mechanical math1

8/4/2019 Mechanical Math1

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SCREW EXTRUSION

Intern. Polymer Processing IX (1994) 1 Hanser Publishers, Munich 1994 11

H. Potente*, J. Ansahl and B. Klarholz

Department of Plastics Technology, University of Paderborn, Germany

Design of Tightly Intermeshing Co-Rotating

Twin Screw Extruders

Composite models for the calculation of the filling levelprofiles, the pressure profiles, the melting profiles, the resi-dence time distributions, the temperature profiles, the shear

stress profiles, and the power consumption in modular tightlyintermeshing co-rotating twin screw extruders (ZSK) aredeveloped. A complex systematic design procedure was com-

piled, which is explained in part in this paper. The simulation

of the intermeshing co-rotating machine involves both screwand kneading disc elements, including left- and right-handedsections. Kneading blocks were approximated by a screw ofequivalent pitch with making allowance for the leakage

flow across the flights from one channel to the adjacentchannel. The mathematical treatment of co-rotating twin

screw extruders has been based up according to the theory ofsingle screw extruders. There was seen to be a good correla-tion between calculated and experimental results.

1 Introduction

In the design of tightly intermeshing co-rotating twin screw

extruders, which will simply be referred to as co-rotatingtwin screws in what follows, the screw design is drawn up asa function of the requirements imposed by the compound-ing functions and the compounding properties of the mate-rial. In practice, design is almost always performed on anempirical basis, and past developments have not culminatedin so-called all-round screws, which would be equally suit-able for compounding a wide range of thermoplastic mate-rials. This is why all the machine builders include a numberof different screw configurations in their ranges, which aretailored to different needs.

Taking the idea put forward by J. Prause in [1], it ispossible to divide the available configurations into four basic

categories. The compression that the material is subjected toin the machine plays an essential role in drawing thisdistinction. Since, with co-rotating twin screws, the meanchannel depth is laid down by the distance between the axesand the outside diameter of the screw, the material beingcompounded is compressed through a reduction in the pitchand/or through backwards-conveying elements (such asscrew elements or kneading blocks) or even through axialand radial flow restrictors. The four basic categories, which

* Mail address: Prof. Dr. Helmut Potente, University of Pader- born, Dept. of Plastics Technology, Pohlweg 47-49 D-33098

Paderborn, Germany.

can be combined at random in machines with modular-typescrews and barrels, will be briefly set out below [1]:

Low-work screws

These screws exert no compression, or only a very low levelof compression. They are used preferentially for heat-sensi-tive materials which require gentle compounding.

Standard screws

Standard screws are designed with a standard compressionratio and are used both for melting and compoundingtasks. The screw design of so-called standard screws variesfrom one machine builder to the next.

High-shear screws

These screws incorporate elements which generate highshear rates in order to improve dispersion behaviour. In thecase of co-rotating twin screws, this is achieved through theapplication of backwards-conveying elements and/or

kneading blocks. Kneading blocks create high shear rates,particularly with partial filling. In principle, the mean resi-dence time is reduced with partial filling, which comesabout with an increase in the speed for a constant materialthroughput. The lower residence time, in turn, represents adisadvantage when it comes to the shear deformation thatis caused, this being the product of shear rate and residencetime. Backwards-conveying elements affect the back-pres-sure length respectively the residence time in the up-channeldirection so that the shear deformation increases.

Vented screws

Vented screws are used to degas the polymer melt whenresidual monomers or other ancillary products of the reac-tion have to be removed from the flow of melt.

The screw categories set out above only represent abreakdown according to the principle involved. In practice,there is a wide range of screw configurations supplied bythe machine builders. Virtually all manufacturers have athree-flighted machine series with relatively low flightdepths and another series with medium and large flightdepths [2, 3]. No single-flighted series or series with four ormore flights is known in practice [2, 3]. There are, however,combinations of single-flighted elements in the feed zonewith two- or three-flighted elements in the plasticization and

processing zone. The problem that the machine builders


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H. Potente et al.: Co-rotating Twin Screw Extruders

12 Intern. Polymer Processing IX (1994) 1

face here is that they have to guarantee to their customers,in advance, that the type of compounding that these cus-tomers require will succeed using the screw geometry thatthey design.

Since a large number of machine builders generally em- ploy modular-type designs for the screw and barrel, inorder to permit greater flexibility, the machine can be

modified in so many different ways that there are an infinitenumber of possible variations for the screw configuration.If an attempt is now made to obtain an overview of the

capacity of individual screw configurations or, going intogreater detail, of individual screw elements, then empiricalstatements do not provide sufficient assistance. In order to

be able to analyse the influence of the screw geometry, thematerial and the process parameter, it is necessary to em-

ploy quantitative methods as well, which make statementsabout the compounding or extrusion process.

2 The geometry of co-rotating twin screw extruders2.1 Screw geometry elements

A screw configuration is essentially made up of conveyingelements (positive helix angle) and backwards-conveyingelements (negative helix angle). Different conveying condi-tions thus exist as a function of the size of the pitches,which, in theory, can assume any value between plus andminus infinity. The individual conveying elements can bestandard screw elements (Fig. 1, top) or kneading blocks(Fig. 1, bottom). Kneading blocks are combinations of anydesired number of kneading disks of different widths andoffset angles.

