Method for Simulating Gas Permeability of a Coke Bed Including Fines Based on 3D Imaging on the Coke Particle Morphology

Abstract

In ironmaking blast furnaces, the particle size distribution and voids in the coke bed affects the upward flow of gas, and consequently, the efficiency of the combustion reaction. To clarify the influence of coke pulverization on the packing structure of the coke bed, the permeability of the bed was evaluated using detailed dynamics simulation and geometric data analysis. To obtain detailed 3D morphology of the coke, we derived digital geometric data using rotational strength tests. Using the Euler–Lagrange coupling approach with the multisphere discrete-element method, the effect of the volume fraction of fines and distribution in the coke bed on the gas flow was analyzed. The void shape in the 3D coke bed structure was quantified using geometric data and simulated gas flow distributions. Although a continuous void network was observed in the packed bed before pulverization, areas of highly restricted (or no) gas flow were observed after pulverization. The dominant effect of coke degradation on the packed bed structure was the disruption of the gas flow path because of fines clogging the pores and narrowing the gas flow path. The developed simulation method can comprehensively analyze the effects of coke degradation on the gas flow distribution in the coke bed and can be used to analyze and control the instability of industrial blast furnaces.

1. Introduction

In an ironmaking blast furnace, a coke packed bed descends slowly as the coke is consumed by combustion in the raceway. In addition to the downward flow of solids, there is an upward flow of reducing gas in the coke bed. Although the motion of the solids is negligible on the timescale of gas flow, the ever-changing packing structure makes characterization of the gas flow difficult. Furthermore, the coke bed has a complex structure that cannot be characterized as a packed bed of simple spheres.^1,2) Because the gas stagnation induced by local bed clogging can cause a significant pressure drop, a detailed understanding of the process and internal furnace control are necessary to avoid operational instabilities.

The dynamic behavior of fine particles (fines) flowing in the packed coke bed (including their degradation, i.e. breakdown into smaller particles) has been the subject of intense study in blast furnace operation because fine particles get detached from the coke particles and affect the permeability of the coke bed.^3,4,5) Several researchers have studied the governing mechanisms of coke degradation and clogging in packed beds both experimentally^6,7) and numerically,^8,9,10,11) and more recently, using machine learning.¹²⁾ Most attempts to assess multiphase flow (including fines and gas) in packed beds have involved the correlation of the holdup with the pressure drop or other variables through the application of empirical or semi-empirical approaches.¹³⁾ The most significances of these studies involved the development of empirical correlations for specific ranges of conditions, which are unreliable outside of these ranges as they omit vital information. A typical example is the neglect of coke degradation dynamics at high temperatures, as most existing empirical correlations for this fail to consider the effect of 3-dimensional (3D) individual coke shape changes. Therefore, local fines clogging must be evaluated using an averaged continuum approximation.

In this study, we propose a new holistic permeability evaluation scheme that uses detailed dynamics simulation and geometric data to accurately model the experimental 3D degradation behavior of coke particles. When using a multisphere discrete-element method (MS-DEM) treatment, both the coke particles and the dispersed phase can be directly tracked in packed bed structures with different cokes morphology.^15,16,17) Moreover, the Eulerian–Lagrangian (E–L) coupling scheme, also known as the computational fluid dynamics-DEM (CFD-DEM) approach that enables the investigation of the interactions between gaseous and solid phases, has been used to solve the gas flow field in various coke bed structures.¹⁸⁾ Using the E–L coupling approach, the two-phase fines–gas flow in a coke bed can be directly determined in accordance with the coke degradation behavior, which is used as an input. Rotational strength tests allow changes in the shape of coke particles, owing to the external force to be tracked by a 3D scanning technique. This measurement was applied to several coke samples having different strength indices, namely the coke strength after reaction. The measured 3-D shapes of individual coke particles were converted to the shapes of coke particles, their changes, and the fine-particle generation condition in MS-DEM. The coke bed structures containing fine particles were generated by MS-DEM, and then, the gas flow behaviors within these artificial beds were numerically analyzed. Thus, coke bed permeability values can be predicted using different coke degradation behaviors, i.e., the pressure drop disordering can be predicted by considering the motion for all coke particles and generated fine particles. Moreover, the relationship between the 3D void-scale structure of the coke bed, including fines, and the corresponding gas flow distribution was investigated using geometric data analysis based on digital images. Consequently, the relationship between the pore sizes determined from 3D microscopic structure and coke degradation can be quantified. This study aimed to establish a seamless physical model for studying the effect of the coke degradation behavior and 3D pore structure on gas flow characteristics.

2. Methodology

The gas flow in the packed bed, which was affected by the settling of coke due to gravity, coke degradation, and fine-particle generation, was calculated using the E–L coupling numerical scheme. Numerical analyses using the Lagrangian approach, which can track moving calculation points, were applied for the motions of coke and fine particles. The grid-based Eulerian continuous fluid model was used for calculating the gas flow.

