On the Question of the Equation for the Size Characteristic of Hard Coals
Abstract Generated abstract
The paper examines analytical equations used to describe the particle size distribution of hard coals and argues that established formulas by Andreev, Rosin and Rammler, Chernyi, and Beloglazov do not adequately represent the coarser size classes. Using logarithmic transformations of experimental size characteristic curves for coals from different basins, the author derives a modified equation that straightens the curve over a wider range and satisfies the limiting conditions at zero and maximum particle size. Comparison with experimental data indicates that the proposed general form gives closer agreement for hard coal, especially when using a material-dependent coefficient, with a suggested value of about 1.11 for hard coals and 2.0 for quartz and feldspar.
Full Text
On the Question of the Equation for the Size Characteristic of Hard Coals
B. A. Zemlyakov
(Presented by Prof., Doctor of Technical Sciences I. V. Gebler)
The granulometric composition of loose materials plays an important role in many processes of their processing and in their use in a number of branches of industry. Therefore it is necessary to know the regularities of change in the sieve composition of materials and to express these regularities in the form of equations.
In the literature there is a large number of works by various authors devoted to the search for an analytical expression of equations for the size characteristic. The equation of S. E. Andreev [1], [4] and the exponential equation of P. Rosin and E. Rammler [5] have become most widespread.
In S. E. Andreev’s equation
\[ Y = AX^{k} \tag{1} \]
\(X\) is the particle size of the product to which the weight cumulative yield in minus \(Y\) corresponds;
\(A\) and \(k\) are experimental constants.
\[ A = \frac{1}{M^{k}}, \]
where \(M\) is the size of the largest particles of the product.
Equation (1) reflects well the distribution of particles by size classes only for 60% of the finest classes of the product and becomes relatively little suitable when it is necessary to estimate the composition of the coarsest classes. The value of the parameter \(M\), obtained from the equation \(M_T\), always proves to be less than its actual value \(M_0\) (Fig. 1).
In the equation of P. Rosin and E. Rammler, proposed by the authors for coal,
\[ Z = e^{-Rx^{m}}, \tag{2} \]
where
\(X\) is the size of the product to which the weight cumulative yield in plus equal to \(Z\) corresponds;
\(e\) is the base of the natural logarithm;
\(R\) and \(m\) are constants determined experimentally,
\[ R=\frac{1}{X_e^m}, \]
where \(X_1\) is the size to which the cumulative oversize yield of the product is equal to 0.368.
Equation (2) well reflects the distribution of 80% of the finest particles by size and, like equation (1), may lead to substantial errors in estimating the composition of the coarsest size classes of the material. Equation (2) does not satisfy the limiting conditions (at \(X=M,\ Z\ne 0\)). Calculation by this equation is considerably more complicated than by equation (1).
L. M. Chernyi [6] proposed an equation for the size characteristic that gives good agreement with experimental data, but because of its complexity, the presence of three empirical constants, and the need to use special tables, this equation cannot be considered entirely suitable for practical calculations.
N. K. Beloglazov proposed an equation for the size characteristic of monomineral ores [2]:
\[ Y=\frac{2bX^n}{1+bX^n}. \tag{3} \]
Fig. 1. Size characteristic of hard coal in logarithmic coordinates \(\lg Y,\ \lg X\).
The coefficients \(b\) and \(n\) entering into the equation are determined experimentally. Equation (3) satisfies the limiting conditions (\(X=M,\ Y=1;\ X=0,\ Y=0\)) and is applicable in the range of values \(Y=0.9\), i.e., for 90% of the product. Calculation by equation (3) is considerably simpler than by the equation of P. Rosin and E. Rammler and of L. M. Chernyi. In the same work N. K. Beloglazov points out that, when it is necessary to characterize the size of all 100% of the material, one should use the equation he proposed
\[ Z=e^{-\frac{bX^n}{1-bX^n}} \tag{4} \]
The investigations we carried out of the size characteristics of hard coals showed that none of the equations considered fully reflects the actual distribution of coal particles by size (Table 1). Curves constructed on the basis of calculations by the equations and on the basis of experimental data differ significantly from one another. This discrepancy, apparently, was to be expected, since coal is not a homogeneous mass and in its properties is analogous rather to polymineral than to monomineral ores, for which the equations listed above were mainly derived.
