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Research on Extended Transfer Matrix of Centrifugal Pumps
Abstract: Based on the research on the transmission characteristics of pure hydraulic system, the torque parameters of rotating shaft are introduced to study the transmission characteristics of centrifugal pumps. Focusing on the pump shaft torque excitation response, the transfer matrix of the numerical expression and the evaluation of the system disturbance sources, discusses the pump dynamic transmission characteristics of the basic characteristics. The results show that it is feasible to apply the basic experimental method, data processing and the mode identification of transmission coefficient. The response of pump shaft torque shows other influencing factors but still shows the symmetrical distribution similar to water pressure. The extended transmission The matrix shows that the transmission characteristics of torque and hydraulic system parameters have different characteristics, and the related transmission coefficients show a certain linear relationship. The analysis of the disturbance sources verifies the correctness of the transmission characteristics determination method from the application point of view. Keywords: Centrifugal Pump System Extended Transfer Matrix The research results of associated pulsating propulsion (POGO) phenomenon in aerospace engineering have greatly promoted the transmission matrix of hydraulic machinery. However, the research of hydraulic machinery transfer matrix is ​​an in-depth study on the dynamic stability of hydraulic machinery Especially in the frequency domain). In the research work, the following assumptions are commonly used in the calculation of dynamic fluctuations: Uniform flow and positive pressure assumptions, neglecting the assumption of flow velocity (because the flow velocity is much less than the transmission velocity) and the plane wave assumption (because the duct cross-section size is smaller than the wave length More) and the assumption of linear transmission [1]. In addition, the acoustic and excitation techniques used also provide technical assurance of fast and accurate experimentation. In general, much of the past research has been conducted from transfer matrices that contain flow pressure and flow fluctuation vectors or the like. This method grasps the important aspects of the problem, and obtains the research results that reflect the basic dynamic transmission characteristics of hydraulic machinery. It is a reasonable and feasible choice in the initial stage of the research work, and can be called the basic matrix. As research progresses, it is not only necessary but also feasible to extend the above transfer matrix and introduce other parameters related to dynamic characteristics, such as pump shaft torque, rotational speed, and water guide vane opening [2]. Of course, the resulting experimental research will be complicated. Therefore, the wider the considerations are, the more mechanical properties are known and the more in-depth research is conducted, but a step-by-step approach to research should be adopted based on the problem objectives, test facilities and data processing conditions. This article presents an experimental study on the dynamic transmission characteristics of pump shaft torque carried out on a specially designed EPFL_IMHEF Institute of Hydraulic Engineering and Mechanics of the Federal Institute of Engineering in Lausanne, Switzerland, focusing on the fundamental characteristics of torque-actuated response, the transfer matrix The reason for the expression and internal disturbance sources. Among them, discussing the correlation between pump shaft torque fluctuation and hydraulic system parameter fluctuation, and determining their characteristic transmission relations are the basic contents. 1 Experimental setup and data 1.1 Experimental setup Figure 1 shows the PF4 bench for the EPFL_IMHEF study. The test machine is a Francis pump turbine. The specific speed of the pump is nq = 39 (power ratio speed ns = 3.13nq), the diameter of the impeller is 152mm and the number of blades is seven. Pump drive side of a 30kW DC motor, the motor speed control by the governor to maintain a substantially constant rotation angular velocity. When the pump speed is 2000r / min, the optimal conditions of flow and specific energy were Q = 9.5L / s and E = 85J / kg. The hydraulic pipeline of the experimental device includes two test pressure steel pipes, the inner diameter of the high pressure side pipe is 100mm, and the low pressure side is 150mm. Three pressure sensors are respectively installed on the pipe walls on both sides of the high pressure and low pressure. Two water acoustic filters are respectively connected to the outside of the test pressure pipe, which can guarantee zero impedance boundary conditions over a wide frequency band and are relatively separated from the rest of the system by the rubber expansion ring. Valve actuators were installed in the pressure pipe 6 measuring points 500mm and 1 outside the measuring point 1000mm, the experimental operation using one of them. The pressure sensor is a quartz piezoelectric type whose signal gain can be adjusted according to the sensor sensitivity to give a voltage output corresponding to the fluctuation of the water pressure. Rotor torque fluctuation of pump turbine is measured and measured by the strain gauge calibrated on the pump shaft. 1.2 Important Concepts Design appropriate experimental methods to ensure the correct processing and application of experimental data, we must accurately grasp some of the basic concepts. Here are special ways to identify the excitation shock and transmission of the wave method. 