IMPROVING THE EFFICIENCY OF THE COAL GRINDING PROCESS IN BALL DRUM MILLS AT THERMAL POWER PLANTS

Copyright © 2022, Authors. This is an open access article under the Creative Commons CC BY license 11. Vainer, L. G., Flusov, N. I. (2013). Geometrical modification of the face-grinding wheels in the dressing. Vestnik Tihookeanskogo gosudarstvennogo universiteta, 17. Available at: http://science-bsea.bgita.ru/2013/mashin_2013_17/vainer_geometr.htm 12. Kalchenko, V., Kalchenko, V., Sira, N., Yeroshenko, A., Kalchenko, D. (2020). Three-Dimensional Simulation of Machined, Tool Surfaces and Shaping Process with Two-Side Grinding of Cylindrical Parts Ends. Advanced Manufacturing Processes, 118–127. doi: https://doi.org/10.1007/978-3-030-40724-7_12 13. Kalchenko V., Kalchenko V., Slednikova O., Kalchenko D. (2016). Modular 3D modeling of ends bilateral grinding process by wheels with conical calibrating sections. Scientific Journal of the Ternopil National Technical University, 4 (84), 82–92. Available at: https://visnyk.tntu.edu.ua/pdf/84/337.pdf 14. Kalchenko, V., Kalchenko, V., Kalchenko, O., Sira, N., Kalchenko, D., Morochko, V., Vynnyk, V. (2020). Development of a model of tool surface dressing when grinding with crossed wheel and cylindrical part axes. Eastern-European Journal of Enterprise Technologies, 3 (1 (105)), 23–29. doi: https://doi.org/10.15587/1729-4061.2020.202441 15. Li, H. N., Axinte, D. (2016). Textured grinding wheels: A review. International Journal of Machine Tools and Manufacture, 109, 8–35. doi: https://doi.org/10.1016/j.ijmachtools.2016.07.001


Introduction
Worldwide, TPPs are facilities that are used for the simultaneous production of thermal and electric energy. Different types of primary energy carriers such as natural gas, coals, biomass are used to run facilities. Some of them are di-rectly used but the others must be initially processed before combustion. The fuel preparation process is energy-intensive, which results in an increased final energy price. Following this, to reduce final costs for energy production, mechanisms are being sought to improve the energy efficiency of each unit of the plant.
The possibilities of using solid fuels, in particular coals, in thermal power plants are far from being exhausted at present. In addition, due to the significant increase in the price of natural gas, the possibility of fuel base change in TPPs seems difficult to implement.
Coal is considered the second most important primary energy source in the world after fuel oil, as it provides about a quarter of the world's energy needs. Many countries of the world have coal reserves, and the average supply of coal is about 230 years. The Republic of Kazakhstan, which has huge reserves of this type of fuel, according to some forecasts, will be provided with it for more than 300 years [1].
At present, there are so many TPPs worldwide working on coals and using inefficient technology for grinding fuels. Almost all of the countries in the Former Soviet Union use similar fuel grinding technology. Due to low energy prices over the last few decades, no attention has been paid to the share of energy used in coal preparation. Nowadays, this fuel preparation system mostly affects the final price of energy produced.
Much of the research examines single parameters affecting the energy efficiency of the coal grinding system, while there is still no information for a comprehensive assessment.

Literature review and problem statement
Depending on the type of coals and the process of their combustion, the specific energy consumption for coal grinding at various TPPs range from 14.45 to 55.57 kWh/t. High energy needs refer to fuels with high resistance to grinding, low volatiles yield (less than 15 %), and the presence of liquid ash removal [2]. The study reveals that in recent decades, the share of volatiles in coals, as well as liquid ash, grows leading to increased energy consumption for grinding. A change in the composition of the raw material is the reason for the increased consumption of energy for grinding, which will affect all thermal power plants running on ball drum mill grinding systems. In [2], it is stated that this share of energy is classified as very significant compared to the modern ones. Since investment costs for the replacement are very significant, new methodologies and approaches for improving the efficiency of single items in the systems have been sought in this study. Different approaches for the analysis of the operation of fuel preparation systems at TPPs are presented in [3]. A thermodynamic analysis was performed to determine the efficiency of the fuel preparation system. The results are not supported by experimental results, and they have limited practical application.
The efficiency and reliability of the equipment for pulverizing power plants mostly depend on the physical and mechanical properties of the processed fuel and its further combustion.
