h potente* j klarholz design of tightly intermeshing co ......twin screw extruders composite models...

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 level profiles, the pressure profiles, the melting profiles, the resi-dence time distributions, the temperature profiles, the shear stress profiles, and the power consumption in modular tightly intermeshing co-rotating twin screw extruders (ZSK) are developed. 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 screw and kneading disc elements, including left- and right-handed sections. Kneading blocks were approximated by a screw of “equivalent pitch” with making allowance for the leakage flow across the flights from one channel to the adjacent channel. The mathematical treatment of co-rotating twin screw extruders has been based up according to the theory of single 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-rotating twin screws in what follows, the screw design is drawn up as a 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 an empirical basis, and past developments have not culminated in 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 number of different screw configurations in their ranges, which are tailored to different needs.

Taking the idea put forward by J. Prause in [1], it is possible to divide the available configurations into four basic categories. The compression that the material is subjected to in the machine plays an essential role in drawing this distinction. Since, with co-rotating twin screws, the mean channel depth is laid down by the distance between the axes and the outside diameter of the screw, the material being compounded is compressed through a reduction in the pitch and/or through backwards-conveying elements (such as screw elements or kneading blocks) or even through axial and 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-type screws and barrels, will be briefly set out below [1]:

Low-work screws

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

Standard screws

Standard screws are designed with a standard compression ratio and are used both for melting and compounding tasks. The screw design of so-called standard screws varies from one machine builder to the next.

High-shear screws

These screws incorporate elements which generate high shear rates in order to improve dispersion behaviour. In the case of co-rotating twin screws, this is achieved through the application 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 comes about with an increase in the speed for a constant material throughput. The lower residence time, in turn, represents a disadvantage when it comes to the shear deformation that is caused, this being the product of shear rate and residence time. Backwards-conveying elements affect the back-pres-sure length respectively the residence time in the up-channel direction so that the shear deformation increases.

Vented screws

Vented screws are used to degas the polymer melt when residual 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 a breakdown according to the principle involved. In practice, there is a wide range of screw configurations supplied by the machine builders. Virtually all manufacturers have a three-flighted machine series with relatively low flight depths and another series with medium and large flight depths [2, 3]. No single-flighted series or series with four or more flights is known in practice [2, 3]. There are, however, combinations of single-flighted elements in the feed zone with two- or three-flighted elements in the plasticization and processing zone. The problem that the machine builders

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 that they design.

Since a large number of machine builders generally em-ploy modular-type designs for the screw and barrel, in order to permit greater flexibility, the machine can be modified in so many different ways that there are an infinite number 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 into greater detail, of individual screw elements, then empirical statements do not provide sufficient assistance. In order to be able to analyse the influence of the screw geometry, the material and the process parameter, it is necessary to em-ploy quantitative methods as well, which make statements about the compounding or extrusion process.

2 The geometry of co-rotating twin screw extruders 2.1 Screw geometry elements

A screw configuration is essentially made up of conveying elements (positive helix angle) and backwards-conveying elements (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 and minus infinity. The individual conveying elements can be standard screw elements (Fig. 1, top) or kneading blocks (Fig. 1, bottom). Kneading blocks are combinations of any desired number of kneading disks of different widths and offset angles.

Figure 1 bottom shows two-flighted kneading blocks with 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 a kneading block. Kneading blocks can be arranged with other 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 of ϕs can be defined for a kneading block, by analogy to the observation of a standard screw element.

2.2 Theoretical self-wiping profile

Tightly intermeshing co-rotating twin screw extruders are extruders with two parallel-axis screws of identical geome-try, which rotate in the same direction and with the same angular velocity. As a rule, both screws possess the same outside diameter over the entire length of the screw and each point on the surface of one screw is scraped by the other 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 screw elements 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 Θ

22 2S SD D

h( ) (1 cos ) a sin2 2

Θ = + Θ − − Θ

(1)

and describes the theoretical self-wiping profile in polar coordinates. It was formulated by M. L. Booy [4] and, within the validity range 0 ≤ Θ ≤ Ω, can be replaced in an excellent approximation (Taylor) by a parabola of the fourth order for the machine series a/DS ≤ √2/2 encoun-tered in practice:

4 21 2 Sh( ) a a D aΘ = ⋅ Θ + Θ + − (2)

where 2

S S1 S

D D1 1a Da 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 Θ [4]


Intern. Polymer Processing IX (1994) 1 13

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

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

S maxt cos ex .

