Motohiko NOHMI*
Satoshi YAMAZAKI**
Shusaku KAGAWA*
Byungjin AN*
Donghyuk KANG***
Kazuhiko YOKOTA****
*
Technologies, R&D Division
**
Fluid Machinery & Systems Company
***
Aoyama Gakuin University (Currently studying in Saitama University)
****
Aoyama Gakuin University
The cavitation surge of a hydraulic pump is characterized by the interaction of unsteady cavitation inside the pump and the piping system surrounding the pump. For computation of the cavitation surge, it is mandatory to consider piping system dynamics. There are two methods for calculating piping system dynamics with low computing costs. One is lumped parameter system calculation that assumes the incompressibility of fluid. The other is a one dimensional distributed parameter system calculation that takes into consideration wave propagation around the piping system. Ohashi and Akimoto individually suggest criteria for choosing calculation methods for the objective piping system. This study evaluates Ohashi and Akimoto’s criteria. In preliminary research, the dynamics of simple hydraulic oscillation system without cavitation are calculated using both the lumped parameter system method and the distributed parameter system method. From the quantitative comparison of the results of two methods, it was clear that Ohashi and Akimoto’s criteria are reasonable, after the residue of initial condition is totally dissipated by the pipe friction.
Keywords: Surging, Cavitation Surge, Lumped Parameter System, Distributed Parameter System, Method of Characteristics, Resonator
It is widely known that when cavitation is firstproduced and enlarges with the reduction of NPSH, unstable phenomena such as rotating cavitation and cavitation surge occur. Surging is a self-excited vibration phenomenon that has been widely observed and researched in fluid machineries handling gas such as compressors. The continuance of the vibration require the capacitance elements (spring effect) in the system. In gas machines, a gas with compressibility plays this role. In pumps, since liquids exhibit small compressibility, surging hardly occurs compared to gas machines.However, if there are capacitance elements such as tanks or, if there is cavitation occurring inside a pump, the cavitation itself becomes a capacitance element and will enter into a vibration state. The latter case is called the cavitation surge of pumps. When a cavitation surge generates, in principle it may have a wide frequency range, and human ears can easily detect low-frequency fluctuations of 10 Hz or below. It generates a characteristic noise called chugging noise. Controlling this noise is an important issue in pump design1). Surging is related not only to the main components in the fluid machinery system, such as pumps and compressors, but also to all elements that comprise the liquid feeding channels, including surrounding piping, valves, tanks, etc. Therefore the behavior of the whole system, including these elements, must be analyzed.
Lately, it has been reported that CFD is being applied to these analyses. However, two problems arise: first, the low frequency of the phenomenon compared to the rotation speed of impellers makes the calculation time long; second, piping and others elements contained in the analysis increase the calculating area making the analytical grid scale enormous. To solve these problems, proposals2)-6) have been made to link two types of analysis: a three-dimensional fluid calculation applied to fluid machinery vicinities, and, for the other elements, either a simplified one-dimensional distributed parameter system model calculation or a lumped parameter system fluid model calculation. The onedimensional distributed parameter system model analyzes the transmission of the pressure wave inside piping considering the compressibility of liquid.On the other hand, the lumped parameter system fluid model (the so-called rigid-body approximation) neglects the compressibility of liquid and analyzes the motion equation of a fluid column inside piping. To efficiently analyze the piping system in an adequate accuracy, either the lumped parameter system calculation or distributed parameter system calculation must be properly selected. In relation to this, the following Ohashi Criteria is known7), 8).
L> c/8f Must consider compressibility
L< c/8f Possible with incompressible fluid
Here, L is piping length, c is the sound speed of wave motion propagating through a piping system, and f is the frequency of the phenomenon in question. Akimoto has proposed similar criteria9). Akimoto’s coefficients are 1/10 instead of 1/8 of Ohashi’s, but the decades of each show resemblance. In this article, they are collectively named Ohashi and Akimoto’s criteria. However, the validity and scope of Ohashi and Akimoto’s criteria have not been well enough defined. Therefore in this study, as a preparation phase for the technological development of cavitation surge analysis including the whole pump water supply piping system, targeting a one-dimensional hydraulic oscillating system that does not include cavitation, the lumped parameter system calculation and distributed parameter system calculation have been implemented. The two analyses are compared to verify the effectiveness of Ohashi and Akimoto’s criteria.
