Shock and Vibration

Shock and Vibration / 2021 / Article
Special Issue

Vibration and Control of Fluid Machinery and Systems

View this Special Issue

Research Article | Open Access

Volume 2021 |Article ID 5581684 | https://doi.org/10.1155/2021/5581684

Wen Song, Chenshi Yang, Xiaoyi Zhang, Yongdong Li, "Mathematical Modelling and Dynamic Analysis of a Direct-Acting Relief Valve Based on Fluid-Structure Coupling Analysis", Shock and Vibration, vol. 2021, Article ID 5581684, 11 pages, 2021. https://doi.org/10.1155/2021/5581684

Mathematical Modelling and Dynamic Analysis of a Direct-Acting Relief Valve Based on Fluid-Structure Coupling Analysis

Academic Editor: Ling Zhou
Received25 Jan 2021
Revised11 Mar 2021
Accepted31 Mar 2021
Published12 Apr 2021

Abstract

To explain the sudden jump of pressure as the variation of water depth for a direct-acting relief valve used by torpedo pump as the variation of water depth, a 2-DOF fluid-structure coupling dynamic model is developed. A nonlinear differential pressure model at valve port is applied to model the axial vibration of fluid, and a nonlinear wake oscillator model is used to excite the valve element in the vertical direction; meanwhile, the contact nonlinearity between the valve element and valve seat is also taken into consideration. Based on the developed dynamical model, the water depths for the sudden jumps of pressure can be located precisely when compared with the experimental signals, and the corresponding vibration conditions of the valve element in both the axial and vertical directions are explored. Subsequently, in order to eliminate the sudden jumps of pressure, different pump inlet pressure was tested experimentally; when it was decreased to 0.4 MPa, the pressure jumps ever appeared during the dropping and lifting processes were removed, and the numerical simulation based on the developed mathematical model also verified the experimental measurements.

1. Introduction

Direct-acting relief valve is widely applied for pressure adjustment in fuel pump [15]; when it is used underwater for torpedo, its stability is generally affected by the ocean environment. Specifically, as there is pressure fluctuation at the relief valve due to the influences of water depth, the valve element can lose its stability via Hopf bifurcation [68] and can further enter into chaos via grazing bifurcation due to the axial impact between the valve element and valve seat [6, 914]. Based on the previous investigations, the contact nonlinearity [1518] is the match point for understanding the dynamic behaviors of this or similar systems. In addition, all the predecessors’ researches focused on the one-degree-of-freedom (1-DOF) vibration, i.e., the axial vibration of the valve element, without considering the influence of the vibration in the longitudinal direction. However, based on the field experimental test, the traces of collision between the valve element and the valve seat were explored in both the axial and longitudinal directions. Under such circumstances, a two-degree-of-freedom (2-DOF) vibration model for the relief valve should be taken into consideration.

In order to accurately describe the longitudinal vibration of the valve element, the origin of the longitudinal excitation acting on the valve element should be determined primarily. For a direct-acting relief valve with a narrow flow channel, the fuel flowing within it will be accelerated quickly to reach a high speed at the outlet of the flow channel, under such circumstances, vortex shedding appears when the fuel flows across the valve element which is thus excited to vibrate in the longitudinal direction due to the lift force originated from the vortex-induced vibration (VIV) [19]. Meanwhile, as there is increase of the vibration amplitude of the valve element, the intermittent impacts between the valve element and the valve seat will occur also in the longitudinal direction, which will further intensify the VIV [20]. Generally, the VIV is described by a wake oscillator model mathematically [21], based on which, Lin et al. [22] studied the correlation between the fluid flow in the axial direction and the lift force acting on the structure in the longitudinal direction. Chen and Li [23] presented a fluid-structure coupling model to investigate the influences of the VIV on the structure, in which a modified van der Pol nonlinear equation was employed to simulate the vortex shedding. Doan and Nishi [24] applied the wake oscillator model to quantify the vortex-induced lift force acting on the structure. According to their works, the wake oscillator model will also be considered to be applied for the following mathematical modelling in the present work. By this way, a 2-DOF fluid-structure coupling dynamic model will be proposed in the present work to investigate the correlation between the vibration condition of valve and the sudden jump of pressure of the pump observed in experiments.

