Summary of the selection of ball size of ball mill

(I) Factors Affecting the Size of Steel Balls The grinding process is a complex dynamic process with influencing factors. The factors affecting the size of the steel ball are various. From the principle analysis of the crushing process, the mechanical essence of the broken ore or ore of the steel ball is to exert a crushing force on the ore or the ore to overcome the cohesive force of the ore or the ore and destroy it, so it may affect the crushing process. The factors are divided into two categories: one is the factor of the broken object; the second is the factor of the broken power.
Factors that break the object include the mechanical strength of the rock and the geometry of the nugget or ore. The cohesive force of nuggets or ore particles is determined by their internal particle bonding mode and strength. Macroscopically, the hardness of rock ore is often used to characterize its mechanical strength, which is to characterize the rock's ability to resist damage. China often uses the Platt hardness coefficient as the classification coefficient of the relative robustness of rock ore, which is used to characterize the mechanical strength of rock ore. The greater the mechanical strength of the nuggets or ore, the greater the crushing force required for crushing, and naturally requires a large steel ball size. When the geometry of the nuggets or ore is the same, the size of the steel ball required for the mechanical strength of the nuggets or ore particles is larger than the size of the steel ball required for the mechanical strength. Larger nuggets require larger steel ball sizes when the mechanical strength of the rock is constant. However, it should be noted here that the mechanical strength of the nuggets or ore particles increases as their geometry decreases. Therefore, when determining the crushing resistance of the ore or ore, the mechanical strength σ pressure or the geometric size d of the ore or ore should be considered. If we want to consider the impact on grinding, the density of the ore and even the mineral composition of the ore will have an impact on the grinding. Large-density minerals tend to have a relatively high hardness. When grinding, they sink into the bottom of the mill with strong grinding action and are easily subjected to strong crushing. Minerals with low density are less affected by the grinding action. When the ore contains mineral components such as coal and talc , the steel ball is often difficult to bite the ore, which reduces the probability of breakage of the broken ore of the steel ball, thereby increasing the power consumption of the grinding product. The type of minerals of mica flakes is difficult to grind, which also increases the electricity consumption of grinding products.
There are many factors of crushing force, such as steel ball filling rate φ, ball density ρ, ball effective density ρ ^e , mill diameter D, mill rotation rate Φ, grinding concentration R, mill liner shape and structure. Wait.
The mill rotation rate Φ and the steel ball filling rate φ are combined to determine the movement state and energy state of the mill steel ball. In addition to the function of the protection cylinder, the mill liner also affects the coefficient of friction of the cylinder wall to the ball load. , which in turn affects the motion state of the steel ball. When the ball is put into a throwing motion state, the ball rises at a high height, the ball's position energy is large, and the striking force when falling is also large. When the ball is used as a sloping motion state, the height of the ball rises is not large, the ball's position energy is not large, and the ball has little impact force when the ball rolls down the slope.
The density of the ball naturally affects the mass m of the ball, which also affects the amount of energy carried by the ball, that is, the impact force of the ball. When the size is the same, the ball having a high density has a large striking force and a high productivity, and the ball having a small density has a small impact force and a small productivity. Mill productivity increases almost linearly as the density of the steel ball increases. Commonly forged steel balls of density 7.8g / cm ^3, the density of the steel balls only 7.5g / cm ^3, the density of the lower cast iron ball, only 7.1 ~ 7.3g / cm ^3. In the past, the development and testing of tungsten carbide balls have been carried out. The density of the balls is as high as 13.1 g/cm ³ , which is 1.68 times that of forged steel balls, and the productivity is 90% higher than that of forged steel balls. In general, the density of rolled or forged balls is greater than that of casting, and some unfinished air is left in the casting. Since the ball falls into the slurry, the pulp has resistance to the ball, or the ball is subjected to buoyancy in the slurry. What really matters is the effective density of the ball, that is, the density after subtracting the density of the pulp. In the coarse grinding, the concentration of the slurry is large, the buoyancy of the slurry is large, and the impact on the ball is also large. The concentration of the slurry in the fine grinding is smaller, and the influence of the buoyancy of the slurry is relatively smaller. It should be said that the density of several commonly used ball steels is not too large, and the impact on grinding is not too great, but this effect can not be ignored. In severe cases, the productivity can be reduced by 10% to 15%.
The inner diameter D of the mill mainly affects the height of the steel ball rising, which in turn affects the potential energy and the striking force of the steel ball. In the large-size mill, the height of the steel ball rises is large, the ball's position energy is large, and the striking force when falling or rolling down is also large, and even the large steel ball position in the large mill can make up for the insufficient size of the ball. In the small-size mill, the height of the ball rises is small, the ball's position energy is small, and only the larger-sized ball is required to meet the crushing force requirement. The specifications of foreign mills are generally larger than those of domestic ones, and the transfer rate is also low. The size of steel balls used is also smaller than that of domestic ones. This phenomenon has a bearing on the diameter of the mill.
The effect of slurry concentration on grinding is complicated. Generally speaking, when the concentration of pulp is large, the buffering effect on the steel ball is large, which weakens the striking force of the steel ball, which is unfavorable for grinding; however, when the concentration is large, the ore is easy to adhere to The steel ball and the surface of the lining plate are advantageous for the crushing of the ore particles. Similarly, the low concentration of the slurry has a small buffering effect on the steel ball, but it is not conducive to the adhesion of the ore particles to the surface of the steel ball and the lining. Moreover, the influence of slurry concentration on coarse grinding and fine grinding is also different, even related to the properties of ground ore, and the effects under different ore properties are also different. Since the influence of slurry concentration on grinding is complicated, the suitable slurry concentration can only be determined through experiments. As mentioned above, the liner can affect the movement state of the steel ball in addition to the protection cylinder. In general, the degree of unevenness of the surface of the lining plate has different frictional effects on the ball load. The unevenness of the unevenness is called the non-smooth lining. The friction coefficient of the ball load is large, and the ball load is also increased higher, so that there is a large striking force, so almost no smooth lining is used for rough grinding. The small unevenness is called a smooth lining. The friction coefficient of the ball load is small, the ball load is lifted lower, and the striking force is also small. Therefore, a smooth lining is often used for fine grinding. In the self-grinding mill and the gravel mill, the situation is different, the nugget is large, and in order to upgrade the larger nugget, a lifting lining is specially set, and the nugget can be mentioned to a higher position. However, in the self-grinding mill and the gravel mill, the function of the lining plate is still to protect the cylinder and affect the movement state of the medium, but the lining plate has a greater influence on the motion state of the medium.
The above analysis shows that the factors affecting the size of the steel ball are more than ten kinds, which are complicated and complicated, which brings great difficulty in determining the size of the steel ball. [next]
(2) Process and method for determining the size of the steel ball
Since the size of the steel ball is critical to the impact of grinding, long-term dressing and crushing workers are studying how to accurately determine the size of the steel ball. However, this problem is difficult to solve because there are too many parameters affecting the size of the steel ball. Despite this, people continue to explore and seek to find a scientific way to accurately determine the size of the steel ball.
Initially, people considered the simplest method and tried to find a single proportional relationship between the diameter of the steel ball and the grain size of the mill. Therefore, the investigation of more than 50 working ball mills shows that the ratio of the diameter of the steel ball to the maximum grain size of the ore is 2.5~130, that is,

