Method of calculating ball / sag mill steel ball diameters

Abstract

The present application relates to a kind of methods for calculating the diameter of ball / semi-autogenous mill steel ball, belong to the technical field of grinding in beneficiation.The same two kinds of ore samples are selected, one is standard mechanical rock sample, the basic physical mechanics parameters of its Poisson's ratio, elastic modulus are obtained, the other is mill feed ore sample, the fracture energy of mill feed each size fraction ore is obtained;The Poisson's ratio μ of standard rock sample and elastic modulus Y (GPa) are determined and the stiffness k of ore is calculated p (GPa);The fracture energy E c of each size fraction ore in mill feed is determined;The fracture energy of the size fraction when the fracture probability is 95% is selected as the limit fracture energy of the size fraction, and the limit fracture energy E b实验 of 5 size fractions is obtained;The relationship between limit fracture energy and ore particle size is established and the limit fracture energy of the grinding size fraction is calculated;The limit fracture energy of the grinding size fraction is brought into the theoretical formula of ball diameter, and the theoretical diameter of the steel ball required by the mill is calculated.The present application can calculate the accurate ball diameter required by ball / semi-autogenous mill;The present application has the advantages of convenient and fast, simple calculation.

Description

TECHNICAL FIELD

[0001] The application relates to a method for calculating the diameter of a ball / semi-autogenous mill steel ball, and belongs to the technical field of grinding in mineral processing. BACKGROUND

[0002] Grinding is an energy-intensive operation, and the operation of a ball / semi-autogenous mill is accompanied by a large amount of energy consumption, which is a heavy disaster area of energy saving and emission reduction in mines. Therefore, even a small improvement can bring great benefits to industrial production. For a semi-autogenous mill, although the semi-autogenous mill replaces the conventional medium and fine crushing and coarse grinding, the grinding energy consumption of the semi-autogenous mill is slightly higher than that of the conventional three-stage closed circuit. For a ball mill, on the one hand, a large amount of energy is consumed for grinding, and secondly, the product particle size composition of grinding will restrict the subsequent separation. Therefore, grinding quality improvement and energy consumption reduction have become a technical problem for the concentrator.

[0003] The crushing of brittle particles is a complex process, and the result is affected by factors such as loading rate, particle strength, shape, particle size and internal structural defects. The nature of the failure mechanism is determined by the material properties of the particle material and the nature of the stress field around and inside the individual particle. It is generally believed that the deformation of the particle in the initial stage can be described by the Hertz model. When the brittle cracks inside or on the surface of the particle meet the failure criterion, the crack will rapidly expand in an unstable manner and may or may not branch. When the crack or cracks appear from the particle, several particle fragments are formed. The number and size of the fragments depend on the size and location of the initial defects and the extent of crack branching. That is, for a semi-autogenous / ball mill, the grinding medium is the implement of crushing, which determines whether and how crushing occurs, and determines the grinding energy consumption and product particle size distribution of the mill.

[0004] There are many factors affecting the grinding effect and energy consumption, including the mechanical properties and particle size of the ore, the specifications of the mill, the shape and parameters of the liner, the diameter and ratio of the steel ball, the grinding concentration, the steel ball filling rate and the ratio of material to ball. For the site conditions, the diameter and ratio of the steel ball are often the most easily adjusted and the most obvious factors. The diameter of the steel ball is affected by the strength and particle size of the ore, the diameter of the mill, the rotational speed, the filling rate, the grinding concentration and the liner parameters. Foreign commonly used formulas for calculating the diameter of the steel ball include the F.C.Bond formula, the Rowland (Allis-Chalmers) formula and the Azzaroni formula. These three formulas are all based on the Bond work index for calculation, and consider parameters such as the diameter of the mill, the density of the ore, the feed size and the rotational speed of the mill. Professor Duan Xixiang in China defines the strength of the ore through the fracture stress of a standard mechanical test piece, calculates the kinetic energy of the steel ball using the Davis steel ball motion theory, and deduces the ball diameter half theoretical formula by comprehensively considering many factors. At present, there is no clear theoretical formula to calculate the diameter of the steel ball of the mill. SUMMARY

[0005] In view of the problems and deficiencies of the prior art, the present application provides a method for calculating the diameter of steel balls of a ball / semi-autogenous mill. The present application measures the fracture energy of ore (the fracture energy required for breaking when the loading rate of ore is consistent with the normal rate of steel balls impacting ore in the mill) by using an ultrafast load cell (UFLC), and theoretically derives a theoretical formula of the steel balls required by the ball / semi-autogenous mill, so as to calculate the accurate ball diameter required by the ball / semi-autogenous mill. The present application is realized by the following technical scheme.

