## 3 Results and Discussions
### Pressure Loss and Energy Dissipation Due to the Bend
For example, [PERSON] and [PERSON] gave the expressions of the friction coefficient: \(\frac{f_{c}}{f_{p}}=\left[1-\left\{1-\left(\frac{11.6}{D\varepsilon}\right)^{0.45}\right\}^{2.222}\right]^{-1}\) and \(\frac{f_{c}}{f_{p}}=1+0.033(logDe)^{4.0}\) for Newtonian fluids [46]. [PERSON] concluded \(\frac{f_{c}}{f_{p}}=1+K_{1}(logDe)^{K_{2}}\) for power-law fluids [41]. Here, \(f_{p}\) is the friction coefficient of the straight pipe with a same length. \(De=Re\sqrt{R/R_{c}}\), in which \(R_{c}\) is the curvature radius. However, in the present study, we obtained the converse relation between \(f_{c}\) and \(De\), as observed more clearly in the plot of Figure 14. We speculate that this is due to two effects: one is the special rheology of our fluid, particularly the plasticity-yield stress; the other is the not fully developed flow in the strong curvature. At the lower flow rate, the yield stress fluid flows with an almost unyielded plug at the central part of the pipe, resulting in a nearly infinitely large viscosity [45]. Nonetheless, there is still a weak Dean vortex secondary flow inside the plug [47]. As the flow rate increases, the width of the plug shrinks, and in the places where the plug locates at the lower flow rate, the infinite viscosity sharply reduces to an intermediate value. In contrast, the magnitude of the squared vorticity increases by a smaller magnitude. Remember that in our case, the vorticity always weighs more on the inner-wall side of the channel (Figures 8 and 13). Compared to the outer-wall-weighted vorticity in the fully developed flow in a smaller curvature, the vorticity in our case is relatively smaller and hindered due to the outer-wall pointed fluid velocity. Therefore, the decrease in the viscosity is greater than the increase in the squared vorticity, and the variation of the viscous energy dissipation rate \(\varepsilon\sim\eta|\overset{\rightarrow}{\omega}|^{2}\) with the flow rate follows that of the viscosity, which decreases as the flow rate increases. At least, this is true in the range of the flow rates in the present study. It is uncertain to predict if the relation between the energy loss and the flow rate would return to the positive correlation at even higher flow rates, at which the decrease of the viscosity might be less than the increase in the squared vorticity magnitude. Because in engineering, the coal slurries are not conveyed at higher rates, that range is not discussed here. Moreover, the turbulence begins to arise at the higher rates, and the energy loss would be not only due to the viscosity, which would make the problem more complicated. We also plot the friction coefficient \(\frac{f_{c}}{f_{p}}\) as a function of the Nusselt number \(Nu\) that represents the ratio of the convective heat transfer to the conductive transfer, shown in Figure 14. When calculating \(Nu\), similar to \(\varepsilon\), we selected the horizontal radius near the \begin{table}
\begin{tabular}{c c c} \hline \(\xi\) & **0.5 m/s** & **1 m/s** \\ \hline
300/300 (K) & 39.29 & 11.18 \\
283/300 (K) & 44.18 & 13.03 \\
273/300 (k) & 47.23 & 14.18 \\ \hline \end{tabular}
\end{table} 
Table 1: Pressure loss coefficient in different cases. outer-wall side of the cross section at the middle of the U-shaped channel as representative of the whole bend part:  
\[Nu=\frac{|\rho C_{p}\Delta T\overrightarrow{u}|}{|k\
abla T|}. \tag{9}\]  
\(\overrightarrow{u}\) is the average velocity in the main heat transfer direction (positive x) outside the temperature-plug region (for example, Figures 6 and 11). \(\Delta T\) is the temperature difference between the fluid body and the pipe wall. \(\
abla T\) is the temperature gradient in x direction. As shown in Figure 14, \(\frac{f_{c}}{f_{p}}\) again decreases as \(Nu\) increases, since \(Nu\) is similar to \(De\), which is proportional to the flow rate.