ACTION AND CALCULATION OF REDUCED BLASTING CHARGES AND LOOSENING CHARGES

S. D. OSNOVIN

Fulfillment of the five-year plan for the development of the USSR in 1951–1955 requires the introduction of advanced technology in all branches of the national economy, including mining operations.

Blasting operations, the scale of which increases more and more with each year, are of very great importance in mining operations.

For the correct and effective performance of blasting operations, an adequate theoretical basis and a sufficient level of scientific development of charge calculation are necessary.

Meanwhile, as was noted at the conference on drilling-and-blasting operations convened by the All-Union Scientific Engineering-Technical Mining Society in Moscow in 1939, and then at the conference on drilling-and-blasting operations held by the All-Union Scientific Engineering-Technical Mining Society in Moscow from June 20 to 24, 1952, questions of charge calculation still do not have the necessary theoretical basis, and the level of charge calculation lags behind the rate of development of drilling-and-blasting operations; this lag is becoming more and more perceptible. In the resolution adopted at the last conference, the scientific development of this problem was set forth as one of the main and priority tasks.

In the present work, the calculation of reduced blasting charges and loosening charges, which are most widely used in mining operations, is based on a different principle, a different explanation of the physical essence of the phenomena arising during the explosion of a charge, and a different method of calculating them than has been the case up to now. The history of science and technology shows how fruitful it sometimes is to approach from a new point of view some previously known phenomenon or established view in one or another field of our knowledge.

Methods Used for Calculating Concentrated Charges

As the basis of the formulas for calculating reduced blasting charges and loosening charges that have become most widespread, all researchers used a formula of general form for calculating charges of normal throw, proposed almost one hundred years ago. This formula is as follows:

\[ L = q \cdot W^{3}\,\kappa z, \tag{1} \]

where \(L\) is the weight of the charge of normal throw in kg;
\(q\) is the specific consumption of explosives in kg/m\(^3\);
\(W\) is the least resistance line in m.

In this formula it is assumed:

1) proportionality of the magnitude of the charge to the weight of the medium torn away or, in other words, to the volume of the crater;

2) under otherwise equal conditions—the constancy of the specific consumption of explosives with a change in the LLR.

Later studies and practical observations showed that, in the performance of blasting operations in mining, when only fragmentation of the rock is required, without any significant throw, or only its loosening, the specific consumption of explosives does not remain constant and, as the depth of placement of the charge increases, an ever smaller specific consumption of explosives may be used. In this connection, new formulas were proposed for calculating charges, which received the name formulas for calculating charges of reduced throw and loosening charges.

All these formulas differ in outward form from the general formula for calculating charges of normal throw only in that a multiplier \(f(n)\) has been introduced into them, taking into account the change in the specific consumption of explosives as a function of the depth of placement of the charge¹. In introducing \(f(n)\) into the general formula for calculating charges of normal throw, all researchers pursued one aim—to establish the regularity of the change in the specific consumption of explosives with a change in the depth of placement of the charge, but did not reveal the reason why this occurs.

All formulas for calculating charges of reduced throw and loosening charges can be represented in the general form:

\[ Q = f(n)\cdot q \cdot W^3 = f(n)\cdot L \kappa^2, \tag{2} \]

where \(Q\) is the weight of a reduced-throw charge or loosening charge, in kg;
\(f(n)\) is the function of the throw index (charge action);
\(q\) is the specific consumption of explosives, in kg/m\(^3\);
\(W\) is the LLR, in m;
\(L\) is the magnitude of the charge of normal throw with LLR \(= W\).

The task of all authors consisted mainly in deriving a mathematical expression for \(f(n)\), using their own personal experience and experimental data.

Many formulas were proposed for expressing \(f(n)\). A considerable part of these formulas did not come into practical use and at present is only of historical interest. In recent years formulas for determining \(f(n)\) have been proposed by S. D. Osnovin (1939), N. V. Melnikov (1944), V. E. Filippovich (1944), V. K. Karchevskii (1945), and S. D. Osnovin (1948).

The numerical values of \(f(n)\) and the regularity of their variation according to the different formulas diverge sharply from one another, depending on what empirical data and what method of processing them the authors used.

A characteristic feature inherent in all formulas for expressing \(f(n)\) is that all researchers who proposed one or another mathematical expression for \(f(n)\), proceeding from the principle adopted in constructing formulas for calculating charges of normal throw, assumed a single regularity of variation of \(f(n)\), along a single curve, so that \(f(n)\) could have only one order of numerical values, regardless of the physicomechanical properties of the medium being blasted. Only the formula of S. D. Osnovin occupies a separate position.

However, despite the large number of formulas proposed for expressing \(f(n)\), with diverse magnitudes and orders of numerical values, there is still no generally accepted formula; and the application of these formulas in practice—in rocks with different physicomechanical properties—leads to unsatisfactory, contradictory results.

Such, in brief outline, is the state of the question with the methods currently used for calculating concentrated reduced blasting charges and loosening charges. We see the reason for this situation in the incorrectness of the fundamental assumptions adopted in constructing formulas for calculating reduced blasting charges and loosening charges.

Proposed methods for calculating concentrated charges

When a charge is detonated, its energy is expended in two main directions—on acting upon the rock being detached and on deformation toward the body of the mass. The work of the charge in the first direction is useful, since in this case the charge fulfills its direct purpose. The work of the charge in the second direction only causes an increase in the weight of the charge, an expenditure of excess energy not used for useful work, and therefore it may be regarded as useless.

The calculation of each charge must be carried out so that it can perform the useful work required of it. Otherwise the rock being detached will retain all the properties of the mass, and the deformations toward the free surface will be the same as those toward the mass.

For a charge to perform useful work, according to the fundamental proposition we put forward, it must at least:

1) destroy the bond between the rock being detached and the mass;
2) overcome the resistance of the force of gravity and the inertia of the rock being detached;
3) overcome the resistance due to the pressure of the air column;
4) crush and displace the entire volume of the detached rock.

Considering the action of a charge during an explosion, we introduce two concepts—the separation crater and the ejection crater.

By the separation crater we shall mean the crater bounded by the surface of separation.

By the ejection crater we shall mean the crater within whose limits the ejection of rock takes place.

By the normal specific consumption of explosives is meant that specific consumption of explosives, independent of the degree of rock ejection, which is adopted when the line of least resistance is equal to 1 m.

It should be noted that, at values of the ejection (charge-action) index \(n \leq 0.75\), as practical observations have shown, the rock is not ejected, and the external action of the charge is expressed only in loosening and swelling of the rock (loosening charges).

The surface of the ejection crater cannot lie outside the surface of the separation crater.

Let us now suppose that a concentrated charge placed at a depth of 1 m acts upon one free surface and ejects the detached rock as a single whole, i.e., without crushing it. In this case the charge must overcome the resistance along the surface of the separation crater, the resistance of the force of gravity and the inertia of the detached rock, and the resistance due to the pressure of the air column.

