A steel ball size optimization method based on determination of impact speed of a mass center of a steel ball group in a mill

Abstract

The application relates to a steel ball size optimization method based on determination of impact speed of a steel ball group centroid in a mill, and belongs to the technical field of grinding in mineral processing. The centroid of the steel ball group in the mill is determined, a ball mill model is established, a filling rate, a steel ball number and a critical rotation are calculated according to mill parameters in production, the ball mill model is input for simulation, centroid coordinates are obtained, and the distance R from the centroid to the center of the mill is calculated c , then the ratio lambda of the distance from the mill center to the effective radius is calculated, and the steel ball diameter required for grinding a specific particle size is calculated according to a formula D b . The method aims to overcome the problem that the calculated speed value is large due to the fact that the steel ball speed in the intermediate polycondensation layer represents the steel ball group speed in the prior art, and provides a more accurate steel ball size optimization scheme. The centroid position of the steel ball group in the mill is determined through numerical simulation, and the steel ball diameter is derived based on the steel ball speed of the centroid layer.

Description

Technical Field

[0001] This invention relates to a method for optimizing steel ball size based on determining the impact velocity of a group of steel balls inside a mill, belonging to the field of grinding technology in mineral processing. Background Technology

[0002] Grinding is one of the most critical and energy-intensive operations in mineral processing. Especially during the operation of ball mills and semi-autogenous mills, energy consumption remains a significant challenge for energy conservation and emission reduction in mines. For ball mills, the grinding process not only consumes a large amount of energy, but the particle size distribution of the ground product directly affects subsequent beneficiation processes. Therefore, improving grinding efficiency and reducing energy consumption have become urgent technical challenges for the mineral processing industry.

[0003] The motion of steel balls within a ball mill is crucial, determining whether ore crushing occurs and the manner of crushing, thus directly impacting the mill's energy consumption and product particle size distribution. The crushing of brittle particles is a complex process influenced by various factors such as loading rate, particle strength, shape, particle size, and internal structure.

[0004] In existing technologies, the selection of steel ball diameter typically employs a series of formulas, such as the FC Bond formula, the Rowland (Allis-Chalmers) formula, and the Azzaroni formula. These formulas are based on the Bond work index and consider parameters such as mill diameter, ore density, feed particle size, and mill speed. However, these formulas consider relatively few factors and are rarely used in China. In contrast, Duan Xixiang defined ore strength using the fracture stress of standard mechanical specimens, derived the steel ball kinetic energy using Davis's steel ball motion theory, and proposed a semi-theoretical calculation method for steel ball size by integrating multiple factors. Xiao Qingfei defined ore strength using the fracture energy of irregular ore, similarly derived the steel ball kinetic energy using Davis's steel ball motion theory, and further derived a theoretical calculation method for steel ball size (a method for calculating the diameter of steel balls in ball / semi-autogenous mills, CN202410968325.2).

[0005] Although various formulas exist for calculating the diameter of steel balls, most current methods for optimizing steel ball size rely on Davis's theory of steel ball motion, which was proposed in the 1920s. This theory does not consider the influence of the liner on the motion state of the steel ball, and the calculation results are often too large. As the grinding process has become increasingly refined, this theory is no longer sufficient to meet current needs, and the calculation results are not accurate enough. Summary of the Invention

[0006] This invention provides a method for optimizing steel ball size based on the impact velocity determined by the centroid of a steel ball group within a mill. This method aims to overcome the problem in existing technologies where the calculated velocity value is too large due to using the velocity of steel balls within the intermediate condensation layer to represent the overall steel ball group velocity, and provides a more accurate steel ball size optimization scheme. The centroid position of the mill's steel ball group is determined through numerical simulation, and the steel ball diameter is derived based on the steel ball velocity in the centroid layer.

