DETERMINATION OF THE OPTIMAL CROSS-SECTIONAL DIMENSIONS OF ENTRIES DRIVEN IN COAL

V. N. Leont’ev
Docent, Candidate of Technical Sciences

The task of strengthening the economy of the coal industry and reducing the cost of production is closely connected with questions of driving preparatory workings. The economic costs of driving workings depend first of all on the successfully selected or calculated dimensions of their cross section. It is generally accepted to determine the cross-sectional dimensions of workings on the basis of the requirements of mine transport and safety regulations. But a cross section calculated solely according to technical requirements is not always optimal, since the economic side of the question is not taken into account. The question of determining the cross-sectional dimensions of workings may also be approached from another side, namely, these dimensions may be selected in such a way that the total costs of driving, maintenance, and ventilation are the smallest. The stated problem has repeatedly been solved with varying degrees of approximation to actual mine conditions. In general form, according to what has been said, the cross section of any mine working will be a function of three dependent variables

\[ S=f(A;\ B;\ C), \tag{1} \]

where \(A\) is the cost of driving and timbering the working;
\(B\) is the cost of maintaining (repair and upkeep of) the working;
\(C\) is the cost of energy for ventilating the working.

The most advantageous cross section according to economic factors will be the one for which the sum of the enumerated costs is minimal. But a cross section of a working calculated only according to economic factors does not always satisfy safety requirements or the conditions for the free placement of transport means and equipment. Therefore only such a cross section of a working should be considered optimal in which both the former and the latter requirements are observed to the fullest extent.

As regards calculations of the cross sections of workings made according to technical factors, they are well known and do not present special difficulties. On the contrary, calculations made according to economic factors are distinguished by their complexity, have not been fully studied, and therefore have become the subject of the present work, as has the mutual coordination of technical and economic calculations in determining the optimal dimensions of entries driven in coal. The question of determining the optimal cross-sectional dimensions of entries is of not only theoretical but also great practical interest. Work carried out in the mines of the Kuzbass on the standardization of cross sections of workings was usually limited to a one-sided solution of the question: the cross sections were determined on the basis only of satisfying the requirements of transport and the permissible velocities of air movement through the workings. Checks of the correctness of the established cross-sectional dimensions of workings according to

economic considerations and without mutual coordination of the calculations. This caused the appearance in the mines of Kuzbass of a large number of different types of entry cross sections, led to difficulties in establishing standards, and to excessive material expenditures.

Let us proceed to an analysis of the components of the general cost equation.

I. On the cost of driving and supporting entries

The cost of driving and supporting entries is approximately equal to

\[ A=(K-k)SL, \tag{2} \]

where \(K\) is the cost of driving and supporting 1 cubic meter of the working, in rubles;
\(k\) is the cost of 1 cubic meter of coal, in rubles;
\(S\) is the cross-sectional area of the working, in square meters;
\(L\) is the length of the working, in meters.

In solving questions in general form, it has been customary to assume that the cost of 1 cubic meter of horizontal and inclined workings, depending on the dimensions of the section and the strength of the rocks being driven through, changes relatively insignificantly. Likewise, the magnitude of undercutting and the thickness of the seam along which the working is driven have an insignificant effect on the cost of driving them. At the same time, many believe that the type of support and the purpose of the working substantially affect its cost [1]. On the other hand, the question of the cost of 1 cubic meter of a working is decisive in calculating the expenditures for driving and supporting workings; therefore an exact expression for the value \(K\) in formula (2) is highly desirable. Serious doubts are raised by the generally accepted proposition that the value \(K\) does not depend, or almost does not depend, on the dimensions of the section of the working and the strength of the surrounding rocks. The proponents of this view resolved the question too generally—mainly on the basis of comparing the estimated costs of different workings driven under different conditions in thin coal seams (with undercutting), taking into account the techniques and methods of work chiefly in the pre-Stakhanovite period. Taking into account that we are considering the question of the cost of driving entries in thick seams, with respect to which analyses of this kind have not been carried out, and that the Stakhanovite movement introduced substantial corrections into output norms, the techniques, and the methods of driving them, we believe that the question of the cost of 1 cubic meter of workings in general, and of entries in particular, must be fundamentally reviewed and refined.

Let us analyze the cost of driving and supporting 1 cubic meter of an entry, using for this purpose the norms and rates of the standard-setting handbook for underground work, compiled in 1940 by the Kuzbass Combine on the basis of the actual labor productivity and output norms at 33 mines of Kuzbass. In the course of the analysis we shall try to trace how the value \(K\) changes, and whether it changes depending on the section of the working, the strength of the coal, the technique of drivage work, etc.

In the development of thick seams in Kuzbass at the present time, the following four basic methods of driving entries are used, if they are considered from the point of view of coal extraction at the entry face.

First method. Coal extraction at the entry face is carried out exclusively with explosives (VV); shot-hole drilling is done with hand-held electric drills.

Second method. Coal extraction is carried out exclusively with pneumatic picks, without the use of explosives.

Third method. Driving of entries and coal extraction are carried out by using a BSh undercutting machine and subsequent breaking down of the undercut coal with pneumatic picks.

Fourth method. The driving of entries is carried out according to the preceding scheme, but the undercut coal is broken not by cutting machines, but by explosives.

