Computational Fluid Dynamics (CFD) Technology Methodology and Analysis of Waste Heat Recovery from High-Temperature Solid Granule: A Review

Abstract

High-temperature solid granules are by-products produced by various industrial processes and contain an obvious quantity of waste heat. Therefore, recovering their heat can not only reduce energy costs but also prevent polluting the environment, which has a significantly valuable sense of sustainable development. Computational fluid dynamics (CFD) technology is widely used to solve challenges involving heat recovery, which can simulate the heat and mass transfer processes of the gas–solid two-phase flow. Herein, a review about the mass flow analysis methods, including the Euler–Euler and Euler–Lagrange methods, as well as heat transfer mechanisms, covering heat conduction, heat convection and heat radiation, is made. Meanwhile, the bases of numerical models, mass flow and heat transfer are also summarized. In addition, at the end of the paper, a prospect about this research field is proposed. This article not only reviews common research methods but also summarizes relevant new models and methods that have emerged in recent years. Based on existing work, it both fully demonstrates the widespread application of CFD technology in the field of recovering heat from high-temperature solid granule fields and summarizes the development trends and further utilization prospects of the technology.

1. Introduction

It has become a common agreement that waste heat recovery is an important technology for the full utilization of resources and sustainable development [1,2]. High-temperature solid granules, widely produced in many industries, often contain considerable amounts of heat, such as steel slags, an over 1400 °C by-product of the steel industry [3]. The recovery of 70% of the waste heat during the cooling down process of the high-temperature steel slags from 1400 °C to 200 °C can lead to the reclamation of over 950 kJ of heat per ton of slag. In 2023, over 101.9 million tons of rough steel were produced in China [4]. Given this, it will recover more than 116.1 × 10³ GJ of heat if heat recovery systems are implemented across all steel slags. The same or similar heat recovery potential also appears in cokes [5,6], blast furnace slags [7], cement sinkers [8], or other high-temperature solid granules [9]. Therefore, heat recovery technology is gaining more and more attention in academic and industrial fields. For example, Jiang et al. [10] in the tubular heat exchanger, noted that the gas–solid system heat recovery efficiency is improved by the increasing of these two coefficients.

There are many mature technologies for recovering waste heat from solid waste nowadays, most of which rely on the approaches such as the rotating cup granular method [11] and rotating drum granular method [12] to preliminarily cool and solidify molten solid material into relatively stable particles. M. Barati et al. gave a detailed review of the most common heat recovery from the high-temperature solid granule method [13]. However, the shapes and the diameters of the granular particles are complex (generally depending on the granular condition [14]), which makes the heat transfer condition inside the granular bulk also complex. In other words, a solid granular heat recovery system always means a high cost of time and funding. Therefore, in different conditions, we need to verify which design or process is more suitable.

The numerical approach is a much lower-cost method compared to experimental approaches. All we need is to set an accurate and credible model to find suitable heat recovery systems and corresponding conditions for specific application scenarios. During this and the past century, computational technology has rapidly developed. Computational fluid dynamics (CFD), a powerful technology combined with mathematics, engineering and computer science, was first used in the aerospace field [15]. In recent years, CFD technology has become widely used in various fields such as industrial process analysis [16], machine and vehicle design [17], civil and environmental engineering [18], chemical analysis [19], waste recycling [20], materials preparation [21] and so on. The foundation of CFD technology is based on three fundamental physics laws: the law of conservation of mass, Newton’s second law and the law of conservation of energy. These three laws are described as the continuity equation, momentum conservation equation and energy conservation equation. The parameters or the form of these equations may vary because of the difference in fluid characteristics or flow condition [22]. Due to this, these important characteristics such as fluid density and viscosity, flow channel and others should be confirmed before certain equations, which are needed in certain CFD calculations. In 1883, O. Reynold found, through the fluid flow through glass tube experiment, that the character of the motion of flow is related to not only the velocity of fluid, but also the product of the linear dimension of the space of the fluid (i.e., hydraulic diameter ($D_h$)) [22]. The feature size is different in different flow channels; for example, if the fluid flows inside a circular tube in a certain direction, $D_h$ is equal to the diameter of the tube. For complex flow channel, here is the equation to calculate $D_h$: $D_h={\displaystyle \frac{4A}{P}}$, where $A$ and $P$ are, respectively, the cross-sectional area and the perimeter of flow channel. As is well known, Reynolds number ($Re={\displaystyle \frac{\rho v{D}_h}{\mu }}$) is a standard non-dimensional number to judge whether the flow is laminar or turbulent. The experiments about the flow in a limit area under different flow conditions have been widely performed in the last century [23,24,25,26,27], but the value of Reynolds number is only distributed within a small range. Currently, with the development of computational technology, CFD can operate within a wider range of Reynolds numbers, including extremely high or low Reynolds numbers [28,29,30,31,32,33], which a benefit from the development of the computational calculation power, as well as the optimization of the calculating model and the grid structure. In light of this, the mentioned optimization is detailed, introduced and reviewed in the context. It can be clearly seen from these references that in different flow conditions such as different flow media, flow channel, or velocity, the Reynolds numbers are quite different, so in different CFD cases, the conservation equations used are also different.

In addition to the above, CFD and CFD coupling technology are also used for the analysis of mass flow and heat transfer in processes involving high-temperature solid granule flows, which are rarely comprehensively reviewed. Given this, this paper intends to review the application of CFD technology in the field of waste heat recovery from high-temperature solid waste. This paper summarizes and organizes recent advancements in CFD technology, including new theories such as model comparison and new grid partitioning methods, as well as empirical formulas proposed by researchers. As for the context of this paper, a brief introduction of using the CFD method to analyze the mass flow and heat transfer process will be made in Section 2, then the review will be divided into two major parts, mass flow introduction and heat transfer introduction, which are, respectively, introduced in Section 3 and Section 4 and the summary and prospects about the technology are made in the last chapter. This article provides a comprehensive and detailed summary of the existing methods for analyzing mass flow and heat transfer, which can help the researchers provide certain reference values in model selection, empirical formula selection, grid partitioning and other aspects.

2. Computer Fluid Dynamics (CFD) Method and CFD Coupling Method

CFD technology supports coupling with heat transfer process in the gas–solid two-phase system, which is an important step in solving such cases. The coupling between mass flow and heat transfer often requires researchers to customize program coupling interfaces through computer programming techniques. For example, the coupling of commercial software Altair^© EDEM and Ansys^© Fluent requires user-defined interfaces [34]. Researchers can also achieve CFD-DEM coupling by directly writing programs based on open-source code such as OpenFOAM.

