On the Operating Mode of an Unbalanced Skip Hoisting Installation

I. K. Khrustalev

(Presented by Prof., Doctor of Technical Sciences I. A. Balashev)

A rational operating mode of a hoisting installation must satisfy the following conditions.

1. For given productivity and hoisting height, the hoisting installation must have maximum efficiency, or, what is the same thing, minimum energy consumption.

2. Under the same conditions, it must have the minimum installed power of the hoisting electric motor.

A brief analysis of these requirements makes it possible to judge their inconsistency.

The efficiency of a hoisting installation may approximately be determined from the following relation [10]:

\[ \eta_y=\frac{\eta_g\cdot \eta_{\mathrm{zp}}}{\alpha k}, \tag{1} \]

where \(\eta_g\) is the efficiency of the electric motor, \(\alpha=\dfrac{v_{\max}}{v_{\mathrm{av}}}\) is the speed factor,

\(\eta_{\mathrm{zp}}\) is the efficiency of the gear transmission, and \(k\) is the coefficient of shaft resistances.

From (1) it is clear that the efficiency of the hoisting installation increases as \(\alpha\) decreases, but at the same time the starting period during the run of the hoisting installation is shortened and, consequently, the acceleration \(j\) of the moving system increases. The latter, in turn, leads to an increase in the installed power of the hoisting electric motor, since the maximum starting force increases \((F_{\max}=j)\).

With increasing \(\alpha\), the accelerations of the moving system decrease and, consequently, the maximum driving forces in the starting period also decrease, which leads to a reduction in the installed power of the hoisting electric motor, since

\[ \gamma=\frac{F_{\max}}{F_{\mathrm{str}}}\leq \gamma_g, \tag{2} \]

where \(F_{\max}\) is the maximum starting force,

\(F_{\mathrm{str}}\) is the installed force of the hoisting electric motor,

\(\gamma_g\) is the permissible overload of electric motors.

But with increasing \(\alpha\), the maximum speed of motion of the conveyances \(V_{\max}\) increases, and this causes an increase in the installed power of the electric motor, since

\[ P_{\mathrm{str}}\geq P_{\mathrm{ef}}=\frac{F_{\mathrm{ef}}V_{\max}}{102\eta_{\mathrm{zp}}}, \tag{3} \]

where \(F_{\mathrm{ef}}\) and \(P_{\mathrm{ef}}\) are the effective driving force and power.

Thus, in analyzing the operating modes of a hoisting installation, it is necessary to find such a rational mode under which the stated conditions will be satisfied to the greatest possible extent, as a whole.

The operating mode of a hoisting installation is determined chiefly by the velocity diagram of the conveyances; therefore the solution of the problem of the most advantageous operating mode reduces to the selection of a rational velocity diagram.

The author has analyzed the operating modes of a large number of skip hoists at mines of the Kuzbass \[4; 7\], and not a single hoist, in our opinion, fully satisfies the above conditions. Thus, for example, at one of the skip hoisting installations of the Anzherugol’ trust, for the specified capacity, hoisting height, and conveyance capacity, the average speed of skip motion is \(V_{cp}=6\ \text{m/sec}\). There the accepted maximum speed is \(V_{max}=10.6\ \text{m/sec}\), and a hoisting electric motor with a construction power of \(P_{cmp}=1050\ \text{kW}\) has been installed. Here it would have been possible to adopt \(V_{max}=7.8\ \text{m/sec}\); in that case \(P_{cmp}=800\ \text{kW}\).

At one of the skip hoisting installations of the Kaganovichugol’ trust, with \(V_{cp}=2.45\ \text{m/sec}\), \(V_{max}=5.2\ \text{m/sec}\) and \(P_{cmp}=1200\ \text{kW}\) were adopted. In this case it would have been necessary to adopt \(V_{max}=3.2\ \text{m/sec}\) and \(P_{cmp}=750\ \text{kW}\). These two examples show how uneconomically hoisting installations are operated. Therefore the question of choosing a rational mode is a matter of primary importance.

In designing a hoisting installation, its annual output and the depth of the mine are usually specified. From the specified values \(A_{god}\), \(T\), and \(H_{sh}\), the hourly output of the hoisting installation is determined; in this case

\[ A_{chas}=\frac{cA_{god}}{bt}\ [\text{t/hour}], \tag{4} \]

where \(c=1.15\text{--}1.25\) is the reserve coefficient,
\(b\) is the number of working days per year,
\(t\) is the number of hours of hoist operation per day.