Coal dust prepared in drum mills must have a polydisperse structure that meets the needs of the specific thermal power plant and must comply with the following important indicators such as fineness (R 90 ), average dust particle size (x), specific dust surface (S), and others.
In [4], the impact of the drum length to diameter ratio and also particle filling rate in the process of segregation of particles in a rotating drum is presented. The outputs from the study showed that short and long drums can be successfully used for the separation of the raw material (coals) both in radial and axial directions. This leads to better final grinding of the raw material but still, single parameters have been observed. The disadvantage of [4] is that no mathe-matical model is presented to determine different waterfall modes of the material in the drum in terms of the diameter of the balls. In addition, the relationship between the size of the balls and the size of the raw material is not specified.
Mathematical modeling is successfully used to optimize the performance of the ball drum mill when grinding brown coals [5,6]. The segregation index provided in [5] is a result of the experimental study for three different diameters of the drum mill as well as three particle sizes. The amount of volatiles and ash content have a significant impact on the distribution in a ball drum mill, respectively for fuel grinding, and this was not discussed in this paper. In [6], both grinding and transportation processes are modeled with respective equations. A close look at the equations reveals that energy consumption for grinding is contrariwise proportional to air consumption. The error between experimental and numerical data amounted to ±10 %. The paper presents the impact of two main parameters on mill productivity. However, there are other parameters that must be considered in terms of the entire unit efficiency. No other parameters influencing the grinding process are also considered.
The degree of grinding in the drum can be improved by knowing the solid particles' pathlines in the drum or their collision with the wall. The performed CFD study with the adopted Eulerian approach mechanism showed that at a high rotational speed of the BDM, the impact of the specularity (specular reflection) and restriction coefficient are the most significant [7]. The results show that the specularity and restitution coefficient mostly affect the wall of the ball drum mill. Still, only single operational parameters have been analyzed in terms of the grinding process.
The methodology of determining the power consumed by the ball drum mill electric motor in terms of ball load and mill grinding capacity is presented in [8]. One of the conclusions is that the degree of filling the drum with balls significantly affects the grinding capacity of the mill. The size of the balls and rotational speed were not observed and discussed as the significant parameters of the grinding process.
The mechanism of the total energy can be successfully used in terms of the evaluation of crushing energy [9]. The different energy distribution in a BDM was due to different ball sizes. The Monte-Carlo method was used to predict the total regional energy distribution in a drum mill. The study is interesting to predict distribution in a ball drum mill but still not so useful to predict the amount of energy (specific energy consumption) for fuel grinding.
In recent studies, computer modeling has been used to visualize the behavior of the balls in a ball drum mill. A 3D modeling was proposed as a mechanism for an efficient and convenient way of feeding the mills with balls. In [10], it was found that the broken energy is in relation with the concentration of the balls in a ball drum mill. As a disadvantage of the paper, it can be pointed out that the distribution of balls by size in the drum is not considered. The energy exchange device proposed in [11] uses the pre-grinding of coals. Still, the energy consumption process is considered only in terms of the balls in a ball drum mill. There is also no information about the details of the performed experimental study.
The visual representation can be successfully used to improve the design of the grinding facility. Moreover, the theoretical interpretation of factors affecting the quality and productivity of the BDM is presented in [12]. The drawback of the study is that the results have not been validated and the reliability of the obtained dependencies is doubtful.
Based on the performed literature review, it is quite clear that scientific studies in this area are limited where a comprehensive approach to the parameters affecting the fuel grinding must be sought.

The aim and objectives of the study
The aim of the study is to determine the significant parameters improving the energy efficiency of the coal grinding system that uses an individual closed circuit for preparing coal dust. This will allow managing the grinding process in the most effective way in terms of the TPP needs. Also, this will enable to determine the optimal performance of the ball drum mill (improved energy efficiency) at different characteristics of the inlet material.
To achieve the aim, the following objectives are accomplished: -to define all possible parameters that could affect the efficiency of the ball drum mill during fuel grinding; -to analyze the significant parameters affecting the operation of the ball drum mill, and perform a multiscale experiment; -to obtain regression equations for the significant parameters, and define the acceptable region of operation of the drum mill.

