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2021 PROCEEDINGS

MODELING FULLY INTERMESHING CO-ROTATING TWIN-SCREW EXTRUDER

KNEADING-BLOCKS: PART B. POWER CONSUMPTION AND VISCOUS

DISSIPATION

Wolfgang Roland1,2, Ursula Stritzinger1, Christian Marschik1, and Georg Steinbichler1

1Institute of Polymer Extrusion and Compounding, Johannes Kepler University Linz, Linz, AUSTRIA

2Pro2Future GmbH, Linz, AUSTRIA

Abstract

Modeling twin-screw extrusion is commonly based on

significant geometric simplifications such as the

representation of the flow domain as flat channels.

Furthermore, the prediction of the conveying

characteristics and power demand of kneading blocks is

typically based on their approximation as conveying

elements. Considering the accurate flow geometry of fully

intermeshing co-rotating twin-screw extrusion kneading

blocks we analyzed the power characteristics by means of

three-dimensional numerical simulations for Newtonian

flow. Therefor we first conducted a dimensional analysis to

identify the dimensionless characteristic influencing

parameters. Next, we derived novel dimensionless power

parameters and then conducted a parametric design study.

Our proposed power parameters are capable to

simultaneously cover conveying and non-conveying screw

elements. The results provide new insights in the power

characteristics of kneading blocks and are fundamental for

screw design, screw simulation, and scale-up. In Part A. [1]

of this work we focused on the conveying parameters.

Introduction

Fully intermeshing co-rotating twin-screw extruders

with parallel screws are commonly used for compounding

applications because they show a very good mixing

efficiency. Because of the versatile compounding demands

twin-screw extruders consist of modular screws which can

be configured for each application using different screw

elements. The most important screw elements are

conveying elements and kneading blocks.

In twin-screw extruders large proportions of the screw

are partially filled. However, back-conveying elements,

kneading blocks, and the extrusion die located at the screw

tip are consuming pressure and hence, are completely filled

with polymer. To assure continuous processing the

elements that are located before these pressure-consuming

sections must provide sufficient pressure, additionally

resulting in a fully filled region right before the pressure

consuming screw sections. Besides the conveying

characteristics that determine the screw filling, power

demand and viscous dissipation are of major importance

for twin-screw extrusion machineries.

Modeling twin-screw extrusion mainly focuses on the

fully filled screw region. Most of the approaches to

modeling conveying elements are based on a flattened

axially open screw channel [2]. A first analysis of the flow

through co-rotating twin-screw extruders is provided by

Erdmenger [3]. Based on a Newtonian flow Denson and

Hwang [4] provided throughput-pressure relationship for

Newtonian flows, being closely related to that of single-

screw extruders. In contrast to single-screw extruders the

melt-channel is deviating considerable from a rectangular

cross section – the channel depth is a function of the cross-

channel direction – as considered by Booy [5].

Furthermore, for more detailed analyses the intermeshing

region was taken into account by Szydlowski and White

[6]. Considering the non-Newtonian viscosity behavior

Szydlowski and White [7], and Mans-Zloczower [8]

conducted numerical simulations to derive non-linear

pumping curves.

Kneading blocks have attained much less attention as

conveying elements. Nevertheless, they build the heart of

co-rotating twin-screw extruders as they are responsible for

the dispersion of fillers and additives. Moreover, kneading

blocks are typically operated fully filled determining the

back-pressure length of the preceding conveying elements

and, hence, the global filling ratio of the machinery.

Potente et. al [9] proposed to model kneading blocks as

conveying elements, where the staggering angle and the

kneading disc width form an apparent screw pitch. A

different approach that is based on the flattened screw

channel was presented by Szydlowski et. al [10] conducting

numerical analyses including the staggering angle and nip

clearance. Some studies [e.g., 11, 12] also apply complex

3D CFD calculations to predict the flow field in kneading

block segments.

Besides the conveying behavior, the power

consumption and viscous dissipation are of major

importance. The introduced mechanical power over the

screw shaft is transformed into: (i) viscous dissipation

heating up the material, and (ii) pressure generation. Some

fundamental relationships for the power characteristics of

twin-screw extrusion conveying elements are given by

Kohlgrüber [13]. Furthermore, curves for the power

parameters for conveying elements and the application to

predict the axial temperature profile are shown. Schöppner

et. al [14] presented an approach to determine the axial

temperature profile via average shear rates.

In this paper we analyze the power characteristics of

kneading discs for fully intermeshing co-rotating twin-

screw extruders. We applied the theory of similarity to

determine the characteristic dimensionless influencing

parameters. For conveying elements Kohlgrüber [13]

presented power parameters, based on the linear behavior

for Newtonian fluids. We re-formulated the analysis and

introduced adapted power parameters that are capable to

cover all kneading blocks. Next a comprehensive 3D CFD

parametric design study by varying the dimensionless

influencing parameters within a wide range of application

is conducted. To consider the periodic fluctuations of the

power parameters caused by the changing geometry due to

the screw rotation, we identified characteristic angular

screw positions that represents the time averaged behavior.

