Numerical Research on Hydraulically Generated Vibration and Noise of a Centrifugal Pump Volute with Impeller Outlet Width Variation

The impeller outlet width of centrifugal pumps is of significant importance for numbers of effects. In the paper, these effects including the performance, pressure pulsations, hydraulically generated vibration, and noise level are investigated. For the purpose, two approacheswere used to predict the vibration and sound radiation of the volute under fluid excitation force. One approach is the combined CFD/FEM analysis for structure vibration, and then the structure response obtained from the FEM analysis is treated as the boundary condition for BEM analysis for sound radiation. The other is the combined CFD/FEM/BEM coupling method. Before the numerical methods were used, the simulation results were validated by the vibration acceleration of the monitoring points on the volute. The vibration and noise were analyzed and compared at three flow conditions. The analysis of the results shows that the influences of the sound pressure of centrifugal pumps on the structure appear insignificant.The relative outlet width


Introduction
The volute pump type is perhaps the most common type of centrifugal pumps in the world.It is widely applied in industrial and civilian use.In these applications, the vibrations and noise problems are getting more and more attention.Both vibrations and noise can affect the centrifugal pump performance and its life.In centrifugal pumps, the sources of vibrations and noise may lie in hydraulic or mechanical aspects [1].But, under normal operating condition, the blade passing frequency is the most usual excitation of vibrations and noise.The blade passing frequency (BPF) is represented as the product of the number of blades and rotation speed.Large BPF amplitude (and its harmonics) can lead to a lot of noise and vibrations, which may be the source of components wear and bearing failure [2].This frequency is a consequence of the nonuniformity of the flow at the impeller outlet which is caused by the effects of the rotor-stator interaction.
The nonuniformity of the flow exiting the impeller is greatly affected by the impeller outlet width, according to the literature [3].The turbulent dissipation losses in the collector increase with the nonuniformity of the flow at the impeller outlet.As a result, the pump performance and shaft power are affected.Studies have been conducted in order to improve the pump performance through optimizing the outlet width [4][5][6].In respect of vibrations and noise, although these characteristics are expected to vary with the outlet width, the detailed studies have not been conducted.The current work is mainly to investigate the effects of the impeller outlet width on hydraulically generated vibration and noise of a centrifugal pump.

Experimental Setup
In this study, a single entry, single volute centrifugal pump with 5 blades was used as the experimental machine.The impeller is designed to operate at 2900 rpm.The designed flow rate is 50 m 3 /h, and the designed head is 30 m.And its corresponding specific speed   (SI) is 26.7.The study comprises four impellers with different outlet width.These impellers will be termed A, B, C, and D, respectively.General pump geometric values for A, B, C, and D are identified in Table 1.
The experiments were carried out in a close hydraulic test rig, as shown in Figure 1.A more detailed description of the test facility and the experimental procedure can be found in our previous work [28].In present study, the vibration measurements are carried out by using four PCB 352A60 accelerometers with the sensitivity of 10 mv/(m/s 2 ).These accelerometers were fixed on four positions of the volute exterior surface, as shown in Figure 2.

Fluid Simulation
The pump model details are exposed in [28].Figure 3 presents an example of the mesh and the interfaces of the pump.The commercial software CFX was applied to solve the transient fully 3D Reynolds-averaged Navier-Stokes equations in the whole pump.Turbulence was simulated with a k- SST model.The standard wall function was used to calculate boundary layer variables.In the study, the  + on the impeller and volute surfaces is well below 16, satisfying the requirements of the turbulence model and wall function.The boundary conditions are set as a constant total pressure at the inlet and a mass flow rate at the outlet.The transient calculations were initialized from the steady solutions.The average residual convergence criterion was set to be 1E-5.The CFD results were recorded after five impeller revolutions to achieve a stabilized solution.The simulations in this study were carried out on a cluster of twelve Intel Xeon 5600 nodes.
The grid dependence study was carried out through five grid topologies.Further details about the CFD model can be found in our previous work [28].