CFD technology supports coupling with other regions. For example, in a transient reactor test facility reactor, CFD coupling methods can be separated in two ways: internal coupling and external coupling. Internal coupling exhibits a higher efficiency in message transfer; however, a requirement of coupling codes to exchange the data of these solvers always exists [35]. There are also several successful coupling methods examples in different research fields, such as the Reactor Monte Carlo code (RMC)–CFD software coupling [35], the High Order Spectral (HOS) and potential-meshfree CFD coupling [36], CFD–control system design (CSD) loose coupling [37] and CFD–Finite Element Method (FEM) coupling [38]. However, the Directly Numerical Simulation method (DNS method), CFD–two fluids method (TFM) and CFD–discrete element method (DEM) and multiphase particle-in-cell method (MP-PIC) are the most used to analysis the gas–solid two-phase system, which will be described in the next two sections. CFD and CFD coupling methods are widely used in heat recovery process analysis of high-temperature solid granules. Chen et al. [39] used the Reynold-Average Navier–Stokes method (RANS), one kind of method belong to the TFM, to analyze the dust deposition process during long-term heat recovery process in the tube-and-shell exhaust heat recovery exchanger. The result shows that both the relative heat resistance and the normal heat resistance of the heat recovery system would increase after a four-hour generation. Feng et al. [40] analyzed the heat recovery process moving a bed heat exchanger with embedded agitation by an extended CFD-DEM, and the result shows that the embedded agitation can enhance the efficiency of the heat recovery, with the proportion of the increase around 20%. Our last research was also focused on the heat transfer model analyzed by the CFD-DEM [41]. Yang et al. [42] investigated the mass flow and heat transfer process used the MP-PIC method; the simulation model shows a satisfied result corresponding with the experimental result. Feng et al. [43] used the TFM to study the gas–solid heat transfer process in a vertical tank for sinter waste heat recovery; in addition to verifying that the model is similar to the experimental results, it was also found that the air outlet temperature decreased with the increase in air flow rate and the decrease in inner diameter and height of cooling section, and there was a balance point between various parameter settings.

The development of more efficient CFD and CFD coupling technologies are one of the significant directions in the numerical simulation of gas–solid two-phase systems. For example, M.F. Diba et al. [44] propose simplifying both gas-phase and solid-phase CFD coupling approaches, which shows a realistic result on capturing the particulate phase in the dense stagnant regions of a bubbling, circulating fluidized bed.

To simulate the heat recovery process from high-temperature solid granules to cooled down media, a mass flow analysis and a heat transfer analysis are indispensable. As shown in Figure 1, to deal with a case about waste heat recovery from high-temperature solid granules or other gas–solid two-phase heat transfer cases, the suitable mass flow models and equations should be confirmed by considering the requirement, the complexity of the case and the computational resources at first; the calculation model for the heat transfer also needs to be determined, especially the heat radiation model. Then, an appropriate grid structure is needed for the computer to perform iterative calculations. Afterward, it is necessary to couple the flow and heat transfer processes, which are often achieved through user-defined code or programs.

3. Mass Flow Analysis

3.1. Solve Model Selecting

In Section 2, the fluid phase can be treated as a continuous body for further research based on the three conservation equations. However, for gas–solid two-phase flow, the solid phase requires different solution methods and models. Some models treat the solid phase as a continuous phase, tracking its flow trend and using the same or similar equations as those for the fluid; others, particularly when each solid particle is relatively large, prefer to discretize the solid phase and treat it as an element phase, with each element’s flow traced using Newton’s second law [45,46]. These two models are called the Euler–Euler model and the Euler–Lagrange model, respectively.

3.1.1. Two Fluids Method

The two fluids method is one of the most representative Euler–Euler methods, which treats both granular phase and fluid phase into continuum phases and uses similar equations to solve. The main equations for the solid phase in the TFM are quite similar to those for the fluid phase. In a given volume, the sum of the proportions of the solid phase and the gas phase equals 1, as described by the following equation:

$${\alpha }_s+{\alpha }_g=1$$

(1)

${\alpha }_s$—the proportions of the volume of solid phase; ${\alpha }_g$—the proportions of the volume of gas phase.

As the solid phase is treated as a continuum in the TFM, the solid phase conservation equations are similar to those of the gas phase. The mass and the moment conservation equations are

$${\displaystyle \frac{\partial }{\partial t}}({\alpha }_g{\rho }_g)+▽·({\alpha }_g{\rho }_g{\mathit{u}}_{\mathit{g}})=0$$

(2)

$${\displaystyle \frac{\partial }{\partial t}}({\alpha }_s{\rho }_s)+▽·({\alpha }_s{\rho }_s{\mathit{u}}_{\mathit{s}})=0$$

(3)

$${\displaystyle \frac{\partial }{\partial t}}\left( {\alpha }_g{\rho }_g{\mathit{u}}_{\mathit{g}}\right)+▽·\left( {\alpha }_g{\rho }_g{\mathit{u}}_{\mathit{g}}{\mathit{u}}_{\mathit{g}}\right)=-{\alpha }_g▽P_g+▽·\stackrel{̿}{{\mathit{\tau }}_{\mathit{g}}}+{\alpha }_g{\rho }_g\mathit{g}-\beta ({\mathit{u}}_{\mathit{g}}-{\mathit{u}}_{\mathit{s}})$$

(4)

$${\displaystyle \frac{\partial }{\partial t}}\left( {\alpha }_s{\rho }_s{\mathit{u}}_{\mathit{s}}\right)+▽·\left( {\alpha }_s{\rho }_s{\mathit{u}}_{\mathit{s}}{\mathit{u}}_{\mathit{s}}\right)=-{\alpha }_s▽P_g-▽P_s+▽·\stackrel{̿}{{\mathit{\tau }}_{\mathit{s}}}+{\alpha }_s{\rho }_s\mathit{g}-\beta ({\mathit{u}}_{\mathit{s}}-{\mathit{u}}_{\mathit{g}})$$

(5)

$\rho$ is the density of the phase; $\mathit{u}$ is the velocity vector; $P$ is the inside pressure of the two phases; $\beta$ is the moment exchange coefficient between the gas phase and solid phase; and $\stackrel{̿}{\mathit{\tau }}$ is the viscosity stress tensor (described by Equations (6) and (7)):

$$\stackrel{̿}{{\mathit{\tau }}_{\mathit{g}}}={\alpha }_g{\mu }_g\left(▽{\mathit{u}}_{\mathit{g}}+{{\mathit{u}}_{\mathit{g}}}^{\mathit{T}}\right)-{\displaystyle \frac{2}{3}}{\alpha }_g{\mu }_g▽·{\mathit{u}}_{\mathit{g}}\stackrel{̿}{\mathit{I}}$$

(6)

$$\stackrel{̿}{{\mathit{\tau }}_{\mathit{s}}}={\alpha }_s{\mu }_s\left(▽{\mathit{u}}_{\mathit{s}}+{{\mathit{u}}_{\mathit{s}}}^{\mathit{T}}\right)-{\alpha }_s{({\lambda }_s-{\displaystyle \frac{2}{3}}\mu }_s)▽·{\mathit{u}}_{\mathit{s}}\stackrel{̿}{\mathit{I}}$$

(7)

${\lambda }_s$ is the volume viscosity coefficient; $\mu$ is the viscosity coefficient.

