filament
to the die mandrel. With this type of die, it is possible to have ready
access to the upstream end of the die mandrel where heating or cooling or
other control or manipulation of this member is easily effected. Owing
to the use of side feed, there is no need for a spider assembly in the
production of hollow extrusions, but the unbalanced feed does introduce its
own problems.
Offset dies have been developed from crossheads. They are arranged
so that the material is made to change direction twice in an attempt to
compensate as far as possible for the imbalance resulting from the single
direction change which occurs in the ordinary crosshead. They are popular
for the production of pipe where the lack of a spider and also the ease of
applying temperature control to the mandrel improve the quality of the
product.
Although the systems of die attachments have been classified above
into three categories, each presumably being suited to one general type of
extrusion, it is possible that all three systems may be adapted for the
production of any type of extruded product. The decision as to which
type is to be used in a particular case is usually determined by questions of convenience, space, the availability of equipment and the previous
experience of the operator. In this text, the term cable covering
equipment refers to cable covering dies. Although the melt flow analysis
has been made for crosshead dies in particular, the analysis should also
apply to other kinds of dies in general. For further descriptions of
extrusion and cable covering equipment and processing, the reader is
referred to the texts of Renfrew & Morgan (1960), McKelvey (1962), Griff
(1962), Jacobi (1963), Schenkel (1966), Tadmor &Klein (1970), Fenner
(1970), Bernhardt (1974), and Fisher (1976).
1.2 PROPERTIES OF POLYMERIC MATERIALS
The design of manufacturing and processing equipment requires
considerable knowledge of the processed materials and related compounds.
The continuous development of the modern process industries has made it
increasingly important to have information about the properties of
materials, including many new chemical substances whose physical properties have
never been measured experimentally. This is especially true of polymeric
materials. Polymers are composed of large molecules, usually consisting
of repeating chemical units called mers. In this context, the term
polymer refers primarily to thermoplastics and cross-linked polymers
(thermosets). Despite the macro-molecular nature of polymers, in this
work their melts are treated as continuous media and the continuum
mechanics equations are applied to the analysis of melt flow.
A distinctive feature of polymeric materials is that the properties
can be influenced decisively by the method of manufacturing and by
- 12 -
processing. The properties are sensitive to the processing conditions;
for instance, they are greatly dependent on the degree of orientation
imparted. This is a direct consequence of the elasto-viscous nature of
the starting material. Due to the high viscosity of polymer melts, all
the molecular processes are greatly retarded. This means poor heat
conductivity and slow relaxation. There is a certain sensitivity to
thermal, mechanical or chemical degradation during processing.
In practice, most polymers are processed via a melt. The processing
technique involves four phases which are often closely connected:
1. 'Transportation' of the material to the forming section of the
processing machinery where transport properties are important.
2. 'Conditioning' of the material, by heating, to the forming process
where thermal properties are important.
3. 'Forming' of the material where rheological properties are
important.
4. 'Fixation' of the imposed shape where thermal, rheological and
especially transfer properties, like thermal conductivity, rate
of crystallization, etc., are important.
In each of these phases, the material is subject to changing
temperatures, changing external and internal forces and varying retention
times, all of which contribute to the ultimate structure. It is this
fluctuating character of the conditions in processing which makes it so
difficult to choose criteria for the processing properties.
The melt flow analysis of cable covering equipment involve mainly the
'forming' and 'fixation' phases where thermal, rheological and transfer
properties are of great importance. Unlike most conventional Newtonian
- 13 -
fluids whose viscosities only vary with temperature and pressure, polymer
melts are non-Newtonian. Methods for estimating the properties of
polymers, in the solid, liquid and dissolved states, in cases where
experimental values are not available have been provided in the book of
van Krevelen (1972). This book also contains various tables on the
properties of polymers.
1.3 THEORETICAL ANALYSIS IN RELATION TO PRACTICAL DIE DESIGN
Practical die design is concerned with the construction of dies for
commercial extrusion where such factors as die cost in relation to the
length of run, ease of construction of the die and adapting it for the
production of other sections when it becomes redundant, ease of
dismantling for cleaning and cost of replacement of component parts, are
of great importance. The ease of handling Of an extrusion die and of
effecting the necessary adjustments, such as concentricity in a tube die,
have a great effect on the saleability of extrusion equipment.
Most cable covering equipment is operated and designed on the basis
of past experience. This approach is often satisfactory for known
materials and applications. With the introduction of new polymers and
processes, and the trend towards better quality products, trial-and-error
methods of design often prove to be extremely costly and time-consuming.
Many thermoplastic materials are heat sensitive and too high a
temperature or too long a residence time in a process will cause them to
degrade. The control of the temperature of a polymer melt involves, in
general, the prevention of any sharp temperature gradients. The residence
time is dependent on the design and construction of the extrusion equipment,
including the screw termination, the breaker plate and screen pack, the
die adaptor system and the die itself. Owing to the slowness of movement
- 14 -
of the layer of melt which is in direct contact with the die walls, there
is a pronounced tendency for it to adhere to the metal surfaces and
eventually to degrade with dire consequences to the product. In order
to minimise this danger, it is desirable to design the die with narrow
channels so that the effects of the residence time distribution gradient
will be reduced. It is highly important that all changes of shape and
dimensions take place gradually to avoid steps and shoulders and other
obstructions to flow near the channel walls.
The theoretical analysis of melt flow in cable covering equipment is
quite complex due to the non-Newtonian flow properties and the coupled
thermal effects. Apart from predicting the output of the die, given the
general die geometry, theoretical analysis also serves to design a die to
give a required uniform output which is usually only achieved by means of
die-gap adjustment. The early attempts to analyse melt flow in cable
covering equipment were based on many simplifying assumptions which were
far from reality. Only with the aid of digital computers have more
realistic treatments been possible. New materials and processes can now
be examined far more rationally than by the practical trial-and-error
approach.
1.4 OBJECTIVES OF THIS WORK
The object of the work described in this thesis is to examine, both
theoretically and experimentally, polymer melt flow in cable covering
equipment. The emphasis is on the applications of the analysis of flow
of molten polymers. The aim is to obtain efficient methods of solving
practical industrial problems of die design, in particular the design of
multi-layer crosshead dies.
Experiments on a three-layer crosshead are reported which provide
- 15 -
practical verification of the theoretical analyses. The predicted shape
of the extruded cable and the pressure drop in the die are compared with
the experimentally measured results and improvements made to the flow
analysis.
In addition to the treatment of melt flow, deflector distortion due
to the hydrostatic pressure in the die channel walls is analysed
theoretically. The problem of combining such an analysis with that of
melt flow is also considered. An attempt is made to investigate
temperature development in pressure flows. Finally, the use of the method
of analysis in die design is presented.
- 16 -
CHAPTER 2
PHYSICAL PROPERTIES OF POLYMERS
2.1 IMPORTANCE OF POLYMER PROPERTIES IN MELT FLOW
It is essential to know certain fundamental physical properties,
i.e viscosity, density, enthalpy and thermal conductivity, ideally as
functions of temperature and pressure, when analysing polymer melt flow.
Owing to the non-Newtonian flow properties of polymers, viscosity must
also be obtained as a function of the rate of deformation.
There is a considerable need for accurate polymer property data for
use in the analysis of extrusion and forming processes. As polymer
properties change with improved manufacturing processes and as new grades
replace old ones, these data must be revised. The current wide range of
commercially available polymers have made property measurement a major
industrial activity.
The accuracy and consequent usefulness of all melt flow calculations
are directly dependent on the quality of the property data used. It is
particularly important to use absolute property values which are
independent of the method of measurement and viscosity data frequently
suffer from this defect.
2.2 MEASUREMENT OF POLYMER VISCOSITY
Many methods have been devised to study the viscous properties of
polymers and they are discussed in the texts of Ei ri ch (1960) , Van Wazer et a1
(1963), Fredrickson (1964), Brydson (1970), and Walters (1975). Some of
the methods are highly empirical and do not lead to the quantitative
determination of fundamental data, whilst others are of little value with
polymer melts. For such melts, two types of instrument are of particular
- 17 -
interest:
(a) Rotational rheometers
(b) Capillary rheometers
The most widely used for polymer melts under conditions relevant to
extrusion and cable covering processes is the capillary rheometer.
Briefly, this consists of a heated barrel at the bottom of which is
fitted a small die containing a capillary tube. The melt is forced
through this capillary by a piston, which is driven either at constant
speed or under constant pressure, inside the barrel. The apparatus is
maintained at a constant temperature, which is assumed to be the uniform
temperature of the melt. Measurements are made of volumetric flow rate,
Q, of the melt and reservoir pressure, P. In analysing the flow through
the capillary viscometer, the following assumptions are made:
i) There is no slip at the wall. There is evidence that, at high
shear rates and with high molecular weight polymers, slip
conditions prevail. Benbow et al (1963) and Westover (1966)
have shown that, under such conditions, the conventional analysis
is useless and flow curves derived from them have no validity.
Useful analyses for capillary flow in conditions where slip
occurs have, however, been proposed by Lupton & Regester (1965).
ii) The fluid is time-independent. If a material is time-
dependent, then it would be expected that, in passing through a
tube, the apparent viscosity of the melt would change.
iii) The'flow is steady and laminar.
iv) The melt is incompressible. At the low pressures used in most
(2.1)
(2.2)
Ya 32Q
Tr D3
P D T = 4L
- 18 -
commercial viscometers (1500 lbf/in2 or less), the
compressibility of the polymer melt appears to be negligible.
At higher pressures, however, this effect can be more important,.
and the dependence of melt viscosity on pressure becomes quite
significant (see Section 2.3.2).
v) The flow is isothermal. Viscous drag will cause a frictional
heat build-up which will be proportional to the product of shear
stress and shear rate. Thus, under high shear conditions, such
as occur in injection moulding, a rise in temperature could be
as high as 20°C (see Brydson (1970)).
Using the assumptions above, the apparent shear rate and shear stress
at the wall can be determined, respectively, by the following expressions:
where D is the diameter, and L is the length of the capillary. The
apparent viscosity, pa, is then given by the simple ratio:
T
Ya
(2.3)
2.2.1 Errors Involved in Viscosity Data
The viscosity data obtained from capillary rheometer
experiments involve the following two errors:
- 19 -
i) Reservoir pressure drop: Since the reservoir pressure is
measured at the piston, it depends on the volume of melt
remaining in the barrel. Therefore, to overcome any errors,
P should always be measured at the same position of the piston,
so that the reservoir pressure drop may be treated as part of
the capillary end correction. This procedure requires that
the reservoir be recharged after each test. This type of
error has been discussed in the literature by Metzger & Knox
(1965) and Ballman (1963).
ii) Capillary end correction: The mean pressure drop per unit
capillary length, P/L, should be the uniform pressure gradient
for the assumed fully-developed isothermal flow. The total
pressure drop, P, includes losses at the capillary inlet and
exit.
One way of avoiding end errors is to use a very long capillary,
which has a very large length-to-diameter ratio of at least 100. Such a
long capillary, however, requires high reservoir pressures, which influence
melt viscosity. A better method is to use two or more capillaries of the
same diameter but of different lengths. End errors may then be determined
by plotting reservoir pressure against capillary length at constant flow
rate and extrapolating to either zero length or zero pressure. Ram &
Narkis (1966) and Bagley (1957) have reviewed the literature on end
corrections in great detail. Duvdevani & Klein (1967), Van Wazer et al
(1963) and Schreiber (1966) have also considered these and other sources
of error.
20 -
2.2.2 Heat Effects in Capillary Rheometers
High rates of shear cause a high energy dissipation and
therefore a temperature rise in the capillary. These temperature rises
caused in the melt give rise to local variations in viscosity. Brinkman
(1951) studied heat effects in capillary flow and predicted that the
temperature is highest near the walls of the capillary where the rate of
shear is highest. Martin (1967) has presented some analytical solutions
for viscometric flows with heat generation and showed that the viscosity
calculated from capillary rheometer measurements gives an accurate estimate,
even when viscous heating is significant.
2.2.3 Rabinowitsch Shear Rate Correction
Equation (2.1) gives the true shear rate at the capillary wall
for a Newtonian melt. Rabinowitsch (1929) derived a correction for non
Newtonian flow behaviour which is given by several authors, including
Van Wazer et al (1963), Schreiber (1966), Duvdevani & Klein (1967),
Ballman (1963), Metzger & Knox (1965), Ram & Narkis (1966) and Bagley
(1957). The correction is based on the following assumptions:
(a) The flow is axial with a velocity which is a function of
radial position only.
(b) There is no slip at the wall.
(c) The shear rate at any point is a function only of the shear
stress at that point.
The true shear rate at the wall is then given by:
Y (31i' f .1)
Ya (2.4)
4n'
- 21 -
where: d(ln t) d(ln ya)
(2.5)
This result requires the evaluation of n at each shear rate
and, since differentiation of experimental results is an extremely
unreliable procedure, true shear rates are rarely quoted. If, however,
the melt obeys the power-law equation, then n' = n and the correction
factor is constant over reasonably wide ranges of shear rate, which
greatly simplifies its use.
The true viscosity may be determined as:
u 411 1 p
a
3n'+1 (2.6)
2.2.4 The Power-Law Equation
It has been experimentally observed that, for most polymers,
the relationship between shear stress and shear rate can be represented,
over limited ranges of shear rate, by a power-law equation of the form:
= C n Y (2.7)
or: p _ C yn-1 (2.8)
where C, and to a much lesser extent n, are functions of temperature and
pressure. The above two equations are normally valid over shear rate
ranges of at least one decade. When considering a specific process, the
range of interest is usually limited and, therefore, no serious errors
would be introduced. For example, the relevant shear rate range in
- 22
extrusion equipment is 10-100 s 1.
2.3 FACTORS AFFECTING POLYMER VISCOSITY.
The ease of flow depends on the mobility of the molecular chains and
the forces holding the molecules together. It is widely known that an
increase in temperature will reduce viscosity. The influence of
environmental pressure and shearing history of the polymer melt are much
less well known and it is only in recent years that their importance has
begun to be appreciated. These influences are discussed in some detail
below.
2.3.1 Effect of Temperature
It is well established that, for liquids which show Newtonian
behaviour, viscosity and temperature may be related by an Arrhenius
equation of the form:
u = A eE/ (2.9)
where A is a constant which is a function of pressure and shear rate, E is
the activation energy, R is the universal gas constant, and T is the
absolute temperature. It has been concluded by Porter & Johnson (1966),
after reviewing the existing literature, that, for many polymers, equation
(2.9) agrees well with experimental results over ranges of temperature of
about 30°C. Over this range, the following more empirical equation
describes the experimental results which is also more convenient to use
than equation (2.9):
n-1 = uo J~
Yo Y exp (— b Or — T0)) (2.10)
-23-
where uo is the effective viscosity at reference shear rate, Yo, and
reference temperature, To, and b is the temperature coefficient of
viscosity at constant shear rate.
It has been shown by Martin (1967) that, if the melt obeys
equation (2.10), then capillary rheometer results provide accurate
estimates of n and b, even when viscous heating is significant.
2.3.2 Effect of Pressure
The dependence of melt viscosity on pressure is normally
approximated by:
u = B exp (a p) (2.11)
where p is the pressure, a is the pressure coefficient of 'viscosity, and
B is a function of temperature and shear rate. Therefore the power-law
equation would become:
n-1 Y exp (- b (T -
Y0
uo 0 4-a p) (2.12)
where a is defined at constant shear rate.
The experimental determination of a is difficult and,
consequently, it has been attempted by only a few people, including
Maxwell & Jung (1957), Westover (1960), Semjonow (1965) and Hellwege et
al (1967). Duvdevani & Klein (1967) have also attempted to obtain values
of a from conventional capillary rheometer results and they quote a as
approximately 4.4 m2/GN for polyethylene. It is evident, even from the
very limited results quoted, that the pressure dependence of viscosity
should be included in the analysis of extrusion processes (with pressures
up to, say, 35 MN/m2), particularly for the more pressure-sensitive polymers.
-24-
2.3.3 Effect of Shear History
The viscosity of certain polymer solutions changes with time
as the liquids are stirred at constant shear rate. The possibility of
shear history effects, both irreversible and reversible, was recognised
by early workers in the field of polymer melt flow measurement, for
example, the change in viscosity on shearing natural. rubber. Irreversible'
effects are usually due to either cross-linking or chain scission. Such
changes may be caused by oxidation or mechanico-chemical processes. In
general, to minimise these effects, it is desirable to use samples which,
prior to being charged into the rheometer barrel have had similar shear
and heat histories.
It has been suggested by some workers that (see Brydson (1970)),
if a capillary experiment is carried out and the extrudate collected, cut
up and re-extruded, in the case of polyethylene there is a negligible
change in flow properties. Whilst this may be true at shear rates below
the critical shear rate, at higher shear rates, it has been noted by
Howells & Benbow (1962) that, in the case of polyethylene, pre-shearing
reduces the amount of elastic turbulence and also reduces the viscosity by
a factor of two. Since corresponding decreases in molecular weight were
not observed, the indications were that this was a chain disentanglement
effect. This explanation is supported by the fact that heating of the
polyethylene samples for several hours at 190°C causes reversal of the
effect (Brydson (1970)). Such heating would be expected to cause
molecular movements, leading eventually to a state of entanglement similar
to that of the original melt.
-25-
2.4 EFFECT OF TEMPERATURE AND PRESSURE ON OTHER PROPERTIES OF POLYMERS
Andersson & Backstrom (1973). have shown that the thermal conductivity
of polyethylene increases with pressure, while the specific heat decreases
at low pressures. There is very little published data on the influence
of temperature on thermal conductivity, although, for a number of polymers,
the temperature dependence appears to be small (Bil' & Avtokratova (1966)).
The densities of polymer melts depend on both temperature and pressure.
Kanavets & Batalova (1965) have studied the thermal expansion and
compressibility of thermoplastics. They showed that the compressibility
of amorphous and crystalline polymers, between pressures of from 50 to
1200 kgf/cm2 with rise in temperature, increases particularly above the
glass transition temperature. For example, in polyethylene at 100°C and
500 kgf/cm2, the compressibility is about 0.5%, and at 150°C about 3%.
Under processing conditions, however, in particular extrusion and cable
covering, the variation of density with pressure and temperature is usually
small.
Neglecting pressure dependence, the enthalpy of most polymer melts is
an approximately linear function of temperature over quite wide
temperature ranges. Enthalpy is an important property which is relevant
to thermal convection in the conservation of energy equation.
2.5 A CONSTITUTIVE EQUATION FOR POLYMER MELTS
The relationship between shear stress and shear rate is known as the
constitutive equation for the material, and involves the influence of
temperature and pressure. The most general relation for an inelastic,
homogeneous and isotropic fluid (i.e. its properties are not explicitly
dependent on position and direction, respectively), the Reiner-Rivlin
fluid, can be shown under isothermal conditions (see, for example, Serrin
-26-
(1959)) to have the form:
t. = — p aid + n1 ein + n2 eik e (i, = 1, 2, 3) (2.13)
kj
where p is the isotropic dynamic pressure, n1 and n2 are generalised
viscosity and cross-viscosity, respectively, and 6i is the Kronecker
delta defined by:
a
i3
= 1 if i = j
a = o if i j ij
A precise thermodynamic definition of p is difficult; for a fluid at rest
it becomes the hydrostatic pressure. For an incompressible fluid, p is
arbitrary as far as the constitutive equation is concerned (see Aris
(1962)).
The material properties n1 and n2 are arbitrary scalar functions of
temperature, pressure and the three principal invariants of ei:
= eli
I2 = e13 ei3
I = det le..) sj
(2.14)
(2.15)
(2.16)
where 'det' means the determinant of the enclosed matrix. Therefore, the
parameters n1 and n2 can be represented as:
= n1(II , 12 , 13 , T , p) (2.17)
n2 = n2(I1 ,
i2 , I3 , pJ (2.18).
(2.21) g = ij
expressed as the sum of symmetric and anti-sryntiuet
-27-
Equation (2.13) describes inelastic polymer melt flow behaviour,
taking account only of the response to the local instantaneous rate of
deformation. Melt elasticity is essentially a 'memory' effect, depending
on the deformation history of the particular fluid element. In extrusion
processes, however, elasticity is generally not important as the melts are
subjected to large rates of deformation for relatively long times.
Pearson (1966) has proposed more complex constitutive equations to take
account of elastic effects, but these are difficult if not impossible to
use.
Having stated a general constitutive equation (equation (2.13)) for',
polymer melts, the generalised viscosities n and n2 must be correlate
ar as possible with the viscosity data obtained from the capilla
rheometer experiments. Assuming the polymer melt to be locally
incompressible, equation (2.14) becomes:
I I = 0
Neglecting the pressure dependence of viscosity, equations (2.17) and
(2.18) become:
n1 = n 1(I2, I3, T) (2.19)
n2 = n2 (I2, I3,T1 (2.20)
The local velocity gradient tensor gi can be expressed in Cartesian
coordinates as:
-27a-
= , (gi gni) fJē (gam
= wij
where wij
is the rate of rotation tensor, and ei is the rate of
deformation tensor:
ei. = (gi0 g~ i)
g '0 2
(2.22)
(2.23)
The only non-zero velocity gradient in capillary flow is
gzr = dvz/dr, where r and z are the radial and axial coordinates,
respectively. Therefore, from equation (2.16), I3 = 0, and from equations
(2.15) and (2.23):
dv = ( z) 2
at the capillary wall
(2.24)
12 dr
= (y)2
Using equation (2.13), the wall shear stress is given by:
T = tzr = n1 (*Y2, 0,T) Z (2.25)
Comparing this result with the power-law equation (2.10):
n-1
n1(I23 0, T) = 2uo exp (- b or — T0)) (2.26) Yo
In the subsequent analysis, the melt flows are predominantly shear flows,
in which 13 = 0 and n2 is not important. Hence,-equation (2.26) is an
appropriate form of constitutive equation.
-28-.
CHAPTER 3
MATHEMATICAL MODEL FOR MELT FLOW IN CROSSHEAD DIES
3.1 INTRODUCTION
The type of polymer melt flow encountered in cable covering crossheads
is essentially flow in relatively shallow channels. In this chapter, an
attempt will be made to set up a mathematical model which will describe the
flow. Having selected a suitable coordinate system, it is possible to
write down the governing flow equations. Given the relevant boundary
conditions and the constitutive equation. of the melt, in principle, a
solution of these fully three-dimensional equations can be found. In
practice, however, we must simplify the problem by making many assumptions.
3.2 CROSSHEAD DIE GEOMETRY
The particular problem to be considered is that of flow in crosshead
dies used in the covering of high-voltage electrical cables. It is not
uncommon for a single head unit to be used to apply two or three layers of
different materials during one pass of the conductor. For example, in
the trials used to test the present method of analysis, explained in
Chapter 5, three layers were applied to a tape-covered stranded copper
conductor. For each layer, the problem is to design a system of flow
channels which accepts a side-fed supply of melt and distributes it into
a tube of uniform thickness which is then extruded as part of the cable.
Figure 3.1 shows one commonly used form of arrangement for attempting
to achieve the desired uniformity. A narrow radial gap between concentric
cylindrical and conical surfaces, the outer members of which have been
removed for illustration purposes, serves to distribute the melt. As the
melt tends to take the shortest path from the inlet to the channel exit,
- 29 -
this path is deliberately blocked by a.heart-shaped area which fills the
radial gap and forces the melt flow to follow longer paths of more uniform
length. The cylindrical portion of the component shown is known as the
deflector, while the subsequent conically tapered portion is termed the
point. Channel geometry and therefore the flow are intended to be
symmetrical about the centre line of the heart-shaped blockage.
Assuming that the channel depth, h, is small compared to the mean
radius of the deflector, it is often reasonable to assume that the channel.
may be unrolled. Figure 3.2 shows the shape of one half of the flow
channel unrolled and plotted on the z,e plane, z being the axial coordinate
and a the angular coordinate measured from the line of symmetry through
the flow inlet. The region bounded by points A, B, C and D is on the
deflector, while that bounded by C, D, E and F is on the point, as
indicated in Figure 3.1. Also shown in the flow channel are some
triangular finite elements which are discussed later. Clearly, flow
paths between the inlet boundary AB and outlet EF are of reasonably
uniform length. It should be noted that the channel depth is often
reduced by tapering in the axial direction in both the deflector and point
regions. Indeed, in the deflector region, the channel depth may also be
varied in the circumferential direction to improve the flow distribution.
Figure 3.3a shows the cross-section of a typical die in the r-z plane
where r is the mean radius of the flow channel and h is the local channel
depth normal to the channel boundaries. Ld and p are the lengths of the
deflector and point regions, respectively, L being the overall length and
a the angle of inclination of a typical portion of channel to the z-axis.
-30-
3.3 THE LUBRICATION APPROXIMATION
Provided a channel containing a flowing melt can be described as
narrow, the analysis of the flow may be treated in a relatively simple
manner. A narrow channel is one in which one of the channel dimensions,
normal to the direction of flow, is small compared to the other two
dimensions, and only varies slowly over the region of interest. In the
present context of cable covering crossheads, the radial depth of the
channel is small compared to its axial and circumferential dimensions, and
only varies slowly in these directions.
Figure 3.3a shows a typical axial cross-section through the flow
channel. The channel depth, h, is small, such that:
h « r h « L (3.1)
r and L being the mean radius and overall axial length, respectively, and
h is subject to only small local variations:
laand « 1 Ir DO 1 (3.2)
Pearson (1962,1966,1967) has shown that such conditions are necessary for
the lubrication approximation to be applicable, which means that the flow
can be treated as locally fully developed between flat parallel surfaces,
provided the Peclet and Reynolds numbers which are based on h are small.
As far as channel taper is concerned, Benis (1967) investigated isothermal
flows in channels of varying gap, both theoretically and experimentally,
and concluded that the lubrication approximation holds for taper angles up
to 10° ph/az = 0.2).
Pearson (1967) formalised the approximation procedure by a
- 31 -
perturbation approach, and showed that the resulting uniform flow is a
valid first approximation to the flow of an elastic non-Newtonian fluid
in.a narrow channel. Neglecting geometric terms, including curvature and
gap variation, melt inertia and elastic effects (Pearson (1967)), the only
effect likely to invalidate the lubrication approximation is that due to
thermal convection. For present purposes, however, the flow is assumed
to be isothermal in the sense that any temperature variations within the
flow do not affect velocity profiles. Justification for this assumption
is provided in Chapter 7.
Application of the full lubrication approximation in the z-direction
assumes that velocity and temperature profiles are fully developed (see
Fenner (1970)). The first condition in equation (3.2) makes this
assumption reasonable for velocity profiles (except for their dependence
on developing temperature profiles); however, it is not generally valid
for temperature profiles. Pearson (1967) and Yates (1968) showed that the
full lubrication approximation is only marginally justified in many
extruders. At worst, the full lubrication. approximation provides.
estimates for the maximum temperature rise at any point in the flow and
the minimum rate of heat generation. In contrast, the "isothermal
condition provides an estimate of the maximum rate of heat generation.
It assumes that velocity profiles are independent of temperature profiles.
3.4 LUBRICATION APPROXIMATION APPLIED TO CONSERVATION EQUATIONS
The continuum mechanics equations governing melt flow are those of
conservation of mass, momentum and energy. These can be concisely
expressed in tensor notation, as shown in the texts of McKelvey (1962)
and Pearson (1966). The rate of deformation tensor may be represented
by ei j, where•
-32-
1 av. av . ei j = .
v. being the local velocity component in the x. coordinate direction.
Assuming that the density of the melt is locally constant, the continuity
equation for conservation of mass reduces to:
e.. 3.4)
Using, for example, cartesian coordinates x, y and z with corresponding
velocities u, v and w, respectively, and applying the lubrication .
approximation in the z-direction to velocities, which assumes that
velocities are fully developed in the z-direction, then:
= u(x,y). , v = v(x,y) , w = w(x,y) (3.5)
and the continuity equation (3.4) becomes:
au ax
4.av o ay (3.6)
The conservation of momentum involves a balance between inertia,
viscous, pressure and body forces. In comparison with viscous and
pressure forces, it can be assumed that body forces, such as those due to
gravity, are negligible. Due to the low Reynolds numbers associated with
melt flows, it may also be assumed that inertia effects are likewise
negligible. The equations for conservation of momentum therefore reduce
to:
ap ax . 1
3.7)
3.3) axe ax2
ax ax ay
= aT aT
XX f xY (3.8
-33-
where p is the local hydrostatic pressure, and T2. the viscous stress
tensor. Applying the lubrication approximation in the z-direction to
velocities, the equilibrium equations (3.7) become:
aT aT _ + YY (3.9)
ay ax ay
aT aT 22.= Pry = zx f zy (3.10) az
ax ay
where the pressure gradient, z, is independent of x and y.
The conservation of energy equation, for steady flow and locally
constant thermal conductivity, k, and specific heat at constant pressure,
Cp, can be written as:
DT _ z
p p v. p k a 2 t T. e. 2 ax. la 2
(3.11)
where T is temperature, and p is density. Owing to the presence of
significant convection terms in equation (3.11), it is much less reasonable
to apply the lubrication approximation in the z-direction to temperatures.
If, however, it is assumed that temperatures are fully developed in the
z-direction, then:
T = T(x,y) (3.12)
and the energy equation reduces to:
- 34 -
aT aT k a2T a2T au ~+ p C
p (u - + v -J = (-+-J + 1" - + 1"
ox oy ox2 ay2 xx ox yy ay
(o u + ~J ow aw (3.13) 1" + 1" - + 1" xy ay oX yz ay zx ax
The equations can be further simpl ified if the lubrication
approximation is applied in the x-direction. Thus, equations (3.5) and
(3.12) reduce to:
u = u(yJ v = 0 w = wry) (3.14)
T = T(y) (3.15)
and therefore the equilibrium equations (3.8) to (3.10) and the energy
equation (3.13) reduce to:
?£ oX
£ ay
p z
= p x
= 0
= ~ (3.16) ay
(3.17)
(3.18)
au aw -1" --1" -
xy ay yz ay (3.19)
-35-
3.5 GOVERNING FLOW EQUATIONS FOR MELT FLOW IN CROSSHEAD DIES
The analysis which follows is similar to that originally presented by
Pearson (1962) extended to allow for conical channel geometry. At a
particular point in the flow channel, the local velocity profile is of the
form shown in Figure 3.3b. Let this resultant profile be in the direction
s, which in general is neither axial nor circumferential, and let v(y) be
the velocity. The local radial coordinate, y, is measured from the mid-
surface of the channel, itself a distance r from the axis. If V is the
local mean velocity and Qs
is the volumetric flow rate per unit width
normal to the s-direction, then:
Qs = f v dy = h V - h
(3.20)
r.
The form of the velocity profile depends on the non-Newtonian viscous
properties of the melt concerned. For practical purposes, a power-law
constitutive equation relating shear stress, T, to shear rate, y, is
generally the most useful (see equations (2.7) and (2.8)):
T = uo Yo (3.21)
where n is the power-law index, and uo is the reference viscosity at the
processing temperature and reference shear rate, Such Such a relationship
provides a good fit of rheological data over comparatively wide ranges of
shear rate. Given the constitutive equation and using the lubricatiofl
approximation, the relationship between flow rate and pressure gradient
the 8-direction may be d *tve __ ws:
v n-1 dv = y~ = PO (Y (3.22a)
-35a-
For fully developed flow:
a P = = _SE s as
By
Therefore, since the velocity profile is symmetrical about y = 0:
Tsy = s y
Combining equations (3.22a) and (3.22c):
h o dv n-1 dv _ n-1 141 s Y
Yo
(3.22b)
(3.22c)
(3.22d)
Considering the region y 0, dv/dy < 0 and Ps < 0. Hence:
n-1
dv _ - (yo ( s))1/n 111/n = - C y1/n (3.22e)
uo
where:
n-1
PO
Integrating equation (3.22e) gives:
1/n+I _ v = 1 y + B
-- + 1 n
where B is the constant of integration. Assuming that there is no slip
at the wall:
v = 0 at
Therefore:
1 C ((~)1/n+1 - y1/n+1)
—+1 n
(3.22g)
_ h 2
- 35b -
Using equation (3.20):
h/2 = 1 2C h) 1/nr-2 = 2 f v ~
~
o n +2 (3:22h)
Substituting equation (3.22f) into equation (3.22h):
n-1 2n2nn 1
Qs 6) 1/n+ 0 2 = ~ " 8)) 1/n
- Ps
- P s
=
=
u0
2n+1 2n + 1 n n 1 1 2 2n ! Qs uo h
+1 /2n + 1)n ( Qs 22n Q u
n-1 yo 0
)n-1 1
2n s o h2 h3
Therefore: Ps = as
Qs u
h3 (3.22)
-36-
where:
1 ( '2n )n 22n+1 2n + 1
and iris the viscosity at the .mean shear rate, V/h:
( Qs )n-1 h2 Y
0
(3.23)
(3.24)
Figure 3.3c shows a typical small portion of the flow channel
inclined at an angle a to the axis of the conductor. Let x be the
coordinate along the channel in the axial plane, and let Qx and Qe be the
volumetric flow rates in the rand circumferential directions, per unit
length in the circumferential and x directions, respectively.
Conservation of mass in incompressible steady flow requires that:
(Qx + dQx) (r + dr) de - Qx r de + (Q0 + dQe) dx - Q0. dx =
a ax
aQ r Qx) +
6 = 0 (3.25)
.
ae
which is essentially an integral form of equation (3.6). This equation
is automatically satisfied by the following stream function, 11,:
ae Qe ax 3.26)
.Now, using equations (3.22) to (3.24), the pressure gradients in the x and
e directions are given by:
Qs u ax h3
_ ae
r Qe u
h3
(3.27)
-37-
where the resultant flow rate used in evaluating u is given by:.
Qs 2 Qx 2+ Q B2 (3.28)
From the fact that the pressure, p, must satisfy the mathematical identity:
36 (2x) āx (aē) = 0
can be derived the result:
3 ( u
- .
arm) } a (r u
-
arm,) 30 r h3 ae ax
h3 3x
(3.29)
(3.30)
The most convenient coordinates to use for the analysis of a complete
flow channel are z and 0 rather than x and e. As z = x cos a, equations
(3.30) and (3.28) become:
( u
- a
") + cos a a
(r u cos a arm') = 0 (3.31)' ae r h3 ae az h3 az
Q (r aē2 + āz s2 =
)2 ) (cos a (3.32)
3.6 BOUNDARY CONDITIONS
In order to solve the conservation equations, the. appropriate boundary
conditions must be specified. Applying the lubrication approximation,
there is only one non-zero velocity component, namely, v in the s-direction
(see Figure 3.3b), which is a function of y only. Assuming the melt does
not slip at the channel walls, the velocity boundary conditions are:
- 38 -
at = r-2 and y = r2 (3.33)
Slip is likely to occur when the shear stress at the interface exceeds a
certain value (Fenner (1970)). Shear stresses generated in cable covering
dies are relatively low as compared to wire coating dies where the speeds
are much higher. In view of the very high shear stresses generated in
wire-coating dies, it might be concluded that slip often occurs. However,
the probable connection between slip and melt fracture and the fact that
successful die designs are those that avoid melt fracture, as discussed by
Hammond (1960), means that slip is presumably also avoided.
Thermal boundary conditions may involve both temperatures and first
derivatives of temperature with respect to the coordinate normal to the
boundary (see Figure 3.4). At the die wall, it is reasonable to assume
that there is a good thermal contact between the melt and metal surfaces:
T = Tb(z) on Y = ±2
(3.34)
where Tb is the temperature of the inner surface of the die wall.