The absence of an equation for the size characteristic that would make it possible, with a sufficient degree of accuracy, to characterize the distribution
particles of hard coal by size led us to the search for such an equation. In the new equation it was necessary, on the one hand, to use the property of equations (1) and (2) of giving good agreement between experimental data and calculated data in the region of small values of \(X\), and, on the other hand, to try to extend this region to coarser particles of the material.
If the experimental data are plotted in the coordinate system \(\lg X\) and \(\lg Y\), then, according to equation (1), expressed in logarithmic form,
\[ \lg Y = k \lg X + \lg A , \tag{5} \]
the points obtained should lie on one straight line, the tangent of whose angle of inclination to the \(\lg X\) axis is numerically equal to the value of the parameter \(k\). The noted condition remains valid only for part of the experimental data in the region of small values of \(X\).
If all the experimental points lay on a single straight line, this would mean that the equation of this straight line reflects the actual size distribution of the product particles. Then, from the equation obtained, it would be possible to determine the value of the parameter \(k\), the size of the coarsest particles of the product \(M\), and the yield of the individual size classes of the product.
In order for all the experimental points to fall on one straight line (Fig. 1), it is necessary that the experimental curve \(ABED\), in the region of large values of \(X\), deviate to the left and coincide with the straight line \(AG\). In establishing the corresponding dependence between \(X\) and \(Y\), this condition proved to be essential.
If equation (5) is transformed,
\[ \lg \frac{1}{Y} = - k \lg X - \lg A \tag{6} \]
and the property of logarithms is used, according to which the absolute value of the difference \(\lg v - \lg (v-p)\), for \(v > p\), will be the greater the smaller the value of \(v\), then it follows that the value of \(Y\) will be the greater, the greater the difference between \(\lg \frac{1}{Y}\) and \(\lg \left(\frac{1}{Y} - p\right)\). At the same time, as in equation (1), the curve of the function \(\lg \left(\frac{1}{Y} - p\right)\) must lie below the curve of the function \(\lg \frac{1}{Y}\).
The analysis carried out of the size characteristics of coals from various coal basins showed that, by selecting the value of the coefficient \(p\), it is possible to achieve a significant change in the character of the experimental curve. The straightening of the curve of the size characteristic of hard coals in logarithmic coordinates is obtained over a broader range at the value \(p = 0.9\), that is, in the coordinates \(\lg \left(\frac{1}{Y} - 0.9\right)\) and \(\lg X\) (Fig. 2). Consequently, the equation of this straight line will have the form
\[ \lg \left(\frac{1}{Y} - 0.9\right) = - n \lg X - \lg A, \]
whence
\[ Y = \frac{A X^n}{1 + 0.9 A X^n}. \tag{7} \]
In the equation obtained:
\(X\) is the size of the product particles to which the cumulative weight yield of undersize \(Y\) corresponds;
\(A\) and \(n\) are constants determined from experiment
\[ A=\frac{1}{0.1M^n}, \]
where \(M\) is the mean size of the largest product particles.
The parameter \(n\) reflects the character of the distribution of particles by size classes, or the degree of curvature of the size-characteristic curve in the coordinates \(X\) and \(Y\).
Equation (7) satisfies the boundary conditions (for \(X=0,\ Y=0\); for \(X=M,\ Y=1\)) and has the advantage that it is applicable in the range of values of \(Y\) from zero to 90–95%.
If in the equation obtained we take \(A=\dfrac{C}{0.9}\) (where \(C\) is an experimental constant), then equation (7) assumes the form
\[ Y=\frac{1.11CX^n}{1+CX^n}. \tag{8} \]
Fig. 2. Straightening of the size-characteristic curve in logarithmic coordinates.