Adding to the mine depth \(H_{sh}\) the approximate height from the shaft collar to the unloading device \(h_{z}=15\div30\ \text{m}\) and the height from the loading device to the working horizon \(h_{z}=15\div30\ \text{m}\), we obtain the approximate hoisting height \(H\) in meters.

From the hourly output and the hoisting height, the skip capacity is determined.

To determine the most advantageous skip capacity, we have proposed the formula \[3\]:

\[ Q_{n}=0.005A_{chas}H^{0.237}. \tag{5} \]

The most advantageous skip capacity in tons, obtained from (5), is rounded to the nearest standard value. Figure 1 shows a chart for selecting a skip, compiled on the basis of (5).

From \(A_{chas}\) and \(Q\), the cycle time of the hoisting installation \(T_{c}\) and the time of motion of the conveyances \(T\) are determined:

\[ T_{c}=\frac{3600Q}{A_{chas}} \tag{6} \]

and

\[ T=T_{c}-\Theta, \tag{6a} \]

where \(\Theta\) is the pause in the operation of the hoisting installation. It may be taken \[4\]:

for skips of capacity up to \(4\ \text{t}\) — \(7\ \text{sec}\),
\(6\ \text{t}\) — \(8\ \text{sec}\),
\(8\ \text{t}\) — \(11\ \text{sec}\).

Next, the average reference speed and the maximum speed of movement of the vessels are determined. In this case,

\[ V_{\mathrm{av}}=\frac{H}{T} \tag{7} \]

and

\[ V_{\max}=\alpha V_c=\alpha\frac{H}{T}. \tag{8} \]

Fig. 1. Chart for determining skip capacity

In the technical literature [1; 2; 5; 6; 9 and others], it is recommended that the most advantageous values of the speed factor be taken as \(\alpha=1.1 \div 1.3\).

As we see, the range of recommended speed factors is rather broad, and therefore, in different cases of designing operating modes, the performance indicators of hoisting installations will also vary rather widely.

In fact, suppose that in the calculation we obtained \(V_{\mathrm{av}}=8\) m/sec. In this case, for \(\alpha=1.1\), \(V_{\max}=8.8\) m/sec, while for \(\alpha=1.3\), \(V_{\max}=10.4\) m/sec. All the economic indicators will also be different.

Proceeding from what has been said, it is necessary to establish narrower limits for the variation of the most advantageous values of \(\alpha\).

Let us first consider a three-period diagram of the speed of movement of the vessels.

Taking \(\eta_g = 0.9\), \(\eta_{\mathrm{p}} = 0.93\), and \(k = 1.15\), on the basis of (1) we obtain

\[ \eta_y = \frac{0.73}{\alpha}, \]

whence \(\eta_y = \varphi(\alpha)\) will have the following values:

\[ \alpha = 1.1;\qquad 1.2;\qquad 1.3;\qquad 1.4;\qquad 1.5;\qquad 1.8;\qquad 2.0; \]

\[ \eta_y = 0.663;\qquad 0.61;\qquad 0.56;\qquad 0.52;\qquad 0.487;\qquad 0.405;\qquad 0.365. \]

On the basis of (2), the approximate power of the hoisting electric motor may be determined from the expression

\[ P = \frac{F_{\max} V_{\max}}{102 \eta_{\mathrm{э.п}} \gamma_g} = \alpha \frac{F_{\max} V_{\mathrm{cp}}}{102 \eta_{\mathrm{э.п}} \gamma}. \tag{9} \]

Here \(V_{\mathrm{cp}}\) is determined on the basis of expression (7).

\[ F_{\max} = KQ + pH + mj. \tag{10} \]

The mass \(m\), reduced to the circumference of the drum, may be replaced by the product of the mass coefficient \(\mu\) of the hoisting installation and the capacity of the vessel \(Q\), i.e.

\[ m = \mu Q. \tag{11} \]

In preliminary calculations for skip hoists at \(H \le 400\ \text{m}\), without large errors one may take \(\mu = 1\); then

\[ F_{\max} = (k + j)Q + pH. \tag{12} \]

In Fig. 2 we have plotted graphs for determining the weight per running meter of rope \(p\), kg/m, as a function of \(Q\).

