Determination of the Radius of the Vortex Center of the Circulating Flow in a Fluid Coupling
Abstract Generated abstract
The paper addresses the determination of the vortex center radius in the circulating flow of a fluid coupling, a parameter needed for designing special hydrodynamic couplings whose structural elements depend on this flow position. It argues that estimates based on constant meridional velocity can deviate from experiment, and derives relations that account for variation of meridional velocity through continuity and energy equations under stream theory assumptions. Because the resulting differential equation is impractical for direct design use, the paper proposes a successive approximation method using an equivalent stream and a velocity variation coefficient. A numerical example and comparisons with experimental results under varied operating conditions indicate that the method can be used for engineering calculation of the vortex zone radius.
Full Text
Determination of the Radius of the Vortex Center of the Circulating Flow in a Fluid Coupling
Yu. I. Kapustin
(Presented by the scientific seminar of the Department of Applied Mechanics)
In the calculation and design of hydrodynamic couplings that have certain structural elements connected, to one degree or another, with the vortex center of the flow, it is necessary to know its position in the cavity of the fluid coupling, which is usually specified by the magnitude of its radius. Such fluid couplings include safety fluid couplings [1], which have a vertical axis of rotation, traction fluid couplings with a horizontal axis of rotation [2], and certain other special designs of fluid couplings.
Determining the radius of the vortex center of the flow from the condition of constancy of the meridional velocity of the flow over the cross section of the fluid coupling gives results that differ somewhat from experimental ones. This may lead to incorrect placement of certain elements of the fluid coupling and to obtaining an unsatisfactory external characteristic. The discrepancy between the calculated and experimental values of the radius of the vortex center of the flow may be explained by the fact that the flow in the fluid coupling differs substantially from one of uniform velocity [3]. The flow velocity at the pump outlet exceeds the flow velocity at the inlet.
Let us determine this radius, taking into account the variation in the magnitude of the meridional velocity in the working cavity of the fluid coupling. In doing so, we shall make the following assumptions:
1) for the flow in the working cavity, we assume the validity of stream theory;
2) the number of blades of the working wheels is taken to be infinitely large;
3) the blade thickness is neglected.
Introduce the notation (Fig. 1):
\(x\) — radius at the inlet of an elementary stream tube,
\(y\) — radius at the outlet of the same stream tube,
\(dx, dy\) — width of the elementary stream tube, respectively, at the outlet from and inlet to the turbine wheel,
\(W_x, W_y\) — meridional velocity of the flow in the elementary stream tube at the inlet and at the outlet.
In the calculation we shall consider the following quantities known: \(R_a\) — the active radius and \(R_0\) — the initial radius of the fluid coupling, \(\omega_n\) — the angular velocity of the pump wheel, \(\omega_t\) — the angular velocity of the turbine wheel, as well as the law of distribution of the meridional velocities of the flow in the inlet section of the turbine wheel, i.e., the dependence
\[ W_x = f(x). \]
To determine the radius of the vortex zone in general form, let us write the continuity equation for the selected elementary stream tube:
\[ W_y \cdot F_y = W_x \cdot F_x, \tag{1} \]
Fig. 1
where \(F_y\) and \(F_x\) are the flow areas in the elementary stream tube, respectively at the outlet and inlet of the turbine wheel, which are equal to:
\[ \begin{aligned} F_y &= \pi (y+dy)^2 - \pi y^2,\\ F_x &= \pi x^2 - \pi (x-dx)^2. \end{aligned} \tag{2} \]
Substituting expressions (2) into (1), carrying out the obvious simplifications and neglecting \(dy^2\) and \(dx^2\) as quantities of second order of smallness in comparison with the remaining quantities, we obtain
\[ W_y \cdot y \cdot dy = W_x \cdot x \cdot dx. \tag{3} \]
Solving the differential equation obtained under the boundary conditions: a) \(x=R_a,\ y=R_0\) and b) \(x=R_{\mathrm{ц}},\ y=R_{\mathrm{ц}}\), one can determine the radius of the vortex zone of the flow. For its solvability it is necessary to know the magnitude of the change in meridional velocity at the outlet from the turbine wheel, i.e., the dependence \(W_y=f(W)\).
