*2.3. Model Validations*

In order to validate the reliability and accuracy of present computational model and methods, the heat transfer coefficients obtained with DEM simulations are compared with the experimental results of Liu et al. [1]. The variations of heat transfer coefficient (*h*) for the particle flow through a tube row are presented in Figure 5. It is found that the variation trend of the heat transfer coefficients obtained by the DEM simulation is similar to the experimental results. The maximum deviation between the simulation and experimental results is 16.3%, and the average deviation is 11.8%. In the present simulation, the particle is simplified as sphere and the particle outlet velocity is kept constant for each certain case, while the particles used in the experiments [1] were irregular and the distribution of particle outlet velocity were not uniform. Therefore, the deviations exist between present simulation results and experimental results [1].

**Figure 5.** Comparison of heat transfer coefficients of simulation results and experimental measurement.

## **3. Results and Discussion**

The velocity vector distributions of particle flowing around different tubes at a certain time are presented in Figure 6. This shows that the effect of tube shape on the particle flow at both upstream (Zone 1) and downstream (Zone 3) regions of different tubes is remarkable. It is noted that the particle velocity at the upstream region (Zone 1) of different tubes is very small and a particle stagnation zone is formed. Meanwhile, it is noted that, at the downstream region (Zone 3) of different tubes, a particle cavity zone is formed, where particles are almost untouched on the tube wall. For particle flow around different tubes, it is found that both the stagnation and cavity zones for the circular tube are the largest, and they are the smallest for the elliptical tube. This may indicate that, for the elliptical tube, particles can renew faster at the upstream region (Zone 1) of the tube and the tube wall can be touched with more particles at the downstream region (Zone 3) of the tube. Therefore, the heat transfer of particle flow around an elliptical tube would be better. For the elliptical-flat elliptical tube and flat elliptical-elliptical tube, the characteristics of the stagnation zone and cavity zone are consistent with the tube with the same shapes at upstream region and downstream region.

**Figure 6.** Velocity vector distributions of particle flow around different tubes (*v*out = 0.5 mm/s, *d*p = 1.72 mm): (**a**) Circular tube; (**b**) Flat elliptical tube; (**c**) Elliptical tube; (**d**) Elliptical-Flat elliptical tube; (**e**) Flat elliptical-Elliptical tube.

The variations of time-averaged particle contact number with *v*out for different tubes are presented in Figure 7. The contacting situation between particles and tube wall is well reflected by using time-averaged particle contacting number on the unit area of tube surface. In the present study, the tube wall is considered to be touched by particles as the gas film surrounding the particle surface (δ = 0.1 *d*p) touches the tube wall. As shown in Figure 7, when the particle outlet velocity (*v*out) varies from 0.5 mm/s to 8 mm/s, the particle contact numbers at different zones of all the tubes change very little with *v*out. The particle contact numbers at Zone 1 of all the tubes are the highest and they are the lowest at Zone 3. At Zone 1 and Zone 2, the particle contact numbers of elliptical tube, flat elliptical tube, elliptical-flat elliptical tube and flat elliptical-elliptical tube are close to each other. At Zone 1, the

particle contact number of circular tube is a little higher than that of the elliptical tube, flat elliptical tube, elliptical-flat elliptical tube and flat elliptical-elliptical tube, while at Zone 2, the particle contact number of circular tube is a little lower. As for Zone 3, big differences of particle contact number existed for different tubes. As compared with the circular tube, the particle contact number at Zone 3 of the elliptical tube and flat elliptical tube can increase by 112.3% and 53.5%. Due to the same tube shape at the downstream region, the particle contact numbers at Zone 3 of the elliptical-flat elliptical tube and the flat elliptical-elliptical tube are quite close to those of the flat elliptical tube and elliptical tube.

**Figure 7.** Variations of time-averaged particle contact number with *v*out for different tubes (*d*p = 1.72 mm).

In order to compare the particle renewal situations at the stagnation zone (Zone 1) of particle flow around different tubes, the variations of particle contact time with *v*out at Zone 1 of different tubes are analyzed, as presented in Figure 8. The contacting time of particles with tube wall for unit length of tube is a cumulative variable. The contacting time of particles is accumulated when the tube wall is touched by the gas film surrounding the particle, which is the average result for particles contacting the tube at each time step. As shown in Figure 8, it is found that, as particle outlet velocity (*v*out) increases from 0.5 mm/s to 8 mm/s, the particle contact time with the tube wall at Zone 1 of different tubes decreases rapidly, and the particle flow at the stagnation zone of all the tubes is accelerated. Furthermore, it is found that under the same particle outlet velocity (*v*out), the particle contact time at Zone 1 of the circular tube is highest and it is the lowest for the elliptical tube, which means the particle flow renewal situation at the stagnation zone (Zone 1) of elliptical tube is the best. Due to the same tube shape at Zone 1, the particle contact times at Zone 1 of the elliptical-flat elliptical tube and the flat elliptical-elliptical tube are almost the same to those of the elliptical tube and flat elliptical tube. As compared with the circular tube, when *v*out = 0.5 mm/s, the particle contact time at Zone 1 of the elliptical tube and flat elliptical tube can decrease by 39% and 21%.

