Dynamic Characteristic Analysis of Centrifugal Pump Impeller Based on Fluid-Solid Coupling

Abstract

Purpose: Centrifugal pumps are prone to vibration problems during operation due to poor dynamic characteristics of its impellers that serve as the only running parts of such devices; the dynamic characteristics of the impeller during operation are the main reasons for the vibration of the centrifugal pump. Therefore, it is important to study the internal fluid flow and its influence on the dynamic characteristics of the pump impeller and to explore the causes of vibration during the transient start-up process. The understanding of such phenomena may lead to better design of such impellers. Methods: The geometry of the flow channel inside the centrifugal pump is established using Creo 4.0 software (American PTC company). The internal fluid flow computer simulation is carried out using Flomaster V9 software (UK Flowmaster company) to obtain the variation law of speed and flow during the start-up of the centrifugal pump, which is loaded into the simulation calculation of the centrifugal pump. The variation of speed and flow during the start-up process was further processed using the fluid-structure coupling method, and the structural vibration characteristics of the impeller under transient radial force are obtained by harmonic response analysis. Results: During the starting process of the centrifugal pump, the speed and flow first increased sharply and then decreased until reaching a stable process. During this period, the impeller vibration changed sharply; the overall vibration amplitude increased and fluctuated stably at the amplitude of 0.01 mm. In the unsteady numerical simulation of the centrifugal pump, the radial force on the impeller changes periodically. The time domain signal is transformed into a frequency domain signal, and the fundamental frequency of the impeller and the passing frequency of the blade are 101.67 Hz and 610 Hz, respectively. Conclusions: The radial force is the main cause of impeller vibration, and the transient radial force has the least dynamic impact on the impeller structure under the design condition and has a relatively large impact under the off-design condition. In order to ensure the stable operation of the centrifugal pump, it is necessary to avoid the centrifugal pump working under the non-standard flow condition, especially the small flow condition.

1. Introduction

The centrifugal force generated by the centrifugal pump when the impeller rotates makes the fluid obtain energy, so that the pressure energy and kinetic energy of the fluid after passing through the impeller are increased, so as to transport the fluid [1,2]. The common centrifugal pump is shown in Figure 1; the impeller is installed in the spiral casing, and when the impeller rotates, the fluid is transported through the suction chamber axial flow and then turns 90° into the impeller channel and flows out radially to reach the pressure chamber through the diffusion tube discharge. During the starting process of the centrifugal pump, the speed and flow do not directly reach the rated value but are accompanied by a rising process. In this process, the force on the impeller is uneven, which is very prone to vibration instability. In the normal operation of the centrifugal pump, its internal flow field is not uniform flow. The fluid force generated in the flow field, including flow pulsation, rotating cavitation, etc., the fluid force caused by the interference of the suction chamber and discharge chamber, and the fluid force caused by the dynamic and static interference between blade and diaphragm all force the impeller to vibrate. These fluid forces, which will impact the impeller at any time, induce the impeller to vibrate.

