Burden Particle Contour Extraction from Digital Elevation Model of Blast Furnace Rough Surface

Abstract

This paper reports new research progress on characterizing the burden surface particles of the blast furnace. An algorithm is proposed to extract the contour of the particles on the burden surfaces from their digital elevation models. The statistical distributions of particle size corresponding to the coke and sintered ore burden surfaces are counted from the extraction results of the particle contours. The statistical results obtained in the former research are compared with those of here. The particle surface height distributions can be approximated based on the burden particle size distributions. The peak positions of the estimated particle surface height distribution are consistent with that of the burden surface height distribution.

1. Introduction

In the blast furnace (BF), burden particle size is an essential factor affecting its working state.^1,2) The particle size distribution on the burden surface reflects the granularity in the top burden layers to a certain extent. It can be a basis for evaluating the permeability of the burden.^3,4)

The early particle statistics are mainly obtained through the sieving statistical method,^5,6) which belongs to invasive detection because the particle spatial positions in the sieving process are changed. Different than the sieving statistical method, the optical particle statistical method can keep the original packing morphology of particles (e.g., the granular packed bed). The optical sensor performs non-intrusive detection of the stacked particles to obtain an image containing the shape information of the particles. The particle statistical method based on image processing has been used in ironmaking for pellet particle measurement and pellet particle size statistics.^7,8) However, the particle shape in the captured image is a two-dimensional representation projected from a three-dimensional particle shape under the illumination of a light source, which directly affects the sharpness of the particle edge in the image. Therefore, vital experimental conditions are needed to obtain high-quality images that are suitable for analysis. Compared with the particle statistical method, which is based on image processing, methods based on the digital elevation model (DEM) directly process the three-dimensional particle shape. The obtained particle contour is barely affected by the light source, and the extraction of the particle shape is more stable and accurate. According to the three-dimensional particle shape in the DEM, one can directly define the index to analyze, breaking the limitation of the particle shape definition and analysis in the two-dimensional image. Through the most recent literature review, the research of particle statistical methods based on DEM is rarely reported. Thurley^9,10) used a triangular scanning system composed of the line laser and the optical camera to achieve the 3D contour scanning of pellet particles on the conveyor belt, and proposed edge detection and image segmentation algorithms to extract the particle shape. Very recently, Matsuo¹¹⁾ adopted a medical X-ray computed tomography scanner to obtain the 3D shapes of coke particles and evaluated particle shape by image analysis.

In our former research,¹²⁾ an RGBD camera (Asus Xtion Primesense Carmine) was adopted to detect the rough surface texture of the simulated cold-state tiled burden belts. The DEMs and roughness statistics of the burden surface were obtained, as demonstrated in Fig. 1.

Fig. 1. Burden belt DEM constructed from RGBD measurement. (Online version in color.)

In this paper, an algorithm is further designed to extract the particles’ contours on the burden surface. The proposed algorithm deviates from the image processing framework, which adopts edge detection and image segmentation to calculate the particle shape. It directly extracts particle contours from the DEM, which is simple, fast, and has a clear physical meaning. Particle size distributions of two tiled burden surfaces are counted from the extracted burden particle contours. Moreover, the relationship between the particle size distribution and the height distribution of the rough burden surface is discussed.

2. Definition of Burden Particle Contour

A peak on a rough burden surface is defined as the highest point of its surrounding local area. The surrounding neighborhood of a peak is referred to as a mountain, in which all ascending paths meet and terminate at the peak. Conversely, a pit is defined as the lowest point of its surrounding local area. The surrounding neighborhood of a pit is referred to as a valley, in which all descending paths meet and terminate at the pit. The mountain’s formation depends on the size and shape of the particles on the rough surface, which are formed by spreading the burden particles.

An area of the rough surface containing a mountain is denoted by Ω. The contour of a mountain is a closed curve, denoted as C_h⊂Ω, which is defined as

  

$$C_h=\left\{(x,y)|(x,y)\in \mathrm{\Omega },\mathrm{\Phi }(x,y)=h\right\}$$(1)

where h is the height difference between the plane where the contour C_h is located and the reference plane. Φ(x,y) is a Lipschitz function that satisfies

  

$$\{\begin{array}{cc}\mathrm{\Phi }(x,y)-h<0& \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{\hspace{0.17em} } (x,y)\mathrm{\hspace{0.17em} } \mathrm{o}\mathrm{u}\mathrm{t}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{\hspace{0.17em} } C_h,\\ \mathrm{\Phi }(x,y)-h>0& \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{\hspace{0.17em} } (x,y)\mathrm{\hspace{0.17em} }\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{\hspace{0.17em} } C_h.\end{array}$$(2)