The last method of driving is used in practice less often than the others; therefore, in our analysis we consider it possible to confine ourselves to the first three methods. In addition, in driving entries, various methods of loading the broken coal into mine cars may be used, which must also be taken into account. The typical methods should be considered to be: loading by hand, loading with the aid of a “Pit-Car-Loader” type loader and, finally, manual loading onto a shaking conveyor. As for the remaining operations in heading work—supporting, the nearest haulage of the loaded coal, etc.—they usually remain unchanged under different driving methods and therefore, in the proposed analysis, are taken into account by us in precisely this form. Taking the indicated methods of work as a basis, we calculated the cost of \(1\ \mathrm{cu.\ m}\) of entry for each of these methods, but for different cross sections of workings. It was assumed that horizontal workings are driven at the same rate exclusively in coal (without ripping the side rocks), and that the other operations during driving, apart from cutting and loading the coal, i.e. supporting, haulage of the broken coal, etc., remain unchanged for identical conditions. Thus, for example, the method of supporting an entry with a clear cross section inside the support of \(4\ \mathrm{sq.\ m}\), under all possible methods of cutting and loading the coal, remains unchanged and is the one provided for in the mine passport for this working, etc. The calculations were based on the method of computing the cost of \(1\ \mathrm{cu.\ m}\) of working used by the estimating department of the Kuzbassugol combine. The calculations were made for working sections ranging from 3 to \(14\ \mathrm{sq.\ m}\) and were summarized in 18 calculation tables (216 variants in all). The final data for each of the variants are given in Table 1. The results of the calculations made allow the following conclusions to be drawn.

1. The dynamics of the cost of \(1\ \mathrm{cu.\ m}\) of an entry are influenced chiefly by labor costs for coal extraction, for supporting the working, and by the cost of the materials consumed.

2. The cost of \(1\ \mathrm{cu.\ m}\) of entry depends on the cross-sectional area of the working; specifically, as the cross-sectional area of the working increases, the cost at first decreases (for workings with sections from 3 to approximately \(5\ \mathrm{sq.\ m}\)), and then begins to increase (for workings with sections greater than \(5\)–\(6\ \mathrm{sq.\ m}\)).

3. Fluctuations in the cost of \(1\ \mathrm{cu.\ m}\) of entry far exceed the \(\pm 5\%\) to which proponents of the independence of the cost of \(1\ \mathrm{cu.\ m}\) of a working from the dimensions of its section usually refer; for the sections considered, the cost of \(1\ \mathrm{cu.\ m}\) varies within limits of up to 400 and even more percent.

4. The cost per cubic meter of entry also depends on the hardness of the coal. For the cases considered and for entry sections up to approximately \(10\ \mathrm{sq.\ m}\), as the hardness of the coal decreases, the cost per cubic meter likewise decreases. For workings with sections greater than \(10\ \mathrm{sq.\ m}\), on the contrary, as the hardness of the coal decreases, the cost of \(1\ \mathrm{cu.\ m}\) of working increases.

5. The method of driving also affects the cost of \(1\ \mathrm{cu.\ m}\) of working. An insufficiently detailed analysis of this factor does not allow us to say exactly what this influence is; one thing remains beyond doubt: this factor too exerts its influence on the cost per cubic meter of working.

The principal conclusion from the analysis carried out should be considered to be that the cost of \(1\ \mathrm{cu.\ m}\) of a working depends on the dimensions of its section, the hardness of the coal being driven, and the method of driving.

Table 1

Cost of 1 cubic meter of entry \((K\)—in rubles, kopecks)

The analysis presented of the principal elements (see Table 1 on p. 186) making up the cost of a cubic meter of an entry once again confirms the correctness of the conclusions drawn.

Let us first consider the question of the cost of cutting coal. It follows from the calculation tables that labor costs in extracting coal with explosives decrease as the cross-section of the working increases and as the hardness of the coal decreases, which must be regarded as a quite normal phenomenon. Indeed, the expenditure of labor and the costs associated with it must undoubtedly be proportional to the number of blastholes per cubic meter of the working in the solid. The latter quantity, as is not difficult to show, will decrease with an increase in the cross-section of the working and with a decrease in the hardness of the coal, and, consequently, labor costs, at least for drilling blastholes, must also decrease. The following calculation well confirms what has been said. The number of blastholes per 1 sq. m of the face plane may be calculated with sufficient accuracy by the empirical formula of Prof. M. M. Protodyakonov:

\[ n=\frac{2.7\sqrt{f}}{\sqrt{S}}, \tag{3} \]

where \(n\) is the number of blastholes per 1 sq. m of the face plane,
\(f\) is the coefficient of rock (coal) hardness according to Protodyakonov,
\(S\) is the cross-sectional area of the working in the solid, in sq. m.

In order to pass from one square meter of area in the solid to one cubic meter of area in the clear, which is necessary for further calculations, it is sufficient to multiply the preceding expression (3) by a certain coefficient \(i\), representing the ratio of the cross-sectional area of the working in the solid to its cross-sectional area in the clear (for a working length equal to one), \(i=\frac{S}{S_0}\), where \(S_0\) is the cross-sectional area of the working in the clear.

