Melting in single screw extruders: models, calculations, screw design

Melting in single screw extruders

Citation for published version (APA):

Meijer, H. E. H. (1980). Melting in single screw extruders: models, calculations, screw design. Technische

Hogeschool Twente.

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Published: 22/05/1980

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melting in single screw extruders

models, calculations, screw design

1. De geringe waarden van de optimale gangdiepte en de extreme drukken die gevonden worden bij gebruik van gegroefde invoerzones, zijn voor een belangrijk deel te wijten aan een gebrek aan smeltkapaciteit van de tot nu toe gebruikte schroeven. Het rendement van de extruder wordt er onnodig door verlaagd.

2. Transportprestaties van gegroefde invoerzones hangen direkt af van de korrelvorm. Dit is in de praktijk pas merkbaar zodra de smeltkapaciteit van de schroef voldoende is en de invoerkapaciteit beperkend wordt. Het feit dat producenten de korrelvorm van het te levere~ materiaal naar willekeur veranderen geeft aanleiding tot de veronderstelling dat zijzelf nog nooit een goed ontworpen extruder in gebruik hebben gehad.

3. Schroefontwerpen van Barr, Dray & Lawrence en Kim zijn gebaseerd op een vergroting van het werkzame warmteoverdragend oppervlak en verhogen de smeltkapaciteit ten hoogste met een faktor twee ten opzichte van de niet erg gebruikelijke standaardschroef met konstante kanaaldiepte. Een verderE verbetering, door vermindering van de gemiddelde smeltfilmdiktes, is slechts te verwachten als deze f ilmdiktes inderdaad een funktie van de dwarskanaalrichting zijn.

4. In de modellen die zijn opgesteld voor het smeltproces in extruders en in de interpretatie van meetresultaten via koelproeven bestaat nog zoveel onzekerheid dat theorie en experiment steeds in overeenstemming gebracht blijken te kunnen warden. Dit lijkt een vrij zinloze bezigheid. Eerlijker en nuttiger is het de totale smeltkapaciteit van verschillende schroeven onderling te vergelijken.

5. Een meteringzone mengt slecht, pompt slecht en homogeniseert slecht, doseert helemaal niet en is in veel extruders feitelijk overbodig.

6. Slechts vanuit analytische beschouwingen zijn fundamentele verbeteringen in schroefontwerp te verwachten.

7. Een verbeterd numeriek programma is slechts iets minder fout dan het vorige.

8. Een student die zijn baccalaureaatsexperiment zo zou afronden en evalueren zoals de THT dat gedaan heeft, werd zonder pardon verwezen naar het kandidaats.

9. De eenvoud van de konstruktie en de vrijheid in gebruik van de stoelen in japanse sneltreinen moet haast wel verbluffend zijn voor N.S. konstrukteurs.

10. Als de THT kon wegpromoveren zoals het bedrijfsleven dan hadden we hier minstens 50 campusdekanen.

11. Iedere student zou de resultaten van zijn werk, behalve zijn afstudeer-verslag, kolloquium en examen, in een (konsept) artikel van ten hoogste tien pagina's moeten presenteren.

12. De dikte van een verslag is hopelijk niet altijd omgekeerd evenredig met het resultaat van het onderzoek.

Models, calculations, screw design

PROEFSCHRIFT

ter verkrijging van de graad van doctor in de tech-nische wetenschappen aan de Techtech-nische Boaeschool Twente, op gezag van de rector magnificus, prof .dr.ir. H.H. van ·den Kroonenberg, volgens besluit van het College van Dekanen in het openbaar te verdediaen op

donderdag 22 mei 1980 te 16.00 uur

door

Henricus Eduard Hubertus Meijer

geboren op 15 mei 1949 te Amsterdam

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR

vooral via de afronding van een van de extrusievakken. Dank aan George Perera. Zonder Willem Ottink, die ver-antwoordelijk is voor ontwerp en fabrikage van de ma-chines, speciale meetapparatuur en schroeven, en voor het uitvoeren van talrijke experimenten, was dit onderzoek niet mogelijk geweest.

Summary

In this work three main aspects are distuinguished. The theory of melting is represented by an evaluation of various models which have been proposed over the years. The models of Tadmor and Lindt are recognized as two extremes, intermediates between these two are possible. All improvements reported in literature are based on Tadmor's model. As a further improvement the in-fluence of convective heat transport in the upper layer of this model is taken into account and investigated. The original Lindt model is extended to include a non linear viscosity and convection in the meltlayers and inside the solid bed. The relation between channel tapering, solid bed velocity and cal-culated down channel pressure gradient is investigated as well as the influence of the heating of the solids on calculated melting lengths.

The second aspect is encountered in calculations comparing the different screw designs which have been proposed to increase melting capacity: compression and multichannel screws, screws by Maillefer, Barr, Dray & Lawrence and Kim. The comparison is mainly based on analytical calculations and only the important screws are compared on the basis of a more elaborate numerical

calculation.

The third part contains the calculation, design, testing and development of a new screw concept which is superior in theory and practice to all others, at least when combined with a cooled, grooved feed section.

These three aspects are not contained in distinct parts. A different division into chapters has been preferred. After the introduction and a description of the assumptions which have to be made, all aspects of the melting process are treated in a comprehensive, analytical approach. Thereafter a numerical treatment is given. Finally the results of experiments are presented and a discussion follows.

Introduction 1. 1.1 1. 2 1. 3 General

assumptions and simplifications Introduction

Extruder geometry

The liquid phase: melt flows 1.3.1 Conservation laws

1.3.2 Constitutive equation 1.3.3 Boundary conditions

1.3.4 Two dimensional flow and lubrication approximation 1.3.5 Isothermal newtonian approach

1.3.6 Dimensionless form 1.4

1. 5

2.