[0006] A method for calculating the diameter of steel balls of a ball / semi-autogenous mill, comprising the following steps:

[0007] Step 1: selecting representative ore samples

[0008] Two kinds of ore samples are selected, one is a standard mechanical rock sample to obtain the basic physical and mechanical parameters of Poisson's ratio and elastic modulus, and the other is a mill feed ore sample to obtain the fracture energy of each particle size of the mill feed ore;

[0009] Step 2: measuring the Poisson's ratio μ and elastic modulus Y (GPa) of the standard rock sample and calculating the stiffness k of the ore p (GPa) of the ore;

[0010] The Poisson's ratio μ and elastic modulus Y of the standard rock sample are measured by uniaxial compression test, and the stiffness k of the ore is calculated by the following formula p :

[0011] k p =Y / (1-μ 2 );

[0012] Step 3: measuring the fracture energy E of each particle size of the mill feed ore c

[0013] The ultrafast load cell (UFLC) is used to perform impact test on five particle sizes of the mill feed ore, and the fracture energy is calculated by integrating the force-displacement curve of the test, and the calculation formula is as follows:

[0014]

[0015] In the formula, α is the deformation variable, α c is the deformation variable when breaking, and F is the force;

[0016] Step 4: establishing the relationship between fracture energy and fracture probability and obtaining the limit fracture energy of each particle size

[0017] For limited test sample data, the fracture probability P of ore particles can be calculated by a probability estimation factor:

[0018]

[0019] Where i is the ranking 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;

[0020] The fracture probability is taken as the ordinate, and the fracture energy is taken as the abscissa. The Logistic model is used to fit the data, and the expression is

[0021]

[0022] Where a and b are the model fitting parameters;

[0023] The fracture energy of the particle size when the fracture probability is 95% is selected as the limit fracture energy of the particle size, and the limit fracture energy E of the five particle sizes is obtained b实验 , J;

[0024] Step 5: Establish the relationship between the limit fracture energy and the particle size of the ore and calculate the limit fracture energy of the particle size to be ground

[0025] The fracture energy of brittle materials has a power function relationship with the particle diameter. The relationship between the limit fracture energy and the particle size of the ore is established by Step 4, and the expression is

[0026] E b =α1d β

[0027] Where: α1 and β are model fitting parameters, and α2 and α3 are model fitting parameters, and v n is the normal velocity of the steel ball impacting the ore in the mill, The limit fracture energy of any particle size of the ore can be accurately calculated by the formula;

[0028] Step 6: Calculate the diameter D b

[0029] The limit fracture energy of the particle size to be ground is brought into the theoretical formula of the ball diameter to calculate the theoretical diameter of the steel ball required by the mill, and the formula is

[0030]

[0031] Where: D b is the diameter of the steel ball required for grinding a certain particle size, m;

[0032] ψ is the rotational speed of the mill, %;

[0033] E b is the limit fracture energy of the ore, J;

[0034] k p is the stiffness of the ore, GPa, and k s is the stiffness of the steel ball, which is 230 GPa;

[0035] p e ρ is the effective density of the steel ball, kg / m 3 ;

[0036] D m is the diameter of the "ball load intermediate condensation layer", m;

[0037] g is the acceleration of gravity, m / s 2 .

[0038] Each same ore sample in Step 1 is 6-8 representative large ore blocks taken from different ore pits and ore sections of the mining site, and a uniaxial compressive strength test is performed on the ore blocks with a standard rock sample height-diameter ratio of 2:1 to obtain the Poisson's ratio and elastic modulus basic physical and mechanical parameters.