2 2Θ ϕ

= +π

(5)

Some essential equations for establishing the theoretical self-wiping profile are listed in Table 1. By taking this profile as a basis, it is possible to calculate its free axial cross-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 )21= −

Intermeshing angle S

aarccosD

2 Ω

=

Flight angle iπ

φ Ω= −

Kneading block pitch (equivalent screw pitch) Kn

Kn

2t Lj

πα

=

Pitch angle SS

tarctanD

ϕ

π =

Maximum flight width Smax

te2

cos( )φπ

ϕ=

Maximum channel width Smax max

t cos( )b ei

ϕ= −

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 a2 D

2

1 h(x)

5 3 a

+

−

Ω Ω

Ω φ

=

+= + −

+

∫

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

Fig. 4. Free axial cross-sectional area on a two-flighted screw element profile Table 2. Free axial cross-sectional area (theoretical self-wiping profile (D = Ds))

shows the cross-sectional area of a two-flighted screw profile. The partial surfaces A1 – A4 and hence also the free cross-sectional areas can be readily calculated. Table 2 contains 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 between the axes of a = 37.5 mm. The maximum channel depth is hmax = 2.5 mm in all three cases and is thus constant. Al-though the mean channel depth (see Table 1) is reduced with an increasing number of flights, the free cross-sectional area (see Table 2) becomes greater.

2.3 Practical self-wiping profile

In order to guarantee practical relevance, however, it is necessary to proceed from the self-wiping profile in prac-tice, i.e. the true contour of the screw elements for all the calculations. 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 = φ) 2

1 S81A Dφ=

Circle section of ground (radius = a – Ds/2, angle = φ) 2 S

1 D8

A a 2φ(2 )= −

Substitute circle section (radius = a, angle = Ω/2) 2

3 41A A a4

Ω+ =

Triangular area (base line = a, high = Ds/2 · sin(Ω/2)) 4 S

1A aD sin4 2

Ω

=

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

Area of pierce barrels 2Bar S 4A (2 )D 2A

41

π Ω= − +

Free cross-sectional area fr Bar PrA A 2A= −



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 in process engineering terms, such as flight gap SF, radial gap SR and 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 generally operated from a full hopper but via a metering unit. This means that it is possible for the individual screw elements to be partially filled if the screw elements can convey more material than is supplied to them via the metering unit. The filling level of individual screw elements is quantified through 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 the leading edge of flight and is transported in the direction of the 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 as follows in accordance with Fig. 7:

f

max

x

fmax b

f 2

1h h(f) h(x ) h x dxbx

2−

= = = ⋅ ( )+

∫ (7)

maxf f

bb b(f) b(x ) x .

2= = = + (8)

In order to correctly calculate the overall conveying system of a co-rotating twin screw, it is necessary to have physico-mathematical models that make allowance for the three mechanisms 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. In the groove model, the screws are pictured as being fixed and the 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 projected in a single plane. For co-rotating twin screws, this gives a physical substitute model as shown in Fig. 8a and b with k parallel channels over which a plate moves with a velocity of 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

π − Ω ϕ φ= = − +

+ π (9)

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



Fig. 8a. Groove model for standard screw elements

conveyed through the intermeshing zone if a feed flow develops in the intermeshing zone which is equivalent to the drag flow in the free channel sections upstream [6]. The concept “free channel section” is taken to mean the channel section that is located outside the intermeshing zone. On the basis of the above statement, the application of kine-matic reversal by contrast to numerical methods makes no difference 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 at bottom right)

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

To ensure that the conveying model comes as close as possible to reality, particularly when the kneading blocks are observed, allowance is made especially for the leakage flows via 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.



