Fractal Dimension Analysis and TOPSIS Method for Comprehensive Evaluation of Slagging Tendency of High-Alkali Coal from Xinjiang

Abstract

High-alkali coal can cause slagging and fouling and impact the operational lifespan of the boilers. Traditional single-indicator methods often yield inconsistent results when evaluating the slagging risk of high-alkali coal. In this study, six coal samples were selected and systematically analyzed for their slagging characteristics using scanning electron microscopy (SEM), X-ray fluorescence (XRF), X-ray diffraction (XRD), and ash morphology analysis. Furthermore, a comprehensive evaluation model was constructed by integrating the technique for order preference by similarity to ideal solution (TOPSIS) with the entropy weight method. Additionally, based on images of ash morphology, the fractal dimension (D) was introduced as a quantitative indicator to predict slagging tendency through crack characteristics. The results show that TF, ZD, and KB samples, which are rich in alkaline oxides (CaO, Fe₂O₃, Na₂O, K₂O), form low-melting-point eutectic silicates during combustion, resulting in significant melting and agglomeration with wide cracks between aggregates, indicating a strong slagging tendency. Their fractal dimensions (D) range from 1.81 to 1.92. In contrast, HM and WQ samples, dominated by SiO₂ and Al₂O₃, form high-melting-point mullite and quartz, showing loose ash morphology with uniformly distributed cracks and a weak slagging tendency, with D values of 1.68 and 1.75, respectively. A significant negative correlation was observed between D and the E-TOPSIS model (y = 3.54 − 1.72x). Therefore, fractal analysis allows for rapid assessment of slagging risk without the need for complex chemical testing. This study provides valuable insights for predicting the slagging tendency of high-alkali coal during combustion.

1. Introduction

Coal remains the dominant energy source in China’s energy mix and is expected to maintain a central role in the country’s power generation for the coming decades [1]. Among China’s major coal-producing regions, some areas are increasingly acknowledged as a strategic hub for the development of both fossil fuels and renewable energy [2,3]. Owing to their unique depositional environment and geological evolution, coal from these regions is typically enriched with alkali and alkaline earth metals, particularly sodium and calcium [4]. The high concentrations of these elements substantially increase the risk of slagging during combustion, which negatively impacts heat transfer in boilers, accelerates fouling, and causes unplanned shutdowns, thereby compromising the stable and large-scale utilization of high-alkali coal resources [5,6,7].

Slagging is controlled by complex interactions among ash chemistry, mineral transformations, and combustion dynamics [8]. Coals from various regions exhibit notable compositional differences; for instance, the ash of Zhundong coal contains up to 6% Na₂O and more than 20% CaO (relative to ash mass), whereas the ash of Hami coal contains more than 70% of refractory oxides such as SiO₂ and Al₂O₃ (relative to ash mass). This variability complicates the prediction of slagging behavior [9,10,11]. Notably, the reliability of traditional methods for slagging risk prediction is often questionable, with empirical studies reporting predictive accuracies below 50% in certain cases [12]. For instance, several studies have reported that the ash fusion temperature (AFT) of many Zhundong coals exceeds 1350 °C, which suggests weak slagging and coking tendencies. However, this contradicts practical observations, as these high-AFT coals exhibit severe deposition issues [13]. Zhu et al. [14] reported that seven of the eight indices failed to accurately predict the slagging and fouling behavior of Zhundong coal in China, with only the alkali–acid ratio (R_B/A) yielding accurate results. Yang et al. [15] also demonstrated the limited accuracy of slagging indices for Zhundong coal. Their results indicate that the silica ratio (G) and the composite alkali–acid ratio (R_Z) are the most reliable indicators, whereas the iron/calcium ratio (R_Fe/Ca) and the silicon/alumina ratio (R_S/A) are the least accurate indicators. These limitations underscore the need for a more comprehensive and robust evaluation approach.

Recent studies have explored artificial neural networks (ANN) and machine learning methods for predicting coal slagging behavior, demonstrating improved accuracy over traditional indices [16]. These methods often require extensive datasets and lack interpretability regarding morphological features, which motivates the complementary use of image-based fractal analysis in the present study. Fractal theory has become a powerful tool for quantifying complex structures that exhibit irregularity and self-similarity [17,18,19]. In coal science, it has been successfully used to characterize pore networks and surface textures [20,21,22,23]. The fractal dimension (D) is a scale-invariant parameter of structural complexity that offers more comprehensive morphological insights than conventional chemical analysis. In the context of ash slag formation, the highly porous and uneven surfaces formed through the melting and recrystallization of minerals at high temperatures are well suited for fractal analysis. For microstructural characterization, scanning electron microscopy (SEM) provides high-resolution surface morphology images that capture the intricate fracture networks and agglomerated structures formed during ash melting. For fractal dimension calculation, optical photographs of ash slag are employed, as they clearly capture the macroscopic fracture networks and surface features that are directly indicative of slagging severity. The box-counting method, a widely used fractal analysis technique, is particularly suitable for binary images because it quantifies structural complexity through a simple and robust algorithm, allowing for objective comparison of fracture network characteristics across different coal ash samples.

The technique for order preference by similarity to ideal solution (TOPSIS) is a multi-criteria decision analysis method that ranks alternatives based on their closeness to the ideal solution [24]. Each alternative’s performance is compared with the ideal solution to obtain a relative closeness value, which is then used to determine the optimal choice. Weight determination is a key step in TOPSIS. EW calculates attribute weights based on data diversity, quantifying the uncertainty and information content of indicators without relying on subjective judgment, making it widely applicable [25].

