ON DETERMINING THE HEIGHT OF LEVELS IN STEEPLY DIPPING DEPOSITS WITH LIMITED STRIKE DIMENSIONS

G. E. BAKANOV

The results of the fourth five-year plan and the plans outlined by the directives of the Nineteenth Congress of the CPSU for the fifth five-year plan show that our mining industry is developing at a gigantic pace \[1; 2; 3 and 4\]. Soviet miners, enthusiastically carrying out the instructions of the Party and the Government, are at present conducting grandiose work on the construction of new mining enterprises and the reconstruction of existing ones. In carrying out this work, the miners of the Soviet Union set themselves the task of creating socialist enterprises fully meeting the requirements of the basic economic law of socialism.

The essential features and requirements of the basic economic law of socialism are the assurance of the maximum satisfaction of the constantly growing material and cultural needs of the whole of society through the continuous growth and improvement of socialist production on the basis of higher technology \[5, 22\].

The continuous growth and improvement of socialist production on the basis of higher technology are the principal tasks that confront our mining industry, as they do the other branches of the Soviet socialist economy.

The dimensions of mine levels substantially influence the efficiency of mining enterprises. Therefore, studying the question of establishing their rational dimensions is one of the important measures in improving the production of extraction of useful minerals, which constitute the basis of the industrial might of the Soviet Union.

1. The height of mine levels and its influence on the efficiency of mining steeply dipping deposits

In the mining of ore deposits, the dimensions of mine fields, especially their length, are very often determined by the dimensions of the ore deposits themselves. Consequently, under such conditions the height of mine levels is a fundamental element both of the mine itself and of the mining systems applied in it. Therefore, it is very important here to adopt this element in such a way as to ensure the highest performance indicators of the mine. The height of mine levels has analogous significance in the mining of hard-coal deposits as well, when the boundaries of its field along the strike have already been established at the mine (the mine fields are bounded by fault lines, the opening of new levels of existing mines, etc.).

The height of mine levels influences the continuity of mine operation, the cost of the mineral being extracted, the intensity of mining of the deposit or mine field, the productivity of mine workers-

their condition and, in a number of cases, the completeness of extraction of useful minerals and the safety of work in stopes.

The height of levels has a great influence on the costs of driving and equipping capital and preparatory workings (near-shaft workings and chambers, crosscuts, field and ore main drifts, etc.). In this respect the importance of level height is especially great in thick ore deposits, where the absolute magnitude of the indicated costs often reaches several million rubles within a single level. In this case, increasing the height of mine levels substantially reduces the expenditures for driving and equipping workings on the level horizons.

On the other hand, in highly disturbed deposits, thin veins that often pinch out, or poorly explored veins, it is often only with a successfully chosen level height that the real possibility is determined of cutting the vein or bed by a level crosscut. Otherwise, the crosscuts that are driven may pass through barren strata without encountering the deposit being opened. In such a case, with an unsuccessfully chosen level height, additional costs may arise and additional time may be required for driving new opening workings. Naturally, under such conditions it will be more expedient to carry out mining with lower levels than would be the case with a more regular occurrence of the deposits. The additional costs that would arise from reducing the height of the levels would obviously be partly offset by savings in the preliminary and detailed exploration of the deposit.

In mining systems that do not permit caving of the wall rocks before the complete extraction of all sublevels in the block, as, for example, in a system with sublevel ore extraction, the height of levels and the length of blocks are mutually dependent on the stability of the wall rocks.

From what has been said it is clear that the height of mine levels influences the operating efficiency of a mine or ore mine. In this connection, it is of great importance to be able, in each specific case, to approach correctly the question of selecting the height of mine levels. This is especially important at the present time, when in the USSR large-scale work is being carried out for the further expansion of the mining industry, requiring the deepening of many operating mines and the construction of new enterprises for the extraction of various useful minerals.

Nevertheless, the methodology for determining the rational height of mine levels for steeply dipping veins and bedded deposits has as yet been little studied and is theoretically insufficiently developed. Therefore, in practice, under such conditions (with steep dip), the height of levels is usually adopted on the basis of the operating experience of other ore mines, which by no means always work deposits of an analogous character.

