Determination of output and mine car weight by blocks at iron ore deposits in the absence of weighing facilities at the mine
Abstract Generated abstract
The paper addresses the problem of determining monthly ore output and mine car weights by production block at iron ore mines that lack weighing facilities. It argues that the common practice of distributing the discrepancy between statistical and actual mine output in proportion to recorded block output can give substantial errors, because it does not account for differences in ore grade and car loading. A correction method is developed using simultaneous balances of ore mass and metal, with least squares allocation of block corrections, and an example shows much closer agreement with actual block output than the mass balance method alone. The corrected block outputs are then used to estimate average mine car weights and to derive an empirical relation between car weight and metal content.
Full Text
Determination, for Iron-Ore Deposits, of Output and Mine-Car Weight by Blocks in the Absence of Weighing Facilities at the Mine
V. I. Akulov
(Presented by the scientific seminar of the Department of Mine Surveying and Geodesy)
In the absence of weighing facilities at a mine, the output of ore mass for the reporting month by individual blocks is determined either from statistical accounting or from mine-survey measurements. By mine-survey measurements it is possible to determine output only for those blocks in which instrumental surveying of the stoping face is possible.
Determination of output from statistical accounting is based on daily shift-bureau records of the number of mine cars received from each block.
In statistical accounting it is necessary to know the weight of a mine car for each block; this is usually established by direct weighing once a month or once a quarter, and sometimes after an even longer interval of time.
Extending the mine-car weight obtained from sample weighing to all mine cars received from a block during the month is one of the reasons for the considerable deviations observed in practice between statistical output for the mine as a whole and actual output. Therefore, in the absence of weighing facilities, the problem of correcting the statistical output of blocks in accordance with the actual output of ore mass for the mine as a whole practically always arises.
The actual output of ore mass for the mine as a whole during the reporting month, in the absence of weighing facilities, can be determined with sufficient accuracy from the balance of ore-mass movement on the surface in the case where it is not beneficiated, i.e., is marketable ore; or from the movement balances of marketable ore and tailings, or from the marketable ore obtained during the reporting month, in the case where the ore mass is beneficiated.
When determining, from the marketable ore obtained during the reporting month, the actual output of ore mass for the mine as a whole, one proceeds from the following system of equations:
\[ Q_T + Q_x = Q_0; \]
\[ Q_T C_T + Q_x C_x = Q_0 C_0, \tag{1} \]
where \(Q_T\) is the output of marketable ore for the month;
\(Q_x\) is the amount of tailings for the month;
$Q_0$—the actual production of ore mass for the mine as a whole for the month;
$C_T$, $C_x$, and $C_0$—the metal content, respectively, in commercial ore, tailings, and in the ore mass brought to the surface by the mine.
Solving the system of equations (1) with respect to $Q_0$, we obtain:
\[ Q_0 = Q_T \frac{C_T - C_x}{C_0 - C_x}, \tag{2} \]
Determination of the actual production of ore mass for the mine as a whole by formula (2) is carried out at some mines, for example, at the Temir-Tau mine of the Kuznetsk Metallurgical Combine.
Adjustment of the statistical production of blocks in accordance with the actual production of ore mass for the mine as a whole may be performed in two ways:
1) by the ore-mass balance,
2) by the balance of ore mass and metal.
When adjusting the production of individual blocks by the ore-mass balance, the discrepancy between the statistical production for the mine as a whole and the actual production is distributed in proportion to the statistical production. This method of adjusting the statistical production of blocks takes into account neither the degree of loading of mine cars from different blocks nor the correctness of determining their weight. As a result, the production of blocks established on the basis of the ore-mass balance may differ considerably from the actual production.
A close agreement of the adjusted production with the actual production, as our studies have shown, is obtained when the production of blocks is established from the balances of ore mass and metal brought to the surface for the mine as a whole. The method of adjusting the statistical production of blocks by the balances of ore mass and metal brought to the surface has not yet been developed in the literature; therefore, below we shall discuss its theoretical aspect.