The normal specific consumption of explosives for the indicated charge, \(q^1_{\text{norm.}}\), theoretically, in differentiated form, may be represented as the sum of three quantities:

\[ q^1_{\text{norm.}} = q_{\text{surf.}} + q_{\text{grav.}} + q_{\text{air}}, \tag{3} \]

where \(q_{\text{surf.}}\), \(q_{\text{grav.}}\), and \(q_{\text{air}}\) are the corresponding differentiated specific consumptions of explosives for overcoming the resistance along the surface of the separation crater, the resistance of the force of gravity and inertia of the detached rock, and the pressure of the air column.

Let us further suppose that, under the same conditions, the charge produces not only the ejection of the rock as a single whole, but also its crushing. In this case the specific expenditures of explosive—\(q_{\text{surf.}}\), \(q_{\text{grav.}}\), and \(q_{\text{air}}\)—can no longer retain their former values, since the energy expended on crushing the rock will not be used to overcome the principal resistances indicated above; they must be increased, and the normal specific explosive consumption in this case, in differentiated form, will be expressed as follows:

\[ q_{\text{norm.}}=(q_{\text{surf.}}+q'_{\text{crush.}})+(q_{\text{grav.}}+q''_{\text{crush.}})+(q_{\text{air}}+q'''_{\text{crush.}}), \tag{4} \]

where the terms on the right-hand side of the equation are the corresponding differentiated specific explosive expenditures for overcoming the resistance along the surface of the crater of rupture, the resistance of gravity and inertia of the detached rock, and the pressure of the air column, taking into account the expenditure of their energy on crushing the rock.

Introducing abbreviated notation, we obtain

\[ q_{\text{norm.}}=q_w+q_v, \tag{5} \]

where

\[ q_w=(q_{\text{surf.}}+q'_{\text{crush.}})+(q_{\text{air}}+q'''_{\text{crush.}}), \tag{6} \]

\[ q_v=(q_{\text{grav.}}+q''_{\text{crush.}}). \tag{7} \]

Whether in this case a greater, equal, or smaller share of the explosive is expended on crushing the rock than on overcoming the principal resistances is of no importance for our subsequent conclusions.

The normal specific explosive consumption—\(q_{\text{norm.}}\)—for given blasting conditions may be calculated by using the available methods for this purpose or, better, determined experimentally.

Let us now consider what will happen to the specific explosive consumptions \(q_w\) and \(q_v\) as the depth of charge placement increases. To this end, first of all, we shall determine how the change in the depth of charge placement will affect the magnitude of the individual resistances overcome by the charge.

We shall call the number of units of rupture surface per unit volume of the rupture crater the specific rupture surface, and the number of units of base area of the rupture crater per unit of its volume the specific base area.

Let us initially take the rupture crater in the form of a cone of revolution. The specific rupture surface for a crater in the form of a cone of revolution will be:

\[ \frac{S}{V}=\frac{\pi r l}{\frac{1}{3}\pi r^2 W}=\frac{3}{W\sin \alpha}, \tag{8} \]

where \(S\) is the rupture surface of the crater, \(m^2\);
\(V\) is the volume of the rupture crater, \(m^3\);
\(r\) is the radius of the crater base, \(m\);
\(l\) is the generatrix of the crater, m;
\(\alpha\) is the angle of the crater half-opening, in degrees.

It follows from expression (8) that, at a constant angle \(\alpha\), i.e., for similar craters, the specific rupture surface of craters in the form of a cone of revolution will vary inversely proportionally to the line of least resistance.

Table 1 gives the results of calculations by formula (8) of the relative change in the specific rupture surface of craters in the form of a cone of revolution as a function of the magnitude of the line of least resistance, for similar craters.

Table

Relative change in the specific breakaway surface for craters in the form of a cone of revolution as a function of the line of least resistance

LLR, m	0.01	0.1	1.0	2.0	5.0	10.0	20.0	30.0	40.0
Relative change in the specific breakaway surface	100	10	1.0	0.5	0.2	0.1	0.05	0.033	0.025

On the basis of Table 1 it can be established that the specific breakaway surface for craters in the form of a cone of revolution:

1) decreases sharply for an LLR \(> 1\) m;
2) increases sharply for an LLR \(< 1\) m.

By investigating breakaway craters in the form of a paraboloid of revolution, a truncated cone of revolution, etc., one can be convinced that, for any of them, the specific breakaway surface, when similarity is preserved, will also vary inversely proportional to the LLR.

Let us prove the correctness of this proposition not for individual geometric forms of breakaway craters, but in a general form.

The breakaway surface of a crater in the form of a cone of revolution is

\[ S=\pi \cdot r \cdot l, \tag{9} \]

the volume is

\[ V=\frac{1}{3}\pi \cdot r^2 \cdot W. \tag{10} \]

Let us denote the breakaway surface of a crater of any geometric form by \(S^1\), and its volume by \(V^1\).

The breakaway surface \(S^1\) can be expressed through the breakaway surface \(S\), by changing it by some unknown \(n\)-fold number of times, and the volume \(V^1\) through the volume \(V\), by changing it by some unknown \(n_1\)-fold number of times; then we obtain:

\[ S^1=n \cdot \pi \cdot r \cdot l, \tag{11} \]

\[ V^1=n_1 \cdot \frac{1}{3}\pi \cdot r^2 \cdot W. \tag{12} \]

The specific surface will be expressed as

\[ \frac{S^1}{V^1} = \frac{n \cdot \pi \cdot r \cdot l} {n_1 \cdot \frac{1}{3}\cdot \pi r^2 \cdot W} = \frac{n}{n_1}\cdot \frac{3}{W \cdot \sin \alpha}. \tag{13} \]

If the breakaway crater of one or another geometric form, with a change in LLR, preserves its similarity, then the ratio \(\frac{n}{n_1}\) and the angle \(\alpha\) in expression (13) will be constant quantities, and its specific breakaway surface will vary in exactly the same way as that of a breakaway crater in the form of a cone of revolution, i.e., inversely proportional to the LLR.

This will also be true in the case when the resistance along the breakaway surface of some geometric body and along the breakaway surface of a crater in the form of a cone of revolution is expressed in equivalent units.

Thus, everything that has been said about the change in the specific breakaway surface with a change in LLR with respect to breakaway craters

in the form of a cone of revolution also fully apply to breakage funnels of any geometrical form.

By the same method as above, we prove that for breakage funnels of any geometrical form the specific area of the base will also vary inversely proportionally to the line of least resistance.