2. Heights of mine levels used in practice in the USSR, and observed tendencies with respect to their change

In mining steeply dipping deposits, the heights of mine levels used in practice vary within the range of 30–120 m. Most often, mine levels are made 40–45–100 m high. In the majority of cases coal deposits are mined with higher levels than ore deposits. If at large coal mines the levels have a height of 70–100 m [6], then at large ore mines they are more often made 60–80 m high [7]. At gold-ore, rare-metal, and po-

at polymetallic enterprises the heights of mine levels are rarely greater than 40–60 m.

Gold-ore and rare-metal deposits are often represented by thin, insufficiently regular, disturbed veins. The content of useful components in the ore changes constantly. Enterprises usually have small dimensions and a small cubic volume of capital and development workings on the level horizons. Therefore, the arrangement here of levels of such height is often caused by the specific features of the deposits themselves. As for large mining enterprises developing thick, steeply dipping ore bodies, such as, for example, many enterprises of the iron-ore industry (Krivbass, the Urals, Gornaya Shoriya), the use at them of levels 50–60 m high is not always justified by sufficiently well-founded considerations. Coal deposits, often occurring in more complex geological conditions than iron-ore deposits, are usually worked by higher levels.

In the postwar period some mines of Krivbass have already shifted to working with levels 90 m high and more [7].

During the years of the five-year plans in the USSR, a fundamental reconstruction of mining enterprises was carried out. Ore mines and coal mines were transferred to new, advanced Soviet equipment, which made it possible to mechanize all processes of extracting minerals, to raise sharply the productivity of miners, and to make their labor safe and easier. During this time there was also a continuous increase in mine levels, which provided more effective conditions for working mineral deposits. Before 1930 the height of mine levels in Krivbass did not exceed 21–30 m. At present, in the mines of the basin the levels are made no less than 50–110 m.

At the Ural copper mines (Kalatinsky, Karabash, Belorechensky, Karpushinsky, and others), until 1929 the heights of levels were, as a rule, 32 m. At present the copper-ore deposits of the Urals are worked mainly by levels of 40–60 m [8].

The giant mines built in South Kuzbass during the first five-year plan at first worked the mine fields allotted to them with levels of 50 m; at present, in these mines, except for some of them, coal is mined in levels as high as 100 m. At mine “A” the level height adopted is 80 m. At mine “B” 50-m levels were used from the 265 horizon to the 165 m horizon. Further, at this mine they began to carry out extraction with 100-m levels. Also at mine “V,” all levels from the 225 m horizon began to be made 100 m high; before the indicated horizon, the levels were 50 m each.

Such a situation with respect to the height of mine levels is observed at almost all coal mines and ore mines of the Soviet Union. This makes it possible to conclude that, together with the improvement in the technology of working steeply dipping coal and ore deposits, carried out in the USSR during the years of the five-year plans, there also gradually occurred an increase in mine levels at mining enterprises developing such deposits. This was especially noticeable during this time at mining-ore enterprises. In Russia before the Great October Socialist Revolution, at ore mines, levels of 32 m were considered high; at present in the USSR, mine levels of such dimensions may be considered low, up to 60 m—medium, and more than 60 m high.

3. The Concept of the Minimum and Optimum Height of Mine Levels

To ensure the continuous operation of a mining enterprise (coal mine, ore mine), it is necessary to adopt mine levels of such dimensions that, under conditions—

which would ensure the timely preparation of new levels for extraction.

Such conditions can be achieved by choosing the appropriate height of a mine level. The smallest level height \((h_m)\) at which these conditions begin to be ensured is called the minimum height of a mine level.

As the height of levels changes, the costs per ton of mineral reserves in the level change for driving crosscuts and drifts, for driving and equipping shaft stations, for hoisting the mineral and rock, for drainage, etc. Under some mining systems, as the height of levels changes, losses of useful minerals increase or decrease, depending on how the level height itself changes (increases or decreases).