Let us introduce the notation:
$q_1, q_2, \ldots, q_n$—the production of ore mass, respectively, from blocks $1, 2, \ldots, n$ according to statistical records;
$(q_1), (q_2), \ldots, (q_n)$—the correction to the statistical production of ore mass, respectively for blocks $1, 2, \ldots, n$, due to adjustment;
$Q_1, Q_2, \ldots, Q_n$—the adjusted production of ore mass, respectively, of blocks $1, 2, \ldots, n$;
$C_1, C_2, \ldots, C_n$—the metal content in the ore mass mined, respectively, from blocks $1, 2, \ldots, n$;
$Q_0$—the actual production of ore mass for the mine as a whole;
$C_0$—the metal content in the ore mass brought to the surface for the mine as a whole;
$n$—the number of operating blocks.
The adjusted production of blocks must satisfy the following two equations:
\[ Q_1 + Q_2 + \ldots + Q_n = Q_0, \tag{3} \]
\[ Q_1 C_1 + Q_2 C_2 + \ldots + Q_n C_n = Q_0 C_0. \tag{4} \]
Equation (3) is the balance of ore mass, and equation (4) is the balance of metal brought to the surface for the mine as a whole.
Substituting the statistical output into equations (3) and (4) instead of the corrected output, we obtain
\[ q_1+q_2+\ldots+q_n-Q_0=\Delta Q. \tag{5} \]
\[ q_1C_1+q_2C_2+\ldots+q_nC_n-Q_0C_0=\Delta M, \tag{6} \]
where \(\Delta Q\) is the difference between the statistical output of ore mass for the mine as a whole and the actual output, or the discrepancy in the ore-mass balance;
\(\Delta M\) is the discrepancy in the metal balance.
Since
\[ \begin{aligned} Q_1&=q_1+(q_1),\\ Q_2&=q_2+(q_2),\\ &\ldots\\ &\ldots\\ Q_n&=q_n+(q_n), \end{aligned} \tag{7} \]
then, substituting the corrected output from (7) into equations (3) and (4) and taking into account equalities (5) and (6), we obtain
\[ (q_1)+(q_2)+\ldots+(q_n)+\Delta Q=0, \]
\[ (q_1)C_1+(q_2)C_2+\ldots+(q_n)C_n+\Delta M=0. \tag{8} \]
The system of equations (8), assuming that the metal content in the ore mass both for the mine as a whole and for individual blocks has been determined from sampling with sufficient accuracy, contains \(n\) unknown corrections \((q_1), (q_2), \ldots, (q_n)\) to the statistical output of the blocks. Consequently, when \(n>2\), the system of equations (8) is indeterminate; for the corrections \((q_1), (q_2), \ldots, (q_n)\), an infinite set of values can be obtained.
Among all corrections \((q_1), (q_2), \ldots, (q_n)\) satisfying the system of equations (8), we shall find those corrections that satisfy the condition:
\[ \sum_1^n \frac{(q_i)^2}{q_i}=\text{minimum}. \tag{9} \]
The corrections \((q_1), (q_2), \ldots, (q_n)\) satisfying the system (8) and condition (9) are calculated by the formulas:
\[ \begin{aligned} (q_1)&=q_1\kappa_1+q_1C_1\kappa_2,\\ (q_2)&=q_2\kappa_1+q_2C_2\kappa_2,\\ &\ldots\\ &\ldots\\ (q_n)&=q_n\kappa_1+q_nC_n\kappa_2, \end{aligned} \tag{10} \]
where \(\kappa_1\) and \(\kappa_2\) are undetermined multipliers, or correlates. The correlates \(\kappa_1\) and \(\kappa_2\) are determined from the system of normal equations:
\[ \sum_1^n q_i\kappa_1+\sum_1^n q_iC_i\kappa_2+\Delta Q=0, \]
\[ \sum_1^n q_iC_i\kappa_1+\sum_1^n q_iC_i^2\kappa_2+\Delta M=0, \tag{11} \]
where \(\sum_{1}^{n} q_i\) is the production of ore mass for the mine as a whole according to statistical records.