Figure 1 bottom shows two-flighted kneading blockswith five kneading discs of a constant width, staggered at

plus 45, 90 and minus 45. The kneading disc width and

the angle of stagger will generally be constant within akneading block. Kneading blocks can be arranged withother screw elements in any desired combination. A knead-

Fig. 1. Screw configuration elements

ing disc is a screw element with an infinitely high pitch. In principle, a helix angle ofs can be defined for a kneading block, by analogy to the observation of a standard screwelement.

2.2 Theoretical self-wiping profile

Tightly intermeshing co-rotating twin screw extruders areextruders with two parallel-axis screws of identical geome-try, which rotate in the same direction and with the sameangular velocity. As a rule, both screws possess the sameoutside diameter over the entire length of the screw andeach point on the surface of one screw is scraped by theother screw. This applies for screw elements and kneading

blocks. The axial profile obtained from this motion princi-ple is shown in Fig. 2. The profile will be identical for screwelements and kneading discs when observing a single ma-chine size and the same number of flights.

The channel depth h is a function of the angular coordi-nate

2

2 2S SD Dh( ) (1 cos ) a sin2 2

= +

(1)

and describes the theoretical self-wiping profile in polarcoordinates. It was formulated by M. L. Booy [4] and,within the validity range 0 ,can be replaced in anexcellent approximation (Taylor) by a parabola of thefourth order for the machine series a/DS2/2 encoun-tered in practice:

4 2

1 2 Sh( ) a a D a = + + (2)

where

2

S S1 S

D D1 1a D

a 128 a 48

1 =

24 (3)

and

S2 S

D1a D .

8 a

1 = 4

(4)

When the screw channel and the barrel are laid out flat(Fig. 3), it is possible to describe the screw channel geome-

Fig. 2. Channel depth h as a function of the polar coordinate [4]


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Intern. Polymer Processing IX (1994) 1 13

Fig. 3. Screw channel and housing laid out flat (SR= 0: theoreticalself-wiping profile)

try in Cartesian coordinates x,y,z by employing the follow-ing coordinate transformation:

S maxt cos e

x .2 2

= +

(5)

Some essential equations for establishing the theoreticalself-wiping profile are listed in Table 1. By taking this

profile as a basis, it is possible to calculate its free axialcross-sectional area in a simple manner, because the limit-ing arcs of the profile are circular arcs in each case. Fig. 4

Table 1. Geometric relationships (theoretical self-wiping profile)

Radial gap R SS (D D )2

1=

Intermeshing angleS

aarccos

D2

=

Flight anglei =

Kneading block pitch

(equivalent screw pitch)Kn

Kn

2t

L

j

=

Pitch angle SS

tarctan

D

=

Maximum flight width Smaxt

e2

cos( )

=

Maximum channel width Smax maxt cos( )

b ei

=

Maximum channel depth max Sh D a=

Mean channel depth

max

max

b /2

max b /2

5 3

1 2

S

h dxb

a a

2 D2

1h(x)

5 3 a

+

=

+= +

+

where: a1,a2 see Eqs. (3), (4)

Fig. 4. Free axial cross-sectional area on a two-flighted screwelement profile

Table 2. Free axial cross-sectional area (theoretical self-wipingprofile (D = Ds))

shows the cross-sectional area of a two-flighted screwprofile. The partial surfaces A1 A4 and hence also the freecross-sectional areas can be readily calculated. Table 2contains the necessary equations. Fig. 5 shows the cross-section of a 1-, 2- and 3-flighted twin screw for an outside

screw diameter of DS = 40 mm and a distance betweenthe axes of a = 37.5 mm. The maximum channel depth ishmax = 2.5 mm in all three cases and is thus constant. Al-though the mean channel depth (see Table 1) is reducedwith an increasing number of flights, the free cross-sectionalarea (see Table 2) becomes greater.

2.3 Practical self-wiping profile

In order to guarantee practical relevance, however, it isnecessary to proceed from the self-wiping profile in prac-tice, i.e. the true contour of the screw elements for all thecalculations. This contour, which is shown for screw ele-

ments in Fig. 6, is referred to as a practical self-wiping

Circle section of flight

(radius = Ds/2, angle = )2

1 S8

1A D=

Circle section of ground

(radius = a Ds/2, angle = ) 2 S1

D8

A a 2(2 )=

Substitute circle section

(radius = a, angle = /2)2

3 4

1A A a

4+ =

Triangular area(base line = a,

high = Ds/2 sin(/2)) 4 S1

A aD sin4 2

=

Area of one profile Pr 1 2 3A (A A )i A 2i= + +

Area of pierce barrels2

Bar S 4A (2 )D 2A4

1 = +

Free cross-sectional area fr Bar Pr A A 2A=


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Fig. 5. Cross-section of a 1-, 2- and 3-flighted screw element profile(theoretical self-wiping profile (SR= 0))

Fig. 6. Tightly intermeshing screws (practical self-wiping profile

(sR > 0)) [6]

profile in what follows and contains all the gaps required inprocess engineering terms, such as flight gap SF, radial gapSRand the flight land gap Sw.

3 Conveying model for co-rotating twin screws

One of the chief features of co-rotating twin screws is that,contrary to single-screw extruders, they are not generallyoperated from a full hopper but via a metering unit. Thismeans that it is possible for the individual screw elements to

be partially filled if the screw elements can convey morematerial than is supplied to them via the metering unit. Thefilling level of individual screw elements is quantifiedthrough the introduction of the degree of filling

f

max

A b hf .

