Research on the Detection Principle of Coal Ash by X-Ray Transmission Based on FLUKA

Abstract

This study addresses the timely and accurate measurement of coal ash content by proposing a detection model based on nuclear science technology, which is validated using FLUKA 4-4.0 simulation software. The background provided highlights the fact that coal ash content is a critical sales indicator, and its precise measurement is essential for adjusting production parameters in coal preparation plants. In terms of methodology, this study employs the widely used FLUKA4-4.0 software in the field of nuclear physics to simulate X-ray transmission through coal, investigating the impact of changes in coal type on the accuracy of ash measurements. The results indicate that, when the proportions of high-atomic-number elements in coal remain constant, the ash measurement results are accurate and reliable. However, significant fluctuations occur when these proportions change. The conclusion emphasizes the fact that variations in coal type are the primary cause of inaccuracies in ash measurement, particularly when the ratios of high-atomic-number elements are altered. This research provides a new perspective on the online measurement of coal ash content and offers theoretical support for improving measurement accuracy.

1. Introduction

Coal is a crucial energy resource and chemical raw material [1]. In the modern coal processing and utilization industry, ash content is a key indicator of coal quality. Traditional measurement methods, such as combustion weighing and quick ash floatation [2], are labor-intensive and time-consuming, often leading to delayed information feedback and coal ash content exceeding standards [3]. These methods are inefficient and cannot achieve the real-time monitoring of coal quality. The accurate measurement of ash content in coal is essential for optimizing production processes, enhancing product quality, and reducing costs. The real-time monitoring of coal quality allows for timely adjustments to production parameters, ensuring that coal products meet customer requirements and environmental standards. Therefore, developing and applying advanced ash measurement technologies, such as online analyzers and automated detection systems, is vital for enhancing the overall competitiveness and sustainable development of the coal processing industry. These technologies provide rapid and precise ash content measurements, enabling companies to achieve refined management and intelligent production, thus maintaining a competitive edge in the market.

Advanced coal ash content detection methods include radioisotopic methods [4], foam image recognition [5], and others. Researchers such as Qiu Zhaoyu have utilized image analysis to estimate the ash content in clean coal, achieving predicted values which match those from rapid ash measurements [6]. Additionally, Zhang Xiufeng and colleagues have developed a real-time ash content analysis method based on pseudo-dual-energy X-ray transmission, verifying the feasibility of this X-ray ash analyzer through the linear relationship between ash content and five characteristic parameters of X-rays [7]. Patra and others have used particle-induced X-ray emission (PIXE) spectroscopy to characterize coal and coal ash samples, quantitatively expressing the major and trace elements in coal [8,9]. Yao Shunchun and team have developed a rapid coal analyzer based on laser-induced breakdown spectroscopy (LIBS) for a quick quality analysis of coal powder [9]. Lv Wenbao and colleagues have studied the mechanism of measuring ash content with low-energy gamma rays under equal coal layer thickness, correlating the attenuation of radioactive source energy before and after transmission with the ash content [10]. These techniques aim to provide more accurate and efficient ash content measurements, which are crucial for optimizing production processes and ensuring product quality in the coal industry.

Among various radiation-based ash measurement techniques, X-ray transmission stands out for its high stability and fewer operational constraints, making it the most widely used and effective method [11]. However, the accuracy of ash content measurements in coal using X-ray transmission can be affected by variations in coal quality, leading to fluctuations in the measurement results [12]. Currently, much of the research focuses on algorithms for handling data fluctuations, with less emphasis on understanding the underlying principles of ash content variation [13,14]. M.E. Medhat and colleagues utilized the FLUKA 4-4.0 software to estimate the mass attenuation of various biological materials at different energy levels [15]. The Monte Carlo method can be employed for additional calculations regarding the photon attenuation characteristics of different soil samples [15,16]. This article investigates the principle of measuring coal ash content using X-ray transmission through the simulation software FLUKA 4-4.0 (CERN, Geneva, Switzerland), which is based on the Monte Carlo method.

