# Optimization of Pin Type Single Screw Mixer for Fabrication of Functionally Graded Materials
## 2 Materials and Methods
### Simulation Settings and Paste Properties  
To obtain a definitive solution for the flow field, it is necessary to establish the initial boundary conditions in the simulation domain. When preparing FGM parts with extrusion technology, a pressure-based solver was chosen to analyze the low-velocity, incompressible flow field. The boundary conditions were set separately for the velocity inlet and pressure outlet. The combined feed rate for the two inputs was held constant at 0.5 mm/s. Therefore, the total volumetric flow rate could be determined by calculating the cross-sectional area of  
\begin{table}
\begin{tabular}{c c c} \hline  & **x-Axis** & **y-Axis** \\ \hline Point 1 & 0.5–2 mm & 0.1–1 mm \\ \hline Point 2 & 0.5–2.5 mm & 3–5 mm \\ \hline Point 3 & 0.5–2.5 mm & 3–5 mm \\ \hline \end{tabular}
\end{table}
Table 1: Range of coordinate values for the three control points.  
\begin{table}
\begin{tabular}{c c} \hline Name & **Value [Unit]** \\ \hline Length of screw & 60 mm \\ \hline Minor diameter of screw & 6 mm \\ \hline Pitch of screw & 8 mm \\ \hline Screw flight width & 1 mm \\ \hline Channel width of screw & 7 mm \\ \hline Screw flight height & 1 mm \\ \hline Pin height & 1 mm \\ \hline Input diameter & 2.5 mm \\ \hline Outlet diameter & 1 mm \\ \hline Contraction angle & 65\({}^{\circ}\) \\ \hline Major diameter of contraction channel & 9 mm \\ \hline Length of forming channel & 6 mm \\ \hline \end{tabular}
\end{table}
Table 2: Structural parameter of mixing chamber mesh model.  
the inlet, resulting in a value of 2.4 mm\({}^{3}\)/s. To simulate the rotating flow field in the mixing zone of the pin-type screw, the transient simulation employed the Rotating Reference Frame (RRF) method. This involved designating Fluid-1 and Fluid-2 (as depicted in Figure 1b) as static and dynamically rotational domains, respectively, with both domains being connected through an interface. In addition, the screw wall was assigned a no-slip boundary condition and was configured as a moving wall to simulate the adjustable operating speed of the screw, ranging from 10 to 50 rpm.  
In this research, we employed two readily available types of calcium carbonate-based toothpaste, designated as Material A (white color) and Material B (green color), with similar non-Newtonian rheological properties to simulate the mixing extrusion process. The densities of Materials A and B were determined to be 1120 kg/m\({}^{3}\) and 1285 kg/m\({}^{3}\), respectively, using the specific gravity method. Both pastes exhibit shear-thinning rheological characteristics. Therefore, the study employed the power-law function, as shown in Equation (3), which is a commonly utilized tool in non-Newtonian fluid modeling, to describe their rheological properties. The rheological parameters for both pastes were assessed using a rotational rheometer (MCR 302, [PERSON], Graz, Austria) at room temperature. The test results are depicted in Figure 2, and the corresponding rheological parameters were derived through the power-law function, as presented in Table 3. Finally, in the confines of a narrow mixing channel, in conjunction with the aforementioned pin-type screw speed, the viscous behavior of the mixing paste was characterized as laminar flow.  
\[\tau=K(\dot{y})^{n} \tag{3}\]  
where is the shear stress with unit of pa, \(K\) represents the flow index with a unit of pa \(\cdot\) s\({}^{n}\), and \(n\) symbolizes the dimensionless flow behavior index, which is less than 1 for shear thinning paste. \(\dot{y}\) is the shear rate with a unit of s\({}^{-1}\).