This dependence can be found as follows. From the equation of total energy in relative motion, taking account of losses in the working wheels, we have:
for the pump wheel
\[ \frac{P_{2x}}{\gamma} + \frac{W_{2x}^{2}-U_{2x}^{2}}{2g} = \frac{P_{1y}}{\gamma} + \frac{W_{1y}^{2}-U_{1y}^{2}}{2g} -\Delta h_{\mathrm{н}}, \tag{4} \]
for the turbine wheel
\[ \frac{P_{3x}}{\gamma} + \frac{W_{3x}^{2}-U_{3x}^{2}}{2g} = \frac{P_{4y}}{\gamma} + \frac{W_{4y}^{2}-U_{4y}^{2}}{2g} +\Delta h_{\mathrm{т}}. \tag{5} \]
Subtracting one from the other, we obtain
\[ = \Delta h_{\mathrm{т}}+\Delta h_{\mathrm{н}} = \frac{P_{3x}-P_{4y}}{\gamma} - \frac{P_{2x}-P_{1y}}{\gamma} + \frac{W_{3x}^{2}-W_{4y}^{2}}{2g} - \]
\[ - \frac{W_{2x}^{2}-W_{1y}^{2}}{2g} + \frac{U_{2x}^{2}-U_{1y}^{2}}{2g} - \frac{U_{3x}^{2}-U_{4y}^{2}}{2g}. \]
In view of the fact that, in steady operation, the change in the magnitude of the meridional velocity in the pump wheel is equal to the change in the magnitude of the meridional velocity in the turbine wheel, the equality holds
\[ W_{3x}^{2}-W_{4y}^{2}=W_{2x}^{2}-W_{1y}^{2}. \]
Since the length of the stream filament in the working wheels is the same, the mean meridional velocities are equal; therefore the pressure differences in the indicated sections of the elementary stream filament must also be equal:
\[ \frac{P_{2x}-P_{1y}}{\gamma}=\frac{P_{3x}-P_{4y}}{\gamma}. \]
This equality is valid for an infinitely large number of blades, since in this case the additional losses associated with the rotation of the working wheels are negligibly small [4].
Then
\[ \Delta h_n+\Delta h_t= \frac{U_{2x}^{2}-U_{1y}^{2}}{2g} - \frac{U_{3x}^{2}-U_{4y}^{2}}{2g}. \]
Substituting
\[ \begin{aligned} U_{1y}&=\omega_n\cdot y, \qquad &U_{2x}&=\omega_n\cdot x,\\ U_{3x}&=\omega_t\cdot x, \qquad &U_{4y}&=\omega_t\cdot y, \end{aligned} \tag{6} \]
we obtain
\[ \Delta h_n+\Delta h_t=\frac{\omega_e^{2}(x^{2}-y^{2})}{2g}, \]
where
\[ \omega_e=\sqrt{\omega_n^{2}-\omega_t^{2}}. \]
Since, by assumption, \(\Delta h_n=\Delta h_t\), it follows that
\[ \Delta h_n=\frac{\omega_e^{2}(x^{2}-y^{2})}{4g}. \tag{7} \]
Substituting (7) into (4), replacing \(U_{1y}\) and \(U_{2x}\) by their expressions (6), and \(P_{1y}/\gamma\) and \(P_{2x}/\gamma\) by their expressions
\[ \frac{P_{1y}}{\gamma}=\frac{\omega_t^{2}y^{2}}{2g}, \qquad \frac{P_{2x}}{\gamma}=\frac{\omega_t^{2}x^{2}}{2g}, \]
and canceling by \(\dfrac{1}{2g}\), we obtain
\[ \omega_t^{2}x^{2}-W_{2x}^{2}-\omega_n^{2}x^{2} = \omega_t^{2}y^{2}+W_{1y}^{2}-\omega_n^{2}y^{2} -\frac{1}{2}\omega_e^{2}(x^{2}-y^{2}). \]
Carrying out the obvious simplifications, we shall have
\[ W_{1y}^{2}=W_{2x}^{2}-\frac{\omega_e^{2}}{2}(x^{2}-y^{2}) \]
or
\[ W_{1y}= \sqrt{ W_{2x}^{2}-\frac{\omega_e^{2}}{2}(x^{2}-y^{2}) }. \tag{8} \]
Substitute expression (8) into (3):
\[ \sqrt{ W_{2x}^{2}-\frac{\omega_e^{2}}{2}(x^{2}-y^{2}) }\cdot y\cdot dy = W_{2x}\cdot x\cdot dx. \tag{9} \]
The nature of the distribution of meridional velocities at the inlet to the turbine wheel has a distinctive form and may be approximated by the following relation:
\[ W_{2x}=a W_{\mathrm{cp}}\left(1-\frac{x-R_{\mathrm{ц}}}{R_a-R_{\mathrm{ц}}}\right)^{\frac{1}{m}}\cdot \frac{x-R_{\mathrm{ц}}}{R_a-R_{\mathrm{ц}}}, \tag{10} \]
Substituting \(W_{2x}\) from (10) into (9), we obtain a differential equation that can be solved only by approximate methods [5], and therefore cannot be used for practical calculations.