Assuming a fully developed temperature profile:
aT = ay
0 on y = 0 (3.35)
Since the convection terms in equation (3.11) allow for development
of temperatures in the directions of flow, it is necessary in general to
specify some initial temperature profile at the beginning of the region
where the melt flow is to be analysed. The simplest form of temperature
profile at inlet can be assumed to be:
- 39 -
(3.36)
where T1 is a constant.
Point
Melt. in
Figure 3.1: Typical deflector and point used inside a cable covering crosshead
=TT
52
27
z=
Figure 3.2: One half of the flow channel plotted on the (z,e) plane, showing a mesh of triangular finite elements
- 42 -
h
\\\\\A\\\\\\ /////11//////,
Lp
(a) Typical axial cross-section
(b) Local velocity profile over the (c) Typical inclined portion of channel depth
Figure 3.3: Flow channel geometry and coordinates
the flow channel
\\\\.\T (x , t 2 . h l t
T b \\\\\\\
T (o,y)= T1
- 43 -
\\\\\ T (X, -2 h) =Tb \\\\
L
Figure 3.4: Flow channel between flat parallel surfaces
-44-
CHAPTER 4
SOLUTIONS OF THE CROSSHEAD DIE FLOW EQUATIONS.
4.1 REVIEW OF EXISTING SOLUTIONS
From the literature, it has appeared that very little published work
exists on the analysis of cable covering die performance. Axisynmetric
wire coating flows have been studied by some authors, including Ferrari
(1964) and Hammond (1960), who described rather empirical approaches to
die design, based largely on the need to avoid melt fracture,
i.e. extrudate roughness. Bernhardt (1974) gave exact solutions for drag
flow in wire coating dies and McKelvey (1962) gave a very simple treatment
for Newtonian fluids. Bagley & Storey (1963) presented exact solutions
for shear rates and velocities in Newtonian flow. Fenner & Williams
(1967) solved simplified wire coating die flow equations with the help of
numerical methods and digital computing. The results gave pressure
profiles, tension in the wire and maximum shear stresses. A finite
element method was used by Fenner (1974) to compute the relationship
between pressure drop and flow rate for a conical wire coating die.
Recently, finite element methods have been applied by various authors
to continuum mechanics problems. Palit & Fenner (1972) used a finite
element approach to solve isothermal slow channel flow of power-law fluids.
The fully developed flow was normal to the channel cross-section. The
method and results were compared with a finite difference method for
rectangular channels and with exact solutions for the Newtonian case.
They also applied the finite element technique to isothermal incompressible
two-dimensional slow flows of power-law fluids. Examples considered were
rectangular and axisymmetric converging channel flows, recirculating flows
in rectangular channels and flow round cylinders. Results were
-45-
successfully compared with both finite difference and analytical solutions.
Hami (1977) has studied non-Newtonian, isothermal flows for extruder .
channels, using a power-law model for shear-rate dependence of viscosity.
He has also considered the non-isothermal flow problem involving thermal
conduction.and heat dissipation.
In the limited amount- of literature available on the solution of die
flow equations, the most successful attempts were made by Pearson '(1963),
Gutfinger, Broyer & Tadmor (1974) and Ito (1974). Pearson (1962,1963,
1966) provided a mathematical analysis of the die design problem for a
power-law fluid. He solved for two-dimensional model by introducing a
stream function which led to a partial differential equation of the form
shown in equation (3.31). He adopted the shallow channel approximation
and developed a numerical technique for designing a crosshead die to give
uniform outflow at the die lips. The flow was assumed to be isothermal
and incompressible and the channel depth around the inner die body was
taken to be the variable at the designer's disposal. Laminar flow of
power-law fluids through shallow three-dimensional channels of varying gap
was considered by Benis (1967). By means of a perturbation'scheme, he
showed that the shallow channel approximation.will begin to break down
when the local channel angle becomes of the order of l0°. The theory
used by Benis (1967) for computing flow patterns in non-uniform channels
follows closely that presented by Pearson (1962,1966). Gutfinger et al
(1974) analysed a crosshead die, which had already been analysed by Pearson
(1963), using the. Flow Analysis Network (FAN) method. FAN is a finite
element method which was developed to solve isothermal two-dimensional
viscous non-Newtonian flow problems in relatively narrow gaps with smoothly
varying separation. Numerically, it involves only the simultaneous
solution of sets of linear algebraic equations. The method is described
- 46 -
in detail in the paper of Broyer, Tadmor & Gutfinger (1974). Given the
polymer rheological properties and the die geometry, the flow streamlines
in the die and the flow rate uniformity at exit can be calculated for any
given head pressure. Ito (1974) has also analysed the flow of polymer
melt inside a crosshead die and presented a method of designing a die to
extrude products of uniform thickness. He obtained the shear rate at the
die lip wall which he related to the quality of the extrudate which was to
be within specified limits.
The procedures for solving numerically laminar flow problems that may
be described by the Navier-Stokes equations are discussed by Atkinson et
al (1967). Atkinson et al (1969,1970) used a FE method to analyse two-
dimensional Newtonian flows. While they used triangular elements, they
assumed a modified cubic variation of stream function over each element.
As Zienkiewicz (1971) has pointed out, such a formulation may give rise to
convergence difficulties as the size of the element is reduced. In
general, the velocity components are not continuous across the element
boundaries, resulting in infinite velocity gradients, which may invalidate
the summation in equation (4.21). Atkinson et al apparently had no such
difficulties. Three point triangular elements are by far the simplest to
use, particularly for non-Newtonian flow and Palit & Fenner (1972) have
shown that, at least for applications similar to the present one,
satisfactory solutions are obtained. Caswell & Tanner (1978) have shown
how flow patterns can be traced numerically for arbitrary axisymmetrical
geometries in wire coating dies using finite element methods. From the
streamlines, they showed that the usual lubrication theory holds only to
within one or two gap widths of the region where the fluid meets the wire.
The work done by Pearson (1962,1963,1966) in analysing mathematically
-46a-
the numerical technique used in solving the quasi-harmonic partial
differential equation (3.31). Pearson used a FD approach to solve this
equation as an alternative to the FEM adapted in this work. He used the
shallow channel approximation and, having shown that it is possible to
predict streamlines given the general geometry of the die, he attempted to
reverse the process. In inverting the method of analysis described, the
flow was assumed to be isothermal and incompressible and the channel depth
around the inner die body was taken to be the variable at the designer's
disposal, similar to the analysis described in Chapter 8. Streamlines in
the flow domain were prescribed such that the design would give uniform
outflow at the die lips. The method used to define the streamlines and
the orthogonal isobars and the technique employed in solving the resulting
equations described in Chapter 8 differ a great deal from Pearson's
approach. In order to define a system of flow lines, Pearson considered
the two-dimensional irrotational flow of an ideal fluid caused by a line
source at the origin. He obtained a solution to this problem using the
method of images, the image distribution of sources and sinks which he
illustrated, where the rows of sources and sinks extend to infinity in both
directions. He described a stream function as the mass flow of a
Newtonian fluid in the case of uniform channel depth, caused by a unit
positive source at the origin, and represented it as a convergent double
series. He solved the equations numerically which gave the channel depth
distribution for a crosshead die.
Gutfinger et al (1974) used a very simplified FE mesh as compared
with the one used in this analysis. They divided the flow field into an
Eulerian mesh of square cells and for each cell the local field variable
was averaged and centred. The mesh was numbered by indices i and å in the
two directions which counted cell centre positions or nodes. Gutfi nger et
- 46b -
al used the conservation of mass principle that the net outflow from all
field nodes should equal zero. Hence, they wrote simplified equations
for each nodal point in the flow domain in terms of the nodal point
pressures. Solution of these simultaneous equations gave the pressures
at all the nodes in the flow domain. Once the pressure distribution was
known, the flow rate distribution was calculated using the equations of
motion.
- 47 -
4.2 THE SOLUTION PROCEDURE USED IN THIS WORK
This section describes some original work on the analysis of polymer
melt flow in crosshead dies. Some results of this work have been accepted
for publication (Fenner & Nadiri (1978)) and a copy of the paper is
attached to the back of this thesis.
The method described here is specifically designed to solve the cable
covering crosshead problem governed by equation (3.31). The same approach
is also applicable to other shallow channel flows, and indeed to other
problems governed by mathematically similar equations. A finite element
method has been applied to the solution of the equations. Although finite
element methods were originally developed for digital computer use in the
stress analysis of solid structures and components (Zienkiewicz (1971)),
they have also been applied to fluid mechanics and heat transfer problems,
including the slow non-Newtonian flows encountered in polymer processing
operations (Palit & Fenner (1972)).
The first step towards the finite element formulation is to represent
the problem region with a finite number of elements interconnected at a
discrete number of points, called nodal points. In general, the elements
can be of any shape but, for the purpose of this work, triangular elements
have been chosen.
Figure 3.2 shows one half of the complete solution domain irr the
(.,e) plane divided into triangular finite elements. Although the
straight sided elements cannot follow the curved boundaries AC and BD
exactly, with a reasonable number of elements the maximum deviations are
acceptably small. It should be noted that the number of elements across
the width of the flow is constant, which means that the elements near the
narrow inlet boundary are much smaller than those near the deflector and
point outlets, CD and EF, respectively. It is the ability to fit complex
-48-
geometric boundary shapes and to allow varying densities of elements
within the solution domain that makes finite element methods attractive.
Palit & Fenner (1972) have compared and contrasted finite element and
finite difference methods for problems of the present type, and discussed
the advantages of using simple triangular elements for non-Newtonian flow
problems. They defined a stream function for incompressible two-
dimensional flow which was assumed to vary quadratically over each element.
This had the advantages of resulting in a constant viscosity distribution
over each element, even for non-Newtonian fluids, and avoided the time-
consuming necessity of numerically integrating power-law functions over
the surfaces of the elements. An additional advantage of using a
formulation which results in constant element viscosities is that
axisymmetric flow problems can be solved with only trivial modifications
to the method for rectangular flows. It is unnecessary to resort to the
full cylindrical polar coordinate system. The price which must be paid
for these simplifications is a loss of accuracy for a given number of
nodal points as compared with more sophisticated formulations. For the
problems considered by Palit & Fenner (1972), however, the accuracy of
their method, which is similar to the one used in the present analysis,
using a reasonable number of nodal points, is satisfactory for most
practical applications. Although the ultimate test is through a
comparison of computing times for comparable accuracy, even this must be
qualified by a consideration of the relative flexibility and adaptability
of the basic computer program to a wide range of practical problems.
In order to solve the problem by a finite element method, it is
required to establish a variational formulation for equation (3.31) (see
Fenner (1975)) which is of the form:
- 49 -
a aē (k1 aēJ k2 z (k3 a -) = p (4.1)
where k1 = k1(e,z, derivatives of 1)
k3 = k3(0,z, derivatives of *)
k2 = k2 (z)
6 is the angular coordinate of the channel
p is the stream function where r Qx = alp/ae; Q0 = — Wax
and Qx,Qe are the volumetric flow rates defined in terms of local lengths
along the channel
The general variational approach to the solution of a continuum
mechanics problem is to seek a stationary value (often a minimum) for a
quantity x which is defined by an appropriate integration of the unknowns
over the solution domain. Such a quantity x is often referred to as a
"functional". When such a principle is used in a finite element analysis,
the, variation of x is carried out with respect to the values of the
unknowns at the nodes of the mesh.
In order to find the required functional, it is convenient to let:
a0 (k1 aē) ~ k2 az (k3 az) = o (4.2)
Now, p is a continuous function of position which, in general, can only be
defined exactly in terms of an infinite number of parameters, such as
values of the function at particular points in the solution domain. The
object of the present analysis is to provide a means of determining an
approximate form of p in terms of a finite number of parameters. Let n
be a typical such parameter, and multiply the above definition of x by the
50 -
derivative of j with respect to n to give:
a. = at a al, al a an an ae (k (k 1 aē) an x2 ā 3 ā) = az
As x is zero everywhere within the solution domain:
ff .x = de dz
where the integration is performed over the entire domain.
Now, considering the terms in equation (4.3):
a" a (k a~) = a (
arm' k arm') — k a a211,
an ae 1 ae ae an 1 ae 1 ae ae an
a (arm, k arm) - 1 k a (a P) 2 ē an. 1 ae 2 1 ān 30
(4.3)
(4.4)
(4.5)
Similarly:
k e a lk
arm) k a (a,* k a,4) 1 2 ān āz 3 az 2 āz an 3 az 2
a 2 k3 a n (a*az). (4.6
Therefore, equation (4.4) becomes:
ff {3 ka
62±)2 + 1 k k
a (4)2} de dz — I 0 (4.7)
2 1 an ae 2 2 3 an az
where:
ff {aē (21. k1 ae + k2 az (ā 7<3 az)} de da (4.8)
Applying Green's theorem:
1 aJ, rae -rcos2 a az de 4.12)
- 51 -
arm, arm, any arm, I (k an ae dz - k2 k3 an az de) (4.9)
where the line integration is performed around the boundary of the solution
domain. The direction of this integration has. not been defined because,
at least in this instance, solutions where I = 0 are sought, i.e. where,
on the domain boundary:
= 0 (4.10) an .
or,: k1 -a-Lk dz - k2 k3 az de = o
(4.11)
Now, equation (4.10) occurs whenever the boundary distribution of . is
prescribed, and equation (4.11) implies that (from equation (3.31)):
Figure 4.1 shows a small portion of the boundary in the plane of the
channel. If n is the direction of the outward normal to the boundary at
a particular point as shown:
-41 an = Doc
y t ra9 sin y (4.13)
where y is the angle between the normal and the x-axis. Now:
cos y = r ds , sin y .— ds
(4.14)
where s is distance along the boundary measured in the anti-clockwise
direction, and the negative sign is due to the fact that, for positive
-52-
sin y, x decreases as s increases. Hence:
ds al) r de — ~dx
an ax ae r
cos a āzr de aē r cos «
1
cosa [tI COS2 a atil
az Z alp r ae JJ (4.15)
Hence, if equation (4.12) holds, then:
an - 4.16)
To summarise, if the conditions on the boundary are such that either
the value of ip is prescribed and therefore independent of n, or the value
of its first derivative normal to the boundary is zero, then I = 0. More
general boundary conditions of the form:
a1 a an
+ a2 # a3 =
(4.17)
can, if necessary, be handled by a suitable use of non-zero I, where a1,
a2 and a3 are constants, Wan is the stream function gradient normal to
the boundary, and p is the stream function.
Provided I = 0, the solution of the governing differential equation
is obtained when the value of the following functional derivative with
respect to n is zero:
a a12 ? k k3 an 4a (a~) 2} de dz o (4.18) 1 an ae 2 az an = ff {
-53-
Returning to the actual equation (3.31), the required stationary condition
is obtained when:
ā ff {L u
-
a A2 + 1ru
-
cos 2 a a (a'~)2}do dz = o (4.19) 2 r h3 an . ao 2 h3 an az
holds for all the unknowns, n, required to be found. In the present
method, the values of stream function, p, at the corners or nodes of all
the triangular elements are chosen as the unknowns. The only restriction
on the validity of equation (4.19) is that on the boundaries either the
value of i must be prescribed or its first derivative with respect to
distance normal, to the boundary must be zero. This restriction is
satisfied in the present problem which is subject to the following boundary
conditions:
* = 0 on BDF , p = 1 on ACE
DO = 0 on AB az
= 0 on EF (4.20)
Provided the inter-element boundaries make no contribution to the integral
expressed in equation (4.19), (automatically satisfied by triangular
elements with linear distributions of * - i.e. conforming elements.) the
overall integral may be found by summing the integrations performed over
all the individual elements:
DX (m)
an an (4.21)
where x(m)is
the contribution of typical element m to the total value of
x. With the linear distribution of p over the element with respect to z
and e, the following approximation may be used:
ax (m)
an a* a2*
27,723 3e an 30
-54-
r u cos2 a any 32*
T2-3 az an az}
(4.22)
where Fuld Tare the mean values of radius and .channel depth over the
element, and om is the area of the element in the (z,e) plane represented
by equation (4.28). In practice, the variations of r and h over an
element are sufficiently small for mean values to be used for the present
purpose. While the angle a also varies along the flow channel, the mesh
is chosen such that the slope of the channel is constant over any one
element.
Figure 4.2 shows a typical triangular element, numbered m, in the
solution domain. It has nodes at its corners numbered, in an anticlockwise
direction, i, j and k and dimensions as shown. Local coordinates z and
0' are parallel to z and 0, but have their origin at node is Assuming a
linear distribution of stream.function over the element:
*(z'3 e') - C1 + C
2 z' + C3 0' (4.23)
C1, C2 and C3 can be found in terms of the nodal point values of stream
function iv ., lyj and ipk. The stream function values at the nodes i, j and
k can be written using equation (4.23) as follows:
*i = Cl (4.24)
*j = Cl 4- C2 ak - C3 bk (4.25)
*k = C1 - C2 aj C3 bj (4.26)
Rearrangement of these. equations (see Appendix Al) gives the parameters in
equation (4.23):
1 (B) 2Am l
2A
where dm is the area of the element:
[
b. b. bk
ai aj ak
(B)
Using equation (4.23), equation (4.22) becomes:
(m) — — aC - a 2n m {r u 0032 a C2 f u C3
h3 an r h3
Om u aC' DC {c2 2 C3 3}
an. an
-55-
and (B) is a matrix of element dimensions:
(B) (a)m
(4.27)
(4.28)
ac 3}
(4.29)
an
(4.30)
where C2 = r cos a C2.
The derivatives of x(m) with respect to the three nodal point values
of ,p associated with element m may therefore be expressed as:
- 56 -
3C2 ac3
aI'i
act aC3
aqv a1pj
aC2 ac3
ak a*k
C'" 2
c3,
Am.
(.4.31)
Considering the terms on the right hand side of equation (4.31):
fc"
2 1
C3 2Am B') (o)m (4.32)
where:
bi r cos a
ai
• b, r cos a
a. a
ac2 aC3
(BuT (4.34)
a,y2
act
a*i
ac3
act ac3
2a m
47(
where the superscript T indicates a matrix transposition. Hence, equation
(4.31) reduces to:
- 57 -
A u m 1 T 1 ') (d)m u (B') (d) 4A 2,— m (4.35) r h3 2pm 2Am
Combining equations (4.21) and (4.35):
1z3 (B
')T (B') (6)m = (i)m (6)m = (x) (6) = 0 (4.36)
4o r m
where (k)m is the individual square element stiffness matrix, and (6) is a
vector containing the stream function values for all the nodal points in
the mesh. Square matrix (x), which in the finite element method context
is often referred to as the overall stiffness matrix, contains coefficients
assembled from the properties and dimensions of the individual elements
(Fenner (1975)).
Before equations (4.36) can be solved for the unknown values of ,y, the
boundary conditions defined by equations (4.20) must be imposed by
appropriately modifying equations associated with boundary nodes at which
the value of p is prescribed. The equations are not linear because the
element mean viscosities, }.c., are dependent upon the local gradients of tp..
Using equations (3.24), (3.32) and (4.23):
(C2 2 + C3 2/21
r h2 yo
(4.37)
- 58
4.3 THE METHOD OF SOLUTION
The problem considered in this work is a nonlinear one in the sense
that it involves the flow of a non-Newtonian fluid. The viscosity of a
non-Newtonian fluid depends on the local rate of deformation in a
prescribed manner. Such behaviour can be accommodated in the finite
element analysis by treating element viscosities not as constants but as.
functions of the element strain rates (in this work, functions of the local
gradients of 1), which therefore need to be up-dated during the solution
process. For this reason, direct elimination methods, e.g. Gaussian
elimination, are not very suitable for solving nonlinear overall equations
(Fenner (1975)). The total number of equations that have to be solved is
equal to the number of nodal points. In most cases, this is in the
region of 250 equations. Therefore, efficient methods of solution, to
minimise the computing time, are sought. In general, the non-zero
coefficients of the square overall stiffness matrix do not lie very close
to the diagonal. To be able to save storage and economise on computing
time, the non-zero coefficients of the square stiffness matrix are
transformed into a rectangular stiffness matrix. A pointer matrix is
then introduced, indicating the column location of a particular term.
The method of solution used here is that described by Palit & Fenner
(1972) in which the iterative Gauss-Seidel successive over-relaxation
approach is employed. An advantage of iteration, for this particular
problem, is that the element viscosities are up-dated during the solution
process, while in using direct elimination the whole of the solution
process has to be repeated a few times with the up-dated viscosities,
which might be less efficient. The equations are first linearised by
assuming suitable constant values for the element viscosities, five
iterations are then performed to estimate the nodal point values of stream
-59-
function. Using these values to up-date the viscosities, the process is
repeated until satisfactory convergence is achieved. The choice of
starting values for the unknowns does not normally affect whether the
Gauss-Seidel process converges, and often has comparatively little effect
on the number of iterations required. It is possible to predict whether
convergence is likely to be achieved with a particular set of linear
equations. .Varga (1962) has stated the sufficient condition for
convergence as that of "diagonal dominance" of the coefficient matrix (K).
If (K) is diagonally dominant, then:
(kii( >. / (k-.I for i = 1, 2, ., n (4.38) i=/
~ 1
and the inequality is satisfied for at least one row. While diagonal
dominance is sufficient to ensure convergence in the case of linear
equations, it may not be necessary, provided these conditions are only
mildly contravened. In the case of nonlinear equations, provided the
non-linearity is not very high, it can be assumed that for convergence the
diagonal dominance condition might still be applicable.
It is often possible to improve the rate of convergence by a technique
which is generally known as over-relaxation. If the set of equations to
be solved can be expressed in matrix form as:
'k11
k21
kn1
k12 ....... k2n
k22 2n
•
• •
kn2 ....... nn
• • • •
(4.39)
(x) (s) = (F) (4.40)
-60-
then each unknown can be expressed as a function of the others as follows:
6 (m) k1 (f . — Z 1
k2. 6. (m) — k2~ 6 (m-1) (4.41)• 22 j=1 j=2+1
where the superscripts denote iteration numbers. Clearly, it is essential
for all the diagonal coefficients, k2., to be non-zero. Equation (4.41)
provides new estimates, ts." 1, which, provided the process is convergent,
are closer to the required solutions than the s2(m-1). Over-relaxation
applies a limited amount of extrapolation from these two sets of estimates
towards the final solutions. Thus, if 62(m) are the values obtained from
equations (4.41), the extrapolated values after the mth iteration'are:
62(m) = 6.(m_1I
+ w (62(m) - 6.(m-1) )
4.42)
where w is an over-relaxation factor, which is the same for all the
equations. For a particular set of linear equations, there is an optimum.
value of w, normally in the range 1 < w < 2. The optimum value varies
according to the size of mesh used. The purpose of over-relaxation is to
accelerate convergence, rather than to promote convergence in an otherwise
divergent iteration scheme. The use of too large a value of w can cause
divergence.
According to the degree of accuracy of the results, a convergence
criterion is required. This can take the form of a "relative error", er:
(4.43) .
- 61 -
where is the change between successive iterations,
(m) (m-1). i.e. osi = 6 — si , n is the number of cycles of iteration, and c
is the prescribed value of the degree of accuracy, often called the
tolerance limit. For the problems investigated in this thesis, the
tolerance, c, is chosen to be 10-6.
Equation (4.36) can thus be solved to give the stream function values.
at all the nodal points. The flow rate of molten polymers passing the
outlet boundary between any adjacent pair of nodal points is equal to the
difference between the stream function values at these points, off,.
Assuming that there is no subsequent circumferential redistribution of
material, the corresponding thickness of the polymer layer on the finished
cable will be proportional to off;/oe, where AO is the difference in e
coordinate between the two nodes. Hence, the ratio of local to mean
thickness can be computed as a function of angular position around the
cable. Given the cable speed and the total flow rate of polymer forming
a particular layer, its mean thickness can also be determined if required.
4.4 SOLUTION PROCEDURE USED FOR PRESSURE DISTRIBUTION
Having computed the stream function distribution over the solution
domain in terms of values at the nodal points of the mesh, other results
may be derived as required. For example, the pressure distribution and
hence the overall pressure difference between flow inlet and outlet may be
computed as follows. For each element, the mean pressure gradients in.
the z and e directions can be found with the aid of equations (3.27),
(3.26) and (4.23) as:
_ 8z
C3
8 —1-1-72; cos a C2 (4.44) 0 h3 4, h3 r cos a
mm
rn=m1
a
(6)m 4.45) (ia)m , ( ē) n
-62-
where C2 and C3 can be found from the nodal point stream function values
using equations (4.27). The pressure gradients at the nodal points may
then be found by averaging the values of the gradients associated with the
elements having a particular point as a node. Figure 4.3 shows six
typical elements sharing one particular node. The axial and angular
pressure gradients at this node may be represented as:
where (ap/ez) and (ap/ae)n are the axial and angular pressure gradients
of the node n, respectively, and the m. are the numbers of adjacent
elements, the total number of elements being M (in this case 6). The
pressure difference, dp, between two adjacent nodal points, 1 and 2, say
(see Figure 4.4), can be written as:
dp
(z,0 - p2 ()A dz f (aē).~ de (4.46)
where A is the mid-point between points 1 and 2, (ap/ez)A and (ap/ae)A are
the pressure gradients at A, and the distances between the two nodes in the
z and a directions are dz and de, respectively. The pressure gradients
at A can be represented in terms of the pressure gradients at the two
nodes 1 and 2 as follows:
a
( ~A 2 1(az)1 (az)2). (4,47)
1 (ae~A ā
PE) (22 (ae)2
Using equation (4.46) and assuming extrusion into atmosphere at zero
-63-
pressure, it is possible to calculate the values of the pressures at all
the nodes.
64 -
Figure 4.1: Part of a typical two-dimensional solution domain boundary
1
-bk
z,
Figure 4.2: Typical triangular finite element
-65-
Figure 4.3: Elements sharing a typical nodal point
Figure 4.4: Pressure at two adjacent nodal points
- 66 -
CHAPTER 5
MULTI-LAYER CABLE COVERING EXPERIMENTS
5.1 INTRODUCTION
In this chapter, experiments are reported concerning the performance
of an extrusion crosshead used in the three-layer covering of high
voltage electrical cables. Both extrusion pressure requirements and
circumferential distributions of polymer layer thicknesses in the finished
cable were measured and they will be compared later on in this chapter
with the theoretical results of the finite element method of melt flow
analysis within the crosshead, which was described in Chapter 4. Some
of the experimental results on extrusion pressures and circumferential
cable layer thickness distributions have been presented in a paper by
Nadiri & Fenner (1978) which has been submitted for publication.
There appears to be very little other published work on the
experimental results concerning the performance of extrusion dies. Palit
(1972) performed some experiments on the flow of molten polyethylene and
polystyrene for converging cylindrical dies and a silicone polymer in a
wire-coating die. He then compared the results with the FE predictions
in terms of dimensionless pressure drop and wire tension difference (in
the case of the wire-coating die) and showed that they agree reasonably
well.
The cable covering process is used to coat, continuous lengths of
either solid or stranded metallic conductors with one or more layers of
polymeric material. Polymer is supplied in molten form, normally by a
screw extruder, to a die, through the centre of which is passed the
conductor. Owing to the presence of the conductor, it is not convenient
to align the axis of the extruder with the direction of cable production.
-67-
Instead, the screw axis is usually at right angles to the cable, the
change in the direction of flow being effected by a crosshead. If more
than one layer of different polymers are to be applied simultaneously, the
design of the crosshead is further complicated by the need to accept melt
from, say, two extruders and to form the required layers with as nearly
uniform thicknesses as possible. An important advantage of such a
simultaneous multi-layer covering process is that the risk of contamination
between successive layers is minimised, thus enhancing the electrical
properties of the finished cable. This is particularly important for
large high-voltage cables.
A very important factor in determining cable quality is the uniformity
of the thicknesses of the polymer layers. In order to satisfy electrical
performance criteria, each layer must be of 'a prescribed minimum thickness.
Uniformity along the cable is obtained by drawing the conductor through
the crosshead at a constant speed, at the same time maintaining constant
melt flow rates from the extruders. Uniformity in the circumferential
direction in any cable cross-section is more difficult to achieve because
it depends on the ability of the crosshead to distribute side-fed supplies
of melt into concentric circular tubes to be applied to the conductor.
If the actual layer thicknesses vary significantly, then, because the
minimum thicknesses are prescribed, excess material will be contained in
the thicker parts of the layers, adding to the cost and bulk of the cable.
Designs for the internal geometries of cable covering crossheads are
often developed by slow and expensive trial-and-error methods. A
rational design procedure is therefore highly desirable, and the
theoretical basis of such a method has been proposed in Chapter 4, using
a finite element technique of melt flow analysis. In this chapter, the
experimental results will be presented on circumferential cable layer
- 68 -
thickness distributions and extrusion pressures, the main object being to
compare with the finite element computations rather than to optimise the
particular cable-making operation. Once thickness distributions produced
by a given crosshead design can be predicted, it is possible to invert the
analytical procedure to use it as a design tool.
5.2 THE CABLE COVERING PROCESS IN GENERAL
The cable covering process is used to coat continuous lengths of cable,
tube and a variety of products with a layer of extruded thermoplastic
material. The tremendous demand for covered cable in the radio and
electrical industries makes this one of the most important of the'many
extrusion processes. Plastic coatings range in size from PVC covering on
tiny telephone wires to flexible jacketing several inches in diameter over
huge bundles of wire. They also include many domestic and industrial
electrical wiring applications. Some wires receive coverings of two
different plastics, each offering specific advantages. Plasticised PVC
can offer flame resistance and high flexibility, nylon gives extreme
strength, abrasion resistance and resistance to certain chemicals, and
polyethylene offers exceptional insulating ability and chemical resistance
at reasonable cost. All offer moisture resistance.
Figure 5.1 shows the basic equipment in a typical cable extrusion
line. Drums of uncovered wire are mounted on a payout stand, which may
be free to rotate, friction braked or power driven, depending on their
size and on the requirements of the system. The payout is essentially a
reel stand and on rudimentary set-ups the reel of uncovered wire is merely
supported on centres so that it may revolve as the wire is taken off and
some form of simple braking is provided to prevent the inertia of the drum
from affecting the tension of the wire. In more complex systems, and for
- 69 -
fine wires, the payoff reel is driven so that a greater degree of control
may be exercised on the tension of the wire throughout the system and to
minimise the tension on wires of very small diameter. The payout drum
may be mounted with its axis at 90° to the line or parallel toit. A
desirable feature on a payoff unit is the facility for mounting two reels
simultaneously so that wire from a fresh reel may be introduced to the
system with the minimum speed change when the preceding reel runs out.
From the payout drum, the wire is led through the die where it is
coated with polymer. The polymer is fed as granules into the extruder
barrel from a hopper. The purpose of the barrel and screw of the extruder
is to deliver a correctly conditioned polymer melt at a constant rate into
the head. To condition the polymer, heat must be supplied either by
conduction from the heated barrel or by mechanical work done on the polymer.
In the cable covering process, in general, two types of crosshead dies
are used. One type is the tubing die where the conductor gets covered
with the plastic after it has emerged from the head, unlike the case of a
pressure die where the cable leaves the die already coated. In the tubing
die, the tube is drawn onto the conductor just after the die face by a
vacuum drawn through the same passage in which the conductor travels.
Tubing dies require a larger clearance than pressure dies between conductor
and passage because vacuum must be drawn through that clearance. A good
figure is 0.5 mm all around between the conductor and the passage. There
is no danger of plastic flowing up through this clearance because the
plastic is extruded through a separate annulus. The ratio of the area of
this annulus to the final plastic cross-sectional area is called the "draw
down ratio". If this ratio is too high, a rough surface and/or internal
strains in the coating will result. Typical draw down ratio for LDPE is
1.3.
-70-
On leaving the die, the covered conductor is cooled by passing
through a water trough. The troughs are small in cross-section, but can
be very long, especially for polyethylene. This plastic not only needs
much cooling because of its heat of crystallisation, but also requires
slow cooling to prevent the development of voids in the shrinking coating.
Troughs for polyethylene are at least 10 metres long and can go up to
35 metres or more. For large undersea cables, the trough is often
85 metres long. Thin-walled insulation can be cooled with tap water.
However, for medium and large cross-sections, the surface cannot set too
quickly, or the plastic will shrink inside and separate from the conductor.
Accordingly, the water temperature is controlled. For polyethylene, the
coated wire must enter a bath at 140° to 180°F for the best results.
Often the trough is divided into several compartments of successively
decreasing temperatures, ending with water at ambient temperature.
After leaving the water bath, the cable is drawn through the line by
a capstan. This is either single or double-drum and is grooved to hold
the wires that are wrapped around it Four or five wraps are customary.
Diameters range from 12 to 20 inches for small wires through 32 inches
for intermediates. Large cables use "tractor" pullers or "caterpillar"
capstans.
From the pulling capstan, the wire is picked up on rotating take-up
reels that are similar to the rotating payoff reels. Automatic devices
are again used at high speeds to switch from a full reel to an empty one.
The controls and their synchronisation, for a large and complex line
such as for undersea cable, are very intricate and expensive equipment.
In general, wire and cable covering is a high-quality operation and this
is reflected in costs and quality of equipment.
- 71 -
5.3 THE CABLE COVERING PROCESS EMPLOYED IN THE TRIALS
Figure 5.2 shows the general arrangement of the equipment used in the
cable covering trials. The 22 mm diameter stranded copper conductor was
taken from a storage drum through a vertical distance of some 24 m before
being passed over a large capstan wheel and then into the extrusion
crosshead between two screw extruders supplying the insulation and screen
materials (see Figure 5.3). Curing of the crosslinking polymers was
achieved by high temperature steam applied to the cable in a long catenary-
shaped tube starting at the crosshead. The overall length of the
catenary was some 85 m, the lower end of the tube containing water for
cooling purposes. After leaving the catenary via a sealing gland, the
cable passed through a caterpillar haul-off (see Figure 5.4) and was wound
onto a drum. The speeds of the capstan wheel and haul-off were controlled
and coordinated in such a way as to prevent the cable touching the inside
of the narrow catenary tube, at least until it had entered the cooling
water.
In the cable-making trials described here, three layers of polymer
were applied to a tape-covered stranded copper conductor. The thin inner
and outer layers, which served as screens, were of the same material
supplied by one screw extruder. The much thicker intermediate layer,
which provided the required electrical insulation was of another material
supplied by a second machine. The two extruders were connected to
opposite sides of the crosshead with their screw axes at right angles to
the direction of motion of the conductor. Although not directly relevant
to the present investigation, the particular polymers used were basically
crosslinking polyethylenes. Crosslinking materials have the advantage of
allowing higher working temperatures for the finished cable, but need to
be cured after extrusion.
-72-
The polymers used in the trials were a white crosslinking low. density
polyethylene (HFDB 4201) for the insulation, and a black semi-conducting
crosslinking ethylene/vinyl acetate copolymer (HFDA 0580) for the screens.
Nominal radial thicknesses were 0.7 mm for each of the two screens and
8 mm for the insulation giving an overall cable diameter of 42 mm,
including the 0.5 mm conductor tape. The black polymer was extruded from
a small Andouart machine, 60 mm in diameter and the white polymer from a
larger Fawcett Preston machine, 4i" in diameter. Both extruders are
shown in Figure 5.3.
5.3.1 Crosslinking and Semi-Conducting Materials
Polyethylene can be crosslinked either with chemical agents or
by irradiation. The latter method is too costly for large-scale operation.
