LOAD CHARACTERISTICS AND THEIR APPLICATION FOR EVALUATING THE OPERATING EFFICIENCY OF PNEUMATIC MOTORS

S. I. Shubovich

(Presented by Prof. V. T. Yurinskii, Doctor of Technical Sciences)

In the literature on questions of the pneumatic drive of mining mechanisms and machines, a very detailed description is often given of the design of pneumatic motors; the parameters and most important indicators for the nominal conditions of their operation are presented, but little attention is paid to the dependences of effective power, torque, and compressed-air consumption on the shaft rotational speed or on changes in the load of these motors—the so-called speed and load characteristics.

In some works [1—3], speed characteristics of pneumatic motors are given (not always sufficiently complete [1]); however, load characteristics, which are of special value in studying the economic and tractive indicators of pneumatic motors used to drive mining mechanisms and machines operating with variable loads, are not mentioned at all. The literature also does not consider methods for analyzing and evaluating the economy of pneumatic motors operated under conditions of variable loads. In this connection, operating engineers and designers often encounter difficulties in selecting a rational type of pneumatic motor for a particular mining machine with a specified load schedule.

The purpose of the present article is partly to fill this gap.

Speed characteristics

In bench tests of pneumatic motors, speed characteristics are usually taken; these are graphs of effective power $N_e$, torque $M_e$, and the minute and specific air consumptions $Q$ and $q$ as functions of the shaft speed $n$. Speed characteristics taken at the nominal (design) pressure of compressed air supplied to the motor, by analogy with the terminology adopted in studying the operation of internal-combustion engines, may be called external characteristics, while characteristics taken at any pressure lower than the nominal pressure $P_{\text{nom}}$ (at “partial” pressure) may be called partial characteristics (Fig. 1).

If a pneumatic motor is equipped with a speed governor, and if this governor was not disconnected when the characteristics were taken¹, then

external characteristics will have the form, for example, shown in Fig. 2. In the range of rotational speeds from \(n=0\) to the rotational speed at which the governor comes into operation, \(n_A\), the changes in effective power, torque, and air consumption with speed are determined only by the properties of the motor, since in this speed range the parameters of the compressed air supplied to the motor remain unchanged. In the interval of rotational speeds

Fig. 1

from \(n_A\) to \(n_B\), the changes in the motor indicators are determined by the properties and adjustment of the speed governor, since the latter, in this speed range, as the speed regime changes, acts on the spool installed in the air-supply channel and changes the parameters of the air entering the working cavity of the motor. Therefore, the portions of the speed characteristics within \(n_A \div n_B\) may be called governor characteristics. The initial points of the governor characteristics \(A_1\) and \(A\) (Fig. 2) lie on the external characteristics, and their terminal point \(B\) corresponds to the idling of the motor.

As is known, the mechanical load of the motor is determined by the magni-

Fig. 2

of the external torque applied to the motor shaft. In the case of the drive of a mining machine, the external torque with respect to the motor will be the resistance torque \(M_c\) on the drive shaft of this machine. Usually the speed governor of a pneumatic motor is adjusted so that it comes into operation at a speed \(n_d = n_{\text{nom}}\), corresponding to the nominal (rated) power. Therefore, when the value of the torque \(M_c\) is less than the nominal torque \(M_{\text{nom}}\) (underloading of the motor), the operation of the motor will proceed along the governor characteristic. If it turns out that \(M_c\) is greater than \(M_{\text{nom}}\), but less than the maximum torque of the motor \(M_{\max}\) (overloading of the motor), then in this case the motor will operate along the external characteristic.

Mining machines and mechanisms in most cases operate with loads that vary over very wide limits and, consequently, under operating conditions the motors of these machines may work both on the governor characteristic and on the external characteristic.

