**1. Introduction**

In the recent years, the world's energy and environmental problems have become more and more prominent, which makes it urgent to improve energy efficiency and develop sustainable energy techniques. Moving bed heat exchanger (MBHE), with the advantages of low cost, clean energy and wide particle size adaptation, is gradually applied in many high temperature waste heat recovery industries [1], such as slag [2], cement [3], coke [4] and concentrated solar power (CSP) [5]. For the MBHE, the granular flow inside is driven by the gravity, and the heat transfer is relatively lower [6] as compared with packed bed [7] or fluidized bed [8]. Therefore, it is important to improve the heat transfer of particle flow in the MBHE.

The type of heat transfer element in the MBHE may include horizontal tubes, vertical tubes and parallel plates. For the MBHE with vertical tubes [9] or parallel plates inside [10], the solid particles move vertically along the tubes or palates without fully mixing, which would limit the heat transfer of particle flow along the tubes or plates in the MBHE. However, the particle flow around a horizontal tube should be disordered and fully mixed [11], and the heat transfer around the horizontal tube should be better. Many researchers have studied the effects of particle material, particle size, particle velocity, finned tube and layout of horizontal tubes in the MBHE. Qoaider et al. [12] found that the heat transfer performance of MBHE with mixed particles of 50% sand and 50% basalt particles were 30% better than that of MBHE with 100% sand particles by experiment. Increasing particle velocity can significantly improve the heat transfer coefficient, Al-Ansary et al. [13] found that the heat transfer coefficient would increase from 80 W/(m2·K) to 160 W/(m2·K) as the sand velocity increased from 1 mm/s to 3 mm/s. Nguyen et al. [14] found that, when the particle velocity changes from 3 mm/s to 10 mm/s, finned tubes can enhance heat transfer as compared with the smooth tubes. Liu et al. [1] analyzed the effects of particle diameter, particle velocity and tube arrangement on the performances of heat exchanger by experiment. They found that the heat transfer performance is better in staggered heat exchanger and the heat transfer coefficient increased as the particle size decreased or as the particle velocity increased. However, when the particles flowed around the horizontal tube, a stagnation zone would form at the upstream region of the tube and a cavity zone would form at the downstream region of the tube [15], which would seriously affect the flow and heat transfer of particles [16]. The particles in the stagnation zone move slowly or even remain stationary, while the particles in the cavity zone are almost untouched with the tube wall. Bartsch and Zunft studied the dense granular flow around horizontal tubes by CFD model [17], discrete element method (DEM) model and experiment [18]. The simulation results could well capture both the stagnation zone and cavity zone at the upstream and downstream of the tube, which was consistent with the experimental results. The heat transfer coefficient in these two zones was the lowest, and the heat transfer mainly occurred on the side region of the tube wall [19]. Therefore, it is an effective way to improve heat transfer performance of particle flow in the MBHE by reducing both the stagnation zone and cavity zone around the tube, such as using vibration or profiled tubes. Guo et al. [20] found that vibration could accelerate the renewal of particles at the upstream region of the tube, strengthen particle contact at the downstream region of the tube and increase the heat transfer coefficient of particle flow around the tube. However, the vibration may also significantly increase the wear rate of the tube. As for profiled tubes, Morris et al. [21] developed a continuum model for flowing particles heat transfer and studied the particle flow in the hexagonal tube array in the solar receiver [22], where the particle inlet velocity was up to 1 m/s and the thermal performance was obviously affected by the particle size. They found that the overall heat transfer increased as particle mass flow rate increased. Furthermore, Takeuchi [11] experimentally studied the heat transfer of particle flow around a circular tube, an elliptical tube and a lenticular tube. The study showed that the effect of tube shape on the heat transfer performance of particle flow around the tube was remarkable, and the performance of lenticular tube was the best.

According to the above studies, it should be noted that the effect of tube shape on the heat transfer of particle flow around tube is remarkable. In order to further understand the mechanism of this effect and improve the heat transfer performance of particle flow in the MBHE, the heat transfer of gravity-driven dense particle flow around different tubes was numerically investigated in the present study, including circular tube, elliptical tube, flat elliptical tube and the combination of elliptical tube and flat elliptical tube. A variety of factors, such as velocity vector, particle contact number, particle contact time and heat transfer coefficient of particle flow at different particle zones around the tube are carefully analyzed. Meanwhile, the effect of particle diameter on heat transfer is discussed. The results of the present study would be important and beneficial for the optimal design of MBHEs.