The present study aims to establish a comprehensive evaluation framework for assessing the slagging characteristics of high-alkali coal. The slagging tendency of each coal sample was comprehensively analyzed using scanning electron microscopy (SEM) with energy-dispersive spectroscopy (EDS), X-ray fluorescence (XRF), X-ray diffraction (XRD), and ash morphology analysis. The entropy weight (EW) method was employed to determine the weights of eight individual slagging risk indicators (B/A, S/A, S, F/C, Rs, RFu, DT, and ST), which were then integrated using the technique for order preference by similarity to ideal solution (TOPSIS) to establish a new slagging evaluation model, referred to as E-TOPSIS. Based on ash morphology image analysis, the fracture features of ash slag were extracted, and the corresponding fractal dimension values were calculated to quantitatively assess the slagging tendency. The results indicated that the fractal dimension method showed strong correlation with the E-TOPSIS model, enabling rapid assessment of slagging risk without the need for complex chemical testing. The findings provide a reference and guidance for assessing the slagging tendency of high-alkali coal.

2. Experimental Section

2.1. Sample Preparation

The six coal samples were collected from major mining regions in Xinjiang, China, including Yili (YL), Hami (HM), Urumqi (WQ), Turpan (TF), Zhundong (ZD), and Kubai (KB). These samples were selected to represent the typical compositional range of high-alkali coals encountered in practice, with varying contents of alkaline and acidic oxides, allowing the proposed methodology to be tested across a spectrum of slagging behaviors. Before testing, all samples were naturally air-dried, finely ground to a particle size of less than 0.1 mm, and stored in sealed containers. The proximate analysis, ultimate analysis, and measurement of the higher heating value (Q_gr,d) of the raw coals were performed using a proximate analyzer (5E-MAG6700, Changsha Kaiyuan Instruments Co., Ltd., Changsha, China), an elemental analyzer (Vario EL III, Elementar Analysensysteme GmbH, Hanau, Germany), and an automatic calorimeter (SDC715, Hunan Sundy Science and Technology Co., Ltd., Changsha, China), respectively. The coal properties are summarized in Table 1.

Table 1. Proximate and ultimate analysis, and heating value.

Sample	Proximate Analysis (wt%)			Ultimate Analysis (wt%)					Qgr,d (MJ/kg)
	Mad	Ad	Vdaf	Cdaf	Hdaf	Odaf *	Ndaf	St,daf
YL	17.12	18.18	48.21	69.42	3.00	24.93	1.51	1.14	19.81
HM	6.00	42.26	55.21	66.03	5.92	26.11	1.46	0.48	15.53
WQ	2.94	36.93	46.41	75.76	5.45	16.82	1.43	0.54	19.25
TF	2.52	5.45	34.17	78.12	5.34	14.47	1.46	0.61	30.53
ZD	9.57	6.52	32.41	77.17	4.36	16.92	0.76	0.79	27.03
KB	10.71	5.70	39.31	77.98	4.89	14.81	1.40	0.91	29.74

Note: ad—air-dried basis; daf—dry ash free basis; *—by difference.

Ash samples were prepared in a muffle furnace as follows. A sample of 2 g was first heated at 250 °C for 30 min, followed by heating to 450 °C at a rate of 10 °C/min with a holding time of more than 3 h. After air cooling, the samples were weighed and reheated at 450 °C until the weight difference between consecutive measurements was less than 0.001 g.

High-temperature ash residue samples were prepared in a tubular furnace as follows. Slagging tests were performed at 1100 °C with an airflow rate of 1.0 dm³/min using the raw coal samples. Each sample was placed in an alumina crucible and combusted for 2 h. After combustion, two sets of parallel samples were subjected to different treatments: one set was naturally cooled, after which the ash was ground to a fineness of below 200 mesh and sealed for further analysis; the other set was used for image acquisition under standardized optical conditions. To minimize the matrix effects, a dynamic weighing method based on the predicted ash yield was employed to maintain a consistent slag mass of (0.40 ± 0.01) g across all samples.

2.2. Ash Analyses

The elemental composition and specific composition of the ash samples were determined through XRF (BRUKER S8 TIGER, Bruker AXS GmbH, Karlsruhe, Germany). Each sample was measured three times, and the results are expressed as the mean value. Ash fusion characteristics, including deformation temperature (DT), softening temperature (ST), hemispherical temperature (HT), and flow temperature (FT), were measured using an automatic ash fusion analyzer (TJHR-6000, Hebi Tianjian Electronic Technology Co., Ltd., Hebi, China).

The microstructural features and slagging morphology were analyzed using SEM-EDS (ZEISS Gemini SEM 300 and Sigma 300, Carl Zeiss AG, Jena, Germany), with images captured at a magnification of 10,000× and a resolution of 5 μm. XRD (Bruker D2 Phaser, Bruker AXS GmbH, Karlsruhe, Germany) was used to identify the main crystalline compounds of the samples over a 2θ range of 10° to 80° with a step size of 0.02°.

2.3. Construction of Evaluation Model

In assessing slagging tendency, a range of empirical indices have been advanced, such as the acid ratio (B/A), silicon/alumina ratio (S/A), slag viscosity index (S), iron/calcium ratio (F/C), sulfur index (Rs), fouling index (R_Fu), deformation temperature (DT), and softening temperature (ST). These indices are commonly employed in engineering contexts [26,27,28,29,30]. However, each single index reflects only a limited aspect of slagging tendency, and the predictive results often vary when applied to coals with complex compositions and significant regional differences. Therefore, single-index methods exhibit certain biases and limitations in evaluating slagging risk.

To address these limitations, a comprehensive evaluation framework was developed by integrating multiple slagging-related parameters using the E-TOPSIS approach. The E-TOPSIS approach employs the EW method for weight determination and the TOPSIS method for index synthesis, representing an improvement relative to conventional single-index methods. In the selection of indices, various slagging evaluation indices and their combinations were calculated and tested. After continuous screening, eight typical slagging indices were selected to build the model: B/A, S/A, S, F/C, Rs, R_Fu, DT, ST. The mathematical expression of the slagging index and the rules for determining the slagging degree are presented in Table 2.