Fig. 9. Area of one triangular gap

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

Area similar to triangle caused by the offset angle of the considered kneading block

[ ]

S1 (d )2

S

55 1

33 2

S

A (D h( )) d

2 DA I ( ) I4

1 aI ( )2 5

1 a( )2 3

1 (d ) D a2

h( )Ω

∆

Ω − − φ

∆

θ θ θ

α φ α φ

Ω Ω α ω

Ω Ω α φ

φ

= −

= − −−

= − − −

+ − − −

+ − ⋅ −

∫

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

Number of area (see above)

S, KnThread

Kn

Kn

sin ( )j b 1L

J

∆

ϕ= −

where: bThread = (2π – Ω) ⋅ DS ⋅ cos (ΨS)

Corrected radial gap R,corr R

Thread

j AS Sb

∆ ∆= +

3.1 Throughput equations

In the case of the observation of two-dimensional flow, the correlation shown in Fig. 10 applies between the dimension-less volume flow and the dimensionless pressure difference for an equivalent rectangular channel (Eqs. 7 and 8, where: f = 1). If this correlation is described by the equations listed in Tables 4, 5 and 6 [13] then the approximation plotted with a dotted line on Fig. 10 is obtained. The equations are valid in the following ranges: a) Conveying elements:

v0.5 n < 1 for 0.1 < 1.9< < π (10) or

v0.2 n < 0.5 for 0.3 < 1.7< < π (11)

and

S Sa/D 0.85; t/D 1.6.≥ ≤ (12)

Fig. 10. Dimensionless volume flow as a function of the dimensionless pressure gradient for conveying and backwards-conveying elements border of validity of approximated function

Table 4. Simple channel geometry (equivalent rectangular channel)

b) Backwards-conveying elements: v0.2 n 1 for 0 1.≤ ≤ < π ≤ (13)

The following procedure is employed in order to make allowance for the leakage flows via the radial gap:

Finitely small sections of a pair of screw elements are observed (i > 1). The pressure at a random point (Fig. 11) can be calculated from the pressure gradient in the direction of the channel. Point 1 is located above the centre of the screw thread at point z1. Parameter p1 is the pressure at point 1, and ∆p/∆z the pressure gradient in the channel direction in the environment of 1. Pressure p1 works out at:

1 1pp z .z

∆=

∆ (14)

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

2 2pp z .z

∆=

∆ (15)

The pressure difference between points 2 and 1 is then:

x 1 2p p p∆ = − (16) respectively,

x 1 2pp (z z )z

∆∆ = −

∆ (17)

axial unwounded

Length of channel outside the intermeshing zone

frtXL (2 )2

π Ωπ

= − frfr

S S

L t(2 )X( ) 2 sin ( )

Z sinπ Ω

ϕ π ϕ−= =

Length of channel inside the intermeshing zone

eitXL2

Ωπ

= eiei

S S

L tX( ) 2 sin ( )

Z sinΩ

ϕ π ϕ= =

Length of element pair

Ele fr eiL L L tX= + =

Ele fr ei

Ele

S S

Z Z LL tX

sin ( ) sin ( )ϕ ϕ

= +

= =



Table 5. Throughput functions (conveying elements)

General function v pf( )π π=

Conveying elements:

Function for the unidimensional case pv 0.941

nπ

π = −

General two-dimensional function of Potente where and

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

n2,

2, 912, 1,

φ π φ φ φ π

= − + + −

0.94 n 1S S S

0.94P

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

−

1,

ϕ ϕ ϕ− − +φ = n 1

SPcos ( )−

,2φ ϕ=

General two-dimensional function of Fornefeld

where and

v F F p, ,1 2π φ φ π= −

(1 n)n

SFcos ( )

−

1,φ ϕ=

n 1S

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

cos ( )n

−

− ϕ ϕ2,

ϕφ =

Backwards-conveying elements:

General two-dimensional function (ϕS = 11.72) where and

v PA A, ,1 2π φ φ π= −

(0.31 0.69n)1,Α 2,Αφ φ= − +

(1 n)1 en

−2,Αφ =

respectively,

max maxx

S

b epp .z tan

+∆∆ = − ⋅

∆ ϕ (18)

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

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

z xV k V V .= ⋅ ∓ (19) The negative leading sign is valid for conveying screws and the positive leading sign for backwards-conveying screws. Here, V is the volume flow which is imposed on the system through the metering unit. This volume flow passes each perpendicular line of the groove model (vertical dotted line, AC, in Fig. 8, of length (2 ⋅ π – Ω) ⋅ Ds)). The length bthread, which is decisive for the leakage flow in the x direction, is described by the adjacent side AB to the helix angle ϕs.

In cases in which the principal flow in the z direction is not essentially influenced by the leakage flow in the x direction, similar throughput equations can be written for these two flows, like those in the groove without leakage flows. The equations for the leakage flow model in Table 7 are obtained with the dimensionless parameters in Table 8.

3.2 Degree of filling

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

1v f .π = ⋅φ (20)

Table 6. Dimensionless variables for the equations of Table 5

Channel outside intermeshing zone

k,frv,fr

0z max

1 nfr

v,fr n0z fr

V

v b h2h

6Kv

1

pZ

+

π

∆π

=

=

Channel inside intermeshing zone

k,eiv,ei

0z max max

1 nei

Pfr n0z ei

V

v (b e )h2h

6Kv

1

pZ

+

π

∆π

=−

=

Pressure differential for one element pair

p,Ele p.ei p.frπ π π= +

One element pair v,Ele

0z max

1 nEle

p,Ele n0z Ele

V Fkv b h

2h

6Kv

1

pZ

+

π

∆π

=

=

Factor of geometry maxei fr

ei fr max max

bFb e

1 Z ZZ Z

= ++ −

Pressure gradient for several element pairs

p,ges p,Ele ii

( )π ∑ π=



Fig. 11. Model for making allowance for the radial gap leakage flows Table 7. Dimensionless variables for the leakage flow model

Channel volume flow channelv

max 0z1 b hv2

Vπ =

Gap volume flow gapv

thread R 0x1 b s v2

Vπ =

Channel pressure gradient channel

channel channel

1 + n

P nchannel 0z fr

h p6K v Z

∆π =

Gap pressure gradient gap

gap gap

1 + nR

p n 0.94frgap 0x gap

s pZ6K v n∆

π =

Factor of geometry max maxgeo S

max

e cot ( )e

bπ ϕ

+=

Table 8. Throughput functions in order to respect leakage flow

Total channel volume flow channel channelv 1 2 pY Yπ π= −

Factor in order to respect gap volume flow gap

z1 v

1fY k1φ

π=

Factor in order to respect gap volume flow gap

z2 v

1fY k2φ

π=

Factor in order to respect gap volume flow gap1 geo Pgap vf 1± ± π π π= +

Eq. 20 makes allowance with factor φ1 for the fact that the circulation flow leads to a lower conveying capacity in the element than when a purely drag flow is observed. In the case of a purely unidimensional flow, φ1 = 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 calculations that are linked with the degree of filling calculation. The variation of the degree of filling has a decisive influence on the operating behaviour of the extruder, e.g. on the resi-dence time behaviour and on the shear load acting on the material (key word: temperature). The contour of the screw flight means that, as the degree of filling falls, the mean effective channel depth to be taken into the calculation also becomes smaller (Fig. 7). A higher mean shear rate thus results from a lower degree of filling [5].

Fig. 12 shows the correlation between the degree of filling and the mean shear rate for different geometries when 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 and the 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 local degrees of filling over the length of the screw configuration. A screw element pair of constant geometry is observed. A distinction 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 through the addition of the values for all screw elements:

F i A i S ii i i

m (m ) (m ) (m ) .= ∑ + ∑ + ∑ (24)

4 Melting profile

It 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



backwards-conveying elements on the melting process needs to be described.

