3.2.2. Mesh Generation and Boundary Conditions

The structured mesh is applied for the computational domains by using ANSYS ICEM version 14.1 (ANSYS, Inc., Commonwealth of Pennsylvania, USA). As shown in Figure 4, a special refinement is applied around the blades and vane diffusers to improve the accuracy of the simulation. In addition, six groups of computational grids have been chosen to analyze the influence of the mesh size on the prediction of the hydraulic performance of the model pump. Figure 5 presents the analysis of mesh independence verification, the total element number is chosen as 6,680,830 by considering the simulation accuracy and efficiency. In addition, boundary conditions concern a bar at the inlet and the mass flow rate at the outlet of the computational domain. All the other walls are treated as non-slip boundaries. The maximum nondimensional wall distance, *y* + < 10, was obtained in the complete flow field, which could satisfy the requirement of all turbulence modeling methods used in this paper. The pre-converged steady flow field (based on the SST turbulence model) obtained is accepted as the initial condition followed by the unsteady simulation (based on the DES model). The time step for the unsteady simulation is set as 1.7857 <sup>×</sup> <sup>10</sup>−<sup>4</sup> s.

**Figure 4.** Mesh of the multi-stage centrifugal pump.

**Figure 5.** Analysis of mesh independence.

#### 3.2.3. Flow Field Results

The global performance of the model pump is shown in Figure 6. The experimental and simulated data of the total head agree well with nearby design flow rates. The calculated head is slightly different from the measured value at the part-load and overload condition. The value of calculated efficiency is lower than the measured one and the differences become bigger with the increment of the flow rate. While the error of the head between the experiment and the simulation is lower than 4.5% and the error of the efficiency is lower than 3.8%, the established calculation model and the selected number of grids could fully support the next acoustic calculations. By the way, the highest efficiency working condition of the model pump appears in the 0.8*Q*d. This might be related to subsequent acoustic characterization.

**Figure 6.** External characteristics of numerical calculation and experiment.

Figure 7a–d describes the velocity distribution on the mid-span of the impeller and the diffuser. At the 0.6*Q*<sup>d</sup> condition, the velocity distribution in the impeller is not uniform and the separation is detected at the inlet of the diffuser. With the increase of the flow rate, the flow field becomes better distributed while the diffusers are impugned by the high-velocity flow. The wake flow induced by the impingement can generate higher pressure pulsations. As pointed out by Gülish [36], the wake flow and the separation in the centrifugal pump gives rise to the pressure pulsations and the subsequent radiated noise.

(**a**) 0.6*Q*d at impellers and positive vanes (**b**) 0.8*Q*d at impellers and positive vanes

(**c**) 1.0*Q*d at impellers and positive vanes (**d**) 1.2*Q*d at impellers and positive vanes

**Figure 7.** Velocity distribution in the mid-scan of the pump's first stage at different flowrates.

Considering the influence of the upstream flow, the flow in the returning vane diffuser passages is more complex, as shown in Figure 7e–h. At the part-load condition, several separations block several passages. Additionally, the flow is greatly deaccelerated in the diffuser passages at the overload condition and the separation flow is detected near the suction side of the diffuser. When the flow reaches 0.8*Q*d, the number of separations is the fewest. This might indicate that minor noise would produce at this flow condition.

In order to understand the pressure pulsation characteristics of the flow in the model pump at various stages of the multistage centrifugal pump, 12 monitoring points at every stage, totaling 36 points, are set on the cross-section of the impeller, the positive vane, and the reverse guide vane. As shown in Figure 8, the first letter in each monitoring point name is: *F* means the first level, *S* means the second level, and *T* means the third level. Moreover, the second letter in each monitoring point name is: *Y* means the impeller, *D* means the positive vane, and *F* means the reverse guide vane. The dimensionless pressure pulsation coefficient *Cp*\* as shown in Formula (20) is used for further data reduction.

$$\text{Cp}^\* = \frac{(p - \overline{p})}{0.5\rho v\_2^2} \tag{20}$$

where *p* is the static pressure of the monitoring point, *p* is the average value of the static pressure, ρ is the fluid density, and *v*<sup>2</sup> is the circumferential velocity at the impeller outlet. Afterward, standard deviation CPS was used to characterize the pressure pulsation intensity of each monitoring point, and fast Fourier transform processing was used to obtain the frequency spectrum of the flow pressure fluctuation.