The center position $(\overline{x},\overline{y})$ of the contour is defined as $\overline{x}=M_{10}/M_{00},\overline{y}=M_{01}/M_{00}$, where $M_{ij}={\int }_{C_h}{x}^i{y}^{j}ds$ and ds denotes the differential length of C_h. The longest radius R_max and the shortest radius R_min of the contour are defined as $R_{max}=\underset{(x,y)\in C_h}{max}\Vert (x,y)-(\overline{x},\overline{y})\Vert$, $R_{min}=\underset{(x,y)\in C_h}{min}\Vert (x,y)-(\overline{x},\overline{y})\Vert$, respectively, where ||‧|| is the Euclidean norm. The aspect ratio of the contour is defined as

  

$$M_{AR}=R_{max}/R_{min}.$$(3)

The average radius of the contour is defined as

  

$$R_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}=\frac{1}{l_h}{\int }_{C_h}\Vert (x,y)-(\overline{x},\overline{y})\Vert ds$$(4)

where l_h is the length of C_h that calculated as $l_h={\int }_{C_h}ds$.

3. Contour Extraction of Burden Surface Particles

An extraction algorithm of the particle contours on rough burden surfaces is proposed. The flow of the algorithm is demonstrated in Fig. 2. In extracting the particle contour, the height of the contour surface increases in a range from the lowest point to the highest point of the burden surface. That is to obtain the contours at every height. The four essential steps of the proposed algorithm marked as ①–④ in Fig. 2 are explained below.

Fig. 2. Flow chart of the contour extraction algorithm.

Step ①: The height at any position (x,y) on the rough burden surface is denoted by z(x,y). Let h be a given height of the contour surface, the bunch of corresponding contours in an implicit form of the Lipschitz function Φ is expressed as C_i:Φ_i(x,y)=h,(i=1,…,n), which satisfies z(x,y)=h, $\forall (x,y)\in \underset{i\in \{1,\mathrm{...},n\}}{\cup }{C}_i$, where n is the number of contours.

Step ②: Since $\underset{i\in \{1,\mathrm{...},n\}}{\cup }{C}_i$ may contain contours corresponding to both the mountain and valley types of area, the contours of the mountain area corresponding to the individual burden particles need to be screened out. Given that the mountain area is convex and the valley area is concave, there is an inclusion relationship for the contours at different heights for the same mountain or valley. For the mountain area, the higher the contour surface is, the smaller the contour is, which is the opposite for the valley area. Define a small and positive increment Δh, and extract the points on the corresponding contour curves at the heights of h+Δh or h−Δh as $(x^{+},y^{+})\in \underset{i=\{1,\mathrm{...},n\}}{\cup }{C}_i^{+}$ or $(x^{-},y^{-})\in \underset{i=\{1,\mathrm{...},n\}}{\cup }{C}_i^{-}$, respectively. For each contour to be judged specified by a subscript i, examine its points using the non-zero winding rule, which is a standard criterion in computer graphics.¹³⁾ Take $(x^{+},y^{+})\in C_i^{+}$ as an example. In a polar coordinate, with (x⁺,y⁺) being the origin, calculate the winding number of C_i around (x⁺,y⁺) by Eq. (5).¹⁴⁾

  

$$w_{C_i}(x^{+},y^{+})=\frac{1}{2\pi }{\int }_{C_i}d\theta$$(5)

where θ is the polar angle. If $w_{C_i}(x^{+},y^{+})\ne 0$ holds, according to the non-zero rule, it means that (x⁺,y⁺) locates inside C_i. Since $(x^{+},y^{+})\in C_i^{+}$, it indicates that $C_i^{+}$ is inside C_i. Either $C_i^{+}$ being inside C_i or $C_i^{-}$ being outside C_i manifests that the original contour C_i corresponds to the mountain area. In the opposite situation, C_i corresponds to the valley area.

Step ③: By applying the non-zero winding rule again, the inclusion relationship among the contours at different heights of a mountain can be examined. Contours inside other contours are omitted, and the largest contour corresponding to each mountain is obtained.

Step ④: Due to the influence of the burden particles’ preparation and transportation process, the particle shapes are approximately spherical and blocky.¹¹⁾ Therefore, the aspect ratio of the particle contour, defined in Eq. (3), is adopted as the screening condition by setting a threshold M_AR < m. The screened individual contours embody the size of the corresponding individual burden particles.

4. Results and Discussions

Figure 3 shows the contour extraction results of the burden particles from the sections of the DEM of the tiled burden surface. The azimuth and range resolutions in the DEM sections are 1.7 mm and 2 mm, respectively. This tiled burden surface is formed by two kinds of burden materials: coke and sintered ore.