Materials and methods of research
The fuel grinding system subject of the study is part of the TPP-2, Temirtau. Currently, the TPP-2 is experiencing the following main problems: -physical aging of equipment; -decrease in the share of electricity generation based on thermal energy consumption; -decrease in the efficiency of combined generation of heat and electricity, and deterioration of technical and economic indicators; -reduced level of automation of technological process control; -relatively high negative environmental impact; -increased operational and maintenance costs. One of the solutions to existing problems is to improve the efficiency of boiler units and auxiliary equipment of power plants, which include coal grinding mills [13].
The schematic diagram of the coal dust preparation system at the TPP-2 site is shown in Fig. 1. This system differs from the traditionally accepted individual scheme of dust preparation with an intermediate hopper for the ballast machine. The main distinguishing features of the solid fuel preparation system at the TPP-2 include: -defrosting of fuel is carried out only in exceptional cases via hot air produced in a steam boiler at very low air temperatures (during winter season); -in other cases, the coals are dried with hot air from the air heater of the steam boilers right before the fuel is supplied to the BDM, and thus the balance of the supplied primary and secondary air is disturbed; -currently, preliminary stages of coal crushing are not provided, as a result, coal with large lumps (more than 25 mm) is fed into the mill.
The specified features of the TPP-2 mostly have a negative impact on the energy performance of the plant, for example, the gross efficiency of power boilers is about 86 up to 87 %, the specific power consumption for grinding is overestimated at 39.86 Wh per kg.
The identified drawbacks of the pulverization system show that the system does not allow the preparation of high-quality coal dust, since the ventilation mode has a significant impact on the operation of a ball drum mill. At low air velocities, dust removal is troubled, because of lower lift ability. Increasing the air velocity leads to a removal of large particles, electricity consumption for grinding is significantly increased, due to the lack of automation, the intermediate hopper is often filled, leading to the shutdown of the mills. Taking into account the noted shortcomings, the study focuses on the preparation of a mathematical model and proposes the methodology for the selection of important parameters affecting the operation of the BDM. The proposed dust system is installed at each of the six boiler plants (TP-81) at the site of the TPP. For a single steam boiler (TP-81), two individual closed dust systems equipped with ball drum mills are mounted. The concerned dust preparation system at the TPP-2 contains the following items: an Sh-25А ball-drum mill (Sh-320/570), one mill fan of the ВМ 18А type with a capacity of 108 thousand m 3 /h and a screw conveyor with a capacity of 60 t/h [11].
Coals from the bunkers are transported to the raw coal feeder (RCF), which controls the degree of loading of the mill. From the RCF, coal enters the throat of the Sh-25A mill, where hot, cold, and weakly heated air is supplied from an air heater, intended for drying coals [14] and transporting them throughout the dust system, dust hopper, and burners.
The process of drying coals in the mills is regulated by the weakly heated and cold air supplied through inlets of each dust system. The air recirculation system additionally facilitates the drying of coals. In the mill, the coals are grounded, and the mixture of dust and air is sucked into the separator through the outlet of the mill by a mill fan. In the separator, the reduction of the flow velocity is observed, as well as the direction change. Because of the swirl motion of the flow, large particles are separated from the flow. The particles trapped in the separator are discharged through the return pipe to the mill [15].
The gas-solid mixture from the separator is directed and enters into the cyclone tangentially. Coal dust particles due to centrifugal force deflect to the wall of the cyclone, losing speed, slide down the wall and crumble into the dust bin or reversible screw conveyor. Via the screw conveyor, the dust is sent to the dust bin located nearby the boiler. The air cleaned from solid particles is sucked out from the cyclone by a mill fan and supplied to the burners through the dust duct. From the industrial bunker, the dust is mixed with air supplied by the mill fan and blown into the boiler furnace [16].
The automatic regulation system [17] installed at the TPP-2 has poor design and is also weakly automated, as the process of regulating the operation and protection of the dust preparation system is performed by integrated block regulators [3]. The scheme of the dust preparation system with a ball drum mill is shown in Fig. 1.
At the same time, there are significant drawbacks in the operation of the power plant's fuel preparation system. Structural changes were made in the fuel path of the TPP-2 in Temirtau, JSC «AMT» due to the deterioration of the crushing equipment, which led to an increase in the energy consumption of the dust preparation process.
In the classical scheme of the dust preparation system available at TPPs, a two-stage coal crushing system is used with a grinding coefficient kg > 1.1 [9]. The fractional composition of the Karaganda coals contains particles with average diameters from 40 mm to 3 mm. This heterogeneous composition of the coal particles when entering the mill complicates its operation, increases the grinding time, and leads to increased energy consumption for grinding [18]. Also, a destructive impact on the walls of the mill and grinding balls could be observed. In order to reduce this negative impact on the BDM, it was proposed to install a crushing press with a studded surface, which allows crushing large pieces of coal to the optimal size (25 mm). Such an approach could be applied in case of the absence of a preliminary crushing process at the TPP-2 [19].
