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Citation: Vidal, J.; Nóbrega, J.M. An
Enhanced Temperature Control
Approach to Simulate Profile
Extrusion. Polymers 2024,16, 904.
https://doi.org/10.3390/
polym16070904
Academic Editor: Keon-Soo Jang
Received: 17 February 2024
Revised: 11 March 2024
Accepted: 16 March 2024
Published: 25 March 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
polymers
Article
An Enhanced Temperature Control Approach to Simulate
Profile Extrusion
João Vidal 1,* and João Miguel Nóbrega 2
1Soprefa-Componentes Industriais SA, R. Alfredo Henriques, 4520-909 Mosteiró, Portugal
2Institute for Polymers and Composites, University of Minho, Campus de Azurém, 4800-058 Guimarães,
Portugal; mnobrega@dep.uminho.pt
*Correspondence: jpovidal@gmail.com
Abstract: Thermoplastic extrusion, a widely used method for processing thermoplastic materials,
requires precise temperature control to ensure product quality. However, existing computer-aided
engineering tools often oversimplify the temperature distribution calculations, leading to addi-
tional discrepancies between simulations and the actual processes. This study introduces a novel
multi-region modeling approach to address this issue. By employing realistic temperature control
conditions, the methodology overcomes the limitations of current numerical modeling tools. The
key contributions include the development of a transient, incompressible, non-isothermal solver inte-
grated into the OpenFOAM computational library and the implementation of a specialized boundary
condition that emulates Proportional-Integral-Derivative (PID) control using real-time thermocouple
measurements. The findings highlight temperature deviations at the flow channel walls and total
pressure drop while demonstrating a smaller impact on velocity and flow uniformity at the outlet
under steady-state conditions. This research substantially advances the understanding of thermal
dynamics in extrusion processes, offering crucial insights for enhancing temperature control and
laying the groundwork for more effective and precise operational strategies.
Keywords: profile extrusion dies; temperature control; boundary condition; OpenFOAM
1. Introduction
The polymer extrusion process is a critical industrial manufacturing technique used
across various economic sectors to produce profiles for construction, automotive, and
other applications [
1
]. The process involves five main components: the extruder, die,
calibration and cooling, haul-off, and cutting, each with a defined function in the extrusion
process [
2
]. Temperature plays a crucial role in achieving high-quality output products [
3
],
particularly in the die, where effective heating is essential. Cartridge heaters and band
heaters are commonly used to achieve temperature control in the die [
4
,
5
]. To achieve stable
temperature fields in the die, various control algorithms have been developed, including
Proportional-Integral-Derivative (PID) and fuzzy logic algorithms [6,7].
Computational simulation is an invaluable tool in understanding and optimizing the
extrusion process. Open-source software, such as OpenFOAM
®
, has enabled companies
and researchers to perform cost-effective numerical modeling, implementing complex
physical models with ease [8,9].
For modeling single-screw extruders, researchers have used the Finite Element Method
(FEM) and imposed temperatures at the barrel and screw walls [
10
,
11
]. Similarly, for twin-
screw extruders, models incorporating heat transfer coefficients and temperature at the
barrel and screw materials have been developed [12,13].
The extrusion die, being the crucial component in shaping the molten material, has
obviously garnered attention for simulation. Researchers have developed methodologies
imposing boundary conditions for temperature at the outer surface of the flow channel and
Polymers 2024,16, 904. https://doi.org/10.3390/polym16070904 https://www.mdpi.com/journal/polymers
Polymers 2024,16, 904 2 of 14
considering the torpedo as insulated [
14
,
15
]. More recent work explored complex three-
dimensional shapes but did not consider temperature effects on the flow [
16
]. Extrudate
swell, a significant phenomenon in polymer extrusion, has also been studied using methods
like the Boundary Element Method (BEM) [17].
Design and optimization of polymer profile extrusion dies traditionally relied on
trial-and-error based on designers’ expertise, leading to increased costs due to numerous
iterations [
1
,
18
]. Numerical tools have been proposed to automate die design optimization,
combining Finite Element Analysis (FEM) and Flow Analysis Network [
19
]. Studies
using the Finite Volume Method (FVM) aimed to optimize slit extrusion dies, considering
temperature effects by imposing temperatures at the inlet and flow channel walls [
20
,
21
].
Commercial software like PolyXtrue and Polyflow has been used to simulate polymer
extrusion die flow channels [
22
,
23
], but the calculations are limited to the flow channel,
with certain assumptions at its surface. The open-source OpenFOAM
®
computational
library has also been employed in modeling the polymer extrusion process in a similar
manner and considering temperature on torpedo walls, or utilizing insulation [
24
,
25
]. This
assumption has been defined based on intuition, but its validity was never verified before.
In the context of modeling profile extrusion, all available tools typically assume that
either the temperature or the heat flux is imposed at the flow-channel surface. However,
in practical applications, temperature control in extrusion is often achieved using thermo-
couples that measure the temperature in the metallic tool at a finite distance from the flow
channel surface. Indeed, Abeykoon [
6
] introduced a novel method for measuring die melt
temperature profiles. Yet, this method affects the melt flow and is only applicable in very
specific open locations, which excludes the extrusion die flow with varying flow channels.
As a result, it cannot fully replace the currently available temperature control systems.
Given these considerations, to improve the usefulness of computational simulation
tools in understanding and optimizing the profile extrusion process, it is critical to recognize
and address the limitations that arise from oversimplified temperature modeling. Such
simplifications in temperature modeling may lead to inaccuracies in predictions and limit
the effectiveness of efforts to enhance existing temperature control strategies. In this work,
the authors present a new simulation code, which can model an incompressible, transient,
multi-region solver and a realistic temperature control boundary condition implemented in
the OpenFOAM computational library that replicates the real systems behavior.