Fig. 2. Rope type as a function of skip capacity and hoisting height.

To determine the acceleration \(j\), we shall use the following relation

\[ [8]:\quad V_{\max}^{2}-2TV_{\max}\frac{j_1j_3}{j_1+j_3}+2H\frac{j_1j_3}{j_1+j_3}=0. \tag{13} \]

Taking \(j_1=j_3\), we obtain:

\[ j=\frac{a^{2}V_{\mathrm{avg}}}{T(a-1)}. \tag{14} \]

Taking for the calculation \(k=1.15\); \(\eta_{\mathrm{gp}}=0.93\), \(\gamma=1.8\), and substituting (14) and (12) into (9), we obtain:

\[ P=aV_c\frac{\left(1.15+\frac{a^{2}V_{\mathrm{avg}}}{T(a-1)}\right)Q+pH}{170}. \tag{15} \]

On the basis of (15), the dependence \(P=\varphi(a; F_{\max}; V_{\mathrm{avg}})\) has been represented; it is given in Table 1 and in Fig. 3.

It is evident from Fig. 3 that the most rational operating modes are represented in graphs 1, 2, 4, 7, 9, 11, 14, 17, 20, 26, 29, 32, 34, and 37. Here we have the minimum approximate power of the hoisting electric motor and \(\eta_y>60\%\).

Of the most advantageous operating modes presented, the average rational speed multiplier for a skip \(Q=3\ t\) is \(a=1.2\); for \(Q=4\ t\), \(a=1.19\); for \(Q=6\ t\), \(a=1.18\); for \(Q=12\ t\), \(a=1.19\).

Hence we have the average most advantageous speed multiplier for skip hoists with a three-period speed diagram \(a_n=1.19\simeq1.2\). In this case the rational maximum accelerations do not exceed the value \(j=0.7\ \mathrm{m/sec^2}\).

With the existing loading and unloading devices, skip hoists cannot operate according to a three-period speed diagram. They operate according to a five-period diagram or, in the best case, can operate according to a four-period speed diagram [4, 7]. In this case the economic indicators of the installation’s operation are considerably reduced.

A skip hoisting installation, owing to the presence of loading and unloading devices, cannot have identical laws of motion both during the starting period and during the stopping period. To avoid accidents, the speeds of motion of the vessels in the unloading and loading devices must be limited by specified bounds.

Usually in the technical literature [5, 6, 8, etc.] the limiting speed for entry of the skip into the unloading curves and exit from them is given as \(V\leq1.5\ \mathrm{m/sec}\). However, operating practice for skip hoists in Kuzbass shows that the speeds of entry of the skip into the unloading curves and the speed of exit from them are not the limiting factor in the operating mode of skip hoisting installations [4]. On the basis of the author’s investigations it may be concluded that the limiting factor in the operation of the installation is the motion of the skips in the loading device. In this case the speed of seating the skip on the feeder carriage (the speed of entry into the loading device) with the existing rigid seating system must be \(V_k\leq0.5\ \mathrm{m/sec}\), and with elastic seating (using at least rubber packet shock absorbers on the carriage), \(V_k\leq1\ \mathrm{m/sec}\) [4, 7]. The speed of exit of the skip from the loading device must be \(V_z\leq1.2\ \mathrm{m/sec}\).

Let us establish the rational value of the speed multiplier for a five-period speed diagram. In doing so, to simplify the calculations we shall adopt a symmetrical diagram, for which \(V_z=V_k\) and \(j_1=j_3\).

Fig. 3. Dependence of the approximate power of the hoisting electric motor and the efficiency of the hoisting installation on the speed multiplier \(\alpha\).

Vertical axis: \(\alpha\).
Horizontal axes: \(P\), kW; \(\eta_{\text{у}}\), %.

Table 1

Dependence of the estimated power of the hoisting electric motor on $\alpha$, $F_{\max}$, and $V_{\mathrm{av}}$

The figures set in bold type indicate the numbers of the curves in Fig. 3.