**Figure 8.** Variations of particle contact time with *v*out at Zone 1 of different tubes (*d*p = 1.72 mm).

The variations of local heat transfer coefficients of particle flow with *v*out for different tubes are presented in Figure 9. The local heat transfer coefficient of particle flow at Zone 1 of elliptical tube is a little higher than that of the flat elliptical tube, while the local heat transfer coefficient of circular tube is the lowest, as shown in Figure 9a. At Zone 1, the difference of particle contact number between different tubes is small (see Figure 7), while the particle contact time of the circular tube is obviously higher (see Figure 8); therefore, the local heat transfer of particle flow at Zone 1 on the circular tube is lower. When the particle outlet velocity (*v*out) changes from 0.5 mm/s to 8 mm/s, as compared with the circular tube, the local heat transfer coefficient of particle flow at Zone 1 on the elliptical tube and flat elliptical tube can increase by 19.7% and 16.9% on average, respectively. As Zone 2 is concerned, it shows that the local heat transfer coefficients of particle flow on different tubes are quite similar, which may indicate that the particle contact situation at the side region of particle flow on different tubes should be similar, as shown in Figure 9b For the circular tube, although the particle contact number at Zone 2 is the lowest when compared with other tubes, there are more new particles contacting with the tube wall at Zone 2 due to the highest contact time at Zone 1, which would lead to similar heat transfer coefficients to those of other tubes. At Zone 3, it shows that the difference of local heat transfer coefficients of particle flow for different tubes is relatively large, as shown in Figure 9b. The local heat transfer coefficient of particle flow at Zone 3 for the elliptical tube is the highest and it is the lowest for the circular tube. The cavity zone formed at downstream region of the elliptical tube is the smallest (see Figure 6) and particle contact number at Zone 3 for the elliptical tube is the highest (see Figure 7); therefore, the local heat transfer of particle flow at Zone 3 for the elliptical tube is the highest. When the particle outlet velocity (*v*out) changes from 0.5 mm/s to 8 mm/s, as compared with the circular tube, the local heat transfer coefficient of particle flow at Zone 3 for the elliptical tube and flat elliptical tube can increase by 210.0% and 130.4% on average, respectively. Finally, it is found that the local heat transfer coefficients of the elliptical-flat elliptical tube and the flat elliptical-elliptical tube at Zone 1 and Zone 3 are almost the same to those of the tubes with the same shape at these zones.

The variations of heat transfer coefficients of particles flow with *v*out for different tubes are presented in Figure 10. It shows that as the particle outlet velocity (*v*out) increases, the heat transfer coefficient of particle for all the tubes increases gradually. The heat transfer coefficient of particle flow around elliptical tube is the highest and it is the lowest for the circular tube. As the particle outlet velocity (*v*out) changes from 0.5 mm/s to 8 mm/s, when compared with the circular tube, the heat transfer coefficient of particle flow for the elliptical tube and flat elliptical tube can increase by 20.3% and 15.0% on average. As compared with the flat elliptical tube, the elliptical-flat elliptical tube would enhance the heat transfer at the upstream region of the tube, and the flat elliptical-elliptical tube would improve the heat transfer at the downstream region of the tube, as shown in Figure 9. The heat transfer coefficient of the flat elliptical-elliptical tube is higher than that of the elliptical-flat

elliptical tube, which shows that the optimization of particle flow and heat transfer at the downstream region of the tube can improve the overall heat transfer performance more efficiently. The heat transfer coefficients of these two shapes of tubes are higher than that of the flat elliptical tube but lower than that of the elliptical tube.

**Figure 9.** Variations of local heat transfer coefficients of particle flow with *v*out for different tubes (*d*p = 1.72 mm): (**a**) Zone 1; (**b**) Zone 2 and Zone 3.

**Figure 10.** Variations of heat transfer coefficients of particle flow with *v*out for different tubes (*d*p = 1.72 mm).

Finally, the variations of heat transfer coefficients for the particle flow around an elliptical tube with different particle diameter (*d*p) are presented in Figure 11. It shows that, as the particle diameter decreases from 2.5 mm to 1 mm, the heat transfer coefficients of particle flow around the elliptical tube increase, which is consistent with the experimental results of Liu et al. [1]. When *v*out = 2 mm/s, as the particle diameter decreases from 2.5 mm to 1 mm, the local heat transfer coefficients of particle flow at the upstream region (E\_Zone 1), side region (E\_Zone 2) and downstream region (E\_Zone 3) of an elliptical tube can increase by 61.1%, 59.4% and 102.6%, and the heat transfer coefficient around the elliptical tube can increase by 63.6% on average.

**Figure 11.** Variations of heat transfer coefficients of particle flow around an elliptical tube with different diameters (*v*out = 2 mm/s).