The rest of this paper is organized as follows. Section 2 introduces the mathematical modelling of the direct-acting relief valve, in which a vibration model, a fluid differential pressure, a nonlinear wake oscillator model, and a hard impact model are integrated to develop a whole dynamic model, which is further nondimensionalized for the following numerical simulations. In Section 3, the comparisons between the experimental measurements and the numerical simulations are carried out to verify the developed mathematical model. At last, concluding remarks are provided in Section 4.

2. Mathematical Modeling

In this section, a 2-DOF mathematical model about the fluid-structure coupling system of a direct-acting relief valve is developed. Primarily, the physical model is simplified from the schematic of the direct-acting relief valve, based on which, the dynamic model is developed and its corresponding nondimensionalization is conducted.

2.1. Physical Model

Based on the schematic of the direct-acting relief valve shown in Figure 1(a), the physical model can be simplified [10, 11] as shown in Figure 1(b), in which two Kelvin–Voigt models () are utilized to describe the vibrations of the valve element () in both the axial and vertical directions. The axial vibration originates from the interaction between the force of the fluid pressure difference at the valve port () and the force () acting on the back pressure rod. While the vertical vibration of the valve element is determined by both the inlet pressure force () and the lifting force caused by the vortex shedding in vortex-induced vibration (VIV). In addition, and are used to represent the displacements of the valve element in the axial and vertical directions, respectively.

2.2. Dynamic Model of Relief Valve

In order to develop the whole dynamic model of a relief valve, based on the physical model shown in Figure 1, a basic vibration model is deduced aswhere

The explanations of all the system parameters can be found in Table 1.


ParametersSignsUnitsValues

Rod body’s total masskg13.7e−3
Oriented seat’s spring stiffnessN/m1.266e5
Spring precompressionm2.227e−3
Overflow amountm3/s7.67e−5
Density of waterkg/m31000
Density of fuelkg/m31232
Pump inlet pressurePa0.4e6∼0.8e6
Pump outlet pressurePa10e6
Radius of back pressure ejector pinsm3.0e−3
Radius of ball valvem2.3812e−3
Radius of valve seat channelm2.2e−3
Radius of pump inletm8e−3
Atmospheric pressurePa1e5
Washer’s damping30
Vertical vibration damping of the rod string30
Fluid damping10.9
Vertical stiffness of the rod stringN/m1e6
Fluid compression modulusPa5e9
Pore throat’s fluid volumem31.1404e−7
Fluid resistance coefficient11.0
Fluid lift coefficient10.2
Added mass coefficients11.0
Strouhal number10.15∼0.2
Restitution coefficient of impact10.2∼0.4

In equation (3), the fluid differential pressure is determined via a nonlinear fluid differential pressure model at valve port [10,11].where

In addition, in order to obtain the additional fluid mass that vibrates with the structure , the additional fluid damping, and the lifting force due to the vortex shedding , which are included in equation (1), a nonlinear wake oscillator model is used as [14,15]where

Meanwhile, the contact nonlinearity between the valve element and valve seat is also taken into consideration as

Based on the above analyses, a compact form of the dynamic model of the direct-acting relief valve system can be described as

2.3. Nondimensionalization of Dynamic Model

Before the numerical simulation, the developed dynamic model is further nondimensionalized to obtain the dimensionless equations. Firstly, three basic reference parameters of the studied system are defined, including the natural frequency , the reference time , and the reference displacement aswhereis the atmospheric intensity of pressure. Based on the above three basic reference parameters, eight dimensionless variables can be obtained as

Therefore, the dynamic model of the direct-acting relief valve system can be transformed into the dimensionless equations aswhere is the nondimensional acceleration of valve element in the vertical direction, and is generally applied to describe the influence of the structure on the vortex-induced vibration [15, 16].

