## 5 Conclusions  
In this work, a dynamic coarse grain DEM-CFD method for gas-solid fluidized beds is presented by considering the particle-group crushing and polymerization. Compared with the original coarse grain DEM-CFD method, the coarse graining system is updated at every simulation step. It allows a dynamic adjustment of the number and components of the particle-groups. Big particle-groups can be crushed into several small particle-groups or original particles. Conversely, several small particle-groups or original particles can be polymerized into a big particle-group. That is, the number and size of the particle-groups are not invariable but change dynamically during the simulation. The size of each particle-group relies on its equivalent diameter, which is measured by an EMMS type model. If the actual diameter of a particle-group is greater than its equivalent diameter, then crushing of this particle-group takes place. As to the polymerization of two colliding particle-groups, their velocity difference after collision is chosen as a criterion. If the velocities of the two particle-groups after collision are fairly close, then the two particle-groups can be regarded as one new particle-group. Due to the diversity of particle-group sizes, the resulting particle-group heterogeneity as well as the flow heterogeneity induced by the particle cluster formation should be considered. For this reason, the EMMS drag force model is adopted in this work. A non-staggered FVM and the SIMPLER algorithm  
Figure 14: Snapshots of particle distribution obtained by the PCPM at different coarse grain ratio.  
(**a**) \(t=0.40\)s; (**b**) \(t=6.44\)s. (**a**) \(l=1\), \(l=2\), \(l=3\). (**b**) \(l=1\), \(l=2\), \(l=3\).  
Figure 13: Snapshots of particle distribution obtained by the PCPM for a two dimensional fluidized bed at the later stage: the phenomenon of bubble merging along the transverse direction. (**a**) 7.72 s. (**b**) 7.80 s. (**c**) 7.88 s. (**d**) 7.96 s. (**e**) 8.02 s. (**f**) 8.10 s. (**g**) 8.18 s. (**h**) 8.26 s. (**i**) 8.32 s. (**j**) 8.40 s. (**k**) 8.48 s.  
are used to perform the simulations. Flow patterns and pressure drop predicted by the current dynamic coarse grain DEM-CFD method agree better with the experimental results than the original method. For fluidized beds involving more particles, the new method can reveal the complex fluidization pattern evolution in dense particle flow systems. The simulation results also suggest that excessive coarse grain will make the coarse grain system severely deviate from the original system and proper coarse grain will greatly reduce the simulation time. In conclusion, the highlights of this work can be summarized as follows:  
* A dynamic particle-group crushing and polymerization procedure is considered in the coarse grain DEM-CFD model.
* The dynamically coarse graining is based on an EMMS type model.
* Heterogeneities are considered in the coarse graining systems by using the EMMS drag force model.
* The method is validated by comparing the simulation results with the experimental ones.  
Conceptualization, X.W. and J.O.; methodology, X.W. and J.O.; software, K.C. and T.K.; formal analysis, K.C. and T.K.; writing--Original draft preparation, X.W.; writing--Review and editing, J.O.; funding acquisition, X.W. and J.O. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant No. 11671321, 11971387), the Major Research Plan of the National Natural Science Foundation of China (Grant No. 91434201) and the Young Talent Fund of the University Association for Science and Technology in Shaanxi, China.  
The authors would like to thank the anonymous referees, whose valuable comments helped to improve this paper.  
The authors declare no conflict of interest.