Second, the verification analysis of the semi-theoretical formula of the ball diameter
Formula (17) is derived directly from the principle of fracture mechanics, and some assumptions have been made for certain conditions and some necessary empirical corrections have been made to the grinding conditions. Whether this formula is correct and of practical value must be tested by practice. The results will be verified later, and some preliminary verification analysis will be done here.
(1) Comparative analysis of calculation results with various common ball diameter formulas
The preliminary verification analysis of the semi-theoretical formula of the spherical path can take the method of comparative analysis of the calculation results. Since the common ball diameter empirical formulas have been used by the ore dressing workers, and some conclusions have been made on their accuracy and scope of application, the semi-theoretical formula of the ball diameter and the common ball diameter empirical formula can be used simultaneously. Calculate the required ball diameter for the same grinding conditions. From the comparative analysis of the calculation results of each formula, the semi-theoretical formula of the spherical diameter can be verified initially.
Now take the mill and very large common Size of an example of a ball mill 3.2m × 3.1m, Φ take 80%, φ take 45%, Ï taken 7.8g / cm 3, Ï t take 3 g / cm 3 , R d taken 75% of the metal ores is typically σ pressure: 80 ~ 160MPa (800 ~ 1600kg / cm 2), averaged 120MPa1200kg / cm 2. The above conditions are calculated according to the semi-theoretical formula of the spherical diameter to calculate the steel ball size required for each particle size. China's mill feeding ore has a coarser grain size, some small plants reach 50mm or thicker, and the finely grounded ore is considered to be 0.15mm. The actual calculation results are listed in Table 1. In order to compare with the calculation results of the common empirical ball diameter formulas abroad, the simple formula for calculating the list is D b = 25.4. 
, Rasumov simple calculation formula Davis formula D b =k And Olevsky formula Etc., calculate according to the meaning and regulations of each empirical formula. The calculation results are also listed in Table 1 to compare with the calculation results of the semi-theoretical formula. In order to compare with the actual data of our country, we will include the practical data of our country in Table 1. Most of them are the experience values ​​of the various mineral processing plants over the years. Finally, the experimental research value is added. The ball diameter of the ore below 3mm is the experimental research value of the author. The ore feeding of more than 5mm is the production test result of reducing the ball diameter of some factories and mines in recent years. Through comparison, it is also possible to judge the accuracy and reliability of the calculation result of the semi-theoretical formula of the ball diameter, which is a relatively comprehensive and comprehensive inspection and verification method.
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From the results listed in Table 1:
(1) The calculation result of the semi-theoretical formula of the ball diameter is more accurate than the calculation result of any empirical formula. This conclusion is obtained from the following results: 1 The calculation results of the Davis sphere formula are generally too large, and both coarse and fine grinding are the same, and the results of the formula (17) are different than the Davis formula. The result is small and proved to be more precise than the Davis formula. 2 The Olebissky formula is a formula that is well-recognized by the ore dressing industry, and the ball diameter value under the formula (17) is larger than the Olevsky formula, which proves that it is better than Olevsky. The formula is accurate. 3 The results of the simple calculation formula of the list are generally considered to be low under the coarse-grained level and more accurate at the medium-fine-grain level. It is reflected in the table that the result of formula (17) is larger than that of the List formula when the ore is thicker than 10 mm, and the results of the two formulas are close when the particle size is below 10 mm. This also proves that the formula (17) is more accurate than the simple calculation formula of the list, and can overcome the shortcoming of the smaller ball diameter of the Bunde formula. 4 The disadvantage of the simple calculation formula of Rasufov is that the calculation result under the coarse grain level is low, and the middle and fine grain size is still basically accurate. The results of formula (17) under coarse particles are larger than those of the Rasummo formula, and the results of the two formulas under medium and fine particles are similar, which also proves that the simple calculation formula of Rasufov is accurate. 5 Domestic and foreign experts believe that the ball diameter of the mill in China's concentrator is too large, and the ball diameter under the formula (17) is smaller than that of the ball, which overcomes the problem of large ball diameter in production. 6 Compared with the test ball diameter, the result of formula (17) is very close to the test result, especially the ore size is less than 3mm. The above 6-point analysis results show that the calculation result of the semi-theoretical formula of the spherical diameter is more accurate than the calculation result of any empirical spherical diameter formula in Table (17).