Volute Structural and Acoustical Simulation Method
The methods for vibration and sound simulation used in the study are illustrated in Figure 4.The approaches consist of the following steps.
A The fluid model and structure model are prepared.
B CFD computation is carried out by using CFX and the nonlinear fluid excitation force on the pump is stored in a time-series.C The structure model is imported into Ansys software.The modal analysis is carried out under constraint.D The time-series fluid excitation force is imported into Ansys by using the APDL tool in the software.The time-series force is treated as the boundary condition.
And the structure response is simulated.
Steps (A)∼(D) are the FEM analyses for vibration.
E The outer surface of the pump structure is extracted and meshed, which is treated as BEM mesh for acoustic simulation.F The BEM mesh is imported into the Sysnoise software.
The pump structure response and modal data are imported into the Sysnoise software.G Define the sound material properties and constraint.H In Sysnoise, the normal velocity distribution on the outer surface nodes of the structure is transferred to the surface nodes of the BEM model.
0 The vibration velocity data on the BEM mesh is set as boundary condition, and then the acoustic simulation is carried out by using Sysnoise.
Steps (E)∼(0 ) are the BEM analyses for acoustic.But for the coupled FEM-BEM algorithm, more steps are need on the basis of steps (A)∼(0 ).For the coupled method, the additional steps are as follows.
1 Import the FEM mesh into the Sysnoise software.
2 Define the solid material properties and constraint for the FEM model in Sysnoise.3 In the software Sysnoise, the pump outer surface is treated as a coupled surface.This surface is used to exchange data between the FEM model and BEM model.In this case, the coupled surface is the same with the BEM mesh; 4 Calculate the response of the coupled surface under vibration excitation from the FEM model and sound pressure excitation from the BEM model.
From the contents mentioned above, the difference between the two approaches is that the coupling surface between the FEM model and BEM model was built in the coupled FEM-BEM methods in order to consider the feedback effects of the sound pressure on the pump structure.

Volute Vibration Simulation Method.
The equation that governs the dynamic response of the structure can be written in the following form: where [M] is the mass matrix, [C] is the damping matrix, [K] is the stiffness matrix, {} is the nodal structural displacement vector, and {P(t)} is the external excitation force vector applied on the nodal structure which is a function of time.
The damping matrix [C] can be a linear combination of the stiffness and mass matrices according to Rayleigh's theory.The formula is where ,  represent the mass and stiffness proportional damping constants, respectively.The two constants can be given as a function of the natural frequency and the damping ratio, as the following equations: where   and   are the th and th mode natural frequency, respectively;   and   are the th and th mode damping ratio, respectively.In present study, both modes are assumed to have the same damping ratio ( =   =   ) and the damping ratio was estimated to be 0.04 [29,30]; then; (3) can be written as The excitation force {P(t)} is calculated by transferring the hydrodynamic load on the CFD mesh to the structure mesh.To realize this process, a discrete data transfer including three steps was developed in this study, as shown in Figure 5.
Step 1 is mesh searching used to find matching CFD surface elements for each structure node, which can be very timeconsuming.To get over this problem, a bucket algorithm developed by Bonet and Peraire [31] was used to reduce the search complexity.The step 2 is mesh matching, which is to find a nearest CFD mesh element to a structure mesh node.The CFD mesh shape can be defined as where r E , represents CFD mesh element nodes, B  (, ) is the element basis functions, and (, ) is the CFD mesh element coordinates.The surface elements for CFD calculations are quadrilateral, as shown in Figure 6.The distance from a structure node (N S  ) to any node on the CFD mesh element can be written in the following formula: The nearest CFD mesh element node (  ,   ) should meet the following equation: Step 3 is the data transfer from fluid surface elements to structure nodes.The data transfer function was defined as [32] where  E  is the pressure on the CFD mesh element nodes and  S  (  ,   ) is the pressure transferred from fluid to the structure nodes.Figure 7 shows the CFD mesh used in the data transfer process.Figure 8 presents the structure mesh of the pump volute, which consists of 59201 elements and 15785 nodes.
The material used in the simulation is iron, with the properties elastic module  = 211 GPa, the density  = 7870 kg/m 3 , the poisson ration ] = 0.29.The excitation force {P(t)} was set as the boundary condition.The constraints were imposed as follows: the nodes of the foundation bolt hole were completed fixed, with   =   =   = 0; the displacement of nodes on the bearing holes was   =   =   = 0;   = 0 on the inlet flanges;   = 0 on the outlet flanges.The boundary set was shown in Figure 8.