The article is organized as follows: in Section 2, the governing equations, the solver,
and the boundary condition implementation are presented. In Section 3, the industrial
case study and different possible approaches for modeling are presented. In Section 4, the
result presentation, analysis, and discussion are performed. Finally, in Section 5, the main
conclusions of the work are presented.
2. Computational Framework
2.1. Solver Implementation
The multi-region solver implementation draws its foundation from pimpleFoam [
26
],
a solver that employs the PIMPLE method for coupling pressure and velocity while seam-
lessly integrating PISO and SIMPLE methodologies [27].
The equations solved are the momentum conservation (1),
∂u
∂t+∇·(ρuu) = −∇p+∇·τ(1)
The mass conservation (2),
∇·u=0, (2)
To consider the influence of temperature, it is imperative to incorporate the energy
conservation equations, both in the fluid (3) and within the solid (4),
∂T
∂t+∇ · (uT)−∇·(α∇T)=1
cp
τ:∇u, (3)
Polymers 2024,16, 904 3 of 14
∂T
∂t−∇·(α∇T)=0, (4)
In the given equations, the variables hold the following meanings:
T
represents the
temperature,
ρ
the fluid density, uthe velocity vector,
p
the pressure,
τ
the deviatoric stress
tensor,
cp
the specific heat, and
α
symbolizes thermal diffusivity. Within Equation (3), the
last term on the right-hand side (τ:∇u) represents the viscous dissipation contribution.
As illustrated in the flowchart presented in Figure 1, for each time step, the developed
transient solver solves the momentum balance equations, mass conservation, and energy
conservation equations for the fluid and, subsequently, the energy conservation equation
for the solid. The process is iteratively repeated until the convergence, measured by the
residuals of each equation, is achieved.
Polymers 2024, 16, x FOR PEER REVIEW 3 of 16
, (3)
(4)
In the given equations, the variables hold the following meanings:
represents the
temperature,
the fluid density, u the velocity vector,
the pressure,
the deviatoric stress
tensor,
the specific heat, and
α
symbolizes thermal diffusivity. Within Equation (3), the
last term on the right-hand side (
) represents the viscous dissipation contribution.
As illustrated in the flowchart presented in Figure 1, for each time step, the developed
transient solver solves the momentum balance equations, mass conservation, and energy
conservation equations for the fluid and, subsequently, the energy conservation equation
for the solid. The process is iteratively repeated until the convergence, measured by the
residuals of each equation, is achieved.
Figure 1. Solver flowchart.
2.2. Boundary Condition Implementation
The implementation of the heater control boundary condition performed in this work
was based on the externalWallHeatFluxTemperature [28] boundary condition, which im-
poses a heat flux at the external wall.
The implementation of the PID algorithm [29] is based on the following equation:
!
"
#
"
"
(5)
Figure 1. Solver flowchart.
2.2. Boundary Condition Implementation
The implementation of the heater control boundary condition performed in this
work was based on the externalWallHeatFluxTemperature [
28
] boundary condition, which
imposes a heat flux at the external wall.
The implementation of the PID algorithm [29] is based on the following equation:
PID(t)=Kpe(t)+KiZe(t)dt +Kd
de
dt (5)
where,
Kp
,
Ki
and
Kd
are, respectively, the proportional, integral, and differential gains.
Moreover,
e(t)
, the difference between the probe and objective values is given by
the following:
e(t)=Tprobe −Tobj (6)
Subsequently, following the incorporation of the PID equation, a conditional response
was introduced to the boundary condition. Specifically, when the PID(
t
) value is below 0, a
fixed Gradient boundary condition is employed to impose the input power from the heater.
Polymers 2024,16, 904 4 of 14
Conversely, if the PID(
t
) function value equals or surpasses 0, a Robin boundary condition
is employed, which represents the heat flux through natural convection.
3. Case Studies
The assessment of the developed code was performed with an industrial case study,
which focuses on the production of an LED encasing profile, whose cross-section, which
is approximately 40 mm wide, is shown in Figure 2. The extrusion die employed in this
investigation comprises two heaters to control the temperature in two different regions
of the die, the Adapter and the Die Land, as illustrated in Figure 3(left). Furthermore, as
shown in Figure 3(right), each heater is equipped with an independent thermocouple that
controls the temperature in the corresponding region.
Polymers 2024, 16, x FOR PEER REVIEW 4 of 16
where,
,
and
#
are, respectively, the proportional, integral, and differential gains.
Moreover,
, the difference between the probe and objective values is given by the fol-
lowing:
$%&'
%&(
(6)
Subsequently, following the incorporation of the PID equation, a conditional re-
sponse was introduced to the boundary condition. Specifically, when the PID(
) value is
below 0, a fixed Gradient boundary condition is employed to impose the input power
from the heater. Conversely, if the PID(
) function value equals or surpasses 0, a Robin
boundary condition is employed, which represents the heat flux through natural convec-
tion.
3. Case Studies
The assessment of the developed code was performed with an industrial case study,
which focuses on the production of an LED encasing profile, whose cross-section, which
is approximately 40 mm wide, is shown in Figure 2. The extrusion die employed in this
investigation comprises two heaters to control the temperature in two different regions of
the die, the Adapter and the Die Land, as illustrated in Figure 3 (left). Furthermore, as
shown in Figure 3 (right), each heater is equipped with an independent thermocouple that
controls the temperature in the corresponding region.
Figure 2. Industrial case study profile cross-section.
Figure 3. Extrusion die regions (left) and die heating control elements (right).
The LED encasing profile in the industrial case study is made of polycarbonate ma-
terial (Trirex 3027 U[M1], supplied by Samyang Corporation, Seoul, Republic of Korea).
For the constitutive model, the Bird–Carreau model coupled with the Arrhenius Law,
given by Equations (7) and (8), was employed. The former considers the shear rate,
)*
,
dependence while the laer the temperature,
, dependence.
Figure 2. Industrial case study profile cross-section.
Polymers 2024, 16, x FOR PEER REVIEW 4 of 16
where,
,
and
#
are, respectively, the proportional, integral, and differential gains.
