# Influence of Fractal Disc Filter Flow Channel Parameters on Filtration Performance
## 2 Materials and Methods
### The Cross Section Shape  
In order to increase the amount of sediment and intercept sediment of different particle sizes, the entrance section of the laminated flow passage is larger than the exit section. The inner circle shape of the exit section determines the filtration accuracy of the lamination. Each section shape of the laminated flow passage is an isosceles triangle of different shapes, and the size of the base and height of the triangle determine the shape of the isosceles triangle. Therefore, the flow passage of triangles with different internal sections is simulated without changing the filtering accuracy or taper.  
The bottom length of the inner section is 0.24 mm minimum and 0.64 mm maximum, and the gradient is 0.08 mm. Before optimization, the bottom side length of the fractal disc filter was 0.36 mm. Observe Table 1 and Figure 4 for the specific parameters and dimensions of the triangle in the inner section.  
The maximum cross-sectional area of a single flow passage does not correspond to the cross-sectional area of the whole filter element. With the decrease in bottom side length, the triangular area of a single inner section decreases first and then increases. The minimum area is 0.050 mm\({}^{2}\), and the corresponding bottom side length is 0.32 mm.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**External Section (mm)**} & \multicolumn{2}{c}{**Internal Section (mm)**} & \multicolumn{2}{c}{**1–1\({}^{\prime}\) Section (mm\({}^{2}\))**} \\ \hline \multirow{2}{*}{**External Base**} & \multirow{2}{*}{**External Height**} & \multirow{2}{*}{**Internal Base**} & \multirow{2}{*}{**Internal, Height**} & \multirow{2}{*}{**Area of Single Section**} & \multicolumn{2}{c}{**Area of the Whole**} \\  & & & & & & **Filter Element** \\ \hline
0.86 & 0.26 & 0.64 & 0.20 & 0.076 & 20,659.33 \\
0.74 & 0.27 & 0.56 & 0.20 & 0.067 & 19,425.07 \\
0.62 & 0.28 & 0.48 & 0.21 & 0.060 & 20,055.80 \\
0.54 & 0.30 & 0.40 & 0.23 & 0.055 & 20,558.50 \\
0.44 & 0.32 & 0.36 & 0.24 & 0.050 & 19,619.86 \\
0.42 & 0.35 & 0.32 & 0.26 & 0.050 & 19,926.86 \\
0.32 & 0.55 & 0.24 & 0.41 & 0.059 & 19,421.28 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dimensions of the triangle of the section.  
Figure 4: Triangular shape of the cross-section.  
The maximum area is 0.076 mm\({}^{2}\), and the corresponding bottom side length is 0.64 mm. The minimum cross-section area of the whole filter element is 19,421.28 mm\({}^{2}\), and the corresponding bottom surface length is 0.24 mm. The maximum value is 20,558.50 mm\({}^{2}\), and the corresponding bottom side length is 0.40 mm.  
On the premise that the diameter of the tangent circle remains unchanged, the height of the triangle increases continuously with the decrease in base length. The triangular base of a single flow passage section decreases at the cost of increasing the section height. The larger the section height is, the larger the lamellar thickness is. And the height of the filter element is certain; the greater the thickness of the lamination, the smaller the number of laminations. The formula is as follows:  
\[S=2\times S_{1}\times n\times m \tag{1}\]  
where \(S_{1}\) is the section area of a single flow passage; the unit is mm\({}^{2}\); \(n\) is the number of flow passages on the stack; \(m\) is the number of stacked pieces installed on the filter element.