[0007] The technical solution of this invention is: A method for optimizing steel ball size based on determining the impact velocity from the centroid of a group of steel balls inside a mill, the specific steps of which are as follows: (1) Determine the centroid of the steel ball group inside the mill Establish a ball mill model; The filling rate, number of steel balls, and critical rotation speed are calculated based on the mill parameters in production. The mill model is then input for simulation to obtain the centroid coordinates and calculate the distance Rc from the centroid to the mill center. Finally, the ratio λ of the distance to the mill center to the effective radius is calculated. (2) When a steel ball impacts ore, the critical condition for crushing is eEn≥Eb, that is... ; The theoretical formula for the diameter of a steel ball is derived from this: in D b The required steel ball diameter (m) for grinding a specific particle size; ψ The mill speed is %; l This represents the ratio of the distance from the centroid to the mill center to the effective radius; E b Let J be the ultimate fracture energy of the ore. k p For ore stiffness, GPa, k s The stiffness of the steel ball is 230 GPa. r e The effective density of the steel ball is kg / m³. 3 ; D 0 represents the effective diameter of the mill, in meters (m). g is the acceleration due to gravity, m / s² 2 ; π is the mathematical constant pi. The required steel ball diameter for grinding a specific particle size can be calculated by using the ratio λ of the distance from the mill center to the effective radius. D b .

[0008] The specific steps for determining the centroid of the steel ball group inside the mill in step (1) are as follows: ① Use SOLIDWORKS to create a 3D model of the ball mill and export the .stl file, then import it into EDEM; ② Calculate the filling rate, number of steel balls, and critical speed according to the mill parameters in production; set the filling rate, number of steel balls, speed, and direction of rotation in EDEM, where the speed is the product of the rotational speed and the critical speed; ③ Based on the material properties of the steel balls and liners during the grinding process, set the relevant material properties and contact parameters in the EDEM; ④ Set other simulation parameters: Set the time step to 10s and the cell size to 2Rmin~3Rmin; ⑤ Start the simulation, enter the Analyst tree, right-click Setup Selections, select Add Selections, select Grid Bin Group, click the newly created Grid Bin Group 01, scroll down to Grid Bin Group, record the coordinates of the Center, find Number of Bins, set the number of grids in the X direction to 50, the number of grids in the Y direction to 1, and the number of grids in the Z direction to 50; ⑥ After the simulation, the centroid coordinates are obtained as ( x 1, y 1), The center coordinates of the mill are ( x 2, y 2) Through calculation formula R c =(( y 1- y 2) 2 +( x 1- x 2)) 1 / 2 The distance from the center of mass to the center of the mill is obtained. R c Then through the calculation formula l = R c / R 0, R 0 represents the effective radius of the mill. R 0= D 0 / 2 gives the ratio λ of the distance to the mill center to the effective radius.

[0009] The ultimate fracture energy Eb of the ore in step (2) is obtained through the following steps: ① Take two types of identical mineral samples: one is a standard mechanical rock sample, used to obtain the basic physical and mechanical parameters such as Poisson's ratio and elastic modulus; the other is a mill feed mineral sample, used to determine the fracture energy of each particle size ore. ②Determine the Poisson's ratio μ and elastic modulus Y of a mineral sample and calculate the ore stiffness kp; The Poisson's ratio μ and elastic modulus Y of a mineral sample were determined by uniaxial compression test, and the ore stiffness kp was calculated by the following formula. ; ③ Determine the fracture energy of each particle size of another ore sample in the mill feed. E c ; Pressure tests were conducted on the ore of each particle size in the mill feed using a uniaxial pressure testing machine. The fracture energy was calculated by integrating the force-displacement curves from the tests. The calculation formula is as follows: In the formula: α For deformable variables, α c The deformation at the point of breakage; ④ Establish the relationship between fracture energy and fracture probability, and derive the ultimate fracture energy for each grain size. Given limited experimental sample data, the probability of ore particle fracture... P This can be calculated using probability estimation factors: In the formula, i is the rank of the fracture energy of a certain ore in the ascending order of the strength of all samples, and n is the number of samples. Plotting fracture energy on the x-axis and fracture probability on the y-axis, a Logistic model is used to fit the model, expressed as follows: In the formula, a and b are the model fitting parameters.