Moreover, in order to obtain the simplified form of equation (13), 14 new dimensionless parameters, such as damping ratio, spring precompression ratio, system stiffness ratio, dimensionless depth, and dimensionless pump displacement, are defined as

In summary, a complete dynamic model about the fluid-structure coupling system of the direct-acting relief valve has been developed.

3. Comparison of Experiments and Numerical Simulations

Based on the developed mathematical model in Section 2, the numerical simulation of the studied system about the relief valve can be carried out, and thus the comparison between the numerical simulation and the experimental measurements. Specifically, the differential equations for the developed dynamic model were coded in MATLAB, which were then solved by the 4th Runge–Kutta algorithm. For this purpose, the actual parameters of an experimental device are used as the system parameters for numerical simulations, see Table 1.

3.1. Condition One: 0.8 MPa Inlet Pressure

Primarily, the 0.8 MPa inlet pressure was considered since the unstable vibration of the relief valve was explored experimentally in such condition, see Figure 2(a). Figure 2(a) shows the variations of the pump inlet pressure, outlet pressure, and back pressure from top to bottom. Moreover, the back pressure is determined by the hydraulic pressure; hence, the water depth can be calculated according to the back pressure, and the corresponding result is depicted as the green curve in the third subplot of Figure 2(a). The corresponding schematic of the experimental setup is displayed in Figure 3. During the experimental test, the pump was gradually sunk to the water depth of 300 m and then slowly lifted to the water surface. It can be seen from Figure 2(a) that an obvious pressure step appears at 20 s, whose corresponding water depth is around 100 m where the back pressure rod was in contact with the guide holder. From then on, the vibration condition of the relief valve can be simulated by the developed mathematical model, since the valve element () simplified in the physical model consists of both the back pressure rod and the guide holder, see Figure 1. Therefore, the range of the water depth for the numerical simulation was set as [100 m, 300 m] with the depth interval 1 m, and the obtained result is shown in Figure 2(b). Specifically, the differential pressure at the valve port reflects the correlation between the inlet pressure and the outlet pressure, which is basically consistent with the trend of the outlet pressure measured during the experimental tests, since the outlet pressure is much greater than the inlet pressure. In addition, due to the narrow flow area at valve port, its differential pressure is larger than the pressure measured in both the pump outlet and the pump inlet.

In addition, it is found from Figure 2 that the relief valve is unstable which experienced two sudden jumps of pressure during either the dropping process or the lifting process. Specifically, during the process of dropping the relief valve from 100 m to 300 m, the experimental signals showed the first sudden jump of pressure at 137 m where an obvious pressure pulse can be observed. With the further increase of the water depth, the experimental signals showed another sudden jump of pressure at 265.3 m, whose inlet pressure was further magnified. Similarly, during the process of lifting the relief valve from 300 m to 100 m, two sudden jumps of pressure appeared at 267.9 m and 129.5 m, respectively.

When comparing the numerical simulations with the corresponding experimental measurements, the same phenomenon can be observed. Specifically, during the dropping process, the water depths of the two pressure jumps appeared at 141 m and 274 m, respectively. While in the lifting process, the water depths for the two pressure jumps appeared at 270 m and 135 m, respectively. Furthermore, according to the error calculation, see Table 2, comparing with the experimental measurements, the errors of the predicted depths for the four pressure jumps by the numerical simulations are limited in 5%, which demonstrates the correctness and accuracy of the developed mathematical model effectively.