The X-ray transmission method for detecting ash content is an advanced coal quality analysis technique. Its fundamental principle relies on the differences in mass attenuation coefficients of various elements. In coal, elements with higher atomic numbers, such as silicon (Si) and aluminum (Al), have larger mass attenuation coefficients, whereas elements with lower atomic numbers, like carbon (C), have smaller mass attenuation coefficients [17]. The ash remaining after coal combustion primarily consists of inorganic salts with high atomic numbers. Therefore, by measuring the difference in energy attenuation before and after X-ray transmission, a correlation can be established with the coal’s ash content, enabling the non-destructive detection of coal ash.

This paper presents a theoretical derivation for a mathematical model that calculates the attenuation ratio of X-rays penetrating coal layers to determine the ash content. The FLUKA 4-4.0 program was utilized to compute the attenuation ratios of six types of coal with varying ash contents but identical compositions of high-atomic-number (Z) elements, correlating these ratios with the coal’s ash content. The linear correlation of the results validates the accuracy of the proposed ash calculation model. Additionally, the study calculated six coal samples with varying amounts of silicon (Si), the primary contributor to inorganic content, while keeping the proportions of other elements constant. This investigation explores the reasons for the inaccuracy of the online ash measurement model when there is a change in coal type.

2. Materials and Computational Methods

The Monte Carlo method [18] is a computational approach based on random sampling or statistical trials, which approximates solutions through the calculation of a large number of random samples. It is particularly suitable for simulating complex systems and analyzing problems with multiple variables and high uncertainty. In this paper, we employ the FLUKA 4-4.0 software, a powerful tool for Monte Carlo particle transport simulation, commonly used in the fields of nuclear science and technology and nuclear medicine [19,20,21]. This study attempts to apply this software in the field of coal ash content detection.

2.1. Relationship Between Inorganic Content in Coal and Elements in Ash

Assuming that the total mass fraction of elements with higher atomic numbers in the coal sample is C_z, and the total ash content is A_d, let us briefly discuss the relationship between C_z and A_d [22]. The mass fraction of each element in the coal ash is shown in

Table 1. Mass fractions of coal ash constituents.

Element Name	Si	Al	Fe	Ca	Mg	S	Other
Oxide	SiO2	Al2O3	Fe2O3	CaO	MgO	SO3	—
Mass fraction of various oxides in coal ash /%	Q1	Q2	Q3	Q4	Q5	Q6	—
Elemental mass fraction in oxides ηi /%	0.4674	0.5293	0.6994	0.7147	0.6030	0.4005	—
The mass fraction of each element in the coal ash /%	q1	q2	q3	q4	q5	q6	—

.

From the definition of mass fraction as C_z, the total value of C_z can be obtained as follows:

$$C_z={\displaystyle \sum _{i=1}^6{q}_i}$$

(1)

Given that the total ash content A_d is 1, the relationship between the mass fractions of oxides and the elements in the coal ash is as follows:

$$q_i=Q_i\cdot {\eta }_i$$

(2)

The relationship between the ash content A_d and the mass fractions of the elements, C_z, is as follows:

$$A_d=k{C}_z={\displaystyle \frac{{\displaystyle \sum Q_i}}{{\displaystyle \sum q_i}}}{C}_z$$

(3)

In this study, coking coal from a particular mine is employed as a case example for computational purposes. The mass fraction of oxide components in its coal ash is shown in

Table 2. Oxide composition in the coal ash of coking coal from a specific mine.

Element Name	Si	Al	Fe	Ca	Mg	S	Total
Oxide	SiO2	Al2O3	Fe2O3	CaO	MgO	SO3	—
Mass fraction of various oxides in coal ash /%	51.93	30.5	6.24	2.49	0.74	1.36	93.26
Elemental mass fraction in oxides ηi /%	0.4674	0.5293	0.6994	0.7147	0.603	0.4005	—
The mass fraction of each element in the coal ash /%	24.2721	16.1437	4.3643	1.7796	0.4462	0.5447	47.55

:

At this juncture, the relationship between A_d and C_z is characterized as follows:

$$A_d={\displaystyle \frac{93.26}{47.55}}{C}_z=1.96C_z$$

(4)

It can be observed that the relationship equation varies for different types of coal, yet it generally approximates to a value of 2. Consequently, should the coal sample type change, the relationship must be recalculated. For simplicity in calculating the coefficients, a few minor components were not taken into account, although the approach remained consistent.