Between the specific explosive consumptions:

\[ q_{\mathrm{surf.}},\quad q_{\mathrm{grav.}},\quad q_{\mathrm{air}} \]

and

\[ q'_{\mathrm{crush.}},\quad q''_{\mathrm{crush.}},\quad q'''_{\mathrm{crush.}} \]

there must exist a direct proportional relationship. And since this is so, the specific explosive consumption \((q_{\mathrm{surf.}} + q'_{\mathrm{crush.}})\) will vary in the same way as the specific breakage surface, i.e., inversely proportionally to the line of least resistance, or to the depth of charge placement.

In the same proportion the specific explosive consumption \((q_{\mathrm{air}} + q'''_{\mathrm{crush.}})\) will also vary.

The specific explosive consumption \((q_{\mathrm{grav.}} + q''_{\mathrm{crush.}})\), with a change in the depth of charge placement and with other conditions being equal, as earlier researchers assumed and as we assume, will remain constant.

Proceeding from these three propositions, we establish the following dependence between the specific explosive consumption for a line of least resistance not equal to \(1\ \mathrm{m}\) and the normal specific explosive consumption \((\mathrm{LLR} = 1\ \mathrm{m})\):

\[ q = \frac{(q_{\mathrm{surf.}} + q'_{\mathrm{crush.}}) + (q_{\mathrm{air}} + q'''_{\mathrm{crush.}})}{W} + (q_{\mathrm{grav.}} + q''_{\mathrm{crush.}}). \tag{14} \]

In accordance with the abbreviated designations adopted earlier:

\[ q = \frac{q_w}{W} + q_v, \tag{15} \]

where \(q\) is the specific explosive consumption for \(\mathrm{LLR} \ne 1\ \mathrm{m}\), in kg.

Expression (15) shows that, as the depth of charge placement increases, the specific explosive consumption \(q_w\) will vary inversely proportionally to \(W\), while the specific explosive consumption \(q_v\) will remain constant. This means that, as the depth of charge placement increases, the breakage surface and the pressure of the air column will exert an ever smaller relative influence on the magnitude of the total specific explosive consumption \(q\), while the weight or volume of the rock being broken away will exert an ever greater influence, and conversely.

In addition, we introduce the very important relation

\[ \frac{(q_{\mathrm{surf.}} + q'_{\mathrm{crush.}}) + (q_{\mathrm{air}} + q'''_{\mathrm{crush.}})} {(q_{\mathrm{grav.}} + q''_{\mathrm{crush.}})} = \frac{q_w}{q_v} = k . \tag{16} \]

The coefficient \(k\) is an abstract number representing the ratio of the total specific explosive consumption for overcoming resistance along the breakage surface of the funnel and from the pressure of the air column to the specific explosive consumption for overcoming the resistance of gravity and the inertia of the rock being broken away, for a charge with \(\mathrm{LLR} = 1\ \mathrm{m}\), taking into account the expenditure of their energy on crushing the rock. The law by which the specific explosive consumption \(q\) changes with a change in the depth of charge placement will depend on the value of the coefficient \(k\).

Let us now express \(q_w\) and \(q_v\) through the normal specific consumption of explosive—\(q_{\text{norm.}}\) (LLR \(=1\) m), using for this the following system of previously established relationships:

\[ q_{\text{norm.}} = q_w + q_v; \tag{17} \]

\[ q = \frac{q_w}{W} + q_v; \tag{18} \]

\[ \frac{q_w}{q_v}=k. \tag{19} \]

Hence:

\[ q_v=\frac{q_{\text{norm.}}}{k+1}; \tag{20} \]

\[ q_w=\frac{k\cdot q_{\text{norm.}}}{k+1}. \tag{21} \]

After substituting the values for \(q_w\) and \(q_v\) into expression (18), we obtain:

\[ q=\frac{k\cdot q_{\text{norm.}}}{k+1}:W+\frac{q_{\text{norm.}}}{k+1}= \]

\[ =\frac{1}{k+1}\cdot\left(\frac{k}{W}+1\right)\cdot q_{\text{norm.}}\ \text{kg} \tag{22} \]

or

\[ q=f(k)\cdot q_{\text{norm.}}\ \text{kg}, \tag{23} \]

where

\[ f(k)=\frac{1}{k+1}\cdot\left(\frac{k}{W}+1\right), \tag{24} \]

\(f(k)\) is a correction to the normal specific consumption of explosive when LLR \(\ne 1\) m.

As is evident from the formula, the specific consumption of explosive—\(q\)—depends on two quantities:

1) the value of \(f(k)\);
2) the value of the normal specific consumption of explosive—\(q_{\text{norm.}}\).

The theoretically possible numerical values of \(f(k)\) are given in Table 2.

Theoretically, the value of \(f(k)\) may vary within the limits from 1 (\(k=0\)) to \(\frac{1}{W}\) (\(k=\infty\)). For rocks these limits will be narrower.

The principal properties of \(f(k)\) are as follows:

1) when LLR \(=1\) m and for all values of \(k\), \(f(k)\) will be equal to 1;
2) for LLR \(\ne 1\) m, depending on the value of the coefficient \(k\), the change in the numerical values of \(f(k)\) will occur not according to a single law, but according to different laws, and the more sharply the larger the value of the coefficient \(k\);
3) there is a sliding scale of variation of the numerical values of \(f(k)\) for LLR \(\ne 1\) m as a function of the value of the coefficient \(k\).

Table 3 gives comparative data on the numerical values of \(f(k)\) and the values of \(f(n)\) according to the formulas of other authors. For LLR \(=30\) m the values of \(f(k)\) are taken as equal to the values of \(f(n)\).

The values of \(f(n)\) according to the formulas of N. V. Melnikov and V. E. Filippovich are very close to the values of \(f(k)\) at the coefficient \(k=1.9\). The values of \(f(n)\) according to the formula of V. K. Karchevsky completely coincide with the values of \(f(k)\)

Table 2

Numerical values of \(f(k)\) for various values of the coefficient \(k\)

