ON THE FLOW CHARACTERISTICS OF SERVO SPOOL VALVES IN THE REGION OF SMALL AXIAL OPENINGS AND OVERLAPS

I. A. NEMIROVSKII, A. M. GURTYAKOV

(Presented by the scientific-technical seminar of the Department of Mining Machines and Mine Transport)

For calculations and design of servo hydraulic drives it is necessary to know the flow characteristics of the control distributor spool valves. In most studies, the working window of a servo spool valve is regarded as an orifice in a thin submerged wall, which leads to the use of the classical formula

\[ Q_z = f \mu \sqrt{\frac{2}{\rho}\Delta p}, \tag{1} \]

where

\(Q_z\) — flow rate through the working window of the servo spool valve;
\(f\) — flow area of the working window;
\(\mu\) — discharge coefficient;
\(\rho\) — density of the working fluid;
\(\Delta p\) — pressure drop across the working window.

When formula (1) is used, the quantity is usually taken as a linear function of the axial opening of the spool valve [2], i.e.

\[ f = h \cdot b, \]

where

\(h\) — axial opening of the spool valve (Fig. 1),
\(b\) — width of the working window.

In the region of small openings and overlaps of the servo spool valve, comparable with the radial clearance, the actual value of the flow-area cross section of the working window may differ substantially from the values obtained by formula (2).

The studies of V. A. Leshchenko [3] made it possible to propose an empirical formula that takes into account the radial clearance in the spool valve

\[ Q_z = \frac{ 0.87 \cdot b \, (h^2+\delta^2)^{2/3} \, p^{\frac{2+ah^2}{2+2ah^2}} }{ 60 \gamma \left(1+2.94 \cdot 10^{-3} l^{3/2}\right) }, \tag{3} \]

where

\(\gamma\) — specific weight of the oil in \(g/cm^3\);
\(b\) — length of the working window in \(cm\);
\(h\) — magnitude of the working-window opening in \(mm\);
\(l\) — magnitude of the working-window overlap in \(mm\);
\(\delta\) — radial clearance in \(mm\);
\(p\) — pressure drop in \(kg/cm^2\);

\[ a = 7.5 \cdot 10^{-4}; \]

\(\gamma\) is the weight of a unit volume of oil in \(\mathrm{kg/cm^3}\);

Fig. 1. Diagram for the graphical calculation of the area of the working ports of tracking spool valves

E. M. Khaimovich, having investigated the flow characteristics of tracking spool valves, proposed the following formula, which takes into account both the radial clearance and the viscosity of the working fluid [4]:

\[ Q = \frac{ 0.04\, b \left[ \frac{4000 + 2.5h^2 + 0.1l^2}{4000 + h^2 + 0.3l^2} \right] (h^2 + \delta^2)^{1/2} (\Delta p)^{\frac{2+2ah^2}{2+2ah^2}} }{ \left[ 1 + \frac{4l}{\delta} \right] \left[ \frac{\nu}{\nu_{\mathrm{n}}} \right]^{\frac{18+0.08h}{18+h}} } \ \mathrm{cm^3/sec}, \tag{4} \]

where

\[ \begin{aligned} b &\text{ is a length dimension in } \mathrm{cm};\\ h \text{ and } \delta &\text{ are the slot opening and the radial clearance in microns;}\\ p &\text{ is pressure in } \mathrm{kg/cm^2};\\ a &\text{ is a coefficient equal to } 7.5 \cdot 10^{-4};\\ l &\text{ is the length of overlap of the slot in microns in the axial direction;}\\ \nu_{\mathrm{p}},\ \nu_{\mathrm{n}} &\text{ are the actual and normal calculated kinematic viscosities in } \mathrm{cm^2/sec}. \end{aligned} \]

V. A. Khokhlov, using the results of experiments by V. A. Leshchenko, showed the dependence of the coefficient \(\xi\) of hydraulic losses in the working port of a spool valve on the axial distance between the working edges [5], under the assumption that the edges are absolutely sharp and that the spool valve is positioned concentrically with respect to the sleeve opening. At the same time, it is known that, for technical reasons, chamfers and roundings are present on the working edges of tracking spool valves and sleeves.

Thus, G. I. Kamenetskii gives data indicating that the working edges of tracking spool valves have the form of a chamfer \(0.001\ \mathrm{cm}\) long, arranged at an angle of \(20^\circ\), and recommends carrying out a graphical calculation of the actual area of the live cross section [1].

