EMPIRICAL FORMULAS EXPRESSING THE PRESSURE OF AN AIR FLOW ON PLUMB-LINE WIRE THROUGH THE VELOCITY OF AIR MOTION AND THE DIAMETER OF THE WIRE

G. F. LYSOV

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

The pressure of an air flow on the wire of a plumb line when orienting a mine shaft is determined by the following formula (1.67):

\[ p = cqs, \tag{1} \]

where \(p\) is the pressure of the air flow in \(\text{kg}/\text{m}\) per unit length of wire,
\(c\) is a certain coefficient,
\(q\) is the dynamic pressure of the air flow in \(\text{kg}/\text{m}^2\) per unit area of a flat plate,
\(s\) is the area in \(\text{m}^2\) of the so-called “midship” section of the body, i.e. its projection onto a plane perpendicular to the direction of motion of the air flow.

The coefficient \(c\) in formula (1) is determined from a special graph (1.68) as a function of the Reynolds criterion \(R\), the value of which is found from the expression:

\[ R = \frac{vd}{f}, \tag{2} \]

where \(d\) is the diameter of the plumb-line wire in \(\text{cm}\),
\(v\) is the velocity of the air flow in \(\text{cm}/\text{sec}\),
\(f\) is the kinematic viscosity of air in \(\text{cm}^2/\text{sec}\).

The magnitude of the kinematic viscosity \(f\) depends mainly on the temperature of the air. Maerx, who investigated this question, provided a graph (1.68) for determining the kinematic viscosity as a function of air temperature. It follows from this graph that when the air temperature varies within the range from 0 to \(10^\circ\), the kinematic viscosity changes its value from 0.140 to \(0.146\ \text{cm}^2/\text{sec}\), i.e. very insignificantly. Therefore, in calculations it is quite justified and recommended (1.68) to consider the kinematic viscosity constant and equal to \(1/7\).

The dynamic pressure of the air flow \(q\) is calculated by the formula:

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

where: \(\gamma = 1.296\) is the weight of \(1\ \text{m}^3\) of air in kg,

\(g = 9.81\) — acceleration due to gravity, in \(m/sec^2\);

\(v\) — velocity of the air flow, in \(m/sec\).

The area of the “midship” section \(s\) of one running meter of the wire of the plumb line is determined by the formula:

\[ s = d \cdot 1\,m = d, \tag{4} \]

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

Analyzing formulas (1), (2), (3), (4), it may be said that, for a constant kinematic viscosity, the quantity \(p\) is a function of the air-flow velocity \(v\) and the diameter of the plumb-line wire \(d\). In order to establish the functional dependence of \(p\) on \(v\) and \(d\), we calculated a series of values of \(p\) (in g/running m) according to the formulas given above, varying the value of \(v\) within the limits from 0.1 to 5.0 \(m/sec\) in steps of 0.3–0.2 \(m/sec\), and \(d\) within the limits from 0.5 to 3.0 \(mm\) in steps of 0.5 \(mm\). The results of the calculation are presented in Table 1.

Table 1

Note: The values of \(p\) in Table 1 are given in g/running m.

Using the data of Table 1, we constructed a series of graphs of the dependence of \(p\) on the value of the wire diameter \(d\) at constant air-flow velocities \(v\) (Fig. 1). Considering Fig. 1, it may be concluded that, for a constant value of \(v\) within the range from 1.0 to 5.0 \(m/sec\), the dependence of the quantity \(p\) on the diameter of the plumb-line wire \(d\) has a lin-

curvilinear character. In the same way, a series of graphs was subsequently constructed for the dependence of the quantity \(p\) on the velocity of the air stream \(v\), for constant diameters of the wires of the feelers within the range from \(0.5\) to \(3.0\) mm. Considering the graphs shown in Fig. 2, one may conclude that in this case the dependence of \(p\) on \(v\) will be curvilinear, more precisely, parabolic.

Thus, it seems possible to express the pressure of the air on the wire of the feeler directly analytically, through the velocity of motion of the air and the diameter of the wire, in the form of one or several empirical expressions of the following structure:

\[ p = xdv^y, \tag{5} \]

\[ p = xdv^y + z, \tag{6} \]

\[ p = (xd + z)v^y, \tag{7} \]

\[ p = (xd + z)v^y + \zeta. \tag{8} \]

Of the empirical formulas given for the dependence of \(p\) on \(v\) and \(d\), the most suitable from the standpoint of physical meaning is the expression of the form (5), for which the value \(p\) becomes equal to zero when \(v\) or \(d\) is equal to zero. However, taking into account the empirical character of the formulas presented and the fact that, in reality, the dependence of \(p\) on \(v\) and \(d\) is probably a more complex function, it was also necessary to investigate formulas (6), (7), (8), and, on the basis of the results of these investigations, draw the appropriate conclusions.

We determined the parameters \(x\) and \(y\) of formula (5) twice by the method of least squares (the method of indirect observations), i.e., with the following conditions satisfied:

\[ \begin{aligned} \text{a)}\quad &[\varepsilon\varepsilon] = \min,\\ \text{b)}\quad &[P\varepsilon\varepsilon] = \min, \end{aligned} \tag{9} \]

where \(\varepsilon\) is the deviation of the value \(p\), calculated by the established formula (5), from the corresponding value of this quantity recorded in Table 1;

\(P\) are weights introduced for the purpose of obtaining an empirical formula (5) equally suitable for determining the quantity \(p\) at large and small values of \(v\) and \(d\).