Table 2. Slagging, fouling, abrasion, and corrosion tendencies based on ash composition.

Parameter	Formula	Criteria
		Low	Medium	High
B/A	(CaO + MgO + Fe2O3 + Na2O + K2O)/(SiO2 + Al2O3 + TiO2)	<0.206	0.206–0.400	>0.400
S/A	SiO2/Al2O3	<1.87	1.87–2.65	>2.65
S	(SiO2·100)/(SiO2 + CaO + MgO + Fe2O3)	>78.8	66.1–78.8	<66.1
F/C	Fe2O3/CaO	<0.3	0.3–3	>3
Rs	(B/A)·St,d	<0.6	0.6–2.6	>2.6
RFu	(B/A)·(Na2O + K2O)	<0.6	0.6–40	>40
DT	-	>1289 °C	1108–1288 °C	<1107 °C
ST	-	>1390 °C	1260–1390 °C	<1260 °C

2.3.1. Determination of the Index Weights

The EW method was employed to determine the weights of the evaluation indices in the model. It effectively captures data uncertainty and reduces subjective bias [31]. By calculating the information entropy of each index, the method quantifies the variability of indices into objective weights, enabling comprehensive evaluation and decision-making. The specific computational steps are as follows [32].

In step 1, the initial decision matrix is constructed:

$$X=\left[\begin{array}{cccc}{x}_{11}& x_{12}& \dots & x_{1n}\\ x_{21}& x_{22}& \dots & x_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ x_{m1}& x_{m2}& \dots & x_{mn}\end{array}\right]$$

(1)

where X is the initial decision matrix, m is the number of samples, and n is the number of evaluation criteria.

In step 2, the initial decision matrix is normalized. To eliminate the effects of differing dimensions among the evaluation indices, the initial decision matrix was normalized. For cost-type indicators (where lower values are preferred), normalization was performed using Equation (2); for benefit-type indicators (where higher values are preferred), Equation (3) was used. The resulting normalized decision matrix is shown in Equation (4).

$$y_{\mathit{ij}}= {\displaystyle \frac{{{{(x}_{ij})}_{max}^j-x}_{ij}}{{{(x}_{ij})}_{max}^j-{{(x}_{ij})}_{min}^j}}$$

(2)

$$y_{\mathit{ij}}= {\displaystyle \frac{x_{ij}-{{(x}_{ij})}_{min}^j}{{{(x}_{ij})}_{max}^j-{{(x}_{ij})}_{min}^j}}$$

(3)

$$Y=\left[\begin{array}{cccc}{y}_{11}& y_{12}& \dots & y_{1n}\\ y_{21}& y_{22}& \dots & y_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ y_{m1}& y_{m2}& \dots & y_{mn}\end{array}\right]$$

(4)

where x_ij is the value of the j-th criterion for the i-th sample and y_ij is the corresponding normalized value; here, i = 1, 2, 3, … m, j = 1, 2, 3, … n; and Y is the normalized decision matrix.

In step 3, entropy is calculated as follows:

$$e_j=-{\displaystyle \frac{1}{In\left(m\right)}}{\sum }_{i=1}^m{P}_{ij}In(P_{ij})$$

(5)

$$P_{ij}=y_{ij}/{\sum }_{i=1}^m{y}_{ij}$$

(6)

where e_j is the entropy of the j-th criterion, $-{\displaystyle \frac{1}{In\left(m\right)}}$ is the entropy coefficient, and P_ij is the proportion of the value of i-th sample in the j-th criterion.

In step 4, the weight of each criterion is calculated as follows:

$$w_j=(1-e_j)/{\sum }_{i=1}^m(1-e_j)$$

(7)

where w_j is the weight of each factor.

2.3.2. TOPSIS Method

The TOPSIS method is widely used for multicriteria decision-making [33]. The core idea of this method is to identify alternatives that are closest to the positive ideal solution and farthest from the negative ideal solution. In this study, benefit-type indicators (DT, ST, S) are those for which higher values indicate lower slagging risk, while cost-type indicators (B/A, S/A, F/C, R_s, R_Fu) are those for which higher values indicate higher slagging risk, based on standard coal ash chemistry principles (Table 2). The positive ideal solution maximizes benefit criteria and minimizes cost criteria, whereas the negative ideal solution maximizes cost criteria and minimizes benefit criteria. The detailed procedure is as follows.

In step 1, the normalized decision matrix is constructed as shown in Equations (1)–(4).

In step 2, the weights obtained in the previous step are applied to the normalized decision matrix, expressed as follows:

$$Z=\left[\begin{array}{cccc}{z}_{11}& z_{12}& \dots & z_{1n}\\ z_{21}& z_{22}& \dots & z_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ z_{m1}& z_{m2}& \dots & z_{mn}\end{array}\right]$$

(8)

$$z_{ij}=W_j\times y_{ij}$$

(9)

where Z is the weighted decision matrix.