A b h(f 1)

= =

= (6)

Fig. 7. Partially filled screw channel [5] outside the intermeshing zone

Fig. 7 shows the partially filled state of the channel cross-

section of a screw element. The material accumulates at theleading edge of flight and is transported in the direction ofthe screw channel. The definition (6) is based on a perpen-dicular material/gas phase boundary (compare with [5]).

The mean effective channel depth and width is defined asfollows in accordance with Fig. 7:

f

max

x

fmax b

f2

1h h(f) h(x ) h x dx

bx

2

= = = ( )+

(7)

maxf f

bb b(f) b(x ) x .

2

= = = + (8)

In order to correctly calculate the overall conveying systemof a co-rotating twin screw, it is necessary to have physico-mathematical models that make allowance for the threemechanisms of

feeding and conveying of solid material, melting, conveying of melt,whilst simultaneously taking the filling level into account.

The groove model forms the basis of all the models. Inthe groove model, the screws are pictured as being fixed andthe screw barrel surface as rotating around the screws(kinematic reversal). The screw channels and the screw

barrel surface are taken as being laid out flat and projectedin a single plane. For co-rotating twin screws, this gives a

physical substitute model as shown in Fig. 8a and b with kparallel channels over which a plate moves with a velocityof v0, the rotational velocity of the screw barrel.

The number of parallel channels is calculated using [4]:

S S

max max

(2 )D sin ik 2i 1 .

b e

= = +

+ (9)

The intermeshing zone is included in the conveying modelin the form of a narrowing of the channel. This is done onthe basis of the principle that, viewed in physical terms,when the screw is only partially full, material can only be


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Fig. 8a. Groove model for standard screw elements

conveyed through the intermeshing zone if a feed flowdevelops in the intermeshing zone which is equivalent to thedrag flow in the free channel sections upstream [6]. Theconcept free channel section is taken to mean the channelsection that is located outside the intermeshing zone. Onthe basis of the above statement, the application of kine-matic reversal by contrast to numerical methods makes nodifference if mean values are used for all physical values.

Naturally, it is not possible to describe the processes in

differentially small regions in the intermeshing zone, since

Fig. 8b. Groove model for kneading blocks (substitute channel atbottom right)

this, after all, can only be achieved with a three-dimensionalconveying model. Further the influence of the circulation ofthe flow near by the thread is neglected.

To ensure that the conveying model comes as close aspossible to reality, particularly when the kneading blocks areobserved, allowance is made especially for the leakage flowsvia the radial gaps. When the kneading blocks are observed,this includes the leakage flows that pass the gaps (Figs. 8b,9), which are brought about through the angle of stagger of

the discs. The equations listed in Table 3 are obtained.


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Fig. 9. Area of one triangular gap

Table 3. Equations for the corrected radial gap in respect of thegroove model for kneading blocks

Area similar to

triangle caused by

the offset angle ofthe considered

kneading block

[ ]

S

1(d )

2

S

5

5 1

3

3 2

S

A (D h( )) d

2 DA I ( ) I

4

1 aI ( )

2 5

1 a( )

2 3

1(d ) D a

2

h( )

=

=

=

+

+

where: a1,a2 see Eqs. (3), (4)

Number of area (seeabove)

S,Kn

ThreadKn

Kn

sin ( )

j b 1L

J

=

where: bThread = (2 ) DS cos (S)

Corrected radial gapR,corr R

Thread

j AS S

b

= +

3.1 Throughput equations

In the case of the observation of two-dimensional flow, thecorrelation shown in Fig. 10 applies between the dimension-

less volume flow and the dimensionless pressure differencefor an equivalent rectangular channel (Eqs. 7 and 8, where:f = 1). If this correlation is described by the equations listedin Tables 4, 5 and 6 [13] then the approximation plottedwith a dotted line on Fig. 10 is obtained. The equations arevalid in the following ranges:a) Conveying elements:

v0.5 n 1). The pressure at a random point (Fig. 11)can be calculated from the pressure gradient in the directionof the channel. Point 1 is located above the centre of thescrew thread at point z1. Parameter p1 is the pressure at

point 1, and p/z the pressure gradient in the channeldirection intheenvironmentof 1. Pressure p1 works out at:

1 1

pp z .

z

=

(14)

Point 2 is located in the centre of the same screw thread atpoint z2. Pressure p2 is as follows at this point:

2 2

pp z .

z

=

(15)

The pressure difference between points 2 and 1 is then:

x 1 2p p p = (16)

respectively,

x 1 2

pp (z z )

z

=

(17)

axial unwounded

Length of

channeloutside theintermeshingzone

fr

tXL (2 )

2

= frfrS S

L t(2 )X

( ) 2 sin ( )

Z

sin

= =

Length ofchannelinside theintermeshingzone

ei

tXL

2

= eiei

S S

L tX

( ) 2 sin ( )Z

sin

= =

Length of

element pairEle fr ei

L L L tX= + =

Ele fr ei

Ele

S S

Z Z L

L tX

sin ( ) sin ( )

= +

= =


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Table 5. Throughput functions (conveying elements)

General function v pf( ) =

Conveying elements:

Function for the unidimensional casep

v 0.941

n

=

General two-dimensional function of Potente

where

and

p

v 1 2 3 P p0.P Pc (1 n) (c c n)

n

2,2, 912, 1,

= + +

0.94 n 1

S S S

0.94P

n (2 cos( )) cos ( ) cos( )

n

1,

+ =

n 1

SPcos ( )

,2 =

General two-dimensional function of Fornefeld

where

and

v F F p, ,1 2 =

(1 n)

nSF

cos ( )

1, =

n 1

S

(1 sin ( S) cos( S))F

cos ( )

n

2,

=

Backwards-conveying elements:

General two-dimensional function (S = 11.72)

where

and

v PA A, ,1 2 =

(0.31 0.69n)1, 2, = +

(1 n)1 en

2, =

respectively,

max maxx

S

b epp .

z tan

+ =

(18)

This pressure difference, in conjunction with the drag flowacting in the x direction, causes the leakage flow from onechannel to the neighbouring channel.