2.1

The solid phase Conclusions

The melting zone. Analytical treatment Different models 2.1.1 Introduction 2.1.2 Tadmor model 2.1.3 Lindt model 2.1.4 Other models 2.1.5 Discussion 2.2 Effective melting 2.2.1 Compression screw 2.2.2 Multichannel screw 2.2.3 Discussion

2.3 Improved screw design 2. 3 .1 Introduction

2.3.2 .Maille fer screw 2.3.3 Barr screw

2.3.4 Dray & Lawrence screw

2.3.5 Kim screw

2.3.6 In gen Housz screw 2.3.7 Discussion 1.1 1.1 1. 1 1. 3 1. 3 1.6 1. 9 1.10 1.14 1.15 1.17 1.1 g 2. 1 2.2 2.2 2.5 2 .12 2.22 2.27 2.29 2.29 2.33 2.35 2.36 2.36 2.38 2.44 2.45 2.46 2.49 2.57

2.4 2.5 3. 3. 1 3.2 3.3 3.4 3.5 4. 4. 1 4.2 4.3

Influence of flight clearance Conclusions

The melting zone. Numerical treatment Introduction

Tadmor model Lindt model

Use of improved theory for different screw designs Conclusions

Experiments Introduction Cooling experiments

Testing of new screw designs 4.3.1 The first prototype

4.3.2 The second prototype 4.3.3 More detailed investigation 4.4 4.5 5. 5. 1 5.2 5.3 5.4 Feed section Conclusions Discussion Models Calculations Screw design Suggestions

Material properties used

List of symbols References Samenvatting 2.61 2.67 3.1 3 1 3.4 ;3. 17 3 '3 3.36 4.1 4.1 4.4 4 .14 4. 14 4. 18 4 .19 4.22 4.28 5.1 5. 1 5.2 5.4 5.5

INTRODUCTION

The single screw extruder is the most used and universal machine in the polymer processing industry. The plasticating extruder has three essential functions:

(1) Transport of solid material that is fed through a hopper in granu-lar form or as a powder.

(2) Melting of the solids.

(3! Pumping of the melt to dies which form the endproduct.

Often three functional zones are distinghuished as is shown in figure 0. (1) The solids conveying zone which also COillJ'resses the material to let

the air out between the particles. (2) Melting zone.

(3) Pumping zone where also homogenisation takes place.

Figure 0 An extruder

Cbung (1970) correctly remarks that this division into zones may cause conf1JSion because for instance in the melting zone all three functions are present. Figure 0 shows a rather old fashioned extruder with a con-ventional feed section and a short compression screw. Somewhat more recent developments start with the design of effective feed sect' ons, extremely cooled with grooved barrels, which increase the throughput and pressure

build up 0£ this part in such a way that melting and pumping zones are

completely overfed. At the same time the pumping zone is changed. Its two functions pumping and homogenisation are separated. Specially design~d

homogenisation parts arise together with various shear elements. In the effective feedzone enough pressure is generated and therefore no pumping

0.2

-task remains for this last part of the screw which now consists only of passive (pressure using) elements, better fitted to their more special tasks. In the last few years much attention has been given to the melting zone. Special designs aim to increase melting capacity or meltquality. We will compare and discuss them here and give a possible start to new developments by consistent screw design: When functional tasks are done by functional screw parts the pumping capacity of the melting zone can be decreased to increase its melting capacity, a change similar to the one realised in the pumping zone.

Theory of the extruder reaches back more than half a century to the analysis of the screwpump (1922, anonymous), Rowell and Finlayson (1928). Nowadays the conventional pumping zone is rather accurately calculated, Yates (1967), Martin (1969) and in review Fenner (1977). Analyses of flows through dies are easily deduced from these calculations by a si1aple change of boundary conditions. For a meltpump the results are within suf-ficient accuracy, however for plasticating extruders the assumption P

0 ; 0, at the start of the pumping section is not permitted.

Therefore attention is given to the feed section because of the simple reason that everything that goes into the extruder must come out. Calculation of the inletzone of the extruder as a solid-pump based on friction behaviour was introduced by Darnell and Mol (1956) and sub-sequently improved, e.g. Lovegrove and Williams (1974), Ingen Housz and Gorissen (1976), but still not sufficiently. The need for a model des-cription of the remaining part in between them, becomes stronger and Tadmor (1966) presents the first analysis of the complex two phase flow here. Of course his work is improved by many other authors, see e.g. Shapiro (1971).

There are only few attempts to couple the analyses of all three zones to one big extruder model (Tadmor and Klein (1971)). ~he two coupling quan-ti quan-ties in every puP1p are through.out and pressure gradient. Problems arise mainly due to the lack of accuracy in the determination of the pressure gradient in the melting zone and to the complexity of most recent model descriptions for the separate parts. The lack of an accurate model for the effective grooved barrel feed zone, where throughput is undependent of demanded pressure in a very wide range -only a first attempt by Grun-schloss (1979) - prevents a descriptive theory for the above mentioned actual developments in practice.

Theory usually follows extrusion practice only from a distance. A further disadvantage exists in the inaccessibility of recent theoretical analyses. One often must be a numerical expert to understand newly published mate-rial. Yet theoreticians maintain that they try to come to new screw de signs. We will attempt to avoid the danger of getting lost in complexity and will try to argue that the outcomes of theory with respect to screw design are only possible by a thorough theoretical understanding of the local process, understanding of which can only come from analytical treat-ment.

An analytical approach requires a multitude of simplifying assumptions. This report starts with a review of these in chapter 1. The main line of this chapter is based on a lucid review article of Fenner (1977). The second chapter contains the main part of this work: i t deals with melting in single screw extruders, comparing different models which are proposed, and different screws which are designed, in a simple way. Of course one can only permit oneself the luxury of analytical treatment when one is also capable of overcoming the disadvantages and simplifications thereof, by numerical calculations. Chapter 3 reports the derivation of the pro-grams and the results of improved calculations for the models and screws of chapter 2. Theory and experiment meet in chapter 4. The melting capa-cities of some conventional screws are compared with those of two proto-types which are examples of the result of a development that is essen-tially based on theory. Lastly chapter 5 gives a discussion of the re-sults and some suggestions.

- 1.1

-1. GENERAL DESCRIPTION OF FLOWS IN EXTRUDER CHANNELS ASSUMPTIONS AND SIMPLIFICATIONS

1.1. Introduction

Especially as an introduction to those readers who are not familiar with the theoretical aspects of polymer processing problems, we will shortly discuss the assumptions which are generally made to make calculations possible. Fenner (1977) will be our guide in this chapter that does not pretent to be original. We will start with the simplification of extruder geometry. Then the liquid phase follows, with first, the conservation laws with which flows can be described, then the constitutive equation which tries to describe the behaviour of molten plastic and lastly the boundary conditions which determine the solution. The lubrification approximation simplifies the equations considerably. We will look at the consequences of a Newtonian appr.oach, which is always very useful and stimulates the search for better insight. The last paragraph gives the dimensionless form of the resulting equations for melt flows. Then the solid phase. As only very little is known of the behaviour of solids every description until now is rather speculative. Knowledge of all this will reduce our preten-tions and force us to be carefull with conclusions.

1.2. Extruder geometry

Figure (1.1) shows a part of the extruder in which the main dimensions: internal barrel diameter D, channel width W, flight width e, channel depth H, flight clearance

o

and pitch angle ~' are shown.