The crushing press provides optimum crushing of oversized coal lumps due to the impact of its hydraulic hammer on the ramming plate. As a result of using this press, coal with a sieve size distribution from 25 mm to 3 mm will be supplied to the grid and hence to the receiving raw fuel hopper.
The main task of this study is to show the influence of the fractional composition of the fuel material at the inlet, which is further processed by the crushing device [19] on the final efficiency of the grinding process in a ball drum mill [17]. The efficiency of the proposed dust system will be verified also with the means of mathematical modeling that determines the quality of the grinding process.
The process of mathematical modeling will take into account the entire complexity of the technological process and meet a wide range of requirements for the high-quality production of coal dust [20].
To identify the main factors that could affect the grinding process in a ball drum mill [21] and search for their optimal values, an experimental study was performed by [22].
The experimental setup for studying the influence of operating parameters (drum speed, ball loading degree, etc.) on the efficiency of the grinding process is shown in Fig. 2. The experimental setup includes a ball drum mill that operates in a continuous mode.
An experimental ball drum mill with a standard size of 0.35×0.85 m, which was designed for the purpose of the study, is presented in Fig. 2. The control panel includes the «Start» button, which starts and stops the drum feeder, centrifugal fan, laboratory BDM. The power consumed is measured by the «OVEN» PChV102-1K5-A frequency converter. The multimeter (IMS-F1) allows measuring the parameters of the power supply network: voltage (U, V), current (I, A). The rotational speed of the mill is recorded with the «OVEN» TX01 frequency tachometer. The output current ranges from 4 up to 20 mA. The raw material from drum feeder 1 is transported into hopper 2 and through the pipe screw of the hollow trunnion 1 -raw coal hopper; 2 -gate of a tape feeder; 3 -belt fuel feeder (BFF); 4 -ball drum mill; 5 -separator; 6 -cyclone; 7 -dust bunker; 8 -mill fan; 10 -case of throttle converters; 9, 11, 12, 13 -dampers; 14 -vacuum regulator at the inlet of the mill; 15 -temperature regulator of the air mixture behind the mill; 16 -fuel regulator; 17 -regulator of an emergency cold air supplier; CU -manual control unit of the loading, cover enters the cavity of drum 3. The drum 3, complete with covers, rotates, resting on bearings with hollow trunnions, made a single unit with the covers. The material in the cavity of drum 3 is subject to fine grinding because of the impact of the balls (Fig. 3). The type and size of these balls are shown in Table 1.  The filling factor of the drum volume with grinding balls is determined by eq. (1): where m t is the total mass of balls, loaded into the drum, kg; V is the volume of the drum ball mill, m 3 ; ρ g b is the bulk density of grinding balls, kg/m 3 ; L is the length of the drum, m; R is the radius of the drum, m.
The total mass of balls loaded into the drum is according to [23].
To understand the grinding process taking place inside the mill, as well as to obtain numerical values of the kinematic characteristics of grinding bodies, a numerical study was performed. Numerical and experimental studies complement each other and provide high accuracy of the output data. An important aspect of a numerical experiment is the ability to visualize the calculation results. The results obtained in the course of the numerical analysis can be successfully used to optimize the performance of the on-site ball drum mills.
The experimental study will show the main technological and operational characteristics of ball drum mills affecting the efficiency of dust preparation systems. The following directions can be identified in terms of the performance of the experimental study: -conducting experimental research, obtaining experimental data necessary for mathematical modeling; -determination of factors affecting the efficiency of the mill; -selection of evaluation criteria characterizing the efficiency of the grinding process; -performing a multifactorial experimental study with the accepted range of variation of the considered parameters of the grinding process; -determination of performance indicators of the ball mill.
The study of the processes occurring during grinding in a ball drum mill was carried out in two stages.
Stage 1. The analysis of the performance indicators of the ball mill at the TPP-2 of JSC «AMT» is made in order to determine the quantitative effect of various factors on the performance indicators of the BDM: the load factor of the grinding chamber with grinding bodies; the rotational speed of the mill drum; weighted average ball diameter.