Then per 1 cu. m of working in the clear there will be

\[ n_1=i\frac{2.7\sqrt{f}}{\sqrt{S}}. \tag{4} \]

It follows from the formula given that, with an increase in the cross-section \(S\) of the working and a decrease in the hardness of the coal \(f\), the number of blastholes per 1 cu. m of working in the clear must decrease and, consequently, the labor costs for drilling blastholes must also decrease.

In addition to drilling blastholes, the duties of the face worker also include the breaking up of the blasted coal. Unfortunately, we do not have data by which it would be possible to judge precisely the degree of influence of this factor on the cost of the operation of interest to us. Only one thing remains unquestionable: the cost of breaking up blasted coal must also decrease with a decrease in the hardness of the coal and an increase in the cross-section, since with an increase in the cross-section of the working the degree of confinement of the blasted medium (the face) is also reduced. Thus, we may consider that the nature of the change in labor costs when cutting coal with explosives is correctly reflected in the calculation tables.

Exactly the same nature of change in labor costs also occurs when coal is extracted exclusively with pneumatic hammers. This is understandable, since the influence of the working cross-section and the hardness of the coal is likewise manifested here and, undoubtedly, has the same character as in the preceding case.

When coal is extracted by a BS cutting machine in combination with pneumatic hammers, the labor cost of extracting 1 cu. m also tends—

...toward a decrease with an increase in the cross section of the working, but only up to a certain limit; in workings of large dimensions, when, evidently, operation of the BSh machine becomes somewhat more complicated (more difficult installation, relocation), the cost of 1 cubic m of coal increases.

The next characteristic element of the cost of work is timbering. The labor costs for timbering must, obviously, be proportional to the amount of timbering. This latter quantity, i.e., the amount of timbering, is undoubtedly a function of the cross section of the working and of the strength of the rocks; consequently, labor costs for timbering must also depend on the strength of the rocks and the cross section of the workings.

In fact, for a given thickness of support, the number of door sets falling per 1 linear m of working can be calculated by the formula of Prof. P. M. Tsimbarevich [2]:

\[ N=\frac{1}{\dfrac{0.24\,d^3\cdot \tg \beta_0 \cdot R_b}{a^3\cdot \gamma}} =\frac{a^3\cdot \gamma}{0.24\,d^3\cdot \tg \beta_0 \cdot R}, \tag{5} \]

where \(N\) is the number of door sets per 1 linear m of working,
\(d\) is the diameter of the support crosspiece,
\(a\) is the half-span of the working,
\(\beta_0\) is the angle of internal resistance of the roof rocks,
\(R_b\) is the permissible bending stress in the support material,
\(\gamma\) is the unit weight of the roof rocks.

For given \(d\), \(\gamma\), and \(R_b\), the value of \(N\) will vary depending on changes in the quantities \(a\) and \(\tg \beta_0\), i.e.,

\[ N=f(a;\tg\beta_0). \]

Then we can write that

\[ N=C\cdot\frac{a^3}{\tg \beta_0} \quad \text{and} \quad C=\frac{\gamma}{0.24\,d^3\cdot R_b}. \]

If we turn to the workings under consideration, it is not difficult to conclude that the areas of their cross sections vary chiefly because of changes in the widths of the workings and only slightly because of changes in their heights. On average, the height of the workings in the clear within the support may be taken as equal to 2 m. In this case the dependence of the half-span on the cross-sectional area of the workings follows a straight-line law of the following form:

\[ a=25S-18. \tag{6} \]

Substituting its value for \(a\) in the expression for \(N\), we obtain

\[ N=C\frac{(25S-18)^3}{\tg \beta_0}. \]

Per 1 cubic m of working in the clear there will be the following number of door sets:

\[ n=i\cdot\frac{N}{S} =i\cdot C\frac{(25S-18)^3}{S\cdot \tg \beta_0}. \tag{7} \]

It follows from this that the amount of timbering per 1 cubic m of working, and consequently also the labor costs for timbering, depend on the cross-sectional area of the working and the strength of the rock. Other conditions being equal, these costs are the greater, the larger the cross-sectional area of the working \(S\) and the lower the rock strength \(\beta_0\). This conclusion is well confirmed by the data of all the calculation tables.

The remaining labor costs—costs for loading the broken coal, for its nearest transportation, etc.—also often depend on the cross-sectional dimensions of the working and account for a significant share of the cost of one cu. m of excavation. Thus, for example, labor costs for loading coal in narrow workings, i.e., where there is a constricted work front, will always be higher than in spacious workings, etc. We believe that the latter is easily explained and requires no special proof.

As for expenditures on materials, here too there is a certain dependence between the cost of the materials consumed, the strength of the coal, and the cross section of the workings. We have just established such a dependence for support materials, showing that the consumption of the latter and their cost per 1 cu. m of working, in general, will increase as the cross sections of workings increase. With regard to the consumption and cost of explosives, this dependence also exists and is expressed in the fact that, as the cross-sectional dimensions of the working increase, the specific consumption and cost of explosives decrease and, conversely, increase as the strength of the coal increases. The validity of this is easy to understand if one takes into account that the magnitude of the specific consumption of explosives (and, therefore, also their cost) depends on the confinement of the medium being blasted. As the cross-sectional area of the working increases, the degree of confinement of the medium being blasted decreases, and the specific consumption of explosives also decreases.