The lumped parameter system calculation and distributed parameter system calculation are implemented to a simple water supply system in Figure 1.
The system in Figure 1 has a pressure oscillation source on the inlet end of a right circular pipe with a fixed cross-sectional area and has a tank with a constant static pressure on the outlet end. Its unsteady flow is evaluated by a numerical analysis, under the condition of sine wave shaped pressure variation being applied from the pressure oscillation source. The motion equation for the lumped parameter system of Figure 1 is:
Here, P is static pressure, U is cross-sectional average flow velocity inside the pipe, L is piping length, D is piping diameter, ρ is the density of water, λ is the Darcy pipe friction factor, t is time, the subscript E is the oscillation source on the upstream side, and T is the tank on the downstream side. Since the cross sectional area of the pipe is fixed and water is assumed to be incompressible, U (t) is constant everywhere inside the pipe, and is a function of time only.
Next, if the density change of water is assumed to be very small, the law of conservation of mass and the momentum conservation law for the distributed parameter system of Figure 1 are:
Here, x is position coordinates of the axis direction of the pipe, c is sound speed, p and u is static pressure and distribution of cross-sectional average flow velocity inside the pipe, respectively. Since the compressibility of water is considered, p (x, t), and u (x, t) will be the function of position x and time t. U (t) and u (x, t) agree for steady flow. Several equations relating to the pipe friction of one-dimensional unsteady flow inside a pipe have been proposed, but in this study, to make it as simple as possible, pipe friction equations for a steady flow are used in equations (1) and (3). The boundary conditions of the pressure oscillation source and tank are:
Here, ω is an angular frequency and fluctuation frequency f =ω/2π. The subscripts AE, CE, and CT are the amplitude of the variation in static pressure of an oscillation source, a component of the static pressure of an oscillation source, and a component of static pressure in the tank, respectively.
Setting the initial conditions as a steady flow at a constant flow rate, in lumped parameter system:
In distributed parameter system:
Here, the subscript i represents initial conditions. The equation (10) refers to a so-called hydraulic gradient. To this steady-state, application of sine wave shaped pressure variation from the pressure oscillation source will be started, and the transient change after that will be evaluated by a numerical analysis.
The values are: pipe length L=10 m, pipe diameter D=0.1 m, the density of water ρ=1000 kg/m3 , and with consideration of the elastic deformation of the pipe, the sound speed inside the piping system c=1000 m/s. Pipe friction factor λ=0.01. Static pressure of the tank PCT
= 0Pa. In this study, regardless of a static pressure value inside a pipe, either cavitation to a pressure oscillation source, or water column separation inside a pipe are assumed not to occur. The initial conditions are Ui
=1 m/s for the lumped parameter system and ui
=1 m/s for the distributed parameter system. The pressure amplitude of the oscillation source PAE
=250 Pa. This value is the half of 500 Pa, which is pipe friction loss when letting 1 m/s of water flow into a pipe with the above dimension value given.
From the above values, the frequency using the Ohashi criteria will be determined by the following.
In this study, the lumped parameter system and distributed parameter system are numerically analyzed by the explicit second-order Runge-Kutta method and method of characteristics, respectively.In the calculation of the distributed parameter system, pipe friction is discretized by a semi-implicit method. In the distributed parameter system, to evaluate the grid dependency of a solution, two grid types that divide piping into 25 or 250 are used. Since these two agreed well, the result divided by 25 is represented in this study.