Solving the system of normal equations (11), we obtain:
\[ K_1 = - \frac{\Delta Q}{\sum_{1}^{n} q_i} - \frac{\sum_{1}^{n} q_i C_i}{\sum_{1}^{n} q_i}\,K_2, \tag{12} \]
\[ K_2 = \frac{ \dfrac{\Delta Q \sum_{1}^{n} q_i C_i}{\sum_{1}^{n} q_i} - \Delta M }{ \sum_{1}^{n} q_i C_i^2 - \dfrac{\left(\sum_{1}^{n} q_i C_i\right)^2}{\sum_{1}^{n} q_i} }. \tag{13} \]
Let us denote:
\[ \overline{C} = \frac{\sum_{1}^{n} C_i q_i}{\sum_{1}^{n} q_i}, \tag{14} \]
where \(\overline{C}\) is the metal content in the ore mass delivered to the surface for the mine as a whole according to statistical records. In the notation (14), formulas (12) and (13) take the form
\[ K_1 = \frac{\Delta Q}{\sum_{1}^{n} q_i} - \overline{C}K_2, \tag{15} \]
\[ K_2 = \frac{\Delta Q \overline{C} - \Delta M} { \sum_{1}^{n} q_i C_i^2 - \overline{C}^{\,2}\sum_{1}^{n} q_i }. \tag{16} \]
Formula (16), taking into account equalities (5), (6), and (14), takes the form:
\[ K_2 = \frac{Q_0\left(C_0-\overline{C}\right)} { \sum_{1}^{n} q_i C_i^2 - \overline{C}^{\,2}\sum_{1}^{n} q_i }. \tag{17} \]
For convenience of calculation, it is necessary to express the statistical output of the blocks, i.e., \(q_1, q_2,\ldots,q_n\), in formulas (10), (15), and (17) in tens of thousands of tons; in this case the indicated formulas take the form:
\[ \begin{aligned} (q_1) &= q_1'\kappa_1 + q_1'C_1\kappa_2,\\ (q_2) &= q_2'\kappa_1 + q_2'C_2\kappa_2,\\ &\ .\ .\ .\ .\ .\ .\ .\\ &\ .\ .\ .\ .\ .\ .\ .\\ (q_n) &= q_n'\kappa_1 + q_n'C_n\kappa_2, \end{aligned} \tag{10a} \]
\[ K_1 = -\,\frac{\Delta Q}{\displaystyle\sum_1^n q_i'} - \overline{C}K_2, \tag{15a} \]
\[ K_2 = \frac{Q_0(C_0-\overline{C})} {\displaystyle\sum_1^n q_i'C_i^2-\overline{C}^{\,2}\displaystyle\sum_1^n q_i'}, \tag{17a} \]
\[ q_i'=\frac{q_i}{10000}. \tag{18} \]
The correctness of determining the corrections \((q_1),(q_2),\ldots,(q_n)\) is checked by the equality:
\[ (q_1)+(q_2)+\ldots+(q_n)=-\Delta Q. \tag{19} \]
The adjusted output for individual blocks is calculated by formulas (7). The correctness of determining the adjusted output of the blocks is checked by equalities (3) and (4).
Let us consider the adjustment of the statistical output of individual blocks for the case when, throughout the mine as a whole, all mine cars have the same loading by volume.
With the same loading by volume of all mine cars in the mine, the actual output of ore mass for individual blocks will be:
\[ \begin{aligned} Q_1^{\text{ф}} &= tq_1,\\ Q_2^{\text{ф}} &= tq_2,\\ &\ .\ .\ .\\ &\ .\ .\ .\\ Q_n^{\text{ф}} &= tq_n, \end{aligned} \tag{20} \]
where \(t\) is the coefficient of loading of the mine cars by volume.
