# An Outline of Fused Deposition Modeling: System Models and Control Strategies
## 2 Systems
### Deposition
#### 2.2.1 Strand Geometry  
[PERSON] et al. [38] presented a geometrical model of the filament's trajectory during deposition. After the extrudate reaches the substrate, it is possible to identify two different deposition conditions based on the ratio between nozzle-to-substrate distance \(z\) and nozzle diameter \(d\), defined as the normalized gap  
\[G=\frac{z}{d}. \tag{11}\]  
In the case \(G\geq 1\): the exiting flow is free to deposit without any interference, reaching a free-standing height \(h_{f}\)  
\[h_{f}=2\sqrt{\frac{\gamma_{SE}}{g\rho}} \tag{12}\]  
which is only a function of the material's properties at the given temperature (\(\gamma_{SE}\) is the surface energy, \(g\) the standard gravity, \(\rho\) the density). In the case \(G<1\): the printing head prevents the melt from expanding upwards and constrains it to spread horizontally. The strand height \(h\) is often approximated to be equal to the nozzle-to-substrate distance \(z\). A simple model of strand geometry at steady-state can be computed using the mass conservation principle while assuming the shape of the cross-section to be oblong (see Figure 4), which results in a strand width given by [39; 40]  
\[w=\frac{\pi}{4}\frac{v_{e}}{v_{h}}\frac{d^{2}}{z}+z\Big{(}1-\frac{\pi}{4} \Big{)}, \tag{13}\]  
where \(v_{e}\) is the extrusion speed, \(v_{h}\) the printing speed, \(d\) the nozzle diameter and \(z\) the distance between nozzle and substrate. Alternatively, a different analytical model of width and height of the deposited strand has been obtained in the case of nozzle equipped with a spreading head [41], as depicted in Figure 5. The analytical model has been compared with the simulation results under the assumptions that the fluid is isothermal and Newtonian. Figure 4: Scheme of the assumed cross-section of the deposited strand. \(v_{e}\) is the extrusion speed (i.e., the exiting velocity of the extrudate from the nozzle), \(v_{h}\) the printing speed (i.e., the linear velocity of the EE), \(d\) the diameter of the nozzle, and \(h\) the high of the deposited strand, which is often assumed to be equal to the distance between the nozzle and the substrate \(z\). The strand geometry obtained is given by  
\[w =2(r_{i}+L),\] \[L =\frac{1}{2}\Bigg{(}-r_{e}+\sqrt{(r_{e}-2r_{i})^{2}+\frac{4}{v_{h}} \frac{q}{z}(r_{e}-r_{i})}\Bigg{)}, \tag{14}\] \[h =\frac{q}{wv_{h}}=\frac{z}{2}+\frac{zL}{2(r_{e}-r_{i})},\]  
where \(w\) and \(h\) are the strand width and height, respectively, \(r_{e}\), \(r_{i}\) the external and internal radii, respectively, and \(q\) is the output flow rate. Moreover, the average temperature of the deposited strand is solved as  
\[T=T_{\text{inlet}}+\frac{K\big{(}\frac{v_{h}}{z}\big{)}^{n-1}v_{h}^{2}}{3k} \bigg{(}1-\frac{3}{4\text{Br}}\bigg{)}(1-\exp{(-3\text{Ca})}), \tag{15}\]  
where Br is the Brinkman number, Ca the Cameron number, \(K\) the material consistency and \(k\) the thermal conductivity. The authors concluded that the hypothesis of isothermality holds at a substrate temperature higher than 100 \({}^{\circ}\)C and high printing speeds. CFD software makes it possible to derive qualitative considerations by simulating the situation of interest. For example, [PERSON] et al. [39] performed simulations with the intent of studying the morphology of the strand under different printing conditions as a function of two factors: the normalized gap \(G\) and the velocity ratio \(v_{r}\)  
\[v_{r}=\frac{v_{h}}{v_{e}}, \tag{16}\]  
which is defined as the ratio between linear velocity of the printing head \(v_{h}\) and extrusion speed \(v_{e}\). As presented in their work, variations of the above factors strongly impact the geometry of the strand cross-section but not the cross-sectional area (Figure 4). In general, both a reduction in the gap or the printing speed strengthen the side flow, thus increasing the width of the strand. The strand height, on the other hand, has been shown to shrink or swell depending on the velocity ratio.