Rotor strength and critical speed analysis of a vertical long shaft fire pump connected with different shaft lengths

The vertical long shaft fire pump (VLSFP) is mainly used in fire-fighting places far away from land and lacking large amounts of water supply. The paper selected the XBC18-178-240LC3 model of VLSFP as the research object. First, the experimental–numerical hydraulic performance of the single-VLSFP was carried out, and then the hydraulic performance of the multi-VLSFP was analyzed by the same numerical simulation method as single-VLSFP. After that, three rotor models (Z4 model, Z5 model-original model and Z6 model) were designed by modeling software, connected by different length and number of the shaft section under the same total length of the intermediate shafts. Finally, the rotor’s strength and critical speed of three models were analyzed and checked via the CFD simulation and the Workbench software. The study mainly found: (1) Through the strength check of the impeller, maximum equivalent stress of the three models was less than the allowable stress of the rotor material, which indicated the structural design of them met the safety requirement; (2) Through the critical speed check of the shafting rotor, the working speed of the VLSFP was lower than 0.8 times the first-order critical speed of the three models, which indicated the rotor could avoid the resonance and the structure of the three models met the dynamic design requirement. According to the stress check of the impeller and the critical speed check of the shafting rotor, combining the time and labor cost when the VLSFP was installed and disassembled many times before and after the test or operation, the paper selected the Z4 model to be the optimal model, which could provide a theoretical support for the subsequent structure design optimization of the vertical long shaft fire pump.


Scientific Reports
| (2022) 12:9351 | https://doi.org/10.1038/s41598-022-13320-z www.nature.com/scientificreports/ dynamic behaviors of rotors. Castillo et al. 21 confirmed that the impact test was a useful method for modal parameters identification of electrical submersible pump. Minette et al. 22 investigated the dynamic behavior of an electrical submersible pump under operational conditions installed in a test well by identification of its natural frequency and damping parameters, using the Least Square Complex Exponential method. Huang et al. 23 studied the modeling method of the rotor blade modes of the turbo-molecular pump, proposing a simplified method of the blade modified model based on the basic invariance principle of the mass and the moment of inertia before and after simplification. However, there is very little literature about the analysis of the rotor modal and critical speed of the shafting of the VLSFP. In addition, in practical engineering project, the VLSFP, which is connected with more shaft segments, needs to take more time and the labor cost when it is installed and disassembled many times before and after the test or operation. Therefore, the paper selected the XBC18-178-240LC3 VLSFP as the research object. First, the experimental-numerical hydraulic performance of the single-VLSFP was carried out, and then the hydraulic performance of the multi-VLSFP was analyzed by the same numerical simulation method as single-VLSFP. After that, three rotor models (Z4 model, Z5 model and Z6 model) were designed by modeling software, connected by different length and number of the shaft section under the same total length of the intermediate shafts. Finally, the rotor's strength and critical speed of three models were analyzed and checked via the CFD simulation and the Workbench software, and the optimal solution was selected from the three models, providing a theoretical support for the subsequent design optimization of the vertical long shaft fire pump.