Some costs, such as those for driving main drifts, crosscuts, and others, decrease as the height of mine levels increases; others (hoisting, drainage) increase. Owing to this character of the variation in costs that are functionally related to the level height \(h\), it is evident that in each particular case there may be found a value of \(h\) at which the sum of these costs will have the least value. Such a height of mine levels, at which this is achieved, is called the optimal, or economically expedient, level height.

If the optimal level height \((h_0)\) for certain conditions is not less than the minimum height \((h_m)\), then, for reasons of economy, it is desirable to bring the level height close to the value \(h_0\). Sometimes, depending on the geological structure of the deposit and the nature of the mining system used, it becomes necessary to make levels lower than optimal, i.e., to take \(h < h_0\). In such a case the value of \(h\) must not be less than \(h_m\). Thus, the technically expedient height of mine levels \((h)\) may be equal to or less than the optimal \((h \leq h_0)\), and equal to or greater than the minimum \((h \geq h_m)\).

4. Determination of the Minimum Height of Mine Levels

The minimum height of a level may be found as follows. Let

\(Q_e\) — reserves of the useful mineral in the level, \(t\);

\(k\) — coefficient of their recovery during mining;

\(\rho\) — coefficient of dilution of the mineral during extraction;

\(t_e\) — time required to mine the level;

\(t_p\) — time required to prepare one level;

\(t_r\) — time required to create a reserve of prepared reserves;

\(V\) — annual lowering of the level of mining operations in the mine, \(m/\text{year}\);

\(k_0\) — coefficient of advance of preparatory work, equal to \(\dfrac{t_e}{t_p}\);

\(\gamma\) — bulk density of the ore, \(t/m^3\);

\(A\) — annual productivity of the mine, \(t/\text{year}\);

\(h\) — vertical height of the mine level, \(m\);

\(S\) — area of the horizontal section of the deposit within the mine field;

\(\alpha^\circ\) — dip angle of the mineral deposit.

Then

\[ A=\frac{kQ_e}{t_e}=\frac{kQ_e}{t_p+t_r}=\frac{kQ_e}{k_0 t_p}. \]

On the other hand,

\[ t_{\mathrm{э}}=t_n+t_p=k_0 t_n=\frac{h}{V}=\frac{kQ_{\mathrm{э}}}{A}. \]

Hence the minimum level height is

\[ h_m=k_0 Vt_n. \tag{1} \]

Since

\[ V=\frac{A(1-\rho)}{kS\gamma}, \]

then

\[ h_m=k_0 Vt_n=\frac{Ak_0t_n(1-\rho)}{kS\gamma}, \tag{2} \]

The time for preparing a level [9] is

\[ t_n=\frac{l_1}{v_1}+\frac{l_2}{v_2}+\frac{l_3}{v_3}+\frac{l_4}{v_4}, \tag{3} \]

where \(l_1\) is the length of the mine shaft within one level, m;
\(l_2\) is the length of the crosscut of the level being designed, m;
\(l_3\) is the length of the drift in the wing required for placing the stoping front; when extraction proceeds from the boundary of the field toward the shaft, it is the length of the larger wing, m;

\(l_4\) is the length of the raise (or winze), equal to \(\dfrac{h}{\sin\alpha}\), m;

\(v_1, v_2, v_3\), and \(v_4\) are, respectively, the rates of driving these workings.

For vertical shafts

\[ l_1=h;\qquad l_2=l_0\pm h\operatorname{ctg}\alpha, \]

\(l_0\) is the length of the crosscut of the level being developed, or the greatest distance of the mine shaft from the upper boundary of the field, if the time for preparing the first level is being determined. The plus sign is used when the crosscuts increase with the depth of the shaft, and the minus sign when they decrease.

