Single Screw Extrusion Modeling with Slip Effects

REGULAR CONTRIBUTED ARTICLES
A. Lewandowski, K. Wilczyński*
Faculty of Production Engineering, Polymer Processing Department, Warsaw University of Technology, Warsaw, Poland
Global Modeling of Single Screw Extrusion
with Slip Effects
An extensive study is presented on the polymer melt flow with
slip effects in a single screw extrusion process. Fully three-dimensional non-Newtonian FEM computations are performed
to design the screw pumping characteristics and the die characteristics which may be implemented into the composite
model of the process. ANSYS Polyflow software is applied to
model the melt flow in the extruder. An analysis is performed
for the flow of polymers with slip effects both in the screw
(on the screw and barrel surfaces) and in the die. Screw
pumping characteristics and die characteristics are calculated and modeled for various power law indices, and various
slipping parameters. The effect of slipping on extruder operation is discussed.
1 Introduction
One of the fundamental assumptions of fluid mechanics is the
non-slip condition which implies that at the fluid-solid interface the fluid velocity is equal to the solid surface velocity.
This assumption is broadly accepted in modeling of polymer
extrusion (Tadmor and Klein, 1970; 2006; White, 2003; Rauwendaal, 2013) although its validity is not always obvious.
It is generally assumed that flowing materials in the screw
extruders and dies adhere to the wall, and several models of
extrusion have been developed using this approach (Agur and
Vlachopoulos, 1982; Vincelette et al., 1999; Potente et al.,
1993; Wilczyński, 1999). Recently, an extensive review on
modeling of the single screw extrusion has been presented by
Ilinca and Hetu (2010) as well as by Altinkaynak et al. (2011),
and for co-rotating twin screw extrusion by Malik et al.
(2014). A review on global modeling of screw processing has
been presented by Teixeira et al. (2012) and Wilczyński et al.
(2012).
There are several materials like filled polymers (e. g. wood
plastics composites), elastomers, pure polymers like poly(vinyl
chloride) (PVC) and high-density polyethylene (HDPE), polymer suspensions, as well as ceramic materials and foodstuffs
like meat and dough which display wall slippage under certain
* Mail address: Krzysztof Wilczyński, Warsaw University of Technology, Faculty of Production Engineering, Polymer Processing
Department, 02-524 Warsaw, Narbutta 85, Poland
E-mail: [email protected]
Intern. Polymer Processing XXXIV (2019) 1
conditions. Therefore, a lot of research on the description of
wall-slippage has been done.
The phenomenon of wall slippage was studied for the first
time by Mooney (1931). A number of different approaches to explain this phenomenon have been later presented in the literature.
It is generally accepted that, for highly entangled linear polymers, wall slippage can occur above a certain wall shear stress.
The basic mechanisms for the development of a slippery behavior have been described by Georgiou (2003), Dimakopoulos and Tsamopoulos (2006), and Tsouka et al. (2014). These
are the wettability of the solid substrate to the fluid (e. g. solid-liquid surface tension), local migration phenomena even in
melts, and the presence of a lubricant.
A number of papers have been published to answer the question how best to consider wall slippage when designing extruder machines. However, it was not possible to satisfactorily predict the process behavior of wall-slipping materials. An
extensive review on this subject has been presented by Potente
et al. (2009a).
Worth and Parnaby (1977) presented a theoretical analysis
of the effects of wall slip on the throughput rate and power consumption for a one-dimensional isothermal Newtonian case,
and suggested that the power consumption of the extruder is reduced as a result of wall slip.
Meijer and Verbraak (1988) performed two-dimensional
Newtonian isothermal studies of single screw extrusion with
slip boundary conditions, and showed the possible influence
of slip on the velocity profiles and pumping characteristics of
the extruder. Thus it has become clear that slip boundary conditions can change the extrusion behavior significantly.