Operating the extruder with a metering unit makes it more 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 for describing 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. The groove models shown in Fig. 8 are taken as a basis in all cases. In the observation of kneading blocks, however, no allowance is made for the fact that the existing layers of melt and solids are rearranged by the staggered kneading discs and by the radial gap leakage flows. The radial gap leakage flows themselves, however, are not neglected. As in the conveying model, a corrected radial gap width is em-ployed for the calculation. In this way, the radial gap leakage 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, which makes 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 the observation of tightly intermeshing screws. If non-tightly intermeshing screws are observed [10], then rearrangement takes place in the intermeshing zone on account of the considerably greater flight gap.

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

F F F 1 0x Rd 1( v x h ) k v ( S )dz 2

− ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ δ − (25)

can no longer be solved. Because of this, numerical meth-ods have been used to calculate the length of the solid material bed to date. An analysis of Eq. 25 shows that, on a 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

RS

0 0 0

S x X; ; y ;cb

δ δ δ( )= = = = δ =

δ δ δ 1 + (27)

and where δ0 is the initial melt film thickness at the barrel wall for X b,= and the ratio of the solid material with x to the screw channel width b is the standardised solid bed with y. The standardised melt film thickness is formed from the quotient of the melt film thickness δ to the starting melt film thickness δ0. Eq. 25 can be solved with this function. The equations set out in Table 9 are obtained as the solution.

For purposes of simplification, the start of melting (= the point of meltpool formation (PMF)) can be equated with 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 the dimensionless 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 discs in addition to once in spite of radial gap; b) radial gap of common (backwards-) conveying elements

1 const.δ = (T) f (T)η = Rs 0>

2

2

1 S 0X

S Z F1 R1 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

−+

ρ ∆ δ λ 1

δ δ λ 1

− += = −+ − +

3

cS

0 S

S

2 S

1

2

c0 1 R 1 R

y

lgc

ylgy

( s )y s

1∗

1

−

ψ ψδψ ψ

δ ψ

ψ ψ

ψ ψ

δ δ

−= = =−−−=

= − +

4

1 nA

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

β

+

= + = − −− −

= −

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( )

ψ

ρϕ

1

1

π ζ

δπ ζ

= = − − − −= = =

SR = SR,corr (if kneading block)

5 Material temperature

The material temperature calculation is performed on the basis of the groove model. The following assumptions are made 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 be neglected in this way,

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

(c = cv = cp),



Table 10. Dimensionless variables for the temperature calculation

0 Z0

Z

2 1 n 1 n0 0

p

T TT h Z

( ) KvBr

c VGzk

y z

h h

hbZ

+ −

Ζ Ζ

Θ ξ ζ

τγ≈

λΤ λΤ

λ

−= = =

=

=

mean dissipative specific powerτγ =

– 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 and channel depth.

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

0

2T

z 02cv ( ) ez y

−β(Τ − )δΤ δ Τ= λ + τγ .

δ δ (28)

Introducing the dimensionless parameters summarised in Table 10, we obtain

TzGz Br e .

−β ⋅ ⋅ Θ2

2

δΘ δ Θ⋅ = +

δζ δξ (29)

On account of the exponential function, the differential equation can no longer be solved. If it is assumed that a calculation is performed on a section by section basis, then the 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 that

T 0.β ∆ → (30) The following simplified differential equation is obtained:

Gz r.2

2

δ Θ δΘ− = − Β

δζδξ (31)

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

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

a mean value per coordinate unit ∆ζ and volume unit for ζ > 0.

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

The set of solutions in Table 11 is obtained with the assumptions 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 which molten plastic is first present. This is at the first point of meltpool formation (PFM).