[0010] The fracture energy at which the fracture probability of this particle size is 95% is selected as the ultimate fracture energy of this particle size, and the ultimate fracture energy Eb of each particle size is obtained. ⑤ Establish the relationship between ultimate fracture energy and ore particle size and calculate the ultimate fracture energy of the particle size to be ground. The relationship between ore grain size and ultimate fracture energy is established, expressed as follows: In the formula: α and β are the model fitting parameters; Based on the particle size selected in step ④ and the ultimate fracture energy Eb corresponding to each particle size, the model fitting parameters α and β are obtained, and then the ultimate fracture energy of ore of any particle size can be accurately calculated. ⑥ Calculate the required steel ball diameter Db for each particle size in the mill feed. Where: Db is the diameter of the steel ball required to grind a specific particle size, in meters; ψ is the mill speed, % λ represents the ratio of the distance from the centroid to the mill center to the effective radius; Eb is the ultimate fracture energy of the ore, in J; kp is the stiffness of the ore, in GPa, and ks is the stiffness of the steel ball, in 230 GPa. ρe is the effective density of the steel ball, kg / m3; D0 is the effective diameter of the mill, in meters; g is the acceleration due to gravity, m / s²; π is the mathematical constant for a circle.

[0011] The standard mechanical rock sample collection process is as follows: collect 3 to 5 representative large blocks of ore from different mine pits and sections, select standard rock samples with a height-to-diameter ratio of 2:1 for uniaxial compressive strength tests, and obtain the basic physical and mechanical parameters such as Poisson's ratio and elastic modulus.

[0012] For the semi-autogenous mill, the fracture energy of the ore of each particle size was determined by selecting five particle sizes: -25+10mm, -35+25mm, -45+35mm, -60+45mm, and -80+60mm. The representative ore blocks of each particle size were of similar quality and numbered no less than 30.

[0013] For the ball mill, the fracture energy of the ore of each particle size was determined using five particle sizes: -6.7+4.75mm, -9.5+8mm, -12+10mm, -15+13.2mm, and -19+16mm. The representative ore blocks of each particle size were of similar quality and numbered no less than 30.

[0014] The working principle of this invention is: (1) No. k Derivation of the impact kinetic energy of a layered steel ball like Figure 2 As shown, the grinding steel balls inside the mill consist of several layers, each with a detachment point A. k and a landing point B k They satisfy the same geometric conditions. In the diagram, the outermost grinding steel balls of the spherical load undergo a throwing motion at point A0 with radius R0, and the throwing velocity is... v 0, landing point is B0; A k Indicates a radius of R k place k The drop point of the steel balls in the grinding mill, with a drop velocity of v. k The landing point is B. k A n The innermost grinding steel ball drop point has a drop velocity of [missing information]. v n The landing point is B. n .by k Taking grinding steel balls as an example, grinding steel balls in A k The angle of fall of the point in the projectile motion is α k At landing point B k The angle of return at that location isβ k .

[0015] k The steel ball falls back to point B k The normal impact velocity at that time is: (1) In the formula, ψ Indicates the mill rotational speed. l k express k The ratio of the distance from the steel balls to the center of the mill to the effective radius, i.e. l k = R k / R 0, R 0 represents the effective radius of the mill. R 0= D 0 / 2.

[0016] Select diameter as D b (m), mass is m (kg), density is r (kg / m 3 The study focuses on a single steel ball with a mass of: (2) k A steel ball from the layer falls back to point B. k The normal impact kinetic energy is: (3) Mining is often done wet. Considering that the slurry will absorb some of the kinetic energy of the steel balls, the effective density of the steel balls is taken. r e (kg / m 3 The calculation formula is: (4) In the formula, r t Density of ore, kg / m³ 3 , C The concentration is expressed as a percentage by weight of the slurry.

[0017] at this time, k The impact kinetic energy of the steel ball impacting the ore is: (5) (2) The velocity of the steel ball in the center of mass layer represents the velocity of the group of steel balls. The motion states of steel balls within the mill vary considerably. Using the velocity of the steel balls in the centroid layer as a representative velocity of the entire steel ball group provides a reasonable average value, accurately reflecting the overall behavior of the group. Therefore, using the velocity of the steel balls in the centroid layer for steel ball size derivation simplifies the calculation process, reduces errors, and provides a more precise steel ball diameter optimization scheme.

[0018] If the distance from the centroid to the mill center is R c The effective radius of the mill is R 0, the ratio of the distance from the centroid to the mill center to the effective radius of the mill. l = R c / R If 0, then the impact kinetic energy of the steel ball in the center of mass layer is: (6) (3) Derivation of the theoretical formula for the diameter of a steel ball When a steel ball impacts ore, the proportion of the energy carried by the steel ball that is distributed to the ore and can be used for fracture can be determined based on the elastic constant of the contacting object. e The estimate is given by Hertzian contact theory: (7) In the formula: k s For the stiffness of the steel ball, take 230 GPa. k p Let GPa be the ore stiffness, and the calculation formula be... (8) In the formula: Y It is Young's modulus. m It is Poisson's ratio.