In step 3, the positive and negative ideal solutions are defined. Z⁺ represents the optimal case (positive ideal result) and Z⁻ represents the worst case (negative ideal result). These are calculated as shown in Equations (10) and (11).

$$Z^{+}=\left\{Z_1^{+},Z_2^{+},{\dots Z}_n^{+}\right\}=\{max{Z}_{ij}\left|j=\mathrm{1,2},\dots ,m\}\right.$$

(10)

$$Z^{-}=\left\{Z_1^{-},Z_2^{-},{\dots Z}_n^{-}\right\}=\{min{Z}_{ij}\left|j=\mathrm{1,2},\dots ,m\}\right.$$

(11)

In step 4, the Euclidean distances between each alternative and both the positive and negative ideal solutions are calculated.

$$S_i^{+}=\sqrt{{\sum }_{i=1}^n\left(Z_{ij}-Z_j^{+}\right)}, (i=1, 2, \dots ,n)$$

(12)

$$S_i^{-}=\sqrt{{\sum }_{i=1}^n\left(Z_{ij}-Z_j^{-}\right)}, (i=1, 2, \dots ,n)$$

(13)

In step 5, the relative proximity (C_i) between the evaluation object and the ideal solution is determined, with larger C_i values indicating greater proximity.

$$C_i={\displaystyle \frac{S_i^{-}}{S_i^{-}+S_i^{+}}}$$

(14)

2.4. Calculation Method of Fractal Dimension

During combustion experiments using a crucible, coal ashes with different slagging tendencies exhibited distinct behaviors. For coal with a high slagging tendency, the ash typically appeared in blocky, molten or aggregated forms, with irregular edges and a tendency to adhere and form clumps. In contrast, for coal with a low slagging tendency, the ash was relatively loose, with uniformly distributed particles, more regular shapes, and little tendency to stick together. Based on these observed characteristics, fractal theory was employed to quantitatively characterize the surface morphology of coal ash, revealing the slagging tendencies of different coals.

With the evolution of fractal theory, several well-established fractal calculation methods have been proposed. Among them, the box-counting method, as a more intuitive and user-friendly calculation method, was applied in this study to characterize slagging tendency. The calculation process of the box-counting method used the collected surface images as experimental samples. During the calculation, the images were segmented using boxes, and the number of fracture boxes was counted to obtain the relationship between S and N, as shown in Equation (15) [34].

$$N=C\times S^{-D}$$

(15)

where N is the number of boxes, S is the boxes of size, C is the fitted constant, and D represents the fractal box-counting dimension. To calculate the fractal dimension more intuitively, both sides of Equation (15) were logarithmically transformed, yielding the coefficient relationship equation shown in Equation (16).

$$\mathit{Iog}\left(N\right)=-D\mathit{Iog}\left(S\right)+\mathit{Iog}\left(C\right)$$

(16)

In the present study, high-resolution images of the coal ash slag surfaces after combustion were captured under standardized lighting conditions using a precision imaging system. A series of digital operations were performed in the software ImageJ 1.54d to enhance the surface image quality and convert it into a binary image required for fractal calculation. The image processing workflow for the coal ash surface is roughly depicted in Figure 1.

Figure 1. Image digital processing workflow.

Firstly, the original surface image of the coal ash slag (Figure 1a) was converted to a grayscale image as shown in Figure 1b. Then, the median filtering operation was applied to eliminate interference elements such as scratches and noise dots. The processed grayscale image was then transformed to obtain the final binary image, as shown in Figure 1c. In the binary image, the black regions represented slagging areas with a pixel value of 0, while the white regions represented fractures with a pixel value of 1.

In Figure 1c, the fractal dimension analysis was conducted using the Fractal Box Counter plugin in ImageJ software. According to the image resolution, the subdivision factors were set as [2, 4, 8, 16, 32, 64, 128, 256], meaning that the number of boxes used to divide the image doubled in each iteration. The partial subdivision process of the image is illustrated in Figure 2a. After eight levels of subdivision, evaluation, and counting, the software automatically recorded the corresponding scale factors and the number of fracture boxes (sub-matrices). By linearly fitting the logarithm of the obtained data, the results were obtained as shown in Figure 2b, where the slope of the fitted line, 1.75, represented the fractal dimension of the fractured image. Here, the fractal dimension (D) reflects the complexity and self-similarity of the coal ash surface. The D value closer to 2 indicates a surface with prominent cracks and blocky aggregates, corresponding to coal ash with a high slagging tendency. In contrast, the D value closer to 1 indicates a surface with uniformly distributed particles, corresponding to coal ash with a low slagging tendency.

Figure 2. Fractal dimension calculation process: (a) schematic diagram of the binary image subdivision; (b) linear regression fitting for a representative ash sample.

3. Results and Discussion

3.1. Ash Properties and Slagging

3.1.1. Combustion Experiment Results

Based on the microstructure of ash residues after high-temperature combustion (Figure 3) combined with the ash fusion temperatures (AFT) (Figure 4), the slagging tendency and agglomeration behavior of six coal samples were analyzed. According to AFT, the samples were classified into three slagging risk levels: high risk (DT < 1107 °C, ST < 1260 °C), medium risk (DT: 1108–1288 °C, ST: 1260–1390 °C), and low risk (DT > 1289 °C, ST > 1390 °C).

Figure 3. The degree of agglomeration of the coal samples: (a) YL, (b) HM, (c) WQ, (d) TF, (e) ZD, and (f) KB.

Figure 4. Ash fusion temperatures of the samples.

For WQ and HM, both the deformation temperature (DT) and softening temperature (ST) indicate medium to low risk, which is highly consistent with their loose, powdery ash morphology, uniform particle distribution, and few cracks observed in the ash images, collectively confirming their weak slagging tendency and minor agglomeration. In contrast, TF, ZD, and KB show medium risk based on DT but high risk based on ST. Their ash exhibits partial melting, particle adhesion, and compact structures, indicating more pronounced slagging and agglomeration than predicted by DT alone, suggesting rapid development of the molten phase once softening begins. YL shows low risk according to DT but medium risk based on ST, with its morphology displaying transitional characteristics between loose and compact structures. Notably, the spherical molten clumps with smooth, dense surfaces observed in Figure 3f visually reflect the severe slagging and complete agglomeration corresponding to ST high-risk samples. Overall, the evaluation based on AFT and microstructural observations complement each other: AFT provides a theoretical risk classification, while the morphology reveals the actual extent of melting and agglomeration.