Moreover,
, the difference between the probe and objective values is given by the fol-
lowing:
$%&'
%&(
(6)
Subsequently, following the incorporation of the PID equation, a conditional re-
sponse was introduced to the boundary condition. Specifically, when the PID(
) value is
below 0, a fixed Gradient boundary condition is employed to impose the input power
from the heater. Conversely, if the PID(
) function value equals or surpasses 0, a Robin
boundary condition is employed, which represents the heat flux through natural convec-
tion.
3. Case Studies
The assessment of the developed code was performed with an industrial case study,
which focuses on the production of an LED encasing profile, whose cross-section, which
is approximately 40 mm wide, is shown in Figure 2. The extrusion die employed in this
investigation comprises two heaters to control the temperature in two different regions of
the die, the Adapter and the Die Land, as illustrated in Figure 3 (left). Furthermore, as
shown in Figure 3 (right), each heater is equipped with an independent thermocouple that
controls the temperature in the corresponding region.
Figure 2. Industrial case study profile cross-section.
Figure 3. Extrusion die regions (left) and die heating control elements (right).
The LED encasing profile in the industrial case study is made of polycarbonate ma-
terial (Trirex 3027 U[M1], supplied by Samyang Corporation, Seoul, Republic of Korea).
For the constitutive model, the Bird–Carreau model coupled with the Arrhenius Law,
given by Equations (7) and (8), was employed. The former considers the shear rate,
)*
,
dependence while the laer the temperature,
, dependence.
Figure 3. Extrusion die regions (left) and die heating control elements (right).
The LED encasing profile in the industrial case study is made of polycarbonate material
(Trirex 3027 U[M1], supplied by Samyang Corporation, Seoul, Republic of Korea). For the
constitutive model, the Bird–Carreau model coupled with the Arrhenius Law, given by
Equations (7) and (8), was employed. The former considers the shear rate,
.
γ
, dependence
while the latter the temperature, T, dependence.
η.
γ,T=aTη∞+aT(η0−η∞)
1+aTλ.
γ21−n
2
(7)
aT=expE
R1
T−1
T0 (8)
For the simulation, the physical parameters provided in Table 1were obtained from
Aali et al. [
30
], where the same material was characterized with capillary and parallel plate
rheometries. The die material properties are provided in Table 2.
Polymers 2024,16, 904 5 of 14
Table 1. Polycarbonate melt (fluid) properties.
Property Description Value Units
η0Viscosity at null shear rate 5376 Pa·s
η∞Viscosity at infinite shear rate 0 Pa·s
λModel parameter 0.0013 s
nPower-law index 0.35
E
R
Ratio between activation energy
and the universal gas constant 13,951.59 K
T0Reference temperature 518.15 K
αThermal diffusivity 1.46 ×10−7m2/s
cpSpecific heat capacity 1200 J/(kg·K)
KThermal conductivity 0.21 W/(m·K)
Table 2. Extrusion die (solid) properties.
Property Description Value Units
αThermal diffusivity 3.33 ×10−6m2/s
KThermal conductivity 16 W/(m·K)
To assess the flow distribution at the flow channel outlet, and following the method-
ology proposed by Rajkumar et al. [
24
], the cross-section was divided into elemental (ES)
and Intersection (IS) sections, as illustrated in Figure 4. The extrusion die performance
was quantified by an overall objective function, given Equation (9), that combines the
contribution of the individual sections’ objective function (Equation (10)), each of which is
weighted based on the corresponding outlet section area (
Atarget,i
), being
Atarget,tot
the total
cross-section area. In Equation (10),
Qi
and
Qtarget
, represents, respectively, the actual and
target flow rates in each section [24].
Fobj =
∑ES+IS
Fobj,i
Atarget,i
Atarget,tot (9)
Fobj,i=
Qi
Qtarget −1
maxQi
Qtarget , 1(10)
To assess the implemented code in detail, three different modeling approaches were
employed, which are described in the following subsections.
Polymers 2024, 16, x FOR PEER REVIEW 6 of 16
Figure 4. Flow channel outlet cross-section division in elemental (ES) and Intersection (IS) sections.
3.1. Conventional Approach
This case study mimics the conventional simulation approach, described in Section
1, usually employed to model these cases. To simulate the process with the approach, just
the flow channel is considered, and its external surface is divided into different regions
(named as patches), as shown in Figure 5, on which boundary conditions are applied in
accordance with Table 3 for all the fields being calculated: velocity, pressure, and temper-
ature. In this case, following the conventional modeling approach described in Section 1,
the Temperatures imposed at the patches Inlet, Adapter, and Die Land were the ones set
up in the real system. Moreover, the Inlet velocity was defined to achieve an average ve-
locity of 3 m/min at the Outlet, which was the actual production velocity.
Figure 5. Conventional approach flow channel geometry with the indication of the considered
boundary patches.
Table 3. Conventional approach boundary conditions.
Patch Pressure Velocity Temperature
Inlet Null Normal
Gradient
Fixed Value
(0.282 m/min) Fixed Value (245 °C)
Adapter Null Normal
Gradient No Slip Fixed Value (228 °C)
Die Land Null Normal
Gradient No Slip Fixed Value (220 °C)
Torpedo
Null Normal
Gradient No Slip Zero Gradient
Figure 4. Flow channel outlet cross-section division in elemental (ES) and Intersection (IS) sections.
Polymers 2024,16, 904 6 of 14
3.1. Conventional Approach
This case study mimics the conventional simulation approach, described in Section 1,
usually employed to model these cases. To simulate the process with the approach, just
the flow channel is considered, and its external surface is divided into different regions
(named as patches), as shown in Figure 5, on which boundary conditions are applied
in accordance with Table 3for all the fields being calculated: velocity, pressure, and
temperature. In this case, following the conventional modeling approach described in
Section 1, the Temperatures imposed at the patches Inlet, Adapter, and Die Land were
the ones set up in the real system. Moreover, the Inlet velocity was defined to achieve an
average velocity of 3 m/min at the Outlet, which was the actual production velocity.