The helix coordinate system is the natural one in extruders. Yet this system is only used when other, more simple, coordinate system lead to questionable theories and possible mistakes e.g. in the calculation of the pressure build up in the feed section. Generally the complicated form of the conservation equations in orthogonal curved coordinate systems is being avoided by taking a local cartesian system on the screw. For this reason i t is more convenient to consider the screw stationary and the barrel rotating (in opposite direction). This is possible under the fol-lowing assumptions:

(gl) Body forces such as those due to gravity and centrifugal forces are

neglirible in comparison with viscous and pressure forces.

The second generally made assumption, because of H << D, is.

(g2) Channel curvature is negligible.

Now we can consider the screw unrolled. The system degenerates Jnto a rectilinear channel, over which an infinite flat plate moves, at an angle with the length direction equal to the pitch angle of the screw.

(g3) The unrolled channet is prismatic and rectangular.

The flanges of the flight are perpendicular to the screw. Radii are neglected. The height H is constant over the width W.

Fisure (1.2) gives an idea of the new situation.

Figure 1.2 Geometry and coordinatesystem in the unwrapoed channel

By the choice of the coordinate system in this manner we have created a left handed system. Unfortunately this is a historically grown matter.

- 1 3

-Pearson loathes this inconsistency and interchanges the x and z direc-tions. Becduse he is the only one who does so, i t leads to confusion. To avoid this we will comply with the usual practice: z is channel di-rection.

1.3. The liquid phase: melt flows

In comparison with the poor description of the solid phase (1.4), the liquid phase may be relatively accurately described. With aid of sim-plified forms , the conservation laws, the constitutive equation and the r'.:>undary conditions can be described in this simplified geometry.

1.3.1. Conservation laws

The continuQ~ mechanics equations governing melt flows are those of

conservation of mass, momentum and energy. They can be concisely ex-pressed in tensor notation, but we will also give a more convenient form. The viscous stress tensor will be represented by the symbol~ or Tij ,

the rate of shear tensor by e or eij

e or e .. lJ 1. 2 0v.

(.,,.-2

o'X. J

av.

+

_k

J ( 1.1) l

~ is the symmetrical part of the rate of deformation tensor

Vy_ ,

from which the rotation tensor~ is split-off. Velocities vi are the local

components of v in the xi direction. The continuity equation

reduces with

(g4) The density of thn melt

7AJ

(locally) constant.

to

0 or e .. 0 lJ

The momentum equation

- \IP + \l•J: + F (1. 2)

reduces with (gl) (F = 0) and (because Re << 1):

(g5) Inertia effects are negligible in comparison with viscous and pressure

forces.

to: \IP ()p dX, l (lT . . l J

Tx-:-J ( 1. 3)

P represents the local hydrostatic pressure. Finally the energy equa-tion

D

Dt

reduces with (g4) and

(g6) The flow is steady everywhere.

\Iv

(g?) Thermal conductivity Am and specific heat cm are (locally) constant.

to: Pm c (v•\l)T _{m -} A \72T _m + J: \Iv _or p c 3T A 3 2 T ( 1. 4) v. dX. -2- + Tij e .. mm l m J l l dX. l

where T is the temperature.

It may be useful to realise the consequences of these approximations of constant material properties. Figure (1.3) is borrowed from Gos-linga (1977) and shows the differences in accuracy in the solid and liquid phase. It gives

A(T), p(T), c(T) and, finally, a(T) A(T) p (T) c(T)

'i'\

1.00 p [_lo KG!M3j 30 C (KJ/KG°I<]

_I

2

!

\:IT~rs] I 2Q

~

L~

..I

II HDPE .35 .20 HOPE

o •0 _eo - - T ' _{ii20 I} r1~ci _\ .1s!o r

LOPE I I I I HOPE 200 140 I o~ -I

'°

BO I -xi!o 140 100 I 1;;-T1~1 '200

u,

25 1..0 2.0-WM2;s] A[W1M'K)

_I

lp~JKGtM3j ) C [KJ/KG0_'9

I

1

J

I 1.351 \ 1.5 PVc I 1.3cl PVC \ I 11 PVC/ _{I I} PVC .1ol -r['Gi

'200

1.251 _i12~qtc( 1,00

J

--T['c] I .o~ t4or --rrcJ '8(j I 1120 I 1150 I ro

_''°

r00-r Q="'[-~f iao I 1120 I 1150 I _'200 60 1120 i 1130 1200

1.3.2. Constitutive equation

With this last equation a description is given of the material which flows through the extruder. It is known that (molten) plastic under influence of stresses at different velocities, temperatures, pressures and during shorter or longer times shows a rather complicated beha-viour. A correct representation thereof in a constitutive equation is not yet found and even if t~is could be i t would be of such complexity that i t could not be used in deriving analytical solutions of the ba-lance equations for the extrusion situation. Assuming that:

(g8) I'olymer melt ca:n be regarded as a:n inelastic fluid.

An important simplification is already made. Nevertheless we can only work with a relative simple empirical form of the remaining viscous behaviour:

(g9) The power law approximation is valid.

In this approximation the shear rate dependency of melt viscosity ('shear thinning', which occurs because of a certain way of orientation of the very long molecules in flow direction) is described with a simple power. The temperature and pressure dependency is being approximated by an exponential (Arrhenius-type) equation:

~ 2~ or T . . _2µeij lJ (1.5) ~ with -b(T-T 0) S(p-po) 2-2a -a µ _{µo e} e _I2 (1.6) µ

0 is a reference viscosity measured at T0, P0 and shear rate 1;

I

2 is the second invariant of the reate of shear tensor:

tr ('"-'"-) ~e .. e .. lJ lJ

because

~

is a symmetric tensor; 2-2a is a constant necessary to norma-lize the power law description, some authors do not use i t . Furthermore:

- 1. 7

-a gives the shear dependency; the power law index which is responsible for the non linear behaviour.

b gives the temperature dependency, which couples momentum and energy equations such that numerical solutions become inevitable.

B

gives the pressure dependency.

values for a are between .2 and .4 while the shear rate can accept

2 -2 0 -1

high values (of order 10 ) ; b is of order 10 C , wh:i.le the occuring temperature differences are of order 102• I t is clear that their influences on viscosity are considerable. Because the lo-cal pressure differences in extruders are generally not too large

(of

or~er

102) and

S

is small (of order 10-3bar-1) i t is usually

assumed that

(glO) Pressu:r>e dependency of viscosity is negligible.

