CONSTRUCTION OF A CIRCULAR NOMOGRAM FOR CALCULATING THE DEPRESSION OF MINE WORKINGS

A. F. KARATAEV

A linear instrument for calculations by formulas with many variables, constructed on the principle of a slide rule, becomes large and cumbersome when the ranges of variation of the variable quantities are wide.

Placing the scales of the variable quantities on a circle makes the instrument more compact and convenient for carrying out calculations. Therefore we develop a method for constructing circular instruments for calculations by formulas with many variables.

If the circumference of a fixed circle is divided into two semicircles and uniform divisions are marked on the outer side of these semicircles, the values of which increase clockwise, and on a second, movable circle a diameter is drawn, one end of which is thickened and the other provided with an arrow, then, after aligning the thickened end of the diameter of the movable circle with the zero division of the fixed circle and moving it 3 divisions clockwise, the opposite end of the diameter with the arrow will indicate 3 (Fig. 1). When the thickened end of the diameter of the movable circle is moved another 4 divisions clockwise, the opposite end of the diameter with the arrow will indicate the sum of the divisions (segments), \(3 + 4 = 7\) (Fig. 1). When the thickened end of the diameter of the movable circle is moved counterclockwise by 5 divisions, the opposite end of the diameter of the movable circle will indicate the difference of the divisions, \(7 - 5 = 2\) (Fig. 1).

Fig. 1.

This property of the circumference of the fixed circle and the diameter of the movable circle is taken as the basis for the method of constructing circular instruments (nomograms).

A circular instrument (nomogram) for calculations by a formula of the form

\[ Z=x+y \tag{1} \]

can be constructed by placing the variable quantities on the semicircles of the fixed and movable circles.

Placing the scale \(Z\) on the left semicircle of the fixed circle, the scale \(x\) on the right, and the scale \(y\) on the right semicircle of the movable circle, and arranging the increasing values of the variables clockwise, we obtain a circular instrument (nomogram) for addition (Fig. 2). Then

Fig. 2.

the calculation by formula (1) may be performed as indicated in Fig. 2, i.e., for \(x=4\) and \(y=3\),

\[ Z=x+y=4+3=7, \]

we add the segment of the scale \(y=3\) to the segment of the scale \(x=4\). The opposite end of the diameter of the movable circle of the scale \(y\) (with the arrow), corresponding to the value of the variable \(y=3\), will indicate the answer on the scale \(Z\) of the movable circle.

If the increasing values of the scale \(y\) on the movable circle are placed counterclockwise on the left semicircle of the movable circle, then we obtain a circular instrument (nomogram) for subtraction (Fig. 3) according to a formula of the form

\[ Z=x-y. \tag{2} \]

The calculation may be performed as indicated in Fig. 3, i.e., for \(x=5\) and \(y=3\),

\[ Z=x-y=5-3=2, \]

from the segment of the scale \(x=5\) we subtract the segment of the scale \(y=3\).

The calculation by formulas (1) and (2) is obtained still more simply if, in the instruments (nomograms) (Figs. 2, 3), the scale \(y\) is arranged conversely, i.e., in the instrument (nomogram) for addition the scale \(y\) is placed on the left semicircle of the movable circle and the increasing values of the variables are placed counterclockwise (Fig. 4), while in the instrument (nomogram) for subtraction

scale \(y\) should be placed on the right-hand side of the semicircumference of the movable circle, with the increasing values of the variable clockwise (Fig. 5).

Fig. 3.

Visible labels in Fig. 3: \(z=x-y\), \(2=5-3\); fixed circle; movable circle; scale \(z\); scale \(x\); scale \(y\); diameter indicating the answer.

The calculation in this case will consist in aligning the marks of the variable quantities \(x\) and \(y\), while the arrow of the diameter of the movable circle corresponding to the zero value of the variable \(y\) on scale \(Z\) will indicate the answer.

Fig. 4.

Visible labels in Fig. 4: \(z=x+y\), \(7=4+3\); fixed circle; movable circle; scale \(z\); scale \(x\); scale \(y\); diameter indicating the answer; key for using the nomogram: \(x-y \rightarrow z\).

Using the first method of construction (Figs. 2, 3) and the second method (Figs. 4 and 5), one can construct a circular instrument (nomogram) for calculations by the formula:

Fig. 5.

\[ Z = x + y + l + C - k. \tag{3} \]

Constructing the circular instrument (nomogram) by the first method alone led to the necessity of having five circles (Fig. 6), whereas the combination of the first and second methods makes it possible to place the variables on three circles.

Fig. 6.

Figure 7 shows the construction of a circular instrument (nomogram) for calculation by formula (3) with \(x = 2,\ y = 3,\ l = 2,\ C = 5,\ k = 4\), and \(Z = 8\).