The metal content in the ore mass actually delivered to the surface for the mine as a whole will be:
\[ C_0= \frac{\displaystyle\sum_1^n Q_i^{\text{ф}}C} {\displaystyle\sum_1^n Q_i^{\text{ф}}} \tag{21} \]
or, taking into account equalities (20), we obtain:
\[ C_0=\frac{\displaystyle\sum_{1}^{n} q_i C_i}{\displaystyle\sum_{1}^{n} q_i}. \tag{22} \]
Comparing formulas (14) and (22), we note that \(C_0=\overline{C}\). When \(C_0=\overline{C}\), the correlates \(K_1\) and \(K_2\), on the basis of formulas (15) and (17), are equal to:
\[ K_1=-\frac{\Delta Q}{\displaystyle\sum_{1}^{n} q_i}, \tag{23} \]
\[ K_2=0. \tag{24} \]
Substituting \(K_1\) and \(K_2\) from (23) and (24) into (10), we obtain:
\[ \begin{aligned} (q_1)&=-\frac{\Delta Q}{\displaystyle\sum_{1}^{n} q_i}\,q_1,\\[4pt] (q_2)&=-\frac{\Delta Q}{\displaystyle\sum_{1}^{n} q_i}\,q_2,\\[4pt] &\ldots\ldots\\ &\ldots\ldots\\[4pt] (q_n)&=-\frac{\Delta Q}{\displaystyle\sum_{1}^{n} q_i}\,q_n. \end{aligned} \tag{25} \]
Consequently, with the same loading by volume of all mine cars in the mine (provided that the bulk density of the ore mass for individual blocks has been established with sufficient accuracy and does not change during the month), i.e., when \(C_0=\overline{C}\), the discrepancy between the statistical output of ore mass for the mine as a whole and the actual output is distributed in proportion to the block output according to statistical accounting, which corresponds to the method used in practice. In this case, the corrected output of individual blocks will satisfy both the ore-mass balance (3) and the metal balance (4).
In all other cases, i.e., when the metal content \(\overline{C}\), calculated by formula (14), is not equal to the actual \(C_0\) obtained from sampling, the correction of the statistical output of the blocks must be carried out on the basis of the balances of ore mass and metal delivered to the surface for the mine as a whole, i.e., on the basis of formulas (10a), (15a), and (17a).
Let us show by an example the advisability of correcting the statistical output of blocks on the basis of the balances of ore mass and metal delivered to the surface for the mine as a whole.
A mine extracting iron ore has four blocks in operation. The weight of a mine car, depending on the iron content in the ore mass, is expressed by the equation
\[ d = 0{,}8 + 0{,}035C, \tag{26} \]
where \(C\) is the iron content in the ore mass, in %.
On the basis of equation (26), in column 3 of Table 1 the actual weights of mine cars for the different blocks have been calculated.
Let us assume that, in the statistical accounting of output, the weight of a mine car has been adopted for the mine as a whole as equal to \(2{,}75\,t\). It is necessary, on the basis of the actual output of ore mass for the mine as a whole \((Q_0 = 76{,}040\,t)\) and its iron content \((C_0 = 52{,}45\%)\), to establish the most probable output for the individual blocks, assuming that the actual weights of the mine cars (column 3, Table 1) are not known to us.
APPENDIX
Table 1
| No. of blocks | Number of mine cars, \(N\) | Actual weight of mine car, \(t\) | Actual output, \(Q^\circ\) \((t)\) | Iron content, \(C_i\) (%) | Calculation of \(C_0\) (%): \(\dfrac{Q_0^i}{Q_0}\) | Calculation of \(C_0\) (%): \(\dfrac{Q_i^0}{Q_0} C_i\) | Quantity of iron, \(t\), \(P = 0{,}01 Q_i^0 C_i\) |
|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 1 | 6,000 | 2,55 | 15,300 | 50 | 0,201 | 10,05 | 7,650 |
| 2 | 5,000 | 2,90 | 14,500 | 60 | 0,191 | 11,46 | 8,700 |
| 3 | 10,000 | 2,72 | 27,200 | 55 | 0,358 | 19,69 | 14,960 |
| 4 | 8,000 | 2,38 | 19,040 | 45 | 0,250 | 11,25 | 8,568 |
| 29,000 | 76,040 | 1,000 | 52,45 | 39,878 |
In Tables 1 and 2, respectively, the actual and statistical output of ore mass by blocks and for the mine as a whole has been calculated, as well as the iron content \(C_0\) (columns 6 and 7, Table 1) by formula (21) and \(\overline{C}\) (columns 6 and 7, Table 2) by formula (14).