2.2. The Principle of X-Ray Attenuation

X-rays, like gamma rays, are forms of electromagnetic radiation and are classified as uncharged particle photons. There are three primary mechanisms by which photons deposit energy in a medium: photoelectric absorption, Compton scattering, and pair production. The occurrence of these effects is probabilistic, and we use the cross-section σ to denote the likelihood of these interactions. Information published by the National Institute of Standards and Technology (NIST) indicates the following [23]:

$${\mu }_m={\displaystyle \frac{{\sigma }_{TOT}}{\mu A}}$$

(5)

The atomic mass unit (u), defined as 1.6605402 × 10⁻²⁴ g, represents one twelfth of the mass of a carbon-12 atom, A denotes the relative atomic mass of the target element, and σTOT refers to the total cross-section for photon interactions with atomic matter, typically expressed in barns per atom (1 barn = 10⁻²⁴ cm²). The attenuation coefficient, photon interaction cross-sections, and related quantities are functions of photon energy.

Badawi et al., in their radiometric analysis of samples using gamma-ray spectroscopy, observed that, when X-rays penetrate a material under collimated conditions, the energy of the rays decays exponentially [24]. The attenuation diagram is shown in Figure 1. The attenuation formula is as follows:

$$I(x)=I_0{e}^{-{\mu }_m\rho x}$$

(6)

Utilizing the attenuation formula enables the determination of the mass fraction of elements with higher atomic numbers, C_z, in coal samples. By leveraging the previously established relationship between Ad and C_z, the ash content of the coal can be derived. The total mass attenuation coefficient (μ_m) represents the coal sample’s density (ρ), the distance (x) that the X-ray penetrates, and the initial X-ray intensity (I₀) before it passes through the coal sample. The intensity of the X-ray at a distance x is denoted as I(x).

The formula for the mass decay coefficient can be expressed as follows [25]:

$$\begin{array}{ll}{\mu }_m& =C_z{\mu }_z+(1-C_z){\mu }_c\\ & ={\mu }_c+C_z({\mu }_z-{\mu }_c)\end{array}$$

(7)

The total mass attenuation coefficient for elements with higher atomic numbers is ${\mu }_z$, and, for those with lower atomic numbers, it is ${\mu }_c$. Let us assume that the mass fractions of the elements in the coal sample are as shown in

Table 3. Definitions of mass fraction symbols for each element.

Element name	C	H	N	O
Low-Z element mass fraction	r1	r2	r3	r4
Element name	Si	Al	Fe	Ca	Mg	S
High-Z element mass fraction	p1	p2	p3	p4	p5	p6

.

Here, it should be noted that the mass fraction r_i pertains to the collective mass fraction of all low-Z elements, while the mass fraction p_i pertains to the collective mass fraction of all high-Z elements.

The two mass attenuation coefficients in the aforementioned equation can be expressed as follows:

$${\mu }_c={\displaystyle \sum _{i=1}^4{\mu }_{ci}{r}_i}$$

(8)

$${\mu }_z={\displaystyle \sum _{i=1}^6{\mu }_{zi}{p}_i}$$

(9)

Substituting these values, we obtain the following:

$${\mu }_m={\displaystyle \sum _{i=1}^4{\mu }_{ci}{r}_i}+C_z({\displaystyle \sum _{i=1}^6{\mu }_{zi}{p}_i}-{\displaystyle \sum _{i=1}^4{\mu }_{ci}{r}_i})$$

(10)

Once the total mass attenuation coefficient is determined, taking the logarithm of both sides of the attenuation equation and rearranging yields we obtain the following:

$${\mu }_m={\displaystyle \frac{1}{\rho x}}(\ln I_0-\ln I_1)={\mu }_c+C_z({\mu }_z-{\mu }_c)$$

