ON THE PROBLEM OF DETERMINING THE DESIGN ERROR, OPTIMAL LOAD VALUES, AND WIRE DIAMETER IN MINE ORIENTATION

A. P. Kazachek, G. F. Lysov

(Presented by the scientific seminar of the Department of Mine Surveying and Geodesy)

The magnitude of the pressure of the air flow on the plumb-line wire \(P\) \((g/m)\), needed for a preliminary calculation of the expected design error in mine orientation, is computed by the formula

\[ P = cqS_m, \tag{1} \]

where \(c\) is a certain coefficient;
\(q\) is the dynamic pressure of the air flow per unit area of a flat plate, \(kg/m^2\);
\(S_m\) is the area of the “midship” section of the wire, \(m^2\).

The coefficient \(c\) in formula (1) is determined from a special graph as a function of the Reynolds criterion \(R_e\), whose value is found from the expression

\[ R_e = \frac{Vd}{f}, \tag{2} \]

where \(d\) is the diameter of the plumb-line wire, \(cm\);
\(V\) is the velocity of the air flow, \(cm/sec\);
\(f\) is the kinematic viscosity of air, \(cm/sec\) (in calculations it is recommended to take the value of \(f\) at a temperature of \(20^\circ\), equal to \(1/7\)).

The dynamic pressure of the air flow \(q\) \((kg/m^2)\) is calculated by the formula

\[ q = \frac{\gamma}{2g} V^2, \tag{3} \]

where \(\gamma = 1.296\) is the weight of \(1\,m^3\) of air, \(kg\);
\(g = 9.81\) is the acceleration of gravity, \(m/sec^2\);
\(V\) is the velocity of the air flow, \(m/sec\).

The area of the “midship” section \(S_m\) \((m^2)\) of one running meter of plumb-line wire is calculated by the formula

\[ S_m = d, \tag{4} \]

where \(d\) is the diameter of the plumb-line wire, \(m\).

Next, the deflection of the plumb line by the air flow \(a\) \((mm)\) is determined, according to the hypothesis of Prof. D. N. Ogloblin [1], by the formula

\[ a = \frac{PhH}{Q}, \tag{5} \]

where \(h\) is the length of the lower part of the plumb line subjected to air pressure, m;

\(H\) is the length of the entire plumb line (shaft depth), m;

\(Q\) is the acting load, kg.

Finally, the magnitude of the expected error in projecting the line of the plumb bobs \(\Theta''\) (sec) is found from the expression

\[ \Theta''=\frac{2a}{c}\rho, \tag{6} \]

where \(c\) is the distance between the plumb bobs, m;

\(\rho=206265''\).

The described procedure for the preliminary calculation of the expected projection error when orienting through a single vertical working is rather complicated and requires a considerable expenditure of time. In this connection, in practice mine surveyors rarely carry out a preliminary calculation of the projection error. Meanwhile, the orientation of mine workings without a preliminary calculation of the expected projection error, which is the principal source of errors in orientation through a single vertical working, may lead to an error in selecting the magnitude of the load or the wire diameter and, consequently, does not guarantee a correct solution of the projection problem.

Fig. 1.

Analyzing formulas (1), (2), (3), (4), one may conclude that, for a constant kinematic viscosity of air, the magnitude \(P\) is a function of the air-flow velocity \(V\) and of the diameter of the plumb-line wire \(d\). Considering the dependence of the magnitude \(P\) on the wire diameter at a constant air-flow velocity \(V\) (Fig. 1), it may be said that, for \(V\) values ranging from 0.1 to 5.0 m/sec, this dependence is rectilinear. At the same time, the dependence of the magnitude \(P\) on the air-flow velocity at a constant diameter of the plumb-line wire in the range from 0.5 to 3.0 mm (Fig. 2) is curvilinear, more precisely parabolic.

The stated conditions are met by the following empirical expression for the dependence of \(P\) on \(V\) and \(d\):

\[ P=(0.079d+0.0133)V^{1.791}+0.0005. \tag{7} \]

In the empirical formula (7) given above, the value of the plumb-bob diameter \(d\) must be substituted in mm, and the air-velocity magnitude \(V\) in m/sec. In this case \(P\) will be calculated in g/m.

Fig. 2.

Formula (7) is inconvenient for calculating the value of \(P\), but it can be used to construct a nomogram, by means of which it is possible quite simply and with sufficient accuracy to find the value of \(P\). The nomogram and the scheme for using it are presented in Fig. 3.

Using the proposed nomogram, one can find values of \(P\) over wide ranges, i.e., with \(d\) varying from 0.5 to 3.0 mm and \(V\) from 0.1 to 5.0 m/sec.