2.1. Numerical Scheme of Solid Motion Tracking by MS-DEM

MS-DEM is a method to handle the motion of irregularly shaped solids. To express complex shapes, numerous small spherical computational elements are accurately arrayed to enable low calculation costs and intuitive mounting. In this study, the coke particles and container wall as the array of the elements and the fine particles as a single element were treated as the solid. The MS-DEM uses an ordinary DEM contact-force model and its fundamental. Calculation procedure followed previously described methods.^19,20) The elements existing inside the solid conform to the equations governing the respective translational and rotational motions of solid. When the solid-solid contact force F_c and gravity are considered as the forces acting on each individual element (the effect of the gas drag force on the particle motion is negligible for simplicity), the governing equations for the solid are written as follows:   

$$m_g\frac{d{\mathbf{v}}_g}{dt}={\displaystyle \sum }_{i=1}^{Nc}{\mathbf{F}}_{c,i}+m_g\mathbf{g}$$(1)

  

$${\mathbf{I}}_g\frac{d{\omega }_g}{dt}={\displaystyle \sum }_{i=1}^{Nc}{\mathbf{T}}_{c,i}$$(2)

Here, i is the element index, m_g is the mass of the solid, v_g is the translational velocity of the center of the mass, N_c is the number of particles in solid, g is gravity, I_g is the moment of inertia, ω_g is the angler velocity, and T_c,i is the torque around the solid. For F_c of the element i, the conventional contact-force model used in the DEM²¹⁾ was applied. In this study, the solid was assumed as rigid body, namely no special technique for binding among the elements was used. The temporal position and velocity for each element is calculated by using v_g and ω_g, to satisfy the following relations:   

$${\mathbf{v}}_g=\frac{1}{k}{\displaystyle \sum }_{i=1}^k{\mathbf{v}}_i$$(3)

  

$${\omega }_g={\mathbf{I}}_g{}^{-1}{\displaystyle \sum }_{i=1}^k\left\{({\mathbf{x}}_i-{\mathbf{x}}_g)\times {\mathbf{v}}_i\right\}$$(4)

  

$${\mathbf{x}}_g=\frac{1}{k}{\displaystyle \sum }_{i=1}^k{\mathbf{x}}_i$$(5)

Where, x_g is position vector of the rigid-body center of mass, and k is the number of particles in a solid. The variation of relative position vector of each element from the center of solid mass, q_i = x_i − x_g, can be related with solid rotation. Considering finite time step Δt, quaternion Q_i^t = (q₀, q_i) = (q₀, q_x, q_y, q_z) is related to the conjugate quaternion Q* by Q_i^t = QQ_i^t−ΔtQ*. Here, the quaternion around the rotational axis ${\omega }_g/\left|{\omega }_g\right|$ is expressed as the rotational angle $\theta =\left|{\omega }_g\right|\mathrm{\Delta }t$, at ${\text{q}}_0=cos\frac{\theta }{2}$, ${\mathbf{q}}_i=\frac{{\omega }_g}{\left|{\omega }_g\right|}sin\frac{\theta }{2}$, and the transformation to the rotation matrix R is described using quaternion components.²²⁾ Therefore, the transformation from the quaternion to the element position vector x_i^t, comprising the rigid body, is expressed as follows:   

$${\mathbf{x}}_i{}^t={\mathbf{x}}_g+\mathbf{R}{\mathbf{q}}_i{}^{t-\mathrm{\Delta }t}$$(6)

Using this technique, the structures of packed coke beds are obtained. The solid shapes reflect the actual shapes of coke particles; the measurement method is explained later. The solids are randomly dropped into a rectangle container of predetermined shape. The motion of the solids is tracked until the static state is reached. Then, coke degradation and the accompanying fine particle generation are represented in the following manner. This calculation scheme is robust in the range of k ≥ 1, where a solid with k = 1 is defined as “fine,” and k ≥ 1 is defined as “coke.” Some elements are detached from the surface of the solid, and the individual detached elements are treated as fine particles similar to that observed in previous reports.²³⁾ In this study, the rotation of fine particles is ignored, and the fines diameter is constant.

2.2. Gas Flow Calculation

To evaluate the gas permeability of a packed bed structure, a CFD simulations of the void shape of each coke bed were performed. The entire calculation domain, which corresponded the inner shape of the container, was discretized into uniform cubic lattices, where a cell was occupied by a gas or solid (coke or fine) phase. The CFD cell size was set equal to the element size.¹⁶⁾ The phase boundary shape, namely the void shape, was then obtained from the locations of solid elements. The CFD cells that included the coordinate of the element center were marked as “filled” and treated as being occupied by the solid. The gas flow behavior within the “non-filled” cells was calculated. As the boundary condition, the velocity on the surfaces of the “filled” cells was set to zero. Although this simple classical boundary condition has a lower resolution than a body-fitted mesh, it has a significant advantage in terms of performing 3D geometric data analysis (see Section 2.5).

The gas flow distribution was then estimated by solving the continuity and the Navier–Stokes equations assuming an incompressible flow and constant properties:   

$$\nabla \cdot \mathbf{u}=0.$$(7)

  

$$\rho \frac{\partial \mathbf{u}}{\partial t}+\rho (\mathbf{u}\cdot \nabla )\mathbf{u}=-\nabla P+\mu {\nabla }^2\mathbf{u}.$$(8)

where u and P are the gas velocity and the pressure, respectively. A staggered-grid finite-difference method was then applied using a Cartesian coordinate system to discretize Eq. (8) based on the unsteady simplified marker and cell method.²⁴⁾ The non-steady, advective, pressure gradient, and viscosity terms were discretized using the second-order Adams–Bashforth method, third-order upwind difference (UTOPIA scheme), first-order forward difference scheme, and center difference schemes with second-order accuracy, respectively. To determine the pressure of each CFD cell, the Poisson equation of the pressure correction at each time step derived from the continuity equation required iterative calculation, and was solved by applying the Bi-CGStab method.²⁵⁾ As mentioned above, for simplicity of the dynamics calculations, the influence of the gas flow on the solid motion was ignored because the average gas flow velocity in this study is assumed to be sufficiently small; that is, the influence of momentum exchange from the gas to the other phases must be small enough to avoid the fluidization of fines.