Substituting into equation (1) the value of \(t_n\), expressed through \(l_{1-4}\) and \(v_{1-4}\), we obtain:

\[ h_m=k_0V\left(\frac{h_m}{v_1}+\frac{l_0}{v_2}\pm\frac{h_m\operatorname{ctg}\alpha}{v_2}+\frac{l_3}{v_3}+\frac{h_m}{v_4\sin\alpha}\right). \]

Hence the minimum level height is

\[ h_m= \frac{ k_0V\left(\dfrac{l_0}{v_2}+\dfrac{l_3}{v_3}\right) }{ 1-k_0V\left(\dfrac{1}{v_1}\pm\dfrac{\operatorname{ctg}\alpha}{v_2}+\dfrac{1}{v_4\sin\alpha}\right) }. \tag{4} \]

Expressing in equation (4) the quantity \(V\)—the annual lowering of the level of mining operations—through \(A\)—the annual mine output, we obtain:

\[ h_m= \frac{ \dfrac{l_0}{v_2}+\dfrac{l_3}{v_3} }{ \dfrac{kS\gamma}{k_0A(1-\rho)} -\left(\dfrac{1}{v_1}\pm\dfrac{\operatorname{ctg}\alpha}{v_2}+\dfrac{1}{v_4\sin\alpha}\right) }. \tag{5} \]

For inclined shafts

\[ l_1=\frac{h}{\sin\alpha}; \qquad l_2=l_0. \]

If the shaft is driven through the useful mineral, then \(l_2=0\).

The minimum height of a level is

\[ h_m=\frac{k_0 V\left(\dfrac{l_0}{v_2}+\dfrac{l_3}{v_3}\right)} {1-\dfrac{k_0 V}{\sin\alpha}\left(\dfrac{1}{v_1}+\dfrac{1}{v_4}\right)} \tag{6} \]

or

\[ h_m= \frac{\dfrac{l_0}{v_2}+\dfrac{l_3}{v_3}} {\dfrac{kS\gamma}{k_0 A(1-\rho)}-\dfrac{1}{\sin\alpha}\left(\dfrac{1}{v_1}+\dfrac{1}{v_4}\right)}. \tag{7} \]

If the inclined shaft is driven through the useful mineral, then, when determining the minimum height of a level, it is necessary in equation (6) or (7) to exclude the term \(\dfrac{l_0}{v_2}\).

When finding \(h_m\) for flank shafts, the value \(l_3\) should be understood as the full length of the front of stoping operations in the level, or the full length of the shaft field if extraction is carried out on retreat.

In practice, as noted above, the height of mine levels must be taken no less than \(h_m\). Otherwise the enterprise will not prepare new levels in time, since they will be worked out faster than they are prepared.

If, for any reason, the level height must be taken less than the found \(h_m\), then it is necessary to develop a special procedure for preparing levels, allowing certain workings to be driven not sequentially but in parallel.

In this case the possible time for preparing a level may be found from equations (1) and (2):

\[ t_n=\frac{h}{k_0V}=\frac{kS\gamma h}{k_0A(1-\rho)}. \tag{8} \]

The question of the possible time for preparing levels is solved in the same way also in the case when, for a given level height, it is required to find the time when preparatory work must begin. Parallel driving of preparatory workings is of great importance, since it makes it possible to carry out the preparation of levels at a faster rate.

The necessary reserve of prepared reserves can also be created with lower levels if two levels are prepared simultaneously [9]. In this case the minimum height of levels is

\[ h_m= \frac{\dfrac{l_0}{v_2}+\dfrac{l_3}{v_3}} {\dfrac{2kS\gamma}{k_0A(1-\rho)}-\left(\dfrac{2}{v_1}+\dfrac{2\operatorname{ctg}\alpha}{v_2}+\dfrac{1}{v_4\sin\alpha}\right)} \tag{9} \]

or

\[ h_m = \frac{\dfrac{l_0}{v_2}+\dfrac{l_3}{v_3}} {\dfrac{2kS\gamma}{k_0 A(1-\rho)}-\dfrac{1}{\sin\alpha}\left(\dfrac{2}{v_1}+\dfrac{1}{v_1}\right)} \tag{10} \]

Equation (9) applies when the deposit is opened by vertical shafts, and equation (10) when it is opened by inclined shafts.

Comparing equations (4–7) with equations (9–10), it is easy to see that, all other conditions being equal, the minimum level height found from equations (9–10) will be less than that given by the corresponding equations (4–7).