As can be seen from Fig. 7, we first add \(x\) and \(y\), using the second method of constructing a circular instrument (nomogram) for three variables (Fig. 4). From the pointer of the diameter of the middle movable circle, which indicates the reading on the lower fixed circle, by the first method of constructing circular instruments (nomograms) for three variables (Fig. 3) we construct the scale \(k\). Next, using the second method, we place the scale \(l\) on the upper movable circle and, using the first method, the scale \(C\).

Fig. 7.

The indicated method of constructing a circular instrument (nomogram) for calculations by formula (3) leads to the following key for using the instrument (nomogram), placed on the blank field of the upper movable circle: \(x — y\), \(k — l\), \(C — Z\), i.e., in order to find \(Z\), it is necessary first to bring together the marks of the scales \(x\) and \(y\), and then, while holding the middle and lower circles, bring together the marks of the scales \(k\) and \(l\); the mark of the scale \(C\) will indicate the answer on the scale \(Z\).

Formula (3) shows that the answer (i.e., \(Z\)) does not change with the order in which the addends \(x\), \(y\), \(l\), \(C\) are added and the subtrahend \(k\) is subtracted. Consequently, in constructing a circular instrument (nomogram), any arrangement and combination of the scales of the variables may be made.

The rational arrangement of the scales should be based on:

1. Achieving the minimum diameter of the circular instrument (nomogram) for the adopted limits of the variables.

2. Maximum utilization of the circumferences of the lower fixed circle, the middle movable circle, and the minimum utilization of the upper movable circle. This makes it possible, without adding circles, to use the circular instrument (nomogram) for calculation by other formulas as well.

Positive variables, in the construction and use of the circular instrument (nomogram), are summed in one direction, and negative variables in the opposite direction; therefore, in transferring in formula (3) the positive variables to one side and the negative variables to the other, i.e., excluding the negative variables, the minimum diameter will be determined.

The rational diameter of the circular nomogram will be when

\[ Z+k=x+y+l+C. \tag{4} \]

For \(x=4,\ y=5,\ l=4,\ C=3,\ k=2\) and \(Z=14\), the rational diameter of the circular nomogram is determined from the expression

\[ \pi D_{\text{rat}}=l_{\text{rat}}=Z+k=x+y+l+C=14+2=4+5+4+3=16, \]

whence

\[ D_{\text{rat}}=\frac{l_{\text{rat}}}{\pi}=\frac{16}{\pi}. \tag{5} \]

The foregoing is confirmed by Figs. 7 and 8, in which two circular nomograms have been constructed with the same division value and the same diameters of the circles, but the first (Fig. 7) is for \(Z=8\), and the second (Fig. 8) is for \(Z=14\).

Fig. 8.

If, instead of uniform scales, logarithmic scales are plotted on the constructed circular instruments (nomograms) (Figs. 7 and 8), then the readings will be related by the equation:

\[ \lg Z=\lg x+\lg y+\lg l+\lg C-\lg k \tag{6} \]

or

\[ Z=\frac{x\cdot y\cdot l\cdot C}{k}. \tag{7} \]

Consequently, replacing uniform scales by logarithmic ones makes it possible to convert instruments (nomograms) for addition and subtraction into instruments (nomograms) for multiplication and division.

The lengths of the scales of the variables are then determined by the formulas:

\[ l_h=\lg h\cdot M_h, \tag{8} \]

\[ l_x=\lg x\cdot M_x, \tag{9} \]

\[ l_y=\lg y\cdot M_y, \tag{10} \]

\[ l_l=\lg l\cdot M_l, \tag{11} \]

\[ l_c=\lg c\cdot M_c, \tag{12} \]

\[ l_k=\lg k\cdot M_k, \tag{13} \]

where \(l_h,\ l_x,\ l_y,\ l_l,\ l_c,\ l_k\) are the lengths of the scales of the variables, and \(M_h,\ M_x,\ M_y,\ M_l,\ M_c,\ M_k\) are the scales of the variables.

Then formula (6) takes the form:

\[ l_h=l_x+l_y+l_l+l_c-l_k . \tag{14} \]

For all formulas that can be reduced to the form of formula (14), the described method of constructing circular instruments (nomograms) is applicable. The reduction of other formulas to the form of formula (14) is set forth below.

The formula

\[ h=\frac{x^2}{y^3} \tag{15} \]

after logarithmization will take the form:

\[ \lg h=2\lg x-3\lg y. \tag{16} \]

In order that the length of the scale \((l_h)\) of the variable \(h\) be equal to the algebraic sum of the lengths of the scales \((l_x-l_y)\) of the variables \(x\) and \(y\), and that equality (16) take the form

\[ l_h=l_x-l_y, \]

it is necessary to select the corresponding scale factors of the scales.