It is also obvious that, as the strength of the coal increases, the specific consumption of explosives and their cost per 1 cu. m increase.

The specific consumption of detonators is proportional to the specific consumption of blast holes. Since the latter decreases as the cross section of the working increases and, conversely, increases as the strength of the coal increases, as was established earlier, it follows that the specific consumption of detonators will likewise decrease as the cross section of workings increases or as the strength of the coal decreases.

If we trace the cost of 1 cu. m of entry according to the data of the old Kuzbass norm books (1936–1939), here too it is found that this quantity does not remain constant and depends, at least, on the cross-sectional dimensions of the workings.

Thus, in the overall balance of costs making up the cost of 1 cu. m of working, some decrease as the cross section of workings or the strength of coal increases, while others, on the contrary, increase. As a result, the costs are balanced in such a way that, in general, the cost of 1 cu. m of entry does not remain constant, but changes, especially with the cross-sectional dimensions of the working and the strength of the coal.

The law of variation of the cost \(K_S\) of 1 cu. m of working as a function of its cross-sectional dimensions can be represented graphically. For workings with cross sections up to 10 sq. m, the change in this quantity follows a curve close to a parabola. The equation of this curve has the form

\[ K_S = a - bS + cS^2. \tag{8} \]

For workings with cross sections greater than 10 sq. m, the law of variation of the quantity \(K_S\) proceeds along a straight line having the equation

\[ K_S = aS - b. \tag{9} \]

Changes in the cost \(K_f\) of 1 cu. m of entry as a function of coal strength \(f\) and for different cross sections of workings may be expressed by a general equation of the following form:

\[ K_f = p + m f^n, \tag{10} \]

where \(p\), \(m\), and \(n\) are constants depending on the method of driving.

For entries with a cross section greater than 10 sq. m, the exponent \(n\) assumes negative values. This indicates that, as the strength of the coal increases, the cost of 1 cu. m of entry will decrease, rather than increase, as is observed in the case of entries of smaller cross section.

And, finally, the quantity \(K\) can be expressed analytically as a function of two variable quantities, namely: the cross-sectional area of the entry \(S\) and the strength of the coal \(f\). For this it is sufficient to multiply the partial dependences of the quantity \(K\) and divide by the arithmetic mean of all observations, i.e.,

\[ K=\frac{\psi(S)\varphi(f)}{K_0}=\frac{K_S K_f}{K_0}, \]

where \(K_0\) is the arithmetic mean cost of 1 cu. m of entry according to the calculation tables.

It is not difficult to see that, for entries with a cross section up to 10 sq. m, the following will be the general cost equation:

\[ K=(a-bS+cS^2)(p+mf^n). \tag{11} \]

For entries with a cross section greater than 10 sq. m,

\[ K=(aS-b)(p+mf^n). \tag{12} \]

The values of the constant coefficients \(a, b, c, p, m\), and \(n\) entering equations (11) and (12) are given in Table 2.

Table 2

| No. | Method of driving | \multicolumn{6}{c}{Entries with cross section up to 10 sq. m: values of coefficients} | \multicolumn{5}{c}{Entries with cross section greater than 10 sq. m: values of coefficients} |
|---:|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| | | \(a\) | \(b\) | \(c\) | \(p\) | \(m\) | \(n\) | \(a\) | \(b\) | \(p\) | \(m\) | \(n\) |
| 1 | Driving by means of explosives and hand loading | 1.26 | 0.17 | 0.02 | 11.0 | 4.0 | 1.0 | 3.33 | 12.0 | 0.847 | 0.19 | −2.9 |
| 2 | Driving by means of explosives with loading by “pit-car loader” or onto a conveyor | 1.23 | 0.17 | 0.02 | 14.4 | 0.53 | 3.46 | 3.34 | 12.5 | 0.84 | 0.19 | −2.85 |
| 3 | Driving by means of pneumatic picks and hand loading | 1.16 | 0.16 | 0.02 | 2.9 | 8.57 | 1.0 | 2.99 | 10.73 | 0.95 | 0.09 | −3.11 |
| 4 | Driving by means of a BSh drilling machine, pneumatic picks, and hand loading | 1.54 | 0.3 | 0.037 | 7.94 | 3.22 | 2.08 | 3.37 | 14.46 | 0.97 | 0.037 | −1.96 |

II. On the Cost of Maintenance

“The question of the maintenance of entries,” wrote Prof. M. M. Protodyakonov, “is one of the least developed questions of mining, despite its extraordinary importance for the prime cost of the mineral and the most decisive influence that maintenance exerts on the choice of the development system, the size of mine fields, the distances between bremsbergs, etc.” [3]. Usually, for calculations, average bookkeeping figures are taken, which equalize entries that are under entirely different conditions with respect to support, cross section, surrounding rocks, whether the mine works several seams, etc. Using such figures, Prof. B. I. Bokiy, Prof. L. D. Shevyakov, Prof. A. S. Popov, and others multiply the average maintenance cost \(r\)

per unit length of the working per unit time by its length \(l\) and by the service life of the working \(t\), and obtain the expression for the total cost of maintenance \(rlt\). To determine \(r\) in this case, average figures for maintenance cost, as given by accounting reports, are used.