Fundamentals

Screw Geometry and Process Parameters

The geometry of the kneading discs is based on the self-

wiping Erdmenger profile [15] as shown in Figure 1. The

Erdmenger profile builds the axial cross section of the

kneading discs with the barrel diameter , the screw outer

diameter , the screw core diameter , the center line

distance , the screw clearance , and the nip clearance

as the main geometry parameters. Several discs are placed

consecutively with the width and a staggering angle .

The axial distance between the discs is given by .

As the major processing parameters, we identified the

screw speed , viscosity of the polymer melt, flow-rate

, pressure gradient , power introduced by the screw

shaft , and viscous dissipation .

Figure 1: Geometry of a 60° kneading block: Top, the cross

section of an Erdmenger profile is depicted. Bottom, the

staggering angle and the axial arrangement are depicted.

Theory of similarity

For obtaining generalized results we apply the theory of

similarity and transform our problem into a dimensionless

representation. Therefor we need to identify the basic units

of our geometry and processing parameters, which have

been identified in the previous section. In our case the

dimensions of mass [kg], length [m], and time [s] appear.

According to the Buckingham П-Theorem [16] a

dimensional matrix can be created (cf., Table 1 and Table

2, respectively showing the part of the geometry and

processing parameters).

Table 1: Dimensional matrix for kneading blocks: part

geometry parameters

0

0

0

0

0

0

0

1

1

1

1

0

1

1

0

0

0

0

0

0

0

Table 2: Dimensional matrix for kneading blocks: part

processing parameters

0

1

1

0

1

1

0

-1

-2

3

1

1

-1

-1

-2

-1

3

3

Table 1 and Table 2, in combination, build a

dimensional matrix with a rank of , and

dimensional parameters. As reference parameters we chose

the barrel diameter , the screw speed , and the polymer

melt viscosity . Hence, we derive dimensionless

parameters, which we subdivided into dimensionless

independent influencing parameters, and dimensionless

target parameters as listed in Table 3.

Instead of describing the cross-section of the

Erdmenger profile by , , and we will use the

diameter ratio (Eq. (1)), the dimensionless clearance

(Eq. (2)), and the dimensionless nip gap (Eq. (3)) for the

following analyses. Furthermore, the dimensionless

undercut will be reformulated and based on the disc width

(Eq. (4)).

(1)

(2)

(3)

(4)

Table 3: Dimensionless independent influencing

parameters and dimensionless target parameters.

Influencing parameter

Target parameter

Using these alternative dimensionless parameters for

the geometry description prevents for obtaining parameter

combinations that are geometrically not possible in

practice. Otherwise , , and would need to be

compatible in order to avoid collisions of the two screws

and to avoid penetration of the screws into the barrel. The

original and alternative dimensionless parameters are

related according to:

(5)

(6)

(7)

(8)

Analytical Modeling

In this analysis we assume a Newtonian flow behavior

of the polymer melt. In this case a linear relationship

between the throughput and the pressure gradient is derived

for fully filled screw sections. In terms of the dimensionless

parameters obtained above this is generally written as:

(9)

with the dimensionless drag flow-rate , and the

dimensionless element conductivity . Figure 2

schematically shows the characteristic curve. Note, for co-

rotating twin-screw extruders the flow rate is determined

by the feeder, hence, it is depicted on the abscissa. For

details regarding the throughput-pressure relationship see

Part A. of this work [1].

Figure 2: Characteristic curve for the dimensionless

pressure-throughput relationship of co-rotating twin-screw

extruders.

Additionally, when considering a Newtonian fluid, the

dimensionless power-consumption is linearly related

with the dimensionless volume flow-rate , as well as the

dimensionless pressure-gradient . Considering the

screw channel being similar as for single-screw extruders

this was proven in [17, 18]. Hence, in terms of the

dimensionless parameters the power characteristics can be

written as:

(10)

(11)

with the energy that is absorbed due to total flow

restriction and the dimensionless turbine parameter that

gives the ratio between power change caused by a flow-rate

change at identical screw speed. The characteristic line for

the dimensionless power as function of the dimensionless

flow-rate is given in Figure 3.

Figure 3: Dimensionless power line as function of the

dimensionless volume flow-rate for co-rotating twin-screw

extruders.