[0019] When a steel ball impacts ore, the critical condition for crushing is: eE n ≥E b ,Right now (9) Substituting equation (7) into equation (9), we obtain the theoretical formula for the diameter of the steel ball: (10) In the formula: D b The required steel ball diameter (m) for grinding a specific particle size; ψ The mill speed is %; l This represents the ratio of the distance from the centroid to the mill center to the effective radius; E b Let J be the ultimate fracture energy of the ore. k p For ore stiffness, GPa, k s The stiffness of the steel ball is 230 GPa. r e The effective density of the steel ball is kg / m³. 3 ; D 0 represents the effective diameter of the mill, in meters (m). g is the acceleration due to gravity, m / s² 2 ; π is the mathematical constant for a circle.

[0020] The beneficial effects of this invention are: This invention addresses the problem of overestimating the calculated velocity values ​​caused by using the velocity of steel balls within the intermediate condensation layer to represent the velocity of the steel ball group. It provides a more accurate steel ball size optimization scheme. The centroid position of the mill's steel ball group is determined through numerical simulation, and the steel ball diameter is derived based on the steel ball velocity in the centroid layer. Attached Figure Description

[0021] Figure 1 This is a schematic diagram of the process of this invention; Figure 2 This is a diagram showing the trajectory of the grinding steel balls during their impact in the working principle of this invention. Figure 3 This is a diagram of the limiting fracture energy for each particle size in this invention; Figure 4 This is a graph showing the power function relationship between particle diameter and fracture energy in this invention. Detailed Implementation

[0022] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0023] A method for optimizing steel ball size based on determining the impact velocity from the centroid of a group of steel balls inside a mill, the specific steps of which are as follows: (1) Determine the centroid of the steel ball group inside the mill Establish a ball mill model; The filling rate, number of steel balls, and critical rotation speed are calculated based on the mill parameters in production. The mill model is then input for simulation to obtain the centroid coordinates and calculate the distance Rc from the centroid to the mill center. Finally, the ratio λ of the distance to the mill center to the effective radius is calculated. (2) When a steel ball impacts ore, the critical condition for crushing is eEn≥Eb, that is... ; The theoretical formula for the diameter of a steel ball is derived from this: , Where Db is the diameter of the steel ball required to grind a specific particle size, in meters; ψ is the mill speed, % λ represents the ratio of the distance from the centroid to the mill center to the effective radius; Eb is the ultimate fracture energy of the ore, in J; kp is the stiffness of the ore, in GPa, and ks is the stiffness of the steel ball, in 230 GPa. ρe is the effective density of the steel ball, kg / m3; D0 is the effective diameter of the mill, in meters; g is the acceleration due to gravity, m / s²; π is the mathematical constant pi. The required steel ball diameter Db for grinding a specific particle size can be calculated by using the ratio λ of the distance from the mill center to the effective radius.

[0024] The specific steps for determining the centroid of the steel ball group inside the mill in step (1) are as follows: ① Use SOLIDWORKS to create a 3D model of the ball mill and export the .stl file, then import it into EDEM; ② Calculate the filling rate, number of steel balls, and critical speed according to the mill parameters in production; set the filling rate, number of steel balls, speed, and direction of rotation in EDEM, where the speed is the product of the rotational speed and the critical speed; ③ Based on the material properties of the steel balls and liners during the grinding process, set the relevant material properties and contact parameters in the EDEM; ④ Set other simulation parameters: Set the time step to 10s and the cell size to 2Rmin~3Rmin; ⑤ Start the simulation, enter the Analyst tree, right-click Setup Selections, select Add Selections, select Grid Bin Group, click the newly created Grid Bin Group 01, scroll down to Grid Bin Group, record the coordinates of the Center, find Number of Bins, set the number of grids in the X direction to 50, the number of grids in the Y direction to 1, and the number of grids in the Z direction to 50; ⑥ After the simulation, the centroid coordinates are obtained as ( x 1, y 1), The center coordinates of the mill are ( x 2, y 2) Through calculation formula R c =(( y 1- y 2) 2 +( x 1- x 2)) 1 / 2 The distance from the center of mass to the center of the mill is obtained. R c Then through the calculation formula l = R c / R 0, R 0 represents the effective radius of the mill. R 0= D 0 / 2 gives the ratio λ of the distance to the mill center to the effective radius.