The SEM images of the six coal samples after combustion at 1100 °C are shown in Figure 5. The YL, HM, and WQ samples exhibit relatively fine and flocculent particles with rough surfaces and well-developed porous networks, suggesting the presence of a porous carbonaceous framework formed during combustion. This morphology correlates with a lower tendency for slagging, as the porous structure inhibits dense fusion. In contrast, TF, ZD, and KB samples display more compact and fused agglomerates with smoother surfaces, indicating a higher degree of melting and increased slagging potential. The dense structure of these samples facilitates particle bonding during high-temperature exposure, which can promote the formation of larger slag aggregates. This is consistent with the ash morphology observed in the crucibles.

Figure 5. SEM images of the coal samples after combustion: (a) YL, (b) HM, (c) WQ, (d) TF, (e) ZD, and (f) KB.

EDS analysis showed that the ash samples were primarily composed of Fe, Al, Ca, and Si (Figure 6), with their total content exceeding 85% in all samples (Figure 7). YL was notably enriched in Ca (37.83%) and Fe (17.03%), whereas HM and WQ contained high levels of Si (47.47–54.34%) and Al (24.46–37.11%). TF and ZD exhibited elevated contents of Ca (20.39–39.26%) and Fe (18.86–20.58%), whereas KB showed high contents of Si (37.45%) and Al (28.15%) and a notable Na content (8.55%). These patterns reflect variations in the original mineralogy, including silicates and oxides, and imply different melting behaviors at elevated temperatures. Samples rich in Ca and Fe (YL, TF, and ZD) tended to form low-melting-point eutectics, which corresponds to a higher slagging risk. Conversely, the Si- and Al-rich samples (HM and WQ) tended to form higher-melting-point phases, indicating a lower slagging risk. The high Na content in KB may lower the AFT, thereby facilitating slag formation. These insights enhance the understanding of the coal ash melting characteristics and slagging behavior under high-temperature conditions.

Figure 6. Element mapping of the coal ash samples after combustion: (a) YL, (b) HM, (c) WQ, (d) TF, (e) ZD, and (f) KB.

Figure 7. Elemental content of the coal ash samples determined by EDS: (a) YL, (b) HM, (c) WQ, (d) TF, (e) ZD, and (f) KB.

Figure 8 presents the XRD patterns of the raw coal and the ash samples after combustion at 1100 °C, highlighting considerable differences in mineral compositions. These mineralogical changes are closely linked to slagging risk. YL raw ash is dominated by high-melting minerals such as quartz (SiO₂) and calcite (CaCO₃), which result in a loose ash structure and low slagging tendency. At high temperatures, these phases transform into calcium silicate (Ca₂SiO₄), magnesia (MgO), and other silicate phases; the formation of low-melting eutectics and sintering promotes slagging. HM raw ash is mainly composed of kaolinite (Al₂Si₂O₅(OH)₄), muscovite (KAl₂(AlSi₃O₁₀) (OH)₂), and low dauphinee twinned (SiO₂). Dehydration and layered structures promote the formation of porous ash, reducing the risk of slagging. Upon heating, mullite (Al₄Si₂O₈), quartz (SiO₂), and other refractory phases appear; mullite enhances refractoriness through sintering, while cristobalite may cause localized melting, yet the overall slagging tendency remains low. For WQ, raw ash is rich in quartz (SiO₂), forming a stable skeleton that limits slagging. The high-temperature ash shows recrystallized mullite, elevating the melting temperature. TF raw ash contains quartz (SiO₂), iron-bearing silicates, and other aluminosilicate minerals, showing moderate slagging risk. High-temperature ash features sodium-rich anorthite (Ca,Na) (Al,Si)₄O₈), and titanomagnetite (Fe–Ti oxides), where alkali and Fe–Ti oxides synergistically enhance slagging. ZD raw ash, dominated by quartz (SiO₂), remains relatively stable. Upon heating, high-temperature ash contains akermanite (Ca₂MgSi₂O₇), which acts as a flux to densify ash and lower its melting point, thereby promoting slagging. KB raw ash contains quartz (SiO₂) and muscovite (KAl₂(AlSi₃O₁₀) (OH)₂), indicating a low slagging risk. After combustion, anorthite (CaAl₂Si₂O₈) and related fluxing minerals promote sintering and ash melting through Na–Ca–Mg interactions.

Figure 8. XRD patterns of the samples: (A) raw coal, (B) ash samples after combustion at 1100 °C; (a) YL, (b) HM, (c) WQ, (d) TF, (e) ZD, and (f) KB. [1. calcium silicate (Ca₂SiO₄), 2. magnesia (MgO), 3. quartz (SiO₂), 4. calcite (CaCO₃), 5. mullite (Al₄Si₂O₈), 6. cristobalite (SiO₂), 7. kaolinite (Al₂Si₂O₅(OH)₄), 8. muscovite (KAl₂(AlSi₃O₁₀)(OH)₂), 9. low dauphinee twinned (SiO₂), 10. sodium-rich anorthite ((Ca,Na) (Al,Si)₄O₈), 11. titanomagnetite (Fe–Ti oxides), 12. lithium cuprate (LiCuO), 13. akermanite (Ca₂MgSi₂O₇), 14. anhydrite (CaSO₄), 15. anorthite (CaAl₂Si₂O₈), 16. muscovite 2M1 (KAl₂(AlSi₃O₁₀)(OH)₂)].