Where k is the proportionality factor, ranging from 2.5 to 130.
The scale factor k is as wide as 2.5~130, which is simply unusable, which proves that this simple method is not acceptable. The reason why it is not because: 1 steel ball diameter D _{b is} affected by many factors, it is unscientific to grasp only one feeding particle size and throw away various factors. The channel is opened for a wide range of error generation. 2 The relationship between the diameter D _{b of the} steel ball and various influencing factors is complicated. There is no basis to indicate that there is a direct and single proportional relationship between the diameter D _{b of the} steel ball and the particle size d of the ore. Since this is the case, it is still necessary to find This proportional relationship, the method itself is unscientific, and the resulting relationship can only be false, and it is impossible to have application value.
Later, on the basis of summarizing the previous lessons, people took a step forward and no longer looked for a direct proportional relationship. Instead, they considered that the ball diameter D _b (mm) is proportional to the square root of the maximum grain size d of the ore, and the factors considered are Increased and included factors not considered in the scale factor. Since each researcher considers the starting point of the problem differently and the experience of each person is different, there are many empirical formulas for the ball path. The following are some empirical formulas often used in the ore dressing industry:
Rasufov formula:

Where i is the spherical diameter coefficient;
n — mineral material properties parameters;
d — the maximum particle size of the ore, ie 95% sieve size, mm.
Equation (2) cannot be used directly, and two sets of tests must be performed for a specific ore. Two equations are listed in a group, and i and n are solved from the equations to obtain a specific spherical path equation. For the convenience of application, KA Rassumov proposed that the medium hard ore can be directly calculated using the following simple calculation formula D _b (mm):