Lawal and Kalyon (1993; 1994) developed an analytical
model that described single screw extrusion of viscoplastic
fluids subjected to different slip coefficients at screw and barrel
surfaces, and later Kalyon et al. (1999) and Malik et al. (2014)
studied numerically co-rotating twin screw extrusion subjected
to wall slip at barrel and screw surfaces.
Potente and his co-workers (2002a; b; 2003; 2005; 2006)
performed very extensive studies on modeling of single screw
extrusion which were based on the works of Hatzikiriakos and
Dealy (1992), Hatzikiriakos et al. (1993) and Hatzikiriakos
(1994). This approach enables a description of the wall-slipping velocities with an aid of a mathematical model based on
an energy balance.
First, Potente et al. (2002a; b) developed an analytical model
for the flow of wall-slipping polymers which described the
one-dimensional isothermal Newtonian case, and this mod-
Ó Carl Hanser Verlag, Munich
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A. Lewandowski et al.: Global Modeling of Single Screw Extrusion with Slip Effects
eling was extended to the two-dimensional flows using numerical methods. Later, Potente et al. (2009a; b) presented an analytical model to calculate the pressure/throughput and drive
power behavior of wall-slipping polymers in the melting section of a single screw extruder, and they also developed a model (Potente, 2009c) for describing the melt temperature development in a single screw channel for wall-slipping polymers.
Recently, several studies have been performed on modeling
of flow of wall-slipping polymers in the dies, e. g. by Hatzikiriakos and Mitsoulis (2009), Gupta (2011) and Ferras et al.
(2007).
An effect of viscoelasticity which is critical for a slippery
behavior was the subject of studies performed by Dimakopoulos and Tsamopoulos (2009) and Pettas et al. (2015). An effect
of surface properties on the polymer melt slip and extrusion defects was discussed by Kissi et al. (1994). The relation between
slip and extrusion instabilities were comprehensively studied
by Denn (2001).
Now, it is well established that extrusion of wall-slipping
polymers results in reduction of pressure which is necessary
to push the polymer through the die. The screw characteristics
also change for wall-slipping polymers. This consequently affects the operating point of the extruder, and a comprehensive
description of extrusion process of wall-slipping polymers requires developing models both for the screw (plasticating unit)
and for the extrusion die.
In this paper, we have used ANSYS Polyflow CFD software
(Ansys, Inc., Canonsburg, Pensylvania, USA, 2017) for modeling the polymer melt flow with slip effects in a single screw extruder to develop screw pumping characteristics which may be
implemented into the composite (global) model of the process.
According to our knowledge, it is the first fully three-dimensional non-newtonian modeling for screw pumping characteristics with slip effects both in the screw (on the screw and the
barrel surfaces) and in the die. This allows global studying of
an operation (operating point) of an extrusion of wall-slipping
polymers. Modeling has been performed in the true 3D geometry of the screw and the die, and the effect of slipping parameters on the extrusion performance has been studied.
For vwall = 0 and eslip = 1 we have
fs ¼ Fslip vs ;
ð3Þ
and
vs ¼ ð1=Fslip Þfs :
ð4Þ
For generalized Newtonian flow, eslip = 1 preserves the linear
character of the flow problem. When eslip < 1, the law becomes
nonlinear and requires an iterative solution technique.
It is important to note that using these equations we have
wall slip from the very beginning, so there is no critical shear
stress up to which we have no slip.
The constitutive equation of power-law fluids has been used
for flow modeling. This may be expressed as
s ¼ mc_ n ;
ð5Þ
where s is the shear stress, c_ is the shear rate, m is the consistency coefficient and n is the power-law exponent. A Newtonian fluid is obtained by m = l (Newtonian viscosity) and
n = 1.
