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This paper proposes an energy-based approach for modeling a screw extruder used for 3D printing. This approach was used due to the difficulty in measuring the salient variables associated with regulation of the process state. The control-oriented steady-state model for the screw extruder is based on the reliably available process variables of heater current and screw speed, which constitute the manipulated variables. The controlled variable for this extrusion process is the extrusion rate. This model is based on balancing the energy between the work done by the screw, the heat delivered by the heater at the nozzle, and the enthalpy of the extruded product stream. The fine measurement available is the current commanded by the heater control system to maintain a fixed temperature at the nozzle. An array of thermistors are used as feedback for the temperature profile along the extruder. The screw speed is calibrated for a stepping motor used for conveying the material. This steady state model can then be helpful for developing a dynamic model for a controller capable of accurate flow control based on preview of the extrusion rate but with a simple yet robust hardware requirement. Copyright © 2016 by ASME Country-Specific Mortality and Growth Failure in Infancy and Yound Children and Association With Material Stature Use interactive graphics and maps to view and sort country-specific infant and early dhildhood mortality and growth failure data and their association with maternal
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Proceedings of the ASME 2016 Dynamic Systems and Control Coference
October 12-14, 2016, Minneapolis, Minnesota, USA
Dylan Drotman
Department of Mechanical and Aerospace Engineering
University of California, San Diego
San Diego, CA 92093
Mamadou Diagne
Department of Mechanical Engineering
University of Michigan
Ann Arbor, MI 48109
Robert Bitmead
Department of Mechanical and Aerospace Engineering
University of California, San Diego
San Diego, CA 92093
Miroslav Krstic
Department of Mechanical and Aerospace Engineering
University of California, San Diego
San Diego, CA 92093
This paper proposes an energy-based approach for model-
ing a screw extruder used for 3D printing. This approach was
used due to the difficulty in measuring the salient variables asso-
ciated with regulation of the process state. The control-oriented
steady-state model for the screw extruder is based on the reliably
available process variables of heater current and screw speed,
which constitute the manipulated variables. The controlled vari-
able for this extrusion process is the extrusion rate. This model
is based on balancing the energy between the work done by the
screw, the heat delivered by the heater at the nozzle, and the en-
thalpy of the extruded product stream. The fine measurement
available is the current commanded by the heater control system
to maintain a fixed temperature at the nozzle. An array of ther-
mistors are used as feedback for the temperature profile along the
extruder. The screw speed is calibrated for a stepping motor used
for conveying the material. This steady state model can then be
helpful for developing a dynamic model for a controller capable
of accurate flow control based on preview of the extrusion rate
but with a simple yet robust hardware requirement.
Polymer extrusion is an unpredictable process and hence it
is highly prone to fluctuations. The process parameters are cou-
pled in a complex manner and hence difficult to set-up and con-
trol [1]. More precisely, screw extrusion processes exhibit fluc-
tuations due to external disturbances entering the process from
various sources, as well as internal flow instabilities which cause
undesirable effects such as surging or spurt flow, and other forms
of product nonuniformity [2]. Internal instabilities such as cyclic
breakup and buildup of the solid bed in the melt, flow impuri-
ties, and nozzle flow instabilities are also well-known potential
sources of process variability [2–6].
The difficulty in representing such phenomena occurring
during the heat and mass convection in the extruder stands as
a major roadblock in designing high performance controllers to
ensure fast extrusion and a quality final product. Several physical
models based on fluid dynamics have been proposed in the liter-
ature. The models involve significant mathematical complexities
in the process characterization and lead to complicated control
design methodologies [7–17]. Energy balance modeling com-
bined with experimental data has been developed to overcome
such issues. A simplified control-oriented model is constructed
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using an energy conservation argument, balancing the heat power
injected into the extruder chamber, the heat losses and the ejected
heat energy through the nozzle [1].
The present work is dedicated to the study of screw extrusion
as a new research avenue for 3D printing processes. A rotating
screw is exploited to enable a continuous feeding mechanism and
to build a sufficiently high pressure in the extruder chamber in
order to improve the printing speed and resolution compared to
Fused Deposition Modeling [18–20] and syringe-based 3D print-
ing processes [21, 22]. In addition, the screw motion extends
the mixing capabilities of the system, thereby improving the ho-
mogeneity of the extruded material [22–27]. Consequently, it
enables the processing of a broader range of raw materials and
permits an easy recycling of wasted plastic during extrusion.