From this it may be concluded that, by applying such a method of preparing levels (two levels at a time), it is possible to ensure a high intensity of deposit mining and, if necessary, to carry on stoping operations with lower levels.

5. Determination of the Optimum Level Height

In determining the optimum height of mine levels, in addition to the structural features of the mining system itself, the arrangement of preparatory level horizons is of great importance. From this point of view, modern mining systems may be divided into three groups:

a) without leaving interlevel pillars at the main horizons (slice caving, sublevel caving, and others);

b) with the temporary leaving of interlevel pillars (sublevel extraction from the horizon of drifts and crosscuts, level extraction with ore broken by deep vertical boreholes, mining with ore storage, level caving, and other systems used chiefly in mining thick ore deposits);

c) mining systems in which extraction of the mineral can be carried out with the leaving of pillars near the drifts and without pillars (continuous mining systems, room-and-pillar mining systems, used mainly in thin beds, veins, etc.). The nature of extraction (with pillars or without pillars near the drifts) has a substantial influence on the choice of the optimum level height. If in the first case the height of the levels does not affect the completeness of extraction of the mineral, then in the second case the latter depends to a large extent on the dimensions of the levels. In connection with this, the method of solving the stated problem also changes.

A. Mining without Pillars near the Level Drifts

For systems of this group, the optimum level height is determined as a function of unit costs [10–13]:

a) for driving and equipping crosscuts, shaft-station yards, and shaft-bottom chambers;

b) for driving and equipping the workings of the preparatory horizon (main and parallel drifts, crosscuts, raises, and other workings necessary for creating the preparatory horizon);

c) for hoisting the rock mass, drainage, delivery of materials to the stoping workings, and the movement of face workers to the place of work;

d) for maintaining raises, manways, and winzes.

In addition to the factors listed above, the level height is also influenced by the costs of haulage of the rock mass in crosscuts, by mainten—

the sinking of raises and mine shafts. However, these costs are relatively small and, in determining the level height, may be disregarded.

Let it be necessary to determine the optimal height \(h_{\text{opt}}\) of a level (Fig. 1). Then the costs depending on its height may be represented as follows.

Sinking and equipping raises

If \(h\) is the height of the level being designed; \(l_{\text{kv}}\) is the length of its raise; \(c'_{\text{kv}}\) is the cost per linear meter of raise; \(\alpha^\circ\) is the dip angle of the deposit; \(H_0\) is the depth of the main horizon of the level being developed; \(V_1\) is the volume of the raise \(l_0\), and

Fig. 1. Diagram of level arrangement

\(c_1\) is the cost of sinking the raise referred to a unit of its volume; then the costs of sinking the raise are:

\[ C_{\text{kv}} = c'_{\text{kv}} l_{\text{kv}} = c'_{\text{kv}} (l_0 \pm h \operatorname{ctg}\alpha) = c'_{\text{kv}} l_0 \pm c'_{\text{kv}} h \operatorname{ctg}\alpha = c_1 V_1 \pm c'_{\text{kv}} h \operatorname{ctg}\alpha . \]

Putting

\[ \frac{kS\gamma h}{1-\rho}=qh, \]

we find that the costs of sinking raises per tonne of mineral are

\[ C_1=\frac{c_1 V_1}{qh}\pm\frac{c'_{\text{kv}}\operatorname{ctg}\alpha}{q}. \]

Shaft stations, chambers, and workings of the preparatory horizon

Let \(V_2, V_3, V_4, \ldots\) be the volumes of workings driven in waste rock within one level horizon (apart from raises), \(c_2, c_3, c_4, \ldots\) respectively the costs per m\(^3\) of their driving, \(R_1, R_2, R_3, \ldots\) the volume of workings driven in the useful mineral, and \(r_1, r_2, r_3, \ldots\) their costs per m\(^3\) (excluding the value of ore recovered incidentally). Then the unit costs of driving these workings

\[ C_2=\frac{c_2 V_2+c_3 V_3+c_4 V_4+\ldots}{qh}+ \]

\[ +\frac{r_1 R_1+r_2 R_2+r_3 R_3+\ldots}{qh}. \]

Hoisting of the rock mass and drainage

Costs of hoisting the rock mass [14]

\[ C_3=k_n c_n(H_0+h)=k_n\left(c'_n+\frac{c''_n}{H_0+h}\right)(H_0+h)= \]

\[ =k_n c'_n(H_0+h)+k_n c''_n, \]

where
\(k_n\)—coefficient allowing for the yield of rock;
\(c_n\)—total cost of hoisting, rub/t;
\(c'_n\)—cost of “pure” hoisting, rub/t;
\(H_0+h\)—hoisting height.

