A Review of the Continuum Theory-Based Stress and Drag Models in Gas-Solid Flows

Abstract

The continuum theory-based models, which include solid stress models and gas-solid drag models, are required for the modeling of gas-solid flows in the framework of the Eulerian–Eulerian method. The interactions among particles are characterized by their diverse behaviors at different flow regimes, including kinetic motion, particle–particle collision and enduring friction. It is difficult to describe the particle behaviors at various regimes by mathematical methods accurately. Therefore, it is very important to develop proper solid stress models that can capture the inherent characteristics of the flow behaviors. In addition, the gas-solid fluidization system is a typical heterogeneous system, which exhibits locally inhomogeneous structures such as bubbles or particle clusters with different shapes and sizes. Due to these inhomogeneous characteristics, the gas-solid drag model has become one of the key challenges in the simulation of gas-solid flows. Various forms of constitutive relations for solid stress models and gas-solid drag models have been reported in the literature. In this paper, we reviewed the solid stress models crossing various flow regimes and drag models in both micro- and mesoscales, which provide a useful reference for model selection in simulating gas-solid flows.

1. Introduction

Gas-solid flow systems are found in many industrial applications, such as the conversion of biomass [1,2], polymerization [3],metallurgy [4], etc. When it comes to the simulation of industrial-scale gas-solid flow systems, the Eulerian–Eulerian method, which models gas and solid phases as interpenetrating continua, has an advantage over the Eulerian–Lagrangian method, in which particles’ trajectories and particle–particle collisions are resolved. The Eulerian–Eulerian model, however, needs solid stress models and gas-solid drag models to close the continuous momentum equation of the solid and gas phases, as shown in Table 1.

Flows in gas-solid fluidization systems can be typically classified into three different regimes according to the interactions among particles [5,6], as shown in Figure 1. The first regime is the inertial regime, in which flows are rapid and particles interact with each other primarily through collisions. The second regime is the quasi-static regime, in which deformations of the solid phase are slow and particles interact with each other through enduring frictional contacts. The third regime is the intermediate regime, in which particles transmit momentum through both instantaneous collisions and frictional contacts. In the literature, various forms of constitutive relations for these regimes have been reported. In the inertial regime, solid stress is generally closed by the kinetic theory of granular flows (KTGF), which was developed in analogy with the kinetic theory of gases. In the quasi-static regimes, solid stress is often closed by the frictional stress model, which has the capacity to deal with enduring contacts and frictions. In recent years, a η(I_s)-rheology model, which is based on the experimental or Discrete Element Method (DEM) results, seems to provide a better transition for solid stress from the inertial regime to the quasi-static regime.

In the simulation of gas-solid flows, drag force is the dominant force between the gas and the particles. The gas-solid fluidization system is a typical inhomogeneous system, which includes gas bubbles and particle clusters. The drag model has become one of the key challenges in the simulation of gas-particle flows due to its inhomogeneous characteristics. Generally speaking, interphase drag models can be divided into two categories: microscopic drag models for a homogeneous grid scale and mesoscale drag models for an inhomogeneous grid scale. The microscopic drag model, which is applicable to the uniform system, cannot fully consider the mesoscale effect and the drag drop phenomenon due to the limitation of computing capacity [7]. However, the mesoscale drag model can capture the mesoscale inhomogeneous structure by considering the structural anisotropy, but there are still some problems with the accuracy and universality of this model. The mesoscale drag models are divided into filtered drag models based on turbulence modeling and energy minimization multi-scale (EMMS) models, which are based on the minimum energy principle.

In this study, we present a critical review of the various reported closure equations for solid stress and interphase drag force. The solid stress models are summarized in Section 2, which includes the kinetic-collisional law for dilute rapid flows (Section 2.1), the frictional law for dense slow flows (Section 2.2), the transition from dilute to dense flows (Section 2.3) and the wall boundary condition (Section 2.4). The interphase drag models are summarized in Section 3, which includes the microscopic drag models based on experiments and DNS (Section 3.1) and mesoscale drag models based on the filtered method and EMMS (Section 3.2). Various expressions for solid stress and interphase drag force are listed in tables and compared in figures in each section.

2. Solid Stress Models

To close Equation (4), expressions for the solid pressure p_s and stress tensor τ_s are required. In the past, there were many empirical models [8,9,10,11] with expressions as a function of solid concentration ε_s. However, the major disadvantages of these empirical correlations are that the physical origin of solid pressure and viscosity are unclear and the effective transport properties are difficult to determine. Currently, the commonly used methodology is the kinetic theory in the dilute regime and the frictional stress model in the dense regime.

2.1. Kinetic-Collisional Law

When the solid flow is dominated by particle collisions, which is true for a rapidly sheared flow in the inertial regime, the kinetic theory may be employed to formulate the stresses acting on the averaged flow [12,13,14]. Originally, the kinetic theory was developed by Chapman and Cowling [15] for gases, to predict the behavior of molecules whose interaction energy is conserved. Considering the energy dissipation during particle collisions, this theory was used to model the rapid flow of particles.

2.1.1. Solid Pressure and Viscosity

The kinetic-collisional particle pressure closed by the kinetic theory is constituted by both the kinetic and collisional contributions,

(5)$p_{kc}=p_k+p_c$

with the kinetic contribution,

(6)$p_k={\rho }_s{\epsilon }_s\theta$

and the collisional contribution,

(7)$p_c=2{\epsilon }_{\mathrm{s}}(1+e)g_0{\rho }_{\mathrm{s}}{\epsilon }_{\mathrm{s}}\theta$

The kinetic-collisional particle stress is expressed as,

(8)${\tau }_{kc}={\mu }_b\left(\nabla \cdot \overrightarrow{u_s}\right)I+2\left({\mu }_k+{\mu }_{\mathrm{c}}\right)D_s$

Here, μ_k is the kinetic viscosity (kinetic contribution), μ_c is the collision viscosity (collisional contribution) and μ_b is the bulk viscosity (dilatation or contraction contribution). It is necessary to note that the expression of solid pressure is almost the same in different versions of kinetic theory, while the expressions of solid viscosity vary in different versions due to different considerations. The expressions of kinetic viscosity μ_k, collision viscosity μ_c and bulk viscosity μ_b deduced by different authors are summarized in Table 2, Table 3 and Table 4, respectively. The different contributions of solid pressure and viscosity are compared in Figure 2. It shows that the region with ε_s < 0.2 is dominated by the kinetic contribution, while the collisional and bulk contributions dominate the regions with ε_s > 0.2.

Based on the kinetic theory of gases proposed by Chapman and Cowling [15], many models for granular flow were subsequently developed by analogy. Lun et al. [12] reported the most widely used comprehensive model for slightly inelastic granular flow supported by the corresponding constitutive relations for the energy balance. The model incorporates both kinetic and collisional contributions to particle stress, conduction and dissipation. Gidaspow [16] improved the kinetic theory by incorporating fluid–particle interactions into the macroscopic equations. Garzó and Dufty [17] further proposed a generalized but formally complicated theory by employing the enhanced Enskog theory, while their results on the shear viscosity and thermal conductivity are numerically not very different from those of Lun et al. [12]. Additionally, particles suspended in fluids experience a random force due to the thermal fluctuations in the fluid around them, in addition to the average hydrodynamic force. Brownian motion may take place for those sub-micron/nanoscale particles, which was studied by Chamkha and co-workers [18,19,20,21,22,23] deeply. In this review, we do not intend to discuss such Brownian motion of very fine particles, nor the agglomeration effect [24] and tribocharging effect [25,26,27] due to van der Waals forces and charge transfer between particles, although they do exist in nature and in some industrial applications.

2.1.2. Equation for Granular Temperature

The solid pressure and viscosity described previously depend on the granular temperature θ, which is defined as the random kinetic energy of the particles.

(17)$\theta =\frac{1}{3}⟨{\left(\overrightarrow{u_s}-⟨\overrightarrow{u_s}⟩\right)}^2⟩$

The granular temperature θ is solved by a balance equation summarized in Table 5. These expressions can mainly be divided into two categories: 1. Granular energy produced by stress is assumed to dissipate locally by collisions among particles, which can be solved by the algebraic method. 2. The non-local effects, which include convection and diffusion, are considered, and then a partial differential equation (PDE) is needed. Based on the expression of Lun et al. [12], Gidaspow [16] later extended the source term by considering the interactions between gas and solid particles, f_gs.