Depth (m)1st pressure jump2nd pressure jump3rd pressure jump4th pressure jump

Experimental measurements137.0265.3267.9129.5
Numerical simulations141.0274.0270.0135.0
Errors (%)2.923.270.784.25

In order to understand the variation of the dynamic responses of the valve element in different vibration phases, Figure 4 displays three typical vibration conditions of the valve element; they represent the vibration conditions of the valve element in the three depth regions divided by the two pressure jumps. Specifically, at the water depth of 100 m, no sudden change of pressure was experienced by the valve element; hence, its vibrations in both the axial and longitudinal directions show regularly periodic motions. However, after the first sudden jump of pressure, the quasiperiodic vibration of the valve element appeared at the water depth of 150 m; furthermore, when the second sudden change in pressure was crossed, the valve element vibrated chaotically in both directions at the water depth of 300 m. Therefore, the change of the vibration condition of the valve element is strongly related to the sudden jump of pump pressure.

3.2. Condition Two: 0.4 MPa Inlet Pressure

For this studied relief valve, the sudden jumps of pressure appeared in the case with 0.8 MPa inlet pressure bring negative influences on the system stability; hence, the main purpose of this study is to eliminate these pressure jumps. Comparing with the modification of structure parameter which needs to remanufacture the relief valve, the control of the pump inlet pressure is the most convenient way in the practical operation to analyze the influence of system parameter on the dynamic responses of the relief valve. Therefore, different pump inlet pressures were tested experimentally, and their measurements are displayed in Figure 5. As observed in the subplots in Figures 5(e)-5(f), when further increasing the pump inlet pressure to 0.9 MPa or 1.0 MPa, the fluctuation of pump pressure is more drastic; while if the pump inlet pressure is decreased, see the subplots in Figures 5(c)-5(d) and 5(b), the amplitude of the pressure jump is also decreased. Finally, when the pump inlet pressure is dropped down to 0.4 MPa, see subplots in Figure 5(a), the pressure jumps are suppressed effectively. Therefore, 0.4 MPa inlet pressure was explored as the ideal parameter, which is also verified by the corresponding numerical result, see Figure 6. Specifically, there is only one pressure step which has appeared at the water depth of 59.56 m. This is the position where the back pressure rod was in contact with the guide holder. Except that, no other pressure jump was observed, namely, the relief valve maintained stable condition during both the dropping and lifting processes, and this is the expected working condition of the relief valve.

In addition, during the numerical simulation, the multistability coexistence for the vibration of the valve element was explored at the water depth of 66 m, and its details are shown in Figure 7. The first coexisting vibration condition of the valve element was obtained when increasing the water depth to 66 m, see the subplots in Figure 7(a), where the three-dimension phase portraits for both the axial and longitudinal vibrations, and the time series for both the flow area and the seven system variables shown in equation (13), are displayed. In the first stable vibration condition, the time series of the flow area presents an obvious periodicity, and both the axial and longitudinal vibrations of the valve element are also periodic motions. However, when decreasing the water depth to reach 66 m, the second coexisting vibration condition appears, see the subplots in Figure 7(b). Specifically, the time series of the flow area is coupled by multifrequency, and the duration required for the valve element undergoing a complete cycle is longer. Based on the above comparison, it can be concluded that although both the vibrations occurred at the same water depth of 66 m, their significant differences in vibration conditions may trigger completely different vibration evolution paths and pressure change trends, which may bring potential risks for the application of torpedo; therefore, such a multistability coexistence should be taken into consideration seriously for this or similar dynamic systems.

4. Concluding Remarks

In this present work, a 2-DOF dynamic model was developed to simulate the fluid-structure system of a direct-acting relief valve. During the mathematical modelling, a vibration model, a fluid differential pressure, a nonlinear wake oscillator model, and a hard impact model were integrated to develop a whole dynamic model, which was further transformed as a nondimensional model. Such a nondimensional dynamic model can be applied for the analysis of the direct-acting relief valve with the same or the similar structures.

Compared with the experimental measurements, the prediction errors of the water depths for sudden jumps of pressure from the numerical simulation were less than 5%. Therefore, the results of numerical simulation were in good accordance with the experimental measurements, which verified the correctness and accuracy of the developed mathematical model.