Let us use the following relations [9]:

\[ T=\frac{H_0}{V_{\max}}+\frac{V_{\max}}{2}\left(\frac{1}{j_1''}+\frac{1}{j_3'}\right) +V_3\left(\frac{1}{j_1'}-\frac{1}{j_1''}\right)+ \]

\[ +V_k\left(\frac{1}{j_3''}-\frac{1}{j_3'}\right) \tag{16} \]

and

\[ H_0=H-2h_k+\frac{V_3^2}{2j_1''}+\frac{V_k^2}{2j_3'}. \tag{17} \]

Here \(j_1''\) and \(j_3'\) are the maximum accelerations and retardations, \(j_1'\) and \(j_3''\) are the acceleration and retardation during motion of the skip in the loading device,

\[ h_k=h_z-\text{the distance traveled by the skip in the loading device.} \]

Taking \(h_z=1.5\ \text{m}\); \(j_1''=j_3'\) and \(j_1'=j_3''\), on the basis of (16) and (17), for \(V_3=V_k=1\ \text{m/sec}\) we have:

\[ j_{\max}= \frac{\alpha V_{\mathrm{cp}}+\dfrac{1}{\alpha V_{\mathrm{cp}}}-2} {T-\dfrac{H+3}{\alpha V_{\mathrm{cp}}}-6} \tag{18} \]

and for \(V_3=V_k=0.5\ \text{m/sec}\),

\[ j_{\max}= \frac{\alpha V_{\mathrm{cp}}-1+\dfrac{0.25}{\alpha V_{\mathrm{cp}}}} {T-\dfrac{H+3}{\alpha V_{\mathrm{cp}}}-6}. \tag{19} \]

Next, on the basis of relations (1), (9), (10), (12), and (15), we find \(\alpha=f(A_{\text{hour}};\ Q;\ H)\) and \(j_{\max}=\psi(\alpha)\), \(P=p(\alpha)\), \(\eta_y=\eta(\alpha)\).

The indicated dependence is given in Table 2.

It follows from Table 2 that, in most cases, the minimum power of the hoisting electric motor is obtained at \(\alpha=1.4\). However, in this case \(\eta_y\) decreases significantly. Considering a reduction of the installation efficiency below 56% to be irrational, the most advantageous speed multiplier will be

\[ \alpha_n=1.3. \]

Calculations show that the rational speed multiplier will be exactly the same for a four-period speed diagram as well.

Table 2 and the calculations show that the maximum acceleration under the rational operating mode for a skip installation does not exceed \(j_{\max}=0.85\ \text{m/sec}^2\), while the most advantageous maximum speed is considerably lower than the limiting speed \(0.8\sqrt{H}\ \text{m/sec}\) permitted by the safety rules.

The actual maximum speed, as is known, is determined after establishing the standard number of revolutions of the electric motor \(n_g\) and the gear ratio of the reducer \(i\). In this case, the approximate number of revolutions of the motor is calculated by the formula \(n=\dfrac{60V_{\max}i}{\pi D_b}\) (20), and then the nearest higher standard number of revolutions is adopted, and thereafter the actual maximum speed is determined as \(V_{\max}=\dfrac{\pi D_b n_g}{60i}\), (21), where \(D_b\) is the drum diameter.

Table 2

Dependence of \(j_{max}\), \(P\), and \(\eta_y\) on \(\alpha\)

¹) \(A_{\text{hour}}\) has been taken for all skips on the basis of Fig. 1.

²) The figures in parentheses are for the five-period velocity diagram at \(V_3 = V_k = -0.5\) m/sec.

For rapid determination of a rational transmission ratio of the reducer \(i\), and to facilitate the calculations, we have constructed the graphs shown in Figs. 4 and 5.

The calculations show that, with the existing standard \(n_0\), \(D_b\), and \(i\), one can select \(V_{max}\) differing very little from \(V_{max} = \alpha V_{\text{avg}}\).

When designing the operating mode, it is important for the designer to know what maximum acceleration it is rational to assign to a given skip hoisting installation.

For this purpose, in our view, it is first of all necessary to establish the approximate motor power and to make a preliminary selection of it.

Fig. 4. Maximum speed as a function of the number of motor revolutions

The approximate power is determined by the formula

\[ P_o = \rho \, \frac{k Q H}{102 T}, \tag{22} \]

where \(\rho\) is the characteristic of the dynamic operating mode. It can be determined from formula [10]:

\[ \rho = \alpha \sqrt{ 1 + 2 \left( 1 - \frac{1}{\alpha} \right) \left( \frac{pH}{kQ} + \frac{\mu H}{kT^2} \cdot \frac{\alpha^2}{\alpha - 1} \right)^2 }. \tag{23} \]

For

\[ k = 1.15, \qquad \mu = 1 \quad \text{and} \quad \alpha = 1.3 \]

\[ \rho = 1.3 \sqrt{ 1 + 1.01 \left( 5.63 \frac{H}{T^2} + \frac{pH}{Q} \right)^2 }. \tag{24} \]

The values of \(\rho\) calculated from (24) for the most advantageous values of \(A_{\mathrm{chas}}\), \(H\), and \(Q\) encountered in practice in unbalanced skip hoisting installations are given in Table 3.