As shown in Figure 8b, the pressure pulsation intensity of the monitoring points in the multi-stage centrifugal pump shows a periodic change, and it always became larger around the positive guide vane flow channel, indicating that the flow between the stages is similar and the biggest hydraulic exciting force appears at the positive guide vane. The maximum value of the pressure pulsation intensity appears on monitoring points near the throat of the positive guide vane, indicating that the rotor-stator interaction between impeller and positive guide vane makes the greatest contribution to the pressure pulsation intensity. Other flow conditions show the same results. For more detail, the maximum pressure pulsation intensity appears in the first-stage positive guide vane flow channel, and the secondary and final guide vanes make less difference. Then, frequency analysis of the pressure fluctuation at different flow rates is processed on the monitoring points of the first positive guide vane, as shown in Figure 8c–f. Since the sampling time step of the numerical calculation is 1.7857 <sup>×</sup> <sup>10</sup>−<sup>4</sup> s, the sampling frequency is 5600 Hz. According to the Nyquist sampling theorem, the maximum frequency that can be analyzed is 2800 Hz.

(**b**) amplitude distribution of pressure fluctuation at 1.0*Q*<sup>d</sup>

**Figure 8.** *Cont.*

(**c**) spectrum of pressure fluctuation at 0.6*Q*<sup>d</sup>

(**d**) spectrum of pressure fluctuation at 0.8*Q*<sup>d</sup>

(**e**) spectrum of pressure fluctuation at 1.0*Q*d

**Figure 8.** Pressure fluctuation of monitoring points.

The results of the frequency spectrum show that the pressure pulsation is mainly concentrated in the low-frequency region within 1000 Hz, and the main frequency is the blade passing frequency (327 Hz) and its multiple. The monitoring point with a large peak is also located near the throat of the positive guide vane at all flow rates, which corresponds to the pressure pulsation intensity results. Under the small flow conditions such as 0.6*Q*<sup>d</sup> and 0.8*Q*d, there are many low-frequency pulsations smaller than the blade passing frequency for all monitoring points. It means the periodic pressure pulsation caused by the rotor-stator interaction between impeller and positive guide vane is the major source of the pump unstable operation. The 0.8*Q*<sup>d</sup> has the lowest amplitude of the frequency spectrum, which corresponds to the phenomenon that this flowrate present the highest efficiency.

### *3.3. Radiated Noise Analysis*

### 3.3.1. Computational Domain

The computational domain for the acoustic simulation includes the structural domain and air-borne domain, as shown in Figure 9. The inner surface of the vane diffuser of the structural domain is loaded with information of the unsteady flow to get the sound source in the subsequent simulation, and the definition of the air-borne domain is to get the distribution of the radiated noise in the subsequent simulation.

(**c**) the structure of the vane diffuser

**Figure 9.** Calculation domains of radiated noise simulation.

#### 3.3.2. Mesh Generation and Boundary Condition

Unstructured mesh, which has better adaptability to the geometry, is applied in the acoustic simulation. To guarantee the precision of the acoustic computation, the maximum mesh size should meet the Formula (21). In this study, the maximum mesh size is 0.008 m. And the mesh for acoustic simulation is shown in Figure 10.

(**a**) mesh for the structural domain (**b**) acoustic simulation mesh

**Figure 10.** The sound field calculation mesh of multistage centrifugal pump.

The detailed information of the unsteady flow is extracted as the sound source near the surface of the vane diffusers. In addition, the material properties of the structure domain are shown in Table 2. Considering the total time of the unsteady CFD simulation and the time step, the frequency range of the acoustic simulation is set from 0 Hz to 2800 Hz, and the resolution is set as 5.3 Hz. The data transmission of the interface between the structural domain and the air-borne domain is finished with the integral interpolation method. In order to analyze the properties of the radiated noise, 60 monitor points are mounted equally in the mid-span surface of the second impeller and distances between the diffuser surfaces of these points are 1 m, as shown in Figure 11.


**Figure 11.** The pre-setting of radiated noise calculation.

#### 3.3.3. Acoustic Field Results

Figure 12 shows the predicted sound pressure spectra at different flow rates. As can be seen, the dominant frequency of the multi-stage pump is the blade passing frequency (327 Hz). Furthermore, the sound pressure level (SPL) of the radiated noise at the low frequency increases with the increase of the flow rate. In contrast, the SPL above 2700 Hz has the opposite trend. Additionally, the SPL at the blade passing frequency decreases slightly before 0.8*Q*d, and increases afterward. In addition, there is a slight increase in the SPL around 1500 Hz, which is assumed to be linked with the vibration mode of the pump system.

**Figure 12.** Frequency response curves of radiated noise.

Figure 13 shows the sound pressure level contour at blade passing frequency. The sound pressure level reaches the highest at the 0.6*Q*d, while the sound pressure level reaches the highest near the vane diffuser of the first stage. This is due to the fact that the intensity of the pressure fluctuation at the first stage is highest and the structural strength is relatively lower around the first stage.