Moreover, it is necessary to understand the drag force model and internal stress distribution of the solid phase when using the TFM to solve the problem [47,48,49]. The kinetic theory of granular flow (KTGF) was proposed to deal with these problems. In this theory, Ding and Gidaspow, based on the kinetic theory of dense gases, proposed a method that replaced the usual thermal temperature with a granular flow equation for which a differential equation is derived with the utilization of the kinetic theory [50]. Gidaspow [51] used an empirical correlation of elasticity between two particles to analyze element velocity vectors in a 2D gas–solid fluidized bed, such as pressure porosity fields, and flow patterns. Particle dynamics model equations are also wildly used in analyzing the internal properties of the solid phase [52,53]. The models mostly used in TFM analyzing are Sankaran Sundaresan’s models [54,55]; the Syamlal–O’Brien model [56,57], Wen–Yu model [58] and Gidaspaw model [59] are also commonly used in TFM analyzing, and some similar research also directly set the drag coefficient [60]. In recent years, many new drag force models in different conditions have been proposed [47], such as Zhang’s theory for the burning slags [61] and Li’s theory for heterogeneous theory [62]. It should be noticed that the coefficient $C_d$ is the “drag coefficient”, which is described differently in various references, but is always related to the Reynolds number of the gas phase [63]. The internal drag coefficient of particles with Reynolds numbers less than 200,000 had been reviewed by Walter [64]; from this reference it can be found that both the Reynolds number and drag coefficient calculating equation should be selected based on a specific analyzed method, as does the drag force model. There are many modifications that are suitable for different working conditions. Uwitonze introduced the volume fraction volume into the calculation of $C_d$ in the study of heat transfer in circulating fluidized beds [65]. The energy-minimization multi-scale (EMMS) drag force model is a model that focuses on the meso-scale phenomenon of a particle cluster, which can predict the saturation carrying capacity and the steady states at both the top and the bottom of an S-shaped axial profile.

The TFM is widely used in many kinds of big industrial mass flow (and heat transfer) cases. For example, Lu et al. [66] use the TFM to simulate the granular velocity distribution and granular pressure distribution in a 3D industrial-scale circulating fluidized bed furnace, and Shah [67] studied the prediction of mass flow behavior and the interphase force characteristics of a 3D large-scale circulating fluidized bed furnace. These two studies both compared the Wen–Yu drag force model and energy-minimization multi-scale (EMMS) drag force model, and the results showed that the EMMS model is more suitable for multi-scale fixing solid granule flow analysis than the Wen–Yu drag force model.

Besides traditional TFM analyzing, the Large Eddy Simulation method (LES) is also used in analyzing gas–solid two-phase flow, especially when used for analyzing the time-varying turbulent flow cases such as swirling flow, and the results always have a better performance because of the use of sub-grid scale (SGS) stress, which has been developed in recent years [68,69]. However, even though LES has been used in analyzing the mass flow characteristics of gas–solid two-phase flow for a long time, heat transfer analysis via this method underwent little research until the 2020s [70,71], which means that there is a research gap in this region.

3.1.2. CFD–Discrete Element Method

The CFD-DEM coupling method is predominantly used due to its higher accuracy. DEM is an analysis method suitable for discontinuous research objects. Cundall and Strack [72] first proposed this method and employed it to calculate the force–displacement relationship within a granular accumulation system. The following describes the force–displacement relationship between two contacting disks.

From Figure 2, it is evident that shear force and normal force are present at the contact area between two disks. These disks can be modeled as two soft, elastic objects where contact occurs over a period rather than at a single time point (i.e., using the soft ball model). This interaction includes three stages, contact, extrusion and restoration, so that the contact forces are directly related to the characteristics of the contact surface such as the contact area and friction coefficient and so on. Two disks (two particles in 3D) using the contact force calculate method were described in detail by Hertz [73], Deresiewicz [74] and R.D. Mindlin [75,76]. K.L. Johnson et al. [77] discovered that an adhesion energy exists between the two contact bodies, which would prevent the contact bodies from being separated.

The fluid phase equation is the same as the TFM, and the normal force and the tangential force can be calculated by the following equation [78]:

$${\mathit{F}}_{\mathit{n}}=-k_{\mathit{n}}{\mathit{\delta }}_{\mathit{n}}+C_n{\mathit{v}}_{\mathit{n}}$$

(8)

$${\mathit{F}}_{\mathit{t}}=\mathrm{m}\mathrm{i}\mathrm{n}(\mu {\mathit{F}}_n,\left|-k_t\int {\mathit{v}}_{\mathit{t}}dt{-C}_t{\mathit{v}}_{\mathit{t}}\right|)$$

(9)

$k_{\mathit{n}}{,k}_{\mathit{t}}$ is the normal (tangential) spring coefficient; ${\mathit{\delta }}_{\mathit{n}},{\mathit{\delta }}_{\mathit{t}}$ is the normal (tangential) spatial overlap distance; $C_n,C_t$ is he normal (tangential) damping coefficient; and ${\mathit{v}}_{\mathit{n}},{\mathit{v}}_{\mathit{t}}$ is the normal (tangential) relative velocity.

R.D. Mindlin described the tangential force in detail in ref. [75]. The following figures illustrate the variation trend of the eccentric distance of the contact surface as the tangential force T changes.

From Figure 3 we notice that during the process of P—S and S—P, at the point of tangential force back to 0, there still exists an eccentric distance because of the self-equilibrating distribution [75].

Based on the above result, the total force and total moment on a single disk (or particle) in a granular accumulation system can be calculated using the following equations:

$$m_i{\displaystyle \frac{d{v}_i}{dt}}={\mathit{F}}_{\mathit{g}}+{{\sum }_{j=1}^n\mathit{F}}_{nij}$$

(10)

$$I_i{\displaystyle \frac{d{\omega }_i}{dt}}=+{{\sum }_{j=1}^n\mathit{r}\times \mathit{T}}_{ij})$$

(11)

${\mathit{F}}_{\mathit{g}}$ is the gravity of the particle; ${\mathit{F}}_{nij}$ is the contact force between particle i and j; $\mathit{r}$ is the distance between the contact point and the center of the particle; and ${\mathit{T}}_{ij}$ is the tangential force between particle i and j.

Oda, Konishi and Iwashita [62] proposed a modified DEM (MDEM) based on the DEM. In the MDEM, rolling generated with the direction perpendicular to the contact surface as the axis is considered, compared to the traditional DEM where rolling freely exists with no resident force. The rolling resident force in the MDEM is related to rotational stiffness and relative rotation due to rolling. The total MDEM force–displacement model is shown in Figure 4.

Compared to other CFD coupling methods, CFD-DEM and TFM coupling is much more suitable for gas–solid two-phase flow, especially in circulating fluid bed mass flow and heat transfer analysis. In recent years, a large quantity of papers about the CFD-DEM coupling model have been developed and verified.

There have been many innovative studies about CFD-DEM itself in recent years; for example, coarse-grained (CG) CFD-DEM, which is proposed in this century as a CFD-DEM simulation method that ensures computational accuracy while reducing computational power requirements by merging particles that are close in position and parameters into coarse particles [80,81], is wildly used in many simulations. Chu et al. used this method in analyzing gas cyclone gas–solid two-phase flow [82,83]. However, the research in using CG CFD-DEM to analyze the heat transfer process started in the 2020s, so there is still a significant research vacancy in this region [84], as well as in the comparation of CG CFD-DEM and traditional CFD-DEM in different mass and heat transfer cases. De Munck et al. [85] compared the accuracy of the CG CFD-DEM model with the traditional CFD-DEM in a fixed bed case, the result shows a very well agreement in the dimensionless temperature profiles, with the quantity of coarse-grain ratios at 1–2.5, which is shown in Figure 5. De Munck also verified that during the fluidization process of particles in a bubble fluidized bed at different gas flow velocities, the temperature of particles in the CG CFD-DEM and CFD-DEM are almost the same. Furthermore, other optimization models from the DEM have also appeared, for example, the dense discrete element model.

