Proposal for an Improvement of Hard Coal Enrichment Accuracy in Systems with Secondary Beneficiation

Abstract

There are various ways to improve the accuracy of hard coal enrichment. This article discusses the possibility of improving the accuracy of beneficiation by using the same enrichment machines located in the processing plant, introducing only the multiple (secondary) enrichment of raw coal feed. Secondary enrichment is a technology that allows for an improvement of the accuracy of mineral separation. The main advantage of this solution compared to that currently used is the lack of the necessity of introducing the crushing process. Secondary enrichment without crushing is a simple solution, and at the same time beneficial from the point of view of obtaining a commercial product with specific quantitative and qualitative parameters. This article shows the effect of this type of enrichment on the improvement of the shape of partition curves in an example of systems with two-product jigs. On the basis of the results of simulation studies, it was shown that the use of secondary enrichment results in obtaining a higher yield of concentrate of a given quality and thus increasing production value. The article uses the method of digital modelling of coal processing operations and technological systems, which should be treated as one of the possible methods of creating new knowledge, in particular knowledge about the behavior of concentrator group systems under different production assumptions and with different raw coal characteristics.

1. Introduction

It is essential to obtain maximum production value from raw coal during the implementation of various commercial contracts; therefore, it is crucial to look for different ways to increase it [1,2,3]. The production volume depends on the method of coal mining [4,5,6,7] and coal enrichment [8,9]. Until now, many models and methods of coal separation have been used, for example gravity separation, centrifugal separation, and electrostatic separation [10,11,12,13,14,15]. Secondary enrichment systems are known in the theory and technology of mineral resource processing, but in the case of coal beneficiation, they are used relatively rarely [16,17,18,19,20].

The quality—and therefore also the production value—of enriched product yields depends on the enrichment of raw coal, the degree of complexity and configuration of the coal-processing system, and the parameters of the separation processing operations.

The comparison of the effects of raw coal enrichment in various technological systems, especially the comparison of the production value of a given quality, allows us to assess the suitability of individual enrichment variants [20]. So far, it has been shown [21] that in various systems of gravity enrichment groups it is possible to obtain a significant increase in the value of concentrate production compared to a single concentrator—this increase occurs in the production of concentrates with good quality parameters. The study [22] showed that the use of jig systems with secondary enrichment is economically viable with a low set ash content in the concentrate, and therefore with a good-quality product.

In the processing of minerals, when the proportion of the useful component is small and there are numerous intrusions and adhesions with the gangue grains, multistage enrichment and secondary enrichment of middlings with prior loosening of the grains by crushing are used. In the flotation processing of metal ores, the enrichment of some products [3], i.e., product recirculation, is used; in the case of coal flotation, it is also possible to obtain optimal beneficiation results by using an appropriate configuration of flotation machines and product recirculation [23,24].

Post-beneficiation indicates essentially the re-enrichment of the middlings, which are the adhesions of the useful component and gangue. Prior to re-enrichment, the middling concentrate is most often crushed to separate the grains of the useful mineral (which is very often in the form of injected grains) and gangue. In the case of hard coal, where the proportions of combustible organic substances and mineral substance in the adhesions can be comparable, apart from secondary enrichment with crushing the simplest form of secondary enrichment without crushing can also be used.

In three-product enrichment, the middling concentrate can be enriched immediately after leaving the concentrator in its natural state, so it is called a natural middling concentrate. It can be re-enriched, and it is also possible to recirculate the natural middling concentrate during enrichment, e.g., in a three-product jig or two two-product jigs.

The use of different versions of enrichment in groups of concentrators—secondary enrichment in different concentrators or enrichment with recirculation—may lead to an improvement in the quality of the carbon concentrates. In coal processing plants, installing an additional jig (or jigs) in the enrichment process line is the easiest way to improve the accuracy of the enrichment. The use of such systems may be justified in the case of changes in enrichment or changes in the required quality of enrichment products, resulting from the implementation of variable commercial contracts.

The proposed method of secondary enrichment allows omitting the crushing process, which is always a source of undersize grains in the final product. Thus, commercial products with a specific grain composition can be obtained. The use of crushing would require rescreening of the concentrate.

This article is limited to an analysis of the effects that can be achieved in the systems of enrichment groups with secondary enrichment of the middling concentrate without recirculation. Studies where the issues of gravitational secondary enrichment are addressed to some extent include [2,19,20,21].