Stage 2. Experimental studies were carried out on grinding coals in a ball mill, in order to determine the impact of the following factors on the mill performance: the rotational speed of the mill drum; weighted average diameter of the ball; the coefficient of loading the grinding chamber with grinding bodies.
To assess the grinding efficiency, a standard technique was used based on the sieve method for determining the particle size distribution [18,24]. The laboratory installation for determining the particle size distribution of coal dust includes a «Vibrating screen AS200» screening machine from RETSCH (Germany), electronic scales, and auxiliary equipment.
The study of the grain size composition and the separation of particles into classes in laboratory conditions was carried out using a «MicroSizer 201» laser analyzer [26] (Fig. 4). With the specified equipment, it is possible to classify the coal dust grains in the range from 0.2 up to 1 000 microns. The outputs represent the relation of the weight fraction of particles to their diameter in the form of histograms and tables.
During the experimental study, the process of grinding fuel in the ball drum mill was improved in the following way: the volume of the drum of the ball mill was loaded by 20 % with a standard set of grinding bodies (steel balls) simultaneously with the fuel to be ground. The mass of the crushed fuel should be up from 14 to 18 % of the total mass of the grinding bodies. The standard set of grinding bodies (balls) used in the experimental mill is presented in Table 1. The total ball number is 96 pieces, made of low-alloy carbon steel (8HF). The mass of the ball loading is 7.791 kg.
To ensure the optimal intensity of the grinding process, the mass of additional grinding bodies should be between 30 and 35 % of the mass of a standard set of grinding bodies [27]  During the grinding process, the fuel, together with additional grinding bodies, moves along the mill drum, passing the distance from the inlet neck to the outlet grate, after which the finished grinding product is discharged from the mill together with additional grinding balls. The described improvement of the grinding process in ball drum mills makes it possible to increase the efficiency of the grinding process in a ball drum mill by increasing the uniformity of the fractional composition, due to the use of grinding bodies of various sizes [28][29][30].
The mathematical modeling of the motion of a ball in a waterfall mill operation mode was described in detail in [31]. In a waterfall mode of operation of a ball mill, the balls, together with the fuel, can make a circular motion caused by the gravity force (Fig. 5).
With a uniform spherical loading, a layer of fuel and balls forms a layer-by-layer ordered system consisting of «k» layers. The number of layers can be determined by the following relationship [29]: where b 1 is the parameter of the semi-axis of the elliptical trajectory; r is the weighted average radius of the grinding load ball, m; The efficiency of the grinding process in a ball drum mill [32,33] by increasing the uniformity of the fractional composition is confirmed by the theory of the mechanism of material destruction in a waterfall mode of the mill charge motion. According to this theory, the amount of energy transferred to the material because of the action of the force F 1 , for a specific «k i » layer of the ball charge j is determined by the ratio [22,30,31]: where Q is the productivity of the mill, kg/h; F is the force, N; h is the maximum distance covered by the force, m. The average size of spherical fuel particles formed as a result of destruction of the volumes of deformation zones V 0i and V 0(i-1) , and inlet coal particles located between (i-1) and i layers of the ball charging the mill drum [31] in terms of the cascade mode of motion, and in the case of loading the considered volumes by the value of the optimal load (P opt ), is presented with the following relation: .
where μ is the Poisson's ratio (values between 0.14 and 0.16 are accepted for coals); r is the weighted average radius of the grinding load ball, m; ρ b and ρ f are the balls and fuels density, kg/m 3 ; ω is the rational speed of the drum ball mill, s -1 ; f is the friction coefficient for fuel and balls (the values for coals range from 0.3 up to 0.35).
The close look at eq. (4) shows that the dependence between d i and ω is linear.
The same approach is used to determine the diameter of d i-1 (i-1) particle layer of the ball charge, and also assess the average diameter (d avg ). The following assessment is made [34]: (5) shows also that the average diameter is in a linear relation with the rotational speed of the ball drum mill. For each specific k layer, there is a specific diameter d i , and also a resulting force, which determines the exact amount of energy transferred through the layer.

