Effects of feed rates on temperature profiles and feed forces in material extrusion additive manufacturing☆
Introduction
Material extrusion (MatEx) additive manufacturing of thermoplastics is one of the most widely used approaches to fabricate 3-D plastic structures [[1], [2], [3], [4]]. In MatEx, also known as fused deposition modeling or fused-filament fabrication, polymer filament is forced by pinch wheels into a heated cylindrical tube, where it is melted and then extruded through a nozzle to produce desired plastic parts and products. MatEx is finding applications across numerous sectors, including biomedical and aviation, due to its ability to customize and iterate. However, one primary limitation of MatEx is the limited overall rate of production, as compared to traditional screw extrusion and injection molding.
A primary reason for the slower manufacturing rate is the thermal process history and associated thermal gradients during the process. In traditional polymer extrusion, solid pellets are heated in the conveying and melting zones of a single (or twin) screw extruder and forced under pressure into the extruder nozzle or mold, which are maintained at high temperatures. Much research has been done to find the corresponding temperature profiles. The melted polymers have been modelled as either Newtonian [5] or non-Newtonian fluids [[6], [7], [8], [9], [10], [11], [12]]. In the case of a Newtonian fluid, it is a so-called Graetz–Nusselt problem [5]. In these thermal models [[5], [6], [7], [8], [9], [10], [11], [12]], polymers are assumed to be completely molten. Experimentally, temperature gradients on the order of 3 °C / mm have been reported in [13].
The thermal history of a polymer undergoing MatEx is vastly different. Prior work on thermal gradients can roughly be divided into two categories: gradients in the extruded piece, and those in the extrusion head. In the former category, theoretical models have been developed [[14], [15], [16], [17], [18], [19]] along with experimental measurements [20,2]. It is clear from work on the welding strength in parts produced by MatEx, that temperature of the extrudate is a key parameter in determining the weld strength of the part [2]. For semi-crystalline thermoplastics, recent theoretical work shows that flow and temperature play key roles in the crystallization kinetics of extrudates [21]. If there are radial temperature gradients, they would influence such kinetics. Furthermore, large thermal gradients of extruded polymers may lead to warping and distortion of a final product [4].
The temperature fields in the extruded polymer find their origin in the MatEx extruder head, and research indicates these can be significant. Compared to traditional polymer extrusion, large thermal gradients in the polymer can exist in the extrusion head during MatEx [22]. Here, large gradients and the overall temperature field affect the melting and flow of the polymer in the extrusion head, and this directly impacts the maximum allowed feed rate of the process. In the extruder head, there are limited measurements of temperature profile [23] and so modelling becomes critical.
Graetz has previously considered a heat-transfer problem of a Newtonian flow in cylindrical tubes for two cases [24]: i) the flow rate is constant and uniform throughout the flow, and ii) it has a Poiseuillean distribution over a cross-section of the tube. The model of the first case has been used in MatEx to determine temperature profiles in the solid portion of the filament [25,3,12,22]. Meanwhile, based on the assumption that a filament melted as soon as it entered an extruder, the model of the second case was also applied to find the temperature distribution of the polymer [26]. In addition, Osswald et al. [27] investigated a limiting case, where a solid filament only began to melt when it reached near the end of the extruder.
It is known that the front portion of the filament inside an extruder may be melted, while its back part may still be solid (Fig. 1). Thus, in exploring the temperature profiles inside the extruder, these two portions of the filament should be considered, separately. Furthermore, the models for these two portions should be combined to give good prediction of temperature profiles at the nozzle exit. In addition, Graetz focused on a Newtonian flow, and he did not consider viscous heating in his models [3]. Once melted, the filament materials commonly used in MatEx, such as acrylonitrile butadiene styrene (ABS), poly(lactic acid), and poly(lactic acid)–polyhydroxybutyrate copolymer, should be considered as non-Newtonian fluids [3]. Due to these concerns, based on the previous studies, particularly that of Graetz [24], this study aims to determine the temperature distributions in solid and melted portions of the filaments, using two different models. The polymer melt is treated as a non-Newtonian fluid, and the corresponding viscous heating is considered. The applicability of existing pressure-speed relations is also explored, which were originally derived for pure melt flows.
Section snippets
Temperature distribution
A typical extruder used in MatEx includes three parts [1]: a cylindrical liquefying tube with a length of l1 and a radius of , a conical connecting tube with a length of l2, and a cylindrical print nozzle with a length of l3 and a radius of (Fig. 1). For reference, their dimensions in our extruder are also given in Fig. 1. In addition, pinch wheels and a cylindrical tube are often used to move and guide a filament into the cylindrical liquefying tube. Since the cylindrical tube is much
Temperature distribution for the melt portion whose Z ranges from Zs to 1
The equation of the motion for a polymer melt is [[5], [6], [7]]:where p is pressure and is the shear stress. For a power-law non-Newtonian fluid,where denotes dynamic viscosity and n is power-law index. When n = 1, Eq. (11) is reduced to be a constitutive relation for a Newtonian fluid. is normally temperature-dependent. For simplifying the problem while still obtaining a good approximate solution, it is set to be a constant and equals , which is the
Relation between feed force and feed rate
As discussed in Sec. 3.3, varies with U. The feed force (F) increases with U. The maximum allowed feed rate also depends on the maximum allowed feed force of a printer [3,22]. Therefore, it is important to know the relation between F and U. This relation has been previously derived for traditional extrusion [31], which involves only pure melt flows. We want to examine its applicability to MatEx.
Let denote the pressure drop on the solid portion of the filament. Set , , and
Summary and conclusions
In this work, we have explored polymer extrusion in additive manufacturing, and derived theoretical expressions (i.e., Eqs. (9),(23),(28) and (33)) to determine temperature distributions during the extrusion process. For different , we find the corresponding sizes of solid and melted regions of a polymer filament, as well as the temperature profiles in these regions. For , the majority of ABS filament is melted, and the thermal gradient in the extruded filament is small. Furthermore, for
CRediT author statement
Cheng Luo: Theoretical models, Experimental Work, Writing. Xiang Wang: Numerical Calculation, Presentation of Results in Figures. Kalman B. Migler: Writing- Reviewing and Editing, Technical Input and Discussions. Jonathan E. Seppala: Assistance in Experiments, Technical Input and Discussions.
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