Drainage costs [14]

\[ C_4=\mu c_v(H_0+h)=\mu\left(c'_v+\frac{c''_v}{H_0+h}\right)(H_0+h)= \]

\[ =\mu c'_v(H_0+h)+\mu c''_v, \]

\(\mu\)—water-abundance coefficient;
\(c_v\)—total cost of drainage, rub/t;
\(c'_v\)—cost of “pure drainage” (by analogy with hoisting).

Delivery of workers and materials to stoping workings, maintenance of chutes and raises

Costs of delivering workers

With mechanized delivery of workers to the stoping workings, the costs per one ton of mineral are

\[ C_5=c_p\frac{h}{2}, \]

where \(c_p\)—the cost of raising and lowering workers along raises through a height of 1 m, referred to one ton of mineral;

\[ \frac{h}{2} \]
—the average hoisting height.

When workers travel on foot, these costs may be represented as follows. Let \(t\) be the time spent by a worker on ascending and descending for a hoisting height of one meter, in minutes; \(T\) the duration of the shift in hours; \(\eta\) a coefficient allowing for the time allotted to work, in fractions of unity (excluding rest); \(p\) the productivity of face workers in tons per shift; \(\omega\) a coefficient for determining the average height of ascent–descent, taken as equal to 0.25 or 0.5 depending on the order of movement of the workers [13]; and \(c\) the average shift wage of face workers with all additions. Then the costs associated with trans-

with the movement of workers, when walking on foot, per one ton of mineral will be

\[ C_5=\frac{c\omega th}{60\eta pT}. \]

Taking

\[ \frac{c_p}{2}=\frac{c\omega t}{60\eta pT}=\frac{\Delta_1}{2}, \]

we obtain the general expression for these costs for both cases

\[ C_5=\frac{\Delta_1 h}{2}. \]

Similarly, the costs for delivering timbering and other materials to stoping workings will be

\[ C_6=\frac{\Delta_2 h}{2}. \]

Likewise, for the repair of raises [13],

\[ C_7=\frac{\Delta_3 h}{2}. \]

Putting

\[ \frac{\Delta_1}{2}+\frac{\Delta_2}{2}+\frac{\Delta_3}{2}=\frac{\Delta}{2}, \]

we obtain

\[ C_5+C_6+C_7=\frac{\Delta}{2}h. \]

Differentiating

\[ \sum_{i=1}^{i=7} C_i \]

with respect to the variable \(h\), setting the first derivative equal to zero, and solving the resulting equation with respect to the same variable, we find that the optimal level height is

\[ h_0= \sqrt{ \frac{\sum c_i V_i+\sum r_i R_i} {q'\left(k_n c'_n+\mu c'_в+\dfrac{\Delta}{2}\right)} }, \tag{11} \]

\[ \sum c_i V_i \]

is the cost of level workings driven in rock;

\[ \sum r_i R_i \]

is the cost of workings of the level horizon driven in the mineral (excluding the cost of the useful mineral extracted in the process).

As noted above, when calculating \(\sum c_i V_i\), the volume of the crosscut \(l_0\) of the level being developed is taken. When determining the optimal height of the first level, the length of the crosscut is taken equal to the distance from the apex of the mine field to the mine shaft. If a zero crosscut is driven, then its length is adopted for the first level.