1. Determination of the significant parameters af fecting the efficiency of the ball drum mill
Four parameters affecting the operation of the ball drum mill have been identified -rotational speed of the mill, degree of filling of the drum with balls, average diameter of the balls, and velocity of the supplied air. Fig. 6 is a representation of the change of average diameter of the dust (d avg,d ) in terms of drum rotational speed (ω). The relations are obtained by the calculations of the average particle size (5) due to the performed experimental study.   6 shows that the increase of the rotational speed of the ball drum mill increases the average size of dust particles. Statgraphics 19 software is used to perform the analysis. For each of the three regression lines, three statistic values such as t-statistics, p-value, and R-squared are presented. The analysis shows that R-squared is approximately 99 %, t-statistic is more than 3, and p-value is close to 0. The correlation coefficient is approximately 99 %. All statistical values are strong, therefore the standard error is approximately zero. Line d 1 is a regression line of d 1 vs. ω. The equation of the fitted model is: Since the P-value in the ANOVA table is less than 0.05, there is a statistically significant relationship between d 1 and ω, s -1 at the 95.0 % confidence level. The R-squared statistic indicates that the model as fitted explains 99.9437 % of the variability in d 1 . The correlation coefficient equals 0.999719, indicating a relatively strong relationship between the variables. The standard error of the estimate shows the standard deviation of the residuals to be 0.000127. This value can be used to construct prediction limits for new observations by selecting the Forecasts option from the text menu. The mean absolute error (MAE) of 0.0000645161 is the average value of the residuals. The Durbin-Watson (DW) statistic tests the residuals to determine if there is any significant correlation based on the order in which they occur in your data file. Line d 2 is a regression line of d 2 vs. ω. The equation of the fitted model is: Since the P-value in the ANOVA table is less than 0.05, there is a statistically significant relationship between d 2 and ω, s -1 at the 95.0 % confidence level. The R-squared statistic indicates that the model as fitted explains 99.9437 % of the variability in d 2 . The correlation coefficient equals 0.999719, indicating a relatively strong relationship between the variables. The standard error of the estimate shows the standard deviation of the residuals to be 0.000127. The mean absolute error (MAE) of 0.0000645161 is the average value of the residuals. The Durbin-Watson (DW) statistic tests the residuals to determine if there is any significant correlation based on the order in which they occur in the data file. Line d 3 is a regression line of d 3 vs. ω. The equation of the fitted model is: .
. ω The same conclusion can be made for this model. The correlation coefficient equals 0.999719, indicating a relatively strong relationship between the variables.
The standard error of the estimate shows the standard deviation of the residuals to be 0.000127. The mean absolute error (MAE) of 0.0000645161 is the average value of the residuals.
The relation is linear for each specific value of the drum load factor. For example, at the rotational speed of the mill drum ω = 0.89 s -1 , the average particle size of the final product is d i = 0.00852 mm. With an increase in the rotational speed of the drum mill to ω = 1.2 s -1 , the average diameter increases to d i = 0.019 mm.
At j = 0.35 and the rotational speed of the mill drum ω = 0.98 s -1 , the average particle size of the final product increases to d i = 0.019 mm. Further increase in the rotational speed of the mill drum to ω = 1.2 s -1 leads to an increase in the diameter to d i = 0.020 mm.
On the other hand, at each fixed value of the rotational speed, when the load factor increases, the average particle size of coal dust increases nonlinearly, and apparently, at the load factor j→1, the average size of dust particles tends to the particle size of the initial fuel material. Loading the mill with coals during the operation of a ball mill is also of great importance. With a small amount of coal, the number of idle blows of the balls increases. With a large amount, the impact force is softened by a layer of coal of increased thickness. Fig. 7 shows the relations of the experimental ball drum mill at different loading levels with grinding balls (j = 14, 27, 41, 55 and 68 %) and the amount of the initial fuel. The highest productivity was obtained with a load of balls j = 41 % and a weight load of the mill with coal of 1500 g, equal to 187 g/h.
The results provided in Fig. 7 were obtained through the experimental setup of the ball drum mill, where the maximum fuel load is 2800 g. The purpose of the experiments was to determine the degree of ball loading ensuring maximum productivity.
The same statistical analysis in terms of Fig. 6 was performed. For all the dependent variables (14,27,41, 55 and 68 %), the regression lines were prepared: With the degree of loading the drum volume with steel balls by 41 % (consisting of various balls with diameters of 15, 20, 25, and 40 mm) and fuel by 59 %, the mill productivity increases from 50 g/h to 187 g/h, i.e. almost 3.74 times. This is a prove not for a stable mode, but the intense nature of the dependence of the productivity (Q) and the degree of loading the mill with grinding balls and fuel.
