A Probabilistic Statistical Method for the Determination of Void Morphology with CFD-DEM Approach

Abstract

Voids that are formed by gas injection in a packed bed play an important role in metallurgical and chemical furnaces. Herein, two-phase gas–solid flow in a two-dimensional packed bed during blast injection was simulated numerically. The results indicate that the void stability was dynamic, and the void shape and size fluctuated within a certain range. To determine the void morphology quantitatively, a probabilistic method was proposed. By statistically analyzing the white probability of each pixel in binary images at multiple times, the void boundaries that correspond to different probability ranges were obtained. The boundary that was most appropriate with the simulation result was selected and defined as the well-matched void boundary. Based on this method, the morphologies of voids that formed at different gas velocities were simulated and compared. The method can help us to express the morphological characteristics of the dynamically stable voids in a numerical simulation.

1. Introduction

Voids that are formed by gas injection in a packed bed are critical in various applications, such as moving-bed coal gasifiers [1], blast furnaces [2] and COREX melting gasifiers [3]. For example, the tuyere raceway in the lower part of the blast furnace is exactly the void that is formed by a high-pressure air blast, which provides heat and a reducing gas for reaction in the upper part. The void morphology (i.e., shape and size) is believed to be an important factor to determine the distribution of the reducing gases and heat [4]. Therefore, an exploration of the void morphology has important implications for controlling the gas distribution and reactions [5].

In past decades, void formation has been investigated extensively by different experimental techniques and by numerical simulation [4,6,7,8,9]. For example, based on a dimensional analysis, Rajneesh et al. [10,11] established a cold model experiment and proposed two void correlations to determine the void depth and height. X-ray computed tomography (CT) was used by Nogami et al. [12] to measure the three-dimensional (3D) shape and structure of the void by evaluating the differences between two successive CT images. Di et al. [13,14] established a semicircle model with a 1:20 scale according to the prototype of one factory, and derived the formula to determine the void surface area on the basis of fractal theory and coke particle velocity contour. Lu et al. [15] used an optical method to study the void formation in a 3D packed bed, and concluded that the void was closer to an ellipsoid.

Because of experimental limitations in terms of high construction costs, a numerical simulation that uses a continuum approach or a discrete-element method (DEM) has become necessary to study the void formation process for realistic processes in advance. In the continuum approach, the fluid and particle phases are considered continuous media. Rangarajan et al. [16] used a two-fluid modeling method to study the void formation, and reported that the gas velocity influences the void size. In general, the continuum approach is suitable for large-scale computations, but finds it difficult to obtain detailed microscopic dynamic information on the void. The discrete-based approach can solve this difficulty. For example, Tsuji et al. [17,18,19] proposed a coupling method using DEM and a continuum approach to study plug conveying in a pipe. Xu et al. [20] used combined continuum and discrete modeling to simulate gas–solid flow in packed beds by establishing a two-dimensional (2D) slot model. Miao et al. [21] coupled computational fluid dynamics (CFD) and DEM to investigate the formation of voids in a full-scale 2D slot model, and found that three void types formed at different gas velocities, such as a clockwise circulating void (at a low gas velocity), an anti-clockwise circulating void (at a medium gas velocity) and a plumelike void (at a high gas velocity). With CFD–DEM, Sun et al. [3] studied the void formation in COREX by analyzing the movement and phase volume fraction of particles, and reported that it took 4 s for the void to reach a steady state.

Previously, the void morphology was usually determined as a static instantaneous value in literature [22]. In reality, however, the morphology of void changes with time. Umekage et al. [23] reported that the void morphology changed before steady state was reached. Santana et al. [24] observed that the void oscillated alternately from t = 1.3 s to t = 3 s, and stabilized after t = 4.0 s macroscopically for a 2D rectangular geometry model. Although the void tended to be stable macroscopically after a period, the void is likely to fluctuate and maintain a dynamic stability. However, the void morphology during the macroscopic stable stage remains unclear.

In this work, we established a 2D two-phase gas–solid flow model based on CFD–DEM to simulate void formation in a packed bed. A cold experiment was conducted to verify the modeling. In the simulation, we found that the void morphology was under a dynamically stable condition even after the macroscopically stable stage from our simulation results. We proposed a probabilistic method to determine the exact void morphology by analyzing each pixel in the obtained simulation images with time. To analyze the feasibility of this probabilistic method, the voids that formed for a series of gas velocities were studied. This work provides an important strategy for determining the void morphology in various applications.