[0025] The ultimate fracture energy Eb of the ore in step (2) is obtained through the following steps: ① Take two types of identical mineral samples: one is a standard mechanical rock sample, used to obtain the basic physical and mechanical parameters such as Poisson's ratio and elastic modulus; the other is a mill feed mineral sample, used to determine the fracture energy of each particle size ore. ②Determine the Poisson's ratio μ and elastic modulus Y of a mineral sample and calculate the ore stiffness kp; The Poisson's ratio μ and elastic modulus Y of a mineral sample were determined by uniaxial compression test, and the ore stiffness kp was calculated by the following formula. ; ③ Determine the fracture energy Ec of each particle size of another mineral sample in the mill feed; Pressure tests were conducted on the ore of each particle size in the mill feed using a uniaxial pressure testing machine. The fracture energy was calculated by integrating the force-displacement curves from the tests. The calculation formula is as follows: In the formula: α is the deformation, and αc is the deformation at the point of breakage; ④ Establish the relationship between fracture energy and fracture probability, and derive the ultimate fracture energy for each grain size. For limited experimental sample data, the fracture probability P of ore particles can be calculated using a probability estimation factor: In the formula, i is the rank of the fracture energy of a certain ore in the ascending order of the strength of all samples, and n is the number of samples. Plotting fracture energy on the x-axis and fracture probability on the y-axis, a Logistic model is used to fit the model, expressed as follows: In the formula, a and b are the model fitting parameters.

[0026] The fracture energy at which the fracture probability of this particle size is 95% is selected as the ultimate fracture energy of this particle size, and the ultimate fracture energy Eb of each particle size is obtained. ⑤ Establish the relationship between ultimate fracture energy and ore particle size and calculate the ultimate fracture energy of the particle size to be ground. The relationship between ore grain size and ultimate fracture energy is established, expressed as follows: In the formula: α and β are the model fitting parameters; Based on the particle size selected in step ④ and the ultimate fracture energy Eb corresponding to each particle size, the model fitting parameters α and β are obtained, and then the ultimate fracture energy of ore of any particle size can be accurately calculated. ⑥ Calculate the required steel ball diameter Db for each particle size in the mill feed. Where: Db is the diameter of the steel ball required to grind a specific particle size, in meters; ψ is the mill speed, % λ represents the ratio of the distance from the centroid to the mill center to the effective radius; Eb is the ultimate fracture energy of the ore, in J; kp is the stiffness of the ore, in GPa, and ks is the stiffness of the steel ball, in 230 GPa. ρe is the effective density of the steel ball, kg / m3; D0 is the effective diameter of the mill, in meters; g is the acceleration due to gravity, m / s²; π is the mathematical constant for a circle.

[0027] The standard mechanical rock sample collection process is as follows: collect 3 to 5 representative large blocks of ore from different mine pits and sections, select standard rock samples with a height-to-diameter ratio of 2:1 for uniaxial compressive strength tests, and obtain the basic physical and mechanical parameters such as Poisson's ratio and elastic modulus.

[0028] For the semi-autogenous mill, the fracture energy of the ore of each particle size was determined by selecting five particle sizes: -25+10mm, -35+25mm, -45+35mm, -60+45mm, and -80+60mm. The representative ore blocks of each particle size were of similar quality and numbered no less than 30.

[0029] For the ball mill, the fracture energy of the ore of each particle size was determined using five particle sizes: -6.7+4.75mm, -9.5+8mm, -12+10mm, -15+13.2mm, and -19+16mm. The representative ore blocks of each particle size were of similar quality and numbered no less than 30.

[0030] Example: Calculations were performed using a Φ3.2×4.5 m ball mill. (1) Determine the centroid of the steel ball group inside the mill ① Use SOLIDWORKS to create a 3D model of the ball mill and export the .stl file. Calculate the filling rate, number of steel balls, and critical speed according to the mill parameters in production. The filling rate of the Φ3.2×4.5 ball mill is 40%, the speed rate is 77%, and the critical speed is 23.19 rpm.

