The effect of the impeller eccentricity on the hydrodynamic characteristics of a centrifugal pump

To explore the effect of impeller eccentricity caused by shaft parallel misalignment on the hydrodynamic characteristics of a centrifugal pump, a numerical simulation model based on computational fluid dynamics (CFD) is established. The SST k-ω turbulence model is used to describe the flow field in centrifugal pump. The eccentric motion of the impeller is described by the sliding mesh method (SMM). The results indicates that the nonuniformity and instability of flow field will be aggravated with the increment of impeller eccentricity, which then reduce the pump head and efficiency to a certain extent. The radial force exerted on the impeller is directly proportional to the eccentricity of the impeller. In addition, the impeller eccentricity has a significant impact on the peak rotation frequency of shaft frequency. This study can provide a numerical reference for eccentricity fault diagnosis and vibration control of centrifugal pump impellers to carry out optimal design and vibration reduction.


Introduction
The centrifugal pump is extensively utilized as a transfer device in engineering applications.Owing to the particular structure characteristics and working environment, the centrifugal pump is prone to failure, which affects the stability of operation.To improve the reliability and service life of centrifugal pumps, scholars have extensively researched the mechanism and performance of centrifugal pump failures.
Yao et al. [1] calculated the fluid excitation force on the impeller by transient state and derived the law of influence of unbalanced mass on the rotor vibration characteristics.Gao et al. [2] used theoretical equations to analyze the bending vibration characteristics of rotor systems; Fu et al. [ 3 ] used an experimental method to analyze the vibration signals of a centrifugal pump under abrasion failure; Zhou et al. [4] investigated the influence of the radius of impeller rear shroud on the operating characteristics of pumps; Wang et al. [5] studied the influence of uneven gap distribution on impeller fluid thrust caused by different radial mounting deviations of the impeller.
As a core component of centrifugal pump, the impeller has a significant impact on the hydraulic performance.In recent years, a series of studies on impeller failure are carry out.Qian et al. [6 ] established a multivariate regression theoretical model of impeller parameters and multistage pumps to study hydraulic performance and axial force; Li et al. [7] studied the eccentric rotor's influence on the excitation force of the sealing ring of the nuclear main pump; Zhou et al. [8] investigated the influence of different pump flow parameters on the fluid-induced-force of the impeller; Chen et al. [9] analyzed the effect of impeller wear ring clearance on the hydraulic of pumps.
Based on the above analysis, most centrifugal pump failures have been thoroughly investigated.However, due to the motion modeling of the impeller and the complexity of the internal flow of the centrifugal pump, the effect of the impeller eccentricity on the flow characteristics of the internal flow field has not been systematically explored.In this study, the characteristics of internal flow in the centrifugal pump under impeller eccentricity failure is investigated by the computational fluid dynamics (CFD) method.The effect of eccentricity failure on the radial force on the impeller surface and outlet pressure pulsation is further studied, which are directly related to the hydrodynamic performance of pumps.For the CFD modeling of pumps with impeller eccentricity, and the exploration of hydrodynamic performance, this study can provide a guidance.

CFD model of Pump
The parameters of the centrifugal pump model are shown in Table 1.The geometry model of fluid consists of five parts, (1) inlet, (2) impeller, (4) clearance (4) volute, and (5) outlet, as shown in figure 1.In order to simplify the study, the clearance leakage at the front and back shrouds of the impeller are ignored, and the conveying medium is incompressible clear water.The solid part is regarded as a rigid body, which is not deformed during the movement.To ensure the full development of internal flows, the inlet and outlet portions are extended properly.show the fluid domain of impeller, with the interface mesh being refined at nodes for improved computational precision.And the boundary layer mesh was divided to ensure the accuracy of the numerical simulation in capturing and reading the wall data.The mesh size at the inlet and outlet is moderately enlarged to save the number of meshes and at the same time ensure the accuracy of the calculation.The SST k-ω turbulence model, which is commonly used in the centrifugal pump simulation analysis, was used in this study for CFD analysis.It can accurately describe the internal flow conditions and correctly predict the liquid flow.Use the SIMPLEC method and the Second Order Upwind format to solve the model and handle the discrete terms.Model calculation convergence accuracy 1e-5, the pressure outflow is set as outlet boundary to simulate the state of the pump outlet condition.
Considering the sensitivity of the computational model to the boundary flow, and the accuracy of the blade surface data readings required in this study, a dimensionless wall distance  + is proposed to evaluate the accuracy of the boundary mesh. + is defined as follows:

Volute
Outlet = √ (2) Where y is the thickness of the first layer of grid,   is the velocity of wall shear stress，  is the wall shear stress.As shown in figure 3, the  + value is close to 1 in most of the regions, where the maximum value is near the outlet at 2.9, which satisfies the requirements of the SST k-omega model calculation.are finally used.