An open-cycle operating ball mill is used during the experimental study in order to determine the following main factors affecting the performance of the mill: -rotational speed of the ball drum mill; -weighted average diameter of the ball; -velocity of the supplied air; -coefficient of loading the grinding chamber with grinding bodies [21].
During a ball charging process, industrial balls are used, the technical specification of which is provided in Table 1. The qualitative characteristics of coal dust have been determined: -fineness of grinding R 90 % ; -productivity of the mill Q, kg/h; -power consumed by the mill P, W; -specific surface area of the final product S, cm 2 /g. The fineness of grinding and specific surface area of the final product were determined by the method of sieve analysis by sieving a sample of coal dust [35] using vibrating screens of various diameters 1 000, 500, 200, 90, 71 μm.
The mill operating mode is characterized by grinding time and drum rotational speed.
The milling product itself, all other things being equal, is characterized by its specific surface (S), largest size (d out ), and in some cases, by its grain size composition (granulomet-ric characteristics R 90 % ) [15]. Granulometric distribution is presented in Fig. 8.
On the left vertical axis, the total output R x is presented, on the horizontal axis -the values of the lower size limits of particles, in microns. Specific power consumption for fuel grinding is associated with many factors, such as changes in the grindability coefficient kg of fuel, fineness of grinding dust R 90 , mill productivity, drying agent feed rate, drying agent temperature and ball diameter [19,20,31].

2. Analyzing significant parameters and performing a multifactorial experiment
The results of the experiment study have been used to prepare the factorial experiment [36]. Table 2 shows the averaged results from the experimental study that allows compiling a design matrix for a full-factor experiment. In terms of the results from Table 2 and their statistical processing using commercial software, regression equations in the form of Q = Q(x 1 , x 2 , x 3 , x 4 ), P = P(x 1 , x 2 , x 3 , x 4 ), and S = S(x 1 , x 2 , x 3 , x 4 ) are obtained. The regression equation obtained [37] shows the behavior of the technological, energy, and economic indicators such as Q -mill productivity, P -power consumed by the mill, and S -specific surface area of the final product. The regression equation is valid only in a narrow range of variation of the processed parameters.
The regression equation (9) is derived based on the data obtained in accordance with 24 on-site trials (experimental tests) [35]:  Table 2 Summarized results from the numerical study Equation (9) gives valuable information about the main parameters affecting the grinding process. The productivity, power consumed and surface area can be independently adjusted to have the optimal operation of the drum mill.

3. Obtaining regression equations for the significant parameters affecting the operation of the ball drum mill
The significance of the regression coefficients of the relations obtained was determined by calculating the Student's criterion [30]. Validation of the regression model for adequacy was performed by calculating the Fisher criteria, which reflects how well this model explains the total variance of the dependent variable, and compares with the table values.
The regression equation characterizing the relation of the mill productivity Q in terms of the factors x 1 , x 2 , x 3 , x 4 in coded form is expressed with: x x x x − With some transformations, taking into account the levels of variation of the main factors, the following regression equation describing the grinding process is as follows: . .
Some main conclusions can be made based on the performed regression analysis: factor x 1 , the relative rotational speed of the mill drum has the greatest impact on the positive change in mill productivity. The coefficients at x 1 and x 1 2 are both positive and have the highest specific impact on the mill productivity. When the relative rotational speed of the drum mill increases, the productivity also increases. This is due to the fact that the balls and material are centrifuged at high relative rotational speeds of the drum, as a result, the finished product decreases and becomes coarse. Then coefficients x 2 , x 3 , and x 4 are negative. At x 2 (weighted average diameter of the ball), x 3 (speed of the supplied air in the mill drum), and x 4 (load factor of the grinding media), the overall productivity of the mill decreases [21].
The regression equation presenting the relation of the power consumed by the mill in terms of the factors obtained in the current study has the following explicit form: The distribution of the response functions from the relative rotational speed of the mill drum is presented in Fig. 9.
The obtained regression equations (10)-(13) allow optimizing the grinding process. With the proper value selection of the input parameters (ω, j, v, d avg ), it is possible to adjust the process running most efficiently.
When increasing the relative rotational speed of the drum mill from 0.76 to 0.92 (increasing the total mill power by 17.9 %), the energy consumption increases from 1170 W to 1225 W or by 4.5 %. At the same time, the production capacity of the mill increases from 65 kg/h to 72 kg/h, i.e. by 9.7 %, and the specific surface area of the final product decreases from 2 439 cm 2 /g to 2 238 cm 2 /g, which is 8.2 % less. Fig. 9 shows that the largest specific surface area of the final product and low power consumed by the mill drive are achieved at a relative rotational speed of the mill drum amounted to ω = 0.76. In this case, the productivity of the mill has a minimum value, therefore, this option is preferable in the case where the finest final product is required.