LNS, m	\(k=0\)	\(k=0.1\)	\(k=0.5\)	\(k=1.0\)	\(k=1.2\)	\(k=1.4\)	\(k=1.6\)	\(k=1.8\)	\(k=2.0\)	\(k=3.0\)	\(k=4.0\)	\(k=5.0\)	\(k=6.0\)	\(k=7.0\)	\(k=8.0\)	\(k=9.0\)	\(k=9.5\)	\(k=10.0\)	\(k=10.5\)	\(k=15.0\)	\(k=\infty\)
1.0	1	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
2.0	1	0.95	0.83	0.75	0.73	0.71	0.69	0.68	0.66	0.63	0.60	0.58	0.57	0.56	0.56	0.55	0.55	0.54	0.54	0.53	0.50
3.0	1	0.94	0.78	0.67	0.64	0.61	0.59	0.57	0.55	0.50	0.47	0.44	0.43	0.42	0.41	0.40	0.40	0.39	0.39	0.38	0.38
4.0	1	0.93	0.75	0.63	0.59	0.56	0.54	0.52	0.50	0.44	0.40	0.37	0.36	0.34	0.33	0.32	0.32	0.31	0.31	0.30	0.25
5.0	1	0.93	0.73	0.60	0.56	0.53	0.51	0.49	0.47	0.40	0.36	0.33	0.31	0.30	0.29	0.28	0.28	0.27	0.27	0.25	0.20
6.0	1	0.92	0.72	0.58	0.54	0.51	0.49	0.46	0.44	0.37	0.33	0.31	0.29	0.27	0.26	0.25	0.25	0.24	0.24	0.22	0.17
7.0	1	0.92	0.71	0.57	0.53	0.50	0.47	0.45	0.43	0.36	0.31	0.29	0.27	0.25	0.24	0.23	0.23	0.22	0.22	0.20	0.14
8.0	1	0.92	0.71	0.56	0.52	0.49	0.46	0.44	0.42	0.34	0.30	0.27	0.25	0.24	0.22	0.21	0.21	0.20	0.20	0.18	0.12
9.0	1	0.92	0.71	0.56	0.51	0.48	0.45	0.43	0.41	0.33	0.29	0.26	0.24	0.22	0.21	0.20	0.20	0.19	0.19	0.17	0.11
10.0	1	0.92	0.70	0.55	0.51	0.47	0.45	0.42	0.40	0.32	0.28	0.25	0.23	0.21	0.20	0.19	0.19	0.18	0.18	0.16	0.10
15.0	1	0.91	0.70	0.54	0.50	0.46	0.43	0.40	0.38	0.30	0.25	0.22	0.20	0.18	0.17	0.16	0.16	0.15	0.15	0.13	0.07
20.0	1	0.91	0.69	0.53	0.48	0.45	0.42	0.39	0.37	0.29	0.24	0.21	0.19	0.17	0.16	0.145	0.140	0.136	0.133	0.11	0.05
30.0	1	0.90	0.68	0.52	0.47	0.44	0.40	0.38	0.36	0.28	0.23	0.19	0.17	0.15	0.14	0.130	0.125	0.121	0.118	0.094	0.033

Table 3

Comparative values of \(f(n)\) according to various authors and \(f(k)\)

LNS, m	\(f(n)=(0.2+0.8n^3)\)	\(f(k)\) at \(k=4.8\)	\(f(n)=(\sqrt{1+n^2}-0.4)^3\)	\(f(k)\) at \(k=4.2\)	\(f(n)=(0.4+0.6n^3)\), Boreskov	\(f(k)\) at \(k=1.6\)	\(f(n)=(0.5+0.5n^3)\), Frolov	\(f(k)\) at \(k=1.1\)	\(f(n)=\left(\dfrac{4+3n}{7}\right)^3\)	\(f(k)\) at \(k=3\)	\(f(n)=\left(\sqrt{\dfrac{w+1}{w}}-k\right)^3\)	\(f(k)\) at \(k=7\)	\(f(n)=\dfrac{(w+1-k)\sqrt{\left(\dfrac{w+1}{w}\right)^3}}{2\sqrt{2w^3}}\), Melnikov	\(f(k)\) at \(k=1.9\)	\(f(n)=\left(\dfrac{w+1}{2w}\right)^{3/2}\), Filippovich	\(f(k)\) at \(k=1.9\)	\(f(n)=\left(\dfrac{0.9}{w}+0.1\right)\), Karpovskii	\(f(k)\) at \(k=9\)
1.0	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
1.5	0.64	0.72	0.68	0.73	0.73	0.80	0.77	0.84	0.80	0.75	0.62	0.71	0.76	0.78	0.76	0.78	0.70	0.70
2.0	0.50	0.59	0.53	0.60	0.62	0.69	0.68	0.74	0.68	0.63	0.49	0.56	0.68	0.67	0.65	0.67	0.55	0.55
2.5	0.40	0.50	0.46	0.51	0.55	0.64	0.63	0.70	0.59	0.55	0.38	0.48	0.58	0.60	0.58	0.60	0.46	0.46
3.0	0.36	0.45	0.39	0.46	0.52	0.59	0.60	0.65	0.55	0.50	0.33	0.42	0.55	0.56	0.54	0.56	0.40	0.40
4.0	0.30	0.38	0.35	0.39	0.48	0.54	0.56	0.61	0.49	0.44	0.28	0.34	0.53	0.51	0.50	0.51	0.32	0.32
5.0	0.27	0.34	0.33	0.35	0.45	0.51	0.55	0.58	0.45	0.40	0.24	0.30	0.47	0.48	0.46	0.48	0.28	0.28
6.0	0.26	0.31	0.30	0.33	0.44	0.49	0.53	0.56	0.42	0.37	0.22	0.27	0.44	0.45	0.45	0.45	0.25	0.25
7.0	0.24	0.29	0.29	0.31	0.43	0.47	0.53	0.55	0.40	0.36	0.21	0.25	0.43	0.44	0.44	0.44	0.23	0.23
8.0	0.23	0.28	0.27	0.29	0.43	0.46	0.52	0.54	0.37	0.34	0.19	0.24	0.42	0.43	0.42	0.43	0.21	0.21
9.0	0.23	0.27	0.26	0.28	0.42	0.45	0.52	0.53	0.36	0.33	0.18	0.22	0.41	0.42	0.41	0.42	0.20	0.20
10.0	0.23	0.26	0.26	0.27	0.42	0.44	0.52	0.53	0.36	0.32	0.17	0.21	0.40	0.41	0.40	0.41	0.19	0.19
15.0	0.21	0.24	0.25	0.25	0.41	0.43	0.51	0.51	0.32	0.30	0.16	0.18	0.39	0.39	0.39	0.39	0.16	0.16
20.0	0.21	0.21	0.23	0.23	0.41	0.42	0.51	0.50	0.30	0.29	0.15	0.17	0.38	0.38	0.38	0.38	0.14	0.14
30.0	0.20	0.20	0.22	0.22	0.40	0.40	0.50	0.50	0.28	0.28	0.15	0.15	0.37	0.37	0.37	0.37	0.13	0.13

Note: the values of \(f(n)\) according to the formulas in columns 2, 4, and 10 of Boreskov and Frolov have been recalculated for the same throw indices as those adopted in the formula in column 12 for LNS values of different magnitude.

at the coefficient \(k=9\). The values of \(f(n)\) according to the formulas of other authors are close to the values of \(f(k)\) at the values of the coefficient \(k\) indicated in Table 3.

From the properties of \(f(k)\) it follows that:

1) with an increase in the LNS—for all values of the coefficient \(k \ne 0\)—the specific consumption of explosive will decrease;

2) the decrease in the specific consumption of explosive will occur not according to one law, but according to different laws;

3) the decrease in the specific consumption will occur the more sharply, the greater the value of the coefficient \(k\).

Let us now return to formula (22).

