*2.1. Governing Equations for Solid Particles*

There are two types of motion for a particle in moving bed—translation and rotation—and the motion of individual particles is determined by Newton's second law of motion, while the Hertz–Mindlin contact theory is adopted for the interaction between particles. So, the governing Equations of particle *i* with mass *mi* and moment of inertia *Ii* can be written as

$$m\_i \frac{d\overrightarrow{v\_i}}{dt} = \sum\_{j=1}^{k\varepsilon} \left(\overrightarrow{F}\_{c,ij} + \overrightarrow{F}\_{d,ij}\right) + \overrightarrow{F}\_{f,i} + m\_i \overrightarrow{g} \tag{1}$$

$$I\_i \frac{d\overrightarrow{\omega\_i}}{dt} = \sum\_{j=1}^{kc} \left(\overrightarrow{M}\_{t,ij} + \overrightarrow{M}\_{r,ij}\right) \tag{2}$$

where <sup>→</sup> *<sup>v</sup> <sup>i</sup>* and <sup>→</sup> *ω<sup>i</sup>* are the translation velocities and rotation velocities of the particle *i*, and *kc* is the number of particles interacting with particle *i*. → *Fc*,*ij* and <sup>→</sup> *Fd*,*ij* are the contact force and non-contact force respectively, <sup>→</sup> *F <sup>f</sup>* ,*<sup>i</sup>* is the particle–fluid interaction force acting on the particle *i*, *mi* → *g* represents the gravitational force. <sup>→</sup> *Mt*,*ij* and <sup>→</sup> *Mr*,*ij* are the torques generated by tangential force and rolling friction force. In this work, the expressions of forces and torques have been listed in the literature [20].

Contacts between particles are significant for dense phase such that conductive heat transfer must be taken into account. The heat flux between the particles is defined as Equation (3), and the contact area is incorporated in the heat transfer coefficient *hc*, as shown in Equation (4). The temperature change over time of each particle is updated explicitly by Equation (5).

$$Q\_{\vec{i}\vec{j}} = h\_{\vec{c}} \Delta T\_{\vec{i}\vec{j}} \tag{3}$$

$$h\_c = \frac{4k\_ik\_j}{k\_i + k\_j} (\frac{3F\_N r^\*}{4E^\*})^{1/3} \tag{4}$$

$$m\_i \mathcal{C}\_p^i \frac{dT\_i}{dt} = \sum\_{j=1}^{ki} Q\_{ij} + Q\_{if} + Q\_{i,rad} \tag{5}$$

where Δ*Tij* is the temperature difference between particle i and particle j, *ki* and *kj* are the thermal conductivity of particle i and j, respectively, and *FN* is the contact normal force. *r*∗ is the geometric mean of the particles radii according to the Hertz–Mindlin contact theory and *E*∗ is the effective Young's modulus. The heat flux *Qij* is the heat conduction flux between particles i and j, and *Qi f* is the convection heat flux between fluid and solid particles. *Qi*,*rad* is the radiation heat flux between particle i and its surrounding environment.

#### *2.2. Governing Equations for Gas Phase*

The gas is treated as a continuum phase, and for the incompressible fluid, the fluid field can be solved by continuity, momentum, and energy conservation equations. In the current work, the standard *k* − *ε* model is adopted to solve fluid flow, and the governing Equations are given as follows.

Continuity Equation:

$$\frac{\partial(\rho\_f \varepsilon\_f)}{\partial t} + \nabla.(\rho\_f \varepsilon\_f \vec{u}) \tag{6}$$

Momentum Equation:

$$\frac{\partial(\rho\_f \varepsilon\_f \stackrel{\rightarrow}{u})}{\partial t} + \nabla.(\rho\_f \varepsilon\_f \stackrel{\rightarrow}{u} \stackrel{\rightarrow}{u}) = -\varepsilon\_f \nabla p + \nabla.(\mu\_f \varepsilon\_f \stackrel{\rightarrow}{\nabla} \stackrel{\rightarrow}{u}) + \stackrel{\rightarrow}{S}\_m \tag{7}$$