The starting temperature startΘ is worked out from the energy 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

1Z

Z 0

Br Br( ) erfGz 2

Gz

BrBr e2

( )d

GzT T

T

2

2

2

− ξ ζ 2 ξ

ξ Θ ζ,ξ Θ ζ ξ ξ

ζξ

Θ Θ ζ,ξ ξ

= + +

+ −π

−= = ∫

j2

vS yi2

dd T 0.dydy

λ + τ = (32)

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

1 nA

FL Z F1 2 A

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

+

ξ ξ = + − ξ + − −

A1 e−ξ( − )]

(33)

where 1 n 1 n

FL rel

S Z FL

K(T )vBr

(T T )

+ +δ=

λ − (34)

yξ =

δ (35)

Z FLA .n

β(Τ − Τ )= (36)

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

1

0 start Zstart start1

Z

0

v( )T( )dT T

T T ;T

v( )d

ξ ξ ξ−

= = Θ =ξ ξ

∫

∫ (37)

with allowance for the velocity profile: A

rel A

e 1v( ) ve 1

ξ −ξ =

− (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 the solution. The mean temperature for the melt film is addi-tionally the starting temperature for the temperature profile calculation. Assuming Br = 0, the starting temperature can be established with the simplified solution [11]:

A

start FL Z FL A

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

e A 1

− + − = + − ⋅

− − (40)



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

2A 2Ani 3 4T [B T(A 1)2e (AB Te A A2)]A∆ ∆= − − + −∓

1 n

Z F12 A

1 AB Br T T T A TA e 1 n

+ β

∆ ∆

= = − =−

[ ]

21

A2 F1

2 A3 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 )

11

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 to make for easier handling. This “DSE” software package, which is currently available as a prototype (only internal use), makes it possible for the entire process to be simulated from 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 temperature profile from the melt temperature (die) to the solid material temperature (hopper) is assumed. In order to make al-lowance for the true temperature profile, it is necessary to adopt an iterative approach. The degree of convergence is high, the calculation results are obtained after only four steps.

The calculation results for a process divide up into values 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 time of approximately 1 minute on a personal computer, if the necessary data are available.

7 Experimental investigations

The experimental investigations were conducted with a fast-running, tightly intermeshing co-rotating twin screw extruder of type ZSK 30. Results from tests with higher screwdiameters were also available. The test set-up used in this paper is shown in Fig. 14. The material data used is to be found in [6, 13]. The examples selected are repre-sentative of a population of more than 300 operating points.

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 for three 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 and the degree of filling in terms of the centre of the screw element. The calculated values for the integral mean degree of filling and the screw channel mass have been entered in the legend. The designation PMF marks the point at which the meltpool first forms.

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

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

Fig. 14. Test set-up



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 measured mass in the screw channels. The level of agreement is good, only when there are small masses in the screw channels is a systematic deviation seen, which is caused mainly by the three numbered elements 8, 9 and 10.

7.3 Melting profile

The melting profile experiments and calculations show that the melting lengths in the considered co-rotating twin screws (ZSK 30) are short on account of what, in relation to 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) for which the equations listed in Table 5 are somewhat outside of the 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 terms of 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 filled regions is considerably poorer than that in (completely) full areas. 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.



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 melt temperatures are fixed in the intermeshing zone in each case, 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. When assessing the profile, it must be borne in mind that the experimental 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 listed in Table 5 are somewhat outside of the valid range

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

Fig. 20 shows the correlation between the calculated and the experimentally determined melt temperatures at the screw tips. 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 and material and process parameters on the compounding pro-cess in tightly intermeshing, co-rotating twin screw extrud-ers. A complex, systematic design procedure was compiled to this end, which is explained in part in this paper. The systematic design procedure includes a melting model as one of its basic models, which can be used both for stan-dard screw elements and for kneading blocks.

The comparison of experimental and theoretical results shows that, in the practical range of process-relevant parameters, there is a good level of agreement. This applies for 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 and simple basis. Using a software package of this type, the machine manufacturer can analyse different screw configu-rations and process conditions as to their suitability for the compounding of polymers. This means that an instrument is available for supplementing what has generally been the empirical design of screw configurations to date with a quantitative method.

