# An Outline of Fused Deposition Modeling: System Models and Control Strategies
## 2 Systems
### Extrusion
#### 2.1.1 Single-Screw  
The technologies used to extrude powder or pellet involve a screw mechanism. Several different screw configurations have been developed [17], including single, twin or multiple screws. This article focuses on the single screw version since it is the most studied and widespread.  
Figure 1 depicts a scheme of a single screw extruder and highlights its components: hopper, endless screw, barrel, heaters and nozzle. The pellet or powder is introduced through the hopper into the first section of the screw. The rotation presses the material toward the tip. Along the way, friction and heat melt the material. In the final section of the screw, the nozzle directs the flow outwards. The phase shift from solid to liquid occurs gradually from hopper to nozzle, forming four sections based on the percentage of melted material: solid conveying zone [18; 19; 20; 21], pre-melting zone, melting zone [22; 23; 24; 25] and melt conveying zone [26; 27; 28; 29]. Continuously modeling the interaction between solid granulate and fluid-like melt is not trivial, and to date, there exists no global model that describes the complete screw extrusion process. The authors refer to this article [30] for a comprehensive review of the mathematical models of both single-screw and twin-screw extrusion configurations, grouped by melting section.  
Two operating conditions of the screw can be identified: (i) flood fed and (ii) starve fed. The former is achieved at a filling factor (FF) of 1, defined as [31]  
\[FF=\frac{V_{fill}}{V}, \tag{1}\]  
where \(V_{fill}\) is the volume of the screw channel occupied with material, while \(V\) is the volume of the screw channel. A filling factor lower than one leads to starvation. Starvation occurs when the screw rotation is so high that the output flow rate exceeds the input flow rate provided by the hopper. When air bubbles form during a flood fed extrusion, these can cause discontinuities in the flow that lead to defects of the part. A controlled starve fed extrusion, however, shows several advantages, such as a decrease in the chances of clogging and an acceleration of the melting process [31].  
Figure 1: Scheme of the single-screw extruder: 1—solid polymer; 2—hopper; 3—barrel; 4—screw; 5—heaters; 6—die; 7—extrudate; A—solid conveying zone; B—pre-melting zone; C—melting zone; D—melt conveying zone; E—melt flow zone. Image from reference [30].  
The key variable for control purposes is the volumetric flow rate \(q\) exiting from the nozzle. As a first approximation, the flow rate \(q\) of single-screw extrusion can be written as [31]:  
\[q=q_{d}-q_{p}, \tag{2}\]  
where \(q_{d}\) is the volumetric drag flow and \(q_{p}\) the volumetric pressure flow, which can be defined as  
\[\begin{split} q_{d}&=\alpha\omega,\\ q_{p}&=\beta\frac{\Delta p_{c}}{\eta L_{f}},\end{split} \tag{3}\]  
where \(\Delta p_{c}\) is the pressure increase in the screw channel, \(\eta\) the viscosity of the material, \(L_{f}\) the fully filled length of the screw channel and \(\alpha\) and \(\beta\) represent geometrical constants.  
Equation (3) represents a static model that is valid at steady-state, whereas it does not describe the dynamics of the flow rate during transient states. The dynamical behavior of the extruder is usually studied through demanding software simulations based on discrete element methods, often under strong assumptions, e.g., the fluid being isothermal or Newtonian. Moreover, the coupling between the flow rate and temperature is often neglected in practice, and the model of the extruder is usually considered only as a function of the rotation speed of the screw. The common approach consists of parametrizing the transfer function between the rotation speed and flow rate as a first-order transfer function with delay and then selecting the parameter values that best fit the data, which are collected as step responses of the rotation speed at a constant temperature.