# Pressure Fluctuation and Flow Characteristics in a Two-Stage Double-Suction Centrifugal Pump

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## Abstract

**:**

## 1. Introduction

## 2. Research Object and Numerical Method

#### 2.1. Two-Stage Double-Suction Centrifugal Pump

_{q}= 26, which is defined as,

^{3}/s; n is the rotating speed, 750 rpm; H is the head of the pump, m. The pump consists of two suction chambers and two parallel two-stage impellers with a common shaft in the same casing. The horizontal split pump case is used. A cross-section view of the pump is shown in Figure 1. The two-stage double-suction centrifugal pump has two semi-spiral suction chambers symmetrically distributed on the left and right sides, leading the fluid to the first-stage impellers on each side, respectively.

#### 2.2. Turbulence Model and Boundary Conditions

_{t}is the turbulence eddy viscosity, β = 0.075, β* = 0.09, σ

_{k}= 0.5, σ

_{ω}= 0.5, σ

_{ω}

_{2}= 0.856 are the model coefficients, and F

_{1}is the blending function, which is used to blend the k-ω model (F

_{1}= 0) and k-ε model (F

_{1}= 1).

_{2}(where D

_{2}is the diameter of the impeller) upstream away from the inlet of the suction chamber with the mass flow rate condition. The static pressure condition is given at 1.5D

_{2}downstream away from the volute outlet. The transient rotor-stator approach with moving mesh was used for the simulation of the domain including both the rotating impeller and stationary part. The no-slip wall boundary condition was set at all the walls. Information exchanged between the rotating part and the stationary part was settled by the rotor–stator interface. Four typical flow conditions, 0.6Q

_{n}, 0.8Q

_{n}, 1.0Q

_{n}, 1.2Q

_{n}, (Q

_{n}is the rated flow rate of the pump) were studied in the unsteady simulation. The simulation domain is shown in Figure 2.

#### 2.3. Monitor Points

_{3}–I

_{5}were set around the baffle as shown in Figure 3.

_{1}–A

_{8}were set at the exit section. Monitor Points B

_{1}–B

_{8}were set at the corner section. Monitor points C

_{1}–C

_{6}were set at the confluence section. A

_{1}–C

_{6}are continuously distributed from the upstream to the downstream.

_{1}is on the head of the volute tongue, and Point D

_{1}is on the head of the partition, as shown in Figure 5.

_{11}–P

_{19}. They are from P

_{21}–P

_{29}for the second-stage impeller. The monitor points on the suction side of the first-stage impeller are from S

_{11}–S

_{19}. They are from S

_{21}–S

_{29}for the second-stage impeller. All the monitor points are rotating with the impeller.

#### 2.4. Grid Convergence and Time Step Analysis

#### 2.4.1. Grid Convergence Analysis

_{i}is the cell volume and N is the total number of the cells in the grid generation. Three sets of grids with 9.5 × 10

^{5}, 2.1 × 10

^{6}, and 4.8 × 10

^{6}elements were generated to ensure that the grid refinement factor was greater than 1.3. The velocity of 100 points in the middle section of the volute was chosen as a key variable for the grid convergence analysis. Fixed-point iteration was used to calculate the apparent order, p. The expression is shown below,

_{32}= ϕ

_{3}− ϕ

_{2}, ε

_{21}= ϕ

_{2}− ϕ

_{1}, ϕ

_{k}denotes the solution on the kth grid.

^{6}elements) is small as shown in Figure 7a. The maximum discretization uncertainty is 6.1%, which is lower than the 10% recommended [20]. Figure 7b shows that the extrapolated value is close to that of the fine grid solution. The whole apparent order, p, is between 0.019 and 19.48 with an average value of 4.03. Both the GCI and the apparent order satisfy the requirement of grid convergence analysis. Therefore, a grid with 4.8 × 10

^{6}elements was used in the following study. Figure 8 shows the final grid resolution.

#### 2.4.2. Time Step Analysis

_{1}at the head of the volute tongue. Both the time domain and frequency domain distribution were obtained. The pressure coefficient C

_{p}is defined as:

_{2}is the circumferential velocity at the impeller outlet, n is the rotation speed, and ∆p is the difference between the transient pressure and the time-averaged value.