3 Flow Simulation
The ANSYS Polyflow v.17.0 CFD program has been used for
the simulations. The geometrical model of the screw flow has
been built using tetrahedral elements for the moving part (screw)
and hexahedral elements for the flowing material. The geometrical model of the die flow has been built using tetrahedral elements for the flowing material in the first section of the die,
and hexahedral elements in the second and third sections. The
meshing procedures have been described in Wilczyński and Lewandowski (2014a). The computation model has been discretized using 420 441 elements with 498 771 nodes for the screw
(7 441 tetrahedral elements and 413 000 hexahedral elements),
and 117 027 elements with 41 984 nodes for the die (108 678 tetrahedral elements and 8 349 hexahedral elements). The powerlaw model defined by Eq. 5 has been used for modeling. The parameters of the model were as follows: the consistency coefficient m = 30111 Pa sn and the flow index n = 0.24.
2 Slip Analysis
For the slip analysis, a nonlinear slip velocity law is usually assumed that approximates the actual slip behavior of several
fluids, including molten polymers and polymer solutions (Hatzikiriakos and Mitsoulis, 2009). This may be written as
us ¼ bsbw ;
ð1Þ
where us is the slip velocity, b is the slip coefficient, b is the
slip-law exponent, and sw is the wall shear stress. The Generalized Navier’s law (ANSYS Polyflow, 2017) is given by
fs ¼ Fslip ðvwall vs Þjvs vwall jeslip 1 ;
ð2Þ
where fs is the shear stress, vs is the tangential velocity of the
fluid, vwall is the tangential velocity of the wall (it is assumed
to be zero by default), Fslip and eslip are the model parameters.
The full slip is obtained for Fslip = 0. The law is linear when
eslip = 1, and corresponds to a power law when 0 < eslip < 1.
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Fig. 1. Boundary conditions for the die flow
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A. Lewandowski et al.: Global Modeling of Single Screw Extrusion with Slip Effects
The scheme of modeling is shown in Figs. 1 and 7. The following flow boundary conditions define the process for the
die flow (Fig. 1) and the screw flow (Fig. 7):
BS1 – Inflow (Qin) = (Q0), the flow rate Q0 is imposed at the
inlet of the domain,
BS2 – Vanishing normal forces and tangential velocities are
imposed (fn & vs) = (0, 0),
BS3 – Slip condition, i. e. vanishing normal velocity is imposed, while the tangential force is a function of (vs – vwall),
i. e. fs = f (vs – vwall), (vn & fs) = (0, f (vs – vwall) & vwall = 0),
BS4 – Slip condition, i. e. vanishing normal velocity is imposed, while the tangential force is a function of (vs – vwall),
i. e. fs = f (vs – vwall), (vn & fs) = (0, f (vs – vwall) & vwall = (Cartesian velocities vx & vy & vz = N)).
Boundary conditions BS1 and BS2 imply that the pressure
may develop along the screw and the die. Since the pressure
at the end of the modeled element is unknown for the screw
(for the die it is equal to zero), the pressure gradient may be
computed in this case, only. The possible meaningless negative
pressures which may be observed when modeling an extrusion
process have been recently discussed by Goger et al. (2014),
and Wilczyński and Lewandowski (2014a).
The simulations have been performed using HP/Supermicro Hydra computer (Sci. Linux 6.0 operation system) of x
86 architecture, equipped with eight 2.8 GHz processors, and
12 288 to 40 960 MB of Random–acces memory (RAM). Simulations were time consuming, and took 8 to 20 h for the
screw and 1 to 6 h for the die (for non-Newtonian case). The
simulations have been performed at the Centre for the Computational Science (Interdisciplinary Centre for Mathematical
Modeling, ICM) at the University of Warsaw, Poland, and
were supported by PL-Grid Infrastructure.
A)
B)
Fig. 2. Effect of the Fslip parameter on the velocity profile and pressure profile (Fslip = 0 – full slip) for eslip = 1 (linear law): A) pressure
profile, B) velocity profile
4 Die Flow
In this section the effect of wall slippage on the polymer melt
flow in the extrusion die is presented. We have simulated the
flow through the rod cylindrical die. This is a typical 3-sectional die of 45/5 mms in diameter, and 180/30/60 mms in
length. The geometry of the flow and the boundary conditions
are depicted in Fig. 1.