[0031] ②Based on the material properties of the steel balls and liners during the grinding process, set the relevant material properties and contact parameters; ③ Import the ball mill .stl model file, set the material properties, rotation speed and rotation direction. The rotation speed is the product of the rotational speed and the critical speed, and the rotation speed is 17.86 rpm.

[0032] ④ Create a pellet factory to produce steel balls; ⑤ In the simulation settings, set the time step to 10s and the cell size to 2Rmin~3Rmin; ⑥ Start the simulation. After completion, go to the Analyst tree, right-click Setup Selections, select Add Selections, select Grid Bin Group, click the newly created Grid Bin Group 01, scroll down to Grid Bin Group, record the center coordinates, find Number of Bins, and set the number of grids in the X direction to 50, the number of grids in the Y direction to 1, and the number of grids in the Z direction to 50. Record the center coordinates as (1590, 1590).

[0033] ⑦ Click File, select Export, select Hyper Mash Data. After entering the window, set StartTimestep to 5s and End Timestep to 10s. Click Material Centre of Mass Export. In Selections, select Grid Bin Group 01, set Filename to COM, and click Export.

[0034] ⑧ The coordinates of the centroid can be obtained from the exported COM.csv file. From these coordinates, the distance from the centroid to the mill center can be calculated. R c Further calculations were performed to determine the ratio of the distance from the centroid to the mill center to the effective radius. l The coordinates of the centroid are obtained as (1047.34, 1115.26), and the calculation is... R c The effective radius of the mill is 721.01 mm. R 0 = 1550mm, then l It is 0.47.

[0035] (2) Select representative mineral samples Select 40 ores from each of five similar particle sizes: -6.7+4.75mm, -9.5+8mm, -12+10mm, -15+13.2mm, and -19+16mm.

[0036] (3) Determine the Poisson's ratio of standard rock samples m and elastic modulus Y (GPa) and calculate the ore stiffness. k p (GPa) The Poisson's ratio of standard rock samples was determined by uniaxial compression test. m and elastic modulus Y， The measurement results are shown in Table 1.

[0037]

[0038] The ore stiffness is calculated using the following formula. k p .

[0039] Ore stiffness k p =91.0 / (1-0.31 2 =100.67 GPa (4) Determine the fracture energy of each particle size of ore in the mill feed. E c A uniaxial pressure testing machine was used to conduct pressure tests on five different ore sizes in the mill feed, and their fracture energy was calculated by integrating the test force-displacement curves.

[0040] (5) Establish the relationship between fracture energy and fracture probability and derive the ultimate fracture energy for each grain size. Given limited experimental sample data, the probability of ore particle fracture... P This can be calculated using probability estimation factors: In the formula, i It is the ranking of the fracture energy of a certain ore among all the test specimens arranged in ascending order of strength. n This refers to the number of samples.

[0041] Plotting fracture energy on the x-axis and fracture probability on the y-axis, a Logistic model is used to fit the model, expressed as follows: like Figure 3 As shown, the ultimate fracture energies for the five particle sizes -6.7+4.75mm, -9.5+8mm, -12+10mm, -15+13.2mm and -19+16mm are 0.54 J, 1.21 J, 2.69 J, 4.83 J and 8.26 J, respectively.

[0042] (5) Establish the relationship between the ultimate fracture energy and the ore particle size and calculate the ultimate fracture energy of the particle size to be ground. The fracture energy of brittle materials exhibits a power-law relationship with particle diameter. By deriving data, the relationship between ore particle size and ultimate fracture energy can be established, expressed as follows: In the formula: α and β The model fitting parameters are used, and the ultimate fracture energy of ore of any particle size can be accurately calculated using this formula.

[0043] Establish the relationship between particle diameter and fracture energy as follows: Figure 4 As shown, the power function relationship between particle diameter and fracture energy is obtained as follows: E b =0.0035 d 2.7535Therefore, the ultimate fracture energy of any particle size can be calculated.

[0044] (6) Calculate the diameter of the mill steel balls D b The mill has a diameter of 3.2 m and a rotational speed of [missing information]. ψ It has a content of 77% and an ore density of 4000 kg / m³. 3 The density of the steel ball is 7800 kg / m³. 3 The grinding concentration is 75%, and the effective density is calculated using equation (4). r e 5514 kg / m 3 , l The ore stiffness is 0.47. k p The value is 100.67. Substitute the diameter of each particle size into the formula. E b =0.0035 d 2.7535 The ultimate fracture energy of each particle size can be calculated. Substituting the above parameters into formula (10), the required steel ball diameter for each particle size of ore can be calculated. Specific data are listed in Table 2.