We shall next use the empirical expression for the dependence of \(P\) on \(V\) and \(d\) to determine the magnitude of the expected design error; for this purpose we substitute the value of \(P\) from formula (7) into expression (5), and by formula (6) determine the design error

\[ \Theta''=\frac{2\rho hH\left[(0.079d+0.0133)V^{1.791}+0.0005\right]}{cQ}. \tag{8} \]

The term 0.0005 appearing in the numerator of formula (8) is significant when the value of \(P\) is very small (less than 0.01), i.e., when the air-flow velocities are less than 0.3 m/sec and the selected plumb-bob diameter is less than 0.8 mm. In these cases the expected design error must be calculated by formulas (5) and (6), determining the value of \(P\) from the nomogram proposed above. In the overwhelming majority of cases \(P\) has a value greater than 0.01; therefore the term 0.0005 in formula (8) may be neglected (the error in determining \(\Theta''\) that we allow by omitting the value 0.0005 will not exceed 5%) and formula (8) may be written in the following form:

\[ \Theta''=\frac{2\rho hH(0.079d+0.0133)V^{1.791}}{cQ}. \tag{9} \]

Fig. 3.

Labels in the figure:

• \(V\) — air-flow velocity, m/sec
• \(P\) — air pressure on the wire, g/m
• \(d\) — wire diameter, mm
• Diagram for use:
• \(V\)
• \(P\)
• \(d\)
• “given”
• “answer”

Formula (9) is also inconvenient for a direct calculation of the design error \(\Theta''\), but it may be used to construct a nomogram intended for determining the value of \(\Theta''\) from given values of \(H, h, V, d, Q\), when the distance between plumb lines, \(c\), is equal to 1 meter. This nomogram and the scheme for using it are presented in Figs. 4 and 5.

Fig. 4.

According to the proposed nomogram, the value of \(\Theta''\) can be found with an accuracy of one unit in the second decimal place when \(H\) varies within the limits \(30\text{–}600\) m, \(h\)—from 1 to 5 m, \(V\)—from 0.1 to 5.0 m/sec, \(d\)—from 0.5 to 3.0 mm, and \(Q\)—from 6 to 400 kg.

In order to determine the design error \(\Theta''\) for any value of \(c\), the found value of \(\Theta''\) must be divided by the specified value of \(c\) in meters.

We shall now use formula (8) to solve the question of choosing the optimal values of the load and the diameter of the plumb-line wire when orienting a mine through a single vertical shaft. Taking into account that, in order to reduce the design error, it is necessary for each—

of the wire take the maximum load, equal in magnitude to 60% of the tensile strength of the wire, express the value \(d\) in formula (8) by the magnitude of the load \(Q\), for which purpose we shall use the data on the permissible weight of loads given in the technical instruction (3). Let us construct a graph of the dependence between the diameter of class “P” wire and the maximum permissible weight of the load (Fig. 6). Obviously, an analogous graph can also be constructed for the characteristic of the dependence between the diameter of the plumb-line wire and the weight of the load for class “V” wire.

Fig. 5.

Considering the graph presented, one may say that the dependence between the wire diameter and the maximum permissible load can be expressed by the formula

\[ d = yQ^{x}. \tag{10}^{1)} \]

Substituting into this formula the load values \(Q\), equal to 26 and 330 kg, and the values of the corresponding diameters of class “P” wire (0.5 and 2.0 mm), and solving two logarithmic equations, we find

\[ d = 0.0844Q^{0.516}, \tag{11} \]

where \(d\) is the wire diameter, mm,
\(Q\) is the magnitude of the maximum load, kg.

\(^{1)}\) It would be more correct to express the dependence of \(d\) on \(Q\) by the formula \(d = aQ^{\frac{1}{2}}\), but, taking into account the recommendations of the technical instruction, it is better to retain the given formula (10).

Using the values of the maximum load \(Q\) given in the technical instruction for class “P” wire of diameter \(0.8,\ 1.0,\ 1.5\) mm, one can compute from formula (11) the values of the indicated diameters and verify the good convergence of the results.

Fig. 6.

In the same way, the dependence between \(d\) and \(Q\) for class “B” wire can be established, expressed by the formula

\[ d = 0.0744 Q^{0.555}. \tag{12} \]

Substituting the value of \(d\) found from formula (11) into expression (8), we obtain

\[ \Theta'' = \frac{2\rho hH}{c} \left[ \frac{(0.00666 Q^{0.546} + 0.0133) V^{1.791}}{Q} + \frac{0.0005}{Q} \right]. \tag{13} \]

Neglecting the very small term \(\dfrac{0.0005}{Q}\), we simplify formula (13)

\[ \Theta'' = \frac{2\rho hH V^{1.791}}{c} \times \left( \frac{0.00666 Q^{0.546} + 0.0133}{Q} \right). \tag{14} \]

Formula (14) expresses the relation between the projection error, the objective “parameters” \(H,\ h,\ V\), the selected distance between plumb lines \(c\), and the value of the load \(Q\) for class “P” wire.

