Two-phase flow-induced vibration fatigue damage of tube bundles with clearance restriction
Introduction
The flow-induced vibration fatigue damage of a steam generator has been an industry-wide concern since the 1970 s, which can become a significant mechanism impacting the reliability of the steam generator. Turbulence forces induced by the two-phase flow and the fluidelastic forces are of major significance for tube fatigue. Hence, increasing attention has been paid to the prediction of the flow-induced vibration for the tube bundles subjected to two-phase flow. The main flow excitation mechanisms of the tube bundles subjected to two-phase flow are turbulence and fluidelastic instability. Several studies have been performed to investigate the two flow excitation mechanisms.
Fluidelastic instability is an important excitation mechanism that may cause a short term failure of the steam generator tubes. Since the 1980 s, to predict the critical velocity of the fluidelastic instability of the tube bundles subjected to cross-flow, several fluidelastic force models were proposed by the scholars. Tanaka and Takahara [1] proposed the unsteady fluid force model which can be regarded as three kinds of forces: the inertia force due to the added mass of the fluid; the damping force due to the fluid; the stiffness force due to the dynamic pressure and cylinder displacement. This unsteady fluid force model can be utilized to predict the critical velocity of the fluidelastic instability of the tube bundles in cross-flow. The major drawback of the unsteady fluid force model is that all the relevant fluid force coefficients must be measured for each tube array pattern. Lever and Weaver [2] developed a simple first principle model to predict the threshold of the fluidelastic instability of a full tube array in cross-flow, requiring no measured fluid force coefficients. Paidoussis et al. [3] developed a full linear unsteady potential-flow solution for fluid flowing across the tube bundles which is consistent with certain basic physical checks. An improved quasi-static fluid force model was developed by Price and Paidoussis [4] to investigate the fluidelastic instability of cylinder rows subjected to cross-flow. For the high mass-damping parameters, the fluidelastic instability of the tube bundles is controlled by the stiffness mechanism. Paidoussis et al. [5] presented a semianalytical model to predict the threshold of the fluidelastic instability controlled by fluid-dynamic stiffness terms.
Turbulence is another important excitation mechanism that may cause the fatigue and fretting wear of the steam generator tubes. Taylor et al. [6] performed two experiments to determine the characteristics of the random excitation forces acting on the tube bundles in two-phase cross-flow. Their experimental results indicated that the flow rate and the void fraction have a great effect on the excitation forces. Based on the available experimental data, Langre and Villard [7] proposed an upper bound on the magnitude of the random buffeting excitation forces that apply to tube bundles in two-phase cross-flow. They found that the dynamic pressure, viscosity, or surface tension were not relevant to the dimensionless spectra of the random fluid forces. Taylor and Pettigrew [8] carried out several experiments to measure the random fluid forces that apply to tubes in two-phase cross-flow. The data from these experiments were utilized to determine the suitable guidelines for random excitation forces. Zhang et al. [9] measured the vibration excitation force acting on a rotated triangular tube array subjected to two-phase cross-flow. Some unexpected quasi-periodic forces were observed in the experiments. Xu et al. [10], [11] used an inverse analysis method based on the oscillation displacements to investigate the hydrodynamic force acting on the cylinders.
Along with the development of the computer industry, numerical calculation is widely used in the computation hydrodynamics. Pedro et al. [12] used a computational fluid mechanics (CFD) methodology involving the tube motion and dynamic re-meshing to simulate the unsteady flow in a normal triangular tube array. Parrondo et al. [13] calculated the fluctuating flow in a parallel triangular tube array with the pitch-to-diameter ratio of 1.57. The simulations were developed with a commercial code to solve the 2D-URANS equations. Lai et al. established the CFD models to determine the slip ratio between the water phase and the air phase [14], and the periodic fluid force in a rotated triangular tube array [15], respectively. The correctness of the CFD calculation was verified by the experimental results.
Based on these proposed fluid force model, the fluidelastic instability and the nonlinear dynamics of the tube bundles subjected to two-phase flow were investigated. Rottmann and Popp [16] used a simple “tube-in-channel-flow” model to investigate the fluidelastic instability of a parallel triangular tube bundle. Feenstra et al. [17] conducted several experiments to measure the flow-induced vibration responses and the threshold of the fluidelastic instability of the tube bundles in a cross-flow of refrigerant 11. They found that the vortex shedding could be disrupted by a small amount of bubbles in the flow. An experimental study was carried out by Chung and Chu [18] to investigate the fluidelastic instability of the normal square tube array and the rotated square tube array with the same pitch-to-diameter ratio of 1.633 in air–water two-phase cross-flow. The experimental results indicated that the vibration characteristics of the rotated square tube array are quite different from those of the normal square tube array in the two-phase flow. Mahon and Meskell [19] investigated the interaction between the acoustic resonance and the fluidelastic instability in a normal triangular tube array. They found that the time delay between the flow field and the tube motion may be modified by the acoustic resonance. Zhao et al. [20] investigated the dynamics of the tube bundles subjected to cross-flow. The vibration characteristics of two tubes with in-line and parallel configurations were discussed. Palomar and Meskell [21] used a theoretical-CFD hybrid methodology to investigate the fluidelastic instability of a normal triangular tube array subjected to cross-flow. The numerical results illustrated that the Reynolds number has a great effect on the threshold of the fluidelastic instability. In our previous studies, a series of experimental and theoretical studies have been carried out to investigate the fluidelastic instability of a rotated triangular tube array subjected to two-phase flow [22], [23], [24], [25], and the mathematical models were presented to study the nonlinear dynamics of the tube bundles subjected to two-phase flow and loose support [26], [27], [28].