(2) The semi-theoretical formula of the spherical diameter is more accurate in the calculation of the spherical diameter of the coarse, medium and fine-grained grades, and the application has a wide range of particle sizes. The formulas of List and Rassum are more accurate when they are in the middle and fine-grained grades, and the deviations are larger under the coarse-grained grades. Although the formula (17) is more accurate than the other formulas at the coarse-grain level, it has been confirmed from recent research by the author that the calculation result of the semi-theoretical formula of the ball diameter under coarse grinding is still too large and needs further correction, which will be followed. Discussion.
(3) The semi-theoretical formula of the ball diameter is perfect, the meaning of the formula is clear, and the relationship between the ball diameter and each influencing factor is consistent with the law of the grinding process. Other empirical ball diameter formulas are far from being able to do this. Although they are simple in structure and convenient in calculation, the calculation results are highly biased, and some can not be used. This is the biggest problem. For the ball diameter parameters that have a great influence on the grinding, it is obvious that the accuracy of the results should be placed at the top of the evaluation formula.
The comparative analysis of the above results has in fact verified the accuracy and applicability of the semi-theoretical formula of the ball diameter, which also proves that the various assumptions and empirical parameters made in the formula derivation process are reasonable.
(II) Adaptability analysis of semi-theoretical formula of ball diameter
In theory, the semi-theoretical formula of the ball diameter is not summarized from some specific empirical data, but is derived from the fracture mechanics of the rock and is derived using the strict Davis and Levinson theory, and The value of the empirical parameters is mostly derived from experimental research data. Therefore, it should not have any limitations, but should have wide applicability and be suitable for ball diameter calculation in various occasions. The following is a detailed analysis of its adaptability:
(1) The essence of the formula is to select the crushing energy from the actual needs of rock ore crushing, that is, to select the size of the steel ball. Therefore, there is no limiting property on the grain size of the rock, so that under various feeding sizes That is, it can be applied under coarse, medium and fine particles. This has been proven in the previous section of the analysis.
(2) The formula includes the strength factors of rock ore, so it can be applied to ores with different strengths and different hardnesses, that is, hard ore, medium hard ore and soft ore can be applied, even non-metallic ore can be applied. The σ pressure of non-metallic ore and general stone is more than 80 MPa (800 kg/cm 2 ), and the ball diameter calculated under the same grain level is 15% to 20% lower than that of metal ore. This is consistent with the actual application in industrial production.
(3) In the process of derivation of the formula, the two important factors of the diameter and the rotation rate of the mill are also included. In the calculation of D0, the ball loading rate φ is also taken into account, so different specifications, different speeds and different The ball loading rate can be adapted. At the same time, its calculation results can also reflect the difference between different specifications, different transfer rates and ball diameter under the ball. Experts at home and abroad believe that the use of the ball in China is too large, but it is too large and why it is too large, but no one has made it clear. This formula can answer this question. The ball mills in foreign countries are generally large, with a diameter of 4~6m, and there are not many 4m or less. Due to the large size of the mill, the ball loading rate is low, and the ball loading rate of the 6m diameter mill is as low as about 20%. At the same time, the transfer rate Also lower, mostly 65% ​​to 75%. In the large-diameter mill, the height of the ball rises, the position of the ball is large, and the positional energy can make up for the lack of the ball diameter. Due to these circumstances, plus the foreign ball mill feeding ore size is less than 15mm, the largest ball size in foreign ball mills is more than 75~80mm, Φ100mm balls are rarely used, and Φ120mm balls are not used at all. However, China's national conditions are different. The largest diameter of domestic grinding machine is 3.6m, and the diameter of 3.2m diameter is mostly in the factory. In order to ensure the steel ball has a large impact energy, the transfer rate is higher, more than 80%~85%. The ball loading rate is 40%~50%, plus the coarser grain size of China's mills is 15~25mm, which makes the ball diameter in China's mills bigger than foreign ones. The ball diameter of the mill is 120~127mm, and a few coarse ore mills use the Φ100mm ball. Therefore, when the national conditions are different, there will be differences in the path of the ball. This is a normal phenomenon. Regardless of the national conditions, it is unclear whether the size of the ball is used in terms of the size of the steel ball. If the ball diameter semi-theoretical formula is used for calculation, the result can reflect the difference between the ball of the mill and the ball at home and abroad, and even the magnitude of the ball is large. Indeed, the ball used in China's mills is larger. If we use the semi-theoretical formula of ball diameter to accurately select the ball diameter, it can solve the problem of large ball diameter in China. In this respect, the author has done industrial tests in several factories and mines, which has solved the problem of large ball diameter of the mill.