Future developments are to lead on to a commercial software package for analysing and simulating the pro-cesses that take place in co-rotating twin screws so as to help the machine manufacturer and/or user out of the dilemma 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, München, Wien (1989)

3 Hensen, F., Knappe, W., Potente, H.: Handbuch der Kunst- stoff-Extrusionstechnik, Bd. 2: Extrusionsanlagen. Hanser, Munchen, Wien (1989)

4 Booy, 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 Flüssigkeiten. University, Munich (1976)

6 Potente, H., Ansahl, J.: Optimierung von Schneckenpaaren für die 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-Doppelschneckenmaschinen für die Verarbeitung von Polyolefinen. DFG-Project, Po 171/ 16-2; Analyse der Plastifiziervorgänge 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)



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

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

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

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

13 Potente, 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, and BASF, Ludwigshafen who provided us with test material free of charge.

Date received: November 16, 1992 Date 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 curve offset 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 coefficient K 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 profile K0T 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 depth TB,T0,1 reference temperature h enthalpy T1 reference temperature hmax maximum channel depth Ti melt temperature at location i h(θ) channel depth as a function of polar coor- TF solid material temperature dinate Θ 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

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

channel flow model frVΚ,

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



k1,k2 constants ξ dimensionless coordinate in y direction m material throughput π0,1 dimensionless characteristic value m material in screw channel πgeo dimensionless geometry variable for the mA material in screw channel in melting zone leakage flow model mF material in screw channel in solid zone πp dimensionless pressure gradient mS material in screw channel in melt zone πp,gap dimensionless pressure gradient in the gap n flow law exponent πp,channel dimensionless pressure gradient in the ngap flow law exponent in respect of gap screw channel nchannel flow law exponent in respect of channel πp,gas dimensionless pressure gradient in total n0 screw speed πp,Ele dimensionless pressure gradient in respect p pressure, melt pressure of a pair of elements pi melt pressure at location i πp,ei dimensionless pressue gradient inside the ∆p pressure difference intermeshing zone ∆px pressure difference in x direction πp,fr dimensionless pressure gradient outside sF flight gap width the intermeshing zone sR radial gap width Vπ dimensionless volume flow sR,corr corrected radial gap V,gapπ dimensionless drag flow in respect of the sW flight land gap width gap t pitch V,channelπ dimensionless drag flow in respect of the v velocity channel vrel relative velocity mπ dimensionless material flow v0 circumferential velocity of the screws V,eiπ dimensionless volume flow in the inter- v0x x component of circumferential velocity meshing zone v0z z component of circumferential velocity V,Eleπ dimensionless volume flow in respect of an vF mean solid bed velocity element pair vj velocity in j direction V,frπ dimensionless volume flow outside the vx velocity in x direction intermeshing zone vz velocity in z direction ,S density x Cartesian coordinate at right angles to τ shear stress channel direction τ mean shear stress xf position of the material front in the screw Φ flight land angle channel 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,Α 2,Α

1, 2,

1, 2,

φ ,φ

φ φ φ φ φ φ φ φ

Greek symbols Ω intermeshing angle α angle of stagger of kneading discs ϕ helix angle β temperature shift (Arrhenius formulation) ϕS helix angle at outside screw diameter γ shear rate ϕS,Kn characteristic helix angle at outside screw γ mean shear rate diameter in respect of a kneading block δ mean melt layer thickness ψ standardised melt layer thickness δ melt layer thickness ψs standardised radial gap thickness δ0 starting melt layer thickness

0δ mean starting melt layer thickness Dimensionless characteristic values

ζ dimensionless coordinate in z direction Br Brinkmann number η melt viscosity Gz Graetz number θ polar coordinate θ dimensionless temperature Indices

0θ mean temperature calc. calculated startθ dimensionless mean starting temperature exp. experimental

λ,λS thermal conductivity F solid

h potente* j klarholz design of tightly intermeshing co ......twin screw extruders composite models...

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