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MODELING FULLY INTERMESHING CO-ROTATING TWIN-SCREW EXTRUDER KNEADING-BLOCKS: PART B. POWER CONSUMPTION AND VISCOUS DISSIPATION

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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.
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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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Article
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An issue of modeling of twin-screw extrusion of polymeric materials is reviewed. The paper is written in honor of Prof. James L. White who was a pioneer in studying this issue. A global approach to process modeling is presented which includes solid polymer transport, polymer plasticating, and the flow of molten polymer. The methodology of CFD modeling of twin-screw extrusion is presented as well as the examples of this modeling which show the details of the process. Optimization and scaling of twin-screw extrusion are also covered. And finally, the future prospects of developments and research of twin screw extrusion is discussed.
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The flow in an intermeshing kneading disk region of a co-rotating twin-screw extruder is simulated by a model, based on the hydrodynamic lubrication theory for a Newtonian fluid. The pressure distribution and flow field in the disk region are calculated by numerical methods. The effect of various parameters is investigated, including staggering angle, nip clearance, screw speed, and throughput rate.
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Mit mehr als einer Schneckenwelle arbeitende Systeme erweitern den Anwendungsbereich bekannter Einwellen-Schnecken erheblich. Je nach der Geometrie der zusammenarbeitenden Teile sind diese Geräte mehr zur Förderung von Gasen und Flüssigkeiten oder mehr zur thermischen und mechanischen Bearbeitung zäher, plastischer oder gekörnter Güter geeignet. Auf die Schubgeschwindigkeit der Schnecken bezogene Stoffgeschwindigkeiten lassen eine Kennzeichnung und Ordnung typischer Vorgänge zu. Maschinen mit auswechselbaren Arbeitselementen können sehr verschiedenen verfahrenstechnischen Aufgaben angepaßt werden.
  • U Stritzinger
  • W Roland
  • H Albrecht
  • G Steinbichler
  • Spe Antec Tech
U. Stritzinger, W. Roland, H. Albrecht, G. Steinbichler, SPE ANTEC Tech. Papers (2021).
  • D D Denson
  • B K Hwang
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