2.1.3. Restitution Coefficient and Radial Distribution Function

The restitution coefficient e (0 < e < 1) included in the solid pressure and viscosities represents the energy dissipation during particle collision. The limit values of e = 0 and e = 1 represent complete and no energy dissipation during particle–particle collisions. It may be necessary to point out that the conventional kinetic theory considers only the inelastic effect on the normal component of the relative velocity during particle collisions, while the effect of the tangential contacts is usually neglected. To include both effects, the restitution coefficient e should be replaced by its effective value e_eff (e_eff ≤ e) [28,29,30], which is a function of the interparticle friction coefficient μ_p. The expressions of the effective restitution coefficient proposed by Jenkins and Zhang [28] and Chialvo and Sundaresan [31] are summarized in Table 6 and compared in Figure 3a. It is necessary to note that Equation (21) is a simplified expression of Jenkins and Zhang [18], which will yield non-physical results when μ_p > 0.2. After that, a more complicated expression was proposed by Duan and Feng [32] and Yu et al. [33]. However, their results are similar in the law and mediate the interparticle friction coefficient.

The radial distribution function g₀ takes into account the spatial correlation of particles with finite volumes. In the dilute limit, no spatial correlations exist and g₀ = 1. As the particle concentration increases, a spatial correlation develops due to volume exclusion effects, and g₀ increases accordingly. The expressions of the radial distribution function proposed by different authors are summarized in Table 7 and compared in Figure 3b. These expressions can mainly be divided into two categories: 1. the expressions that didn’t consider the maximum packing effect in the dense limit, such as Carnahan and Starling [34]. 2. the expressions that consider the maximum packing effect in the dense limit, such as Bagnold [35], Ahmadi and Ma [36], Savage [37] and Chialvo and Sundaresan [31]. The radial distribution function g₀ will tend to infinity in the dense limit when the maximum packing effect is considered, as shown in Figure 3b. It is necessary to note that Chialvo and Sundaresan [31] have correlated the maximum packing limit ε_s,max with the interparticle friction coefficient μ_p.

2.2. Frictional Law

When a granular flow is dominated by enduring contacts among particles, the relationship between the solid stress τ_s and the deformation rate D_s becomes inherently different from a kinetic-collisional law. It cannot even be treated as an extreme case of the kinetic theory. Two ways are often used to establish the frictional law so far. The first one (the well-known frictional stress model) is based on the critical state theory of soil mechanics [38,39,40], and the other one is based on the η(I_s)-rheology method.

2.2.1. Frictional Stress Model

Several frictional stress models have been proposed to handle enduring contacts and frictions among particles at high solid volume fractions, including the Syamlal model [38,39] and the Srivastava-Sundaresan model [40,41]. In both models, the frictional pressure is a function of the solid volume fraction ε_s, while the frictional stress is expressed as:

(28)${\tau }_f=2{\mu }_f{D}_s$

Syamlal et al. [39] proposed that frictional stresses need to be considered only for regions where the solid volume fraction is higher than the maximum packing limit, ε_s,max. Srivastava and Sundaresan [40] proposed a frictional stress model to account for the strain rate fluctuations based on the viscosity expression of Schaeffer [38]. In this model, the frictional stress affects the granular flow at a minimum frictional solid volume fraction, ε_s,min, that is below the maximum packing limit, ε_s,max. The expressions of the Syamlal model and the Srivastava–Sundaresan model are summarized in Table 8. The frictional pressures of the Syamlal model and the Srivastava–Sundaresan model are compared in Figure 4. A substantial amount of research has been conducted to compare these two models in bubbling fluidized bed [42,43,44,45,46,47] and bin discharge flow [48]. Most of these results show that the Srivastava–Sundaresan model is more physical in dense granular or gas-solid flows.

2.2.2. η(Is)-Rheology Model

There have been numerous experiments or DEM (Discrete Element Method)-based studies to determine the rheological correlation for dense granular flows under simple shear. In these studies, the experimental or DEM results were obtained to build up the rheological relationship between shear stress and solid pressure, which is expressed as

(33)${\tau }_s=\eta \left(I_s\right)p_s^{}$

Most of these empirical results indicate that the shear stress in granular flows depends on the inertial number [50,51,52].

(34)$I_s=\frac{{\dot{\gamma }}_s{d}_s}{\sqrt{p_s^{}/{\rho }_s}}$

The inertial number can be interpreted as the ratio between the macroscopic deformation timescale $1/{\dot{\gamma }}_s$ (time for a particle to move one particle’s diameter under the action of an applied shear rate) and the inertial timescale $\sqrt{d_s^2{\rho }_s/p_s^{}}$ (time for a particle to move one particle’s diameter under the unbalanced action of the solid phase’s pressure) [53]. Based on this method, the expressions of shear stress and inertial number have been extended to a 3D frame by Zhao et al. [45].

(35)${\tau }_s=\eta (I_s)p_s\frac{D_s}{\Vert D_s\Vert }$

(36)$I_s=\frac{2\Vert D_s\Vert d_s}{\sqrt{p_s/{\rho }_s}}$

The ratio of shear stress to solid pressure, also the so-called internal friction coefficient η, is a function of the inertial number. This parameter decides the rheological relationship to a large extent. The expressions of the internal friction coefficient proposed by different authors are summarized in Table 9. Figure 5 shows the comparison of these expressions across various flow regimes. In the quasi-static regime, all expressions are almost the same as η → η_yield, which reflects a main characteristic of granular flow, also called the yield criterion: flow is not possible below a critical shear stress [50]. The internal friction coefficient η increases in the intermediate regime and achieves a steady value in the inertial regime, according to the expressions of Jop et al. [50] and Chialvo et al. [54]. However, the expression of da Cruz et al. [55] keeps increasing in the inertial regime because they simplified the particles into 2D disks. Unlike Jop et al. [50], Chialvo et al. [54] considered the effect of the particle friction coefficient μ_p.

2.3. Combination of Kinetic-Collisional Law and Frictional Law

In gas-particle flows, as the concentration of particles increases, the mechanism of momentum transfer among particles changes from kinetic-collisional to frictional gradually. The commonly used methodology is to combine the kinetic-collisional law and the frictional law to form a complete theory that applies to both the dilute and dense regimes. According to Johnson and Jackson [41], the kinetic-collisional stress and the frictional stress should be added directedly:

(40)${\tau }_s={\tau }_{kc}^{}+{\tau }_f^{}$

Then the frictional effect will be considered when ε_s > ε_s,max for the Syamlal model and ε_s > ε_s,min for the Srivastava–Sundaresan model. Although this method performs well in many simulations [40,42,43,56,57], the transition from the dilute regime to the dense regime still has unresolved issues that significantly affect the bubble [42,45]. Based on this, many researchers tried to propose a blending function to combine these regimes:

(41)${\tau }_s=\chi {\tau }_{kc}+(1-\chi ){\tau }_f$

Liu et al. [58] proposed an inverse trigonometric function-based blending function that needs to specify the boundary between inertial and quasi-static regimes. Chialvo and Sundaresan [31] proposed a blending function without the specification of an arbitrarily chosen boundary based on both the inertial number and solid concentration. Their expressions are summarized in Table 10 and compared in Figure 6. By contrast, the expression of Chialvo and Sundaresan [31] is more physical for the transition from inertial to intermediate and then quasi-static regimes [45].

2.4. Wall Boundary Condition

It is widely acknowledged that the no-slip wall boundary condition is a suitable choice for the gas phase in the continuum modeling of gas-solid fluidization since the Knudsen number of the gas phase is typically low. However, the wall boundary condition for the particle phase is not well understood. Although no-slip, partial-slip and free-slip wall boundary conditions are all used in CFD simulations for particle phase, the partial-slip one is the most popular choice [41,59,60,61].

Johnson and Jackson [41] proposed the well-known partial-slip model by equating the tangential force per unit area exerted on the boundary by the particles to the corresponding stress within the particle assembly close to the boundary. After that, Li and Benyahia [62] revised the model of Johnson and Jackson [41] by adopting the classic rigid-body theory for particle-wall collision. Schneiderbauer et al. [53] proposed another type of wall boundary condition for predicting the particle slip velocity and the pseudo-thermal energy flux using measurable quantities as inputs. The model of partial-slip models is summarized in Table 11. Numerous numerical studies on the impact of various particle-wall boundary conditions on the hydrodynamics of gas-solid flow [63,64,65], as well as on the impact of the input parameters in a particular wall boundary condition [66,67,68], have been investigated. It has been demonstrated that the model formulations and input parameters both significantly affect the bed hydrodynamics. Although there exist some models for the partial-slip condition currently, the exact form of boundary conditions for the particle phase at a surface is currently unknown. Some works about the conditions borrowed from rarefied gas dynamics, such as the works of Chamkha and co-workers [69,70,71,72], may be helpful to develop new models of the wall boundary condition in the future.