[0039] When calculating the diameter of the steel ball of the semi-autogenous mill, the ore of each particle size of the mill feed obtained in Step 1 is -25+10 mm, -35+25 mm, -45+35 mm, -60+45 mm and -80+60 mm, respectively.

[0040] When calculating the diameter of the steel ball of the ball mill, the ore of each particle size of the mill feed obtained in Step 1 is -2+1 mm, -5+2 mm, -8+5 mm, -10+8 mm and -12+10 mm, respectively.

[0041] The specific process of Step 5 for establishing the relationship between the limit fracture energy and the ore particle size and calculating the limit fracture energy of the to-be-ground particle size is as follows:

[0042] Step 5.1 A fitting relationship is established between the limit fracture energy E b实验 of the five particle sizes obtained in Step 4 and the particle diameter d:

[0043] E b实验 = α2d β , wherein α2 and β are fitting parameters, and d is the particle diameter;

[0044] Step 5.2 A fitting relationship is established between the limit fracture energy and the loading rate:

[0045] wherein α3 and β1 are fitting parameters, v n is the normal velocity of the steel ball impacting the ore in the mill,

[0046] That is

[0047] Step 5.3 When the expression in Step 5.1 is substituted into Step 5.2, the relationship between the limit fracture energy and the ore particle size is obtained: E b = α1dβ

[0048] wherein: a1 and b are model fitting parameters, wherein a2, a3 are model fitting parameters, v n is the normal velocity of the steel ball impacting the ore in the mill,

[0049] The working principle of the present application is:

[0050] (1) Derivation of kinetic energy of grinding medium

[0051] A single medium with a diameter of D b (m), a mass of m (kg), and a density of p (kg / m 3 ) is selected for study. Taking a steel ball as an example, the mass of a single steel ball is

[0052]

[0053] Let the effective radius of the mill be R1 (m) and the diameter be D (m). According to the Davis steel ball motion theory, the normal velocity of the outermost steel ball when it falls back is

[0054]

[0055] The impact kinetic energy of the steel ball on the cylinder liner is

[0056]

[0057] Grinding is usually wet grinding. Considering that the ore slurry will absorb part of the kinetic energy of the steel ball, the effective density of the steel ball is taken as p e (kg / m 3 ), and the calculation formula is

[0058]

[0059] wherein p t is the ore density, kg / m 3 , and C is the weight percentage concentration of the ore slurry, %.

[0060] At this time, the kinetic energy of the steel ball impacting the ore is

[0061]

[0062] The above study is based on the outermost ball, but the outermost ball cannot represent the entire ball load in the mill. It is assumed that the mass of the entire ball load is concentrated in a certain ball layer, i.e., the "middle condensation layer", and the diameter of the medium in this layer to the center of the mill is set as D m , then:

[0063]

[0064] R1, R2 are the radii of the medium layers which are the outermost and innermost layers respectively; k = R2 / R1, k is related to the rotation rate ψ and the filling rate φ of the semi-autogenous mill Table 1 Values of parameter k for various filling rates φ and rotation rates ψ

[0065] Table 1 Values of parameter k for various filling rates φ and rotation rates ψ

[0066]

[0067] Note: When calculating the diameter of the steel ball in the ball mill, the steel ball filling rate is taken; when calculating the diameter of the steel ball in the semi-autogenous mill, the total filling rate is taken.

[0068] The angle of separation α is related to the rotation rate ψ, cos α = ψ 2 Then:

[0069]

[0070] (2) Steel ball diameter theoretical formula derivation

[0071] Research shows that the loading rate will affect the crack propagation behavior of the particles, and the particle fracture energy is related to the loading rate. Tavares found that brittle materials exhibit higher particle strength at higher rates, and his experimental data show that the particle fracture energy increases with the increase of the loading rate.

[0072] If the normal velocity of the steel ball impacting the ore in the mill is v n (m / s), the loading rate is v n , the energy required to break the ore with a diameter of d (m) is E b .