Polymers 2024, 16, x FOR PEER REVIEW 6 of 16
Figure 4. Flow channel outlet cross-section division in elemental (ES) and Intersection (IS) sections.
3.1. Conventional Approach
This case study mimics the conventional simulation approach, described in Section
1, usually employed to model these cases. To simulate the process with the approach, just
the flow channel is considered, and its external surface is divided into different regions
(named as patches), as shown in Figure 5, on which boundary conditions are applied in
accordance with Table 3 for all the fields being calculated: velocity, pressure, and temper-
ature. In this case, following the conventional modeling approach described in Section 1,
the Temperatures imposed at the patches Inlet, Adapter, and Die Land were the ones set
up in the real system. Moreover, the Inlet velocity was defined to achieve an average ve-
locity of 3 m/min at the Outlet, which was the actual production velocity.
Figure 5. Conventional approach flow channel geometry with the indication of the considered
boundary patches.
Table 3. Conventional approach boundary conditions.
Patch Pressure Velocity Temperature
Inlet Null Normal
Gradient
Fixed Value
(0.282 m/min) Fixed Value (245 °C)
Adapter Null Normal
Gradient No Slip Fixed Value (228 °C)
Die Land Null Normal
Gradient No Slip Fixed Value (220 °C)
Torpedo
Null Normal
Gradient No Slip Zero Gradient
Figure 5. Conventional approach flow channel geometry with the indication of the considered
boundary patches.
Table 3. Conventional approach boundary conditions.
Patch Pressure Velocity Temperature
Inlet Null Normal Gradient Fixed Value (0.282 m/min) Fixed Value (245 ◦C)
Adapter Null Normal Gradient No Slip Fixed Value (228 ◦C)
Die Land Null Normal Gradient No Slip Fixed Value (220 ◦C)
Torpedo Null Normal Gradient No Slip Zero Gradient
Symmetry Symmetry Symmetry Symmetry
Outlet Fixed Value (0 Pa) Null Normal Gradient Null Normal
Gradient
3.2. Multi-Region Approach
The novel approach implemented to simulate the flow considers both the metallic
tool and the flow channel and a larger number of patches at the geometry surface together
with the interface, as shown in Figure 6. The corresponding boundary conditions are
presented in Table 4. In this case, both at the Adapter Heater and Die Land Heater
patches, the novel boundary condition, described in Section 2.2, was imposed. Accordingly,
the operation of both heaters is controlled by the average temperature calculated at the
respective thermocouple patches: Adapter Thermocouple and Die Land Thermocouple.
Polymers 2024,16, 904 7 of 14
Polymers 2024, 16, x FOR PEER REVIEW 7 of 16
Symmetry Symmetry Symmetry Symmetry
Outlet Fixed Value (0 Pa)
Null Normal
Gradient
Null Normal
Gradient
3.2. Multi-Region Approach
The novel approach implemented to simulate the flow considers both the metallic
tool and the flow channel and a larger number of patches at the geometry surface together
with the interface, as shown in Figure 6. The corresponding boundary conditions are pre-
sented in Table 4. In this case, both at the Adapter Heater and Die Land Heater patches,
the novel boundary condition, described in Section 2.2, was imposed. Accordingly, the
operation of both heaters is controlled by the average temperature calculated at the re-
spective thermocouple patches: Adapter Thermocouple and Die Land Thermocouple.
Figure 6. Multi-region approach geometry with the indication of the considered boundary patches.
Table 4. Multi-region approach boundary conditions.
Patch Pressure Velocity Temperature
Inlet
Null Normal
Gradient
Fixed Value
(0.282 m/min) Fixed Value (245°)
Adapter Heater N/A N/A
externalWallHeatFluxTempera
turePID
Target T = 228 °C
Die Land Heater N/A N/A
externalWallHeatFluxTempera
turePID
Target T = 220 °C
Adapter
Thermocouple N/A N/A Null Normal Gradient
Die Land
Thermocouple N/A N/A Null Normal Gradient
Die Wall
N/A
N/A
Natural Convection
Adapter Wall N/A N/A Null Normal Gradient
Interface Null Normal
Gradient No Slip Mapped Wall
Symmetry Symmetry Symmetry Symmetry
Outlet Fixed Value
(0 MPa) Zero Gradient Zero Gradient
3.3. Mixed Approach
The third and last case study considered aims at following the conventional approach
(Section 3.1), but with adjusted temperature on patches Adapter and Die Land. For that,
Figure 6. Multi-region approach geometry with the indication of the considered boundary patches.
Table 4. Multi-region approach boundary conditions.
Patch Pressure Velocity Temperature
Inlet Null Normal
Gradient
Fixed Value
(0.282 m/min) Fixed Value (245◦)
Adapter Heater N/A N/A
externalWallHeatFluxTemperaturePID
Target T = 228 ◦C
Die Land Heater N/A N/A
externalWallHeatFluxTemperaturePID
Target T = 220 ◦C
Adapter
Thermocouple N/A N/A Null Normal Gradient
Die Land
Thermocouple N/A N/A Null Normal Gradient
Die Wall N/A N/A Natural Convection
Adapter Wall N/A N/A Null Normal Gradient
Interface Null Normal
Gradient No Slip Mapped Wall
Symmetry Symmetry Symmetry Symmetry
Outlet Fixed Value (0 MPa) Zero Gradient Zero Gradient
3.3. Mixed Approach
The third and last case study considered aims at following the conventional approach
(Section 3.1), but with adjusted temperature on patches Adapter and Die Land. For that,
based on the calculations performed with the multi-region approach (Section 3.2), the
average temperature at those patches was computed after reaching quasi-steady state
conditions, being the obtained values imposed at the respective patches, as presented in
Table 5. The geometry used in the case study is the same as the one used for the conventional
approach, which is illustrated in Figure 5.
