I. Overview figure 1 As shown in Fig. 1, the spherical diameter of the spherical particles of the same diameter is different on the light (thin) sheet, and the rounded diameter is different when cut at the waist of the ball, and the small round is obtained when the top of the ball is cut. . The rounding of spherical particles of different diameters is more different. So discuss ... Since d(v)j and d(A)i are continuously changed in size, the above relationship can be established. figure 2 Third, measurement and calculation methods image 3 Table 1 CACалтъΙκοв calculates all α ji coefficients when the number of groups k ≤ 15 (Table 1). When calculating the coefficients in the reference table, it should be noted that it is only applicable to the case where the diameter of the mineral particle sphere (or the plane truncation on the light (thin) sheet) is grouped by the phase synchronization length △. With this coefficient table, when (Nv)1 in the first example is calculated by the equation (10), the following formula can be used: Example galena in the lead zinc ore, as measured by using a particle diameter of the cross-sectional diameter d (A) i and (N A) i as shown in Table 2, the test meter - Table 2    Galena diameter method measurement results Serial number Section perimeter diameter interval d (A)i /mm Number of sections in 1mm 2 Particle diameter Number of spherical particles in 1mm 3 N (V)j /mm -3 N (A)i /mm -2 D (V)j /mm 1 0～0.018 0 0.018 0 2 0.018～0.036 18 0.036 198.71 3 0.036～0.054 twenty four 0.054 98.83 4 0.054～0.072 43 0.072 438.98 5 0.072～0.090 50 0.09 601.53 6 0.090～0.108 44 0.108 619.05 7 0.108~0.126 19 0.126 281.42 8 0.126～0.144 2 0.144 28.69 total 　 200 　 2267.21 Here k = 8; △ = 0.018 mm; α ji lookup table (table in mineral particle size measurement (2)) and Nv calculation data are listed in Table 2. An example of the calculation of (Nv)j is as follows: In summary, the measurement and calculation steps for solving the particle size of spherical mineral particles by the diameter method can be summarized as follows: table 3  Galena area method measurement results Serial number Particle cross-sectional area interval Ai(μ 2 ) Number of sections in 1mm 2 Particle diameter Number of spherical particles in 1mm 3 N (V)j /mm -3 N (A)i /mm -2 D (v)j /mm 1 9500～5994.5 40 0.10998 598.69 2 5994.5～3782.53 57 0.08736 865.2 3 3782.53～2386.78 45 0.0694 625.78 4 2386.78～1506.06 15 0.05513 -74.67 5 1506.06～950.32 19 0.04379 368.76 6 950.32～599.65 12 0.03478 177.54 7 599.65～378.38 10 0.02763 245.27 8 378.38～238.76 2 0.02195 -198.42 total 　 200 　 2608.15 Long U-Bolts, Custom U-Bolts, Extended U-Bolts Ningbo Brightfast Machinery Industry Trade Co.,Ltd , http://www.brightfastener.com
The mineral particles in the ore have a long-term extension, a two-way extension and a three-way equal length in terms of their monomer crystal habits. However, the situation changed when examining their “process granularityâ€. Because in the crushing, grinding and separation, whether the mineral in the ore is a monomer or an aggregate, as a geometric element that cannot be ignored in the process, the minerals are mostly irregular grains of equal length. Works with machinery and pharmaceuticals. Therefore, in the establishment of the basic theory and method of mineral particle size measurement, for the sake of simplicity, it can be assumed in principle that: (1) the mineral "standard particle size" of the particles is spherical, and its cross section on light and flakes is circular; (2) If the diameter d (v) of the spherical particles is sized into d (v)1 (d (v)min ), d (v) 2 , d (v) 3 ,..., d (V)j ,...,d (v)n (d (v)max ) The cross-sectional circle diameter d (A) on the light (thin) sheet can also be classified into d (A) 1 (d (A) min ), d (A) 2 by size . d (A) 3 ,...,d (A)i ,...,d (A)n (d (A)max ). 
