Bulletin of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Volume 173
1970

Evaluating the Influence of the Position of the Rotation Axis of a Hydraulic Coupling on Its External Characteristic

Yu. I. Kapustin, A. V. Murin

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

The use of hydraulic couplings with a vertical axis of rotation in various machines and units [1], [2] raises the question of considering the acceptability of known methods for calculating hydraulic couplings with a horizontal axis of rotation for determining the basic parameters of hydraulic couplings with a vertical axis. Such hydraulic couplings, in comparison with ordinary ones, as studies [3] have shown, have certain special features. The safety hydraulic coupling with active diameter \(D_a = 280\) mm, with the lower arrangement of the pump wheel, when transmitting a torque equal to 1.5 kgm, had a slip of 6.5% with the horizontal position of the axis, and 7.5% with the vertical position. The efficiency of the hydraulic coupling at this torque, when the position of the axis was changed, decreased from 93.5% to 92.5%, which is equivalent to an increase in losses in the hydraulic coupling by 15.4%.

As is known, most hydraulic couplings have a slip at nominal torque of 2–3%. An increase in slip by 1% is equivalent to an increase in losses in the hydraulic coupling by 30–40%. Therefore, determining the basic dimensions of hydraulic couplings with a vertical axis of rotation by methods for hydraulic couplings with a horizontal axis of rotation may lead to a significant discrepancy between the calculated and actual characteristics of these hydraulic couplings. In this connection, it became necessary to investigate the influence of the position of the rotation axis of the hydraulic coupling on its external characteristic and to evaluate it when calculating hydraulic couplings with a vertical axis of rotation.

For this purpose, an experimental hydraulic coupling was manufactured in the laboratory of the Department of Applied Mechanics (Fig. 1). Its design made it possible to carry out tests with different positions of the working wheels relative to one another, namely: with the lower arrangement (Fig. 1,a) and the upper arrangement (Fig. 1,b) of the pump wheel. The profiles of the working wheels were chosen according to the recommendation of the firm “Vulcan.” The axial clearance between the wheels was 8 mm. The tests of the hydraulic coupling were carried out at a constant speed of the pump wheel. Water was used as the filling fluid; the filling of the hydraulic coupling was about 100%.

The external characteristics of the experimental hydraulic coupling, obtained for different positions of the axis and different relative arrangements of the working wheels, are shown in Fig. 2. Here curve 1 is the external characteristic of the hydraulic coupling with a horizontal axis of rotation, curve 2 is the external characteristic of the hydraulic coupling with a vertical axis

Fig. 1

b: \(\omega_{\text{т}}\), \(\omega_{\text{н}}\)

a: \(\omega_{\text{т}}\), \(D_a 210\), \(\omega_{\text{н}}\), \(\infty\)

rotation, and curve 3 is with the pump wheel in the upper position. The course of these characteristics shows that the position of the axis of rotation affects the torque transmitted by the hydraulic coupling in different ways for different positions of the working wheels. With the pump wheel in the lower position, the hydraulic coupling transmits a considerably smaller torque, whereas with the pump wheel in the upper position it at first transmits less torque than with the horizontal position of the axis, and then exceeds it.

Fig. 2

To clarify the reasons causing such a change in the transmitted torque, let us consider the scheme of the circulation flow in the cavity of the hydraulic coupling for different positions of its axis of rotation. The scheme of the fluid flow with the horizontal position of the axis is shown in Fig. 3, a. As the studies of U. Oprecht [4] showed, at small slips the vortex center of the circulation flow in hydraulic couplings having the profile recommended by the firm “Vulkan” is not located in the gap between the wheels, but is displaced into the pump wheel. Measurements of the flow-velocity angles in the pump wheel, carried out by the authors, showed that the magnitude of the displacement of the vortex center from the edge of the pump wheel in the range of slips from 0 to 10% is on the order of 5 mm.

The scheme of the circulation flow with a vertical axis and the pump wheel in the lower position is shown in Fig. 3, b. Because in this case the gravitational forces are directed parallel to the axis of rotation, they change the magnitude of the meridional velocities in sections I—I and II—II. The meridional velocity in section I—I increases, whereas in section II—II it decreases, which leads to an even greater displacement of the vortex center into the pump wheel. This causes a decrease in the flow rate of liquid passing through the turbine wheel and, accordingly, a reduction in the torque transmitted by the hydraulic coupling.