Calculation of the size characteristics of hard coal, carried out by equation (8), showed that for almost all values of \(X\) the results obtained are closest to the experimental ones. This situation is easy to trace in Table 1. At small values of \(X\), the quantity \(CX^n\) may be neglected, and then equation (8), like equation (3), be-
Table 1
Comparison of experimental data with calculated data obtained by various equations of size characteristics
| Size, mm | Cumulative yield of product in undersize — \(Y\), %: actual | Cumulative yield of product in undersize — \(Y\), %, calculated by equations: S. E. Andreev (1) | Cumulative yield of product in undersize — \(Y\), %, calculated by equations: Sh. Rozin and E. Rammler (2) | Cumulative yield of product in undersize — \(Y\), %, calculated by equations: N. K. Beloglazov (3) | Cumulative yield of product in undersize — \(Y\), %, calculated by equations: (9) |
|---|---|---|---|---|---|
| 150 | 100,00 | 292,00 | 99,03 | 140,50 | 99,30 |
| 80 | 89,12 | 188,00 | 94,04 | 118,00 | 91,50 |
| 60 | 87,53 | 153,00 | 89,46 | 106,70 | 86,80 |
| 40 | 80,90 | 115,00 | 78,95 | 90,70 | 78,80 |
| 25 | 65,24 | 82,80 | 67,50 | 72,70 | 67,80 |
| 14 | 53,22 | 50,20 | 50,50 | 53,00 | 52,90 |
| 10 | 43,66 | 43,35 | 41,85 | 43,20 | 44,25 |
| 6 | 31,26 | 30,60 | 30,50 | 31,10 | 32,20 |
| 3 | 18,56 | 18,50 | 19,30 | 19,10 | 19,40 |
| 1 | 8,41 | 8,65 | 8,20 | 8,47 | 7,82 |
| 0 | 0,00 | 0,00 | 0,00 | 0,00 | 0,00 |
| \(K=-0,705\) \(A=-8,55\) \(M_{\mathrm{t}}=32,70\) |
\(m=0,790\) \(R=0,0888\) \(x_e=21,45\) |
\(n=0,794\) \(b=0,0442\) \(M_{\mathrm{t}}=51,30\) |
\(n=0,942\) \(c=0,0755\) \(M_{\mathrm{t}}=166,00\) |
passes into the equation of S. E. Andreev. This indicates that equation (1) is an approximate expression of the given equation.
The equation obtained can, in general form, be expressed as follows:
\[ Y=\frac{BCX^n}{1+CX^n}, \tag{9} \]
where \(B\) is a certain coefficient constant for the given material.
If in the equation obtained we take \(B=2.0\), then equation (9) coincides with equation (3). However, N. K. Beloglazov gives this coefficient only a formal character and does not connect it with the physical properties of the product. Meanwhile, according to our observations, the value of the coefficient \(B\) was found to be equal to two for monomineral ores (quartz, feldspar) and \(B=1.11\) for hard coals. Consequently, the more universal form, in comparison with other forms, of the equation for the size characteristic should be equation (9), in which the coefficient \(B\) may take different values depending on the physical properties of the product. For hard coal, some deviation of the value of the coefficient \(B\) from \(B=1.11\) is sometimes observed, within the limits from \(B=1.0\) to \(B=1.25\). It is quite possible that these deviations are a consequence of differences in the petrographic composition of coals from individual deposits and seams.
The circumstance noted makes it possible to raise the question of the possible existence of a regularity in the change of the coefficient \(B\) for different products. Establishing such a regularity would considerably facilitate the task of determining the size characteristic of different products.
Conclusions
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The size characteristics of dispersed products can be represented most fully by equation (9).
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The coefficient \(B\) of equation (9) may take different values depending on the physical properties of the products. For example, for hard coals \(B=1.11\), and for quartz and feldspar \(B=2.0\).
Literature
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Andreev S. E. Mining Journal, No. 6, GONTI, 1959.
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Beloglazov N. K. “Equation of the size characteristic of monomineral ores.” Proceedings of the Leningrad Mining Institute, vol. 32, issue 3, Metallurgizdat, 1956.
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Egorov I. G. Theory of Crushing and Fine Grinding. GONTI NKTP USSR, 1938.
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Razumov K. A. Design of Concentrating Plants. Metallurgizdat, 1952.
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Rosin P. and Rammler E. Journal of Fuel Inst. 7, X, 33, No. 31--29, 1933.
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Chernyi L. M. “Regularities of the granulometric composition of products of crushing and grinding.” Beneficiation of Ores of Mining-Chemical Raw Materials, issue 1, Proceedings of GIGKhS, 1950.