From the melt flow balance (see the dotted triangle withcorner points ABC in Fig. 8), it follows that

z xV k V V .= (19)

The negative leading sign is valid for conveying screws andthe positive leading sign for backwards-conveying screws.Here, V is the volume flow which is imposed on the systemthrough the metering unit. This volume flow passes each

perpendicular line of the groove model (vertical dotted line,AC, in Fig. 8, of length (2 ) Ds)). The length bthread,which is decisive for the leakage flow in the x direction, isdescribed by the adjacent side AB to the helix angle s.

In cases in which the principal flow in the zdirection isnot essentially influenced by the leakage flow in the xdirection, similar throughput equations can be written forthese two flows, like those in the groove without leakageflows. The equations for the leakage flow model in Table 7are obtained with the dimensionless parameters in Table 8.

3.2 Degree of filling

A pure drag flow in the direction of the screw, channel isassumed to exist for the conveying mechanism:

1vf . =

(20)

Table 6. Dimensionless variables for the equations of Table 5

Channel outsideintermeshing zone

k,fr

v,fr

0z max

1 n

frv,fr n

0z fr

V

v b h

2h

6Kv

1

p

Z

+

=

=

Channel insideintermeshing zone

k,ei

v,ei

0z max max

1 n

eiPfr n

0z ei

V

v (b e )h2

h

6Kv

1

p

Z

+

=

=

Pressure differential forone element pair

p,Ele p.ei p.fr = +

One element pair v,Ele0z max

1 n

Elep,Ele n

0z Ele

V F

kv b h

2

h

6Kv

1

p

Z

+

=

=

Factor of geometry max ei fr ei fr max max

bF

b e

1Z Z

Z Z

= ++

Pressure gradient for

several element pairsp,ges p,Ele i

i( ) =


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Fig. 11. Model for making allowance for the radial gap leakage flows

Table 7. Dimensionless variables for the leakage flow model

Channel volume flowchannelv

max 0z

1b hv

2

V =

Gap volume flowgapv

thread R 0x

1b s v

2

V =

Channel pressure gradientchannel

channel channel

1+ n

P n

channel 0z fr

h p

6K v Z

=

Gap pressure gradientgap

gap gap

1 + n

Rp n 0.94

frgap 0x gap

s pZ6K v n =

Factor of geometrymax max

geo S

max

ecot ( )

e

b

+=

Table 8. Throughput functions in order to respect leakage flow

Total channel volume flow channel channelv 1 2 pY Y =

Factor in order to respect

gap volume flow gapz

1 v

1fY k1

=


gap volume flow gapz

2 v

1fY k

2 =


gap volume flow gap1 geo Pgap vf 1 = +

Eq. 20 makes allowance with factor1for the fact that thecirculation flow leads to a lower conveying capacity in theelement than when a purely drag flow is observed. In thecase of a purely unidimensional flow, 1 = 1 applies.

The actual melt distribution may deviate from the ide-alised distribution in Fig. 7. That is one reason why only

Fig. 12. Mean shear rate as a function of the filling level

integral mean values can be assessed in all the calculationsthat are linked with the degree of filling calculation. Thevariation of the degree of filling has a decisive influence onthe operating behaviour of the extruder, e.g. on the resi-dence time behaviour and on the shear load acting on thematerial (key word: temperature). The contour of the screw

flight means that, as the degree of filling falls, the meaneffective channel depth to be taken into the calculation also becomes smaller (Fig. 7). A higher mean shear rate thusresults from a lower degree of filling [5].

Fig. 12 shows the correlation between the degree offilling and the mean shear rate for different geometrieswhen observing an isothermal, non-Newtonian melt flow.The calculation was performed with and without observa-tion of the leakage flows via the radial gap. There is a

pronounced correlation between the mean shear rate andthe degree of filling when low degrees of filling are observed(f < 0.5).

3.3 Mass in the screw channel

The mass in the screw channel is a function of the localdegrees of filling over the length of the screw configuration.A screw element pair of constant geometry is observed. Adistinction is drawn between three ranges here:

Feed or solid conveying zone

F fr Ele Sm A L f .= (21)

Melting zone:

A fr Ele Sm A L f(y (1 ) (T)).y= + (22)

Melt zone:

S fr Elem A L f (T).= (23)

Finally, the mass in the screw channel is obtained throughthe addition of the values for all screw elements:

F i A i S ii i i

m (m ) (m ) (m ) .= + + (24)

4 Melting profileIt is only possible to arrive at a sufficiently global assess-ment of the extrusion process if the melting profile is

known. The influence of the kneading blocks and/or the


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backwards-conveying elements on the melting processneeds to be described.