Table 5. Mixed approach boundary conditions.
Patch Pressure Velocity Temperature
Inlet Null Normal Gradient Fixed Value (0.282 m/min) Fixed Value (245 ◦C)
Adapter Null Normal Gradient No Slip Fixed Value (232 ◦C)
Die Land Null Normal Gradient No Slip Fixed Value (230 ◦C)
Torpedo Null Normal Gradient No Slip Null Normal Gradient
Symmetry Symmetry Symmetry Symmetry
Outlet Fixed Value (0 MPa) Null Normal Gradient Null Normal Gradient
3.4. Computational Mesh
Prior to initiating the numerical studies, a mesh sensitivity study was conducted
to determine the necessary refinement level. The mesh generation process was carried
Polymers 2024,16, 904 8 of 14
out using the OpenFOAM utility snappyHexMesh for both the conventional and multi-
region approaches. The meshes selected for carrying out the studies are presented in
Figures 7and 8.
Polymers 2024, 16, x FOR PEER REVIEW 8 of 16
based on the calculations performed with the multi-region approach (Section 3.2), the av-
erage temperature at those patches was computed after reaching quasi-steady state con-
ditions, being the obtained values imposed at the respective patches, as presented in Table
5. The geometry used in the case study is the same as the one used for the conventional
approach, which is illustrated in Figure 5.
Table 5. Mixed approach boundary conditions.
Patch Pressure Velocity Temperature
Inlet
Null Normal
Gradient
Fixed Value (0.282
m/min)
Fixed Value
(245 °C)
Adapter
Null Normal
Gradient No Slip
Fixed Value
(232 °C)
Die Land Null Normal
Gradient No Slip Fixed Value
(230 °C)
Torpedo Null Normal
Gradient No Slip Null Normal
Gradient
Symmetry Symmetry Symmetry Symmetry
Outlet Fixed Value (0
MPa)
Null Normal
Gradient
Null Normal
Gradient
3.4. Computational Mesh
Prior to initiating the numerical studies, a mesh sensitivity study was conducted to
determine the necessary refinement level. The mesh generation process was carried out
using the OpenFOAM utility snappyHexMesh for both the conventional and multi-region
approaches. The meshes selected for carrying out the studies are presented in Figures 7
and 8.
Figure 7. Conventional and Mixed approaches mesh.
Figure 7. Conventional and Mixed approaches mesh.
Polymers 2024, 16, x FOR PEER REVIEW 9 of 16
Figure 8. Multi-region approach mesh.
4. Results and Discussion
Figure 9 illustrates the average temperature evolution of the heaters and their respec-
tive thermocouples in both the Adapter and Die Land regions for the multi-region ap-
proach case study. These results clearly demonstrate the influence of the heater’s state on
the temperature of both the heaters and their respective thermocouples. When the heater
is turned on, the temperature in that area increases immediately, while the opposite occurs
when it is disconnected, while at a lower cooling rate, since the heat removal occurs by
natural convection. Due to the finite distance between the thermocouple and the heater,
there is a time delay between the actions of the heater and the corresponding effect meas-
ured by the thermocouple.
Figure 9. Evolution of the average temperature at the heaters and respective thermocouples for the
multi-region approach case study.
The evolution of the objective function, Equation (10), which measures the flow dis-
tribution uniformity, is illustrated in Figure 10 for the multi-region approach case study.
As shown after a period of 300 s, it becomes clear that the objective function stabilizes,
reaching almost steady-state conditions, and the influence of the unsteady heater opera-
tion diminishes.
Figure 8. Multi-region approach mesh.
4. Results and Discussion
Figure 9illustrates the average temperature evolution of the heaters and their respec-
tive thermocouples in both the Adapter and Die Land regions for the multi-region approach
case study. These results clearly demonstrate the influence of the heater’s state on the tem-
perature of both the heaters and their respective thermocouples. When the heater is turned
on, the temperature in that area increases immediately, while the opposite occurs when
it is disconnected, while at a lower cooling rate, since the heat removal occurs by natural
convection. Due to the finite distance between the thermocouple and the heater, there is
a time delay between the actions of the heater and the corresponding effect measured by
the thermocouple.
The evolution of the objective function, Equation (10), which measures the flow
distribution uniformity, is illustrated in Figure 10 for the multi-region approach case
study. As shown after a period of 300 s, it becomes clear that the objective function
stabilizes, reaching almost steady-state conditions, and the influence of the unsteady heater
operation diminishes.
Polymers 2024,16, 904 9 of 14
Polymers 2024, 16, x FOR PEER REVIEW 9 of 16
Figure 8. Multi-region approach mesh.
4. Results and Discussion
Figure 9 illustrates the average temperature evolution of the heaters and their respec-
tive thermocouples in both the Adapter and Die Land regions for the multi-region ap-
proach case study. These results clearly demonstrate the influence of the heater’s state on
the temperature of both the heaters and their respective thermocouples. When the heater
is turned on, the temperature in that area increases immediately, while the opposite occurs
when it is disconnected, while at a lower cooling rate, since the heat removal occurs by
natural convection. Due to the finite distance between the thermocouple and the heater,
there is a time delay between the actions of the heater and the corresponding effect meas-
ured by the thermocouple.
Figure 9. Evolution of the average temperature at the heaters and respective thermocouples for the
multi-region approach case study.
The evolution of the objective function, Equation (10), which measures the flow dis-
tribution uniformity, is illustrated in Figure 10 for the multi-region approach case study.
As shown after a period of 300 s, it becomes clear that the objective function stabilizes,
reaching almost steady-state conditions, and the influence of the unsteady heater opera-
tion diminishes.
Figure 9. Evolution of the average temperature at the heaters and respective thermocouples for the
multi-region approach case study.