3. Gas-Solid Drag Models

Gas and particle phases are coupled through an interphase drag force term in momentum equations, Equations (3) and (4). The drag force correlations that cannot be derived directly are acquired by fitting the results of experiments and simulations. These models are commonly divided into two groups: the microscale drag model for a homogeneous system and the mesoscale drag model for an inhomogeneous system.

3.1. Microscale Drag Model

3.1.1. Experiment-Based Drag Model

Early research developed the expressions for the average drag force by measuring the pressure drop of fixed bed and the terminal settling velocity of particles [73,74]. Considering the effect of the surrounding flow field, the volume fraction was introduced based on the single-particle drag coefficient. It is necessary to note that this model does not account for the effect of mesoscale inhomogeneity on the drag force.

The Wen–Yu drag model [74] was developed based on the experiment data in a liquid-solid settling system, which limits this model to be valid in the dilute regime only. The Ergun model [73], which was developed from the experiments of fixed bed, is suitable for dense regimes. The Gidaspow model [16] adopted the Wen–Yu model [74] in dilute regimes $\left({\epsilon }_g\ge 0.8\right)$ and the Ergun model in dense regimes $\left({\epsilon }_g<0.8\right)$, which combined the advantages of both models but left a gap at the void fraction of 0.8. Physically, the interphase drag force should be continuous functions of solid volume fraction and a Reynolds number. Different methods have been proposed to address this discontinuity. Di Felice [75] took into account the effect of the Re number and avoided this inconsistency. Lu et al. [76] proposed a blending function by using an inverse trigonometric function. Dahl et al. [77] addressed this problem by using linear interpolation. Recently, Liu et al. [78] introduced a dimensionless parameter, the inertial number, to solve this gap with a dynamic method. These experiment-based drag models are summarized in Table 12 and compared in Figure 7.

3.1.2. DNS-Based Model

The DNS technique can resolve the gas phase on a time and space scale and estimate the interaction force between gas and solid phases by integrating the stress on the gas-solid interface. Currently, DNS is acknowledged as the most accurate modeling technique since it does not require a model to close the gas-solid momentum transfer. Therefore, researchers have employed this method to build a variety of drag models [80,81,82].

Hill et al. [83] developed the first expression with particles randomly fixed based on the DNS method. Van der Hoef et al. [84] subsequently investigated the drag force of monodisperse and bidisperse systems in the condition of Stokes flow. Beetstra et al. [85] then derived the drag correlation for monodisperse and bidisperse systems at high Reynolds numbers (up to 1000). Sarkar et al. [86] and Yin and Sundaresan [87] further studied the drag force in a bidisperse system to develop new drag models. Rong et al. [88] developed a drag relation for a monodisperse system and then extended it to a polydisperse system with particle concentration $0.1\le {\epsilon }_s\le 0.6$ and Reynolds number $0.01\le Re\le 3000$. Rubinstein et al. [89] explored particle motion at low and high Stokes number limits with low particle Reynolds numbers and found that fluctuations in particle concentration should be included to account for the effects of heterogeneity. Tang et al. [90] proposed a drag correlation that takes particle fluctuation into account based on the finding that the drag in the dynamic granular system is obviously higher than that in the static system. By simulating the settlement of particles in a periodic domain with an applied gradient, Tavanashad et al. [91] investigated the effect of particle velocity fluctuations and particle aggregation on drag force, which covers a wide range of density ratios. These DNS-based drag models are summarized in Table 13.

Compared with the experiment-based drag model, the DNS-based drag model provides a more effective way to couple momentum transfer between gas and particle phases. However, the DNS-based method still has some deficiencies, such as limited range of flow parameters and strict grid resolution requirements. Future development and enhancement of the DNS-based method to study gas-solid flow is still required.

3.2. Mesoscale Drag Model

The complex and variable gas-solid flow behaviors in gas-solid fluidization systems present a significant challenge to the development of a suitable drag model due to the spatio-temporal gap between the phenomena on a macroscopic scale and the physical laws on a microscopic or mesoscopic scale. In particular, the mesoscale structure of the particle cluster is very distinct in circulating fluidized beds, which captures the heterogeneous characteristics of gas-solid flows. The prediction of drag force will be inaccurate due to the neglect of mesoscale inhomogeneous structures. Some research [97,98] demonstrated that the non-uniform characteristics of “Up-dilute-down-dense” cannot be predicted in the Euler–Euler simulation of circulating fluidized beds, especially for the fine particles of Geldart A, because the drag between the gas and solid phases is significantly overestimated by the uniform drag model. In fact, the drag force will drop by several orders of magnitude due to the emergence of mesoscale cluster structures. These drag models must be corrected at the sub-grid level in order to accurately reflect the non-uniform structure within the grid. The mesoscale drag model of the filtered mesh method and energy minimum multiscale method (EMMS) will be reviewed herein.

3.2.1. Filtered Drag Model

With the assumption that particles follow the law of the Maxwell velocity distribution function, the filtered grid method can deal with the non-uniform distribution throughout the grid cells. The simulation of fine grid TFM provides a detailed analysis of the flow structure in the gas-solid flows. Based on this, a correlation that can be applicable to the coarse grid simulation is established.

The filtered drag model was initially proposed by Agrawal et al. [99] based on a fine grid simulation where the grid scale affects the effective drag force, particle viscosity and pressure. A lack of time-scale separation between the resolved and unresolved structures was found by Andrews et al. [100]. Igci et al. [98,101] estimated the average drag force and particle stress on a filtered mesh, and then connected the sub-grid drag coefficients with the average voidage. After this, Igci et al. [97] further introduced the effect of edge wall. The filtered grid method employed by Igci et al. [97,101] is summarized in Table 14. The later researchers [86,102,103] attempted to extract grid information which is linked to drag correction. They found that the sub-grid drag model that considered the average voidage and average slip velocity improved the accuracy of the filtered model. Zhu et al. [104] found that the wall effect should be taken into consideration in the drag model based on the simulation of the bubbling fluidized bed.

Recent studies have shown that some feature variables related to drag correction must be incorporated into the new model. Similar models were employed by Parmentier et al. [105] and Ozel et al. [106] to determine the drag coefficient based on drift velocity. Ozel et al. [107] claim that by incorporating the variance of particle concentration as a characteristic variable, it is possible to characterize the heterogeneity of particle distribution on the grid scale more effectively. Schneiderbauer [108] developed the equations of pulsation velocity and particle volume fraction variance to decrease the drag force between the solid and gas phases. By analyzing the correlation between the characteristic variables and the drag force from the fine grid, the filtered drag force model is able to reflect the mesoscale flow structure in the gas-solid fluidized bed.

3.2.2. EMMS Drag Model

Based on the assumptions of scale decomposition and stability condition, the EMMS model was first proposed by Li and Kwauk [109] for the heterogeneous structures in the riser. The original EMMS model can be used to predict the global heterogeneity of gas-solid fluidization, such as choke phenomena, by establishing mesoscale stability conditions. The EMMS model can be used to calculate the fluid dynamics of axial and radial inhomogeneity in a straight tube of a circulating fluidized bed by extending the stability conditions from the local structure to the global distribution. The basic idea of the EMMS drag model is to obtain the structural parameters by solving the EMMS equations at the sub-grid scale level and then determine the non-uniform drag by considering the effect of flow structures on the drag force. The characteristics of gas and particles as well as the operating conditions can be used to predict the effective interphase drag force. Yang et al. [110] incorporated the EMMS drag model of Li and Kwauk [109] into the two-fluid model and proposed a semi-empirical improvement. The equations in the EMMS drag model of Yang et al. [110] are listed in Table 15.

Later on, the EMMS drag model was further extended to bubbling fluidized beds, which considered bubbles as the mesoscale structure instead of clusters in more flow states [111,112]. By taking into account the effects of the structure in the conservation equation, Hong et al. [111] and Song et al. [113] constructed structure-dependent multi-fluid models, which can be simplified into the EMMS model to represent the overall flow behaviors in fluidized beds. Du et al. [114] decomposed the energy term of the EMMS model and discussed the various extreme conditions. The EMMS drag model can reasonably characterize the mesoscale inhomogeneous structure and the drag drop caused by such structures. Therefore, it can be widely used in the simulation of bubbling fluidized beds and circulating fluidized beds [115].