Operating the extruder with a metering unit makes itmore difficult to record the melting profile and, in particu-lar, to determine the start of melting. According to J. L.White [7], too, there are no known models to date fordescribing the melting profile in co-rotating twin screws.

One reason for this is the complexity of kneading blocks.A melting model is presented in what follows which is

suitable for quantifying melting in conveying and back-wards-conveying elements and/or kneading blocks. Thegroove models shown in Fig. 8 are taken as a basis in allcases. In the observation of kneading blocks, however, noallowance is made for the fact that the existing layers ofmelt and solids are rearranged by the staggered kneadingdiscs and by the radial gap leakage flows. The radial gapleakage flows themselves, however, are not neglected. As inthe conveying model, a corrected radial gap width is em-

ployed for the calculation. In this way, the radial gapleakage flows are included in the descriptive differential

equation (Eq. 25) for the solid bed profile.In order to calculate the melting length and the solid bed

profile, a modified Tadmor model is employed, whichmakes allowance for the location-dependent melt film thick-ness. Other pushed melting models [8, 9] will not be consid-ered in this paper. The Tadmor model can be used in theobservation of tightly intermeshing screws. If non-tightlyintermeshing screws are observed [10], then rearrangementtakes place in the intermeshing zone on account of theconsiderably greater flight gap.

The differential equation that describes the changingsolid bed profile in the screw channel direction

F F F 1 0x R

d 1

( v x h ) k v ( S )dz 2 = (25)can no longer be solved. Because of this, numerical meth-ods have been used to calculate the length of the solidmaterial bed to date. An analysis of Eq. 25 shows that, ona double logarithmic plot [11], straight lines are obtained

by way of a good approximation (Fig. 13).These can be described by the simple function [11]:

CS

S

y1

= =

(26)

where

R

S0 0 0

S x X

; ; y ; cb

( )

= = = = = 1 + (27)

and where 0 is the initial melt film thickness at the barrelwall for X b,= and the ratio of the solid material with x tothe screw channel width b is the standardised solid bed withy. The standardised melt film thickness is formed fromthe quotient of the melt film thickness to the starting meltfilm thickness 0. Eq. 25 can be solved with this function.The equations set out in Table 9 are obtained as thesolution.

For purposes of simplification, the start of melting(= the point of meltpool formation (PMF)) can be equatedwith the first point of (complete) filling if, as with co-rotat-

ing twin screws, there is a long, partially filled feed zone.

Fig. 13. Dimensionless melt layer thickness as a function of thedimensionless solid bed width

Table 9. Solutions for the melting model [11] considering the leak-age flow through: a) area caused by the offset angle of kneading discsin addition to once in spite of radial gap; b) radial gap of common(backwards-) conveying elements

1 const. = (T) f (T) = Rs 0>

2

2

1 S 0X

S Z F1 R

1 n1 n

2 S F1 rel

S Z F1

k v h

(T T )b c sXy 1

b k K (T )v2

(T T ) c

+

1

1

+= = + +

3

cS

0 S

S

2 S

1

2

c

0 1 R 1 R

y

lg

cy

lgy

( s )y s

1

1

= = ==

= +

4

1 n

A

1 2A 2 A

Z F1

1 1 2 Ak 2 k (e A )

1 e A A e 1

A (T T )n

1

+

= + =

=

5

S

S 1 0 0 S

S S S

X 1y [1 (1 c)(1 ) ]

b 1 c

k v D z L2m D D sin( )

1

1

= =

= = =

SR= SR,corr(if kneading block)

5 Material temperature

The material temperature calculation is performed on thebasis of the groove model. The following assumptions aremade for the temperature calculation: the screw channel is observed in the form of a flat

channel, i.e. b h . The influence of the thread can beneglected in this way,

the melt adheres to the wall, the flow is laminar and sluggish and incompressible

(c = cv = cp),


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Table 10. Dimensionless variables for the temperature calculation

0 Z0

Z

2 1 n 1 n

0 0

p

T T

T h Z

( ) KvBr

c VGzk

y z

h h

hbZ

+

= = =

=

=

mean dissipative specific power =

the flow behaviour of the melt obeys the power law, mean values established over individual sections are used

for all the material values (with the exception of viscosity), allowance is made for the filling degree profile through the

introduction of the effective (mean) channel width andchannel depth.

Under these simplifying assumptions, the descriptive differ-ential equation for an individual zone of constant geometryis as follows:

0

2T

z 02cv ( ) e

z y

( ) = + .

(28)

Introducing the dimensionless parameters summarised in

Table 10, we obtain

Tz

Gz Br e .

2

2

= +

(29)

On account of the exponential function, the differentialequation can no longer be solved. If it is assumed that acalculation is performed on a section by section basis, thenthe equation can be further simplified if it is assumed that

the temperature changes in the direction of the screw chan-nel are small. It then follows thatT 0. (30)

The following simplified differential equation is obtained:

Gz r.2

2

=

(31)

The differential equation was solved in [12]. The followingassumptions were made to this end:a) For the region > 0 the mean starting temperature

start( 0) = = applies.b) The heat produced is introduced into the calculation as

a mean value per coordinate unit and volume unit for

> 0.c) The barrel wall temperature TZ is constant.The temperature is approaching a limit value with increasingdistance perpendicular to the barrel wall (semi-infinite space).