(11)

By performing algebraic manipulations, we can derive the following relationship:

$$C_z={\displaystyle \frac{\ln I_0-\ln I_1}{\rho x({\mu }_z-{\mu }_c)}}-{\displaystyle \frac{{\mu }_c}{{\mu }_z-{\mu }_c}}$$

(12)

From the previously established relationship between ash content and the mass fraction of elements with higher atomic numbers, as given by Equation (3), the expression for the ash content calculation model can be formulated as follows:

$$A_d=k({\displaystyle \frac{\ln I_0-\ln I_1}{\rho x({\mu }_z-{\mu }_c)}}-{\displaystyle \frac{{\mu }_c}{{\mu }_z-{\mu }_c}})$$

(13)

where $k={\displaystyle \frac{{\displaystyle \sum Q_i}}{{\displaystyle \sum q_i}}}$, ${\mu }_c={\displaystyle \sum _{i=1}^4{\mu }_{ci}{r}_i}$, and ${\mu }_z={\displaystyle \sum _{i=1}^6{\mu }_{zi}{p}_i}$, it can be observed that, if the proportion of coal components remains constant, the value of $\ln I_0-\ln I_1$ will vary according to the different ash contents. The primary data detected by the detector are I₀ and I₁. When the type of coal sample and the mass thickness (ρ_x) remain unchanged, which means the proportion of each high-Z element remains constant, this model of the relationship between ash content and the content of high-Z elements can be effectively applied.

2.3. Construction of Coal Materials

2.3.1. Construction of Coal Materials with Unchanged Elemental Proportions

Utilizing X-ray fluorescence analysis data from a coal mine on the chemical composition of ash from coal with different ash contents after combustion, a coal sample model can be constructed. The mass fractions of elements with higher atomic numbers can be calculated according to Table 1. The elemental content before combustion can be inferred from the oxide content in the ash. Taking sample 1 as an example, the calculation results are shown in

Table 4. Mass fractions of elements in the ash of sample 1 coal.

Oxide	SiO2	Al2O3	Fe2O3	CaO	MgO
Mass fraction of various oxides in coal ash /%	9.92	3.53	0.67	1.06	0.20	15.38
Element name	Si	Al	Fe	Ca	Mg
Elemental mass fraction in oxides	0.4674	0.5293	0.6994	0.7147	0.603	(Fixed value)
The mass fraction of each element in the coal ash /%	4.6352	1.8682	0.4701	0.7574	0.1206	7.85

.

Assuming that the ratios of various elements in the organic matter of coal sample 1 (with an ash content of 15.38%) remain consistent, their relative mass fractions are as shown in

Table 5. Organic element content in coal.

Element Name	C	H	N	O
Low-Z element mass fraction	79.99	4.69	1.49	5.97

.

Following the same methodology, the mass fractions of each element in samples 1 to 6 are shown in

Table 6. Distribution of element mass fractions in coal samples 1 to 6.

	C	H	N	O	Si	Al	Fe	Ca	Mg	Ash
Sample 1	79.9900	4.6927	1.4931	5.9726	4.64	1.87	0.47	0.76	0.12	15.38
Sample 2	77.6364	4.5547	1.4492	5.7969	6.24	2.51	0.63	1.02	0.16	20.69
Sample 3	75.1292	4.4076	1.4024	5.6096	7.94	3.2	0.81	1.3	0.21	26.34
Sample 4	73.5855	4.3170	1.3736	5.4944	8.99	3.62	0.91	1.47	0.24	29.83
Sample 5	71.2633	4.1808	1.3302	5.3210	10.57	4.26	1.07	1.73	0.27	35.07
Sample 6	69.4298	4.0732	1.2960	5.1841	11.82	4.76	1.2	1.93	0.31	39.21

.

