## 3 Experimental Results
### Steady-State Tests
The particle superficial velocity inside the tubes is calculated as  
Figure 8:0  
where \(N\) is the number of tubes, \(\rho_{p}\) is the material density of particles, and \(A_{i}\) is the cross-sectional area of each tube. Table 7 shows the results of the conducted tests. The results include the heat transfer rates, the LMTD, the heat exchanger effectiveness, the particle velocity inside each tube, and the overall heat transfer coefficient. It should be noted that the overall heat transfer coefficient was calculated based on the heat transfer area of the particle side. The propagation of the bias error of the measured variables into the calculated quantities was estimated using the Engineering Equation Solver (EES) software package as  
Figure 8:1  
where \(u_{b}\) refers to the bias uncertainty, \(Y\) is the dependent variable, and \(X_{i}\) are the independent variables. Figure 10 shows a graphical representation of the variation of the overall heat transfer coefficient and effectiveness with the particle mass flow rate. It can be seen that as the particle mass flow rate increases, the \(U\) and \(\varepsilon\) increase. By observing the amount of collected heat (\(\dot{Q}_{p}\)) compared to the supplied heat (\(\dot{Q}_{a}\)), it is noticeable that the \(\dot{Q}_{p}\) approaches \(\dot{Q}_{a}\) at higher flow rates. This might be the main reason of such increase in the values of \(U\) and \(\varepsilon\). Moreover, as can be seen from Figure 10, when the mass flow rate increases, both \(U\) and \(\varepsilon\) approach a plateau, i.e., a constant value, since \(\dot{Q}_{p}\) gets almost equal to \(\dot{Q}_{a}\). The temperature distributions of air and particles across the heat exchanger are shown in Figure 8:2 It should be noted that the particles temperatures, other than the inlet and outlet, have been estimated by applying a heat balance between air and particles. It is evident from Figure 11 that more than half of the heat transfer occurred at the lower section of the heat exchanger where the temperature difference is the largest. It could also be noted that these results are based on test conditions under a particle heating scenario, and it could be better in a particle cooling situation especially if the particle flow rate is balanced with the air flow rate so that the approach temperature is maximized throughout the exchanger instead of having a pinch point at the top of the exchanger, thereby, enhancing the overall heat transfer coefficient. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Test** & \(\dot{Q}_{a}\) & \(\dot{Q}_{p}\) & \(LMTD\) & \(U\) & \(\varepsilon\) & \(V_{p}\) \\  & [**kW**] & [**kW**] & [\({}^{\circ}\)C] & [**W/m\({}^{2}\)**-\({}^{\circ}\)C] & [\({}^{\circ}\)C] & [**mm/s**] \\ \hline
1 & 32.06 \(\pm\) 0.34 & 23.63 \(\pm\) 0.43 & 19.45 \(\pm\) 0.47 & 86.28 \(\pm\) 2.24 & 0.728 \(\pm\) 0.005 & 4.4 \\
2 & 31.44 \(\pm\) 0.34 & 26.80 \(\pm\) 0.54 & 16.93 \(\pm\) 0.52 & 103.66 \(\pm\) 3.31 & 0.812 \(\pm\) 0.007 & 5.7 \\
3 & 31.41 \(\pm\) 0.34 & 28.80 \(\pm\) 0.66 & 15.93 \(\pm\) 0.57 & 113.96 \(\pm\) 4.16 & 0.861 \(\pm\) 0.008 & 6.8 \\
4 & 31.48 \(\pm\) 0.34 & 29.91 \(\pm\) 0.76 & 15.13 \(\pm\) 0.65 & 122.33 \(\pm\) 5.19 & 0.896 \(\pm\) 0.009 & 7.9 \\
5 & 32.01 \(\pm\) 0.33 & 31.31 \(\pm\) 0.84 & 14.49 \(\pm\) 0.77 & 131.70 \(\pm\) 6.81 & 0.932 \(\pm\) 0.010 & 8.8 \\
6 & 29.16 \(\pm\) 0.33 & 29.25 \(\pm\) 0.95 & 13.03 \(\pm\) 0.83 & 135.11 \(\pm\) 8.32 & 0.950 \(\pm\) 0.013 & 10.0 \\
7 & 28.36 \(\pm\) 0.33 & 28.80 \(\pm\) 1.05 & 12.51 \(\pm\) 0.90 & 137.72 \(\pm\) 9.52 & 0.962 \(\pm\) 0.015 & 11.1 \\
8 & 28.16 \(\pm\) 0.33 & 28.77 \(\pm\) 1.16 & 12.33 \(\pm\) 0.91 & 139.18 \(\pm\) 9.87 & 0.965 \(\pm\) 0.016 & 12.2 \\
9 & 28.29 \(\pm\) 0.33 & 28.53 \(\pm\) 1.26 & 11.27 \(\pm\) 1.18 & 151.97 \(\pm\) 15.20 & 0.976 \(\pm\) 0.018 & 13.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results of the steady-state tests. Figure 11: Temperature distributions of air and particles across the heat exchanger, Test no. 3.