Formula (16) shows that the scale factors of the scales must be directly proportional to the exponents of the variables. Denoting by \(a\) the exponent of the variable \(h\), by \(b\) that of the variable \(x\), and by \(c\) that of the variable \(y\), and by \(M_h,\ M_x,\ M_y\) the scale factors of the scales of the variables, we obtain:

\[ \frac{a}{b}=\frac{M_h}{M_x}, \tag{17} \]

\[ \frac{a}{c}=\frac{M_h}{M_y}. \tag{18} \]

The lengths of the scales of the variables are determined by the formulas:

\[ l_h=\lg h\cdot M_h, \tag{19} \]

\[ l_x=\lg x\cdot M_x, \tag{20} \]

\[ l_y=\lg y\cdot M_y, \tag{21} \]

and formula (16) takes the form:

\[ l_h=l_x-l_y . \tag{22} \]

Formula (22) is analogous to formula (14). Consequently, the above-described method of constructing circular nomograms is also applicable to formula 15.

Using the above-described method for constructing a circular instrument from formulas with many variables, we construct a circular instrument (nomogram) for calculating the depression of mine workings.

The depression of individual mine workings that constitute a ventilation stream is currently determined by the formula

\[ h=\alpha \frac{LP}{S^3}Q^2, \tag{23} \]

where \(h\) is the depression of the working, in mm water column;
\(\alpha\) is the coefficient of friction, depending on the degree of roughness of the working;
\(L\) is the length of the working, in m;
\(P\) is the perimeter of the working, in m;
\(S\) is the cross-sectional area of the working, in m\(^2\);
\(Q\) is the quantity of air passing through the working, in m\(^3\)/sec.

The dependence between the cross section of the working \((S)\) and its perimeter \((P)\) is expressed by the formula:

\[ P=C\sqrt{S} \qquad \text{or} \qquad C=\frac{P}{\sqrt{S}}, \tag{24} \]

where \(P\) is the perimeter of the working, in m;
\(C\) is a coefficient depending on the shape of the working;
\(S\) is the cross-sectional area of the working, in m\(^2\).

This makes it possible, in formula (23), to replace the perimeter \((P)\) by the cross-sectional area \((S)\) according to formula (24). Then formula (23) will take the form:

\[ h=\alpha \frac{LC\sqrt{S}}{S^3}Q^2 \qquad \text{or} \qquad h=\alpha \frac{LC}{S^{2.5}}Q^2. \tag{25} \]

To eliminate the need for arithmetical calculation of the value \(C\), and also to establish average values of \(C\), depending on the shape in approximate calculations, we compile a table for standard cross sections of mine workings.

For a square cross-sectional shape with side of the square equal to \(a\), the perimeter \(P\) will be equal to \(4a\), i.e. \(P=4a\), and the area of the square \(S\) is equal to \(a^2\), i.e. \(S=a^2\). By formula (24) we determine the value \(C\), substituting \(P\) and \(S\), expressed through the side \(a\) of the square,

\[ C=\frac{P}{\sqrt{S}}=\frac{4a}{\sqrt{a^2}}=4, \qquad C_{\mathrm{sq}}=4. \]

For a circular section \(P=\pi D\), and \(S=\dfrac{\pi D^2}{4}\) (where \(D\) is the diameter of the circular working); substituting these values into formula (24), we obtain:

\[ C=\frac{\pi D}{\sqrt{\dfrac{\pi D^2}{4}}}=3.55, \qquad C_{\mathrm{cir}}=3.55. \]

For a rectangular section with sides \(a\) and \(b\), \(P=2a+2b\), and \(S=a\cdot b\).

\[ \text{For } \frac{b}{a}=4,\qquad b=4a, \]

\[ P=2a+2(4a)=10a,\qquad S=a\cdot 4a=4a^2. \]

Substituting these expressions for \(P\) and \(S\) into formula (24), we obtain:

\[ C=\frac{10a^2}{\sqrt{4a^2}}=5. \]

For

\[ \frac{b}{a}=3,\qquad b=3a, \]

\[ P=2a+2(3a)=8a,\qquad S=a\cdot 3a=3a^2. \]

Substituting these expressions for \(P\) and \(S\) into formula (24), we obtain:

\[ C=\frac{8a}{\sqrt{3a^2}}=4.63. \]

For

\[ \frac{b}{a}=2,\qquad b=2a, \]

\[ P=2a+2(2a)=6a,\qquad S=a\cdot 2a=2a^2. \]

Substituting these expressions for \(P\) and \(S\) into formula (24), we obtain:

\[ C=\frac{6a}{\sqrt{2a^2}}=4.25. \]

Using the obtained values of the coefficient \(C\) for various forms of the cross section of workings, we compile the table.

