Instruments for Measuring Liquid Flow Rate in the Study of Hydraulic Systems of Metal-Cutting Machine Tools
Abstract Generated abstract
This paper addresses the need for a fast, low resistance flowmeter for experimental studies of hydraulic systems in metal cutting machine tools, where dynamic flow measurement is less well served than pressure measurement. It describes an axial impeller flowmeter with a photoelectric transducer, outlines methods for static and dynamic signal registration, and presents a theoretical calculation of impeller geometry, including relations among flow rate, rotational speed, pressure drop, torque, and relative rotational error. Tests of a prototype operating over 3 to 90 l/min showed a nearly linear calibration characteristic and a maximum relative error not exceeding 1%, supporting the instrument’s use in both static and dynamic hydraulic investigations.
Full Text
INSTRUMENTS FOR MEASURING FLUID FLOW RATE IN THE STUDY OF HYDRAULIC SYSTEMS OF METAL-CUTTING MACHINE TOOLS
Yu. Ya. Komisarenko, P. Ya. Krauinsh
(Presented by the scientific seminar of the Departments of Machine Tools and Metal Cutting and of Mechanical Engineering Technology)
When carrying out experimental studies of hydraulic systems, one most often has to deal with the measurement of the pressure of the working fluid and its flow rate.
For measuring pressure, a great variety of instruments have been developed and are produced serially, suitable both for static and for dynamic tests.
The same cannot be said of flowmeters suitable for the investigation of hydraulic systems of metal-cutting machine tools; therefore, the creation of a flowmeter meeting the present-day level of development of hydraulic drives is a very pressing task.
At present, when conducting static tests, i.e., when studying established non-oscillatory processes, the flow rate is judged by the weight or volume of the liquid filling a measuring tank over a known interval of time. For dynamic investigations, low-inertia hydraulic motors find application. A sufficiently accurate investigation of nonlinear processes occurring in hydraulic systems requires the use of an instrument that would possess sufficient speed of response, low resistance to the flow of the liquid passing through it, constancy of the flow characteristic irrespective of the rate of change of the flow rate in the system, a sufficiently large measurement range, and the ability to withstand peak pressures in the hydraulic system. The characteristics of the instrument must not vary depending on the pressure of the measured fluid stream.
A number of flowmeter designs are presented in [8], [5], [6], [7].
Very accurate and at the same time sufficiently simple are instruments in which the flow velocity is converted into the rotation of an axial impeller. The rotational speed of this impeller can be measured by photoelectric, inductive, or capacitive transducers.
The authors have developed and tested an instrument for measuring flow rate with an axial impeller and a photoelectric transducer.
The block diagram of the electronic part of the instrument is shown in Fig. 1a. The light flux coming from the light source \(S\) through condenser 1 to photoelement 3 is interrupted by the blades of impeller 2. The signal, in the form of an alternating voltage, is amplified by amplifier 4.
Static measurement of flow rate can be carried out either by recording the number of pulses over a certain interval of time, or by comparing on an electronic oscilloscope the frequency of the ...
Fig. 1. Diagram of operation of the flowmeter.
coming from amplifier 4 with a reference frequency generated by the generator; in addition, the pulse repetition frequency can be measured by means of a frequency meter (for example, Ch3-6).
When dynamic processes are being investigated, the signal from the output of amplifier 4 is fed to the loop of a loop oscillograph.
The fabricated and tested prototype of the instrument has the following data: impeller diameter \(D = 20\) mm, number of blades \(z = 2\), impeller length \(l = 10\) mm, permissible pressure \(P_0 = 150\) kg/cm\(^2\), impeller shaft diameter \(d_1 = 1\) mm, outside diameter of the thrust bearing \(d = 3\) mm, pitch of the helical spiral of the impeller \(T = 60\) mm, impeller weight \(G = 1.450\) g, diametral clearance \(\Delta D = 0.5\) mm. The impeller rotates in sliding bearings made of fluoroplastic.
The flowmeter operates stably when the flow rate varies from 3 l/min to 90 l/min.
The relationship between the rotational speed of the impeller \(\Theta\) and the flow rate \(Q\) has the form: \(\Theta = Q_n + aQ\), which is also confirmed in [5].
The quality of operation of the instrument depends to a large extent on the proper choice of the geometric dimensions of the impeller.
The outside diameter of the rotor is selected on the basis of the maximum permissible velocity of oil motion.
\[ V = 3 \div 5\ \text{m/sec}. \tag{7} \]
The impeller shaft diameter and the bearing diameter are selected for technological reasons. The diametral clearance between the impeller and the tube is recommended to be selected within the range of \(2 \div 5\%\) [7]. The length of the impeller is selected on the basis of the required overlap coefficient:
\[ p = \frac{lz}{T}. \tag{1} \]
For normal operation of the impeller it is necessary that \(0.3 < p < 1.2\).
Then
\[ l = 0.3 \div 1.2 \frac{T}{z}. \]
When measuring steady-state flow rates, there is an optimal value of the ratio of the impeller diameter to its pitch, \(\xi = \dfrac{D}{T}\), which can be determined by considering the operation of the rotor in a fluid stream.