When discussing the characteristic parameters of mineral particle size , at least the following three questions need to be answered: (1) the average diameter d (V) of spherical particles of different sizes;
(2) The number of mineral particles in a unit volume is d (v)j (N v ) I ; (3) The volume particle size distribution of a mineral in a mineral per unit volume is d (v)j F (v)j . These characteristic parameters can all be calculated from the two-dimensional characteristic parameters measured on the light (thin) sheet. The important problem is to determine the stereological relationship between the above three-dimensional feature parameters and the two-dimensional feature parameters, and to find a simple measurement and calculation method.
Second, the basic relationship - (NV) J 's formula
For ores composed of multiple minerals, we can assume that the diameter of the spherical particles of the mineral to be measured is d (v)1 (d (v)min ), d (v) 2 ,...,d (v)j ,..., d (v)max , their circular diameter on the light (thin) sheet is d (A)1 (d (A)min ), d (A) 2 ,...,d (A)i ,...,d (A )max . According to the previous assumptions, d (A)1 and d (A)max =d (V)max . If the particle diameter d (v)j and the plane circular diameter d (A)i are divided into the same number of intervals by size, then
d (A)1 =d (v)1
d (A)2 =d (v)2
d (A)max =d (v)max[next]
The following characteristic parameter list can be listed for this. d ( v ) j d ( v ) 1 d ( v ) 2 d ( v ) 3 d ( v ) 4 d ( v ) 5 d ( A ) i d ( A )1 d ( A )2 d ( A )3 d ( A )4 d ( A )5 ( N v ) j ( N v ) 1 ( N v ) 2 ( N v ) 3 ( N v ) 4 ( N v ) 5 ( N A ) i ( N A ) 1 ( N A ) 2 ( N A ) 3 ( N A ) 4 ( N A ) 5
It can be seen from Fig. 1 that the truncated circle with a diameter d(A)i on the light (thin) piece is not always the truncation circle of d(V)j=i ball, but includes d(V)j when j ≥ i The rounding of all the balls, ie [next]
1. Diameter measurement The diameter measurement method is first proposed by XA Щварц and then refined by CA CалтъΙκοв. So also known as
The diameter method is to measure the maximum cross-sectional diameter d (A)max on the light (thin) sheet, and then divide the rounds of different sizes into k groups, and the group distance (also called step size) is â–³. Ie d (A)max
△ = ——————
k
From this, d (A) 1 = 1 △, d (A) 2 = 2 Δ, ... d (A) i = i Δ..., d (A) max = k Δ. According to the previous assumptions, the spatial diameter d (V) of the spherical spherical particles can also be divided into k groups, and there are d (V)max = d (A)max and d (V)1 = 1 △, d (V ) 2 = 2 △, ..., d (V)j = j △...d (V)max = k △.
As previously stated, the number of spherical particles (Nv) j with a diameter equal to d (V)j (j = 1, 2, ..., k) per unit volume can be calculated by equation (5)
(N A ) ij
(N V ) j = —————
F ij d (V)j
Therefore, the formula for measuring and calculating (N A ) ij and F ij needs to be derived.
F ij is the probability that a d (A)i truncation is obtained when a light (thin) slice cuts a d (V)j ball. As shown in Fig. 3, let d (V)j = 2r max and d (A)i = 2r. Available from the map [next]
[next]
1
(N v ) 1 = ——[ α 1-1 (N A ) 1 + α 1-2 (N A ) 2 + α 1-3 (N A ) 3 ]
â–³
In the formula, the coefficient α ji , except for the first term (ie, when ji) is positive, the other items are negative values ​​(the table has been marked with a positive sign (+) or minus sign (- in front of the corresponding coefficient value). )).
Where j is the number of groups of mineral particle diameter d(v);
i - the number of groups of circular (d) diameters on the light (thin) sheet;
(N v ) 1 - the number of particles of the spherical particle diameter d (v) 1 = 1 Δ per unit volume ;
1
—— α 1-1 (N A ) 1 —— The spherical diameter from the plane truncation of d (A)1 = 1 △ can be calculated as d (V)1 = 1 △, d (V) 2
â–³
= 2 â–³, d (V) 3 = 3â–³ total number of particles.