2. Characteristics of the Mathematical Model

For an ideal enrichment process, fractions with densities below the separation density pass entirely to one product and fractions with higher densities entirely to the other. This ideal mineral separation process is illustrated by the ideal concentrate partition curve, which has the shape of a step response (Figure 1). This characteristic can be described by Equation (1). The values of the function describing the partition curve are defined as the partition numbers f(ρ). Therefore, the partition number for the concentrate f(ρ_s) would take only two values, depending on the density ρ of the elementary fractions of the feed and the density of the ideal separation (ρ_s):

$$\begin{array}{l}f(\rho )=1\hspace{1em}\hspace{1em}\hspace{1em}dla\hspace{1em}\rho \le {\rho }_s\\ f(\rho )=0\hspace{1em}\hspace{1em}\hspace{1em}dla\hspace{1em}\rho >{\rho }_s\end{array}$$

(1)

The actual enrichment process is characterized by several partition curves, the shape of which depends on the type of the concentrator and the grain class of the enriched coal. These curves always differ in shape from the ideal curve (Figure 1). Unlike the ideal partition curve, which is a discontinuous function (1), empirical curves can be described by monotonic functions. In the case of the gravity enrichment of hard coal, the partition curves for the concentrate are monotonically decreasing functions.

The partition numbers for individual elementary carbon fractions are determined from the partition curve depending on their density and grain class. For a jig (and any other type of concentrator), several partition curves are defined for the different particle size classes that the feed consists of. In theoretical terms, the partition number statistically determines the probability of the transition of a feed grain with a specific density to a concentrate. In quantitative terms, it means the relative share of a given elemental density fraction in the concentrate. It follows that the difference 1–f(ρ) reflects the probability that some of the grains pass to the waste. The partition curve shows that the separation density parameter is defined as the density of the elemental carbon fraction, which goes 50% to concentrate and 50% to waste. Erroneous grains appear in the enrichment products as a result of the non-ideal course of the enrichment processes. These are grains of waste fractions with a density higher than the separation density, going to the concentrate and constituting its contamination, and coal grains with a density lower than the separation density, ending up in waste and constituting carbon losses. Both effects are very unfavorable. It would be advisable to carry out the coal enrichment process in such a way that its partition curve has a shape as close to the ideal as possible. The accuracy of enrichment is determined by the shape of the partition curves; therefore, the improvement of enrichment accuracy should be equated with the improvement of the shape of the partition curve, which results in a smaller amount of incorrect grains in the concentrate and waste.

In the simulated model of coal processing [21,25,26], generalized partition curves (discussed below) were used for the concentrate; the coordinates of the points are given in tabular form [27]. Since the course of the gravity enrichment processes also depends on the grain size, the mathematical model for each enrichment operation specifies several generalized partition curves (from 4 to 7), identified for the respective grain classes, which increases the accuracy of simulation forecasts. The method of determining the partition numbers for a given grain grade of feed for specific elementary density fractions of the feed is given below.

The selected point of the partition curve determines the relative share in the concentrate (or in the waste) of the density fraction with the density ρ_j of the enriched material. Since the partition curve is a function of the partition density, in order to become independent from this value the so-called generalized partition curve, which is defined for the reduced density, was introduced:

$${\overline{\rho }}_{lj}=\frac{{\rho }_j}{r_l\cdot {\rho }_{50}}$$

(2)

where:

• ${\overline{\rho }}_{lj}$—reduced density of the j density fraction in grain class l, where the partition curve is defined;
• ρ_j—density of fraction j;
• $r_l$—correction factor for grain class l, where the partition curve was determined; in the case of a jig, it ranges from 0.940 (for the dimensionally largest grain class) to 1.302 (for the finest grain class);
• ${\rho }_{50}$—separation density.

The generalized partition curve is not a function of the separation density. Its use significantly simplifies the algorithm of numerical determination of the partition numbers. In the calculations performed, generalized partition curves determined for the concentrate were used; they are presented in Figure 2. They concern various grain classes: curve 1 concerns the finest grains, curve 7 concerns the largest grains (in the calculations in point 4, the jig model practically uses only this curve); curve Av is the average curve (used for the analysis in point 3).