2.3. Digitalization of the Shape and Deformation Behavior of Actual Coke

To obtain the detailed 3D shape information for the representative coke samples, the surfaces profiles of the coke particles were converted to the digital geometric data corresponding to coke surfaces using the following procedure. Figure 1 shows a diagram of acquisition of 3-D geometrical data of the shape variation in a coke sample, and its transmission to the coke degradation model. The 3D morphology of representative metallurgical coke samples was obtained using a NextEngine 3D scanner Ultra HD (with a resolution of 0.1 mm). The obtained surface coordinates were converted into standard triangulated language (STL), in other words, the surface shape was polygonated with a triangular mesh.²⁶⁾ To apply this shape information to the MS-DEM, the spherical particles in the model were filled to conform to the shape. With these procedures, the shape variations of the individual coke samples with degradation can be obtained. The shape of the coke particle is described as an array of the element in MS-DEM. The obtained coke surface shell that is described with STL polygons was filled with the spherical elements arranged in a cubic lattice according to a level set function.²⁷⁾ That is related to the distance from the center position of each particle arranged in a square lattice at the interval constant particle diameter (d = 1.5 mm) to the surface of the triangular polygon embedded in the volume. Figure 1(b) shows the concept of the coke degradation model. The broken line illustrates the initial shape of a coke sample, and the solid line is the shape of the same sample after the degradation test. The volume between these two lines is disintegrated due to mechanical stress during the degradation test. The right figure in Fig. 1(b) depicts an arrangement of the spherical elements in a region of the coke sample. Initially, all the elements constitute the coke sample, and the white elements are detached from the coke particles during the degradation test. The released elements can move individually according to the equation of translational motion (Eq. (1)). This procedure describes the generation and flow of fine particles.

Fig. 1.

Schematic of the coke deformation model. The volume fraction of fines generated from the broken coke is consistent with the conservation of volume. (a) Changes of coke particle shape by tumbler test and tracking by 3D-scanning. (b) Fine particles generation at 50 rev. by multi-sphere model. (Online version in color.)

To track the deformation behavior of each coke samples, the tumbler test (JIS K 2151–1977), which is one of the rotational strength test methods, was adopted.^28,29) The coke samples were prepared by the following procedure. First, 10 kg of metallurgical cokes were sieved to obtain the particle size of 25–38 mm. The sieved coke was gasified at 1373 K for 40 min in a CO₂ atmosphere. Then, nine portions of each of the coke samples were selected from the gasified coke so that the average particle size of 35 ± 0.05 mm. The shapes of these representative coke samples were scanned by the above-mentioned method. On each representative coke, a 1-mm-diameter hole was made in each representative coke sample, and a thread was tied through the hole as a marker.¹⁸⁾ The coke samples were introduced into the tumbler test machine and it was rotated for 50 revolutions. The representative coke samples were collected from the machine and 3D surface shapes were scanned again. In this study, 4 types of coke were evaluated. Table 1 shows the changes in the sphere-equivalent particle diameter of each coke sample before and after the tumbler test. The decreases in average diameter were in the range of 2.87 to 4.78 mm, and the average degradation ratios [= 1-(coke volume after degradation)/(volume before degradation)] were obtained as 0.226, 0.262, 0.355, and 0.357 [-] for the coke types A to D, respectively.

Table 1. Average particle diameters (mm) of the four types of coke before and after pulverization.

Sample #	A, 0 rev.	A, 50 rev.	B, 0 rev.	B, 50 rev.	C, 0 rev.	C, 50 rev.	D, 0 rev.	D, 50 rev.
1	36.46	35.04	31.24	26.72	37.54	33.51	37.50	32.65
2	29.30	27.35	29.93	28.23	43.85	33.05	43.36	39.68
3	39.53	34.75	33.82	32.35	36.56	33.22	31.14	29.06
4	36.33	34.52	37.34	34.94	34.84	31.92	31.59	25.18
5	29.98	27.35	40.65	36.51	32.72	26.80	37.84	31.54
6	35.45	34.65	29.90	25.32	32.89	30.86	31.75	28.43
7	34.64	28.43	36.63	34.90	29.31	20.95	34.62	27.74
8	39.90	35.76	31.82	27.13	35.45	32.47	29.44	25.19
9	30.19	28.25	37.77	32.16	25.90	24.14	32.74	25.98
Average	35.05	32.18	34.74	31.40	35.01	30.25	34.95	30.17

