High-temperature molten salt circulating cold salt pump shaft system prestress response analysis research

To investigate the response characteristics of the rotor system in a high-temperature molten salt pump, a specific model of a cold salt pump was selected as the research object. The natural frequencies and mode shapes, critical speeds, and harmonic response analysis based on modal analysis of the rotor system with and without prestress were computed. The results indicate that with prestress, the natural frequencies of the rotor are higher compared to the rotor without prestress, and there is no clear pattern among the various vibration modes. The calculated critical speed for the second mode is much higher than the rotor’s operating speed. This suggests that the rotor system will not experience resonance at its operating speed, meeting design requirements and ensuring stable operation. When the rotor of the cold salt pump is subjected to unbalanced forces, vibrations occur in the x, y, and z directions. The region with the largest vibration amplitude is located at the fifth-stage impeller. At a frequency of 52 Hz, the corresponding harmonic response displacement reaches its maximum value, and this frequency remains unchanged with variations in the magnitude and position of the unbalanced force. Therefore, it is advisable to avoid operating the rotor system at the critical speeds corresponding to the first and second modes. Furthermore, the operating speed of the rotor system is much lower than the critical speeds corresponding to the first and second modes, so axial unbalanced forces generally do not affect the normal operation of the rotor system.


Introduction
As the main pump in solar thermal power systems, high-temperature molten salt pumps play a crucial role in the thermal cycle and medium transportation of high-temperature molten salt.They are considered as core equipment in solar thermal power systems [1].High-temperature molten salt pumps consist of two types: the cold salt pump, which transports cold salt from the cold salt storage tank to the hot salt storage tank, and the hot salt pump, which transfers hot salt from the hot salt storage tank to the cold salt storage tank.Compared to the hot salt pump, the cold salt pump has characteristics such as high flow rate, high head, long submerged shaft, and multiple impeller stages, requiring higher performance and reliability.Due to factors such as uneven mass distribution of the impeller, blade damage, uneven stress during rotor design, processing, and assembly, or installation deviations, the cold salt pump may experience unbalanced forces.This can lead to problems such as vibration, noise, and even damage to critical components, which can significantly affect the performance and safety of the pump [2].Therefore, modal analysis and response analysis of the cold salt pump rotor system are of great importance.
As a core equipment in solar thermal power generation systems, high-temperature molten salt pumps have been extensively studied by researchers both domestically and internationally.Guo [3] analyzed the hydraulic performance, structural strength, and structural dynamics of molten salt pumps.Cheng et al. [4] numerically analyzed the solid-liquid two-phase flow inside molten salt pumps with different blade numbers, particle sizes, and particle volume fractions.Wang et al. [5] combined numerical methods with experimental research to optimize and improve the hydraulic structure, thereby enhancing the hydraulic efficiency of liquid molten salt pumps.Li et al. [6] made adjustments to the hydraulic structure of long-axis molten salt pumps to reduce the radial forces on the volute.Zhu [7] compared and analyzed the distribution of static pressure in molten salt pumps under different operating conditions, explaining the influence of particle size and volume fraction on the pressure pulsation characteristics in the time and frequency domains within the molten salt pump.Shao et al. [8] conducted tests on the external characteristics of the pump using water as the test medium, validating the numerical simulation results by obtaining results such as the trajectory of fluid motion, absolute velocity, and performance curves in the flow field.Li [9] conducted numerical simulations of the turbulent flow field inside the molten salt pump and found periodic variations in the impeller and spatial guide vane.Wang [10] performed numerical calculations and analysis of the external characteristics of molten salt pumps using different media and found that the density and viscosity of the molten salt medium have minimal influence on the external characteristics, with results from water testing serving as a reference for the operational performance of high-temperature molten salt pumps.Jin [11] used a fluid-structure-thermal coupling method to analyze the structure of molten salt pumps at different temperatures and discovered the stress distribution patterns in the impeller components.
However, in the current research on molten salt pumps, there is limited consideration for the impact of internal flow on the rotor and the analysis of the pump shaft system response.This paper focuses on the cold salt pump and utilizes ANSYS CFX software and the ANSYS Workbench platform to investigate the modal characteristics, critical speed, and harmonic response of the cold salt pump rotor.By studying these aspects, a better understanding of the performance and behavior of the cold salt pump can be achieved.