It is known that the fatigue breakage is one important damage style induced by the flow-induced vibration. Forecasting the tube bundles’ fatigue life is very important to the security of a steam generator. As mentioned above, despite a lot of theoretical and experimental studies have been performed to analyze the flow-induced vibration and the fluidelastic instability of the tube bundles in cross-flow, few have considered the influence of the flow-induced vibration on the fatigue damage of the tube bundles. Especially, the tubes of a steam generator are always constrained by the support structures. In general, there is a gap between the tubes and the support structures which may lead to a collision. The interacton forces between the tube bundles and the support structures may affect the fatigue life of the tubes. Therefore, it is still a problem deserving further investigations for the flow-induced vibration fatigue damage of the tube bundles subjected to two-phase flow and the clearance restriction.
In this study, considering the effect of the fluidelastic force and the turbulence random force, a mathematical model of the tube bundles subjected to two-phase cross-flow and the clearance restriction was developed. Based on S-N curves by using the numerical results of the vibration responses, the flow-induced vibration fatigue damages of the tube bundles within different void fraction conditions were estimated. More importantly, the effects of the clearance restriction, the flow pitch velocity, and the void fraction of the two-phase flow on the fatigue damage were discussed.
A detailed implementation procedure for the fatigue evalutaion methodology is as follows:
① To determine the equivalent power spectral density (EPSD) of the turbulent excitation force acting on the tube bundles by experimental measurement.
② To calculate the vibration responses of the tube bundles system with clearance restriction subjected to fluidelastic force and turbulent force. The fluidelastic force is produced by a coupling between the tube motion and the flowing fluid. The turbulent force is transformed from the EPSD curve into a force–time record.
③ To evaluate the fatigue damage of the tube bundles system based on the S-N approach. As the flow-induced stress cycles are random, the Rainflow Cycle Counting method is used to calculate the distribution of stress amplitudes.
Section snippets
Theoretical analysis
In a steam generator, the tube bundles are threaded through the support structures such as the tube support plate (TSP) and the anti-vibration bars (ATVs). Typically, to avoid the effect of the thermal expansion or contraction, there would be a small gap between the tube bundles and the support structures, as shown in Fig. 1. Therefore, when the vibration amplitude of the tube is larger than the gap, a collision occurs. In the present study, we treated the flexible tube as a simply-supported
Methodology for fatigue damage estimation
The S-N curve is one of the most extensively used method to estimate the fatigue damage of the tube bundles of a steam generator in nuclear engineering. In practical applications, the total fatigue damage can be estimated by the linear accumulation rule. Hence, in the present study, the Palmgren-Miner rule was employed to calculate the total fatigue damage [32], which can be expressed as:
where Damagetotal is the total fatigue damage, Ni is the permissible number of cycles, ni
Discussion
According to the numerical results obtained in the last sections, it is obvious that the void fraction and flow pitch velocity are the important parameters to the flow-induced vibration fatigue damage of the tube bundles subjected to two-phase flow and the clearance restriction. Thus, the influences of the flow pitch velocity and the void fraction of the two-phase flow on the fatigue damage were discussed quantitatively in this section.
The variation of the fatigue damage and the RMS stress of
Conclusion
The flow-induced vibration fatigue damage of a flexible tube in a rotated triangular tube array subjected to the two-phase flow and the clearance restriction was studied based on S-N curves. A fatigue damage evaluation of the tube bundles was made using the time history of the stress response considering the effects of the turbulence random force and the fluidelastic force. The concluding remarks are as follows.
- (1)
For the tube bundles with a clearance restriction, the gap between the tube and the
CRediT authorship contribution statement
Jiang Lai: Conceptualization, Funding acquisition, Investigation. Shihao Yang: Writing – original draft. Lingling Lu: Methodology, Writing – review & editing. Tiancai Tan: Validation. Lei Sun: Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China: [Grant Number 12072336].
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2022, International Journal of Pressure Vessels and PipingCitation Excerpt :As mentioned above, random excitation is an important mechanism of the tube bundles when the flow pitch velocity is lower than the critical velocity of the fluidelastic instability. Considering the effect of the loose support such as the tube support plate and anti-vibration bar, turbulence-induced vibration may cause potential tube fretting [22] and fatigue damage [23]. Thus, the dynamic characteristics of the tube bundles subjected to turbulence excitation force and loose support is an important issue on the security of a steam generator.