2.4. Calculation Procedures and Conditions

In this study, the numerical simulations were performed according to the following procedures and conditions. First, the packed bed of coke samples before the degradation test was formed in a rectangular container with square horizontal cross-section of side 0.32 m. Four packed beds were formed for the individual coke types. The container was filled by dropping 1296 pieces of coke particles (144 pieces of each of the nine representative samples). The initial position and azimuthal angles of each dropped coke particles were determined using a pseudorandom number. The coke particles and the container walls consisted of the MS-DEM elements of diameter d = 1.5 × 10⁻³ m. The following physical properties were applied to the coke and the fine particles; density = 1050 [kg/m³], dynamic friction coefficient = 0.43, Young’s modulus = 2.4 [GPa], and Poisson’s ratio = 0.35 [-]. The coke particle motion was tracked with the time step of Δt = 1.0 × 10⁻⁵ [s] until steady state was achieved. The obtained packing structures were treated as the initial structures. Images of the packing process and the initial structure of coke A are shown in Fig. 2. The bed height in this case was approximately 0.6 m, and the 3D coke bed structure using the actual coke shape was obtained with sufficient resolution.²⁶⁾ Once the initial structure was obtained, the gas flow simulation was performed till the flow velocity reached a steady state. The gas flow was uniformly introduced from the bottom of the bed structure with an upward velocity of 0.1 [m/s]. For the gas flow simulation, cubic lattices were used at uniform intervals that were the same as the element diameter d = 1.5 [mm], and no-slip condition was imposed on the solid surface for gas flow as mentioned above.

Fig. 2.

MS-DEM simulation results for coke A charging process, where 1296 coke particles of identical strength were dropped into a container with a square cross-section of 0.32 × 0.32 m². The same calculation was performed for coke B, C, and D, and the height of each layer was approximately 0.60 m. (Online version in color.)

Subsequently, as explained in Section 2.3, the solid elements in the detached volume during 50 revolutions of the tumbler test were released from each coke particle. The released solid elements fell down within the coke bed and corruption of the packing structure occurred. Thus, the movements of coke particles and fine particles were tracked again while gas flow simulation continued. To maintain equivalent fine-particle distribution in the entire calculation region, only the fine particles were allowed to flow out from the bottom of the container, and the discharged fine particles were re-introduced on the upper part of the packed bed according to the periodic boundary condition. This fine-particle circulation operation was performed every 0.1 s. The simultaneous computation of the gas flow and solid behavior was continued till the average velocity of the fine particles reached a stable value.

2.5. Evaluation of the Void Shape in the 3D Coke Bed Structure

In this study, the packed coke bed structure that reflected the actual coke particle shape was numerically generated, and the gas flow behavior including the pressure drop was analyzed. As the gas phase flowed inside the void space, it was deemed necessary to formulate a method to evaluate the void shape, which could be associated with the gas flow characteristics. The Burn algorithm^30,31) was applied to evaluate the 3-D shape of voids and investigate the relationships between the results of the coke bed structure simulation and void shapes. Figure 3 shows a schematic diagram of “Burn algorithm.” For the evaluation of the void shape with this algorithm, the same cubic lattice voxels that was used for the flow analysis were used. Note that Fig. 3 is displayed in 2-D for simplicity, calculations were carried out in 3-D. First, all voxels that were occupied by the solid phase (coke or fine particle) were labeled as “0.” Second, the gas phase voxels adjacent to the solid phase voxels were labeled as “1.” It may be noted that we labeled voxels that were attached to the solid voxel through the side face and those attached through the side line or apex. Third, the gas phase voxels next to the voxels with label “1” and without their own labels were labeled as “2.” These rules were applied to the surrounding 3 × 3 × 3 – 1 = 26 cells in 3-D for each step update, where the minimum number of cells plus 1 was stored. Then, this labeling process was repeated by increasing the labeling numbers till all voxels were labeled. The labeling number is also known as the “Burn number.” A larger Burn number indicates a location where the void is thick because the location is far away from the solid surface. The frequency distribution and maximum value of the “Burn number” determine the geometric characteristics of the void shape. Thus, the relationships between the Burn number distribution and coke bed permeability have been evaluated by calculating the frequency distribution of the Burn number, and examining its relationship with the flow velocity value stored in each voxel.

Fig. 3.

Schematic of the algorithm used to evaluate the void structure by the Burn method, where each cell was assigned an integer value depending on its composition (solid phase = 0) and proximity to the solid (gas phase ≥ 1). (Online version in color.)

3. Results and Discussion

3.1. Changes in Coke Bed Structure and Gas Flow by Coke Degradation

In this section, we discuss the variation in gas flow behavior arising from coke degradation and the accompanying fine-particle generation. Figure 4 compares the coke bed structures of the four types of coke before and after degradation. The figure shows the calculated distributions of coke, fines, and gas over the vertical cross-section of the center of the packed coke bed, shown at black, red, and white areas, respectively. As mentioned above, the solid elements near the coke particle surface detach from the coke particle by degradation and turn into fine particles. At the same time, each coke particle shrinks, i.e., during degradation, the outer parts of the coke particles were fractured and removed as fines, resulting in a change in the bed structure. The coke A has the highest strength and generates the lowest number of fine particles. The fine particle generation amount increases in the order of coke A, B, C, and D. In all the cases, the bed height decreases after degradation. This disorder is caused by random packing of the initial bed and minor difference in strength. The fine particles distribute unevenly, and precipitate intensively above the coke particles and at bottleneck parts⁹⁾ among coke particles. Although the average particle size ratio of coke to fines is d_coke/d_fine ≥ 10 and it indicates that the fine particles can penetrate through the coke bed by the gravitational force,¹¹⁾ in some heavy precipitation region, the clogging occurs since the fine particles are supplied one after another.^8,9) The comparison of the four figures after degradation shows that the vacant space tends to decrease with an increase in the amount of fine particles, and more concentrated clogging occurs.