Polymers 2024, 16, x FOR PEER REVIEW 10 of 16
Figure 10. Evolution of F
obj
, Equation (10), along with the simulation for the multi-region approach
case study.
The results from Figure 11 show the temperature distribution across three different
case studies: the conventional (CONV), multi-region (CHT), and mixed (CONV+CHT) ap-
proaches. It is observed that the multi-region (CHT) approach produces a higher and more
uniform temperature distribution along the outer wall of the flow channel compared to
the conventional (CONV) approach. Additionally, the temperature distribution on the
outer surface of the mandrel is slightly higher in the CHT approach, especially near the
inlet. This difference is aributed to the higher temperature of the adapter walls, which
leads to an elevated polymer temperature.
Interestingly, despite these differences, the results indicate that heat transfer within
the mandrel is negligible. This is likely due to minimal heat loss through the front surfaces
of the mandrel and low heat transfer through the spider legs, the connecting elements
between the mandrel and the die. As a result, assuming an insulated patch in the conven-
tional (CONV) and mixed (CONV+CHT) approaches seems to be a reasonable approxi-
mation. While this assumption may not exactly match the predictions of the CHT ap-
proach, it provides a close approximation.
Figure 10. Evolution of Fobj , Equation (10), along with the simulation for the multi-region approach
case study.
The results from Figure 11 show the temperature distribution across three different
case studies: the conventional (CONV), multi-region (CHT), and mixed (CONV+CHT)
approaches. It is observed that the multi-region (CHT) approach produces a higher and
more uniform temperature distribution along the outer wall of the flow channel compared
to the conventional (CONV) approach. Additionally, the temperature distribution on the
outer surface of the mandrel is slightly higher in the CHT approach, especially near the
inlet. This difference is attributed to the higher temperature of the adapter walls, which
leads to an elevated polymer temperature.
Interestingly, despite these differences, the results indicate that heat transfer within the
mandrel is negligible. This is likely due to minimal heat loss through the front surfaces of
the mandrel and low heat transfer through the spider legs, the connecting elements between
the mandrel and the die. As a result, assuming an insulated patch in the conventional
(CONV) and mixed (CONV+CHT) approaches seems to be a reasonable approximation.
While this assumption may not exactly match the predictions of the CHT approach, it
provides a close approximation.
Polymers 2024,16, 904 10 of 14
Polymers 2024, 16, x FOR PEER REVIEW 11 of 16
Figure 11. Comparison of temperature field for the three case studies considered.
In Figure 12, two important observations can be made about the pressure field anal-
ysis. Firstly, it is clear that the higher temperatures predicted in the CHT approach are
associated with a lower pressure drop. This effect is especially noticeable in the Adapter
region of the flow channel. The Adapter cross section is thicker, and therefore, the fluid
has a lower velocity magnitude, resulting in a very small pressure gradient along this re-
gion due to continuity. As the flow channel becomes narrower, the velocity increases, re-
sulting in a greater pressure drop. As a result, the most significant differences in the pres-
sure field are primarily observed in the Adapter region.
Figure 12. Comparison of pressure field for the three case studies considered.
Figure 11. Comparison of temperature field for the three case studies considered.
In Figure 12, two important observations can be made about the pressure field analysis.
Firstly, it is clear that the higher temperatures predicted in the CHT approach are associated
with a lower pressure drop. This effect is especially noticeable in the Adapter region of
the flow channel. The Adapter cross section is thicker, and therefore, the fluid has a lower
velocity magnitude, resulting in a very small pressure gradient along this region due to
continuity. As the flow channel becomes narrower, the velocity increases, resulting in a
greater pressure drop. As a result, the most significant differences in the pressure field are
primarily observed in the Adapter region.
Polymers 2024, 16, x FOR PEER REVIEW 11 of 16
Figure 11. Comparison of temperature field for the three case studies considered.
In Figure 12, two important observations can be made about the pressure field anal-
ysis. Firstly, it is clear that the higher temperatures predicted in the CHT approach are
associated with a lower pressure drop. This effect is especially noticeable in the Adapter
region of the flow channel. The Adapter cross section is thicker, and therefore, the fluid
has a lower velocity magnitude, resulting in a very small pressure gradient along this re-
gion due to continuity. As the flow channel becomes narrower, the velocity increases, re-
sulting in a greater pressure drop. As a result, the most significant differences in the pres-
sure field are primarily observed in the Adapter region.
Figure 12. Comparison of pressure field for the three case studies considered.
Figure 12. Comparison of pressure field for the three case studies considered.
As evidenced by the velocity contours shown in Figure 13, at the outlet of the flow chan-
nel, the predicted flow fields were mostly similar among the three approaches, with some
minor differences, especially between the CHT approach and the CONV and CONV+CHT
approaches. When analyzing sections ES1, ES2, and ES3, it is evident that the CONV ap-
proach predicted higher velocities, while the CHT approach predicted the lowest velocities.
Polymers 2024,16, 904 11 of 14
This observation may initially seem counterintuitive, as higher temperatures in the CHT ap-
proach would typically promote higher velocities due to lower viscosities. However, upon
closer examination of other sections, it becomes apparent that the slightly higher velocities
predicted by the CONV approach in certain sections are offset by lower velocities in other
sections. This effect is likely influenced by temperature’s impact on flow distribution. Since
the flow rate remains constant in all cross sections, any increase in flow in certain sections
must be compensated for by a decrease in flow in other sections. Therefore, while the CHT
approach may result in lower velocities in some sections due to higher temperatures, it
can also lead to slightly higher velocities in other sections, resulting in an overall balanced
flow distribution.
