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Dieser Artikel geht auf die Konstruktion und Charakterisierung eines kompakten Extruders ein, der mit einer chaotischen Mischstrategie ausgestattet ist und auf die Verarbeitung recycelter Kunststoffe zugeschnitten ist. Die Studie unterstreicht die Fähigkeit des Extruders, selbst mit recyceltem Polypropylen einen stabilen Durchsatz und eine effiziente Energienutzung zu erreichen. Zu den Schlüsselthemen zählen die Konstruktionsabsichten des Extruders, der sich auf die Aufrechterhaltung eines laminaren Schleppströmungsregimes konzentriert und gleichzeitig Merkmale wie konische Mischschlitze und ein nichtlineares variables Pitchprofil integriert, um das Mischen und die Homogenisierung zu verbessern. Der Artikel untersucht auch den Simulationsansatz mithilfe der computergestützten Strömungsdynamik (CFD) zur Modellierung nicht-isothermaler, nicht-newtonscher Schmelzeflüsse, was entscheidende Einblicke in Massendurchflussraten und Temperaturfelder innerhalb der kompakten Geometrie bietet. Experimentelle Methoden und Ergebnisse werden diskutiert und die Leistungsfähigkeit des Extruders über verschiedene Prozessbedingungen hinweg demonstriert. Die Studie schließt mit einer detaillierten Analyse des kurzlebigen Verhaltens und des Energieverbrauchs des Extruders und bietet wertvolle Daten zur Optimierung seines Einsatzes in Anwendungen der additiven Fertigung. Die Ergebnisse unterstreichen das Potenzial kompakter Schneckenkonstruktionen für stabilen Durchsatz und vorhersehbare Temperaturregelung, was sie zu vielseitigen Plattformen für eine nachhaltige Polymerextrusion macht.
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Abstract
This study presents the pilot-scale implementation of a compact extruder designed to process post-consumer recycled plastics using chaotic mixing theories. The system features a novel mixing screw with an exceptionally short 8:1 length-to-diameter ratio and counter-rotating mixing slots cut into double flights of variable pitch. These features induce chaotic advection to enhance dispersive and distributive mixing by disrupting solid bed formation, improving thermal and compositional homogeneity in the extrudate through the baker’s transformation. A model-based screw design methodology using non-isothermal, non-Newtonian computational fluid dynamic (CFD) simulations was developed to optimize geometry for energy efficiency and throughput. Significant experimental validation employed a custom-built extruder with a 31 mm screw and an instrumented nozzle equipped with melt pressure and temperature sensors. Throughput and melt temperature were characterized as functions of screw speed, barrel temperature, nozzle geometry, and extruder orientation. Multiple regression models for simulated and experimental data showed high fidelity (R² ≈ 0.99) and strong correlation (r ≈ 0.95). Discrepancies in mass output were attributed to lower packing density of pelletized feedstock relative to the continuum assumption in simulations. The extruder demonstrated near-linear control of melt temperature (1.1 °C per unit setpoint) and mass flow rate (0.84 g min⁻¹ per RPM) with minimal cross-effects. A specific energy consumption of 0.254 kWh kg⁻¹ at 200 °C and 60 RPM corresponded to 76% efficiency, representing a 26% improvement over a previous design. The system offers reduced energy use, short residence times, and enhanced processing of recycled and bio-based polymers, advancing sustainable manufacturing in polymer processing and additive manufacturing.
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1 Introduction
Single screw extrusion remains the most widely used method for processing thermoplastic polymers, owing to its relatively simple design, scalability, and ability to continuously convert solid feed into a homogeneous melt. Over decades of development, screw designs have evolved to improve plastication, melt uniformity, and energy efficiency through modifications to flight geometry, channel depth, and compression zones [1]. However, most conventional single-screw extruders are designed for large-scale industrial processing and therefore employ long length-to-diameter (L/D) ratios (typically between 20:1 and 40:1) optimized for virgin polymers with consistent rheological properties [2, 3]. Although effective for stable, well-characterized resins, these configurations can be inefficient or unsuitable when processing heterogeneous, recycled, or bio-based materials, where shorter residence times are critical to minimizing thermal degradation [4] and more aggressive mixing features are required to homogenize varying feedstocks [5, 6].
Reducing the L/D ratio offers the potential to lower equipment size, weight, and residence time, but introduces significant challenges in melting and homogenization. With less axial distance available, short screws must incorporate various mechanisms such as enhanced mixing elements, variable pitch, or flight modifications to ensure adequate plastication while avoiding excessive shear or thermal gradients [7‐11]. This trade-off has motivated recent studies applying computational fluid dynamics (CFD) to model non-Newtonian polymer flow in compact screw geometries, aiming to identify screw features that improve temperature uniformity and compositional homogeneity [12, 13]. Despite this progress, relatively few published designs achieve both compactness and robust processing capability, particularly for recycled feedstock where variability in particle size, contamination, and melt viscosity complicates extrusion [14].
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Fused filament fabrication by material extrusion is the most accessible and prevalent additive manufacturing process but remains limited in production speed by the melting rate of the filament. Single-screw extrusion offers significant advantages for 3D printing by enabling direct processing of pellets, powders, and recycled feedstocks while providing improved mixing and higher deposition rates compared to filament-based approaches [15‐19]. However, compact screw extruders tailored to the dynamic conditions of additive manufacturing remain scarce. Designing such systems requires balancing size, weight, and energy efficiency with the need for stable melt quality and throughput under variable screw speeds and heterogeneous materials. This gap highlights the need for new extruder architectures that combine compactness with robust processing performance.
To address these limitations, the present study introduces a compact, double-flighted screw extruder with an exceptionally short L/D ratio, developed through a model-based, simulation-driven design methodology. By integrating CFD analysis with experimental validation, the work demonstrates that short screws can achieve reliable throughput, stable melt temperatures, and efficient energy usage, even when processing recycled polypropylene [20]. These results provide new insight into the trade-offs and opportunities associated with low L/D single-screw extrusion and establish a pathway toward sustainable, high-performance extruder designs for polymer processing and additive manufacturing applications.