3.1.2. Ash Composition Analysis

Based on their acid–base characteristics, oxides can be divided into acidic oxides (SiO₂, SO₃, P₂O₅, and TiO₂), basic oxides (Na₂O, CaO, K₂O, MgO, and Fe₂O₃), and amphoteric oxides (Al₂O₃ and ZnO) [29]. Table 3 presents the oxide contents of ash samples obtained via XRF. The results indicate that the oxide compositions vary among the different ash samples. HM and WQ showed high Al₂O₃ contents (33.14% and 20.71%, respectively), and Al₂O₃ can react with SiO₂ to form high-melting-point aluminosilicates, thus inhibiting slagging. In contrast, YL and ZD showed higher concentrations of basic oxides, with combined CaO and Fe₂O₃ contents of 61.94% and 51.24%, respectively. When the CaO content exceeds 30% (44.29% in YL and 34.46% in ZD), its interaction with Fe₂O₃ facilitates the formation of low-melting-point eutectics, thereby increasing the risk of slagging. The TF sample exhibited an abnormally high SO₃ content (21.28%), and SO₃ tends to form low-melting-point compounds. The KB sample exhibited elevated SiO₂ (55.08%) and Na₂O (6.75%) contents, indicating a tendency to form sodium silicate glass phases. The ZD sample also showed a relatively high chlorine content (4.36%), and chlorine may react with alkali metals and hydrogen to generate volatile aerosols. These aerosols may accumulate on the boiler surfaces, leading to the formation of adhesive slag layers.

Table 3. Oxide composition of ash from different coal samples.

Sample	SiO2	Al2O3	Fe2O3	Na2O	SO3	K2O	CaO	Cl	TiO2	MgO	P2O5	Others
YL	10.06	5.91	17.65	5.87	4.12	0.21	44.29	2.03	0.63	7.93	0.10	1.20
HM	53.33	33.14	4.09	0.95	1.81	0.47	3.17	0.53	1.58	0.52	0.17	0.24
WQ	59.77	20.71	4.35	1.10	3.67	3.12	3.21	0.07	1.48	1.58	0.43	0.52
TF	33.09	14.68	14.34	0.74	21.28	1.25	11.77	0.11	0.95	1.04	0.23	0.52
ZD	16.03	6.97	16.78	5.77	8.13	0.40	34.46	4.36	0.79	4.28	0.22	1.81
KB	55.08	13.94	6.95	6.75	6.27	3.86	3.44	1.31	1.05	0.21	0.05	1.09

Table 3 shows that high-slagging samples (YL, ZD) are enriched in basic oxides, while low-slagging samples (HM, WQ) are dominated by acidic oxides. TF has exceptionally high SO₃, and KB contains significant Na₂O and SiO₂, both of which can promote slagging. One-way ANOVA on total basic oxides (CaO + Fe₂O₃ + Na₂O + K₂O + MgO) between high-slagging (TF, ZD, KB) and low-slagging (HM, WQ) groups showed a significantly higher mean in the high-slagging group (p = 0.031), confirming that alkaline oxide enrichment is statistically linked to increased slagging risk.

3.2. E-TOPSIS for Evaluation of Ash Sample Slagging

Based on a comprehensive analysis of coal properties, key factors affecting high-temperature slagging tendency were identified. Eight coal quality indicators were selected to establish a unified evaluation system, reflecting coal slagging tendency from ash composition, high-temperature phase transformation, and slagging/fouling potential. The B/A, S/A, silica ratio (S), and F/C ratio characterize compositional and adhesive effects, while Rs, RFu, DT, and ST evaluate overall slagging/fouling risk and ash fusion behavior.

Table 4 presents the eight major indicators and the slagging tendencies of the six coal samples, with the risk levels visually represented through color coding. The results reveal substantial discrepancies among different evaluation indices when applied to the same sample. For instance, the YL sample exhibited a high slagging risk based on the B/A ratio and S index, whereas its S/A ratio suggested a low slagging risk. A similar pattern was observed for the KB sample, where the S and R_s indices indicated low risk, whereas the S/A ratio suggested a high risk. In contrast, the evaluations for ZD and TF were relatively consistent. The WQ sample, which contains both a high S/A ratio (tending to lower slagging) and a certain content of basic oxides (increasing fusibility), received a medium risk rating—representing a compromise among conflicting indicators. These inconsistencies highlight the necessity of a multi-criteria approach such as E-TOPSIS.

Table 4. Eight major indicators based on experimental data.

Samples	DT	ST	B/A	S/A	S	F/C	Rs	RFu
YL	1360	1380	4.575	1.701	12.582	0.399	4.255	27.836
HM	1521	1535	0.105	1.609	87.266	1.289	0.029	0.149
WQ	1250	1330	0.163	2.885	86.736	1.356	0.055	0.688
TF	1140	1160	0.598	2.254	54.943	1.218	0.347	1.188
ZD	1140	1150	2.593	2.301	22.405	0.487	1.919	15.986
KB	1120	1140	0.300	3.950	84.135	2.021	0.258	3.176

Note: Color Represent: Low Slagging; Medium Slagging; High Slagging.