Various indicators were adopted as a measure of the effectiveness of the dense-medium separators. Here, the probable dispersion E_p was assumed:

$$E_p=\frac{{\rho }_{25}-{\rho }_{75}}{2}$$

(3)

where:

• ${\rho }_{20}$—density of the fraction where 25% of grains go to the concentrate;
• ${\rho }_{75}$—density of the fraction where 75% of the grains go to the concentrate.

The higher the E_p value, the greater the enrichment imprecision, and yet probable dispersion is equated with enrichment imprecision. Changes in probable dispersion have a very significant impact on the concentrate yield of a given quality, and thus on the production value. Based on the previous analyzes [21], it can be concluded that:

• the increase in probable dispersion E_p significantly deteriorates the quality of the concentrate at the same separation density (while increasing the losses of combustible parts in waste [3]),
• if the unit price of coal (per 1 Mg) depends on its quality parameters, an increase in the E_p index causes a decrease in the unit price of coal at the same distribution density,
• the optimal quality parameters of the concentrate, at which the maximum production value is obtained, deteriorate with the increase of the E_p index,
• changes in probable dispersion significantly influence the obtained maximum production value; in the case of a jig, these may be changes of up to 20%,
• the simulation model of partition curves enables forecasting the influence of the E_p index on the production value.

All of the generalized partition curves presented related to the concentrate i are defined with dozens of points. The partition numbers corresponding to the specific reduced densities of the density fractions are calculated from the fourth-order Lagrange interpolation polynomial. Linear interpolation is used for the fraction densities close to the extreme values of the generalized partition curves. As a rule, however, the boundaries of the grain classes in which the partition curves have been identified do not coincide with the boundaries of the grain classes of the feed. For this reason, there are always cases where one feed grain class includes several grain classes in which separate partition curves are defined. Generally speaking, the grain size classes of the feed and the grain classes in which the partition curves are defined are usually different. The partition numbers for a given grain class of feed are therefore a weighted average:

$$f_{ij}({\rho }_{50})={\displaystyle \sum _{l=1}^{m_i} f_{lj}}({\overline{\rho }}_{lj} )\cdot u_{li}$$

(4)

where:

• f_ij(ρ₅₀)—partition number of the density fraction j in the grain grade of feed i;
• f_lj(ρ₅₀)—partition number of the density fraction j in grain class l, where the partition curve was determined;
• u_li—share of the grain class l, in which the partition curve was determined, in the grain grade of feed i;
• m_i—number of particle classes where the partition curves were defined, which are contained wholly or partially in the grain grade of the feed i.

The structure of the mathematical model of the two-product enrichment operation is given in Figure 3. The model assumes that the discrete values of the quality parameter in the grain class i and the density fraction j of the yield streams of the enrichment operation are the same as in the input stream. The same assumption applies to the models of all other treatment operations. Therefore, such a simplification ignores, for example, the effects of grain crushing in processing and during transport.

Each process stream in the model in Figure 3 is represented by several arrays of values. These are discrete yield values in the grain class i and density fraction j, and discrete values of the quality parameters ash content, pyrite sulfur, total sulfur, calorific value, and moisture content (depending on which of the abovementioned parameters are available in the quantitative and qualitative characteristics of the feed).

The symbols in Figure 3 are as follows:

γ_{ij F}, γ_{ij C}, γ_{ij W}—discrete yields in the grain class i and density fraction j in the input stream of the feed and in the concentrate and waste streams (0 ≤ γ_ij ≤ 1);

λ_{ij F}, λ_{ij C}, λ_{ij W}—values of the quality parameter in discrete yields of the input stream of the feed and in the streams of concentrate and waste (e.g., ash content aij);

f_ij—partition number of grain class i and density fraction j, determined from the dependence (4) (0 ≤ f_ij ≤ 1).

The total yield of the concentrate (expressed in % of the feed) from the gravitational enrichment operation is calculated using the relationship

$$Y_C={\displaystyle \sum _{i=1}^{Nc}{\displaystyle \sum _{j=1}^{Nf} f_{ij}( {\rho }_{50})\cdot {\gamma }_{ij F}}}\cdot 100={\displaystyle \sum _{i=1}^{Nc}{\displaystyle \sum _{j=1}^{Nf} {\gamma }_{ij C}}}\cdot 100$$

(5)

and the total waste yield (also expressed in % of the feed):

$$Y_W={\displaystyle \sum _{i=1}^{Nc}{\displaystyle \sum _{j=1}^{Nf} [1-f_{ij}({\rho }_{50}) ]\cdot {\gamma }_{ij F}}}\cdot 100={\displaystyle \sum _{i=1}^{Nc}{\displaystyle \sum _{j=1}^{Nf} {\gamma }_{ij W}}}\cdot 100$$

(6)

where:

• Y_C, Y_W—total concentrate and waste yield from the gravity enrichment operation;
• N_c—number of grain classes in feed;
• N_f—number of density fractions in the feed.