2 Methodology
2.1 Screw design intent with chaotic mixing strategy
The primary objective of the extruder design was to develop a compact, lightweight, and high-throughput screw extrusion system suitable for integration into additive manufacturing platforms. The design of the compact extruder screw challenges the basic design intent in single-screw extruders to maintain a laminar drag-flow regime. In standard screw geometries, the processed material follows predictable streamlines according to the well-known Tadmor melting model [21], resulting in low thermal homogeneity and inefficient dispersive mixing.
To achieve higher throughput and melt uniformity within a constrained 8:1 L/D ratio, the investigated screw design employs a “chaotic mixing” strategy [5] implemented via two geometric mechanisms. First, the inclusion of tapered mixing slots along a counter-rotating helix serves to physically interrupt the screw channels. This geometry forces the melt to undergo repeated splitting and reorientation (i.e. the “baker’s transformation” [22]). Theoretically, this shifts the deformation mechanism from pure shear toward elongational flow. The Manas-Zloczower mixing index [23] defined as \(\:{\lambda\:}_{MZ}=\left|\mathbf{D}\right|/\left(\left|\mathbf{D}\right|+\left|\varvec{\Omega\:}\right|\right)\) where \(\:\mathbf{D}\) is the rate of deformation tensor and \(\:\varvec{\Omega\:}\) is the vorticity tensor, is effectively increased by these slots. By promoting elongational flow \(\:{\lambda\:}_{MZ}\sim\:1\) over simple shear flow \(\:{\lambda\:}_{MZ}\sim0.5,\:\)the design achieves superior dispersive mixing of heterogeneous feedstock at lower energy inputs than conventional screw designs.
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Second, the mixing screw features a non-linear variable pitch profile (Table 1). This design introduces higher compression ratios and axial pressure gradients that are intended to continuously destabilize the velocity profile, preventing the formation of a static solid bed and enhancing the “field synergy” between the velocity vector and the temperature gradient. As described by Li et al. [5], maximizing this synergy angle is critical for enhancing the local Nusselt number and convective heat transfer in polymer melts where thermal conductivity is low (e.g. ~ 0.2 W/mK).
Table 1
Screw design details by axial position measured in number of screw diameters
Axial length, D: Purpose
Flight Pitch, mm
Flight Width, mm
Channel Width, mm
Channel Depth, mm
Compression Ratio (%Feed)
1: Feed
37
3.00
15.50
5.29
100
2: Feed
37
3.00
15.50
5.29
100
3: Plastication
34
2.85
14.15
4.87
119
4: Plastication
31
2.70
12.80
4.35
147
5: Plastication
29
2.55
11.95
3.83
179
6: Plastication
27
2.40
11.10
3.31
223
7: Metering
25
2.25
10.25
2.50
320
8: Metering
24
2.10
9.90
2.50
331
2.2 Machine design details
The system was constrained to a total mass under 10 kg, enabling compatibility with motion systems such as gantry-style 3D printers and robotic arms, where payload capacity is a limiting factor. The final assembly achieved a weight of 5.5 kg, well below the design threshold. A minimum target mass flow rate of 3 kg/hr was established to ensure sufficient material throughput for medium- to large-scale additive manufacturing operations. To achieve this output goal, a 31 mm diameter was selected to balance the melting capacity with torque and thermal requirements. A short length-to-diameter (L/D) ratio of 8:1 was adopted, informed by a first proof of concept design [12] that demonstrated that such reduced geometries can offer effective melting and homogenization when tailored appropriately. Compared to that initial work, the present research investigates a scaled version tailored to 3D printing with improved melt pressure and temperature sensing, a larger motor for operating at higher torque and speed, improved controls providing multiple temperature zones and transient control, and more thorough performance characterization.
The extruder assembly design, shown in Fig. 1, is comprised of an extrusion screw residing within a barrel that is attached to front and rear mounts. Axial thrust loads applied to the screw by the melt pressure are supported by a needle roller thrust bearing (McMaster 5909K15) located behind the screw shank that is located in the rear mounting block. The front and rear mounts were made of stainless steel (SS) 304 while the screw and barrel were made of SS 17-4PH hardened to 60 HRC for high strength and wear resistance; extended tip set screws in the mounting plates provide a lightweight and compact method to retain the barrel. An aluminum base plate with rubber feet supports the extruder for bench top testing. Power and torque are delivered to the screw by a planetary gearbox (Stepper Online 60PE025-SSAF1) having a 20:1 gear reduction coupled to brushless DC motor (Applied Motion HT23-SS4DGA) providing 1 Nm torque at 1800 RPM.
At the heart of the design is a 31 mm diameter double-flighted mixing screw, shown in Fig. 2. The screw features a variable-pitch profile to balance volumetric intake and progressive compression over its 8:1 L/D ratio. The screw geometry is divided into three key functional zones that are detailed in Table 1 as a function of the axial length measured by the number of screw diameters. The feed zone comprises the first two turns of the screw length and features a channel depth of D/6. The initial helix pitch of 37 mm supports a large inlet volume per revolution for increased feeding. The screw in the feed portion is hollowed out providing a “heat break” to reduce the heat transfer and temperatures in the feed throat, thrust bearing, gearbox, and motor.
The transition zone occupies a majority of the screw length with a compression angle of 0.76°, enabling gradual pressure build-up and efficient melting with minimized risk of shear degradation. The specific compression angle of 0.76° was derived geometrically from the target compression ratio and the constrained transition length. Standard design guidelines for single-screw extrusion of semi-crystalline polyolefins recommend a compression ratio (CR) between 2.5:1 and 4.0:1 to ensure adequate melting and melt conveying [24, 25]. To maximize melting capacity within the limited residence time of the compact 8:1 L/D format, a target CR of ~ 3.3:1 was selected, defined by a feed channel depth, \(\:{H}_{F}\), of 5.29 mm and a metering depth, \(\:{H}_{M}\), of 2.50 mm. Given the fixed axial length of the transition zone, \(\:{L}_{T}\), dictated by the compact machine envelope, the channel taper angle is solved as \(\:\varphi\:=\text{atan}\left(\left({H}_{F}-{H}_{M}\right)/{L}_{T}\right)\). This angle represents the maximum gradual transition possible within the compact footprint without necessitating an abrupt, high-shear barrier section.