Load characteristics

For the analysis of the economic and dynamic indices of a pneumatic motor operating under variable-load conditions, when the main factor determining its operating regime is not the speed (number of revolutions) but the load, speed characteristics are not very convenient and are insufficiently illustrative. In such cases it is more convenient to use load characteristics, i.e., such characteristics for which the independent variable is taken to be a parameter evaluating the motor load, for example, the magnitude of the effective torque \(M_e\). An approximate form of the load characteristics of a pneumatic motor is shown in Fig. 3. Load characteristics may

Fig. 3

be obtained experimentally\(^{1}\) or may be constructed from the data of speed characteristics, for example, by the graphical method described below.

\(^{1}\) As far as the author knows, such experimental characteristics of pneumatic motors are not given in the literature.

It is obvious that each constant value of the compressed-air pressure will correspond to a quite definite load characteristic. Therefore, the load characteristics must indicate those constant pressures in the air network at which these characteristics were obtained. Load characteristics obtained at the normal air pressure \(p_{\text{nom}}\) may be called nominal load characteristics, while characteristics obtained at pressures below the nominal pressure may be called partial load characteristics.

For a comparative evaluation of the efficiency of pneumatic motors of different types, it is expedient to construct the load characteristics as functions of a dimensionless load parameter. As such a dimensionless parameter one may take the motor load factor

\[ K_n=\frac{M_c}{M_{\text{nom}}}. \]

When the motor operates on the governor characteristic, \(0<K_n<1\); on the external characteristic, \(1<K_n<K_{n\max}\).

The maximum value of the torque of a pneumatic motor occurs at \(n=0\)—at the moment of motor starting (more precisely, with the motor stalled). Therefore

\[ K_{n\max}=\frac{M_{\max}}{M_{\text{nom}}} =\frac{M_{\text{start}}}{M_{\text{nom}}} =K_p. \]

Here \(M_{\text{start}}\) is the starting torque of the motor;

\(K_p\) is the starting-torque coefficient, which is usually given in the passport data as an indicator of the traction properties of pneumatic motors.

In Fig. 4, as an example, the economic load characteristics \(q=\varphi(K_n)\) are presented for two pneumatic motors: the gear motor PRSh-10 \((N_e=10\ \text{hp})\) and the turbine motor DTR-20 \((N_e=20\ \text{hp})\). According to

Fig. 4

from these characteristics it is easy to determine not only the specific air consumptions for a given degree of load \(K_n\) of the motors being compared, but also those load ranges in which the greatest economic effect in air consumption can be achieved from the use of one or another motor. Thus, for example, it follows from the graphs (Fig. 4) that a comparatively greater economic effect from the use of a turbine motor instead of a gear motor can be obtained when the load of the driven machine is expressed by the coefficient \(K_n = 0.4\text{–}0.5\). From these same graphs it can be seen that, when the motors operate under overload, the turbine and gear motors are practically equivalent in air consumption only at \(K_n \simeq 1.25\).

Graphical Method for Constructing Load Characteristics

For greater clarity and for convenience in the subsequent analysis of the influence of individual factors on the course of the curve of specific consumptions, the graphical constructions of the load characteristics are carried out below separately: for operation of the motor within the limits of the governor (Fig. 5) and external (Fig. 6) speed characteristics.

Along the four coordinate axes of the diagrams (Figs. 5, 6), the quantities \(q\), \(M_e\), \(K_n\), and \(n\) are plotted to the appropriate scales. The scales of these quantities, except \(K_n\), are chosen arbitrarily, according to the desired dimensions of the diagrams. The scale of the quantity \(K_n\) is found from the condition that the segment \(M_{\text{nom}}\) on the ordinate axis of the diagram corresponds to \(K_n = 1\).