Expanding the parentheses on the right-hand side of equation (22), we obtain:

\[ q=\frac{k}{(k+1)\cdot W}\cdot q_{\text{norm.}}+\frac{1}{k+1}\cdot q_{\text{norm.}} . \]

At an infinitely large value of LNS:

\[ q=\frac{1}{k+1}\cdot q_{\text{norm.}} . \tag{25} \]

The specific consumption of explosive—\(q\)—will tend to these limiting-minimum values, for all numerical values of the coefficient \(k\), as LNS increases:

1) the limit toward which the specific consumption of explosive will tend as LNS increases will depend on the magnitude of the normal specific consumption of explosive—\(q_{\text{norm.}}\)—and on the value of the coefficient \(k\);

2) at a very large value of LNS the relative influence of the resistance along the free surface and of the pressure of the air column will be negligible, and the specific consumption of explosive will essentially be expended on overcoming the resistance of the forces of gravity and inertia of the rock being detached, and on its crushing;

3) at LNS \(=30\) m, the specific consumption of explosive, for any value of the coefficient \(k\), will be very close to its limiting-minimum value.

As an example, it may be noted that at the coefficient \(k=10\) the limiting-minimum value of the specific consumption of explosive is \(0.091\,q_{\text{norm.}}\), at \(k=1\) it is \(0.50\,q_{\text{norm.}}\), and at \(k=0.1\) it is \(0.9\,q_{\text{norm.}}\).

For LNS \(<1\) m, the specific consumption of explosive, as LNS decreases, for each value of the coefficient \(k\), will, on the contrary, increase, tending toward its maximum limit.

The reasons for the change in the specific consumption of explosive with a change in the depth of charge placement had not been clarified. The author of the work, on the basis of an analysis of those resistances which arise and are overcome by the charge at the moment of explosion when it performs useful work, identifies these reasons and gives them a physical explanation.

We derived the formulas:

\[ q=f(k)\cdot q_{\text{norm.}}; \]

\[ q_{\text{norm.}}=(q_{\text{surf.}}+q'_{\text{crush.}})+(q_{\text{grav.}}+q''_{\text{crush.}})+ \]

\[ +(q_{\text{air}}+q'''_{\text{crush.}}). \]

After their transformation we obtain:

\[ q=f(k)\cdot(q_{\text{surf.}}+q_{\text{grav.}}+q_{\text{air}})+ \]

\[ +f(k)\cdot(q'_{\text{crush.}}+q''_{\text{crush.}}+q'''_{\text{crush.}}). \tag{26} \]

From the last formula the following is established:

1) between the specific consumption of explosives for overcoming the principal resistances and the specific consumption of explosives for crushing the rock there is a directly proportional dependence;

2) with an increase in the depth of emplacement of the charge, the value \(f(k)\) will decrease; in proportion to the latter, the specific consumption of explosives for crushing the rock will also decrease, and, consequently, the relative size of the fragments will increase in inverse proportion;

3) with an increase in the depth of emplacement of the charge, the size of the rock fragments will increase the more sharply, the smaller the value of \(f(k)\), or, in other words, the larger the value of the coefficient \(k\);

4) with an increase in the depth of emplacement of the charge and in the size of the rock fragments, the degree of rock throw will decrease; the radius of the throw crater

\[ \frac{r}{W} \]

will decrease, while the radius of the rupture crater

\[ \frac{r_1}{W} \]

will remain unchanged;

5) at a least line of resistance \(=30\ \text{m}\), the values of \(f(k)\) and of the specific consumption of explosives will be very close to the limiting minimum; the throw of rock will also be close to the limiting minimum.

For least lines of resistance \(<1\ \text{m}\), as they decrease, the opposite phenomena will be observed.

Table 4 gives, by way of example, calculated data on the relative increase in the size of rock fragments as the least line of resistance increases, for single concentrated charges acting on one free surface, for values of the coefficient \(k=1\) and \(9.5\).

Table 4

Relative size of rock fragments after blasting

Least line of resistance, m	Value of the coefficient \(k\)	Value of the coefficient \(k\)
	\(k=1\)	\(k=9.5\)
1	2	3
0.25	0.40	0.27
0.50	0.67	0.53
1.00	1.00	1.00
2.00	1.33	1.82
3.00	1.50	2.50
4.00	1.59	3.12
5.00	1.66	3.57
6.00	1.70	4.00
7.00	1.73	4.55
8.00	1.75	4.76
9.00	1.79	5.00
10.00	1.82	5.26
15.00	1.85	6.25
20.00	1.89	7.14
30.00	1.92	8.00

The weight of the charge will be equal to the product of the specific consumption of explosives, \(q\), and the volume of the rupture crater.

Formula (23), for determining the magnitude of the specific consumption of explosives, \(q\), as a function of the depth of emplacement of the charge, is applicable for any geometric form of the rupture crater.

For calculating the magnitude of the charge, a rupture crater of any geometric form may likewise be adopted. It is important only that the geometri-

the conical form of the crater and its similarity would be preserved both in determining the normal specific consumption of explosives and in subsequent calculations.

For a simpler calculation of the crater volume, its geometrical form is usually taken as a cone of revolution with a semi-apex angle of \(45^\circ\), which is the form we also adopt.

Then the formula for calculating charges will be:

\[ Q=\frac{1}{k+1}\cdot\left(\frac{k}{W}+1\right)\cdot q_{\text{norm.}}\cdot W^{3}\ \text{kg} \tag{27} \]

or

\[ Q=f(k)\cdot q_{\text{norm.}}\cdot W^{3}\ \text{kg}. \tag{28} \]

According to the formula, the magnitude of the charge depends on three quantities:

1) the normal specific consumption of explosives—\(q_{\text{norm.}}\);
2) the depth of emplacement of the charge, or the LNS;
3) the value of the coefficient \(k\).

Characteristic features of the formula:

1) it is a formula of general form;
2) it follows from it that, with a change in the depth of emplacement of the charge, its magnitude will change according to different laws, depending on the value of the coefficient \(k\);
3) at values of the coefficient \(k=0\), it is transformed into the formula \(Q=q_{\text{norm.}}\cdot W^{3}\);
4) at values of the coefficient \(k=\infty\)—into the formula \(Q=q_{\text{norm.}}\cdot W^{2}\);
5) the formulas proposed up to now for calculating concentrated charges of reduced throw and loosening charges are particular solutions with respect to the proposed formula.

For calculating charges by formula (28), the normal specific consumption \((\text{LNS}=1\ \text{m})\) may be determined at different values of the throw index \(n\leq 1\). The regularity of the change in the magnitude of the charge with a change in the depth of its emplacement will depend on the throw index \(n\) at which the normal specific consumption of explosives—\(q_{\text{norm.}}\)—has been determined for a given rock. If loosening charges are required for all LNS, then the normal specific consumption of explosives—\(q_{\text{norm.}}\)—must be determined for a loosening charge. In the same case, when the normal specific consumption of explosives—\(q_{\text{norm.}}\)—is determined for a charge of normal throw, and \(f(k)=1\), then for all LNS these will be charges of normal throw.

