ON THE ISSUE OF DETERMINING THE POWER OF THE ELECTRIC MOTOR OF A HOISTING INSTALLATION

I. K. KHRUSTALEV

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

Effective Power

During operation, a hoisting installation performs a series of alternating operations for raising or lowering a load. Upon completion of each hoisting operation, the moving system stops, and then, after a pause, its motion proceeds in the opposite direction.

Consequently, for each operation the speed of the vessels changes from zero to a maximum value, and then, remaining constant for a certain interval of time, falls from this maximum value to zero. This change in speed leads to a change in the driving force developed by the hoisting electric motor (Fig. 1). Thus, the hoisting electric motor operates for a long time under a repeated short-time variable load.

For selecting the power of electric motors operating under a prolonged variable load, there are several methods. The essence of all these methods comes down to ensuring that the selected motor, operating according to the given load diagram, does not heat above the permissible temperature. In this case, the method most often used is that of determining the power of the electric motor by the root-mean-square, or effective, current.

Fig. 1. Diagrams of speed and driving forces of a skip hoisting installation.

The effective, or root-mean-square, current, as is known, is determined by the formula

\[ I_{\mathrm{eff}}=\sqrt{\int \frac{I^{2}\,dt}{T_{\mathrm{e}}}}. \tag{1} \]

In practice it is more convenient to use calculated load diagrams \(M=f(t)\) or \(F=\varphi(t)\). In fact, with sufficient accuracy for practice, it may be assumed that the motor current is proportional to its torque \(M\), and for hoisting installations with a constant rope-winding radius \(I \equiv M \equiv F\). For this case, using the load diagram \(F=f(t)\), we obtain an expression for determining the effective driving force developed by the motor at the circumference of the drum. In this case

\[ F_{\mathrm{eff}}=\sqrt{\frac{\sum\limits_{1}^{n} F^{2}\,dt}{T_{\mathrm{e}}}} \tag{2} \]

and

\[ P_{\mathrm{eff}}=\frac{F_{\mathrm{eff}}V_{\max}}{102\,\eta_{\mathrm{zp}}} =\frac{V_{\max}}{102\,\eta_{\mathrm{zp}}} \sqrt{\frac{\sum\limits_{1}^{n}F^{2}\,dt}{T_{\mathrm{e}}}}. \tag{3} \]

Here \(V_{\max}\) is the maximum velocity of motion of the conveyances,
\(\eta_{\mathrm{zp}}\) is the efficiency of the gear transmission from the motor to the drums,
\(T_{\mathrm{e}}\) is the effective time of the operating cycle of the electric motor.

To determine \(T_{\mathrm{e}}\), the technical literature \([1,2,3,4,5,6,7,8,9,10,11\) et al.] gives the following expressions:

\[ T_{\mathrm{e}_1}=t_1+t_2+t_3+\Theta, \tag{4} \]

\[ T_{\mathrm{e}_2}=t_1+t_2+t_3+\frac{1}{3}\Theta, \tag{5} \]

\[ T_{\mathrm{e}_3}=\frac{2}{3}(t_1+t_3)+t_2+\frac{1}{2}\Theta, \tag{6} \]

\[ T_{\mathrm{e}_4}=\frac{2}{3}(t_1+t_3)+t_2+\frac{1}{3}\Theta, \tag{7} \]

\[ T_{\mathrm{e}_5}=\frac{1}{2}(t_1+t_3)+t_2+\frac{1}{3}\Theta, \tag{8} \]

\[ T_{\mathrm{e}_6}=\frac{1}{2}(t_1+t_3)+t_2+\frac{1}{4}\Theta, \tag{9} \]

where \(t_1\) is the starting period, \(t_3\) is the stopping period,
\(t_2\) is the period of steady-state motor operation, and \(\Theta\) is the pause.

Let us make a brief comparative analysis of formulas (4)–(9) for determining \(T_{\mathrm{e}}\).

Cooling of the motor during the pause, during the acceleration period, and during the stopping period takes place under more difficult conditions than at constant speed; therefore the terms \(t_1+t_3\) and \(\Theta\) must be preceded by coefficients less than unity. From this point of view formulas (6)–(9) are the most acceptable.