The approximate motor power may also be determined from the formula:

\[ P_{op} = \frac{F_{\max} V_{\max}}{102 \eta_{\mathrm{zp}} \gamma} = \frac{\left[(k + i)Q + pH\right] V_{\max}} {102 \eta_{\mathrm{zp}} \gamma}. \tag{9a} \]

From \(n_0\) and \(P_{op}\), determined by (22), a standard electric motor is first selected, and then \(\gamma_d\) is established.

Fig. 5. Maximum speed as a function of the motor speed.

In the nameplate data of motors, as is known, the value is given

\[ m=\frac{M_{\max}}{M_n}, \quad \text{whence} \quad M_{\max}=mM_n . \tag{25} \]

Taking into account the voltage drop during the starting of powerful motors by \(5\div 7\%\), \(M_{\max}\) will decrease by \(10\div 14\%\). Further, bearing in mind the starting of the hoisting electric motor by means of a metallic rheostat in the rotor circuit with \(5\div 9\) resistance steps, in order to obtain the average overload capacity it is also necessary to reduce the maximum torque by \(10\div 15\%\). Consequently, the permissible calculated overload of the electric motor will be

\[ \gamma_g=(0.75-0.8)m=(0.75\div 0.8)\frac{M_{\max}}{M_n}. \tag{26} \]

Table 3

Values of \(\rho\) for the most advantageous values of \(A_{\mathrm{hour}}, H\), and \(Q\)

Equating (22) and (9a) and carrying out the transformations, we obtain:

\[ j_{\max}= \frac{k}{\alpha} \left( \rho \gamma_g \eta_{\mathrm{sp}}-\frac{1}{a} \right) - \frac{pH}{Q} \left[ \mathrm{m/sec^2} \right]. \tag{27} \]

For \(k=1,15\) and \(\alpha=1,3\) we have:

\[ j_{\max}=0,88\rho \gamma_g \eta_{\mathrm{sp}}-\frac{pH}{Q}-0,68. \tag{28} \]

Calculations show that the magnitudes of the maximum accelerations in all cases do not exceed the limiting standards established by the safety regulations.

Conclusions

1. On the basis of the material presented, the rational maximum speed for an unbalanced skip hoisting installation should be taken as \(V_{max}=\alpha_n V_{av}\), where \(\alpha_n=1.3\).

2. The rational maximum acceleration is the acceleration determined by formula (28).

3. To improve the operating mode of the installation and to bring its operation closer to a three-period speed diagram, it is necessary to reconstruct the loading devices; above all, the use of rubber packet shock absorbers on the dispenser carriage is highly desirable.

References

1. Fedorov M. M. The most advantageous dynamic mode in certain types of mine hoisting installations. Ugol’ i zhelezo, Nos. 11, 12, 14, 1926.

2. Elanchik G. M. The most advantageous dynamic operating mode and the principles of rational calculation of electric mine hoisting machines. Gornyi zhurnal, Nos. 7, 8, 9, 1935.

3. Khrustalev I. K. The most advantageous skip capacity. Izvestiya TPI, vol. 78, 1955.

4. Khrustalev I. K. Diagram of the velocity of motion of vessels in skip hoisting installations. Izvestiya TPI, vol. 78, 1955.

5. Kiselev V. I. Mining Mechanics. Metallurgizdat, 1947.

6. Elanchik G. M. Mine Hoisting Installations. Gostoptekhizdat, 1941.

7. Khrustalev I. K. Dynamics of skip loading. Izvestiya TPI, vol. 78, 1955.

8. Khadzhikov R. N. Mine Hoisting Installations. Ugletekhizdat, 1956.

9. Kiselev V. I. Hoisting Installations for Deep Mines. Metallurgizdat, 1954.

10. Nesterov P. P. et al. Drivage Hoisting Machines. Kiev, 1953.