2. Materials and Methods

2.1. Modeling

2.1.1. Particle Movements Described in DEM

Particle movement in the packed bed was simulated by DEM (EDEM^TM, version 2018, DEM solutions, Edinburgh, UK), and the collision between particles was considered with the soft ball model that was proposed by Cundall [25]. The governing equations for translational and rotational motion of particle i are [26],

$$m_i\frac{d{v}_i}{dt}=F_{pf,i}+(F_{cn,ij}+F_{ct,ij}+F_{dn,ij}+F_{dt,ij})+m_{i}g$$

(1)

$$I_i\frac{d{\omega }_i}{dt}={\displaystyle {\sum }_j(M_{t,ij}+M_{r,ij})}$$

(2)

$$I_i=0.4m_i{R}^2$$

(3)

where m_i, v_i, ω_i and R are the mass, translational velocities, rotational velocities and radius of particle i, respectively. F_pf,i, F_cn,ij, F_ct,ij, F_dn,ij, F_dt,ij and m_ig are the particle fluid interaction force, normal contact force, normal damping force, tangential contact force, tangential damping force and gravitational force of particle i, respectively. I_i is the moment of inertia of particle i. The torque acting on particle i includes two components: torque by tangential forces M_t,ij and rolling friction torque M_n,ij [26]. The detailed information of the forces and torques acting on particle i can be found in Appendix A.

2.1.2. Governing Equations for Gas Phase in CFD

For the gas phase, the CFD method (FLUENT) is used to determine the flow field. The gas flow is described by Navier–Stokes equations, including the standard k-ε turbulence model [27], which can be written as,

$$\frac{\partial \epsilon \rho }{\partial t}+\nabla \cdot (\rho \epsilon u)=0$$

(4)

$$\frac{\partial \epsilon \rho u}{\partial t}+\nabla \cdot (\rho \epsilon uu)=-\nabla p+\nabla \cdot (\mu \epsilon \nabla u)+\rho \epsilon g-S$$

(5)

$$\epsilon =1-\frac{{\displaystyle {\sum }_i{V}_i}}{V_{\mathrm{cell}}}$$

(6)

where ε is the void fraction, ρ, u, μ and p represent the bulk density, velocity vector, viscosity and pressure of the fluid, respectively; V_i represents the volume of particle I; V_cell represents the volume of a CFD mesh cells and S is the momentum sink, which is the sum of all gas drag forces in the grid.

2.1.3. The Coupling of Particle and Gas

The CFD–DEM coupling module was used to couple the DEM simulation with CFD. At each time step, the DEM transmits information to the CFD model, including the volume fraction, position and velocity of the individual particles. In the CFD simulation, the momentum sink S (the sum of all gas drag forces in the grid) is added to each of the mesh cells to represent the effect of momentum transfer from the DEM particles. The momentum sink can be expressed as:

$$S=\frac{{\displaystyle \sum _i^n{F}_D}}{V_{cell}}$$

(7)

where F_D is the force on particle i in an iteration from the fluid and n is the number of particles in a single CFD mesh cell.

Momentum coupling would lead to an additional gas drag force on the particles. The drag force on particles (F_D) was calculated according to Ergun [28] and Wen & Yu [29] in the simulation,

$$F_D=\frac{\beta V_i\left|u-v_i\right|(u-v_i)}{1-\epsilon }$$

(8)

where v_i denotes the solid velocity vectors and β is described by,

$$\beta =\{\begin{array}{l}{\beta }_{\mathrm{Ergun}}=150\frac{{(1-\epsilon )}^2\mu }{2\epsilon R}+1.75(1-\epsilon )\frac{\rho }{2R}\left|u-v_i\right|,\epsilon <0.8\\ {\beta }_{\mathrm{Wen}\&\mathrm{Yu}}=\frac{3}{4}{C}_D\rho {\epsilon }^{-1.65}(1-\epsilon )\left|u-v_i\right|,\epsilon \ge 0.8\end{array}$$

(9)

$$Re=\frac{2R\rho \epsilon \left|u-v_i\right|}{\mu }$$

(10)

$$C_D=\{\begin{array}{l}\frac{24}{R},Re\le 0.5\\ \frac{24}{R}(1.0+0.15R{e}^{0.687}),0.5<Re\le 1000\\ 0.44,Re>1000\end{array}$$

(11)

where C_D is the resistance coefficient, which can be determined based on the Reynolds number Re.

2.1.4. Simulation Conditions and Procedure

A 2D slot model was used to simulate the void formation in a packed bed via the CFD–DEM approach. As shown in Figure 1, the 2D slot model is composed of a rectangular solid slot and a tube at the left wall (i.e., the tuyere, which is simplified as a square tube here). The cell size is 44.3 mm. The 88.6 mm × 88.6 mm square tube with a depth of 300 mm was inserted as a gas entrance.

During the simulation, a certain number of 40-mm-diameter spherical particles was pre-charged first from the top of the model. The particle/cell size ratio is about 1.1/1, and the particle size is slightly smaller than the cell size. The density, material shear modulus and Poisson’s ratio of the particle were 1000 kg/m³, 0.1 GPa and 0.21, respectively. Other parameters that were used in the simulation are shown in

Table 1. Parameters used in the simulation.