The effect of the Fslip parameter on the velocity profile and
pressure profile (Fslip = 0 – full slip) for eslip = 1 (linear law) is
depicted in Figs. 2 and 3, and the effect of the Fslip parameter
on the die characteristics is shown in Fig. 4. It is clearly seen
that the velocity profile drastically changes and pressure substantially drops when slipping increases (Figs. 2 and 3). It results from the die characteristics that the same flow rate requires lower die pressure (Fig. 4).
The effect of the eslip parameter on the velocity profile and
pressure profile for Fslip = 0,002 (eslip = 1 – linear law, 0 < eslip
< 1 – nonlinear power law) is depicted in Fig. 5, and the effect
of the eslip parameter on the die characteristics is shown in
Fig. 6. Here (Fig. 5) it is seen that pressure substantially increases when nonlinearity increases (slipping decreases), and
the same flow rate requires higher die pressure (Fig. 6). This
means that nonlinearity decreases slip effects.
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5 Screw Flow
In this section the effect of wall slippage on the polymer
melt flow in the extruder screw is presented. We have simulated the flow through the conventional screw element with
3 mms in channel depth, and 180 mms in lenght (4 turns).
The geometry of the flow and the boundary conditions are depicted in Fig. 7.
The flow in the screw channel was simulated at different slip
conditions, with slip at the screw surface, slip at the barrel surface and slip at the screw and barrel surfaces. The results of the
simulations are presented in Fig. 8. The velocity profile in the
screw channel changes drastically when slipping at the walls
is allowed. Velocity at the barrel surface increases while at
the screw surface decreases.
The effect of the Fslip parameter on the velocity profile and
pressure profile (Fslip = 0 – full slip) for eslip = 1 (linear law) is
depicted in Figs. 9 to 11. It is clearly seen that the velocity profile drastically changes when Fslip changes. When Fslip decreases (higher slip), the velocity at the barrel surface increases, and at the screw surface decreases (Fig. 9).
83
A. Lewandowski et al.: Global Modeling of Single Screw Extrusion with Slip Effects
A)
B)
C)
Fig. 4. Effect of the Fslip parameter on the die characteristics
(Fslip = 0 – full slip) for eslip = 1 (linear law)
84
Fig. 3. Effect of the Fslip parameter on the
velocity profile and pressure profile (eslip = 1,
linear law): A) no slip, B) Fslip = 0,002, C)
(Fslip = 0 – full slip)
The effect of the Fslip parameter on the pressure profile is
clearly shown in Fig. 10. The simulations were performed
at different inflow conditions (BS1). First, the outflow condition has been applied which means drag flow only, without
pressure generation (pressure flow). Then the pressure and
drag flow were allowed (Q = 0,001 kg/s), and finally the pressure flow balanced the drag flow which means that the flow
in the screw channel has been throttled (Q = 0). The pressure
gradient was positive, which means that the pressure flow
decreased the drag flow. However, for full slip condition
(Fslip = 0) the pressure gradient changed sign and became
negative. When Fslip decreased (higher slip), the pressure gradient in the screw channel also decreased. The effect of slip
conditions (no slip, Fslip = 0,005, Fslip = 0 – full slip) on the
velocity profile and pressure profile are also illustrated in
Fig. 11.
The effect of the Fslip parameter on the screw characteristics
is shown in Fig. 12. The drag flow decreases when Fslip decreases (higher slip), and at the same flow rate the pressure
gradient decreases.