To promote in-situ distributive and dispersive mixing, a set of tapered mixing slots are incorporated using a sweep along a counter-rotating helix having a variable-pitch. These mixing slots have a length equal to one-sixth of the screw circumference and an initial depth of 2.65 mm (half the feed channel depth) with the depth decreasing along the counter-rotating helix having an outward taper angle of 1.2°. The dual-flighted design with the single counter-rotating sweep for the mixing slots ensures that the flight opposing every mixing slot continuously wipes the barrel surface to prevent stagnation. Finally, the metering zone provides a consistent melt flow through a channel depth of D/10 and a helix pitch reduced to 24 mm. As shown in Table 1, the flight thickness, channel depth, and channel width are gradually reduced along the length of the screw resulting in a compression ratio in the final metering zone of 331% relative to the volume in the feed zone. This compression ratio is greater than traditional general-purpose screw designs (employing a 200% compression ratio) to force recirculation through the mixing slots to enhance melting and homogeneity.
Screw speed was controlled via a motor driver (Applied Motion SSDC10-IP StepSERVO) connected to an Archimajor 1.0 microcontroller board (UltiMachine, South Pittsburg, TN, USA) operating a custom Klipper firmware developed by re:3D. This controller was calibrated to send the motor steppings to the DC brushless motor to achieve desired screw speeds (e.g. 30 and 60 RPM). The same controller provided power to the band heaters (IMS Company 134621), each having a diameter of 38.1 mm and a length of 50.8 with a power output of 300 W, in response to k-type thermocouple feedback indicated as TCF, TCT, and TCM for the feed, transition, and metering zones as indicated in Fig. 1. Two instrumented nozzles were designed, each with a ¾−16 thread into the front of the barrel, and machined with orifice diameters of 3.1 and 6.2 mm. These instrumented nozzles incorporated an intrusive melt thermocouple (Tempco TMB00238) protruding 3.2 mm into the melt stream prior to the final orifice as well as a pressure transducer (Dynisco PT4676 with 20 MPa range). Processing data was also acquired via the Archimajor controller at a rate of 0.25 Hz and subsequently analyzed using MATLAB. This acquisition rate is much lower than desired but sufficient in characterizing the transient melt temperature behavior as subsequently presented.
2.3 Simulation approach
To engineer the extrusion screw design for optimal thermal and flow behavior under realistic processing conditions, a three-dimensional, non-isothermal, non-Newtonian computational fluid dynamics (CFD) simulation was conducted using SolidWorks Flow Simulation. This simulation employs a cell-centered Finite Volume Method (FVM) to solve the steady-state Navier-Stokes equations for conservation of mass, momentum, and energy [26]. The formulation treats the polymer melt as a weakly compressible fluid, where density changes are coupled to the local pressure field via the material’s bulk modulus. To ensure high fidelity in predicting viscous dissipation and pressure drop, the rheological behavior of the polymer was not approximated by a standard power-law equation. Instead, the viscosity was modeled using second-order interpolation of tabulated shear-viscosity data obtained at multiple characterization temperatures. This approach reduced curve-fitting errors often found in standard constitutive models.
One of the key simplifications in the simulation was the use of a rotating barrel with a stationary screw, which is a mathematically equivalent representation of relative motion between screw and melt. This choice enables the use of a fixed mesh, eliminating the computational burden and complexity associated with remeshing in rotating domains, while preserving the essential shear and velocity gradient characteristics of the system.
The simulated geometry consisted of a full 3D representation of the screw-barrel assembly, with the feed throat section of the barrel and screw removed to provide an axisymmetric fluid inlet adjacent the downstream location of the feed throat. The inlet and outlet pressures were both fixed to atmospheric, allowing a pressure-driven solution to emerge based on shear-driven melt flow. The barrel surface temperatures were defined to reflect the experimental heating setup, with the heated surfaces corresponding to the band heater locations held at a uniform temperature of 220 °C, representing the fully developed thermal conditions in the metering zone. The inlet melt temperature was set to 140 °C, simulating the initial thermal state of the polymer as it enters the screw. The primary limitation of this approach is the assumption of a continuum in the processed material while the actual feedstock (pelletized recyclates) varies in composition and is not fully dense in the screw’s feed zone. Accordingly, the simulations are likely to overestimate the mass throughput of the extrusion process.
A cross-section of the mesh is shown in Fig. 3. The computational domain for the CFD simulations consisted of 829,524 finite volume cells, with 401,758 fluid cells and 427,766 solid cells representing the screw and barrel components. Of the fluid domain cells, 260,619 cells were positioned at fluid-solid interfaces, which are critical for accurately capturing heat transfer and shear interactions near surfaces. These mesh parameters were selected as the standard meshing level, balancing computational efficiency and numerical accuracy. All the reported simulation results were produced with this mesh density, requiring approximately 2,745 CPU seconds (~ 46 min) per simulation run on average. To verify solution accuracy, a mesh independence study was conducted by refining the global mesh density by a factor of 2.2 (resulting in ~ 1.8 million cells). The predicted mass flow rate and bulk melt temperature differed by less than 1.2% compared to the standard grid (829,524 cells). Consequently, the standard mesh was adopted for all subsequent simulations as it provided a converged solution with optimized computational efficiency.
The polymer used in the simulation was Borealis RD208CF, a copolymer polypropylene (CoPP) with an MFI of 8 g/10 min, selected for its rheological similarity to the characterized recycled PP feedstock (MFI 8.63 g/10 min as detailed in [27]). To ensure high fidelity in predicting viscous dissipation, the viscosity utilized tabulated shear-viscosity data (provided in Table 2) with second-order polynomial interpolation. This approach accurately determines the local dynamic viscosity, \(\:\eta\:\), as a continuous function of shear rate, \(\:\dot{\gamma\:}\), and temperature, \(\:T\), without the fitting errors associated with simplified constitutive models.