Chemical crosslinking, usually with organic peroxides, is less expensive
and is being used to some extent today. The crosslinked wire is more
resistant to solvents, chemicals and heat; in fact, it can be so
effective that the insulation does not ever melt, but rather becomes
rubbery and tacky at very high temperatures. Service temperatures for
crosslinked polyethylene are above 135°C. Often, great quantities of
carbon black are included in crosslinking compounds, as it aids and
participates in the crosslinking reaction:
Crosslinking by irradiation is done on the finished wire after
it leaves the extruder, but before reeling. Chemical crosslinking is
done by incorporating the agent in the compound and extruding it at such a
temperature as to prevent the reaction in the extrusion. die. High
pressures are thus generated and must be taken into account in die design.
The insulation is then "cured" in a steam-heated chamber.
Chemical crosslinking often can be achieved by passing the
-. 73 -
coated wire through a bath containing the reactive agent. This is done
most conveniently at the first stage of the cooling trough as the plastic
is hottest and most reactive there. In this method, the agent is used
only where needed most at the surface.
Where some degree of electrical conductivity is required, semi-
conductive jacketing is customarily made by adding large amounts of
certain carbon blacks to the compound, similar to the carbon filled
crosslinked compounds mentioned in the first paragraph of this section.
5.3.2 The Extrusion Crosshead
The crosshead used was of the tubing type, in which the three
layers of melt were first extruded together to form a single tube which
was then applied to the conductor as it left the head. Once in contact
with the conductor, the molten tube was stretched in the direction of
motion, thereby reducing its external diameter by some 30%. Figure 5.5
shows the axial cross-section through the crosshead assembly and melt
inlets from the extruders. The melt to form a given layer entered a
narrow gap between concentric cylindrical and conical surfaces. The
cylindrical region was formed by a flow deflector having a blockage to
flow contoured in such a way as to promote uniformity of the circumferential
distribution of flow leaving the deflector. To the end of the deflector
was screwed a tapered point, the dimensions and angle of which could be
chosen to suit the particular cable specification. The three pairs of.
deflectors and points may be distinguished as the inner (screen),
intermediate (insulation) and outer (screen). As can be seen in Figure
5.5, the flow channel for the insulation layer, for example, was contained
between the intermediate deflector and point, and the inner surfaces of
the outer deflector and point. The inner point was bored to suit the siie.
-74-
of the tape-covered conductor. All components of the crosshead were made
to fine tolerances and clearances to promote accurate alignment.
Figure 5.6 shows a view of the intermediate deflector as seen
from the melt inlet side. As the melt would tend to take the shortest
path from the inlet to the deflector exit, this path was deliberately
blocked by a heart-shaped area which filled the radial flow gap and forced
the melt to follow longer paths of more uniform length. Flow channel
geometry was symmetrical. about the centre line of the heart-shaped area,
and the deflector created some throttling by a slight tapering of the
channel depth in the axial direction. Further throttling was provided by.
the tapering conical flow channel between points. Figures 5.7 and 5.8
show .views of the unrolled flow channels of the inner and outer deflectors,
respectively. The unrolled view of the intermediate deflector flow
channel has been shown in Figure 3.2.
5.3.3 Experimental Measurements and Instrumentation
In order to be able to determine the angular orientation of
any cross-section of the finished cable in relation to the extrusion
crosshead, a small V-shaped nick was made in the edge of the outer die
which left a longitudinal mark on the surface of the cable. The need for
this was emphasised when the mark was observed to emerge from the catenary
at approximately 180° to the position of the nick, due to twisting of the
cable.
Positions along the cable at which extrusion conditions were
changed were also marked when the cable left the catenary. The
appropriate times to make such marks, many minutes after the particular
section had been extruded, were determined from the cable speed and
catenary length. Cable speed was measured both by a tachometer
- 75 -
registering at the capstan wheel and more accurately by recording on a
stopwatch the time taken for a mark on the conductor entering the crosshead
to move through a fixed distance. Screw speed was also measured using a
stopwatch as well as being recorded from the speed indicators built into
the system.
When manufacturing crosslinked polyethylene, the cable is
closed up in the steam tube where it is surrounded by high pressure steam
at 210 to 220°C. This effectively precludes access to the extruded
product as it issues from the die; hence, under normal manufacturing
conditions, it is not possible to measure melt temperatures directly while
making cable. Such measurements were, however, made with the aid of a
Cr/Al thermocouple probe during preliminary bleed trials under equivalent
extruder operating conditions. A number of measurements of temperature
of the extruder barrels and at several points within the body of the head .
were recorded using Cu/Con and Fe/Con thermocouples. These could well be
different from the temperatures at the barrel/melt interface.
Melt pressures were detected with the aid of two Dynisco
Model PT420 transducers, one in each melt inlet to the crosshead. The
amplified signals from these instruments were monitored by an ultra-violet
recorder. One transducer was fitted into a large inlet which was
machined in the 4i" Fawcett Preston extruder. A "blanking" plug was used
on the head when the transducer was not fitted. The second transducer was
fitted into a smaller inlet in the 60 mm Andouart machine.
Further instrumentation was also available for a number of
other process parameters, such as steam pressure and temperature, and
power consumptions.
76 -
5.3.4 Experimental Procedure
Before making cable, bleed trials were carried out in which
the melt, either insulation layer alone or insulation plus screens, was
extruded into air in the absence of conductor (see Figure 5.9). The
tests were carried out at various extruder screw speeds and timed samples
were taken by applying the bleed to a metre long free running silicone
greased tube. These trials served both to check layer concentricities
and thicknesses, and to permit melt temperatures to be measured. As
already indicated, a thermocouple probe was used to sample mainly
insulation layer extrudate temperature. This was significantly higher
than any indicated extruder barrel or head temperature, and increased with
extruder speed, and hence melt flow rate. The relationship between melt
temperature and screw speed so obtained was used to estimate temperatures
in the subsequent cable making trials. A knowledge of melt temperatures .
was necessary to determine viscosities used in predicting pressure
distributions in the crosshead flow channels.
Temperatures of extruder barrels and head could be varied only
within very narrow limits. Experience had demonstrated that, with a
complicated exercise of this kind, the machines needed to be precisely
set at empirically determined optimum conditions, otherwise quality will
suffer.
On completion of the bleed trials, the conductor was
introduced, the catenary tube sealed at the crosshead, and steam applied
to cure the resulting cable. . The process was run at.a series of four
cable speeds from 1.45 m/min to 2.43 m/min, extruder speeds being adjusted
to give the required cable dimensions. The Fawcett extruder speed varied
between 30 and 45 rpm, and the Andouart extruder speed between 20 to 25 rpm.
The machines were not run too quickly since at high screw speeds there is a
77 -
risk of pre-cure or "scorch" in the barrels and head.
Although the cable dimensions were approximately those of a
33 kV application, the purpose of the experiments was to evaluate
crosshead performance rather than try to meet a precise cable specification.
At each speed, the line was run at least long enough for the position of
the speed change on the cable to emerge from the catenary and be marked.
Once sufficient time had been allowed for extrusion conditions to become
steady at a given speed, measurements of all the process variables were
made. In particular., ultra-violet recordings of the pressure transducer
signals were made for sufficiently long times to detect not only
fluctuations associated with extruder screw rotation, but also any longer
term variations due to extruder surging.
Samples of both the insulation and screen materials were
retained for rheological tests. These allowed polymer viscosities to be
measured as functions of shear rate and temperature. Both viscosities and
power-law indices, which determine how viscosities vary with shear rate,
are needed in the analysis of flow within the crosshead (see Chapter 4). `
The finished cable was sectioned at a number of positions to
permit polymer layer thickness distributions to be examined. For each
cable speed, five sections were taken at intervals from a region of the
cable where nominally steady running conditions had been achieved. At
each section, a thin parallel slice of the covering was cut at right angles
to the axis of the cable, and the thicknesses of the layers measured along
a series of radii using a travelling microscope. These radii were taken
at 15° intervals around the section, the imperfection introduced in the
outer screen by the die nick serving as a reference point for angular
measurements. Some small longitudinal variations in layer thicknesses
between different sections at the same cable speed were detected. The
-78-
fact that the periods of such variations correlated closely with the
periods of small fluctuations in the measured extrusion pressures showed
that they were due to slight extruder surging (see Section 5.4.2).
5.4 EXPERIMENTAL RESULTS
In this section, the experimental results obtained during the trials `
for the 300 mm2, 33 kV, 3-layer tool set will be discussed in detail.
Tables 5.1a, 5.1b and 5.1c show the measurements taken during the
bleed tests. The first bleed test was made at a Fawcett speed of 30 rpm,
when only the white insulation layer was extruded. Then the Fawcett
speed was gradually increased to 40 rpm and finally both extruders were
run together, the Andouart speed reaching approximately 21 rpm.
During the bleed trials, the melt was extruded in the absence of the
conductor, hence the line speed was zero. Altogether, 3 bleed runs were
made and, once steady running conditions were achieved, for each run at
least two readings were taken of the variables, including speed,
temperature, and. pressure.
In Table 5.1a, the extruder speeds, together with the currents
supplied to them and the time when the readings were taken, are presented.
Two sets of screw speeds have been tabulated; one set corresponding to
the readings on the tachometer and the other set obtained using a
stopwatch.
The barrel temperature at four zones of the Fawcett extruder, and at
three zones of the Andouart extruder, are shown for each of the three
bleed runs in Table 5.1b.
Finally, in Table 5.1c are tabulated the head temperatures at three
zones, the throat and melt temperatures and the pressures at the melt
inlets to the two extruders.
TABLE 5.1a
Time Run
Number
Fawcett** Screw Speed
(rpm)
Andouart** Screw Speed
(rpm)
Fawcett* Screw Speed
(rpm)
Andouart* Screw Speed
(rpm)
Fawcett Ammeter (amps)
Andouart Ammeter (amps)
10.25 10.34 10.37
10.45 10.52
11.15 11.23
1 1 1
2 2
3 3
29.3
39.3
39.1
0
0
22.5
30.0 30.0 30.0
40.0 40.1
40.0 40.1
0 0 0
0 0
20.7 21.0
85-90 85-90 85-90
83-87 83-87
82-85 82-85
0 0 0
0 0
16.4 15.6
** Screw speeds measured using a stopwatch
* Screw speeds reading on the tachometer
to
TABLE 5 l b
Time Run
Number
Fawcett Barrel Temperatures (°C)
Andouart Barrel Temperatures (9C)
Zone 1 Zone 2 Zone 3 Zone 4 Zone 1 Zone 2 Zone 3
10.25 1 121 123 122 126 100 103 100 10.34 1 121 123 122 126 102 103 100 10.37 1 121 123 122 126 102 103 100
10.45 2 126 127.5 124 127 100 • 104 100 10.52 2 126 128 125 127 102 104 100
11.15 3 127 129 126 127 100 115 105 11.23 3 127 129 126 128 100 115 107
TABLE 5.1c
Time Run
Number
Head Temperatures (°C) Fawcett Throat
Temp rature (~ C)
Melt Temperature
(C)
Fawcett Pressure (lbf/in2)
Andouart Pressure (1bf/int) Zone 1
(Bac(} Zone 2
Zone 3 (Die)
10.25 1 125 130 125 123
10.34 1 137 130 125 128 124-129 800
10.37 1 145 132 129 128
10.45 10.52
2 2
135 130
133 134
129 125
130 . 131
136-138 940
11.15 3 130 134 125 135 138-140 1000 5100 . 11.23 3 125 134 125 135
-82-
The results for the cable making trials are presented in Tables 5.2a,
5.2b and 5.2c. Four runs were made at line speeds of 1.45, 1.72 ,1.95 and
2.43 m/min, as measured using a stopwatch. The Fawcett speed was
gradually increased from 30 rpm to 45 rpm and the Andouart speed from
20 rpm to 25 rpm.
In Table 5.2a are tabulated the line speeds corresponding to the two
screw speeds and the currents supplied to the extruders, for each cable run,
including the time when the readings were taken. Two sets of line and
screw speeds have been presented, one set being the one measured by a
tachometer registering at the capstan wheel and the other. measured using a
stopwatch.
The Fawcett and Andouart barrel temperatures for each cable run are
shown in Table 5.2b.
The throat and head temperatures and the steam pressures, together
with the readings of the two pressure transducers fitted to the Fawcett
and Andouart extruders, are presented in Table 5.2c.
5.4.1 Polymer Properties and Data Processing
In order to be able to compare the theoretical FE predictions
with the experimental results, the relevant physical properties of the .
polymers are required for use in the flow analysis computer program. The
viscosity data were obtained from capillary rheometer measurements, as
described in Chapter 2. The machine used was an Instron Capillary.
Rheometer (see Van Wazer et al (1963)), with two capillaries of lengths
and 2" and nominal diameter 0.05".
The white crosslinking polyethylene of the insulation layer
was tested at three temperatures, namely 115°C, 120°C and 130°C, and
measurements were made over a wide shear rate range of 14.57 to 2914 s-1.
TABLE 5.2a
Time Run
Number
Fawcett** Screw Speed
(rpm)
Andouart** Screw Speed
(rpm)
Fawcett* Screw Speed (rpm)
Andouart* Screw Speed
(rpm)
Line Speed** (m/min)
Line Speed (m/min)
Fawcett Ammeter. (amps)
Andouart Ammeter (amps)
12.50 13.00 13.17 13.50
14.05 14.20
14.45 15.00
15.30 15.45 15.58
1 1 1 1
2 2
3 3
4 4 4
29:0
34.3
39.5
44.0
22.3
23.0
24.9
27.0
30.0 30.0 30.1 30.0
35.7 35.7
40.3 40.3
45.2 45.2 45.2
20.3 20.3 20.3 20.3
20.5 20.9
23.2 23.0
25.3 25.3 25.3
1.45
1.72
1.95
2.43
1.3 1.3 1.3
1.6 1,6
1.9 1.9
2.45 2.45 2.45
85-90 83-88 83-88
82-85 82-87
82-84 81-85
78-83 78-83 78-83
15.~ 15.2 15.0
15.0 15.0_
15.5 15.5
15.5 15.2 15.3
** Line and screw speeds measured using a stopwatch
* Screw speeds reading on the tachometer
TABLE 5.2b
Time Run Number
Fawcett Barrel Temperatures (°C)
Andouart Barrel Temperatures (°C)
Zone 1 Zone 2 Zone 3 Zone 4 Zone 1 Zone 2 Zone 3
12.50 1 121 123 121 125 100 115 107 13.00 1 121 123. . 121 125 100 110 100 13.17 1 119 123 122 126 100 111 100 13.50 1 119 123 122 125 101 112 100
14.05 2 121 127 123 127 101 112 100 14.20 2 123 127 123 127 • 101 • 112 100
14.45 3 127 131 125 127. 101 115 102 15.00 3 128 132 126 127 102 115 102
15.30 4 131 133 126 127 105 118 • 105 15.45 4 131 133 127 127 105 118 106 15.58 4 131 133 127 127 105 119 106
TABLE 5.2c
Time
Head Temperatures (°G) Fawcett Throat
Tempe~Oature ("C)
Steam Pressure (Ibf/int)
Fawcett Pressure (Ibf/int)
Andouart Pressure (lbf/int)
Run Number ZBāc~l
( ) Zone 2
Zone 3 (Die)
12.50 1• 129 129 126 130 205
13.00 1 135 125 125 128 195 1250 5450-5600 13.17 1 132 126 124 128 181
13.50 1 129 126 130 125 220
14.05 2 127 126 • 125 128 210 1300 5490=5640
14.20 2 130 127 126 130 215
14.45 3 135 126 125 135 218 1270 5575-5770 15.00 3 136 127 125 135 205
15.30 4 138' 126 125 135 218
15.45 4 125 '126 125 135 220 1300 5620-5630
. 15.58 4 132 126 125 135 218
86 -
Allowances were made for reservoir pressure drop, capillary end
corrections and the Rabinowitsch shear rate correction (see Chapter 2).
Curve-fitting of the empirical power-law equation (2.10) over the shear
rate range for dies, 14.57 to 145.7 s-1, gave the following data for the
crosslinking low density polyethylene:
Power-law index n = 0.391
Effective viscosity at reference shear rate
(pyo) 1 s-1 and reference temperature 120°C uo = 22.56 kNs/m2
Temperature coefficient of viscosity : b = 0.0285 °C-1
The black ethylene/vinyl acetate copolymer for the screens was
also tested at three different'temperatures of 110°C, 120°C and 130°C,
using two capillaries. Curve-fitting of the empirical power-law equation
over the shear rate range 14.57 to 145.7 s-1 gave the following parameters:
Power-law index n = 0.317
Effective viscosity at reference shear rate
(y0)1 s-1 and reference temperature 120°C : po = 85.62 kNs/m2
Temperature coefficient of viscosity b = 0.0166 °C-1
During the tests, it was noticed that the white polyethylene
would not melt at 110°C and at temperatures above 130°C, both polymers
started showing signs of crosslinking.
Timed samples of both polymers were taken at known piston
speeds of the Instron, in order to be able to calculate their melt
densities. They were then weighed on a chemical balance and, knowing the
diameter of the rheometer barrel, their specific gravities were calculated.
-87-
This gave for the low density polyethylene, a specific gravity value of
0.813 and for the ethylene/vinyl acetate copolymer a specific gravity
value of 1.102.
Timed bleed samples of the insulation layer and insulation
plus screen layers were taken during the cable covering trials. These
samples were weighed and knowing their densities, the mass flow rate of .
the intermediate layer was worked out. Alternatively, the mass flow rate .
of the insulation layer was worked out from the cable samples, knowing the
line speed, the cross-sectional area of the intermediate layer in the
cable and the density of the polymer in this layer. Figure 5.10 shows
the mass flow rate of polymer from the Fawcett machine as a function of
extruder screw speed. Two sets of data have been plotted, one
corresponding to the output rate calculated using the bleed samples, and
the other to the cable samples. Both sets seem to lie on straight lines
through the origin, which means that the melt flow rate varies linearly
with screw speed. However, the output in the case of the bleed trials is.
slightly higher than in the case of the cable trials, which could be due
to back pressure onto the extruder.
5.4.2 Analysis of the Experimental Results
A study of the pressure charts obtained from the ultra-violet
recorder showed that the transducer situated at the melt inlet to the
crosshead fromthe Andouart machine was measuring sinusoidal pressure
fluctuations. During the cable trials, with the small variations in
Andouart speed, the amplitude and period of these fluctuations remained
almost constant. The amplitude was approximately 150 lbf/in2 and the
period was about 2 minutes. The pressure charts corresponding to the
bleed tests were also checked and it was found that, during the bleed
-88-
trials, the pressures remained fairly constant.
An examination of the pressure charts corresponding to the
transducer fitted at the melt inlet to the crosshead from the Fawcett
machine showed that the pressures in this case remained constant, during
both the cable and bleed trials.
In order to be able to compare the predicted coating
thicknesses of the extruded cable with the experimental results, cable
samples were taken at the four different line speeds. At each speed,
within one complete cycle of the Andouart pressure fluctuation, five
samples were taken as shown in Figure 5.11. If T is the period of one
complete cycle, then the five samples were taken at positions corresponding
to 0, T/4, T/2, 3T/4 and T. Hence, twenty cable samples were measured
and compared with the predicted results.
To be able to see the effect of Andouart pressure fluctuations
on the overall shape of the cable, the outer diameters of the twenty cable
samples were measured using a micrometer at four diameters of 45° intervals
around the circumference. The outer diameters were then made
dimensionless by dividing through by the mean outer diameter of the twenty
samples. Figure 5.12 shows the variation of the dimensionless outer
diameter round the circumference for three samples at each of the line
speeds of 1.72, 1.95 and 2.43 m/min. Results for the 1.45 m/min cable
run lie quite close to the ones for the 1.72 m/min and for this reason they
have not been plotted to avoid overlapping curves. From this graph, it
can be concluded that, with varying Andouart pressure, the shape of the
cable remains the same but its size varies, which is due to slight extruder
surging (see Section 5.3.4; Fenner (1970); Fisher (1976)). If, for
instance, insufficiently molten or plasticated material is conveyed into
the metering section, the machine will extrude irregularly. It is then
-89-
said that the output from the die "surges". Surging is reduced by using
a low molecular weight material or by an increase of the melt temperature.':
Fisher (1976) has obtained an equation for surging which describes the
stability of the system for any degree of disturbance. He predicted that
an increase in die pressure and an increase in the rate at which melt
enters the filled channel will decrease the amount of surging and increase
the melt channel length.
The actual radial thickness distributions round the
circumference of each of the three layers of the twenty cable samples have
been plotted on Figures 5.13 to 5.32. The radial thickness distributions
for three samples taken from the third bleed trial, with both extruders
running, have also been plotted on Figures 5.33 to 5.35. A close study
of the experimental radial thickness distributions shows that at any
particular cable speed, there is hardly any variation in the shape of the
radial thickness distribution graphs, but the curves are slightly shifted
in the vertical direction.
5.5 FINITE ELEMENT FORMULATION OF THE PROBLEM
The method of solution for computing the predicted performance of the
extrusion crosshead used in the experimental three-layer covering of high
voltage electrical cables is described in Chapter 4. In this section,
the type of mesh used will be discussed, together with the calculation of
residence time distribution, and it will be shown how to plot streamlines
and isobars within the flow domain.
5.5.1 Mesh Generation
Detailed engineering drawings were obtained from AEI Cables
Limited of the three-layer crosshead. A finite element mesh was
-90-
constructed for the analysis of flow in the channels for each of the three
layers. The geometric flexibility of the finite element method makes it
easy for a mesh of triangular elements to be constructed to suit the
complex shape of the flow channels. The FE mesh used here is chosen such
that element angles exceeding 90° are omitted as much as possible. Fenner
(1975) has shown that the absence of obtuse-angled elements is sufficient
to ensure convergence of the Gauss-Seidel method for problems of the
harmonic type.
The FE mesh for the flow channels, which is shown in Figure 3.2,
is generated by the computer after feeding in the coordinates of the nodal
points along the boundaries BDF and ACE. The actual number of nodes
employed along AC and BD is 14 and along ACE and BDF is 19. Each point
on the lower boundary is joined to the corresponding point on the upper
boundary by a straight line, e.g. B to A, D to C and F to E. Each of the
19 lines so formed are then divided into 12 equal intervals to give 13
nodal points on each line (e.g. AB, CD, EF) and hence 247 nodes altogether.
The neighbouring nodes are then appropriately joined together to form
quadrilaterals. Each quadrilateral is divided into two triangles using a
diagonal in such a way as to omit obtuse angles as much as possible. The,
total number of triangular elements used for the present mesh is 432. As
shown in Figure 3.2, the numbering of the nodes starts at point B and
increases along the rows, and similarly for the elements.
Tests were made to see the effect of mesh refinement on the
accuracy of the results and it was found that, as the number of elements
and nodal points were increased the computed solution approximated more
closely to the true solution, which was unique. Due to increasing
computing time, it was found that it would not be advisable to use a very
fine mesh. As a result of a compromise between accuracy and cost of
- 91 -
computation, a mesh of 19 x 13 was chosen. Further refinement of the
mesh causes only insignificant changes in the final results. Another
method of achieving more accurate results instead of using a finer mesh is
the h2-extrapolation technique which is described in Appendix A2.
5.5.2 Convergence and Accuracy
It is well known that the over-relaxation factor has a very
significant effect on the rate of convergence of the iterative Gauss-Seidel
method used in this investigation. Using (6) = 0 as the initial
approximation of the solution vector (see equation (4.36)), and taking
e = 10-6 as the tolerance limit, the effect of w on the number of
iterations to convergence has been studied.
In Figure 5.36, the total number of cycles of iteration, N,
for a convergent solution is plotted against the values of w between 1 and
2, for different size meshes. It can be seen that N varies considerably
with w and therefore it is desirable to determine the value of w at which
N is a minimum to economise on the computing time This value of w is
termed the optimum over-relaxation factor, apt, and it varies according
to the size of mesh used. For our particular problem with a 19 X 13
mesh, wopt was taken to be 1.75.
5.5.3 Computation of Stream Function Values and Pressure Distribution
A computer program was used for the solution of the set of
linear equations (4.36) to yield the stream function values. In the
program, the area of the two-dimensional solution domain was divided into
triangular three-node elements, over each of which linear •variations of the
unknown were assumed. The elements may therefore be said to be of the
constant strain triangular type, although the term is only strictly valid
- 92 -
if the unknown is a displacement. In this sub-section will be described
how the computer program is organised to solve for the stream function
values.
Initially, the coordinates of the nodal points along the lower
and upper boundaries are fed into the program as data and then the rest of
the node coordinates, nodal point numbers, element node numbers and element
numbers are generated by the program. Once the mesh generation is
complete, the material properties, such as the viscosity at reference shear
rate, yo = 1 s-1, and the power-law index are input as data. After this,
the boundary conditions are prescribed and, for the purpose of the present
analysis, the stream function values along the lower boundary (ACE; see
Figure 3.2) are chosen to be ii = 0 and for the upper boundary (BDF) p = 1.
The input data necessary to control the Gauss-Seidel solution process are
the maximum number of cycles of iteration, the over-relaxation factor and
the convergence tolerance which are also specified.
Next, the values of the unknowns, i.e. stream function, are
initialised to zero and the element stiffness matrix is formed. For non-
Newtonian flow, the element viscosities are not constant and, in this
analysis, they are represented by equation (4.37) where the viscosities
are up-dated at every five cycles of iteration. Palit (1972) studied the
convergence rates for downstream and recirculating flows with viscosities
being up-dated after varying numbers of iterations during the solution
process. He found •that up-dating viscosities after about 4-6 cycles of
iteration gave very fast convergence rates. Once the iterations converge
to the specifiedtolerance limit, the stream function value at each nodal
point is known.
The computation of pressure distribution follows the
description given in Section 4.4. Once the stream function distribution
n=N Txy _ (dT)
n=1 n.
-93-
and the pressure distribution are known,. it is possible to draw streamlines
and isobars through the solution domain, as shown in Figure 5.7. In
general terms, the streamlines define flow lines and material never crosses
a flow line. The isobars are supposedly orthogonal to the streamlines,
as can be seen.
5.5.4 Computation of Residence Time Distribution
This subsection describes how the distribution of residence
time of polymer passing through the crosshead is worked out. The time
taken for the polymer to travel along a streamline XY, say, (see Figure
5.7) is calculated by summing the time taken to cross each triangular
element that the streamline passes through. For example, the residence
time along streamline XY, can can be represented as:
where N is the total number of elements that the streamline crosses, n = 1
being the first element and (dT) the time taken to cross the nth element
along the line.
Figure 5.37 represents a typical element that a streamline
crosses. In order to be able to calculate (dT) , the coordinates of the
two points where the streamline crosses the sides of the triangle must be
known. Once these are known, the distance, D, that the polymer travels.
across the element can be worked out The next step is to find the mean
velocity of the polymer within the element, V. From equation (3.20):
h V (5.2)
r ae) 2 f .lax.
i C , V = (r) 2 . (C2 cos a)2 ./h
J
or:
h2 v2 = 2
(5.3)
94 -
where h is the local channel depth normal to the channel boundaries, and
Qs is the, local volumetric flow rate per unit width normal to the
s-direction. Using equations (3.26) and (3.28), it can be written that:
Since a linear distribution of p is assumed over each element, the velocity
within each element is a constant, i.e. V = constant. Now, (dT)n can be
written as
(dT) n = 5.4)
Hence, the amount of time the polymer spends travelling along any set path
within the die can be calculated.
Figure 5.38 shows the residence time distribution within the
flow channels of the inner, intermediate and outer layers. The time taken
for a particle of polymer melt at a total flow rate of 1 in3/sec to travel
along each of the 9 streamlines has been plotted. The standard deviations
of the residence time distribution graphs are 0.809, 0.917 and 1.626 for
the inner, outer and middle layers, respectively. The more uniform the
residence time distribution, the more uniform is the coating thickness.
5.6 COMPARISON OF THEORY WITH THE EXPERIMENTAL RESULTS
In this section, the experimental results on extrusion pressures and
circumferential cable layer thickness distributions will be compared with
the finite element computations.
-95-
5.6.1 Comparison of Theoretical and Experimental Thickness
Distributions
Although some longitudinal variations in mean layer thicknesses
were detected for a given cable speed (see Sections 5.3.4 and 5.4.2), the
relative circumferential distributions did not change significantly. As
it is, these circumferential variations that are of primary interest in
the present work, it is sufficient to consider dimensionless thickness
distributions, which depended only on cable speed. Dimensionless
thickness in the present context is defined as the ratio between the.
local radial thickness of the particular layer and its mean thickness
obtained by averaging over all the thickness measurements made at the
particular section.
Figures 5.39, 5.40 and 5.41 show dimensionless thickness ,
plotted against angular position, (I), around the cable for the inner screen,
insulation layer and outer screen, respectively. The origin for in
each case corresponds to the position of the nick which is the lowest
point of the cable as it emerged from the crosshead, and c is measured in
the clockwise sense looking along the cable in the direction of motion.
For the screens, the thickness distributions are not significantly
dependent on cable speed and results for only two speeds are shown to
avoid overlapping curves. In the case of the insulation layer, however,
all four speeds are shown 'separately. Corresponding to each cable speed,
the thickness distribution for only one of the five cable samples has
been plotted.
Also shown in Figures 5.39 to 5.41 are the thickness
distributions predicted by the finite element analysis technique described
in Chapter 4. These predictions are for the particular material
properties and crosshead geometries used in the experiments. Perfect
-96-
alignment and concentricity of all crosshead components is assumed.
Hence, the theoretical distributions are symmetrical about 0 = 90° and .
2700, corresponding to the axial plane through the melt inlets to the
deflectors. There is apparently very little agreement between theory and
experiment. In the case of the screens, while the theory predicts very.'
good uniformity of thickness due to a high degree of throttling by the
narrow tapering channels over the points, the distributions achieved in
practice were much worse and of different shapes for the two layers.
Similarly, for the insulation layer, although the measured and predicted
deviations from perfect uniformity are of similar magnitude, they do not
correspond in terms of angular position. Reasons for the observed
discrepancies will be elucidated in the following subsections.
5.6.1.1 Effects of gravitational forces
Taking the insulation layer first, Figure 5.40 shows
that the trend in the measured thickness distribution with increasing
cable speed is towards increased thickness at 0 = 0° and reduced thickness
at 0 = 180°. In other words, the polymer layer became thicker on the
underside of the cable and thinner on top. Temperature measurements
recorded during the bleed trials showed that the melt became significantly.
hotter and less viscous with increasing output rate from the extruder.
The less viscous the melt, the more effect gravity could have had before
curing and cooling stiffened the insulation layer. The resulting
distortion is sometimes known as "peardropping", after the shape produced.
In order to estimate the possible extent of this effect, a, typical
viscosity for low shear rate deformation can be selected from the measured
rheological data. This viscosity can be converted into an equivalent
elastic modulus with the aid of a characteristic time for the deformation,
-97-
in this case selected as 50 seconds. Hence, the deformation of the cable
cross-section, allowing for the presence of the very much stiffer
conductor, can be predicted by a finite element method which is described-
in Chapter 6.
Figure 5.42 shows the predicted insulation layer
thickness distribution due to peardropping at the melt temperature
associated with the highest cable speed. Although shown for a
characteristic time of 50 seconds, the distribution according to this
simple model is proportional to time. Clearly, the maximum changes in
thickness are of a similar order of magnitude to those shown in Figure
5.40. It therefore appears that much of the discrepancy between theory
and experiment for the insulation layer can be explained in terms of
peardropping, the magnitude of which changes with cable speed.
5.6.1.2 Effects of geometric imperfections
With the thin inner and outer screens, there were
apparently negligible gravitational effects, although Figures 5.39 and
5.41 show the thickness deviations to be much greater than predicted.
The thickness of the outer screen was a minimum on the underside and a
maximum on the top of the cable. Similarly, the thickness of the inner
screen was greatest and least at = 270° and 90°, approximately, at the
sides of the cable. These deviations suggest geometric imperfections in
the form of distortion or misalignment of the crosshead components in the
vertical and lateral planes through the cable axis, respectively.
Leaving aside for the moment the possibility that
extrusion pressures caused significant distortion of the deflectors, it is
unlikely that misalignment of the deflectors was responsible for the
observed effects, because the deflectors fitted inside each other with
-98-
only very small clearances. This was not so with the points, however,
and these were screwed into the ends of the deflectors, but were otherwise
unconstrained. In view of the small sizes of the final radial channel
depths associated with the screens, the slightest imperfections in the
alignment of the points would have had considerable effects on thickness
distributions in the finished cable. Misalignment is discussed in more
detail in Chapter 6.
Supposing that there were angular misalignments, 6,
between the axes of the points forming the flow channels for either screen
layer. Provided the direction of this misalignment is known, the method
of flow analysis can accommodate the resulting geometric assymmetry. In
general, it is no longer possible to assume a plane of symmetry between
the two identical halves of a deflector, the only exception being when the
misalignment of the points is in the same plane. Considering the present
experimental results, it is possible to deduce the plane of point
misalignment for the screens as already indicated. Applying the method
of flow analysis with various values of the angle s in these planes until
reasonable agreement between the theoretical and experimental results is
achieved, Figures 5.43 and 5.44 are obtained for the inner and outer
screen thicknesses, respectively. Clearly, the agreements are now very
satisfactory and misalignments would appear to be responsible for the
earlier poor correlations. Note, however, the very small values of a,
0.03° and 0.10°, respectively, involved.
5.6.1.3 Thickness tolerances for high voltage cables insulated
with crosslinked polyethylene
Figures have been agreed internationally (in
Industrial Engineering Chemistry (IEC)) on thickness tolerances for high
-99-
voltage cables having XLPE insulation. The Specification (IEC Pub. 502-1)
lists nominal insulation thicknesses which, for the voltage ranges that
this work is concerned with, i.e. 6/10, 8.7/15, 12/20, 18/30 kV (the'
first figure in each case is the single phase figure and the second is the
three phase, i.e. x single phase),. are independent of conductor size
and are:
kV Nominal Insulation Thickness (mm)
6/10 3.4
8.7/15 4.5
12/20 5.:5
18/30 8.0
The nominal thickness'of a cable sample is arrived at
by measuring the thickness in six positions equi-spaced around the sample
(preferably measured optically using a thin slice of insulation) making
sure that one measurement is at the thinnest place. The nominal thickness
is the average of the six readings. The nominal value so obtained must
not be less than the appropriate, value on the abovetable.
Additionally, the thickness at any place may be less
than the specified nominal value provided that the difference does not
exceed 0.1 mm + 10%.of.the specified nominal value.
The table may therefore. be expanded as follows:
kV Nominal Insulation Thickness (mm) Minimum at a Point (mm)
6/10 3.4 2.96
8.7/15 4.5 3.95
12/20 5.5 4.85
18/30 8.0 7.1
- 100 -
It is reasonable to assume that such a tolerance
would apply to higher voltage cables (up to 20 mm radial thickness has
been used in the USA and Japan for working voltages in the 100 to 130 kV
range).
5.6.2 Comparison of Theoretical and Experimental Pressure
Distributions
In addition to predicting polymer layer thicknesses in the
finished cable, the finite element method of analysis is also able to
provide pressure distributions throughout the melts flowing in the
crosshead. Although no pressures within the head were measured during
the experiments, the pressures at the flow inlets for the two streams of .
polymer were recorded. Knowing the steam pressure in the catenary, the
pressure differences across the head were found for comparison with the
computed figures. Table 5.3 shows the results for the four cable speeds
tested. As the measured pressures were subject to small periodic
fluctuations, the relevant ranges are shown. For the inner and outer
screens, although there was only one pressure measured, theoretical
pressure differences were obtained from the separate flow analyses for the
two layers. It was noted from the results that the predicted pressure
drop for the outer screen is slightly higher than the pressure drop for
the inner screen for each cable speed. This difference in pressures can
be explained if the sketch in Figure 5.45 is examined. In the theoretical
analysis, for the inner screen, the pressure drop between the points A and B is
calculated, while, for the outer screen, the pressure drop between the
points A.and D is calculated. The difference in the predicted pressures
of the screens, therefore, could well account for the pressure drop between
the points B and D. An exact theoretical analysis to calculate the
TABLE 5.3
Measured and Predicted Pressure Differences Across the Crosshead
Cable Speed (m/min)
INSULATION LAYER SCREENS
Theoretical (lbf/in2)
Measured (lbf/in2)
Theoretical Measured (lbf/in2) Inner
(lbf/in2) Outer.
