Mathematical modeling of processes of technogenic deposits development

The article aims to substantiate the tendency to solve the problem of the mineral and raw material base of non-ferrous metals due to the involvement of technogenic origin in substandard metal-containing raw materials. Critical analysis and systematization of new information on the current state of the mineral and raw materials base of non-ferrous metals and the environmental problems of mining associated with it. New data on the recovery and loss of metals in the process of ore dressing are presented. It is shown that the total value of metals in waste is comparable to the value of potential mineral resources in the bowels. The problems of the use of mining waste should be solved in a single package with environmental problems by creating a single technological cycle for the extraction and processing of industrial minerals of industrial waste, the use of which can make the development of industrial deposits economically viable. The practical significance of the work lies in the possibility of using best practices for non-ferrous metallurgy enterprises that are searching for ways to survive in the conditions of the establishment of market relations.


Introduction
In the context of increasing demand for non-ferrous metals, the issues of strengthening the state of the mineral resource base due to the involvement of substandard metal-containing raw materials of technogenic origin are of particular relevance [1,2]. Studies on this topic use information about the current state of the issue and use the methods of analysis and systematization of new information for proof. The scientific novelty of research on this topic consists of considering the issue of involving the production of substandard metal-containing raw materials of technogenic origin simultaneously with environmental issues. Studies on this topic indicate the need to use best practices for non-ferrous metallurgy enterprises in the context of market relations with partial exhaustion of available deposits.

Materials and methods
The provision of basic sectors of the economy with reserves of the main types of minerals seems satisfactory, but recently the urgency of the problem of the mineral and raw materials base of non-ferrous metals has been increasing [3]. In the developed industrial countries of the world, the level of use of industrial waste reaches 70-80%, while in Ukraine and neighbouring countries it does not exceed 12-15%. In the USA, for example, 20% of all aluminium, 33% of iron, 50% of IOP Publishing doi: 10.1088/1755-1315/1049/1/012030 2 lead and zinc, and 44% of copper are obtained from industrial waste [4]. A similar trend in the use of secondary resources is observed in Canada, Great Britain, South Africa, Spain and other countries. For example: • In the state of Montana (USA), the Mandiski mine dumps annually produce 2 tons of Au and 4 tons of Ag with a gold content of 0.84 g/t and silver of 2.8 g/t; • In the state of Michigan (USA), 60% copper recovery has been achieved from the beneficiation of tailings containing 0.3% Cu; • In South Africa, from the dumps of gold recovery plants with a gold content of 0.53 g/t and uranium -40 g/t, 3.5 tons of gold and 696 tons of uranium per year are obtained with a productivity of 50,000 tons/day.
The development and creation of an information retrieval system for the processing and integrated use of mineral raw materials from technogenic deposits is a cardinal step towards a waste-free technology. In the Kryvyi Rih iron ore basin, according to various estimates, dumps contain up to 13 billion tons of overburden, and tailings contain up to 6 billion tons of waste from the enrichment of low-grade iron ore. In recent years, the possibility of using the mineral mass accumulated in dumps and tailings of the Kryvyi Rih basin has been increasingly studied. Obviously, the lack of methods for assessing man-made deposits adopted at the state level hinders their commissioning [5,6]. Currently, about 25 billion tons of solid waste alone has been accumulated on the territory of Ukraine.
These wastes negatively affect natural landscapes and environmental conditions, occupying an area of about 150 thousand hectares of fertile land and worsening the human environment. Technogenic deposits lead to the exclusion from economic circulation of large areas of land occupied by production waste. In addition, there is destruction or a decrease in the quality of land due to dust drifts from dumps and tailings [4]. The problem of industrial waste disposal is of paramount importance. An important circumstance is that the cost of commercial products from industrial waste is 5-15 times less than from traditionally mined ores from mineral deposits. The tendency to develop technogenic reserves is growing, which is becoming the main, and sometimes the only supplier of raw materials. The amount of non-recoverable metals is characterized by (figure 1).
The extraction rate of the main minerals is 65-78%, and associated elements in the extraction of non-ferrous metals -10-30%. Elements such as In, Ga, Tl, Bi, Hg are almost entirely lost in the flotation tailings. Losses of other metals are characterized by (figure 2). In the copper sub-sector about 220 million tons of tailings have been accumulated, in which the copper content (0.34-0.37%) is close to the condition (0.35-0.50%). Sulfur accounts for 30-50% of the cost of tailings of ores beneficiation, precious metals -25-45%, copper -10-20%, and zinc -10-15%.
The beneficiation tailings of copper-nickel ores contain industrial concentrations of platinum, gold, and silver, which are already available to modern processing technologies. When tungstenmolybdenum ores are enriched, up to 60% copper, up to 81% bismuth, up to 62% tantalum, gold, silver, and other elements are not extracted. The metal content exceeds 0.04% with the condition during production >0.1% WO3 (Tungsten trioxide).
Using new processing technologies, substandard ore dumps are a technogenic deposit suitable for mining at a lower cost than when mining metals from primary ore. In the tungstenmolybdenum sub-sector, the flotation and flotation-gravity concentration tailings contain about 400 thousand tons of molybdenum and more than 100,000.0 tons of tungsten. The metal reserves in the processing waste are equivalent to the reserves of new deposits (figure 3).

