# Optimization of Pin Type Single Screw Mixer for Fabrication of Functionally Graded Materials
## 3 Results and Discussion  
### Numerical Validation  
Three different screw configurations were compared to further verify the efficacy of the optimized pinned screw mixer in reducing transition delay time. In addition, the mixing performance of these three screw setups was assessed based on the uniformity of the extruded material and the Lyapunov index. The first screw, illustrated in Table 2 and Figure 9a, represents the original screw without any pins, with its structural parameters, including screw length, pitch, minor screw diameter, and screw flight height, remaining unchanged. Figure 9b showcases the second screw, utilized to demonstrate that the volume in the mixing chamber does not play a decisive role in transition delay times. Custom pins were transformed into cylinders, maintaining the same cross-sectional area and height as the optimized pin-type screw mixer, with a diameter of 2.4 mm and a height of 1 mm. The center of the circle was positioned at the midpoint of the y-axis at an applicable height of (0, 4). The third configuration was the pin-type screw optimized with RSM, as depicted in Figure 8.  
This study then compared the transition delay times of three screw mixers based on the simulation setup detailed in the preceding section. The total flow time was set at 400 s for this analysis. In Figure 10a, we have marked the initial moments when the material A component at the outlet shifted to 0.86 and the termination moments when it reached 0.14. This time interval is defined as the transition delay time. The results reveal that the transition delay time for the optimized pin-type mixer, cylindrical pinned mixer, and pilless mixer stands at 113 s, 158 s, and 169 s, respectively. Notably, compared to the pilless screw mixer, the transition delay time reduced by approximately 33.1%. Moreover, it is evident that, when comparing mixers with the same volume, optimizing the pin shape still reduces the transition delay time by 28.5%. Figure 10b illustrates the probability density curve of the residence time distribution, exhibiting the rate of change of material A's volume fraction at the outlet. The results indicate that the maximum rate of change for the optimized pin-shaped screw mixer reaches 2%/s. In contrast, the corresponding maximum rates of change for the cylindrical pin and the pilless screw are only 1.5%/s and 1.3%/s, respectively. In order to ensure the homogeneity of the printed material, the assessment and monitoring of mixing uniformity at the outlet play a crucial role. The uniformity at the outlet was assessed using the coefficient of variance (COV), defining the extruded mixing material as homogeneous when its value exceeded 95% [51]. As depicted in Figure 11, all three mixers were able to meet the criteria for mixing uniformity at the ratio conversion where mixing difficulty is greatest. However, it is worth noting that the change in uniformity during extrusion was more pronounced for the cylindrical pinned screw and the pilless screw. Figure 9: Comparison of screw mixers. Five randomly selected particles' trajectories were tracked from the two inlet points of the three screw types using ANSYS FLUENT Vision 2020R2 (fluid simulation software) to evaluate and compare the efficacy of the optimized pin shape in enhancing the chaotic mixing of the two paste-like substances in the chamber. This tracking yielded the spatial coordinates of the particles as they evolved over time. Analysis of the particle trajectories, as depicted in Figure 12, reveals a higher degree of chaos associated with the optimized pin-type mixer. Moreover, the level of chaotic mixing was quantified using Equation (6)'s Lyapunov exponent. \[\lambda=\lim_{ \begin{array}{c}t\rightarrow\infty\\ \Delta x\to 0\end{array} }\frac{1}{t}\ln\frac{|\Delta x(t)|}{|\Delta x(0)|} \tag{6}\]  
where \(\Delta x(0)\) is the separation of the two particles at the initial time and \(\Delta x(t)\) represents the separation at time \(t\). Figure 11: Uniformity curve at the outlet.