[0073] When the steel ball impacts the ore, the proportion of the energy carried by the steel ball that is allocated to the ore and can be used for breaking can be estimated according to the elastic constant e of the contact object, which is given by the Hertz contact theory:

[0074]

[0075] In the formula: k s is the stiffness of the steel ball, which is 230 GPa, k p is the stiffness of the ore, GPa, and the calculation formula is

[0076] k p = Y / (1-μ 2 )(9)

[0077] In the formula: Y is the Young's modulus, and μ is the Poisson's ratio.

[0078] When the steel ball impacts the ore, the critical condition for breaking is eE n ≥ E b , that is

[0079]

[0080] Put formula (8) into formula (10), and arrange to obtain a theoretical formula of the steel ball diameter:

[0081]

[0082] In the formula, D b is the steel ball diameter required for grinding a specific particle size, m;

[0083] ψ is the grinding mill rotation rate, %;

[0084] E b is the fracture energy required for breaking when the ore loading rate is consistent with the normal rate of the steel ball impacting the ore in the mill, J;

[0085] k p is the stiffness of the ore, GPa, k s is the stiffness of the steel ball, which is 230 GPa;

[0086] ρ e is the effective density of the steel ball, kg / m 3 ;

[0087] D m is the diameter of the "intermediate condensation layer", m;

[0088] g is the acceleration of gravity, m / s 2 ;

[0089] π is the circular constant.

[0090] The beneficial effects of the present application are:

[0091] The present application can calculate the accurate ball diameter required for the ball / semi-autogenous mill; the present application has the advantages of being convenient, fast, and simple to calculate. BRIEF DESCRIPTION OF DRAWINGS

[0092] Figure 1 is a flowchart of the present application;

[0093] Figure 2 is a graph of the limit fracture energy of different particle sizes measured by the ultrafast load cell (UFLC) when the present application is loaded statically;

[0094] Figure 3 is a fitting graph of the limit fracture energy of different particle sizes and the particle diameter of the present application;

[0095] Figure 4 is a fitting graph of the fracture energy of the ore changing with the loading rate of the present application. DETAILED DESCRIPTION

[0096] The application will be further described in connection with the accompanying drawings and specific embodiments.

[0097] Embodiment 1

[0098] A method for calculating the diameter of a ball / semi-autogenous mill steel ball, the steps of which include:

[0099] Step 1: Select a representative ore sample

[0100] Two identical ore samples are selected, one of which is a standard mechanical rock sample to obtain its Poisson's ratio, elastic modulus and basic physical and mechanical parameters, and the other is a mill feed ore sample to obtain the fracture energy of each particle size of the mill feed ore;

[0101] Step 2: Measure the Poisson's ratio μ and elastic modulus Y (GPa) of the standard rock sample and calculate the stiffness k of the ore p (GPa) of the ore;

[0102] The Poisson's ratio μ and elastic modulus Y of the standard rock sample are measured by uniaxial compression test, and the stiffness k of the ore is calculated by the following formula: p

[0103] k p =Y / (1-μ 2 );

[0104] Step 3: Measure the fracture energy E of each particle size of the mill feed ore c

[0105] An ultra-fast weighing sensor (UFLC) is used to perform impact tests on five particle sizes of the mill feed ore, and the fracture energy is calculated by integrating the force-displacement curve of the test, and the calculation formula is as follows:

[0106]

[0107] In the formula, α is the deformation variable, α c is the deformation variable when breaking, and F is the force;

[0108] Step 4: Establish the relationship between fracture energy and fracture probability and obtain the limit fracture energy of each particle size

[0109] For limited test sample data, the fracture probability P of the ore particle can be calculated by a probability estimation factor:

[0110]

[0111] In the formula, i is the ranking of the fracture energy of a certain ore in the ascending order of all sample strengths, and n is the number of samples;

[0112] Taking the fracture energy as the horizontal coordinate and the fracture probability as the vertical coordinate, the Logistic model is used to fit it, and the expression is​

[0113]

[0114] where a and b are model fitting parameters;

[0115] The fracture energy of the particle size fraction selected at 95% fracture probability is the ultimate fracture energy of the particle size fraction, and the ultimate fracture energies of the five particle sizes are obtained b实验 , J;

[0116] Step 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

[0117] The fracture energy of brittle materials has a power function relationship with the particle diameter, and the relationship between the ore particle size and the ultimate fracture energy is established by the data obtained in Step 4, and the expression is