The profiles of the SPL of the directivity field shown in Figure 14 demonstrate the dipole characteristic behavior. It is found that two SPL valleys appear around at θ = 120◦ and 300◦ and fluctuate slightly with the change of the flowrate. This phenomenon illustrates that the rotor/stator interaction is the main source of the fluid-induced noise.

(**a**) 0.6*Q*d (**b**) 0.8*Q*<sup>d</sup>

(**c**) 1.0*Q*d (**d**) 1.2*Q*<sup>d</sup>

**Figure 14.** The directivity of the radiated noise field at blade passing frequency.

#### **4. Experimental Verification**

The experimental research was done in a closed-loop test rig beside the semi-anechoic room of the National Research Center of Pumps, China. The test rig shown in Figure 15 for pump performance measurement meets the Grade 2 accuracy based on the ISO 9906.2012 standard. The accuracy of the flow rate measurements is ±2.5%, the head is ±3%, the torque is ±2.5%, and the rotation is ±1%. In addition, to verify the validity of the acoustic simulation results, the radiated noise of the multi-stage pump is measured in the semi-anechoic room built with 15 dB background noise and 50 Hz cut-off frequency. As shown in the above figure, the five walls of this semi-anechoic room are equipped with an anechoic wedge that could form a semi-free acoustic field. In order to improve the accuracy of the measurement, the multi-stage pump is installed on a damping base and the pipelines are supported with the damping disc. In addition, an anechoic tank is arranged between the water tanks and the multi-stage pump to reduce the flow-induced noise inside the pipelines. All the data collection equipment is put in the monitoring room to minimize the influence of the accuracy of the measurement.

The data collection system is composed of the module to measure the static characteristic and the other module to capture the radiated noise of the model pump. The magnetic inductive flowmeter performs the measurements of the volume flow, pressure sensors capture the static pressure at the inlet and outlet of the multi-stage pump and monitor the torque value of the model pump. The noise is measured with the PCB 14043 type microphone (PCB Piezotronics Inc., New York, USA) and processed based on the LMS test Lab platform. The accuracy of the SPL of the radiated noise is ±1 dB. The acquisition sampling frequency is set to 6000 Hz, and the resolution frequency is set to 0.5 Hz. Every test used 120 s for the signal acquisition and treated frequency resolution by the Hanning window in general.

(**a**) test rig **Figure 15.** *Cont.*

(**b**) model pump and the layout of the pipelines (**c**) monitoring room

**Figure 15.** Test of the multi-stage centrifugal pump.

### *4.1. Radiated Noise at the Di*ff*erent Flow Rate*

To investigate the relationship between the radiated noise and flow rate, the flow rate is regulated by the butterfly valve to detect the radiated noise between 3 m3/h–11.6 m3/h (0.375*Q*d–1.45*Q*d). Five positions set for the microphones followed the Chinese standard GB/T 29529-2013 [37], as shown in Figure 16a, consistent with the position of the monitoring point set in the numerical simulation. Figure 16b shows the setup with enclosures. Figure 17 shows the radiated noise of the model pump at different flowrates concerning the above two conditions and the comparison with the numerical simulation results. In order to analyze the variation of the total sound pressure level of the radiated noise of the multi-stage centrifugal pump, the average total sound pressure level *LP* is expressed as follows. Where *N* represents the number of sound monitoring points and *L*pi is the total sound pressure level of each sound monitoring point.

$$\overline{L}\_P = 10 \cdot \lg \left[ \frac{1}{N} \sum\_{i=1}^{N} 10^{0.1L\_{Pi}} \right] \tag{22}$$

(**a**) location of the microphone (**b**) with acoustic enclosure

**Figure 16.** Experiment method.

**Figure 17.** The radiated noise of the multi-stage centrifugal pump at different flowrates.

As seen in Figure 17, the radiated noise of the model pump at all flow rates with acoustic enclosure is 8 dB lower than the case without the enclosure. This fact proves that the enclosure is necessary for the experiment. On the other hand, the simulated total sound pressure level is consistent with the experiment that the total sound pressure level fluctuates with the increment of the flow rate and the minimal value emerges around 0.8*Q*<sup>d</sup> where the efficiency is the highest. As pointed out by Gülish [24], the sound pressure level of the induced noise is in the inverse relationship with the efficiency. The differences between the simulation and experiment are within the order of 10 dB because the background noise inside the anechoic chamber still provides some disturbance. Although we used sound elimination materials on the pipe system, the motor still emits a strong radiated noise even when we used an enclosure.