It should be noted that in works [3] and [4] no quantitative information is given on the condition of the working edges of the tracking spool valves of the experimental setup. It is of interest to analyze the influence

Fig. 2. Dependence of the area of the working port of a follow-up spool on the axial opening or overlap. Radial clearance \(\delta = 0.0015\) cm. Curve 1—analytical calculation by method 5. Curve 2—graphical calculation for the working edges of the spool and bushing with chamfers of \(0.0004\) cm at an angle of \(20^\circ\). Curve 3—graphical calculation for the working edges of the spool and bushing with chamfers of \(0.0010\) cm at an angle of \(20^\circ\).

Fig. 3. Dependence of the velocity of flow of the working fluid through the effective cross section of the port of a follow-up spool on the axial opening or overlap. Dotted line—calculation from curve 1 in Fig. 2; dash-dot line—calculation from curve 2 in Fig. 2; solid line—calculation from curve 3 in Fig. 2.

1. \(\Delta p = 3\ \text{kg}/\text{cm}^2\); 2. \(\Delta p = 5\ \text{kg}/\text{cm}^2\); 3. \(\Delta p = 10\ \text{kg}/\text{cm}^2\).

quality of the working edges on the result obtained by V. A. Khokhlov under the assumption of absolute sharpness of these edges.

Figure 1 gives a diagram of the graphical calculation of the flow area of the metering orifice of the tracking spool valve with a radial clearance \(\delta = 0.0015\ \text{cm}\) for various shapes of the working edges.

As can be seen from Fig. 2, for working edges having the form of chamfers \(0.001\ \text{cm} \times 20^\circ\), the values of the flow area (curve 3) differ substantially from the values obtained by calculation according to the method of [5] (curve 1). For chamfers \(0.0004\ \text{cm} \times 20^\circ\) this difference is not so significant, but its influence on the value of the loss coefficient \(\xi\) is quite appreciable.

Figure 3 gives the results of calculating the mean velocity in the flow area of the working orifice of the spool valve according to method 5 (dotted curves) and taking into account the values of the flow areas obtained by graphical calculation for two shapes of working edges (dash-dotted curves for a chamfer \(0.0004\ \text{cm} \times 20^\circ\); solid curves for a chamfer \(0.0010\ \text{cm} \times 20^\circ\)) and for three values of \(\Delta p\). In all these calculations, the experimental data of V. A. Leshchenko [3] were used.

Fig. 4. Dependence of the local resistance coefficient \(\xi\) of the metering orifice of the tracking spool valve on axial opening or overlap. Dotted line—calculation according to curve 1 in Fig. 2; dash-dotted line—calculation according to curve 2 in Fig. 2; solid line—calculation according to curve 3 in Fig. 2.

1. \(\Delta p = 3\ \text{kg}/\text{cm}^2\); 2. \(\Delta p = 5\ \text{kg}/\text{cm}^2\); 3. \(\Delta p = 10\ \text{kg}/\text{cm}^2\).

The values of the loss coefficient $\xi$ corresponding to various shapes of the working edges are given in Fig. 4, from which it is evident that, even with a very small chamfer on the working edges of the spool and sleeve ($0.0004\ \text{cm}$) and a fairly large radial clearance ($0.0015\ \text{cm}$), analysis of the flow characteristics in the region of axial distances between the working edges of the spool and sleeve must be carried out with allowance for the actual condition of the edges.

REFERENCES

1. G. I. Kamenetskii. Hydraulic follow-up drives of copying machine tools. In: Automation of Technological Processes in Mechanical Engineering. Drive and Machine Control. Moscow, Academy of Sciences of the USSR, 1956.

2. B. L. Korobochkin. Study of hydraulic follow-up systems of copying machine tools. Dissertation for the degree of Candidate of Technical Sciences. Moscow, 1955.

3. V. A. Leshchenko. Hydraulic Follow-up Drives for the Automation of Machine Tools. Mashgiz, Moscow, 1962.

4. E. M. Khaimovich. Hydraulic Drive and Hydraulic Automation of Metal-Cutting Machine Tools. Mashgiz, Moscow, 1959.

5. V. A. Khokhlov. Electrohydraulic Follow-up Drive. Nauka, Moscow, 1966.