The set of solutions in Table 11 is obtained with theassumptions listed above [12]. Each heating zone is allo-cated a constant wall temperature. The temperature calcu-lation starts at the point in the screw channel at whichmolten plastic is first present. This is at the first point ofmeltpool formation (PFM).

The starting temperature start is worked out from theenergy equation for the melt film at the barrel wall at this

point. The energy equation for the melt film at the barrel

wall [11] runs:

Table 11. Solutions for the axial temperature development

0

Gz 4

1

Z

Z 0

Br Br ( ) erf

Gz 2Gz

BrBr e

2

( )d

GzT T

T

2

2

2

2

,

,

= + +

+

= =

j

2v

S yi2

dd T0.

dydy + = (32)

Under the boundary conditions T(0) = TFL; T( ) = TZ(compare to figure in Table 9), the following solution is

obtained:1 n

A

FL Z F1 2 A

1 A

T( ) T (T T ) Br [1 eA e 1

+

= + +

A1 e

( )]

(33)

where

1 n 1 n

FL rel

S Z FL

K(T )vBr

(T T )

+ +=

(34)

y =

(35)

Z FL

A .n

( )= (36)

It is then possible to calculate a mean temperature for themelt film:

1

0 start Z

start start1

Z

0

v( )T( ) dT T

T T ;T

v( )d

= = =

(37)

with allowance for the velocity profile:

A

rel A

e 1v( ) v e 1

= (38)

and the relative velocity:

2 2

rel 0Z FZ 0Xv (v v ) v .= + (39)

The equations listed in Table 12 are obtained by way of thesolution. The mean temperature for the melt film is addi-tionally the starting temperature for the temperature profilecalculation. Assuming Br = 0, the starting temperature can

be established with the simplified solution [11]:

A

start FL Z FL A

1 A 1e 1

A 2 A

T T (T T ) e A 1

+

= + (40)


11/15



Table 12. Initial temperature for the calculation of the axial temper-ature development

2A 2A

ni 3 4T [B T(A 1)2e (AB Te A A2)]A = +

1 n

Z F12 A

1 AB Br T T T A T

A e 1 n

+

= = =

[ ]

21

A

2 F1

2 A

3 1 F1

4 A

A A (B ) 3AB 2(B 1)

A T(A(B ) 2B 1) AT 2 e

A A B Te TA 2AT (A )

1A

2A(e A )

1

1

1

1

= + + + = + + +

= + + +

=

In this way, all the equations for calculating the tempera-ture along the screw configuration are known.

6 Software package

On account of the complexity of the calculation equations,these were brought together into a software package tomake for easier handling. This DSE software package,which is currently available as a prototype (only internaluse), makes it possible for the entire process to be simulatedfrom the angles mentioned: solid material conveyance,melting and melt conveyance.

The calculation is performed on an element-by-element basis from the die to the hopper. A linear temperatureprofile from the melt temperature (die) to the solid materialtemperature (hopper) is assumed. In order to make al-lowance for the true temperature profile, it is necessary toadopt an iterative approach. The degree of convergence is

high, the calculation results are obtained after only foursteps.

The calculation results for a process divide up intovalues along the screw configuration:

filling degree profile, pressure profile, temperature profile, melting profile, mean wall shear stress profile, mean shear stress profile, power requirements,

and the following scalar results: mean degree of filling, mass in screw channels, mean and shortest residence time, residence time distribution, variance of residence time distribution, self-cleaning coefficient,

axial mixing coefficient.Calculating a complete process calls for a calculation timeof approximately 1 minute on a personal computer, if thenecessary data are available.

7 Experimental investigations

The experimental investigations were conducted with afast-running, tightly intermeshing co-rotating twin screwextruder of type ZSK 30. Results from tests with higherscrewdiameters were also available. The test set-up usedin this paper is shown in Fig. 14. The material data usedis to be found in [6, 13]. The examples selected are repre-

sentative of a population of more than 300 operatingpoints.

7.1 Pressure and degree of filling

The pressure and degree of filling curve is presented in Fig.15. The calculations were performed with the actual values,i.e. with the measured values for the throughput and forthree screw speeds. The melt temperature curve was calcu-lated (Fig. 19). In addition, the pressure was always ex-

pressed in terms of the start or end of the screw element andthe degree of filling in terms of the centre of the screwelement. The calculated values for the integral mean degree

of filling and the screw channel mass have been entered inthe legend. The designation PMF marks the point at whichthe meltpool first forms.

The agreement between the calculated and measured pressures in the metering zone is good. The experimentalpressures at the discharge of the machine are nearly con-stant and result from the melt temperature, the throughputand the die geometry.

Fig. 16 shows calculated melt pressures by comparisonto experimental values at different points along the screws.The deviations are less than 10% on average.

Fig. 14. Test set-up


12/15



Fig. 15. Melt pressure and filling level as a function of the dimen-sionless screw length

Fig. 16. Comparison of calculated and measured melt pressures

7.2 Mass in the screw channels

Fig. 17 shows the comparison of calculated and measuredmass in the screw channels. The level of agreement is good,only when there are small masses in the screw channels is asystematic deviation seen, which is caused mainly by thethree numbered elements 8, 9 and 10.