1.2.1 Excitation Oscillation Figure 2 is a schematic illustration of a rotary valve actuator that creates perturbations on the hydraulic system of an experimental setup. The rotary valve is driven by an induction motor connected to a frequency converter, which generates an excitation signal to the hydraulic system when its cock periodically blocks the passing jet at a given frequency of change, with the air slot on the water line Play a buffer role. The exciter is small, easy to install and has negligible reaction to the system. Fig. 3 Water pressure response during one stimulus cycle Fig. 3 shows the frequency stimulated frequency of the channel 4 and the corresponding amplitude of the signal change during one stimulus cycle. The abscissa shows the excitation length of 120s, expressed as time t. The ordinate shows the frequency change (denoted by f in Hz) of the channel 4 pressure surge and the change in amplitude (ΔP4, unit: 102 Pa). The test data measured conditions: rotor speed 2000r / min, flow 11.4L / s, specific energy 80.2J / kg. 1.2.2 Identification of transmission wave Fluctuation of flow in a hydraulic pipe Parameters are obtained indirectly by using the characteristic acoustic impedance of the pipe, which means that an important acoustic parameter such as wave velocity is to be used. This parameter is calculated using the oscillatory pattern of a set of signals taken from the pressure signals of three isometric pressure sensors on a homogeneous pipe in the device [3]. If p1, p2 and p3 are three sensor pressure fluctuation signals and L is the distance between adjacent pressure sensors, the wave velocity a of the driving wave can be calculated by the following transfer function equation: This equation is assumed to be a plane wave. In the above identification process Fluctuations handled less than 250Hz, consistent with the assumption. 1.3 Data Processing Computer data acquisition equipment to obtain the test data recorded in the time domain is the original data. If the sampling frequency is 1024 Hz, the excitation perimeter is taken as 120 s and 240 analysis windows are defined on it, each analysis window lasts 0.5 s and occupies 512 sample data. If the Hanning window-weighted Fourier transform is used to process each analysis window, the transfer function of the discrete Fourier transform coefficients in the frequency domain of the water pressure data in the time domain for the general hydraulic system pressure fluctuations can be used to organize the wave velocity of the transmission wave. And indirect calculation of traffic complex coefficient qx (n). Data processing to reduce the calculation error is also used as a test point for the signal reference channel, usually selected at the opposite side of the excitation test machine side. The Fourier transform of the time domain moment data records is the same as the water pressure signal. 2 Incentive Response of Moment Torque fluctuation of drive shaft and pressure fluctuation of hydraulic system are interlinked by pump impeller. As the two signals from different physical media, when the experimental operation in the face of the same incentive disturbance, the two responses are different, resulting in different characteristics. Figure 4 shows the response recorded under the same excitation conditions as in Figure 3. The ordinate shows both the change in frequency f and the change in amplitude (expressed as ΔT in Nm / 40) of the shaft torque. As can be seen from the two plots, the magnitude of this torque variation also shows a nearly symmetrical distribution trend at the corresponding frequency location, but the stimulated frequency distribution is no longer as consistent with the original excitation frequency as the hydraulic pressure (the moment is not directly stimulated by the water pressure Influence), but also related to other factors. The factor discussed here is actually the pump shaft rotation frequency during the experiment. The figure shows a concentrated frequency band of about 33 Hz, which is exactly the rotational characteristic frequency of the experimental apparatus. If you look at the power spectrum of an on-line data logger, you will see that the torque signal has a triangular distribution like water pressure, but at a much weaker amplitude than the response at the rotation frequency. If you choose to use high-pressure or low pressure side of the exciter position in the test, the corresponding torque response will be mainly in the range of differences. The use of low-pressure side of the actuator, the response amplitude is greater than the use of high-pressure side of the situation, it is because the use of the low-pressure side of the high-pressure side of the water intake more affect the suction head. Again, this phenomenon reminds us that the response of the moment of inertia is closely linked to the energy conversion characteristics of the flow inside the pump impeller. 