There are also many comparation research results on the TFM and CFD-DEM: N. Almohammed [86] compared the TFM and CFD-DEM in gas–solid two-phase flow by a lab-scale spouted fluidized bed and compared the simulate result with the experimental one. The result shows that with a flow rate of 0.005 kg/s, both the TFM and CFD-DEM can achieve an acceptable simulate result, but with the flow rate increasing to 0.006 kg/s, the accuracy of TFM result decreases fast, and the CFD-DEM achieves a better simulate result. In addition, N. Almohammed also summarized formal research about the comparation of these two methods [63,87,88], and the result reflects that the CFD-DEM mostly performs better agreement with experimental results than the TFM, but the cost of the simulating time necessary to perform the numerical simulation with the CFD-DEM model is much larger than to perform a simulation with the TFM, even four orders of magnitude higher. M. Lungu [45] also compares the TFM and CFD-DEM in a small-scale fluidized bed (0.075 m × 0.23 m × 1.22 m) using open source code MFIX; in this simulator, the basic flow features are well predicted by both modeling approaches, such as the core-annulus structure and slugging dynamics, and the result reflects that the accuracy of the CFD-DEM is higher than TFM. The reason is the overprediction of the mean pressure drop by the TFM is likely due to the relatively larger predicted values of the mean voidage. P. Ostermeier et al. [46] compared the TFM and CFD-DEM by a pilot plant reactor scale simulator. The result indicates that the TFM is rather suitable for monodisperse particles in small and complex geometries with small computational cells. In contrast, the CFD-DEM can be applied for strongly polydisperse particles and to depict cohesion effects. In addition, for selecting grid and time step independence, the results of the CFD-DEM depend to varying degrees on the parcel number and particle time step size, while the TFM is totally independent in the frame considered. It is worth noting that with the development of computer technology [89,90], the simulation of gas–solid two-phase flow is increasingly inclined to use the Euler–Lagrange model compared to the Euler–Euler model. In recent years, especially since 2020, the difference in the number of articles published using different models for numerical calculations can to some extent verify this statement.

3.1.3. Direct Numerical Simulation Method

The Direct Numerical Simulation (DNS) method permits researchers to set the equations and the relationship directly based on different research situations. The boundary condition can also be directly defined in this method, which leads to an extremely high accuracy in each case. Unlike the TFM and CFD-DEM, DNS is independent of any empirical assumptions and can directly solve all parameters in the flow field. However, to obtain an accurate result, the size of the grids used in DNS is quite smaller than particle size, and a large number of fine grids cause an enormous computational load; because of that, most early numerical simulations by the DNS method are limited in a small space (mostly smaller than 0.1 m³) and a small quantity of particles (less than 100,000) [91].

Table 1. DNS research and quantity of particles used in research.

Researchers	Quantity of Particles
Höfler and Schwarzer (2000) [92]	10,000
Yin and Koch (2007) [93]	8000
Yin and Sundaresan (2009) [94]	5208
Jin et al. (2009) [95]	21,336
Pan et al. (2001) [96]	1000

lists some early studies that used the DNS method and the quantity of particles used by the research institute:

The equations of the fluid phase and solid particle phase are mostly the same as the CFD-DEM, but the equations that can be defined rely on the users’ request; for example, in Tang’s interactive immersed boundary DNS method, the fluid phase momentum equation around the fluid-solid boundary can be described as

$${\displaystyle \frac{\partial }{\partial t}}\left( {\rho }_g{\mathit{u}}_{\mathit{g}}\right)+▽·\left( {\rho }_g{\mathit{u}}_{\mathit{g}}{\mathit{u}}_{\mathit{g}}\right)=▽P+{\mu }_g{▽}^2{\mathit{u}}_{\mathit{g}}+{\mathbf{f}}^{IB}$$

(12)

${\mathbf{f}}^{IB}$ is the force density in the contact area.

The numerical simulation result showed good accuracy in intermediate-${Re}_g$ flows (${Re}_g$ = 50, 100) [97]; this model is also used in a pseudo-2D gas fluidized bed numerical simulation [98].

The DNS method has also been used in 2D and 3D heat transfer research in recent years [99,100,101]. Huang et al. [102] investigated the flow and heat transfer in bidisperse gas–solid systems with particle diameter ratios ranging from 1:2 to 1:4 and pursued the overall Nusselt number, and based on this, a predict model was formulated to predict the overall Nusselt number through a correction to the monodisperse heat transfer correlations. Chandil M et al. [103,104] studied the heat transfer process under different ${Re}_g$ particle-resolved DNS methods. However, although there are currently some large-scale numerical simulations, the amount of research on using DNS to study in this field is still relatively limited.

Overall, the DNS method has a higher calculated accuracy compared to the TFM and CFD-DEM, but the relative computational cost is also much larger than the former due to an extremely large quantity of grid in each gas–solid or liquid–solid two-phase flow case. Based on this, optimizing the grid structure to reduce computational complexity and thus reduce computation time will be a research and development direction for the DNS method. Meanwhile, due to the large computational complexity of DNS methods, establishing a database and utilizing machine learning to improve efficiency is also a development direction [105].

3.1.4. Multiphase Particle-in-Cell Method

Due to the numerous computational power costs of the CFD-DEM, many optimization methods are developed based on it, such as dense discrete element model (DDPM) and multiphase particle-in-cell (MP-PIC) method. The DDPM is more suitable for these simulations which have dense solid phase and only focus on the collision between solid granules. P. Ostermeier [46] also mentioned an analysis method called the Dense Discreate Phase Method (DDPM). The characteristics of the DDPM are roughly between the TFM and CFD-DEM, so that the DDPM is intended to be applied to large poly-disperse particulate systems with lower computational demand than the CFD-DEM under the same simulated setting. Chen et al. compared the result of process analyzing impinging gas–solid jets in a channel, and the result reflects that the DDPM method cannot predict the impinge process very well due to the over-simplified treatment of two particles’ interaction [106].

The MP-PIC method is a novel discrete particle simulation approach developed in recent years, optimized from the DEM and more commonly used than the DDPM in analyzing gas–solid two-phase flow because of higher accuracy and a more detailed result of granules’ sliding and rolling. This method uses particle phase pressure gradient force instead of particle collision to characterize the interaction force between particles and calculates the particle motion trajectory. This method was proposed in the 1960s as a single-phase particle-in-cell (PIC) method and has been utilized in multiphase particle analyzing in the 1990s and this century [107,108,109]. Same as the CG CFD-DEM, the MP-PIC method also treats particles which have the same position, velocity, volume and density as a single particle cell. However, different from the CG CFD-DEM, in the calculation process, the parameters of the particle cells are consistent with other normal particles, and only the number of particles contained in these particle cells is considered when calculating the solid-phase breadth at the corresponding position (shown in Figure 6). By introducing particle phase pressure to replace the complex particle collision calculation in the CFD-DEM, this simplified method can also significantly reduce the computational workload related to particle collisions during the calculation process.