Olevsky formula:
Where d _k - the product size of the grinding, μm.
Davis formula:

Where d - 80% sieved feed size;
K—experience correction coefficient, taking different coefficient values ​​for different hardness: hard ore, taking k=35; soft ore, taking k=30.
The simple formula of the experience of the list:
Where d - 80% sieved feed size, mm
In China, some engineers have adopted the method of optimal number selection and relied on the Rasuf ball diameter empirical formula to solve the derivation and proposed the following empirical formula:

Where d - 95% sieved feed size, mm.[next]
Despite this, the above empirical formula still has a big problem: First, the factors considered are still too small, and second, it is very difficult to include the remaining factors with an empirical coefficient. Therefore, the error of these empirical formulas is still large. The author proves through experiments that the results of the Olevsky formula are generally much smaller; the results of the Davis formula are generally too large; the calculation formula of Rasmov simply calculates the coarse level when the ball diameter is used. Too many, the required diameter of the fine-grained grade is still basically feasible, but it is also slightly larger; the simple calculation formula of List has similar problems with the simple formula of Rasufov, and so on. Even so, these formulas can still be used, but the errors are large. If you know their faults, the corrections are still available.
Due to the large error in the calculation results of the empirical ball diameter formula, this will inevitably affect their application. In this case, people simply determine by experiment. Although the method of determining the ball diameter is more accurate than the empirical formula, the test workload is large, time-consuming and costly. The fine-grained test is better, and it can be tested on a laboratory mill. The workload is not very acceptable. For the coarse grinding machine, because the mass of the ore is large, it can only be tested on the industrial mill. This workload is too large, the test period is also very long, the human and material consumption is large, and there are few factories and mines willing to do this work. It is. Therefore, although the method of determining the ball diameter is more reliable, it is difficult to apply more due to the above problems.
People always want to use the formula to directly calculate the ball diameter. In recent years, I have been working on the empirical formula. Since the previous empirical formula has too few factors to consider and the error is large, then the factors to be considered are increased. There are also many researches in this area. Several empirical formulas including the ball diameters are also proposed. The typical two empirical formulas widely used in Europe and the United States and the region are: Aris Chalmers The ball diameter empirical formula and Knox-Rod's ball diameter empirical formula. The formula of Aris Chalmers is:

Knox-Rod’s ball diameter D _b empirical formula is

Where D _b — the diameter of the required steel ball, in;
F—80% sieved feed size, gm;
S _S - ore density, t / m ³ ;
W _i — ore work index to be milled, kW•h/t;
D — mill inner diameter, ft;
C _S - mill transfer rate, %;
K _m — the empirical correction factor, which is selected as shown in the following table.

The ball diameter empirical formulas of the above two companies consider up to five factors, and the empirical correction coefficient k _m value indicates other factors that are not considered. Therefore, it should be said that they consider the main factors affecting the ball diameter, and for some factors Theoretical derivation has also been made, and it should be said that the calculation results are more accurate than those of the previous empirical formulas. Because of this, these two empirical ball diameter formulas are currently widely used in Europe and the United States. [next]
However, the above two empirical ball diameter formulas are not convenient for use in factories and mines in China. First, their formulas are inclusive of active index W _i, the majority of our concentrator does not work index data W _i, to make such information more time and cost, the majority of our processing plant only Platts hardness coefficient values. Second, their ore-feeding particle size F is 80% sieved particle size, the unit is μm, while China has used 95% sieved particle size for a long time, the unit is mm or cm. Moreover, their empirical coefficients are summed up in foreign experience. Foreign mills have large diameters and large diameter steel balls have large potentials, which can make up for the lack of ball diameter. China's mill has a small diameter and requires a large ball diameter. Therefore, foreign experience may not be suitable for China's selection of factories. In view of the above situation, the author starts from China's national conditions, using the theory of fracture mechanics and the theory of Davis et al. to derive a semi-theoretical formula of the ball diameter D _b (cm):