By adopting, for a given LNS, intermediate values of the specific consumption of explosives between the specific consumptions of explosives for a charge of normal throw and for a loosening charge, it is possible to regulate the throw of rock between a charge of normal throw and a loosening charge.

When calculating by formula (28) charges arranged in rows, the distance between charges must be taken according to the parameters existing on this point. In the case of two or more free surfaces, it is necessary additionally to introduce into the formula a correction coefficient that takes into account the degree of confinement of the medium being blasted.

Applied methods for calculating elongated charges

In developing the question of calculating elongated charges of reduced throw and loosening charges, all authors proceeded from the same principle and methods of constructing formulas as for concentrated charges.

The following formulas are most widespread:

1) single charges of normal throw

\[ L=q\cdot W^{2}\cdot l\ \text{kg}; \tag{29} \]

2) single charges of reduced throw

\[ Q=f(n)\cdot q\cdot W^2\cdot l\ \text{kg}; \tag{30} \]

3) single charges in benches (overdrilling)

\[ Q=q\cdot W^2\cdot H\ \text{kg}; \tag{31} \]

4) when blasting in series (overdrilling)

\[ Q=q\cdot a\cdot W\cdot H\ \text{kg} \tag{32} \]

or

\[ Q=0.25\,m^2\cdot q\cdot W^2\cdot H\ \text{kg}, \tag{33} \]

where \(q\) is the specific consumption of explosives, \(\text{kg}/\text{m}^3\);

\(W\) is the LLR, or, in benches, the calculated line of resistance (along the bench toe), if the bench is not vertical, m;

\(l\) is the charge length, m;

\(f(n)\) is a function of the throw index (charge action);

\(H\) is the bench height, m;

\(a\) is the distance between boreholes in a row, m;

\(m\) is the distance between the axes of charges, referred to the LLR \(=\dfrac{a}{W}\).

The formulas for calculating elongated charges have the same shortcomings as the formulas for calculating concentrated charges, and to the present time none of them is generally accepted.

Proposed Methods for Calculating Elongated Charges

The calculation of elongated charges is based by us on the same principle as for concentrated charges.

Proceeding from this principle, each elongated charge, when blasted, must perform the same useful work as a concentrated one, i.e.:

1) break the bond of the rock being detached with the massif;

2) overcome the resistance of the force of gravity and the inertia of the rock being detached;

3) overcome the resistance due to the pressure of the air column;

4) fragment and displace the entire volume of detached rock.

For elongated charges, just as for concentrated charges, whether the break-out crater is in the form of a triangular prism, or with an elliptical base, or the detachment occurs in a plane along the line of boreholes and in the plane of the bench toe, as takes place in series blasting of charges in benches, and in general for break-out craters of any geometric form, the specific consumption of explosives:

1) for overcoming resistance along the surface of the break-out crater will vary inversely proportional to the LLR;

2) for overcoming the resistance of the force of gravity and the inertia of the rock being detached, other conditions being equal, will remain a constant quantity;

3) for overcoming the resistance due to the pressure of the air column will vary inversely proportional to the LLR.

But, in contrast to concentrated charges, the last three propositions will be valid only in the case where, with an increase in the depth of charge placement, not only the similarity of the break-out craters is preserved, but the ratio also remains constant:

\[ \frac{l}{W}=m_1 \]

or

\[ \frac{H}{W}=m_2, \]

where \(l\) is the charge length, and \(H\) is the bench height.

Using the method developed earlier, we have derived the following formulas for calculating elongated charges as a function of their position in the object being blasted and of the blasting sequence.

1. For calculating single elongated charges parallel to a free surface, with unconstrained action:

\[ Q=\frac{1}{k+1}\cdot\left(\frac{k}{W}+1\right)\cdot q^0_{\text{norm.}}\cdot W^2\cdot l\ \text{kg} \tag{34} \]

or

\[ Q=f(k)\cdot q^0_{\text{norm.}}\cdot W^2\cdot l, \tag{35} \]

where \(W\) is the LLR, m;
\(q^0_{\text{norm.}}\) is the normal specific consumption of explosive (LLR \(=1\) m), in kg;
\(l\) is the length of the charge, m.

1. For calculating charges in benches (overbreak surface) in the presence of one free surface, formula (34) may be used, as well as the formula:

\[ Q=\frac{1}{k+1}\cdot\left(\frac{k}{W}+1\right)\cdot q^0_{\text{norm.}}\cdot W^2\cdot H\ \text{kg} \tag{36} \]

or

\[ Q=f(k)\cdot q^0_{\text{norm.}}\cdot W^2\cdot H\ \text{kg}, \tag{37} \]

where \(W\) is the LLR, or the design line of resistance (along the bench floor), if the bench is not vertical, m;
\(H\) is the height of the bench, m.

Formulas (35) and (37) should give identical results in the calculation; the difference lies only in the method of determining the normal specific consumption of explosive, \(q^0_{\text{norm.}}\). According to formula (35), this consumption is determined on the basis of the volume \(W^2\cdot l\), and according to formula (37), \(W^2\cdot H\).

1. For calculating charges in benches (overbreak surface) when they are arranged in rows and blasted simultaneously:

\[ Q=\frac{1}{k+1}\cdot\left(\frac{k}{W}+1\right)\cdot q^1_{\text{norm.}}\cdot a\cdot m\cdot W^2\ \text{kg}, \tag{38} \]

or

\[ Q=f(k)\cdot q^1_{\text{norm.}}\cdot a\cdot m\cdot W^2\ \text{kg}, \tag{39} \]

or

\[ Q=f(k)\cdot q^1_{\text{norm.}}\cdot a\cdot W\cdot H\ \text{kg}, \tag{40} \]

where \(a\) is the distance between charges, m;
\(W\) is the LLR, or the design line of resistance if the bench is not vertical, m;

\[ m \text{ is the ratio } \frac{l}{W}. \]

For elongated charges, the expression for \(f(k)\) is identical to the expression \(f(k)\) for concentrated charges, and its values should be taken from Table 2.

Everything that was said about the properties of \(f(k)\), the specific consumption of explosives, its limiting minimum and limiting maximum values, the degree of rock fragmentation, the relative size of the rock fragments with a change in the depth of placement of the charge, etc., with respect to concentrated charges, applies fully also to elongated charges.