Hence the power characteristic is fully described by the

two parameters , and , which depend on the

influencing dimensionless geometry parameters defined

above. A similar approach was presented by Kohlgrüber

[13] for conveying elements. In this approach the turbine

point as the intercept with is used. In our

analysis for kneading blocks we had to choose a different

way using so that we are capable to cover also non-

conveying kneading blocks. For a non-conveying element,

e.g., a 90° kneading block, the turbine parameter will

become , and hence the turbine point is not

defined.

Combining the throughput-pressure relationship and

the power-characteristics the dimensionless dissipation

can be determined. Base on the energy balance the

dissipated energy is determined with the drive power and

the pressurization power by:

(12)

In terms of the dimensionless parameters this can be re-

written by:

(13)

Applying the dimensionless throughput-pressure

relationship (Eq. (9)), and the dimensionless power-

characteristics (Eqs. (10) and (11)) the dimensionless

dissipation is determined by the conveying and power

parameters according to Eqs. (14) and (15) as function of

the dimensionless volume flow-rate and dimensionless

pressure gradient, respectively. As a result, a quadratic

relationship is obtained as schematically depicted in Figure

4.

Figure 4: Dimensionless dissipation as function of the

dimensionless volume flow-rate for co-rotating twin-screw

extruders.

(14)

(15)

Parametric Design Study

Based on the dimensional analysis we identified a set of

dimensionless influencing and a set of dimensionless target

variables. In the analytical modeling section, we revealed

the correlation between the dimensionless drive power and

the dimensionless flow-rate, which is fully described by the

two power parameters and . These are kneading block

specific parameters which depend on following

dimensionless influencing parameters:

• dimensionless diameter ratio ,

• dimensionless disc width ,

• dimensionless kneading disc distance ,

• staggering angle ,

• dimensionless clearance ,

• dimensionless nip clearance .

A parametric design study is carried out by varying

these dimensionless influencing parameters as shown in

Table 4 and Table 5, resulting in a set of 1,536 independent

design points. Note, is replaced with

for the parametric design study considering the ZSE

MAXX series of Leistritz. The effect of varying gap

regions is omitted, and we chose common values with

and .

Table 4: Variation of the dimensionless input parameters

, , and .

parameter

min

max

increment

1.45

1.8

0.05

0.05

0.4

0.05

0.1

0.6

0.10

Table 5: Variation of the staggering angle .

parameter

values

30°

45°

60°

90°

Flow Simulations

For solving the flow field in twin-screw extrusion

kneading blocks we used the 3D FEM simulation software

ANSYS Polyflow [19]. Following flow-assumptions are

made: (i) the flow is stationary and isothermal, (ii) the fluid

is incompressible, (iii) the fluid sticks to the wall, and (iv)

gravitation and inertia forces are ignored due to the low

Reynolds numbers in polymer melt flows. With these

assumptions the macroscopic conservation equations of

mass and momentum are given respectively by Eqs. (16)

and (17) with the velocity vector , the hydrostatics

pressure , and the stress tensor [20].

(16)

(17)

The stress tensor is determined by Eq. (18) with the

viscosity and the rate-of-deformation tensor , which is

obtained by the velocity gradient tensor according to Eq.

(19).

(18)

(19)

Based on the velocity and pressure field obtained by the

numerical simulation the drive power can be determined.

The drive power is given according to Eq. (20) by the screw

torque and the angular speed .

(20)

Evaluating the stress and the pressure fields the total

stress tensor can be determined (Eq. (21)). With the total

stress tensor it is possible to calculate the torque vector

according to Eq. (22). The surface specific torque is

given by the cross product of the location vector of the

surface with respect to the rotation axis with the surface

force , with as the unit normal vector of the surface.

The total screw torque is then obtained by building the

surface integral over the screw surface. Note, only the

torque around the axis of rotation contributes to the

power demand. The torque is determined separately for

each of the two shafts and then summed up.

(21)

(22)

For discretizing the flow domain, a tetrahedral mesh

was generated. We refined the mesh in the screw clearance

and in the intermeshing region. The length of the

kneading element, respectively the number of kneading

discs was chosen to obtain a periodic section. In addition,

for the first and last kneading disc we modeled a disc with

the half width only. This means that the inlet and outlet

geometry are identical and we could apply periodic

boundary conditions between the inlet and outlet. Hence,

the length of the flow domain for double-flighted twin-

screw extruders is given by Eq. (23). To account for the

periodic fluctuations caused by the screw rotation, we first

evaluated a representative angular position for each

staggering angle (cf. Part A. [1]) before conducting the

parametric study. The resulting mesh of one configuration

is depicted in Figure 5. Depending on the length of the

resulting flow domain the number of elements is around 1

to 1.2 million. Further details on the solver settings and

iteration schemes are provided in Part A. [1].

(23)

Figure 5: Mesh of the flow domain.