Table 2
Shear-viscosity data for Borealis RD208CF at 200, 230, and 260 °C
Shear Rate (s⁻¹)
Viscosity
200 °C (473 K)
Viscosity
230 °C (503 K)
Viscosity
260 °C (533 K)
0.1
3060
—
—
0.4
2950
1490
744
1.0
2710
1420
736
4.0
2180
1230
677
10.0
1670
1010
589
39.8
1000
672
428
100–107
617
440
300
287–398
273
247
188
1000
—
—
91
2.4 Experimental methods
To evaluate the performance of the built extruder, a structured series of experiments were performed following a carefully planned Design of Experiments (DOE) approach. Since the primary objective of developing this compact extruder was to efficiently process and thoroughly mix a variety of polymeric materials, the material selected for the benchtop evaluation was a post-consumer recycled polypropylene (rPP) grade designated as rPP, characterized by a melt flow rate (MFR) of 8.63 ± 0.05 g/10 min. This melt flow rate was measured according to standard ASTM D1238 conditions, confirming its suitability for extrusion-based additive manufacturing processes. The specific grade, rPP, was identified as EEI H700-12 polypropylene resin, manufactured by EEI Recycling (St. Petersburg, FL) and commercially provided by Geon Performance Solutions (Avon Lake, OH). As shown in Appendix Fig. A1, the resin predominantly consists of black polyolefin with notable proportions of gray, bright (deep) white, and several other colors. Extensive characterization and detailed physical, rheological, and thermal property data for this specific recycled polypropylene grade is documented in a previous study [27] indicating that the various colors have different viscosities and levels of contamination with polyethylene (PE) as typical of PCR recyclates. The material was employed in its as-received condition without additional preprocessing or special treatments such as drying, pelletizing, or filtration.
The DOE consisted of 11 experimental runs as detailed in Table 3. The DOE is predominantly a full-factorial design with augmented runs that were added to investigate interaction effects and best/worst case scenarios. Specifically, the effect of screw speed is investigated by comparing runs 1 and 2 for 220℃, runs 3 and 4 and 5 for 240℃, and runs 6 and 7 for 200℃. A second-order model for the effect of temperature can then be estimated by multiple regression of the complete data set. Additional runs were performed to assess the effect of barrel temperature profiling (via run 8) as well as the effect of extruder orientation (via runs 9 and 10). A replicate run (number 11) was also performed of the reference run (number 1) to assess the consistency of the apparatus and experimentation.
Table 3.
Experimental DOE matrix for extruder characterization
Each DOE experimental run was conducted at defined and controlled process parameters to investigate the effect of screw speed (RPM), barrel temperature, and extruder incline angle. Every experimental run followed a standardized protocol. First, the extruder was allowed to heat and stabilize at its processing temperature for 30 min. Then, the extruder was operated continuously for a total of seven minutes at the specified screw speed. The first minute of this operation was used exclusively to allow the system to achieve a stable equilibrium in temperature and melt flow. Afterwards, the extrudate from the nozzle was wiped from the nozzle face collected every minute. Collection continued until the completion of the sixth minute, yielding five distinct mass flow samples per run. Each extrudate sample was subsequently cooled to ambient temperature, weighed individually, and the mass recorded, providing values for steady-state mass flow rates in grams per minute. The reported mass flow rates represent the mean of the five discrete weight measurements with statistical analysis providing the confidence intervals and p-values of the resulting models. Total system power consumption was monitored using an inline digital power meter (Kuman KW47; resolution 0.1 W), which sampled voltage and current at the main supply. For each run of the experimental design, the meter was reset prior to the steady-state window, and the average power draw was recorded alongside the temperature and mass flow metrics.
To quantify the stability of the temperature control system, the Root Mean Square Error (RMSE) was calculated for each heating zone. The RMSE serves as a standard metric for control deviation and is defined as:
where \(\:n\) is the number of data points in the sampling window, \(\:{T}_{obs,i}\) is the measured wall temperature the i-th observation, and \(\:{T}_{Set,i}\) is the setpoint temperature at the same instant. A lower RMSE indicates tighter process control and higher thermal stability. In the context of estimating the fidelity of the statistical models, the RMSE is also later presented based on the difference between the observed and modeled state of interest wherein a lower RMSE indicates higher model fidelity.
3 Results and discussion
3.1 Characterization of simulation results
Simulations corresponding to each of the DOE runs (Table 3) were conducted. Representative particle traces showing the temperature histories for runs of particular interest are shown in Fig. 4. All the temperature distributions show the predominant effect of the heating of the cool polymer feedstock entering at right as it is conveyed towards the nozzle outlet at left. Further, all the mixing slots are performing the baker’s transformation to a portion of the processed material when the flows from the forward channel cross above the flows in the bottom of the preceding channel.
Fig. 4
Particle traces and temperatures for (a) DOE run 1 – reference condition at 220℃ and 30 RPM, (b) DOE run 7 – 200℃ and 60 RPM, and (c) DOE run 8 – profiled barrel temperatures and 30 RPM. The baker’s transformation is observed occurring when the flows from the forward channel crosses above the flows in the bottom of the preceding channel
The top plot (panel a) for the reference run at 220℃ and 30 RPM shows that the material is quickly heated by the center of the screw’s transition zone. The melt exiting the nozzle achieves a temperature around 222℃, higher than the set-point. By comparison, the middle plot (panel b) of Fig. 4 is for DOE run 7 at 200℃ and 60 RPM, which corresponds to the highest torque condition of the DOE. The higher speed results in faster convection of the cool polymer melt with cooler temperatures penetrating further into the transition zone of the screw. However, the melt temperature exiting the nozzle orifice attains a temperature of 205.7℃, an even higher relative difference to its set temperature than observed in the reference run. This temperature overage occurs because the lower temperature causes a higher viscosity in the processed material while the higher screw speed causes higher shear rates. The viscous dissipation, being proportional to the product of the shear stress and the shear rate, thus causes a significant increase in the melt temperature. Overall, these simulation results predict that the compact 8:1 L/D mixing screw geometry provides sufficient thermal residence time for the melt to equilibrate with the barrel set temperature. The thermal gradients observed in the early to mid-screw region underscore the role of viscous dissipation and mixing features in driving homogenization.
The bottom contour plot (panel c) of Fig. 4 corresponds to a typical profiled barrel temperature of 220℃ in the nozzle and metering zones, 200℃ in the transition zone, and 175℃ in the feed zone. This temperature contour shows that the temperature of the material being processed varies gradually from the inlet to the outlet even when operated at the low speed of 30 RPM. The maximum temperature of 220.4℃ also indicates that the processed material is just reaching the target level as it is exiting the extruder. Accordingly, the simulation results suggest that profiling the barrel temperature may result in lower output and less process stability when operated at higher speeds.
