2025 Volume 65 Issue 11 Pages 1760-1765
Refined control at the burden particle scale is a critical advancement in modern blast furnaces. This study explores the characterization of burden surface roughness using fractal analysis. An improved triangular prism method algorithm is proposed, which optimizes the calculation of grid center heights and introduces a surface area ratio to effectively mitigate computational errors caused by center height deviations and grid coverage discrepancies in classical methods. These refinements significantly improve the stability of fractal dimension calculations. Using digital elevation modeling data from prior research, this study calculates, for the first time, the fractal dimensions of three typical burden surfaces: large-granularity coke, small-granularity coke, and sintered ore. The results reveal fractal dimensions of 2.1453, 2.0597, and 2.0152 for these surfaces, respectively. Combined with previously reported statistical roughness parameters, these fractal dimensions provide foundational support for subsequent research in blast furnace radar signal interpretation and rough burden surface simulation.
In the precise process control of modern blast furnaces (BFs), the structure of the burden layer is a critical research focus, as it directly affects BF permeability and serves as a decisive factor in regulating furnace conditions. Refined control at the burden particle scale has become a key developmental trend. Advancements in modeling, simulation, and measurement technologies have facilitated in-depth investigations into the impact of burden particles on layer structure and furnace conditions, establishing this as a prominent research focus. From a numerical simulation perspective, Jiao et al.1) developed a three-dimensional multiphase flow model to analyze the impact of burden particle size on BF efficiency. Considering the irregular shapes of actual burden particles, Jiang et al.2) utilized 3D scanners to capture particle morphology and constructed discrete element simulation models to study the effects of particle morphology on bed porosity. From an experimental measurement perspective, Matsuo et al.3) employed X-ray computed tomography (CT) to characterize the 3D morphology of coke particles, while Wu et al.4) analyzed fine particles sampled from a cold-state BF using XRD and SEM-EDS techniques, revealing their properties and impacts on furnace conditions. However, the harsh internal environment of blast furnaces—marked by high temperatures, high pressures, and heavy dust—limit existing studies to numerical simulations or cold-state measurements, making real-time online monitoring highly challenging.
BF radar is one of the few advanced technologies capable of overcoming these extreme conditions, allowing long-term online measurement of stockline level and burden surface shape.5) Nevertheless, burden surface roughness, influenced by the particle, leads to complex microwave scattering during radar wave propagation, degrading measurement accuracy. A thorough understanding of burden surface roughness is essential for quantifying and mitigating its adverse effects on measurement accuracy. In our prior works,6,7) we introduced a RGBD camera-based optical 3D scanning system and conducted preliminary explorations by performing non-contact measurements on tiled burden belts with varying particle types and sizes, establishing corresponding digital elevation models (DEMs). Statistical parameters describing surface roughness (e.g., root mean square height) were extracted for typical surfaces composed of coke and sintered ore particles, revealing correlations between particle size and surface roughness. However, these statistical parameters alone are insufficient to fully characterize the roughness of stochastically distributed burden surfaces.
Traditional roughness characterization methods lack scale-dependent analysis. This necessitates further investigation into the multiscale characteristics of BF burden surfaces. Fractal analysis, a primary technique for studying scale-dependent surface roughness, employs fractal dimension (FD) as a core metric to quantify roughness complexity. As presented in predecessors’ research,8) there are profound connections between the FD and the microwave scattering characteristics of the rough surface. They may serve as prior knowledge in resolving the microwave scattering results or estimating the burden surface roughness characteristics, boosting the performance of burden surface radar detection in the future. As a preliminary, the FDs of rough burden surfaces are important foundation for such approaches. Existing methods for calculating FDs include Box Counting (BC), Differential Box Counting, Covering Method, Triangular Prism Method (TPM), and etc. Nayak et al.9) comprehensively reviewed these methods and their application areas which include medical science, remote sensing, material science, and etc. So far, no studies have analyzed the FDs of the rough BF burden surfaces.