To simplify the mathematical derivations, the following assumptions have been made:
1) the velocity field of the fluid in the tube is uniform;
2) the magnitude of the swirl of the flow \(\omega\) is small in comparison with the rotational speed of the rotor \(\Theta\).
If there were no frictional forces in the impeller supports, and also no forces resisting the rotation of the impeller in the fluid, then the following relationship would hold between the velocity of fluid flow \(V\) and the rotational speed of the impeller \(\Theta\):
\[ \Theta_0 = \frac{2\pi V}{T}. \tag{2} \]
Under real operating conditions, the impeller rotates at a lower speed \(\Theta < \Theta_0\), as a result of which the flow, passing through the impeller, acquires a certain angular rotational velocity
\[ \omega = \Theta_0 - \Theta. \tag{3} \]
And since the axial flow velocity \(V\) remains unchanged, the kinetic energy of the flow increases, which, in accordance with Bernoulli’s law, leads to a pressure drop between sections I--I and II--II (Fig. 1b)
\[ \Delta p_k=\frac{1}{16}\rho D^2\omega^2, \tag{4} \]
where \(\Delta p_k\) is the component of the pressure drop at the impeller caused by the increase in the kinetic energy of the flow;
\(\rho\) is the density of the liquid.
As a result of the frontal resistance of the impeller blade, a moment of resistance to impeller rotation appears, as well as an axial force pressing the impeller against the thrust bearing.
To determine these forces, let us isolate on the blade an element \(dr\), located at a distance \(r\) from its axis of rotation, and consider the velocity diagram for this element (Fig. 1c). The force acting on this element is:
\[ dX=\frac{1}{2}C_xF_1\rho V_{\Sigma}^{2} =\frac{1}{2}C_xdrS\rho\left(V^2+\Theta_0^2r^2\right), \tag{5} \]
where \(C_x\) is the coefficient of frontal resistance,
\(F_1\) is the midsection area of the blade element,
\(V_{\Sigma}\) is the velocity of flow around the given element,
\[ V_{\Sigma}=\sqrt{V^2+\left(\Theta_0 r\right)^2}. \]
This force is resolved into components \(dX_{oc}\) and \(dX_n\), respectively parallel and perpendicular to the axis of rotation of the impeller:
\[ dX_{oc}=dX\sin\beta =dX\frac{T}{2\pi r\sqrt{1+\dfrac{T^2}{4\pi^2r^2}}}, \tag{6} \]
\[ dX_n=dX\cos\beta =dX\frac{1}{\sqrt{1+\dfrac{T^2}{4\pi^2r^2}}}, \tag{7} \]
where \(\beta\) is the angle of rise of the impeller blade at radius \(r\); moreover,
\[ \tan\beta=\frac{T}{2\pi r}. \]
The force \(dX_{oc}\), pressing the impeller against the thrust bearing, produces a moment of resistance:
\[ dM_{tr_2}=dX_{oc}df. \]
The force \(dX_n\) also produces a moment of resistance to rotation of the impeller:
\[ dM_{tr_3}=2dX_nr. \]
The total moments of these forces are determined as:
\[ M_{tr_2}=fd\int_{d_i}^{D/2}dX_{oc}; \qquad M_{tr_3}=\int_{d/2}^{D/2}rdX_n. \]
Taking into account that \(d\) is small in comparison with \(D\), we integrate within the limits from \(0\) to \(D/2\); the expression under the radical is expanded in a Taylor series.
and take the first two terms. In carrying out the expansion it was taken into account that
\[ \frac{T^2}{4\pi^2 r^2} > 1. \]
Performing the integration and making the necessary transformations, we obtain:
\[ M_{tr}=\frac{1}{2} C_x f d \delta \rho \left\{ \frac{\Theta_0^2 \zeta^2 T^3}{8\pi^2} -\frac{\Theta_0^2 T^3 \zeta}{2\pi^2} +\pi^2 V^2 T +\Theta_0^2 T^3 \ln \left( T+\frac{\pi D}{2} \right) \right\}. \tag{8} \]
\[ \begin{aligned} M_{tr} = 2 C_x \delta \rho \Bigg\{& \frac{\Theta_0^2 T^4 \zeta^2}{32\pi^2} -\frac{\Theta_0^2 T^3 \zeta}{8\pi^3} +\frac{\Theta_0^2 T^4}{4\pi^4} \ln \left( T+\frac{\pi D}{2} \right) +\frac{\Theta_0^2 \zeta^4 T^4}{64} \\ &-\frac{\Theta_0^2 T^4 \zeta^3}{24\pi} +\frac{\Theta_0^2 T^4 \zeta^2}{8\pi^2} +\frac{\Theta_0^2 T^4 \zeta}{2\pi^3} +\frac{\Theta_0^2 T^4}{\pi^4} \ln \left( T+\frac{\pi D}{2} \right) \Bigg\}. \end{aligned} \tag{9} \]
The moment of resistance to rotation of the spinner, \(M_{tr_2}\), \(M_{tr_3}\), and \(M_k\), also causes a pressure drop across the spinner:
\[ \Delta P_0=\frac{2\pi}{FT}\left(M_{tr_2}+M_{tr_3}+M_k\right), \tag{10} \]
where
\[ F=\frac{\pi D^2}{4}, \]
\[ M_k=\Delta p_k \frac{\pi D^2}{4} f \frac{d}{2}, \]
where \(M_k\) is the moment of resistance caused by the pressure drop \(\Delta p_k\). The total moment of resistance to rotation of the spinner will then be:
\[ M_{tr}=F f \frac{d}{2}\left(\Delta p_0+\Delta p_k\right). \tag{11} \]
Solving the system of equations (10) and (11), we have:
\[ M_{tr}\left(1-\frac{d}{2}\frac{2\pi f}{T}\right) = \frac{d}{2}Ff \left[ \Delta p_k+\frac{2\pi}{FT}\left(M_{tr_2}+M_{tr_3}\right) \right]. \tag{12} \]
The twisting of the flow as it passes through the impeller leads to the appearance on the latter of a torque, which can be determined from the law of conservation of momentum [9]:
\[ M_{dv}\cdot \Delta \tau=\Delta(I\omega), \tag{13} \]
where \(I\) is the moment of inertia of the column of liquid;
\(m\) is the mass of liquid that has passed through the impeller during the time interval \(\Delta \tau\).