At present, many scholars have studied the dynamic characteristics of centrifugal pump impellers. Pavesi et al. [3] and Yuan et al. [4] analyzed the frequency characteristics of pressure pulsation of unsteady flow in centrifugal pumps by an experimental method. Abdolahnejad et al. [5] studied the influence of the slip coefficient of a centrifugal pump on its performance characteristics and design parameters by means of numerical simulation and test. Derakhshan et al. [6] optimized the impeller shape of centrifugal pump based on artificial bee colony algorithm, and the results showed that the efficiency of the Berkeh 32–160 centrifugal pump increases by 3.59% when the total differential pressure increases by only 6.89 m. Based on CFD-PBM model, Ge et al. [7] studied the influence of inlet gas volume fraction on void fraction in centrifugal pumps. Gu et al. [8] discussed the influence of solid particle concentration on the performance of an MP80-160 centrifugal pump, and the results showed that the higher the solid particle concentration was, the greater the centrifugal pump head was and the lower the efficiency was. Hermez et al. [9] used the standard k–s turbulence model to study the centrifugal pump impeller, and the results showed that under low flow, the total differential pressure of the centrifugal pump increased by 22.1% and the maximum efficiency could reach 92.3%. Ji et al. [10] used the Wray Agarwal (WA) turbulence model to simulate the internal and external characteristics of centrifugal pump, and the results showed that the WA model could effectively calculate the energy performance of centrifugal pump under various working conditions, and had high accuracy. Jia et al. [11] through research showed that with the increase of flow, the stability of the external characteristics of centrifugal pump decreases, the stability of open impeller was the worst, while the stability of closed impeller under large flow was the best, and the shell vibration of closed impeller was the smallest. Based on the N-S equation, Zhang et al. [12] proposed a design method of a high-speed centrifugal pump. Khoeini et al. [13] studied the effects of diffuser blade geometric parameters and impeller micro groove depth on the performance of a vertical suspension centrifugal pump through the combination of tests and numerical simulation. Using a large eddy simulation turbulence model, Kuang et al. [14] analyzed the effect of impeller inclination on the internal flow characteristics of a centrifugal pump under design Q) and off design (0.55 Q). Li et al. [15] and Shi et al. [16] numerically simulated the three-dimensional turbulent flow field in the tandem vane centrifugal pump, gave the critical geometric parameters of the combined impeller and volute, and clarified the influence of the combined impeller on the flow characteristics of the volute of the centrifugal pump. Lu et al. [17] carried out the numerical calculation of the whole flow channel of the centrifugal pump under the conditions of different flow and effective net positive suction head. The results show that cavitation occurs near the tongue of the centrifugal pump under the condition of large flow. Quan et al. [18] believed that the inducer could offset the axial force of the centrifugal pump and improved the stability of the pump. The research results of Wu et al. [19] showed that for a pump with a broken impeller, under the condition of Q, the head decreases by 9.85%, the efficiency decreases by 1.06%, and the vibration at the outlet flange increases the most. Due to the blade fracture, the amplitude between the peak value of pressure fluctuation at the pump tongue and the pump outlet increases by 4.7% and 9.5% respectively. Wang et al. [20] used a laser vibrometer to measure the vibration of the centrifugal pump and analyzed the distribution law of impeller vibration and pressure pulsation. Wu et al. [21] studied the influence of blade pressure profile on the hydraulic and dynamic performance of a low specific speed centrifugal pump through experiment, and the result indicated that the unsteady pressure pulsation of a centrifugal pump can be effectively alleviated by blade PS modification. Yuan et al. [22] calculated the turbulent flow and structural response of three impellers with different parameter designs by using CFX and ANSYS Workbench and found that the closed impeller had the worst stability and the best hydraulic performance, while the split impeller had the worst stability and the best hydraulic performance. According to the research of Adamkowski et al. [23], Song et al. [24], and Jaiswal et al. [25], with the occurrence of cavitation, the flow pattern at the impeller hole of the centrifugal pump deviates from the ideal situation, and vibration will occur on the blade, resulting in noise in the pump. Cui et al. [26] used Fluent to conduct unsteady numerical simulation of a double-volute multistage centrifugal pump, and the results showed that in the three flows of 0.6 Q, Q, and 1.2 Q, the maximum radial force appeared at 0.6 Q, and the balance of radial force should be considered in the design of multistage centrifugal pump. Nan et al. [27] studied the effects of flow, number of blades, outlet installation angle, and impeller outer diameter on the dynamic response of centrifugal pump, and gave the optimal design parameters of centrifugal pump from the perspective of vibration reduction.

In general, scholars have studied the dynamic characteristics of centrifugal pump widely. Cavitation mainly occurs at the front and back of the impeller and the inner surface of the front cover. During centrifugal pump startup, cavitation is due to the fact that when the fluid flows through the flow passage parts of the pump, with the decrease of fluid pressure, when the fluid pressure near the blade is lower than the saturated vapor pressure of the fluid itself, the fluid will vaporize and produce bubbles. As the bubble continues to increase, it collapses and disappears under external conditions such as gas dissolution and steam condensation, thus causing local water hammer and cavitation. When the centrifugal pump is cavitating, it will cause the change of the external characteristics of the pump and produce vibration. There are many sources for centrifugal pump vibration, such as bent shafts, lack of balance, misalignments, reaction forces, and contact between components. Due to the asymmetry of the worm casing structure of the centrifugal pump, when the impeller drives the fluid work, the complex unsteady flow will produce radial force on the impeller, which will make the impeller vibrate and deform. As a result, the hydraulic performance of the centrifugal pump decreases.