4. Summary and Perspectives

In this paper, we reviewed the solid stress models crossing various flow regimes and drag models in micro- and mesoscales, which provide a useful reference for model selection in simulating gas-solid flows. In the inertial regime, different versions of the kinetic theory, restitution coefficient models and radial distribution functions could be used for the kinetic-collision law. In the intermediate and quasi-static regimes, the frictional stress model, which is based on soil mechanics, and η(I_s)-rheology model are well established for the prediction of enduring contacts. The combination methods for different flow regimes and wall boundary conditions need further development with a deep physical basis. Different drag models for gas-particle fluidizations, including experimental data-based empirical models, DNS-based drag models, as well as those meso-scale drag models that considered non-uniform structure effects in the fluidization process, are well used in industrial and academic studies. With the development of computer science as well as machine learning algorithms, accurate prediction models that generate from well calculated DNS data could be further applied for industrial scale numerical simulations with higher accuracy. Further trends of the drag models may depend more on the machine learning-based techniques based on accurate analysis of particle scale flow structures. Our review could provide a valuable reference for the model selection of gas-solid flows across varied flow regimes, such as chute flow, bubbling fluidized beds and circulating fluidized beds with risers and downers, etc.

Author Contributions

J.Z.: Conceptualization, Methodology, Software, Writing—Original draft (Solid stress models), Investigation, Visualization. X.G.: Investigation, Writing—Original draft (gas-solid drag models). G.L.: Resources, Formal analysis, Validation, Methodology, Writing—review and editing, Supervision. R.W.: Data curation, review and editing. H.L.: Writing—Review and editing, Supervision. All authors have read and agreed to the published version of the manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

Latin symbols
${\overrightarrow{u}}_g,{\overrightarrow{u}}_s$	Velocity vector for gas and solid, respectively
${\tilde{u}}_g,{\tilde{u}}_s$	Favre filtering gas and solid velocities in Filtered drag model
$\overrightarrow{u_{sw}}$	Slip velocity between particles and wall
$\Phi$	Specularity coefficient
$t$	time
$p_g,p_s$	Pressure for gas and solid
$\overline{p}$	Resolved gas pressure
$d_s$	Particle diameter
$p_{kc},p_f$	kinetic-collision pressure and frictional pressure for solid phase
$C_D$	Drag coefficient for single particle
$Re$	Particle Reynolds number
Fr	Froude numbers
$g_0$	Radial distribution function
$\beta$	Gas-solid drag coefficient
$I$, Is	Unit tensor, inertial number
$e,e_{eff}$	Restitution coefficient and effective restitution coefficient
$D_s$	Deviatoric rate-of-strain tensor of solid phase
$N_{st}$	Mass specific energy consumption
$U_{g,}{U}_s$	Superficial fluid and particle velocity in EMMS drag model
$U_{\mathrm{mf}}$	Superficial fluid velocity at minimum fluidization
Greek symbols
${\epsilon }_g,{\epsilon }_s$	Concentration for gas and solid phases, respectively
${\overline{\epsilon }}_g,{\overline{\epsilon }}_s$	Resolved concentration for gas and solid phase, respectively
${\rho }_g,{\rho }_s$	Density for gas and solid phase, respectively
${\tau }_g,{\tau }_s$	Stress tensor for gas and solid phase, respectively
${\tau }_{kc},{\tau }_f$	Kinetic-collisional and the frictional stress tensor
${\mu }_g$	Gas viscosity
${\mu }_b,{\mu }_{kc}$	Bulk, kinetic-collisional and frictional viscosity
${\epsilon }_{s,\max }$	Maximum solid volume fraction at packing
${\eta }_{yield}$	Yield friction coefficient
$\eta$	Internal friction coefficient
$\theta$	Granular temperature
${\kappa }_s$	Conductivity of granular fluctuating energy
${\gamma }_s$	Collisional dissipation
${\mu }_p$	Particle friction coefficient
$\chi$	Blending function
$\omega$	Drag factor
Subscript
$g$	Gas phase
$s$	Solid phase
c	Particle in dense phase
f	Particle in dilute phase
i	Cluster phase
eff	Effective variable

Footnotes

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Figures and Tables

Figure 1. Interaction mechanisms among particles at different flow regimes.

Figure 2. Dimensionless solid pressure and viscosity variation with different contributions, (a) Solid pressure, (b) Solid viscosity (This figure is plotted based on the model of Lun et al. [12]).

Figure 3. Effective restitution coefficient and radial distribution function, (a) Effective restitution coefficient, (b) Radial distribution function [28,31,34,35,36,37].

Figure 4. Frictional pressure predicted by the Syamlal model and Srivastava–Sundaresan model, with εs,min = 0.5 and εs,max = 0.65.

Figure 5. η(Is) relationship at various flow regimes [50,54,55].

Figure 6. Blending function proposed by different authors, (a) Liu et al. [58] with εs,min = 0.5 and εs,max = 0.65, (b) Chialvo and Sundaresan [31].

Figure 7. Comparison of experiment-based drag models in conditions of (a) Re = 100, (b) Re = 500, (c) Re = 1000 and (d) Re = 2000.

Table 1

Governing equations.

Continuity Equations:
$\frac{\partial }{\partial t}\left({\epsilon }_g{\rho }_g\right)+\nabla \cdot \left({\epsilon }_g{\rho }_g\overrightarrow{u_g}\right)=0$	(1)
$\frac{\partial }{\partial t}\left({\epsilon }_s{\rho }_s\right)+\nabla \cdot \left({\epsilon }_s{\rho }_s\overrightarrow{u_s}\right)=0$	(2)
Momentum Equations:
$\frac{\partial }{\partial t}\left({\epsilon }_g{\rho }_g\overrightarrow{u_g}\right)+\nabla \cdot \left({\epsilon }_g{\rho }_g\overrightarrow{u_g}\overrightarrow{u_g}\right)=\nabla \cdot {\tau }_g-{\epsilon }_g\nabla p_g+{\epsilon }_g{\rho }_g\overrightarrow{g}-\beta \left(\overrightarrow{u_g}-\overrightarrow{u_s}\right)$	(3)
$\frac{\partial }{\partial t}\left({\epsilon }_s{\rho }_s\overrightarrow{u_s}\right)+\nabla \cdot \left({\epsilon }_s{\rho }_s\overrightarrow{u_s}\overrightarrow{u_s}\right)=-\nabla p_s+\nabla \cdot {\tau }_s-{\epsilon }_s\nabla p_g+{\epsilon }_s{\rho }_s\overrightarrow{g}+\beta \left(\overrightarrow{u_g}-\overrightarrow{u_s}\right)$	(4)

Table 2

Kinetic viscosity.

Lun et al. [12]:
${\mu }_k=\left(\frac{2+\alpha }{3}\right)\left[\begin{array}{c}\frac{5\sqrt{\pi }}{24(1+e)(3-e)g_0}\left(1+\frac{2}{5}(3e-1)(1+e){\epsilon }_s{g}_0\right)\end{array}\right]{\rho }_s{d}_s{\theta }^{1/2}$with α=1.6	(9)
Gidaspow [16]:
${\mu }_k=\frac{10\sqrt{\pi }}{96\left(1+e\right)g_0}{\left[1+\frac{4}{5}\left(1+e\right){\epsilon }_s{g}_0\right]}^2{\rho }_s{d}_s{\theta }^{1/2}$	(10)
Garzó and Dufty [17]:
${\mu }_k=\frac{5\sqrt{\pi }[1-\frac{2}{5}(1+e)(1-3e){\epsilon }_s{g}_0]}{96\left[1-\frac{1}{4}{(1-e)}^2-\frac{5}{24}\left(1-e^2\right)\right]g_0}\rho d_s{\theta }^{1/2}$	(11)

Table 3

Collision viscosity.