To construct the load characteristics when the motor operates on the governor, in quadrant III of the diagram (Fig. 5) the curves \(M_e = f_1(n)\) of the external, partial, and governor characteristics are plotted. The compressed-air pressures at which the external and partial characteristics were obtained, \(p_{\text{nom}}, p_1, p_2, p_3 \ldots\), are indicated respectively above each curve. In quadrant II the curves of specific consumptions \(q = f_2(n)\) are plotted for the same compressed-air pressures (\(p_{\text{nom}}, p_1, p_2, p_3 \ldots\)). The coordinates of the points of intersection of the external and partial characteristics with the governor characteristic \(A, 1, 2, 3 \ldots\) determine the speeds and the values of the torques developed by the motor under steady-state operating conditions at the working air pressures \(p_{\text{nom}}, p_1, p_2, p_3 \ldots\). By transferring the ordinates of these intersection points from quadrant III to the horizontal axis of quadrant I, the load coefficients \(K_n, K_{n1}, K_{n2}, K_{n3}\ldots\) are found, corresponding to the steady-state regimes at air pressures \(p_{\text{nom}}, p_1, p_2, p_3\ldots\). In quadrant II, on the verticals drawn through the points \(A, 1, 2, 3\ldots\), the segments bounded by the abscissa axis and the corresponding curves \(q = f_2(n)\) determine the values of the specific air consumptions \(q_{\text{nom}}, q_1, q_2, q_3\ldots\) at the load coefficients \(K_n, K_{n1}, K_{n2}, K_{n3}\ldots\). Finally, by intersecting the horizontals \(q_{\text{nom}}, q_1, q_2, q_3\ldots\) with the verticals \(K_n, K_{n1}, K_{n2}, K_{n3}\ldots\) in quadrant I, the points of the desired load characteristic \(q = \varphi(K_n)\) are obtained.

The method of constructing the dependence of the motor speed on the load coefficient is clear from the drawing (Fig. 5).

To construct the load characteristics when the motor operates at speed regimes covered by the external characteristic from \(n = 0\) to \(n = n_A\), in quadrant III of the diagram (Fig. 6) only the external characteristic \(M_e = f_1(n)\) is plotted, and in quadrant II, correspondingly, one curve \(q = f_2(n)\). On the external characteristic \(M_e = f_1(n)\), an arbitrary number of points \(A, 1, 2, 3\ldots\) is selected, and the constructions are carried out in the same sequence as described above.

Using the curves \(n=\varphi_1(K_n)\), \(q=\varphi(K_n)\) and the known relations

\[ N_e=\frac{M_e\cdot n}{716.2};\qquad Q=qN_e \]

one can also construct the characteristics \(N_e=\varphi_2(K_n)\) and \(Q=\varphi_3(K_n)\).

Fig. 5

Influence of Certain Factors on the Form of the Economic Load Characteristic

From the diagram (Fig. 5) it can be seen that the course of the economic characteristic with respect to load \(q=\varphi(K_n)\) depends strongly on the law according to which the specific consumptions increase as the working pressures of the compressed air supplied to the motor are reduced \((q=f(p)\) at \(n=\mathrm{const})\).

The more rapidly the specific consumptions increase as the compressed-air pressure is reduced, the more strongly the economy of the pneumatic motor decreases when the load is reduced within the limits of the governor characteristic.

From the same diagram it may be noted that the shape of the governor characteristic \(AB\) cannot substantially affect the form of the economy characteristic \(q=\varphi(K_n)\). Therefore, in the absence of an experimental governor characteristic, with sufficient accuracy for practical purposes it may be replaced by the straight-line segment \(AB\), as is shown by the dashed line in Fig. 2¹.

Fig. 6

The course of the curve of specific consumptions \(q=\varphi(K_n)\), when the motor operates under overload, is substantially influenced by two

factor: the form of the curve of specific air consumptions \(q=f_2(n)\) and the slope of the external characteristic of the effective torque to the abscissa axis (Fig. 5). A flat course of the curve \(q=f_2(n)\) ensures a flat course of the economy characteristic \(q=\varphi(K_n)\), and, consequently, the economy of an engine possessing such a characteristic will change little when the operating mode deviates from the nominal one toward overloads.

An increase in the slope of the torque curve \(M_e=f_1(n)\) to the abscissa axis leads to a flatter course of the economy characteristic with respect to load. But the slope of the torque characteristic \(M_e=f_1(n)\) is determined, as is known, by the value of the starting-torque coefficient. Therefore pneumatic motors with a large starting-torque coefficient possess an economic characteristic that is more favorable in form; overloading such motors is accompanied by a relatively smaller decrease in economy.