If, however, high requirements are not imposed on the quality of the final product and it is necessary to increase the productivity of the mill, then increasing the relative rotational speed of the mill drum is recommended. In this case, operational parameters from ω = 0.81 up to 0.87 should be adopted.
Considering the values of the mill productivity, the power consumed, and the specific surface of coal dust, the weighted average diameter of the balls was determined -it should be in the range of d avg = 33.5 up to 34.5 mm.
For the optimal performance of the ball drum mill, the load factor of the grinding media should be j = 0.325 up to 0.335. For the velocity of air supplied, the acceptable region is v = 0.2 up to 0.3 m/s. Experimental studies of the grinding process in a ball drum mill performed by [21,[38][39][40] were used to support and verify the data obtained during the regression analysis.

Discussion of the results obtained via experimental and numerical studies
Grinding is the most important technological process in the production of coal dust. The finer the dust, the faster it burns in the furnace of the boiler unit. To obtain finely dispersed coal dust, continuous drum mills are most often used. The fineness of fuel grinding in a ball mill is determined by: -the properties of the material to be ground, more precisely, its grindability coefficient; -the type and design of the mill, the degree of filling of the drum volume with grinding media, as well as the characteristics of the grinding media: their configuration, material, porosity, etc.; -the operation mode of the mill, which is characterized by the grinding time and drum speed.
Therefore, the purpose of this work is the development of a methodology for selecting a rational composition of grinding media; operation modes of the mill, ensuring the maximum efficiency of the coal grinding process, taking into account its physical and mechanical properties and the development of a mathematical model for determining the optimal parameters for the destruction of material in ball drum mills.
The main objective of this work is to study the effect of the fractional composition of the initial fuel on the efficiency of the grinding process in a ball mill [5]. The initial coal delivered to the thermal power plant passes the stage of fine crushing in a crusher, the design of which is proposed by the authors [8] and has dimensions up to 2.5 mm. The effectiveness of the dust system in the new conditions will be proved by the creation of a mathematical model of the BDM characteristics, which determine the quality of the grinding process. The ball mill mathematical model should ensure the preparation of coal dust with a grinding fineness for the initial coal R 90 = 18 up to 22 %, which is an indicator for industrial mills. The boundary condition for the mathematical modeling is that the coal dust obtained should have a size of less than 90 microns when performing a sieve analysis. Only from 18 up to 22 % of the composition of the sample taken for analysis can have dimensions of more than 90 microns. The results of the analysis of the coal dust composition with a Microsizer 201 laser analyzer confirm the accepted boundary conditions. The proof is the graph in Fig. 6, where the minimum average particle sizes reach even 25 μm to 5 μm. The minimum average particle size of coal dust depends on the nature of the movement of fuel particles in the mill, on the density of the fuel and grinding balls, and most importantly, on the relative speed of the mill drum. The graph shows the nature of the increase in the average size of dust particles with increasing drum speed and under certain conditions it reaches 90 microns.
Based on the results of the experimental study, a mathematical model was selected that shows improved characteristics of the BDM. These mill features include: -productivity Q, kg/h; -power consumed by the mill drive P, W; -specific surface area of the finished product S, cm 2 /g. The mathematical model was obtained using the theory of a full-factor experiment [26]. This theory uses the central compositional rotatable plan of the full factorial experiment. Table 2 presents the averaged results of experimental studies, which made it possible to compile a plan matrix for a full-factor experiment. Based on the results of the experiments and their statistical processing using commercial software, we have obtained regression equations Q = Q(x 1 , x 2 , x 3 , x 4 ), P = P(x 1 , x 2 , x 3 , x 4 ), S = S(x 1 , x 2 , x 3 , x 4 ). Multifactorial analysis shows the relationship between such economic indicators as Q -mill productivity, P -power consumed by the mill, and S -specific surface of the final product in terms of ω, d avg , v, j.
In the current study, the general purpose is to optimize the operation of the BDM at the TPP-2 JSC «AMT» in Temirtau. For this purpose, preliminary experimental studies of the BDM operation were carried out and the relations between the main technological parameters of the grinding