Lun et al. [12]:
${\mu }_c=\left(\frac{2+\alpha }{3}\right)\left[\begin{array}{c}\frac{\sqrt{\pi }{\epsilon }_s}{6(3-e)}\left(1+\frac{2}{5}(3e-1)(1+e){\epsilon }_s{g}_0\right)+\frac{4}{5\sqrt{\pi }}(1+e){\epsilon }_s^2{g}_0\end{array}\right]{\rho }_s{d}_s{\theta }^{1/2}$with α=1.6	(12)
Gidaspow [16]:
${\mu }_c=\frac{4}{5\sqrt{\pi }}\left(1+e\right){\epsilon }_s^2{g}_0{\rho }_s{d}_s{\theta }^{1/2}$	(13)
Garzó and Dufty [17]:
${\mu }_c=\frac{4\sqrt{\pi }[1-\frac{2}{5}(1+e)(1-3e){\epsilon }_s{g}_0](1+e){\epsilon }_s}{96\left[1-\frac{1}{4}{(1-e)}^2-\frac{5}{24}\left(1-e^2\right)\right]}\rho d_s{\theta }^{1/2}$	(14)

Table 4

Bulk viscosity.

Lun et al. [12] and Gidaspow [16]:
${\mu }_b=\frac{4}{3\sqrt{\pi }}\left(1+e\right){\epsilon }_s^2{g}_0{\rho }_s{d}_s{\theta }^{1/2}$	(15)
Garzó and Dufty [17]:
${\mu }_b=\frac{4}{5\sqrt{\pi }}\left(1+e\right){\epsilon }_s^2{g}_0{\rho }_s{d}_s{\theta }^{1/2}$	(16)

Table 5

Granular temperature equation.

Algebraic Equation:
$(-p_{kc}{\rm I}+{\tau }_{kc}):\nabla \overrightarrow{u_s}={\gamma }_s$	(18)
Lun et al. [12] (PDE)
$\frac{3}{2}\left[\frac{\partial }{\partial t}\left({\epsilon }_s{\rho }_s\theta \right)+\nabla \cdot \left({\epsilon }_s{\rho }_s\theta \overrightarrow{u_s}\right)\right]=(-p_{kc}{\rm I}+{\tau }_{kc}):\nabla \overrightarrow{u_s}+\nabla \cdot \left({\kappa }_s\nabla \theta \right)-{\gamma }_s$	(19)
Gidaspow [16] (PDE)
$\frac{3}{2}\left[\frac{\partial }{\partial t}\left({\epsilon }_s{\rho }_s\theta \right)+\nabla \cdot \left({\epsilon }_s{\rho }_s\theta \overrightarrow{u_s}\right)\right]=(-p_{kc}{\rm I}+{\tau }_{kc}):\nabla \overrightarrow{u_s}+\nabla \cdot \left({\kappa }_s\nabla \theta \right)-{\gamma }_s+f_{gs}$	(20)

Table 6

Effective restitution coefficient accounting for particle friction.

Jenkins and Zhang [28]:
$e_{eff}=e-\frac{\pi }{2}{\mu }_p+\frac{9}{2}{\mu }_p^2$	(21)
Chialvo and Sundaresan [31]:
$e_{eff}=e-\frac{3}{2}{\mu }_p\exp (-3{\mu }_p)$	(22)

Table 7

Radial distribution function.

Carnahan and Starling [34]
$g_0=\frac{1-0.5{\epsilon }_s}{{\left(1-{\epsilon }_s\right)}^3}$	(23)
Bagnold [35]
$g_0={\left[1-{\left({\epsilon }_{\mathrm{s}}/{\epsilon }_{s,max}\right)}^{1/3}\right]}^{-1}$	(24)
Ahmadi and Ma [36]
$g_0=1+4{\epsilon }_s\frac{1+2.5{\epsilon }_s+4.5904{\epsilon }_{\mathrm{s}}^2+4.515439{\epsilon }_{\mathrm{s}}^3}{{\left[1-{\left({\epsilon }_s/{\epsilon }_{s,max}\right)}^3\right]}^{0.678021}}$	(25)
Savage [37]
$g_0=\frac{1-7{\epsilon }_s/16}{{\left(1-{\epsilon }_s/{\epsilon }_{s,max}\right)}^2}$	(26)
Chialvo and Sundaresan [31]
$g_0^{}=\frac{1-0.5{\epsilon }_s}{{\left(1-{\epsilon }_s\right)}^3}+\frac{0.58{\epsilon }_s^2}{{\left[{\epsilon }_{s,max}({\mu }_p)-{\epsilon }_s\right]}^{3/2}}$	(27)

Table 8

Frictional stress models.

Syamlal model
Frictional pressure (Syamlal et al. [39])$p_f=\{\begin{array}{ll}0\hfill & \mathrm{if} {\epsilon }_s<{\epsilon }_{s,max}\hfill \\ {10}^{25}{\left({\epsilon }_s-{\epsilon }_{s,max}\right)}^{10}\hfill & \mathrm{if} {\epsilon }_s⩾{\epsilon }_{s,max}\hfill \end{array}$	(29)
Frictional viscosity (Schaeffer [38])${\mu }_f=p_f\frac{\sqrt{2}\sin \varphi }{2\sqrt{D_s:D_s}}$	(30)
Srivastava–Sundaresan model
Frictional pressure (Johnson and Jackson [41])$p_f=\{\begin{array}{l}0 ,{\epsilon }_s\le {\epsilon }_{s,min}\hfill \\ Fr\frac{{\left({\epsilon }_s-{\epsilon }_{s,min}\right)}^r}{{\left({\epsilon }_{s,max}-{\epsilon }_s\right)}^s},{\epsilon }_s>{\epsilon }_{s,min}\hfill \end{array}$with Fr = 0.05 N/m2, r = 2 and s = 3 (Ocone et al. [49])	(31)
Frictional viscosity (Srivastava and Sundaresan [40])${\mu }_f=p_f\frac{\sqrt{2}\sin \varphi }{2\sqrt{D_s:D_s+\frac{\theta }{d_s^2}}}$	(32)

Table 9

Internal friction coefficient.

Da Cruz et al. [55]:
$\eta (I_s)={\eta }_{yield}+b{I}_s$with ηyield = 0.25 and b = 1.1	(37)
Jop et al. [50]:
$\eta (I_s)={\eta }_{yield}+\frac{{\eta }_c-{\eta }_{yield}}{I_0/I_s+1}$with ηyield = tan (21°), ηc = tan (33°), I0 = 0.3 (Forterre and Pouliquen [42])	(38)
Chialvo et al. [54]:
$\eta (I_s)={\eta }_{yield}({\mu }_p)+\frac{{\alpha }_1}{{\left(I_0/I_s\right)}^{{\beta }_1}+1}$with α1 = 0.37, β1 = 1.5, I0 = 0.32	(39)

Table 10

Blending function for kinetic-collisional law and frictional law.

Liu et al. [58]:
$\chi \left({\epsilon }_{\mathrm{s}}\right)=0.5-\frac{\arctan \left[25\left({\epsilon }_s-{\epsilon }_{s,min}\right){\left({\epsilon }_{s,max}-{\epsilon }_{s,min}\right)}^{-2}\right]}{\pi }$	(42)
Chialvo and Sundaresan [31]:
$\chi ({\epsilon }_s,I_s)=\frac{1}{{\left(0.2{\epsilon }_s/I_s\right)}^{1.5}+1}$	(43)

Table 11

Models of wall boundary condition.