The proposed control-oriented model describes the output
flow rate at the extruder nozzle as a function of the screw speed
and the temperature distribution of the melting material along
the extruder barrel. A relationship between the output flow rate
and known quantities can be determined by the power balance
of the system. The work done on the plastic by the motor and
the heater is directly related to the energy required for extrusion.
More specifically, the energy injected into the chamber by the
heater and energy generated by viscous dissipation due to the ro-
tating screw will be used to infer the output energy at the nozzle
exit. Heat losses are also accounted for due to the interaction
with the external environment. This study stands as a first step
toward the construction of a dynamical feedback law which can
stabilize the output flow rate without requiring a pressure mea-
surement as in all of the current existing results.
This paper is organized as follows: The screw extrusion
process is described in Section 2. In Section 3, the overall en-
ergy balance approach is presented to derive the control-oriented
model. In Section 4, we discuss the design of a small-scale screw
extruder for 3D printing and present experimental data with an
estimate of the output energy based on the contributions from the
motor and the heater. Concluding remarks and future directions
are given in Section 5.
The pellets flow into the extruder screw channel through the
inlet and are transported by the screw towards the nozzle opening
(Figure 1). The screw extruder is divided into conveying zones
(transport zones), melting zones (for material fusion) and mixing
zones where the extruded melt is exposed to high pressure before
it is ejected through the nozzle [13–17]. The heater near the noz-
zle heats up the barrel and screw which liquefies the polymer as
the pellets are transported. Once the material has liquefied and
exits the screw channel, the material enters the buffer chamber
which is the empty space between the bottom of the auger and
the nozzle. The rotation of the motor transports and compresses
the material, forcing it through the small opening in the nozzle.
FIGURE 1. Depiction of the screw extrusion process for 3D printing.
Heat sources and sinks affect the consistency of the material
at the output and the consistency of flow at the inlet. Convection
from ambient temperature and conduction from the barrel and
screw play a large role in the material melting mechanism inside
the screw channel. The amount of time the material is exposed to
these heat sources (Residence Time Distribution (RTD)) depends
on the screw speed. If the temperature is too high, the material
may melt at the inlet and jam the entire system. Too high a tem-
perature can also cause the material to burn and become less vis-
cous at the outlet. Too low a temperature will prevent extrusion,
causing a jam due to the slow melting rate. One major contribu-
tion to the heat generation is the temperature developed locally in
the extruder due to the rotating screw. In some cases, the internal
heat generation in the polymer can become larger than the heat
required to maintain temperature along the barrel. Because of
this effect, many extrusion processes are equipped with cooling
systems [28, 29]. The rotating screw which enables the transport
of the material affects the output flow rate in many ways as a de-
termining factor of the viscous heat generation (shearing effects)
and the RTD which are difficult to measure.
Energy losses play a role in the total energy consumption
in the system due to the barrel exposure to ambient temperature.
This detracts from the viscous heating term. Without heat losses,
the mechanical energy generally contributes 70 to 80 % of the to-
tal energy and the barrel heaters contribute about 20 to 30 % [30].
Power losses to the system can be associated with the mechani-
cal aspect of the extruder such as in the gearing system or as heat
convection on the extruder barrel and heat losses from the motor.
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FIGURE 2. Extruder diagram illustrating power entering and leaving
the system from Stage 1 (top of the barrel) to Stage 2 (bottom of the
This analysis will be centered around the change in energy
from the top of the extruder barrel at Stage 1 to the bottom of the
barrel at Stage 2 (Figure 2).
This modeling approach uses the thermodynamic properties
of the extrusion process. By calibrating the extrusion process
at steady state, the flow rate of the extruded material is deter-
mined at different screw speeds and temperatures. The energy
balance for the extruder starts from the first law of thermodynam-
ics. Since we are interested in output flow rate, we determined
each term as a power contribution
KE +˙
PE +˙mh=˙
where ˙
KE is the rate of kinetic energy change across the barrel,
PE is the rate of potential energy change, ˙mis the output mass
flowrate, his the specific enthalpy change, ˙
Qis the rate of heat
change, and ˙
Wis the change in mechanical power.