In order to eliminate the pressure jumps explored during experimental tests, the pump inlet pressure was adjusted; when it was decreased from 0.8 MPa to 0.4 MPa, the pressure jumps ever appeared during the dropping and lifting processes were removed totally, and the numerical simulation based on the developed mathematical model also verified the experimental measurements. Therefore, decreasing the pump inlet pressure is an effective way to suppress the sudden jump of pump pressure.

In addition, the multistability coexistence was explored for the pump inlet pressure of 0.4 MPa with the water depth of 66 m. The significant differences in vibrations for the different stable conditions may trigger completely different vibration evolution paths of the valve element, and thus the different pressure change trends of the relief valve may bring potential risks for the application of torpedo; therefore, such a multistability coexistence should be taken into consideration seriously for this or similar dynamic systems.

For the future work, according to the verified mathematical model for the direct-acting relief valve developed in this paper, the optimization of both the structure parameters and the control parameters can be carried out to maintain the stability of the pressure system even under the varied working condition, and this part of work will be discussed in the following papers.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

  1. J. V. Bukowski, W. M. Goble, R. E. Gross, and S. P. Harris, “Analysis of spring operated pressure relief valve proof test data: findings and implications,” Process Safety Progress, vol. 37, no. 4, pp. 467–477, 2018. View at: Publisher Site | Google Scholar
  2. K. Hyunjun and k Sanghyun, “Optimization of operation parameters for direct spring loaded pressure relief valve in a pipeline system,” Journal of Pressure Vessel Technology, vol. 140, no. 5, Article ID 051603, 2018. View at: Google Scholar
  3. J. Lei, J. Tao, C. Liu, and Y. Wu, “Flow model and dynamic characteristics of a direct spring loaded poppet relief valve,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 232, no. 9, pp. 1657–1664, 2018. View at: Publisher Site | Google Scholar
  4. W. Wu, G. Tian, and Q. Hao, “Analysis and optimization of abnormal opening of main valve of relief valve under alternating pressure,” Journal of Huazhong University of Science and Technology (Natural Science Edition), vol. 46, no. 7, pp. 78–83, 2018. View at: Google Scholar
  5. L. Gabor, C. Alan, and H. Csaba, “Nonlinear analysis of a single stage pressure relief valve,” Iaeng International Journal of Applied Mathematics, vol. 39, no. 4, pp. 1–14, 2009. View at: Google Scholar
  6. C. Bolin and A. Engeda, “Analysis of flow-induced instability in a redesigned steam control valve,” Applied Thermal Engineering, vol. 83, pp. 40–47, 2015. View at: Publisher Site | Google Scholar
  7. S. Wu, C. Li, and Y. Deng, “Stability analysis of a direct-operated seawater hydraulic relief valve under deep sea,” Mathematical Problems in Engineering, vol. 2017, Article ID 5676317, 11 pages, 2017. View at: Publisher Site | Google Scholar
  8. Y. Li, “Hopf bifurcations in general systems of Brusselator type,” Nonlinear Analysis: Real World Applications, vol. 28, pp. 32–47, 2016. View at: Publisher Site | Google Scholar
  9. Q. Hao, W. Wu, X. Liang, and Z. Liu, “Effects of structure parameters on abnormal opening of pilot-operated relief valve under alternating pressure,” IEEE Access, vol. 7, pp. 33932–33942, 2019. View at: Publisher Site | Google Scholar
  10. C. J. Hős, A. R. Champneys, K. Paul et al., “Dynamic behavior of direct spring loaded pressure relief valves: III valves in liquid service,” Journal of Loss Prevention in the Process Industries, vol. 43, pp. 1–9, 2016. View at: Google Scholar