For a final comparison of formulas (4)–(9), we shall make several clarifications.

It is known that the final selection of a hoisting electric motor is made after checking the motor selected according to the effective power

for overload capacity. In this case \(P_{\mathrm{n}} \geq P_{\mathrm{eff}}\) and \(\gamma=\dfrac{F_{\max}}{F_{\mathrm{n}}}<\gamma_0\), where:

Table 1

Data on skip hoisting installations and the designed mode of their operation

\(^{1)}\) The notation for the quantities is the same as in Fig. 1.

$P_n$ is the rated power of the adopted electric motor, $F_{\max}$ is the maximum driving force calculated from the conditions of the specified operating mode of the installation, $F_n$ is the rated force of the motor, and $\gamma_0$ is its permissible overload.

Taking into account that an asynchronous electric motor is used as the machine drive, and that during starting and stopping it operates with a metal rheostat connected in the rotor circuit, the permissible overload should be [12] $\gamma_0 = (0.75 \div 0.8)m$, where

\[ m=\frac{M_{\max}}{M_n} \]

is the overload capacity of the motor according to its nameplate data.

In view of the fact that, for asynchronous hoisting motors, usually $m=1.95 \div 2.3$, $\gamma_0=1.5 \div 1.8$.

Next we shall carry out comparative calculations for specific examples, taking for this purpose various operating conditions of skip hoists encountered in practice. Data on these hoists are given in Table 1.

Comparative calculated data on $T_{\mathrm{e}}$, $P_{\mathrm{eff}}$, and

\[ \gamma=\frac{F_{\max}}{F_{\mathrm{eff}}} \]

are given in Table 2.

Table 2

$^{1)}$ The overload capacity of the motor should be determined from the expression

\[ \gamma=\frac{F_{\max}}{F_n}, \]

but since we are not dealing with specific motors, we therefore do not know $F_n$; the effective force, however, is $F_{\mathrm{eff}}\leq F_n$. Consequently, with accurate selection of the motor,

\[ \frac{F_{\max}}{F_{\mathrm{eff}}}\simeq \frac{F_{\max}}{F_n}. \]

The data of Table 2 show that, for the same conditions, the effective motor power calculated with the various values of \(T_{\mathrm{e}}\) recommended in the literature differs by \(13 \div 19\%\). It follows from this that the standard powers \(P_{\mathrm{n}}\) adopted on the basis of \(P_{\mathrm{ef}}\) will also be different. Thus, for example, for case I, according to \(P_{\mathrm{ef}1}\) one may take \(P'_{\mathrm{n}} = 320\ \text{kW}\), while according to \(P_{\mathrm{ef}6}\), \(P''_{\mathrm{n}} = 400\ \text{kW}\). In this case a motor of power \(P'_{\mathrm{n}}\) will have an inadmissibly large overload \(\gamma = 1.84\).

Correspondingly, for case II \(P'_{\mathrm{n}} = 600\ \text{kW}\), \(P''_{\mathrm{n}} = 710\ \text{kW}\), and \(\gamma' = 1.87\).

It is also clear from Table 2 that the most acceptable overloads of the hoisting electric motor are obtained when \(T_{\mathrm{e}}\) is calculated by formulas (8) and (9), and that the overloads of the hoisting motor, as a rule, never exceed the permissible limits when \(T_{\mathrm{e}}\) is calculated by formula (9).

Proceeding from the above, in our opinion it would be useful to discontinue the use of formulas (4)–(8) for determining the effective time in calculations of the power of hoisting electric motors, since the abundance of formulas giving widely differing results in determining one and the same quantity misleads designers and engineering personnel of mines and ore mines. All the more so because all these formulas are empirical. For determining \(T_{\mathrm{e}}\), it is advisable to use formula (9).

\[ T_{\mathrm{e}}=\frac{1}{2}(t_{1}+t_{3})+t_{2}+\frac{1}{4}\Theta . \tag{9} \]

Approximate Power

When designing a hoisting installation, it is necessary to have calculations for determining the approximate power of the motor, on the basis of which its preliminary selection from catalogues is made. Without a preliminarily selected hoisting electric motor, the designer cannot specify the operating regime and carry out a calculation of the hoist dynamics, since in determining the dynamic forces it is necessary to know the mass of the moving parts of the installation reduced to the circumference of the drum; the latter, however, cannot be determined without knowing the flywheel moment of the electric motor.

