## 1 Introduction  
Inside a gas-solid fluidized bed, particles driven by a gas flow exhibit complex and intriguing flow patterns. The mutual interaction between the discrete particles and continuum gas brings desirable features for many industrial applications, which include rapid heat and mass transfer, good mixing of solids and fast chemical reaction [1; 2; 3]. Over the past few decades, many attempts have been made to model the complex gas-solid flow behavior in fluidized beds [4; 5; 6; 7]. The discrete element method (DEM) which was first described by [PERSON] and [PERSON] [8] is one of the most used simulation tools for dense granular flows [9; 10]. In the DEM, each particle is tracked individually based on [PERSON]'s second law of motion. The particle-particle and particle-wall interactions are modeled by using springs, dashpots and friction sliders. To be able to model a gas-solid fluidized bed, the DEM is coupled with computational fluid dynamics (CFD) to set up a DEM-CFDmodel, where the motion of individual solid particles is calculated by DEM and gas flow is described by a local volume averaged continuum approach [11; 12; 13; 14]. In this model, the gas phase gives information for calculating the fluid drag force acting on individual particles, and meanwhile the solid phase gives information for determining the porosity from the coordinates of individual particles. Without DEM, CFD alone can also be used to simulate the gas-solid fluidized beds. The corresponding method is often referred to as direct numerical simulation (DNS). The DNS method has the advantage of high precision but encounters unaffordable computing overhead. By contrast, the DEM provides an efficient way for calculating the contact force between fluid and particles. It uses empirical drag force formulas instead of direct calculations. The computational grid size can be much larger than the particle size. Due to the use of much coarser computational grid compared to DNS, the CFD simulation of fluid phase becomes feasible for the DEM-CFD method. During the past two decades, the DEM-CFD method has been extensively applied to gas-solid flows [15]. [PERSON] et al. [11] first used the DEM-CFD method to predict the complex gas-solid flow behavior inside a two-dimensional fluidized bed, and then they [16; 17] extended the method to simulate the flow patterns inside a large-scale bubbling fluidized bed. [PERSON] et al. [18] validated the two-dimensional DEM-CFD simulations of gas-solid fluidized beds by comparing the results with experimental data. They compared three different porosity estimation procedures which transform the results in two dimensions to three dimensions. [PERSON] [19] investigated the phenomenon of bubbling by using DEM-CFD simulations with different sizes of particles. The study revealed different types of fluidization regimes as stated in the literature. In another paper, [PERSON] [20] studied the phenomenon of minimum fluidization by using DEM-CFD simulations with different sizes of particles. The relationship between the pressure drop across the bed and the fluid superficial velocity was presented. So far, the DEM-CFD method has been deeply developed in simulating the fluidized beds. Many satisfactory results have been reported qualitatively or quantitatively agreeing with experiments [21; 22]. The application of the DEM-CFD method has been extended to a large variety of dense gas-solid flow problems such as multiphase flow in dense medium cyclones [23], wood gasification in bubbling fluidized beds [24], char combustion in fluidized beds [25], melting of the granular packed bed [26], simulation of three-phase flows [27], etc. Now the DEM can handle more complex particles, including non-spherical particles [28] and cohesive particles [29]. Although the DEM is well established, the number of the tracked particles is severely restricted by the limited computing power. This is especially critical because applications of the DEM to device scale systems are hardly to be finished in practical time scales.