Only in the calculations of injection moulding machines where extreme-ly high local pressure differences can be reached, this simplification is obviously not admissible.

The material constants of the power law equation (1.6) b, µ

0 and a are

found from measurements of viscosity made with e.g. a rheogoniometer or a (capillary) viscometer by curve fitting. It so happens that this fit is rather good between well defined limits. Constant values of µ

0 and a

can be used when no.extreme shear rates are present, while b itself in fact still contains the absolute temperature which limits the validity of the approximation to a temperature difference (T-T

0) of approximately

40 °c. Inaccuracies are introduced because of the broad field of shear rates and temperatures which occur in the melting zone. An illustration thereof we find in figure (1 . 4) . Figure (1 . 4a) gives the relation be-tween sl1ear stresses and shear rates as measured for low density polye-thylene. From these measurements we get figure (1.4b) which should show straight lines according to the power law approximation. Figures (1.4c) to (1.4f) give an indication of the viscosity changes.

Figure 1.4a

Measurements of shear stress-shear rate relation of LDPE Stamylan 1510, on a capillary viscometer. Parameter is temperature

Figure 1.4c

Shear rate dependency of viscosity

Figure 1.4d

Temperature dependency of viscosity

Figure 1.4e

As 1.4a, plotted on log-log paper

Figure 1.4e

As 1.4c, plotted on log-log paper

Figure 1.4f

- 1.9

-We should realise that the reference viscosity µ (of order 10 4 ) and

. 0

the actually occuring viscosity µ (of order '10 2 ) do not have much in common, because of the definition of µ

0 at y = 1. This can be

unplea-sant in testprocedures of numerical programs, therefore i t would be better to define a reference viscosity at a representative average shear rate y

0, similar to the definition of µ0 at the average

tempe-rature T

0. Eq. (1.6) needs a little change in this case by replacing

1.3.3. Boundary conditions

In the end every flow problem is fully determined by its boundary con-ditions. Here too some assumptions must be made in order to come to solutions. The most important one is:

(gll) There is no slip at the walls.

Fenner indicates that Kennaway (1957) has stated that slip between polymer melt and metal surfaces occurs as soon as a limitating shear

stress is reached. It seems that this value normally exceeds the occur-ring shear stresses in extruder channels with an exception for unplas-ticized PVC. Also in the leakage gap 8 between screw flight and barrel this critical shear stress could be exceeded under special conditions. The investigation of this assumption, without which calculation possi-bilities decrease rapidly, is still going on, see e.g. Uhland (1977). For the thermal boundary condition we have two extremes:

1. perfect temperature prescription T = Twall and

2 • per ect iso ation f . 1 .

a;-

3T

=

0

~-,-""!f.

..z__

By the definition:

3. 0

(which in dimensionless form leads to the Biot number) the whole field between these two may be reached with 0 ~ H ~ oo.

In the numerical calculations 3 can be built in because there it gives no other technical problem then the choice of the value for H. There is not much evidence of acceptable values. To achieve analytical results one must choose either 1 or 2. It has become usual to choose the sim-plest one, 1 for the barrel wall, because the barrel is normally kept at constant temperature, by assuming:

(g12) There is good thermal contact beb.i!een the melt and the barrel surface,

:t

so T

=

Tb when

y

=

H.

The screw is normally neutral (no heating, no cooling). Martin (1969) pleads for the application· of boundary condition 2 here, but this per-fect isolation leads to an axial temperature difference inside the screw. This will cause an axial heatflux which continuously must be supported when steady state is reached. Therefore condition 2 does not seem real, 3 is suggested.

Nevertheless we will at first assume (to avoid the necessity of a total heat balance on the screw) :

(g13) There is also a good thermal contact beb.i!een melt and screw surfaces,

so T = Tse for y = O, x = 0 and x = W.

For Ts~ a typical measured temperature profile may be choosen. Because

the convection terms in (1.4) allow for a development of the temperature profile in flow direction we will need starting conditions for this para-bolic equation. These must be given separately for each problem.

1.3.4. Two dimensional flows and lubrication approximation

The remaining equations are still of such complexity that we will need some other assumptions.

1.3.4.1. Two dimensional flows

Often i t is possible by the choice of a suitable coordinate system to reduce the fully three dimensional problem to one which is to a good

approximation two dimensional. We can reject the less important

direc-tion often by rotating the coordinate system such that one of its axis is in the same direction as the main flow. In the melting zone this is the relative direction of the barrel movement to the solid bed.

~

index b for barrel

- 1.11

-To distinguish this direction from x we will introduce the symbol

x

see figure (1.5).

Figure 1.5 Rotated coordinate system in the melting zone

With the third direction 1;; less important, so w "' 0 and u, v and T not being a function of 1;; any more, we have

u u(x, y) v v(x, y) w

The continuity equation (.1.2) reduces to

dU + dV Q

ax.

ay

=

The momentum equation (1. 3) to

dT dT

3P _ M + _rr

ax-=

3X 3Y +

and the energy equation (1.4) becomes

p c (u

~T

+ v 3T ) m m

ax

3y 0 h yy T

8Y

T(x, y) 3P

aT

= 0 ( 1. 7) (1. 9) ( 1.10)

1.3.4.2. Lubrication approximation

As Fenner defines this approximation, i t involves in essence the local replacement of the actual flow in the parallel or nearly parallel gap between smooth surfaces by uniform flow between plane parallel surfaces. We need two assumptions to be able to apply the lubrication approxima-tion:

(g14) The characteristic height Hof the problem involved is ma:ny times

smal-ler than the characteristic length L.

H -2

(L

or order 10 ) and therefore also

(g15) The influence of the flanges of the flight on the flow is negligible.

With these two assumptions and use of a dimensionless form there follows:

au

au

- <<

ax

3y

so u ~ u(y) , and

av

av

-

_ax

<<

_ay

so v ~ v(y).

Hence with eq. (1.8):

dV <<

au

ay

ay

With this there results for eq. (1.9), after inspection of eqs. (1.5) and (1.6): dP dx dT _I;L dy

ap

ay

~

0

T

XY

µ du µ dy combination gives: T

XY

- 1.13 -l'i'E T - T 0

The dimensionless form of the energy eq. (1.10) shows that in the conduction term

du

and in the dissipation

TXY

dy is dominant.