The mass flow rate and bulk temperature of the extrudate were estimated by taking the surface integral of the velocity and temperature contours at the outlet. Multiple regression was then performed using the function fitlm in MATLAB 2025 (MathWorks, Cambridge, MA) for these values as a function of the factors specified in Table 3 with the results provided for mass flow rate in Table 4 and for bulk temperature in Table 5. In this method, the vector of coefficients, \(\:\beta\:\), in the linear equation \(\:\mathbf{y}=\mathbf{X}\beta\:+\:\varvec{\epsilon\:}\) are chosen to minimize the residual errors, \(\:\varvec{\epsilon\:}\), using the method of least squares given the response vector (mass flow or temperature), \(\:\mathbf{y}\), as a function of the design matrix of predictors (RPM, Set Temperature), \(\:\mathbf{X}\). The resulting coefficients represent the estimated value of the model parameter, \(\:\beta\:\), representing the change in the response variable per unit change in the predictor. The standard error (SE) represents the standard deviation of the sampling distribution of the model coefficient, indicating the certainty of the estimate relative to its estimated value.
Table 4
Multiple regression results for simulation predictions of mass flow rate [g·min⁻¹] as a function of set temperature and screw speed (Number of observations: 8, Error degrees of freedom: 5, RMSE of 2.02 g·min⁻¹, R2 = 0.995)
Factor
Coefficient
SE
t-statistic
p value
Intercept, g·min⁻¹
−56.32
10.24
−5.599
0.00272
Set temperature, ℃
0.2386
0.0474
5.034
0.00399
Set screw speed, RPM
1.173
0.0423
27.71
1.15E-6
Table 5
Multiple regression results for simulation predictions of bulk temperature as a function of set temperature and screw speed (Number of observations: 8, Error degrees of freedom: 5, RMSE of 0.303℃, R2 > 0.999)
Factor
Coefficient
SE
t-statistic
p value
Intercept, ℃
7.1726
1.533
4.679
0.00544
Set temperature, ℃
0.9681
0.00709
136.5
4.01E-10
Set screw speed, RPM
0.0345
0.00633
5.443
0.00284
The primary result of Table 4 (with a p-value of 1.15E-6) is that the mass output increases 1.173 g per minute per unit increase in the screw speed (RPM). The negative value for the intercept suggests that the screw must rotate with a high set temperature to get any flow, while the positive but small value for the set temperature suggests that the mass flow rate increases slightly with the set barrel temperature. The results for bulk temperature in Table 5 are also statistically significant. The main result (p-value of 4.01E-10) is that each degree increase in the set temperature is expected to result in a 0.968℃ increase in the output extruder temperature. The effect of the screw speed on shear heating via viscous dissipation is also evident with each unit increase in RPM causing a 0.034℃ rise in melt temperature (p-value of 0.00284). The intercept of 7.2℃ is the least significant effect (p-value of 0.00544) and driven by the air convection in the CFD simulation. Both models have a high coefficient of determination (R² > 0.99) without any outliers identified. Residual analyses showed no strong curvature or interaction trends, supporting the adequacy of a linear model for this operating window. Accordingly, the regression confirms that each process variable primarily influences its intended outcome (e.g. melt temperature driven by set temperature and mass flow rate driven by screw speed) with only minor cross-effects showing strong axiomatic control [28].
Figure 5 plots the mass flow rate and bulk temperature results along with the regression models provided in Tables 4 and 5. While the figure visualizes these three discrete levels, the linearity of the behavior is supported by the statistical strength of the regression across the full dataset. For bulk melt temperature, the model shows a near-unity sensitivity to set temperature (≈ 0.97 °C per 1 °C change) and only a weak dependence on screw speed (≈ 0.035 °C·RPM⁻¹). Accordingly, the top-left plot (panel a) rises almost perfectly linearly with the set temperature while the top-right plot (panel b) is nearly flat as a function of the screw speed. For mass output, the regression indicates throughput is dominated by screw speed (≈ 1.17 g·min⁻¹·RPM⁻¹), with a smaller but positive effect from set temperature (≈ 0.239 g·min⁻¹·°C⁻¹). The lower-right plot (panel d) exhibits strong linear growth with RPM (R2 ~ 0.995), consistent with the theoretical expectation for drag-induced flow in the laminar regime (Q ~ N) where pressure backflow effects are linear or negligible.
Fig. 5
Simulation predictions of melt bulk temperature and mass flow rate as a function of set temperature and screw speed including modeled main effects with uncertainty bands at the 95% confidence levels
Together, the fits capture the simulations well and again highlight a desired decoupling of melt temperature and mass flow rate. Both models have a high coefficient of determination (R2 > 0.99) without any outliers identified. Residual analyses showed no strong curvature or interaction trends, supporting the adequacy of a linear model for this operating window. Accordingly, the regression confirms that each process variable primarily influences its intended outcome (e.g. melt temperature driven by set temperature and mass flow rate driven by screw speed) with only minor cross-effects showing strong axiomatic control [28].
3.2 Characterization of experimental results
Each of the run conditions in Table 3 were also performed for the built extruder. The bulk melt temperature was calculated as the average temperature reading from the intrusive melt thermocouple while the mass flow rate was calculated as the average mass collected per minute with five replicates per run condition. Multiple regression models were then performed also using the fitlm function in MATLAB for the factors including the set temperature, screw speed, and extruder orientation. Tables 6 and 7 provide the respective models for the mass flow rate and bulk temperature. The primary result of Table 6 (p-value < 0.0001) is that the mass flow rate increases 0.8388 g·min⁻¹ per unit increase in set RPM. Furthermore, all other factors, including the intercept are not statistically significant for the mass flow rate. Comparison of the coefficients modeling the observed behavior (Table 6) with the simulated behavior (Table 4) shows that every factor is qualitatively consistent. The observed mass flow sensitivity of 0.84 g·min⁻¹·RPM⁻¹ is approximately 72% of the predicted simulation value (1.17 g·min⁻¹·RPM⁻¹) made assuming a fully dense feedstock. This discrepancy aligns with standard tribological properties of polymer pellets, where the bulk density is significantly lower than the solid density. Rauwendaal [24] as well as Campbell and Spalding [29] note that the packing factor for standard polyolefin pellets typically ranges between 0.5 and 0.6 in a loose state, increasing slightly under feed-throat compaction. The observed 0.72 ratio indicates that the simulation’s assumption of a 100% dense fluid continuum at the inlet overestimates the conveying capacity by exactly the margin of the feedstock’s free volume.
