*2.1. Physical Model*

The physical model for particle flow around a single tube is presented in Figure 1. As shown in Figure 1a, the spherical particles flow around a single tube from the top to bottom in the particle flow channel and exchange heat with the tube wall. The dimensions of the particle flow channel are *L* (length) × *W* (width) × *H* (height), and the particle diameter (*d*p) is mainly fixed at 1.72 mm. The tube is located in the channel center. All the vertical channel walls are adiabatic, and the temperature of tube wall is kept at *T*tube = 287 K. Particles are generated at the entrance of the flow channel, and the particle inlet temperature is kept at *T*in = 773 K. All particles are driven by the gravity and have a dense filling in the flow channel. The particle velocity in the channel is controlled at the outlet [5], and the particle outlet velocity is kept constant along the vertical direction (*z*). As shown in Figure 1b, five different tubes are adopted for the study, including circular tube, elliptical tube, flat elliptical tube, elliptical-flat elliptical tube and flat elliptical-elliptical tube, where the perimeters of different tube cross sections are kept the same. Typical geometric parameters of the particle flow channel and tubes are presented in Table 1. In the present study, the particle flow around tube wall is divided into three zones, including the upstream region of the tube (Zone 1), the side region of the tube (Zone 2), and the downstream region of the tube (Zone 3). The geometric divisions of different particle zones are also presented in Figure 1b.

**Figure 1.** Physical model and cross section zones for particle flow around different tubes: (**a**) Physical model; (**b**) Cross section zones for different tubes.

**Table 1.** Typical geometric parameters of particle flow channel and tubes.