There results: p c (u

~

+ v

~)

m m

ax

:Jy :J2T du A - - + T -m

ai

XY

dy

Both convective terms are needed, because generally u >> v - see (1.8) - but also

:JT

<<

:Jy

Because the Peclet numbers are large (of order 103-105) the

con-( 1.12)

( 1.13)

( 1 .14)

( 1. 15)

vection as a whole is certainly not negligible. For flow in the plli~ping

zone or in the leakage gap i t is often possible a posteriori to prove that vertical convection is small compared with the horizontal one, so there (1.15) reduces to:

dT d2T du

p C u - = ; \ - - + T

m m :Jx

:ii

XY

dy ( 1.16)

In the meltfilms underneath and above the solid bed this is certainly not possible because freshly molten (rather cold) material is fed into this flow and introduces vertical velocities.

Despite this, there are only a few attempts to take convection into account: Griffin (1974). Until now both terms are always neglected in the analysis of the melting zone because they introduce severe problems in the calculations. Pearson (1976) and Shapiro (1q71) quite rightly question the correctness of this simplification but they nevertheless assume:

(g16) Temperatures are fully developed.

Herewith (1.16) reduces still further, to

0 f.

ID

du

+T

-XY dy ( 1.17)

As Pearson (1976) points out the loss of the effect of convective heat transport may be counteracted by adding an extra term to the heat balance. This helps somewhat but is still incorrect. Tadmor et al. (1967) introduced this term, not so much in order to improve the heat balance but as a result of faulty argumentation, Pearson supplied an extra multiplication factor k (0 2-_ k 2-_ 1) to diminish its effect and Edmondson and Fenner (1q75) are not the only ones who are confused so that they use i t completely wrongly.

1.3.5. Isothermal newtonian approach

Two more assumptions are made in this special case:

(g17) Viscosity is independent of temperature.

-bi'IT

This implies that the material constant b in e equals zero. Momentum and energy eqs. are uncoupled by this, velocity profiles can be calculated independent of the temperature profilPs.

(g18) Viscosity is independent of shear rate.

-a The power law index a in 1

2 equals zero. The material behaves as a newtonian fluid. Both assumptions together make for a constant viscosity. Analytical solutions become possible and that is the strength of this

approach which guarantees clarity and provides an understandable

quali-tative description of the problems. When more exact quantiquali-tative re-sults are needed one must of necessity make more complicated calculations without

(g17)

and

(gl8).

As said before i t is rather advisable to use a constant viscosity which is based on averaged values of temperature and shear rate because otherwise completely faulty conclusions could be drawn. Eqs. ( 1. 11) and ( 1. 15) reduce in this case to

dP

dX

1.3.6. Dimensionless form - 1.15 -2 µ(du) dy (1.18) ( 1.19)

There are different ways to obtain dimensionless forms of eqs. (1.11),

(1.14) and (1.15) or (1.18) and (1.19). We will use the following dimen-sionless parameters:

x

L

y_

H rv u :'::!.

v

rv v v v

e

Substitution of (1.14) in momentum eq. (1.11) gives with (1.20)

rv dP A di; d

-e

d;'{ 1-2a dn (e (dn) With A a constant, A

Substitution of (1.14) in energy eq. (1.15) gives with (1.20)

rv 3

e

Gzu ~ + Pe v rv 38

an

-e

d;'i

1-2a Gre _(dn)

in which we recognize the important numbers:

Gz Pe H

L

,

the Graetz number

VH A

the Peclet number, which m

Pe

,

in a

a m p c

m mm

]J b v2-2aH2a

Gr 0 the Griffith number

A

,

(1.20)

(1. 21)

As said before the Peclet number is of order 103 - 105 and under-lines as does the Graetz number (of order 10 103 because H is

L

of order 10-2) the importance of convection in comparison to con-duction. The Griffith number is of order 1 - 10 and gives the ratio between dissipation and conduction. It also tells something about the coupling between momentum and energy equations because of the presence of b. In the isothermal newtonian approach we can not use the dimen-sionless parameter 8

=

b(~-T

₀

) of (1.20), because there b

=

0. In this case a characteristic temperature difference

8 T-T

SC T -T

b SC

is used, which leads to the Brinkman number which has the same meaning and order as the Griffith number. Eqs. (1.18) and (1.19) get the fol-lowing form: 'V dP A di; 2'V d u dn2 with A p H 0 µ v 0

a dimensionless constant times H

L Gz 'V _u ()8 _{<lt; + Pe v} 'V ()8 _<ln with Brinkman Br 2 µ v H L (1. 23) (1.24)

- 1.17

-1.4. The solid phase

The knowled ge of the solid phase, the compressed granules in the solid bed, is little. One cannot even speak of one kind of solid phase because not only granules but also powders are used as feedstock. Powder exist in different particle sizes, granules differ in size and shape, in strength and deformability. One can assume that not only the absolute size of the granules but also-their size relative to the dimensions of screw channel geometry will determe its beha.viour. Despite these con-siderations normally i t is always ·assumed (and mostly implicit) that:

(g19) The powder

or

granule sol{d bed is to be described as if it were a

continuum.

The main properties of this continuum are absolutely unclear. The only serious experiments with packed granules were aimed at the problems in the feed zone: determination of friction coefficients, measurement of pressure anisotropy·. The latter appears to be rather important, (Schnei-der 196.8). A description of the properties of the solid bed is nor-mally evaded for lack of elementary knowledge and - adjusted to the aims of an accidental model vaguely paraphrased as

(g20) The solid bed is easy to deform,

or just the opposite, cannot resist

big forces (due to shear stresses) or can. Anyhow the same assumptions ar·e always made, about constant material properties, as in the liquid phase. ! t was illustrated in figure (1.3) that the incorrectness of these assumptions are larger below the melting point.

( g21) Coefficient of conduction

f.

*

is constant.

s

(g22) Density

ps

is constant.

(g23) Specific heat cs is constant.

and finally

(g24) There exists a fixed, sharp melting point Tm;

the latent heat of fusion

is A. This is a simplification of the c(T) curve in figure (1.3) to what is known as the Stefan problem, in which the heat of fusion is concen-trated at one defined temperature, the melting point. For PE this appro-ximation is rather reasonable, because of its. cristalline character, for PVC much less so. Concerning assumption

(g22)

i t might be interes-ting to gl~nce at figure (1.6) which presents two proposals for the pressure dependency of the density of a bed of granules:

Tadmor & Broyer (1972): Hegele (1972): -cP e c p Po+ clp 2 (0 < c2 < 1)

Figure 1.6 Pressure dependency of solid bed density ps' according to Hegele and to Tadmor

It is clear that the difference is small if the solid bed is suffi-ciently compressed in the feed zone.