Fig. 4.

Simulation results of the spatial distribution of coke (black), fines (red), and voids (white), shown as 2D vertical cross-sections at the center of the packed beds before and after pulverization of coke (a) coke A, (b) coke B, (c) coke C, and (d) coke D. (Online version in color.)

Figure 5 shows the gas velocity distributions for each steady-state condition. It may be noted that the gas flow behavior within the void space is analyzed in this study, and the solid part is colored white in this figure. Network-like gas flow paths are formed in the bed and the gas velocity distributes on reflecting from the packing structures. The gas velocity distributions clearly show the different structures before and after the degradation. Before degradation, the velocity distributions are relatively uniform, although high velocity zones appear along the side wall due to the wall effect on the void distribution. After degradation, the white-colored regions, indicating the solid component, become continuous, indicating that the generated fine particles stack among the coke particles and restricted the flow path. At the same time, the higher velocity region colored red appears locally in the middle part of the bed. With increase in the amount of fine particles, the higher velocity regions tend to expand and connect to each other. As the average degradation ratio increases, more regions of high gas flow velocity are observed. In the case of coke D, these areas have a continuous string morphology. The bed height decreases by degradation, while the amount of the solid (coke and fine particles) is constant, resulting in a decrease in the voidage of the bed. Due to the clogging of the void space in the bed, the gas flow paths were narrowed and limited. The strong concentration of the gas flow occurs in the barely kept small voids. As the gas inflow is constant, the increase in the flow velocity in a narrow flow path is reasonable.

Fig. 5.

Simulation results of the gas velocity distributions (constant gas flow of 0.1 m/s) through the packed beds consisting of coke (a) coke A, (b) coke B, (c) coke C, and (d) coke D. The white regions indicate the solid phase, while the flow velocity of the gas phase increases from blue to red. (Online version in color.)

With these changes in the gas flow pattern by degradation, the pressure distributions in the bed change as shown in Fig. 6. Due to the gas flow simulation method, namely, gas flow analysis within the void space of randomly packed irregular-shaped solids, the isobars fluctuate mildly. Before degradation, the isobars are practically located horizontally with constant spatial intervals. This implies that the pressure changes linearly with height, and the gas phase macroscopically flows one-dimensionally that is a typical flow field in the packed bed.^32,33) After the degradation, the width between the isobars decreased, indicating a higher pressure drop compared to before degradation. In addition, the increase in isobar distortion after degradation indicated the effect of gas flow drifting.

Fig. 6.

Simulation results of the pressure distributions inside the packed beds of coke (a) coke A, (b) coke B, (c) coke C, and (d) coke D. The increase in pressure in the beds is visualized by a change in color from blue to red, where each color is divided by isobar lines. Each pressure was normalized by the maximum pressure (red). (Online version in color.)

The four types of coke with different strengths showed different packed bed structure and gas flow distribution. These differences were quantified by analyzing the 2D sections at the same position and comparing the pressure drops. Figure 7 shows the evaluation procedure and relationship between the volume fraction of fines and the pressure drop. As shown in Fig. 5, there is a wall effect on the packing structure near the wall region. Thus, the cubic portion of the side of dimension 0.15 m is cut out from the middle part of the packed bed as shown in Figs. 7(a) and 7(b). It may be noted that the location of the cut-out portion is common throughout all conditions. As shown in Fig. 7(c), the average pressures on the top and the bottom faces of the cut-out cubic portion are calculated, and then the pressure drop [Pa/m] is obtained. Figure 7(d) summarizes the variations in the fine-particle volume fraction and pressure drop with degradation. The fine-particle volume fraction increases with increase in the fine-particle generation quantity. The change in the void fraction associated with degradation was obtained from the particle volume in the examined coke variants: 0.4741→0.4123 (coke A), 0.4557→0.3424 (coke B), 0.5004→0.3430 (coke C), 0.4496→0.2967 (coke D). Notably, the pressure drop clearly increased with the increasing volume fraction of fines. As mentioned above, the pressure drop increases by coke degradation, and larger increment is observed in the bed with higher fine-particle volume fraction.

Fig. 7.

Quantitative evaluation of the pressure drop as a function of the volume fraction of fines. Generation of analysis cubes with side lengths of 0.15 m for the states (a) before and (b) after pulverization. (c) The average pressure difference (P1–P2) was calculated using a representative cube from the packed bed. (d) The pressure drop as a function of the amount of generated fines. (Online version in color.)

3.2. Change in the Coke Bed Structure by Coke Degradation

As noted in a previous study,³⁴⁾ the most prominent gas flow behavior in a packed bed is channel-like flow, and concentrated gas flow channels can be observed in the coke bed after degradation (Fig. 5). Some channels are partially or completely clogged with the fine particles generated by coke degradation, and this narrows the channels within which the gas can flow. It is considered that the increase in the pressure drop by degradation is resulted from the changes in the void structure and gas flow inside. In other words, the multiple effects of choking and thinning of the flow channels, the connection of the open channels, gas flow concentration, and other factors determine the pressure drop of the packed bed with degradation. Thus, the coke bed permeability might be correlated to the 3-D structure of the coke bed based on the actual particle shapes. In this section, the “Burn algorithm,” as explained in Section 2.5, is introduced to characterize the void structure.