To overcome the limitations of individual indicators, these parameters were selected to construct a comprehensive discriminant index, and the TOPSIS method was applied to perform an integrated slagging risk evaluation for the six coal samples (YL, HM, WQ, TF, ZD, and KB). The data presented in Table 3 and the calculation procedure of the entropy method are also introduced. First, the initial evaluation matrix X was constructed as shown in Equation (17). Subsequently, matrix X was normalized: B/A, S/A, F/C, R_s, and R_Fu were processed using Equation (2), whereas DT, ST, and S were normalized employing Equation (3). As a result, the normalized matrix Y (Equation (18)) was obtained. The entropy method was then used to determine the weight of each index. The entropy values were calculated via Equations (5) and (6) (Equation (19)), followed by weight determination as per Equation (20).

$$X=\left[\begin{array}{cccccccc}4.58& 1.70& 0.40& 4.26& 27.84& 1360& 1380& 12.58\\ 0.10& 1.61& 1.29& 0.03& 0.15& 1521& 1535& 87.27\\ 0.16& 2.89& 1.36& 0.06& 0.69& 1250& 1330& 86.74\\ 0.60& 2.25& 1.22& 0.35& 1.19& 1140& 1160& 54.94\\ 2.59& 2.30& 0.49& 1.92& 15.99& 1140& 1150& 22.40\\ 0.30& 3.95& 2.02& 0.26& 3.18& 1120& 1140& 84.13\end{array}\right]\begin{array}{c}\mathrm{Y}\mathrm{L}\\ \mathrm{H}\mathrm{M}\\ \mathrm{W}\mathrm{Q}\\ \mathrm{T}\mathrm{F}\\ \mathrm{Z}\mathrm{D}\\ \mathrm{K}\mathrm{B}\end{array}$$

(17)

$$Y=\left[\begin{array}{cccccccc}0.00& 0.96& 1.00& 0.00& 0.00& 0.60& 0.61& 0.00\\ 1.00& 1.00& 0.45& 1.00& 1.00& 1.00& 1.00& 1.00\\ 0.99& 0.45& 0.41& 0.99& 0.98& 0.32& 0.48& 0.99\\ 0.89& 0.72& 0.49& 0.92& 0.96& 0.05& 0.05& 0.57\\ 0.44& 0.70& 0.95& 0.55& 0.43& 0.05& 0.03& 0.13\\ 0.96& 0.00& 0.00& 0.95& 0.89& 0.00& 0.00& 0.96\end{array}\right]\begin{array}{c}\mathrm{Y}\mathrm{L}\\ \mathrm{H}\mathrm{M}\\ \mathrm{W}\mathrm{Q}\\ \mathrm{T}\mathrm{F}\\ \mathrm{Z}\mathrm{D}\\ \mathrm{K}\mathrm{B}\end{array}$$

(18)

$$e=\left\{e_{B/A},e_{S/A},e_{F/C},e_{Rs},e_{Fu},e_{DT},e_{ST},e_S\right\}=\{0.8865,0.8870,0.8666,0.8942,0.8851,0.6886,0.6898,0.8307\}$$

(19)

$$w=\left\{w_{B/A},w_{S/A},w_{F/C},w_{Rs},w_{Fu},w_{DT},w_{ST},w_S\right\}=\{0.0757, 0.0754, 0.0890, 0.0706, 0.0767, 0.2078, 0.2070, 0.1130\}$$

(20)

The normalized matrix Y was then weighted using Equation (8) to generate the weighted decision matrix Z (Equation (21)), and the positive and negative ideal values (Equation (22)) were determined using Equations (10) and (11).

$$Z=\left[\begin{array}{cccccccc}0.00& 0.07& 0.09& 0.00& 0.00& 0.13& 0.13& 0.00\\ 0.08& 0.08& 0.04& 0.07& 0.08& 0.21& 0.21& 0.11\\ 0.08& 0.04& 0.04& 0.07& 0.08& 0.07& 0.10& 0.11\\ 0.07& 0.06& 0.04& 0.07& 0.07& 0.01& 0.01& 0.07\\ 0.03& 0.05& 0.09& 0.04& 0.03& 0.01& 0.01& 0.02\\ 0.07& 0.00& 0.00& 0.07& 0.07& 0.00& 0.00& 0.11\end{array}\right]\begin{array}{c}\mathrm{Y}\mathrm{L}\\ \mathrm{H}\mathrm{M}\\ \mathrm{W}\mathrm{Q}\\ \mathrm{T}\mathrm{F}\\ \mathrm{Z}\mathrm{D}\\ \mathrm{K}\mathrm{B}\end{array}$$

(21)

$$\left\{\begin{array}{c}{Z}^{+}=\left\{Z_{B/A},Z_{S/A},Z_{F/C},Z_{Rs},Z_{Fu},{wZ}_{DT},Z_{ST},Z_S\right\}\\ =\left\{0.0765, 0.0762, 0.0899, 0.0713, 0.0774, 0.2099, 0.2091, 0.1141\right\}\\ Z^{-}=\left\{Z_{B/A},Z_{S/A},Z_{F/C},Z_{Rs},Z_{Fu},{wZ}_{DT},Z_{ST},Z_S\right\}\\ =\left\{0.0008, 0.0008, 0.0009, 0.0007, 0.0008, 0.0021, 0.0021, 0.0011\right\}\end{array}\right.$$

(22)

The Euclidean distance formulas (Equations (12) and (13)) were applied to calculate the distance between each sample and the positive and negative ideal solutions. The relative proximities of the ash samples to the ideal solution were calculated using Equation (14) and ranked. The relative proximity (C) ranges from 0 to 1, with higher values indicating lower slagging risk. The C can be divided into three grades to characterize the degree of slagging of the evaluation object (C < 0.5 high; 0.5 ≤ C ≤ 0.6 medium; C > 0.6 low).

The EW method takes into account the correlation between the indices and determines the weight of each index based on the degree of data discretization. Then, the TOPSIS method is applied to synthesize multiple indices through the obtained weights, ultimately providing a comprehensive evaluation using the E-TOPSIS method, as shown in Table 5. The slagging tendency of the six coal samples from low to high is HM, YL, WQ, TF, KB, and ZD. It can be seen that TF, KB, and ZD exhibit severe slagging, while HM has the highest C value, indicating minimal slagging risk; WQ and YL are classified as medium risk. The results are generally consistent with the trends of ash composition and fusion behavior presented in Section 3.1. The slight deviation observed for WQ may be attributed to its ash containing both a high S/A ratio, which tends to lower slagging, and a certain content of basic oxides, which increase fusibility, resulting in a medium risk rating rather than a purely low risk. Overall, the E-TOPSIS method effectively reduces the impact of anomalous indicator values, improving the accuracy of slagging risk evaluation.

