Numerical modeling of the polymer flow through the hot-end in filament-based material extrusion additive manufacturing
Introduction
Performance of material extrusion Additive Manufacturing (AM), also known as Fused Deposition Modeling (FDM), or Fused Filament Fabrication (FFF), depends on the hot-end design, a device that is responsible for melting and extruding the polymeric filament. The essential elements of the hot-end are the liquefier that delivers heat to the polymer and the extrusion nozzle that deposits the material as thin strands, which build three-dimensional objects. This manufacturing process differs substantially from the conventional screw extrusion, where a significant amount of heat is generated by viscous heating. In material extrusion AM, the polymer has a short residence time in the hot-end and most of the heat is delivered by conduction from the walls of the liquefier. Furthermore, the flow is driven by the pressure of the solid filament that acts as a piston before melting, thereby realizing a continuous extrusion process that is also different from conventional ram extrusion. Thus, specific computational models and experimental tests are needed to understand and predict the material extrusion AM process [1].
The experimental analysis of the polymer flow through the hot-end is challenging due to the small diameter of the channel. Phan et al. [1] investigated the heat transfer effects in the hot-end and presented the correlation between the Nusselt number and the Graetz number. They measured the pressure inside the extrusion nozzle by monitoring the power consumption of the filament feeding system. Direct measurements of pressure were done later by Anderegg et al. [2], where a transducer was introduced to the flow channel. The measured pressure was around 30 % larger than predicted by the theoretical model of Bellini et al. [3]. Nienhaus et al. [4] quantified experimentally the influence of the nozzle geometry on the force required to feed the filament through the hot-end. They showed that above a critical feeding rate, the extrusion becomes unstable and the feeding force increases abruptly. The feeding force was also measured by Serdeczny et al. [5], and similarly, the unstable extrusion regime was identified. Moreover, a model to predict the maximum feeding rate of stable extrusion was presented. Osswald et al. [6] proposed an analytical model of the hot-end flow for a limiting case scenario, where the polymer melts at the bottom surface of the liquefier. They assumed that the heat is conducted to the polymer only from that surface. Peng et al. [7] introduced a pigment to the filament to visualize the velocity profile inside the hot-end. They concluded that the flow in the hot-end differed from the fully developed velocity profile of a power-law fluid, because of non-isothermal effects.
Computational Fluid Dynamics (CFD) simulations are an attractive tool to study and optimize the process. Alternative geometries of the flow channel could be investigated to improve the heat transfer and increase the maximum feeding rate of stable extrusion [4,5]. Reduction of the pressure drop inside the nozzle will lead to smaller force required to feed the filament, hence reducing the torque requirements for the stepper motor. Optimized size of the stepper motor is necessary to reduce the inertia of the print head, thereby enabling larger accelerations and faster printing. Furthermore, the knowledge gained from high fidelity flow simulations could lead to a better control of the extrusion rate, which would contribute to reducing geometrical inaccuracies due to over- or under-extrusion [[8], [9], [10]].
Many numerical models have been used to investigate the deposition of the extrudate [[10], [11], [12], [13], [14], [15], [16], [17], [18]] and the evolution of the temperature field after the material has been placed [[19], [20], [21], [22], [23]]. Most of these simulations only include the tip of the extrusion nozzle and assume a steady extrusion rate. The flow through the complete geometry of the hot-end channel has been simulated in [[24], [25], [26], [27], [28]]. The focus of these simulations was mainly to predict the pressure drop in the hot-end as well as the extrudate surface quality. However, the simulations lack quantitative comparison with experiments, so it is unknown whether their modeling assumptions are correct and their predictions accurate. In particular, these works assume that the molten polymer fills the entire liquefier and a perfect contact is established between the wall of the channel and the fluid. Pigeonneau et al. [29] argue that the assumption of the perfect contact is valid based on their numerical simulations of the heat transfer, which were compared to the measurements of the maximum temperature registered during the flow through the hot-end by Peng et al. [7]. On the other hand, the measurements presented in [1,5] show that the polymer temperature at the outlet is lower than the liquefier temperature, suggesting the presence of a thermal resistance between the channel wall and the fluid [28]. Moreover, during the actual process, there is a gap between the incoming filament and the wall of the liquefier channel that has a larger diameter. It has not been fully explained in the existing literature whether the molten polymer fills this gap completely and how the position of the melt zone inside the hot-end depends on the printing parameters. This element is however crucial to calculate the heat flux between the hot-end wall and the filament, as well as the shear forces that ultimately determine the filament feeding force.
In this work, we present CFD simulations of the polymer flow through the hot-end channel. Heating of the filament from the solid state (room temperature) to the melt state (extrusion temperature) is included in the model. We investigate two modeling alternatives: Model 1, which is a simplified approach, where the entire domain is assumed to be fully filled with the polymer; and Model 2, which resolves the free surface of the fluid inside the hot-end. Model 2 gives an insight into how the material fills the channel. Both shear thinning and temperature dependency of the polymer viscosity are included. The modeling results are compared with the experimental measurements of the force required to feed the filament through the hot-end. The simulations are tested for different extrusion temperatures and hot-end geometries. The remaining of the paper is organized as follows. Section 2 introduces the numerical model and methodology of the study. Section 3 discusses the simulation results and compares the model predictions with the experimental measurements. We conclude our study in Section 4.
Section snippets
Physics of the numerical simulations
Fig. 1a illustrates a typical hot-end assembly, which includes the nozzle that extrudes the material; the liquefier that melts the polymer; a heat break that reduces heat transfer from the liquefier to the upper parts of the printing head; and a heat sink that dissipates the excess of heat to the environment. The numerical model includes the internal channel of the hot-end, as shown in the Fig. 1b. A solid filament with a diameter DF enters the computational domain with a constant velocity V
Preliminary results - polymer melting and filling of the empty liquefier
As a starting point, both Models 1 and 2 are used to simulate the polymer flow inside the hot-end at low filament feeding rate. In Model 1 (no free surface), as an initial condition, the entire channel is filled with molten polymer at a temperature equal to the liquefier wall temperature (). In Model 2 (free surface resolved), the channel is initially empty. Next, a cold solid filament ( is inserted at the inlet boundary, with a constant feeding rate . For both of
Conclusions
This work presented CFD simulations of the flow of polymer inside the hot-end channel. Two alternative modeling approaches were investigated: Model 1, where the entire domain was fully filled with the polymer, and a novel Model 2, where the free surface of the polymer inside the liquefier was resolved. It was shown that at low and moderate filament feeding rates, the melt zone extends up to the heat break and a recirculation region forms between the incoming filament and the channel wall. The
CRediT authorship contribution statement
Marcin P. Serdeczny: Conceptualization, Methodology, Investigation, Formal analysis, Writing - original draft. Raphaël Comminal: Writing - review & editing, Formal analysis, Supervision. Md. Tusher Mollah: Writing - review & editing, Formal analysis. David B. Pedersen: Supervision, Resources, Funding acquisition. Jon Spangenberg: Writing - review & editing, Supervision, Project administration, Funding acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors would like to acknowledge the support of the Danish Council for Independent Research (DFF), Technology and Production Sciences (FTP) (Contract No. 7017‐00128). Moreover, the authors thank FLOW-3D for their support in regards to licenses as well as setting up the models.
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