The equations of the fluid phase and solid phase are also quite close to the CFD-DEM, but the inside stress needs to be further described due to the close correlation between stress parameters and actual particle collisions. Snider [108], based on Lun’s model and Harris–Crighton’s [110,111] models, proposed the calculation equation:

$$p_s={\displaystyle \frac{p_s\times {{\epsilon }_s}^{\alpha }}{\max \left[{\epsilon }_{s,cp}-{\epsilon }_s,\delta (1-{\epsilon }_s)\right]}}$$

(13)

${\epsilon }_{s,cp}$ is the proportion of solid under the densest packing condition; $\alpha , \delta$ is the dimensionless parameters ($\alpha$ and $\delta$ rely on many characteristics of the simulation cases).

There have been many works using the MP-PIC method to study gas–solid two-phase in recent years: Li et al. [112] used MFIX to couple the EMMS and MP-PIC to calculate the flow of particle flow in the ascending pipe of a 2D meso-scale circulating fluidized bed, and the results were similar to the experimental results. Recently, there have been many studies using the MP-PIC method to simulate particle flow and heat transfer processes in large circulating fluidized beds [113,114,115,116].

Li et al. [117] used the DEM and MP-PIC to numerically simulate the fluidization process of the same biomass fluidized bed. The results showed that compared with the MP-PIC, the CFD-DEM performed poorly in dealing with complex flow regime changes and non-steady state characteristics of the reactor. This may be because CFD methods typically use discrete grids to simulate flow fields, which may result in a loss of accuracy when simulating nonlinear reactors. However, there is still a gap in the comparative study of the CFD-DEM and MP-PIC under different case conditions.

3.2. Grid (Mesh) Building

Essentially, computer solving process is an iterative process, and it is necessary to build a suitable grid (synonymous with mesh) construct to address various engineering problems. The structure of a grid construct is closely related to the complexity of simulated geometry and boundary conditions. The grid size, especially the grid size of the DNS method, CFD-DEM and MP-PIC should be determined based on the case type and particle size, and the grid size of the TFM is not directly related to particle size. The grid size of the CFD-DEM and MP-PIC is bigger than the particle size while the DNS method’s gas-phase grid size is much smaller than the particle size; moreover, the grid partitioning method varies depending on the selected model [118].

The number and quality of grids directly impact the simulation accuracy of simulation. Finer grids structures lead to more accurate results but demand more time and computational power. Grid constructions are generally categorized into structured and unstructured grids. Structured grids offer advantages like faster construction, quicker calculations and better grid quality but are less suited for complex models compared to unstructured grids (see Figure 7) [119,120,121,122,123].

However, as the flow channel becomes more complex, a simple grid divide method cannot adapt more sophisticated channels and boundaries, and local grid refinement method (LGR) is proposed to fit this situation and obtain more accurate data in the CFD-DEM and MP-PIC method. For example, B. Blterge [121], based on Heinemann’s LGR method [120], proposes an adaptive static and dynamic local grid-refinement technique, which can be used in the solution of nonlinear parabolic differential equations. The technique can detect the boundary data gradient level and auto-refine the grid. Furthermore, it can define a nonhomogeneous Dirichlet-type boundary condition from the coarse grid solution to fine grids at the interfaces of coarse/fine grid, which ensures the data accuracy and reduce the quantity of computer calculations. In recent years, many 2D or 3D physical field CFD studies have also used local mesh refinement techniques, whether using the TFM or CFD-DEM [124,125,126,127,128,129].

As for the CFD-DEM, it deals with non-sphere gas–solid flow condition, the solution method can be separated into a resolved CFD-DEM class and unresolved CFD-DEM class, and the ratios of fluid grid size to particle size are quite different [130,131]. The resolved CFD-DEM applies to any shape particles, but it requires the fluid grid to accurately resolve the particle shape because the accuracy of the particle shape directly determines the accuracy of the fluid flow and particle motion solutions; in this method, the grid size is always smaller than solid granules [132]. The resolved CFD-DEM can also deal with non-spherical particles [133]; for non-spherical particles, a grid construct with single grid size of ${\displaystyle \frac{1}{10}}$ of the granular volume equivalent diameter can ensure about 1% of the calculation accuracy [134]. The unresolved CFD-DEM calculates the interaction between particles and the surrounding fluid by a drag force model (mostly an experimental empirical formula) and because of that, it does not need to describe the particle boundary by the fluid grid, so relatively large fluid grids can be used, but the drag force becomes more important in this method, and it needs a large number of experiments to verify a drag force model or make certain a coefficient [131,134,135,136].

The advancement of grid partitioning technology mainly brings two contributions: improving computational accuracy while reducing computational power consumption. Local mesh refinement is a method for dealing with complex flow fields or complex parts of flow fields. Local grid refinement always appears at the inlet, outlet [134] and complex area (such as the corner and bend area) [137,138,139,140] of the gas–solid two-phase granular flow channel.

In addition, local mesh refinement can also occur at complex flow boundary conditions due to not only the friction between the coarse wall and the gas phase or solid phase (if using TFM) but also the dissipation from particle–wall collisions; the later one is influenced by the particle–wall restitution coefficient. Worth being noticed, the dissipation of collision between the particles and wall can also occur although the wall is smooth [141,142]. Both of these factors make the calculation of mass flow near the wall more complex, therefore grid refinement is required for positions near the wall [143,144,145,146]. Nuno M.C. Martins et al. [143] made an effort to find “the most efficient” grid structure of a 4 m long, 0.02 m diameter water pipe under two kinds of flow condition (laminar flow and turbulent flow), which means the maximum accuracy and the minimum computational calculate effort. The result indicated that in a water pipe, it is important to keep the meshes thin enough near boundaries where velocity gradients are higher, which means the length of circumferential-direction and the radical-direction of the grid should be the smallest, but on the other hand, the length of the axial-direction of the grid seems that the impact to the result is not significant.

Considering the characteristics of structured and unstructured grids, structured grids are much more suitable for regular geometry such as cylinders or rectangles and unstructured grids opposites. However, common CFD solvers can only be applied to one grid form such as ANSYS^© Fluent which prefer unstructured grid structure to reduce the number of grids in the structure. Nowadays, Candence^© provides a solver which can calculate a mixed grid construct [147], but so far, no reliable articles or reports have described the Candence Fidelity CFD solver dealing with complex construct flow channel problems.

Generally speaking, for different particle sizes and research models, it is necessary to set up grids of different sizes. However, compared to the size of the entire flow channel, the size of a single grid is not significantly different from the particle size. Therefore, if it is necessary to simulate actual working conditions such as industrial production conditions, a huge grid structure is required to achieve it, and this structure also requires a huge amount of computer calculating power [134]. In terms of reducing computational power consumption, Qiu et al. [122] proposed a local grid refinement method to solve wide particle size distribution gas–solid two-phase flow problem, as shown in Figure 8; this method is different from Blterge’s method, as this method detects the distribution of coarse particles and fine particles. When it detects a fine particle in a coarse grid, the coarse grid will be separated into several fine grids regularly, and this method keeps the ratio of grid size and particle size as 3:1 to ensure the accuracy of calculation; but, if there is less grid to compare than only fine grid is constructed to reduce computing cost. In addition to the improvement methods mentioned for grid building, many other grid-construction methods have also been proposed in recent years, and this aspect has become one of the future development directions of this technology [89,90].