In this paper, the structure of the centrifugal pump and its flow channel model are established by Creo 4.0 software, and the centrifugal pump simulation experiment platform is built by Flomaster V9 software to obtain the variation law of speed and flow during the start-up of the centrifugal pump, which is loaded into the simulation calculation of centrifugal pump, and the impeller dynamic characteristics during the start-up of centrifugal pump are analyzed by the fluid-structure coupling method. Then, the steady-state radial force of centrifugal pump under different working conditions is calculated and analyzed, the variation law of radial force and flow of impeller is obtained. The variation law of transient radial force and time of impeller under unsteady flow is analyzed, the vibration characteristics of centrifugal pump impeller under the action of radial force are studied, and the frequency of vibration instability of impeller under the action of radial force is obtained.

2. Fluid-Solid Coupling Theory of Centrifugal Pump

The dynamic equation of centrifugal pump impeller is:

$$M_s\ddot{X}+C_s\dot{X}+K_{s}X=F_s(t)$$

(1)

In Equation (1), M_s is the mass matrix of centrifugal pump, C_s is the damping matrix of centrifugal pump, K_s is the stiffness matrix of centrifugal pump; $\ddot{X}$ is the acceleration vector, $\dot{X}$ is the velocity vector, and X is displacement vector; F_s(t) is the prestress of centrifugal pump, namely: fluid force, self-gravity, and self-rotating centrifugal force.

The mode of centrifugal pump impeller in air is dry mode and undamped mode. Therefore, C_s = 0, F_s(t) = 0, and the dynamic equation of Equation (1) can be simplified as:

$$M_s\ddot{X}+K_{s}X=0$$

(2)

For the case of linear small disturbance, the ideal fluid is inviscid, uniform, and irrotational. According to Euler equation, the dynamic balance equation or motion equation of fluid can be derived as follows:

$$\left.\begin{array}{l}\rho \frac{\delta v_x}{\delta t}=-\frac{\partial p}{\partial x}\\ \rho \frac{\delta v_y}{\delta t}=-\frac{\partial p}{\partial y}\\ \rho \frac{\delta v_z}{\delta t}=-\frac{\partial p}{\partial z}\end{array}\right\}\mathrm{or}\left.\begin{array}{l}\rho \frac{{\partial }^{2}u}{\partial t^2}=-\frac{\partial p}{\partial x}\\ \rho \frac{{\partial }^{2}v}{\partial t^2}=-\frac{\partial p}{\partial y}\\ \rho \frac{{\partial }^{2}w}{\partial t^2}=-\frac{\partial p}{\partial z}\end{array}\right\}$$

(3)

where, ρ is the density of the fluid, u, v, and w are the displacement components of the fluid particle respectively, v_x, v_y, and v_z are the velocity components of the fluid particle respectively, and p is the pressure of the fluid particle.

Set k as the compression modulus of the fluid, and the continuity equation of the compressible fluid is:

$$-\left(\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z}\right)=\frac{1}{k}\frac{\partial p}{\partial t}$$

(4)

From Equation (3) to Equation (4), the derivative of time t can be obtained, and the fluid motion control equation is:

$${\nabla }^{2}p=\frac{1}{c^2}\frac{{\partial }^{2}p}{\partial t^2}$$

(5)

In Equation (5), c is the velocity of sound in the fluid, and its magnitude is determined by the formula c = (k/ρ_f)^1/2, ρ_f is the density of the fluid. ${\nabla }^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}$${\nabla }^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}$ is Laplace operator.

The effect of fluid medium on impeller structure can be expressed as:

$$n\nabla P=-{\rho }_{f}n\ddot{u}$$

(6)

In Equation (6), ∇P is the pressure gradient along the normal vector n, and $\ddot{u}$ is the velocity vector of a liquid particle.