From the perspective of fractal analysis, the burden surfaces exhibit two defining characteristics: (1) They are three-dimensional rough surfaces that can be represented in DEM form; (2) Their roughness originates from granular burden particles. While multiple FD calculation methods may be applicable to burden surfaces, the TPM is prioritized for the following reasons: (1) TPM was initially proposed by Clarke10) for calculating the FD of terrain surfaces using DEM data acquired via remote sensing; (2) It has been successfully applied to study the FD of sand particle surfaces.11) The similarities in both object features (rough 3D surfaces) and application domains (granular systems) prioritized the adoption of TPM for rough burden surfaces. However, the classical TPM suffers from two limitations: inaccuracies in grid data heights and discrepancies in total surface coverage area across scales. To address the first issue, Sun et al.12) enhanced grid data precision through three computational strategies. Ju et al.13) tackled the second limitation by eliminating coverage area discrepancies across varying discrete scales.
This study calculates the FDs of BF burden surfaces using an improved TPM algorithm tailored for processing burden surface DEM data. Building on classical TPM results and leveraging detection data characteristics, the proposed algorithm enhances computational stability by optimizing grid center height calculations and introducing a surface area ratio. For the first time, FDs are calculated for three typical burden surfaces (large-granularity coke, small-granularity coke, and sinter), establishing a foundation for subsequent research.
In fractal analysis, the calculation of FDs follows a standardized framework:
Step 1: Measure the characteristics of the object at different scales.
Step 2: Plot the measured characteristic quantity against the scale using the data from Step 1.
Step 3: Derive the slope of the resulting curve to obtain the FD of the object.
For TPM-based fractal analysis, the above framework is implemented as follows: First, the total surface coverage area (SCA) of the rough burden surface is calculated at various discrete scales using the TPM algorithm. Repeated calculations generate a relationship curve between SCAs and discrete scales. Finally, the FD of the rough surface is determined by calculating the slope of the SCA-scale curve.
2.1. Classical TPM Algorithm10)Let ϵ denote the discrete scale. The burden surface is discretized into square grids with side length ϵ. For each grid unit, let a, b, c, and d represent the surface heights at the four vertices of the grid, as illustrated in Fig. 1 (left: 3D view; right: top view).
In Cartesian coordinates, let h denote the surface height function. For a grid vertex at position (i, j), the heights are correlated as:
(1) |
The height at the grid center e is estimated as:
(2) |
A quadrangular pyramid is constructed using the above five points a to e. The surface coverage area of the grid unit Si,j is approximated by summing the areas of the four triangular faces (A, B, C, D) of the pyramid:
(3) |
The areas of the triangular faces are computed using Heron’s formula:
(4) |
where o, p, q, r, w, x, y, z represent edge lengths, and sa, sb, sc, sd are the semi-perimeters of the triangles:
(5) |
The edge lengths are calculated using the Pythagorean theorem as:
(6) |
The SCA of the rough burden surface, denoted by S, is a function of discrete scale ϵ, can be calculated by summing the surface coverage areas of all grid units as,
(7) |
The SCA at every discrete scale within the analyzing range can be calculated by repeating the above calculations, yielding a SCA-scale relationship curve. Denote by D, the FD of the rough surface, it related to the discrete scale as,
(8) |
The FD, D, is then derived from the slope of the SCA-scale curve in logarithmic plot.
2.2. Improved TPM AlgorithmThe classical TPM suffers from two limitations:
1. Center Height Bias: The grid center height in Eq. (2) is estimated from its surrounding vertex heights rather than directly measured, introducing calculation errors.
2. Grid Coverage Discrepancies: Edge regions narrower than the grid size remain uncovered, causing inconsistencies in SCA across different scales.
In the classical TPM, fixed-size rough surfaces exhibit unequal residual areas in edge regions when discretized at different scales. These residual areas directly impact the total coverage area computed by the TPM. This study utilizes rectangular burden belt measurement data from prior research.6) For such rectangular regions, as the grid size continuously changes during discretization, the residual areas not only fluctuate but also exhibit abrupt jumps, significantly affecting TPM results. Three representative cases are illustrated in Fig. 2. In each subfigure, the dashed contour denotes the rectangular burden belt, while small grids (similar to the top-view grid in Fig. 1) represent discrete units. Subfigure 2(a) demonstrates abrupt jumps in total coverage area as the grid size increases from 77 to 79, driven by changes in widthwise partitioning. Subfigure 2(b) shows similar jumps during grid size increments from 94 to 98, caused by lengthwise partitioning variations. Subfigure 2(c) depicts simultaneous changes in both width and length partitioning as the grid size grows from 108 to 112, leading to coverage area discontinuities.