Figure 8 shows the Burn number distribution on the same vertical cross-section as Figs. 5 and 6. The void space is colored with the Burn number, while the solids are colored as white. The side length of the cubic voxels to evaluate the Burn number has the same length of the solid element, namely it coincides with the cells for gas flow analysis. The Burn number corresponds to the distance from the solid surface. All the voxels with Burn number greater than five are colored red. In the packed bed before degradation, a continuous void network, which is indicated by the spatial connections of voxels with the Burn numbers of two or three can be observed, even in the 2D cross-section images. In this cross-section, the volume occupied by the coke particles (Burn number = 0, white) is spatially dispersed. In contrast, larger continuous white regions that consist of the coke and fine particles are observed in the packed structure after degradation, and the voids become discontinuous. In the packed beds before degradation, the voxels colored yellow and red, for which the Burn number is larger than four, are distributed over the packed bed. After degradation, such voxels seem to decrease, and the voids in the bed in coke A are generally wider than that in coke D.

Fig. 8.

Simulation results of the Burn number distribution in the packed bed. The white areas are the solid phase, while the colors change from blue to red with increasing Burn number. (a) coke A, (b) coke B, (c) coke C, (d) coke D before and after pulverization. (Online version in color.)

For more quantitative analysis, the frequency distributions of the Burn number before and after degradation are summarized in Fig. 9. In this figure, the frequency of Burn number is plotted against the square of the Burn number. From the geometrical relation, for a single cylindrical void, the Burn number distribution is linear to the Burn number, and linear to the square of the Burn number for a single spherical pore.³⁰⁾ In all packed bed structures, the frequency decreases exponentially with increase in the Burn number. With degradation, the gradient of frequency decline increases, and this tendency is more remarkable in the cases with higher fine-particle generation. In the beds of coke C and D, the maximum Burn numbers decrease. These trends suggest that the large void space in the bed decreases with degradation, or the narrowing of the gas flow paths is evident. It may be noted that the frequency of the cells with the value of unity corresponds to the effective interfacial area at the CFD cell scale, and not to the actual interfacial area. The frequency distributions generally show linear relationship with the square of the Burn number with slight deviation of the downward convex. The plot for the maximum Burn number tends to deviate from the overall tendency due to the spatial resolution. It is considered that the void shape in the coke bed generally has tank (spherical) shape, and the tank-type voids are connected with tube type voids.

Fig. 9.

Frequency distribution of the square of the Burn numbers for coke (a) coke A, (b) coke B, (c) coke C, (d) coke D before and after pulverization.

From the geometrical consideration, in the packed bed of mono-dispersed spheres, the tube type flow path is mainly formed among three particles with triangle arrangement, and the tank-type void is formed among more than four particles in an arrangement such as tetrahedral.²⁾ The shape of the actual coke particles is irregular, and has edges, faces, and vertices. With such shapes, the tube type flow paths among three particles are shorter compared to the sphere bed because the particles can touch one another through the edges and vertices. At the same time, the void space surrounded by flat or concave faces can be formed and such space volume may be larger in the bed with actual-shaped coke.¹⁾ Therefore, the contribution of tank-type voids appears in the Burn number frequency.

In the packed bed without the existence of fine particles, the void size distribution shifts to a smaller value with decrease in the particle size. In the degradation condition in this study, the decrease in the coke particle size and the formation of the fine particles occur at the same time. The decrease in the void size enables the clogging of the flow paths, and narrow tube and tank-type voids are filled with the fine particles. The size shrinkage and the clogging of the voids together changes the Burn number distribution as shown in Fig. 9. Each maximum value of Burn number varies with coke A, B, C, D. This indicates that the initial void structure is different in each case. On the other hand, the larger size of the void shows insignificant change between 0 and 50 rev in each case. This suggests that some arch structure of the coke particles forms during variation in the packing with degradation, and a few void spaces with the size similar to as the coke particles can be observed in the packed bed after degradation in Fig. 8.

3.3. Void Morphology and Gas Permeability

In this section, the gas permeability characteristics of the void structures in the coke beds is considered. The dominant effect of coke degradation on the packed bed structure leads to narrowing of the gas flow path and decrease in void homogeneity, which is enhanced by an increase in the amount of fine particles in the bed. Although the frequency distribution of the Burn number expresses the overall morphological changes of the gas flow path, it is also necessary to relate the fluid dynamics information to the morphology of the flow path shape. This can help understand the relationship between the change in the microscopic structure and the permeability of the bed. In most transport phenomena analyses, the void space in the packed bed is assumed to have a continuous structure of circular tubes with various diameters in the flow direction.^16,35) The pressure drop, ΔP, per unit length, L, is equivalent to the energy loss due to the shear stress acting on the tube wall surface. The shear stress corresponds to the frictional force per unit surface area applied to the fluid on the tube wall surface. The relationship between the frictional force of the fluid flowing through the pipeline and the kinetic energy is $f\frac{1}{2}\rho {\mathbf{w}}^{2}A$. Here, f is the friction coefficient of the tube (unknown factor), and w is the streamwise flow velocity. As discussed in the previous section, the typical void shapes are tank and tube. The flow condition in this study is laminar and the permeability can be characterized by the viscous friction on the solid surface. In this study, the resolution is limited to the cell size d, thus it is impossible to evaluate the gas flow at a scale below 1.5 mm. However, this study uses computational cells with constant size, thus the flow velocity in the cells with the Burn number of unity corresponds to the velocity gradient on the solid surface dw/dn. Here, n is the distance from the solid surface in the normal direction, and Δn is fixed as d/2. Figure 10 shows the relationship between the pressure drop, ΔP/L, and the average gas velocity, w_ave². The pressure drop in the coke bed is strongly correlated with the gas velocity. The differences in the initial packing structures are insignificant, while the effect of fine-particle generation has a significant influence on the increase in both w_ave and ΔP. The correlation $\mathrm{\Delta }P/L\propto {\mathbf{w}}_{ave}{}^2$ indicates that ΔP is strongly dependent on the increase in the fraction of fine particles.