Of course the conservation laws are also applicable to the solid con-tinuum. Under the predescribed assumptions they equal eqs. (1.2), (1.3) and (1.4) when the indexes mare replaced by s and the dissipation term in (1.4) is set to zero. The continuity eq. (1.2) and the momentum eq. (1.3) - if used - are only needed in the integral form of an over-all mass balance and force equilibrium. The energy eq. (1.4) is nor-mally the only one that is used, namely for calculating a developing temperature profile inside the solid bed which is necessary to determine

p c s s

- 1.19

-( u 3T + 3T 3T

3x v

ay

+ w ~ A _s (1.25)

Eq. (1.25) can often be simplified. E.g. to a two dimensional form by choosing a coordinate system that moves with the (non deforming) solid bed in channel direction (z) or by neglecting conduction and convection in x direction. The simplest one-dimensional form is reached by under-taking both actions.

1.5. Conclusions

With the aid of many simplifications which could introduce as many inaccuracies we have adapted the complex reality to our limited calcu-lation possibilities. With these tools we will try to describe the two phase flow in the melting zone. we shall see that in addition the assumption of a very specific simplifying model is required in order to come to conclusions and predictions. More than one model is possible.

-2. THE MELTING ZONE. ANALYTICAL TREATMENT

This chapter presents some simplified melting zone calculations. We will make as many simplifications as are needed in the basic formulae, in trying to come to s.ome understanding. At first the different models proposed for a screw with constant channel depth will be treated in paragraph (2.1). In (2.2) we glance at melting in the normally used compression screw and the theoretically interesting multichannel screw and come to a number of new screw designs in (2.3). The effect of their different geometry on melting, is compared and discussed. The leakage flow through the gap between screw flight and barrel is ne-glected here because its influence on melting rate can only numeri-cally be taken into account in some of these screws. In (2.4) we look at this effect before conclusions are drawn in (2.5).

- 2.2

-2.1. Different models

2.1.1. Introduction

Since the first qualitative description by Maddock in 1959 of the two phase. flow in the melting zone, with special attention to the place of the solid melt interface and the progression of the melting proces, a more thorough investigation of this zone has been started .. This has resulted in a number of melting models which will be discussed in this paragraph.

,---

. -- - - · -- - · - - - -· - - - ·

I

I

L.

·----··---Figure 2.1 An idea of the melting process according to Maddock

Tadmor is the first who proposed a melting model, (1966) based on the observations of Maddock (1959) and Street (1961). During the years this model has been changed, improved and adjusted to newer insights and better calculation possibilities with digital computers, by him-self and many other authors. The most accurate extension has been developed in England under Pearson by a.o. Shapiro and Halmos.

A new discussion starts because of the observations of Dekker (1976) x His findings deny the still remaining basic concept of.the Tadmors model, that of the deforming solid bed and the existence of a meltpool. Lindt t1976) analyses this situation and proposes another basic model, the consistency of which is indeed questionable. Before, Edmondson

(1973) had tried to leave the Tadmor-Pearson line on a critical point in an attempt to present another model that could be understood as an intermediate between Tadmor and Lindt. However he fails to draw the consequences of his own description and makes a serious mistake. Still other models will be possible. Lindt's attempt (1976) to let the pres-sure gradients cause the occurance of his model, seem doomed to failure by lack of accuracy. Our own observations Meyer (1975), Gieskes (1976) showed that Tadmor and Lindt models could be following each other under normal operating conditions. Gieskes also observed that a pure Tadmor model only occurs when screw cooling is used. Fenner confirmed this in 1977. We can conclude that both extremes, Tadmor and Lindt, can occur in practice depending e.g. on the properties of the solid material, the operating conditions and the screw geometry. In a tapered screw the utter differences between the models will vanish, specially when mel-ting on the screw surface is taken into account.

For maximal clearness we will start now with the simplest models for a constant depth screw by adding seven commonly used assumptions to the general ones

(gl)

to

(g24)

of chapter 1. We need them here to obtain analytical results, and they will be discussed in more detail later.

(al) The most important melting mechanism takes place between the hot barrel

and the solid bed.

Because of the thin layers and the big velocity

differences, conduction and dissipation (which cause the heat flux to the solid bed for melting) are dominant here. Melting on the screw sur-face and on the flanges of the screw flight are only secondary processes.

(a2) The influences of pressure gradients on the velocity profiles are

ne-glegible.

(highly viscous flow in very thin layers).

z

He proposes another screw geometry for the extrusion of polypropylene

on a 90 mm extruder. During the experimental evaluation of this

con-cept he pulled the screw several times. Although the results there

did not agree with the Maddock model he did not report on this

as-pect in more detail.

- 2.4

-(a3) The solid bed is semi-infinite.

There is no heating up. The temperature

in the middle remains constant at hopper temperature.

(a4) Heat convection is neglegible.

The flow is thermally fully developed.

(a5) The velocity of the solid bed in channel direction is constant.

(a6) Screw and barrel terrrperatur•e remain ccnstant

in the melting zone.

(a?) The channel depth is constant

in the melting zone.

Before switching to more assumptions which still have to be made to achieve a complete model description we will give some special atten-tion to a small but essential problem that arised in Tadmor's first mathematical model and has been solved already long ago. It dealt with the meltfilm thickness that was supposed to be constant above the solid bed and of the same order as the flight clearance. This suggestion followed from the observations of Maddock, despite the fact that exact measurements are impossible. Because of the all de-terminating roll of the thickness of this layer on the heatflux to the solid bed and therefore on the calculated melting length, most care has to.be given to its magnitude.

The drag flow with capacity l;;Vh must carry all the molten mass away. During this transport the total mass flow increases because of the feeding with freshly molten material from beneath. The suggestion of constant meltfilm thickness, demands an increasing negative pressure gradient toe.transport the extra mass. This violates assumption

(a2).

Shapiro (1971) investigates this problem with extreme accuracy and comes to the same. conclusion that Vermeulen c.s. (1971) gets in a very simple way: the pressure gradients needed in these thin layers are unrealistically high, the meltfilm thickness cannot be constant and will increase. This diminishes the heatflux to the solid bed.

L

--~-/

Figure 2.2 The meltfilm between solids and barrel, with constant or increasing thickness

There are several steps to be distinghuished in the calculation of each model:

1. Determination of the temperature distribution in the meltfilm, to give the local heatflux from the melt to the interface. 2. Determination of the temperature distribution in the solid bed,

to give the local heatflux from the interface.

3. A heat balance on the solid melt interface, to give the local melting velocity.

4. A mass balance in the meltfilm, to give the local meltfilm thick-ness.

5. A mass balance in the solid bed, to give the solid bed profile. From this last step the melting length is easily calculated, inclu-ding the correction of the increasing meltfilm thickness.