Table 6
Multiple regression results for observed mass flow rate as a function of set temperature and screw speed (Number of observations: 8, Error degrees of freedom: 4, RMSE of 5.51 g·min⁻¹, R2 = 0.926)
Factor
Coefficient
SE
t-statistic
p value
Intercept, g·min⁻¹
−4.944
27.92
−0.1771
0.866
Set temperature, ℃
0.03122
0.1291
0.2418
0.819
Set screw speed, RPM
0.8388
0.1215
6.901
0.000979
Orientation, °
0.01128
0.06793
0.1661
0.8746
Table 7
Multiple regression results for observed bulk temperature as a function of set temperature and screw speed (Number of observations: 8, Error degrees of freedom: 4, RMSE of 4.9℃, R2 = 0.957)
Factor
Coefficient
SE
t-statistic
p value
Intercept, ℃
−2.247
25.53
−0.08800
0.934
Set temperature, ℃
1.104
0.1185
9.317
0.000739
Set screw speed, RPM
−0.3011
0.1152
−2.614
0.0592
Orientation, °
0.06392
0.06176
1.035
0.3592
The regression results for outlet temperature, shown in Table 7, are also very consistent with the modeled simulation results (Table 5). The main result (p-value of 0.000739) indicates that the set temperature has a dominating positive effect with a 1.1℃ increase in the output temperature per unit change in set temperature, confirming that the temperature setpoint almost entirely determines the output melt temperature. Another difference between the observed and simulated behaviors is that the screw speed showed a statistically weak but potentially negative effect (coefficient = −0.30℃·RPM⁻¹, p ≈ 0.06). This negative coefficient suggests that there is a thermal contact resistance that inhibits the heating of the processed material and results in lower melt temperature with increasing screw speed. In other words, the negative coefficient suggests that the viscous heating term is dominated by the convection of the cooler material being processed to the outlet with lower residence times at higher screw speeds.
Figure 6 provides the mass flow rate and melt temperature of the extrudate wherein the points represent the average of the observed values for each run condition, the error bars represent the standard deviations of the observed values for each run condition, the dashes represent the model predictions at the process settings, and the shaded regions represent the span of the 95% confidence level. Similar to Fig. 5, the top-left plot (panel a) of Fig. 6 again demonstrates the desired controllability of the outlet melt temperature as a function of the set temperature while the bottom-right plot (panel d) demonstrates the correlation of the observed mass flow rate with the set screw speed.
Fig. 6
Observed outlet melt temperature and mass flow rate as a function of set temperature and screw speed including multiple regression models and confidence intervals at the 95% confidence level
From a practical standpoint, these findings highlight the value of treating screw speed and barrel set temperature as largely independent control levers. Barrel temperature should be prioritized when targeting melt quality, as it overwhelmingly dictates the temperature of the extrudate and properties related to bond strength in 3D printing [30]. Meanwhile, the screw speed should be adjusted to control the output flow rate and road width as a function of the print velocity. The minor cross-effects—slight cooling at higher speeds and small flow gains at higher temperatures—can be leveraged for fine-tuning but do not compromise the independence of the two main variables. This decoupled control framework simplifies operation and provides a clear basis for scaling or adapting the extruder to different recycled polymer feedstocks and manufacturing applications.
3.3 Transient behavior
The coefficients of determination for the fitted multiple regression models are 0.926 for mass flow rate (Table 6) and 0.957 for melt temperature (Table 7). While both models provide accurate estimates of their responses, the statistical significance is lower than the fitted models for simulations due to observed variation in the melt temperature and mass flow rate as a function of time during each extrusion run in Table 3. Figure 7 provides a plot of the transient behavior for run 5 that was performed at the highest temperature (240℃) and screw speed (75 RPM).
Fig. 7
Observed transient behaviors for sensed melt temperature and mass of collected extrudate samples for a set temperature of 240℃ and screw speed of 75 RPM (run 5 of Table 3)
As shown in the top let plot (panel a) of Fig. 7, the extrudate melt temperature initially overshoots the 240 °C setpoint due to the shear heating of the material in the metering section of the screw. After 50 s, however, the convection of the cooler melt behind the transition zone causes a drop in the melt temperature down to, and then below, the setpoint. As shown in the bottom-left plot (panel c) of Fig. 7, the mass flow rate is also highest at the start of the extrusion when there is the least resistance to flow due to the higher melt temperatures. As the melt temperature drops and the viscosity increases, the mass flow rate drops from 70 to 45 g per minute. The coupled behavior of the melt temperature and mass flow rate is consistent with the rheology of the melt and the machine’s thermal inertia: as the polymer cools, its viscosity rises, die pressure loss increases, and the attainable mass flow rate at the fixed screw speed drops.
The rightmost plot (panel b) of Fig. 7 provides the average extrudate temperature (error bars indicating the standard deviation of the acquired melt temperatures during each 60 s interval) as a function of the observed mass flow rates across the experimental run. The plotted data indicates that hotter melt temperatures coincide with higher mass flow rates while cooler melt temperature corresponds to lower mass flow rates. While the observed melt temperatures are quite close to the 240℃ setpoint, the error bars show that the variation is greatest at the extreme melt temperatures. In the context of the 3D printing, this figure suggests that feed-forward adjustments of zone temperatures (adding heating power to the barrel as a function of screw speed) can help hold both melt temperature and output steady.
The slow drift and mild cycling observed in Fig. 7 calls into question the thermal time constants of the barrel/nozzle and potential issues as to controller stability. To investigate these effects, the transient temperatures of the various barrel zones and intrusive melt thermocouple are plotted in Fig. 8. Inspection of the temperature traces shows that the transition and metering zones are particularly prone to oscillation, while the feed and nozzle regions remain comparatively stable. The barrel temperature in the transition zone oscillates ~ 6℃ about the setpoint with a periodicity of ~ 100 s, which likely increases the extrudate melt temperature and mass flow rate variations. These oscillations can be attributed to thermal inertia in the heater bands and local melt shear heating, which act on similar timescales. Despite these issues, the melt temperature stabilizes near the 240 °C setpoint.