In summary, each numerical simulation method for particle flow has its domain of adaptation. The TFM uses empirical models to analyze the internal stress of the solid phase and neglects collisions and other behaviors between particles because it treats all particles as an independent phase. As a result, the TFM allows for the simulation of large-scale cases with lower computational complexity. The CFD-DEM can obtain results with higher accuracy compared to the TFM by studying the interaction between each particle separately. However, the increased computational power requirements and computation time under the same cases will also significantly impact performance. The MP-PIC method combines the advantages of the TFM and CFD-DEM by employing particle-in-cell methods, which significantly reduce computational complexity while maintaining accuracy. In comparison to traditional CFD-DEM methods, this approach allows for the calculation of particle interactions under larger operating conditions. The DNS method exhibits high degrees of user control in gas–solid contact areas without relying on empirical formulas or models, resulting in highly accurate solution results. However, despite its potential, numerical simulations for larger operating conditions still require significant computing power resources (

Table 2. The comparation of four major CFD gas–solid two-phase analysis methods.

Simulation Method	Advantages	Disadvantages	Recommend Application Region
TFM	Lowest computational power cost. Treats the whole solid phase as a fluid phase.	Ignores the interaction of solid granules inside the phase. Most cases need two grid constructions to satisfy the accurate requirement.	The large-scale gas–solid two-phase systems without the need to study the interactions between particles, for example, circulating fluidized bed.
CFD-DEM	The interaction information of particles is calculated by the selected contact force model. Higher calculation accuracy of the same case than the TFM.	Significantly bigger computational cost than the TFM in calculating the same case (can be optimistic by new technology such as the CG CFD-DEM).	Mostly a laboratory-scale or proportionally reduced model. Sometimes also used in large-scale systems, but not as frequently as the TFM due to high computational power cost.
DNS	Highest computational accuracy. Can customize boundary conditions without being constrained by existing models.	Highest computational cost due to the extremely high quantity of grid.	Little-scale gas–solid two-phase flow with a very limited number of particles.
MP-PIC	Uses the particle-in-cell method to significantly reduce the computational cost with a satisfied accuracy.	Only approximate particle interaction data due to the ignoration of the characteristics of individual particles.	Suitable for larger systems with a bigger quantity of granules than CFD-DEM model.

).

4. Heat Transfer Analysis

This section provides a comprehensive review of studies investigating heat transfer behaviors in the context of waste heat recovery from high-temperature solid wastes. The primary objectives are to (1) briefly introduce mainstream simulation methods for resolving particle-scale heat transfer, (2) provide a foundation for selecting appropriate heat transfer models based on specific research problems, and (3) outline the boundary and initial condition settings for these models.

In the research field involved in this review, heat exchangers operating in packed bed, moving bed and fluidized bed states [9] are typically employed to recover the waste heat from high-temperature solid wastes. The simulation research concerning heat transfer in waste heat recovery systems tends to adopt methods based on only the DEM, TFM and CFD-DEM [148]. Herein, only DEM refers to a method only using DEM to characterize the heat transfer phenomena related to particles. The DEM is an efficient technique for accurately simulating particle flow behaviors at all scales by treating particle materials as a discrete phase rather than a continuous one. In fact, the characteristics of granules have influence on the heat transfer [149]. As mentioned in Section 2, more physically realistic contact models for obtaining contact forces, as well as motions equations based on Newton’s second law for further capturing motion information, are used in the DEM. However, the initially proposed DEM cannot resolve the particle-scale heat transfer due to the lack of sub-models. Therefore, many researchers endeavor to come up with their particle-scale heat transfer sub-models, integrate them into the framework of the DEM, and utilize them to further study the specific heat transfer problems. Based on these, several enhanced DEM models integrated with heat transfer sub-models have been developed, enabling the simulations involving particle-scale heat and mass transfer. The most commonly used heat transfer sub-models are compressively reviewed in the following section. Many scholars have employed this approach to address particle-scale heat transfer while neglecting the involvement of the fluid [150,151]. Nevertheless, due to the neglect of fluid participation, the simulations based on only DEM are not suitable for scenarios where gas–solid heat transfer phenomena cannot be ignored, such as a case introducing strong gas–solid interactions into the heat transfer systems. Additionally, when a large number of particles must be considered—particularly when the count reaches tens of millions or more—the computational cost becomes prohibitively high.

Given these drawbacks, the TFM serves as a good alternative to using only DEM to analysis heat transfer processes. It effectively considers the presence of interstitial fluid surrounding the particles and significantly reduces computational costs by regarding the solid phase as a quasi-fluid phase. A detailed introduction to the mass transfer modeling of the TFM can be found in Section 2. Moreover, the modeling of heat transfer in the TFM primarily focuses on characterizing the heat transfer coefficients between particles and particles, particles and fluids, as well as fluids and fluids. In other words, the inter-particle effective thermal conductivity, the interphase heat transfer coefficient, and the inter-fluid thermal conductivity must be characterized during the simulation. Due to its unique approach to handling the solid phase, these parameters may not always reflect physical reality entirely, which often necessitates the use of specific empirical or semi-empirical correlations to model and calculate them. For instance, gas–solid interphase heat transfer coefficients are typically predicted by employing empirical correlations of the Nusselt number ($\mathrm{N}\mathrm{u}$) [152]. In fact, the so-called heat transfer coefficients and effective thermal conductivities are comprehensive reflections of several heat transfer processes, including conduction, convection and radiation, for the convenience of particle-scale heat transfer modeling. In return, the TFM is expected to be most cost-effective compared to other simulation methods. However, it is challenging to quantify the respective contributions of conduction, convection and radiation to the overall intensity of heat transfer. Meanwhile, in studies based on the TFM, the mean particle size must be kept small enough to maintain the validity of the approximate continuity assumption for the particle phase. This requirement significantly limits the method’s applicability when dealing with large particles.

Based on these disadvantages of the TFM, some scholars have made many attempts with the use of the CFD-DEM. As mentioned above, the CFD-DEM, which inherits all the features of the DEM, can provide a comprehensive understanding of mass transfer characteristics. Moreover, with the introduction of the heat transfer sub-model, the CFD-DEM emerges as the most physically realistic method for investigating the heat and mass transfer characteristics in the gas–solid systems. However, it is foreseeable that the computational cost of the CFD-DEM is the highest among these methods. In addition to these mainstream methods, researchers have developed several simplified approaches to investigate the heat transfer in the waste heat recovery systems. These include, but are not limited to, treating the packed-state particles as a porous media entirety [43], grouping similar particles into one parcel (coarse-grained CFD-DEM) [153], coupling DEM with fast transform spectral method [154], adopting plug/viscous flow simplification [155] and employing energy conservation analysis [156,157]. A detailed introduction to the additional simulation method can be found in the corresponding source literature.

In this section, the common heat transfer sub-models used in only DEM, the TFM and CFD-DEM are reviewed following the general classification, which is heat conduction, heat convection and heat radiation. Herein, the particle-scale sub-models are primarily utilized to obtain the heat fluxes inside the particle, as well as between particles and other particles, and between particles and fluids. The heat fluxes between particles and walls, as well as between fluids and walls, are ignored due to their similarity to the previously mentioned fluxes related to particles.

4.1. Particle-Scale Heat Conduction

Particle-scale heat conduction will occur when an obvious temperature difference exists within a particle (internal heat conduction), between two adjacent particles in contact (contact heat conduction) or among two nearby particles and their interstitial fluid (gas film heat conduction).