Johnson and Jackson [41]
Particle slip velocity at the wall:$\frac{\overrightarrow{u_{sw}}\cdot {\tau }_s\cdot \overrightarrow{n}}{\left|\overrightarrow{u_{sw}}\right|}+\frac{\Phi \sqrt{3\theta }\pi {\rho }_s{\epsilon }_s{g}_0\left|\overrightarrow{u_{sw}}\right|}{6{\epsilon }_{s,max}}+p_s^{}\tan {\varphi }_w=0$	(44)
Granular energy at the wall:$-\overrightarrow{n}\cdot \overrightarrow{q_s}=\frac{1}{4}\pi {\rho }_s{\epsilon }_s\theta \left(1-e_w^2\right)g_0\frac{\sqrt{3\theta }}{{\epsilon }_{s,max}}-\overrightarrow{u_{sw}}\cdot \frac{\Phi \sqrt{3\theta }\pi {\rho }_s{\epsilon }_s{g}_0\overrightarrow{u_{sw}}}{6{\epsilon }_{s,max}}$	(45)
Schneiderbauer et al. [53]
Particle slip velocity at the wall:$\frac{\overrightarrow{u_{sw}}\cdot {\tau }_s\cdot \overrightarrow{n}}{\overrightarrow{u_{sw}}}+\frac{1}{2}{\mu }_w\left(1+e_w\right){\rho }_s{\epsilon }_s{g}_0\theta \mathrm{erf}\left(\sqrt{\frac{3}{2}}\frac{r}{{\overline{\mu }}_0}\right)+p_s^{}\tan {\varphi }_w=0$where $\mathrm{erf}(x)=\frac{2}{\sqrt{\pi }}{\displaystyle {\int }_0^x\exp }\left(-{\xi }^2\right)d\xi$	(46)
Granular energy at the wall:$\begin{array}{l}-\overrightarrow{n}\cdot \overrightarrow{q_s}=\frac{\sqrt{2\theta }\left(1+e_w\right){\rho }_s{\epsilon }_s{g}_0}{4\sqrt{\pi }{\overline{\mu }}_0^2}\exp \left(-\frac{3r^2}{2{\overline{\mu }}_0^2}\right)\times \{{\mu }_w[2{\mu }_w{\left|\overrightarrow{u_{sw}}\right|}^2\left(1+e_w-{\overline{\mu }}_0\right)+\theta (7{\mu }_w\left(1+e_w\right)\\ -4{\overline{\mu }}_0\left(1+{\mu }_w\right)-3{\mu }_w{\overline{\mu }}_0^2\left(1+e_w\right))]+{\overline{\mu }}_0^2\sqrt{\theta }\exp \left(-\frac{3r^2}{2{\overline{\mu }}_0^2}\right)\\ \times \left[\sqrt{\theta }\left(2\left(e_w-1\right)+3{\mu }_w^2\left(1+e_w\right)\right)-\sqrt{2\pi }{\mu }_w\left|\overrightarrow{u_{sw}}\right|\mathrm{erf}\left(\sqrt{\frac{3}{2}}\frac{r}{{\overline{\mu }}_0}\right)\right]\\ +\frac{1}{2}{\mu }_w\left(1+e_w\right){\rho }_s{\epsilon }_s{g}_0\theta \mathrm{erf}\left(\sqrt{\frac{3}{2}}\frac{r}{{\overline{\mu }}_0}\right)\left|\overrightarrow{u_{sw}}\right|\end{array}$	(47)

Table 12

Experiment-based drag models.

Wen and Yu [74]
$\beta =\frac{3}{4}{C}_D\frac{{\rho }_g{\epsilon }_g{\epsilon }_s\left|\overrightarrow{u_g}-\overrightarrow{u_s}\right|}{d_s}{\epsilon }_g^{-2.65}$where $C_D=\{\begin{array}{ll}\frac{24}{Re}\left(1+0.15R{e}_{}^{0.687}\right)\hfill & Re\le 1000\hfill \\ 0.44\hfill & Re>1000\hfill \end{array}$,$Re=\frac{{\rho }_g{\epsilon }_g\left|\overrightarrow{u_g}-\overrightarrow{u_s}\right|d_s}{{\mu }_g}$	(48)
Syamlal and O’Brien [79]
$\beta =\frac{3}{4}{C}_D\frac{{\rho }_g{\epsilon }_g{\epsilon }_s\left|\overrightarrow{u_g}-\overrightarrow{u_s}\right|}{d_s}\frac{1}{V_r^2}$ $C_D={\left(0.63+4.8\sqrt{\frac{V_r}{Re}}\right)}^2$ $V_r=0.5A-0.03Re+0.5\sqrt{{\left(0.06Re\right)}^2+0.12Re(2B-A)+A^2}$ $A={\epsilon }_g^{4.14},\hspace{1em}B=\{\begin{array}{ll}0.8{\epsilon }_g^{1.28}\hfill & {\epsilon }_g\le 0.85\hfill \\ {\epsilon }_{\mathrm{g}}^{2.65}\hfill & {\epsilon }_g>0.85\hfill \end{array}$	(49)
Gidaspow [16]
$\beta =\{\begin{array}{ll}\frac{3}{4}{C}_D\frac{{\rho }_g{\epsilon }_g{\epsilon }_s\left|\overrightarrow{u_g}-\overrightarrow{u_s}\right|}{d_s}{\epsilon }_g^{-2.65}& {\epsilon }_g\ge 0.8\\ 150\frac{{\epsilon }_s^2{\mu }_g}{{\epsilon }_g^{}{d}_s^2}+1.75\frac{{\rho }_g{\epsilon }_s\left|\overrightarrow{u_g}-\overrightarrow{u_s}\right|}{d_s}& {\epsilon }_g<0.8\end{array}$	(50)
Di Flice [75]
$\beta =\frac{3}{4}{\left(0.63+\frac{4.8}{\sqrt{Re}}\right)}^2\frac{{\rho }_g{\epsilon }_g{\epsilon }_s\left|\overrightarrow{u_g}-\overrightarrow{u_s}\right|}{d_s}{\epsilon }_g^{1-\phi }$ $\mathrm{with} \phi =3.7-0.65\exp \left[-\frac{{(1.5-\log Re)}^2}{2}\right]$	(51)
Lu and Gidaspow [76]
$\beta =\chi \left[\frac{3}{4}{C}_D\frac{{\rho }_g{\epsilon }_g{\epsilon }_s\left|\overrightarrow{u_g}-\overrightarrow{u_s}\right|}{d_s}{\epsilon }_g^{-2.65}\right]+\left(1-\chi \right)\left[150\frac{{\epsilon }_s^2{\mu }_g}{{\epsilon }_g^{}{d}_s^2}+1.75\frac{{\rho }_g{\epsilon }_s\left|\overrightarrow{u_g}-\overrightarrow{u_s}\right|}{d_s}\right]$ $\mathrm{with} \chi =\arctan \left(\frac{150\times 1.75\left(0.2-{\epsilon }_s\right)}{\pi }\right)+0.5$	(52)
Liu et al. [78]
$\beta =\chi \left[\frac{3}{4}{C}_D\frac{{\rho }_g{\epsilon }_g{\epsilon }_s\left|\overrightarrow{u_g}-\overrightarrow{u_s}\right|}{d_s}{\epsilon }_g^{-2.65}\right]+\left(1-\chi \right)\left[150\frac{{\epsilon }_s^2{\mu }_g}{{\epsilon }_g^{}{d}_s^2}+1.75\frac{{\rho }_g{\epsilon }_s\left|\overrightarrow{u_g}-\overrightarrow{u_s}\right|}{d_s}\right]$ $\mathrm{with} \chi =\frac{1}{{(0.2{\epsilon }_s/I_s)}^{1.5}+1}$	(53)

Table 13

DNS-based drag models.