(lbf/in2) Mean
(lbf/in2)
1.45 945 1030-1069 5211 5368 5290 5230-5419
1.72 1020 1085-1090 5217 5225 5221 5275-5430
1.95 1061 1052-1065 5496 5437 5467. 5357-5565
2.43 1105 1080-1082 5577 5536 5557 5400-5412
- 102 -
pressure drop between points B and D is quite complicated since the
boundaries are moving and there is more than one material involved.
The theoretical pressure predictions given in Table 5.3 are not
exactly related to the experimentally measured ones, since the predictions
do not include the pressure drop, dp, between the transducer and the melt
inlet to the deflector (see Figure 5.46a). The magnitude of dp is in the
region of 3% of the whole pressure drop for that layer. The analysis
used for the calculation of dp is explained in detail in Appendix A3.
The experimental data was not sufficient to tell the exact
melt temperature of each layer. Thus, it was decided to base the
theoretical pressure calculations on a melt temperature of T = 135°C.
The pressure was initially predicted at a reference temperature of
To = 120°C and then modified using:
u = po exp (— b (T — To)) (5.5)
where the pressure is directly proportional to u, uo being the effective .
viscosity at reference temperature To and reference shear rate yo, and b
being the temperature coefficient of viscosity at constant shear rate.
The dependence of viscosity on pressure has been ignored in this analysis,
since the exact value of a, the pressure coefficient of viscosity, is not
known for the materials used in the experiments. However, it is assumed
that this effect will be negligible.
On the whole, the agreement between predicted and measured
pressure differences is very satisfactory and provides further confirmation
of the validity of the finite element method of flow analysis.
-103-
5.6.2.1 Deflector distortion
While only overall predicted pressure differences
have been compared with measured data, the predicted pressure profiles
within the flow channels can be used to study distortion of the deflectors.
Owing to the large difference between the inlet pressures for the screens
and insulation layer, the deflectors were subject to considerable
differences in pressure between their inner and outer surfaces, which
caused some deformation. This deformation was estimated for a typical
deflector and its effect on melt flow studied with the aid of the finite
element analysis, and found to be negligible. The analysis is described
in greater detail in Chapter 6.
5.7 CONCLUSIONS
The performance of an extrusion crosshead used in the three-layer
covering of high voltage electrical cables has been studied experimentally.
Measurements of overall pressure differences across the head and
circumferential distributions of polymer layer thicknesses were compared
with theoretical finite element analyses of melt flow within the crosshead.
While the agreement on pressures was good, attempts to correlate the
thickness distributions were initially unsuccessful. It was found that,
in the case of the relatively thick polymer insulation layer, the thickness
distributions were influenced as much by gravity after the cable left the
crosshead as by inadequacies in the flow channel design. While the theory
predicted excellent thickness distributions for the thin inner and outer
screen layers, assuming perfect alignment of the crosshead components, the
measured distributions were much less uniform. It was found, however,
that the discrepancies could be explained by very slight misalignments,
typically 0.1° or less, of the deflectors and points at the threaded joints
- 104 -
between them. This finding emphasises the need for good mechanical
design and very accurate construction of crosshead components if geometric
imperfections are not to seriously affect performance. The ability of .
the flow analysis technique to predict final thickness distributions,
particularly under conditions of unsymmetrical geometry, justify its use
in designing crosshead deflectors in preference to laborious trial-and-
error methods.
PAY-OFF ..CONDUCTOR-+EXTRUDER--►WATE.R I3A7H ---.► CAPSTAN--,.CAC3LE WIND 11..1)
Figure 5.1: Equipment used in general cable covering process
Figure 5.2: The equipment used to extrude and cure a three-layer cable
n ------ STORAGE D R.U, M
SEALING GLPF ND
CAPSTAN WHEEL
EXTF,U.510k CROSS HEAD ( XTR..U.D 'R5 MOT 5i+0WN)
5TEAM
CON9u.CTOR CATS N A RY T1A,13 E
WATER Dft,U,M
CRIERA1LLA1a. HALLL--OFF
ō rn
Figure 5.3: The Fawcett and Andouart screw extruders
Figure 5.4: Sealing gland and caterpillar haul-off
INSULATION MELT IN
~
SCREEN MELT IN
OUTER DEFLECTOR
INTERMEDIATE DEFLECTOR
DEFLECTOR
. " ---+----" .
Figure 5.5: Cross-section through the crosshead assembly
I J
l
Figure 5.6: The intermediate deflector
5TaEAMI.►►J i5
)1,0 IS 4,100 $44114 ""%oit
titre axraine X Y
X
Figure 5.7: One half of the inner deflector flow channel plotted on the (z,o) plane
Figure 5.8:' One half of the outer deflector flow channel plotted on the (z,9) plane
Figure 5.9: Sample being collected during bleed trials
so
70
u 50
W Lf0
30
0 OLE.1D T65T j
0 cAlbLE 1t21A1.5
20
1 0
114 -
10 /0 30
FAWC677 5PEF..D (no m,) 0 >♦.0
50
Figure 5.10: Variation of mass flow rate with Fawcett extruder screw speed during bleed tests and cable trials
- 115 -
VILE
55 U
. 1Z E
.
T,
7/4 7/2 3714.
TIM E
5am.pl,e6 viers taken, at posi.tion:s Ouch, have been, ma.rk.ed, with, `x' .
Figure 5.11: One complete cycle of Andouart pressure fluctuations
- 116 -
o , 0 corre5pond to aferen,t positions on the, Andott.art rre s5u.re c cLe.
0325 011-t9'd° 447 ,454 gOD-23o r35=3l5°
pO51TIOH or. `fl{ CIRCUMFJr FLENCg. (clocKun5E. FROM THE NICK)
Figure 5.12: Variation of outer diameter round the circumference corresponding to three different stages of the Andouart pressure cycle
INNER
o
-117-
OUTER
c1.00 6b.00 120.00 180.00 240.00 3b0.00 360.00 POS I T I.ON ON THE CIRCUMFERENCE (DEGREES ),0°
Figure 5.13: Circumferential thickness distributions of the three layers of the first sample taken at a cable speed of 1.45 m/min
0 O 0
O O
OUTER
INNER
- 118
0
cb.00 60.00 120.00 180.00 240.00 300.00 360.00 POSITION ON THE CIRCUMFERENCE (DEGREES),4°
Figure 5.14: Circumferential thickness distributions of the three layers of the second sample taken at a cable speed-of 1.45_mLmi.n ___...
0 C)
0
O 0
MIDDLE
INNER
0 • 0
.-,
OUTER
119 -
93.00 60.00 120.00 180.00 2'40.00 3100. .00 360.00 POSITION ON THE CIRCUMFERENCE (DEGREES),cp°
Figure 5.15: Circumferential thickness distributions of the three layers --`_,
of the third sample taken at a cable speed of 1.45-m/min
0 N1 RIDDLE
CD
CO
INNER
OUTER
0
gratl
0
- 120 -
93.00 6b.00 120.00 1b0.00 240.00 300.00 360.00 POSITION ON THE CIRCUMFERENCE (DEGREES l , °
Figure 5.16: Circumferential thickness distributions ..of _the-three layers of the fourth sample taken at a cable speed of .1.45 m/min
0 N-
INNER
OUTER
- 121
c.00 60.00 120.00 180.00 240.00 3100.00 360.00 POSITION ON THE CIRCUMFERENCE (DEGREES) 0°
Figure 5.17: Circumferential thickness distribution of the three layers of the fifth sample--taken at a cable speed -of-t.45 m/min
INNER
OUTER
0 0 m
- 122 -
93.00 60.00 120.00 180.00 240.00 300.00 360.00 POSITION ON THE CIRCUMFERENCE (OEGREES), 6°
Figure 5.18: Circumferential thickness distributions of the three layers of the first sample taken at a cable speed- of- 1.72 m/min
- 123 -
co
0 0 CO CO
oC W I-- Luc
,
J _J
O O
(ID •- W
U I-4 pp Z •
-J CC C3 o CCO iX N
INNER
OUTER
O O
97.00 60.00 120.00 160.00 240.00 300.00 360.00 POSITION ON THE CIRCUMFERENCE (DEGREES)7 4°
Figure 5.19: Circumferential thickness distributions of the three layers. of the second sample taken at a cable speed of 1.72 m/min
- 124 -
0 0 (D(O
W
W ō
—I in J
0 CO •
W
C.)o
• Cr)
J Q ' S
O0 QO
f~N
O
INNER
OUTER
O O .--
91.00 60.00 120.00 180.00 240.00 300.00 POSITION ON THE CIRCUMFERENCE (DEGREES)
360.00
Figure 5.20: Circumferential thickness distributions of the three layers of the-third sample taken at a cable speed of 1.72 m/min
- 125 -
INNER
0 0 •
OUTER
9).00 60.00 120.00 1160.00 2140.00 300.00 360.00. POSITION ON THE CIRCUMFERENCE (DEGREES) 0°
Figure 5.21: Circumferential thickness distributions of the three layers of the fourth sample taken at a cable speed of 1.72 m/min
0 0 co
0 0 .-4
OUTER
INNER
0 O
- 126 -
9l•00 6b.00 120.00 ib0.00 240.00 .3b0.00 POSITION ON THE CIRCUMFERENCE (DEGREES)
360.00 o
Figure 5.22: Circumferential thickness distributions of the three layers of the fifth sample taken ata cable speed of-1.72 m/min
b O h
INNER
0 CD
-I
OUTER
- 127 -
0 t7
91.00 60.00 120.00 180.00 240.00 300.00 360.00 POSITION ON THE CIRCUMFERENCE ( .DEGREES),9s°
Figure 5.23: Circumferential thickness distributions of the three layers of the first sample taken at a cable speed of 1.95 m/min
0 0
INNER
- 128 -
O
OUTER
O O
c1.00 60.00 120.00 180.00 240.00 300.00 360.00 POSITION ON THE CIRCUMFERENCE (OEGREES)4
Figure 5.24: Circumferential thickness distributions of the three layers —of--the—second sample taken at a-cable speed of 1.95 m/min
O 0 .r
OUTER
INNER
a
-129-
0
9Ū .00 6b .00 120 .00 180.00 240.00 300.00 3b0.00 POSITION ON THE CIRCUMFERENCE (OEGREES),0°
Figure 5.25: Circumferential thickness distributions of the three. layers of the third sample taken at a cable speed of 1.95 m/min
N
INNER
OUTER
- 130 -
O
9..00 60.00 120.00 180.00 240.00 300.00 360.00 POSITION ON THE CIRCUMFERENCE (DEGREES),0°
Figure 5.26: Circumferential thickness distributions of the three layers of the fourth samplē taken at a cable speed of 1.95 m/min
0 0 co
0 N~
INNER
- 131 -
OUTER
91•00 60.00 120.00 180.00 240.00 300.00 360.00 POSITION ON THE CIRCUMFERENCE (DEGREES3,0°
Figure 5.27: Circumferential thickness distributions of the three layers of the fifth sample taken at a cable speed of 1.95 m/min
MIDDLE
-♦
A
INNER
O
OUTER
N
- 132 -
cb.00 6b.00 120.00 180.00 240.00 3b0.00 3b0.00 POSITION ON THE CIRCUMFERENCE (DEGREES) p°
Figure 5.28: Circumferential thickness distributions of the three layers of the first sample taken ata cable speed of 2.43 m/min
0
Orr
0 0 U 1
NICOLE
a a
•
OUTER
INNER
-133-
O O
9 .00 613.00 1'20.00 180.00 240.00 300.00 360.00 POSITION ON THE CIRCUMFERENCE (DEGREES) , °
Figure 5.29: Circumferential thickness distributions of the three ]avers of the second sample taken at a cable speed of 2.43 m/min
0 0 r' MIDDLE
.-4
OUTER
INNER
- 134 -
O 0 93.00 60.00 120.00 180.00 240.00 300.00 360.00
POSITION ON THE CIRCUMFERENCE (OEGREES),cfr°
Figure 5.30: Circumferential thickness distributions of the three layers of the third sample taken at a cable speed of 2.43 m/min
NIODLE •
A
INNER
O O •
OUTER
C3
03
0
- 135 -
9.00 60.00 .120.00 180.00 2140.00 300.00 360.00 POSITION ON THE CIRCUMFERENCE ( DEGREES ),0
Figure 5.31: Circumferential thickness distributions of the three layers of the fourth sample taken at a cable speed of 2.43 m/min
0 O
OUTER
INNER
-136-
0 O co
0 N MIDDLE
9.00 60.00. 120.00 ibo.0o 240.00 300.00 360.00 POSITION ON THE CIRCUMFERENCE (OEGREES),c°
Figure 5.32: Circumferential thickness distributions of the three layers of the fifths sample-taken at a cable speed of 2.43 m/min
I
-137-
6
2
4
0 60 120 IWO 210 300 360
POSITION ON TNE CIRC u,M FEU NCE , 40
Figure 5.33: Circumferential thickness distributions of the three layers of the first bleed sample taken during the third bleed trial
138 -
60 120 180 240 20O
p05ITION ON 1HE CIRCU,MFE.MNCE ,
Figure 5.34: Circumferential thickness distributions of the three layers of the second bleed sample taken during the third bleed trial
360
Figure 5.35: Circumferential thickness distributions of the three layers of the third bleed sample taken during the third bleed trial
- 13 9 -
MIDDLE
6
2
1
INN Ok • •
CATER
'0 60 120 180 240 300 360
POSITION ON TNE CIRCLI,MFER£NCE ,
500
440
~00
350
00 ,
0 6x 6
a TI XII o2Ix21 50
- 140 -
1.2. 1. y. , 1.6 f.8 2.0 ovsk-REI.A xATIoN FACTOR
Figure 5.36: Effect of over-relaxation on convergence to a tolerance of 10-6
- 141 -
Figure 5.37: A streamline crossing a typical element.
- 14 2 -
12.8
q.6
oU.TER . St. C,eVLcL:ti0It o.Rrr
INNER
5t. deviation. 0.80q
3.2
1.6
I I I I I I i 1 l
0.1 0.. 0.3 o.4. o.5 0.6 0.7 0.8 OA
STREAM FUNCTION VALUES CORRESPONDING TO THE STREAMLINES
Figure 5.38: Residence time along the 9 streamlines in the inner, intermediate, and outer deflector flow channels
TH
ICKN
ESS
THICKN
ESS
•45m /min. CABLE SPEED ❑ 2.43m/min. CABLE SPEED. O THEORY
180 240 0
300 360 60 120
- 143 -
Figure 5.39: Circumferential thickness distribution for the inner screen
--2.43m/min CABLE • SPEED
0.9
I.95 m/min CABLE SPEED
z wO.9
I • I -= 1.72 m/min CABLE SPEED
V 1.0 2
A • EXPERIMENT O THEORY
cc Lt-1
I • I —1•45m/min CABLE SPEED 0 a
I.O
- 144 -
0.9 0 60 120 180 240 300 360
co o
Figure 5.40: Circumferential thickness distribution for the insulation layer
1.501—
1.00
A 1•45 m/min CABLE SPEED O 2.43m/min CABLE SPEED O THEORY
CKN
ESS
w 0.50 2 >- J 0 a
60 120 180 0 240 300 360
145 -
Figure 5.41: Circumferential thickness distribution for the outer screen
- 146 -
I.25
1.00
w 2
0.90 cn• w z U I ~ 0.75 a w 2 >- J 0 a
0.50. 0 60 120 J80 240 300 360 c O
Figure 5.42: Estimated insulation layer thickness distribution due to gravitational forces
A I •45 m/ min CABLE SPEED 0 PREDICTED FOR 11=0.03°
.180 240 I
300 360 . 60 120
- 147 -
Figure 5.43: Measured and predicted thickness distributions for the inner ,. screen, allowing for misalignment of the points
TH
ICK
NESS I MEAN
A I •45m/min CABLE SPEED
PREDICTED FOR fl =0•IO°
0 60 120 180 240 300 360
- 148 -
Figure 5.44: Measured and predicted thickness distributions for the outer screen, allowing for misalignment of the points
INSULATION MEL 1 IN
1 OU1ER DEFLEC10R
\NIERM£D1ATE 1)EFLECTOR
I NNE R DE FLE C10R
Figure 5.45: Sketch of crosshead ass~mbly
- 150 -
ANNULAR FLAW ANflLY5I5
t1A~Jr It' THIS RECTION
PiPE; F)..oW ANALY515 MATE IN THIS REGION
r
MELT INLET 10 TNE 'EFI_!`.CTOR
Figure 5.46a: Pressure drop between transducer and melt inlet to the deflector
\\\\\\\\,,
\\\\\\\\\\ \ \ \\\\\
Figure 5.46b: Flow in a cylindrical pipe
151 -
CHAPTER 6
POSSIBLE CAUSES OF CABLE ECCENTRICITY
6.1 INTRODUCTION
Among the many quality specifications an insulated cable has to meet,
cable eccentricity is perhaps the most important. This is necessary to
ensure that nowhere will the insulation thickness be below a certain
minimum value. Any improvement of concentricity implies, therefore, an
equivalent reduction of outer insulation diameter requirements, resulting,
in economising on the coating material. Concentricity, as well as other
cable qualities, depends primarily on die design.
In view of these facts, it is surprising to find that no significant
attempts appear to have been made in analysing the process fundamentally,
except by Tadmor & Bird (1974) who analysed the stabilising forces in wire
coating dies. They derived expressions for the lateral force which the
polymeric fluid exerted on an off-centred wire due to secondary normal
stress difference and due to hydrodynamic effects. The lateral force due
to normal stresses (with no axial pressure gradient) tends to restore the
wire to its central location, and the numerical example carried out by
Tadmor & Bird seems to indicate that this force might be significant.
The hydrodynamic stabilising effect comes into action when the wire moves
at an angle to the die. Like lubrication effects, it is due to viscosity.
Contrary to the normal stress effect, the hydrodynamic effect will tend
only to restore the wire axis into a position parallel to the die axis.
Eccentricity will be reduced only if the guider tip, through which the bare
wire passes immediately before getting coated with polymer, is centred.
This, of course, emphasises the need for a perfect mechanical centering of
the guider tip.
- 152
In this chapter, possible causes for cable eccentricity are discussed,
bearing in mind the experimental results obtained during the trials
described in the previous chapter.
6.2 HEAD MISALIGNMENT IN RELATION TO THE CATENARY
During the trials described in Chapter 5, in order to accommodate the
straight portion encompassing the splice box and housing, the head on the
catenary plant was slightly "off-line" by a angle a, as shown in Figure
6.1. Due to this head misalignment, a lateral force, F, will be exerted-
by the copper conductor on the tip of the inner point, which will tend to
deflect it downwards. It was decided to calculate the magnitude of this
lateral force in order to see whether or not it will deflect the point
down by a significant amount. If this deflection is significant, then it
is possible that it will result in an eccentric cable.
In order to be able to calculate this lateral force, F, the theory of
a catenary has to be studied. The cable is considered to be attached to
two fixed points A and B and carrying a distributed load along the cable,
as shown in Figure 6.2a. In the case of a cable carrying a distributed
load, the internal force at a point D is a tensile force, T, directed
along the tangent to the curve (see Beer & Johnston (1976)). Figure 6.2b
shows the free body diagram of the portion of the cable extending from the
lowest point C to a given point D of the cable. The forces acting on the
free body are the tension force, To, at C, which is horizontal, the tension
force, T, at D, directed along the tangent to the cable at D, and the
resultant W of the distributed load supported by the portion of cable CD.
Horizontal and vertical equilibrium give:
T cos (6.1) To
- 153 -
and:
T sin e= W= w s (6.2)
where w is the load per unit length of cable, s. Therefore:
T = ,JTo2 + w2 62 (6.3)
In order to simplify the subsequent computations, a constant c is
introduced where:
c = To/w (6.4)
Therefore, equation (6.3) can be re-written as:
T = w ie2 + s2 (6.5)
The free body diagram of the portion of cable CD (Figure 6.2b) cannot be
used to obtain directly the equation of the curve assumed by the cable,
since the horizontal distance from D to the line of action of the
resultant W of the load is not known. To obtain this equation, the
horizontal projection of a small element of cable of length ds will first
be written down:
dx = ds cos e (6.6)
From equation (6.1), cos e To/T. Therefore, equation (6.6) can be
re-written using equations (6.4) and (6.5) as:
dx = Too ds = wc ds - ds
T w ✓e2 f 32 1/1 s2/e2 (6.7)
- 154 -
Selecting the origin 0 of the coordinates at a distance c directly below
C, as shown in Figure 6.2c, and integrating from C(0,c) to D(x,y):
x = fs = c [s ink-1 (-),s = c sinh-1. (-) o v1 + s 2/c2 ` o
(6.8)
This equation, which relates the length s of the portion of cable CD and
the horizontal distance x, may be written in the form:
s = c Binh () 6.9)
The relation between the coordinates x and y may now be obtained by
writing:
dy = dx tan e = T dx = dx = sinh ( ) dx (6.10)
Integrating from C(0,c) to D(x,y),
y-c = f sinh O dx = c [cash Cx) I= c (cosh — I) o Jo
Therefore: y = c cosh (c)
(6.11)
This is the equation of a catenary with vertical axis, y.
Using equation (6.2), the tension, T, at point B can be calculated,
provided the length of the cable CB, i.e. s, is known. The value of s can
be worked out using equations (6.1), (6.2), (6.4) and (6.9) as follows:
From equations (6.1) and (6.2):
tan 8 - (6.12) ws T o .
155 -
But, from equation (6.4), T0 = w c; therefore:
tan = s c
(6.13)
Substituting into equation (6.9):
tan e = sieh (—) (6.14)
Since a and x are known, a can be worked out and, using equation (6.9), s
can be calculated. Therefore, the stress in the copper conductor due to
the tension T can be found. The value of this stress at the top of the
catenary, at point B, is approximately 13.9 MN/m2 which is approximately
6.4% of the ultimate tensile strength of copper (UTS of copper = 216 MN/m2).
The lateral force onto the conductor, F, works out to be 81.3 N. The
deflection of the tip of the inner point due to this lateral load can be
calculated assuming that the point acts as a cantilever (see Figure 6.3)
The end deflection, 6, can then be represented as:
_ F L3 3 E
(6.15)
where L is the length of the inner point, E is Young's modulus for mild
steel, which is what the inner point is made of, and i is the mean second
moment of area of the point. The fixed edge of the cantilever shown in
Figure 6.3 corresponds to the threaded part of the inner point where it is
screwed into the end of the inner deflector. The end deflection, 6, at
the tip of the point works out to be 0.0011 mm which is negligible, even
compared to the manufacturing tolerances which are ± 0.0254 mm.
-156-
6.3 MISALIGNMENT OF THE CROSSHEAD COMPONENTS
Comparison of the experimental and theoretical thickness distributions,
discussed in the previous chapter, for the thin inner and outer screens
suggested geometric imperfections in the form of distortion or
misalignment of the crosshead components in the vertical and lateral
planes through the cable axis, respectively. In this section, the exact
interpretation of these misalignments and how they are incorporated into
the flow analysis program will be described. Deflector distortion will
be discussed in Section 6.4.
In the case of the inner layer, the misalignment of the points was in
the plane of symmetry and therefore only one half of the flow channel had
to be studied. The experimental thickness distribution graph for the
inner layer (Figure 5.39) suggested that either the inner or outer body
of the inner layer is off-set in the lateral plane. For this case study,
it was assumed that the inner point is off-set in the negative x-direction,
as illustrated by Figures 6.4a and 6.4b, so as to produce the effect of a
thick polymer layer at the melt inlet side, i.e. at = 270°. Angle 0.,
in Figure 6.4a, represents the angular off-set between the axes of the
inner and intermediate points. The. only modification that has to be made
to the flow analysis computer program is to up-date the channel depth in
the tapering region of the tool set. The next paragraph describes in
detail how the new channel depth in the tapering region is worked out.
Figure 6.4a represents a plan view of the cross-section through the
middle of the inner point cut by a lateral plane. Let us suppose that
ABC and A'B'C' represent the outer surface of the inner point before and
after being off-set, and OD and OD' are the axes of the inner point before
and after the misalignment, respectively. If the broken line in Figure
6.4b represents the inner surface of the intermediate point, then the new
- 15 7 -
channel depth distribution, H(z,4), at a given axial distance, z, can be
worked out as a function of (I), using:
H(z,(p) = EG(z) - EF'(z,(1)) (6.16)
where EG is the inner radius of the intermediate point which is a function
of z only. Once the new channel depth distribution which is due to off-
setting of the points is obtained, the flow is analysed to predict the
inner screen thickness distribution round the cable. From Figure 5.39,
it is apparent that the misalignment of the inner point by an angle
a = 0.03° is sufficient to produce the experimental results. Such an
angular off-set will reduce the channel depth right at the tip of the
points, at = 90° (see Figure 6.4b), from 0.051" to 0.049".
In the case of the outer layer, the misalignment of the points was
in the vertical plane, which meant that the flow was no longer symmetrical
about the lateral plane and, therefore, the whole of the flow domain had
to be considered. Figure 6.5 shows the complete flow channel unrolled in
the z,e plane with a few triangular elements and nodal point numbers.
The numbering of the nodes in this case follows a different pattern to
the previous one shown in Figure 3.2. The only difficulty in analysing
the flow for this mesh is the boundary conditions that have to be
prescribed for the heart shape, i.e. for the nodes 8 to 31. It is no
longer valid to assume that the stream function values along the boundaries
of the heart shape are zero, since the off-set is not in the plane of
symmetry. Due to the direction of this misalignment, the flow rate in one
half of the flow channel is greater than the flow rate in the other half.
In order to be able to prescribe the correct stream function values for
the inner boundary, a condition was introduced which stated that the total
- 158 -
pressure drop for the upper half of the mesh (in Figure 6.5) should be
equal to the total pressure drop for the lower half. Assuming that the
stream function values for nodes 184 to 202, inclusive, are ~Uoub = 1, and
for nodes 203 to 221, inclusive, '~oZb -
1, the equal pressure criterion
gives rise to a stream function value along the inner boundary of
'iib = -0.21. This value of l'ib
was obtained by trying various different
non-zero values of ip ib
and working out the difference in pressure drops
(tip) for the upper and lower halves of the mesh. It was noticed that
t'ib varied nearly linearly with (op) and, by suitable linear extrapolation,
the corresponding i'ib
value for (tip) = 0 was worked out.
As in the case of the inner screen, the outer screen channel•depths
in the tapering region of the tool set were up-dated, accordingly. The
experimental thickness distribution graphs for the outer layer (see Figure
5.41) suggested that either the outer point or outer die is off-set. In
this case, it was assumed that the outer die is off-set in the positive
y-direction, as illustrated by Figures 6.6a and 6.6b, so as to produce a
thick polymer layer on top of the cable, i.e. at = 180°. Angle s, in
Figure 6.6a, represents the angular off-set between the axes of the outer .
point and outer die. Figure 6.6a shows a view of the cross-section
through the middle of the outer die cut by a vertical plane where ABC and
A'B'C' represent the inner surface of the outer die before and after being
off-set, respectively. Assuming that OD and OD' are the axes of the outer
die before and after the misalignment and the broken line in Figure 6.6b
represents the outer surface of the outer point, then the new channel
depth distribution, H(z,8), at a given axial distance, z, can be written
as:
Hlz, e) = EF'(z, e) — EK(z) (6.17)
- 15 9 -
where EK is the outer radius of the outer point. • Once the new.channel
depth distribution is obtained, using the suitable boundary conditions,
the flow is analysed to predict the outer screen thickness variation round
the cable. From Figure 5.41, it can be deduced that a misalignment of the
outer die by S = 0.10° is sufficient to produce the experimental results.
Such an angular off-set reduces the channel depth at exit from the die,
i.e. at point C, corresponding to = 0°, from 0.022" to 0.019".
(see. Faye t5cia.)
6.4 THEORETICAL ANALYSIS OF DEFLECTOR DISTORTION
The object of this analysis is to calculate the magnitude of
deformation of the outer surface.of the outer deflector due to a net
hydrostatic pressure of more than 4000 lbf/in acting on it (see Table
5.2c) and comparing the thickness distribution of the outer screen due to
the distorted deflector with the undistorted one. Deformation takes
place only at the deflector channel walls which are treated as thin
cylindrical shells with fixed edges, corresponding to the island
boundary in the middle and the upper boundary, as shown in Figure 6.7.
The theoretical analysis follows very closely that presented by Timoshenko
& Woinowsky-Kreiger (1959) for the general case of deformation of a
cylindrical shell.
In order to be able to establish the differential equations for the
displacements u, v and w which define the deformation of a shell in three
dimensions, the equations of equilibrium of an element cut out from the
cylindrical shell by two adjacent sections perpendicular to the axis of
the cylinder and by two adjacent axial sections (Figure 6.8) are required.
The corresponding element OABC of the middle surface of the shell after
deformation is shown in Figures 6.9a and 6.9b with the resultant forces and
moments, respectively. Before deformation, the axes x, y and z at any
- 159a -
In practice, it was not possible to measure point misalignment to
justify these results since the crosshead die assembly was dismantled soon
after the experiments were carried out. Offsetting of points has only
been inferred by the analysis. However, its practical significance is
extremely important in that if misalignment cannot be avoided then its
effects may be more serious than any deficiencies in deflector design.
- 160 -
point 0 of the middle surface had the directions of the generatrix, the
tangent to the circumference, and the normal to the middle surface of the
shell, respectively. After deformation, which is assumed to be very
small, these directions are slightly changed. The z axis is then taken
to, be the normal to the deformed middle surface, the x axis in the
direction of a tangent to the generatrix, which may have become curved,
and the y axis perpendicular to the x,z plane. The directions of the
resultant forces will also have been slightly changed accordingly, and
these changes must be considered in writing the equations of equilibrium
of the element OABC. The derivation of the equilibrium equations is
described in detail in Appendix A4 which can be expressed in their three-
dimensional form as follows:
a x N a
ax DO
a2w a Qx
ax2 —a x4 ax2 a2v
av a2w a2v aw, ax ax a cp) N (
ax ac āx) -
aN axcp 32v ay a2w
a¢+ a
ax + a Nx
ax2 Qx (Ti ax 34)
(6.18)
a2v
N4)x ax a 4 aa ax) Q~ (1 + a 4
, aaej a
0
a aQx f aQ~ x~ fax ax a~) + a N a22 ax a~ ax
2
w
2 +N !~ +
aaad a 2~ +N4x ~āx ax acp) f qa =
a a~
where Qx and Q are the shearing forces parallel to the z axis, and Nx, N
and x1
are the membrane forces per unit length of axial section and a
section, perpendicular to the axis of the cylindrical shell; q is the
- 161 -
intensity of a normal pressure acting on the shell; M and x are the
bending moments per unit length of axial section and a section
perpendicular to the axis of the cylindrical shell, respectively; Mx is
the twisting moment per unit length of an axial section of the cylindrical
shell; and a and h are the mean radius and thickness-of the cylindrical
shell, respectively.
The three equations of rotational equilibrium with respect to the x,
y and z axes (Figure 6.9b) can be derived as before for the equilibrium
equations, by taking into consideration the small angular displacements of
the sides BC and AB with respect to OA and OC, respectively. These
equations are:
%V aM ~
a2v a2v aw = a
ax — a Mx
axe M# f ax 4 ax ) a
~~
2 OV a x f a x
? ~
v a. ax ax2
M a 2v aw ax act)
ax) a Qx = 0
(6.19)
2 2 2 MX ( ax ax a~J f a Mx
a 2 M~x 11 + aaa
4 + a w2)
ax a a~
av 2 — M (ax f ax a~2 4- a (x~ — N~ x ) = 0
In the derivation of equations (6.18) and (6.19), the change of
curvature of the element OABC was taken into consideration. This
procedure is necessary if the forces N , N~ and N
are not small in
comparison with their "critical" values, at which lateral buckling of the
shell may occur. If these forces are small, their effect on bending is
negligible and, therefore, all terms containing the products of the
resultant forces or resultant moments with the derivatives of the small
displacements u, v and w can be omitted from equations (6.18) and (6.19).
aN a al ▪ x0
1 DAY aN
0 4-a
34) ax ax a
34
(6.21)
-162-
Thus, equations (6.18) and the first two equations of (6.19) reduce to:
a x aN - a
ax a4)
aN
0
DN
f a x4) QA 0
34) ax
a aQxaQ~+Nfga = 0 ax a(1)
9M aM ~x +a
x aQx = 0
a(15 3x
(6.20)
Eliminating the shearing forces Qx and Q, equations (6.20) simplify to:
ax aN▪ (Px
a -
0 ax
a2M(1)x a2m a2
(1) 1 a2M~
f a 4- + ax 30 ax2 ax a0 a 342
q a =
Finally, the three equations in (6.21) can be expressed in terms of the
displacements u, v and w as follows:
a?u 1— v a2u + 1 f v a2v v aw =
ax2 2 2a ax aci) a ax
1+ v a2u 1- v a2v ▪ 1 a2v 1 aw 2 ax a4 + a
2 ax2 a.2 `E''.57-1) a~
(6.22)
h2 l
a3w +
a3w 1 +
h2 (1 v) a
2v a2v 12a
8x2 a~ a2 De12a ax2 a2 42
and:
- 16 3 -
v au f 37,w _ h2 (a D 4 w + 2 a4w + @44,
ax a 4 a 12 ax4 a ax2 42 a3 44
— h2 (2 — v a 3v a 3v ) = a q (1 — v2)
12 a ax2 a~ a3 43 E h
where E is the modulus of elasticity in tension and compression, and v is
Poisson's ratio. The intermediate steps between equations (6.21) and
(6.22) are shown in Appendix A5. Timoshenko & Woinowsky-Krieger (1959)
have stated that other investigators have shown that the last two terms on•
the left hand side of the second equation in (6.22) and the last term on
the left hand side of the third equation are small quantities of .the same
order as those which were disregarded by assuming a linear distribution of
stress through the thickness of the shell and by neglecting the stretching
of the middle surface of the shell. Therefore, it will be logical to omit.
the above-mentioned terms and use the following simplified system of
equations in the analysis of thin cylindrical shells:
32u 1—v32u1fv 32v _ v3w = ax2 2a2 42
2a ax 4 a ax 0
1 f v a2u 1 - 2 ax a. ~ a 2
a2v 1 a2v
ax2 a ae
1 3w a a cp
0 (6.23).