It follows from formula (14) that, in order to reduce the projection error under any conditions, it is advantageous to increase the magnitude of the load and the diameter of the wire. Thus, Drumm’s remark, given without proof in work (2), is confirmed: “a large load cannot spoil the orientation, whereas a small one can certainly do so.” It also follows from formula (14) that the magnitude of the projection error depends on the totality of five factors

\((H, V, c, Q \text{ and } h)\) and the design conditions (the influence of dripping water, the elasticity of the plumb-line wire, secondary oscillations, etc.). Consequently, ignoring any of the named factors when determining the expected orientation error, or when choosing the load and wire diameter, may lead to incorrect conclusions. Below we shall dwell on this assumption in greater detail.

Taking into account, however, that an increase in the load and in the wire diameter may, to some extent, cause an increase in the contact error (this question requires additional investigation), it may be said that the most justified choice should be considered to be such a choice of the magnitude of the load \(Q\) and of the diameter of the plumb-line wire corresponding to this load, for which (for a given orientation error \(\Theta''\), objective “parameters” \(H, h, V\), and accepted \(c\)) equality (14) is satisfied, i.e.

\[ \frac{0.00666Q^{0.546}+0.0133}{Q} = \frac{\Theta''c}{2\rho HhV^{1.791}} \tag{15} \]

or, for \(\Theta''=90''\),

\[ \frac{0.00666Q^{0.546}+0.0133}{Q} = \frac{45c}{\rho hHV^{1.791}}. \tag{16} \]

An analogous equality may be written for a load suspended on a class “B” wire:

\[ \frac{0.00587Q^{0.555}+0.0133}{Q} = \frac{45c}{\rho hHV^{1.791}}. \tag{17} \]

Formula (16) is inconvenient for direct calculation of the load magnitudes \(Q\), but it can be used to construct a nomogram by means of which it is possible to determine the magnitude of the load \(Q\) and the wire diameter \(d\). This nomogram and the scheme for its use are presented in Fig. 7.

Using the proposed nomogram, one can find the values \(Q\) and \(d\) over wide ranges, i.e. with \(H\) varying from 30 to 1000 m, \(h\) from 1 to 10 m, \(V\) from 0.1 to 5.0 m/sec, and \(c\) from 0.5 to 10 m.

There is no need to present here another nomogram for determining the load magnitude for a class “B” wire, since in this case the described nomogram may be used. Having determined from the nomogram the load magnitude \(Q\) for a class “P” wire and reducing it by 15%, we obtain the optimal value of the load when designing with a class “B” wire. As for the wire diameter, it can easily be found by interpolation from the table given in the technical instruction (3).

Let us now use the given nomogram and verify the correctness of the recommendation of § 220 of the surveying instruction (3), in which only three factors are taken into account (wire diameter, depth of the mine shaft, and velocity of the air current) out of the five factors indicated earlier that influence the orientation error.

Let \(H=250\) m, \(V=0.6\) m/sec, \(h=6\) m, \(c=2\) m.
From the nomogram we find: \(Q=180\) kg, \(d=1.4\) mm.
The instruction, however, taking into account only \(H, V\), and \(d\), recommends for these conditions, i.e. for \(H\) less than 300 m and \(V\) less than 0.7 m/sec, that a thin wire with diameter up to 1 mm inclu-

but, consequently, a load of no more than 92 kg. In fact, the use of wire thinner than 1.4 mm and a load of less than 180 kg will not, under the given conditions, ensure a design error within 90″, i.e., will not ensure the correct solution of the orientation problem.

Fig. 7.

Conclusions

1. It should be regarded as established that, in order to reduce the design error caused by the action of the air flow on mine plumb lines, it is advantageous under all conditions to increase the magnitude of the load.

2. The magnitude of the design error should be taken into account from the combined effect of all five factors (shaft depth, air-flow velocity, distance between plumb lines, load magnitude, and length of the lower part of the plumb line subjected to air pressure), with allowance for the design conditions (the influence of dripping water, secondary oscillations, etc.). Thus, the recommendations of § 220 of the technical instruction [3], in our opinion, should be regarded as insufficiently substantiated.

1. In the work a new method has been developed for determining the optimal value of the load (wire diameter) on the basis of the aggregate of design conditions, and nomograms have been provided that make it possible to determine with sufficient accuracy the expected design error and the optimal value of the load (wire diameter) when carrying out orientations.

2. Experimental studies must be conducted for the final verification of the proposed method for selecting the value of the load and the wire diameter when orienting through a single vertical mine shaft.

REFERENCES

1. D. N. Ogloblin. Mine-surveying work in the underground development of deposits. Part III, Metallurgizdat, 1953.

2. D. N. Ogloblin. Orientation of an underground survey through one vertical shaft, ONTI, 1938.

3. Technical instruction for the performance of mine-surveying work, Ugletekhizdat, 1959.