On the Theory of a Gear-Type Pneumatic Motor

S. I. Shubovich, G. N. Garashchuk

(Presented by the scientific seminar of the Department of Applied Mechanics)

In the literature on pneumatic gear-type motors, the theoretical working process is examined in sufficient detail, while little attention is paid to the actual process taking place in a real motor with spur and helical rotors. In [1], for example, the basic relations for the indices of the theoretical process are derived, and the transition from these indices to effective indices is carried out by introducing into the corresponding formulas a correction factor, erroneously called the mechanical efficiency, since this factor accounts not only for friction losses in the mechanism, but also for losses caused by deviations of the actual working process from the theoretical one, i.e., indicator losses. It is quite obvious that such a method of accounting for indicator losses does not reveal the nature of the phenomena affecting the indices and characteristics of the motor. In [2] it is indicated that the actual process differs significantly from the theoretical one, and it is proposed that the losses associated with this be taken into account by the so-called coefficient of fullness of the indicator diagram; however, the explanations given in that work for the nature of the indicator losses cannot, in our opinion, be regarded as correct.

The absence in the literature of a more detailed analysis of the actual working process of gear-type motors with spur and helical rotors served as the basis for the publication of the present article, whose aim is partially to fill this gap.

In deriving the equations for operation and power on the toothed rims of the rotors [1], the action of air-pressure forces on the lateral surfaces of the tooth spaces is considered under the assumption that the pressures at any point of the inlet cavity and the outlet cavity are constant within the boundaries of these spaces and do not change with variation of the rotor speed. Such conditions do not exist in a real motor, since, as the rotors rotate, the air, filling the spaces in the inlet cavity and releasing them in the outlet cavity, is throttled in the flow sections between the teeth of the meshing rotors; as a result, the pressure at different points of the space in the meshing zone of the toothed rotors will have different magnitudes. In the inlet cavity the air pressure will decrease as it approaches the pitch point of the mesh, and in the outlet cavity it will increase. Since the change in pressure is caused by the hydraulic resistances of the peculiar air passage formed by the tooth surfaces in the meshing zone, then, as the air velocity in this passage increases and, consequently, as the rotor speed increases, the pressure gradient will increase. These phenomena confirm

are confirmed by experiments [3, 4]. Thus, for example, measurements carried out on the PRSh-10 motor [3] showed that the pressure in the gaps located near the meshing pole on the inlet side is, under the nominal speed regime, approximately 8–10 percent of the pressure measured in the inlet cavity directly at the inlet opening for compressed air. These experiments also established that the pressure gradient increases in proportion to the air density and to the rotor speed to the power 1.60–1.65. Thus, the dependences obtained in [1] may be regarded only as the basic dependences of the theoretical working process of a gear motor.

In order to take into account the influence of the hydraulic resistances that occur in the meshing zone of the rotors of a real motor on the amount of work performed by the air, let us consider the action of the forces on the lateral surfaces of the gaps, taking into account the fact that the pressures in the inlet and outlet cavities vary in the directions toward the meshing pole. Let us assume conditionally that, for some fixed position of the rotors (Fig. 1), the air is throttled only in sections I—I, II—II, and III—III, and also that the pressures in the volumes bounded by these sections and by the contact lines (Fig. 1) have the values \(p\), \(p_1\), \(p_2\), \(p_c\), \(p_{01}\), and \(p_0\).

Fig. 1. Meshing of the rotors of a gear motor

Using the well-known proposition that the resultant of the projections of the pressure forces on the lateral surface of a gap is equal to the pressure multiplied by the area of the projection of this surface, one can obtain an expression for the circumferential force, reduced to the initial circle of one of the rotors, which, in accordance with the notation adopted in Fig. 1, is written in the following form:

\[ P=\frac{b}{2r}\bigl[p\left(R_e^2-r_{y_4}^2\right)+p_1\left(r_{y_4}^2-r_{x_3}^2\right)+p_2\left(R_e^2-r_{y_3}^2+r_{x_2}^2-r_{x_3}^2\right)- \]
\[ -p_c\left(R_e^2-r_{y_3}^2+r_{x_1}^2-r_{x_2}^2\right)-p_{01}\left(r_{y_1}^2-r_{x_1}^2\right)-p_0\left(R_e^2-r_{y_1}^2\right)\bigr], \tag{1} \]

where \(b\) is the length of the rotor teeth.

Analyzing equation (1), one may observe that the circumferential force, which ultimately determines the amount of work on the toothed rims of the rotors, i.e., the amount of indicated work, is the sum of the tangential components of the forces of air pressure on the lateral surfaces of those spaces into which the teeth of the mating rotor partially or completely enter (Fig. 1). It follows from this that the amount of indicated work depends on the magnitude of the pressures only in those spaces in which the teeth of the mating rotor are located at the given instant. The pressure in all other spaces has no effect on the amount of work. Taking these properties into account, spaces of the first kind may conventionally be called working spaces, and those of the second kind—nonworking spaces.