DV w h2 ( a4w + 2 a4w f a4w a ) =
x a a a 12 ax4 a ax2 42 a3 44
a q (1 - v2) Eh
Assuming that there are no variations in the x-direction, i.e. Ox = 0,
equations (6.23) further reduce, and the first equation becomes:
1 — v a2u
2a2 a~2 0
- 16 4 -
Therefore:
u = A 4 + B
where A and B are the constants of integration. For a cylindrical shell
with fixed edges (see Figure 6.10):
= 0 ; therefore B = '0
= 0 ; therefore A = 0
Therefore:
(6.24)
The second equation in (6.23) reduces to:
1 a2v 1 aw _ ā 42 a 0
Therefore: a2v
ac2
av ' w +A
where A is the constant of integration.
av = 0 and therefore A =
Therefore: av 7 = w (6.25)
Finally, the third equation in (6.23) reduces to:
1 av — w _ h2 aYw _ a q (1 — v2)
a 147 a 12a3 ati)4 E h
at
at
= e
at = 0
- 165 -
Substituting from equation (6.25), for lav/a0), the simplified equation
below is obtained:
_ 12 a4 q (1 — v2) (6.26)
E h3
Let: 12 a4 q (1 — v2)
(6.27) E h3
Therefore: a4w = z (6.28)
which is the governing equation for the deformation of the deflector
channel walls in the radial z-direction. Integrating equation (6.28)
gives:
a4w a44
w = e c3 e
Z 24" 6 tB 2 fC~ f D (6.29)
where A, B, C and D are constants of integration. From Figure 6.10, there
are 4 boundary conditions, i.e.
= 0 , 9w _
(6.30)
at c = 0 , DW _ a 0
Thus, the 4 unknowns A, B, C and D can be found and, substituting into
equation (6.29), give:
w = 24 (~4 — 20 0+ .02 .~2) (6.31)
Using this equation, the deflection of each triangular element in the flow
domain was calculated and it was found that, for a net hydrostatic pressure
difference of about 4300 lbf/in2, the mean deflection was 1.2% of the mean
- 166 -
channel depth. With the new channel depth distribution of the distorted
deflector, the flow was re-analysed in order to predict the outer layer
thickness distribution. Figure 6.11 compares the dimensionless thickness
round the circumference of the outer screen for the distorted and
undistorted deflectors. Results show that hardly any difference exists
between the two distributions. Hence, it was concluded that effect of
deflector distortion on melt flow uniformity was negligible.
6.5 CABLE ECCENTRICITY DUE TO GRAVITATIONAL FORCES
The effect of gravitational forces on cable eccentricity was
demonstrated earlier on in Section 5.6.1.1. In this section, the method
of analysis used for examining. the peardropping effect will be described.
Figure 6.12 shows a cross-section of the cable which is a solid conductor
surrounded by a thick layer of solid polymer on which gravity acts. The
object of this analysis is to predict the cross-sectional shape of the
cable as a function of time using the finite element method. In this
analysis, the cable is treated as a plane body of uniform thickness and
its elastic behaviour is studied. The body is assumed to have isotropic,
though not necessarily homogeneous, elastic properties and to deform under
the condition of plane strain. The in-plane area of the body is divided
into triangular three node elements which have distinct elastic properties.
The method used for generating the circular mesh is described in detail in
Appendix A6. For the purpose of this analysis, 217 nodal points and 384
elements were used and the numbering of the nodes and elements are shown
in Figure 6.12. In-plane displacement components, which are assumed to
be small, were used as the nodal point variables. Linear displacement
fields are assumed over each element, implying constant element strains
and stresses.
- 16 7 -
If inertia forces are negligibly small, the equilibrium equations
for the three coordinate directions can be expressed as:
aa aa aa xx + xy + xz
ax ay az
Da Da aa yx + ~ f yz
•
Y -r- = 0 ax ay az
3a aa aa zx +
z y,
zz
ax ay az + Z
-
=
(6.32)
(6.33)
(6.34)
where 7, -land z are the local components of the body forces per unit
volume acting on the continuum in the coordinate directions. In •the
present analysis, 7 = z = 0 and 7 represents the gravitational forces.
Due to the slow fluid flow which is dominated by pressure and viscous
forces, the inertial effects are neglected. Assuming the surface
tractions applied to the body are in the x-y plane, the resulting state of
strain at such a section is two-dimensional, being independent of z and
with w = 0, where w is the displacement in the z-direction. Therefore,
for the two-dimensional analysis, equations (6.32) to (6.34) reduce to:
aa aa xx + xy _ 0
ax ay
Da aa xy# yy + y = 0
ax ay
Da zz = 0
az
(6.35)
(6.36)
(6.37)
and the direct and shear components of strain or strain rate may be
defined as:
- 168 -
0
au exx = ax
= = au a
xy eyx ay 3x
av aw e
=
= zy az ay
aw au = exz ax āz
ezz = āz = 0 (6.38)
(6.39)
(6.40)
(6.41)
av eyy
āy
Provided there are no temperature changes, for a material which is
homogeneous, isotropic and linearly elastic, the constitutive equations
are:
Ē {axx - v (ayy +
azz) J
1 {a y - v (azz xx) J
ezz = Ē [6zz - v (a XX
+ ayy ) J
exy axy _ 2 (1 + v) a
G E
xy
(6.42).
(6.43)
(6.44)
(6.45)
where E is Young's modulus, G is the shear modulus, and v is Poisson's
ratio. For plane strain cases, equation (6.44) reduces to:
azz = v (xx + ayy ) (6.46)
- 169 -
6.5.1 Finite Element Formulation of a Two-Dimensional Biharmonic
Equation
In this sub-section, the formulation of a finite element
analysis for two-dimensional problems of the plane strain type will be
described. Since displacements are treated as the unknowns, the method
is unsuitable for plane strain problems involving incompressible materials.
Figure 6.13 shows the nodal point displacements for a typical element.
The displacements at points within the element are given by:
u (x,y) = C1 f C2 x + C3 y (6.47)
v(x,y) = C4 + Cs x + C6 y (6.48)
where C1 to C6 are constant for the particular element, and x and y are
local coordinates with the origin at node i. Now, C1 = u2, C4 = vi and
the remaining parameters can be found with the aid of equations (4.27) and
(4.29) as:
IC2 C5 1 b . b~ bk
C3 C6 2~m
a. a. ak ,
u . V Z . 2
U. v
•uk vk,
(6.49)
where the definitions of element dimensions and areas are exactly the same
as described in Section 4.2.
The analysis of plane strain involves only the strain
components exx, eyy and exy. Using the strain definitions given in
equations (6.38) and (6.39), these may be re-expressed as:
a.
b.
b. o b.
2 0
0 a2 0
.az b2 aj
-170-
Du ax
xx
yy
C2
1 2 m
(6.50)
C3 +C
where B is a dimension matrix:
and d2 is a subvector of displacements:
a. = (6.52)
Since the analysis is formulated with displacements as the
unknowns, which are continuous across the inter-element boundaries,
compatibility of strains is automatically satisfied within each element
(see Fenner (1975)) .
The strains, and therefore the corresponding stresses ate, ay
,
and axy, are constant over each element. Figure 6.14a shows these
stresses acting on a rectangular prism of unit thickness. Their effects
can be expressed in terms of equivalent forces acting at the mid-points of
the sides of the element, as shown in Figure 6.14b. These forces at the
mid-points can further be replaced by an equivalent set acting at the nodes,
as shown in Figure 6.14c. In order to maintain the same resultant force
- 171
and moment about any point on a side of the element, the force at the mid-
point must be replaced by two equal forces of half the magnitude at the
relevant nodes. For example:
U. = - 2 (aXx
axy ak) -2 (a b. + x
r~ a~)
and: V. _ - 2 (a ak b ) -32 (a a.+a b.)
~y k k yU d xy i
Using the definition of element dimensions; ak + a~ - a2,
bk + b. _ - b.. Therefore, the above two expressions become:
U. = 2 (agi b.+ a a.)
V. = 2 (ayy a .+ a
b.)
(6.53)
(6.54)
Similar expressions can be obtained for the other force components acting
on the element at its nodes to give:
R. axx BT aYU
Rk, .arm,
(6.55)
where BT is the transpose of the dimension matrix defined in equation (6.51), and the force terms, such as R., are subvectors:
U. R. 2 (6.56)
V. 2
1 eYY = —E xy ,
(1 - 2)
—v (1 +v)
0
— v (1 + v)
(1 — v2)
0
crYY
2 (1 + v)) .6xY:
(6.58) 0
- 172 -
In the present analysis, forces acting on the element include
those due to gravity. Using the definition given in Section 6.5 for
the total body force acting on the element in the y-direction is Yam.
This force acts at the centroid of the element and it is equivalent to
3 Y om acting at each of the three nodes of the element. Hence, the body
force at a node i, due to element m, can be represented as:
G(m) = m 3 11 6.57)
The relationships between stresses and strains can be obtained
with the aid of the constitutive equations (6.42) to (6.45). Assuming
the plane strain condition and using equation (6.46), the constitutive
equations become:
Equation (6.58) may be inverted to give stresses in terms of strains as
follows:
D e (6.59)
where D is the elastic property matrix, defined as:
1 v*
~* 1
0 0 . (1 — v *)
D = E*
1 —
(6.60)
- 173 -
and the modified material properties are:
E v 1 — v
- v2 6.61)
Substituting equation (6.59) into equation (6.55) and using
equation (6.50) for the definition of strains, the forces acting on the
element at the nodes due to the internal stresses can be expressed in
terms of the corresponding displacements as:
R . 2
R.
Rk
4 BTD.B m
S. 2
(Si (6.62)
•k
This result may be expressed as:.
Rm = kn Sm (6.63)
where sm is the element displacement vector, and km is the element stiffness
matrix given by:
BT
11
k21
-k31
k12
k22
k32
k13,
k23
k33- m
D B 4~ m
(6.64)
where each of the coefficients is a 2 x 2 submatrix which can be expressed
as:
krs
k k xx xy
kyx kyy rs
6.65)
- 174 -
For example, k
in this submatrix can be interpreted as the force which
must be applied in the x-direction to the element at the node corresponding
to the rth row of R (that is, i, j or k, according to whether r is 1, 2
or 3) to cause a unit displacement in the y-direction at the node
corresponding to the sth row of m. The element stiffness matrix in
equation (6.64) in its full form is, therefore, a 6 x 6 matrix.
The overall stiffness matrix is assembled using the direct
equilibrium method. The actual internal stresses and body forces acting
on individual elements have been replaced by the equivalent forces acting
at the nodes of the mesh. The conditions required for equilibrium can be
expressed as:
externally applied forces on the elements L (forces at the nodes) L ( at these nodes )
Therefore, for equilibrium of forces acting at node i:
F G(m) = R(m) (6.66)
where the subvector F. represents the forces applied externally at the
node, which is not applicable in the present analysis since F. = 0. The
summations indicated in the equation above are performed for elements which
share the node i. The set of equations for all the nodes can be
expressed as:
K S = G (6.67)
where K, s and G are the overall stiffness matrix, displacement vectors,
and body force vectors, respectively. Equation (6.67) can also be derived
using a variational formulation of the finite element analysis which
-175-
provides a more general approach than the above direct equilibrium method
(see Fenner (1975)).
In order to be able to solve the linear algebraic equations
represented by equation (6.67), the iterative Gauss-Seidel successive over-
relaxation approach is used which is similar to the method used for
solving equation (4.36) for harmonic problems. The sufficient condition
for convergence of the Gauss-Seidel method is diagonal dominance of the
overall stiffness matrix. For harmonic problems, this condition is
achieved if there are no obtuse-angled elements so that every element
stiffness matrix is diagonally dominant (see Fenner (1975)). For
biharmonic problems, however, the 6 x 6 element stiffness matrix, km,.
defined by equation (6.64) is never diagonally dominant. Without
attempting to present a detailed analysis of the stiffness matrix, Fenner
(1975) has stated that the best conditions for convergence of the Gauss-
Seidel method applied to biharmonic problems are obtained when the elements
are as nearly equilateral as possible. Provided long thin elements and.
angles greater than 90° are avoided, convergence is generally satisfactory.
The rate of convergence was increased with the use of an over-relaxation
factor. For the present analysis, with 217 nodal points, an over-
relaxation factor of 1.75 was used, and the tolerance limit was chosen to
be 10'6.
Before the overall linear algebraic equations (6.67) can be
solved, boundary conditions have to be applied which could be in the form
of either externally applied forces or restraints on the nodal point
displacements. For example, the points on a particular boundary may be
given prescribed displacements, or may be allowed to move freely in a
prescribed direction. In the present analysis, node number 44 which
corresponds to the top of the copper conductor is fixed so that both
- 176 -
deflections in the x and y-directions are zero and node numbers 56, 182
and 206 are allowed to move freely in the y-direction only.
After prescribing the boundary conditions, equation (6.67) can
be solved to give the u and v deflections corresponding to each of the
nodal points. Hence, the drooped shape of the cable cross-section can be
worked out as a function of time. In order to simplify the analysis,
since the effect of gravity is relatively more significant in the case of
the thick insulation layer, the section was assumed to consist of only two
materials, i.e. the copper conductor and the surrounding crosslinking low
density polyethylene. Body forces were assumed to act only on the polymer
such that:
P1 = 0 and p2 =
0.0293 lb/in3
where the subscripts 1 and 2 refer to the two materials, copper and polymer,
respectively, and p is material density. Given a melt viscosity, u2,
which may be assumed constant at the very low rates of deformation
involved, and assuming that with small strains the deformation varies
linearly with time, an effective modulus for deformation after a time, t,
can be defined as: Pr
G2 = t
Therefore, assuming a characteristic deformation time of 50 seconds for
drooping, the effective shear modulus for the polymer works out to be:
G2 = — = 3•50 3 = 0.06546 lbf/int 50
Assuming the polymer to be incompressible, i.e. v = , its Young's modulus
can be calculated as follows:
E2 = 2 G2 (1 + v2) = 3 G2 = 0.196 l bf/i n2
In the present finite element method, however, v, cannot be made exactly
equal to , since this will mean that the common factor, (E*/1-v*2),
- 17 7 -
involved in the coefficients of the elastic property matrix in equation
(6.61) will become infinite. For the purpose of the present analysis,
was chosen to be:
= 0.49
The elastic properties used in the analysis for the very much stiffer
copper conductor were:
= 1010 lbf/int
vi 0.3
Figure 6.15 shows the cable cross-section before and after a
characteristic time of 50 seconds of drooping and the new position that
each nodal point has moved to. Since this is a linear elastic problem,
the deflections u and v vary linearly with the drooping time. The
experimental results for the insulation layer (Figure 5.40) suggest that
the maximum changes in thickness are of similar order of magnitude with
the predicted thickness distribution due to peardropping after a
characteristic time of 50 seconds at the melt temperature associated with
the highest cable speed.
6.6 MELT ELASTICITY EFFECTS IN POLYMER FLOW
The most important melt elasticity effects that may be significant to
the present analysis are die swell and melt fracture. In the next two
sub-sections, these effects will be discussed briefly.
6.6.1 Die Swell
Under most conditions, if a viscoelastic fluid is extruded from
-178-
a die into air without subsequent drawing, the cross-sectional area of the
extrudate will exceed that of the die exit. This phenomenon is usually
called die swell. In the most common case, that of the circular die,
diameter ratios of extrudate to die in the range of 2 or 3 are often
observed (Middleman (1977)). In general, it is agreed that die swell is
an elastic stress relaxation phenomenon. However, no single theory of
die swell seems to be generally accepted, each theory being based on some
assumption regarding the effect of stress relaxation on the dynamics of
the extruded jet. Middleman summarised some of the difficulties
associated with developing a rational basis for understanding the die
swell phenomenon and stated that no one is yet near a rational and
comprehensive understanding of this important phenomenon. Although it is
assumed that die swell normally only occurs with elastic materials,
Batchelor et al (1969) have encountered and explained situations where die
swell occurs in the absence of conventional elastic effects.
Bagley et al (1963) have shown that die swell decreases with
increase in the shear strain in the capillary. This is one of the most
important factors determining the die swell of a given polymer which
indicates that continuing strain causes disentanglement of the molecular
chains. They have also showed that, at a fixed shear rate, die swell
decreases with an increase in the length of the die. Die swell increases
with shear up to a limit which is near to the critical shear rate, beyond
which it starts decreasing. According to Metzger et al (1968), the
greater the residence time in the capillary, the less is the die swell.
Experimental studies have been made on this subject by various people and
Brydson (1970) presents some of their important conclusions.
Die swell is not directly relevant to cable covering since the
polymer is extruded onto a conductor which is drawn down in the catenary.
-179-
Therefore, even if there is any swelling immediately after the cable leaves
the die, it will disappear on emerging from the catenary.
6.6.2 Melt Fracture
When operating an extruder at a high output rate, the
extrudate takes on a rough, irregular appearance which is attributed to.
the physical breakdown of the melt. This phenomenon, known as melt
fracture or, more generally, extrudate distortion, occurs when the shear
stress of the melt exceeds its shear strength, such as for example in an
extrusion die where, due to a substantial reduction in channel width, a
sudden increase occurs in the shear rate:
A recent, comprehensive and critical review of the melt
fracture phenomenon by Petrie & Denn (1976) shows that melt fracture is
still poorly understood. Results are often contradictory and
generalisations of a quantitative and predictive nature are rare. Petrie
& Denn suggest that two different phenomena occur, both of which are of
the nature of elastic instabilities. In linear polymers, the instability.
probably occurs in the shear flow of the die. In branched polymers, the
converging entry flow is probably unstable and leads to unsteady flow and
melt fracture. A good understanding of the nature of these instabilities
does, not yet exist.
- 180 -
o of COnci.u.ctor
Figure 6.1: Sketch showing head misalignment in relation to catenary
- 181 -
Figure 6.2a: Cable fixed at two ends with a distributed load
Figure 6.2b: Free body diagram of a portion of the cable
0(o .o
Figure 6.2c: Catenary in relation to coordinate axes
182 -
L
Figure 6.3: Cantilever with a point load
Figure 6.4a: Misalignment of axes of inner and intermediate points
'Figure 6.4b: Inner screen .channel depth distribution at exit from the points before and after the misalignment
► 2O 2. z
243
Figure 6.5: Complete flow channel plotted on the (z,e) plane, showing. some of the triangular finite elements and nodal points
0 p- vie
- 184 -
Figure 6.6a: Misalignment of axes of outer point and outer die
Figure 6.6b: Outer screen channel depth distribution at exit from the points before and after the misalignment
- 185 -
Figure 6.7: Boundary conditions imposed for analysing deflector distortion
Figure 6.8: Element in cylindrical shell formed by two adjacent sections perpendicular to the axis of the cylinder and by two adjacent axial sections
- 186 -
Figure 6.9a: Typical element from the middle surface of the shell after deformation with the resultant forces
- 187 -
Figure 6.9b: Typical element from the middle surface of the shell after deformation with the resultant moments
Figure 6.10: Cylindrical shell with fixed edges
188 -
1.00 x ~ x RA
DIA
L ī
HICK
NE
55
0:15
Z 0.50
0 z U x r a
0.25
0
X U.N-DI5TOR.Tet DEFLgCTOR. "
1)15TOR.TFD ~1✓FLJCTOR
I I I I 1 1
30 Go ao 120 150 I6O
1)051110N ON TH6 CIrtct MFzitemc6 0
Figure 6.11: Circumferential thickness distributions for the distorted and undistorted deflectors
- 189 -
Figure 6.12: Circular finite element mesh representing a cross-section of conductor and polymer
- 190 -
uk
iv4
Figure 6.13: Displacement of a typical element
Figure 6.14a: Stresses acting on an enclosing prism
Figure 6.14b: Forces acting at the mid-points of the element sides
Figure 6.14c: Forces acting at the nodes
- 191 -
_ _ __ PEFoRE Z110O7I106- Act' EFL, 2 ooPi NG
Figure 6.15: Cable cross-section before and after a characteristic time of 50 seconds of drooping
192 -
CHAPTER 7
TEMPERATURE DEVELOPMENT IN PRESSURE FLOWS
7.1 INTRODUCTION
The analysis considered so far for narrow channel flow assumes that
the flow is isothermal. This means that any temperature variations within
the flow, due to either the dissipation of mechanical energy or to heat
conduction at the flow boundaries have negligible effects on the viscous
properties of the melt. In order to confirm the validity of this
assumption, a somewhat simplified form of the actual melt flow in the
crosshead will be considered in this chapter, to compute the temperature
development in the flow channels.
In extrusion, a molten polymer is forced through a die by a pressure
gradient. The energy required to maintain this flow is equal to the
pressure drop in the die flow channels. Most of this energy is being
dissipated in areas of high shear stress which are close to the walls.
Therefore, in designing an extruder die, it is important to know how much
the temperature field in the melt is influenced by the dissipation of flow
energy and by heat conduction towards the walls, and how the temperature
field can be changed through geometry and through the thermal boundary
conditions. The developing temperature and velocity fields are coupled
and, once the temperature field is known, it is possible to calculate the
temperature effects on the velocity field, on the shear stresses and on the
pressure drop.
Brinkman (1951) is one of the - first authors who studied the developing
temperature field in a capillary, analytically, for a Newtonian flow.
Bird (1955) extended Brinkman's method of analysis to describe the heat
effects for the flow of non-Newtonian fluids which obey.a power-law
- 193
relation between the coefficient of viscosity and the shear stress. He
made calculations for developing temperature profiles, assuming that.
(i) the capillary walls are maintained at the temperature of the feed, and
(ii) the capillary walls are thermally insulated. Martin (1967) has
given exact solutions for flows of power-law fluids with heat generation
and temperature dependent viscosity for pressure flow through a circular
tube, shear flow between rotating concentric cylinders and shear flow
between parallel plates. Yates (1968) has treated developing non-
isothermal flow in extruder channels. Forsyth & Murphy (1969) developed
a generalised method for calculating temperature profiles of flowing power-
law fluids during heating, cooling and isothermal flow. They solved the
transport equations of continuity, momentum and energy for a compressible
fluid with temperature dependent power-law viscosity and temperature
dependent density and thermal properties, and showed that their predictions
lie within 6% of the experimental results. Winter (1971) calculated
developing temperature and velocity fields for a Newtonian fluid in plane
Couette flow with a temperature dependent viscosity assuming a locally and
timewise constant wall temperature. Experimental studies have been made
on the effects of temperature on the pressure drop and the average
temperature increase by Gerard & Philippoff (1965) and Cox & Macosco (1974).
The radial temperature distribution has also been studied experimentally
by Forsyth &'Murphy (1969) and Griskey et al (1973). The experimental
results seem to support the corresponding analytical studies quite well.
The fluid flow and heat transfer problems are usually solved by using
the traditional finite difference procedures. For example, using
temperature and strain rate dependent viscosity relations, Fenner (1970)
and Martin (1969) illustrated the thermal effects on the flow
characteristics of extruder channels. An investigation of non-isothermal
- 194
fully developed extruder channel flow was also carried out by Dyer. (1969)
using a finite difference formulation in cylindrical coordinates.
Vlachopoulos & Keung (1972) studied heat transfer to a power-law fluid
flowing between parallel plates and solved the equations numerically using
the finite difference method. Winter (1975) analysed numerically the
development of the temperature fields in extruder dies with circular,
annular and slit cross-section, using the dimensionless parameters Na
(Nahme number) and. Gz (Graetz number). He used a power-law fluid with a
temperature dependent viscosity described by an exponential function.
There seems to be very little published work in the field of
convective heat transfer using the finite element method. Some Of the
few workers in this field include Tay & Davis (1971) who demonstrated the
application of the FE technique to a convection heat transfer problem,
which consisted of determining the temperature distribution for a fluid of
constant physical properties flowing between two infinite parallel planes.
There was no internal heat generation and the velocity profile was fully
developed. Palit (1972) also applied the finite element approach to
extruder channel flow with conduction, convection and viscous dissipation
effects. Nebrensky et al (1973) applied a variational analysis in helical
coordinates to the problem of a developed, steady-state flow in a screw
extruder. They derived a functional in a general form, including inertia,
convective and conductive heat transfer, and pressure and viscous energy
dissipation effects.
7.2 BASIC EQUATIONS DEFINING THE PROBLEM
The object of this analysis is to determine temperature rises in
pressure flows of the type encountered in cable covering crossheads. As
for extruder melt flow analysis, a useful assumption can be adapted which
- 195 -
was first employed by Yates (1968). This assumption treats velocity
profiles as being locally fully developed, i.e. they vary very slowly by
virtue of developing temperature profiles. Figure 7.1 shows a cross-
section through a flow channel formed by two flat stationary surfaces,
where the typical channel depth is h, initial value h1 and the channel
length is L. Melt is admitted to the channel at x = 0 at, say, a constant
temperature T = T1 (where, in general, T1 = TT(y)) and the flow boundaries
are maintained at temperature T. = Tb(x). Let x and y be Cartesian.
coordinates as.shown in Figure 7.1 and let y be always measured from the
mean channel depth. Although h may be allowed to vary in the axial
direction, it is required that:
(8x l « 1
If the volumetric flow rate is maintained at Q per unit width normal
to the plane shown in Figure 7.1, the main interest lies in the magnitudes
of the temperature changes in the melt as it moves a distance L along the
channel. Let u be the melt velocity component in the x direction
(u = u(x)). Assuming the melt density to be constant and the flow to be
steady, the volumetric flow rate per unit width normal to the section shown
is the same at every section along the channel and is given by:
-01/2 h/2 = f u dy 2 f u dy
-h/2 0
(7.2)
the last result only being valid when the boundary temperature profile,
Th(x), is the same for both boundaries.
Using equations (3.8), (3.9) and (3.10), the momentum conservation
equations in the x direction reduces to:
- 196 -
c ax
(7.3)
where p is the pressure, x is the. pressure gradient in the x direction,
and Txy is the viscous stress.
From equation (3.11), retaining x-convection, y-conduction and viscous
dissipation, the energy conservation equation becomes:
DT 2 p Cp u ax = k a
2+ Txy ay
ay 7.4
where p, Cp and k are melt density, specific heat and thermal conductivity,
respectively, and T is the local temperature where T = T(x,y)
The constitutive equation gives:
du = u dry
(7.5)
where u is the melt viscosity which is assumed to be a power-law function
of shear rate and an exponential function of the local temperature, T(x,y),
such that:
Idu 1 2W
)n-1 (- b (T - T0)) (7.6)
where po is the value-of the reference viscosity at reference temperature,
To, yo is the reference shear rate, n is the power-law index, and b is the
temperature coefficient of viscosity.
7.3 DIMENSIONLESS PARAMETERS DESCRIBING MELT FLOW AND HEAT TRANSFER
CHARACTERISTICS
Dimensional analysis can be used to generalise the solutions to the
- 197 -
equations by combining the physical variables to form dimensionless
parameters. At least for the purposes of examining relative magnitudes,
it is convenient to define some dimensionless variables. Starting with
the coordinate system, X and Y can be defined as:
= h1
Y =
In non-dimensionalising x, sometimes L is a more appropriate dimension to
7.7)
use than h1, such that:
A 7.8)
S is the relative channel depth:
S =
U is the dimensionless velocity:
(7.9)
u
V (7.10).
where V i.s the mean velocity, V = Q/h1, and T* is a dimensionless
temperature defined as:
T* = b (T - T1) (7.11)
where T1 is the (assumed) constant melt temperature at x = 0. In order
to make the shear stress, T, dimensionless, a mean shear stress, T > at xY
mean shear rate, y = 17/h1, and temperature T1 has to be defined as:
n-1 V h exp (- b (T1
—T)) 7.12)
-.198
Therefore, T can be made dimensionless by dividing it by the mean shear xy
stress as follows:
_ du 111
dy v
n-1 h1 du (V ay) exp (- b (T - T1 ))
I n-1 n If exp (- T*) (7.13) S
Some important dimensionless ratios which are relevant to heat
generation and heat transfer in polymer melt flow will be defined below:
The Griffith number, G, which is equivalent to the Nahme number, Na,
determines whether heat generation will lead to temperature differences
within the melt sufficient to affect the velocity distribution locally.
This varies in practice from 0 to 200 (see Pearson (1972)), and temperature
independent solutions are only reasonable for G « 1:
(7.14)
From the definition of G, the characteristic temperature rise due to
viscous dissipation is T Y h12/k. Thus, a Brinkman number, Br, can be
introduced which expresses the ratio between this rise and the temperature
difference between the side and inlet boundaries:
T Y h12 Br(x) (7.15)
k (Tb (x) - T1)
Hence, Br determines whether imposed channel wall temperatures or heat
generation are dominant in effective temperature changes in the melt.
This number can vary between 0 to o.
The Peclet number, Pe, can be defined as:
-199-
• P pC
V h1
k 7.16)
Peclet numbers are usually large, which implies that heat conduction along
the streamlines is negligible everywhere except at stagnation points.
The Graetz number, Gz, is an extension of the Peclet number, and it
is defined as: h1 P
p V h12
Gz = L k L
7.17)
The Graetz number expresses the relative importance of thermal conduction
through the depth of the channel and convection in the direction of flow.
h1 is usually a small dimension relative to L and, therefore, although
Peclet numbers are almost always large, Graetz numbers can vary from 10-1
to 104 (see Pearson (1972)).
In order to represent the energy conservation equation in terms of the
dimensionless quantities defined above, equation (7.4) can be re-written as:
P p h1 DT* _ 1 a2T*
b T Y
h12 1 124-1'
k (U DX )
82 31724-
k Sn+1
IdU
( (- T)
* 2 n+1 i.e. Pe U
DX S2 aY2 Sn 1 +dY~
exp (- T*) (7.18)
n4-1 or: Gz U
3A 1
a 2T * + n+
1 ~ ~ exp (- TA) ( 7.19)• S2 aY2 S
To be able to solve equation (7.18), both the distribution of velocity, U,
and the boundary conditions are required which will be discussed in the
next two sections.
-200-
7.4 THERMAL BOUNDARY CONDITIONS
The boundary conditions for the problem considered here are expressed
in dimensionless form. Initial conditions are:
= A = 0 T* = 0 (7.20)
and on.Y = ± Z:
= b (Tb (x) - T1) (7.21)
If symmetry of flow about y = 0 is to be invoked, in order to solve for
only half of the flow domain one of the latter conditions can be replaced
by
Y = 0 (7.22)
Using the definitions of Griffith number and Brinkman number. (equations
(7.14) and (7.15)), and the initial condition given by equation (7.21), a
thermal boundary condition can be prescribed as:
Br on Y = ± (7.23)
Finally, the range of integration of equation (7.18) or equation (7.19) is
from X = A = 0 to X = L/171, A = 1.
7.5 VELOCITY ANALYSIS
In this section, an attempt will be made to derive the function
defining the velocity distribution, U, of the melt. From equation (7.3):
Txy x y (7.24)
- 201 -
since, due to symmetry, at y=.0, T = 0. Equation (7.24) can be made sy
dimensionless by dividing through by the mean shear stress, i,.as follows:
Txy _ x y
T T
7.25)
Equating equation (7.25) with equation (7.13), we get:
P y exp (_ T*) =
x (7.26)
At this stage, it is convenient to define another dimensionless group,
namely the dimensionless pressure gradient, IT , as:
x h1 (7.27)
Therefore, equation (7.26) becomes:
n-1 1
Sn dU dY
n-1
dU dY exp (- T*) = Sn n hl
(7.28)
Due to symmetry, considering only one half of the flow region defined as
0 <. y + h/2, i.e. 0 , y . + i; for negative velocity gradient, i.e.
dU < TY 0 (also
equation (7.28) becomes:
( - 3)n exp (- T*) = - Sn np Y h 1
=n41
Y_ 7 )
- 202
Tp 51+1/n Y1/n exp (T*/n)
Therefore, integrating both sides gives:
(p/n U( — ~rp)1/n S1+1/n exp (T*/n) di '+ A
where A is the constant of integration. Assuming that there is no slip:
at Y = , U = 0
therefore:
U (- IT 51+1/n f2 Y1/n exp (T*/n) dY
Y (7.29)
Provided that, at a particular section, the temperature profile, T*(Y), and
the dimensionless pressure gradient, Gr, are known, the velocity
distribution within the flow can be worked out.
In order to calculate Gr, from equation (7.2):
Q = h/2 2
f udy = 2hV f Udi 0
(7.30)
and from the definition of V (equation (7.10)):
Q = V h1 (7.31)
Dividing equation (7.30) by equation (7.31) gives:
203 -
(7.32)
Substituting for the velocity, from equation (7.29):
= 2 (- wp 1/n S24-1/n
!4- { f
1,1 exp (T */n) dY} dY ( 7.33)
Hence, at a particular section, provided T'E(Y) is known, the integrations
can be performed numerically yielding T. The pressure difference, P,
over the region of interest can then be evaluated as:
j (- Px) dx
which can be non-dimensionalised by dividing by T, as follows:
x=L - P P* _ j _ LdA
x=0 T
L 1
j (- ) dA h1 0
(7.34)
(7.35)
7.6 TEMPERATURE ANALYSIS
The object of the present analysis is to solve equation (7.19) to
give the temperature distribution within the flow. Rearranging equation
(7.19) gives:
dUln+1 exp (- T*) (7.36)
a2 T*
aA aT
aY2
This arrangement serves to make the conduction term independent of,S and
- 204 -
the dissipation term much less dependent on S. Equation (7.36) can only
be solved numerically and, since it is a parabolic equation and the flow
geometry is simple, the finite difference method is the most convenient
to use.
Figure 7.2 shows a finite difference grid in the Y,A plane. By using
Y as a coordinate, the mesh becomes uniform since AY is everywhere a
constant. In the present analysis, AY is chosen to be equal to 0.025 in
order to give reasonable accuracy. This means that the number of nodal
points in the Y direction is 21. The choice of Y as one of the
coordinates makes the real flow be approximated by flow along lines of
constant Y. Although AA need not be constant, in the present work it is
chosen to be 0.02 everywhere in the mesh. This means that the number of
nodal points in the A direction is 51 since 0 . A 4 1.
At a point i,j in Figure 7.2, the dissipation term, Dij, can be
evaluated as:
n+1 exp ( Z~ *)
ij
Dij =5.
-772 G I dY
dU (7.37)
Similarly, the conduction term is:
XZa 8 2T*
ay2
(T1 -2T1 +T Z~ai1 21e 1.-Ja-
(AY)2 (7.38)
and the convection term is.
)
• (U 8A ) • U. ~
. . (7.39)
Therefore, using equation (7.36), for a node i,j the formulation can be
simplified to:
Sm2 GZ (U 8A
)i = z~ + D. j 7.40)
- 205 -
where:
2 (Si S •• ) (7.41)
Equation (7.40) can be solved explicitly for T24.1,j, which only appears in
the convection term. In practice, however, the explicit method is
unstable for all but extremely small AA (see Smith (1975)). Therefore,
in practice, it is more economical to introduce a more stable implicit
method, such as Crank-Nicolson, as shown below.
S 2 Gz (U @T*). = z (C .. t C . ) + D. .
m aA j 24 7,44,a 7 (7.42)
In general, a more arbitrary combination of the conduction terms might be
considered, such as:
a Cid + (1 — a) C.
where: 0 4 a
For the purpose of the present analysis, a has been chosen to be equal to 2•
The method is now implicit in the sense that equation (7.42) involves
three temperatures at (i t 1) in i+1, , and the individual temperatures
at this section must be obtained by solving the set of linear equations
(one for each value of j). Substituting equations (7.37), (7.38) and
(7.39) into equation (7.42), we obtain:
S 2 Gz Ui • m
(Tit1 - Ti ) =
1 (1617)2(TZ +1 — 2 Ti t Ti 1)
2 (p)2 ~~ Jj ,j-
2 (py) 2 2+1, j-1 i (7.43)
-206-
Equation (7.43) can be represented in the following form:
aa 7-1 + Sa a Ya a +1 ōa (7.44)
where: i 51 , 1< j n , n = 21
a• = Y. a
1
2 (AY) 2
Sm2 Gz 1
g. = U. + 2j (AY) 2
S 2 Gz Sa = 2 C. + Dia
U-a
Equation (7.44) is a tridiagonal system of linear equations which can be
easily solved (see Fenner (1974)), provided some boundary conditions are
prescribed.