This formula also takes into account the strength σ _pressure and size d of the ore, taking into account the mill diameter (represented by D ₀ ), the mill rotation rate Φ, and taking into account the effective density σ _{e of} the steel ball, which is comprehensively corrected for unconsidered factors. The coefficient K _c is included, and different particle sizes have different K _c values. Therefore, it can be said that the ball diameter formula derived by the author is the only one and a half theoretical formula in the world, and the factor considered is the most one. Therefore, its calculation result is more accurate than any ball diameter empirical formula.
From the current level of human cognition, it is impossible to derive the theoretical calculation formula of the ball diameter. This is because: 1 The formula that does not consider the mechanical strength of the fractured rock ore is unscientific, and the theoretical formula must consider the broken object. Mechanical strength, but due to the complexity of the mechanical properties of rock ore, the current solid mechanics can not theoretically calculate the mechanical strength of rock ore, but only by the results of engineering measurements, which introduces the measured data of the test. 2 Modern mathematics can't solve more than ten unknown equations. It is impossible to include more than ten factors affecting the ball path. 3 Some influencing factors are currently unable to theoretically make a quantitative description, and cannot be corrected without the help of empirical correction factors.
Therefore, it is impossible to obtain a theoretical formula at present, and at most only a semi-theoretical formula can be obtained. From this point of view, the above semi-theoretical formula is considered to be relatively perfect at present. If it is carefully verified and revised, it should be widely used in China. The author has recently revised this semi-theoretical formula to make it possible to accurately calculate the required ball diameter under certain conditions in a wide range of rough grinding, medium grinding and fine grinding. The industrial test and production application of several selected plants prove that the semi-theoretical formula of the ball diameter can solve the problem of accurate calculation of the ball diameter under each grain level.
(3) The method of determining the ball diameter by experiment The ball diameter error calculated by the empirical formula is large, and the ball diameter has a great influence on the grinding. Therefore, the experiment is directly used to determine the required ball diameter, which must become a certain ball diameter. Important method.
The method of determining the ball diameter is of course influenced by a variety of variable parameters. In order to simplify the problem, only some important variable parameters can be fixed in a certain range of values, and then the test between the ore size and the ball diameter is determined. relationship. In the specific practice, the working parameters that are commonly used or confirmed in the production of the mill to be calculated, such as the transfer rate, the ball loading rate, and the slurry concentration, are selected as fixed values, and then several sets of steel balls are tested according to experience, and the effect is good. A set of balls is the best ball diameter of choice.
The method of determining the ball diameter can be carried out on a laboratory mill or on an industrial mill. Obviously, the tests on laboratory mills are much simpler, and the tests on industrial mills are cumbersome and complex.
The specification of the laboratory mill is small, and the corresponding ore size is also small. The test result can be used as the basis for the ball diameter selection of the medium and fine mill. Because the ore size is usually 3~5mm or less, and the ore size range is narrow, it can be tested by a single ball diameter ball set. The selected ball group should not be lower than 3 groups, preferably 5 or 6 groups. The purpose is to include the best ball diameter to be sought without missing the optimum value. The relationship between the productivity and the ball diameter of the specified grade of the mill when the other parameters are fixed is a unimodal function, as shown in Figure 1. If D ₃ , D ₂ , and D ₃ balls are selected, the productivity curve will rise to D ₃ , and D ₃ cannot be determined, which is the best ball diameter value; likewise, three balls D ₄ , D ₅ , and D ₆ are selected. At the same time, it is impossible to determine that D ₄ is the optimum ball diameter value, and the optimum value can be found only when the productivity curve reaches the peak value and the derivative turns.

Figure 1 Single-peak function curve of mill productivity and ball diameter

An important issue in the test method is how to determine the quality of the grinding effect. In terms of productivity, it is called “thickness level” that does not reach the specified level (such as 0.074mm or other granularity), “fine level” that reaches the specified level and below, and “over-crush” in the fine level. Level, the relationship between several can be shown in Figure 1.
Obviously, the criterion of the amount of processing as the productivity is unscientific, because the purpose of grinding is to make the material have to reach a certain degree of fineness, and only the index to achieve the purpose of this purpose is the science. However, it is still problematic if the productivity is judged only by the content of the fine fraction below the specified particle size, because the higher the fine fraction yield, the larger the yield of the over-comminuted fraction, and the less favorable particles are not obtained. The grade yield is large. Grinding should not only achieve the specified fineness of the product, but also should be as small as possible. Therefore, it is proposed to use the "grinding technology efficiency" index to judge the quality of the mill work, the mill technology efficiency E is: [next]

