Ball mill (1)

First, the semi-theoretical formula derivation of the diameter of the steel ball
The existing ball diameter experience formula has a large limitation and a large error, and some are not very convenient to use in China. Therefore, it is necessary to find a ball diameter formula suitable for China's national conditions. As mentioned before, it is impossible to produce pure theoretical formulas at present, but it is entirely possible to find a framework of theoretical formulas and fill in semi-theoretical formulas with some empirical parameters. In the modern rock mine crushing, the anti-destructive performance of the rock ore can be known by engineering measurement. That is to say, how much damage or destructive force the large nuggets or ore particles need can be calculated. At present, the kinematics of the steel ball in the state of the ball mill is relatively thorough, and how much energy can be calculated when the steel ball falls. Therefore, it is theoretically feasible and scientific to select the steel ball size from the destructive force required for the rock mass to break. The following is a detailed description of how to choose the size of the steel ball according to the anti-destructive performance of the rock, that is, the semi-theoretical formula for deriving the diameter of the steel ball.
As with the derivation of other theoretical formulas, the theoretical formula derivation of the steel ball cannot be made without some allowable assumptions: 1 The mechanics of the steel ball as a drop motion is relatively thorough, and the theory of Davis and Levinson can provide the system. The mathematical method of calculating the motion of the steel ball, so take the steel ball under the throwing motion to study; 2 For the convenience of calculation, take the broken nugget or ore grain as a sphere, that is, a diameter d to represent the nugget or The size of the ore particles; 3 rock ore has a certain brittleness, the damage of the ore block under the impact is brittle failure, that is, the stress should be proportional to; the nugget or ore is uniaxially stressed when subjected to pressure or impact. , that is, the formed rupture surface is parallel to the pressure direction and passes through the center of the ball; the mechanical properties of the 5 rock ore are regarded as uniform, and the required destructive force can be calculated according to the strength limit and the force area; 6 is the dynamic load characteristic when the steel ball breaks the ore block The ability of the nugget to resist dynamic loads is lower than the ability to resist static loads. Compressive ultimate strength σ _pressure (static load) is easy to measure, and the current general information is available in the factory. The ultimate strength of impact resistance (dynamic load) is often lacking in the factory. The large number of data measured practice, generally more than 10 times the σ σ _{punch press,} it is assumed That is, the compressive limit strength σ _{pressure is} used as the calculation basis, and the σ _pressure can be converted into σ _punch . In addition, as with other conventional grinding theory studies, a steel ball is taken from the ball load as a representative for research, and since the steel ball falls at the same speed, the interaction between the steel balls can be ignored. Whether the above assumptions are reasonable can be proved by whether the theoretical and semi-theoretical formulas derived from practice tests are in line with reality.
   (1) Anti-destructive energy E _{-resistance of} ore or ore particles
Nuggets or ore particles are broken objects, and their ability to resist damage is the basis for selecting crushing forces. Obviously, their ability to resist damage is related to their mechanical strength and to the geometry of the nuggets or ore. Here, the compressive ultimate strength σ _pressure of the rock ore is used to indicate the mechanical strength of the rock. Because the Platinum hardness coefficient is commonly used in the ore dressing industry to indicate the strength grade of the rock, and . Since the destruction of nuggets or ore particles is related to their own geometrical dimensions, the greater the damage required for the nuggets or ore particles, the greater the absolute destructive power required for small nuggets or ore particles. The geometrical dimensions are indicated by the diameter d of the spherical ore particles.
In the grinding state of the grinding state, the steel ball is mainly affected by the impact of the crushing of the ore or the ore, and the grinding and stripping is supplemented. Therefore, the nuggets or ore particles are subjected to impact damage, which must be expressed by the impact resistance of the nuggets or ore particles to indicate their resistance to damage. According to the previous assumption, the impact strength of the nugget or ore is only one tenth of the compressive strength, namely:

A spherical ore or ore with a diameter d, as long as it is a section passing through the center of the sphere, its cross-sectional area is:

According to the previous assumption, the damage area of ​​the nugget or ore should be Then, the nugget diameter d or mineral particles can have a destructive impact of _{an anti-F:}

F _resistance = impact strength σ _rush × damage area

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The PA Luo Jin PoдиH hypothesis proposed by pulverization, fragmentation may be calculated in the following way function: d crushing force nugget diameter or mineral particles, which is equivalent to the value equal to the force F acting _anti distance d to make blocks ore or mineral The particle breakage is separated, which means that the required crushing work is:

When destroying a nugget, the required crushing work should be equal to or greater than the anti-destructive energy of the nugget. Therefore, the anti-destructive energy E of the nugget or ore is resistant to:

If the steel ball has a kinetic energy greater than the anti-destructive energy of the ore or ore, the crushing behavior of the ore or ore will occur.
In the derivation process, for the convenience of calculation, the centimeter-gram-second dimension is used, and the calculation result is also convenient for production.
(2) The kinetic energy of the steel ball E _n
The following figure shows the trajectory of the steel ball in the state of the throwing motion. The steel ball rises from the falling point B along with the cylinder to the disengagement point A, and then falls from the point A to the point B to complete a motion cycle.