Table 2
| No. of block | Number of mine cars, \(N\) | Adopted weight of mine car, \(t\) | Output according to statistics, \(q_i\), \((t)\) | Iron content, \(C_i\), (%) | Calculation of \(\overline{C}\) (%): \(\dfrac{q_i}{\sum_1^4 q_i}\) | Calculation of \(\overline{C}\) (%): \(\dfrac{q_i}{\sum_1^4 q_i} C_i\) |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 1 | 6,000 | 2,75 | 16,500 | 50 | 0,207 | 10,35 |
| 2 | 5,000 | ” | 13,750 | 60 | 0,172 | 10,32 |
| 3 | 10,000 | ” | 27,500 | 55 | 0,345 | 18,98 |
| 4 | 8,000 | ” | 22,000 | 45 | 0,276 | 12,42 |
| 29,000 | 79,750 | 1,000 | 52,07 |
From Tables 1 and 2 we have:
\[ Q_0 = 76{,}040\,t; \qquad \Sigma = 79{,}750\,t; \]
\[ C_0 = 52{,}45\%, \qquad \overline{C} = 52{,}07\%. \]
The difference \(\Delta Q\), based on (5), is equal to:
\[ \Delta Q = 79{,}750 - 76{,}040 = 3710\ \text{t}. \]
In Table 3, the statistical output of the blocks has been corrected on the basis of the ore-mass balances, i.e., by the method used in practice. Column 6 of Table 3 gives the amount of iron corresponding to the corrected output of the blocks, which is less than the amount actually issued to the surface for the mine as a whole (column 8, Table 1) by 284 tons. In Table 4, the statistical output of the blocks has been corrected on the basis of the balances of ore mass and metal.
Table 3
| Block No. | Output according to statistics, \(q_i\) (t) | Correction to output, \((q_i)=-\dfrac{\Delta Q}{\sum_{1}^{4} q_i}\,q_i\) | Corrected output, \(Q_i=q_i+(q_i)\), (t) | Iron content, \(C_i\), (%) | Amount of iron, \(P=0.01\,Q_iC_i\), (t) |
|---|---|---|---|---|---|
| 1 | 16,500 | −768 | 15,732 | 50 | 7,866 |
| 2 | 13,750 | −640 | 13,110 | 60 | 7,866 |
| 3 | 27,500 | −1,279 | 26,221 | 55 | 14,422 |
| 4 | 22,000 | −1,023 | 20,977 | 45 | 9,440 |
| 79,750 | −3,710 | 76,040 | 39,594 |
From Table 1 we have:
\[ \sum_{1}^{4} q_i' = 7{,}98;\qquad \sum_{1}^{4} q_i' C_i^2 = 21{,}867. \]
Substituting \(Q_0\), \(C_0\), \(\bar C\), \(\sum_{1}^{4} q_i'\), and \(\sum_{1}^{4} q_i'C_i^2\) into formula (17), we obtain:
\[ k_2= \frac{76{,}040(52{,}45-52{,}07)} {21{,}867-52{,}07^2\cdot 7{,}98} =125{,}09. \]
Substituting \(\Delta Q\), \(\bar C\), \(\sum_{1}^{4} q_i'\), and \(k_2\) into formula (15a), we obtain:
\[ k_1=-\frac{3710}{7{,}98}-52{,}07\cdot 125{,}09=-6978{,}36. \]
Column 9 of Table 4 gives the probable corrections to the statistical output of individual blocks, calculated by formula (10a).
From Table 4, column 9, we have:
\[ (q_1)+(q_2)+(q_3)+(q_4)=-3705. \]
The indicated sum of corrections must be equal to \(-\Delta Q\), i.e., \(-3710\ \text{t}\); therefore the missing five tons, arising because of rounding, are distributed as one ton each to blocks 1, 2, and 4, and two tons to block 3.
Column 10 of Table 4 gives the corrected output of the blocks, calculated by formulas (7).
In column (11), Table 4, the amount of iron corresponding to the corrected output is indicated.
For the mine as a whole, based on the corrected output, 39.877 t of iron were issued to the surface. In actuality, 39.878 t of iron were issued to the surface (column 8, Table 1).
Consequently, the corrected output of the blocks (column 10, Table 4) satisfies both the ore-mass balance (3) and the metal balance (4). Comparing the output of the blocks corrected on the basis of the ore-mass balance (column 3, Table 5) and of the ore-mass and metal balances (column 6, Table 5), we note that the output of the blocks corrected on the basis of the ore-mass balance, i.e., by the method used in practice, deviates from the actual output by as much as 10.2%; whereas the output of the blocks corrected on the basis of the ore-mass and metal balances deviates from the actual output by no more than 0.16%.