Fig. 1 Hydraulic oscillating system of the target to be analyzed
Multiply the frequency of Ohashi criteria f0
=12.5 Hz, by 1/10, 1, 2, 4, and 5 to obtain the respective results 1.25 Hz, 12.5 Hz, 25 Hz, 50 Hz, and 62.5 Hz. For the cases of 1.25 Hz and 12.5 Hz, Figure 2 represents the computed results of the lumped parameter system and the distributed parameter system immediately after an oscillation has started. Figure 2 represents the static pressure PH
and flow velocity UH
at the axial center position of a pipe. They each represent a change of about 2.5 cycles from the start of the oscillation. As shown in Figure 2, in the case of 1.25 Hz, the lumped parameter system and the distributed parameter system agree well with both the pressure waveforms and flow velocity waveforms. This shows that the two lumped parameter systems are in good approximation. In the case of 12.5 Hz, the lumped parameter system and the distributed parameter system of the flow velocity agree relatively well. However, the pressure waveform of the distributed parameter system is superimposed by a sine wave, which is the solution of the lumped parameter system in the form of a serrated high frequency component that reciprocates through piping, suggesting that the two waveforms are not in good agreement.
The said wave that reciprocates through a piping is a transient phenomenon caused by an oscillation against an initial flow that is in a stable state. It is assumed to be damped by pipe friction caused by multiple reciprocation of a pressure wave. We have therefore conducted a 100-second analysis in the 12.5 Hz case.
Figure 3 represents a 100 second time variation for the static pressure and flow velocity at the axial center position of a pipe in the 12.5 Hz case. Figure 3 contains the variation of 12.5 Hz, but since the horizontal axis is as long as 100 seconds, it appears to be a paint out instead of a variation curve.
As shown in Figure 3, 50 seconds of time is required for the static pressure and flow velocity to reach a stable periodic variation state after the variation has started.
Figure 4 represents the change from the final condition at 99.8 to 100 seconds. The serrate components seen in Figure 2 (b) attenuate and dissipate by the pipe friction, as shown in Figure 4. Between 99.8 and 100 seconds, the solution of the lumped parameter system and distributed parameter system is in good agreement.
Next, Figure 5 represents the result of f0
multiplied by 2, 4, and 5. They are all waveforms that have passed the transient phase 100 seconds after the oscillation has started. For the flow velocity, when the frequency is multiplied by 2, the lumped parameter system and distributed system are in good agreement; for the static pressure, however, a significant difference in the amplitude is observed. This indicates that the compressibility of liquid affecting the pressure is becoming non-negligible. For the flow velocity, when the frequency is multiplied by 4, the lumped parameter system and distributed system still show good agreement; for the static pressure, however, the amplitude of the distributed parameter system reaches as high as 500 kPa. Since the amplitude of the static pressure at the axial center position of a pipe for the lumped parameter system remains at 125 Pa, it appears to be a horizontal straight line in Figure 5 (b). When the frequency is multiplied by 5, the phase of the pressure waveforms of the lumped parameter system and distributed parameter system will reverse, and for the velocity waveforms, there are no differences in phase, but the amplitude is significantly different. When the frequency is multiplied by 4, frequency=c/2L. This is the same as the well-known primary resonance frequency of a resonator with free ends on both sides.
For that reason, it is definite that this system has gone into a resonance state and the pressure amplitude became enormous. In Figure 5 (b), the velocity waveforms appear to be in good agreement, but in reality, since the axial center position of a pipe represents a node of the velocity variation, when comparing at a location separated from this position, the difference between the lumped parameter system and distributed parameter system is big. When the frequency is multiplied by 5, since the point is beyond a resonance frequency, phase delay occurs to the oscillation input.
For all of these reasons, in systems where wave motions reciprocate, the values Ohashi and Akimoto have proposed have been shown to set 1/4 and 1/5 of the natural frequency of a system with free ends on both sides as limits that can be regarded as incompressible, which are quite appropriate values as indexes showing the limit of the application of the lumped parameter system. If a system is considered to have a free end or fixed end on one side, the basic frequency will be c/4L. Accordingly, making it half of the above criteria is recommended. These criteria, however, are based on the assumption that the residue of the initial state is dissipated due to pipe friction. In this analysis, after the variation has started, the average flow velocity maintains at approximately 1 m/s, and since pipe friction works stably, the residue of the initial state dissipates in about 50 seconds.