2.3.2. Construction of Coal Materials with Changed Elemental Proportions

In actual ash analyzer measurements of coal samples, the proportion of various elements in the coal may vary. To simulate this variation, the Si element among those with higher atomic numbers can be altered to investigate the accuracy of the calculation model when a specific type of mineral is added alone [26]. It should be noted that increasing the content of the Si element does not change the ratio between other high-Z elements; instead, it reduces the proportion of the total content of other high-Z elements. This allows for the adjustment of elemental ratios without changing the total ash content, simulating the changes in the calculation model when the type of coal changes [27].

Increasing the mass fraction of the Si element by 20%, the resulting mass fractions of the remaining high-Z elements become ${\eta }_1=2.2893$%, down from their original mass fractions of ${\eta }_0=3.2163$%. The original content of the Al element was ${\eta }_{Al0}=1.87$%. The method for calculating the mass fraction of the Al element at this juncture is as follows:

$${\eta }_{Al}={\displaystyle \frac{{\eta }_1}{{\eta }_0}}\times {\eta }_{Al0}=1.3298$$

(14)

The calculation method for the other elements is the same. The new distribution of high-Z element mass fractions for sample 1 is shown in

Table 7. Elemental proportions of sample 1.

Element Name	Si	Al	Fe	Ca	Mg	Subtotal
Original high-Z element mass fraction	4.64	1.87	0.47	0.76	0.12	7.8515
New High-Z quality score	5.5622	1.3298	0.3346	0.5391	0.0858	7.8515

.

The calculation method is the same for other samples. First, the adjusted mass fractions of elements for each sample are presented as shown in

Table 8. Adjusted mass fraction distribution of elements in coal samples 1 to 6.

	C	H	N	O	Si	Al	Fe	Ca	Mg	Ash
Sample 1	79.9900	4.6927	1.4931	5.9726	5.5623	1.3298	0.3346	0.5391	0.0858	15.38
Sample 2	77.6364	4.5547	1.4492	5.7969	7.4827	1.7890	0.4501	0.7252	0.1159	20.69
Sample 3	75.1292	4.4076	1.4024	5.6096	9.5260	2.2781	0.5732	0.9235	0.1503	26.34
Sample 4	73.5855	4.3170	1.3736	5.4944	10.7882	2.5794	0.6490	1.0456	0.1674	29.83
Sample 5	71.2633	4.1808	1.3302	5.3210	12.6869	3.0324	0.7630	1.2293	0.1931	35.07
Sample 6	69.4298	4.0732	1.2960	5.1841	14.1805	3.3902	0.8530	1.3743	0.2189	39.21

.

2.4. Computational Model

In the FLUKA 4-4.0 software, we set up a photon narrow beam with a source energy of 461.7 keV and a diameter of 0.1 mm. The radiation source was located at the coordinates (0,0). We simulated the transmission process with a coal sample thickness of 5 cm, in the form of a cylindrical shape with a diameter of 5 cm. The density of the coal was set at 1.5 g/cm³, and the content of each element was configured according to the aforementioned table. The grid was divided into 100 segments. The Monte Carlo simulation was conducted with 10 million particles [28]. The geometric schematic of the computational model is shown in Figure 2.

For the sake of simplifying calculations and since this study focused solely on the attenuation process of X-rays, no complex geometric models were employed.

3. Results and Discussion

3.1. Analysis of Coal Material Results When Elemental Proportions Remain Unchanged

Following the simulation using particle transport software [29], with distance on the x-axis and energy on the y-axis, the total energy at the current cross-section was projected onto the central axis as the energy at that distance. The attenuation results for each sample are shown in Figure 3.

It can be observed that the relationship between energy and distance for samples 1 to 6 all exhibit exponential decay, yet the differences are not significant. Discerning distinctions from the graphical representation is challenging. Given that the attenuation curves of the various samples are closely similar and interwoven, calculating the relationship between the mass attenuation coefficient and the ash content using Equation (6) would be exceedingly difficult.

Therefore, to investigate the relationship with ash content, the attenuation ratio before and after the transmission of X-rays through a 5 cm coal sample was calculated using Formula (13), mentioned earlier.

