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Experimental and Theoretical Study Focused on Transformation of the Processing Properties of Gold in the Operating Area of a Ball Tube Mill

The article is aimed at studying the process of free gold deformation extracted during grinding in various types of ball mills. Today, ball mills and similar machines are typically used for fine grinding of gold ores. Their working parts include armor plates of a shell, and balls used as grinding media loaded into the shell. Despite the fact that the ball mills dominate over the other fine grinding machines used to extract hard rock gold, they still have serious drawbacks. It is believed that only 2 to 20% of all consumed energy is spent directly on grinding, while the rest of it is used to overcome friction, heat formation, acoustic and mechanical vibrations, etc. In the meantime, mass deformation and degradation of free gold takes place in the mills. The particles degradation has a clearly adverse effect on their further benefication, and their shapes change to various extents as a result of deformation. If the deformation is aimed at forming an isometric shape of particles, their recoverability increases, while the reverse process decreases it.


Introduction
The commercial component destruction, which occurs during grinding in ball tube mills, and the lack of possibility to control this process except for recovery using cheap gravity benefication methods complicates the benefication technology involving cost intensive recovery processes (flotation, cyanidation, leaching, etc.). The data resulting from the study of a mechanism of gold particles deformation during extraction will be applied to improve the ball mill design and to develop new grinding machines and installations for enlarging hydraulic size of particles, which are intended for used at ore-dressing enterprises in the Republic of Sakha (Yakutia).

Relevance
Ball mills are extensively used to reach the highest possible level of commercial component extraction during ore disintegration. When ore is abraded in a ball mill, regrinding of gold occurs, which impairs its recovery using gravity methods. The extent of regrinding depends on the installation operating mode and ore type, and has a considerable importance when treating such mineral deposits as gold, platinum, tin ore, etc. In this vein, the elaborated recommendations apply only to the selection of mills with 'gentle armor' and the preservation of lining coatings [1-6], however, there is no data on studies indicating there was any success in preventing this adverse process. This is highly likely due to the During the shell rotation, the material may break away and drop at different angles.
During the first 2-3 hours of the ball mill operation, some lead flakes, which are less than 0.3 mm thick, swell along the edges (Figure 1).