[0045]

[0046] (7) Design an experiment to verify. The proportion of steel balls was determined based on the mass ratio of each particle size in the mill feed, resulting in a steel ball gradation of Φ90:Φ70:Φ50:Φ40 = 30:20:30:20. Whether this ball ratio is optimal needs to be verified through grinding tests. To conduct thorough comparisons, several other ball ratios were selected for testing. The initial ball ratio on-site was Φ100:Φ80:Φ60 = 30:40:30. A further ratio, slightly larger than the recommended Φ100:Φ80:Φ60:Φ40 = 30:20:30:20, and slightly smaller than the recommended Φ80:Φ60:Φ50:Φ40 = 30:20:30:20, were then proposed. Comparative tests were conducted using these four steel ball gradations. The comparative test was conducted in a laboratory discontinuous ball mill with dimensions D×L450×450mm. The grinding samples were taken from the ball mill belt, with each sample weighing 12kg, a filling rate of 40%, and a ball loading of 104kg. The grinding time for each test was set to 1 hour and 38 minutes to ensure that the fineness of the grinding product (-0.074mm) was controlled at approximately 78-82% (comparable to the fineness required for grinding on-site). The mill speed was also set to 75%, which is comparable to the on-site speed. The test results are shown in Table 3.

[0047]

[0048] As shown in Table 3, the recommended ratio of -0.074mm has the highest yield, which is 5.88 percentage points higher than the on-site ratio. The intermediate optional ratio of -0.15+0.010mm has the highest yield among all ratios.

[0049] Based on the above steps, the ball gradation for a Φ3.2×4.5m ball mill was determined to be: Φ90:Φ70:Φ50:Φ40=30:20:30:20. The accuracy of the invention was verified through experiments.

[0050] The specific embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of the present invention.

Claims

1. A method for optimizing steel ball size based on determining the impact velocity using the centroid of a group of steel balls within a mill, characterized in that... The specific steps are as follows: (1) Determine the centroid of the steel ball group inside the mill Establish a ball mill model; The filling rate, number of steel balls, and critical rotation speed are calculated based on the mill parameters in production. The mill model is then input for simulation to obtain the centroid coordinates and calculate the distance Rc from the centroid to the mill center. Finally, the ratio λ of the distance to the mill center to the effective radius is calculated. (2) When a steel ball impacts ore, the critical condition for crushing is: eE n ≥E b ,Right now ; The theoretical formula for the diameter of a steel ball is derived from this: in D b The required steel ball diameter (m) for grinding a specific particle size; ψ The mill speed is %; λ This represents the ratio of the distance from the centroid to the mill center to the effective radius; E b Let J be the ultimate fracture energy of the ore. k p For ore stiffness, GPa, k s The stiffness of the steel ball is 230 GPa. ρ e The effective density of the steel ball is kg / m³. 3 ; D 0 represents the effective diameter of the mill, in meters (m). g is the acceleration due to gravity, m / s² 2 ; π is the mathematical constant pi. The required steel ball diameter for grinding a specific particle size can be calculated by using the ratio λ of the distance from the mill center to the effective radius. D b .

2. The method for optimizing steel ball size based on determining the impact velocity using the centroid of a group of steel balls within a mill, as described in claim 1, is characterized in that: The specific steps for determining the centroid of the steel ball group inside the mill in step (1) are as follows: ① Create a 3D model of the ball mill and import it into EDEM; ② Calculate the filling rate, number of steel balls, and critical speed according to the mill parameters in production; set the filling rate, number of steel balls, speed, and direction of rotation in EDEM, where the speed is the product of the rotational speed and the critical speed; ③ Based on the material properties of the steel balls and liners during the grinding process, set the relevant material properties and contact parameters in the EDEM; ④ Set other simulation parameters; ⑤ Start the simulation; ⑥ After the simulation, the centroid coordinates are obtained as ( x 1, y 1), The center coordinates of the mill are ( x 2, y 2) Through calculation formula R c =(( y 1- y 2) 2 +( x 1- x 2)) 1 / 2 The distance from the center of mass to the center of the mill is obtained. R c Then through the calculation formula λ = R c / R 0, R 0 represents the effective radius of the mill. R 0= D 0 / 2 gives the ratio λ of the distance to the mill center to the effective radius.