Volute Acoustic Simulation.
The boundary element method (BEM) in the Sysnoise software was applied to perform the volute acoustic simulation.The governing equation for radiated sound pressure in the surrounding air induced by vibration of the volute structure can be written in the following form: where r is a position vector of receiver, r 0 is a position vector on the boundary surface, (r 0 ) is acoustic pressure on the boundary surface,  0 u n (r) is the surface normal The outer surface of the structure was extracted and meshed as the acoustic model used in the BEM calculation, as shown in Figure 9.The acoustical mesh consists of 14706 elements and 13573 nodes.As the maximum valid frequency of the model investigated in this study was 4366 Hz, the mesh was sufficiently fine for the blade passing frequency (242 Hz), according to the literature [33].The normal velocity distribution on the outer surface nodes of the structure was transferred to the surface nodes of the acoustic model, which was set as the boundary condition of the volute acoustic simulation.Then, the sound pressure distribution was solved by using the BEM method.

Volute Structural-Acoustic Coupling Simulation Method.
The acoustic solution is carried out by (9).Considering the effect of the sound pressure on the structural vibration, (1) can be written in the following form: where [M] is the mass matrix, [C] is the damping matrix, [K] is the stiffness matrix, {} is the nodal structural displacement vector, {P(t)} is the external excitation force vector, and [L c ]{p(r 0 )} is the load applied on the structure nodes by the sound pressure.The [L c ] is the coupling matrix, which is defined as where N S  is the structure mesh shape function, N A  is the acoustic mesh shape function, {  } is the normal direction of the coupling surface elements, and   is the number of the coupling surface elements.

Results and Discussions
The vibration and noise induced by inner flow in the pump with different impeller outlet width were analyzed and compared at three flow conditions.Before the numerical methods were used, the simulation results were validated by the vibration acceleration of the monitoring points.The detailed analyses and discussions are as follows.

Experimental Validation.
In this experiment, the noise measurements were not carried out because of a shortage of the anechoic chamber.However, the vibration measurements were used to validate the two methods mentioned above.Figure 10 presents the measured spectra of vibration accelerations of the measuring points.It is found that the synchronous vibration at the shaft frequency Ω of 48 Hz dominates the low frequencies.This can be caused by the excitations from the mechanical unbalance, hydraulic unbalance, bent rotor, and excessive run-out of components.There are noticeable peaks at 3Ω and 4Ω due to loose parts, loose bearing, or rubbing.The peaks at the blade passing frequency (BPF) 5Ω are clearly visible in the figure.The BPF vibrations are excited by the rotor-stator interaction.The peak at labeled A represents the discharge pipe resonance excited by high pressure pulsation at the pump outlet.It is observed that the vibration measurements of the sampling point A2 are greatly affected by the discharge pipe resonance, which may lead to great errors.In order to validate the two simulation approaches, the measured vibration accelerations under the flow condition Φ = 0.162 were used and compared with the simulation results, as shown in Figure 10.As can be seen from this figure, there are no significant differences between   the two vibration simulation approaches.This indicated the influences of the noise of centrifugal pumps on the structure appear insignificant.It is found that the BPF amplitudes at the points A1, A3, and A4 show good agreements with the experimental results.However, at the low frequency components and the A component, the differences between the measured and calculated results are relatively large.That is because the behaviors of the rotor and pipe resonance have been neglected.In the simulation only the dynamic surface pressure on the walls is used.As such, the following discussions focus on the BPF component.

Performance Comparisons.
Through the experiments, the detailed performance of the four pumps with different impeller outlet width is obtained.Figures 11(a  b 2 = 10 mm and b 2 = 12 mm.This is because a sufficiently large outlet width is likely to cause a big increase of the nonuniformity of the flow at the impeller outlet.This can cause more turbulent dissipation losses in the volute.