Using the computational parameters outlined in Section 2.2, the energy decay results calculated in the FlukaFLUKA 4-4.0 software are shown in

Table 9. Energy attenuation results for coal of the same type.

	Ash	I0/MeV	I1/MeV	I0/I1	ln(I0/I1)
Sample 1	15.38	5.02486 × 10−4	2.72601 × 10−5	18.43302116	2.914143684
Sample 2	20.69	5.02230 × 10−4	2.72516 × 10−5	18.42937662	2.913945947
Sample 3	26.34	5.01229 × 10−4	2.72197 × 10−5	18.41420001	2.913122107
Sample 4	29.83	5.00823 × 10−4	2.72196 × 10−5	18.39935194	2.912315443
Sample 5	35.08	4.99829 × 10−4	2.72011 × 10−5	18.37532306	2.911008626
Sample 6	39.21	4.99359 × 10−7	2.72 × 10−8	18.3588048	2.910109285

.

Taking the last column as the x-axis and the measured ash content as the y-axis, a scatter plot was created. A linear regression was then performed, with the following results: the R-value was 0.99068, indicating a very strong reliability of the model. The fitting results of attenuation rate and ash content are shown in Figure 4.

When the composition of the coal remains constant, the relationship between the attenuation coefficient of X-ray transmission through the coal sample and the ash content of the sample is consistent with the results of the model.

3.2. Model Measurement Error When Elemental Proportions of Coal Remain Unchanged

By incorporating the attenuation ratio into the fitted ash calculation formula, the average relative error was found to be 2.09%, and the average absolute error was 0.51%. These values meet the requirements for practical application. The relative and absolute errors of the calculation results are shown in

Table 10. Relative error in ash content calculation for coal of the same type.

	Attenuation Ratio Logarithm	Predicted Value	Actual Ash	Relative Error	Absolute Error
Sample 1	2.915386451	15.11372	15.38	1.73	0.27
Sample 2	2.913940838	21.81687	20.69	5.45	1.13
Sample 3	2.913122107	25.61324	26.34	2.76	0.73
Sample 4	2.912315443	29.35364	29.83	1.6	0.48
Sample 5	2.911008626	35.41321	35.08	0.95	0.33
Sample 6	2.910109285	39.58335	39.21	0.952	0.37

.

3.3. Analysis of Coal Material Results When Elemental Proportions Change

The input card parameters in FLUKA 4-4.0 for this simulation were set to the same conditions as the previous calculations. The simulation results are shown in

Table 11. Energy attenuation results for different types of coal.

	Measured Ash Content	I0/MeV	I1/MeV	I0/I1	ln(I0/I1)
Sample 1	15.38	6.40226 × 10−4	3.47224 × 10−5	18.4384144	2.914436227
Sample 2	20.69	6.39543 × 10−4	3.46705 × 10−5	18.44631603	2.914864678
Sample 3	26.34	6.38267 × 10−4	3.46744 × 10−5	18.4074418	2.912755029
Sample 4	29.83	6.37637 × 10−4	3.46696 × 10−5	18.39181877	2.911905934
Sample 5	35.08	6.36667 × 10−4	3.46675 × 10−5	18.36495277	2.910444107
Sample 6	39.21	6.36302 × 10−4	3.45984 × 10−5	18.39108167	2.911865855

.

Using the last column as the x-axis and the measured ash content as the y-axis, a scatter plot was created. A linear regression was then performed, with the results as follows: the R-value was 0.7494, indicating a poor linear correlation.

When the composition of coal changes, the relationship between the attenuation rate of X-rays passing through the coal sample and the ash content of the coal sample is shown in Figure 5.

When the experimental results from both altered and unaltered coal compositions are compared, it is evident that the scattered data points closely follow the fitted curve if the types of coal constituents remain unchanged. However, a mere change in one element can render the model’s computational accuracy unreliable.

3.4. Model Measurement Error When Elemental Proportions of Coal Change

Upon incorporating the logarithm of the attenuation ratio into the well-fitted ash calculation formula, the average relative error was observed to be 10.72%, and the average absolute error was 2.73%. These results are not satisfactory for practical application needs. The relative and absolute errors of the calculation results are shown in

Table 12. Relative error in ash content calculation for different types of coal.