Figure 2 Granularity of grinding products

Where γ is the yield less than the specified particle size level, %;
γ ₁ — yield of less than the specified particle size level, %;
γ ₂ — ore pulverized particle size grade yield, %;
γ ₃ — the yield of the pulverized particle size grade in the product, %.
It can be seen from the formula (11) that the technical efficiency of the mill is zero when all the products are uniformly pulverized. The efficiency of the mill technology is based on the size of the product to determine the quality of the grinding process. This method is not only computationally complex, but also not closely related to the mechanical processes of grinding.
The author believes that the grinding process is a mechanical process with reduced particle size, then it should be more scientific to use the index of the best particle size reduction as the criterion for the quality of the grinding process. The basic equations for grinding dynamics have been proposed:

Where Q is the amount of coarse fraction remaining after time t;
Q ₀ — the original content of the coarse fraction at the beginning of the grinding;
t — grinding time;
K - a constant determined by the grinding conditions.
It can be seen from equation (12) that the constant K actually reflects the speed of particle size reduction, which can be called the grinding speed constant, which is obtained by equation (12):
or

The coarse-grained content Q ₀ of the raw material is known, and the K value can be obtained by measuring the coarse-grained content Q at the grinding time t. The K values ​​of each group of steel balls at the same grinding time were determined and compared. The ball group with the largest K value had the largest grinding speed, which is obviously the best ball group. The method of selecting the spherical diameter by the value of K is closely related to the purpose of reducing the particle size in the grinding process, and the method of obtaining is simple, which is a scientific method. When I have used finely ground ball diameter in the choice of cloud tin companies, achieved good results.
It is difficult for the laboratory ball mill to give 10~25mm coarse ore, and the laboratory ball mill is difficult to load large steel balls. Therefore, the optimal steel ball size in the rough mill is only determined in the industrial mill. . However, the industrial production mill is a continuous production equipment. To determine which ball or group of balls is good, it is necessary to observe for a long time and separately examine the maximum particle size and average particle size of the mill discharge, the graded overflow, and the graded return sand. The thickness of the two sizes is used to judge that the ball diameter is too large or too small, and the utilization coefficient q(t/m ³ •h) of the mill according to the specified level is used together to determine which group of balls is the best. In this kind of industrial test, it takes 1~3 months to test a ball and more than one year to test five or six kinds of balls. Not only the test period is long, but also the ball is cleaned many times. The workload is large and costly. Therefore, there are not many factories and mines that carry out such system industrial tests, and most of them are concluded through long-term use plus observation and analysis. Of course, this conclusion is not convincing and has strong experience.
In order to solve the problem of long industrial test period, large workload and high cost, the author proposes a method to simplify the industrial test of ball diameter. That is, the intermittent grinding test is carried out in a large laboratory mill with a diameter of 400 mm or more, and the different ball groups are ground to the fineness level of the produced product for comparison, and the optimum ball diameter can also be determined. The determined optimal ball diameter is then verified by observation and analysis in an industrial mill. This not only greatly shortens the test cycle, reduces manpower and material resources, but also eliminates risks for industrial testing. The author has conducted such tests in several factories and mines, proving that the method is successful and the effect is good.
(4) Limitations and errors of the empirical spherical path formula
An empirical sphere formula is a mathematical model that is summarized on the basis of a large amount of experimental data or production materials. This method is still a useful method for the ball milling process in which the influencing factors are intricate and theoretically difficult to make progress. The formula derived from such a method is valuable in that it comes from practice and is higher than practice, both reliability and practicality. In the long years before, the mineral processing workers used these experience ball formulas and their own experience to solve the ball diameter of the mill and solve the problem.
However, it is not difficult to see from the method of the ball diameter empirical formula that it has its own limitations and has a large error. Although the test data or production materials are abundant, they are still limited, or the specifications and forms of the equipment for testing and production are limited, or the type of ore to be tested and produced is limited, or the number of tests and production time is limited. The source of the information is limited. In this way, the scope of the model summarized in the limited data is bound to be limited, and the reliability is lost across this limited range. Therefore, once the empirical formula has taken the scope of the data on which it is summarized, it will inevitably produce large errors.
Even for the same test and production data, different researchers use different mathematical methods, so the mathematical models are different, and the calculated ball diameter results are different.
In addition, the empirical formulas are included in the empirical formula of the ball diameter. Different researchers have different empirical coefficient values ​​according to their respective experiences, and the naturally calculated ball diameter results are also different.
The above analysis shows that the empirical ball diameter formula summed up by the researcher under what conditions is applicable to the conditions defined when summarizing it. If it is applied to it, it will inevitably produce a large error when it is different from the defined conditions. Make an empirical correction. It is necessary to recognize the limitations of the empirical formula, and it is necessary to make empirical corrections for the limitations, otherwise a large error will occur.
The following is an explanation of the limitations and errors of the empirical formula by applying the KA Rasov's empirical ball diameter formula commonly used in China's mineral processing industry.
According to some average conditions, KA Rassum proposes that the diameter D _b of the steel ball required for grinding is proportional to the nth power of the ore particle size d. If the scale factor is i, then: [next]