Table 5. Slagging tendency estimation results of ash samples by E-TOPSIS method.

Subject of Evaluation	S+	S−	C	Rank	Evaluation Results
YL	0.17	0.21	0.55	2	Medium
HM	0.12	0.35	0.74	1	Low
WQ	0.22	0.21	0.50	3	Medium
TF	0.29	0.15	0.35	4	High
ZD	0.29	0.12	0.29	6	High
KB	0.33	0.16	0.33	5	High

S⁺ is the distance between each evaluation index and the positive ideal solution, S⁻ refers to the distance between each evaluation index and the negative ideal solution, and C represents the relative proximity.

3.3. Fractal Dimension of Slag Morphology

Although the aforementioned comprehensive evaluation model generally meets the accuracy requirements for determining the slagging tendency of high-alkali coal, it primarily takes into account the ash chemical composition and fusion characteristics. As discussed in Section 3.1.1, ash residue images can qualitatively reflect slagging tendencies to some extent; for instance, variations in crack density and distribution may be associated with ash composition and fusion behavior. However, conducting a systematic and quantitative comparison among different samples based solely on image observation remains difficult. Therefore, in this study, the box-counting fractal dimension was employed as a quantitative metric for the systematic evaluation of the complexity of coal ash fracture networks and their relationship with slagging behavior.

As shown in Figure 9, the distribution of fractures in the binarized images varies significantly among samples and exhibits distinct fractal dimension values (Figure 10). All fitted curves in Figure 10 have correlation coefficients above 0.99, indicating that the fracture networks on the coal ash surface possess pronounced fractal characteristics.

Figure 9. Coal ash images: (1) Grayscale image; (2) Binary image; (a) YL, (b) HM, (c) WQ, (d) TF, (e) ZD, and (f) KB.

Figure 10. Computational fitting results for fractal results.

The fractal dimension (D) values differ notably among the samples, in the following order: TF (1.92) > KB (1.84) > ZD (1.81) > YL (1.77) > WQ (1.75) > HM (1.68). Samples with higher D values (TF, KB, and ZD) display sparse and simple fracture networks, with particles mostly distributed independently or forming large pores, suggesting that the ash melts easily and agglomerates at high temperatures, corresponding to a high slagging risk. The YL sample, with an intermediate D value, exhibits a hybrid fracture network combining sparse and locally dense regions, indicating a light to moderate slagging risk. In contrast, samples with lower D values (HM and WQ) have highly dense and interconnected fracture networks, with higher ash fusion temperatures and more difficult melting, resulting in a low slagging risk. The results are consistent with the slagging tendency results obtained from experiments, confirming the feasibility of the fractal dimension method.

The evaluation of slagging tendency for the six coal samples indicates a clear overall negative correlation between the fractal dimension (D) and the E-TOPSIS relative proximity (C), as shown in Figure 11. Samples with higher fractal dimensions (TF, KB, and ZD) exhibit sparse fracture networks and a pronounced tendency to melt, corresponding to lower C values and higher slagging risks. In contrast, the sample with the lowest fractal dimension (HM) has a dense fracture network and is more resistant to melting, showing a higher C value and a lower slagging risk. The YL and WQ samples fall at intermediate levels, exhibiting light to moderate slagging tendencies.

Figure 11. Correlation between E-TOPSIS relative proximity (C) and fractal dimension (D).

Although the overall trend is consistent, there are slight differences in the ranking of individual samples between the two methods. The E-TOPSIS assessment indicates that ZD has the highest risk, whereas fractal dimension analysis identifies TF has the sparsest fractures and the most pronounced melting structures; the ranking of WQ also differs between the two methods. These discrepancies mainly arise from differences in methodological focus: the fractal dimension reflects the morphology of the coal ash fracture network after melting, while E-TOPSIS comprehensively considers the coal’s chemical composition, mineralogy, and physical properties. Such a combined strategy is expected to gain broader application beyond the specific samples studied here.

4. Conclusions

This study systematically evaluated the slagging tendencies of six high-alkali coals by integrating advanced characterization techniques with a novel methodology that combines the E-TOPSIS model and fractal dimension analysis. The main conclusions are as follows:

(1)

Coal samples TF, ZD, and KB, which are rich in alkaline oxides such as CaO and Fe₂O₃, tend to form low-melting-point eutectic silicates during combustion, with significant ash agglomeration and high slagging tendency. In contrast, HM and WQ, dominated by acidic oxides (SiO₂ and Al₂O₃), form high-melting-point mullite and quartz, with loose ash and low slagging tendency. The YL sample exhibits balanced acidic and alkaline characteristics, with moderate slagging tendency.

(2)

The established E-TOPSIS model effectively integrated eight common slagging indices, overcoming the limitations and inconsistencies of single-index assessments. The slagging risk was ranked from high to low as ZD, KB, TF, WQ, YL, and HM, with corresponding relative closeness (C) values of 0.29, 0.33, 0.35, 0.50, 0.55, and 0.74. Lower C values correspond to higher slagging tendency.

(3)

Higher fractal dimension values of ash morphology images (ZD = 1.81, KB = 1.84, TF = 1.92) indicate significant melting and agglomeration, with wide fractures and strong slagging tendency. The fractal dimension values are significantly negatively correlated with the E-TOPSIS model results (y = 3.54 − 1.72x). These results demonstrate that fractal analysis can rapidly assess slagging tendency without complex chemical testing, providing valuable reference for predicting the slagging tendency of high-alkali coal during combustion.