If a situation is considered in which the water inside a pipe is at rest initially, if it is analyzed under the condition P
CE =PCT
=0Pa, its flow will be in reciprocation. Thus, since the average flow velocity is almost 0 m/s, the effect of pipe friction is small, and, as a result, the residue of the initial state does not attenuate even after 100 seconds has elapsed (not shown in Fig). Therefore, when transient change is repeated or when pipe friction does not sufficiently work due to slow flow velocity, we should conclude that the lumped parameter system can be only approximated when the representative frequency of the variation is about 1/10 of the frequency proposed in Ohashi and Akimoto’s criteria (about 1/20 for a free end or fixed end system). Even when pipe friction is working sufficiently, to attain a stable periodic variation state for a long piping, it takes as long as a few tens of seconds or more, depending on its length. Thus, it should be noted that the limitation on the computation area still requires a big burden on a three-dimensional CFD in a time domain.
Fig. 2 Time variation of the static pressure and flow velocity at the axial center position of a pipe (when 0.1<em>f<sub>0</sub></em>,<em> f<sub>0</sub></em> after oscillation has started)
Fig. 3 100 second time variation of the static pressure and flow velocity at the axial center position of a pipe (when <em>f<sub>0</sub></em>)
Fig. 4 Time variation of the static pressure and flow velocity at the axial center position of a pipe at a stable period (when <em>f<sub>0</sub></em>)
Fig. 5 Time variation of the static pressure and flow velocity at the axial center position of a pipe at a stable period (when a frequency is higher than <em>f<sub>0</sub></em>)
Surging issues, such as cavitation surge of a hydraulic pump, are characterized by the interaction of unsteady cavitation inside the pump and the piping system surrounding the pump. This study evaluates the validity of Ohashi and Akimoto’s criteria that indicate the scope of a frequency considering the compressibility in the piping system from the comparison of the lumped parameter system analysis and one-dimensional distributed parameter system analysis. It became clear that Ohashi and Akimoto’s criteria are reasonable, after the residue of the initial condition is totally dissipated by the pipe friction. When transient change is repeated or when pipe friction does not sufficiently work due to slow flow velocity, the lumped parameter system can be only approximated when the representative frequency of the variation is about 1/10 of the frequency proposed in Ohashi and Akimoto’s criteria.
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3) An, Kajishima, Numerical Analyses for Cavitation Surge Using an Flow Rate Fluctuation Model in Two-Dimensional Cascades, The 16th Symposium on Cavitation, (2012).
4) Nohmi, Yamazaki, Kagawa, An, Kang, Yokota, Numerical Analyses for Cavitation Surge in a Pump with the Square Root Shaped Suction Performance Curve, ISROMAC2016, (2016).
5) A. Marie-Magdeleine, R. Fortes-Patella, N. Lemoine, N. Marchand, Unsteady flow rate evaluation methodology for identification of the dynamic transfer function of a cavitating Venturi, Proc. CAV2012, (2012).
6) Fujiwara, Nanri, Yoshida, Effect of PSD on Acoustic Cavitation Surge in Inlet Pipe of Turbopump, The 16th Symposium on Cavitation, (2012).
7) Nanri et al., Acoustic Cavitation Surge in a Turbopump, Turbomachinery, Volume 39 Issue 4, (2011), pp. 1-8.
8) Ohashi, Unsteady Problems of Pumps and Fluid Systems Containing the Same, Science of Machine, Volume 22 Issue 4, (1970), pp. 35-42.
9) Akimoto, Water Hammer Effect and Pressure Pulsation [Revised Edition] Volume 1 Fundamental Equation and Various Constants, on demand publishing, (2004), p 1-20 (The first edition was published in 1972 by Nippon Kogyo Shimbun).
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