3 Transfer matrix 3.1 Matrix expression Under the assumption of linear propagation and no external force, parameters of pump shaft torque fluctuation are now embedded in the hydrodynamic wave vector (p, q). Then, the transfer characteristic from one vector to another can be expressed by the following matrix equation: (3) Where, p, q, T are the parameters of water pressure, flow rate and pump shaft torque fluctuation, respectively; The superscripts "4 and 3" indicate that the signals are taken from channels 4 and 3 (see Figure 1), that is, the outlet or inlet of the pump, respectively; [M] is the extended transfer matrix. In equation (3), p, q, T, mi, j are complex coefficients of frequency. Matrix coefficients are named in their respective physical meanings as follows: m11, m22 are water pressure and flow transfer coefficient; m12 is hydraulic impedance coefficient; m21 is hydraulic admittance coefficient; and m31 and m32 are moment admittance coefficients. The coefficients m11, m12, m21, m22 describe the characteristics of the hydraulic system, and the coefficients m31, m32 reveal the energy conversion information between the flow and the mechanical impeller. The coefficients m11 and m22 are dimensionless and the coefficients m12, m21, m31 and m32 can be processed into the dimensionless forms by means of the characteristic impedance on the high-pressure side z4 and the water flow v in the area of ​​the impeller of the pump, ie m12 / z4, m21z4, m31 / ν, m32 / (z4ν). The above complex coefficient matrix equation corresponds to 3 linear equations of 6 unknowns. If there are two sets of independent fluctuation data, for example, separately select the excitation response records on both sides of the pump high and low pressure, then all mi and j coefficients can be calculated according to the linear equation Solve. 3.2 Experimental results Figure 5 shows an experimental calculation results, steady state conditions of the pump test: specific energy E = 80.2J / kg, flow Q = 11.4L / s, speed 2000r / min. The abscissa of the graph is the frequency f, and the ordinates are respectively the complex function matrix coefficients m11, m12, m21, m22, m31 and m32, which are unified into a dimensionless form. The solid line indicates the real part of the complex coefficient and the dotted line indicates the imaginary part. It can be observed that the coefficients m11, m12, m21, m22 show the correspondence with the corresponding elements of the former, not much discussion of them, compared with the basic transfer matrix with only two vectors of water pressure and flow rate. Only torque-related coefficients m31 and m32 are discussed here. Obviously, the m31 and m32 coefficients are much smaller in magnitude than the four coefficients of the hydraulic system and have a small amplitude fluctuation over a large frequency range other than integral multiples of the rotational frequency. This is consistent with the discussion of the torque-inducing response described above. If one looks at a range of frequencies, for example at lower frequencies, and disregards the perturbation of the frequency of rotation, it can be seen that the two coefficients, m31 and m32, are basically linear with respect to frequency. The real part of m31 reveals the inherent relationship between hydraulic energy and moment, while the imaginary part of m32 expresses the inertial effect of water flow to moment. 4 System Disturbance Source Analysis Equation (3) can also be referred to as a homogeneous transmission equation, so it is applied that there is a condition where the internal disturbance sources in the experimental setup are negligible compared to the intensity of external forced excitation. In other words, the equation can also contain perturbation source vectors. If the acoustic transmission of water pump can be expressed by the reflection sources (pS4, qS4) and (pS3, qS3) on both sides of the tested pump, the disturbance of torque fluctuation can be regarded as a disturbance acting on the pump shaft TS, then the above transfer equation is: The location and properties of the disturbance sources are completely random, and the disturbance sources within the system can be analyzed by using the above homogeneous transmission equations. Figure 6 shows the different frequency response of the torque ripple. The solid line corresponds to the external excitation and the dotted line corresponds to the external excitation. Both are theoretical calculations under the experimental conditions described above. In the figure, the abscissa is the frequency f (unit: Hz) and the ordinate is the torque fluctuation amplitude ΔT (unit: Nm) corresponding to different excitation conditions. Figure 7 shows the internal perturbation of the torque as a function of frequency. It is easy to see that all pulses of large magnitude are related to an integer multiple of the shaft rotation frequency, for example at 33 Hz or 7 x 33 Hz (note that the number of pump impeller blades is 7 ). Graphically, the disturbance source TS is a replica of the torque frequency response without external excitation. 5 Conclusion This paper is based on the pressure, flow and hydraulic characteristics of the basic transfer matrix as a starting point, the introduction of pump torque fluctuation parameters to expand the basic transfer matrix experimental study showed that the use of the basic experimental methods, data processing and It is still feasible to pass the pattern recognition of the coefficients. The excitation response of the observed moment shows that although the frequency of fluctuation is clearly related to the other influencing factors, it does exhibit an almost symmetrical distribution like water pressure. The calculated spread transfer matrix shows that there are different characteristics of the torque and the parameter transfer characteristics along the hydraulic pipe. The two factors on torque tend to exhibit a linear relationship over at least a range of frequencies. Disturbance source analysis is not only a simple application of this extended transfer matrix, but also a verification of the correctness of the relevant transmission characteristics determination method. Acknowledgment: This study was completed at the Institute of Hydrodynamics and Hydrodynamics at the Federal Institute of Engineering in Lausanne, Switzerland (EPFL_IMHEF) and would like to thank the co-workers there.
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