In the work Basic Questions of Mining, compiled on the instructions of the Scientific and Technical Council of the coal industry, when considering the cost of maintaining workings, it is assumed that each working passes through three characteristic periods of its service life. The first period applies to workings located in the zone of preparatory operations and enclosed between solid coal pillars. The second period applies to that section along the length which is in the zone of influence of extraction work. And, finally, the third period includes all repairs of workings located behind the extraction work, where the influence of the latter through its pressure in the second period has already ended. In the subsequent calculations the first period is not taken into account because of its insignificant duration and the negligible cost of repairing 1 linear m per year. The most intensive repair belongs to the second period, when a section of the working is subjected to heavy pressure owing to the influence of extraction work. For this period the authors, for rocks of medium stability, adopted four re-settings of the main drift per year. For the section located in the third period, i.e., behind the extraction work, 0.7 re-settings were adopted. These figures were adopted on the basis of the statistical materials of Prof. B. I. Bokii, the small amount of material available in Description of the Donets Basin, and data from practice. Proceeding from the number of re-settings per year, the costs of maintaining 1 linear m of workings for the second and third periods were calculated; using these, the costs of maintaining workings for the entire time of their existence under various conditions were derived.

In addition, in the literature there are attempts to express the cost of repairing workings somewhat differently. Thus, Eng. Krivko [4] proposes calculating the cost of repairing workings by the formula

\[ R = r_1 l + \frac{rlt}{2}, \]

where \(R\) is the total cost of maintaining the working over its entire service life,
\(l\) is the length of the working,
\(t\) is its service life,
\(r_1\) is the cost of repairing 1 linear m of drift after the action of primary pressure,
\(r\) is the cost of repairing 1 linear m of working per unit time from the action of secondary and subsequent pressures, as well as due to rotting of timber and other causes leading to destruction of the supports.

The meaning of Eng. Krivko’s proposal is that, in addition to the element accounted for by the generally known formula for the cost of maintaining workings during their driving (Bokii, Shevyakov, and others)—

\[ R = \frac{rlt}{2}, \]

he introduces a correction \((r_1 l)\), constituting the costs for repairing workings associated with the primary pressure of the rocks on the supports.

Eng. Lipkovich [5], for a working of length \(L = l_1 + l_2 + l_3\), proposes calculating the cost of repair by the formula (with a continuous system of mining)

\[ R = \frac{t}{2}(2r_1 l_1 + 2r_2 l_2 + r_3 l_3), \]

where \(r_1\) and \(l_1\) are the cost of repairing 1 linear m of entry and its length in the face entry;

\(r_2\) and \(l_2\) are analogous data, but in the zone of primary settlement;

\(r_3\) and \(l_3\) are likewise analogous data, but in the part of the entry where rock movement has ceased.

Accordingly, the average annual cost of maintaining 1 linear m of entry is

\[ r_{\mathrm{cp}}=\frac{r_1 l_1+r_2 l_2+r_3 l_3}{l_1+l_2+l_3}. \]

Eng. A. V. Yagupov [6] proposes determining the average annual cost of maintaining 1 linear m of an opening over the entire time it is driven by the following formula:

\[ r=\frac{2S}{LT}. \]

If extraction work does not affect the opening, then the calculation is made by the formula

\[ r=\frac{R}{t_o}. \]

In this case the quantity \(S\) represents the total expenditures for maintaining the entry during the second and third periods over the time \(T\), corresponding to the total duration of its driving.

\[ S=L\left[\frac{R}{2t_o}(T-t_o)+\frac{R'}{t'}(T'-t')\right], \]

where \(L\) is the total length of the entry to the boundary of the field,

\(R\) and \(R'\) are the costs of repairing 1 linear m of entry in the third and second periods,

\(t_o\) is the time during which the newly driven opening does not require repair,

\(t'\) is the average service life of a support frame in the zone influenced by extraction work,

\(T'\) is the time during which an element of the entry remains in the zone influenced by extraction work.

V. B. Komarov and D. F. Borisov [7] propose calculating the cost of repairing an entry over its entire service life by the formula (when driven from the mine toward the boundary)

\[ r=LS\left[a\left(\frac{t_1}{2}+\frac{t_4}{2}+t_2+\frac{t_3}{2}\right)+r_b\right], \]

where \(L\) is the length of the entry,

\(S\) is the cross-sectional area of the opening in sq. m,

\(t_1\) is the period of existence of the entry (years) during development of the level served by it,

\(t_2\) is the period of existence of the entry at full length,

\(t_3\) is the same, but during development of the lower level, when part of the entry is not being maintained and another part is being restored at that time,

\(t_4\) is the same during gradual abandonment of the entry,

\(a\) is an empirical coefficient characterizing the repair costs of 1 linear m of an opening as a function of the dimensions of its cross section,

\(r_b\) is the cost of restoring 1 linear m of entry on the ventilation horizon.