[0118] E b = α1d β

[0119] where: α1 and β are model fitting parameters, and α2 and α3 are model fitting parameters, and v n is the normal velocity of the steel ball impacting the ore in the mill, The ultimate fracture energy of any particle size of ore can be accurately calculated by this formula;

[0120] Step 6: Calculate the diameter D b

[0121] The ultimate fracture energy of the particle size to be ground is brought into the theoretical formula of the ball diameter to calculate the theoretical diameter of the steel ball required by the mill, and the formula is

[0122]

[0123] where: D b is the diameter of the steel ball required for grinding a specific particle size, m;

[0124] ψ is the mill speed rate, %;

[0125] E b is the ultimate fracture energy of the ore, J;

[0126] k p is the stiffness of the ore, GPa, and k s is the stiffness of the steel ball, taken as 230 GPa;

[0127] ρ e is the effective density of the steel ball, kg / m 3 ;

[0128] D m is the diameter of the "ball load intermediate condensation layer", m;

[0129] g is the acceleration of gravity, m / s 2 .

[0130] Example: Φ6.7x3.4m SAG mill is calculated

[0131] (1) Select representative ore samples

[0132] Select 30 ore samples of each of 5 particle sizes of -25+10mm, -35+25mm, -45+35mm, -60+45mm and -80+60mm with similar mass.

[0133] (2) Determine the Poisson's ratio μ and elastic modulus Y (GPa) of the standard rock sample and calculate the ore stiffness k p (GPa)

[0134] Determine the Poisson's ratio μ and elastic modulus Y of the standard rock sample by uniaxial compression test, and the test results are shown in Table 2.

[0135] Table 2 Test results of sample mechanical properties

[0136]

[0137] And calculate the ore stiffness k by the following formula p .

[0138] k p = Y / (1-μ 2 )

[0139] Ore stiffness k p = 52.7 / (1-0.17 2 ) = 54.27 GPa

[0140] (3) Determine the fracture energy E of each particle size of the ore in the mill feed c

[0141] In principle, an ultrafast load cell (UFLC) is required to conduct impact tests on the 5 particle sizes of the mill feed respectively, but when the laboratory conditions are limited, a uniaxial pressure testing machine can be used to determine the fracture energy of the ore under static loading, and a model of the change of fracture energy with loading rate is established, so as to calculate the fracture energy of the ore at the corresponding loading rate.

[0142] (4) Establish the relationship between fracture energy and fracture probability and obtain the limit fracture energy of each particle size

[0143] For limited test sample data, the fracture probability P of the ore particles can be calculated by a probability estimation factor:

[0144]

[0145] 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.

[0146] 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:

[0147]

[0148] The fitting results are shown below. Figure 2 ,Depend on Figure 2 It can be seen that the ultimate fracture energies of the five particle sizes -25+10mm, -35+25mm, -45+35mm, -60+45mm and -80+60mm under static loading are 1.97J, 3.72J, 4.96J, 10.56J and 13.29J, respectively.

[0149] (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.

[0150] The fracture energy of brittle materials exhibits a power-law relationship with particle diameter. Step 4 establishes the relationship between ore particle size and ultimate fracture energy, expressed as follows:

[0151] E b =α1d β

[0152] In the formula: α1 and β are the model fitting parameters, where α2 and α3 are model fitting parameters, v n The normal velocity of the steel balls impacting the ore inside the mill. This formula can be used to accurately calculate the ultimate fracture energy of ores of any particle size.

[0153] Establish the relationship between particle diameter and fracture energy as follows: Figure 3 As shown, the power function relationship between particle diameter and fracture energy is obtained as E. b实验 =0.02637d 1.473 Therefore, the fracture energy of any particle size under static loading can be calculated.

[0154] Tavares obtained data on how the fracture energy of ore changes with the loading rate through experiments, such as... Figure 4 The relationship between the fracture energy change factor and the loading rate was obtained by processing and fitting the data.