In the picture   Speed ​​analysis of steel ball falling

A steel ball with a diameter of D _b (cm) and a volume of V (cm ³ ) was selected for study. When the ball falls falling pulp, pulp by the buffering action of buoyancy, and therefore, the weight of the ball is not functioning, but should be effective by weight C _eO ball of the ball effective mass m _e is:

Where ρ _e — the effective density of the steel ball in the slurry.
will By substituting (6):

When the ball falls off the back point B, which is a speed Ï… _p Ï… _t decomposed to fall back from the mill center point cutting method B direction velocity component tangential to the barrel Ï… _n peripheral velocity. The speed Ï… _{n is} in the impact of the ore block, and the tangential velocity Ï… _tO produces a grinding and stripping action on the ore block or the ore grain along the tangential direction of the tube wall. Here, the impact of the steel ball on the nugget is mainly considered, so Ï… _t is not considered.
According to the theory of Davis and Levinson, Ï… _n is:

also because , will be replaced by (8)

Where R ₁ — the inner radius of the mill. [next]
Thus, with effective mass m _e, normal velocity Ï… _n when a steel ball with a fall back to the point method is the kinetic energy E _n:

Since D=2R ₁ , the above formula is:

Since D=2R ₁ , the above formula is:

Considered in the foregoing is a steel ball on the outermost ball at the R ₁ position. However, the benefit of the outermost ball does not mean that the entire ball load work is beneficial. It is assumed that the total mass of the ball is concentrated on a certain layer of balls. This layer of ball is called the "intermediate polycondensation layer", and the diameter of the layer of this layer is D ₀ . According to the fan shape, the method of calculating the polar moment of inertia of the O point can be obtained:

Where R ₁ , R ₂ are the spherical layer radii of the outermost and innermost spheres. , k is related to the rotation rate Φ and the ball loading rate φ, and k can be directly found in Table 1. And D ₀ = 2R ₀ , then the normal impact kinetic energy E _n of a ball on the "intermediate polycondensation layer" falling onto the landing point liner is:

Table 1 The value of the parameter k when the ball loading rate φ and the rotation rate Φ

( III ) Semi-theoretical workmanship and empirical correction of steel ball diameter
Equation (5) is the energy of the nugget with a particle size d to resist damage, and formula (12) is the kinetic energy of a steel ball with a diameter of Db when striking a nugget. From the analysis of the fracture principle of rock ore, when the impact energy of the ball is greater than the energy of the ore block against the damage, that is, when En≥E is resistant, the ore block is broken, that is:

For the above formula, find the mathematical relationship between the objective function D _b (cm) and other parameters:

It can be known from the theory of steel ball motion that if cosa = Φ ² , then sin ² a = 1 - cos ² a = 1 - Φ ⁴ , sin2acosa = Φ ^{2 -} Φ ⁶ , and the equation (13) is further simplified to obtain:

Formula (14) is derived from the steel ball as a throwing motion, and the mechanics cannot be calculated when the ball is used for sloping motion. The height of the ball rises when the sloping type is not high, and the result calculated by the formula (14) should be appropriately increased. In addition, the breaking effect of the tangential velocity V _{t of the} steel ball is not considered in the previous derivation formula. If the effect of V _t is taken into account, the ball diameter calculated by the above formula should be considered to increase by 15%. Here, considering the limit state, equation (14) becomes:

This formula is derived from the systematic and rigorous Davis and Levinsson steel ball motion theory based on the principle of fracture mechanics. It reflects the steel ball diameter and mill rotation rate Φ steel ball effective density ρ _e mill the diameter D (D ₀ to D directly related, D ₀ also relate to ball ratio φ), anti-rock ultimate strength σ σ _pressure to the relationship between the _pressure and the ore particle size d. It is one of the most considered formulas in the ball diameter formula currently seen. It can be seen from the formula that a large ball diameter is required when the grain size of the ore is large and the ultimate strength of the rock ore is large; the diameter of the mill D also requires a large ball diameter; the effective density of the steel ball is ρ _e hours (that is, the pulp is too thick) The buoyancy is too large and the buffering effect on the ball is large.) A large ball diameter is also required, so the formula reflects the objective law in the process of steel ball crushing. The effect of the rotation rate on the diameter of the steel ball is complicated, and it is difficult to directly see it. It can be seen after calculation. But when Φ = 100%, the formula is calculated , indicating that the diameter of the steel ball is meaningless at this time, because the ball has been centrifuged at this time without the existence of a crushing effect. [next]
The structural framework of this formula is directly derived from the theory, but it includes the compressive ultimate strength σ _{pressure of} rock ore. The e _pressure cannot be calculated theoretically. It can only be obtained by means of actual engineering measurements, and, in consideration of the sloping motion. The ball diameter has been empirically amplified, so this formula can only be regarded as a semi-theoretical formula. If this formula is to be used in practice, some corrections should be made to the place where the derivation is not considered or the assumption is not true.
Equation (15) captures several major factors in the derivation, but some factors are not considered. In addition, in order to derive the formula, it is assumed that the mechanical properties of the rock and mineral are uniform, but in fact the mechanical properties of the rock and mineral are extremely uneven. Moreover, the influence of these factors on the ball diameter is very large, but they are difficult to enter the formula quantitatively. If you do not consider the influence of these factors, formula (15) will also produce large errors, and even make the formula useless. In fact, the various ball diameter formulas proposed by scholars in various countries have empirical correction coefficients. For such complex engineering problems, the purely theoretical formula without empirical correction has no practical value, because simple mathematical formulas cannot include the influencing factors of the grinding process. The following three aspects are corrected for formula (15):
(1) The inhomogeneity of the rock mechanics is corrected. The macroscopic and microscopic cracks in the coarse ore block are many, so the strength is low. When the nugget is thinned, various cracks gradually disappear and the strength gradually increases. The former Soviet scholars have studied the damage strength of different sizes of ore blocks, and the measured data are shown in Table (2). However, the measured data has not been seen when the ore size is less than 1 mm. Prof. Huji of Finland has made the measurement of the grinding power at different particle sizes. The results show that the increase in particle size and the increase in power consumption are not linear. The curve relationship with a similar power function increases, indicating that the power consumption increases much faster than the granularity in fine grinding. According to Hu Ji's research and the fine grinding research conducted by the author, the destructive force values ​​and correction factors below 1 mm are added in Table 2.
Foreign scholars have derived from the FC List theory that the ball mill's ore size is 1/2in (ie 12.7mm). According to European and American technical practice, when the 80% sieve size is 12.7mm, it is equivalent to 15mm of 95% sieve size. Since the formula derivation is carried out under ideal conditions, it can be considered that the optimal spherical diameter obtained is obtained with the lowest energy consumption. Therefore, based on 15mm, the mechanical strength of the nugget increases when it is less than 15mm, and the mechanical strength of the nugget decreases when it is greater than 15mm. In terms of mechanical properties of rock K ₁ represents a non-uniformity correction coefficient, and the reference to 15mm, set to 1, the particle size greater than 15mm in less than 1 K _1, K ₁ are less than 15mm greater than 1, is calculated for each size fraction damaging the K _{1 is} listed in Table 2, respectively.

Table 2   Relationship between failure strength of rock mass and mechanical correction coefficient K ₁ under various granularities

(2) Effective control and correction of the grinding process. During the grinding process, the rough grinding and the fine grinding are quite different. When rough grinding, the steel ball is easy to play with the ore particles, and the ball is easy to bite between the ball to grind, so the probability of grinding is high, the grinding effect is high, and the energy waste is small, that is, the grinding process is easy and effective to control. The fine grinding process is not the case. It is difficult for the steel ball to strike the ore particles between the ore and the ball. The probability of grinding is low, the grinding process is not well controlled, the grinding effect is low, and the energy is wasted.
It is still based on the 15mm with the lowest energy consumption, and K ₂ is the correction coefficient of the grinding process control. K ₂ <1 for those larger than 15 mm and K ₂ >1 for those smaller than 15 mm. Since there is currently unable to present a quantitative basis for amendments in this regard, assuming that only K ₂ and K ₁ equivalent effect.
(3) Correction of the influence of factors such as slurry viscosity. The certain slurry viscosity during rough grinding is beneficial to grinding. The ore particles are easy to adhere to the steel ball and the liner to form a layer of ore. When the steel ball collides with the steel ball or the liner, it can be broken first. The ore particles, the grinding process is effective, and the energy consumption is lower. A large viscosity during fine grinding has a detrimental effect and increases energy consumption. If K ₃ indicates the impact correction in this respect, it is difficult to propose a quantitative correction basis at present, and still follow the K ₂ treatment method, assuming that the influence of K ₃ at the same granularity is the same as K ₁ .
Therefore, let the comprehensive experience correction coefficient be K _C , according to the previous assumptions:

Whether the assumptions of K ₂ and K ₃ are reasonable should be tested by whether the final result is in line with reality.
Thus, after the semi-theoretical formula of the ball diameter is corrected, D _b (cm) becomes:

Equation (17) is the semi-theoretical formula for the ball diameter finally derived. The comprehensive correction factor K _C in the formula can be selected according to the data in Table 3.
Where Φ—mill rotation rate, ;
σ _pressure —rock compressive ultimate strength, kg/cm ² , σ _pressure =100f,
F—rock mass hardness coefficient;
ρ _e — the effective density of the steel ball in the pulp, g/cm ³ ;

ρ _e= ρ-ρ _n
Ρ—steel ball density, g/cm ³ ;
ρ _n — pulp density, g/cm ³ ;

ρ _t — ore density, g/cm ³ ;
R _d - slurry weight concentration (decimal);
D ₀ — ball "intermediate polycondensation layer" diameter, D ₀ = 2R ₀ , R _{0 is} determined by formula (11) and Table 1;
D—mill feed 95% sieve size, cm.
Thus, the required steel ball diameter can be calculated according to equation (17) under the actual given mill operating conditions. The formula is calculated using the cm•g•s system.

Table 3   Comprehensive empirical value positive coefficient K _C value