For each combination of the dimensionless influencing

parameters two simulations were conducted to obtain the

dimensionless conveying parameters (, ) and the

dimensionless power parameters (, ). First, a

simulation with zero pressure gradient was conducted,

hence , which directly gives the drag-flow capacity

. Second, a simulation with non-rotating shafts and a

predefined volume-flow rate was conducted directly giving

the element conductance . For both simulations we

determined the screw torques, respectively, and .

Based on the linear superposition concept for Newtonian

fluids we can calculate the screw torque resulting for a

rotating screw with the summed volume flow rates of both

simulations by:

(24)

Hence, the turbine parameter is obtained by

Eq. (25), and is obtained by Eq. (26).

(25)

(26)

Note, for each of the 1,536 design points of the full-

factorial parametric study that is based on the

dimensionless parameters we created two different

simulation setups in the dimensional representation and

evaluated the flow-rates, screw torques, and drive power.

Then the results were transferred back into the

dimensionless representation in order to obtain the

dimensionless conveying and power parameters.

Results

Our comprehensive parametric design study yields

numerical results for the dimensionless power parameter

and the dimensionless turbine parameter as functions

of the staggering angle , the diameter ratio , the

dimensionless disc width , and the dimensionless disc

distance . Figure 6 depicts the dimensionless power

parameter as function of the staggering angle for various

diameter ratios. It can be seen that for and

the power demand at totally restricted flow is minimum

for the kneading disc with a staggering angle of .

Furthermore, with decreasing diameter ratio the power

demand increases due to increased shear stresses in the

flow channel.

Figure 6: Dimensionless power parameter as function of

for various diameter ratios .

The turbine parameter as function of the staggering

angle is shown in Figure 7 for various diameter ratios. A

kneading block with a staggering angle of is non-

conveying, hence, the power demand does not change with

changing flow rate. Therefore, the turbine parameter is

. Also, a staggering angle of will result in a

non-conveying element with , which is not mapped

by our parametric design study. However, it can be clearly

observed that in between these two limiting cases there is a

staggering configuration which exhibits the highest turbine

effect. From Figure 7 it can be concluded that this

configuration will be approximately around

. Furthermore, with increasing diameter ratio the flow

restriction decreases which in turn results in a reduced

turbine effect.

Taking a look on the effect of the disc width and the

disc distance, in Figure 8 the dimensionless power

parameter is depicted as function of for various

dimensionless disc widths . It is shown that the power

demand at total flow restriction slightly increases with

increasing disc width and with decreasing dimensionless

disc distance. For lower disc sizes the effect of the disc

distance ratio on the power demand decreases too.

Figure 7: Dimensionless turbine parameter as function

of for various diameter ratios .

Figure 8: Dimensionless power parameter as function of

the dimensionless disc width for various undercut ratios

.

The influence of the disc width on the turbine parameter

is given by Figure 9. In terms of the turbine parameter,

again two limiting cases can be identified: for and

the kneading block exhibits conveying angles of

and , respectively. Both cases result in a

non-conveying element, where the power demand will not

change with changing flow rate. Hence the turbine

parameter will tend to for that configurations. For

approximately the maximum turbine

effect is observed which is generally decreasing for higher

undercuts , because the axial flow is less restricted due

to increased leakage flow.

Figure 9: Dimensionless turbine parameter as function

of the dimensionless disc width for various undercut

ratios .

Conclusions

We presented dimensionless power parameters for fully

intermeshing, self-wiping, co-rotating twin-screw

extrusion kneading elements. Based on a dimensional

analysis we revealed seven dimensionless parameters that

fully describe the conveying and power behavior. We have

shown that the power characteristics can be described by

two power parameters: the dimensionless power demand

for complete flow restriction , and the turbine parameter

. These power parameters differ from those available in

literature and are capable to cover screw elements with and

without a conveying effect. The power parameters

available in literature [13], however, are only suitable for

conveying elements. For analyzing the power parameters

an extensive parametric design study with 3,072

simulations was conducted.

The results obtained provide fundamental insights into

the power characteristics of twin-screw extrusion kneading

blocks. For the turbine parameter we identified limiting

cases for the staggering angle and the dimensionless disc

width . Furthermore, the simulation results provide the

basics for screw design, screw simulation, and scale-up.

Together with the conveying parameters that are presented

in Part. A. [1] it is further possible to determine the viscous

dissipation rate and hence estimate the melt-temperature

increase.

Acknowledgements

This work has been supported by Leistritz AG and FFG,

Contract Nr. 854184: Pro2Future is funded within the

Austrian COMET Program under the auspices of BMVIT,

BMDW, and of the Provinces Upper Austria and Styria.

COMET is managed by the Austrian Research Promotion

Agency FFG. The computational results presented have

been achieved using the Vienna Scientific Cluster (VSC).

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