[0155] E b / E b实验 =1.387v 0.136

[0156] Therefore, the fracture energy required to crush the ore under different loading rates can be calculated as follows:

[0157] E b =1.387v n 0.136 E b实验

[0158] Where v n The normal velocity of the steel balls impacting the ore inside the mill is calculated using the following formula:

[0159] (6) Calculate the diameter D of the mill steel balls. b

[0160] The mill has a diameter of 6.7m, a mixing and filling rate of 35%, a rotation speed of 77%, and an ore density of 2670kg / m³. 3 The density of the steel ball is 7800 kg / m³. 3 The grinding concentration is 80%. Referring to Table 1, K is taken as 0.689, and the diameter of the intermediate condensation layer is calculated to be 5.5 m. The semi-autogenous grinding feed has a particle size of 238.5 mm for 95% passing through. From the formula E0 = 0.02637d... 1.473 The energy required to break a 235.88mm ore under static loading can be calculated to be 81.32J. The normal velocity of the steel balls impacting the ore inside the mill is calculated to be 9.98 m / s, derived from E. b =1.387v n 0.136 E b实验 The energy required to crush 235.88mm ore at a loading rate of 9.98m / s is calculated to be 152.11J. Substituting 152.11J into the theoretical formula for ball diameter, the theoretical steel ball diameter of the mill is calculated to be 107.5mm, which can be taken as 110mm in industrial production.

[0161] (7) Design experiments to verify

[0162] The comparative test is carried out in a D x L 450 x 450 mm laboratory discontinuous ball mill, and the grinding sample is taken from semi-autogenous feed, and the calculated steel ball diameter Φ110 is compared with the field steel ball diameter Φ120. The calculated proportion is compared with the field proportion. In order to better simulate production and judge the negative influence of hard stone particles in the semi-autogenous mill, the grinding cycle comparison test needs to be carried out, that is, after grinding for 45 min, the -2 mm particles are discharged, and then the reduced ore amount in the previous batch test is added according to the original ore particle size ratio, and then the ore is ground for 45 min again; the four cycles are continuously carried out. The grinding products +2 mm of various proportions are strictly sieved by a sieve, and then 500 g of the sample is taken after the -2 mm is divided, and then sieve analysis and water analysis are carried out. Several values representing the particle size characteristics are taken out and listed in Table 3, and these indexes are used to judge the advantages and disadvantages of each medium proportion. The representations are as follows: ①-80+25 mm content, used to determine the hard stone accumulation content; ②-2 mm yield, used to determine the generation of qualified particles in the semi-autogenous grinding product; ③-0.074 mm yield, used to determine the fine grinding capacity; and ④-0.20+0.010 mm yield, used to determine the intermediate selected level content.

[0163] Table 3 Grinding index summary table

[0164]

[0165] As shown in the table, the fourth cycle discharge ore under the grading obtained by the present application has good self-grinding capacity, and the hard stone content is reduced, which indicates that the proportion can effectively break the hard stone and eliminate the hard stone accumulation; the -2 mm qualified particle size content is also relatively high, which indicates that the medium system can effectively break the ore, and the -0.074 mm content ratio is 17.00%, which is also higher than the field steel ball grading.

[0166] According to the above steps, it is determined that the best steel ball diameter of a Φ6.7 x 3.4 m semi-autogenous mill is 107.5 mm.

[0167] The specific embodiments of the present application are described in detail above in combination with the drawings, but the present application is not limited to the above embodiments, and various changes can be made within the knowledge of those skilled in the art without departing from the purpose of the present application.