Features of the formulas:

1) they are formulas of a general form;

2) the change in the weight of the charge, calculated from these formulas, with change in the depth of its placement, depending on the value of the coefficient \(k\), will occur not according to one law, but according to different laws;

3) when the value of the coefficient is \(k = 0\), formula (35) is transformed into the ordinary formula for calculating normal-ejection charges \(Q = q^0_{\text{norm.}} \cdot W^2 \cdot l\), and formula (37) into the formula \(Q = q^0_{\text{norm.}} \cdot W^2 \cdot H\);

4) when the value of the coefficient is \(k = \infty\), formula (35) is transformed into the formula \(Q = q^0_{\text{norm.}} \cdot W \cdot l\), and formula (37) into the formula \(Q = q^0_{\text{norm.}} \cdot W \cdot H\);

5) when \(W = l\), formula (35) is transformed into the formula \(Q = f(k) \cdot q^0_{\text{norm.}} \cdot W^3\);

6) when the value of the coefficient is \(k = 0\), formula (39) is transformed into the formula \(Q = q^1_{\text{norm.}} \cdot a \cdot m \cdot W^2\), and formula (40) into the formula \(Q = q^1_{\text{norm.}} \cdot a \cdot W \cdot H\);

7) when the value of the coefficient is \(k = \infty\), formula (39) is transformed into the formula \(Q = q^1_{\text{norm.}} \cdot a \cdot m \cdot W\), and formula (40) into the formula \(Q = q^1_{\text{norm.}} \cdot a \cdot H\).

When calculating charges by the formulas given above, the normal specific consumption of explosives must be determined for the same position of the charge in the object, the same order of blasting, and for the same ratio

\[ \frac{l}{W} = m_1 \]

or

\[ \frac{H}{W} = m_2, \]

at which it is subsequently intended to carry out blasting operations. In the process of carrying out blasting operations, however, it is often necessary to use charges at different values of \(m_1\) or \(m_2\). In the latter case, use of the formulas derived would require, in determining the normal specific consumption of explosives, that this value be determined empirically for all values of \(m_1\) or \(m_2\) that will occur in the conduct of blasting operations. This method is the most accurate, but may not always prove acceptable. Then another method may be used—the method of recalculating the normal specific consumption of explosives, which, though approximate, nevertheless sufficiently meets the requirements that may be imposed in the conduct of blasting operations. For recalculating the normal specific consumption of explosives in this case, Table 5 may be used, which gives data on the relative change in the value of the normal specific consumption of explosives for single charges as a function of the value

\[ \frac{l}{W} = m_1 \]

or

\[ \frac{H}{W} = m_2, \]

compiled on the basis of the corresponding calculations.

Table 5

Relative change in the value of the normal specific consumption of explosives as a function of the value \(\frac{l}{W} = m_1\) or \(\frac{H}{W} = m_2\)

Value of coefficient \(k\)	Value \(\frac{l}{W} = m_1\) or \(\frac{H}{W} = m_2\): 1	Value \(\frac{l}{W} = m_1\) or \(\frac{H}{W} = m_2\): 2	Value \(\frac{l}{W} = m_1\) or \(\frac{H}{W} = m_2\): 3	Value \(\frac{l}{W} = m_1\) or \(\frac{H}{W} = m_2\): 4
\(k = 1\)	1	0.90	0.86	0.85
\(k = 9.5\)	1	0.81	0.75	0.72

Analogously with concentrated charges, for calculating loosening charges the normal specific consumption of explosives must be determined for a charge

Table 6

Magnitude of the specific consumption of explosive substance \(q_1\) for loosening charges

No.	Name of rock	Strength relative to drilling resistance, kg·m/cm³	Crushing resistance, kg/cm²	Impact resistance	Mohs hardness	\(r_w\)—rock resistance	Normal value of \(q_1\), kg
1	Extremely strong, viscous, and dense quartzites	170	3000	30	—	1.3	1.3–1.5
2	Extremely strong, dense basalts, andesites, diabases, diorites, eclogites, granites, graywacke conglomerates	155	2400	24	6.6	1.2	1.2
3	Very strong granites, quartzites, quartz porphyry, siliceous shale	135	2200	22		1.1	1.1
4	Graywacke conglomerates, gneisses, basalts, porphyries, amphibolites	118	2000	20	6.0	1.0	1.0
5	Granite, gneiss, syenite, porphyrite, amphibolite	103	1700	18	—	0.9	0.9
6	The strongest limestones, sandstones, conglomerates	93	1600	16	5.0	0.8	0.8
7	Very dense granites, graywacke conglomerates, very strong siderites, magnesites, sandstones	80	1400	14	—	0.7	0.7
8	Roofing limestone, gneiss, porphyrite, crystalline limestone, very strong shale	70	1200	12	4	0.6	0.6
9	Marble, limestones, dolomite, siderite, magnesite	58	1000	10	—	0.5	0.5
10	Siderite, limestone, sandstone	47	800	8	3	0.4	0.4
11	Sandy shale, bedded sandstone	35	600	6	—	0.3	0.3
12	Strong clay shale, sandstone, gypsum	22	400	4	2	0.2	0.2
13	Hard coal	10	200	2	1	0.1	0.1

Normal value of the specific consumption of explosive substance \(q_1\) in open rock, in kg, when using explosives with a working capacity of 450–480 cm³ and with the value of each of the coefficients \(s\), \(v\), \(\Delta\), and \(d\) equal to unity.

For quartzites that are extremely hard, tough, and dense, according to Table 6, the specific consumption of explosives with a blasting power of 450–480 cm³ is \(q_1 = 1.5\) kg. The average unit weight of quartzites is 2.8.

After substituting the indicated values into formula (42), for quartzites we shall have:

\[ q_v = \frac{1.5}{9.5 + 1} = 0.143 \text{ kg} = 143 \text{ g} \tag{43} \]

or per 1 t of their weight:

\[ a_1 = \frac{q_v}{2.8} = \frac{143}{2.8} \cong 51 \text{ g/t}. \tag{44} \]

For hard coal, the specific consumption of explosives with a blasting power of 450–480 cm³ is \(q_1 = 0.1\) kg. The unit weight of coal is 1.2–1.5. On average, 1.35. Hence

\[ q_v = \frac{0.1}{1 + 1} = 0.05 \text{ kg} = 50 \text{ g} \tag{45} \]

or per 1 t of their weight:

\[ a_2 = \frac{50}{1.35} = 37 \text{ g/t}. \tag{46} \]

Thus, the consumption of explosives with a blasting power of 450–480 cm³ for overcoming the resistance of gravity and inertia of the detached rock, taking into account the expenditure of its energy for fragmentation, is, for quartzites, 51 g/t, and for hard coal, 37 g/t.

The difference between the explosive consumptions \(a_1\) and \(a_2\) is apparently explained mainly by the different toughness and elasticity of these rocks.

According to Table 6, the impact resistance for quartzites is 30, and for hard coal it is 2. In going from quartzites to hard coal, the impact resistance decreases by \(30 - 2 = 28\) units, while the value \(a = a_1 - a_2\) decreases by \(51 - 37 = 14\) g, i.e., with a decrease in impact resistance by one unit, \(a\) decreases by 0.5 g.