Let us show that the remarks made here are of practical significance. The nonworking spaces in the intake cavity, while having no effect on the amount of indicated work, act as capacities carrying compressed air from the intake cavity into the atmosphere; therefore it is desirable to keep the pressure in such spaces to a minimum in order to reduce air consumption. In the working spaces, on the other hand, it is desirable to have the pressure as high as possible in order to obtain the greatest possible useful work. In a gear-type motor, the exact opposite phenomenon is observed. The nonworking spaces are filled without throttling, and therefore by the time they leave the intake cavity the pressure in them reaches the value of the pressure of the air entering this cavity from the line; the working spaces, however, are filled in the presence of throttling and, consequently, the pressure in them will necessarily be lower than in the spaces carrying air out of the motor.

Fig. 2. Diagram of a gear-type motor with compressed-air supply directly into the meshing zone

As a result, this leads to the fact that the weight quantity of air consumed by the gear-type motor (without leaks) is greater than the quantity that performs useful work in it.

The noted inherent shortcoming of the gear-type motor, associated with the fact that the flow of fresh air fills first of all the nonworking spaces, can be eliminated to some extent, for example, by supplying compressed air from the end faces of the rotors directly into the meshing zone (Fig. 2).

On the basis of equation (1), one can write the expression for the elementary indicated work performed by the air when the rotors turn through an angle \(d\varphi\),

\[ \begin{aligned} dL_i &= Pr\,d\varphi = p\,\frac{b}{2}\left(R_e^2-r_{y_4}^2\right)d\varphi + p_1\,\frac{b}{2}\left(r_{y_4}^2-r_{x_3}^2\right)d\varphi +\\ &\quad + p_2\,\frac{b}{2}\left(R_e^2-r_{y_3}^2+r_{x_3}^2-r_{x_4}^2\right)d\varphi - p_c\,\frac{b}{2}\left(R_e^2-r_{y_3}^2+r_{x_1}^2-r_{x_2}^2\right)d\varphi -\\ &\quad - p_{01}\,\frac{b}{2}\left(r_{y_1}^2-r_{x_7}^2\right)d\varphi - p_0\,\frac{b}{2}\left(R_e^2-r_{x_1}^2\right)d\varphi = p\,dV + p_1\,dV_1 +\\ &\quad + p_2\,dV_2 - p_c\,dV_c - p_{01}\,dV_{01} - p_0\,dV_0 . \end{aligned} \tag{2} \]

where \(dV\), \(dV_1\), etc. are the increments of the volumes when the rotors turn through an angle \(d\varphi\) in those spaces in which the pressures denoted by the same subscript occur.

The indicated work performed in the motor during one cycle, i.e., over the angle of rotation of the rotors corresponding to the pitch of engagement, is expressed as follows:

\[ L_{it}=\int_0^t p\,dV+\int_0^t p_1\,dV_1+\int_0^t p_2\,dV_2-\int_0^t p_c\,dV_c-\int_0^t p_0\,dV_{01}-\int_0^t p_0\,dV_0 . \tag{3} \]

Since at the beginning and at the end of the two-pair engagement of the rotors the volume cut off by two contact lines is one and the same, and the heat exchange and leakage of the air contained in this space are negligibly small, then

\[ \int_0^t p_c\,dV_c=0. \]

Taking into account that the first three terms of equation (3) represent positive work performed by the air in the inlet cavity, while the last two terms represent negative work in the outlet cavity, the cyclic indicated work can be represented in the following form:

\[ L_{it}=p_{\mathrm{cp}}\cdot V_t-p_{0\mathrm{cp}}\cdot V_t=p_i\cdot V_t . \tag{4} \]

Here \(p_{\mathrm{cp}}\) and \(p_{0\mathrm{cp}}\) are the mean values of the pressures in the engagement zones, respectively in the inlet cavity and in the outlet cavity;

\(p_i\) is the mean indicated pressure;

\(V_t\) is the change in the total volume of the working spaces during a cycle.

The magnitude of the change in the total volume of the working spaces during a cycle, arising as a result of the entry or exit of the teeth of the mating rotor into or from these spaces, is, in its physical meaning, equivalent to the magnitude of the change in the volume of the cylinder of a piston machine during one stroke of the piston, i.e., to the so-called volume swept by the piston. Therefore the quantity \(V_t\) must not be identified with the volume of a space \(V_{\mathrm{vp}}\), just as, when considering the working process of a piston machine, the volume swept by the piston must not be identified with the volume of the cylinder.

Figure 3 presents a graph of the change in the volume of the inlet cavity, bounded by section IV—IV and by the contact line in the engagement (Fig. 1), as a function of the rotor angle of rotation. The broken line 1 (Fig. 3) represents the graph of the change in the volume of the inlet cavity, and the curves

2—the change in the volume enclosed between two contact lines in double-pair meshing, the so-called trapped space. The angle \(\varphi_1\) determines the position of the rotor at the moment when the trapped space is connected to the inlet cavity; the angle \(\varphi_2\), the position of the rotor at the moment when one nonworking depression filled with compressed air leaves the inlet cavity; \(\varphi_3\), at the moment when the second depression leaves. From the graph (Fig. 3) it can be established that the change in the total volume of the working depressions per cycle, \(V_t\), is equal to two depression volumes minus the volume of the trapped space. If it is taken into account that the useful work per cycle is determined by the quantity \(V_t\), while the consumption of compressed air is \(2V_{\text{in}} = V_t + V_c\), it becomes clear that the trapped space leads to an unproductive consumption of air. From this point of view the trapped space is a harmful space, as is correctly noted in [2]; however, it has no effect on the magnitude of the useful work or the power of the engine.