Evaluation of the Economy of Pneumatic Motors Operating under Conditions of Variable Loads

The data sheet of a pneumatic motor usually gives the value of the nominal specific air consumption. This indicator is quite sufficient for judging the economy of a pneumatic motor used to drive such a machine which, in operation, provides the motor with work at modes close to nominal conditions. The nominal specific air consumption loses its significance as an economy indicator for a motor operating with loads varying over wide limits, since the operating modes of such a motor will not correspond to the working mode at which this indicator was obtained.

As an indicator of the economy of operation of pneumatic motors used to drive machines whose load varies in time according to a known law, one may use the value of the average specific consumption, which is the quotient obtained by dividing the quantity of air consumed over the time interval under consideration \(t_2-t_1\) by the quantity of energy produced by the motor over this same interval of time

\[ q_{\mathrm{cp}}= \frac{\displaystyle\int_{t_1}^{t_2} Q\,dt} {\displaystyle\int_{t_1}^{t_2} N_e\,dt} \tag{1} \]

or

\[ q_{\mathrm{cp}}= \frac{\displaystyle\int_{t_1}^{t_2} \xi(t)\,dt} {\displaystyle\int_{t_1}^{t_2} f(t)\,dt}, \tag{2} \]

also

\[ q_c = \frac{\displaystyle \int_{t_1}^{t_2} \xi_1(t) f(t)\,dt} {\displaystyle \int_{t_1}^{t_2} f(t)\,dt}, \tag{3} \]

where

\[ \xi(t)=Q,\qquad f(t)=N_e \quad \text{and} \quad \xi_1(t)=q. \]

Thus, in order to determine the average specific air consumption of a pneumatic motor operating under variable-load conditions, it is sufficient to measure the areas under the curves \(Q=\xi(t)\) and \(N_e=f(t)\) within the limits from \(t_2\) to \(t_1\), and to divide the first quantity by the second.

For the case of a slowly changing load, when at any instant of time the operating regime of the motor may be regarded as steady, the curves \(Q=\xi(t)\), \(q=\xi_1(t)\), and \(N_e=f(t)\), if the curves \(Q=\varphi_4(K_n)\), \(N_e=\varphi_2(K_n)\), \(q=\varphi(K_n)\), and \(K_n=\xi_2(t)\) are available, can easily be obtained, for example, by a graphical method (Fig. 7).

If the complex curve of load variation with time \(K_n=\xi_2(t)\) can, with a sufficient degree of accuracy, be replaced by a broken line (Fig. 8), then the average specific air consumption can be calculated from the formula

\[ q_{cp} = \frac{\displaystyle \sum_{1}^{m} q_{cpi} N_{cpi}\,\Delta t_i} {\displaystyle \sum_{1}^{m} N_{cpi}\,\Delta t_i}, \tag{4} \]

where \(q_{cpi}\) and \(N_{cpi}\) are, respectively, the average specific air consumption and the average power over the time interval \(\Delta t_i\); \(m\) is the number of selected segments on the approximated load graph.

For time intervals \(\Delta t_i\) in which the load is constant (\(K_n=\mathrm{const}\)), the quantity \(q_{cpi}=q_i\), and therefore it can be found directly from the load characteristic \(q=\varphi(K_n)\) from the known value \(K_{ni}\).

For time intervals \(\Delta t_i\) in which \(K_n\ne\mathrm{const}\), the quantities \(q_{cpi}\) should be calculated by formulas (2–3). In particular cases, when over the considered time interval \(\Delta t_i\) the change in the quantity \(q\) is small, one may use the simplified method for determining the average air consumption \(q_{cpi}\) described below.