The formulas listed, like others, more often relate to the method of calculating \(r_{\mathrm{cp}}\) than to establishing the law governing the variation of this quantity. It is necessary to note the existence of fundamental discrepancies among research...

with regard to the question that interests us, which has given rise to two principal directions of research thought. Thus, Academicians A. M. Terpigorev and L. D. Shevyakov, as well as Prof. B. I. Bokii, in theoretical calculations of the cost of repairing workings, take it to be proportional to the length of the working and to its service life, i.e., express it by a formula of the form \(B = rlt\). The proportionality of the cost of repair to the length of the working and to the time of its existence once gave rise to discussion [8]. A number of researchers on this question, such as, for example, Prof. M. M. Protod’iakonov, and perhaps the majority of recent researchers, do not share the opinion that direct proportionality is admissible here. It cannot be said that this discussion, which has already been going on for about 30 years, has shed light on the actual state of affairs; up to the present time this question still remains open. The main reason for these contradictions must be considered to be the incorrect method or methods applied by the authors in investigating this question. These methods, briefly, amount to the fact that some authors try to give a final solution to the question merely on the basis of speculative conclusions (and here there is especially great diversity), while others do so on the basis of an analysis of crude statistical data. The absence among researchers of factual material (as the result of production observations) gives rise to diversity of opinion, an abundance of “theories,” and the doubtful nature of conclusions suitable only for very rough, purely approximate calculations.

V. I. Lenin, in his work “What the ‘Friends of the People’ Are and How They Fight the Social-Democrats,” speaking of K. Marx’s method of research, notes: “The gigantic step forward made by Marx in this respect consisted precisely in the fact that he... began from the beginning, and not from the end, with an analysis of the facts, and not with final conclusions, with the study of particular, historically determined social relations, and not with general theories as to what these social relations consist of in general” [9].

Undoubtedly, the only correct method of investigating the question touched upon must be considered to be broadly and systematically organized production observations of the repair of workings and an analysis of the factual material accumulated in this way.

What is the situation with factual material?—We know only a small number of works with the results of production observations. These works by no means cover the whole diversity of natural-geological and organizational-technical factors in the service of drifts; they are connected with the mining of only thin coal seams and therefore cannot serve as a basis on which one could build conclusions regarding the cost of repairing drifts when mining thick seams. Since systematic production observations of the repair and maintenance of drifts in the mining of thick seams in the Kuzbass have not been carried out, it does not at present appear possible to approach the determination of the actual cost of maintaining drifts under these conditions with a sufficiently substantiated calculation. It goes without saying that such accounting, with the influence of various and diverse factors insufficiently studied, can for the time being be carried out only very roughly and approximately and on the basis of mine statistical data.

The main haulage drifts in the thick seams of the Kuzbass are driven between intermediate crosscuts for the length of extraction districts, i.e., up to 400 m. The development of extraction districts is normally conceived in the direction from the boundaries toward the mine. In this case, the following two periods of their service are characteristic for the main haulage drifts. The first period relates to the time of driving the drift and the subsequent stoping extraction on the working horizon. During this period the main haulage drifts require very insignificant expenditures for their maintenance, since during the

when seams are worked by a system of intermediate crosscuts, generally speaking, it is not customary to maintain haulage drifts on the working horizon.

The second period is associated with the existence of the drift on the ventilation horizon, when considerable expenditures are required for its restoration (repair of the supports, tracks, etc.), and therefore all expenses connected with maintaining the main haulage drifts over their entire service life should be attributed chiefly to this period.

If, in doing so, we take into account the generally accepted opinion that the cost of repairing a working supported with timber, all other conditions being equal, is directly proportional to the cross section of the working and inversely proportional to the degree of stability of the rocks, then, as a first approximation, the cost of maintaining (repair, restoration) 1 running meter of drift may be expressed as follows:

\[ r = a \frac{S}{f}. \tag{13} \]

Then the cost of maintaining (restoring) the entire working of length \(L\) over its entire service life will be

\[ B = rL = \frac{aSL}{f}, \tag{14} \]

where \(r\) is the cost of repair and maintenance of 1 running meter of drift on the working and ventilation horizons, in rubles,

\(S\) is the clear cross section of the drift, inside the supports, in sq. m,

\(f\) is the coefficient of coal strength according to the scale of Prof. M. M. Protodyakonov,

\(a\) is the coefficient of proportionality.

During seven months of 1939, at the mine named after Comrade Stalin of the Stalinugol trust, 3358 running meters of main drifts were resupported, for which 176,098 rubles 65 kopecks were actually spent. Thus, repair of supports for 1 running meter of main drift cost on average 52 rubles 40 kopecks. If to this is added 8 rubles 49 kopecks, on average attributable to the repair of 1 running meter of track and to drift maintenance, then the total cost of maintaining 1 running meter of a main drift is obtained as approximately equal to 61 rubles. The clear cross section of the main single-track drifts is on average 5.83 sq. m. If the average coal strength according to Protodyakonov is taken as equal to 1.2, and a single resupporting of the drift over its entire service life is assumed both on the working and on the ventilation horizons, then the value of \(a\) in formula (13) is obtained as equal to

\[ a = \frac{rf}{S} = \frac{61 \cdot 1.2}{5.83} \simeq 12.5. \]

During the same seven months, 86 running meters of parallel drifts were repaired at the mine, for a total of 2403 rubles 88 kopecks, which on average amounts approximately to 28 rubles per 1 running meter. If one takes into account that the clear cross section of parallel drifts supported with timber is 3.15 sq. m, then, with an average coal strength \(f = 1.2\), the value of \(a\) for parallel drifts will be equal to

\[ a = \frac{rf}{S} = \frac{28 \cdot 1.2}{3.15} \simeq 10. \]

Of course, these figures (12.5; 10) do not claim great accuracy and are only purely approximate. Academician L. D. Shevyakov, in one of his recent works [10] concerning the repair of workings, proposes, in particular, that the value \(a\) be taken as about 15, noting that the data of Prof. A. S. Popov (\(a = 7.2\)) are now undoubtedly obsolete.