Model of impeller eccentricity
This study investigates the impeller eccentricity due to parallel misalignment failures that occur during the installation and operation of centrifugal pumps.The study focuses on four impeller eccentricity schemes, 0, 0.5, 1.0 and 1.5 mm, respectively.As an example, the impeller eccentricity of 1.5 mm is realized in the impeller eccentricity scheme as shown in figure 5.A parallel misalignment fault resulting in impeller eccentricity causes the geometric center O of the impeller to be offset and not coincide with the center of rotation O'.The center of rotation O' is rotated around the geometric center O of the impeller at a rated speed.At the impeller outlet there is an offset area which connects the impeller outlet to the volute.Adjust the positional relationship between the offset area and the impeller based on the new center.

Hydraulic performance of pump
Figure 6.The effect of eccentricity values on the hydraulic performance.Figure 6 gives the predicted heads and efficiencies under different eccentricity values.The pump heads with eccentricities of 0 mm, 0.5 mm, 1.0 mm and 1.5 mm are 22.756m， 22.105m， 22.083m and 22.057m respectively at the design operating point.And the efficiencies are 49.107%,47.28%, 46.98%, and 46.45%, respectively.It shows that as the flow rate increases from 0.6Qdes to 0.8Qdes, 1.0Qdes, and 1.2Qdes, the head of the pump shows a decreasing trend, same with the characteristics curve of fluid machinery.As the flow increases, the efficiencies improve, indicating that the design point is not the best duty point for this pump.
The maximum deviation occurs between 0mm and 1.5mm, with the maximum difference in flow rate and efficiency being 3.17% and 5.41%, respectively.In figure 6.(a) and (b), the head and efficiency curves drop significantly when there is a parallel misalignment resulting in an impeller eccentricity failure.The pump's performance decreases with increasing eccentricity, but the effect is limited.  =  0.5 2  (3) Figure 7 gives the pressure fluctuation at impeller outlet, plotting the rotational pressure from 0° to 360°.There are six blade numbers on the impeller, the impeller rotates for one week to produce six small peaks and troughs.The failure of impeller eccentricity leads to the generation of an impeller outlet pressure fluctuation.Comparing 0mm-1.5mmoffset distance, the pressure fluctuation increases with the eccentricity distance.In particular, the maximum pressure fluctuations when 0.5mm, 1.0mm and 1.5mm impeller eccentricity failures occur can reach 0%, 3.54%, 6.86% and 10.73% of the pressure fluctuations when no eccentricity failure occurs.

Pressure fluctuations at the impeller outlet
To increase the outlet area and improve hydraulic efficiency, some centrifugal pumps will reduce the thickness of the impeller's outlet blades during the design process.However, this could lead to a certain decrease in the strength of the blades, especially when an eccentric fault of the impeller occurs.Figure 7 and 8 show that the intense pressure wave caused by the eccentric fault at the outlet of the impeller is a challenge to the structural strength of the impeller.
Figure 8 gives the frequency spectrum of pressure fluctuations at the impeller outlet.With an impeller eccentricity failure, the internal fluid domain flow shows significant high frequency fluctuations, especially in the range of 0-20.The frequency amplitude is significantly greater than in the non-eccentric condition, indicating slight vibration and chatter caused by the eccentricity.The BPF is the Blade Passing Frequency, which is 295 Hz in this model.In the event of an impeller eccentricity fault, the BPF does not change significantly, but causes a significant peak in the secondary flap, and the multiplier frequency of the BPF also appears to increase with increasing eccentricity.It is possible to visually determine whether the impeller has an eccentricity fault based on this.This can be used to determine whether there is an eccentricity fault in the impeller.Figure 9 gives the flow velocity fluctuation distribution for four eccentricity values at rated flow.Under the effect of centrifugal force, the flow distribution in each blade channel is similar along the radial outflow without eccentricity failure.From the impeller inlet to the impeller outlet, the flow velocity gradually increases, reaching a maximum at the blade outlet.From the differences in the distribution of velocity clouds, the flow velocity decreases uniformly inside the volute.Impeller eccentricity failure occurs, and the suction chamber and impeller inlet are misaligned by offset.An increase in the lowvelocity region at the impeller inlet was observed from figure 9, while an increase in the high-velocity region at the impeller outlet also occurred.In addition, the low-speed region of the pump where the offset failure occurred was significantly larger than that of the prototype pump, indicating that the eccentricity failure affected the velocity transformation of the fluid, which was not conducive to a smooth and homogeneous change in velocity as the fluid flowed towards the outlet.The low-speed vortex was generated near the outlet, flow irregularities increased, and flow stability was destroyed to some extent; this may be an important reason for the decrease in hydraulic performance such as head and efficiency of the centrifugal pump due to the impeller eccentricity failure.