The centrifugal pump impeller structure is discretized by finite element method to obtain the fluid dynamic equation:

$$M_f\ddot{p}+K_{f}P-\rho R^{T}X=F_f$$

(7)

In Equation (7), M_f and K_f are mass matrix and stiffness matrix of fluid, respectively, R is the coupling matrix between the fluid and the structure, and F_f is the external force acting on the fluid.

For the action of acoustic fluid, the dynamic equation of impeller structure is:

$$M_s\ddot{X}+C_s\dot{X}+K_{s}X-R^{T}P-F_0=F_P(t)$$

(8)

In Equation (8), F_p(t) is the surface force vector on the fluid solid contact surface, and F₀ is the fluid force applied to the impeller structure.

The coupling three-dimensional equation of centrifugal pump impeller structure and flow field can be obtained by combining Equations (7) and (8):

$$\left[\begin{array}{cc}{M}_s& 0\\ M_{fs}& M_f\end{array}\right]\left[\begin{array}{c}\ddot{X}\\ \ddot{P}\end{array}\right]+\left[\begin{array}{cc}{C}_s& 0\\ 0& C_f\end{array}\right]\left[\begin{array}{c}\dot{X}\\ \dot{P}\end{array}\right]+\left[\begin{array}{cc}{K}_s& K_{fs}\\ 0& K_f\end{array}\right]\left[\begin{array}{c}X\\ P\end{array}\right]=\left[\begin{array}{c}{F}_0\\ 0\end{array}\right]$$

(9)

In Equation (9), M_fs and K_fs are equivalent coupling mass matrix and stiffness matrix respectively.

3. Dynamic Characteristics Analysis of Centrifugal Pump Impeller during Start-Up

3.1. Flomaster Simulation Experiment

The centrifugal pump model studied in this paper is Q25H52: Rate of flow Q = 25 m³/h, rotating speed n = 6400 r/min, lift H = 52 m, impeller inlet diameter D₁ = 44 mm, outlet diameter D₂ = 100 mm, outlet width b₂ = 8 mm, number of blades Z = 6, and the medium density of the pump is ρ = 1200 kg/m³, made of titanium alloy, its performance parameters are shown in

Table 1. Impeller material performance parameters.

Parameter	Value
Young’s modulus/GPa	96
Density/(kg/m3)	4620
Poisson’s ratio	0.36
Yield strength/Mpa	800

, and the characteristic curve of Q25H52 centrifugal pump is shown in Figure 2. The full flow channel model of centrifugal pump is shown in Figure 3. The three-dimensional modeling of centrifugal pump is carried out through Creo 4.0 software, and the flow field model is divided by ANSYS Workbench Design Modeler 19.0.

As shown in Figure 4, the simulation experiment platform is built through Flomaster V9 software, which is connected by centrifugal pump, rigid pipe, valve, reservoir, signal generator, and instrument template.

By setting the parameters of each component, the variation data of motor speed and flow with time in the first two seconds of the centrifugal pump start-up process can be obtained. As shown in Figure 5, the speed and flow of the centrifugal pump do not directly reach the rated speed and flow at the beginning but go through a process of change. The speed and flow first increase rapidly until they exceed the rated speed and flow, and then begin to decrease. It presents a wave type change and remains relatively stable near the rated speed and rated flow.

3.2. Vibration Analysis of Impeller during Start-Up

The starting process of centrifugal pump belongs to transient flow. In CFX flow field calculation, set the time step as 0.005 s, and the total calculation time is 2 s. Each time step is saved once, and a total of 400 results are saved. In Figure 6, the fluid domain and impeller solid domain of the centrifugal pump are shown. The time step and total time must be consistent with the setting in the flow field to facilitate the coupling between fluid and solid. The impeller is subject to cylindrical constraints. The setting of speed is the data of speed changing with time obtained from Section 3.1, the formula of speed changing with time is obtained by MATLAB R2018b curve fitting, and the function with time as variable is applied to the simulation of centrifugal pump starting process.