For this process, we can assume potential energy rate and
kinetic energy rate are negligible compared to other terms.
The overall efficiency (ηoverall ) of the system can be deter-
mined by comparing the total power input into the system (Pin)
to the output power used for work (Pout).Plosses represents the
total power losses by the system.
in =˙
out =P
in P
losses (4)
losses =˙
Qlosses +˙
Wlosses (5)
ηoverall =P
in 100 (6)
These terms will be used to define observable system and
material properties in terms of unknown parameters. Abeykoom
et al. demonstrate three power terms dependent on the mass flow:
Energy storage in the material based on temperature change
Energy required to change material state from solid to liquid
Stored energy via compression of the material
The specific enthalpy change between Stage 2 and Stage
1 can be rewritten as the change in enthalpy in melting (con-
stant pressure) and forming (constant temperature) for a semi-
crystalline material. [1].
h=h2h1=hmelt +hf orm (7)
hmelt =¯
Cp(T2T1)+ hf(8)
hf orm =(P
Cp(T2T1)+ hf+(P
where ¯
Cpis the average heat capacity, hfis the specific enthalpy
of fusion, P
2and P
1are the pressure at Stage 2 and Stage 1 re-
spectively and ¯
ρis the average density of the material.
The change in specific enthalpy for forming accounts for less
than 5% of the power contribution [1] therefore it is neglected.
The resulting power contribution from the specific enthalpy is:
out =˙mh=˙mhmelt =˙m(¯
Cp(T2T1)+ hf)(11)
Heat entering the extruder is derived from the energy input
to the heater and from the viscous dissipation from the material.
The power to the heater can be computed using the voltage to the
heater Vand the resistance of the heater R. The viscous dissipa-
tion is dependent on the screw speed N, the temperature through
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the consistency index m, and the filling ratio fin the barrel. We
can write the power contribution from heat as:
Qlosses =V2
R+m f Vbar rel πDN
Qlosses (12)
where Vbarrel is the volume of the barrel, Dis the diameter of
the screw, His the screw channel depth and nis the power law
index [30].
The rate of work done by the system ˙
Wis largely contributed
by the power from the motor. Rather than measuring the current
going into the windings of the motor, a simplified approach can
be used. Since we are only interested in the effects on the output
flow rate, the value for ˙
Wcan be combined with P
losses . The
resulting term can be written as:
losses ˙
= V2
R+m f Vbar rel πDN
Cp(T2T1)+ hf)
The resulting power terms in (13) demonstrate how the out-
put flow rate relates to screw speed and temperature. The left
hand side (P
losses ˙
W) has been lumped into one parameter be-
cause both P
losses and ˙
Ware unknown. The unknown value
losses ˙
Wcan be found since this lumped term is constant for
known values on the right hand side ( ˙
Qand P
out ).
The system is controlled by T2and Nwhere T1is the ambient
temperature. The values determined by experimentation are V
applied to the heater to maintain temperature and the output flow
rate ˙m. The other quantities on the right hand side of (13) that
are known include R,m,f,Vbarrel ,D,H,n,¯
Cp, and hf. For
measured quantities of ˙munder different values of T2and N, a
look-up table can be created to find a relationship between T2,
N, and ˙mbased on the energy in the system defined by the three
power terms P
losses ˙
out , and ˙
Q. If the extruder is exposed to
temperatures or screw speeds not available on the lookup table,
the energy terms are interpolated to find the resulting output flow
rate. For a given screw speed and temperature, ˙
Qand P
losses ˙
are found by interpolation and used to calculate P
out . This is used
to determine the new ˙m.
Using the experimental data captured from the screw ex-
truder, we developed a model demonstrating the effects of the
temperature of the heater and screw speed on the output flow
FIGURE 3. Experimental 3D printer extruder used for printing from
raw pellets.
rate. This data was captured using an experimental extruder we
designed specifically for 3D printing.
Material is stored in an external large hopper and fed into the
inlet of the extruder. To create pellets small enough to flow into
the screw channel, a grinder was used to grind pellets to about 1
mm in diameter. The screw channel in the auger is 2 mm x2 mm
x 6 mm and standard pellets are about 3 mm pellets in diameter.