Effect of the impeller eccentricity on the radial forces
Figure 10.Radial force on eccentric impeller blades.Figure 10 gives the radial force on the eccentric impeller blades.The six blades rotate periodically around the center of rotation.A cycle has six local peaks and troughs, with a small peak and valley at 60°.A sudden change in radial force occurs when the impeller blade outlet edge rotates near the worm gear spacer, similar to the findings in Ref. [10].Without offset, the max value of the impeller radial force is 49.306N, the minimum value is 13.998N, and the change is regular in one rotation cycle.The impeller radial force max value is 67.05N, the min value is 11.26N at 0.5mm offset; the impeller radial force max value is 71.009N, the min value is 3.601N at 1.0mm offset; the impeller radial force max value is 86.456N, the min value is -6.905N at 1.5mm offset.The maximum value of radial force on the blade at 1.0mm offset is 86.456N and the minimum value is -6.905N.
There are two aspects to the eccentricity value of the radial force increase: firstly, the unbalanced radial force on the eccentric impeller increases gradually with the intensification of the eccentricity fault.Especially when the eccentricity distance is 1.5 mm, the eccentric failure causes the extreme value of the radial force of the blade acting on the impeller to increase by 75% compared with that without eccentricity.Secondly, the eccentricity distance increasing, the impeller radial force fluctuates periodically with a larger magnitude.Among them, the original radial force periodic fluctuation is not obvious when no eccentricity occurs, and the radial force fluctuation amplitude increases significantly at 160°-220° when eccentricity occurs, with extreme values appearing, and extreme small values appearing at 0°-30°.

Conclusion
Among the rotor components of centrifugal pumps, the impeller is a key component that affects its hydraulic performance.If failure occurs, the impact on the centrifugal pump is obvious.In this paper, the eccentricity fault of impeller of centrifugal pump is studied by CFD method, and the influence of eccentricity fault on hydraulic characteristics of centrifugal pump is discussed.The internal flow characteristics of a centrifugal pump with varying eccentricity are studied.The main conclusions are as follows: Where shaft failures result in impeller eccentricity failures, the pump head and efficiency will decline to some extent as the eccentricity value increases.An eccentricity failure causes pressure fluctuations at the impeller outlet, and the magnitude of the fluctuations increases as the eccentricity value increases.The shaft frequency is affected by the eccentricity value, and a significant sub-peak appears next to the doubling frequency.The impeller eccentricity fault causes abnormal fluid flow inside the volute.Lowspeed vortices appear near the outlet end of the pump, flow irregularities are exacerbated and flow stability is disturbed to some extent.As the eccentricity value increases, the unbalanced radial force on the eccentric impeller increases.And the eccentric distance increases leading to a greater amplitude of periodic fluctuations in the impeller radial force.

