**4. Conclusions**

In the present study, the heat transfer of gravity-driven dense particle flow around different tubes was numerically investigated with discrete element method (DEM), 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 main findings are as follows:

(1) The effect of tube shape on the particle flow at both upstream (Zone 1) and downstream (Zone 3) regions of different tubes is remarkable. A particle stagnation zone and particle cavity zone are formed at the upstream and downstream regions of all the tubes. Both the stagnation and cavity zones for the circular tube are the largest, and they are the smallest for the elliptical tube. Furthermore, the effect of tube shape on the particle contact number and particle contact time at different zones of particle flow for different tubes is quite different. As for the particle contact number, big differences existed at Zone 3 for different tubes. As compared with the circular tube, when *d*<sup>p</sup> = 1.72 mm, the particle contact number at Zone 3 of the elliptical tube and flat elliptical tube can increase by 112.3% and 53.5%, respectively. When *v*out = 0.5 mm/s and *d*p = 1.72 mm, the particle contact time at Zone 1 of the elliptical tube and flat elliptical tube can decrease by 39% and 21% as compared with the circular tube.

(2) The heat transfer performances of particle flow around different tubes are different. It is found that, at Zone 3, the difference of local heat transfer coefficients of particle flow for different tubes is quite large. When the particle outlet velocity (*v*out) changes from 0.5 mm/s to 8 mm/s at *d*p = 1.72 mm, 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. Furthermore, as the particle outlet velocity (*v*out) changes from 0.5 mm/s to 8 mm/s at *d*<sup>p</sup> = 1.72 mm, 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, respectively. With proper design of the downstream shape of the tube, the overall heat transfer performance can be improved more efficiently. As the particle diameter (*d*p) decreases from 2.5 mm to 1 mm at *v*out = 2 mm/s, the heat transfer coefficient of particle flow around an elliptical tube can increase by 63.6% on average.

The present results show that the heat transfer performance of particle flow around the elliptical tube is better than that of the circular tube and flat elliptical tube. Therefore, the elliptical tube would have a better application prospect in the MBHE. Furthermore, the heat transfer performance inside the tube and the economic analysis for the elliptical tube should be performed in the future.

**Author Contributions:** X.T. performed the calculations and wrote the paper; J.Y. and Q.W. supervised the work and revised the paper; Z.G. and B.S. contributed to revising the paper. All authors contributed to this work. All authors have read and agreed to the published version of the manuscript.

**Funding:** The financial support was provided by National Basic Research Program of China (No. 2017YFB0603500) and China Scholarship Council Fellowship (No. 201806285048).

**Acknowledgments:** We would like to acknowledge financial supports for this work provided by National Basic Research Program of China (No. 2017YFB0603500) and China Scholarship Council Fellowship (No. 201806285048).

**Conflicts of Interest:** The authors declare no conflict of interest.