Symbol	Parameter	Value
e	Restitution coefficient	0.1
μs,p-p	Particle–particle static friction coefficient	0.63
μs,p-w	Particle–wall static friction coefficient	0.56 (particle–wall) 0 (particle–stressless wall)
μr	Rolling friction coefficient	0.32

. When all pre-charged particles had settled, the gas with a density of 0.716 kg/m³ and a viscosity of 1.88 × 10⁻⁵ kg/(m·s) was injected into the packed bed at a constant 232 m/s. The gas outlet at the top was set as the boundary condition of the pressure outlet. The front and rear walls were set with periodic boundary conditions. The left and right walls were solid walls. In the CFD-DEM simulation, the time steps is 5 × 10⁻⁵ s in DEM solution and 4 × 10⁻⁵ s in CFD solution.

2.2. Experiment to Verify Mathematical Modeling

A cold experiment was conducted to verify the modeling. The experimental designs were based on the similarity principle, including the geometry, and flow similarity for the Froude number [30]: $F{r}_{\mathrm{s}}=\frac{{\rho }_{\mathrm{g}}}{{\rho }_{\mathrm{p}}-{\rho }_{\mathrm{g}}}\cdot \frac{u_{\mathrm{s}}^2}{g{d}_{\mathrm{p}}}$ and $F{r}_g=\frac{{\rho }_{\mathrm{g}}}{{\rho }_s-{\rho }_{\mathrm{g}}}\cdot \frac{u_g^2}{g{d}_{\mathrm{p}}}$. On the basis of a geometry similarity principle, a slot experimental model was designed with a 1:10 scale compared with the 2D slot model that was used in the simulation. Based on the flow similarity principle, the gas with a density of 1.23 kg/m³ and a dynamic viscosity of 1.78 × 10⁻⁵ kg/(m·s) was injected into the tuyere at 20 m³/h. Polymethyl methacrylate (PMMA) particles with a stacking angle of 35–45° and a bulk density of 1000 kg/m³ were used as the model particles. The equivalent diameter of the PMMA particles ranged from 3 to 4 mm. Detailed experimental information is provided in Appendix B.

2.3. Probabilistic Method to Determine Void Morphology

Based on the above CFD–DEM approach, void formation in a packed bed can be evaluated by simulation. We propose a new approach to determine the void morphology based on a probabilistic method by analyzing each pixel in the obtained simulation images of the void. As shown in step 1 of Figure 2, at a certain time t = 2.55 s, the simulation images were converted to gray images using the ‘Rgb2gray’ function by MATLAB. Subsequently, the gray images were changed to binary images using the ‘graythesh’ function by finding an appropriate threshold value of the image based on the Otsu method [31].

Because the original image is intercepted according to the resolution of 1280 × 960 of DEM derived image, the derived matrix is 1280 × 960. In these binary images, the black and white areas represent particles and voids, respectively. Afterwards, as shown in step 2, these binary images are exported as matrixes, and contain 0 (black) or 1 (white). Then, the number of 1 (white) at each element in all the matrixes is determined, and the probabilities of the white (P_pix,w) for each pixel in the images is calculated by,

$$P_{pix,w}=\frac{N_{pix,w}}{N_{total}}$$

(12)

where N_pix,w and N_total are the number of whites at each pixel and the total number of samples, respectively.

As shown in step 3, when the images with P_pix,w matched well with the void that was obtained in the simulation, this probability can be used to determine the void size. To verify the probability method, the voids that formed under various gas velocities (i.e., 168 m/s, 184 m/s, 200 m/s, 216 m/s) were studied.

3. Results

3.1. Verification of the Modeling

A mathematical model was verified by a cold experiment. Figure 3 shows the void formation in the simulation (Figure 3a) and experiment (Figure 3b) under the same conditions. As shown in Figure 3a, after gas injection, a void with a plume-like shape (the void height was greater than its depth) formed, which is consistent with previous studies [21]. The packed bed, particle and blast properties were all checked. The blast velocity was more than 1 m/s, and the pore size was more than 0.5 mm. Because the pressure drop and the void morphology can be affected by the conditions of packed bed and flow parameters [32,33], the simulation results need to be verified with the experimental results. Figure 3b shows that a similar plume-like void that formed in the cold experiments under the same conditions. This result indicates that the established mathematical model is suitable for simulating the void formation under realistic conditions.