Figure 2 .
Figure 2. Mesh of fluid domain: (a) Impeller; (b) Boundary layers.The discretization of centrifugal pump CFD model and numerical simulation of flow field are carried out by ANSYS Fluent based on the finite volume method.The unstructured tetrahedral cells are used to mesh the model, having regard to the complexity of the internal structure of the model pump.Figure2(a) and (b) show the fluid domain of impeller, with the interface mesh being refined at nodes for improved computational precision.And the boundary layer mesh was divided to ensure the accuracy of the numerical simulation in capturing and reading the wall data.The mesh size at the inlet and outlet is moderately enlarged to save the number of meshes and at the same time ensure the accuracy of the calculation.The SST k-ω turbulence model, which is commonly used in the centrifugal pump simulation analysis, was used in this study for CFD analysis.It can accurately describe the internal flow conditions and correctly predict the liquid flow.Use the SIMPLEC method and the Second Order Upwind format to solve the model and handle the discrete terms.Model calculation convergence accuracy 1e-5, the pressure outflow is set as outlet boundary to simulate the state of the pump outlet condition.Considering the sensitivity of the computational model to the boundary flow, and the accuracy of the blade surface data readings required in this study, a dimensionless wall distance  + is proposed to evaluate the accuracy of the boundary mesh. + is defined as follows: Figure 2. Mesh of fluid domain: (a) Impeller; (b) Boundary layers.The discretization of centrifugal pump CFD model and numerical simulation of flow field are carried out by ANSYS Fluent based on the finite volume method.The unstructured tetrahedral cells are used to mesh the model, having regard to the complexity of the internal structure of the model pump.Figure2(a) and (b) show the fluid domain of impeller, with the interface mesh being refined at nodes for improved computational precision.And the boundary layer mesh was divided to ensure the accuracy of the numerical simulation in capturing and reading the wall data.The mesh size at the inlet and outlet is moderately enlarged to save the number of meshes and at the same time ensure the accuracy of the calculation.The SST k-ω turbulence model, which is commonly used in the centrifugal pump simulation analysis, was used in this study for CFD analysis.It can accurately describe the internal flow conditions and correctly predict the liquid flow.Use the SIMPLEC method and the Second Order Upwind format to solve the model and handle the discrete terms.Model calculation convergence accuracy 1e-5, the pressure outflow is set as outlet boundary to simulate the state of the pump outlet condition.Considering the sensitivity of the computational model to the boundary flow, and the accuracy of the blade surface data readings required in this study, a dimensionless wall distance  + is proposed to evaluate the accuracy of the boundary mesh. + is defined as follows:

Figure 4 .
Figure 4. Grid independence verification.Figure5.Impeller eccentricity model.As shown in figure4, different numbers of meshes were divided for grid-independent verification.The number of grid cells ranges from 0.15 to 2.81 million.The corresponding head and efficiency of pumps with various mesh specifications are obtained by steady-state numerical simulation.The results indicate that the fluctuations of the simulated head and efficiency decreases with the increment of the number of grids.When the number of grids exceeds 1.06 million, the error fluctuations of efficiency and head are less than 1.5%, which verifies the applicability of the CFD numerical model of centrifugal pump.To balance the accuracy of numerical simulation and computational efficiency, 1.93 million grids

Figure 5 .
Figure 4. Grid independence verification.Figure5.Impeller eccentricity model.As shown in figure4, different numbers of meshes were divided for grid-independent verification.The number of grid cells ranges from 0.15 to 2.81 million.The corresponding head and efficiency of pumps with various mesh specifications are obtained by steady-state numerical simulation.The results indicate that the fluctuations of the simulated head and efficiency decreases with the increment of the number of grids.When the number of grids exceeds 1.06 million, the error fluctuations of efficiency and head are less than 1.5%, which verifies the applicability of the CFD numerical model of centrifugal pump.To balance the accuracy of numerical simulation and computational efficiency, 1.93 million grids

Figure 7 .
Figure 7.The pressure fluctuation at impeller outlet.

Figure 8 .
Figure 8. Frequency spectrum of pressure fluctuations at the impeller outlet.The dimensionless pressure coefficient is defined by equation (3) where u is the impeller tangential velocity.

Figure 9 .
Figure 9.Flow velocity fluctuation distribution for four eccentricity values at rated flow.Figure9gives the flow velocity fluctuation distribution for four eccentricity values at rated flow.Under the effect of centrifugal force, the flow distribution in each blade channel is similar along the radial outflow without eccentricity failure.From the impeller inlet to the impeller outlet, the flow velocity gradually increases, reaching a maximum at the blade outlet.From the differences in the distribution of velocity clouds, the flow velocity decreases uniformly inside the volute.Impeller eccentricity failure occurs, and the suction chamber and impeller inlet are misaligned by offset.An increase in the lowvelocity region at the impeller inlet was observed from figure9, while an increase in the high-velocity region at the impeller outlet also occurred.In addition, the low-speed region of the pump where the offset failure occurred was significantly larger than that of the prototype pump, indicating that the eccentricity failure affected the velocity transformation of the fluid, which was not conducive to a smooth and homogeneous change in velocity as the fluid flowed towards the outlet.The low-speed vortex was generated near the outlet, flow irregularities increased, and flow stability was destroyed to some extent; this may be an important reason for the decrease in hydraulic performance such as head