A mechanical valve is used to control the inlet flow into the
extruder. The valve is actuated using a servo motor and a 3D
printed valve cover. This allows for a range of inlet flow rates to
be achieved. Our inlet has a range from 0 to 0.8 grams per sec-
ond for polylactic acid pellets. To prevent pellets from jamming,
small oscillations were induced into the inlet valve cover. This
directly vibrates the material flowing into the extruder, loosening
jammed particles allowing for consistent inlet flow.
A stepper motor directly drives an auger bit to transport the
pellets into a heated chamber. The heated chamber is comprised
of a aluminum barrel which is insulated with PTFE. Heat is con-
ducted up the heater barrel and auger which melts the mate-
rial. An array of thermistors measure the temperature distribution
along the barrel. A 12V 40W heater is mounted at the bottom of
the aluminum heater barrel. The thermistor closest to the noz-
zle provides feedback for the temperature at the outlet while the
thermistor at the top measures temperature as material enters the
barrel. The rotation of the auger adds pressure onto the liquefied
plastic, forcing the material through the opening in the nozzle
which has been tested on nozzles 0.2mm - 0.4mm in diameter.
The nozzle is threaded into the bottom of the heater barrel.
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Experiments were conducted at ranges of fixed screw speeds
and temperatures with calibration of the integrated steady output
flow rate. By extruding over a period of time, we can measure
the mass of the material at the output at different screw speeds
and different temperatures (Figure 4 and Figure 5). To maintain
steady state temperature for each of the experiments, bang-bang
control was implemented. Material was extruded for one minute
for nine different experiments. Polylactic acid pellets were used
as the raw material for these experiments due to it’s low melting
temperature. The three temperatures tested were 180C, 190C,
and 200C and the three screw speeds tested were 1 rps, 2 rps,
and 3 rps. Each experiment was conducted at a constant fixed
input flow rate. These experiments provide a direct relation be-
tween the screw speed, temperature, and output mass flow rate,
yielding a steady state analysis at a number of operating points.
FIGURE 4. Mass Flow Rate vs. Screw Speed at Different Tempera-
FIGURE 5. Mass Flow Rate vs. Temperature at Different Screw
The modeling to this stage is steady state and invokes a
look-up table for the power contribution of the extruded material
and energy rates attributable to the screw motor. This obviates
the complex modeling and measurement of motor variables and
Our aim is to use the attained experiments in conjunction
with the steady state data to fit a first-order dynamic model to
the mass-flow rate as a function of temperature and screw speed.
Denote ˜
fN,T2as the variation in the mass flow rate from the
steady state value ¯
fN,T2at a nominal temperature T2and screw
speed N. We can then define the model structure as
for small changes ˜
T2and ˜
N. For fixed ˜
T2and ˜
N, this model struc-
ture implies that
This first-order approximation occurs because this result
presumes that κN,T2is a constant. Regarding κN,T2as a constant
κ, the solution of (14) for a fixed speed Nand step change in
nozzle temperature T2at time tswitch is
This dynamic model includes two unknown variables αand
κwhich are determined using experimental data. The measured
variable during step experiments is the total mass of the extruded
material. This value and can be determined based on the inte-
grated mass flow rate.
Mtfinal =¯
fN,T2tfinal +Ztfinal
fN,T2tfinal +˜
κ1eκtfinal +αT2
κ21eκ(tfinal tswitch
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FIGURE 6. Experimental step response for varying temperature at a
set screw speed (1 rps).
Experimental dynamic step responses (Figure 6) are used to
confirm that the integrated flow rate will fit the dynamic model.
The dynamic model can be verified by comparing it to the steady
state model. This validation allows us to look at active controllers
of the extruder using the known preview of the extrusion rate.
The transient behavior is necessary for the development of ef-
ficient feedback control laws. An accurate method for control
can be used when the dynamics of the print head become more
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... The parameters of this process tend to be unstable due to external influences and faults entering the process [1]. These effects of instability of the extrusion process are described in detail in the works [2][3][4][5][6]. In order to achieve fast and continuous extrusion and a quality final product, it is necessary to harmonize the control of the speed ratios on the screw and the temperature ratios in the extruder. ...