Energy Equation:

$$\frac{\partial(\rho\_f \varepsilon\_f \mathbb{C}\_{p,f} T\_f)}{\partial t} + \nabla.(\rho\_f \varepsilon\_f \vec{u} \mathbb{C}\_{p,f} T\_f) = \nabla.(\lambda\_f \varepsilon\_f \nabla T\_f) + \stackrel{\rightarrow}{S}\_{\varepsilon} \tag{8}$$

where *ρ<sup>f</sup>* , → *u*, and *p* are the fluid density, velocity, and pressure. *ε<sup>f</sup>* and *Tf* are the porosity and temperature of the fluid and *Cp*, *<sup>f</sup>* and *λ<sup>f</sup>* are the specific heat capacity and thermal conductivity of fluid. <sup>→</sup> *Sm* is the momentum source term due to the effect of solid particles on fluid motion and <sup>→</sup> *Se* is the energy source term because of the heat exchange between fluids and particles, CFD and DEM are coupled by the source terms <sup>→</sup> *Sm* and <sup>→</sup> *Se*, the expressions of <sup>→</sup> *Sm*, → *Se* are given in detail [45].

#### *2.3. Heat Transfer Models*

The fluid and particles are obtained by solving Equations (1), (2), (6) and (7); while the heat transfer between particles and fluid is governed by Equations (5) and (8). In the present work, the temperature of particles is up to 800 K, so there are three heat transfer mechanisms considered—heat conduction, heat convection, and particle radiation—where the convection heat transfer between fluid and the wall is not considered and the wall is regarded as adiabatic. The models used to describe the different heat transfer mechanisms have been described in detail [33], and for the sake of simplicity, are no longer described here.

In order to better describe the heat transfer performance in the packed beds, several parameters are defined here. The Reynolds number and pore scale hydraulic diameter *dk* are defined as follows.

$$R\_{cp} = \frac{\rho\_f \left(\overrightarrow{u} - \overrightarrow{v}\_i\right) d\_h}{\mu\_f} \tag{9}$$

$$d\_{\rm li} = 4 \frac{\varepsilon\_f}{1 - \varepsilon\_f} (\frac{V\_p}{A\_p}) \tag{10}$$

The heat transfer coefficient and the Nusselt number are defined below.

$$h\_{sf} = \frac{Q\_f}{(\overline{T}\_p - \overline{T}\_f)}\tag{11}$$

$$Nu\_{sf} = \frac{h\_{sf}d\_h}{k\_f} \tag{12}$$

where, *Qf* is total heat flux of the heat convection in the packed beds and *Tp* and *Tf* are the average temperature of the particles and fluids in the beds, respectively. Because of the improvement in heat transfer, at the cost of the increase of pressure drop, in order to compare the overall performance of different configurations, the expression is defined:

$$\frac{\left(\mathrm{Nu}/\mathrm{Nu}\_{0}\right)}{\left(f/f\_{0}\right)^{1/3}}\tag{13}$$

where the *Nu*<sup>0</sup> and *f*<sup>0</sup> is the Nusselt number and friction factor of the packed with uniform particle size distribution.

#### *2.4. Entransy Dissipation*

As described above, the entransy dissipation is introduced to describe the heat transfer ability. Different from the energy destruction minimization [46,47], for the simple heating or cooling process, entransy dissipation is more suitable as a measure of the irreversibility of the heat transfer process. The heat transfer process studied in this paper is to cool the high temperature particles with cold air, so it is suitable to take entransy dissipation as a measure of heat transfer performance.

According to a previously described definition [43], the entransy of flow fluid in the opening system is as follows.

$$G = \frac{1}{2}HT = \dot{m}\mathbb{C}\_p T^2\tag{14}$$

where *<sup>H</sup>* is the enthalpy of the flow fluid and . *m* is the mass flow rate.

Entransy is not conserved and will dissipate during fluid flow, the entransy dissipation rate of unit volume and unit time is

$$
\Phi\_{\mathbf{h}} = -\dot{\mathbf{q}} \cdot \nabla \mathbf{T} = \mathbf{k} (\nabla \mathbf{T})^2 \tag{15}
$$

Equation (15) is derived from the entransy balance equation, which is derived from the thermal energy conservation equation, and the Φ<sup>h</sup> can be considered as the local entransy dissipation. The detailed derivation has been given by the researchers [41], and it is omitted here due to the complexity of the derivation. According to the entransy dissipation extremum principle, when the boundary temperature difference is given, maximizing the entransy dissipation leads to maximum boundary heat flux, which leads to the minimum of equivalent thermal resistance, which is the best performance for the heat transfer [48]. It can be expressed as

$$
\Delta \mathbf{T} \delta \dot{\mathbf{Q}}\_{\mathbf{t}} = \delta \iiint \frac{1}{\mathbf{k}} |\mathbf{q}|^{2} dV = 0 \tag{16}
$$