Hill et al. [83]
${\beta }_{}=\frac{18{\mu }_{\mathrm{g}}{\epsilon }_{\mathrm{g}}^2{\epsilon }_{\mathrm{s}}F}{d_{\mathrm{s}}^2}$$F=\{\begin{array}{ll}1+\frac{3}{8}\left(\frac{Re}{2}\right),{\epsilon }_s\le 0.01 \& \frac{Re}{2}\le \frac{F_2-1}{3/8-F_3}\hfill & \hfill \\ \begin{array}{l}{F}_0+F_1{\left(\frac{Re}{2}\right)}^2, {\epsilon }_{s }>0.01 \& \frac{Re}{2}\le \frac{F_3+\sqrt{F_3^2-4F_1\left(F_0-F_2\right)}}{2F_1}\\ F_2+F_3\left(\frac{Re}{2}\right), \mathrm{Otherwise} \end{array}\hfill & \hfill \end{array}$$F_0=\{\begin{array}{cc}(1-\omega )\left[\frac{1+3\sqrt{{\epsilon }_{\mathrm{s}}/2}+(135/64){\epsilon }_{\mathrm{s}}\ln {\epsilon }_{\mathrm{s}}+17.14{\epsilon }_{\mathrm{s}}}{1+0.681{\epsilon }_{\mathrm{s}}-8.48{\epsilon }_{\mathrm{s}}^2+8.16{\epsilon }_{\mathrm{s}}^3}\right]+\omega \left[\frac{10{\epsilon }_{\mathrm{s}}}{{\epsilon }_{\mathrm{g}}^3}\right],\hfill & \hfill {\epsilon }_{\mathrm{s}}<0.4 \\ \frac{10{\epsilon }_{\mathrm{s}}}{{\epsilon }_{\mathrm{g}}^3},\hfill & \hfill {\epsilon }_{\mathrm{s}}\ge 0.4 \end{array}$$F_1=\{\begin{array}{cc}\frac{\sqrt{2}}{40\sqrt{{\epsilon }_s}}\hfill & \hfill 0.01<{\epsilon }_s\le 0.1\\ 0.11+0.00051\exp \left(11.6{\epsilon }_s\right)\hfill & \hfill {\epsilon }_s>0.1\end{array}$$F_2=\{\begin{array}{cc}(1-\omega )\left[\frac{1+3\sqrt{{\epsilon }_{\mathrm{s}}/2}+(135/64){\epsilon }_{\mathrm{s}}\ln {\epsilon }_{\mathrm{s}}+17.89{\epsilon }_{\mathrm{s}}}{1+0.681{\epsilon }_{\mathrm{s}}-11.03{\epsilon }_{\mathrm{s}}^2+15.41{\epsilon }_{\mathrm{s}}^3}\right]+\omega \left[\frac{10{\epsilon }_{\mathrm{s}}}{{\epsilon }_{\mathrm{g}}^3}\right],\hfill & {\epsilon }_{\mathrm{s}}<0.4\\ \frac{10{\epsilon }_{\mathrm{s}}}{{\epsilon }_{\mathrm{g}}^3},\hfill & \hfill {\epsilon }_{\mathrm{s}}\ge 0.4\end{array}$$F_3=\{\begin{array}{ll}0.9351{\epsilon }_s+0.03667& {\epsilon }_s<0.095\\ 0.0673+0.212{\epsilon }_s+0.0232{\epsilon }_g^{-5}& {\epsilon }_s\ge 0.095\end{array}$ and $\omega =e^{[-10\left(0.4-{\epsilon }_s\right)/{\epsilon }_s]}$	(54)
Beetstra et al. [85]
$\beta =180\frac{{\mu }_g{\epsilon }_s^2}{d_s^2{\epsilon }_g}+18\frac{{\mu }_g{\epsilon }_g^3{\epsilon }_s\left(1+1.5\sqrt{{\epsilon }_s}\right)}{d_s^2}+0.31\frac{{\mu }_g{\epsilon }_{s}Re}{d_s^2{\epsilon }_g}\frac{\left[{\epsilon }_g^{-1}+3{\epsilon }_g{\epsilon }_s+8.4R{e}^{-0.343}\right]}{\left[1+{10}^{3{\epsilon }_s}R{e}^{-0.5-2{\epsilon }_s}\right]}$	(55)
Tenneti et al. [92]
$\beta =18\frac{{\mu }_g{\epsilon }_s\left(1+0.15R{e}^{0.687}\right)}{d_s^2{\epsilon }_g}+104.58\frac{{\mu }_g{\epsilon }_s^2}{d_s^2{\epsilon }_g}+8.64\frac{{\mu }_g{\epsilon }_s^{4/3}}{d_s^2{\epsilon }_g^2}+\frac{18{\mu }_g{\epsilon }_g^2{\epsilon }_s^{4}Re}{d_s^2}\left[0.95+\frac{0.61{\epsilon }_s^3}{{\epsilon }_g^2}\right]$	(56)
Rong et al. [88]
$\beta =\frac{3}{4}{\left(0.63+\frac{4.8}{\sqrt{Re}}\right)}^2\frac{{\rho }_g{\epsilon }_g{\epsilon }_s\left|\overrightarrow{u_g}-\overrightarrow{u_s}\right|}{d_s}{\epsilon }_g^{1-\phi }$with $\phi =2.65(1+{\epsilon }_g)-\left(5.3-3.5{\epsilon }_g\right){\epsilon }_g^2\exp \left[-\frac{{(1.5-\log Re)}^2}{2}\right]$	(57)
Zaidi et al. [93]
$\beta =\{\begin{array}{ll}180\frac{{\mu }_g{\epsilon }_s^2}{d_s^2{\epsilon }_g}+18\frac{{\mu }_g{\epsilon }_g^3{\epsilon }_s\left(\left(1+1.5\sqrt{{\epsilon }_s}\right)\right)}{d_s^2}+\frac{0.612{\mu }_g{\epsilon }_s{\epsilon }_g^{-1.7}Re}{d_s^2}\hfill & Re\le 200\hfill \\ \frac{196.2{\mu }_g{\epsilon }_g^{-0.7}{\epsilon }_s^{1.4}}{d_s^2}+\frac{0.432{\mu }_g{\epsilon }_s{\epsilon }_g^{-1.86}Re}{d_s^2}\hfill & Re>200\hfill \end{array}$	(58)
Bogner et al. [94]
$\beta =\frac{18{\mu }_g{\epsilon }_g^{-3.726}{\epsilon }_s}{d_s^2}\left[1.751+0.151R{e}^{0.684}-0.445{(1+Re)}^{1.04{\epsilon }_s}-0.16{(1+Re)}^{0.0003{\epsilon }_s}\right]$	(59)
Tang et al. [90]
$\begin{array}{l}\beta =180\frac{{\mu }_g{\epsilon }_s^2}{d_s^2{\epsilon }_g}+18\frac{{\mu }_g{\epsilon }_g^3{\epsilon }_s\left(1+1.5\sqrt{{\epsilon }_s}\right)}{d_s^2}\hfill \\ +\frac{18{\mu }_g{\epsilon }_s{\epsilon }_{g}Re}{d_s^2}\left[0.11{\epsilon }_s\left(1+{\epsilon }_s\right)-\frac{0.00456}{{\epsilon }_g^4}+\left(0.169{\epsilon }_g+\frac{0.0644}{{\epsilon }_g^4}\right)R{e}^{-0.343}\right]\hfill \end{array}$	(60)
Sheikh and Qiu [95]
$\beta =\frac{18{\mu }_g{\epsilon }_g^{1.5}{\epsilon }_s}{d_s^2}\left[3^{1.25\left(1-{\epsilon }_g^3\right)}+\mathrm{Re}\left(0.2R{e}^{0.5{\epsilon }_g-1}-0.053{\epsilon }_g+0.073\right)\right]$	(61)
Zaidi [96]
$\begin{array}{l}\beta =\frac{3}{4}{\left(0.63+\frac{4.8}{\sqrt{Re}}\right)}^2\frac{{\rho }_g{\epsilon }_g{\epsilon }_s\left|\overrightarrow{u_g}-\overrightarrow{u_s}\right|}{d_s}{\epsilon }_g^{1-\phi }\\ \mathrm{with} \phi =3.7+\left(6.6\log {\epsilon }_s+5.5\right)\log Re-\left[5.5{\left(\log {\epsilon }_s\right)}^2+15.5\log {\epsilon }_s+9.15\right]\log \left(\frac{{\rho }_s}{{\rho }_g}\right)\end{array}$	(62)

Table 14

The representative filtered drag model by Igci et al. [98].