The total mass of concentrate and waste is calculated from the dependence:

$$M_C=\frac{Y_C}{100}\cdot M_F$$

(7)

$$M_W=\frac{Y_W}{100}\cdot M_F$$

(8)

where:

• M_F, M_C, M_W—total mass (Mg) of the feed to the concentrator, concentrate, and waste from the gravitational enrichment operation, in the assumed period (e.g., hours, shifts, day’s wages).

The weighted average value of Λ_qp for any quality parameter in each product is calculated using the relationship

$${\Lambda }_{qp}=\frac{{\displaystyle \sum _{i=1}^{Nc}{\displaystyle \sum _{j=1}^{Nf}{\gamma }_{ij} {\lambda }_{ij}}}}{{\displaystyle \sum _{i=1}^{Nc}{\displaystyle \sum _{j=1}^{Nf}{\gamma }_{ij}}}}$$

(9)

where:

• γ_ij—discrete yields in the grain class i and density fraction j in the stream under consideration;
• λ_ij—values of the quality parameter in discrete outlets γ_i of the stream under consideration (e.g., ash content aij).

Various other mathematical models are also used to model the enrichment operation [28,29,30,31,32].

3. Improvement of the shape of the partition curves by the secondary coal enrichment method

One way to improve the shape of the partition curves is to use secondary coal beneficiation, or more specifically secondary beneficiation of the middling (or middlings). Figure 4 shows two configurations of separators with secondary enrichment of middlings. In the arrangement of Figure 4a, the concentrate from the first separator is a middling concentrate which is again enriched in the second enrichment. For a given grain class i and the density fraction j of the feed in the first separator s1, working with the partition density ρ_{50 s1}, the partition number takes the value of f_{ij s1}(ρ_{50 s1}). This number expresses the probability with the which the feed grains with fraction density j go into the concentrate and quantitatively denotes the yield of the density fraction j in the concentrate. The middling concentrate is then enriched in the second separator s2 with a partition density ρ_{50 s2}, and the partition number becomes f_{ij s2}(ρ_{50 s2}).

The probability that a given grain will be placed in the final concentrate, provided that it was previously included in the middling concentrate from the first separator s1, is therefore a conditional probability and is expressed as the product of both determined partition numbers f_ij s1(ρ_{50 s1}) and f_ij s2(ρ_{50 s2}). This probability, i.e., the resultant—substitute partition number f_{ij 2sep} for grains of grain class i and the density fraction j of the feed—can be determined using the relationship

$$\begin{array}{ll}{f}_{ij 2sep}& =f_{ij s1}({\rho }_{50 s1})\cdot f_{ij s2}({\rho }_{ 50 s2})=\\ & \left[{\displaystyle \sum _{l=1}^{m_i} f_{lj s1}}({\rho }_{50 s1})\right]\cdot \left[{\displaystyle \sum _{l=1}^{m_i} f_{lj s2}}({\rho }_{50 s2})\right]\cdot u_{li}\end{array}$$

(10)

where:

• f_ij(ρ_s)—partition number of the density fraction j in the grain grade of feed i;
• f_lj(ρ_s)—partition number of the density fraction j in the grain class l, where the partition curve was determined;
• u_li—share of the grain class l, in which the partition curve was determined, in the grain grade of feed i;
• m_i—partition number of particle classes where the partition curves were defined, which are wholly or partially contained in the grain grade of the feed i.