Fig. 2
For practical purposes this same problem can be solved by a simpler method—the method of successive approximations. To this end, let us concentrate the entire flow into an equivalent stream. The radii of its position in the first approximation are found from the condition that the meridional velocity of the flow is constant over the entire section of the fluid coupling (Fig. 2):
\[ r'_{11}=\sqrt{\frac{R_{\mathrm{ц}}^{\prime 2}+R_0^2}{2}}, \qquad r'_{21}=\sqrt{\frac{R_{\mathrm{ц}}^{\prime 2}+R_a^2}{2}}, \tag{11} \]
where
\[ R'_{\mathrm{ц}}=\sqrt{\frac{R_a^2+R_0^2}{2}}. \]
The flow velocity in the equivalent stream at the inlet to the pump wheel is determined from relation (8), which in this case takes the form:
\[ W_1^2=W_2^2-\frac{\omega_{\mathrm{э}}^2}{2}\left(r_{21}^{\prime 2}-r_{11}^{\prime 2}\right). \]
Here \(W_1\) and \(W_2\) are the mean flow velocities in sections I—I and II—II.
Let us carry out some transformations:
\[ (W_2-W_1)(W_2+W_1)=\frac{\omega_{\mathrm{э}}^2}{2}\left(r_{21}^{\prime 2}-r_{11}^{\prime 2}\right). \tag{12} \]
We introduce the coefficient of variation of the meridional velocity by the relation:
\[ \kappa = \frac{W_2}{W_1}. \tag{13} \]
Denote
\[ \frac{W_1 + W_2}{2} = W_{cp}. \tag{14} \]
Substituting expressions (13) and (14) into equation (12) and carrying out the obvious transformations, we obtain
\[ \kappa - 1 = \frac{\omega_{\text{э}}^{2}\left(r_{21}^{\prime 2}-r_{11}^{\prime 2}\right)} {4W_{cp}\cdot W_1}. \tag{15} \]
Since \(W_2=\kappa W_1\) and \(2W_{cp}=W_1+W_2\), replacing \(W_2\) by its expression in terms of \(W_1\), we obtain
\[ W_1=\frac{2W_{cp}}{1+\kappa}. \tag{16} \]
Substituting expression (16) into equation (15), we rewrite it in the form
\[ \kappa - 1 = \frac{(1+\kappa)\cdot \omega_{\text{э}}^{2}\left(r_{21}^{\prime 2}-r_{11}^{\prime 2}\right)} {8W_{cp}^{2}}. \]
Solving with respect to \(\kappa\), we obtain
\[ \kappa = 1+ \frac{2\omega_{\text{э}}^{2}\left(r_{21}^{\prime 2}-r_{12}^{\prime 2}\right)} {8W_{cp}^{2}-\omega_{\text{э}}^{2}\left(r_{21}^{\prime 2}-r_{11}^{\prime 2}\right)}. \]
Dividing the numerator and denominator of the fraction by \(\omega_{\text{э}}^{2}\), we obtain:
\[ \kappa = 1+ \frac{\left(2r_{21}^{\prime 2}-r_{11}^{\prime 2}\right)} {8\left(\dfrac{W_{cp}}{\omega_{\text{э}}}\right)^2-\left(r_{21}^{\prime 2}-r_{11}^{\prime 2}\right)}. \tag{17} \]
Having calculated the coefficient of variation of the meridional velocity \(\kappa\), we find the value of the radius of the vortex zone in the second approximation.
\[ R_{ц}^{\prime\prime} = \sqrt{\frac{R_{0}^{2}+\kappa R_{a}^{2}}{1+\kappa}}. \]
The radii of the equivalent jet in this approximation are determined from relations (11), replacing \(\kappa\) in them by \(R_{ц}\). Repeating calculations analogous to the first approximation, we find the radius of the vortex zone for the third approximation, and so on. If the difference between the values of the radii in the subsequent and previous approximations becomes small enough to be neglected in practical calculations, then the approximate calculation may be ended at this point and the value of the radius of the vortex zone according to the last approximation may be accepted.