7.3 Melting profile

The melting profile experiments and calculations show thatthe melting lengths in the considered co-rotating twinscrews (ZSK 30) are short on account of what, in relationto single-screw extruders, is a relatively small specific

throughput 0m/n . The fact that, in the partially filled range,

Fig. 17. Comparison of calculated and measured mass in the screw channels. * number of screw element (see Fig. 10) forwhich the equations listed in Table 5 are somewhat outside ofthe valid range

Fig. 18. Dimensionless solid bed width as a function of the dimen-sionless screw length

the calculated solid material bed width is expressed in termsof the effective mean channel width, means that the dimen-sionless solid bed width first of all increases there (Fig. 18).In practice, it is seen that the melting in partially filledregions is considerably poorer than that in (completely) fullareas. This can be due to the fact that there is no longer any

pronounced flow in partially filled areas. In these cases,excessively short melting lengths are calculated with the

melting model described.


13/15



Fig. 19. Melt and barrel temperature as a function of the dimension-

less screw length

7.4 Melt temperature curve

Fig. 19 shows a comparison between calculated and experi-mental melt temperatures. The measuring sensors (ther-moelements) for establishing the experimental melttemperatures are fixed in the intermeshing zone in eachcase, since it can then be guaranteed that the melt flows

directly around them in (completely) full regions. The pro-files are reflected in principle in the manner expected. Whenassessing the profile, it must be borne in mind that theexperimental determination of the melt temperatures is

Fig. 20. Comparison of calculated and measured melt temperatures.* number of screw element (see Fig. 10) for which the equation listedin Table 5 are somewhat outside of the valid range

influenced through heat conduction effects. This is the casein the partially filled region and the die region.

Fig. 20 shows the correlation between the calculated andthe experimentally determined melt temperatures at the screwtips. The correlation is satisfactory if it is considered that,with the small size of extruder used, the temperature measure-ment is influenced to a large degree by heat conduction effects.

8 Conclusion

The aim was to establish a quantitative method which could be used to show the influence of the screw geometry andmaterial and process parameters on the compounding pro-cess in tightly intermeshing, co-rotating twin screw extrud-ers. A complex, systematic design procedure was compiledto this end, which is explained in part in this paper. Thesystematic design procedure includes a melting model asone of its basic models, which can be used both for stan-dard screw elements and for kneading blocks.

The comparison of experimental and theoretical resultsshows that, in the practical range of process-relevant

parameters, there is a good level of agreement. This appliesfor processes that can be designated melt-dominated.

Through the implementation of the algorithms in a soft-ware package, which is available as a prototype, it is now

possible to quantitatively assess processes on a rapid andsimple basis. Using a software package of this type, themachine manufacturer can analyse different screw configu-rations and process conditions as to their suitability for thecompounding of polymers. This means that an instrumentis available for supplementing what has generally been theempirical design of screw configurations to date with aquantitative method.

Future developments are to lead on to a commercialsoftware package for analysing and simulating the pro-cesses that take place in co-rotating twin screws so as tohelp the machine manufacturer and/or user out of thedilemma of a purely empirical approach to screw design.

References

1 Prause, J.: Plastics Technology 13, p. 41 (1967); Plastics Tech-nology 14, p. 29 (1968); Plastics Technology 14, p. 52 (1968)

2 Hensen, F., Knappe, W., Potente, H.: Handbuch der Kunst-stoff-Extrusionstechnik, Bd. 1: Grundlagen. Hanser, Mnchen,Wien (1989)

3 Hensen, F., Knappe, W., Potente, H.: Handbuch der Kunst-stoff-Extrusionstechnik, Bd. 2: Extrusionsanlagen. Hanser,

Munchen, Wien (1989)4Booy, M. L.: Polym. Eng. and Sci. 18, p. 973 (1978)5 Werner, H.: Das Betriebsverhalten der zweiwelligen

Knetscheiben-Schneckenpresse vom Typ ZSK bei der Verarbei-tung von hochviskosen Flssigkeiten. University, Munich (1976)

6Potente, H., Ansahl, J.: Optimierung von Schneckenpaaren frdie Aufbereitung und Verarbeitung von vorwiegend Polyolefi-nen auf gleichsinnig drehenden Zweischneckenmaschinen.DFG-Project, Po 171/16-1; Analyse des Leistungs- und Tem-

peraturverhaltens in Gleichdrall-Doppelschneckenmaschinenfr die Verarbeitung von Polyolefinen. DFG-Project, Po 171/16-2; Analyse der Plastifiziervorgnge in Gleichdrall-Dop-

pelschneckenmaschinen bei der Verarbeitung von Polyolefinen.DFG-Project, Po 171/16-3

7 White, J. L.: Twin Screw Extrusion Technology and Principles.Hanser, Munich, Vienna, New York (1990)


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8Pearson, J. R. A.: The Chemical Engineer, p. 91, 2 (1977)9Lindt, J. T.: A Dynamic Melting Model for a Single Screw

Extruder Polym. Eng. Sci. 16, p. 289 (1976)10Hornsby, P. R.: Plastics Rubber Process. Appl. 7, p. 237 (1987)11Potente, H., Wortberg, J., Effen, N., Schppner, V., Stenzel, H.,

Klarholz, B.: Rechnergesttzte Extruderauslegung, Kunststoff-technisches Seminar, University Paderborn (1992)

12 Carslaw, H. S., Jaeger, J. C.: Conduction of Heat in Solids.Oxford University Press, London (1959)

13Potente, H., Ansahl, J., Wittemeier, R.: Int. Polym. Process. 3,p. 208 (1990)

Acknowledgments

We extend our thanks to the Deutsche Forschungsgemeinschaft(DFG) for its financial support as well as to Werner & Pfleiderer,Stuttgart, for making available a ZSK 30 twin screw extruder.Our thanks also go to Werner & Pfleiderer, Stuttgart, andBASF, Ludwigshafen who provided us with test material free ofcharge.