Fig. 3. Diagram of the change in the volumes of the inlet cavity and trapped space with the angle of rotation of the rotors

Therefore, reducing the size of the trapped space, for example by an appropriate choice of the meshing parameters, will increase the economy of the engine but will not change its power.

The mean indicated pressure of a gear-type motor may be represented as the difference between the theoretical pressure drop of the air at the motor inlet and outlet and the pressure lost as a result of throttling in the meshing,

\[ p_i = (p - p_0) - \Delta p . \tag{5} \]

Here \(p\) is the air pressure at the motor inlet;

\[ p_0 \]

is the air pressure at the motor outlet;

\[ \Delta p \]

is the loss of pressure drop associated with throttling in the meshing.

Theoretical and experimental studies [3] have shown that the losses of pressure drop obey the following power-law dependence:

\[ \Delta p = C \cdot p \cdot n^{m}, \tag{6} \]

where \(C\) is a certain constant quantity, depending in a complex way on the parameters of the meshing of the rotors;
\(n\) is the number of revolutions of the rotor per minute;
\(m\) is an exponent varying within the limits from 1.6 to 1.7.

Now, on the basis of (4), (5), and (6), one can obtain an expression for the indicated power of a gear-type motor with spur or helical rotors in the following form:

\[ N_i = \frac{z \cdot n}{60 \cdot 75}(p - p_0)V_t - \frac{z \cdot n}{60 \cdot 75}\Delta p \cdot V_t \ \text{hp}, \tag{7} \]

where \(z\) is the number of rotor teeth.

The first term of this equation represents the theoretical power of the motor, and the second is the power of throttling losses in the meshing of the rotors, i.e., the power of indicated losses.

Taking the ratio of indicated power to theoretical power, one can obtain the expression for the indicated efficiency:

\[ \eta_i = \frac{N_i}{N_T} = 1 - C_1 \frac{p}{p - p_0}\cdot n^{m}, \tag{8} \]

where \(C_1\) is a certain constant quantity for the given motor.

It follows from equation (8) that the indicated efficiency of a gear-type motor with spur or helical rotors varies

Fig. 4. Effective-power characteristics versus revolutions of the PRSh-16 motor

depending on the rotational speed of the rotors not according to a linear law, as is erroneously asserted in works [2, 4], but according to a power curve with exponent \(m = 1.6\text{–}1.7\).

The effective power of the motor can now be represented in the following form:

\[ N_e = N_i \cdot \eta_{\mathrm{M}} = N_{\mathrm{T}} \cdot \eta_i \cdot \eta_{\mathrm{M}} . \tag{9} \]

Here \(\eta_{\mathrm{M}}\) is the mechanical efficiency, taking into account only friction losses in the motor mechanism; \(N_{\mathrm{T}}\) is the theoretical power, which can be calculated with a sufficient degree of accuracy from the known formulas of A. M. Misharin [1].

To verify the relations presented above, we calculated the coordinates of points of the characteristics \(N_e = f(n)\) for the PRSh-16 motor and compared them with the characteristics (Fig. 4) obtained during testing of this motor at the Pneumatika plant [5]. In these calculations the theoretical power was determined by the formula of A. M. Misharin [1], the mechanical efficiency was taken to be constant with respect to speed, and the indicated efficiency was found from equation (8) with coefficient \(C_1\), calculated on the basis of the coordinates of point \(A\) (Fig. 4) of one of the experimental curves. The solid lines in Fig. 4 represent the experimental characteristics, and the points plotted on them are the results of calculations by the formulas mentioned above.

The good agreement of the calculated points with the experimental curves (Fig. 4) is additional evidence that the working hypothesis underlying all our derivations—concerning indicated losses caused by hydraulic resistances in the rotor meshing zone—is in good accord with experimental regularities.

REFERENCES

1. A. M. Misharin. Pneumatic Gear Motors. Ugletekhizdat, Moscow, 1948.

2. K. S. Borisenko. Pneumatic Motors of Mining Machines. Ugletekhizdat, Moscow, 1958.

3. S. I. Shubovich. Investigation of the Working Process and Characteristics of Certain Pneumatic Motors. Dissertation for the degree of Candidate of Technical Sciences, Tomsk, 1956.

4. M. P. Petrov. Investigation of the Principal Parameters of a Pneumatic Gear Motor. Author’s abstract of dissertation for the degree of Candidate of Technical Sciences, Leningrad, 1954.

5. G. I. Kusnitsin. Pneumatic Motors of Mining Machines and Mechanisms. Ugletekhizdat, Moscow, 1953.