#### *4.2. Radiated Noise at the Di*ff*erent Rotational Speed*

The rotational speed of the centrifugal pump has the direct influence of the pressure distribution in the model pump and meantime influence on the radiated noise of the model pump. A frequency converter allows changing the rotational speed of the pump. As seen in Figure 18, the sound pressure level increases almost linearly with the increase of the rotational speed. Processed by the fitting instruments, their relation meets the following formula:

$$y = 0.00545 \text{x} + 66.7 \tag{23}$$

In this formula, *y* represents the sound pressure level, *x* represents rotational speed, and the linear dependence between the fitted curve and the measured data is 0.97894.

**Figure 18.** The radiated noise of the multi-stage centrifugal pump at different speeds.

#### *4.3. Directivity of Radiated Noise at Di*ff*erent Flowrates*

To analyze the contribution of the different types of sound sources, the sound pressure level in the hemispherical surface around the source is measured. Microphones are mounted equally in the mid-span surface of the model pump and these five microphones are 1 m away from the surface of the second diffuser.

Considering the number of the microphones is limited, an interpolation method is used to get the directivity of the radiated noise at different flow rates, as shown in Figure 19. The experimental and simulated data of directivity agree well, considering the effect of background noise. It is obvious that the dipole is the main sound source of the multi-stage centrifugal pump at the different flowrates. Additionally, directivity at the 0.8*Q*<sup>d</sup> is vertical, which explains that the directivity varies with the change of the flowrate.

**Figure 19.** Directivity of radiated noise at different flowrates.

#### **5. Hydraulic Optimization Design**

From the above experimental and numerical analysis, the main frequency of pressure pulsation and radiation noise of the prototype model pump is the blade passing frequency, and the amplitude under it is the largest. The flow passage is too narrow due to the unreasonable design of the number of blades of the impeller and the vane, which leads to poor fluid permeability. There are many vortices in the flow passage of the prototype pump model, and the intensity of its pressure pulsation is large, thereby generating a large radiated noise.

In general, the main measures for reducing the radiated noise of the multi-stage pump include changing the rotational speed and process hydraulic optimization. Optimizing the effect of rotor-stator interaction is the most important region for the hydraulic design of a low noise multi-stage pump because the rotating speed is generally determined by the matching motor. Since the matching between the impeller and the guide vane of the multi-stage centrifugal pump also has a great influence on the pump performance, this study starts with the internal flow improvement and carries out a multi-objective design for the head, efficiency, and noise. The best match between the impeller and the guide vane is derived according to Formula (24) and (25). Finally, the number of impeller blades is six and the vane blade number is nine through mathematical calculation. Other geometric parameters were obtained by the velocity coefficient design method of the centrifugal pump and orthogonal optimization design [1].

$$H\_{\rm impeller} = \frac{\mu\_2}{g} \left( \sigma \mu\_2 - \frac{Q \cot \beta\_2}{\pi D\_2 b\_2 \psi\_2} \right) \tag{24}$$

$$H\_{\text{rature}} = \frac{w}{g} \frac{Q}{a\_3 b\_3 Z\_{\text{rature}}} \frac{D\_3}{2} \tag{25}$$

The numerical calculation process including fluid calculation domain modeling, meshing, boundary condition setting, and flow field calculation is the same as before. Figure 20 gives the simulation results of the optimized pump. Seen from the calculated pump performance, the head and efficiency are higher than the original one after optimization. The efficiency after optimization increased by 12.9% under design flow. At the same time, the optimized multi-stage centrifugal pump achieves the highest efficiency around 1.1*Q*d, and the highest efficiency zone is widened.

The optimized multi-stage centrifugal pump pressure pulsation intensity shown in Figure 20c has the same characteristics as the original pump, that is, the pressure pulsation intensity in the impeller is the smallest, the pressure pulsation intensity in the positive guide vane is the largest, and the maximum value still appears in the positive guide vane passage near the throat at all stages. The pressure pulsation intensity of the optimized multi-stage centrifugal pump is reduced correspondingly to the same position as the original pump. Figure 20d revels that the dominant frequency of the radiated noise of the optimized multi-stage centrifugal pump is still blade passing frequency, but the sound pressure level under it is reduced by 2 dB compared with the original one shown in Figure 12c, which indicates that matching the relationship between the impeller and the guide vane is better after optimization. Reasonably, comparing the spectrum before and after optimization, it can be found that the magnitude of the radiated noise of the optimized multi-stage centrifugal pump in the low-frequency region is significantly reduced.

Finally, the optimized design was manufactured and sent to the product quality inspection center for a pump performance test. The multi-stage centrifugal pump after optimization has improved the head by 4.62 m, the efficiency by 11.57%, and reduced the average total sound pressure level by 2.6 dB at the design flow rate, which indicates that the simulation process and optimization method proposed in this paper is suitable for the pump designer.

(**d**) frequency response curves of the optimized pump noise at 1.0*Q*<sup>d</sup> **Figure 20.** Simulation results of the optimized pump.