Date received: November 16, 1992Date accepted: May 13, 1993

List of symbols

Latin symbols X quotient of screw element pair length to

pitch

A channel cross-sectional area X solid bed width

A parameter for calculating the temperature X1,2 solid bed width

in the melt film Y1 constant for characteristic curve for the

ABar cross-sectional area of pierce barrels throughput

A1 4 area Y2 gradient of throughput characteristic

A area similar to triangle caused by the curveoffset angle of the kneading block Zei length of the intermeshing channel laid

Afr free cross-sectional area out flat

Af filled channel cross-sectional area Zfr length of the free channel cross-section

APr cross-sectional area of one profile laid out flat

D inside barrel diameter Z,z Cartesian coordinates (channel direction),

DS outside screw diameter length of zone laid out flat

F factor of geometry a distance between axes

H enthalpy difference a1,2 coefficientK coefficient of the power law b channel width (rectangular channel)

Kgap coefficient of the power law in respect of b effective mean channel width

gap bmax maximum channel width

Kchannel coefficient of the power law in respect of bthread thread length for leakage flow model

channel c gradient constant for the melting profileK0T coefficient of the power law at the refer- cP specific heat capacity at constant pressure

ence temperature e flight land width

L axial length coordinate emax maximum flight land width

LEle axial length of an element pair f degree of filling outside the intermeshing

Lei axial length of the intermeshing channel zone

Lfr axial length of the free channel (outside f degree of filling of an element pair

the intermeshing zone) f1 dimensionless parameter for the leakage

LKn axial length of kneading block flow model

PMF point of meltpool formation h channel depth (rectangular channel)

T temperature h effective mean channel depthTB,T0,1 reference temperature h enthalpy

T1 reference temperature hmax maximum channel depth

Ti melt temperature at location i h() channel depth as a function of polar coor-TF solid material temperature dinate TFL melting point h(x) channel depth as a function of the Carte-

TM,T material temperature sian coordinate x

TS melt temperature i counting variable

startT mean starting temperature i flight count of elements

TZ barrel temperature j counting variable

V volume flow from metering system j number of triangular gaps in respect of

eiV, volume flow in the intermeshing zone of a kneading block, groove model and leakage

channel flow model

frV, volume flow in the free zone of a channel jKn number of kneading discs

xV volume flow in x direction (leakage flow) k number of parallel channels in groove

zV volume flow in z direction model


15/15


Intern Polymer Processing IX (1994) 1 25

k1,k2 constants dimensionless coordinate in y directionm material throughput 0,1 dimensionless characteristic valuem material in screw channel geo dimensionless geometry variable for themA material in screw channel in melting zone leakage flow model

mF material in screw channel in solid zone p dimensionless pressure gradientmS material in screw channel in melt zone p,gap dimensionless pressure gradient in the gapn flow law exponent p,channel dimensionless pressure gradient in thengap flow law exponent in respect of gap screw channel

nchannel flow law exponent in respect of channel p,gas dimensionless pressure gradient in totaln0 screw speed p,Ele dimensionless pressure gradient in respect

p pressure, melt pressure of a pair of elements

pi melt pressure at location i p,ei dimensionless pressue gradient inside the p pressure difference intermeshing zonepx pressure difference in x direction p,fr dimensionless pressure gradient outsidesF flight gap width the intermeshing zone

sR radial gap width V dimensionless volume flowsR,corr corrected radial gap V,gap dimensionless drag flow in respect of thesW flight land gap width gap

t pitchV,channel

dimensionless drag flow in respect of thev velocity channel

vrel relative velocity m dimensionless material flowv0 circumferential velocity of the screws V,ei dimensionless volume flow in the inter-v0x x component of circumferential velocity meshing zone

v0z z component of circumferential velocity V,Ele dimensionless volume flow in respect of anvF mean solid bed velocity element pair

vj velocity in j direction V,fr dimensionless volume flow outside thevx velocity in x direction intermeshing zone

vz velocity in z direction ,S density

x Cartesian coordinate at right angles to shear stresschannel direction mean shear stress

xf position of the material front in the screw flight land anglechannel

y dimensionless solid bed width

yi mean dimensionless solid bed width for variables for the linear approximation of

element i the throughput equations

z coordinate cut

z z

F F

P P

,

,

,

,

1 2

1 2

1 , 2 ,

1, 2,

1, 2,

,

Greek symbols intermeshing angle angle of stagger of kneading discs helix angle temperature shift (Arrhenius formulation) S helix angle at outside screw diameter shear rate S,Kn characteristic helix angle at outside screw mean shear rate diameter in respect of a kneading block mean melt layer thickness standardised melt layer thickness

melt layer thickness s standardised radial gap thickness0 starting melt layer thickness

0 mean starting melt layer thicknessDimensionless characteristic values

dimensionless coordinate in z direction Br Brinkmann number melt viscosity Gz Graetz number polar coordinate dimensionless temperature

Indices

0 mean temperature calc. calculated

start dimensionless mean starting temperature exp. experimental,S thermal conductivity F solid

mechanical math1

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