Numerical methods
Model introduction. In Table 1, the key design parameters of the XBC18-178-240LC3 VLSFP are shown and in Fig. 1, the drawing of the overall pump is shown. For the VLSFP part, its integral pump shaft consists of 7 single shafts (1 section of impeller shaft, 5 sections of intermediate shaft and 1 section of drive shaft) connected by sleeve couplings. In terms of length, the impeller shaft is 2001 mm, the intermediate shaft is 1848 mm (5 sections are the same), the transmission shaft is 2232 mm, and the shaft section gap is 2 mm, resulting in the total of 13,485 mm for the pump shaft. The material of the pump shaft and impeller is duplex stainless steel (00Cr22Ni5Mo3N), which has the following properties: density = 7850 kg/m 3 , elastic modulus = 2.0 × 10 11 Pa, Poisson's ratio = 0.3, tensile strength = 620 MPa, yield strength = 450 MPa, and allowable stress = 250 MPa.
Three-dimensional modeling. The paper uses Creo5.0 software to carry out three-dimensional modeling of the first-and multi-stage fluid domains of the VLSFP, based on its actual size. The fluid domain refers to the water body flowing through each part of the pump, which is the reason that its shape is similar to the structure of the pump. As shown in Fig. 2, the first-stage fluid domain includes the water bodies flowing through the inlet, first-stage impeller, first-stage space guide vane and outlet, and the multi-stage fluid domain includes the water bodies flowing through the inlet, three impellers, three space guide vanes and the outlet.
Computational domain meshing and independence analysis. Considering the geometric characteristics, computing resources and accuracy of the water model, the ANSYS ICEM17.0 software is adopted to mesh the single channel and full channel of the VLSFP, and the boundary layer is processed to ensure the number and quality requirements of grids. Figure 3a is the single-channel grid model, and Fig. 3b is the full-channel grid model. Figure 4 shows the 5 groups of grid independence analysis results of the single-VLSFP model. As shown in Fig. 4, when the number of grids exceeded 8,782,000, the head and efficiency of the single-VLSFP tend to be stable. Therefore, considering the calculation accuracy and time, the fourth group of grids (8,782,000) is finally selected. flow field is carried out by the ANSYS CFX17.0 software. The followings are the settings: water is used as the medium; the RNG k-ε turbulence model is adopted; the dynamic and static interface is set to Frozen Rotor mode; the boundary conditions are set to pressure inlet and mass flow outlet; the reference pressure selects a standard atmospheric pressure; the automatic wall function is selected to process the near the wall area that is set as a smooth wall; the solution discretization is set to the second-order upwind style; and the convergence residual is set to 10 -4 .   www.nature.com/scientificreports/ that the error of efficiency and head measurements is less than 2%. In this paper, the numerical calculation of the hydraulic performance of single-VLSFP is carried out and compared with the experimental data. After obtaining the reliable calculation method for single-VLSFP, the same numerical settings are applied to the multi-VLSFP. The schematic diagram of the single-VLSFP experimental setup is presented in Fig. 5.     Table 2 are respectively head under experiment, head under simulation, relative error of the head, efficiency under experiment, efficiency under simulation and relative error of the efficiency. As shown in Table 2, the relative error of the head and efficiency at the rated operating point are 1.27% and 2.78% respectively. Further away from the rated operating point, the relative errors of the head and efficiency increase, but the maxima are only about 5%, which is within the normal error range. Figure 6 provides a comparison of the hydraulic performance curve obtained by experiment and simulation for the single-VLSFP. It showed the data agreement is perfect. The above results verify that the numerical simulation method is reliable and can be used to predict the internal and external characteristics of the single-VLSFP.  Table 3 shows the simulated hydraulic performance. With reference to Table 2, the simulated head of the multi-VLSFP is much larger than (about 2.3-2.8 times) that of the single-VLSFP under the same working condition, which corresponds with the actual situation. As shown in Fig. 7, the simulated efficiency of the multi-VLSFP is slightly lower than that of the single-VLSFP, which is also in line with the actual situation of the project.    