At the boundaries, Y = 0 (j = 1) and Y = ? (j = n), no equations can
be obtained since (j — 1) and (j + 1), respectively, correspond to
imaginary points. Therefore, two boundary conditions are required to
solve the system of equations in (7.44). Using an approximation for
aT*/aY which is an order lower than the FD approximation for a2T'*/aY2,
from equation (7.22) the first boundary condition may be imposed as:
X1 = X2 or Ti+1,1 Ti+1,2
A more accurate approximation can be used for T*/aY, although this
involves X3 as well which must be eliminated to make the equations
(7.45)
-207-
tridiagonal (see equation (7.52)). From equation (7.23), the second
boundary condition may be imposed, at Y = I, as:
n G/Br or T14-1 ,n
- G/Br (7.46)
where, in general, Br is a function of. A.
Hence, the problem can now be solved to give the temperature
distribution within the flow.
7.7 SOLUTION PROCEDURE USED
In this section, the outline of the procedure applied in order to
solve the system of equations in (7.44) will be described:
i) The initial temperature profile is set as T = 0 at A = 0 and
the value of the dimensionless pressure gradient, Tr, is
calculated using equation (7.33). Using this value of Trp, the
velocity profile U(Y) is found from equation (7.29).. In both
equations, the integrations are worked out numerically using the
simple trapezoidal rule.
ii) The temperature profile at A = AA is found by solving the set of
equations represented by (7.44), (7.45) and (7.46).
iii) The pressure gradient and velocity profile are found as before
for A= AA.
iv) The temperature profile is found for A = 2M and the same
procedure continued for A = 3AA, 4AA, etc., up to A = 1.
v) The pressure drop, p*, is computed using equation (7.35).
vi) At the end of the analysis, an energy balance is applied as a
check on programming and accuracy (see Section 7.8).
- 208 -
7.7.1 Stability
The solution scheme will become unstable if AA is too large.
The parameter to consider in connection with stability is:
_ M S 2 Gz (AY) 2 m
(7.47)
Experience (Yates (1968)) suggests that, for the extruder problem, the
value of a should be less than 4. However, the same sort of result can
be expected for the present problem. x should certainly be_evaluated for
the minimum value of S.
7.8 ENERGY BALANCE
The application of a simple energy balance to melt flow in the die
flow channels makes it possible to relate mechanical power consumption
with heat output due to convection and conduction. This relationship can
be represented as follows:
Mechanical power input, mech = P Q
(7.48)
Heat output by convection, E~onv
C p Q (ōut in) (7.49)
.where in and out refer to A = 0 and A = 1, respectively, and T is the bulk
mean temperature defined as:
T uTdy (7.50)
Heat output by conduction, Gond = 2 k f (— aTJ __ dx (7.51) ay y=
- 209 -
A good approximation for the derivative at the boundary can be provided
using Taylor's series as follows:
(ayA 2 AY (3 TA — 4 TB + Tc
7.52)
where A, B and C are three consecutive nodal points in a column, as shown
in Figure 7.2, and A lies on the boundary.
Now, from the energy balance, it is expected that:
meth = Econv f Econd
(7:53)
Therefore, any possible error in the temperature analysis may be
represented as a ratio with respect to the total mechanical power output.
as:
error = ( meth Econv - Econd)
7.54)
meth
Non-dimensionalising Econv
using meth' we obtain:
E* Econv _
C P (Tout - Tiny conn meth P
From equation (7.11):
+T*/b
(7.55)
7.56)
Equating equation (7.56) and the non-dimensionalised form of equation
(7.50):
V U (T1 + b ) h dY = T1 +T*/b = (7.57)
From equation (7.31):
210 -
Therefore, using the definition given in equation (7.9), equation (7.57)
reduces to:
= 2S fU(T1 + 1 dY
= 2 S T1 f 2 U dY +- f 2 U T* dY 0 0
7.58)
From equation (7.32), the first term on the right hand side of equation
(7.58) becomes:
2 S.T1 f U dY =
Hence, equation (7.58) can be re-written as:
z T = U
(7.59)
Now, at x = 0, from equation (7.11) :
Therefore, using equation (7.59):
T (7.60)
and: Tout
T1 2 Sout
f (U T*lout di
b (7.61)
Subtracting equation (7.60) from equation (7.61) gives:.
T* out
b
P Q
Econd 2k
Econd meth
ay y= Zh (7.64)
_ 2 k L Econd b h1 P Q o
(h) PU G f 1 0
- 211 -
_ _ 2
~ ~
Tout Tin b Sont 1 (U T )out dY
Using equation (7.32), the above equation reduces to:
- Tout Tin
Substituting this into equation (7.55):
E = C P out Pe Yr*
cony b p* T G p* (7.63)
Similarly, E~ can be non-dimensionalised using mach'
as follci'is:
The temperature gradient can also be put into a dimensionless form as
below:
a' aT a (T T ) DyhaY = hay b 1
1 apt (7.65)
b h aY
Substituting equation (7.65) into equation (7.64) gives:
ā ) Y= 2 dA
3Y)Y=3' d4 (7.66)
- 212 -
Therefore, from equation (7.53) it is expected that:
E* E*E* cond
or: _
Ç)14 1 [ GP*
Pe Tout + o f ~- dA
(7.67)
Thus, a new error term can be defined in terms of the dimensionless
quantities as a percentage as follows:
ERROR =
65,* Econd) -
1 x 100%
IE* I .+} IEcond~ 1
(7.68).
7.9 RESULTS AND DISCUSSION
Taking typical channel dimensions, operating conditions and material
properties for melt flow in cable covering crossheads, the values of the
dimensionless parameters are found to be of the following orders of
magnitude:
0.1
Br >
= 20
Pe = = 50 Gz = 103
While the magnitude of Gz implies that the heat transfer within the flow
is dominated by thermal convection, the size of G ensures that the
resulting temperature changes have only small effects on the velocity
profiles and the isothermal flow assumption is valid. In order to
confirm this conclusion, detailed computations of developing temperature
1
- 213 -
and velocity profiles were made, the results of which will be discussed
in this section.
Heat transfer effects were studied for two materials of power-law
index, n = 0.3 and n = 0.4. For n = 0.3, tests were made for a wide
range of values of the dimensionless parameters. Corresponding to each
test, the total pressure drop, P, and the dimensionless heat output due to
convection, E~onv, and conduction, Eēond' are shown in Table 7.1. Also
shown in Table 7.1 are the results of the energy balance which has been
applied as a check on accuracy. The maximum 'error' is in the order of
1% and the value of the summation, Eēonv + Ego , lies between 0.968 and
1.001 which gives an accuracy within 3.2%.
Figure 7.3 shows two dimensionless velocity profiles at exit from the
channel, i.e. at A = 1, corresponding to G = 0 and G = 2, and for constant
values of Gz = 20 and Br = 106. From the results, it can be deduced that
the velocity distribution is hardly affected by variations of the Griffith
number within the specified range of G. A zero value for the Griffith
number is due to the temperature coefficient of viscosity, b = 0. This
condition gives rise to a temperature independent velocity profile, but no
dimensionless temperature profile. Another method of achieving a velocity
profile which is temperature independent is to make:
exp (b (T — T1)) =
(7.69)
This new condition imposed by equation (7.69) produces exactly the same
velocity profile as for G = 0, and also produces a dimensionless
temperature profile which is shown in Figure 7.4. The E* andand Eēond
- 214 -
TABLE 7.1
Results for Temperature Development in Pressure Flows
n G Gz Br P Econv Eeond E* 4-
+ Econd ERROR
0.3 0.1 20 106 202.5 0.291 0.692 0.983 - 0.879 0.4 0.1 20 106 239.4 0.323 0.668 0.991 - 0.448 0.3 0.0 20 106 203.3* 0.290* 0.691* 0.981* - 0.945* 0.3 0.05 20 106 202.9 0.291 0.691 0.982 - 0.912 0.3 0.2 20 106 201.6 .0.292 0.692 0.984 - 0.816 0.3 0.5 20 106 199.0 0.294 0.693 0.987 - 0.634 0.3 1.0 20 106 203.3* 0.290* 0.691* 0.981* -0.945* 0.3 1.0 20 106 194.7 0.299 0.694 0.993 - 0.364 0.3 2.0 20 106 186.4 0.306 0.695 1.001 0.066 0.3 0.1 1 106 201.7 0.037 0.943 0.980 - 1.005 0.3 0.1 2 106 201.8 0.073 0.908 0.981 - 0.954 0.3 0.1 5 106 202.1 0.155 0.828 0.983 - 0.853 0.3 0.1 10 106 202.3 0.225 0.759 0.984 - 0.816 0.3 0.1 50 106 202.7 0.379 0.598 0.978 - 1.121 0.3 0.1 20 1 188.6 2.487 - 1.519 0.968 - 0.645 0.3 0.1 20 2 195.4 1.346 - 0.369 0.978 - 0.822 0.3 0.1 20 5 199.6 0.703 0.278 0.981 - 0.949 0.3 0.1 20 10 201.0 0.496 0.486 0.982 - 0.909
* Values obtained from test case where G = 1 and exp (b (T - T1)) = 1
- 215 -
values obtained for this temperature profile are tabulated in Table 7.1,
corresponding to both G = 0 and G = 1 (where exp (T') = 1). Also shown
in Figure 7.4 is the temperature profile for G = 2. From these two
curves, it can be deduced that the developing temperature profile hardly
varies for the range of G values considered in this analysis. The overall
pressure drop over the region of interest, P, for the isothermal flow
(G = 0) is found to be within one per cent of the values computed for the
non-isothermal flows. The velocity profiles for the temperature
dependent conditions are also found to be almost identical to the velocity
profile for the isothermal condition. Hence, from these results it can be
concluded that the isothermal flow assumption is valid for the process
investigated in this work. Figure 7.5 shows the dimensionless velocity profile at exit from the
channel for a material with n = 0.3, and 1 < Gz < 50, G = 0.1 and Br = 106 .
From the results, it was noticed that the velocity distribution hardly
changed with varying Graetz number and, therefore, for the range of
1 Gz < 50, only one velocity profile is shown in Figure 7.5. In Figure
7.6, the developing temperature profiles have been plotted for Gz = 1, 20
and 50. The shapes of the profiles describe the effect of dominating
convection relative to conduction, as the Graetz number is increased.
The effect of Brinkman number on the velocity and temperature profiles
has also been investigated and the results are shown in Figures 7.7 and
7.8, respectively. For the wide range of Brinkman numbers used,
1 < Br < 106, and for the constant values of n = 0.3, G = 0.1 and Gz = 20,
it was found that the velocity distributions were hardly affected. High
Brinkman numbers are due to small differences between the inlet and
boundary temperatures, i.e. (Tb — T1), and in such cases conduction tends
to become dominant over convection. The high temperature gradient near
- 216 -
the wall is due to viscous dissipation (see also Figure 7.6).
Figures 7.9 and 7.10 show the velocity and temperature profiles for
two materials of n = 0.3 and n = 0.4, respectively, with G = 0.1, Gz = 20
and Br = 106. From these results, it can be seen that the velocity
profile becomes flatter at the centre with decreasing n, i.e. as the fluid
becomes more non-Newtonian. For n = 1, the velocity profile is nearly
parabolic. The results also show that, as the power-law index increases
from 0.3 to 0.4, the temperatures in the channel also increase somewhat,
but the overall shape of the temperature profile remains the same.
- 217 -
Figure 7.1: Cross-section through a flow channel formed by two flat stationary surfaces
`\ \\\\\\\A \\\\\\\ \\\\\\.\\\\\\\\\ Y= 2,
aY L+14+1
Y= 0
Figure 7.2: One half of the flow channel plotted on the (A,Y) plane, showing a finite difference grid
Y
0 1,0 0.8 0.2
- 218 -
DIMfrNSIONLG55 YEI.00ITY
Figure 7.3: Dimensionless velocity profiles at exit from flow channel for varying Griffith number
0 1
- 219 -
Figure 7.4: Temperature profiles at exit from flow channel for varying Griffith number
- 220 -
05
ai+
0.3
0.2
'1
0.I
I I I I 0.6 0, 6' 1.0 1. z
DIMN5IONL.55 YELOGtty
Figure 7.5: Dimensionless velocity profiles at exit from flow channel for varying Graetz number
0 0
0.L
0.3
0.2,
0.1
i I I 0
- 221 -
0.004, O.004, 0.006 0.O0 d 0.010 0.01.2 0.014 0.016
DINf514510NLe55 TE, MPSRATLUU (i4)
Figure 7.6: Temperature profiles at exit from flow channel for varying Graetz number
- 222 -
0 az 0.6 0.g 1.0 12,
DIM6NSIONZ-E.55 VE,LOCITY
Figure 7.7: Dimensionless velocity profiles at exit from flow channel for varying Brinkman number
0.10 o.0$ I I I t
0.02 0.0it 0.06
DIVIEN510NLE.55 TEtinRATU,RZ. (Tm)
x tR=1 Bft 10
o 6R= 106
- 223 -
Figure 7.8: Temperature profiles at exit from flow channel for varying Brinkman number
- 224 -
DIMENS1ONI.E55 VELOCITY
Figure 7.9: Dimensionless velocity profiles at exit from flow channel for varying power-law index
- 225 -
0.5
0.L
0.3
Y
0.2
0.1
0
0 0.00,2 0.001.1. 0.006 0.008 0.010 0.012,
DIMENSION k5 TEMPERATU.P E
Figure 7.10: Temperature profiles at exit from flow channel for varying power-law index
- 226 -
CHAPTER 8
USE OF THE METHOD OF ANALYSIS IN DESIGN
8.1 INTRODUCTION
It has been shown in the previous chapters how to analyse the polymer
melt flow in cable covering equipment. The extrusion die performance can
be predicted from a knowledge of material properties, flow geometry and
operating conditions. The objective was to be able to predict the cable
covering thickness distribution for a given channel geometry. This means
that 'various different flow geometries have to be tested before arriving
at an optimum die design. An ideal situation, however, would be to be
able to adapt an 'inversion process' which can be directly applied to die
design, i.e. given a required thickness distribution, to predict the
channel geometry. Later on in this chapter, the formulation of the
inversion process will be described and it will be shown how it can be
applied to die design. This direct method will then be compared with the
previous optimisation technique.
A limited amount of literature available on wire and cable die design
is mainly related to wire coating dies. Hammond (1960) studied high speed
wire coating dies and showed experimentally that the critical shear rate
increased by using smaller taper angles in the flow channel. He also
showed that, for successful die design, the effect of land length in
relation to orifice diameter has to be considered. Fenner & Williams
(1967) examined the published literature on existing analytical methods of
designing wire coating dies and made a simple analysis to predict extruder
delivery pressures, wire tensions and maximum shear stresses in the die.
These results were then related to extruder performance, wire strength and
polymer melt fracture. Fenner (1974) analysed the axisymmetric viscous
- 227 -
flow in a wire coating die using finite element methods and computed the
relationship between pressure drop and flow rate. He also described
experiments with a LDPE in which the resulting pipe eccentricity due to
varying die eccentricity was measured. While eccentric pipe is of little
practical use, the ability to predict melt flow patterns and hence final
thickness variations of the extrudate is very important when assymmetrical
dies are to be designed. Caswell & Tanner (1978) used finite element
methods to trace flow patterns for arbitrary axisymmetrical geometries,
such as a wire coating die. They studied the flow patterns in two
different dies and concluded that the function of the "land" in a die is
to provide sufficient pressure loss to overcome the suction effect created
on impact of the fluid with the wire and the taper in the "land" further
increases the overpressure. The term "land" in this context refers to the
region where the fluid is in contact with the wire, inside the die unit.
Finally, they attempted to infer design criteria for wire coating dies
mainly in the region where the polymer meets the wire and upstream of this
region, otherwise known as the "land".
Some of the most successful attempts on die design were made by
Pearson (1962,1963,1964). In the case of narrow channel dies, such as
the crosshead die, melt flow analysis is a valuable aid to the design of
the flow passages and can minimise the subsequent mechanical adjustments
necessary. Pearson showed that the analytical process can be inverted to
predict the channel geometry required to give a specified flow distribution.
Alternatively, the flow analysis can be used to select the best compromise
design using simple component shapes. For example, in a crosshead die,
the optimum offset of the central mandrel (inner body) to counteract the
assymmetrical input can be obtained.
- 228-
8.2 CASE STUDY
Having verified experimentally the method of analysis used in the
previous chapters for the melt flow, it was decided to attempt the'design
of a tool set where the specifications were provided. The exercise was
to determine optimum designs for a three layer crosshead with a solid
conductor area of 400 mm2. The radial insulation thickness for this
arrangement was to be 15 mm with 0.7 mm screen thicknesses for the inner
and outer layers. The nominal diameter of the copper conductor was taken
to be 24.2 mm since it was stranded, with a tape thickness of 0.17 mm.
The overall deflector dimensions and the overall length of the crosshead
were specified by AEI Cables Limited and these dimensions were not to be
changed.
The initial step in the design procedure was to calculate the
diameters at exit of the points and the outer die, i.e. PZ, P2, M1, N1 and
D shown in Figure 8.1. Assuming a diametrical allowance of 1 mm between
conductor and inner point, gives:
= 24.2 + 0.34 + 1.0 = 25.54 mm
Let: P1 + 4.0 29.54 mm
Using a draw-down ratio of 1.3 for each of the three layers, M1 , N1 and D1
can be calculated and their values are given below:
M1 = 31.06 mm
64.73 mm
= 66.79 mm
- 229
Figure 8.1 shows a sketch of the tooling arrangement which has been drawn
to the specifications. As can be seen, the middle layer diverges towards
exit which might not be favourable since throttling is usually expected to
give flow uniformity as opposed to retarding flow in diverging channels.
Thus, the outer point diameter was modified such that the fluid was
flowing in a parallel channel.
Using the flow analysis computer program described in Chapter 4, the
thickness distributions for the three layers were predicted. Figures 8.2,
8.3 and 8.4 show the distributions round the circumference of the inner,
outer and intermediate layers, respectively, of the new tool set and the
experimental tool set (300 mm2, 8 mm insulation thickness). In the case
of the inner screen (Figure 8.2), the predictions for both tool sets are
almost the same. In the case of the outer screen (Figure 8.3), the
experimental tool set seems to give a slightly better thickness
distribution than the new tool set. Finally, the intermediate layer
thickness distributions compared in Figure 8.4 show that the predictions
for the experimental tool set are almost twice as good as the predictions
for the new tool set. Since the channel geometry of the intermediate
layer gave the worst results, it was decided to study this layer in more
detail in order to be able to design a much better channel.
It is important to mention at this point that, so far in the design
of the new tool set, only the point dimensions were modified, the
deflector designs remaining unaltered. One reason for the new tool set
with 15 mm insulation thickness giving a poorer thickness distribution in
relation to the experimental tool set with 8 mm insulation thickness is
the much deeper intermediate channel in the points region of the former
design. In order to be able to improve the design of the new tool set,
before making any alterations to the deflector channel shape, it was
- 230 -
decided, first, to increase the channel depth in the deflector region by a
constant amount in order to decrease the pressure drop over the deflector
region and in return increase the pressure drop over the tapering region
as a proportion of the total. Figure 8.5 illustrates this arrangement
and the slight convergence in channel depth towards exit, resulting from
the increase in the deflector channel depth. Figure 8.6 shows the radial.
thickness distributions for 20% and 100% increases in the deflector channel
depth. From the results, it appears that there has been some improvement
on the previous tool set design, but not very significant.
Next, it was decided to modify the deflector design itself in order
to produce more uniform output. As a first trial, a wedge shape was added
to the intermediate deflector similar to the one in the case of the outer
deflector as shown in Figure 8.7. In this design it was important to
estimate the geometry of the wedge. After fixing the length of the wedge
such that the point P corresponded to a nodal point in the finite element
mesh, the next step was to select the angles a and S. As an initial guess,
it was decided to' take them as a = s = 30°. Initially, the wedge had a
depth of 0.064" at point P, and gradually reduced to zero at the exit due
to the 1.35° angle of taper of the deflector channel depth. The
dimensionless thickness distribution predicted for this design is compared
with the one for the original deflector design with no wedge in Figure 8.8.
It appears that there is a slight improvement in the presence of the wedge.
Another possibility was to make angles a and B not equal, and various tests
were made for varying wedge angles. The best results were obtained for
a = 30° and 0 = 47°. The predictions for this deflector design, together
with the predictions of the experimental tool set design, are shown in
Figure 8.9. As can be seen from the results, the effect of the wedge is
quite significant. Thus, from the results, it can be concluded that using
231 -
the flow analysis described in Chapter 4, it is possible to find optimum
designs for a given set of specifications. The only drawback of this
method is that a few test cases have to be tried out before deciding on
the better design. An ideal method, however, would be the "inversion
process", where the channel geometry is predicted to give the perfect cable
uniformity. In the next section, this method will be analysed in some
detail.
8.3 DEFLECTOR DESIGN BY INVERSION OF THE ANALYSIS
The objective in this analysis is, given a stream function
distribution defining the flow lines in the die channels, to find the
channel depth at any point in the flow region. Since the presence of the
points added onto the deflectors reduces to some extent the non
uniformities in the flow, it is best to find an optimum design for the
deflector alone in order to give good cable thickness distribution. To
illustrate the improvement due to the points, the predictions for the
intermediate deflector with and without the points of the experimental tool
set are shown in Figure 8.10. As can be seen, the shape of the thickness
distribution graphs remain the same but the non-uniformity has been damped
down with the addition of the points.
Given a stream function distribution in the solution domain, such as
the one shown in Figure 8.11, where the streamlines lie along rows of
nodal points, the problem is to find the corresponding channel depth
distribution, h(z,e). The stream function distribution, tp(z,e), must
satisfy the following two conditions:
i) i, must be a constant along boundaries AC and BD. (Let ji = 0
along the lower boundary, BD, and i = 1 along the upper
- 232 -
boundary, AC.)
ii) Streamlines must be orthogonal to boundaries AB and CD,
i.e. flow inlet and exit, which are isobars.
The method which has been adapted in this analysis will be described below:
(a) Information must be given which is equivalent to specifying the
channel depth distribution along one streamline, e.g. the lower
boundary, BD.
(b) Hence, the pressure distribution can be calculated along this
streamline using equation (3.22) as follows:
dp Qs
ds h3
(8.1)
where dp is the pressure drop between two successive nodal points in
a row, ds is the distance between these two nodes, and TI- are
defined by equations (3.23) and (3.24), respectively, h represents
the nodal point channel depths, and Q is the volumetric flow rate
per unit width normal to the s-direction which is the direction of the
streamline. Supposing that the total flow rate coming out at CD is
1 in3/s, then:
where L is the length of CD.
(c) Working from this single distribution of pressure along BD, the entire
pressure distribution, p(z,o), can be determined, bearing in mind
- 233 -
that isobars are orthogonal to streamlines which are already known.
(d) Hence, at any point in the domain 3p/az and ap/ae can be found. But,
using equation (4.44):
aēa ū a ū
- s1 h3 , = s2 h3 (8.2)
where a1 and S2 are locally constant and involve known stream function
gradients. Therefore, the pressure gradient along a streamline,
ap/as, can be represented as:
aP = as
s u 3 h3 (8.3
Hence, the local value of h can be obtained by equating equation (8.3) to
the pressure gradient obtained from the p(z,e) distribution (see item (d)
above). Taking the (resultant) gradient along the local streamline is
merely a convenient way to average ap/ae and ap/z. It should be also
noted that u is dependent on h (see equation (3.24)).
The practical problem is now essentially reduced-to finding p(e,z)
orthogonal to p(e,z). For a typical triangular element, the stream
function distribution can be represented by:
= C1 + C2 z + C3 e (8.4)
which can also be expressed in terms of circumferential dimensions as:
_ 1- C2 z+ r
(8.5)
- 234 -
where r is the mean value of radius for the element., and c = r e. The
stream function gradients can thus be shown as:
ac r (8.6)
If A is the magnitude of the resultant stream function gradient at an
angle a to the z-axis, as shown in Figure 8.12, then:
• = A cos a = az
aI = A sin a =
ac (8.7)
If B is the resultant pressure gradient orthogonal to A, also shown in
Figure 8.12, then:
az = - Bsin a
B C3 Ā — r
C3
ac = B cos a
= 2812
= D C2 (8.8)
Now, assuming a linear pressure distribution over the element:
= E1 + E2 z + E3 6
D C
3 E1 _ z # D C2 r 0 r
8.9)
In equation (8.9), there are only two unknowns, E1 and D. Hence, if two
nodal point pressures are known, the third one can be found.
Since the pressure distribution within an element is assumed to be
a linear one similar to the stream function distribution, using equation
_ - D C3 - 2Q (bi p . + b. p. + bk pk ) r m
= D C2 r = 2Q (a. pi + a. p . + ak a. m
Eliminating D from equations (8.11) and (8.12) gives:
- C3 (a. p + a~ + ak pk ) = C2 r2 (bi p. + b. p. f k Pk )
- 235 -
(4.27), E1, E2 and E3 can be represented in terms of the nodal point
pressures of the element as follows:
Er
2
1 pi ' - 1 o3) pi
'ppj
k ,
(8.10) 2Am
where pi, pi and pk are the pressures at the three nodes of the element.
i, j, k, respectively, numbered in an anticlockwise direction, and am and
B are defined by equations (4.28) and (4.29), respectively. From
equations (8.9) and (8.10):
or:
ai) + pa (C2 1 ha. + C3 a j ) + pk (C2 r2 bk + C3 a
api +13 pi + 1Pk = 0
where: a = C2 b.+C3 a.
C2 r2 b~ + C3 a. (8.15)
Y =
- 236 -
Starting at node B (Figure 8.11), elements with sides along BD have two
pressures prescribed, therefore, using equation (8.14). the third pressure
can be found. This procedure is called the "element by element marching
procedure". Since the boundaries shown in Figure 8.11 are not orthogonal,
the results obtained are not strictly accurate. Due to this non-
orthogonality, the results vary according to 'the sequence of the elements
used to calculate the nodal point pressures. For example, the pressure
at node NN, pNN, in Figure 8.11 can be worked out either using element M1
or M6. If the boundaries were orthogonal, then the two values obtained
for pNN would have been identical.
A second method which is rather more sophisticated involves finding
nodal point pressures by an "averaging procedure". In Figure 8.11, node
NN is shared by six elements, M1, M2, M3, M4, M5 and M6. Assuming that
the number of nodal points in the x-direction is NXPT, then using equation
(8.14):
Y1 pNN =
a2 pi + 2 Pk + Y2 pNN
a3 pNNR3 Pk 13 Pi =
(8.16)
+g p1 +Y4
Pn.f135 PNN 15 Pm =
a6 Pl 136 PAIN 16 pn
where subscripts 1, 2, 3, 4, 5 and 6 stand for elements M1,
and M6, respectively, and:
M4, M5
a1 p .f 0
0
- 237 -
i = NN - NXPT Z = NN + NXPT
j = i+ 1 m= i-
k = NN + 1 n = NN - 1
Adding up all the equations in (8.16) gives, for the node number NN:
Pi (a1 ÷a6) + p j
(61 +a2) +pk (132 + S3) +P1 (13+84) +
pm (y4 + y5) +pn ( (15 +
y6) + p NN 1 + y2 + a3 + a4 + S5 + 86)
8.17)
Similar equations for all the nodal points in the mesh can be written.
In this particular case, a 13 x 13 mesh was chosen with 169 nodal points
and 288 triangular elements. Therefore, altogether 169 linear
simultaneous equations had to be solved to give the pressure distribution
in the field. In order to be able to solve these equations, boundary
conditions were prescribed and the Gaussian direct elimination scheme was
used. The boundary conditions were simply the nodal point channel depths
along BD, which were used to calculate the nodal point pressures along
this boundary. Once the pressure distribution in the solution domain was
worked out, then the element channel depths were calculated using equation
(8.3).
The channel geometries arrived at from the two methods described
above were used to analyse the melt flow in the crosshead channels and
cable thickness distributions were predicted. Theoretically, provided
the orthogonality of boundaries was satisfied, the distributions would
have been perfect. However, as can be seen from Figure 8.13, the
distributions are not as uniform as expected. The predictions in the
- 238 -
case of the "element by element marching procedure" seem to give more
uniform thickness distribution than the "averaging method". The reason
for this is that, since the initial assumption of orthogonal boundaries is
not satisfied, the errors involved in the case of the averaging method
appear to be distributed over the mesh and therefore the results are not
as accurate as the first simpler method. From the results, it appears
that, at the 180° position where the boundaries of the intermediate
deflector are far from being orthogonal, the thickness varies quite
vigorously, the boundary effects being more pronounced in the case of the
latter solution procedure.
In order to be able to demonstrate the importance of orthogonality of
boundaries in the inversion process, two simple meshes which are shown in
Figures 8.14a and 8.14b were analysed. The mesh in Figure 8.14a has
orthogonal boundaries at inlet and exit, while the mesh in Figure 8.14b
has got non-orthogonal boundaries which do not satisfy the initial
condition (ii) specified at the beginning of this section, for this
analysis to be accurate. The channel depth distributions in both of
these meshes were worked out using the element by element marching
inversion procedure and then the flow was analysed. Figure 8.15 shows
the predicted radial thickness distributions for the two meshes and, from
the results, it clearly appears that the orthogonality of boundaries is
essential for the accurate solution of the equations.
8.4 DISCUSSION OF RESULTS
In order to be able to see how much the inversion analysis improved
the intermediate deflector design, the predicted radial thickness
distribution for the original middle deflector alone is plotted in
Figure 8.16, together with the distribution for the deflector design
- 239 -
obtained using inversion. In this same figure is also shown the thickness
distribution for the intermediate deflector with the best wedge shape of
angles a = 30° and s = 47° (see Figure 8.7). The comparisons show that
the middle deflector design obtained using the inversion of the melt flow
analysis gives the most uniform thickness distribution followed by the
deflector with the wedge. It appears that the addition of a wedge shape
to the original deflector design improves the thickness distribution quite
significantly. Figure 8.17 shows the element channel depth distribution
of the intermediate deflector obtained using the inversion analysis.
The channel depth value for each element along the 13 rows of the mesh has
been plotted for altogether 288 elements. In practice, it might not be
very feasible to manufacture a deflector with quite a lot of variation in
channel depth. Therefore, in order to achieve improved deflector design
for practical purposes, the addition of a wedge shape might be preferred
to obtaining a design using the inversion process, since the latter method
produces only marginally better cable uniformity than the former.
One of the important factors in die design is the sensitivity: of cable
thickness uniformity for a given design on material properties, mainly the
power-law index, n. In other words, if an optimum design is made for a
deflector to be used with a particular polymer, it is important to know.
how the thickness distribution will be affected if this deflector is used
with other polymers. For example, the intermediate deflector with the
wedge shape of angles a = 30° and = 47°, shown in Figure 8.7, was
designed for a material of power-law index n = 0.391. The predicted
cable thickness distributions obtained when this design was tested with
two other materials of n = 0.3 and n = 0.5 are shown in Figure 8.18,
together with the results for n = 0.391. It appears that cable uniformity
improves marginally with increasing power-law index, .e. as the material
-240-
becomes less non-Newtonian. Therefore, it can be concluded from the
results that the design should be made for the lowest power-law index
material likely to be processed. The same exercise was also carried out
for the.optimum intermediate deflector design arrived at using the
inversion analysis, for a material of n = 0.391. As before, sensitivity
of the design to different material properties was checked by testing with
two other materials of n = 0.3 and n = 0.5. The polymer thickness
predictions for the three materials are shown in Figure 8.19. Although
there is not much variation in the thickness distributions, the results
again _suggest that the optimum design should be made for the material with
the lowest power-law index.
1 1
N +::a
.-'- f f NI 0, 'P, PL . M, OIA DIA DIA olA OIA
At -::.1.~:31 I I
Figure 8.1: Tooling arrangement for three layer head wfth 15 mm insulation thickness
- 242 -
1
1.05
1.00
0 i
0.95
0.10
• NOW TOOL 5V
x GX P6RIMk.NT AL 'TOOL 5)r1
0.8'5 l I I 1 I 1 I I I
0
$0 120 160 200 21+0 280 320 360
Figure 8.2: Predicted circumferential thickness distributions for the inner screen of the "new tool set" and the experimental tool set
POLY
MET
,. īU
I CKN
E,55
1,05
0.90 NEW TOOL. 5gT
X gXPPRINENTA(. 7001.- 5g.'T
t I i I I i I I
~o SO 120 160 a00 21-o g.50 320 360
1.00
0a5
0
1
- 243
Figure 8.3: Predicted circumferential thickness distributions for the outer screen of the "new tool set" and the experimental tool set
I 1
160 aoo ,214.0 a80 320 360 0 $0 4,0 120
- 244 -
• Nj W ī00L.. 5T X GXPg•RIMENTAI. TOOL 5k,T
Figure 8.4: Predicted circumferential thickness distributions for the intermediate layer of the "new tool set" and the experimental tool set
-245-
INCREP65 DE,FaCT0R,
CI}ANnS 'DEPTH
Figure 8.5: Sketch of a cross-section of the intermediate layer flow channel
360 320 24.0 280
- 246 -
0 iA Z IN A56 IN CHANNEL TAM
X 100 0 IMCREASE IN CHANNEL. DEPTH
• ORIGINAL NEW TOOL 5ET
1 I I I I
0 80 I.0 160 '200 0
Figure 8.6: Predicted circumferential thickness distributions for the intermediate layer with 20% and 100% increases in deflector
channel depth compared with the original "new tool set"
cHANNP. a%p7H OVER,
-OK WkD6E= 0.13e
Figure 8.7: Intermediate deflector with the addition of a wedge shape
0.8' 5 r r
160 100 360 ago I20 320 o 1,0 $o
- 248 -
r,05
I.00
X OZFLCTOR. WrTFi cy' = p= 300 1^IED6-tr • upFLECTO'R. WITHOU.T YJ-DGE
~ORIGINAL. Nf..1.4 TOOL 5E0
0.q0
Figure 8.8: Predicted circumferential thickness distributions for the intermediate layer with and without the addition of a wedge to the deflector
$0 1W 160 a00 40
0° 0 ago .320 360
- 249-
• O£FLECTOR. WITH d= 30° ~ ~.4Ý WE'DG1; O DEFLECTOR. WITH aCr ZS° (3=4.7° WE.DGE X !: XPGRIMENTAL1.'1 1-Z5TE.D TOOL. SET .
Figure 8.9: Predicted circumferential thickness distributions for the intermediate layer with the addition of varying shape wedges to the deflector compared with the "experimental tool set"
250 -
1'4
C C E 0'8
o Deflector only
A Deflector+ point
E 0•2
I 60 120
0 180 240 300 360
Figure 8.10: Computed final thickness distributions for deflector alone, and combined deflector and point
251 -
Figure 8.11: One half of the flow channel plotted on the (z,e) plane, showing a typical nodal point being shared by neighbouring elements
Figure 8.12: Sketch showing orthogonal resultant stream function and pressure gradients
- 252 -
Poo
Hatt 1H
l cK
mE
s5 0.45 _
I.00
0.50 • ELG MGST r ' ELEMENT
MARCHING TROcEDU.R.E. X AVIET AGIN& METHOD
I r I 1 t I r r r 14.0 $0 12.0 160 2200 240 .260 320 360
g°
Figure 8.13: Predicted circumferential thickness distributions for the intermediate layer with the intermediate deflector design obtained using the two different "inversion" approaches
0.25 0
-253-
Figure 8.14a: Finite element mesh With orthogonal boundaries at inlet and exit
Figure 8.14b: Finite element mesh with non-orthogonal boundaries
- 254 -
t
0.5
1.5
2.0
X mil-106-0M At, boU.N ftlES Al INI.GT AND ~XIT
e NoN-01q,7HoG0NA1,. 13O1IPDARIE5
I 1 I I I I I ) 1
o ti.o vo 12.0 160 200 2L.0 c28.0 320 360 0°
Figure 8.15: Predicted circumferential thickness distributions for the two simple meshes with and without orthogonal boundaries
0
- 255 -
1.25
1.00
2
2
`no 0.15 2
ot
D.