The scheme of the circulation flow with the vertical position of the axis and the pump wheel in the upper position is shown in Fig. 3, c. In this case, as a result of the redistribution of meridional velocities, the vortex center of the flow is located in the turbine wheel, and the magnitude of its displacement is somewhat greater than the displacement of the vortex center, with the horizontal position of the axis, into the pump wheel. This also explains the distinctive course of external characteristic 3 (Fig. 2).

Fig. 3

Subfigures: a, b, c.
Visible labels in the diagrams include: \(\omega_{\mathrm{н}}\), \(\omega_{\mathrm{т}}\), \(T\), \(R_0\), \(R_{\mathrm{н}}\), \(R_a\), \(O\), \(a\), \(B\), \(F_3\), \(\Delta T\), \(i-i\), \(i'-i'\).

To estimate the magnitude of the change in torque, let us assume that the form of the meridional section of the hydraulic coupling has been chosen in such a way that the vortex center of the flow in this hydraulic coupling, when the axis is horizontal, is located in the gap between the impellers. Then, so that the results obtained can be used in calculations of hydraulic couplings having the “Vulkan” profile, which is the most widespread at present, we correct the obtained dependences by the amount of displacement of the vortex center when the axis is horizontal.

Euler’s equation, which determines the torque transmitted by the hydraulic coupling, has the form:

\[ M=\frac{\gamma Q}{g}\left(r_{12}^{2}\omega_{\text{н}}-r_{11}^{2}\omega_{\text{т}}\right), \tag{1} \]

where \(Q\) is the liquid flow rate in the circulation circuit,
\(\omega_{\text{н}}\) and \(\omega_{\text{т}}\) are the angular velocities of the pump and turbine wheels,
\(r_{12}\) and \(r_{11}\) are the radii of the equivalent jet at the outlet and inlet of the pump wheel.

Since, when the position of the axis of rotation of the hydraulic coupling changes, we assume that the angular velocities of the pump and turbine wheels remain unchanged, and also, as measurements of the flow-velocity angles have shown, the radius of the vortex zone \(R_{\text{ц}}\) remains constant, then, naturally, the change in torque may be due to a change in the magnitude of the liquid flow rate. The liquid flow rate that determines the torque transmitted by the hydraulic coupling is equal to the flow rate of the liquid passing through the turbine wheel. It can be determined from the relation

\[ Q_{\text{г}}=w\cdot F_{2}, \]

where \(w\) is the mean meridional velocity of the liquid flow in section II—II,
\(F_{2}\) is the flow area in the turbine wheel.

As indicated above, when the axis is vertical, the mean meridional velocity of the liquid in section II—II decreases under the action of gravitational forces. To determine the magnitude of the change in meridional velocity, we write Bernoulli’s equation in relative motion for section II—II for different positions of the axis:

Horizontal position of the axis:

\[ E=z+\frac{p}{\gamma}+\frac{w^{2}-u^{2}}{2g}, \tag{2} \]

vertical position of the axis:

\[ E'=z'+\frac{p'}{\gamma}+\frac{w'^{2}-u'^{2}}{2g}, \tag{3} \]

where \(z\) and \(z'\) are the hydrostatic head of the flow at a point in section II—II,
\(p\) and \(p'\) are the piezometric head,
\(w\) and \(u\) are the meridional and circumferential velocities of the flow.

In view of the fact that, when the position of the axis of rotation changes, the angular velocity of the impellers does not change and, consequently, the total energy of the flow also does not change, and further taking the datum plane \(00\) as shown in Fig. 3,a, we may write:

\[ E=E',\quad p=p',\quad u=u',\quad z=0 \quad \text{and} \]

\[ z'=\frac{h}{2}. \tag{4} \]

Here \(h\) is the distance between the equivalent streamlines in sections I—I and II—II along a line parallel to the axis of rotation and passing through the vortex center of the flow.