${\beta }_e=\frac{3}{4}{C}_D\frac{{\rho }_g{\overline{\epsilon }}_g{\overline{\epsilon }}_s\left|{\tilde{u}}_g-{\tilde{u}}_s\right|}{d_s}{\overline{\epsilon }}_g^{-2.65}{H}_d$ $\mathrm{with} H_d=1-f\left(Fr\right)h_{2D}\left({\overline{\epsilon }}_s\right) \mathrm{and} F{r}_{}^{-1}={\overline{\mathrm{\Delta }}}^{*}=\frac{\left|\overrightarrow{g}\right|\overline{\mathrm{\Delta }}}{v_t^2}$ $\mathrm{where},C_D=\{\begin{array}{ll}24/Re\left(1+0.15R{e}^{0.687}\right)\hfill & Re<1000\hfill \\ 0.44\hfill & Re⩾1000\hfill \end{array}\mathrm{and} Re=\frac{{\overline{\epsilon }}_g{\rho }_g{d}_s\left|{\tilde{u}}_g-{\tilde{u}}_s\right|}{{\mu }_g}$ $f\left(Fr\right)=\frac{F{r}_{}^{-1.3}}{F{r}_{}^{-1.3}+1.5}$ $h_{2D}\left({\overline{\epsilon }}_s\right)=\{\begin{array}{ll}2.7{\overline{\epsilon }}_s^{0.234}\hfill & {\overline{\epsilon }}_s<0.0012\hfill \\ -0.019{\overline{\epsilon }}_s^{-0.455}+0.963\hfill & 0.0012⩽{\overline{\epsilon }}_s<0.014\hfill \\ 0.868\exp \left(-0.38{\overline{\epsilon }}_s\right)-0.176\exp \left(-119.2{\overline{\epsilon }}_s\right)\hfill & 0.014⩽{\overline{\epsilon }}_s<0.25\hfill \\ -4.59\times {10}^{-5}\exp \left(19.75{\overline{\epsilon }}_s\right)+0.852\exp \left(-0.268{\overline{\epsilon }}_s\right)\hfill & 0.25⩽{\overline{\epsilon }}_s<0.455\hfill \\ \left({\overline{\epsilon }}_s-0.59\right)\left(-1501{\overline{\epsilon }}_s^3+2203{\overline{\epsilon }}_s^2-1054{\overline{\epsilon }}_s+162\right)\hfill & 0.455⩽{\overline{\epsilon }}_s⩽0.59\hfill \\ 0\hfill & 0.59<{\overline{\epsilon }}_s⩽0.65\hfill \end{array}$ $\frac{{\overline{p}}_{se}}{{\rho }_s{v}_t^2}=\{\begin{array}{ll}\frac{{\overline{p}}_s}{{\rho }_s{v}_t^2}+0.48F{r}_{}^{-0.86}\left[1-\exp \left(-\frac{F{r}_{}^{-1}}{1.4}\right)\right]\left({\overline{\epsilon }}_s-0.59\right)\left(-1.69{\overline{\epsilon }}_s-4.61{\overline{\epsilon }}_s^2+11{\overline{\epsilon }}_s^3\right)\hfill & {\overline{\epsilon }}_s\le 0.59\hfill \\ \frac{{\overline{p}}_s}{{\rho }_s{v}_t^2}\hfill & {\overline{\epsilon }}_s>0.59\hfill \end{array}$ $\mathrm{where} \frac{{\overline{p}}_s}{{\rho }_s{v}_t^2}=\{\begin{array}{ll}-10.4{\overline{\epsilon }}_s^2+0.310{\overline{\epsilon }}_s\hfill & {\overline{\epsilon }}_s\le 0.0131\hfill \\ -0.185{\overline{\epsilon }}_s^3+0.0660{\overline{\epsilon }}_s^2-0.000183{\overline{\epsilon }}_s+0.00232\hfill & 0.0131<{\overline{\epsilon }}_s⩽0.290\hfill \\ -0.00978{\overline{\epsilon }}_s+0.00615\hfill & 0.290<{\overline{\epsilon }}_s⩽0.595\hfill \\ -6.62{\overline{\epsilon }}_s^3+49.5{\overline{\epsilon }}_s^2-50.3{\overline{\epsilon }}_s+13.8\hfill & {\overline{\epsilon }}_s>0.59\hfill \end{array}$ $\frac{{\tilde{\mu }}_{se}\left|\overrightarrow{g}\right|}{{\rho }_s{v}_t^3}=\{\begin{array}{ll}\frac{{\tilde{\mu }}_s\left|\overrightarrow{g}\right|}{{\rho }_s{v}_t^3}+0.37F{r}_{}^{-1.22}\left({\overline{\epsilon }}_s-0.59\right)\left(-1.22{\overline{\epsilon }}_s-0.7{\overline{\epsilon }}_s^2-2{\overline{\epsilon }}_s^3\right)\hfill & {\overline{\epsilon }}_s\le 0.59\hfill \\ \frac{{\tilde{\mu }}_s\left|\overrightarrow{g}\right|}{{\rho }_s{v}_t^3}\hfill & {\overline{\epsilon }}_s>0.59\hfill \end{array}$ $\mathrm{where}\frac{{\tilde{\mu }}_s\left|\overrightarrow{g}\right|}{{\rho }_s{v}_t^3}=\{\begin{array}{ll}1720{\overline{\epsilon }}_s^4-215{\overline{\epsilon }}_s^3+9.81{\overline{\epsilon }}_s^2-0.207{\overline{\epsilon }}_s+0.00254\hfill & {\overline{\epsilon }}_s\le 0.0200\hfill \\ 2.72{\overline{\epsilon }}_s^4-1.55{\overline{\epsilon }}_s^3+0.329{\overline{\epsilon }}_s^2-0.0296{\overline{\epsilon }}_s+0.00136\hfill & 0.0200<{\overline{\epsilon }}_s⩽0.200\hfill \\ -0.0128{\overline{\epsilon }}_s^3+0.0107{\overline{\epsilon }}_s^2-0.0005{\overline{\epsilon }}_s+0.000335\hfill & 0.200<{\overline{\epsilon }}_s⩽0.6095\hfill \\ 23.6{\overline{\epsilon }}_s^2-28.0{\overline{\epsilon }}_s+8.30\hfill & {\overline{\epsilon }}_s>0.6095\hfill \end{array}$

(63)

Table 15

Equations for the EMMS model. (Yang et al. [110]).

The Drag Expression is Expressed as ($U_g=1.52\mathrm{m}/\mathrm{s},G_s=14.3\mathrm{kg}/{(\mathrm{m}}^2\cdot \mathrm{s})$)
$\beta =\{\begin{array}{cc}\frac{3}{4}{C}_D\frac{{\rho }_g{\epsilon }_g{\epsilon }_s\left|\overrightarrow{u_g}-\overrightarrow{u_s}\right|}{d_s}\omega & {\epsilon }_g\ge 0.74\\ 150\frac{{\epsilon }_s^2{\mu }_g}{{\epsilon }_g^{}{d}_s^2}+1.75\frac{{\rho }_g{\epsilon }_s\left|\overrightarrow{u_g}-\overrightarrow{u_s}\right|}{d_s}& {\epsilon }_g<0.74\end{array}$where$\begin{array}{l}\omega =-0.5760+\frac{0.0214}{4{\left({\epsilon }_g-0.7463\right)}^2+0.0044}\left(0.74\le {\epsilon }_g\le 0.82\right)\\ \omega =-0.0101+\frac{0.0038}{4{\left({\epsilon }_g-0.7789\right)}^2+0.0040}\left(0.82\le {\epsilon }_g\le 0.97\right)\\ \omega =-31.8295+32.8295{\epsilon }_g\left({\epsilon }_g>0.97\right)\end{array}$	(64)
Force balance for particles in the dense phase$\frac{3}{4}{C}_{Dc}\frac{f\left(1-{\epsilon }_c\right)}{d_s}{\rho }_g{U}_{sc}^2+\frac{3}{4}{C}_{Di}\frac{f}{d_{cl}}{\rho }_g{U}_{si}^2=f\left(1-{\epsilon }_c\right)\left({\rho }_s-{\rho }_g\right)(g+a)$Force balance for particles in the dilute phase$\frac{3}{4}{C}_{Df}\frac{1-{\epsilon }_f}{d_s}{\rho }_g{U}_{sf}^2=\left(1-{\epsilon }_f\right)\left({\rho }_s-{\rho }_g\right)(g+a)$
Pressure drop balance equation$C_{Df}\frac{1-{\epsilon }_f}{d_s}{\rho }_g{U}_{sf}^2+\frac{f}{1-f}{C}_{Di}\frac{1}{d_{cl}}{\rho }_g{U}_{si}^2=C_{Dc}\frac{1-{\epsilon }_c}{d_s}{\rho }_g{U}_{sc}^2$
Continuity equation of gas and particle phase$\begin{array}{c}{U}_g-f{U}_c-(1-f)U_f=0\\ U_s-f{U}_{sc}-(1-f)U_{sf}=0\end{array}$
Cluster diameter$d_{cl}=\frac{d_s\left[U_s/\left(1-{\epsilon }_{\max }\right)-\left(U_{\mathrm{mf}}+U_s{\epsilon }_{\mathrm{mf}}/\left(1-{\epsilon }_{\mathrm{mf}}\right)\right)\right]g}{N_{\mathrm{st}}{\rho }_s/\left({\rho }_s-{\rho }_{\mathrm{g}}\right)-\left(U_{\mathrm{mf}}+U_s{\epsilon }_{\mathrm{mf}}/\left(1-{\epsilon }_{\mathrm{mf}}\right)\right)g}$
The global stability condition$N_{st}=\frac{{\rho }_s-{\rho }_g}{{\rho }_s}g\left\{U_g+\left(f{U}_c-U_g\right)\frac{\left({\epsilon }_s-{\epsilon }_c\right)f^2}{1-{\epsilon }_s+\left({\epsilon }_s-{\epsilon }_c\right)f}\right\}\to \min$

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