Mathematical modeling in the organization of the production process of leaching metals

The following will describe the process of how you can use mathematical modeling to organize the production of metal leaching. Mathematical models and results of computational experiments are presented. The results of studying the thickness of films of sulfuric acid-chloride solutions containing the addition of surfactants surface-active substance (SAS), which are formed when they pour out onto a frosted glass surface as a model of the surface of a piece of ore, are presented. A mathematical model how the film thickness changes depending on the feed rate of the solution and its composition, as well as on the angle of inclination of the glass surface has been determined. Underground leaching of metals from ores stimulates the search for ways and means of intensifying the process. Due to the addition of a surfactant, the leaching process is accelerated, since the thickness of the films of leaching solutions flowing over the surface of ore minerals decreases. This direction is relevant in modern science, and research in this direction the growth of indicators allows the national economy to improve and helps to improve and increase knowledge for further development of the metallurgical industry in Russia The article also discusses the results of studies on the electrochemical leaching of metals from polymetallic ore by sulfuric acid-chloride solutions with the addition of a surfactant with asymmetric current pulses. Mathematical models of the dependence of leaching indicators on the density and duration of the flow of direct and reverse polarity are presented. For ease of perception and better visualization, a table was compiled, which reflects the results of comparative analysis to assess the further possibility of processing secondary raw materials.


Introduction
One of the factors that have a significant impact on the course of the process of underground electrochemical leaching of metals from ores is the capacity of the double electric layer, which forms on the surface of ore minerals when flowing around them with leaching solutions [1,4,17]. The literature known to the authors of this work provides information on the magnitude of the Electrical Double Layer (EDL) capacity on the surface of sulfide minerals in sulfuric acid-chloride solutions [2].
Experimental studies to determine the capacity of EDL on the surface of some sulfide minerals in sulfuric-chloride solutions with the addition of surfactants SAS [3,5] were carried out in the Geotechnology laboratory of the NCIMM (STU). The measurements were carried out using a universal meter L, C, R E7-11 at a frequency of 100 Hz in a plexiglass cell, into which the studied mineral electrodes were alternately placed, made of the main ore-forming minerals of the Sadon group of polymetal deposits: galena, sphalerite, pyrite, and chalcopyrite. The dimensions of the studied electrodes were 10x10x20 mm. The working surface was a polished end face of each electrode with dimensions of 10x10 mm. To improve the electrical contact between the mineral electrode and the current-supplying conductor, a contactol layer was applied to a section of this face. After its hardening, this part of the electrode face was electrically and waterproofed, as well as the rest of the electrode faces, so the actual size of the working surface of the mineral electrodes was 10x1 mm. To determine the EDL capacitance on the surface of the studied electrodes, an auxiliary copper electrode 80x80 mm in size was also placed in the measuring cell. The surface area of the auxiliary electrode exceeded the area of the working surface of the studied electrodes by a factor of 640 times [18][19][20].
In the course of the research, the dependence of the EDL capacity on the composition of the sulfuric acid-chloride solution with the addition of surfactants was studied for each of these minerals. The experiment planning method was used (Box-Behnken three-level non-compositional plan). The conditions and results of the experiments are presented in Table 1. Mathematical models of the dependence of the capacity of EDL, formed on the surface of ore minerals, on the composition of the solution with the addition of surfactants SAS. obtained as a result of processing the experimental data. are presented in Table 2. The adequacy of the obtained models to the experimental data is not less than 95%.  Comparison of the results obtained in these studies and presented in [2] indicates that the addition of surfactants according to SAS [3][4][5][6] in sulfuric acid-chloride solutions leads to a decrease in the capacity of DES by 2.5-м3%.

Materials and methods
The thickness of the leaching solution films flowing over the surface of the ore lumps during the infiltration or film-droplet mode of underground and heap leaching is one of the most important factors that determine the distribution of the current between the ore and the solution and, consequently, the efficiency of using electrical process intensifiers. The study of the regularities of the formation of films of solutions directly on lumps of ore and their parameters is complicated by the different wetting of ore and nonmetallic minerals, the changing curvature of the surface of unpolished and unequal surface roughness in different areas of the polished pieces of ore. The difficulty of maintaining the same flow of solutions over the entire surface. and other factors. Therefore. The study of the regularities of changes in the thickness of the resulting films of solutions is carried out by modeling methods using as a model of the surface of ore pieces of a frosted glass surface, as the most wettable material. In the literature known to the authors. there is information about the thickness of the films of binary solutions of underground leaching containing sulfuric acid and sodium chloride [1].
In this work, we present materials for studying the thickness of films formed during the flow of sulfuric acid-chloride solutions containing the addition SAS of surfactants over the matted surface of a glass plate 200 mm long and 30 mm wide. The plate was fixed on one of the short sides on a vertical stand with the possibility of rotation around the axis of attachment. which made it possible to install it with a deviation from the vertical at an angle from 0 to 900 using a micrometric screw with attachments fixed on the same stand. The long sides of the plate were coated with a thin layer of stearin 0.5 mm wide for edge effects, so the width of the solution flow in the experiments was 29 mm. The measurements were carried out by the electro contact method [7]; in order to increase the measurement accuracy, the solution "adhering" to the measuring needle was removed with filter paper after each measurement. The measurement of the film thickness was carried out on the axis of the solution flow. Sue to the action of surface tension forces, it is here that it has its maximum value. When calculating the average value of the film thickness was assumed that the cross-section of the film has the shape of a segment of a circle.
The thickness of the films of sulfuric acid-chloride solutions with the addition of surfactants as a function of the angle of inclination of the surface. The composition and feed rate of the solution were determined in the course of the planned experiment using the Box-Behnken plan [8][9][10][11]. The ranges of variation of the independent variables are given in Table 2 and the experimental values of the thickness of the solution films are given in Table 3. As a result of mathematical processing of experimental data and elimination of statistically insignificant coefficients. the following regression equation was obtained: where δe.av. -the experimentally determined average thickness of the solution film, mm; Vl -the solution feed rate, cm 3 /s; α -the angle of deviation of the plate surface from the horizontal, deg; С1 -the content of sulfuric acid in solution, g/l; С2 -the content of sodium chloride in solution, g/l. The obtained regression equation is adequate to the experimental data with a significance level of 0.05. The reliability of the results obtained is in good agreement with the results of [1] confirmed by the compliance with the calculated values performed according to [9,21], taking into account the data of the study of the physicochemical properties of solutions containing the surfactant additive SAS [2,16] (the deviation does not exceed 3%), and given in Table 2

Results and Discussion
As a result of processing the experimental data using a mathematical apparatus, the authors obtained a regression equation that determines the average film thickness of a solution containing a surfactant. and a mathematical model of the dependence of the EDL capacity [12-14; 15].
An analysis of the obtained equations allows us to conclude that the addition of a surfactant makes it possible to reduce the thickness of the films and leads to a decrease in DEL, which contributes to better leaching of metals.

Conclusion
On the basis of experimental data, regression equations have been established that describe the dependence of the electrode potential of galena, pyrite, sphalerite, and chalcopyrite on the concentration of sulfuric acid-chloride solutions used in underground leaching and containing a surfactant additive. The degree of influence of the reagents on the value of the potential of minerals has been determined on the basis of experimental data, regression equations are synthesized that relate the capacity of the electric double layer on the surface of some sulfide minerals in sulfuric acidchloride solutions with the addition of surfactants.