Equating the corresponding right-hand sides of equations (2) and (3) and replacing \(z'\) by its expression (4), we obtain

\[ w'^2 = w^2 - gh \]

or

\[ w' = \sqrt{w^2 - gh}. \]

The fluid flow rate through the turbine wheel when the axis is in the vertical position will be

\[ Q_{\mathrm{v}} = w' \cdot F_2 = \sqrt{w^2 - gh}\cdot F_2 . \]

Let us introduce a coefficient that accounts for the change in flow rate when the position of the axis of rotation changes, according to the relation

\[ k'_{\varphi} = \frac{Q_{\mathrm{v}}}{Q_{\mathrm{h}}}. \]

Substituting for \(Q_{\mathrm{v}}\) and \(Q_{\mathrm{h}}\) their expressions and carrying out the obvious simplification, we obtain

\[ k'_{\varphi} = \sqrt{1 - \frac{gh}{w^2}} . \tag{5} \]

We shall correct the results obtained for the magnitude of the displacement of the vortex center of the flow into the pump wheel, for couplings having the profile recommended by the Vulkan firm, under the assumption of constancy of the meridional flow velocity over section II—II. In this case, the fluid flow rate in the cavity of the hydraulic coupling, when the vortex center is located in the gap between the impellers, will be written in the form

\[ Q = 2\pi R_{\mathrm{c}}\cdot T\cdot w, \]

where \(T\) is the depth of the meridional section (Fig. 3),
\(w\) is the mean meridional velocity in section II—II.

The useful fluid flow rate in the cavity of a hydraulic coupling having the Vulkan profile, with a horizontal axis, is determined by the expression

\[ Q_{\mathrm{h}} = \frac{1}{1+\alpha_{\mathrm{c}}}\, Q, \tag{6} \]

where \(\alpha_{\mathrm{c}}\) is the coefficient of displacement of the vortex center into the pump wheel, equal to \(\dfrac{\Delta T}{T}\).

The useful fluid flow rate when the axis is vertical and the pump wheel is located below is determined by the relation

\[ Q'_{\mathrm{v}} = \frac{k'_{\varphi}}{1+\alpha_{\mathrm{c}}}\, Q . \tag{7} \]

Expressing the total flow rate \(Q\) through the flow rate \(Q_{\mathrm{h}}\) from expression (6) and substituting it into expression (7), we obtain

\[ Q'_{\mathrm{v}} = k'_{\varphi}\cdot Q_{\mathrm{h}}. \]

The actual fluid flow rate when the axis is in the vertical position, owing to the fact that constancy of velocity over section II—II does not hold, will be considerably greater than that determined from the relation

(7). To account for the nonuniformity of the distribution of the flow velocity over the cross section of the hydraulic coupling, we introduce into expression (5) the coefficient \(\beta_w\).

Then the actual flow rate of liquid through the turbine wheel will be determined by the relation

\[ Q_{\mathrm{в}}=\sqrt{1-\frac{\beta_w \cdot gh}{\omega^2}}\,Q_{\mathrm{г}}. \]

The torque transmitted in this case will be determined by the analogous relation:

\[ M_{\mathrm{в}}=\sqrt{1-\frac{\beta_w \cdot gh}{\omega^2}}\cdot M_{\mathrm{г}}. \tag{8} \]

Figure 4 shows the experimental external characteristics of the hydraulic coupling with a horizontal axis (curve 1) and a vertical axis with the pump wheel located below (curve 2). The dotted line

Fig. 4

denotes the calculated characteristic of the hydraulic coupling, obtained by recalculating the external characteristic for the horizontal axis according to relation (8); in this case the coefficient \(\beta_w\) was equal to 0.5.

In view of the fact that the external characteristic of the hydraulic coupling with a vertical axis and the pump wheel located above differs little from the analogous characteristic with a horizontal axis, in practical calculations the value of the torque transmitted by this hydraulic coupling may be taken as equal to its value for a horizontal axis.

REFERENCES

1. B. A. Gavrilenko, V. A. Minin. Hydrodynamic Couplings. Oborongiz, 1959.

2. V. S. Epifantsev, V. F. Gaidamaka. The Hydraulic Coupling as a Means of Increasing the Durability of Periodically Operating Machines. Hydraulic Machines and Hydraulic Drive. Interdepartmental Republican Scientific and Technical Collection, Kiev, 1965.

3. A. V. Murin. On the influence of the position of the rotation axis of a hydraulic coupling on its characteristics. Izv. TPI, vol. 96, no. 2, Tomsk, 1961.

4. U. Oprecht. Untersuchung von hydrodynamischen Kupplungen. MTZ, Nr. 10, 1955.