Three-dimensional models and mesh partitioning
The text describes a study on a specific cold salt pump.Its main design parameters are as follows: flow rate Q = 820m³ /h, head H = 56m, rotational speed n = 1450r/min.The impeller has an inlet diameter D1 = 300mm, outlet diameter D2 = 462mm, and the number of impeller blades is 7.The fluid domain of the assembled cold salt pump was extracted and divided using Space Claim software, resulting in the fluid domain shown in Figure 1.The calculation domain includes the entire device section from the inlet to the outlet of the cold salt pump.The conveyed medium is 25℃ clear water.To perform the grid independence verification for the cold salt pump under optimal operating conditions and steady-state flow, the fluid domain is divided into grids.As shown in Figure 2, the total number of grids in the fluid domain is 7.84 million.

Figure 2.
The impact of grid resolution on the pump head performance.

Boundary condition settings and constraint settings.
Apply ANSYS CFX software to perform numerical simulation calculations on the entire fluid domain using the standard k-ε model.The boundary conditions are specified as velocity inlet and static pressure outlet.
The structural analysis components of the cold salt pump consist of an impeller and a pump shaft, which are made of material 347H.ANSYS Workbench is used to perform unstructured meshing for the rotor system of the cold salt pump, with a total mesh count of 785,552, as shown in Figure 3.The constraint settings for the rotor system under non-prestress conditions are shown in Figure 4.A cylindrical constraint is applied at the contact interface between the shaft sleeve and the pump shaft to prevent shaft misalignment and constrain the degrees of freedom in the axial and tangential directions of the rotor structure.A fixed support constraint is applied at the end of the shaft to determine the fixed support point of the shaft system.Under prestress conditions, the flow field load needs to be imported from CFX into Mechanical for coupling, and mass force load needs to be included.

Exterior characteristic computation results and analysis.
Numerical calculations were performed on the internal flow field of the cold salt pump, and the resulting pump performance data is shown in Figure 5. From Figure 5, it can be observed that at the design point Q820, the head of the cold salt pump is 454.28m, and the hydraulic efficiency is 78.88%.

Modal analysis.
Modal analysis was performed on the cold salt pump rotor system under prestress and non-prestress conditions, obtaining the first six natural frequencies for comparison.Table 1 shows the mode shapes of the cold salt pump rotor system under non-prestress conditions for the first six modes.It can be observed that due to the periodic symmetry of the rotor structure and boundary conditions, there are cases where the frequency is similar while the vibration direction differs.[12] Additionally, both the natural frequencies and amplitudes tend to increase with the mode order.Figure 6 shows the distribution of the first six mode shapes of the rotor structure under non-prestress conditions.The x and y directions represent the horizontal directions, while the z direction represents the vertical direction.From Figure 6, it can be observed that: Modes 1 and 2 exhibit a swinging mode between the impeller and the rotor, with the maximum displacement occurring between the 5th and 6th stage impellers.Modes 3 and 4 represent first-order bending modes along the x and y directions, respectively, with the maximum displacement occurring at the end of the shaft.2. It can be observed that as the mode order increases, the natural frequencies of the rotor system tend to increase.The natural frequencies of adjacent modes are similar, while the natural frequencies of the first and second mode shapes are relatively lower.Figure 7 shows the mode distribution of the rotor structure under prestress conditions for the first six modes.It can be observed that the natural frequencies and mode shapes under prestress conditions are generally similar to those without prestress conditions.This result aligns with the conclusions reported in existing literature.The first six modes of vibration in the rotor system under prestress conditions.Figure 8 shows the comparison of the first six natural frequencies and amplitudes of the rotor system under prestress and non-prestress conditions.As depicted in Figure 8, the natural frequencies of the rotor system under prestress conditions are higher than those under non-prestress conditions.Among them, the third mode exhibits the most significant increase in natural frequency.This indicates that prestress has an impact on the natural frequencies, and it should be adequately considered when performing harmonic response analysis.