... Several physical models have been proposed in [7][8][9][10][11][12][13][14][15], which mathematically describe this complex extrusion process and result in very complex methodologies for designing its control. The model proposed in the work [5] is focused on control and describes the outlet flow at the extruder nozzle as a function of screw speed and temperature distribution of the melting material along the extruder barrel. ...
... After the liquefied material exits the screw channel, it is compressed into the nozzle and extruded through the small hole in the nozzle. [5,[13][14][15] The key parameters of this technology are the temperature and heating time of the material. The temperature distribution in the extruder as well as its source have a fundamental influence on the melting mechanism of the material, its viscosity, extrusion, flow and printing speed. ...
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Additive manufacturing as the technology of the future brings new challenges. One of them is the economic efficiency of production. This paper focuses on the mathematical analysis and structural design of extrusion screw used for additive manufacturing. The primary objective is to analyze the screw tool geometry and determine a procedure for its design, specifically the theory involved with the pressing tool and force relations which are necessary for the verification of the proposed tool geometry and its strength analysis. Procedures for determining frictional performance of the screw press are used in designing the drive of the screw extruder of 3D printer. Familiarity with the above mentioned procedures forms the basis for research into new tool - screw that will improve the service life and competitiveness of the technology.
... An alternative system to the gear pump and ram extruder to provide the force necessary to extrude semisolid paraffin is a screw pump system. This type of extruder, called screw extruder, is commonly used to print and extrude plastic pellets [25] but also soft ceramic pastes like clay [26]. In this type of extruders, pellets or paste are inserted in a barrel and pushed by an Auger screw towards a nozzle. ...
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Hybrid rocket propulsion systems have proved to be a suitable option for some specific applications in the space transportation domain such as in launch vehicle upper stages, orbit transfer spacecrafts, decelerator engines for re-entry capsules, and small satellites launchers. Part of the renewed interest in hybrid rocket propulsion is due mainly to the safety aspects, cost reduction, and the use of paraffin-based fuel that impacts positively in terms of the solid fuel regression rate. However, paraffin solid fuel grains have poor structural characteristics and sometimes low performance due to the fuel internal ballistics behaviour. More recently, various studies have been carried out to overcome these drawbacks of paraffin-based fuels, such as the addition of energetic nano-sized metallic powder and 3D printing techniques. This study presents a review of the principal concepts of 3D printing processes and extrusion techniques that can be suitable for paraffin grains manufacturing and the conceptual design of a prototype for a 3D printer system under development at the Aero-Thermo-Mechanics Department of Université Libre de Bruxelles.
... The force required by the extruder could be provided by a screw extruder. This type of extruder is commonly being used to print and extrude plastic pellets [11] but also soft ceramic pastes like clay [12]. In this type of extruders, plastic pellets, or paste, are inserted in a barrel, and pushed by an Auger screw towards a nozzle. ...
Conference Paper
Hybrid rocket motors (HRMs) present numerous advantages when compared to other chemical rocket propulsion systems, mainly safety, reliability, and cost. Conventional solid fuels suffer from a low regression rate and poor combustion efficiency, drawbacks that restrained further developments and exploitation of the technology. Two methods are being used to enhance the performance of HRMs, that are the employment of liquefying fuels, such as paraffin wax, which guarantee a high regression rate, and more recently the exploitation of 3D printing to develop complex burning surfaces and turbulence patterns in the combustion chamber to enhance regression rate and mixing of the propellants. A higher performance could potentially be achieved combining the two methods, that is the additive manufacturing of paraffin-based fuel grains. The necessity for a suitable print system and a dedicated temperature control to manage the fast phase transition of the wax makes the 3D printing of this material hard to achieve. This paper presents the first steps conducted at Université Libre de Bruxelles (ULB) towards the creation of an additive manufacturing system for paraffin wax. After the first experiments with the material to assess its printability, three promising manufacturing techniques have been identified and activities are being carried out to research and design the 3D printing setup. To sum up, this paper describes the methods and instrumentation under development at ULB to achieve the manufacturability of paraffin.