Claims

1. A method for calculating the diameter of steel balls in a ball / semi-autogenous grinding mill, characterized by the following steps: include: Step 1: Select representative mineral samples Two identical mineral samples were selected: one was a standard mechanical rock sample, from which the basic physical and mechanical parameters of Poisson's ratio and elastic modulus were obtained; the other was a mill feed mineral sample, from which the fracture energy of each particle size of the mill feed ore was obtained. Step 2: Determine the Poisson's ratio μ and elastic modulus Y of the standard rock sample, and calculate the ore stiffness k. p ; The Poisson's ratio μ and elastic modulus Y of the standard rock sample were determined by uniaxial compression test, and the stiffness k of the ore was calculated by the following formula. p : k p =Y / (1-μ 2 ); Step 3: Determine the fracture energy E of each particle size of ore in the mill feed. c Impact tests were conducted on five particle size fractions of ore in the mill feed using an ultrafast weighing sensor. The fracture energy was calculated by integrating the force-displacement curves from the tests, as shown in the following formula: In the formula: α is the deformation variable, α c F represents the deformation at the point of fracture, and F represents the force. Step 4: Establish the relationship between fracture energy and fracture probability, and derive the ultimate fracture energy for each particle 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; 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 energies E of the five grain sizes are obtained. b实验 J; Step 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. Using data obtained in Step 4, the relationship between ore particle size and ultimate fracture energy is established, expressed as follows: E b =a i d β In the formula: α1 and β are model fitting parameters, and d is the diameter of the particle; where α1 = α2α3v n β2 α2 and α3 are model fitting parameters, v n The normal velocity of the steel balls impacting the ore inside the mill. In the formula: R1 is the radius of the outermost dielectric layer; This formula can be used to accurately calculate the ultimate fracture energy of ores of any particle size. Step 6: Calculate the diameter D of the mill steel balls. b Substituting the ultimate fracture energy of the abrasive particle size into the theoretical formula for ball diameter, the theoretical diameter of the steel balls required for the mill is calculated. The formula is as follows: In the formula: D b The required steel ball diameter (m) for grinding a specific particle size; ψ represents the mill speed, in percentages. E b Let J be the ultimate fracture energy of the ore. k p For ore stiffness, GPa, kJ s For the stiffness of the steel ball, take 230 GPa; ρ e The effective density of the steel ball is kg / m³. 3 ; D m The diameter of the "spherical-loaded intermediate condensation layer" is in meters (m). g is the acceleration due to gravity, m / s² 2 .

2. The method for calculating the diameter of steel balls in a ball / semi-autogenous grinding mill according to claim 1, characterized in that: In Step 1, each identical ore sample consists of 6 to 8 representative large blocks of ore taken from different pits and sections of the mining site. The standard rock sample has a height-to-diameter ratio of 2:1, and a uniaxial compressive strength test is conducted to obtain its Poisson's ratio and elastic modulus, which are basic physical and mechanical parameters.

3. The method for calculating the diameter of steel balls in a ball / semi-autogenous grinding mill according to claim 1, characterized in that: When calculating the diameter of the steel balls in the semi-autogenous grinding mill, in Step 1, the mill feed ore is obtained in five particle sizes: -25+10mm, -35+25mm, -45+35mm, -60+45mm, and -80+60mm.

4. The method for calculating the diameter of steel balls in a ball / semi-autogenous grinding mill according to claim 1, characterized in that: When calculating the diameter of the steel balls in the ball mill, in Step 1, the ore particles of each feed size to the mill are obtained in five sizes: -2+1mm, -5+2mm, -8+5mm, -10+8mm, and -12+10mm.

5. The method for calculating the diameter of steel balls in a ball / semi-autogenous grinding mill according to claim 1, characterized in that: The specific process of establishing the relationship between the ultimate fracture energy and the ore particle size and calculating the ultimate fracture energy of the particle size to be ground in Step 5 is as follows: Step 5.1 Calculate the ultimate fracture energy E for the five particle sizes obtained in Step 4. b实验 Establish a fitting relationship with particle diameter: E b实验 =α2d β , where α2 and β are fitting parameters, and d is the particle diameter; Step 5.2: Establish a fitting relationship between ore fracture energy and loading rate. E b / E b实验 =β3v n β1 Where α3 and β1 are fitting parameters, v n The normal velocity of the steel balls impacting the ore inside the mill. In the formula: R1 is the radius of the outermost dielectric layer; That is, E b =E b实验 α3v n β1 ; Step 5.3: Substituting the expression from Step 5.1 into Step 5.2, we obtain the relationship between the ultimate fracture energy and the ore grain size: W b =α i d β In the formula: α1 and β are model fitting parameters, and d is the diameter of the particle; where α1 = α2α3v n β2 α2 and α3 are model fitting parameters, v n The normal velocity of the steel balls impacting the ore inside the mill. R1 is the radius of the outermost dielectric layer.

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