Figure 1.
A -a flake in the area, where swelling occurs along the edges and out of the rolling balls reach; B -a flake in the area of debris and balls moving in the opposite direction.
It was empirically established that the specific structure of gold grains was formed as a result of microforging of the gold grains edges by sand grains pushed by the air flow [7][8][9][10][11][12][13]. Similar work was carried out in cooperation with research associates from Ammosov North-Eastern Federal University. Swelling at the edges of gold grain flakes is caused by frequent micro-bombardment by small debris. It happens as a result of a considerable lag of the moving gold grains compared to the sand grains pushed by the air flow. Beside, this swelling of the open-worked structure remains due to the fact that large debris do not roll through them.
Consequently, the following conditions are created in the ball mill: the flattened markers are treated by small debris, while the grinding balls and larger debris do not roll through them. This could be explained by the following. Due to their small size and high density, marker particles end up on the mill bottom. Each of them has certain forces acting on it: g -gravitational; FCFcentrifugal; frictional -FFR. Therefore, on the one side, the acting gravitational force pulls the markers down and, on the other, the acting friction force, FFR, carries them away upwards with the moving mill bottom at a linear speed VL. These forces determine the balance of forces.
Sand grains surrounding a piece of gold constantly bombard it and change its shape. Gold has the highest forging properties, so a plastic deformation model could be used in modeling. During air separation, the sand grains near the considered piece of gold change the direction of movement due to the difference in speed, and there occur numerous collisions with empty rocks. At the same time, the piece under consideration also changes its movement direction and tumbles. Thus, an unregulated, random bombardment of the gold piece surface takes place. Therefore, it is fair to say that there happens a consistent isotropic bombardment with grains of sand. If a piece of gold has an irregular volumetric shape, i.e. all three linear dimensions are comparable, then, following the isotropic bombardment, the body is steadily shaped into a ball. The problem changes dramatically if a flat plate is studied. Consider a sufficiently large flat deformable plate exposed to the isotropic bombardment. The thickness of the plate is much less than its length and width ( Figure 2). For simplicity, first, consider a plane problem. When exposed to the isotropic bombardment at the beginning, only those parts of the plate surface that have the least radius of curvature are deformed. Therefore, the deformation starts at the edges. Due to the isotropy of the sand grains stream, flat areas of the plate, i.e. infinite radius surfaces, remain undeformed. The particles flux hitting flat areas of the plate is not taken into account in calculating. After some time, the edge of the plate will have a specific shape ( Figure 3). The plate becomes shorter, but there appears a swelling at the edge, since the volume is retained. Figure 4 shows only that particle flux, which deforms the body. As time passes, the swelling of the plate edge increases and shifts to the left, while the surface restricting the swelling increases. For a dynamic analysis of the swelling surface, it is necessary to solve a problem involving stress and strain tensors, i.e. the elasticity theory. As noted above, if an arbitrary shaped body confined in a certain region of space is exposed to the isotropic bombardment for a long enough period of time it becomes a ball. That is, the body that has the smallest boundary surface when the values of volume are set.
Any plate with the proportionate width and length, following an exposure to the isotropic bombardment for a certain period of time, tends to take the shape of a torus, the inner part of which remains flat (Figure 4).  If a flat disc is considered, the disc edge will take the shape of a torus. It means, that the problem consists in determining the time value, during which the plate takes the shape of a torus of a certain size.
Consider the problem: a plate with an initial shape of a disc having a thickness equal to 2r0 and a radius R0 is exposed to the isotropic bombardment with a uniform particle beam of the given mass m0, average velocity v0, and flux density j0. First, the peripheral part of the disc is deformed, where there are surface areas with the least curvature. As a result, the disc edges swell and its linear size decreases respectively.
During deformation over time dt, the work is done on the body = (2) where is a change in the body surface during deformation, is the proportionality factor, which can be determined through the experiment.
According to the energy conservation law, this work (if energy loss through heat dissipation is ignored) is equal to the total kinetic energy of the particles bombarding the deformable surface, i.e.
is a number of particles hitting the deformable surface over time . A number of particles hitting the surface per unit of time is referred to as a flux Ф; therefore, we introduce the concept of volumetric flux density j0: = = 0 (4) If an isotropic and uniform flux is considered, then 0 is a constant value. From (1), (2), (3) we obtain an ordinary differential equation: (8) The resulting formula holds for a case when the area of the deformable surface gradually increases, for example, for a plane problem. If a flat disc is considered, an increase of the lateral surface radius of curvature leads to a reduction of the disc linear size, which results in a reduction of the lateral surface.
Suppose the initial shape of the plate is a disc with a rounded edge ( Figure 5), then 0 is the area of the rounded part of the initial shape of the disc surface. The disc thickness equals 2 0 , the disc edge in section has a shape of a semicircle with a radius 0 . 0 is a distance from the axis of rotation to the center of the semicircle. Thus, the disc radius equals 0 + 0 .
The disc volume is determined through integration (9) During the isotropic bombardment, only the disc edge is deformed. Besides, the section of the deformable surface becomes larger. The volume limited by this surface also increases. Therefore, the deformable part of the disc becomes larger, while the flat part of the disc is reduced. Considering that the volume of the deformable part (the disc edge) takes the smallest boundary surface under the isotropic action, the deformable area can be approximated by the shape of a dynamically changing torus. During this process, the torus surface increases, and its linear size decreases. The middle part of the disc remains flat.
The volume of the resulting figure is equal to the sum of the torus volumes and the flat area in the middle : 0 = + (10) This volume is equal to the initial disc volume (9). The torus volume is equal to: = 2 2 2 (11) where is the torus edge radius of curvature, is a distance from the torus center to the edge center of curvature.
The volume of the middle flat part is calculated through integration. The obtained formula (14) allows to find the dynamic change of the disc surface deformable part. During the isotropic bombardment, the torus linear size decreases, and the deformable section surface first increases to a certain maximum value, then decreases. This means that there is a peak value. Therefore, it may be said that thin plates take the shape of a torus fast enough, while thick plates tend to take the shape of a ball. After the maximum surface area of the torus is reached, there should be a slowdown in the process of deformation during the isotropic bombardment, since the torus area starts to decrease and its shape tends to take the smallest area, i.e. the body seeks to take the shape of a ball. Therefore, pieces of gold shaped as a torus having certain parameters depending on the initial thickness and size of the plate should often be found in the processed ore. Figure 6 shows a graph of the dynamic development of the disc surface deformation.  The estimates suggest that with an initial disc thickness of 1 mm and a radius of 10.5 mm, the maximum deformable area is achieved when the torus thickness is 2.82 mm and the disc radius is 7.937 mm. Figure 7 shows the disc shape with these parameters.
where is defined by the geometric model (9-14)

Conclusion
Through the example of analyzing the behavior of malleable markers, the process of transforming the shape of tough plastic (malleable) materials during ball grinding is found to require the retention of the initial particle size of commercial components (gold) after extraction from the ore mass, as well as timely removal from the grinding mill. Therefore, an appropriate design of the inner surface (lining) of the ball mill is required to reduce the rolling of gold particles.