The change of radial force on the impeller during the start-up of the centrifugal pump is shown in Figure 7. Within 0.1 s of the start-up process, the transient radial force shows a sharp upward trend as a whole, with the maximum value close to 300 N. Within 0.1 s < t < 0.3 s, the transient radial force shows a short decline. After 0.3 s, the liquid pressure in the impeller channel remains relatively stable as the speed and flow gradually approach the rated working condition, and the transient radial force fluctuates stably around 50 N.

As shown in Figure 8, during the start-up condition, the amplitude of the impeller increases sharply and then tends to be stable, although it decreases slightly in 0.25 s < t < 0.5 s, and then returns to be stable. Within 0.25 s during startup, the overall vibration amplitude of the impeller shows a sharp upward trend, with the maximum value of 0.013 mm. Within 0.6 s, the vibration amplitude of the impeller shows a sharp fluctuation. At 0.25 s < t < 0.5 s, the vibration amplitude of the impeller shows a downward trend. After 0.6 s, as the radial force on the impeller tends to be stable; the vibration amplitude of the impeller gradually remains stable and fluctuates stably at 0.01 mm, with a fluctuation range of 0.009 mm to 0.011 mm.

Figure 9a shows the change of impeller acceleration during the startup of the centrifugal pump. Firstly, the acceleration increases rapidly to mm²/s within 0.2 s. Due to the short time and small speed change, the acceleration decreases rapidly within 0.2 s < t < 0.25 s, and its size is 500 mm²/s. Then, with the stability fluctuation of radial force, the acceleration tends to be more stable, and the fluctuation range is between 0 and 2000 mm²/s.

The change of impeller speed during the start-up of the centrifugal pump is shown in Figure 9b. The speed of the impeller fluctuates violently during the whole start-up process. After the centrifugal pump is started, the speed quickly reaches the maximum value of 10 mm/s within 0.13 s and then rapidly decreases to around 1.8 mm/s within 0.13 s < t < 0.25 s. After 0.25 s, as the acceleration of the impeller tends to be stable, the fluctuation range of the impeller speed decreases, and the fluctuation range is between 0 and 5 mm/s. Through the analysis of the dynamic characteristics of the impeller during the start-up of the centrifugal pump, it can be found that its change law is similar to the change law of the transient radial force of the impeller during the start-up.

4. Analysis of Unsteady Flow Induced Impeller Vibration

4.1. Steady State Radial Force Analysis

According to the simulation calculation, the change trend of radial force with the change of flow under different design conditions is shown in Figure 10a. The radial force is not zero under the design condition, which is due to the asymmetry of the structure of the centrifugal pump itself. After the fluid flows out of the impeller, the pressure is unevenly distributed in the circumferential direction, resulting in the asymmetry of fluid velocity and flow in the impeller, resulting in the non-zero radial force. The radial force is the smallest under the design condition. When it deviates from the design flow, the radial force becomes larger, and the greater the deviation from the design flow, the greater the radial force.

F₁, F₂, F₃, and F₄ in Figure 10b are the radial force direction distribution of centrifugal pump impeller under 0.8 Q, Q, 1.2 Q, and 1.5 Q working conditions, respectively. Under the conditions of large flow of 1.5 Q and 1.2 Q, the vector of radial force during flow is in the fourth quadrant of the rectangular coordinate system. Under the condition of small flow of 0.8 Q, the radial force is in the second quadrant. Under the condition of design flow Q, the radial force is not zero and the direction is in the first quadrant.

4.2. Impeller Vibration Analysis

4.2.1. Calculation and Analysis of Transient Radial Force

Impeller vibration is a dynamic process that changes with time. In the actual working state, the internal flow of the centrifugal pump belongs to unsteady flow, and the radial force is a function of time. Through the unsteady simulation of the centrifugal pump, the results are saved at each time step, and the functional relationship between the radial force and time is obtained.