Proceeding from this, for intermediate groups of rocks between quartzites and hard coal, the following expression may be given for calculating the consumption of explosives with a blasting power of 450–480 cm³ for overcoming the resistance of gravity and inertia of the detached rock, taking into account the expenditure of energy for fragmentation, per 1 t of their weight:

\[ a = a_1 - 0.5(30 - p) = 51 - 0.5(30 - p) \text{ g/t}, \tag{47} \]

where \(p\) is the impact resistance of the rock for which the value of \(a\) is being sought.

When using explosives of another blasting power, and also taking into account the charge density and the quality of stemming, we obtain the following expression for determining the named explosive consumption:

\[ a_3 = \frac{l \cdot d}{\Delta} \cdot a, \tag{48} \]

where
\(l\) is the coefficient characterizing the blasting power of the explosive;
\(d\) is the coefficient characterizing the quality of stemming;
\(\Delta\) is the coefficient characterizing the charge density.

The specific consumption of explosives for overcoming the resistance of gravity and inertia of the detached rock will be:

\[ a_3 \cdot \gamma, \tag{49} \]

where \(\gamma\) is the unit weight of the rock, t/m³.

Specific consumption of explosive for overcoming the resistance along the surface of separation and from the pressure of the air column:

\[ q = a_{3}\cdot \gamma . \tag{50} \]

The normal specific consumption of explosive is

\[ q = q_{1}\cdot S\cdot \vartheta \cdot \frac{l}{\Delta}\cdot d, \tag{51} \]

where \(q_{1}\) is the specific consumption of explosive \((450—480\ \mathrm{cm}^{3})\), adopted from Table 6, in g;
\(S\) is a coefficient characterizing the structure of the rock;
\(\vartheta\) is a coefficient characterizing the degree of confinement of the charge.

Hence

\[ k=\frac{q-a_{3}\cdot\gamma}{a_{3}\cdot\gamma} =\frac{q}{a_{3}\cdot\gamma}-1. \tag{52} \]

After substituting the values for \(q\) and \(a_{3}\), we obtain:

\[ k= \frac{q_{1}\cdot S\cdot \vartheta \cdot \dfrac{l}{\Delta}\cdot d} {\dfrac{l}{\Delta}\cdot d\cdot a\cdot\gamma} -1 = \frac{q_{1}\cdot S\cdot \vartheta}{a\cdot\gamma} -1. \tag{53} \]

The value \(a\) should be calculated by formula (47).

Formula (53) establishes the dependence of the value of the coefficient \(k\) on five quantities. The influence of each of them on the coefficient \(k\) is clear from the formula.

If the normal specific consumption of explosive is determined empirically, then formula (52) should be used to calculate the coefficient \(k\). In this case the value of \(q\) will be equal to the actual normal specific consumption of explosive.

To determine the coefficient \(k\) for elongated charges, the same formulas as for concentrated charges are applicable. The normal specific consumption of explosive for elongated charges is recommended to be determined at the ratio

\[ \frac{l}{W}=m_{1}=1. \]

Conclusion

On the basis of the investigation carried out by him, the author of the work has proposed new formulas for calculating charges.

The distinctive feature of the proposed formulas, following directly from the features of the \(f(k)\) introduced into them, which has one and the same mathematical expression in all formulas, consists in the following:

1) that the change in the \(f(k)\) entering into them, and consequently also in the specific consumption of explosive, with a change in the line of least resistance of the charge, depending on the value of the coefficient \(k\) introduced by us, characterizing the physico-mechanical properties of the rock, will occur not according to one law, as all previous investigators assumed, but according to different laws;

2) that the established sliding scale of values of \(f(k)\), and consequently of the specific consumption of explosive, has such limits beyond which the change of \(f(k)\) and of the specific consumption of explosive with a change in the line of least resistance of the charge cannot lie;

3) that the derived formulas are of the most general form, by which, with proper determination of the normal specific consumption of explosive and the coefficient \(k\), correct results can always be obtained;

4) that the formulas for calculating charges proposed up to now are particular solutions in relation to the proposed formulas.

Many phenomena in blasting, for which no physical explanation had been given—including the reasons for the decrease in the specific consumption of explosives with increasing depth of charge placement—have received their explanation in light of the theoretical propositions developed by the author of this work. The contradiction between the formulas previously proposed by researchers of this question has been eliminated as a contradiction between particular solutions.

Let us now leave the theoretical side and turn to the results that found their final expression in the proposed formulas.

Suppose, nevertheless, contrary to the propositions of the author of this work, that the change in \(f(k)\) and in the specific consumption of explosives with the change in the depth of charge placement occurs not according to different laws, but according to one law. The value of the coefficient \(k\) would then be \(= \mathrm{const}\). Assigning the value \(k = 1.9\), we obtain the same numerical values of \(f(k)\) as \(f(n)\) according to the formulas of N. V. Melnikov and V. E. Filippovich; with \(k = 9\), according to the formula of V. K. Karchevskii, and so on. But then we would again return to the former contradictions, to the former situation, to the former unresolved question: why, despite the diversity of the values of \(f(n)\), is there still no generally accepted formula for expressing it, and why do the values of \(f(n)\), based on the processing of empirical data, differ sharply from one another? In any case, it is better and simpler to investigate a single coefficient \(k\) than to propose ever new solutions and formulas for determining \(f(n)\), based on a principle that has not justified itself.

Suppose now the opposite: that the change in \(f(k)\) and in the specific consumption of explosives with the change in the depth of charge placement occurs not according to one law but according to different laws; however, the limiting bounds for \(f(k)\) and the coefficient \(k\), and those values of \(f(k)\) and the coefficient \(k\) that are obtained for different rocks by the formula proposed for determining them, will not be the true values.

But this is not a fundamental question. The limiting bounds of the values of \(f(k)\) and of the coefficient \(k\) may be taken broader or narrower. The values of the coefficient \(k\) and of the normal specific consumption of explosives for rocks of different physicomechanical properties and blasting procedures, in practical use of the formulas, may be corrected and refined, and the rocks may be classified according to the coefficient \(k\). This is a matter for further work and time.

It is also necessary to note additionally the mathematical property of the expression for \(f(k)\), which consists in the fact that, when the unit on the right-hand side of the equation in the expression for \(f(k)\) is replaced by other numerical values, we shall obtain, for each value of the coefficient \(k\), ever new regularities in the change of \(f(k)\) with the change in the depth of charge placement. Consequently, correction is possible from this side as well.

Thus, in order to obtain correct results in calculating charges by the proposed formulas, it is only necessary to be able to establish the true value of the normal specific consumption of explosives and the value of the coefficient \(k\) for a given rock, and to learn to control the calculation by properly using the possibilities that present themselves.

The checking calculations carried out by the proposed formulas for 13 production blasts with concentrated and elongated charges under different blasting conditions gave entirely positive results.

1. In earlier formulas, the value of \(f(n)\) was made dependent on the value of the throw index (charge action)—\(n\). ↩