To address the above issues, we propose two corresponding improvements to the TPM particularly aiming at our measured DEMs, drawing inspiration by the works of Sun’s12) and Ju’s:13)
2.2.1. Optimized Center Height CalculationThe minimum discrete scale is defined as the horizontal/vertical spacing between adjacent data points in the DEM. Figure 3 demonstrates the grid settings of the improved TPM. For scales that are even multiples of the minimum scale (marked as abcd), as shown in Fig. 3(a), center heights are assigned measured values at the corresponding grid vertices. For odd multiples, as shown in Fig. 3(b), center heights are calculated as the average of the four vertices of the minimum-scale grid (marked as a’b’c’d’) containing the center. Here, we noted that using even multiples reduces errors, as all vertices correspond to measured heights.
Classical TPM does not account for uncovered edge regions, causing abrupt jumps in SCA as grid sizes change. To address this issue, the improved algorithm replaces the SCA, S(ϵ), with the SAR, denoted as Sr(ϵ). The SAR is defined as the SCA divides the sum of grid unit area as below,
(9) |
where n is the number of grid units covering the surface. This ratio eliminates scaling artifacts while preserving the FD calculation, transforming the SCA-scale relationship into an SAR-scale relationship.
The improved TPM algorithm was applied to rectangular burden belt DEM data from prior research.6) By utilizing even multiples and incorporating the SAR, the method ensures stable and accurate FD calculations, eliminating the edge effect and reducing the interpolation error.
To demonstrate the relationships and distinctions in the results and enhance the clarity of our analysis, the computational results are combined and visualized in Figs. 4 and 5, with discussions interleaved between the figures.
Figure 4 presents the relationship between the estimated surface area and the discrete scale for the burden belts. Both horizontal and vertical axes are logarithmically scaled. In Fig. 4(a), the curves derived from the classical TPM for three types of burden belts are plotted. Figure 4(b) shows the results obtained using the improved TPM algorithm, and only grid sizes that are even multiples of the minimum discrete scale were selected for computation. As discussed in the previous section, this approach ensures that all grid height data used in coverage area calculations are measured values, thereby minimizing error introduction.
In Fig. 4, the green and red curves correspond to tiled burden belts composed of large- and small-granularity coke particles, respectively, while the blue curve represents sinter particles. Photographs of the actual burden particles and their corresponding DEMs are shown in Fig. 5. The measurement process and details for the burden belts can be found in our prior work.6)
From the classical TPM-derived SCA-scale relationship curves in Fig. 4(a), it is observed that for discrete scales below 10 mm, the curves for all three surfaces tend to flatten, indicating minimal variation in the SCA with increasing scale. This suggests that surface details are fully captured at scales finer than 10 mm. For scales exceeding 50 mm, the curves exhibit segmented linear behavior. As the discrete scale increases continuously, the SCA initially rises but then undergoes abrupt reductions at specific scales (marked by arrows in Fig. 4(a)). These abrupt changes, which surpass the inherent SCA differences between burden belts, obscure the curves and complicate analysis. This phenomenon corresponds to the second limitation of classical TPM discussed earlier: variations in grid coverage area across scales. At larger scales, discrepancies in grid coverage area dominate the SCA calculations, overshadowing variations caused by intrinsic surface roughness. For intermediate scales (10–50 mm), the SCA calculated by TPM gradually decreases as scale increases. In the lower half of this range, the curves exhibit clearer trends: the SCA of large-granularity coke belts exceeds that of small-granularity coke belts, with steeper trends for the former. When comparing small-granularity coke and sinter belts, the SCA of the former is slightly higher at the same scale, though their trend differences are negligible.
In contrast, the SAR-scale curves obtained from the improved TPM algorithm more distinctly reflect differences in surface roughness, as shown in Fig. 4(b). By replacing the SCA with the SAR, the influence of grid coverage discrepancies is eliminated. Even at scales above 50 mm, differences between curves remain meaningful. Within the 10–50 mm range, the improved TPM curves exhibit reduced local fluctuations and smoother trends compared to classical TPM, facilitating subsequent quantitative analysis. These improvements stem from both the elimination of grid coverage discrepancies and the enhanced precision in discrete grid vertex height calculations.