3.2. Dynamic Stability Characteristics of Void

The void morphologies from 2.55 s to 3.55 s in Figure 4a–f show that the morphology changed with time, and the images were obtained per 0.02 s. The particle positions (i.e., blue, red, green, magenta and dark cyan) move along the void boundary with time, which shows the dynamic stable characteristics of the void. To analyze the boundary change of the void quantitatively, we extracted the particle position at the void boundary when t was 2.55 s, 3.15 s and 3.35 s. As shown in Figure 4g, particles at the left and bottom boundaries were located almost in the same position, whereas for those at the top and right boundaries, the distance deviation was 100 mm and 70 mm, void morphology was not static even after the macroscopically stable stage, that is, the void morphology is in a dynamic stable state with time. Therefore, it may be inaccurate to use the transient morphology of the void as the result during the dynamic stable process.

3.3. Determination of Dynamically Steady Void Morphology

In this work, a mathematic method was proposed to determine the morphology of a dynamically steady void based on the probabilistic method. Figure 5(a1–a7) show an overlay of the binary image and P_pix,w (e.g., P_pix,w > 0%, P_pix,w > 20%, P_pix,w > 40%, P_pix,w > 50%, P_pix,w > 60%, P_pix,w > 80% and P_pix,w = 100%) at t = 2.95 s. The void area that was obtained by this probabilistic method decreased with an increase in P_pix,w and the void area was smaller than that of binary image when P_pix,w was 100%. We also found that when P_pix,w > 50%, the void area matched the void that was obtained in simulation at t = 2.95 s, as shown in Figure 5(a4).

To analyze the morphology of the void areas obtained with different P_pix,w (i.e., P_pix,w > 0% (red), P_pix,w > 50% (purple) and P_pix,w = 100% (green)), the probability boundaries were overlaid in Figure 5b. The region outside the boundary of P_pix,w > 0% with only black samples was denoted as a fixed bed, whereas the region inside the boundary of P_pix,w = 100% with only white samples was denoted as an absolute cavity. The area in the middle was denoted as a particle-transition region. The area obtained at P_pix,w > 50% matched the simulated void, therefore, the boundary from P_pix,w > 50% can be defined as the best matching boundary of the void, with a depth and height of 830 mm and 962.5 mm, respectively. Please refer to Appendix C for details.

To verify that the area obtained at P_pix,w > 50% was suitable for determining the void morphology that formed during the dynamic stability process, we compared voids that were obtained in the simulation at t = 2.55 s, 2.75 s, 3.15 s and 3.35 s with the area from P_pix,w > 50%. As shown in Figure 5(c1–c4), the area of P_pix,w > 50% matched the simulation images at different times well, i.e., 42 images. Hence, P_pix,w > 50% can be applied as the well-matched void boundary. Detailed information on the comparisons between the binary images of the void during the dynamic stable process and images at different probabilities is discussed in Appendix C.

3.4. Void Morphology Formed under a Series of Gas Velocities

Previous studies have reported that the void size increases with an increase in gas velocity [15,20,21,34,35]. Various gas velocities were simulated and compared to verify the reliability of our probabilistic method. Figure 6 shows the void morphology formed for five gas velocities, i.e., 168 m/s, 184 m/s, 200 m/s, 216 m/s and 232 m/s, which were determined by using the probabilistic method with P_pix,w > 50%. When the gas velocities were 168 m/s and 184 m/s, the spherical void was formed near the tuyere, with a 200-mm diameter. When the gas velocity increases to 200 m/s, the void height and depth increased to ~450 mm. Therefore, a positive relationship exists between the gas velocity and the void size.

Figure 6 shows that the void morphology became a plume-like area when the gas velocity increased (i.e., 216 m/s and 232 m/s), and the void size increased visibly. The void height and depth that formed at 232 m/s increased to 962.5 mm and 830.0 mm, respectively. Because the gas velocity influences the blast kinetic energy, the particle penetration depth in the void changes accordingly. Overall, the void height and depth increase with an increase in gas velocity, which is consistent with the results by Santana et al. [26].

Therefore, the probabilistic method is an effective approach to determine the void morphology. The void boundary that was determined by the probabilistic method was continuous but not smooth, which differed from the previous definition of a void boundary. The method provides an effective strategy to solve the deficiency of DEM results, that is, the transience of the DEM results cannot describe the void change during the dynamic stable process, which is important for theoretical and mathematical modeling of the two-phase gas–solid flow and chemical reaction in and around the void.

4. Conclusions

A 2D two-phase gas–solid flow model in a packed bed was established to simulate the void morphology numerically. Using the CFD–DEM numerical simulation, we discovered that the macroscopically stable voids are dynamically stable because of the particle movement around the boundary. We propose a probabilistic method to determine the void morphology during a dynamic stable process. The boundary at P_pix,w > 50% was selected and defined as the well-matched void boundary. The void morphology that formed at different gas velocities was simulated and compared based on this method to verify the reliability of our probabilistic method. With an increase in gas velocity, the void morphology changes from a small circle around the tuyere to a large, tall and thin ellipse.