Usually the approximate power of the hoisting electric motor is determined by the following formula:

\[ P_{\mathrm{or}}=\rho\,\frac{kQH}{102T\eta_{\mathrm{zp}}}, \tag{10} \]

where

\(k\) is the coefficient of mine resistances, \(Q\) is the capacity of the vessel in kg, \(H\) is the hoisting height in m, \(T\) is the travel time determined from the given \(H\), \(Q\), and productivity, \(\eta_{\mathrm{zp}}\) is the efficiency of the gear transmission, and \(\rho\) is the characteristic of the dynamic regime.

To determine \(\rho\), Prof. G. M. Elanchik recommends using the graphs \(\rho=\varphi(T)\) given in his book [2]. However, the values of \(\rho\) adopted from these graphs give extremely overestimated values of the approximate motor power. Much better values of \(\rho\) are obtained from the graphs of V. B. Umansky [12], but here, before determining \(\rho\), it is necessary to find the degree of static imbalance of the installation

\[ \delta=\frac{pH}{kQ} \]

and the constant of the dynamic regime

\[ c=\frac{Hm}{kQT^{2}}. \]

As we see, the latter method of determining \(\rho\) is complicated and does not give accurate results before the mass \(m\), reduced to the circumference of the drum, has been found; this mass, as indicated above, is determined after the preliminary selection of the hoisting electric motor.

In practice, when determining \(\rho\), for the reasons indicated above, neither Umanskii’s charts nor Elanchik’s charts are used; instead, depending on the hoisting depth, \(\rho = 1.6 \div 1.8\) is adopted. This, in turn, while simplifying the calculation, gives inaccurate results.

In our investigation of the operating mode of an unbalanced skip hoisting installation [12], values of \(\rho\) were calculated for the most advantageous values of productivity, hoisting height \(H\), \(Q\), and the speed multiplier \(\alpha\). On the basis of these calculations, the relationship \(\rho = \varphi(H)\) was constructed for non-counterbalanced skips; it is shown in Fig. 2. The values of \(\rho\) obtained from this relationship give better results in determining \(P_{op}\), as is evident from the following comparative calculations (Table 3) for an unbalanced skip hoisting installation.

Fig. 2. Dependence of the dynamic-regime characteristic \(\rho\) on \(Q\) and \(H\) for an unbalanced skip hoist.

Table 3

Comparative calculations for determining \(P_{op}\)

\(^{1)}\) \(\rho'\) and \(P'_{op}\) are the values of \(\rho\) found from the charts of Prof. G. M. Elanchik, and the power calculated from these values.
\(\rho''\) and \(P''_{op}\) are the values of \(\rho\) found from these values.

As we see, the values of the reference powers calculated by formula (10), with values of \(\rho\) found from the graphs in Fig. 2, are close to those calculated by formula (3).

Proceeding from what has been said, it is necessary to recommend that, when determining \(P_{op}\) by formula (10), the values of \(\rho\) be taken on the basis of the graphs shown in Fig. 2.

For purely approximate and comparative calculations, the values of the reference powers of hoisting motors may be determined on the basis of the operating mode of the installation. As is known, the final selection of the hoisting electric motor is made after checking the motor adopted according to \(P_{\mathrm{eff}}\) for overload capacity.

Proceeding from this, the reference power may be determined by the following formula:

\[ P_{op}=\frac{F_{\max}V_{\max}}{102\eta_{\mathrm{eff}}\gamma}, \tag{11} \]

where \(\gamma\) is the motor overload.

The maximum driving force \(F_{\max}\), as is known, occurs during the starting period, and it is equal to:

\[ F_{\max}=kQ+pH+mj_{\max}. \tag{12} \]

Substitute in place of \(m\) its value \(m=\mu Q\), where \(\mu\) is the coefficient of massiveness.