All three curves in Fig. 4(b) exhibit similar overall trends despite shape differences. At small scales (e.g., <10 mm) and large scales (e.g., >100 mm), the SAR stabilizes at higher and lower values, respectively, indicating minimal scale dependency in these regions. At small scales, the calculated SCA closely approximates the true surface area, exceeding the total partitioning area and resulting in SAR values greater than 1. At large scales, the ratio approaches 1, implying that the SCA converges to the total partitioning area as grid cells fail to capture surface roughness. Between these regions, the curves exhibit approximately linear transitions in logarithmic coordinates, suggesting a gradual loss of roughness resolution with increasing scale.
In the fractal analysis workflow, the FD is derived from the slope of the transition interval in these curves. For tiled burden belts, surface roughness originates from burden particles. To determine the transition interval, Fig. 5 illustrates the relationship between the SAR-scale curves (from the improved TPM) and the particle size distributions for three burden belts. Subfigures 5(a)–5(c) correspond to large-granularity coke, small-granularity coke, and sinter ore belts, respectively, with particle photographs and DEMs shown in the insets. Each subfigure employs a linear scale for the horizontal axis (indicating discrete grid size and particle diameter) and dual linear vertical axes for SAR (left) and particle size distribution (right). The curves in Fig. 5 are copied from Fig. 4(b), with particle size distributions sourced from our prior work.7)
For small discrete scales (<10 mm), the particle size distributions of all three belts exhibit a lower cutoff near 10 mm, aligning with the flat trend observed in Fig. 4. Thus, the lower boundary of the transition interval is set at 10 mm. For large scales, the upper cutoff of particle size varies across different burden belts. At these cutoffs, as indicated by the data tips, the SAR stabilizes near 1.01–1.02, approaching unity. Consequently, the upper boundary of the transition interval is defined by the particle size upper limit. The transition intervals are determined as [10, 140] mm for large-granularity coke, [10, 100] mm for small-granularity coke, and [10, 60] mm for sinter.
Linear regression applied to the SAR-scale relationship curves within the above transition intervals yields FDs of 2.1453, 2.0597, and 2.0152 for large-granularity coke, small-granularity coke, and sinter belts, respectively, as determined via Eq. (8). The results suggest that larger coke particles produce higher FDs for their accumulated surfaces. However, surface roughness is influenced by multiple factors, including particle size, shape, and packing density. Experimental limitations prevented the isolation of these factors—for instance, constructing coke and sinter belts with identical size distributions and packing states was not feasible. Nevertheless, the three burden materials used here are representative of real BF operations. Industrially, coke particles are typically larger than sintered ore particles to maintain burden layer permeability, while sinter’s higher density results in denser packing. These factors contribute to the observed lower FDs for sinter surfaces compared to coke, as shown in Fig. 4(b).
Finally, the FDs reported here, combined with statistical roughness parameters from our prior research,6,7) provide a comprehensive parameter set for numerical simulations aimed at modeling realistic BF burden surfaces.14) Such models will support further research on radar signal interpretation and process optimization.
To address the need for multiscale characterization of roughness on BF burden surfaces, this study proposes an improved TPM algorithm for fractal analysis. By optimizing grid center height calculations and introducing a surface area ratio, the proposed algorithm effectively mitigates computational errors caused by grid coverage discrepancies and center height deviations inherent in classical TPM, thereby enhancing the stability and accuracy of FD calculations. Utilizing DEM data of coke and sinter burden surfaces from prior research, FDs were calculated and compared for three typical burden surfaces for the first time. The results reveal FDs of 2.1453, 2.0597, and 2.0152 for large-granularity coke, small-granularity coke, and sintered ore burden surfaces, respectively. These FD parameters, combined with previously reported statistical roughness metrics, form a comprehensive parameter set for quantitatively characterizing BF burden surface roughness. It will provide critical support for advancing research in BF radar signal interpretation, rough surface simulation modeling, and process optimization. The findings establish a methodological foundation for further exploration of dynamic burden surface behavior under operational conditions.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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).