3. The method for optimizing steel ball size based on determining the impact velocity according to the centroid of the steel ball group in the mill, as described in claim 1, is characterized in that: The ultimate fracture energy of the ore in step (2) E b Obtained through the following steps: ① Take two types of identical mineral samples: one is a standard mechanical rock sample, used to obtain the basic physical and mechanical parameters such as Poisson's ratio and elastic modulus; the other is a mill feed mineral sample, used to determine the fracture energy of each particle size ore. ② Determine the Poisson's ratio of a mineral sample. μ and elastic modulus Y And calculate the stiffness of the ore. k p ; The Poisson's ratio of a mineral sample was determined by uniaxial compression test. μ and elastic modulus Y， The ore stiffness is calculated using the following formula. k p . ; ③ Determine the fracture energy of each particle size of another ore sample in the mill feed. E c ; Pressure tests were conducted on the ore of each particle size in the mill feed using a uniaxial pressure testing machine. The fracture energy was calculated by integrating the force-displacement curves from the tests. The calculation formula is as follows: In the formula: α For deformable variables, α c The deformation at the point of breakage; ④ Establish the relationship between fracture energy and fracture probability, and derive the ultimate fracture energy for each grain size. Given limited experimental sample data, the probability of ore particle fracture... P This can be calculated using probability estimation factors: In the formula, i It is the ranking of the fracture energy of a certain ore among all the test specimens arranged in ascending order of strength. n It is the number of samples; Plotting fracture energy on the x-axis and fracture probability on the y-axis, a Logistic model is used to fit the model, expressed as follows: In the formula, a and b are model fitting parameters. The fracture energy at which the fracture probability of this grain size is 95% is selected as the ultimate fracture energy of this grain size, and the ultimate fracture energy of each grain size is obtained. E b ; ⑤ Establish the relationship between ultimate fracture energy and ore particle size and calculate the ultimate fracture energy of the particle size to be ground. The relationship between ore grain size and ultimate fracture energy is established, expressed as follows: In the formula: α and β are the model fitting parameters; Based on the particle size selected in step ④ and the corresponding ultimate fracture energy of each particle size. E b By finding the model fitting parameters α and β, the ultimate fracture energy of ore of any particle size can be accurately calculated. ⑥ Calculate the required steel ball diameter for each particle size in the mill feed. D b In the formula: D b The required steel ball diameter (m) for grinding a specific particle size; ψ The mill speed is %; λ This represents the ratio of the distance from the centroid to the mill center to the effective radius; E b Let J be the ultimate fracture energy of the ore. k p For ore stiffness, GPa, k s The stiffness of the steel ball is 230 GPa. ρ e The effective density of the steel ball is kg / m³. 3 ; D 0 represents the effective diameter of the mill, in meters (m). g is the acceleration due to gravity, m / s² 2 ; π is the mathematical constant for a circle.

4. The method for optimizing steel ball size based on determining the impact velocity using the centroid of the steel ball group within the mill, as described in claim 3, is characterized in that: The standard mechanical rock sample collection process is as follows: collect 3 to 5 representative large blocks of ore from different mine pits and sections, select standard rock samples with a height-to-diameter ratio of 2:1 for uniaxial compressive strength tests, and obtain the basic physical and mechanical parameters such as Poisson's ratio and elastic modulus.

5. The method for optimizing steel ball size based on determining the impact velocity according to the centroid of the steel ball group in the mill, as described in claim 3, is characterized in that: For the semi-autogenous mill, the fracture energy of the ore of each particle size was determined by selecting five particle sizes: -25+10mm, -35+25mm, -45+35mm, -60+45mm, and -80+60mm. The representative ore blocks of each particle size were of similar quality and numbered no less than 30.

6. The method for optimizing steel ball size based on determining the impact velocity according to the centroid of the steel ball group in the mill, as described in claim 3, is characterized in that: For the ball mill, the fracture energy of the ore of each particle size was determined using five particle sizes: -6.7+4.75mm, -9.5+8mm, -12+10mm, -15+13.2mm, and -19+16mm. The representative ore blocks of each particle size were of similar quality and numbered no less than 30.

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