III. On the Cost of Ventilation

Having examined the questions of the cost of driving and maintaining entries, let us calculate the expenses for ventilation; for simplicity we take into account only the expenses for energy.

Per 1 linear m of entry per year they will be equal to [7]:

\[ C=q\,\frac{365\cdot24\cdot Qht}{102\cdot \eta_i} =85.9\,q\,\frac{Qht}{\eta_i}, \tag{15} \]

where \(q\) is the cost of electrical energy in rubles per kWh,

\(Q\) is the quantity of air flowing through the working in \(m^3/\text{sec}\),

\(\eta_i\) is the efficiency of the ventilation installation,

\(t\) is the total service life of the working (years),

\(h\) is the depression per 1 linear m of working in millimeters of water column:

\[ h=\alpha\,\frac{p\cdot Q^2}{2S^3}. \tag{16} \]

Here \(\alpha\) is the coefficient of resistance to the movement of air through the working,

\(p\) is the perimeter of the working in meters.

In developing a mine field of length \(L\) m in the direction from the shaft to the boundaries of the field, the average ventilation length will be equal to \(\frac{L}{2}\) m. Therefore the value of the average depression will be half of its maximum value. This circumstance is taken into account by the fact that a factor of two has been introduced into the denominator of expression (16).

For entries of the dimensions considered, the value of the perimeter can be determined with comparative accuracy from the following expression:

\[ p=4.07\sqrt{S}. \tag{17} \]

Therefore

\[ h=\alpha\,\frac{4.07\sqrt{S}}{2S^3}\,Q^2 =\alpha\,\frac{2.04Q^2}{S^{2.5}} \tag{18} \]

and

\[ C=\frac{175\cdot q\alpha}{\eta_i}\cdot \frac{Q^3t}{S^{2.5}} =m\,\frac{Q^3t}{S^{2.5}}, \tag{19} \]

where

\[ m=\frac{175q\alpha}{\eta_i}. \]

IV. On the Optimal Cross-Sectional Dimensions of Entries

On the basis of the analysis carried out, we may consider that the sum of the listed expenses for driving, maintaining, and ventilating an entry will be

\[ y=A+B+C=(K-k)S_0t+\frac{aS_0}{f^n} +m\,\frac{Q^3t}{S_0^{2.5}}. \tag{20} \]

For \(S>10\) sq. m in equation (20), the value

\[ K=(aS-b)(p+mf^n); \quad \text{for } S<10 \text{ sq. m} \]

\[ K=(a-bS+cS^2)(p+mf^n). \]

To determine the entry cross section corresponding to the minimum expenditure, it is sufficient (provided that the second derivative \(\frac{d^2 y}{dS^2}\) is greater than zero) to determine the variable \(S\) from the equation

\[ \frac{d(A+B+C)}{dS}=0. \]

However, in the present case it is simpler to solve the problem graphically. For the conditions of the mines of the southern Kuzbass, the following approximate values may be adopted for the constant quantities entering the calculation formulas (2) and (20): \(q=0.06\) rubles, \(\eta=0.65\), \(\alpha=0.002\), \(k=15\).

For main single-track entries, \(t=10\) years, \(Q=10\ \text{cu. m/sec.}\), \(i=1.45\). For main double-track entries, \(t=10\) years, \(Q=15\ \text{cu. m/sec.}\), \(i=1.45\). And, finally, for intermediate, parallel entries, \(t=1\) year, \(Q=5\ \text{cu. m/sec.}\), \(i=1.3\).

Having adopted these quantities, we calculate from formulas (11), (12), (2), (14), (19) the costs of driving, maintenance, and ventilation of entries, and, on the same scale, construct in coordinate axes the graphs characterizing these costs. Then, by carrying out graphical addition, we find the total cost curve of the working according to all the indicators mentioned here. The total cost curve makes it possible to determine the entry cross section corresponding to the minimum expenditure.

Thus, for example, Fig. 1 presents the graphical solution of the problem for a main single-track entry driven with the aid of explosives. The series of thin curved lines at the bottom characterizes the cost of driving and timbering 1 running meter of entry in coals of different strength (\(f=0.8\)—\(1.6\)). The bundle of straight lines denotes the cost of repair, the lower curve the cost of ventilation, and, finally, the two thickened curves at the top give the total cost of 1 running meter of entry for the two limiting values of coal strength (\(f=0.8\) and \(1.6\)). The total curves for coals of intermediate strength categories (\(f=1.4;\ 1.2;\ 1.0\)) will be located between the total curves indicated in Fig. 1.

From consideration of the total curves (Fig. 1) it may be established that, in the example considered, the region of minimum expenditure falls within the interval corresponding to working cross sections of \(2.5\)—\(4.0\ \text{sq. m}\).