	Attenuation Ratio Logarithm	Predicted Value	Actual Ash	Relative Error	Absolute Error
Sample 1	2.914436227	19.5198	15.38	26.91679124	4.14
Sample 2	2.914864678	17.53313	20.69	15.25797237	3.16
Sample 3	2.912755029	27.31534	26.34	3.702868375	0.98
Sample 4	2.911905934	31.2525	29.83	4.768673899	1.42
Sample 5	2.910444107	38.03082	35.08	8.411697158	2.95
Sample 6	2.911865855	31.43833	39.21	19.82062038	7.77

.

3.5. Reasons for Errors Caused by Changes in Elemental Proportions in Coal

Comparing these two sets of results, the graphical representation is shown in Figure 6.

When both sets of results and the fitted outcomes are compared, it is evident that the post-modification results exhibit greater dispersion, yet they closely align with the fitted curve. The model’s slope decreases and the intercept diminishes after adjusting the Si content. Analyze using the quality attenuation coefficient and the ash content calculation formula derived earlier. Based on the XCOM photon cross-section database from the NIST official website [23], the following mass attenuation coefficients for various elements at an X-ray energy of 461.7 keV were calculated using coherent scattering data, interpolated in [30]. The mass attenuation coefficients of the main elements in coal ash are shown in

Table 13. Mass attenuation coefficients of inorganic elements in coal. Unit: cm²/g.

Si	Al	Fe	Ca	Mg
0.09079678	0.08763273	0.08792255	0.09062295	0.08970635

.

As deduced from the previously derived mass attenuation coefficients for high-Z elements and the individual elements, the attenuation coefficient for silicon (Si) was the highest at an X-ray energy of 461.7 keV. Consequently, when the content of Si increases, the total mass attenuation coefficient for high-Z elements, denoted as ${\mu }_z$, also increases. Since ${\mu }_z$ is in the denominator of the ash calculation model formula, both the intercept and the slope of the model will increase as a result.

4. Conclusions

To investigate the causes of errors in ash detection when the elemental composition of coal changes, this paper proposes an X-ray transmission model for a constant-thickness coal layer and verifies the accuracy of the ash model using the FLUKA 4-4.0 simulation software, exploring the attenuation principle of X-rays in coal layers, and draws the following conclusions:

• A computational model relating the ash content to the attenuation ratio was proposed, and its reliability was confirmed using simulation software. However, when the composition of coal changes, the method introduced in this paper faces certain limitations.
• This study investigated the impact of changes and non-changes in the elemental composition of coal on the degree of attenuation. It was found that altering the elemental composition affects the overall attenuation coefficient, which can lead to inaccuracies in the ash content model. Specifically, increasing the Si element content results in a decrease in both the intercept and slope of the fitted model.
• Since changes in composition significantly affect the attenuation coefficient, it is not advisable to add other minerals (such as kaolinite, montmorillonite, etc.) for convenience when calibrating ash measurement models in practice. It is preferable to use different ash contents of the same type of coal for experiments.
• Although the X-ray energy used in this experiment was 461.7 keV, attenuated by two orders of magnitude, the attenuation ratio before and after, as seen in Table 9, did not show much difference between samples. After logarithmic operations, the differences between samples become even smaller, reaching three decimal places, but the ash content varies significantly from tens to thirties. This requires an extremely high sensitivity from the sensor. Efforts should be made to explore optimized models or solutions.

The conventional ash fusion and weighing method requires manual operation and has complex steps; hence, its sensitivity is subject to human factors. Other methods, such as dual-energy gamma-ray measurements, require coordination between two detectors and cannot ensure uniform coal layer thickness, making it difficult to accurately measure the ash content. The main reasons for the errors are fluctuations in the proportion of high-atomic-number elements due to changes in coal types and the impact of excess residual magnetite powder in clean coal on the measurement results. We generally adjust a smaller slope to reduce the variability in detection and enhance the stability of the entire system.