... Stringing/Oozing: When nozzle moves to a new location, the molten plastic forms hairy attachments leaving behind is called Oozing. This problem is because of overheating [9]. ...
... For flexible materials, the gear clamping cannot push the molten filament and may cause the filament to bend [10], which results in failure of printing. In contrast, the feeding method of micro-screw extrusion is continuous forced feeding without nozzle clogging [11]. In addition, the allowed layer thickness of micro-screw extrusion is much higher than that of filament extrusion, which has been verified in Section 3.2. ...
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Based on additive manufacturing of wood flour and polyhydroxyalkanoates composites using micro-screw extrusion, device and process parameters were evaluated to achieve a reliable printing. The results show that the anisotropy of samples printed by micro-screw extrusion is less obvious than that of filament extrusion fused deposition modeling. The type of micro-screw, printing speed, layer thickness, and nozzle diameter have significant effects on the performance of printed samples. The linear relationship between the influencing parameters and the screw speed is established, therefore, the performance of printed products can be controlled by the extrusion flow rate related to screw speed.
The so-called Stefan system describes the dynamical model of the liquid–solid phase change in materials ranging from water and ice in the polar caps to metal casting and additive manufacturing (3D printing). The mathematical structure is given by a partial differential equation (PDE) with a moving boundary governed by a scalar ordinary differential equation. Because of the system's moving-boundary nature, control of the Stefan model is unconventional even within the class of otherwise challenging PDE control problems. The second decade of the twenty-first century has witnessed remarkable advances in control design for the Stefan system. Such advances carry significant potential in several areas of technology. In this article, we briefly review the principal literature on control of the Stefan model, along with the associated basics of the PDE analysis of the model and select applications. Principal ideas from our work on control design, stability analysis, and the maintenance of physical phase constraints are given sufficient attention and tutorial treatment so that the article can serve as a self-contained point of entry into the growing subject of boundary control of the Stefan system using the method of PDE backstepping. Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 5 is May 2022. Please see for revised estimates.
In today’s generation 3D printing is one of the most trending technologies, by using 3D printers manufacturing of complicated parts become easy and accurate which is difficult to manufacture by conventional manufacturing methods. 3D printing is based on additive manufacturing, in solid based additive manufacturing two types of processes are used namely Fused Filament Fabrication and pallet extrusion. In this pa-per detailed study of pallet-based screw extruder is carried out, several papers are stud-ied which tells about development of pallet extruder, modelling of screw, design of extruder, controlling the process parameters in pallet extruder, energy-based modelling method for screw used in pallet extruder. Knowledge gained from this study can be used in improving pallet extruder and precise control over process parameters for im-proved printing quality.
Additive manufacturing stands out among manufacturing technologies as a versatile tool for high flexibility and fast adaptability in production. It is applicable in a variety of producing industries, ranging from tissue engineering, thermoplastics, metal and ceramic fabrication. One of the most popular types of 3D printing is Fused Deposition Modeling (FDM), which uses filaments as raw material, which have to be precisely manufactured to achieve a good final product quality.
Volume 24 provides information on the metals, ceramics, and polymers used in additive manufacturing (AM) and how they respond to the transformative forces and energies applied over the course of production. It covers all commercially relevant processes including vat polymerization, material jetting, powder bed fusion, directed energy deposition, binder jetting, material extrusion, and sheet lamination. It describes the production and characterization of powders, resins, and slurries, and the mechanisms by which they are transformed into solid structures and shapes. It explains how subtle differences in the shape, size, or surface chemistry of metal powders can have a profound effect on part quality, and how AM processed materials such as stainless steels, nickel-base superalloys, tool steels, cemented carbides, copper alloys, and precious metals compare with conventionally produced alloys. It discusses safe powder handling techniques, process modeling and simulation, material and manufacturing defects, post-processing, and in-line process monitoring and control. It also covers direct-write processes including microdispensing, aerosol jetting, thermal metal embedding, and laser-induced forward transfer. For information on the print version of Volume 24, ISBN: 978-162708-288-4, follow this link.