The unsteady numerical simulation calculation is carried out under the flow of 0.8 Q, Q, 1.2 Q, and 1.5 Q, respectively, and the variation of radial force within six cycles of impeller rotation is obtained. Set every 15° rotation of the impeller as a calculation time point, that is, a time step. After iteration, it is obtained that each time step converges. Calculate the radial force at each time step and save the results at each time step, so as to obtain the variation relationship between the radial force and time. As shown in Figure 11, due to the dynamic and static interference between the impeller and the volute, there is violent pressure pulsation on the impeller surface, so that the transient radial force on the impeller also has obvious pulsation. The radial force shows periodic changes, with six peaks in a rotation cycle, which is equal to the number of times that the blade sweeps through the volute tongue within one rotation cycle of the impeller, indicating that there is violent dynamic and static interference between the blade and the volute tongue. Under the design flow condition, the radial force, the radial force received by the impeller at each time within one rotation cycle is the smallest, and its size fluctuates between 0 and 80 N. Under non-standard flow conditions of 0.8 Q and 1.2 Q, the radial force fluctuates between 0 and 150 N, while under 1.5 Q flow conditions, the radial force fluctuates between 0 and 300 N. It can be concluded that the greater the deviation from the design flow of the centrifugal pump, the greater the radial force received by the impeller at each time. This is consistent with the variation law of radial force and flow in steady-state calculation. Therefore, in order to ensure the stability and safety of the impeller, the centrifugal pump should try to avoid the flow under non-standard working conditions.

4.2.2. Harmonic Response Analysis

The function of radial force in the frequency domain is obtained by Fourier transform of the function of inner diameter force varying with time in six cycles of impeller rotation, and the obtained function in the frequency domain is loaded into the impeller structure as an external load, as shown in Figure 12.

Calculate the harmonic response analysis under four different working conditions: small flow condition 0.8 Q, standard condition Q, and large flow conditions 1.2 Q and 1.5 Q. According to the calculation results of harmonic response analysis, the amplitude frequency diagrams of the impeller under four different working conditions are obtained in Figure 13. The amplitude of the frequency under four different working conditions is relatively large at 101.67 Hz, which is the fundamental frequency of the impeller; it shows that the vibration of the impeller is closely related to the rotation of the impeller. There is also an obvious peak at the place where the frequency of four different working conditions is 610 Hz, that is, there is a sinusoidal component with the frequency of 610 Hz, which is equal to the passing frequency of the blade of the centrifugal pump. This shows that the interaction between the blade and the diaphragm has an obvious impact on the vibration of the impeller.

By comparing the amplitude of the impeller in Figure 13, the vibration amplitude of the impeller structure is the smallest under design condition Q, the vibration amplitude of the impeller structure is relatively large under small flow condition 0.8 Q and large flow conditions 1.2 Q and 1.5 Q, and the vibration amplitude is the largest under large flow condition 1.5 Q, indicating that the vibration amplitude of the centrifugal pump under rated flow is the smallest, and the greater the deviation from rated flow is, the greater the vibration amplitude is.

5. Conclusions

Taking the Q25H52 centrifugal pump as the research object, this paper studies the dynamic characteristics of centrifugal pump impeller by means of numerical simulation. The conclusions are as follows:

(1)

During the start-up process of centrifugal pump, the speed and flow do not reach the rated value instantaneously but experience the process of rapidly increasing first and then decreasing until reaching stability. During this period, the impeller vibration changes violently; the overall vibration amplitude increases and fluctuates stably at the amplitude of 0.01 mm. The speed and acceleration fluctuate violently and finally stabilizes within a range of fluctuation.

(2)

Under low flow condition 0.8 Q and high flow condition 1.2 Q and 1.5 Q, the radial force on the centrifugal pump impeller increases compared with the design flow Q, and the greater the deviation from the design flow, the greater the radial force.

(3)

In the unsteady numerical simulation of the centrifugal pump, the radial force changes periodically. Every rotation cycle, the transient radial force presents six peaks, which is consistent with the number of times that the blade sweeps through the volute tongue when the impeller rotates one cycle.

(4)

The time domain signal is transformed into the frequency domain signal to obtain the vibration peak at 101.67 Hz and 610 Hz, which are the fundamental frequency of the impeller and the blade passing frequency, respectively. Under the design condition, the dynamic influence of radial force on the impeller structure is the smallest, and it has a relatively large influence under the non-design condition. This is consistent with the variation relationship between radial force and flow of centrifugal pump. Under the design flow, the radial force is the smallest, and the greater the deviation from the design flow, the greater the radial force.