The entransy dissipation is the function of heat flux and temperature gradient, as shown in Equation (11), applying the variational method [49] to the entransy balance equation, Equation (12) can be obtained. At a given temperature difference, the equivalent thermal resistance is defined as the ratio of the square of temperature different divided by the entransy dissipation, which can be written as

$$R\_h = \frac{\Delta T^2}{\Phi} \text{ or } R\_h = \frac{\left(\overline{\Delta T}\right)^2}{\Phi} \tag{17}$$

where, ΔT is the average temperature difference. The smaller *Rh* is, the better the heat transfer performance and the stronger the heat transfer ability is. Due to entransy dissipation is the function of heat flux and the temperature gradient, the total entransy dissipation, because of the finite temperature difference in the packed bed, is defined as follows.

$$
\Phi = \int \left( T\_{\rm h} - T\_{\rm c} \right) dQ \tag{18}
$$

where, *Th* and *Tc* are the temperature of high particles and temperature of cold fluid. So the T–Q graph can be used to visually represent the entransy dissipation of the heating or cooling process.

#### **3. Simulation Conditions**

The physical model of packed bed used in this work and the boundary conditions is shown in Figure 1. There are two sections of the physical model: packed section and outlet section. Considering the complexity of the gas flow in the particle section, and avoiding the influence of the outlet reversed flow, the outlet extends 90 mm downstream. The inlet is specified as a velocity inlet, the outlet is specified as a pressure outlet, and the wall is considered as adiabatic. The packed section is filled with particles with different sizes. Detailed geometric parameters of the packed bed and boundary conditions are shown in Table 1. Firstly, the particles with temperature of 800 K fill with the packed section, until the steady state of the particles is achieved, and then the gas is introduced from the bottom with velocity of 5 m/s, with temperature of 293 K, and after sufficient heat exchange between particles and gas, the gas flows out at the outlet and will be recycled for other use.

**Figure 1.** Computational physical model.


**Table 1.** Detailed parameters of the physical model.

In order to study the strengthening effects of the mixing of particles with different sizes, the packed beds with different particle size distribution are shown in Figure 2. Five kinds of different particle size distributions in the radial direction are studied here, namely, random packing with uniform size of 5 mm (*d* = 5 mm), random mixing distribution with size of 4–5 mm (mixing 4–5 mm), radial distribution of 4–5 mm (*d* = 4–5 mm), random mixing distribution of 3–5 mm (mixing 3–5 mm), and radial distribution of 3–5 mm (*d* = 3–5 mm). When the particles are distributed along the radial direction, the central region is filled with basic particles with diameter of 5 mm, while the near wall region is filled with small particles with diameter of 4 mm or 3 mm, as shown in Figure 2c,e, the average porosity of mixing 4–5 mm and *d* = 4–5 mm, mixing 3–5 mm, and *d* = 3–5 mm are the same, respectively.

**Figure 2.** Radial distribution of different particle sizes: (**a**) random packing with single size particles of 5 mm; (**b**) random mixing of 4–5 mm; (**c**) radial distribution of 4–5 mm; (**d**) random mixing of 3–5 mm; and (**e**) radial distribution of 3–5 mm.

In order to further study the influence of particle size distribution on the heat transfer process and the range of the wall effects in the bed, packed beds with different distribution thickness are studied, as shown in Figure 3. There are three cases with different distribution thickness: case1 (the same as the *d* = 3–5 mm in Figure 2e), case2, and case3, and the near wall region of the three cases are packed with small particles of 3 mm, as shown in Figure 3b–d. Meanwhile, the packed beds filled with big particles with diameter of 5 mm and small particles with diameter of 3 mm are also studied here as comparison, as shown in Figure 3a,e. All the particles in the packed beds are generated by discrete element methods.

**Figure 3.** Radial distribution of different distribution thickness: (**a**) single particle size of 5 mm; (**b**) case1; (**c**) case2; (**d**) case3; and (**e**) single particle size of 3 mm.