The relationship (10) in the case of partition curves averaged for the whole grain composition takes a simplified form:

$$f_{j 2sep}=f_{j s1} ({\rho }_{50 s1})\cdot f_{j s2} ({\rho }_{50 s2})$$

(11)

In the arrangement of Figure 4b, the concentrate from the second separator is the second middling concentrate and is again enriched in the third separator s3 with a separation density of ρ_{50 s3}. The partition number takes the value of f_{ij s3}(ρ_{50 s3}). The replacement partition number f_{ij 3sep} for grains of grain class i and the density fraction j of the feed in the case of the final concentrate is the product of three partition numbers:

$$\begin{array}{ll}{f}_{ij 3sep}& =f_{ij s1}({\rho }_{50 s1})\cdot f_{ij s2}({\rho }_{50 s2})\cdot f_{ij s3}({\rho }_{50 s3})=\\ & \left[{\displaystyle \sum _{l=1}^{m_i} f_{lj s1}}({\rho }_{50 s1})\right]\cdot \left[{\displaystyle \sum _{l=1}^{m_i} f_{lj s2}}({\rho }_{50 s2})\right]\cdot \left[{\displaystyle \sum _{l=1}^{m_i} f_{lj s3}}({\rho }_{50 s3})\right]\cdot u_{li}\end{array}$$

(12)

In the case of averaged partition curves for the entire grain composition, the relationship (12) takes the form

$$f_{j 3sep}=f_{j s1}({\rho }_{50 s1})\cdot f_{j s2} ({\rho }_{50 s2})\cdot f_{j s3}({\rho }_{50 s3})$$

(13)

The resulting partition numbers are determined by assuming the same partition density value in two or three separators. For example, if the partition number in each separator for a certain fraction with a density lower than the separation density is 0.900, then in a system with two separators f_ij zastII = 0.900 · 0.900 = 0.810, and in a system with three separators f_ij zastIII = 0.900 × 0.900 × 0.900 = 0.729. The resultant partition numbers in both systems, with the fraction density equal to the partition density, are respectively 0.250 (0.5002) and 0.125 (0.5003). In the case of a feed fraction with a density higher than the partition density, when the partition number in each concentrator is, e.g., 0.100, the resultant partition numbers are respectively 0.010 (0.1002) and 0.001 (0.1003). The resultant partition numbers are therefore each time smaller than the partition numbers in one enhancer, and the greater the effect, the greater the density of the feed fraction.

Using the dependencies (11) and (13), it is possible to draw graphs of the resultant partition numbers for both systems from Figure 4 for different gravity separators. Figure 5 shows the average curve (identified for the entire grain composition) for a single jig and the resultant curves obtained in systems with two and three jigs; each point of the curves describing systems with two and three jigs was obtained at the same partition density values in two or three jigs.

Figure 5 shows two important properties of the resultant partition curves obtained during the secondary enrichment of the middling concentrate:

(1)

Resultant partition numbers decrease faster at higher feed fraction densities; therefore the shape of the resultant partition curves is closer to the ideal curve, and upgrading accuracy is improved. The effect of improving the shape of the partition curves is greatest in the case of two separators. The third enrichment also improves the shape of the curve, but the obtained effect is not that pronounced.

(2)

The resultant partition curves are also shifted towards lower densities, in particular the resultant partition density is clearly reduced. If the ash content is specified in the final concentrate, then in order to obtain it in systems with secondary enrichment, the separation density in each enrichment has to be increased in relation to the separation density in the system with a single enrichment. Increasing the separation density causes an increase in the yield of the final concentrate and an increase in the production value.

In order to illustrate the effect of the differences in the separation density in the two separators on the shape of the resultant partition curve in the system shown in Figure 4a, two jigs with an averaged partition curve for the entire grain composition were assumed, while the separation density in the first jig was constant and amounted to 1.800 g/cm³. In the second jig, the partition density values were assumed to be smaller, equal, and higher, and the resultant partition curves under these conditions are presented in Figure 6.

The shape closest to the ideal curve is shown by the resultant partition curve at equal partition densities (1.80) in both jigs. The greater the difference in separation density in both jigs, the worse the shape of the resultant partition curve.

Figure 7 shows the changes in the enrichment inaccuracy E_p, with the partition density in the first jig equal to 1.800 g/cm³ and different partition densities in the second jig. The approximating function is a parabola, so the inaccuracy of enrichment grows with the square of the difference in separation density in both jigs. The inaccuracy of enrichment is minimal at equal separation densities. For this reason, when searching for the maximum production value, the production maximization algorithm in secondary enrichment systems selects equal separation densities in two or three separators.

4. Optimization of Production Quantity during Secondary Enrichment in Jigs

Since secondary enrichment without crushing is essential in the case of hard-to-enrich coal, the analysis was limited only to low-enrichment feed with the density and quality characteristics given in

Table 1. Density-quality characteristics of the raw feed stream (8–20 mm).