Boundary conditions setting. ANSYS Workbench18.0 is used to perform statics analysis of the rotor. It is necessary to set the boundary conditions of the rotor according to actual conditions. The boundary conditions refer to the loads and supports. For the VLSFP of this paper, three kinds of loads are considered: the fluid force, the gravitational force and the centrifugal force. The fluid force is loaded to realize unidirectional fluidstructure interaction. Here, the fluid-structure interaction refers to a multi-physical field coupling between the laws describing fluid dynamics and structural mechanics. When a flowing fluid contacts with a solid structure, the structure is subjected to the stresses and strains, and these forces deform the structure. In addition, the gravitational force is loaded by adding a vertical downward gravitational acceleration, and the centrifugal force is loaded by adding the rotation speed. For the choice of constraints, since the shafting rotor is cylindrical, the   Fig. 10e shows the gravitational force and centrifugal force, where "A" represents the centrifugal force and "B" represents the gravitational   Figure 10f shows the support setting of the rotor model, in which the "Cylindrical Supports" are represented by the letters "A-I" and the Remote Displacement is represented by the letter "J". The settings of the above loads and supports effectively ensure the consistency between the static calculation and the actual operating state of the rotor. In addition, factors such as gyro torque and sudden load changes have little effect on the results of rotor statics, which are not considered in this paper. Figure 11a-c show the deformation distributions of Z4, Z5 and Z6 models under the four working conditions of 0.2Q d , 0.65Q d , 1.0Q d and 1.2Q d . The three impellers in the figures are called first-, second-and third-stage impeller from right to left, respectively (same below). According to the fluid-structure interaction theory, due to the action of fluid load on the impeller, the deformation of the rotor mainly occurs on the impeller. As shown in the figure, the deformation of the impeller increases as the number of impeller stage increases, and the maximum deformation occurs at the upper crown edge of the third-stage impeller. The reason is that the fluid gained energy due to the rotation of the impeller, and hence the more stages the fluid passes through the impeller, the more energy it obtains. Therefore, the fluid pressure on the first-stage impeller is the smallest, and as the number of the impeller stages increases, the fluid pressure on the impeller gradually increases. In addition, as the flow rate increases, the overall pressure distribution of the impellers becomes more uniform, because the outlet pressure of the impeller under high flow conditions is lower than that under low flow conditions. With the increase of the flow rate, the deformation of the impeller was continuously reduced and becomes relatively more uniform. Such trend is basically consistent with that of the fluid pressure inside the impeller. www.nature.com/scientificreports/ structure interaction, the equivalent stress of the rotor is mainly reflected in the impeller, the variation trend of which is the same as that of deformation. It can be seen from Fig. 12a  Comparison of static characteristics of the three models. Comparison of change trends of total deformation. In order to intuitively reflect the change trends of the static parameters of three models, Fig. 13 shows the deformation maximum of the rotor under the four working conditions of 0.2Q d , 0.65Q d , 1.0Q d and 1.2Q d for the three models of Z4, Z5 and Z6. It can be seen from the figure that the change trends of the three www.nature.com/scientificreports/ models are basically the same, and the deformation is the largest under the small flow condition of 0.2Q d , which is because that the internal flow of the impeller is not uniform, and then some areas of the impeller would be subjected to larger force under the small flow condition. In addition, compared with Z4 and Z6 models, the deformation value of Z5 model (original model) represented by the red line has a greater downward trend with the increase of flow rate. Under rated conditions and beyond, the maximum deformation of Z5 model is always smaller than Z4 or Z6 model, which indicates that the structural design of Z5 model is more reasonable. However, the maximum deformation of Z4 and Z6 models is within a reasonable range (the rotor deformation grade is generally:10 -2 -10 −1 mm), which also meets the requirements of the rotor structure design.