Results and discussion
The total value of metals in waste according to the indicative estimate is comparable to the value of potential resources of mineral resources in the subsoil and which is several times higher than the value of known reserves in the subsoil, which are not yet used [7][8][9][10]. In most cases, mineral waste is used as raw material for the construction industry (no more than 10%). In technologically developed countries, more than 40% of copper, 35% of gold, and a significant amount of other metals are obtained from waste materials using new technologies, for example, leaching [11,12].
The problems of the use of mining waste should be addressed along with environmental problems. The negative impact of tailings storage facilities on the environment is manifested in the territory, which is 10 times larger than the area occupied by the waste itself. The use of reserves of technogenic deposits is favored by the fact that they are located in habitable areas on the surface of the earth and are crushed, which sharply reduces the cost of obtaining metals. The insufficiency of the undertaken environmental measures is evident from the fact that large areas of land are occupied by waste production of 4th and 5th hazard classes. The environmental impact of waste is regional and global. Soil horizons are enriched by ore components of dumps, in which they are not isolated from water systems and affect the adjacent area.
In this situation, the prospects for reducing the negative impact on the environment are associated with the creation of a single technological cycle for the extraction and processing of minerals of technogenic waste: "ore processing -waste storage -utilization". With the use of new technologies, the development of technogenic deposits can become economically viable production. The main contradictions in the processes of production activity and waste generation of industrial enterprises can be resolved by utilizing technogenic and substandard mineral raw materials in the cycle of integrated development of non-ferrous metal deposits (figure 4). The unified model of leaching takes into account the type of metal leaching, the methods of metal production types of leachable metals. Accounting is carried out with the help of variable "n", that is, each species corresponds to its number: • n = 1 -traditional metal leaching in agitators; • n = 2 -tails activated in the disintegrator are leached in the agitator; • n = 3 -leaching of tailings in the disintegrator combined with activation. • m = 1 -extraction of metals from the balance tailings of coal preparation; • m = 2 -extraction of metals from off-balance tailings of coal preparation.
Accounting for extraction of different metals is taken into account using a variable "p": • p = 1 -corresponds to the zinc leaching model; • p = 2 -corresponds to the leaching model of lead; • p = n -corresponds to the leaching model of another metal.
Lead recovery regression equations for tail leaching in a disintegrator (n = 3; m = 2; p = 2): ε (3; 2; 2) = 39.44 − 1.17X 1 + 16. For each specific process and metal, a single model will be a regression dependence. This single model of metal recovery during leaching is implemented in Maple 9.5, Matlab, Mathcad, etc. During the experiments, the most and least intense impact of all factors, i.e., two series, in one of which the values of all factors were taken as maximum (MAX), and in the second as minimal (MIN). Agitation leaching of crushed materials (AKX). Agitation leaching of materials pre-activated in the dry state (AAX). Leaching of materials at the moment of activation with solutions in a disintegrator (APX). Agitation leaching of materials subjected to activation in a disintegrator together with leaching solutions (PAX). Leaching of materials during activation with their repeated passage together with leaching solutions through a disintegrator (AMX) [13,14] (table 1).
The basis for determining the function between the function and independent factors is the multiple regression method implemented in Statistica 6.0. In the case of a one-factor dependence, the desired function is a one-dimensional surface in two-dimensional space -i.e., the line defined by equation Y = a + a 1 X. According to this, the variable Y can be represented as a function of the constant (a), and the slope coefficient (a 1 ) multiplied by the value of the variable X.
In the case of multiple regression (when multiple predictors are used), the regression surface cannot be displayed in two-dimensional space, but the calculations are practically unchanged. A linear equation describes multiple regression: where k -number of predictors. Regression coefficients (a 1 ...a k ) represent the independent contributions of each independent variable to the prediction of the dependent variable. The regression surface expresses the best-predicted value of the dependent variable (Y ) for the given values of the independent variables (X). Since the task of linear regression procedures is to fit the surface, which is a linear function of the variables X, in accordance with the observed variable Y , the residual values of the observed points can be used to develop the criterion of "best fit". According to the Fisher criterion, hypothetical dependencies at a significance level of 5% are accepted as plausible. Dependencies are sought to take into account linear, quadratic effects and their interactions in the form of a regression equation: where X 1 -dimensionless coefficient characterizing the content H 2 SO 4 ; X 2 -dimensionless coefficient characterizing the content NaCl;  (table 3). Tables 2 and 3 differ in leaching medium. The consumer of secondary processing waste may be the most material-intensive construction industry. Hundreds of millions of tons of raw materials are produced annually in the world for IOP Publishing doi:10.1088/1755-1315/1049/1/012030 9 the production of building materials. The volume of waste generated is commensurate with the demand for the building materials industry for mineral raw materials. However, no more than 10% of the waste is currently used.

Conclusions
The problems of using mining wastes are inextricably connected with environmental problems. They can only be solved by creating a single complex for the extraction and processing of minerals of technogenic origin, which allows the development of technogenic deposits to be economically attractive. The parameters of the leaching of metals from tailings can be determined from the proposed model. A universal mathematical model that describes extracting non-ferrous metals from raw materials of technogenic deposits, considering the impact of various mechanochemical factors on it is proposed. Specific parameters for the extraction of metals from processing tailings can be directly determined from the proposed model, which makes it possible to select the optimal scheme of the technological process and evaluate its technical and economic indicators at the design stage.