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A. Lewandowski et al.: Global Modeling of Single Screw Extrusion with Slip Effects
A)
A)
B)
B)
Fig. 5. Effect of the eslip parameter on the velocity profile and pressure
profile for Fslip = 0,002 (eslip = 1 – linear law, 0 < eslip < 1 – nonlinear
power law): a) pressure profile, b) velocity profile
Fig. 7. Boundary conditions for the screw flow
Fig. 6. Effect of the eslip parameter on the die characteristics for
Fslip = 0,002 (eslip = 1 – linear law, 0 < eslip < 1 – nonlinear power law)
Fig. 8. Velocity profile without slip and with slip: at the screw, at the
barrel, and at the screw/barrel
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A. Lewandowski et al.: Global Modeling of Single Screw Extrusion with Slip Effects
Screw pumping characteristics at different slip conditions
(with slip and without slip) are shown in Fig. 13 for Newtonian
and non-Newtonian cases. Decreasing the power-law exponent
increases the non-Newtonian flow behavior, reduces the screw
pumping capacity in the positive pressure gradient range, and
increases in the negative pressure gradient range. It is also seen
that slipping decreases pressure gradient both when positive or
negative. An effect of the Fslip parameter on the pumping
characteristics is depicted in Fig. 14. When Fslip decreases
(higher slip), the pressure gradient in the screw channel also
decreases.
7 Global Modeling
Fig. 9. Effect of the Fslip parameter on the velocity profile in the screw
channel with slip at the screw/barrel (Fslip = 0 – full slip) for eslip = 1
(linear law)
6 Screw Pumping Characteristics
Screw pumping characteristics are represented in terms of the
dimensionless flow rate and dimensionless pressure gradient
Q ¼ fðDpÞ;
ð6Þ
where Q* is the dimensionless flow rate, and Dp* is the dimensionless pressure gradient.
These characteristics were developed for non-Newtonian
and non-isothermal flows in various screw configurations of
different types, single screw extrusion, twin screw extrusion,
both co-rotating and counter-rotating, e. g. by White and Potente (2003) and Syrjälä (2009).
Screw pumping characteristics may be applied for global
modeling of polymer extrusion which in addition to the melt
flow includes solid conveying and melting of the polymer. It
is time consuming and requires hundreds of iterations to find
the final solution. The pumping characteristics obtained by
advanced simulations may be expressed by some regression
models and implemented into the global model of the process
refining simulations in a reasonable time. This approach has
been recently applied for starve fed and flood fed single screw
extrusion with conventional screws (Wilczyński et al. 2014b,
2018), and with mixing screws (Wilczyński et al., 2015;
2017) as well as for counter-rotating twin screw extrusion
(Lewandowski et al., 2014).
For single screw extrusion with conventional screws the
screw pumping characteristics are defined as
Q ¼
Dp ¼
2Q
;
WHpDNcosu
Hnþ1 sin ’ Dpc
;
6mðpDNcos’Þn Lf
ð7Þ
ð8Þ
where Q is the flow rate, W is the screw channel width, H is the
screw channel depth, D is the screw diameter, N is the screw
speed, u is the angle of the screw flight, Dpc is the pressure
change, Lf is the screw length of the pressure change, m is the
consistency coefficient and n is the power-law exponent.
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Extrusion is a continuous process of pushing of the polymer
through a forming die, and can be considered as a series connection of the extruder and the die. Since the die exerts a resistance
to flow, a pressure is required to push the polymer through the
die. It is important to note that the die pressure is caused by the
die, and not by the extruder. The extruder simply generates pressure to push the polymer through the die. An extrusion process
can be described by the extruder operating charcteristics which
are determined by the screw characteristics (Fig. 12) and the
die characteristics (Figs. 4 and 6). Actual extrusion operating
conditions are defined by the extruder operating point that results from the intersection of the screw characteristics with the
die characteristics. This point determines the flow rate (throughput) and the extrusion pressure. Several examples of extruder
operating characteristics are presented in Fig. 15.