B. Systems of mining with the temporary leaving of interlevel pillars

When extracting interlevel pillars, which in many cases is carried out without prior backfilling of the stopes, considerable losses of useful minerals usually arise. At the same time, the total losses of minerals will be the greater, the thicker such pillars are made and the more often they are left. The thickness of the pillars depends to a certain extent on the thickness of the deposits and on the structural features of the mining systems employed. Mechanized delivery of the mineral to the ore passes makes it possible to reduce the dimensions of the pillars considerably. However, it is not yet possible to avoid them completely in a whole number of mining systems. As for the frequency with which pillars are left, this depends entirely on the height of the mine levels. A high completeness of extraction of the prepared reserves is of great importance under the conditions of our socialist economy. Therefore, when determining the optimum dimensions of mine levels, this factor (the completeness of reserve extraction) must in no way be ignored.

For the time being, without separately taking into account development and extraction workings, the overall coefficient of losses of useful minerals during mining may be represented in the form

\[ k_1=\frac{k'_1 V_k+k'_2 V_c}{V}, \tag{12} \]

\(V_k\)—volume of stope reserves, \(m^3\);
\(V_c\)—volume of inter-stope and interlevel pillars, including the volume of development and extraction workings, \(m^3\);
\(V\)—total volume of the block or extraction area, \(m^3\);
\(k'_1\)—coefficient of mineral losses in extracting stope reserves;
\(k'_2\)—coefficient of mineral losses in extracting pillars (taking into account the completeness of extraction from development and extraction workings).

Substituting the corresponding quantities into equation (12) (Fig. 2), we obtain

\[ k_1= \frac{k'_1 m_0(h-a)(L-c)+k'_2 m_0 aL+k'_3 m_0 c(h-a)} {m_0 hL} = \]

\[ = k'_1+\frac{c}{L}\left(k'_3-k'_1\right) +\frac{a}{h}\left(1-\frac{c}{L}\right)\left(k'_2-k'_1\right), \]

where \(m_0\) is the horizontal thickness of the useful-mineral deposit.

Putting

\[ k'_1+\frac{c}{L}\left(k'_2-k'_1\right)=\lambda_1 \]

and

\[ a\left(1-\frac{c}{L}\right)\left(k'_2-k'_1\right)=\lambda, \]

we obtain

\[ k_1=\lambda_1+\frac{\lambda}{h}. \tag{13} \]

Equation (13) shows that, for the given mining systems, the loss coefficient and the losses of useful minerals, with a change in the level height, vary according to the law of a hyperbola and decrease as the height \(h\) increases.

The mineral recovery coefficient during extraction is

\[ k = 1 - k_1 = 1 - \left(\lambda_1 + \frac{\lambda}{h}\right) = 1 - \lambda_1 - \frac{\lambda}{h} = \lambda_0 - \frac{\lambda}{h}. \]

As is seen from the last equation, with an increase in the height of the levels, the mineral recovery coefficient increases.

Taking into account the conclusions drawn, the recoverable reserves of a level may be represented as

\[ Q_{\mathrm{e}} = \frac{k S \gamma h}{(1-\rho)} = \frac{\left(\lambda_0-\frac{\lambda}{h}\right) S \gamma h}{(1-\rho)} = q_1'(\lambda_0 h-\lambda). \]

Then the costs per ton of mineral, depending on the height of the level, will be:

a) crosscuts

\[ C_1 = \frac{c'_{\mathrm{kv}} l_0}{q_1(\lambda_0 h-\lambda)} \pm \frac{c'_{\mathrm{kv}} h \operatorname{ctg}\alpha}{q_1(\lambda_0 h-\lambda)}. \]

Since the second term, for steep dip, is very small, it may be neglected without appreciable loss of accuracy, and one may take

Fig. 2. Layout of pillars in a block

\[ C_1 = \frac{c'_{\mathrm{kv}} l_0}{q_1(\lambda_0 h-\lambda)} = \frac{c_1 V_1}{q_1(\lambda_0 h-\lambda)}. \]

b) shaft-adjacent yards, chambers, and workings of the development horizon

\[ C_2 = \frac{c_2 V_2 + c_3 V_3 + \ldots}{q_1(\lambda_0 h-\lambda)} + \frac{r_1 R_1 + r_2 R_2 + \ldots}{q_1(\lambda_0 h-\lambda)}. \]

c) hoisting of the rock mass and drainage. The hoisting of the rock mass, as was shown above,

\[ C_3 = k_n c'_n (H_0+h) + k_n c''_n. \]