2.1.2. Tadmor model

The most striking quality of Tadmor's model is its consistency by demanding the property of the deforming solid bed. Total deformation is necessary because of the neglect of the secondary melting processes on screw surface and flanges and of the heating of the solid bed. He replaces 'most important' in assumption

(al)

by 'only' and conti-nues with

(aB) The molten mass is transported by the movement of the barrel relative

to the solid bed,

until i t reaches a screw flight.

(a9) The leading edge of the advancing flight scrapes the melt off the

barrel surface and forces it into a meltpool.

(alO) This meltpool needs place and pushes into the solid bed;

this deforms,

the width decreases, the height is kept constant, solid mass replaces the molten mass.

(all)

(Following this)

The meltfilm thickness profile does not change

in

the melting zone.

(a12) The meltfilm thickness on the passive flange equals the flight

clearance,

This last assumption is made to avoid discontinuity here. In this first analysis the flight clearance 6 is taken zero.

After these limiting assumptions

(gl)

to

(g24)

and

(al)

to

(a12)

2.6

-Figure 2.3 Coordinate system in Tadmor's model

1. In the lubrication approximation adjusted to a (locally) thermal fully developed flow between flat plates without pressure gra-dients, equations (1.18) and (1.19) reduce to the form

dT _:xx_ dy 0 and 0 2 µ(du) dy

Solution with the boundary conditions of figure 2.3:

y 0 u = 0 , T

results in

and

in which Br

y + B y (1 - Y)

h r. h h

, the Brinkman number.

With Fourier's law we find the heatflux to the solid bed:

q - 1' m

dT

dy y=O (2 .1)

Conduction and dissipation both contribute to the heatflU{[;, which

is otherwise proportional to the local height.

y+ Q"

Figure 2.4 Heat fluxes into and from the solid melt interface

2. Inside the solid bed there is no dissipation. Our coordinate system moves with the solids in channel direction. We neglect heat con-vection in x-direction. Eq. (1.25) reduces to the one dimensional form

ll:dT p c v.

-s -s 1- dy

- 2.8

-Solution with boundary conditions y y = - 00 _{: T = Th leads to} T - Th T - T m h from which v. exp (~ y) a s q = - A dT

I

y s dy y=O a s 0 ii s p c s s T T and m (2. 2)

The heatflux into the bed equals the enthalpy difference cs(Tm-Th)

of a solid mass flow

p

_s

v.

between hopver temperature Th and

melt-~

-

.

temperature Tm.

Because of assumptions

(a10)

&

(all)

n Tadmor's model, the inter-face is kept at the same place by a continuous movement of the solid bed in positive y-direction with velocity Vsy

-v ..

l

3. A heat balance on the solid melt interface tells us that the dif-ference between the heatflux into and from this interface is used for melting material with velocity Vi. The heat of fusion necessarly for the phase change is

A.

( _ A dT + A dT ) m dy s dy y=O

with eqs. (2.1) and (2.2) this yields:

MR dQ'm dx - P s v i Am(Tb-Tm) + !:; µVb cs(Tm-Th) +A (2. 3)

The melting rate is proportional to the quotient of the heat

sources (conduction and dissipation) and the heat sink (enthalpy

difference between solids of Th and melt of

~m)

and otherwise

4. The direction of the relative motion Vb is y, see fiqure 2.3. Melting causes an increase of mass flow in the meltfilm. Per unit width ds :

dQ' ' m

The drag flow capacity must increase by an increase of the melt-film thickness: dQ'm Combination leads to dh dx (2. 3)

Integration with boundary condition

x

h

O; h 0, yields:

(2 .4)

The total mass flow

Q' (x)

=

X,r

dQ' ,

tells us

how much mass

m

o m

(per unit width ds) "there flows on a distance

x

from the flight and therefore also

how much is molten over this distance

since the leakage gap

o

= 0.

(2.4)

(2.5)

The appearance of Vbpm in k

2 is understandable. When a certain

drag flow capacity is demanded to transport a given molten mass,

then the higher pmVb gives the lower h and therefore an increa-sing melting velocity.

- 2.10

-v, _ _ _ _ _

Figure 2.5 Transportcapacity of the upper meltfilm

5. Neglecting the meltfilm thickness h in relation to the channel depth H, the mass flow through a solid bed with local width X, which moves with velocity V equals XH ps V This mass flow

sz sz

diminishes in channel direction because of melting. The change

d

over dz equals dz (XH ps Vsz). From (2.5) we know how much melt arises over a length

x

by unit width di;;. With X =

X

siny and di;; = dz siny , see figure 2.3, this can be easily transformed to Q'.m(X) the molten mass (by unit dz) over a distance X:

Q' (X) m with or ~ k 2(siny) ~ i;st (X)

x

Index st for standard

(2 .6)

(see figure (2.3)

.. l

pm v sinq>

J

Combination leads to the mass balance

d

dz (XH Ps vsz)

i., !;st(X)

Solution of (2.7) with boundary condition z in the solid bed profile

x 1/Jst2 2 - = (1 -2H )

w

with 1/Jst !;st

v

wlz

szps (2. 7) 0 x W results (2 .8)

By the definition z

=

Zst : X

=

0 we find from eq. (2.8) the wellknown standard melting length:

2H 1/Jst

(2.9)

With eqs. (2.6), (2.8) and (2.9) we have got a very simple form for the calculated melting length. I t becomes possible to make an estimation of the effects on melting of the different material and process parameters which eventually can be influenced. Inspec-tion of eqs. (2.6) and (2.8) gives another interesting result. Substitution of the last in the first in combination with eq.

(2.9) yields:

Q'm(z) (2 .10)

In other words the melting efficiency (expressed in Q'm) in a constant depth screw decreases linearly in channel direction

~rom ~(W)~

in the b0_{ginning of the melting zone, to zero on the}

end. We will use this conclusion later in the paragraphs 2.2 and 2.3. The (big) influence of a non zero leakage gap between

- 2.12

-flight and barrel will be treated in paragraph 2.4, where we can also find some resulting plots of eqs. (2.4), (2.6) and (2.8). The improvements of this first model, which will shortly be dis-cussed in the introduction of chapter 3, all dealt with assump-tions (al) to (a6) and those of the newtonian fluid (g17) and

(g18). The model description by (a8) to (a12) remained unchanged

until Lindt presented another basic model at first at a confe-rence in 1975.