Critical speed calculation and analysis.
The critical speed of a rotor system depends on factors such as the structural characteristics, stiffness, mass distribution, and operating conditions.When the speed approaches the critical speed, significant changes occur in the dynamic characteristics of the system due to the influence of excitation forces or natural vibration frequencies.The vibration amplitude increases sharply and may exceed the system's design capacity, leading to vibration, noise, fatigue failure, or even catastrophic system failure.Therefore, studying the critical speed of a rotor system is of great importance.The vibration of the rotor system can exhibit swirling motion, which can be categorized as either forward or backward whirl depending on whether the direction of swirling is consistent or opposite to the rotation direction of the rotor system, respectively.When forward whirl occurs, the critical speed increases, while backward whirl leads to a decrease in the critical speed.The critical speed is typically

Harmonic response analysis
To explore the influence of unbalanced response on the frequency and vibration of the cold salt pump, harmonic response analysis is conducted using the modal superposition method based on the previous results of prestress modal analysis.In this section, a qualitative analysis of the cold salt pump is performed using prescribed loads to analyze the effects of different magnitudes and positions of loads on the harmonic response.
In the process of conducting harmonic response analysis using the Ansys Workbench platform, the Harmonic Response Analysis module is selected.The analysis is performed based on the results of the prestress modal analysis.In the analysis settings, the frequency sweep is configured with an initial frequency of 0 Hz and a final frequency of 100 Hz.The number of frequency sweeps is set to 50, which means a sweep is performed every 2 Hz.Monitoring points are placed at the same positions on each blade of the impeller.The objective is to analyze the relationship between the displacement of the rotor system and the excitation frequency, as well as the magnitude and position of the unbalanced response.

Harmonic response analysis results of rotor components with added unbalance forces.
Add an unbalanced force of 1000N along the Z-axis on the eighth-stage impeller.Calculate the curves showing the variation of different rotor displacements with frequency under the influence of the unbalanced force, as shown in Figure 9. Also, compute the response curves illustrating the displacements in the x, y, and z directions with respect to frequency, as depicted in Figure 10.From Figure 10, it can be observed that as the excitation frequency increases, the displacements in all directions on the outer surface of the eighth-stage blade reach their peak at a frequency of 52 Hz.This corresponds to the first and second natural frequencies obtained from the prestress modal analysis.The unbalanced response curves in different directions are approximately the same, indicating that the cold salt pump experiences vibrations in all directions when subjected to unbalanced response, with more pronounced vibrations in the vertical direction.From Figure 9, it can be seen that the maximum amplitude occurs on the outer surface of the fifth-stage impeller, with a corresponding displacement of 0.055mm.This indicates that the fifth-stage impeller is the most sensitive to the unbalanced load response.Therefore, subsequent analyses will focus on the fifth-stage impeller.under the influence of unbalance forces of varying magnitudes.It can be observed that by adding different magnitudes of unbalanced forces on the fifth-stage impeller, the displacement amplitude on the fifth-stage impeller follows the same trend with frequency, reaching its maximum value at a frequency of 52 Hz.Additionally, it can be noted that the maximum displacement amplitude shows a significant variation with the change in unbalanced forces, increasing as the magnitude of the unbalanced forces increases.

Harmonic response analysis of different impellers loaded with the same magnitude of unbalance
forces at the same location.By adding a 200N unbalanced force along the Z-axis on each stage impeller and calculating the response curve of the displacement in the Z-axis direction of the fifth-stage impeller at different locations under the same unbalanced force, as shown in Figure 12, it can be observed that the displacement and acceleration responses on the fifth-stage impeller exhibit the same trend with frequency for different load positions on the rotor components.They both reach their maximum values at a frequency of 52 Hz.The maximum displacement response of the rotor component varies with the position of the load addition.When the unbalanced load is added at the fifth-stage impeller, the maximum displacement response is the highest.On the other hand, when the unbalanced load is added at the first-stage impeller, the maximum response amplitude is the lowest.

4.Conclusion
(1) Through modal analysis and comparative study of the rotor system of the cold salt pump under prestress and non-prestress conditions, it was found that the rotor's natural frequencies under prestress are significantly higher, with an increase ranging from 101% to 130% compared to non-prestress conditions.There is no clear pattern among the mode shapes.Therefore, the influence of prestress should be fully considered in rotor dynamics analysis.
(2) The critical speed of the rotor system was calculated and studied.The operating speed of the rotor is 1450 r/min, with a range of ±10% from 1305 to 1595 r/min.This range is much lower than the critical speed of the second mode.It indicates that the rotor system does not experience resonance during operation at the working speed and meets the design requirements of rotor dynamics, ensuring stable operation.
(3) When the cold salt pump rotor is affected by unbalanced forces, vibrations occur in the x, y, and z directions.The area with the largest vibration amplitude is located at the fifth-stage impeller, followed by the sixth-stage impeller.The corresponding maximum response displacement occurs at a frequency of 52 Hz, which does not change with the magnitude and position of the unbalanced forces.Therefore, it is advisable to avoid operating the rotor system near the critical speeds corresponding to the first and second modes when considering the effect of unbalanced forces.Moreover, the operating speed of the rotor system is significantly lower than the critical speed corresponding to the second mode, so unbalanced forces generally do not affect the normal operation of the rotor system.