Having made analogous solutions for entries driven under different conditions (in the sense of coal strength) and by different methods, we obtain a series of total curves making it possible to determine the most advantageous entry cross section on the basis of economic factors. Proceeding from this, Fig. 2 presents total curves for main single-track entries, which make it possible to establish that for this type of working the economically most advantageous cross sections are those within the limits of \(2.5 \div 4.0\ \text{sq. m}\) in the clear of the timbering. If, further, one takes into account the requirements of the safety regulations concerning the necessity, for main ventilation workings timbered with wood, of having a clear cross section of not less than \(4.5\ \text{sq. m}\), and that a cross section of \(4.5\ \text{sq. m}\) is satisfactory with respect to the placement of transport means (battery electric locomotive on a \(900\ \text{mm}\) track, two-ton car, passage \(700\ \text{mm}\), clear height \(2\ \text{m}\)), we arrive at the conclusion that, for main single-track entries, the optimal cross section is \(4.5\ \text{sq. m}\), measured in the clear of the timbering.

It is not uninteresting to note that in the mines of the Kuzbass, for main single-track haulage entries, the most varied cross sections are used—from 4 to \(10\ \text{sq. m}\) in the clear of the timbering.

As for double-track workings, it turns out that in this case the total curves do not have a region of minimum; the cost of the work—

of this kind grows in proportion to the dimensions of the cross section. Consequently, the optimal section for a two-track entry should be regarded as the minimum section calculated according to technical requirements, i.e., proceeding from the conditions for the convenient placement of mine-haulage equipment and the requirements of safety regulations, both with respect to the velocity of the ventilation current and to the permissible clearances.

Interesting results are obtained with respect to intermediate and parallel entries. An analysis of the cost indicators leads to the conclusion that, as the section decreases, the cost of driving and supporting

Fig. 1

Fig. 2

1 cubic meter of excavation increases, but the cost of driving and supporting 1 linear meter decreases. As the section decreases, the cost of repair and maintenance of 1 linear meter of entry also decreases, \(B = \frac{aS}{f}\), but the cost of ventilation increases. As a result, it turns out that, as the dimensions of the entry section decrease, the total cost of 1 linear meter decreases. If, in addition, one takes into account the observations in the safety regulations that intermediate entries in thick seams with a service life of no more than 3–4 months must have a section of at least 3 sq. m, and that the sections of the remaining workings (crosscuts, raises) must be at least 1.5 sq. m, then we may say that the optimal dimensions for intermediate entries will be sections of 3–3.5 sq. m, and for parallel entries, 1.5–2.0 sq. m. In most cases, in practice, the sections of intermediate

entries of 3–3.5 sq. m and parallel entries of 1.5–2 sq. m also fully satisfy the requirements of mine transport.

Conclusions

1. The studies carried out show that the cost of driving 1 cubic meter of an entry depends, and very strongly, on the dimensions of the cross section of the workings, the strength of the coal, the method of driving, etc.

2. The work indicates a way toward the proper study and determination of expenditures for the repair and maintenance of workings—an extremely important and problematic question. This is the path of a broad organization of observations in production, with subsequent analysis and processing of the factual materials.

3. With reference to the conditions of the mines of southern Kuzbass and analogous conditions, an attempt has been made to determine the optimal cross-sectional dimensions of entries driven in coal. For the mines of Kuzbass such a problem is being solved for the first time.

LIST OF REFERENCES USED

1. Zyagin, P. Z.—Design and calculations of elements for the development of stratified deposits, ONTI, NKTP USSR, 1935.

2. Tsimbarevich, P. M.—Course on mine timbering, ONTI, 1936.

3. Protodyakonov, M. M.—Pressure of rocks and mine timbering. Part II. Mine timbering. Mining Publishing House, 1933.

4. Engineering Worker, No. 10, 1929.

5. Engineering Worker, Nos. 1–2, 1931.

6. Mining Journal, No. 5, 1937.

7. Komarov, V. B., and Borisov, D. F.—Mine ventilation. GONTI NKTP USSR, 1938.

8. Shevyakov, L. D.—On the cost of repairing mine workings. Collection of articles on mining art, issue 1, Donugol Publishing House, 1927.

9. Lenin, V. I.—What are the “friends of the people” and how do they fight the Social-Democrats. 2nd ed., 1931.

10. Shevyakov, L. D.—Outline of an analytical course in mining art. Published by the Sverdlovsk Mining Institute (as a manuscript), 1935.

11. Pokrovsky, N. M.—Driving horizontal and inclined workings. ONTI NKTP USSR, 1937.

12. Group of engineers.—Fundamental questions of mining. NTS KUP, 1933.

13. Bakanov, G. E., and Postoenko, A. N.—Determination of the relative strength of coals by the method of blasting standard charges. Proceedings of the Tomsk Industrial Institute named after S. M. Kirov. Vol. 60, issue II, 1940.

14. Mankovsky, G. I.—Driving vertical shafts. ONTI NKTP USSR, 1935.

15. Lares, G.—A refined formula for calculating charges in mining blasting operations. ONTI NKTP USSR, 1934.

16. Sukhanov, A. F.—Methods for determining the most advantageous magnitudes in the extraction of rocks and useful minerals, GNTIU, Kharkov, 1936.

17. Leontiev, V. N.—Mechanized loading of coal in entry driving. Ugol, No. 9, 1939.