(N A ) 1 ——The number of cut-offs of the diameter d (A)1 = 1△ per unit area on the light (thin) sheet. Mineral particles with a spherical diameter d (V)j ≥ d ( a) 1 may be cut by light (thin) sheets into a truncated circle of d (V) 1 = 1 △. That is, (N A ) 1 contains d
(V)1 = 1 â–³, d (V) 2 = 2 â–³, d (V) 3 = 3 â–³ Three types of spherical mineral particles are cut by light (thin) sheets
(A) 1 = 1 â–³ truncation;
1
—— α 1-2 (N A ) 2 —— The spherical diameter deduced by the plane truncation of d (A) 2 = 2 △ is d (V) 2 = 2 △ , d (V)
â–³
3 = 3 â–³ total number of particles.
(N A ) 2 ——The number of cut-offs of the diameter d (A) 2 = 2 △ per unit area on the light (thin) sheet. Similarly, (N A ) 2 contains d (A) 2 which are cut from light (thin) sheets by spherical mineral particles of d (V) 2 = 2 △ and d (V) 3 = 3 Δ. a circle of =2 △;
1
- α 1-3 (N A) 3 - a d (A) 3 = 3 △ sectional plane circle diameter of the ball can be calculated respectively d (V) 3 = 3 △ , the teeth
â–³
The total number of grains. Conversely, it can be understood that the number of rounds of the mineral particles in the spherical diameter d (V) 3 = 3 Δ , which are cut by light (thin) sheets into d (A) 3 = 3 △;
(N A ) 3 ——The number of cut-offs of the diameter d (A) 3 = 3 △ per unit area on the light (thin) sheet. This kind of rounding can only be done by d (V)3 = 3
The â–³ mineral particle ball is obtained by cutting. [next]
Calculate the (N V ) j and d (V) of the galena.
- 1
d(V) = —— Σ(NV)j·(dv)j = 204.68mm/2267.21 = 0.0903mm
Nv [next]
(1) measuring the perimeter of each particle section of the mineral on the A T area of ​​the light (thin) sheet under an image analyzer;
(2) Converting the section length into the section perimeter diameter d(A)i, and finding d(A)max from it;
(3) Grouping the perimeter diameters of the sections, the number of groups k is preferably not more than 10~15, and the group distance â–³ = d(A)max/k
d (A)i = i â–³,i = 1,2,3...,k.
(4) Measure and calculate (NA)i. That is, the number of d(A)i cross sections in the field of view area AT is measured. Ni.(NA)i = ——. Depending on the density of the mineral particles and the uniformity of the distribution, the image analyzer can be either all-optical (thin) or partially mineral particles. The mineral particles in the AT field are usually not less than 200~300.
k
(5) Calculate (Nv)j, Nv = Σ(N v ) j and (d v ) according to the format of Table 1 of equation (10 ).
j=1
2. The area measurement method CACалтъΙκοв improved the above diameter measurement method, and used the area measurement method to derive a simpler measurement and calculation formula. Because it can be proved that the spatial size distribution of the small 闰 does not obey the normal distribution, but obeys the lognormal distribution. Therefore, for the grouping of particle sizes, the number of equal groups is not used, and the method of equal series is used. For example, when CACалтъΙκοв uses a ratio of the cross-sectional diameter d(A)i or a cross-sectional area Ai of 10 -0.1 or (10 -0.1 ), that is:
d (A)i+1
—————— = 10 -0.1 = 0.7943
d (A)i
Ai+1
Or ————— =( 10 -0.1 )2 = 0.6310 (11)
Ai
From this, the coefficients in the equation (10) can be derived. When k = 12, the formula for (N v ) j is 1
(N v ) j = ————[1.6461(N A ) i - 0.4561(N A ) i - 1 - 0.1162(N A ) i - 2
d (v)j
- 0.0415(N A ) i - 3 - 0.0173(N A ) i - 4 - 0.0079(N A ) i - 5
- 0.0038(N A ) i - 6 - 0.0018(N A ) i - 7 - 0.0010(N A ) i - 8
- 0.0003(N A ) i - 9 - 0.0002(N A ) i - 10 - 0.0002(N A ) i - 11 ] [next]
In the formula, j and i are also group ordinal numbers, except that the particle diameter (or cross-sectional area) at which j (or i) = 1 is the largest. (NA)i is the measured value, i = j of the first term. i = 1...j(i ≤ j); j = 1...k.
The lead-zinc ore in the second example is tested by the area method for the lead ore. The test data and calculation results are shown in Table 3.