The CFD-DEM method is used to study the flow and heat transfer characteristics between gas and particles, to meet the key assumptions, the time step is selected carefully. For the cases with the minimum particle diameter of 3 mm, 4 mm, and 5 mm, the time step is selected as 1e-5s, 2e-5s, and 2e-5s, respectively. In the case d = 5 mm, there are 1927 particles with diameter 5 mm; and in the case *d* = 4–5 mm and mixing 4–5 mm, the number of particles with diameter of 4 mm and 5 mm are 1931 and 953, respectively. In the case *d* = 3–5 mm and mixing 3–5 mm, the number of particles of 3mm and 5 mm are 4723 and 983, respectively. For case2, the number of particles of 3 mm and 5 mm are 6687 and 582, respectively. For case3, the number of particles with diameter of 3 mm and 5 mm are 7870 and 320, respectively. For case *d* = 3 mm, there are 9393 particles with diameter 3 mm.

### **4. Results and Discussion**

#### *4.1. Model Validation*

The packed bed which is randomly packed with uniform size of 5 mm is selected for model validation. And the empirical correlations proposed by Sug Lee [50] are chosen for the validation of friction factor, and the empirical correlations proposed by Demirel [43] are chosen for the validation of heat transfer process, which can be expressed by Nusslet number. The friction factor is calculated by the following expressions [37].

$$
\Delta p = 4f \frac{\rho u^2}{2} \frac{1}{d\_p} \tag{19}
$$

*Energies* **2019**, *12*, 414

$$Af = \frac{12.5(1 - \varepsilon\_f)^2}{\varepsilon\_f^3} (29.32 \text{Re}\_p^{-1} + 1.56 \text{Re}\_p^{-n} + 0.1) \tag{20}$$

where *n* is the factor related to porosity, and can be expressed as

$$n = 0.352 + 0.1\varepsilon\_f + 0.275\varepsilon\_f^2\tag{21}$$

and the Nusslet number concluded by Demirel [43] for spheres can expressed as

$$Nu = 0.217 \text{Re}\_p^{0.756} \tag{22}$$

The CFD-DEM results of friction factor and Nusslet number are compared with the empirical correlation results, which are shown in Figure 4, and the quantitative are given in Table 2. As can be seen from Figure 4a, the numerical simulation results agree well with the results of correlations of Sug Lee [50], and the maximum deviation is 10.25% when Rep number is 6592.9; and, as can be found from Figure 4b, the numerical simulation results are in good agreement with the correlation results proposed by Demirel [49], and the maximum deviation is 6.65% when the *Rp* number is 5405.4. Therefore, it can be concluded that the CFD-DEM approach used in the present work is reliable to simulate the fluid flow and heat transfer in packed beds with low tube-to-particle ratio.

**Figure 4.** Validation of the computation model compared with empirical correlations: (**a**) friction factor and (**b**) Nusselt number.


**Table 2.** Quantitative comparison of model validation.

### *4.2. The Effect of Particle Diameter Distribution*

The radial distribution of particle with different sizes determines the distribution of porosity along the radial direction. To analyze the results conveniently, four typical cross-sections are selected, as shown in Figure 5, the sections of Z = 0.07922 m, Z = 0.06922 m, and Z = 0.05922 m are XY planes close to the outlet, and the plane of Y-section is ZX plane that is at the location of Y = 0 m. In the present work, the tube-to-particle ratio is relatively low (8 < *dt/dp* < 13.3), as discussed above, for low tube-to-particle ratio, the porosity near the wall region is larger than that in the central region when the bed is packed with particles of uniform size; and the gas flows away from the near wall region without sufficient heat exchange with the solid particles, so higher gas velocity can be found near wall region, as shown in Figure 6b, meanwhile, the temperature of the gas is lower than that in the central region, as shown in Figure 6a. Also it can be found from Figure 7 that there is large difference in temperature along the radial direction from the wall to the center of the tube, and the max temperature difference is almost 10 ◦C near the wall region (one diameter from the wall to the center in the radial direction) at the different cross-sections, which means that the wall effects is obvious. So it is important to decrease the porosity near wall region, improve the velocity, and optimize the flow field and temperature field.