The internal heat conduction theory was first deduced by Yovanovich et al. [158] and has been widely used to characterize the heat flux of particle internal heat conduction [159,160,161]. The theory holds the view that (1) the internal heat conduction is induced by the contact of the particle surface with an object at a higher or lower temperature; (2) the temperature distributes uniformly within a certain radius ($R_u$) which is smaller than the particle radius ($R$); and (3) a particle at this radius owns half of total particle mass [159]. (Figure 9) Based on Fourier’s law, the thermal resistance of internal heat conduction ($R_{ic}$) can be deduced, which is given in Equations (14) and (15). For convenience, the equations adopt a form of coupling with gas film heat conduction and let the gas film thickness ($\delta$) be 0 so it can obtain its original form. The internal heat conduction should be considered in cases where this form of thermal resistance cannot be ignored. For instance, Guo et al. [162] adopted the sub-model of internal heat conduction to characterize the temperature gradient inside the particle due to the fact that the criterion (see Equation (16)) related to Biot number ($\mathrm{B}\mathrm{i}$) is not met.

$$R_{ic}={\displaystyle \frac{\sqrt[3]{2}-1}{2\pi k_{p}R(1-\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_1)}}$$

(14)

$$\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_1={\displaystyle \frac{l_{ij}^2+R^2-{(R+\delta )}^2}{2l_{ij}R}}$$

(15)

$$\mathrm{B}\mathrm{i}={\displaystyle \frac{k_f}{{0.085k}_p}}\ll 1$$

(16)

$k_p$ is the particle thermal conductivity; ${\alpha }_1$ is one of the included angle (correspond to Equation (20)); $l_{ij}$ is the particle centroid distance; and $k_f$ is the fluid thermal conductivity.

The contact heat conduction theory was first proposed by Batchelor and O’Brien, which is successfully implemented and further verified by many scholars [163,164,165]. This theory suggests that (1) two contacting particles can conduct heat through their contact surface; (2) the heat flux is proportional to the effective thermal conductivity, temperature difference, and the cube root of the contact area; and (3) the effective thermal conductivity is the harmonic mean of the thermal conductivities of two contacting particles. Based on these, the thermal resistance of contact heat conduction ($R_{cc}$) can be calculated according to Equation (17). Similarly to internal heat conduction, contact heat conduction flux should be considered in cases where this form of thermal resistance cannot be ignored, particularly when the effective thermal conductivity is high enough and particles are in close and compliant contact (Figure 10). Chaudhuri et al. [163] conducted their study on particle-scale heat and mass transfer in rotary calciners and impregnators, focusing solely on contact heat conduction based on their thermal resistance analysis. In their cases, O’ Brien estimation [166] (see Equation (18)) were adopted to support their assumption that only thermal conduction within the solid and thermal conduction through the contact area between two particles in contact were considered, while the former was also disregarded due to the small $\mathrm{B}\mathrm{i}$.

$$R_{cc}={\displaystyle \frac{k_{p1}+k_{p2}}{4k_{p1}{k}_{p2}}}{\left[{\displaystyle \frac{4E}{3F_{n}R}}\right]}^{1/3}$$

(17)

$${\displaystyle \frac{k_p{r}^{*}}{k_{f}R}}\gg 1$$

(18)

$F_n$ is the normal force; $E$ is the particle Young’s modulus.

The gas film heat conduction theory is often used to take into account fluid participation (Figure 11). This theory, initially proposed by Rong and Horio and later modified by Musser [167], assumes that every particle is surrounded by a gas film of limited thickness. According to gas film heat conduction theory, when the surface distance of the binary pair is less than the gas film thickness $\delta$, the heat will be transferred from the high-temperature particle to the low-temperature one in the form of conduction through the gas film overlapping area. Based on Fourier’s law, the thermal resistance of gas film heat conduction ($R_{gfc}$) can be deduced, as shown in Equation (19). As mentioned above, when O’Brien estimation [166] is not satisfied, the gas film heat conduction sub-model should be integrated into the overall heat transfer model. Furthermore, it should be noted that the temperature-dependent property of fluid thermal conductivity should be further considered when the fluid temperature undergoes sharp changes. This, in turn, requires the simulation method to be capable of resolving the fluid information, suggesting that only DEM is not suitable in this case and necessitating the use of CFD. However, the gas film thickness $\delta$, a critical parameter which is usually from 0.2$R$ to 1.0$R$ [168], is often chosen by subjective experience or assumption, lacking a critical selecting criterion. Lu et al. [169] attempted to determine the sensitivity of this parameter to the final heat transfer result, and they found that the result exhibits moderate sensitivity to this parameter compared to other parameters.

$$R_{gfc}={[{\lambda }_f{\int }_{{\beta }_1}^{{\alpha }_1}{\displaystyle \frac{2\pi R\mathrm{s}\mathrm{i}\mathrm{n}\theta d(R\mathrm{s}\mathrm{i}\mathrm{n}\theta )}{l_{ij}-2R\mathrm{c}\mathrm{o}\mathrm{s}\theta }}]}^{-1}$$

(19)

$$\mathrm{c}\mathrm{o}\mathrm{s}{\beta }_1={\displaystyle \frac{l_{ij}}{2R}}$$

(20)

${\beta }_1$ is another included angle smaller than ${\alpha }_1$.

4.2. Particle-Scale Heat Convection

Particle-scale heat convection is typically taken into consideration in cases involving strong gas–solid interactions, such as in countercurrent gas–solid packed beds and fluidized beds (Figure 12). Generally, as mentioned above, scholars tend to adopt specific empirical or semi-empirical correlation to calculate the $\mathrm{N}\mathrm{u}$ of gas–solid heat transfer, further obtaining the thermal resistance for heat convection (see Equation (21)). There are several correlations available for consideration, as comprehensively reviewed in refs. [170,171,172]. Herein three common ones and their application ranges are listed below as shown in Equations (22)–(24). Heat transfer coefficient correlations are constantly being developed, thanks to the continuous efforts of scholars in the field. Several previous common correlations used in packed beds were comprehensively reviewed by Liu et al. [173], such as Galloway correlation [174], and a new correlation was proposed according to their experimental results. Furthermore, predictably, the specific correlation is only suitable for a specific operating condition. Moreover, Aissa et al. [175,176] found that Ranz–Marshall correlation cannot accurately predict the heat transfer between a spherical particle and its surrounding atmosphere at a high temperature for different gases, and further provided a new correlation to obtain better results. At present, there is a lack of a general and universal correlation of gas–solid heat transfer coefficient for establishing particle-scale heat convection sub-model. Therefore, the correlation should be carefully chosen based on the operating condition of the waste heat recovery system.

$$R_{pf}={({\displaystyle \frac{\pi {\lambda }_{f}R\mathrm{N}\mathrm{u}}{2}})}^{-1}$$

(21)

$${\mathrm{N}\mathrm{u}}_{Ranz-Marshall}=2.0+0.6{{\mathrm{R}\mathrm{e}}_p}^{1/2}{\mathrm{P}\mathrm{r}}^{1/3},{\mathrm{R}\mathrm{e}}_p={\displaystyle \frac{2{\rho }_{f}R‖{\mathit{v}}_p-{\mathit{v}}_f‖}{{\mu }_f}}$$

(22)

$$\begin{array}{c}{\mathrm{N}\mathrm{u}}_{Gunn}=(7-10{\alpha }_f+5{\alpha }_f^2)(1+0.7{{Re}_p}^{0.2}{Pr}^{1/3})+(1.33-2.4{\alpha }_f\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+1.2{\alpha }_f^2){{Re}_p}^{0.7}{Pr}^{1/3}, 0<{Re}_p<{10}^5, 0.35<{\alpha }_f<1\end{array}$$

(23)

$${\mathrm{N}\mathrm{u}}_{Wakao}=2+1.1{{\mathrm{R}\mathrm{e}}_p}^{0.6}{\mathrm{P}\mathrm{r}}^{1/3},3<{\mathrm{R}\mathrm{e}}_p<\mathrm{10,000}$$

(24)

${\mathrm{R}\mathrm{e}}_p$ is the particle flow Reynolds number; $\mathrm{P}\mathrm{r}$ is the fluid Prandtl number; ${\rho }_f$ is the fluid density; $\mathit{v}$ is the velocity vector; ${\mu }_f$ is the fluid dynamic viscosity; and ${\alpha }_f$ is the fluid volume fraction.