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The purpose of this work is to present the study of the technical feasibility and functional design of an interchangeable mini head extruder based on a screw with variable section to be used in open-source desktop 3-D printers. The QFD matrix was used during the design planning where the non-commercial users’ requirements were established. Functional structure of the mini head extruder was organized to give support to the mechanical calculation. The choice of a feasible configuration allowed the pre-calculation of the main components of the proposed head which were: screw, barrel and motor system. The materials chosen for the mechanical and rheological calculations were polyamide and the biopolymer PCL, both in powder distribution. Apart from the mechanical design, supported by the QFD, and functional analysis, the mini head extruder was manufactured and assembled, showing its functionality to generate filaments with diameter of 0.7 mm from powder raw material. The complete mechanical and electrical design development and manufacturing of this mini head extruder were made in a period of 24 months. This systematic development demonstrated the importance of the design methodology during the development of the design process and its contribution to the design quality. Besides, the mini head extruder using screw will allow the generation of scaffold structures using biopolymers, as well as to enable comparative studies using different kind of polymers.
This thesis is devoted to the analysis of Partial Differential Equations (PDEs) which are coupled through a moving interface. The motion of the interface obeys to an Ordinary Differential Equation (ODE) which arises from a conservation law. The first part of this thesis concerns the modelling of an extrusion process based on mass, moisture content and energy balances. These balances laws express heat and homogeneous material transport in an extruder by hyperbolic PDEs which are defined in complementary time-varying domains. The evolution of the coupled domains is given by an ODE which is derived from the conservation of mass in an extruder. In the second part of the manuscript, a mathematical analysis has been performed in order to prove the existence and the uniqueness of solution for such class of systems by mean of contraction mapping principle. The third part of the thesis concerns the transformation of an extrusion process mass balance equations into a particular input delay system framework using characteristics method. Then, the stabilization of the moving interface by a predictor-based controller has been proposed. Finally, an extension of the analysis of moving interface problems to a particular class of systems of conservations laws has been developed. Port-Hamiltonian formulation of systems of two conservation laws defined on two complementary time-varying intervals has been studied. It has been shown that the coupled system is a port-Hamiltonian system augmented with two variables being the characteristic functions of the two spatial domains
A model for non-isothermal Newtonian flow startup in an initially empty modular self-wiping co-rotating twin screw extruder is developed. A viscous Newtonian fluid is considered to gradually fill a modular co-rotating twin screw extruder with viscous heating and heat transfer to the barrel (and screw). Thee build up of the axial cup mixing temperature profiles with time calculating for various modular screw configurations.
The existence of multiple stationary values of the fully filled length in reactive extrusion is studied on the basis of a mathematical model describing dynamic extruder behavior. The extruder is modelled as a plug flow type of reactor. Through the introduction of travel time for a fluid element in the extruder, time dependent viscosity models following from rheokinetics can be used. Special attention is paid to ionic and radical polymerization reactions. Hysteresis behavior for stationary values of the fully filled length exists when die resistance and rotational speed are varied. The effects of reaction rate, viscosity levels and degree-of-fill on stable operation at high conversions are studied. The mathematical dynamic model presented here can account for some of the experimentally observed dynamic phenomena in the reactive extrusion of radical polymers.
A qualitative mathematical model on the oscillating behavior of reactive extrusion is presented. Through the introduction of stagnant volume build up in the die as a result of the ongoing reactive extrusion process, a variable time dependent die resistance emerges. A differential equation for the waxing and waning of stagnant volume in the die is developed. This equation is able to give oscillating behavior for certain extruder parameters and viscosity when coupled to an existing dynamic model of the fully-filled length. The effects of reaction rate, rotational speed, feed rate, die resistance and degree-of-fill are studied.
The purpose of this work is to present calculation procedure and functional of an interchangeable head based on variable section applied to portable and open-source 3-D printer. Considering previously studies on demand and design planning it was realized a procedure to the pre-calculation of the main components of the proposed head, which were: screw, barrel and motor system. The materials chosen were polyamide 12 and the biopolymer PCL both powder form. Some rheological properties were obtained from reometry experiments to be included in iterative calculations to final results of critical design parameters of head. As result it was made a mock-up of head to verify its functionality. After this stage, the head with mini pair barrel-screw was manufactured. The preliminary tests showed the capacity of head to carry the powder material generating continuous filaments with average diameter of 0.7mm using a 0.8mm extrusion tip.