Rotor equivalent stress.
Comparison of change trends of equivalent stress. Figure 14 shows the maximum equivalent stress of Z4, Z5 and Z6 under the four working conditions of 0.2Q d , 0.65Q d , 1.0Q d and 1.2Q d . It can be seen from the figure that the variation trends of the three models are basically the same, with the maximum value always occurs under the small flow condition of 0.2Q d . The reason of this phenomenon is also that the internal flow of the impeller is not uniform, and then some areas of the impeller would be subjected to larger force under the small flow condition. In addition, compared with Z4 and Z6 models, the equivalent stress value of Z5 model (original model) is the smallest, which indicates that the structural design of Z5 model is more reasonable. Though the maximum equivalent stress of Z4 and Z6 models is larger than Z5 model, it is still less than the allowable stress of the rotor material, which is also in line with the structural design safety requirements.

Critical speed analysis of three models under different shaft lengths
Rotor natural frequency analysis. Figure 15 shows the first 12 natural frequencies of the three models of Z4, Z5 and Z6 in dry mode. Dry mode refers to the inherent mode of the structure in air, regardless of the influence of the surrounding fluid on the structure mode. Since the water mass of the impeller could be neglected, compared with the mass of the whole shafting rotor, and the pre-stress has little influence on the mode of the rotor, the paper chooses to analyze the inherent characteristics of the shafting rotor in the dry mode. As shown in Fig. 15, with the increase of the order, the natural frequency of the rotor generally presents an increasing  Rotor vibration analysis. The vibration shapes of the rotor can reflect the vibration and torsion amplitudes of each part of it, which is beneficial to find weaker part of the structure design. The first 12-order vibration shapes of three models of Z4, Z5 and Z6 are shown in Fig. 16. Here, the same order of vibration shapes of the three models is put together for analysis and the three models from top to bottom are Z4, Z5 and Z6 respectively. The coordinate system in the lower right corner reflects the actual orientation of the rotor, and the displacement magnification is 100. It can be seen in the figure that for three models, the first-order and second-order vibration shapes are the bending deformation and the largest deformation appears the middle of the first bearing and the first intermediate shaft. However, the higher the order, the more complex the vibration shape. For third-order vibration shapes and above, deformations of the rotor consist of transverse and torsional vibrations, and the  Rotor critical speed analysis. Table 4 shows the critical speeds of the three models. In the table, "WD" represents the Whirl Direction, "CS" represents the Critical Speed, "BW" represents the Backward Whirl, and "FW" represents the Forward Whirl. According to the theoretical basis of rotor dynamics, the critical speeds corresponding to the "FW" are the actual critical speeds of the rotor. It can be seen from the Similarly, for Z6 model, the first-order critical speed, 3488.9 r/min, is even greater, which indicates the structure of Z6 model meets the dynamic design requirement. Figure 17 shows the critical speeds of Z4, Z5, and Z6 models in the first 12 modes. In the figure, the 6 green circles on the black curve represent the first 6-order critical speeds of Z4 model, the 5 purple circles on the red curve represent the first 5-order critical speeds of Z5 model, and the 6 brown circles on the blue curve represent the first 6-order critical speeds of Z6 model. In addition, the first-order critical speeds of three models all appear in the second-order mode. However, the second-order critical speed of Z4 model appears in the fourth-order mode, whereas that of Z6 model appears in the fifth-order mode, and that of Z5 model appears in the sixth-order mode. This indicates that the second-order resonance of Z5 model (original model) is delayed, compared with Z4 and Z6 models. Furthermore, it can be seen from the figure that the critical speed of Z6 model is the largest, and the critical speed of Z4 model is the smallest. The reason for this phenomenon is the same as in "Rotor natural frequency analysis" section.

Optimal model selection.
According to the stress check of the impeller, the critical speed check of the shafting rotor and so on, all three models (Z4 model, Z5 model and Z6 model) meet the requirements of the rotor structure design and the dynamic design. However, in actual engineering project, the VLSFP, which is connected with more shaft segments, needs to take more time and the labor cost when it is installed and disassembled many times before and after the test or operation. To sum up, the paper selects the Z4 model to be the optimal model and provides a theoretical support for the subsequent design optimization of the vertical long shaft fire pump.

Conclusions
The paper selected XBC18-178-240LC3 VLSFP as the research object, using modeling software to design three models (Z4 model, Z5 model and Z6 model) connected by different length and number of the shaft section under the same total length of the intermediate shafts, and then the rotor's strength and critical speed of three models were analyzed and checked via CFD simulation and Workbench software. The main conclusions were as follows: 1. With the increase of the flow rate, the deformation and equivalent stress of the impeller was continuously reduced and became more uniform. www.nature.com/scientificreports/ 2. Through the strength check of the impeller, maximum equivalent stress of the three models was less than the allowable stress of the rotor material, which indicated the structural design of them met the safety requirement. 3. Through the critical speed check of the shafting rotor, the working speed of the VLSFP was lower than 0.8 times the first-order critical speed of the three models, which indicated the rotor can avoid the resonance and the structure of the three models met the dynamic design requirement. 4. According to the stress check of the impeller and the critical speed check of the shafting rotor, combining the time and labor cost when the VLSFP was installed and disassembled many times before and after the test or operation, the paper selected the Z4 model to be the optimal model, which can provide a theoretical support for the subsequent design optimization of the vertical long shaft fire pump. 5. The paper lacked the experimental verification of the dynamics. Due to the complex structure and large volume of the VLSFP, there was no corresponding test site at present. Therefore, in the subsequent research, scaling down the VLSFP model would be considered, and then the rotor dynamics test would be carried out.

Data availability
The datasets generated or analyzed during the current study are available from the corresponding author on reasonable request.