4.3. Particle-Scale Heat Radiation

Generally, when the high temperature particles (e.g., >1000 K [40]) or triatomic gases (e.g., CO₂) exist in a heat transfer system, the heat radiation between particles and between particles and gases should be carefully considered. Herein, the sub-models of particle-to-particle heat radiation are mainly reviewed as the heat radiation between particles and gases is rarely mentioned in the literature. Particle-scale heat radiation sub-models are generally proposed and developed based on the classical radiation laws, such as gray-body radiation law. To our knowledge, there are two mainstream sub-models to characterize particle-scale heat transfer, which are, respectively, the isolated domain heat radiation model and resolved gray-body radiation model. Both of them are deduced based on gray-body radiation law. The basic forms of the particle radiation thermal resistances ($R_{rad}$) of these two sub-models are shown in Equations (25) and (26) [177,178]. According to the isolated domain heat radiation theory, particle heat radiation occurs between a particle and the environment consists of its surrounding particles and fluids, which refers to an isolated domain. The local temperature ($T_{local,i}$) is an abstract temperature contains the temperature information of both the surrounding particles and fluids, which can be calculated by Equation (27). This sub-model is cost-effective to simulate the heat radiation behaviors between numerous particles and their surrounding fluids. Due to the unresolved heat distribution mode in the isolated domain, this sub-model is only suitable for systems where the wall-related heat radiation is not significant. Fortunately, the resolved gray-body radiation model demonstrates a good ability to address this issue. This sub-model offers better physical interpretability since it can be rigorously deduced according to the laws of radiation. However, the key to this sub-model lies in the method used to analyze the view factor ($F_{ij}$) between particle i and j. Several methods are developed by researchers to deal with this trouble. Felske [179] theoretically derived the mathematical expression for calculating the view factor of the two-particle system, which has been widely used in refs. [160,162,180]. Meanwhile, based on the statistics related to the view factor, Antwerpen et al. [181] obtained the relation between view factor and dimensionless distance by curve fitting. Furthermore, some scholars have calculated the real-time view factors of the particle system using numerical simulations through the Monte Carlo sampling method [164,182] (Figure 13 and Figure 14).

$$R_{rad,isolated}={\left(4\pi \sigma R^2\left(T_i^2+T_{local,i}^2\right)\left(T_i+T_{local,i}\right)\right)}^{-1}$$

(25)

$$T_{local,i}={\alpha }_f{T}_f+(1-{\alpha }_f)\overline{T_p}$$

(26)

$$R_{rad,resolved}={({\displaystyle \frac{\sigma (T_1^2+T_2^2)(T_1+T_2)}{\frac{1-{\epsilon }_{p1}}{{\epsilon }_{p1}}\frac{1}{\pi {R_1}^2{F}_{12}}+\frac{1}{\pi {R_1}^2{F}_{12}}+\frac{1-{\epsilon }_{p2}}{{\epsilon }_{p2}}\frac{1}{\pi {R_2}^2{F}_{21}}}})}^{-1}$$

(27)

$\sigma$ is the Stefan–Boltzmann constant; $T_i$ is the particle temperature; $\overline{T_p}$ is the average particle temperature in an isolated domain; and ${\epsilon }_p$ is the particle emissivity.

In recent years, many empirical formulas for radiative heat transfer between particles and particle heat exchangers have been proposed [183,184,185,186], which have also improved the accuracy of CFD technology in simulating particle heat recovery processes and expanded the application scope of the technology.

5. Summary and Prospects

5.1. Summary

In summary, CFD technology has been widely used in the recovering process of high-temperature solid granules since its development, and the result is generally acceptable. To deal with the mass flow questions, CFD technology can be generally divided into two parts, the Euler–Euler method and Euler–Lagrange method, which rely on whether treating solid phase as a continue phase or discrete phase. The TFM, CFD-DEM, DNS method and MP-PIC method are, respectively, the representative methods of these two methods. As for the heat transfer in gas–solid two-phase flow in the process of heat recovery from a high-temperature solid granule, the total process can be separated into three joint processes as a traditional heat transfer problem, but the heat conduction process and heat convection process has a widely permitted model while heat radiation process does not yet. Both mass flow and heat transfer have more detailed variants to deal with complex flow problems, which are partly introduced in the above sections.

5.2. Prospects

As summarized, CFD technology and CFD coupling technology are widely used in analyzing heat transfer processes. Based on the papers published in recent years, we summarize the following research focus and prospects of the high-temperature solid granule heat recovery process analysis using CFD technology.

5.2.1. More Precise Physical Model and Empirical Formula

As mentioned, to verify the solid-phase flow characteristics in different gas–solid flow conditions using the TFM, especially the inside drag force models. In recent years, many new drag force models in different conditions are proposed, as are heat radiation calculation models, which have the benefit of improving the diversity of the models and the accuracy of calculation.

5.2.2. More Suitable Grid Construction for Larger-Scale Simulation as Real Industrial Processes

A large-scale simulation, which has a much larger quantity of grids than the laboratory-scale one, would cost several times more time than the laboratory-scale one, but the result data would also be close to the real industrial processes. So, besides relying on the improvement of computer calculation power, in addition it is also necessary to develop better grid-construct methods and better grid refinement methods.

5.2.3. More Diverse CFD Methods and CFD Coupling Methods for Solving the Problems

In Section 2, Section 3 and Section 4, many CFD coupling methods are represented. Nowadays, many other CFD methods and CFD coupling methods are also developed to solve different problems, and these new methods may also serve the improvement of simulations about heat recovery from high-temperature solid granules.

5.2.4. Other Advanced Technology Development Used in CFD Methods and CFD Coupling Methods

In recent years, the development of advanced technology, such as AI technology and cloud database technology, has played an important role in the development of industrial technology. The establishment of source code libraries and material databases through this technology has played a key role in the development of CFD methods and CFD coupling methods. For example, COMSOL Multiphysics^®, is a software that collects the basic physical properties and interaction relationships of thousands of materials under different conditions, and this technology can also be applied to CFD simulation regions.

Meanwhile, as mentioned in Section 3, integrated machine learning technology can not only reduce computing power and improve efficiency for the DNS method and other mass flow analyzed methods but also help analyze heat transfer processes, especially continuously optimizing empirical formulas for heat radiation.