Then

\[ F_{\max}=kQ+pH+\mu j_{\max}Q=(k+\mu j_{\max})Q+pH. \]

For skip hoists with a hoisting height of \(100 \div 400\) m, one may approximately take \(\mu=1\); then, for an unbalanced skip hoist,

\[ F_{\max}=(k+j_{\max})Q+pH \tag{13} \]

and

\[ F_{\max}=(k+j_{\max})Q \tag{14} \]

for a balanced hoist.

Table 4

Comparative calculations for determining \(P_{op}\)

*) In determining \(P''_{op}\), it was taken from the graphs of Prof. G. M. Elanchik.

From Table 4 it is seen that the values of the approximate powers, calculated by formula (16), are close to those calculated by formula (10), with \(\rho\) taken from the author’s graphs (Fig. 2), and to those calculated by formula (3), with \(T_{\mathrm{e}}\) determined by formula (9).

Taking, for cage hoisting installations at a hoisting depth \(H = 100 \div 400\ \text{m}\), \(\mu = 1.5\), we obtain

\[ F_{\max} = (k + 1.5\, j_{\max})Q + pH. \tag{15} \]

Further, taking \(\eta_{\mathrm{tr}} = 0.92\) and \(\gamma_g^{1*)} = 1.55\), we obtain:

\[ P_o = \frac{F_{\max} V_{\max}}{145}. \tag{16} \]

To determine the suitability of formula (16), we carry out comparative calculations for the examples we have chosen. These calculations are given in Table 4.

Conclusions

1. When selecting a hoist electric motor according to the effective load, it is necessary to take into account the conditions of its worsened cooling during the periods of starting, stopping, and pauses; therefore the effective cooling time must be taken as less than the cycle time, \(T_{\mathrm{e}} < T_{\mathrm{c}}\).

2. For determining \(T_{\mathrm{e}}\) there is a great variety of formulas, giving different results both for \(T_{\mathrm{e}}\) and for \(P_{\mathrm{eff}}\); this, in turn, misleads designers and sometimes leads to errors in choosing the power of the hoist electric motor.

The best results in determining \(T_{\mathrm{e}}\) are given by formula (9), and therefore it should be recommended for use.

1. To determine \(P_{\mathrm{sr}}\) by formula (10), attention must be paid to the accurate determination of \(\rho\). In doing so, we recommend using the graphs shown in Fig. 2.

2. For approximate comparative calculations for determining the power of a hoist electric motor, formula (16) gives good results. It is convenient to use it for calculations when determining the possible maximum acceleration, the required power when changing the operating mode of a hoisting installation already in service, and when determining the motor power when changing the hoisting height and the productivity of the installation.

References

1. Popov V. K. Fundamentals of Electric Drive. Gosenergoizdat, 1951.
2. Elanchik G. M. Mine Hoisting Installations. Gostoptekhizdat, 1941.
3. Vasilevskii M. N. Asynchronous Drive of Mine Hoisting Machines. Ugletekhizdat, 1953.
4. Zhivov A. G. Drive and Automation of Mine Hoisting Machines. GONTI M., 1952.
5. German A. P., Shklyarskii F. N. Mine Hoisting Installations. Ugletekhizdat, 1947.
6. Kiselev V. I. Mining Mechanics. Metallurgizdat, 1947.
7. Pak V. S. Mining Mechanics, Part I. GONTI, 1932.
8. Shklyarskii F. N. Physicomechanical Properties of the Basis of Electric Mine Hoisting. Ugletekhizdat, 1950.
9. Kartashev A. N. Mining Mechanics. Mine Hoisting Machines. ONTI, 1937.
10. Maimin S. R., Shishkov P. F. Collection of Problems and Exercises in Mining Electrical Engineering. Ugletekhizdat, 1955.
11. Khrustalev I. K. On the Operating Mode of an Unbalanced Skip Hoisting Installation. Proceedings of TPI, vol. 88, 1955.
12. Umanskii V. B. Technical-Economic Calculation of Electric Hoisting Machines, in the book by V. Filipp, Electric Hoisting Machines. Donugol, 1929.

\(^{1*)}\) The low permissible overload capacity is adopted on the basis that the specific electric motor and the exact value of \(\mu\) are unknown to us.