0.50
0.25 0 160 20 0
9° 8'0 120 4.0 23'0 320
• ORIGINAL, DEFLLC-ToR.
1)-FI-E.CTOR, WITH WEDGE 0 DE-5iGN FROM INIE,R5101,3
360 2)+0
Figure 8.16: Predicted circumferential thickness distributions for the intermediate layer with different deflector designs
2 3 7 S 5 1f IS IL{- 15 16 10 II 12, 13 20 21 22- 23 24.
t oJ II
R.OW 6
120 ROW 5
ROW 4.
0 vi
i1
0 2.5 1
JL?-1 WIN IbE,R,
60
36
13Z
108
Figure 8.17: Predicted element channel depth distribution for the intermediate layer
- 257 -
0.50-
X MATERIAL WITH 0.3 • MAT✓✓RIAL WITH n= 0.3R1
0 MATZ MAL W11H ri = 0.5
o L 0 80 12.0 160 2,00 240 230 320 360 e°
Figure 8.18: Effect of material properties on the circumferential thickness distributions for the intermediate deflector design with the wedge
- 258 -
1, 00
0.5 In
UI 2
c)
2 d
0.25
o.
X RA -rMR.1AL WITH n= Q.3
• WITH n.- 0.391 o t{ATer.IE1u. WITH
I I t t t 1 1 1 l {4.0 C50 120 160 2001 ,21.-D 2W 320 360
0
Figure 8.19: Effect of material properties on the circumferential thickness distributions for the intermediate deflector design obtained from the "inversion" analysis
- 259 -
CHAPTER 9
GENERAL DISCUSSION AND. CONCLUSIONS
In the previous chapters, FE techniques for the analysis of polymer
melt flow were developed and their applications to die design were
studied. In this chapter, the method and results of the melt flow
analysis and its applications to industrial die design problems are
discussed and the main conclusions drawn from these investigations are
summarised.
9.1 DISCUSSION
The type of polymer melt flow encountered in cable covering crossheads is essentially flow in relatively shallow channels. The general approach
to the analysis of such flows involves the application of continuum
mechanics. Initially, a mathematical model was set up to describe the
non-Newtonian flow in'the crosshead die channels, which was assumed to be
isothermal, steady, laminar, viscous and incompressible. A power-law
constitutive equation was assumed for the melt, and the solutions to the
relevant differential conservation equations were obtained for the
appropriate boundary conditions and melt properties. In view of the
complexity of the conservation equations, many simplifications and
approximations were introduced.
The FEM, due to its geometric flexibility, is suitable for application
to the complex shaped crosshead channel geometry. Although the method
described in this thesis was specifically designed to solve the cable
covering crosshead problem, the same approach is also applicable to other
shallow channel flows and to other problems governed by mathematically
similar equations. In order to solve the problem by a FEM, it was
- 260 -
necessary to establish a variational formulation for the resulting
quasi-harmonic partial differential equation. The general variational
approach to the solution of a continuum mechanics problem involves the
minimisation of a functional for conservation equations with respect to
the field variables. The minimisation gave rise to a set of simultaneous
linear equations which were solved, using the Gauss-Seidel iteration
scheme, for the stream function values. While most of the properties
were treated as locally constant, melt viscosity was non-Newtonian,
depending on the local rate of deformation, also on temperature and, to a
lesser extent, on pressure. Viscosity data were obtained from capillary
rheometer measurements. Although polymer melts are •viscoelastic, the
constitutive equations used in the analysis do not take into account both
the viscous and elastic effects. In extrusion processes, elastic effects
are not predominant since melts are subjected to large rates of deformation
for relatively long times. For isothermal flow in extrusion dies, a
power-law fluid model was therefore adequate to represent the pseudoplastic
nature of polymer melts.
The design of extrusion dies is a matter of sufficient importance to
justify a detailed analysis of the factors involved. In the analysis of
the melt flow, it was assumed that the die can be supplied with a
homogeneous stream of molten material. It was also assumed that the
required physical dimensions of the extruded product and the output rate
were specified. A particular die designed for extrusion of a complex
section can only be used to provide one specified product shape, and that
often in one material only, whereas a tubular film die can, by adjustment
of draw-down and blow-up ratios, be used to provide a variety of film
widths and film thicknesses, often in a variety of materials. Therefore,
design criteria have to be specified, bearing in mind the use of multi-
- 261 -
purpose dies, where, for example, it is easy to adapt an existing die for
the production of other sections when it becomes redundant. Practical
die design is concerned with the construction of dies for commercial
extrusion where such factors as die cost in relation to the length of run,
ease of construction and dismantling for cleaning and cost of replacement
of component parts are of great importance.
Flow control is the primary object of die design. The correct choice
of flow passages from extruder to die-lips is the essential prerequisite
of a successful die. The general channel geometry is governed by certain
overriding mechanical and thermal requirements and mainly by the flow
behaviour of the materials to be extruded. The strength of individual
components to withstand hydrostatic melt pressure and the need to provide
temperature control at all polymer boundaries where heat transfer is to be
expected are important factors to be considered in die design. In
specifying material flow properties for commonly extruded thermoplastics,
many simplifications have to be made for the purpose of analytical
simplicity and to obtain generality. Those properties that are most
widely characteristic of polymer melts must be selected in order that any
design criteria may have maximum applicability.
Some of the common problems industry faces in making cable include
variable coating thickness, surface roughness, off-centre coating,-grooves
parallel to conductor, and separation of covering from conductor.
Variation in coating thickness could be due to surging in the extruder,
unsteadiness of pay-out, capstan or wind-up reels or variation in
temperature, pressure or motor load. Surface roughness might be overcome
by using dies with longer lands or smaller taper angles. Also, running
the extruder slower or hotter might reduce surface roughness. Off-
centering is usually due to die setting not being adjusted properly or
-262-
internal die design not properly compensating for the bend in the
crosshead. Sagging of low-viscosity plastics due to gravity could be
significant when making cable with relatively thick coating. This pear-
dropping effect could be reduced if the machines are run cooler. Grooves
parallel to the conductor usually occur in heavy cables in long troughs
where the cable, while still warm, scrapes against the trough bottom unless
very high tension is applied. Separation of covering from conductor might
be due to poor adhesion or poor contact in a tubing die which could be
overcome by running the polymer hotter. Other. reasons for separation of
covering include dirty or moist conductor, or cooling of the cable too
fast in which case the plastic shrinks and separates from the conductor.
9.2 CONCLUSIONS
The main objective of the work presented in this thesis was to predict
polymer melt flow behaviour in cable covering equipment, for the purpose
of solving practical industrial problems of machine operation and die
design. The particular application considered was flow in cable covering
crossheads. The approach used was to analyse flow patterns and pressure
distributions in shallow channels, also temperature developments in pressure
flows, and to compare them with experimental results. Computed results
obtainable from the analysis included the distribution of polymer layer
thickness on the finished cable, together with the extrusion pressure
required to maintain a given flow rate of melt. Uniformity of coating
thicknesses in the finished cable is of considerable commercial importance.
In order to satisfy electrical performance criteria, each layer of polymer
must be of a prescribed minimum thickness. If the actual thickness varies
significantly around the circumference of the cable, excess material will
be contained in the thicker parts of the layer. A rational method for
263 -
designing deflector flow channels is therefore highly desirable.
In order to illustrate the application of the method of analysis, a
typical deflector and point profile were considered. Two solution
domains were analysed, one with just the deflector region with 169 nodal
points and 288 triangular elements, and the other with the point region
added with 247 nodal points and 432 elements. The nodal point and
element numbers were chosen as a result of a compromise between accuracy
and cost of computation. Further refinement of the mesh caused only
insignificant changes in the final results. The predicted dimensionless
thickness distributions of the polymer layer on the finished cable clearly
showed that the deflector alone gives a poor thickness distribution, but
the addition of the point region results in a considerable improvement.
This is because the flow channel dimensions are such that the pressure
drop over the point is substantially greater than that over the deflector.,
Therefore, the axial symmetry of the tapered point region dominates the
lack of symmetry associated with the deflector to give a reasonably
acceptable thickness distribution.
Full scale extrusion experiments were carried out concerning the
performance of an extrusion crosshead used in the three-layer covering of
high voltage electrical cables. Both extrusion pressure requirements and
circumferential distributions of polymer layer thicknesses in the finished
cable were measured and compared with the results of the finite element
method of melt flow analysis within the crosshead. While agreement on
pressure was good, it was necessary to allow for the effects of both
gravity, in relatively thick layers, and slight misalignments of crosshead
components if the thickness distributions were to be correlated
satisfactorily. The latter effect emphasises the need for a high degree
of accuracy in crosshead design and manufacture.
-264-
An initial assumption of the melt flow analysis was that the flow is
isothermal. In order to confirm the validity of this assumption, a
somewhat simplified form of the actual melt flow in the crosshead was
considered. Taking typical channel dimensions, operating conditions and
material properties for melt flow in cable covering crossheads, the values
of the dimensionless parameters were found. While the magnitude of the
Graetz number implied that the heat transfer within the flow is dominated
by thermal convection, the size of the Griffith number ensured that the
resulting temperature changes have only small effects on the velocity
profiles, and the isothermal flow assumption is valid. This conclusion
was confirmed by detailed computations of developing temperature and
velocity profiles which show that, for example, overall pressure drops are
within one per cent of the values computed for isothermal flows.
The emphasis of the work has been on the application of the results
and methods to solving practical problems. It was shown how the method
of analysis for shallow channels can be applied to crosshead design. The
extrusion die performance could be predicted from a knowledge of material
properties, flow geometry and operating conditions. For a given set of
specifications, it was possible to find optimum designs by means of
modifying the deflector contours, varying the taper angles of the points,
or adding flow obstructions, such as a wedge, to the deflector. The only
drawback of this method was that a few test cases had to be tried out
before deciding on the better design. The alternative was to invert the
method of analysis whereby, given a stream function distribution in the
flow region, the channel depth distribution can be found. From the
results, it was concluded that to achieve improved crosshead design for
practical purposes, the addition of a wedge shape to the deflector is
preferred to obtaining a design using the inversion technique which gives
- 265 -
rise to a deflector design with quite a lot of variation in channel depth,
and which produces only marginally better cable uniformity than the former.
Finally, for a given design, the sensitivity of cable thickness uniformity
to material properties, mainly the power-law index, n, was studied. The
results suggested that, if a design has to be made for use. for a range of
materials, then the optimum design for the material with the lowest power
law index, i.e. most non-Newtonian, should be used.
- 266 -
APPENDIX Al
REPRESENTATION OF C1, C2 AND C3 IN TERMS
OF THE NODAL POINT VALUES OF STREAM FUNCTION
Multiplying equation (4.25) with aj and equation (4.26) with ak,
gives:
a. =
ak ')k
a. + C2 . a
k - C3 a. bk
C1 ak - C2 aj ak + C3 ak b.
Using equation (4.24) and adding equations (A1.1) and (A1.2):
a~ +a — (a. + ak) pZ kJ — a.d bi() .(A1.3)
But, from Figure 4.2:
— a. — a. = ak or a?
+ ak = — a. (A1.4)
Therefore: a2 Z + aj tpj + ak = C3 (ak b. — a. bk) (A1.5)
The area of the element, m, in Figure 4.2 can be worked out as
follows:
(area of surrounding rectangle) - (area of triangles outside element)
- 2 a. b.+2 aj b. a
(ak bj +' ak (b. + bk) — a . b . + a. b .)
-267-
From Figure 4.2:
-b. - bk = b. or bj.+bk = b.
=
(A1.6)
Therefore: 4 lak b bi (ak + a.) + a~ b.)
Om = z (ak b. + a. (b. + b.) )
4 lak b. - a. bk ) (A1.7)
Substituting equation (A1.7) into equation (A1.5):
a. t2+a. ~+ ak lk C3 =
which can be represented in matrix form as:
2A in
C3 = 26, ~ ai aj ak) (A1.8)
Similarly, C2 can be represented in terms of the stream function values at
the nodal points. Multiplying equation (4.25) with bi and equation (4.26)
with bk' gives:
bJ
q = C1 b. + C2 ak bi C3 bi bk (A1.9)
bk ~ = C1 bk - C2 a. bk + C3 b. bk (A1.10)
- 268 -
Using equations (4.24) and (A1.6) and adding equations (A1.9) and (A1.10):
bi bj V + bk Vk = C2 (ak bj — aj bk)
Therefore: b2 V2 + bi Vi+bk 11,k
2A
which can be represented in matrix form as:
C2 20 (bi bj bk)
(A1.11)
Combining equations (A1.8) and (A1.11) gives:
[
C2j 1 b. bij bk
C3 2 m a2 aj ak
(B) 2 m (A1.12)
which is identical to equation (4.27).
-269-
APPENDIX A2
h2-EXTRAPOLATION TECHNIQUE
That it is necessary to use a large number of elements and nodal.
points to achieve satisfactory convergence of finite element methods is
due to the fact that shape functions, such as equation (4.23), provide
only approximate representations of the true variations. A Taylor series
expansion about the origin of the local coordinates shown in Figure 4.2
for the stream function gives:
ip (z', e') = . + lz' az f e' aē ,J ,y f 2 lz' az , f e' a8 J 2 f ( A2.1)
the derivatives being evaluated at the origin. Now, the stream function
at the node j can be written as:
= ak az bk ae'J + 2 (ak az — bk ae 'J2 f ''' (A2.2)
But, from equation (4.23), a4)/az' = C2 and ai,/ae' = C3. Therefore,
equation (4.25) can be re-written as:
z a b 34) = ' ak az' k ae ' (A2.3)
The error involved in using equation (A2.3) as a truncated form of equation
(A2.2) to represent the stream function at the point j is of the order of:
2 2 2
2 (ak az ,2 - 2 ak. bk az' ae' f
bk 2 3132J (A2.4)
This truncation error is of the order of the square of the dimensions of
- 270 -
the element and tends to zero as the element size is reduced.
If solutions ip(1) are obtained when the typical element dimension is
h, and another set tp(2) when it is reduced to zh (that is, when four times
as many elements are used), then if p are the true solutions:
11) tp (1) f e h2 = (2) + 4 e h2 (A2.5)
where e is the constant of proportionality in the error term. Eliminating
this term:
3 i4 — i
(2) (11) (A2.6)
This process of obtaining improved solutions is often referred to as
"h2-extrapolation".
- 271 -
APPENDIX A3
PRESSURE DROP BETWEEN TRANSDUCER AND
MELT INLET TO THE DEFLECTOR
The flow analysis used here for the annular section shown in Figure
5.46a is similar to the one described by Fenner (1970) for an annular
section round the torpedo in a wire coating die unit. A dimensionless
flow rate, nQ, may be defined as:
= Q/C H V (A3.1)
where Q is the volumetric flow rate, H is the mean radial gap of the
annulus:
H =
(A3.2)
2
V is the constant speed of the wire through the die which does not apply
in this case and therefore has to be eliminated, and C is the mean
circumference:
_ (d1 f d2)/2
The dimensionless pressure gradient, Trp, may take the form:
= — z H T 7p
(A3.3)
(A3.4)
where P is the pressure gradient in downstream z-direction, and ;"-- is the
mean shear stress defined at the mean shear rate, V/H:
= u v/H (A3.5)
- 272 -
where Tis the viscosity defined at a mean melt temperature and mean shear
rate, Q/AH, where A is the cross-sectional area of the annulus:
uu ( Q )n-1
m A H y 0
(A3.6)
where Um
is the effective viscosity (in the power-law equation) at the
mean melt temperature and reference shear rate, yo, n being the power-law
index.
,r Q and Trp may be combined to eliminate V using equation (A3.5) and .
define a new annular flow parameter:
7TQ
7r
p
Qu
C H3 (- Pz) (A3.7)
In general, TrA will be a function of n and K, the diameter ratio of the
annulus (K = d2/d1). The numerical solution of this problem reveals that,.
for small values of K, TrA is nearly independent of K (see Fenner (1970)).
The exact solution for K = 1 may therefore be used:
1 2n n nA 22n+1 (2n + 1) (A3.8)
The errors involved in using this expression when K is greater than unity
are less than 1% for K = 2 and less than 2% for K = 3 over the practical
range of power-law index, 0.2 , n , 1. Equation (A3.8) is identical to
the result obtained by McKelvey (1962) based on the work of Fredrickson &
Bird (1958).
Using equations (A3.7) and (A3.8), the downstream pressure gradient,
Pz, can be worked out and hence the pressure drop in the annular section,
(dp)1, (Figure 5.46) can be calculated. In order to be able to calculate
- 273 -
(dp)2, an axisymmetric pipe flow analysis has to be made. Applying the
momentum conservation equation in the z-direction, and assuming a fully
developed flow:
1 a Pz - az = r 3r (r Trz ) (A3.9)
where r is the radial distance from the axis of the pipe, and z is the
distance along the pipe axis, as shown in Figure 5.46b, Trz being the shear
stress component. Integrating equation (A3.9), bearing in mind that, due
to symmetry, Trz = 0 at r = 0:
P P Trz -.
(A3.10) 2
Using the constitutive equation:
Trz = u
dr
dvz (A3.11)
where: l 2 n-1 Y
(A3.12)
12
being the second principal invariant of the rate of deformation tensor.
Fenner (1970) has shown that, for capillary flow:
dv I2
1 l z)2
dr
1 (
1 dvz - ° ~o dr
therefore:
n-1
Substituting (A3.14) into equation (A3.11) and using equation (A3.10):
dvz
dv r P dv )n-1 z = z z 0 (A3.15)
dr 2 dr 110
dr
274 -
n-1 dvz - - (Yo ( z))1/n r1/n = - C r1/n
dr Ito 2 1 (A3.16)
where: n-1
_ (~0 ( Pz))1/n
po 2 (A3.17)
Integrating equation (A3.16):
- C.. r 1/n+1
_ + A (A3.18) 1/n + 1
Applying the no-slip boundary condition, v = 0 at r = D/2, therefore:
vz
Cl {(2)1/n+1 - r1/714-11
1/n+1 (A3.19)
The flow rate: D/2
= 21r f vz r dr
therefore: Q Tr
C1 6-D) 1
/n+3
(1/n + 3) _ 2
(A3.20)
Thus, using equations (A3.20) and (A3.17), the pressure gradient, Pz, in
the pipe section can be worked out and hence the pressure drop, (dp)2, can
be calculated. Therefore, the total pressure drop between transducer and
melt inlet to the deflector is:
dp = (dp)1 + (dp)2 (A3.21)
- 275 -
APPENDIX A4
DERIVATION OF THE EQUILIBRIUM EQUATIONS
FOR A THIN CYLINDRICAL SHELL
The first step in the derivation of the equilibrium equations (see
Timoshenko & Woinowsky-Krieger (1959)) is to establish formulae for the
angular displacements of the sides BC and AB with reference to the sides
OA and OC of the element, respectively (Figure 6.9a). In these
calculations, the displacements u, v and w are considered to be very small.
The angular motions produced by each of these displacements are calculated
and the resultant angular displacements are obtained by superposition.
Let us first consider the rotation of the side BC with respect to
the side OA. This rotation can be resolved into three component rotations
with respect to the x, y and z axes. The rotations of the sides OA and BC
with respect to the x-axis are caused by the displacements vand w. Since
the displacements v represent motion of the. sides OA and BC in the
circumferential direction (Figure 6.8), if a is the radius of the middle
surface of the cylinder, the corresponding rotation of side OA about the
x-axis is v/a, and that of side BC is:
a (v dx) f ax
Thus, owing to the displacements v, the relative angular motion of BC with
respect to OA about the x-axis is•
1 3v a ax
(A4.1)
Because of the displacements w, the side OA rotates about the x-axis
- 276 -
through the angle aw/(a 4), and the side BC through the angle:
a aaa ax (ā4) dx
Thus, because of the displacements w, the relative angular displacement is:
a ax(
aw a a~) dx (A4.2)
Summing up equations (A4.1) and (A4.2), the relative angular displacement
about the x-axis of side BC with respect to side OA is:
1 DV 2
a( axax 4)`x
(A4.3)
The rotation about the y-axis of side BC with respect to side OA is
caused by bending of the generatrices in axial planes and is equal to:
— a2w (A4.4) ax2
The rotation about the z-axis of side BC with respect to side OA is due to
bending of the generatrices in tangential planes and is equal to:
a2v dx ax2
(A4.5)
The three expressions (A4.3), (A4.4) and (A4.5) thus give the three
components of rotation of the side BC with respect to the side OA.
Let us now establish the corresponding expressions for the angular
displacement of side AB with respect to side OC. Because of the curvature
of the cylindrical shell, the initial angle between the lateral sides AB
- 277 -
and.00 of the element OABC is d4. However, because of the displacements
v and w, this angle will be changed. The rotation of OC about the x-axis
is:
v + aw
a a 4
The corresponding rotation for AB is:
ā aad~ + ( ~ā + aad~) d~
Therefore, the relative rotation of AB with respect to OC is:
2
a ~ a ' +
a w) 4 42
Hence, the initial angle 4 must now be replaced by:
4+ ā (ā + az')d~ a~
(A4.6)
(A4.7)
In order to calculate the angle of rotation about the y-axis of side AB
with respect to the side OC, let us first consider the rotation of side OC.
During deformation, OC rotates through an angle equal to -Wax about the
y-axis and through an angle equal to av/ax about the z-axis. Side AB, as
a result of the displacement w, rotates about the y-axis by:
- 3x -ā (ax) d~
Hence, the relative rotation of AB with respect to OC due to the
displacement w about the y-axis is:
278 -
āc (ax) d~ (A4.8)
The rotation of OC in the plane tangent to the shell is:
av + a (av/ax) d ax act,
Because of the central angle do between the two sides, the latter rotation
has a component with respect to the y-axis equal to:
av āx dq (A4.9)
A small quantity of second order is neglected in this expression (see
Timoshenko & Woinowsky-Krieger (1959)). From the two expressions (A4.8)
and (A4.9), the total angle of rotation between the two sides, about the
y-axis is:
D2w Dv (
3x āx) d~ (A4.10)
Rotation about the z-axis of the side AB with respect to OC is caused by
the displacements v and w. Because of the displacement v, the angle of
rotation of side OC is av/ax, and that of side AB is:
9-0 ax a āc ~āx)
a (.4.4)
so that the relative angular displacement is:
a Dv
a 34) ~ax a dcp (A4.11)
-279-
Because of the displacement w, the side AB rotates in the axial plane by
the angle Wax. The component of this rotation with respect to the
z-axis is:
ax 4
•(A4.12)
Joining the two expressions (A4.11) and (A4.12) gives the relative angular
displacement about the z-axis of side AB with respect to OC as:
( a2v a-)d4
aq) ax ax (A4.13)
Having obtained the relative angular displacements, the next step is to
derive the three equations of equilibrium of the element OABC by projecting
all the forces on the x, y and z axes. Beginning with the forces
parallel to the resultant forces, x and Ncpx, and projecting them on the
x-axis, we obtain:
aN aN x dx a 4 ,
(1)x acl) ax ax a4
(A4.14)
Because of the angle of rotation represented by equation (A4.13), the
forces parallel to Ncl) give a component in the x-direction equal to:
— N a2v aw
a ax āx1 4 d (A4.15)
Because of the rotation represented by equation (A4.5), the forces parallel
to x~
give a component in the x-direction equal to:
— Nxq a2v dx a 4ax2
(A4.16)
- 280 -
Finally, because of the angles of rotation represented by equations (A4.4)
and (A4.10), the forces parallel to Qx and Q0 give components in the
x-direction equal to:
- Qx ax2
dx a d0 - Q0 (a92ax
+ 9v)
d0 dx (A4.17) Dx
Summing up all the forces represented by equations (A4.15) to (A4.17), the
equilibrium equation in the x-direction is obtained as:
aN dx a d0 + aN,x
d0 dx — N~ (
a2V
dx 8 ax ax)
d~ ax 30
- N a 2v dx a d0 -x0
ax2 ax2 dx a d0 - Q~ (
a0 a2w
ax) d0 dx =
Similarly, equilibrium equations in the y and z-directions can be obtained.
Assuming that the only external force acting on the element is a normal
pressure of intensity q, the three equilibrium equations can be simplified
to:
a a + aNOx a N a2v - a Q
a2w
ax DO x~
ax2 x
ax2
- N ( 92v aw)
0 a0 ax ax (aV a2w )
(ax a0 ax 0 (A4.18)
DN, aN 2 +a x'+a N a vv
30 ax x
ax2
av a2w Qx ( āx ax a0 )
+ N ( a2v aw) Q (1 + av + a2w ) _ 0x ax a0 ax a a4
a ,a(1)2
(A4.19)
- 281 -
a aQx ~ 3(4 + a N a22
+ x f ax + ax2ā~)
ax D(1) ax
2 2v + N, (1 + aaa~ + a aa 2)
+ N~x fax + ax a~) f q a = 0 (A4.20)
N = E h (ex + v eA) ~ 1 - v2
- 282 -
APPENDIX A5
DERIVATION OF THE DIFFERENTIAL EQUATIONS FOR THE
DISPLACEMENTS OF A THIN CYLINDRICAL SHELL
Timoshenko & Woinowsky-Krieger (1959) have shown that the resultant
forces, x, N~ and x~, a nd the moments x, M an d x~
can be expressed in
terms of the three strain components, ex, e and ex, of the middle surface
of the element OABC and the three curvature changes xx, x4 and xxo as
follows:
E h
1 - v2 (A5.1)
E h Nx~ N.cpx 2 (1 f v) excp
- D (xx t v x ,) Mo _ - D (x( f v xx)
(A5.2)
x~ _ - M x = D (1 - v) xx(P
where D is the flexural rigidity of the shell, defined as:
D E h3
12 (1 - v2) (A5.3)
Substituting equations (A5.1) and (A5.2) into the three simplified
equations of equilibrium and moments represented by equation (6.21), we
have:
a Eh (e +v e) + Eh a (e ) 1 - v2
,,`1. x 2 (1 f v) 8 x~
- 283 -
E v a (e q + v e) + a 1 2 x 2 (1 + v) ax(exp)
D (1 — v) ā (xx(1)) + ā 4 (x( + v xx) = 0
(A5.4)
E h (e, + v ex) — D (1 — v) ax a~ (xx~ ) — a D a (xx v x)
1 v2 ax2
a2 2 — D (1 — v) ax a ( xx(p ) — ā a 2
(Act,
+ v xx) + a
Timoshenko & Woinowsky-Krieger (1959) have shown that the strain
components and curvature changes can be represented in terms of the
displacements u, v and w as follows:
= au ex āx
_ av w _ au av
a acp a exp a 4 ax
(A5.5)
a2w x(P
1 Dv a2w a (aX + ax acp)
axe a2 DO2 ,
Therefore, substituting equation (A5.5) into equation (A5.4), the
differential equations for the displacements of a thin cylindrical shell
can be obtained as shown below:
a au av v 1— v a au av =
ax (ax + v a 4 a w)
+ 2a 4 (a a~ ax) 0
a av w au 1— v a au av aa (a a
4 a + v āx) + a 2 āx (a a + āx)
h2(1 — vJ a av a2w h2 a 1 av 1 a2w pawl
+ 12a ax (ax + ax a~) + 12a ac (a2 aT + a2
8 2 + v ax2
(A5.6)
and:
- 284 -
av w + v Du h2 (1 -v) a2 (av + a2w ) a 4 a ax 12a ax 4ax ax a~
h2 a a2 a2w ____v av 2L.121; _ h2 (1 - v) a2 av 12 axe (axe a2 a~ a2 a~2~ 12a ax a~
(av
a2w ax af
h2 a2 ( Z av 1 a2w v ) a 2w + q a (1 - v2)
12a42 a2 a~ a2 acp2 ax2 E h D
The above equations can further be simplified to give the displacement
equations as follows:
a2u 1+ v a2v v aw 1- v a2u 3032
2a ax 4 a x 2
1 a2v 1 aw 1+ v a2u 4. — a2v4. a 42 a 4 2 ax 4
a 2 ax2
+ h2 (1 — v) D2 + h2 a3w + 1 a3w 12a
3x2 a2 42 12a ax2 4 a2 43,
1 av _ w + au _ h2 (2 - v a3v 1 a 3y) a 4 a ax 12 a ax2 a(1) a3 a~3
h2 (2 a4w + a a4w
1 a4w) + a q (Z - v2)
12 a ax2 a~2
ax4 a3 a~4
E h
ad,2
(A5.7
i r r = R
(n - 1) (A6.2)
-285-
APPENDIX A6
GENERATION OF A CIRCULAR MESH OF TRIANGULAR ELEMENTS
In this appendix, generation of a circular mesh containing triangular
elements of approximately uniform size will be described, similar to the
one shown in Figure 6.12. Let nc be the number of elements at the centre
and nr be the number of nodal points along a horizontal radius. Normally,
n is assigned an integer value between five and eight to ensure that the
element angles at the centre are reasonably close to the angles of an
equilateral triangle of 60°, for reasons of convergence (see Fenner (1975)).
The value of nr is the main parameter determining the total number of
elements in the mesh. Let it be used to count outwards from the centre,
both the rings of elements and the rings of nodal points, ignoring the
centre point, where:
1 . ir nr - 1 (A6.1
If the mesh has a radius of R, and the radial spacing of the rings of
nodes is uniform, then the radius, r, of a typical ring defined by it is:
In order to keep the sizes of the elements approximately uniform, the
number of points per ring, nc ir, is proportional to the radius and hence
to the circumference of the ring. The innermost and outermost rings
contain, respectively, nc and nc (nr, - 1) nodal points and the total
number of nodes, nnp, is:
ne nr r - 1) -t 1 (A6.3)
-286-
The number of elements per ring is nc (2ir - 1) and since the innermost
and outermost rings contain, respectively, nc and nc (2nr - 3) elements,
the total number of elements, nel , is:
ne Z = nc (nr - 1) 2 (A6.4)
Let the origin of the global coordinates X and Y be at the centre of
the circle, and a be the angular coordinate measured in an anticlockwise
direction from the x-axis (see Figure 6.12). Let ie be used to count the
nodes in a particular ring in the anticlockwise direction starting from
e = 0, where:
nc Zr (A6.5)
The angular coordinate, e, of a typical node defined by it and ie can then
be given as:
e = (ie — 1) nti c r (A6.6)
and the global coordinates i and Yi can be expressed as:
. = r cos 6 , I. r sin 6 (A6.7)
ai, aj, ak
a1, a2, a3
B
Br
- 287 -
NOTATION
The mathematical symbols commonly used in this thesis are briefly
defined in alphabetical order in the following list. Where there are two
or more definitions for the same symbol, the relevant chapter or section
is indicated in parentheses. When no such restriction is given for one
of the definitions, it applies everywhere other than for the alternative
definitions. More detailed definitions may be found in or near these
equations. Many of the symbols and definitions used in the Appendices
are the same as in the main text, being derived from the section which
refers to the particular appendix.