Fig. 10.

Relationship between the pressure drop in the coke bed and square of the average streamwise gas flow velocity, 1.5 mm from the solid surface. The line shows a linear best fit of the data. The data are labeled, e.g., and 0 and 50 refer to before and after pulverization, respectively.

Figure 11 shows the fraction of cells with certain gas velocity, w, adjacent to the solid surface. Here, the frequency is normalized by the total number of surface cells because the surface area of each structure is different. Notably, almost of the cells with gas velocity are lower than the average interstitial velocity in the initial condition (approximately 0.22 m/s). In contrast, as the amount of fine particles increases, the number of cells with high w values, especially higher than 0.5 m/s, increases. This suggests that the gas velocity gradient on the solid surface increases. As shown in the previous sections, the gas flow paths narrow down and/or are disrupted with increasing amount of fine particles. Hence, the gas flow velocity through limited gas flow paths must increase to maintain overall constant flow rate. In contrast, the fractions of the low-velocity cells (w < 0.5 m/s) in the cases of after 50 rev show insignificant change with the amount of fine particles, and this trend corresponds to the formation of ineffective voids to the gas flow due to clogging of fine particles.¹⁶⁾ This shows the influence of dynamic arch stability in the coke bed structure³⁶⁾ as mentioned in Section 3.2, where the analysis of the mechanical stability of the structure and the effect of the fine-particle fraction considering the gas drag force³⁷⁾ requires investigation as a future research topic.

Fig. 11.

Streamwise flow velocity distribution of gas phase cells with a Burn number of 1 (in contact with solid phases) for the four coke grades after pulverization.

4. Conclusions

In this study, Eulerian–Lagrangian numerical simulations were performed to analyze the complex packed bed structures of metallurgical blast furnaces, and clarify the influence of coke degradation on the gas flow behavior due changes in the 3D void structure. The main conclusions are as follows:

• The developed simulation method can comprehensively analyze the effects of coke degradation on the gas flow distribution in the coke bed, and can be used to analyze and control the instability of industrial blast furnaces.

• The visualization of the flow velocity and pressure drops showed that higher volume fraction of fines increased the gas flow velocity in the remaining voids, which formed continuous string shaped paths.

• Use of the Burn algorithm showed that the differences in the initial packing structure were insignificant, while the volume fraction of fines had a largest effect on the increase in both the surface cell velocity and pressure drop. The pressure drop was proportional to the square of the surface cell velocity, indicating that the former is highly dependent on the volume fraction of fines.

In this study, a uniform coke degradation is analyzed. However, the actual pulverization behavior in the coke packed bed may be discontinuous and non-uniform. Inhomogeneous degradation behavior can be discussed by using this method and is a future research subject.

Acknowledgment

A part of this study was conducted as a project of the Collaborative Research Division of Advanced Analysis of Iron and Steelmaking Processes, JFE Steel Corporation, and Tohoku University. S. Natsui was supported by an Iron and Steel Institute of Japan (ISIJ) Research Promotion Grant and the Steel Foundation for Environmental Protection Technology (SEPT). We would like to thank Dr. Yasuhiro Saito of the Kyushu Institute of Technology for providing useful contributions regarding the geometric data analysis.