Dewatering

\[ C_4=\mu c'_v(H_0+h)+\mu c''_v . \]

c) Delivery of workers and materials to stoping workings, maintenance of inclines and raises.

\[ C_5+C_6+C_7=\frac{\Delta}{2}\,h . \]

d) Reduction in the economic efficiency of mining the deposit because of losses of the mineral in pillars

\[ C_8=d_0 k_1=d_0\left(\lambda_1+\frac{\lambda}{h}\right). \]

\(d_0\)—the profitability indicator of the industrial use of the mineral [9].

e) Amortization of mine-wide capital expenditures

\[ C_9=\frac{K h}{q_1(\lambda_0 h-\lambda)H_1}, \]

\(H_1\)—the height of the mine field or of the deposit occurrence.

Having differentiated the sum of costs and set the first derivative equal to zero, we obtain the equation

\[ -\frac{\lambda_0\sum c_i V_i}{q_1(\lambda_0 h-\lambda)^2} -\frac{\lambda_0\sum r_i R_i}{q_1(\lambda_0 h-\lambda)^2} -\frac{d_0\lambda}{h^2} -\frac{K\lambda}{q_1H_1(\lambda_0 h-\lambda)^2} +\left(k_n c'_n+\mu c'_v+\frac{\Delta}{2}\right)=0. \]

After the corresponding transformations this equation will have the form:

\[ \begin{aligned} &q_1H_1\lambda_0^2\left(k_n c'_n+\mu c'_v+\frac{\Delta}{2}\right)h_0^4 -2q_1H_1\lambda_0\lambda\left(k_n c'_n+\mu c'_v+\frac{\Delta}{2}\right)h_0^3 \\ &\quad+\left[ q_1H_1\lambda^2\left(k_n c'_n+\mu c'_v+\frac{\Delta}{2}\right) -\lambda_0H_1\sum c_i V_i-\lambda_0H_1\sum r_i R_i -q_1H_1\lambda_0^2\lambda d_0-K\lambda \right]h_0^2 \\ &\quad+2q_1H_1\lambda_0\lambda^2 d_0 h_0 -q_1H_1\lambda^3 d_0=0. \end{aligned} \]

Putting

\[ q_1H_1\lambda_0^2\left(k_n c'_n+\mu c'_v+\frac{\Delta}{2}\right)=A_1; \]

\[ -2q_1H_1\lambda_0\lambda\left(k_n c'_n+\mu c'_v+\frac{\Delta}{2}\right)=A_2; \]

\[ q_1H_1\lambda^2\left(k_n c'_n+\mu c'_v+\frac{\Delta}{2}\right) -\lambda_0H_1\sum c_i V_i-\lambda_0H_1\sum r_i R_i -q_1H_1\lambda_0^2\lambda d_0-\lambda K=A_3; \]

\[ 2q_1H_1\lambda_0\lambda^2d_0=A_4 \quad \text{and} \]

\[ -q_1H_1\lambda^3d_0=A_5, \]

we finally obtain the equation

\[ A_1h_0^4+A_2h_0^3+A_3h_0^2+A_4h_0+A_5=0. \tag{14} \]

This equation is more universal; it can be used to determine the optimal height of mine levels for any mining system. Its approximate solution presents no difficulty.

If, in the first stage, the work is carried out with backfilling in order to extract the pillars more fully, then, when determining the level height, it is necessary additionally to take into account the increased cost of work in extracting the near-level thickness.

After the minimum and optimal height of the levels has been determined, the mining-geological conditions of exploitation are carefully studied and the level height is finally selected.

The main propositions of the present work were presented by the author at a scientific conference of the departments of the TPI in January 1952. At present the author is conducting research aimed at studying the constant quantities entering into equations 11 and 14.

LITERATURE

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14. Popov, A. S. Technical-Economic Analysis in Mining Engineering, 1932.