Symbol Definition
A a function of temperature and pressure in Arrhenius equation (Section 2.3.1)
A cross-sectional area of the annulus shown in Figure 5.46a (Appendix A3)
A dimensionless coordinate along flow channel (Chapter 7)
A magnitude of resultant stream function gradient (Section 8.3)
a mean radius of a cylindrical shell (Section 6.4)
element dimensions (Figure 4.2)
constants in general boundary condition equation (4.17)
a function of temperature and shear rate (or shear stress) in pressure dependence equation (Section 2.3.2)
magnitude of resultant pressure gradient orthogonal to A (Section 8.3)
element dimension matrix
modified element dimension matrix
Brinkman number
- 288 -
Symbol Definition
b temperature coefficient of viscosity at constant shear rate
b., b., bk element dimensions (Figure 4.2)
C "consistency" (viscosity) in power-law equation
C mean circumference of annular gap shown in Figure 5.46a (Appendix A3)
melt specific heat at constant pressure
CZE conduction term defined in equation (7.38) (Chapter 7)
C1 a function of Pz in axisymmetric pipe flow analysis, defined in equation (A3.16) (Appendix A3)
C C2, C3 constants in stream function distribution equation (4.23) (Chapter 4)
C1 to C6 constants in displacement equations (6.47) and (6.48) (Section 6.5)
C2 r - cos a C2 (Chapter 4)
c a constant defined in equation (6.4) which is equal to the distance of the lowest point of the catenary, C, from the origin, 0 (Figure 6.2c)
c circumferential dimension (Section 8.3)
D capillary diameter
distance along streamline crossing a particular element (Section 5.5.4)
D pipe diameter (Appendix A3)
flexural rigidity of a shell defined in equation (A5.3) (Appendix A5)
D elastic property matrix defined in equation (6.60) (Section 6.5.1)
ratio of B to A (Section 8.3)
dissipation term defined in equation (7.37) (Chapter 7)
inner diameter of outer die shown in Figure 8.1 (Section 8.2)
d1, d2 inner and outer diameters of annulus shown in Figure 5.46a (Appendix A3)
D
Dse
D1
-289-
Symbol Definition
E activation energy in Arrhenius equation (Section 2.3.1)
E Young's modulus
E* modified E defined in equation (6.61)
EF' offset outer radius of inner point shown in Figure 6.4b (equation (6.16))
EF' offset inner radius of outer die shown in Figure 6.6b (equation (6.17))
E'F' outer radius of inner point shown in Figure 6.4b. (Section 6.3)
E'F' inner radius of outer die shown in Figure 6.6b (Section 6.3)
EG inner radius of intermediate point shown in Figure 6.4b (Section 6.3)
EK outer radius of outer point shown in Figure 6.6b (Section 6.3)
Econd heat output by conduction defined in equation (7.51)
Econv heat output by convection defined in equation (7.49)
Emech mechanical power input defined in equation (7.48)
E' dimensionless Econd
defined in equation (7.64) cond
Econv dimensionless. Econv
defined in equation (7.55)
E1, E2 Young's moduli for conductor and polymer, respectively (Chapter 6)
E, E2, E3 constants in pressure distribution equation (8.9) (Section 8.3)
element strain vector defined in equation (6.50) (Section 6.5)
er relative error defined in equation (4.43) (Section 4.3)
eT truncation error defined in equation (A2.4) (Appendix A2)
ex, ecb, exo
strain components (Appendix A5)
.. rate of deformation tensor i~
exx, eyy' ezz ,
exy' yz' ezx } direct and shear components of strain or strain rate
(Section 6.5)
- 290 -
Symbol Definition
F lateral force exerted by conductor on tip of inner point (Section 6.2)
vector of overall externally applied forces (Section 4.3)
F. subvector of externally applied force components at node i (Section 6.5.1)
G shear modulus (equation (6.45))
G vector of overall body forces applied to the nodes
G Griffith number (Chapter 7)
Gz Graetz number (Chapter 7)
G. subvector of overall body force components at node i
GZm) body force at node i due to element m
G2 effective shear modulus of polymer (Chapter 6)
H mean radial gap of annulus shown in Figure 5.46a (Appendix A3)
H local channel depth (Section 6.3)
h depth of flow channel
h thickness of a cylindrical shell (Section 6.4)
mean channel depth over an element
h1 channel depth at inlet (Chapter 7)
I integral defined in equation (4.8) (Section 4.2)
I second moment of area for bending
I1, I2, 13 principal invariants of the rate of deformation tensor
i subscript referencing nodal points
i node number defined in relation to NN (equations (8.16) and (8.17))
it counter for circular rings of nodes and elements (Appendix A6)
ie counter for nodes and elements around a circular ring (Appendix A6)
subscript referencing nodal points
- 291 -
Symbol Definition
node number defined in relation to NN (equations (8.16) and (8.17))
(K) overall stiffness matrix
k melt thermal conductivity (Chapter 3, Chapter 7)
k subscript referencing nodal points
k node number defined in relation to NN (equations (8.16) and (8.17))
(k)m element stiffness matrix
krs submatrix of k
k1, k3 functions of e,z and derivatives of ly (Section 4.2)
k2 function of z (Section 4.2)
L capillary length (Section 2.2)
L length of flow channel
L length of inner point (Section 6.2)
Ed axial length of deflector
Lp axial length of point
Z node number defined in relation to NN (equations (8.16) and (8.17))
M number of elements shared by a particular node (Section 4.4)
M 4x bending moments per unit length of axial section and a section perpendicular to the axis of a cylindrical shell, respectively
x~ twisting moment per unit length of an axial section of a cylindrical shell
M1 inner diameter of intermediate point shown in Figure 8.1 (Section 8.2)
M1 to M6 elements sharing node NN shown in Figure 8.11 (Section 8.3)
m subscript or superscript referencing element number
m superscript denoting iteration numbers (Section 4.3)
m node number defined in relation to NN (equations (8.16) and (8.17))
Symbol
N
NN
NXPT
Na
Nci)' x' x(15
- 292 -
n
n
n
n
nc
nel
nnp
P
P
Definition
total number of elements a particular streamline crosses (Section 5.5.4)
number of cycles of iteration for convergence (Section 5.5.2)
a node in the mesh shown in Figure 8'.11 (Section 8.3)
number of points in the x-direction
Nahme number
membrane forces per unit length of axial section and a section perpendicular to the axis of a cylindrical shell
inner diameter of outer point shown in Figure 8.1 (Section 8.2)
power-law index
subscript referencing node numbers
outward normal to the boundary of a solution domain (Figure 4.1) (equations (4.13) to (4.17))
number of cycles of iteration (Section 4.3)
node number defined in relation to NN (equations (8.16) and (8.17) )
gradient of the graph of logarithmic shear stress plotted against apparent shear rate
number of elements at centre of a circular mesh
number of nodes along a horizontal radius of a circular mesh
total number of elements (Appendix A6),
total number of nodal points (Appendix A6)
capillary rheometer reservoir pressure (Section 2.2)
pressure defined in equation (7.34) (Chapter 7)
dimensionless pressure defined in equation (7.35) (Chapter 7)
Pecl et. number
pressure gradient in transverse x-direction
293 -
Symbol Definition
Pz pressure gradient in downstream z-direction
PZ inner diameter of inner point shown in Figure 8.1 (Section 8.2)
P2 outer diameter of inner point shown in Figure 8.1 (Section 8.2)
p pressure
(dp)1 pressure drop in annular section shown in Figure 5.46a (Appendix A3)
(dp)2 pressure drop in cylindrical section shown in Figure 5.46a (Appendix A3)
(dp) total pressure drop between transducer and melt inlet to the deflector, defined in equation (A3.20) (Appendix A3)
Q volumetric flow rate in capillary (Section 2.2)
Q volumetric flow rate in annular section shown in Figure 5.46a (Appendix A3)
Q volumetric flow rate per unit width normal to the section shown in Figure 7.1 (Chapter 7)
Qs, Qx, Qe volumetric flow rates per unit width in the s, x and e directions
Qe Qx shearing forces parallel to z-axis per unit length of an axial section and a section perpendicular to the axis of a cylindrical shell, respectively (Section 6.4)
q intensity of a normal pressure acting on a shell (Section 6.4)
R gas constant in Arrhenius equation (Section 2.3.1)
radius of circular mesh (Appendix A6)
Ri subvector of internally applied forces (and moments) at node i of an element
m vector of internal forces (and moments) applied to an element at its nodes
r radial coordinate
r mean distance of flow channel from axis
mean value of r over an element
- 294 -
Symbol Definition
S dimensionless channel depth (Chapter 7)
Sm average S for two consequent nodes, defined in equation (7.41) (Chapter 7)
Sout value of S at exit from flow channel (Chapter 7)
s resultant direction of flow
s distance measured along the cable (Section 6.2)
T absolute temperature (equation (2.9))
T melt temperature
T period of one complete cycle of the Andouart pressure fluctuation (Section 5.4.2)
T internal tensile force at a point D, directed along the tangent to the cable at D (Section 6.2)
T* dimensionless melt temperature defined in equation (7.11)
bulk mean temperature defined in equation (7.50)
Tb temperature of boundary
TZn bulk mean temperature at inlet to flow channel
Tout bulk mean temperature at exit from flow channel
Tin dimensionless bulk mean temperature at inlet to flow channel
Tout dimensionless bulk mean temperature at exit from flow channel
TXY residence time along streamline XY (Section 5.5.4)
To reference temperature for viscosity determinations
To tension force at point C of the cable shown in Figure 6.2b (Section 6.2)
T1
~dT )n
t1-a U.
melt temperature at inlet
time taken to cross the nth element along a streamline (Section 5.5.4)
stress tensor (Section 2.5)
dimensionless velocity in x-direction (Chapter 7)
- 295 -
Symbol Definition
force component in x-direction applied internally to an element at its node i (Section 6.5.1)
U.
Vi
w
w
w
X, Y
3 s1
xi
velocity component in x-direction
displacement in x-direction (Sections 6.4 and 6.5)
local mean velocity in s-direction (Section 3.5)
mean velocity of melt within the element (Section 5.5.4)
mean velocity (Chapter 7)
force component in y-direction applied internally to an element at its node i (Section 6.5.1)
velocity component in y-direction
velocity in s-direction (Section 3.5)
displacement in y-direction (Sections 6.4 and 6.5)
velocity components in the general x2 and x, coordinate directions., respectively (Section 3.4)
velocity component in the axial z-direction of a pipe (Appendix A3)
resultant of distributed load supported by the portion of cable CD (Section 6.2)
velocity component in z-direction
load per unit length of cable, s (Section 6.2)
displacement in z-direction (Sections 6.4 and 6.5)
dimensionless coordinates along and normal to flow channel (Chapter 7)
local components of the body forces per unit volume acting on the continuum in the coordinate directions (Section 6.5)
global Cartesian coordinates (Appendix A6)
coordinates along and normal to flow channel (Chapter 7)
Cartesian coordinates
coordinate system used for a cylindrical shell (Figure 6.8) (Section 6.4)
general Cartesian coordinate (Section 3.4)
u
V
V
V i., V j
- 296 -
Symbol Definition
z constant defined in equation (6.27) (Section 6.4)
z axial coordinate (Sections 3.2, 3.3, 3.5 and Chapter 4)
z' local axial coordinate (Chapter 4)
Greek Symbols
a pressure coefficient of viscosity (Section 2.3.2)
a inclination of flow channel to axis
a angle by which the head on the catenary plant is off- line (Figure 6.1) (Section 6.2)
a stability term (Section 7.6)
a angle A makes with the z-axis (Figure 8.12) (equations (8.7) and (8.8))
a function of element dimensions (equations (8.14) to (8.17))
a. nodal point constant in temperature development analysis (Section 7.6)
6 angular misalignment between the axes of the points forming the flow channels (Section 5.6.1.2)
function of element dimensions (equations (8.14) to (8.17))
s. nodal point constant in temperature development analysis (Section 7.6)
S1' 2, (33 locally constant functions of stream function gradients (Section 8.3)
shear rate (in simple shear flow)
angle between the outward normal, n, and the x-axis (Figure 4.1) (equations (4.13) to (4.17))
function of element dimensions (equations (8.14) to (8.17) )
mean shear rate in flow channel
apparent shear rate at capillary wall
nodal point constant in temperature development analysis (Section 7.6)
I •
- 297 -
Symbol
Yo
Definition
reference shear rate in empirical power-law constitutive equation
difference operator defining change in the subsequent quantity
Am element area in (z,6) plane
d deflection of tip of inner point (Section 6.2)
(ō) overall vector of nodal point stream function values
ōi unknown such as displacement or stream function at node i
Si subvector of displacements at node i (Section 6.5.1)
si value of S• obtained from equation (4.41) to be substituted into equation (4.42) (Section 4.3)
6.. Kronecker delta se
6 nodal point constant in temperature development analysis. (Section 7.6)
Sm element vector of nodal point stream function values (Chapter 4)
sm element displacement vector (Chapter 6)
a tolerance limit; the prescribed value of the degree of accuracy
constant of proportionality in trucation error term (Appendix A2)
an unknown (nodal point stream function value) (Chapter 4)
111 generalised viscosity (= 2p)
2 generalised cross-viscosity
e angular coordinate
e angle the cable makes at point D, with the horizontal (Figure 6.2b) (Section 6.2)
e angle the cylindrical shell subtends at the centre (Figure 6.10) (equations (6.30) and (6.31))
e' local angular coordinate (Chapter 4)
K diameter ratio of annulus (d2/d1 ) shown in Figure 5.46a (Appendix A3)
- 298 -
Symbol Definition
a parameter defined in equation (4.2)
a stability parameter (Section 7.7.1)
melt viscosity (in simple shear flow)
viscosity at mean shear rate
apparent viscosity in capillary flow
effective viscosity (in the power-law equation) at the mean melt temperature and reference shear rate, y
o
effective viscosity (in the power-law equation) at reference temperature, To, and shear rate, yo
polymer viscosity (Chapter 6)
Poisson's ratio
v* modified v defined in equation (6.61)
v1, v2 Poisson's ratios for conductor and polymer, respectively (Chapter 6)
A annular flow parameter defined in equation (A3.6)
dimensionless pressure gradient
dimensionless flow rate defined in equation (A3.1) (Appendix A3)
melt density
densities for conductor and polymer, respectively (Chapter 6)
element stress vector
u
Pa
um
112
Trp
.JrQ
P
P1. P2
6xx' Qyy' azz' 6xy' yz azx
} direct and shear components of stress (Section 6.5)
Tza
Trz
T xy
shear stress (in simple shear flow)
mean shear stress defined at the mean shear rate (Appendix A3)
viscous stress tensor
shear stress in a pipe (Appendix A3)
viscous stress (Chapter 7)
T
T
- 299 -
Symbol Definition
a function of power-law index, defined in equation (3.23) (Chapters 3 and 4)
angular position around the cable. 4 is measured in the clockwise sense looking along the cable in the direction of motion, and 4) = 0 corresponds to the lowest point of the cable as it emerged from the crosshead
cylindrical shell coordinate shown in Figure 6.8 (Section 6.4)
a functional obtained by integration over the solution domain
contribution of a typical element to x
changes of curvature of a cylindrical shell in axial plane and in a plane perpendicular to the axis, respectively
twist of a cylindrical shell
stream function
nodal point values of stream function
stream function value along inner boundary (Section 6.3)
stream function value along outer lower boundary (Section 6.3)
stream function value along outer upper boundary (Section 6.3)
xxcb
11)
wk
%lb
*oub
w over-relaxation factor
` opt optimum over-relaxation factor
300 -
Abbreviations
AEI Associated Electrical Industries
FD finite difference
FE finite element
FEM finite element method
IEC Industrial Engineering and Chemistry
LDPE low density polyethylene
XLPE crosslinking polyethylene
- 301 -
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FINITE ELEMENT ANALYSIS OF POLYMER MELT FLOW
IN CABLE-COVERING CROSSHEADS.
by
R.T. Fenner and F. Nadiri
(Mechanical Engineering Department, Imperial College of Science and Technology, London SW7 2BX, UK)
Abstract
A finite element method is presented for the analysis of isothermal non-
Newtonian polymer melt flow in narrow channels of complex shape. The
particular application considered is flow in cable-covering crossheads. The
geometric flexibility of the finite element method allows a mesh of triangular
elements to be constructed to suit the shape of the flow channel. Computed
results obtainable from the analysis include the distribution of polymer layer
thickness on the finished cable, together with the extrusion pressure required
to maintain a given flow rate of melt. Some typical thickness distribution
results are presented as an introduction to experimental verification of the
method and its application to crosshead design.
- 2 -
INTRODUCTION
In many types of processing equipment, molten polymers are required to flow
along channels of complex shape. Such melt flows tend to be difficult or
impossible to analyse satisfactorily by simple methods, and successful channel
designs are often developed by slow and expensive trial-and-error methods.
Finite element methods of analysis can be profitably applied to many such
problems to assist in the design process. For example, a method similar to the
one described here has been successfully used to analyse melt flow in pipe
dies (1) .
The particular problem to be considered is that of flow in crossheads used
in the covering of high-voltage electrical cables. It is not uncommon for a
single head unit to be used to apply two or three layers of different materials.
during one pass of the conductor. For example, in the trials used to test the
present method of analysis, three layers were applied to a tape-covered stranded
copper conductor. The thin inner and outer layers, which served as screens,
were of the same material supplied by one screw extruder. The much thicker
intermediate layer, which provided the required electrical insulation was of
another material supplied by a second machine. The two extruders were
connected to opposite sides of the head with their screw axes at right angles
to the direction of motion of the conductor through the head. For each layer,
the problem is therefore to design a system of flow channels which accepts a
side-fed supply of melt and distributes it into a tube of uniform thickness
which is then extruded as part of the cable.
Figure 1 shows one commonly used form of arrangement for attempting to
achieve the desired uniformity. A narrow radial gap between concentric
cylindrical and conical surfaces, the outer members of which have been removed
for illustration purposes, serves to distribute the melt. As the melt tends
to take the shortest path from the inlet to the channel exit, this path is
deliberately blocked by a heart-shaped area which fills the radial gap and
forces the melt flow to follow longer paths of more uniform length. The
cylindrical portion of the component shown is known as the deflector, while the
subsequent conically tapered portion is termed the point. Channel geometry, and
therefore the flow, are intended to be symmetrical about the centre line of the
heart-shaped blockage.
Figure 2 shows the shape of one half of the flow channel plotted on the
z,A plane, z being the axial coordinate and 6 the angular coordinate measured
from the line of symmetry through the flow inlet. The region bounded by points
A, B, C and D is on the deflector, while that bounded by C, D, E and F is on the
point, as indicated in Figure 1. Also shown in the flow channel are some
triangular finite elements which are discussed later. Clearly, flow paths
between the inlet boundary AB and outlet EF are of reasonably uniform length.
It should be noted that the channel depth is often reduced by tapering in the
axial direction in both the deflector and point regions. Indeed, in the
deflector region, the channel depth may also be varied in the circumferential
direction to improve the flow distribution.
Uniformity of screen and insulation thicknesses in the finished cable is of
considerable commercial importance. In order to satisfy electrical
performance criteria, each layer of polymer must be of a prescribed minimum
thickness. If the actual thickness varies significantly around the
circumference of the cable, excess material will be contained in the thicker
parts of the layer. A rational method for designing deflector flow channels is
therefore highly desirable. In this paper, the formulation of the finite
element method of analysis is presented. Details of some experimental
verifications of the method and its application to the design of crossheads will
be published separately.
— 4
ANALYSIS OF NARROW CHANNEL FLOW
Provided a channel containing a flowing melt can be described as narrow,
the analysis of the flow may be treated in a relatively simple manner. A
narrow channel is one in which one of the channel dimensions, normal to the
direction of flow, is small compared to the other two dimensions, and only
varies slowly over the region of interest. In the present context of cable-
covering crossheads, the radial depth of the channel is small compared to its
axial and circumferential dimensions, and only varies slowly in these directions.
Figure 3a shows a typical axial cross-section through the flow channel.
The channel depth, h, is small, such that:
h « r , h «
r and L being the mean radius and overall axial length, respectively, and h is
subject to only small local variations:
ah az 1 (2)
Pearson (2,3,4) has shown that such conditions are necessary for the lubrication
approximation to be applicable, which means that the flow can be treated as
locally fully developed between flat parallel surfaces. As far as channel
taper is concerned, Benis (5) showed that the lubrication approximation holds
for taper angles up to about 100. Neglecting melt inertia and elastic effects
(4), the only effect likely to invalidate the lubrication approximation is that
due to thermal convection. For present purposes, however, the flow is assumed
to be isothermal in the sense that any temperature variations within the flow
do not affect velocity profiles. Justification for this assumption is provided
in the next section.
The analysis which follows is similar to that originally presented by
Pearson (2), extended to allow for conical channel geometry. At a particular
point in the flow channel, the local velocity profile is of the form shown in
Figure 3b. Let this resultant profile be in the direction s, which in general
is neither axial nor circumferential, and let v(y) be the velocity. The local
radial coordinate, y, is measured from the mid-surface of the channel, itself a
distance r from the axis. If V is the local mean velocity and Qs is the
volumetric flow rate per unit width normal to the s-direction, then:
• +zh
Qs = I vdy = h -Zh
(3)
The form of the velocity profile depends on the non-Newtonian viscous
properties of the melt concerned. For practical purposes, a power-law
constitutive equation relating shear stress, T, to shear rate, y, is generally.
the most useful (6):
uo I 0
I n-1
(4)
where n is the power-law index and the reference viscosity at the processing
temperature and reference shear rate, yo. Such a relationship provides a good
fit of rheological data over comparatively wide ranges of shear rate. Given
the constitutive equation, the relationship between flow rate and pressure
gradient in the s-direction may be derived (6) as:
2s Qs
h3
where: =
1 ( 2n in 2 n+1 '2n + 1'
u is the viscosity at the mean shear rate, V/h:
- 6 -
11 = ( Qs ) n-1 h2 yo
(7)
Figure 3c shows a typical small portion of the flow channel inclined at an
angle a to the axis of the conductor. Let x be the coordinate along the channel
in the axial plane, and let Qx and Qe be the volumetric flow rates in the x and
circumferential directions, per unit length in the circumferential and x
directions, respectively. Conservation of mass in incompressible steady flow
requires that: aQ
ex Cr Qx). f 6 0 (8)
This equation is automatically satisfied by the following stream function, 4):
r Qx =ae
Now, using equations (5) to (7), the pressure gradients in the x and 0 directions
are given by:
_ Qx u .~ = rQe u. ax h3 ae h3
where the resultant flow rate used in evaluating u is given by:
,-2 = x 2 + Q
e2
From the fact that the pressure, p, must satisfy the mathematical identity:
ae ( ) - ax ( ) = 0
30
ae
(9)
(10)
(12)
can be derived the result:
7 -
a ( u a.(p) + a (r u 31p) ae r h3 ae 3x h3 ax
(13)
The most convenient coordinates to use for the analysis of a complete flow
channel are z and 8 rather than x and e. As z = x cos a, equations (13) and
(11) become:
a ( u ) + cos a (r u cos a arm') = 0 (14)
ae r h3 ae az h3 az
Qs 2 = (r 30 ) 2 + (cos (15)
THE ISOTHERMAL FLOW ASSUMPTION
An initial assumption of the above analysis of narrow channel flow is that
the flow is isothermal. That is, that any temperature variations within the
flow, due to either the dissipation of mechanical energy or to heat conduction
at the flow boundaries, have negligible effects on the viscous properties of the
melt. In order to confirm the validity of this assumption, it is appropriate
to consider a somewhat simplified form of the actual melt flow in the crosshead.
Figure 4 shows a flow channel formed by two flat parallel stationary surfaces a
distance h apart. Melt is admitted to the channel at, say, a constant
temperature T1, and the flow boundaries are maintained at temperature Tb. If
the volumetric flow rate is maintained at Q per unit width normal to the plane
shown, the main interest lies in the magnitudes of the temperature changes in
the melt as it moves a distance L along the channel.
Let x and y be Cartesian coordinates as shown in Figure 4, and let u be the .
melt velocity in the x direction. Suppose that, in addition to being a power-
law function of shear rate as indicated by equation (4), melt viscosity is also
an exponential function of the local temperature, T(x,y), such that (3,6):
po = p' exp (- b (T - T')) (16)
where b is the temperature coefficient of viscosity, and uō is the value of the
reference viscosity at reference temperature T'. The energy equation governing
heat transfer, and therefore temperature, can then be written in the following
dimensionless form:
714-1 Gz U aX -
a2T* 4-
aY G ll exp (- T*) (17)
8Yz
X and Y are dimensionless coordinates:
X = x/L , Y = y/h (18)
U is dimensionless velocity:
= u/V (19) '
where V is the mean velocity Q/h, and T* is a dimensionless temperature defined
as:
= b (T — T1) (20)
The parameters Gz and G are, respectively, the Graetz number and Griffith
number for the flow (7):
Gz p
p V h2
kL
bTYh2 = k
where p, C and k are melt density, specific heat and thermal conductivity,
respectively. y is the mean shear rate, V/h, and T is the mean shear stress
defined at shear rate y and temperature T1. The Graetz number expresses the
relative importance of thermal conduction through the depth of the channel and
convection in the direction of flow. On the other hand, the Griffith number
determines whether heat generation causes temperature changes within the flow
large enough to alter the velocity profiles. The flow can only be described as
isothermal if G « 1.
In order to solve equation (17), both the distribution of velocity U and
the boundary conditions are required. The former is readily determined and the
latter are given by:
(21)
(22)
- 10 -
T* = 0 at X = 0 = G/Br at Y = ± z (23)
where Br is a Brinkman number for the flow:
Br .7-;17- 7 z2
k (Tb - T1 ) (24)
which expresses the relative importance of heat generation and heat conducted
from the boundaries in changing melt temperatures.
Taking typical channel dimensions, operating conditions and material
properties for melt flow in cable-covering crossheads, the values of the
dimensionless parameters are found to be of the following orders of magnitude:
Gz = 20 , G = 0.1 , Br > 1
While the magnitude of Gz implies that the heat transfer within the flow is
dominated by thermal convection, the size of G ensures that the resulting
temperature changes have only small effects on the velocity profiles, and the
isothermal flow assumption is valid. This conclusion is confirmed by detailed
computations of developing temperature and velocity profiles which show that,
for example, overall pressure drops are within one per cent of the values
computed for isothermal flows.
APPLICATION OF A. FINITE ELEMENT METHOD
Although finite element methods were originally developed for digital
computer use in the stress analysis of solid structures and components (8),
they have also been applied to fluid mechanics and heat transfer problems,
including the slow non-Newtonian flows encountered in polymer processing
operations (9,10). Tadmor et al (11) have already proposed a "Flow Analysis
Network" technique which is a simple form of finite element method applicable
to narrow channel flows. While the method described here is specifically
designed to solve the cable-covering crosshead problem governed by equation (14),
the same approach is applicable to other narrow channel flows, and indeed to
other problems governed by mathematically similar equations.
Figure 2 shows one half of the complete solution domain in the (z,8) plane
divided into triangular finite elements. Although the straight-sided elements
cannot follow the curved boundaries AC and BD exactly, with a reasonable number
of elements the maximum deviations are acceptably small. It should be noted
that the number of elements across the width of the flow is constant, which
means that the elements near the narrow inlet boundary AB are much smaller than
those near the deflector and point outlets, CD and EF, respectively. It is the
ability to fit complex geometric boundary shapes and to allow varying densities
of elements within the solution domain that makes finite element methods
attractive. Palit & Fenner (9) have compared and contrasted finite element and
finite difference methods for problems of the present type, and discussed the
advantages of using simple triangular elements for non-Newtonian flow problems.
In order to formulate a finite element method to solve equation (14), the
fact that it is of the quasi-harmonic type (12) should first be noted. A
method similar to that described by Palit & Fenner (9) is therefore appropriate.
A variational approach can be used to solve the governing partial differential
equation by seeking a stationary value for a functional x which is defined by an
- 12 -
appropriate integration of the unknowns over the solution domain. Following a
derivation similar to that given by Fenner (12), it can be shown that the
required stationary condition is obtained when:
= f j {2 T a (atP)2 ~ r p cost a a (aw)2T do dz = C an r h3 an ao z h3 an az (25)
holds for all of the unknowns, n, required to be found. In the present method,
the values of stream function, *, at the corners or nodes of all the triangular
elements are chosen as the unknowns. The only restriction on the validity of
equation (25) is that on the boundaries either the value of 4' must be prescribed,
or its first derivative with respect to distance normal to the boundary must be
zero (12). This restriction is satisfied in the present problem which is
subject to the following boundary conditions:
iy = 0 on BDF , $ = 1 on ACE
aDIP = 0 -a-1-on AB. , = 0 on EF
Figure 5 shows a typical triangular element, numbered m, in the solution
domain. It has nodes at its corners numbered i, j and k and dimensions as
shown. Local coordinates z' and 0' are parallel to z and 0 but have their
origin at node i. Assuming a linear distribution of stream function over the
element:
11)(z 1,0') = C1 + C2 z' + C3 0'
where C1, C2 and C3 can be found in terms of the nodal point values of stream
function, 4'i, 4j and 4'k, as follows:
(26)
(27)
- 13 -
[c2 - z [ C - 2Am 3
2pm [B] [al m (28).
where Am is the area of the element in the (z,8) plane:
= z (ak bj - aj bk) (29)
and [B] is a matrix of element dimensions:
[B] = bi b j bk
ai a. a k
(30)
Now, because the inter-element boundaries make no contribution to the
integral expressed in equation (25):
aX aX(m) - 3n an mt
0 (31)
where x(m)is the contribution of typical element m to the total value of X.
With the linear distribution of i over the element given by equation (27), the
following approximation may be used:
a (m) r ū DC 2 ū aC3} X = 8 { cos a C t + C an m 3
C2 an r h3 3
an
(32)
where r and h are the mean values of radius and channel depth over the element.
In practice, the variations of r and h over an element are. sufficiently small
for mean values to be used for the present purpose. While the angle a also
-14-
varies along the flow channel, the mesh is chosen such that the slope of the
channel is constant over any one element.
The derivatives of X(m) with respect to the three nodal point values of 4
associated with element m may therefore be expressed as:
aX (m) DC' 2 DC 3_
CB,jT [B'] CSi (33)
31P2
DX (m)
a,Pi
DC' 2
a~,2 3C3
at,j
aX (m)
h3 aip
aC2
a,P. aC3
4am r h3 m
a'Pk a*k
where C2 = r cos a C2:
b2 r cos a b~ r cos a bk r cos a
a. a. ak.
and the superscript T indicates a matrix transposition. Combining equations
(31) and (33):
u _ [B'JT [B] [ ]m = [A.] CS, = [a] m 4A rh3
(35)
where [S] is a vector containing the stream function values for all the nodal
points in the mesh. Square matrix [4], which in the finite element method
context is often referred to as the overall stiffness matrix, contains
coefficients assembled from the properties and dimensions of the individual
elements (12).
Before equations (35) can be solved for the unknown values of , the
boundary conditions defined by equations (26) must be imposed by appropriately
modifying equations associated with boundary nodes at which the value of i is
(34)
( Zl n-1
u = C2 + C3 )
u
(27) :
(36).
- 15 -
prescribed. The equations are not linear because the element mean viscosities,
p, are dependent upon the local gradients of 1i. Using equations (7), (15) and
The method of solution used is that described by Palit & Fenner (9) in which the
iterative successive over-relaxation approach is employed. The equations are
first linearised by assuming suitable constant values for the element viscosities,
a few iterations are then performed to estimate the nodal point values of stream
function. Using these values to update the viscosities, the process is repeated
until satisfactory convergence is achieved.
Having computed the stream function distribution over the solution domain
in terms of values at the nodal points of the mesh, other results may be
derived as required. For example, the pressure distribution and hence the
overall pressure difference between flow inlet and outlet may be computed as
follows. For each element, the mean pressure gradients in the a and 6
directions can be found with the aid of equations (10), (9) and (27) as:
ap - _ u C3 ap r - u cos a C 8 z - (I) 7 3 r cos a ' @6 3 z (37)
where C2 and C3 can be found from the nodal point stream function values using
equations (28). The pressure gradients at the nodal points may then be found
by averaging over the values of the gradients associated with the elements having
a particular point as a node. Therefore, working from a known pressure at
either the inlet or outlet boundary, pressures at all the nodes can be computed
by integrating numerically along lines of nodes joining the inlet and outlet
boundaries.
The flow rate of molten polymer passing the outlet boundary between any
-16-
adjacent pair of nodal points is equal to the difference between the stream
function values at these points, Ai. Assuming that there is no subsequent
circumferential redistribution of material, the corresponding thickness of the
polymer layer on the finished cable will be proportional to AVA8, where AO is
the difference in 0 coordinate between the two nodes. Hence, the ratio of
local to mean thickness can be computed as a function of angular position around
the cable. Given the cable speed and the total flow rate of polymer forming a
particular layer, its mean thickness can also be determined if required.
-17-
TYPICAL PROBLEM AND RESULTS
In order to illustrate the application of the method of analysis to a
typical problem, the deflector and point profile shown in Figure 2 is
considered. The main dimensions (Figure 3) are Ld = 170 mm, Lp = 82 mm,
r (deflector region) = 45 mm, a (point region) = 23.5°. The channel depth,
h, decreases with axial position along the deflector from 6.2 mm at inlet to
3.4 mm, and over the point changes to 9.4 mm at outlet. Across-linking low
density polyethylene having n = 0.39 and uo
= 2.26 x 104 Ns/m2 at yo = 1 s-1 is
processed at a temperature of 120°C, and is supplied at a rate of
10.7 x 10-6 m3/s to each half of the deflector.
Two solution domains are considered, firstly just the deflector region BDCA
(Figure 2), and then the whole region BDFECA, the same constant pressure
condition being used on the outlet boundary in each case. The purpose of
treating these two domains is to compare the melt thickness distribution
obtained using the deflector alone with that achieved when the point region is
added. A finite element mesh of the form shown in Figure 2 is used. The
actual number of nodes employed along AB, CD and EF is 13, along AC and BD also
13, but along CE and DF only 6 are necessary. These numbers are chosen as a
result of a compromise between accuracy and cost of computation. Further
refinement of the mesh causes only insignificant changes in the final results.
Figure 6 shows the predicted dimensionless thickness distributions of the
polymer layer on the finished cable. Clearly, the deflector alone gives a poor.
thickness distribution, but the addition of the point region results in a
considerable improvement. This is because the flow channel dimensions are such
that the pressure drop over the point is substantially greater than that over
the deflector. Therefore, the axial symmetry of the tapered point region
dominates the lack of symmetry associated with the deflector to give a reasonably
acceptable thickness distribution. There is, however, scope for improving the
- 18 -
design of the deflector. Adaptation of the finite element methods of analysis
for this purpose will be described in another paper. Experimental verification
of predicted thickness distributions of the present type will also be presented,
together with comparisons of computed and measured overall pressure differences.
ACKNOWLEDGEMENTS
The authors wish to thank AEI Cables Limited, both for their financial
support of the work described and for their practical advice and many helpful
suggestions. In particular, it has been a great pleasure to work with
Messrs E.T. Lloyd, L.M. Sloman and F.T. White.
- 19 -
NOMENCLATURE
[A] overall "stiffness" matrix
ai, a, ak element dimensions (Figure 5)
[B] element dimension matrix
[B'] : modified element dimension matrix
b temperature coefficient of viscosity
bi, b, bk . element dimensions (Figure 5)
Br Brinkman number
C1, C2, C3 constants in stream function distribution (equation (27))
C2 r cos a C2
Cp . melt specific heat
G Griffith number
Gz Graetz number
h : depth of flow channel
h . mean channel depth over an element
i, j, k subscripts referencing nodal points'
k melt thermal conductivity
L . length of flow channel
Ld axial length of the deflector
Lp . axial length of the point
m subscript or superscript referencing element number
n power-law index
p . pressure
Q
: volumetric flow rate per unit width of channel
Qs, Qx' Q0 . volumetric flow rates per unit width, in the s, x and 0 directions
r mean distance of flow channel from axis
mean value of r over an element
s resultant direction of flow
a
Y
-20-
T . melt temperature
T,
: melt temperature at inlet
Tb : boundary temperature
T* dimensionless melt temperature
T' : reference temperature for viscosity determination
U . dimensionless velocity in x-direction
velocity in x-direction
V local mean velocity in s-direction
V : velocity in s-direction
X, Y dimensionless coordinates along and normal to flow channel
x, y : coordinates along and normal to flow channel
z axial coordinate
z . local axial coordinate
inclination of flow channel to axis
shear rate
mean shear rate in flow channel
reference shear rate for viscosity determination
element area in (z,0) plane
overall vector of nodal point stream function values
element vector of nodal point stream function values
an unknown (nodal point stream function value)
0 angular coordinate
0` local angular coordinate
A0 . difference in angular coordinate between adjacent nodes
viscosity at mean shear rate
Po . reference viscosity (at reference shear rate)
Pc; viscosity at reference temperature T' 0
- 21 -
p melt density
t . shear stress
. .shear stress at mean shear rate
a function of power-law index (equation (6))
X : a functional obtained by integration over the solution domain
(m) X : contribution of a typical element to x
stream function
4'i3, 1Pk • nodal point values of stream function
AI . difference in stream function values between adjacent nodal points
-22-
REFERENCES
1. R.T. Fenner, Plastics & Polymers, 42, 114 (1974).
2. J.R.A. Pearson, Trans. Plastics Inst., 30, 234 (1962).
3. J.R.A. Pearson, "Mechanical Principles of Polymer Melt Processing",
Pergamon Press, Oxford (1966).
4. J.R.A. Pearson, "The Lubrication Approximation Applied to Non-Newtonian
Flow Problems: A Perturbation Approach", Proceedings of the Symposium on
Solution of Non-Linear Partial Differential Equations. (ed. W.F. Ames).
Academic Press. New York. 73 (1967).
5. A.M. Renis. Chem. Eng. Sei.. 22. 805 (1967).
6. R.T. Fenner. "Extruder Screw Design", Iliffe. London (1970).
7. J.R.A. Pearson, "Heat Transfer Effects in Flowing Polymers", in "Progress
in Heat and Mass Transfer", (ed. W.R. Schowalter et al), Pergamon Press,
New York, 5, 73 (1972).
8. 0.C. Zienkiewicz, "The Finite Element Method in Engineering Science",
McGraw-Hill, London (1971).
9. K. Palit & R.T. Fenner, A.I.Ch.E.J., 18, 628 (1972).
10.. K. Palit &R.T. Fenner, A.I.Ch.E.J., 18, 1163 (1972).
11. Z. Tadmor, E. Broyer & C. Gutfinger, Polym. Eng. Sei., 14, 660 (1974).
12. R.T. Fenner, "Finite Element Methods for Engineers", Macmillan, London
(1975).
Figure Captions
Figure 1 : Typical deflector and point used inside a cable-covering crosshead
Figure 2 : One half of the flow channel plotted on the (z,(3) plane, showing a
mesh of triangular finite elements
Figure 3 : Flow channel geometry and coordinates:
(a) Typical axial cross-section
(b) Local velocity profile over the channel depth
(c) Typical inclined portion of the flow channel
Figure 4 Flow channel between flat parallel surfaces
Figure 5 Typical triangular finite element
Figure 6 : Computed final thickness distributions for deflector alone and
combined deflector and point
Point
Melt in
c E . ---~-.......-------,~----. - e = IT
8
A F z
--. ___ -----..:Il.--~-____ - 8 = 0
Z= 0 z=L.
( b) yt h
h
\\\\\\1\\\\\\
(a)
17/ ////,
L
Wm-
•
FIG 3
\\\\\T ( x + 2 h)-T b \\\\\\ 1
~ t
_ i \\\ \\\\\\ T (x , 2 h _) -T L
T (0,y)= T 1
FIG 4
-a i i
bk
r
FIG 5
1.4
C 0 a) E 0.8
U)
C 0.6 U
jO•4
0.2 o Deflector only
o Deflector+ point
60 120 e o 180. 240 300 360 0
FIG 6
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