References

• 1)   S.  Natsui,  A.  Sawada,  H.  Nogami,  T.  Kikuchi and  R. O.  Suzuki: ISIJ Int., 60 (2020), 1453.
• 2)   H.  Nogami and  T.  Yamawaki: ISIJ Int., 60 (2020), 1560.
• 3)   M.  Ichida,  T.  Nakayama,  K.  Tamura,  H.  Shiota,  K.  Araki and  Y.  Sugisaki: ISIJ Int., 32 (1992), 505.
• 4)   J.  Huang,  R.  Guo,  Q.  Wang,  Z.  Liu,  S.  Zhang and  J.  Sun: Fuel, 263 (2020), 116694.
• 5)   X.  Xing,  H.  Rogers,  G.  Zhang,  K.  Hockings,  P.  Zulli and  O.  Ostrovski: ISIJ Int., 56 (2016), 786.
• 6)   H.  Takahashi,  H.  Kawai,  T.  Kondo and  M.  Sugawara: ISIJ Int., 51 (2011), 1608.
• 7)   T. S.  Pham,  D.  Pinson,  A. B.  Yu and  P.  Zulli: Chem. Eng. Sci., 54 (1999), 5339.
• 8)   S.  Remond: Phys. A, 389 (2010), 4485.
• 9)   S.  Natsui,  S.  Ueda,  H.  Nogami,  J.  Kano,  R.  Inoue and  T.  Ariyama: Chem. Eng. Sci., 71 (2012), 274.
• 10)   T.  Lichtenegger and  S.  Pirker: Powder Technol., 325 (2018), 698.
• 11)   Z.  Xie,  S.  Wang and  Y.  Shen: Chem. Eng. Sci., 231 (2021), 116261.
• 12)   S.  Yan,  H.  Zhao,  L.  Liu,  Q.  Sang,  P.  Chen and  J.  Li: Complexity, 2020 (2020), 8785047.
• 13)   S.  Pintowantoro,  H.  Nogami and  J. I.  Yagi: ISIJ Int., 44 (2004), 304.
• 14)   X.  Dong,  A.  Yu,  J. I.  Yagi and  P.  Zulli: ISIJ Int., 47 (2007), 1553.
• 15)   S.  Natsui,  A.  Sawada,  K.  Terui,  Y.  Kashihara,  T.  Kikuchi and  R. O.  Suzuki: Chem. Eng. Sci., 175 (2018), 25.
• 16)   S.  Natsui,  S.  Ishihara,  T.  Kon,  K. I.  Ohno and  H.  Nogami: Chem. Eng. J., 392 (2020), 123643.
• 17)   S.  Natsui,  A.  Sawada,  H.  Nogami,  T.  Kikuchi and  R. O.  Suzuki: ISIJ Int., 60 (2020), 1445.
• 18)   S.  Natsui,  A.  Sawada,  K.  Terui,  Y.  Kashihara,  T.  Kikuchi and  R. O.  Suzuki: Tetsu-to-Hagané, 104 (2018), 347 (in Japanese).
• 19)   P. A.  Cundall: Proc. Int. Symp. on Rock Mechanics, International Society for Rock Mechanics (ISRM), Salzburg, (1971), 1971.
• 20)   D. O.  Potyondy and  P. A.  Cundall: Int. J. Rock Mech. Min. Sci., 41 (2004), 1329.
• 21)   S.  Ueda,  S.  Natsui,  Z.  Fan,  H.  Nogami,  R.  Soda,  J.  Kano,  R.  Inoue and  T.  Ariyama: ISIJ Int., 50 (2010), 981.
• 22)   H.  Kruggel-Emden,  S.  Rickelt,  S.  Wirtz and  V.  Scherer: Powder Technol., 188 (2008), 153.
• 23)   K.  Takahashi,  A.  Yoshino,  T.  Nouchi,  J.  Kano,  S.  Ishihara and  T.  Ariyama: Tetsu-to-Hagané, 105 (2019), 1108 (in Japanese).
• 24)   A. A.  Amsden and  F. H.  Harlow: J. Comput. Phys., 6 (1970), 322.
• 25)   H. A.  van der Vorst: SIAM J. Sci. Stat. Comput., 13 (1992), 631.
• 26)   S.  Natsui,  K. I.  Ohno,  S.  Sukenaga,  T.  Kikuchi and  R. O.  Suzuki: ISIJ Int., 58 (2018), 282.
• 27)   M.  Sussman,  A. S.  Almgren,  J. B.  Bell,  P.  Colella,  L. H.  Howell and  M. L.  Welcome: J. Comput. Phys., 148 (1999), 81.
• 28)   M.  Kondo,  Y.  Konishi and  K.  Okabe: Kawasaki Steel Giho, 6 (1974), 1.
• 29)   T.  Yamamoto,  K.  Hanaoka,  S.  Sakamoto,  I.  Shimoyama,  K.  Igawa and  K.  Takeda: Tetsu-to-Hagané, 92 (2006), 206 (in Japanese).
• 30)   W. B.  Lindquist,  S. M.  Lee,  D. A.  Coker,  K. W.  Jones and  P.  Spanne: J. Geophys. Res.: Solid Earth, 101 (1996), 8297.
• 31)   A.  Uchida,  Y.  Yamazaki,  S.  Matsuo,  Y.  Saito,  Y.  Matsushita,  H.  Aoki and  M.  Hamaguchi: ISIJ Int., 56 (2016), 2132.
• 32)   S.  Natsui,  H.  Nogami,  S.  Ueda,  J.  Kano,  R.  Inoue and  T.  Ariyama: ISIJ Int., 51 (2011), 41.
• 33)   S.  Matsuhashi,  H.  Kurosawa,  S.  Natsui,  T.  Kon,  S.  Ueda,  R.  Inoue and  T.  Ariyama: ISIJ Int., 52 (2012), 1990.
• 34)   Z.  Qi,  S.  Kuang and  A.  Yu: Powder Technol., 343 (2019), 225.
• 35)   K. I.  Ohno,  Y.  Kitamura,  S.  Sukenaga,  S.  Natsui,  T.  Maeda and  K.  Kunitomo: ISIJ Int., 60 (2020), 1512.
• 36)   E.  Goldberg,  C. M.  Carlevaro and  L. A.  Pugnaloni: J. Stat. Mech. Theory Exp., 2018 (2018), 113201.
• 37)   S. A.  Stephan,  Y.  Oishi,  H.  Kawai and  H.  Nogami: ISIJ Int., 60 (2020), 1528.