Figure 1 .
Figure 1.The three-dimensional structure of the fluid domain in a cold salt pump.

Figure 3 .
Figure 3.The mesh for the impeller and rotor.

Figure 4 .
Figure 4. Setting constraints for the structural domain.
Figure6shows the distribution of the first six mode shapes of the rotor structure under non-prestress conditions.The x and y directions represent the horizontal directions, while the z direction represents the vertical direction.From Figure6, it can be observed that: Modes 1 and 2 exhibit a swinging mode between the impeller and the rotor, with the maximum displacement occurring between the 5th and 6th stage impellers.Modes 3 and 4 represent first-order bending modes along the x and y directions, respectively, with the maximum displacement occurring at the end of the shaft.Modes 5 and 6 involve a combined vibration of impeller swinging and blade torsion, with the maximum displacement occurring at the 6th stage impeller.

Figure 6 .
Figure 6.The first six modes of vibration in the non-prestress rotor system.The modal parameters of the first six mode shapes of the cold salt pump rotor system considering the flow field and mass force loads are presented in Table2.It can be observed that as the mode order increases, the natural frequencies of the rotor system tend to increase.The natural frequencies of adjacent modes are similar, while the natural frequencies of the first and second mode shapes are relatively lower.Table2.Modal parameters of the rotor under prestress conditions.
Figure7shows the mode distribution of the rotor structure under prestress conditions for the first six modes.It can be observed that the natural frequencies and mode shapes under prestress conditions are generally similar to those without prestress conditions.This result aligns with the conclusions reported in existing literature.[12][13][14]

Figure 7 .
Figure 7.The first six modes of vibration in the rotor system under prestress conditions.Figure8shows the comparison of the first six natural frequencies and amplitudes of the rotor system under prestress and non-prestress conditions.As depicted in Figure8, the natural frequencies of the rotor system under prestress conditions are higher than those under non-prestress conditions.Among them, the third mode exhibits the most significant increase in natural frequency.This indicates that prestress has an impact on the natural frequencies, and it should be adequately considered when performing harmonic response analysis.

Figure 8 .
Figure 8. Frequency and amplitude distribution of the rotor under non-prestress conditions and under prestress conditions.

Figure 9 .
Figure 9. Harmonic response spectrum plots of each stage impeller under the influence of unbalance forces.

Figure 10 .
Figure 10.Harmonic response spectrum plots for different directions under the influence of unbalance forces.

3. 4 . 2
Harmonic response analysis of the fifth-stage impeller under varying magnitudes of unbalance forces.In the production and installation process of the cold salt pump rotor components, different magnitudes of unbalanced forces can be generated.To investigate the influence of different magnitudes of unbalanced forces on the harmonic response of the rotor components, unbalanced forces of 100N, 200N, 300N, 400N, 500N, 600N, 700N, and 800N are respectively added along the Z-axis direction at the fifth-stage impeller.The analysis is performed to obtain the response graph of the displacement in the Z-axis direction of the fifth-stage impeller with respect to frequency under the influence of different magnitudes of unbalanced forces, as shown in Figure11.

Figure 11 .
Figure 11.Harmonic response spectrum plots of the fifth-stage impellerunder the influence of unbalance forces of varying magnitudes.It can be observed that by adding different magnitudes of unbalanced forces on the fifth-stage impeller, the displacement amplitude on the fifth-stage impeller follows the same trend with frequency, reaching its maximum value at a frequency of 52 Hz.Additionally, it can be noted that the maximum displacement amplitude shows a significant variation with the change in unbalanced forces, increasing as the magnitude of the unbalanced forces increases.

Figure 12 .
Figure 12.Harmonic response spectrum plots of the fifth-stage impeller under the influence of unbalance forces at different positions.

Table 1 .
Modal parameters of the rotor under non-prestress conditions.

Table 2 .
Modal parameters of the rotor under prestress conditions.

Table 3 .
Critical speeds of the first six modes in the rotor system.