Fig. 8
Observed transient behaviors for sensed barrel temperatures and intrusive melt temperatures for the process of Fig. 7 (run 5 of Table 3)
An important operational note is that the extruder could not sustain screw speeds of 75 RPM at lower set temperatures (200 and 220℃) for two reasons. First, the brushless DC motor exhibits a significant drop in output torque with increasing motor speeds. Second, the required torque increases with screw speed given the cooler associated melt temperatures that drive higher viscosity and associated wall shear stresses. In practice, this result means that 3D printing processes will be limited to screw speeds of 60 RPM with mass flow rates around 3 kg/hr for this and similarly behaved materials.
To further investigate the melting kinetics as related to maximum screw speed, screw pulls as described by Maddock [31] were conducted for the rPP as well as a high impact polystyrene (HIPS) having pink and black colorants at the reference condition (run 1 of Table 3). After operating at steady conditions, the extruder was stopped and allowed to cool with the polymer residing within the barrel (i.e., without purging). The screw was then removed from the extruder resulting in the photographs of Fig. 9 in which the first turn of the screw and unmolten pellets are removed for clarity. The rPP begins to melt at around the fourth turn of the screw while the HIPS begins to melt around the fifth turn of the screw. The reason is that rPP has a lower melting temperature around 160℃ while the HIPS begins to flow closer to 180℃. At higher screw speeds, the pelletized materials are conveyed forward and do not have sufficient time to melt without the application of very high shear stresses.
Fig. 9
Screw pulls showing the onset melting location in the transition zone for (top) rPP and (bottom) HIPS
These screw pulls indicate that higher screw speeds can cause unmolten pellets to act like a wedge between the tapered channel in the transition zone and the barrel, thereby causing a motor over-torque condition. Thus, the screw speed and melting capacity can be increased by reducing the taper angle in the transition zone or increasing the temperatures in the feed and early transition zones. The reduced melting capacity associated with lower temperatures in the feed and early transition zones can also be verified by comparing the temperature and mass output of run 1 (reference condition at 220℃) with those of run 9 (having a 220-200-175℃ barrel temperature profile) as shown in Appendix Fig. A1. Even though both conditions have the same nozzle and metering zone temperature, the profiled barrel temperature results in an extrudate melt temperature that is 8℃ lower than the reference condition with a mass output that is 30% lower as well.
The dispersive and local distributive mixing is clearly shown in Fig. 9. Specifically, both materials show smeared, helical streaks and thin tongues of locally smeared polymer bridging adjacent flights that is evidence of surface renewal, flight-tip leakage, and repeated split–recombine events that drive distributive mixing. The double-flight geometry with multiple mixing slots multiplies these events with each screw revolution, forcing cross-channel transport of the process material so that residual solids are laminated into thin striations and rapidly wetted by the melt. This action actively breaks up the solid bed, preventing it from moving as a cohesive plug and instead promoting early and aggressive mixing between the hot melt and the cooler solid particles. The rPP pull (top) displays many fine lamellae and re-wetting bands that disappear by the metering section, whereas the HIPS pull (bottom) shows broader but very quick gradients between the batches of white, pink, and black materials that were successively fed into the extruder. Both screw pulls indicate that the compact mixing screw achieves early melting and effective homogenization within a short L/D, delivering strong distributive mixing with light dispersive shear from the flight clearance.
The transient results previously provided for run 5 underscore the importance of improving thermal management when the screw operates at high and variable speeds, as is common in additive manufacturing applications. In run 5, the melt and zone temperatures exhibited clear oscillations, driven by the increased and fluctuating heating load associated with higher screw speeds. In a typical 3D printing scenario, the screw speed is not constant but instead varies in response to toolpath requirements defined in the G-code, with accelerations and decelerations occurring frequently as the print progresses. This introduces a dynamic heating demand that the current zone control strategy, based on static setpoints, is not optimized to handle. Variations in the mass flow rate will lead directly to variations in the road width and, thus, structural integrity of the printed product. One avenue for improvement is to leverage the knowledge of the pre-planned extrusion profile encoded in the G-code: because the sequence of speed changes is known in advance, the system can be programmed to proactively adjust the heating zones ahead of time. For example, when the screw is scheduled to accelerate, the transition and metering zones could be preheated slightly above their nominal setpoints to counteract the additional cooling effect of higher throughput. Conversely, during low-speed or long dwell periods, temperatures could be adjusted downward to prevent overshoot and thermal degradation. This form of anticipatory, feedforward temperature control would reduce the oscillations observed in run 5, help the extruder achieve equilibrium more consistently during transient operations, and ultimately improve the stability and reliability of melt delivery for additive manufacturing.
The mixing screw and extruder were developed based on CFD flow simulations, an approach of increasing importance for agility and optimality in machinery development. Figure 10 provides correlation plots for the observed and simulated responses for the extrudate temperature and mass flow rate. The temperature has a correlation r ≈ 0.96, with a near-unity slope (y ≈ − 4.13 + 1.037 x, RMSE ≈ 5.5 °C), while the mass flow rate has r ≈ 0.94 with a steady, monotonic relation (y ≈ 6.22 + 0.676 x, RMSE ≈ 5.2 g min⁻¹). The predictability and controllability are both excellent with the primary error in the mass flow rate being driven by the lower packing density of the pelletized feedstock in the real extrusion operation relative to the assumed continuum in the simulation. This level of agreement supports direct use of the simulations for machine and screw-element design.
Fig. 10
Experimental and simulation observations for (left) temperature and (right) mass flow with solid line representing ideal correlation and dashed line representing fitted correlation
The comparison between simulation predictions and physical experiments in Fig. 10 reveals a bifurcation in predictive accuracy of the simulations. While the simulation significantly overpredicts the mass flow rate (by approximately 30%), it predicts the extrudate temperature with remarkable accuracy (within 1.2% of experimental values). This behavior is physically consistent with the modeling assumptions. The mass flow overprediction is a direct consequence of the continuum assumption at the inlet; the CFD model treats the feedstock as a fully dense fluid (density ~ 0.9 g/cm³), neglecting the void volume inherent to the physical pelletized feedstock (density ~ 0.6 g/cm³).However, the strong agreement in outlet temperature confirms the validity of the energy and rheological models. Since the bulk melt temperature is primarily determined by viscous dissipation and conductive heat transfer from the barrel, the match indicates that the simulation correctly resolves the shear heating and thermal gradients within the melt phase, irrespective of the initial solids conveying efficiency.