Pressure Pulsation on the Casing Wall.
The pressure on the casing wall obtained by the CFD calculations was processed by using the Fast Fourier Transform with a Hanning Window.Figure 12 presents the amplitudes of the pressure pulsations at BPF on the casing wall.According to the results, the pressure amplitudes increase with growing outlet width.When the impeller outlet width increases, the nonuniformity of the flow at the impeller outlet gets increased, causing higher pressure pulsations.Significant high levels of pressure amplitudes are mainly detected in three regions, including the region around the tongue, the diffuser wall, and the second hydraulic profile of the volute.This is due to the effect of the blade-tongue interaction at BPF.At BPF, the blade trailing edge just passes the tongue leading edge, which causes strong pressure pulsations around the tongue region.When the impeller outlet width increases, the excessive pressure pulsations at the tongue region propagate into the diffuser and the collector along the flow path.Large vibration displacements are to be expected at these regions because of the excessive pressure pulsations.Figures 13(a) and 13(b) show the pressure amplitudes at BPF under the flow conditions Φ = 0.097 and Φ = 0.130, respectively.It can be found that the pressure pulsations at the second profile get larger with decreasing flow rates.And the pressure pulsations around the tongue regions get larger with growing flow rates.This phenomenon will cause great changes in vibration levels at the two regions over the flow rates.velocities take place at six regions.These regions include the region around the tongue, the diffuser wall, the second and the sixth as well as the eighth hydraulic profile of the volute, and the inlet flange.These regions are expected because of the high pressure pulsations according to Figure 12.Also, an appreciable rise of the vibration velocity is clearly visible when the impeller outlet width increases.Figures 15(a) and 15(b) show the velocity amplitudes at BPF under the flow conditions Φ = 0.097 and Φ = 0.130, respectively.In the figures, the angular coordinate represents the circumferential position of the monitoring points (yellow points located in a circle on the casing wall shown in this figure), and the radial coordinate represents the amplitudes of the vibration velocity in -direction.It can be found that there are two peaks of the velocity amplitudes occurring at the second and eighth hydraulic profile of the volute.As can be seen from Figure 15, obvious reduction at the second hydraulic profile is clearly visible with the increase of the flow rate.This is because the pressure pulsations at the second profile of the volute decreases with growing flow rates according to Figure 13.Also, an appreciable rise of the velocity amplitude is clearly visible when the impeller outlet width increases.

Volute Acoustic Simulation.
A spherical acoustic mesh of radius 0.5 m with the volute at its center was used to calculate the directivity distribution of the sound pressure level that radiated from the pump.Figure 16 shows the sound pressure level at BPF under the flow condition Φ = 0.162.It can be found that as the impeller outlet width increases, the sound pressure level increases, with the maximum magnitude in the vertical direction.The big noise in the vertical direction is caused by the high amplitudes of the pressure pulsations and vibration level at the pump outlet regions.Figures 17(a) and 17(b) show the directivity distributions of the sound pressure level at BPF under the flow conditions Φ = 0.097 and Φ = 0.130, respectively.It is found that the sound pressure level increases with decreasing flow rates.This corresponds with the influence of the flow rates on the pressure pulsations and the vibration.As can be seen from Figure 17, although the sound pressure level increased due to the increase of the impeller outlet width, the SPL value of the pump with b 2 = 10 mm is much larger than the others.This indicates that the impeller outlet width should be selected under a certain level.In this case, the level is expected to be less than 10 mm, corresponding to the relative outlet width  * 2 (b 2 /d 2 ) of 0.06.According to the literature, the relative outlet width  * 2 is commonly selected from empirical data.The  * 2 at   (SI) = 26.7 is 0.083 according to the literature [3].And, when the  * 2 exceeds 0.06, the pump losses are extremely large, according to the performance comparisons in Figure 11.Therefore, the relative outlet width  * 2 at   (SI) = 26.7 in this paper should be less than 0.06, based on an overall consideration of the pump characteristics, pressure pulsations, vibration and noise level.

Conclusion
The effects of the impeller outlet width on the hydraulically generated vibration and noise of a centrifugal pump volute were studied.For this purpose, two approaches were used to predict the vibration and sound radiation of the volute under fluid excitation.Before the numerical methods were used, the simulation results were validated by the vibration acceleration

Figure 3 :
Figure 3: Details of pump mesh and interfaces.

Start Step 1 :Figure 5 :Figure 6 :
Figure 5: Data transfer between fluid and structure mesh.

Figure 7 :
Figure 7: Source mesh of the volute.

Figure 8 :
Figure 8: Structure mesh of the volute.

Figure 9 :
Figure 9: Acoustic mesh of the volute.
simulation by FEM Vibration simulation by FEM-BEM coupling Vibration experiment Vibration simulation by FEM Vibration simulation by FEM-BEM coupling

Figure 10 :
Figure 10: Comparisons between measured and calculated spectra of vibration acceleration of the monitoring points.

Figure 11 :
Figure 11: Performance comparisons between the four pumps with different outlet width.
) and11(b)  present the performance comparisons between the four pumps.According to Figure11(a), the head coefficient increases with the increase of impeller outlet width.A big raise of the head and efficiency is observed when the impeller outlet width changes from 8 mm to 10 mm.This indicates the impeller with outlet width b 2 = 10 mm is more suitable for the pump.As can be seen from Figure11(b), there are no significant changes of the efficiency between the pump with

Figure 12 :
Figure 12: Pressure amplitudes at BPF on the casing wall under flow condition Φ = 0.162.
Figure 14   presents the vibration velocity at BPF under the flow condition Φ = 0.162.It can be found that the largest vibration