Example of calculation: let us calculate the value of the radius of the vortex zone of the flow for a fluid coupling having the following data: \(R_a=10.5\ \text{cm}\), \(R_0=3.7\ \text{cm}\), \(n_{\text{н}}=1070\ \text{rpm}\), with \(S=13\%\), \(M=1.7\ \text{kgm}\). Spindle oil is used as the filling fluid.
First approximation:
- Determine the radius of the vortex zone:
\[ R_{ц}^{\prime} = \sqrt{\frac{R_a^{2}+R_0^{2}}{2}} = 7.85\ \text{cm}. \]
- Determine the radii of the equivalent jet:
\[ r'_{21}=\sqrt{\frac{R_a^2+R_u^2}{2}}=9.25\ \text{cm}. \]
\[ r'_{11}=\sqrt{\frac{R_0^2+R_u'^2}{2}}=6.1\ \text{cm}. \]
- Find the area of the outlet flow from the pump wheel:
\[ F'_{2n}=\pi(R_a^2-R_u'^2)-\delta\cdot n(R_a-R_u')=137.5\ \text{cm}^2. \]
- Calculate the mean meridional velocity:
\[ W'_{\mathrm{av}}=\frac{M\cdot g}{k_f\cdot F'_{2n}\cdot (r_{21}'^{\,2}-r_{11}'^{\,2})\cdot \omega_n\cdot \gamma}=5.5\ \text{m/sec}. \]
- Determine the angular velocities of the pump and turbine wheels:
\[ \omega_n=\frac{\pi n_n}{30}=112\ \text{sec}^{-1}, \qquad \omega_T=\frac{100-S}{100}\cdot \omega_n=98\ \text{sec}^{-1}, \]
\[ \omega_e=\sqrt{\omega_n^2-\omega_T^2}=55.7\ \text{sec}^{-1}. \]
- Determine the coefficient of change of the meridional velocity:
\[ K'=1+\frac{2(r_{21}'^{\,2}-r_{11}'^{\,2})}{8\left(\dfrac{W'_{\mathrm{av}}}{\omega_e}\right)^2-(r_{21}'^{\,2}-r_{11}'^{\,2})}=1.26. \]
- Determine the radius of the vortex zone of the flow for the second approximation:
\[ R_u''=\sqrt{\frac{R_0^2+\kappa'R_a^2}{1+\kappa'}}=8.23\ \text{cm}. \]
The results of the calculations in the subsequent approximations are summarized in the table:
| No. of approximation | \(R_u\), cm | \(r_{1n}\), cm | \(r_{2n}\), cm | \(F_{2n}\), cm² | \(W_{\mathrm{av}}\), cm/sec | \(K\) | \(R_u\), cm |
|---|---|---|---|---|---|---|---|
| 1 | 7.85 | 6.10 | 9.25 | 137.5 | 5.50 | 1.26 | 8.23 |
| 2 | 8.23 | 6.40 | 9.40 | 121.0 | 6.03 | 1.10 | 8.03 |
| 3 | 8.03 | 6.25 | 9.30 | 131.0 | 5.80 | 1.18 | 8.10 |
| 4 | 8.10 | 6.30 | 9.35 | 126.5 | 5.90 | 1.13 | 8.06 |
Thus, within the accuracy of the calculations, the 3rd and 4th approximations gave close results, which may be taken as the value of the radius of the vortex zone. The actual radius, determined experimentally, is equal to 81.0 cm.
Comparison of the calculated and experimental values of the radius of the vortex zone for other slip conditions, different rotational speeds of the pump wheel, and different filling fluids showed that
the proposed method can be used to determine the magnitude of the radius of the vortex zone of the flow in the calculation and design of special hydrodynamic couplings.
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