Polymers 2024, 16, x FOR PEER REVIEW 12 of 16
As evidenced by the velocity contours shown in Figure 13, at the outlet of the flow
channel, the predicted flow fields were mostly similar among the three approaches, with
some minor differences, especially between the CHT approach and the CONV and
CONV+CHT approaches. When analyzing sections ES1, ES2, and ES3, it is evident that
the CONV approach predicted higher velocities, while the CHT approach predicted the
lowest velocities. This observation may initially seem counterintuitive, as higher temper-
atures in the CHT approach would typically promote higher velocities due to lower vis-
cosities. However, upon closer examination of other sections, it becomes apparent that the
slightly higher velocities predicted by the CONV approach in certain sections are offset
by lower velocities in other sections. This effect is likely influenced by temperature’s im-
pact on flow distribution. Since the flow rate remains constant in all cross sections, any
increase in flow in certain sections must be compensated for by a decrease in flow in other
sections. Therefore, while the CHT approach may result in lower velocities in some sec-
tions due to higher temperatures, it can also lead to slightly higher velocities in other sec-
tions, resulting in an overall balanced flow distribution.
Figure 13. Comparison of outlet velocity field for the three case studies considered.
When analyzing the results of the individual objective function (Equation (10)) for
each elemental section, which is ploed in Figure 14, it is clear that in ES1, both the CONV
and the CONV+CHT approaches predicted an excessive flow, while the CHT approach
predicted an insufficient flow rate in that location. The CHT approach also predicted an
insufficient flow rate at ES2, while the CONV+CHT approach predicted an almost perfect
flow rate, and the CONV approach predicted an excessive flow. For ES3, it is noticeable
that the CONV+CHT approach predicted an almost perfect flow rate, while the CONV
approach predicted a significantly excessive flow. The CHT approach’s prediction was in
between the predictions of the other two counterparts. All approaches predicted a lack of
flow in ES4, with CONV+CHT and CONV being almost identical, while CHT was slightly
higher and closer to the target flow due to the higher overall temperature. Similarly, in
ES5, all approaches predicted a lack of flow, indicating the influence of temperature on
flow; the higher temperature in the CHT approach resulted in a higher flow rate. Likewise,
the higher wall temperature in the CONV+CHT approach led to higher flow rates com-
pared to the CONV approach. For ES6, all approaches predicted a slightly higher flow
rate, with CONV and CONV+CHT being almost equal and CHT slightly higher than the
others. In the case of ES7, the mixed approach (CONV+CHT) predicted an excessive flow,
while the other two approaches showed a deficit at the same location.
An overall examination of the temperature distribution at the outlet, shown in Figure
15, reveals a consistent trend across all cases. However, upon closer inspection, noticeable
differences in the temperature distribution among the cases can be observed. In sections
ES1, ES2, ES3, ES4, ES5, and ES9, it is observed that the inner side of the temperature
profile exhibits slightly higher temperatures in the CONV approach and lower tempera-
tures in the CHT approach. This discrepancy arises from the lower temperatures of the
outer walls applied in the CONV and CONV+CHT approaches, which promote higher
viscosities and result in additional viscous dissipation near the inner wall. Figure 16 fur-
ther highlights this effect, particularly in Section ES7, where imposing temperatures on
Figure 13. Comparison of outlet velocity field for the three case studies considered.
When analyzing the results of the individual objective function (Equation (10)) for
each elemental section, which is plotted in Figure 14, it is clear that in ES1, both the CONV
and the CONV+CHT approaches predicted an excessive flow, while the CHT approach
predicted an insufficient flow rate in that location. The CHT approach also predicted an
insufficient flow rate at ES2, while the CONV+CHT approach predicted an almost perfect
flow rate, and the CONV approach predicted an excessive flow. For ES3, it is noticeable
that the CONV+CHT approach predicted an almost perfect flow rate, while the CONV
approach predicted a significantly excessive flow. The CHT approach’s prediction was in
between the predictions of the other two counterparts. All approaches predicted a lack of
flow in ES4, with CONV+CHT and CONV being almost identical, while CHT was slightly
higher and closer to the target flow due to the higher overall temperature. Similarly, in ES5,
all approaches predicted a lack of flow, indicating the influence of temperature on flow;
the higher temperature in the CHT approach resulted in a higher flow rate. Likewise, the
higher wall temperature in the CONV+CHT approach led to higher flow rates compared to
the CONV approach. For ES6, all approaches predicted a slightly higher flow rate, with
CONV and CONV+CHT being almost equal and CHT slightly higher than the others. In
the case of ES7, the mixed approach (CONV+CHT) predicted an excessive flow, while the
other two approaches showed a deficit at the same location.
An overall examination of the temperature distribution at the outlet, shown in
Figure 15
,
reveals a consistent trend across all cases. However, upon closer inspection, noticeable
differences in the temperature distribution among the cases can be observed. In sections
ES1, ES2, ES3, ES4, ES5, and ES9, it is observed that the inner side of the temperature profile
exhibits slightly higher temperatures in the CONV approach and lower temperatures in
the CHT approach. This discrepancy arises from the lower temperatures of the outer walls
applied in the CONV and CONV+CHT approaches, which promote higher viscosities and
result in additional viscous dissipation near the inner wall. Figure 16 further highlights this
effect, particularly in Section ES7, where imposing temperatures on the walls of the flow
channel leads to lower temperatures within the flow channel itself. Finally, for ES8, the CONV
approach predicted a lack of flow due to lower wall temperature, while the CONV+CHT
and CHT approaches predicted excessive flow. At ES9, the flow rates are almost the same for
all approaches.
Polymers 2024,16, 904 12 of 14
Polymers 2024, 16, x FOR PEER REVIEW 13 of 16
the walls of the flow channel leads to lower temperatures within the flow channel itself.
Finally, for ES8, the CONV approach predicted a lack of flow due to lower wall tempera-
ture, while the CONV+CHT and CHT approaches predicted excessive flow. At ES9, the
flow rates are almost the same for all approaches.
In summary, across all elemental sections of the outlet, as presented in Figure 14, the
results in all the different approaches follow the same trend, but some notable differences
were identified. In general, higher temperature leads to a higher flow rate, influenced by
the corresponding lower viscosity. However, due to continuity restrictions, the flow can
only increase in one section if it decreases in another. Moreover, this intricate relationship
may lead to counter-intuitive outcomes.