Figure 15 shows the effect of the slip conditions determined
by Fslip on the extruder operating point. The starting point (1)
has been obtained for slip at the screw (Fslip = 0,01) and no slip
at the die. When slip at the die is allowed (Fslip = 0,002), the operating point moves to the higher flow rate and lower pressure,
i. e. to point (2). When slip at the screw increases (Fslip = 0,003)
and no slip condition at the die is considered, the operating
point moves to point (3), i. e. to lower flow rate and lower pressure to compare with the starting point (1). When slip conditions are allowed at the screw (Fslip = 0,003) and at the die
(Fslip = 0,002), the operating point transfers to point (4). It
means that the flow rate is almost the same as at the starting
point (1) but pressure decreases. Higher slip at the screw
(Fslip = 0,001) with no slip at the die results in operating point
(5), i. e. in lower flow rate and pressure. When slip is allowed
at the die, the operating point moves to point (6). It can be concluded that slip at the screw and at the die have an important
impact on the location of the extruder operating point, that is
on the flow rate and extrusion pressure. When slip at the screw
increases, the flow rate and pressure decrease, when slip at the
die increases, the flow rate increases and the pressure decreases.
8 Conclusions
An extensive study was performed on the polymer melt flow
with slip effects in a single screw extrusion process. The analysis was carried out for the flow with slip effects both in the
screw (on the screw and barrel surfaces) and in the die, and
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A. Lewandowski et al.: Global Modeling of Single Screw Extrusion with Slip Effects
Fig. 10. Effect of the Fslip parameter on the pressure profile in the screw with slip at the screw/barrel at different flow rates (Fslip = 0 – full slip) for
eslip = 1 (linear law)
Intern. Polymer Processing XXXIV (2019) 1
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A. Lewandowski et al.: Global Modeling of Single Screw Extrusion with Slip Effects
A)
B)
C)
Fig. 12. Effect of the Fslip parameter on the screw characteristics
88
Fig. 11. Effect of the Fslip parameter on the
velocity profile and pressure profile
(eslip = 1, linear law): A) no slip, B)
Fslip = 0,005, C) Fslip = 0 – full slip
the effect of slipping on the extruder operation is discussed. It
has been concluded that slip at the screw and at the die have a
substantial effect on the extruder operating point, i. e. on the
flow rate (extrusion throuput) and extrusion pressure. When
slip at the screw increases, the flow rate and pressure decrease,
when slip at the die increases, the flow rate increases and the
pressure decreases.
An effect of slip parameters (Fslip and eslip) on the extrusion
has also been studied. It results from simulations that slipping reduces the screw pumping capacity in the positive pressure gradient range, and increases in the negative pressure gradient range.
It was concluded that slip effects should be taken into account when global modeling the extrusion proces, especially
for materials like filled polymers (e. g. wood plastics composites), elastomers, polymers like poly(vinyl chloride) (PVC)
and high-density polyethylene (HDPE), polymer suspensions,
as well as ceramic materials and foodstuffs like meat and
dough which exibit wall slip effects under certain conditions.
This conclusion is valid not only for the polymer melt flow
but also for melting of the polymer. Slipping may change the
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A. Lewandowski et al.: Global Modeling of Single Screw Extrusion with Slip Effects
A)
Fig. 15. The effect of the slip conditions (Fslip parameter) on the extruder operating point
classical Tadmor melting mechanism which is observed, e. g.
for poly(vinyl chloride) (PVC). FEM modeling of melting
using CFD software will allow to study this problem.
References
B)
Fig. 13. Screw pumping characteristics: A) without slip, B) with slip
(Fslip = 0,01, eslip = 1)
Fig. 14. Effect of the Fslip parameter on the screw pumping characteristics (Fslip = 0 – full slip) for eslip = 1 (linear law) – Newtonian flow
Intern. Polymer Processing XXXIV (2019) 1
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Acknowledgements
The authors would like to acknowledge support from National
Science Center, Poland (DEC-2015/19/B/ST8/00948).
Date received: January 12, 2018
Date accepted: April 08, 2018
Bibliography
DOI 10.3139/217.3653
Intern. Polymer Processing
XXXIV (2019) 1; page 81 – 90
ª Carl Hanser Verlag GmbH & Co. KG
ISSN 0930-777X
Intern. Polymer Processing XXXIV (2019) 1