In the first case we had to solve a system of two normal equations of the following form:

\[ [\lg v \lg v]\,y + [\lg v]\lg x - [\lg v(\lg p - \lg d)] = 0, \]

\[ [\lg v]\,y + n\lg x - [(\lg p - \lg d)] = 0, \tag{10} \]

where \(n\) is the number of all values of \(p\) in Table 1.

Omitting the rather cumbersome calculations, we give the final results for determining the formula of the form (5) by the first method:

\[ p = 0.0929\,dv^{1.755}. \tag{11} \]

In the second case, the weights of the “observations” were preliminarily determined from the data of Table 1, taking into account the orders of the first significant digits

Fig. 1.

Fig. 2.

values of \(p\) (the weights were set inversely proportional to the order of the first significant digit of the values of \(p\) in Table 1). Next, two normal equations of the form were solved:

\[ [P \lg v \lg v]\,y + [P \lg v]\lg x - [P \lg v(\lg p - \lg d)] = 0, \tag{12} \]

\[ [P \lg v]\,y + [P]\lg x - [P(\lg p - \lg d)] = 0. \]

Omitting all the calculations, we give the final results of finding formula (5) by the second method:

\[ p = 0.0947\, d v^{1.714}. \tag{13} \]

The determination of the parameters \(x, y, z, \varphi\) of formulas (6), (7), (8) was carried out in the following order.

The quantity \(x\) was found by solving the normal equation:

\[ [(d_i-d_0)^2]x_0 - [(d_i-d_0)(p_i-p_0)] = 0, \tag{14} \]

for the computation of whose coefficients the data of the row \(v=1.0\ \text{m/sec}\) of Table 1 were used. Here \(x_0\) is the adjusted value of the quantity \(x\), \(p_0\) is the value of the quantity \(p\) at \(v\) equal to \(1.0\ \text{m/sec}\), and \(d_0\) equal to \(0.5\ \text{mm}\). As a result of the computation, the value \(x_0\) was obtained equal to \(0.0790\).

The parameter \(y\) of formulas (6), (7), (8) was found by solving the normal equation:

\[ [\lg v_k \lg v_k]y_0 - \left[ \lg v_k\left( \lg \frac{p_i-p_0}{x_0(d_i-d_0)} \right) \right] = 0, \tag{15} \]

where \(y_0\) is the adjusted value of the quantity \(y\),

\(v_k\) are the air-flow velocities from \(0.1\) to \(5.0\ \text{m/sec}\), recorded in Table 1,

\(p_i\) are the values of \(p\) at velocity \(v_k\) and orifice diameter \(d_i\) equal to \(3.0\ \text{mm}\),

\(p_0\) is the value of \(p\) at velocity \(v_k\) and orifice diameter \(d_0\) equal to \(0.5\ \text{mm}\).

As a result of the computation, the parameter \(y_0\) proved to be equal to \(1.790\).

The adjusted value of the quantity \(z_0\) of formulas (7) and (8) was determined by solving the normal equation:

\[ n z_0 - \left[ \left( \frac{p_k-p_{k-1}}{v_k^y-v_{k-1}^y} \right) -x_0 \right] = 0, \tag{16} \]

where \(n\) is the number of terms of the column \(d=1.0\ \text{mm}\) of Table 1, excluding one \((n=20)\),

\(p_k, p_{k-1}\) are the values of \(p\) of the column \(d=1.0\ \text{mm}\) of Table 1.

The quantity \(z_0\) proved to be equal to \(0.0133\).

The value \(\varphi_0\) of equations (6) and (8) was found by solving the normal equations:

a) for formula (6):

\[ n\varphi_0 - [p_i - x_0 d_i v^y] = 0, \tag{17} \]

where \(p_i, d_i\) are the values of \(p\) and \(d\) of the row \(v=0.1\ \text{m/sec}\) of Table 1.

b) for formula (8):

\[ n\varphi_0 - [p_i - (x_0 d_i + z_0)v^y] = 0. \tag{18} \]

Solving equations (17) and (18), we obtained:

Table 2

a) for formula (6): \(\varphi_0 = 0.0007\),

b) for formula (7): \(\varphi_0 = 0.0005\).

Thus, the dependence of \(p\) (in g/m) on \(v\) (in m/sec) and \(d\) (in mm) can also be represented by the formulas:

\[ p = 0.079\, d v^{1.790} + 0.0007, \tag{19} \]

\[ p = (0.079\, d + 0.0133) v^{1.790}, \tag{20} \]

\[ p = (0.079\, d + 0.0133) v^{1.790} + 0.0005. \tag{21} \]

Next, an analysis was carried out of the established empirical formulas relating the pressure of the air flow on a plumb-line wire to the velocity of the air motion and the diameter of the wire. The results of this work are presented in Table 2.

Conclusions

1. In determining the value of \(p\), there is no need to use formulas (1), (2), (3), (4) and the method described above, since the value of \(p\) can be found directly, with sufficient accuracy, as a function of the given \(v\) and \(d\).

2. Empirical formulas have been established relating the pressure of the air flow on a plumb-line wire to the velocity of the air motion and the diameter of the wire.

3. The most accurate expression for the indicated relation is formula (21), which makes it possible to calculate values of \(p\) with an error of no more than 6%.

4. Formulas (20) and (13) also provide accuracy sufficient for technical purposes; according to them, the value of \(p\) is calculated with an error of no more than 10% and 20%, respectively.

References

Ogloblin D. N. Orientation of underground surveying through one vertical shaft, ONTI, NKTP USSR, 1938.