Figure 14. Comparison of F
Obj,i
, Equation (10), for all elemental sections and the three case studies.
Figure 15. Comparison of outlet temperature field.
Figure 16. Corner temperature field detail.
Figure 14. Comparison of FObj,i , Equation (10), for all elemental sections and the three case studies.
Polymers 2024, 16, x FOR PEER REVIEW 13 of 16
the walls of the flow channel leads to lower temperatures within the flow channel itself.
Finally, for ES8, the CONV approach predicted a lack of flow due to lower wall tempera-
ture, while the CONV+CHT and CHT approaches predicted excessive flow. At ES9, the
flow rates are almost the same for all approaches.
In summary, across all elemental sections of the outlet, as presented in Figure 14, the
results in all the different approaches follow the same trend, but some notable differences
were identified. In general, higher temperature leads to a higher flow rate, influenced by
the corresponding lower viscosity. However, due to continuity restrictions, the flow can
only increase in one section if it decreases in another. Moreover, this intricate relationship
may lead to counter-intuitive outcomes.
Figure 14. Comparison of F
Obj,i
, Equation (10), for all elemental sections and the three case studies.
Figure 15. Comparison of outlet temperature field.
Figure 16. Corner temperature field detail.
Figure 15. Comparison of outlet temperature field.
Polymers 2024, 16, x FOR PEER REVIEW 13 of 16
the walls of the flow channel leads to lower temperatures within the flow channel itself.
Finally, for ES8, the CONV approach predicted a lack of flow due to lower wall tempera-
ture, while the CONV+CHT and CHT approaches predicted excessive flow. At ES9, the
flow rates are almost the same for all approaches.
In summary, across all elemental sections of the outlet, as presented in Figure 14, the
results in all the different approaches follow the same trend, but some notable differences
were identified. In general, higher temperature leads to a higher flow rate, influenced by
the corresponding lower viscosity. However, due to continuity restrictions, the flow can
only increase in one section if it decreases in another. Moreover, this intricate relationship
may lead to counter-intuitive outcomes.
Figure 14. Comparison of F
Obj,i
, Equation (10), for all elemental sections and the three case studies.
Figure 15. Comparison of outlet temperature field.
Figure 16. Corner temperature field detail.
Figure 16. Corner temperature field detail.
In summary, across all elemental sections of the outlet, as presented in Figure 14, the
results in all the different approaches follow the same trend, but some notable differences
were identified. In general, higher temperature leads to a higher flow rate, influenced by
the corresponding lower viscosity. However, due to continuity restrictions, the flow can
only increase in one section if it decreases in another. Moreover, this intricate relationship
may lead to counter-intuitive outcomes.
The above results highlight the importance of accurately calculating temperature dis-
tribution in extrusion processes, particularly in relation to their varying impact on different
sections with varying levels of restriction. Larger, less restricted sections experience dif-
ferent temperature effects compared to smaller, highly restricted sections. Understanding
these temperature dynamics is essential for optimizing profile extrusion processes. Ac-
cordingly, an accurate calculation of the temperature field is invaluable in comprehending
many of the observed effects.
Polymers 2024,16, 904 13 of 14
5. Conclusions
This study represents a significant advancement in the computational modeling of
profile extrusion processes, addressing critical limitations in existing methodologies and
providing valuable insights for process optimization and control. By introducing a novel
simulation code integrated into the OpenFOAM computational library, capable of captur-
ing incompressible, transient flow with a multi-region solver and realistic temperature
control boundary conditions, we have achieved a more comprehensive understanding of
temperature dynamics and flow distribution within extrusion dies.
Through rigorous industrial case studies and comparative analysis of different mod-
eling approaches, including conventional, multi-region, and mixed strategies, we have
demonstrated the effectiveness of the multi-region approach. By incorporating realistic tem-
perature control boundary conditions based on thermocouple measurements and dynamic
adjustment using PID algorithms, our simulations closely mimic real-world extrusion
processes, offering invaluable insights for process optimization and control.
Our findings underscore the critical role of temperature in shaping flow distribution
within extrusion dies, highlighting the importance of accurate temperature modeling for
optimizing product quality and production efficiency. Furthermore, these studies showed
that if the conventional approach usage of insulated boundary conditions for the mandrel
surfaces provides a good approximation to the real case modeled by the CHT approach,
this work provides practical guidance for industry practitioners and researchers, offering
a robust framework for die design modeling and process improvement across various
industrial applications.
Moving forward, future research efforts could focus on further validation and refine-
ment of the proposed methodology, exploring additional case studies and incorporating
more complex geometry and material properties. Additionally, ongoing advancements in
computational techniques and high-performance computing offer exciting opportunities
for enhancing the accuracy and efficiency of profile extrusion simulations.
Author Contributions: Methodology, J.V. and J.M.N.; Conceptualization, J.V. and J.M.N.; Formal
analysis and funding acquisition, J.V. and J.M.N.; Numerical modeling, J.V.; Supervision, J.M.N.;
Writing—original draft, J.V.; Writing—review and editing, J.V. and J.M.N. All authors have read and
agreed to the published version of the manuscript.
Funding: This work was funded by FEDER funds through the COMPETE 2020 Program and Na-
tional Funds through FCT—Portuguese Foundation for Science and Technology under the projects
UIDB/05256/2020/, UIDP/05256/2020.
Institutional Review Board Statement: Not applicable.
Data Availability Statement: Data are contained within the article.
Acknowledgments: The authors acknowledge the support of the Minho Advanced Computing
Center (MACC) computational cluster.
Conflicts of Interest: Author João Vidal was employed by the company Soprefa-Componentes
Industriais SA. The other author, João Miguel Nóbrega, declares that the research was conducted
in the absence of any commercial or financial relationships that could be construed as a potential
conflict of interest.
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