\[ m=\frac{\pi D^3}{4} V\Delta \tau \rho . \tag{14} \]
Expression (13), taking into account (14) and (2), can be transformed as:
\[ M_{dv}=\frac{\rho}{64}D^4\Theta_0 T\omega. \tag{15} \]
Taking into account that in statics \(M_{dv}=M_{tr}\), we substitute equation (15) into expression (12).
\[ \frac{\rho}{64}D^4\Theta_0 T\omega \left(1-\frac{df\pi}{T}\right) = \frac{d}{2}Ff \left[ \Delta p_k+\frac{2\pi}{FT}\left(M_{tr_2}+M_{tr_3}\right) \right]. \tag{16} \]
Substituting, instead of \(M_{mp_2}\), \(M_{mp_3}\), \(\Delta p_k\), their values from equations (8), (9), (4), we obtain the expression relating the relative error \(\dfrac{\omega}{\Theta_0}\) of rotation of the axial impeller to its geometric dimensions:
\[ \frac{\rho}{64}D^4T\omega\left(1-\frac{df\pi}{T}\right) = df\pi\left\{ \frac{\omega^2}{128}\rho^4T^4 + \frac{C_x\delta\rho}{2T} \left[ \frac{fd\Theta_0^2\zeta^2T^3}{8\pi} - \right.\right. \]
Fig. 2. Curves for calculating the optimal value \(\dfrac{D}{T}\)
\[ \left. \left. -\frac{fd\Theta_0^2T^3\zeta}{2\pi^2} + \frac{5fd\Theta_0^2T^3}{4\pi^3} \ln\left(T+\frac{\pi D}{2}\right) + \frac{\Theta_0^2T^4\zeta^2}{8\pi^2} - \frac{\Theta_0^2T^4\zeta}{2\pi^3} + \right. \right. \]
\[ \left. \left. + \frac{2\Theta_0^2T^4}{\pi^4} \ln\left(T+\frac{\pi D}{2}\right) + \frac{\Theta_0^2\zeta^4T^4}{16} - \frac{\Theta_0^2T^4\zeta^3}{6\pi} + \frac{\Theta_0^2T^4\zeta^2}{2\pi^2} + \frac{2\Theta_0^2T^4\zeta}{\pi^3} \right] \right\}. \tag{17} \]
Taking into account that \(d \approx 0.2D;\ f = 0.05\), collecting like terms and discarding terms of the second order of smallness, we solve (17) with respect to \(\dfrac{\omega}{\Theta_0}\):
\[ \frac{\omega}{\Theta_0} = 0.64 \pi \frac{C_x \delta}{D} \left[ 0.0629 \zeta^2 - \frac{2}{6\pi}\zeta + \frac{3}{2\pi^3}\zeta^{-1} + \frac{2}{\pi^4}\zeta^{-2} \ln\left(T+\frac{\pi D}{2}\right) + \frac{0.62}{\pi^2} \right]. \tag{18} \]
Using expressions (17) or (18), for the selected diameter of the impeller \(D\), one can choose the optimal value of the impeller pitch. This problem can also be solved with the aid of Fig. 2, which shows dependence (18).
Thus, an axial impeller flowmeter with a photoelectric transducer is a sufficiently accurate and at the same time simple instrument suitable for the study of metal-cutting machine tools.
The instrument is suitable both for static and for dynamic tests.
Tests of the manufactured instrument showed that its calibration curve \(f = f(Q)\) is approximated by the equation \(f = Q_{\mathrm{n}} + aQ\); the maximum relative error does not exceed 1%. The geometric dimensions can be determined according to the method presented in the article.
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