No. of Dens. Fraction	Fraction Density g/cm3	Yield of Fraction %	Ash Content %	Total Sulphur Content %	Calorific Value kJ/kg
1	<1.30	12.15	4.67	0.84	30,680
2	1.30–1.35	17.96	7.40	0.86	29,630
3	1.35–1.40	10.95	10.99	0.97	28,300
4	1.40–1.50	8.47	17.92	1.10	25,750
5	1.50–1.60	7.43	26.61	1.24	22,550
6	1.60–1.70	7.02	35.81	1.25	19,160
7	1.70–1.80	3.95	43.81	1.13	16,220
8	1.80–1.90	4.04	51.03	1.12	13,560
9	1.90–2.00	2.57	57.08	1.39	11,330
10	>2.00	25.45	75.84	2.75	4420
	Sum	100.00	33.67	1.46	19,960

.

Figure 8 presents the considered enrichment in jig systems. The system with one jig is the reference system, the other two are systems with secondary enrichment of the middling concentrate (middlings), as in Figure 4.

The decision variables in the optimization algorithm described in [21,26] are the separation densities in the jigs. An acceptable range of partition density was assumed in the range of 1300 ÷ 2200 g/cm3. The goal function of the algorithm is the yield of the final concentrate, so the maximum value of the function is sought, depending on the number of jigs:

$$\max M_{Ci}({\rho }_{jig1})$$

(14)

$$\max M_{Ci}({\rho }_{jig1}, {\rho }_{jig2})$$

(15)

$$\max M_{Ci}({\rho }_{jig1}, {\rho }_{jig2}, {\rho }_{jig3})$$

(16)

with an equality limitation of the ash content in each final concentrate:

• A_C = A_Ci
• at successive values of A_Ci, set in steps of 1%,
• where:
• ρ_jig—separation density in jigs, g/cm³;
• Y_C—final concentrate yield expressed in % of the feed yield for enrichment.

Figure 9 shows changes that were determined in the optimal values of separation density in jigs. As expected, in line with the considerations in point 3, the separation densities during re-enrichment of the middling concentrate (middlings) in the two-jig and three-jig systems have the same value in both or all three jigs each time (for each preset ash content in the final concentrate). The corresponding separation density curves ρ_jig1 and ρ_jig2 for the two-jig and ρ_jig1, ρ_jig2, and ρ_jig3 for the three-jig system overlap in Figure 9.

In the systems with re-enrichment of the middlings, the resultant partition curve for the same separation densities shifts to the left, toward lower densities (Figure 5), and the inaccuracy E_p reaches its minimum value (Figure 7). Thanks to this, to obtain the desired ash content in the final concentrate, the optimally determined partition densities in each jig are higher than in the system with one jig (Figure 9), and thus a greater yield of the final concentrate is obtained. The maximum values of the final concentrate yields with different ash contents are shown in Figure 10. This figure shows that in the secondary enrichment systems with a good-quality final concentrate, it is possible to obtain a much higher yield.

The effect of improving the shape of the partition curves (point 3) is most noticeable in the case of using the second separator; the third separator does not bring another large improvement. Clearly, the highest yield occurs in systems with three jigs, and generally in systems with three separators. However, a greater increase in production value is obtained in the second stage of enrichment (Economic analyses of coal enrichment in jigs with secondary enrichment of the middlings, consisting of comparing the increases in production value and increases in operating costs, allow us to conclude that their use is economically viable [22]).

As mentioned above, the optimal separation densities in the two-jig and three-jig secondary enrichment systems of the middling concentrate(s) (middlings) have the same value in both or in all three jigs (for each preset ash content in the concentrate). Therefore, the control of the jigs’ operation (and more generally the concentrators of the same type) should be identical, thanks to which the same enrichment conditions can be obtained (the same separation density). Even with changes in the enrichment characteristics of the feed, online control of the ash content in the concentrate is sufficient; its fluctuations can be corrected by appropriate changes to the separation density in the negative feedback loop, as shown in Figure 11.