(4) Many small and medium-sized factories and mines in China are not perfect due to the crushing process. The mill feeds up to 40~50mm, while the foreign mills generally have finer grain sizes. As seen in Table 1, the foreign ball diameter formula calculates the coarse grinding machine. The ball diameter is generally too low (except for the Davis formula), which is not suitable, and the semi-theoretical formula of the ball diameter can also be applied to domestic small and medium-sized factories and mines.
(5) The efficiency of ore fine grinding is low. B. Bass (Buss et al.) studied the single particle crushing and concluded that under the same stress condition, the larger the particle size, the greater the probability of breaking, natural and fine. Because the crushing particle size is small and the breaking probability is low, the fine grinding power consumption is large, so it is very important to accurately calculate the size of the steel ball under the fine grinding. In the formula derivation, the fine grinding part adopts the author's scientific research data, so the calculation result is very accurate, which is beyond the reach of any ball diameter formula. [next]
(6) At present, foreign concentrators, especially European and American concentrators, are widely used in Ales Chalmers and Knoxlode's ball diameter experience formula. However, the application of these two formulas in China is inconvenient. First, the data of the general reactive power index W i of China's concentrators often only has the Platts hardness coefficient f value data, and the second is that the ore grain size in the formula is 80%. Screen size, while China is customarily using 95% sieve size, so the particle size conversion must be carried out before using these two formulas. Although the author has proposed two specific methods of granularity conversion, but after all, to find data conversion. Moreover, the two empirical formulas themselves are not perfect. Coal , talc , oil shale and mica are recognized as low hardness ores, but their work index W i can be several times the hardest ore work index, because the work index is expressed by the grinding power consumption. Although the hardness of those types of minerals is low, due to the slippery, the ball is not easy to bite the ore particles, and the grinding efficiency is low, so the power consumption is high, and the W i value can reach 100 kW•h/t or more. If you use these two formulas to calculate the diameter of such ore, you will get ridiculous results. In addition, the empirical coefficients in these two formulas are summarized in the case of large foreign mills, and it is not suitable for China's national conditions.
The semi-theoretical formula of the ball diameter is derived from the national conditions of China. The σ pressure or f is owned by the plant. The grain size d can also be expressed by the technical habits of China, so it is convenient to use. Therefore, the semi-theoretical formula of ball diameter will have a good application prospect in China.
(3) Process characteristics of coarse grinding and fine grinding and requirements for ball diameter selection
AF Tagart has divided the grinding into three ranges according to the grain size of the grinding product: coarse grinding (product size 3.3~0.83mm), medium grinding (product size 0.61~0.21mm) and fine grinding (product size 0.015mm or thinner). However, in the actual beneficiation industry production, due to the limitation of the grinding process structure and economic factors, the grinding is usually divided into two stages of coarse grinding and fine grinding. The coarse grinding usually refers to the grinding of the grinding product with a particle size of 0.15 mm or more. The grinding mill has a particle size of 0.15 mm or less. Here we discuss the two stages of rough grinding and fine grinding.
Rough grinding and fine grinding Although the minerals belonging to different particle size ranges are ground, due to the different particle sizes of the grinding, there are many major differences in the grinding process characteristics of the two stages. There are many different choices for the ball diameter. Claim. The process characteristics of the rough grinding and fine grinding stages and the requirements for ball diameter selection are discussed below.
(1) The rough grinding treatment has a wide particle size range, a large grinding ratio, a narrow particle size range of the fine grinding treatment, and a small grinding ratio. China's rough mill processing material has a wide particle size, usually the maximum particle size is about 25mm, and the better is 20~15mm. If it is milled to 0.15mm (-0.074mm is about 80%), the grinding ratio will reach 166. ~100. The fine grinding ore size is calculated according to 0.30mm, the product particle size is 0.074mm, and the grinding ratio is only 4. It can be seen that the coarse grinding is much larger than the fine grinding.
The coarse grinding has a wide range of particle sizes, and in order to improve the grinding efficiency, the steel ball must be multi-staged. Therefore, in addition to ensuring the accurate size of the steel balls of each grain grade, the rough grinding also has an important problem of a reasonable ratio of the ball diameters at all levels. Russian mineral processing scholars have pointed out that the reasonable distribution of the ball in the first coarse grinding machine can increase the production capacity by 30% to 40%. The fine grinding itself has a narrow particle size range and a small grinding ratio. Therefore, the precise selection of the ball diameter is the main problem, but the ratio of the ball diameter is not significant, and one or two balls can be used.