**Figure 5.** Typical cross-sections in the packed bed.

**Figure 6.** (**a**) Temperature distribution at Y-sec and (**b**) velocity distribution at Y-sec.

**Figure 7.** Radial distribution of temperature of particle size 5 mm.

Figure 8 compares the Nusselt number, friction factor at the different Re numbers, as shown in Figure 8a, and the Nu of multiparticle size distributions, which is higher than that of uniform particle size distribution at different Re, meaning that the heat transfer performance is better, and the heat transfer performance of case *d* = 3–5 mm is the best. However, the improvement of heat transfer performance is at the cost of the increase of pressure drop, as shown in Figure 8b, the better heat transfer performance corresponds to the higher friction factor. Figure 8c compares the overall performance of different configurations, with the performance of uniform particle size distribution as a reference. It can be seen that at low Re, the overall performance of multiparticle size distribution is better than reference, but with the increase of Re, the overall performance becomes worse than the reference, which is mainly because the pressure drop increase too fast. So at low Re, *d* = 3–5 mm is recommend because of the better overall performance.

**Figure 8.** Performance comparisons of different particle size distributions: (**a**) Nusselt number; (**b**) friction factor; and (**c**) the overall performance.

To analyze the nature of the different heat transfer performances, the temperature and velocity distributions of different particle size distribution at section Y-Sec are shown in Figures 9 and 10 (Re = 6399). It is found that, due to the small particles are placed in the near wall region, the velocity distribution throughout the bed is more uniform compared with the uniform particle size distribution, and the temperature of the near wall region is close to that of the central region in the case *d* = 4–5 mm, as shown in Figures 9c and 10c. Moreover, with the decrease of particle size near wall region (*d* = 3–5 mm), the porosity decreases further, so the velocity throughout the bed has been increased, so the heat transfer in the core area is enhanced, as shown in Figures 9e and 10e, the temperature at the outlet is higher, as listed in Table 3. The radial temperature distributions of five different particle size distributions at the same section (Z = 0.07922 m) are shown in Figure 11. The uniform particle size distribution *d* = 5 mm, and the random distribution mixing 4–5 mm, mixing 3–5 mm have similar temperature distributions in the radial direction, namely, the temperature in the near wall region is lower than that in the central region, so it indicates the random mixing of big and small particles cannot restrain the wall effects. While the cases of *d* = 4–5 mm and *d* = 3–5 mm can improve the temperature in the near wall region, they also restrain the wall effects and optimize heat transfer in the beds.

**Figure 9.** Local temperature distributions of different particle size distributions (Y-Sec).

**Figure 10.** Local velocity distributions of different particle size distributions (Y-Sec).


**Table 3.** Outlet temperature and average porosity of different distributions (Re = 6399).

**Figure 11.** Radial temperature distribution of different particle size distributions (section Z = 0.07922 m).

The effects on heat transfer also can be found from the change of the isothermal lines, for the uniform particle size *d* = 5 mm, as shown in Figure 9a, the isothermal lines are concave, and both the temperature gradient and heat flux are toward to the center of the tube, but the gas velocity in the center is low, which indicates the heat transfer in the central area is weak; while for *d* = 4–5 mm and *d* = 3–5 mm, the isothermal lines are convex, so both the temperature gradient and heat flow are toward to the near wall region, and the velocity in the near wall region is relatively high, as shown in Figure 9c–e, as a result, the heat transfer throughout the beds are improved. Due to the fact that the velocity of the case of 3–5 mm distribution is higher than that of 4–5 mm, there is more heat flux in the *d* = 3–5 mm, so the outlet gas temperature is higher, as listed in Table 3. The isothermal lines in the cases of mixing 4–5 mm and mixing 3–5 mm are similar to the uniform size distributions, as shown in Figure 9b,d, the heat transfer in the bed is not optimal. So the radial distribution of particle size can obviously enhance the heat transfer in the bed.

In order to evaluate the loss of the heat transfer ability in the process, the T–Q graph and the equivalent thermal resistance is shown in Figure 12. The larger the area surround by the curve is, the greater the entransy dissipation is, and, according to the minimum thermal resistance principle [36], the heat flux is larger, which corresponds to the better heat transfer performance when the boundary temperature is constant, as shown in Figure 12a; the equivalent thermal resistance is smaller, as shown in Figure 12b. It can be seen from Figure 12 that, the area enclosed by the curve corresponding to the distribution of *d* = 3–5 mm is the largest, and the equivalent thermal resistance is the smallest, so for the *d* = 3–5 mm, the heat transfer ability between the gas and solid is utilized in the greatest extent. It should be noticed that there are small difference of the thermal difference between the case of *d* = 3–5 mm and mixing 3–5 mm, *d* = 4–5 mm and mixing 4–5 mm, but the difference among *d* = 5 mm, *d* = 4–5mm and *d* = 3–5 mm is obvious.