Obviously, different grinding conditions have different i and n values. For each specific grinding condition, the i and n values ​​must be determined experimentally, and then formula (14) can be used. This is the limitation of this formula. .
For the solution of equation (14), the equations with two equations must be solved, and the two equations can solve two unknowns. Assuming that the ore particle size d ₁ is determined by experiments to determine that the required ball diameter is D _bl , an equation is obtained:

Then set the ore particle size d ₂ through the test to find that the required ball diameter is D _b2 , then another equation:

Combine the equations (16) and (16) and solve for i and n of this system of equations:

Equation (18) transforms:

Equation (19) takes the logarithm on both sides:

In the formula (20), d ₁ , d ₂ , D _b1 and D _b2 are all known numbers, so n can be obtained. When the value of n is obtained and returned to the equation (17), then i can be obtained.
After i and n are obtained, the general formula between the sphere diameter D and the ore particle size d under the specific conditions is obtained:
D _b =id ⁿ
It is also possible to calculate the spherical diameter required for each ore-feeding particle size under the specific conditions by this formula. If the grinding conditions change, the same method must be used to find a new formula. This is a limitation of this formula and cannot be used across specific conditions when solving equations.
The problem with this formula is that the test with the particle sizes d ₁ and d ₂ makes the formula of the formula more accurate in the range of d ₁ ~ d ₂ , and if it exceeds the range of d ₁ ~ d ₂ , it will inevitably produce a large error. Because the mechanical strength of rock ore increases as the particle size becomes thinner. For example, tests for d ₁ and d ₂ are usually carried out in the laboratory, and the ore size used is usually below 5 mm, the particle size is fine, and the mechanical strength of the ore is high. However, the general formula obtained by the test is obtained in the range of d ₁ ~ d ₂ , and if it is generalized for the range of d=10~25mm, the spherical diameter obtained must be too large, because 10~25mm If the mechanical strength of the nugget is significantly smaller than 5 mm and below, the ball diameter required to calculate the coarse block is necessarily large. For example, some people use 5 and 3mm two-stage test in the laboratory to find the general formula after i and n, and the ball diameter required to calculate the 25mm nugget by the formula is Φ125mm. The author uses his modified semi-theoretical formula of the ball diameter, only Φ100mm is enough. Through one year of industrial tests, it is proved that the use of Φ100mm steel balls is much better than Φ125mm steel balls. It shows that the original calculated ball diameter is too large. This is the reason for the error in the Rasuf's ball diameter formula.
Since the KA Rasufov formula D _b = id ⁿ needs to be tested to determine the parameters i and n, the use is more troublesome. He also suggested that a simple calculation formula can be used directly for medium hard ore:

The range of hard ore, the strength of the ore is twice that of the ore, but the same formula used in the calculation, which will not produce a large error? Moreover, the simplified formula is widely used in different grinding conditions of medium hard ore, and the error is inevitably larger than that of formula (14). [next]
(5) Practice to determine the universality of the ball diameter experience method
Because the factors affecting the ball diameter are complicated, it is difficult to theoretically solve the calculation of the ball diameter. Only the practical method can determine the optimal value of the ball diameter. The above-mentioned methods of determining the ball diameter by experiment and the method of formulating the empirical formula based on the production practice data are all within the scope of practical problem solving. Although the results obtained by such methods have limitations and large errors, they are derived from practice, and they have a real and reliable side. It is still widely used before there is no better way.
The conditions under which different researchers conduct trials vary, and the conclusions drawn must be different. Moreover, different researchers have different production materials based on their research. Even for the same batch of production materials, different researchers use different mathematical methods. Therefore, the empirical formula of the ball diameter obtained by the practical method is various and varied. The former Soviet Union's TK Smeshliayev (CMbІШЛляеB) believed that there is a certain functional relationship between the diameter of the steel ball and the size of the ore being ground, and plots the relationship between the diameter of the steel ball and the ore particle size. Need a ball. Obviously, this method is only suitable for the case where equipment and ore are determined, the condition change is not applicable, and a new curve under new conditions must be drawn.
In cement production, the calculation of the diameter of the ball is often carried out for specific grinding conditions. Therefore, the conclusions drawn by various researchers are often different. HL Starke used the Portland cement slag as the ground material for the grinding test. It is believed that there is a specific particle size that is particularly effective for grinding in terms of the size of the ball. The best conclusion of grinding efficiency. After FW Bowdish tested with high-purity limestone , the conclusion was that the ratio of the ball diameter D _b to the particle size d of the material to be ground There is a maximum grinding rate constant for a certain value. This ratio varies with the grain size of the ore, and the optimum sphere diameter ratio between the stages of 4.699 to 0.15 mm is between 14 and 40. M. Papadakis believes that the ball diameter is too small and too small, there is an optimal ball diameter in the middle, but this value must be determined through experiments, that is, using a laboratory ball mill to test The shortest ball diameter in which large particles can largely disappear in a short period of time. If d ₀ and d ₁ are the maximum particle sizes that should be ground, W ₀ and W ₁ are the functions of the ball, then press The proportion has expanded. According to JN Nijiman's research, there should be an appropriate relationship between the ball diameter D _b and the ore particle size d, and The area of ​​radius is a grinding range, and the combination of D _b and d should ensure that the grinding speed constant has a large value, because the grinding speed constant and the ore size d are a single peak function, and only the ore size is a certain The maximum grinding speed constant is available at the appropriate value. GM-empel believes that the material and diameter of the mill have been fixed, and the rotation speed is maintained, and the filling rate of the ball is optimal. Therefore, the potential energy supplied can be changed as long as the ball diameter is changed, and the maximum initial grinding speed is proposed. The method of determining the best ball diameter. Some researchers believe that in order for the mill to work effectively, it is necessary to have a method of correctly selecting the ball diameter, and it is considered that the previously considered factors are rare for rocks and difficult to measure (such as the pine ratio). A method for determining the ball diameter by considering the influence parameters of the mill and the mechanical properties of the rock is proposed:

Where D _b - the required ball diameter, cm;
d — feed particle size, cm;
K _n — plasticity coefficient, which is the total specific work/elastic deformation ratio work;
D — mill diameter, cm
σ _B — compressive stress, kg/cm ² ;
δ _B — the density of the ball, kg/cm ³ ;
E _p — ore modulus of elasticity, ks/cm ³ ;
F - coefficient , may be taken as 0.15;
K - relative radius, cm;
Φ - transfer rate, %.
In short, in the cement mill, the method of determining the ball diameter mostly adopts a practical method, and the optimum ball diameter under a certain feeding particle size is determined through experiments. The empirical method of determining the ball diameter through practice has universal significance and is applicable to various mineral materials, but this method is cumbersome, and has limited limitations and strong experience.
Most of the empirical ball diameter formulas have the following characteristics: 1 The factors considered are few, only 2~3, which is inconsistent with the actual factors affecting the grinding process; 2 The whole formula uses the same experience when used in different particle size ranges. Coefficient, and the anti-destructive performance of rock ore is different under different block degrees. This kind of unchanging practice is inconsistent with the anti-destructive characteristics of rock and mineral; 3 whether the ore is coarsely ground and finely ground regardless of the mechanics of rock ore There are major differences in the effects of properties, viscosity effects, impact effects, etc., but the empirical formulas do not consider these, which is inconsistent with the actual process of grinding. Due to the above three characteristics, the empirical formulas of each ball diameter will inevitably produce large errors in the calculation, and no further analysis will be made here.

Steel Screws

Our steel screws are also strong and widely used in industry and drone etc. While their color is not colorful as aluminum screws as well. We often offer black,silver, and Zinc-plated color steel screws. Different sizes and types steel screws are avialable. Philip pan screws, button head screws, socket cap screws, countersunk screws and so on are our popular types. Customized CNC steel parts are offered as well. If you need special packing service including special package boxes, it is also ok for us. Any questions or demand, please feel free to contact us.

Steel Thumb Screw,Alloy Steel Set Screw,Steel Wood Deck Screw,Steel Self Drilling Screw

Hobby Carbon CNC Technology(Shenzhen) Limited , https://www.hobbycarbon.com