Fig. 3. Particle contour extraction result from the DEMs of tiled burden surface. (a) coke burden; (b) sintered ore burden. (Online version in color.)

Figures 3(a) and 3(b) correspond to the cases of coke and sintered ore burdens, respectively. The typical particle sizes of each burden are measured and demonstrated on the left of each subfigure. The DEM sections of the burden surface are presented in a vertical view. The contour extraction results of burden particles are demonstrated on the bottom right in the subfigures. The extracted particle contours correspond well with the particle shapes reflected in the DEMs for both burden cases. Additionally, the contours of sintered ore particles in Fig. 3(b) are smoother than that of coke particles in Fig. 3(a), which corresponds well with the actual particle shape. These reveal the validity of the proposed algorithm. Here, we noted that the aspect ratio threshold selection determines the contour’s extraction result. In this research, m is adjusted and selected manually based on the RGB images, which are obtained from the measurement results of the burden surface by the RGBD camera. To get the results presented in Fig. 3, we set m=3.5 for coke burden and m=3 for sintered ore burden.

Based on the contour extraction results, an inference can be made on the relationship between the particle size distribution and the height distribution of the rough burden surface. On the rough burden surface, particles’ size gives rise to different local height fluctuations in their neighborhood. The influence of particle size on local height fluctuations needs to be weighed. Since particle shapes tend to be spherical and blocky after the preparation and transportation process,¹¹⁾ spheres of different radii are adopted to approximate the burden surface particles roughly. A schematic diagram is shown in Fig. 4.

Fig. 4. Particle surface height approximation diagram.

In Fig. 4, the upper hemisphere of the particles contributes majorly to the local height variation. The radius of spherical particles is denoted by R. Δh represents the step of height variation. {S_i|1 ≤ i ≤ k, Δhk ≤ R} is the area of the projected annular regions corresponding to the height variations on the horizontal plane. The difference in area between adjacent regions can be calculated as

  

$$\begin{array}{l}{S}_i-S_{i+1}=\pi \mathrm{\Delta }h[2R-(2i-1)\mathrm{\Delta }h]-\pi \mathrm{\Delta }h[2R-(2i+1)\mathrm{\Delta }h]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=2\pi {(\mathrm{\Delta }h)}^2.\end{array}$$(6)

By Eq. (6), the projected area of a spherical particle on the horizontal plane decreases linearly with the squared height variation. Based on this relation, a statistical result of the surface height of the burden particles can be estimated from the particle size distribution.

Figure 5 shows the statistical results of burden particles. Subfigures (a), (c), and (e) correspond to coke, and subfigures (b), (d), and (f) correspond to sintered ore. Subfigures (a) and (b) are the burden surface height distributions obtained in our former research;¹²⁾ Subfigures (c) and (d) are the distributions of particle contour mean radius defined in Eq. (4); Subfigures (e) and (f) are the estimated distributions of particle surface height converted from the distributions of particle contour mean radius based on the deduced relationship in Eq. (6).

Fig. 5. Burden particle statistics. (a)–(b) measured burden surface height;¹⁰⁾ (c)–(d) mean radius of particle contour; (e)–(f) approximated particle surface height. (Online version in color.)

In Figs. 5(c) and 5(d), the results show that the portion of small contours is greater than the portion of the large ones for both burden materials. However, since the particle sizes on the local height fluctuations were weighted, the bias of the contour size portion is rectified, yielding more central symmetric distributions of the approximated particle surface heights, as indicated in Figs. 5(e) and 5(f). Comparing Figs. 5(e) and 5(f) with 5(a) and 5(b), it shows that the approximated particle surface height distributions reflect the tiled burden surface height distributions to some extent. The peaks’ positions in the distributions of the same burden are consistent. Although using ideal spheres to simulate burden particles is rough, it provides a bridge between the burden surface roughness characteristics and the burden particle size. In-depth relationships are planned to be explored in future research.

5. Conclusion

An algorithm has been proposed by the research team to achieve the contour extraction of burden particles. It is based on the digital elevation model of the burden surface. The extracted contours of coke and sintered ore particles correspond well with their shapes, which are reflected in the measurement results. The burden particle size distributions were counted based on the extracted particle contours. The particle surface height distributions can be approximated based on the burden particle size distributions. It is also found that the peak positions of the estimated particle surface height distribution are consistent with that of the burden surface height distribution. The relationships between the particle size and the burden surface roughness promote fine control of the blast furnace in the future.

Acknowledgment

This research was partially supported by the R&D Program of Beijing Municipal Education Commission (No. KM202310005035) and the Chaoyang District Postdoctoral Research Grant (No. Q1001003202201). We thank Ms. Yue Meng for her assistance in the data processing.

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