The example examined shows that the output of blocks established on the basis of the ore-mass and metal balances approaches the true output. Therefore, in the absence of weighing facilities at the mine, the output of ore mass for individual blocks must be established on the basis of the ore-mass and metal balances.
For the case when instrumental surveying of cleaned stopes is possible, i.e., when, from mine-surveying measurements, it is possible to determine accurately the volume of the excavated space for each block, the output of blocks established on the basis of the ore-mass and metal balances may be used for checking the determination of the volumetric weight of the ore mass in the massif.
When correcting the statistical output of blocks on the basis of the ore-mass and metal balances, the actual weight of the mine cars is taken into account to some extent; this weight, as is known, depends on the metal content in the ore mass and its lump size, on the completeness of filling of the mine car, and on the cleanliness of its emptying at the tipple.
When correcting the statistical output of blocks only on the basis of the ore-mass balance, the actual weight of the mine cars is not taken into account at all; as a result, significant deviations of the corrected output of blocks from the true output are possible.
It should be noted that, when establishing block output on the basis of the ore-mass and metal balances, the degree to which the corrected output of blocks approximates the true output depends on the accuracy of determining the metal in the ore mass of individual blocks and in the mine as a whole.
The accuracy with which the metal content in the ore mass of individual blocks and in the mine as a whole is determined from the data of daily, properly organized sampling is quite sufficient for determining the corrected output of blocks on the basis of the ore-mass and metal balances.
In conclusion, let us consider establishing the weight of a mine car for individual blocks.
The average weight of a mine car over the reporting month for each block may be calculated by the formula:
\[ d=\frac{Q}{N}, \tag{27} \]
where \(Q\) is the corrected output of the block for the reporting month;
\(N\) is the number of mine cars received during the reporting month from the block,
Table 4
| No. of blocks | Ore by statistics, $q_i$ (t) | Iron content, $C_i$ (%) | $q'_i=\dfrac{q_i}{10^4}$ | $q'_i/C_i$ | $q'_i/C_i^2$ | $q'_i k_i$ | $q'_i C_i k_i$ | $(q_i)=-q'_i k_i+q'_i C_i k_i$ (t) | Corrected output, $Q_i=-q_i+(q_i)$ (t) | Quantity of iron, $P=0{,}01Q_iC_i$ (t) | Total cars, $d'_i=\dfrac{Q_i}{N_i}$ (t) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 16,500 | 50 | 1,65 | 82,50 | 4,125 | −11,514 | 10,320 | −1,194−1 | 15,305 | 7,652 | 2,551 |
| 2 | 13,750 | 60 | 1,38 | 82,80 | 4,968 | −9,630 | 10,358 | +728−1 | 14,474 | 8,686 | 2,895 |
| 3 | 27,500 | 55 | 2,75 | 151,25 | 8,319 | −19,191 | 18,920 | −271−2 | 27,227 | 14,975 | 2,723 |
| 4 | 22,000 | 45 | 2,20 | 99,00 | 4,555 | −15,352 | 12,384 | −2,968−1 | 19,031 | 8,564 | 2,379 |
| 79,750 | 7,98 | 21,867 | 55,687 | 5,1982 | −3,705 | 76,040 | 39,877 |
Table 5
Correction of output based on the balance of ore mass and metal
| No. of blocks | Actual output, $Q_i^0$ (t) | Correction of output based on the ore-mass balance | Correction of output based on the metal balance |
|---|---|---|---|
| $Q_i$ (t) | absolute error, $\Delta Q_i=Q_i-Q_i^0$ (t) | ||
| 1 | 15,300 | 15,732 | +432 |
| 2 | 14,500 | 13,110 | −1,390 |
| 3 | 27,200 | 26,221 | −979 |
| 4 | 19,040 | 20,977 | +1,937 |
| 76,040 | 76,040 | +2,369 | |
| −2,369 |
For the example we have considered, column 12 of Table 4 gives, for the blocks, the average weights of mine cars calculated by formula (27). Comparing the calculated weights of the mine cars (column 12, Table 4) with the actual ones (column 3, Table 1), we note their good agreement.
Let us establish, on the basis of the average weights of mine cars for the individual blocks, calculated by formula (27), an empirical formula for determining the weight of a mine car as a function of the metal content in the ore mass.