According to the mean-value theorem, the numerator of equation (3) may be written as follows:

\[ \int_{t_1}^{t_2} \xi_1(t) f(t)\,dt = \xi_1(\tau)\int_{t_1}^{t_2} f(t)\,dt, \]

where

\[ t_1 \le \tau \le t_2. \]

Fig. 7

Visible labels in the figure:

• \(N_e\)
• \(Q\)
• \(k_n\)
• \(k_M\)
• \(O\)
• \(t\)
• \(N_e=\varphi(k_n)\)
• \(Q=\psi(k_n)\)
• \(N_e=f(t)\)
• \(Q=\xi(t)\)
• \(k_n=\xi_e(t)\)
• \(k_n=1\)
• \(k_M=1\)
• \(t_2-t_1\)
• \(45^\circ\)

Point labels visible in the diagram:

• \(1\), \(1'\), \(1''\)
• \(2\), \(2'\), \(2''\)
• \(3\), \(3'\), \(3''\)
• \(4\), \(4'\), \(4''\)
• \(5\), \(5'\), \(5''\)

Therefore

\[ q_{cp i}= \frac{\displaystyle \int_{0}^{\Delta t_i}\xi_1(t)f(t)\,dt} {\displaystyle \int_{0}^{\Delta t_i} f(t)\,dt} =\xi_1(\tau). \tag{5} \]

For small changes of the functions \(\xi_1(t)\) and \(f(t)\) over the time interval \(\Delta t_i\), the value \(\xi_1(\tau)\) can be determined as the mean ordinate of the curve

\[ q_{cp i}=\xi_1(\tau)= \frac{\displaystyle \int_{0}^{\Delta t_i}\xi_1(t)\,dt} {\Delta t_i}, \]

but with a linear change of load with time

\[ \frac{\displaystyle \int_{0}^{\Delta t_i}\xi_1(t)\,dt} {\Delta t_i} = \frac{\displaystyle \int_{0}^{\Delta K_{\mathrm{н}i}}\varphi(K_{\mathrm{н}})\,dK_{\mathrm{н}}} {\Delta t_i} =q_{cp i}. \tag{6} \]

Fig. 8

Thus, the value of the mean specific air consumption \(q_{\mathrm{avg}}\) for individual intervals \(\Delta t_i\) of the approximated load diagram, where \(q=\xi_1(t)\) varies little and \(K_n=\xi_2(t)\) varies little and linearly, can be found as the mean ordinate of the load characteristic \(q=\varphi(K_n)\) within the range of load variation by the amount \(\Delta K_{n i}\).

From the constructions (Fig. 8) it can be seen that the value of the economic performance indicator of the motor, \(q_{\mathrm{avg}}\), depends not only on the form of the characteristics \(q=\varphi(K_n)\) and \(N_e=\varphi(K_n)\) of the given motor, but also on the type of load diagram \(K_n=\xi_2(t)\). For one and the same motor, the value of \(q_{\mathrm{avg}}\) will be different if this motor operates under different load diagrams. Therefore, when using the proposed indicator \(q_{\mathrm{avg}}\), it should be borne in mind that such an indicator evaluates the economy of motor operation only under the given specific operating conditions.

By comparing the values of \(q_{\mathrm{avg}}\) found for various motors when operating the drive of a machine with a specified load diagram, it is possible to establish which of the motors will be the most economical in terms of air consumption under the given specific conditions. Obviously, the most economical motor will be the one for which \(q_{\mathrm{avg}}\) proves to be the smallest.

REFERENCES

1. G. I. Kusnitsin. Pneumatic Motors of Mining Machines and Mechanisms. Ugletekhizdat, 1953.
2. A. M. Misharin. Pneumatic Gear Motors. Ugletekhizdat, 1948.
3. V. M. Balykov. The KKP-1 Coal Combine for Steeply Dipping Seams. Ugletekhizdat, 1952.

1. Such a replacement is possible only in the case where the degree of nonuniformity of the governor
   \[ \delta = 2\frac{n_B-n_A}{n_B+n_A} \]
   does not exceed the usual limits, \(\delta=10\%\). ↩↩