Practically, a model-based design methodology can rely on the CFD results in combination with safety factors derived from the residual spread. Specifically, for temperature decisions, we suggest safety margin of ~ 1×RMSE (~ 6 °C) around limits and setpoints. For throughput sizing and motor/load checks, we suggest a lower confidence bound of the predicted mass flow rate (e.g., the predicted mass flow rate less ~ 2×RMSE) or an equivalent percentage margin consistent with required specifications. Because the control is largely decoupled (in which the set temperature governs melt temperature and the screw speed governs mass flow), we can treat these orthogonal degrees of freedom to tune the designed machinery to compensate for any minor model discrepancies without coupling the two objectives of extrudate temperature and flow rate.
Energy consumption is also of critical importance to achieving process efficiency and cost savings. The specific energy consumption (SEC) for the six operating runs (numbers 1, 2, 3, 4, 6, 7) was calculated by dividing the total energy consumed by the controller and extruder by the total mass of the material collected during that run. The multiple regression results are provided in Table 8. The average energy consumption was 0.372 kW·hr·kg⁻¹. While the model fidelity is limited (R2 = 0.861), the model coefficients of Table 8 suggest that increasing the set temperature increases the specific energy consumption (likely due to heat losses to the environment) while increasing the screw speed reduces the specific energy consumption (likely due to efficiency gains related to higher mass output relative to heat losses to the environment and fixed power consumption of the controller and fans).
Table 8
Multiple regression results for specific energy consumption as a function of set temperature and screw speed (Number of observations: 6, Error degrees of freedom: 2, RMSE of 0.0026 kW·hr·kg⁻¹, R2 = 0.861)
Factor
Coefficient
SE
t-statistic
p value
Intercept, kW·hr·kg⁻¹
0.358
0.257
1.39
0.258
Set temperature, ℃
0.000873
0.00119
0.736
0.515
Set screw speed, RPM
−0.00438
0.00151
2.909
0.062
By comparison with prior works, the lowest SEC for this characterized design was 0.254 kW·hr·kg⁻¹ observed for run 7 of Table 3 that corresponds to the 200℃ operating temperature and screw speed of 60 RPM. Given a theoretical minimum SEC of 0.193 kW·hr·kg⁻¹ for the rPP based on the specific heat and enthalpy [32] the designed extruder was 76% efficient when operating at run 7. This SEC is 26% lower than the previously observed SEC of 0.344 reported for the previous 20 mm design, which itself was limited to a 40 RPM screw speed also due to motor torque limitations.
A process gain matrix, A, can be established from the specified process design parameters (DPs) to the system’s functional requirements (FRs) in accordance with Suh’s axiomatic design methodology [33]. In this application, the DPs are {screw speed, barrel temperature} and the FRs are {mass flow rate, melt temperature}. The coefficients of the gain matrix are imported from Tables 6 and 7 to provide:
Here, the diagonal dominance of this matrix, \(\:{A}_{11}\gg\:{A}_{12}\) and \(\:{A}_{22}\gg\:{A}_{21}\) confirms that the mixing extruder behaves as a decoupled design, allowing for stable, independent control of throughput and temperature. In other words, the diagonal nature of its gain matrix confirms that the system is uncoupled, meaning that mass flow is primarily driven by screw speed and melt temperature by the heater bands with minimal interactions.
4 Conclusions
A compact extruder designed to process post-consumer recycled plastics using chaotic mixing was designed, simulated, implemented, and characterized. The extruder was engineered to remain under 10 kg while maintaining sufficient throughput for medium- to large-scale 3D-printing applications. The final design featured a 31 mm screw with an 8:1 L/D ratio, four independently controlled heating zones, and interchangeable nozzles, supported by a high-torque stepper servo motor and gearbox system. Computational fluid dynamics (CFD) simulations were employed to model non-isothermal, non-Newtonian melt flow, demonstrating that the reduced geometry could still achieve uniform melt conditions under practical operating parameters. These simulations provided critical insight into mass-flow rate and temperature fields within the compact geometry to repeatedly perform the baker’s transformation to the processed material.
The combination of simulation and benchtop testing highlights that compact screw designs can deliver stable throughput, predictable thermal control, and efficient processing even with recycled materials, underscoring their potential as versatile platforms for sustainable polymer extrusion. Benchtop testing using post-consumer recycled polypropylene (rPP) confirmed the extruder’s ability to deliver stable throughput and melt quality across a range of DOE process conditions. The results demonstrated a clear decoupling of control: barrel temperature governed the melt temperature nearly one-to-one, while screw speed determined mass flow at approximately 0.8 g·min⁻¹ per RPM. Cross-effects were present but minor, validating the independence axiom in system design. Regression analysis provided high fidelity models, with R² values exceeding 0.95 for melt temperature and 0.92 for mass flow. The system’s gain matrix was nearly diagonal, indicating intuitive and predictable operation where each process parameter independently regulates its intended outcome. This is a valuable trait for implementation in reducing operator burden and simplifying integration with additive manufacturing platforms.
The primary limitation in operation was transient variations in melt temperature and mass flow rate when starting the extruder from rest. Transient testing highlighted the importance of accounting for dynamic behavior, particularly at high screw speeds. At 75 RPM and elevated set temperatures, the melt and transition zone temperatures exhibited oscillations linked to thermal inertia and fluctuating heat load, while mass output declined as the system settled toward equilibrium. These findings suggest that, for additive manufacturing, where screw speeds vary according to toolpath requirements, preemptive thermal control strategies could enhance stability and output. Because extrusion demands are encoded within the G-code prior to printing, zone temperatures could be adjusted in advance to anticipate higher or lower screw speeds. Such feedforward control, as suggested by Love et al. [34], would minimize oscillations, reduce equilibration time, and improve melt consistency during complex builds.
Acknowledgements
The authors also wish to thank Geon Performance Solutions (Avon Lake, OH) for generously providing the recycled materials used in this study as well as Dynisco (Franklin, MA) for the use of their pressure transducers.
Declarations
Competing interests
The authors have no relevant financial or non-financial interests to disclose.
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Mass output [g·min⁻¹] and melt temperature [°C] of the extrudate using uniform temperature setpoint of 220℃ (run 1 of Table 3) and profiled barrel temperatures (run 9 of Table 3)
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Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.