The weight of a mine car, when it is loaded to the same volume from individual blocks, is determined mainly by the metal content in the ore mass; therefore the empirical formula for determining the weight of a mine car may be represented in the form:
\[ d = A + BC, \tag{28} \]
where \(d\) is the weight of a mine car;
\(A\) and \(B\) are constant coefficients to be determined;
\(C\) is the metal content in the ore mass, in %.
The values of the coefficients \(A\) and \(B\) of formula (8) will be found from the condition
\[ \sum_{1}^{n} (d_i - d_i^{0})^2 = \text{minimum}, \tag{29} \]
where \(d_i\) is the mine-car weight calculated by formula (28) for the \(i\)-th block;
\(d_i^{0}\) is the average mine-car weight for the \(i\)-th block, calculated by formula (27).
Substituting the value of \(d_i\) from (28) into (29), we obtain:
\[ \sum_{1}^{n} (A + BC_i - d_i^{0})^2 = \text{minimum}. \tag{30} \]
Proceeding from condition (30), to determine the coefficients \(A\) and \(B\), we obtain two normal equations of the form:
\[ nA + \sum_{1}^{n} C_i B - \sum_{1}^{n} d_i^{0} = 0, \]
\[ \sum_{1}^{n} C_i A + \sum_{1}^{n} C_i^2 B - \sum_{1}^{n} d_i^{0} C_i = 0. \tag{31} \]
Solving the system of normal equations (31), we obtain the probable values of the coefficients \(A\) and \(B\).
Let us establish, for the mine-car weight in the example we have considered, an empirical formula of the form (28).
The calculation of the coefficients and free terms of the normal equations (31) is given in Table 6.
Column 2 of Table 6 gives the average weights of mine cars for the blocks (column 12, Table 4), calculated by formula (27), based on the corrected output (column 10, Table 4). From Table 6 we have:
\[ n = 4; \qquad \sum_{1}^{4} C_i = 210; \qquad \sum_{1}^{4} C_i^2 = 11{,}150; \]
\[ \sum_{1}^{4} d_i^{0} = 10{,}548; \qquad \sum_{1}^{4} d_i^{0} C_i = 558{,}07. \]
Table 6
| Block No. | Average mine-car weight, $d_i^0$ (t) | Iron content, $C_i$ (%) | $C_i^2$ | $d_i^0 C_i$ | Calculation of mine-car weight by the formula $d_i = 0.831 + 0.0344 C_i$: $0.831$ | Calculation of mine-car weight by the formula $d_i = 0.831 + 0.0344 C_i$: $0.0344 C_i$ | Calculation of mine-car weight by the formula $d_i = 0.831 + 0.0344 C_i$: $d_i$ (t) |
|---|---|---|---|---|---|---|---|
| 1 | 2.551 | 50 | 2500 | 127.55 | 0.831 | 1.720 | 2.551 |
| 2 | 2.895 | 60 | 3600 | 173.70 | 0.831 | 2.064 | 2.895 |
| 3 | 2.723 | 55 | 3025 | 149.76 | 0.831 | 1.892 | 2.723 |
| 4 | 2.379 | 45 | 2025 | 107.06 | 0.831 | 1.548 | 2.379 |
| 10.548 | 210 | 11150 | 558.07 |
Consequently, the system of normal equations (31) will take the form:
\[ \begin{aligned} 4A + 210B - 10.548 &= 0,\\ 210A + 11{,}150B - 558.070 &= 0. \end{aligned} \tag{32} \]
Solving the system of normal equations (32), we obtain:
\[ A = 0.831; \qquad B = 0.0344. \]
Thus, for the example considered, empirical formula (28) will take the form:
\[ d = 0.831 + 0.0344 C. \tag{32} \]
Comparing formulas (26) and (32), we note their good agreement. In column 8 of Table 6, the mine-car weights for the blocks are calculated on the basis of formula (32). The absolute agreement of the mine-car weights calculated by formula (32) with the initial average weights (column 2, Table 6) is explained solely by the fact that the average mine-car weights calculated by formula (27) strictly follow equation (26). In practice, the mine-car weights calculated by the empirical formula will almost always differ from the initial average mine-car weights calculated by formula (27); moreover, significant deviations will indicate unequal loading of the mine cars from different blocks.