2.1.3. Lindt model

Lindt attempts to give a model description of Dekkers findings. Although he was not very succesful in the degree of accuracy and consistency, we shall now give him much attention. Too much in com-parison to the value of his work. There are two reasons for this. The impression exists that every Tadmor model degenerates under nor-mal operating conditions to a kind of Lindt model and secondly the effect of the heating up of the solid bed and several other inte-resting features, such as the calculation possibility of the down channel pressure gradient, are easily shown by extending this sim-plified model.

This brings us to the main properties of Lindt's model:

(a13) Tlw solid bed does not deform, there is no meltpool.

(a14) on the screw is important. The melted mass stays on top of

ar;cl below the solid becl, ancl is transported in channel direction by

drag and pressure flows in meltfilms with continuously growing thickness.

(a15) he the x-ciirectior: nothing happens, the meltfilms stay constant,

drag and pressure flow are neglegible. The model is essential two dimensional.

Of course (a15) is the weakest point. The relative movement of the barrel in x-direction equals Vb sin~ (with sin~ ~ .3 because of the square pitch) and causes a drag flow that needs to be compensated in one way or another. Gieskes (1979) has investigated this problem of mass continuity in x-direction to see under which conditions this

model could be valid. It is not proper to give much attention to Lindt's own attempt (1976) to make the pressure gradient in chan-nel direction responsible. The pressure flow in the usually very small leakage gap plays a determinating roll in his approach. This is questionable because of assumption (a2) which is justified a.o. by Shapiro (1971).

--~-h 10-M

5 6

Figure 2.6a Pressure gradient in cross channel direction x as a function of meltfilm thickness.

Parameter is screw clearence

o.

1~~

[MPa/ni}

10

•2

Figure 2.6b As figure 2.6a, when circulation is possible around the solid bed.

Gieskes' conclusions confirm this point of view. Without the exis-tence of a meltpool the drag flow component must be compensated by a pressure flow, see figure 2.6.a. The pressure gradients needed are excessive unless the meltfilm thickness is either smaller than the flight clearance or of the same order as the channel depth in the metering zone. This means that in the beginning of the melting zone a meltpool formation must take place and the solid bed must deform.

- 2.14

-Everything changes when circulation around the solid bed is possible, see figure 2.6b. Depending on the relative thickness of the layers the magnitude of the pressure gradient becomes realistic. This means that indeed melting according to Tadmor could change to a Lindt model when the meltfilm thickness on the screw and near the flanges of the flight are big enough.

If this change is indeed a degeneration as is supposed in the intro-auction of this chapter and argued by Ingen Housz and Meyer (1978), i t must be confirmed by calculations because there are some coun-teracting effects: In the Tadmor models the heatflux to the interface remains high because of the constant film thickness in channel direction. But the surface to which the heatflux is added becomes smaller because of the decreasing width of the solid bed. The total melting mechanism becomes less effective and decreases

linearly in channel direction. In Lindt's model the width remains unchanged, but here the meltfilm thickness is growing and diminishes the melting capacity.

t '-.

_J

Figure 2.7 The melting process according to Lindt, boundary conditions

v,

Unfortunately calculations using the Lindt model can only be ouasi-analytical. This is mainly due to the need of a down channel pressure gradient to· satisfy the total mass balance if assumption (a?) of constant V holds. This fact seems to violate (a2) but we must

rea-sz

lize that the layers here are much thicker. As stated above one of the advantages of Lindt's model is just the calculation possibility

of this pressure gradient. Again we follow the same steps as in the treatment of the Tadmor model.

1. From eq. (1.18) - in which

x

must be replaced by z in this case with boundary conditions of figure (2.7) we find the velocity profile in the upper layer:

w3

v

_sz + 1Wn₃ (1 + lzp₃ <n₃-1J (2.10) 2 with l!.V = ¥_ and h3 _dP

v

- v

_{n3 P3} bz sz -h3 µLW dz

After substitution of the derivative of eq. (2.10) with respect to n in eq. (1.17), solution with boundary conditions from figure (2.7) yields:

From (2.11) we find the heatflux to the solid bed

q _ A dT

m dy y=O

-

{

(2 .11)

(2 .12)

2. Step (2) remains unchanged, eq; (2.2) gives the heatflux into the solid bed.

3. Step (3), the heat balance on the interface now leads to the expressions for the local melting rate:

4a. - 2.16 -with (2 .13) dP) in which k

3 is not a real constant now, because p3 = p3 (h3, dz . The pressure gradient in the layers 3 and 1 is equal.

We find the melting rate on the screw analogeously:

with 2 p 2 1 P1 m (T -T ) s m + !zµV sz ( - - + 12 -3 + 1) h 2 1 dP P1 µV dz sz and

The local increase of mass flow per unit width dx layers equals this melting rate

dQ' 3 _k3 and dQ' 1 kl dz ps vi3 _h3 dz _h1 in (2 .14) both

The expression for the mass flow Q'

3 is found by intergration of the velocity profile (2.10), Q'

1 follows again analogeously:

+ v _h3 3 Q' _pm ( vbz sz _h3 dP 2

-

12µ and 3 dz (2. 16) vszh1 h1 3 Q' p

- - - -

dP 1 m 2 12µ dz

Sa. Lastly the integral mass balance yields:

(2 .17)

With expressions for p

3, p1, k3 and k1 in eqs. (2.10), (2.13) and (2.14). Equations (2.15), (2.16) and (2.17) must be solved numerically. With aid of a discretisation

Q'

z+dz Q' z +

~

dz dz (2 .18)

these 9 equations with the 9 unknowns h

1, k1,

p

1,

q•

1, h3,

dP

k

3, p3, Q'3, dz can be reduced to one implicit equation for e.g. h

1.

*

A typical solution is shown in figure 2.8.

Pma• o.o-f--~-~~---i---1 ----.- ·- --1---,-0.0 0.1.1 o.o f).6 --,--i '..O z~ 2.31J \m)

··:

:~J

0.0 0.2 O.lJ 0.6 O.B 1.0 l-2.Jll !ml

Figure 2.8 Typical solution of Lindt's model in a constant depth screw. Material LDPE. Throughput 40 kg/h at 80 rpm.

*

At the start h

₁

, h

3

and

j;

must be given. Pith (2.15) z,Je then

find expression for the mass increment from which the mass flows

on z

0 +

dz follow with (2.18). Substitution thereo.f in (2.16)

leads

together

with

(2.17)

to

3

equations with 3 unknowns

h

₁

~

h

3

dP

and

er;;,·

This set can be reduced to one implicit equation (in h

1

for instance) which can be solved with a one-point interaction

method to give the situation at z

0 +

dz. Repetition hereof leads