Table

Coefficient \(C\), accounting for the shape of the working

Taking logarithms of expression (25), we obtain:

\[ \lg h=\lg \alpha+\lg L+\lg C+2\lg Q-2.5\lg S. \tag{26} \]

Formula (26) is analogous to formula (6). Consequently, the described method for constructing circular nomograms is also applicable to formula (25).

Taking the scales of the variable quantities \(h, \alpha, L, C\) to be identical and equal to 10, since their exponents ( \(a=1\) ) are the same, from formulas (17) and (18) we determine the scale \((M_Q)\) of the \(Q\) scale and the scale \((M_s)\) of the \(S\) scale

\[ \frac{a}{b}=\frac{M_h}{M_Q} \quad\text{or}\quad \frac{1}{2}=\frac{10}{M_Q}, \]

whence \(M_Q=20\),

\[ \frac{a}{C}=\frac{M_h}{M_S} \quad\text{or}\quad \frac{1}{2.5}=\frac{10}{M_S}, \]

whence \(M_S=25\).

The limits of the variable quantities entering into formula (25) for calculating the depression of mine workings are taken, according to practical data, as:

\[ \alpha \text{ from } 0.0003 \text{ to } 0.005, \]

\[ L \text{ from } 1 \text{ m to } 100 \text{ m}, \]

\[ C \text{ from } 3.55 \text{ to } 5.0, \]

\[ Q \text{ from } 0.40 \text{ m}^3/\text{s to } 550 \text{ m}^3/\text{s}, \]

\[ S \text{ from } 1.5 \text{ m}^2 \text{ to } 45 \text{ m}^2,\ \text{respectively}, \]

\[ h \text{ from } 0.004 \text{ to } 80 \text{ mm water column}. \]

We construct the circular nomogram (Fig. 9) by the method described above.

Fig. 9.

The circular nomogram for calculating the depression of mine workings consists of three circles lying one on another, with logarithmic scales. The lower fixed circle has the scales \(h\) and \(Q\), the middle one has the movable scales \(S\) and \(\alpha\), and the upper one has the movable scales \(C\) and \(L\). In the free field of the upper movable circle there is printed the key for using the circular nomogram and a table of values of the coefficient \(C\), which takes into account the shape of the cross section of the mine working (Fig. 9).

The key for using the circular nomogram indicates that, in order to calculate the depression of a mine working by formula (25), it is first necessary to align the marks of the scales of the variable quantities \(Q\) and \(S\), and then, having fixed the lower and middle circles, to align the marks of the scales of the variable quantities \(C\) and \(\alpha\); the mark on the scale \(L\) of the upper circle will indicate the answer, i.e., the desired value \(h\) on the lower fixed circle.

The regulations for the technical operation of coal mines require that, before beginning the calculation of the depression of mine workings, the velocity of air movement through them be checked. This makes it possible to establish whether the adopted velocities of air movement through mine workings correspond to the regulations for technical operation, to judge to some extent the expenditure of energy on ventilating the mine workings, and to identify “narrow” places in the adopted ventilation scheme. Therefore, on the middle circle, on a circumference of diameter smaller than the circle itself, logarithmic scales of the variable quantities \(S\) and \(V\) are printed, and on the upper circle, on a circumference of the same diameter as on the middle circle, there is printed a logarithmic scale of the variable quantity \(Q_0 = Q\).

In the field of the upper circle there is placed the key for using the circular nomogram to calculate the velocities of air movement through a mine working, by the formula:

\[ V=\frac{Q_0}{S}=\frac{Q}{S}. \]

This key indicates that, in order to determine the velocity of air movement through a mine working, it is necessary to align the required mark of the scale \(Q_0\) of the upper circle, indicating the quantity of air flowing through the working, with the required mark of the scale \(S\) of the middle circle, indicating the cross-sectional area of the mine working; the arrow located on the upper circle will then indicate the velocity of air movement through the mine working on the scale \(V\) (velocity of air movement through the mine working) of the middle movable circle.

Being simple, the circular nomogram ensures calculation accuracy to the second and even the third significant digit.

While preserving sufficient accuracy in calculating the depression of mine workings, the circular nomogram will make it possible to reduce the time spent on calculating the depression of mine workings by a factor of five or more.

Use of the circular nomogram also considerably reduces the mental effort involved in calculations, as a result of which the probability of error is largely eliminated.

The proposed circular nomogram is applicable for all existing forms of mine workings, with their cross-sectional areas from \(1.5\) to \(45\ \text{m}^2\), any lengths of workings, coefficients of friction from \(0.0002\) to \(0.005\), and velocities of air movement in mine workings from \(0.25\) to \(15\ \text{m/sec}\).

The use of the nomogram for calculating variable quantities by two formulas turns the circular nomogram into a kind of calculating slide rule.