  11. C. Hős and A. R. Champneys, “Grazing bifurcations and chatter in a pressure relief valve model,” Physica D: Nonlinear Phenomena, vol. 241, no. 22, pp. 2068–2076, 2012. View at: Google Scholar
  12. W. Ma, F. Ma, Z. Zhou, and X. Geng, “Bifurcation analysis and experiment of direct acting relief valve,” Vibration Testing and Diagnosis, vol. 36, no. 03, pp. 529–535+606, 2016. View at: Google Scholar
  13. R. D. Eyres, P. T. Piiroinen, A. R. Champneys et al., “Grazing bifurcations and chaos in the dynamics of a hydraulic damper with relief valves,” SIAM Journal on Applied Dynamical Systems, vol. 4, no. 4, pp. 1075–1106, 2005. View at: Publisher Site | Google Scholar
  14. A. Beune, J. G. M. Kuerten, and M. P. C. Van Heumen, “CFD analysis with fluid-structure interaction of opening high-pressure safety valves,” Computers & Fluids, vol. 64, pp. 108–116, 2012. View at: Publisher Site | Google Scholar
  15. M. Liao, Y. Liu, J. Páez Chávez, A. S. E. Chong, and M. Wiercigroch, “Dynamics of vibro-impact drilling with linear and nonlinear rock models,” International Journal of Mechanical Sciences, vol. 146-147, pp. 200–210, 2018. View at: Publisher Site | Google Scholar
  16. M. Liao, J. Ing, J. Páez Chávez, and M. Wiercigroch, “Bifurcation techniques for stiffness identification of an impact oscillator,” Communications in Nonlinear Science and Numerical Simulation, vol. 41, pp. 19–31, 2016. View at: Publisher Site | Google Scholar
  17. M. Liao, J. Ing, M. Sayah, and M. Wiercigroch, “Dynamic method of stiffness identification in impacting systems for percussive drilling applications,” Mechanical Systems and Signal Processing, vol. 80, pp. 224–244, 2016. View at: Publisher Site | Google Scholar
  18. Y. Xu, H. Zhang, and Z. Guan, “Dynamic characteristics of downhole bit load and analysis of conversion efficiency of drill string vibration energy,” Energies, vol. 14, no. 1, p. 229, 2021. View at: Publisher Site | Google Scholar
  19. M. L. Facchinetti, E. de Langre, and F. Biolley, “Coupling of structure and wake oscillators in vortex-induced vibrations,” Journal of Fluids and Structures, vol. 19, no. 2, pp. 123–140, 2004. View at: Publisher Site | Google Scholar
  20. F. He, H. Dai, and L. Wang, “Vortex-induced vibrations of a pipe subjected to unsynchronized support motions,” Journal of Marine Science and Technology, vol. 23, no. 4, pp. 978–990, 2018. View at: Publisher Site | Google Scholar
  21. M. Keber and M. Wiercigroch, “Dynamics of a vertical riser with weak structural nonlinearity excited by wakes,” Journal of Sound and Vibration, vol. 315, no. 3, pp. 685–699, 2008. View at: Publisher Site | Google Scholar
  22. L.-m. Lin, G.-c. Ling, Y.-x. Wu, and X.-h. Zeng, “Nonlinear fluid damping in structure-wake oscillators in modeling vortex-induced vibrations,” Journal of Hydrodynamics, vol. 21, no. 1, pp. 1–11, 2009. View at: Publisher Site | Google Scholar
  23. J. Chen and Q. Li, “Nonlinear dynamics of a fluid-structure coupling model for vortex-induced vibration,” International Journal of Structural Stability and Dynamics, vol. 19, no. 7, pp. 23–38, 2019. View at: Publisher Site | Google Scholar
  24. V.-P. Doan and Y. Nishi, “Modeling of fluid-structure interaction for simulating vortex-induced vibration of flexible riser: finite difference method combined with wake oscillator model,” Journal of Marine Science and Technology, vol. 20, no. 2, pp. 309–321, 2015. View at: Publisher Site | Google Scholar

Copyright © 2021 Wen Song et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Related articles

No related content is available yet for this article.
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views136
Downloads417
Citations

Related articles

No related content is available yet for this article.

Article of the Year Award: Outstanding research contributions of 2021, as selected by our Chief Editors. Read the winning articles.