Enrichment processes have a decisive impact on the real possibilities of selling coal assortments, therefore obtaining the maximum amount of good-quality coal concentrates in the implementation of various commercial contracts is essential. Consequently, it is important to look for different ways to increase the value of production. Secondary processing systems are known in the theory and technology of raw materials processing, however, they are used relatively rarely in coal enrichment. In systems with secondary enrichment of middlings, it is possible to obtain a partition curve shape much closer to the ideal partition curve, which of course significantly improves the accuracy of the enrichment. The effect of improving the shape of the partition curves is greatest in the case of two concentrators. The third concentrator obviously also improves the shape of the curve, but the obtained effect is not that significant (Figure 5). The shape most similar to the ideal curve is assumed by the resultant partition curve at equal partition densities in two (or three) concentrators, respectively. Then the inaccuracy of the enrichment E_p takes the minimum value. The greater the difference in partition density in both jigs, the worse the shape of the resulting partition curve (Figure 6), and E_p increases parabolically (Figure 7), so the upgrading inaccuracy grows with the square of the partition density difference in both jigs. For this reason, when searching for maximum production value, the optimization algorithm selects equal partition densities at a given ash content. The given results of forecasts with maximally wide ranges of the set ash content in the products allow for a comprehensive assessment of the operation of these systems. They are a source of knowledge about the technological system (or its components) for creating more advanced tools for computer-supported decision-making and knowledge engineering. Therefore, the results obtained can support planning and production management processes in the coal processing plant.

5. Conclusions

Gravity separators exhibit some inaccuracy in enrichment. This results from the fact that the partition curves have a shape that deviates from the ideal curve, which is due to the fact that in the process of gravity enrichment, erroneous heavy fraction grains appear in the concentrate, contaminating it, and light fraction grains appear in the waste, which constitute carbon losses.

An innovative approach in the presented research is the new structure of the enrichment systems, which has not been used so far. The use of secondary enrichment allowed us to:

• Obtain an effect equivalent to the improvement of the shape of the partition curves of a single enrichment. Thus, the enrichment system can be treated as a single concentrator, but with better separation efficiency, defined by higher enrichment accuracy (lower probable scatter E_p). This allows us to select a higher separation density for a given ash content in the concentrate, which is associated with a greater yield of the final concentrate (and higher production value).
• The effect of improving the shape of the partition curves is the greater, the higher is the E_p index of a single concentrator. Thus, it is the smallest in dense medium separator systems.
• In the presented forecasts concerning optimal enrichment in jigs, only the best shape separation curve was used (No. 7 in Figure 2), identified for the largest-dimension grains enriched in the jig (the feed in question had grains of 8–20 mm). In the case of finer feed grains, when the partition curves deteriorate in shape, a much greater improvement in the effects of secondary enrichment in the jigs should be expected.
• The resultant partition curves (when enriching the middlings again) were shifted toward lower densities; in particular the resultant partition density clearly decreased (Figure 5). If the ash content is specified in the final concentrate, then in order to obtain it in systems with secondary enrichment, the separation density in each concentrator has to be increased in relation to the separation density in a system with a single enrichment. Increasing the separation density causes an increase in the yield of the final concentrate (and, consequently, an increase in the production value).
• The improvement in the shape of the partition curves resulted in the fact that with the same set ash content in the final concentrate from a group of two or three jigs, it was possible to obtain a greater output of this concentrate than from a single jig—especially in the case of a low set ash content. The amount of feed for the second and third jigs (i.e., middlings) is then significantly smaller, so they can be machines with lower efficiency. It is also possible, if the technological conditions allow using small heavy-liquid cyclones to enrich the middlings, that the increase in the production value will be even greater [21].
• In systems with secondary enrichment, it is possible to obtain a lower minimum ash content in the final concentrate than in the case of a single jig (Figure 10).
• In the case of secondary enrichment of the middlings, it seems particularly easy to optimize the current production, according to the concept in Figure 9. Online measurement of the ash content in the final concentrate is sufficient, and the separation densities in both (or in all three) concentrators should be set at the same value as in Figure 9, because then the resultant inaccuracy of enrichment of the concentrator group is minimal. It is not even necessary to know the enrichment characteristics of raw coal.

Currently, the development of new concepts and inventions depends to a large extent on information processing and knowledge-management technologies. Knowledge is the basis for the development of future enterprises. One of the characteristics of knowledge is that it can be created in a variety of ways. This article uses the method of digital modeling of coal processing operations and technological systems, which should be treated as one of the possible methods of creating new knowledge, in particular knowledge about the behavior of concentrator group systems under different production assumptions (and with different characteristics of raw coal).