As a result, considering the overall performance and the loss and heat transfer ability, it can be concluded that the improvement of heat transfer effect is at the cost of the loss of heat transfer ability. And the radial particle size distribution of 3–5 mm is the optimal configuration.

**Figure 12.** (**a**) T–Q graph of different particle size distributions. (**b**) Equivalent thermal resistance.

#### *4.3. The Effect of Distribution Thickness*

As discussed above, the radial distribution of particle size can obviously improve the heat transfer performance in the bed, in order to study the range of the wall effects the influence of radial distribution thickness on heat transfer is investigated, as shown in Figure 3.

The flow and heat transfer performance are greatly affected by the distribution thickness, as shown in Figure 13, with the increase in distribution thickness, the Nu increases at different Re, but the f also increases. However, with further increase in distribution thickness, the multiparticle size distribution becomes the uniform size distribution of *d* = 3 mm, and the heat transfer performance decrease obviously. The overall performance is shown in Figure 13c, and the performance of the case3 is best, the case *d* = 3 mm is worse than case *d* = 5 mm, which indicates that for the uniform size distribution, the decrease of particle size cannot improve the performance of the beds, and the increase of the outlet temperature is due to the increase of the heat transfer area. The temperature distributions along the radial directions at the same section (Z = 0.07922 m) are shown in Figure 14. It is clearly that the average temperature increases with the increase of distribution thickness, and there is a big jump of temperature when the radial distribution of particles starting from uniform size (from the *d* = 5 mm) to the case1. Moreover, the temperature distribution trends of case1, case2, and case3 are the same: the temperature increases first and then decreases along the wall to the center of the tube, the turning

point of temperature change is one particle diameter away from the wall, so it can be inferred that the wall effects are just one diameter away from the wall

**Figure 13.** Performance comparisons of different radial distribution thickness: (**a**) Nusselt number; (**b**) friction factor; and (**c**) the overall performance.

**Figure 14.** Radial temperature distribution of different radial distribution thickness (section Z = 0.07922 m).

The different performance can be analyzed from the flow and temperature filed, as shown in Figures 15 and 16 (Re = 6399). With the increase of the radial distribution thickness (from Figure 3a–e), the big local flow channels formed in the central region decrease, and the local gas velocity increases, which leads to the improvement of heat transfer throughout the bed, and the outlet temperature is obviously increased, as listed in Table 3. But as the distribution thickness further increases (Figure 3e), the flow and heat transfer characteristics are the same as that of the uniform particle size. For the uniform size of *d* = 5 mm and *d* = 3 mm, the isothermal lines change in the similar trend along the radial direction from the tube wall to the center, namely, the isothermal lines are concave, the wall effects are obvious, the outlet temperature of *d* = 3 mm is higher than that of *d* = 5 mm—because the average porosity of the whole bed is lower and the heat transfer area is larger–but the heat transfer performance in the bed is not improved.

**Figure 15.** Temperature distribution of different radial distribution thickness (Y-Sec).

The entransy dissipation and the equivalent thermal resistance of the five different distributions are shown in Figure 17, the area enclosed by the curve increases gradually with the change of the distribution thickness. So it shows that the entransy dissipation and the heat flux Q increase obviously when the particle size changes from uniform size of *d* = 5 mm to the case1, and the equivalent thermal resistances decrease obviously, as shown in Figure 17b. The obvious increase in entransy dissipation and heat flux from the uniform size of *d* = 5 mm to the case1 occurs because the heat transfer performance is obviously improved. However, the difference of thermal resistance of case1, case2, and case3 is very small, and this is because the change of the distribution thickness do not improved the temperature and flow filed so much, as shown in Figures 14 and 15, the fields of case1, case2, and case3 are similar, and the temperature gradient is similar, so there is a small difference in entransy dissipation and the equivalent thermal resistance.

**Figure 16.** Velocity distribution of different radial distribution thickness (Y-Sec).

**Figure 17.** (**a**) T–Q graph of different radial distribution thickness and (**b**) equivalent thermal resistance.

As a result, considering the overall performance and the entransy dissipation, it can be concluded that the range of wall effects is just one particle diameter away from the wall, the heat transfer effects can be obviously improved by filling small particles in the near wall region, and the increase of distribution thickness can improve the heat transfer performance. The wall effects can be well restrained by the radial distribution of the particle size.