(2) The coarse grain size is coarse when coarse grinding, although the mechanical strength of the coarse ore block is low due to the large number of cracks, but because of the large size, a large absolute impact force is still required. Therefore, the coarse grinding should adopt the throwing work. State, so that the steel ball has a large drop height, to ensure that the steel ball is mainly impact crushing. Fine-grained and fine-grained fine grain size, although the mechanical strength of fine ore particles is high due to less cracks, but the size of the ore particles is small, and the absolute impact force required is small. Therefore, the fine grinding should adopt the sloping working state, so that the steel ball can be fully rolled and peeled, and a slight impact is sufficient. Here you may wish to make a specific calculation to explain the problem more intuitively. According to the data provided in Table (2), the breaking strength of 15 mm coarse particles is 89.15 MPa (891.5 kg/cm 2 ). Calculated, the absolute destructive force required to destroy spherical ore particles of 15 mm in diameter is 15750 N (1575 kg). The 0.15mm ore has a high mechanical strength of 178.5 MPa (1785 kg/cm 2 ), which is twice the strength of the 15 mm coarse grain. However, the absolute destructive force required to destroy spherical ore particles of 0.15 mm in diameter is only 3.15n (0.315kg). The diameter of the 15mm ore is 100 times that of the 0.15mm ore, but the absolute destructive force required for the 15mm ore is 5000 times the absolute destructive force required for the 0.15mm ore. Therefore, rough grinding requires a large impact force to break, while fine grinding requires only a slight impact, mainly by grinding and peeling force. In this way, the rough grinding should ensure that the impact force is sufficient, and the fine grinding should ensure that the grinding and stripping area is large enough, that is, the coarse grinding application of the large ball, the fine grinding application of the small ball, the large ball has a large impact force, and the small ball has a large grinding area. The drop-off work has a large impact, and the sloping work has a strong grinding effect and a slight impact.
(3) Rough grinding depends on the impact, it should ensure that the steel ball and the lining plate are adhered with a layer of coarse particles, which can make the impact more effective during the impact. Therefore, the coarse grinding should ensure a high concentration of the pulp, about 75%~80%. . In the fine grinding, the ore itself is very fine, and some of it is in suspension state. The viscosity of the slurry itself is larger. The higher concentration of the slurry will produce large buoyancy, which has a great influence on the finely ground small diameter steel ball. A relatively thin pulp concentration should be used, about 50% to 65%.
(4) When the fine grinding is performed, the strength of the ore particles is large, the crushed ore particles are small, and the process conditions are difficult to control, resulting in low efficiency of the fine grinding process and high power consumption. Therefore, improving the fine grinding efficiency is of particular importance for fine grinding. The key to improving the fine grinding efficiency is to greatly increase the grinding area. To do this, only reduce the ball diameter and increase the number of balls. Therefore, for fine grinding, the ball diameter should be as small as possible under the premise of ensuring sufficient grinding force. For ore feeding with a particle size of 0.15 mm, a finely ground ball diameter of 8 to 10 mm is sufficient. However, due to the difficulty in production and high cost of such a small steel ball, and easy to discharge from the horizontal mill during use, the minimum ball diameter used in domestic fine grinding production can only be reduced to 30mm, and foreign countries can only be reduced. 15~20mm. If a smaller ball diameter is to be used, only a new ball mill adapted to small ball grinding can be developed. In the current fine grinding production, the diameter of the steel ball can be reduced to 30mm or even to 20 or 15mm. However, as pointed out earlier, fine grinding also requires a slight impact force, and the steel ball also needs a certain quality. The author's experimental research has confirmed that for a certain particle size feedstock, there is a lower limit value of the steel ball size, which is lower than At this critical value, the fine grinding efficiency also decreases.
(5) Regardless of the metal or non-metallic ore, the grinding before ore dressing is a dissociative grinding, and the grinding products are subject to beneficiation. Regardless of re-election, flotation or magnetic separation, it is limited by the particle size, and excessive pulverization is harmful. The over-grinding in fine grinding is more serious. Therefore, it is a significant problem to find a fine grinding efficiency in the fine grinding and to replace the small steel ball with a grinding and light grinding medium. Through many years of research and industrial experiments in a number of factories and mines, the author has come to the conclusion that short cylindrical media is used to replace fine grinding of small steel balls, which is not only efficient but also lightly crushed. It is recommended as a new fine grinding medium and is now in the country. Many factories have been applied. This will be described in detail later.
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