## 3 Numerical Simulation
### The Influence of Inlet and Outlet Conditions and Structural Parameters
To sum up, in the structural design of the specified dimensional parameters within a certain range, in order to achieve uniform flow at each channel and reduce the value of the non-uniformity flow coefficient (\(\Phi\)), the inlet diameter (\(D\)) and channel length (\(L\)) should be increased as much as possible, and the channel diameter (\(d\)) and channel spacing (\(S\)) should be reduced. In all cases, the flow at channel number 1 was the largest, and the flow at channel number 5 was the smallest. The results were different from those of Newtonian fluids, where the minimum pressure was at the channel near the inlet and the maximum pressure was at the end of the cavity region due to the jet effect at the inlet, resulting in minimum flow in the channels at the inlet and maximum flow in the end channels [18]. Pressure in the cavity region and parallel channels is the main factor leading to different flow distribution results. The fluid pressure in an entrance header will change for two reasons [28; 29]: (a) because of wall friction in the straight sections between adjacent side outlets, the fluid pressure will fall in the flow direction; (b) the structural dimensions of parallel channels determine the pressure required for fluid to enter the channel, and fluid always tends to flow preferentially to areas with low flow resistance. When the flow resistance of the channel increases, the pressure at the entrance header of the channel increases, which will promote flow into the channel behind. This explains the effect of changes in structural parameters on the flow distribution. To investigate the strength of the effect of the four structural parameters, \(D\), \(d\), \(S\), and \(L\), on \(\Phi\), we designed the orthogonal test as demonstrated in Table 2, exploring the significance of the influence of the structural parameters. There were four factors to be investigated and each factor had four levels. The L16 (\(4^{4}\)) orthogonal table was selected to schedule the numerical simulation. Table 3 shows the parameters \(SS\), \(df\), \(MS\), and \(F\), which were obtained by calculating the numerical simulation result \(x_{i}\) (in the study, \(x=\Phi\)). \(SS\) is the sum of squares of deviations, \(SS=\sum(x_{i}-\overline{x})^{2}\). \(df\) is the degree of freedom, \(df=k-1\), where \(k\) is the number of factors. \(MS\) is the mean square, \(MS=\frac{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}}{n}\). \(F\) is the F-test, \(F\sim F(k-1,\,n-k)\), where \(n\) is the number of levels. Sixteen groups of simulation experiments were designed and the results were analyzed for variance, as shown in Table 3. The results revealed that the strength of effect on the \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Factor/Level**} & \(D\) & \(d\) & \(S\) & \(L\) \\ \cline{2-5}  & **Inlet Diameter** & **Channel Diameter** & **Channel Spacing** & **Channel Length** \\ \hline
1 & 3 & 0.8 & 2 & 2 \\
2 & 4 & 1.0 & 3 & 4 \\
3 & 5 & 1.2 & 4 & 6 \\
4 & 6 & 1.4 & 5 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Orthogonal parameter table. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Factor**} & \(SS\) & \(df\) & \(MS\) & \(F\) \\ \hline \(D\) & 0.663 & 3 & 0.221 & 2.027 \\ \(d\) & 0.405 & 3 & 0.135 & 1.239 \\ \(S\) & 0.210 & 3 & 0.070 & 0.643 \\ \(L\) & 0.285 & 3 & 0.095 & 0.870 \\ error & 0.327 & 3 & 0.109 & \\ \hlinenon-uniformity flow coefficient (\(\Phi\)) in this experiment was: inlet diameter (\(D\)) > channel diameter (\(d\)) > channel length (\(L\)) \(\approx\) channel spacing (\(S\)).