_{1}with three different time steps show obvious periodicity. All three time-steps can capture enough information in the frequency domain as shown in Figure 10b. The predicted fluctuation components at different frequencies are basically the same, but the amplitude is different. For the pressure distributions in the time domain, the peak value of the pressure coefficient with 180 steps was higher than that of the other two time-steps. For the results in the frequency domain, the peak value of the pressure amplitude at the BPF was the same under the three time-steps. As the time-step decreased, the amplitude of the pressure fluctuation became more consistent. The results in both time domain and frequency domain were almost the same for 360 steps and 720 steps, which means that the numerical solution was temporally stable when the number of the time step was bigger than 360 in one rotation period. Therefore, 360 steps per rotation was used as the final choice for the following unsteady simulation. More than ten rotation periods were calculated until the monitored pressure signals showed distinct periodicity. Then, five more rotation periods were calculated, and the data were sampled for analysis.

## 3. Results and Discussions

#### 3.1. General Fluctuation Performance in the Whole Passage

_{n}, which was 75 Hz. The Y coordinate is the order of the monitor points, which was from I

_{1}–I

_{12}, P

_{11}–P

_{19}, A

_{1}–C

_{6}, P

_{21}–P

_{29}, and F

_{1}–F

_{8}, and the Z coordinate is the pressure coefficient C

_{p}. It can be seen that the pressure fluctuation of the two-stage double-suction centrifugal pump was mainly based on the BPF and its harmonics under the four typical flow rates. BPF and its first harmonic existed at each monitor point of the pump.

_{17}~P

_{19}and P

_{27}~P

_{29}), especially under part load conditions. The bandwidth of the broadband was obviously narrowed as the flow rate increased. The bandwidth of the second-stage impeller was smaller than that of the first-stage impeller. At 1.2Q

_{n}, the broadband almost disappeared in the second-stage impeller, and only BPF and its harmonics remained. Figure 12 shows the pressure fluctuation amplitude distribution at BPF (75 Hz), the first harmonic of BPF (150 Hz), and the first harmonic of RF (25 Hz) along the flow direction in each part of the two-stage double-suction centrifugal pump under four typical flow rates. It can be found that a component with a frequency of 25 Hz showed the strongest fluctuation under each flow rate especially under 0.6Q

_{n}and 1.2Q

_{n}in both the first-stage impeller and the second-stage impeller. A high pressure fluctuation amplitude with BPF was also found in both the inter-stage channel and the volute. Pressure fluctuations with BPF and its harmonics and fluctuations with the first harmonic of the RF are discussed in detail in the following sections.

#### 3.2. Pressure Fluctuation with BPF and Its Harmonics

_{11}–P

_{16}) were the same as that of the suction chamber. In order to explore the relationship between the suction chamber, the leading edge, and the middle of the impeller, the time domain characteristic, the frequency domain characteristic, and coherence analysis were conducted. Figure 13 shows the comparison under the four typical flow rates of the monitor point I

_{1}at the suction chamber and the monitor point P

_{11}at the leading edge of the first-stage impeller. The coherence coefficient C

_{1}is defined as,

_{xy}is the cross power spectral density of the x and y signals, while R

_{xx}and R

_{yy}are the power spectral densities of x and y, respectively.

_{1}was dominated by fluctuation with BPF and its harmonics. In addition to BPF and its harmonics, components with RF and its first harmonic also appeared at Monitor Point P

_{11}. However, the fluctuation amplitude at the two monitor points was the same at BPF and its harmonics. The coherence analysis in Figure 13 shows that the coherence coefficient of P

_{11}and I

_{1}was close to unity at the BPF and its harmonics, which means a strong correlation between those two monitor points.

_{11}at the leading edge of the first-stage impeller.

#### 3.3. Pressure Fluctuation with 25 Hz

#### 3.3.1. Fluctuation at the Leading Edge

_{11}, P

_{21}) near the shroud side of the leading edge. The amplitude of the 25 Hz fluctuation rose up as the flow rate increased. The fluctuation at the second-stage impeller was more obvious than that at the first-stage impeller. Under the 1.0Q

_{n}and 1.2Q

_{n}condition, this component had the largest amplitude among all the monitor points.

_{n}as shown in Figure 14a. However, as the flow rate gradually increased, the symmetrical high velocity distribution disappeared, and a high-speed zone appeared in the direction of the suction chamber inlet. The high-speed zone also appeared in front of the second-stage impeller. The area of the high-speed zone increased as the flow rate increased, but the distribution of the high-speed zones was not the same as that in the first-stage impeller. There was a pair of symmetrically-distributed high-speed zones in front of the second-stage impeller inlet. Under 0.6Q

_{n}, as shown in Figure 14b, the area of the symmetrically-distributed high-speed zone around the impeller inlet was small, and the velocity was relatively lower. With the increase of the flow rate, starting from the 0.8Q

_{n}condition, the area and the velocity gradient of the symmetrically-distributed high-speed zone increased gradually.

_{11}in the first-stage impeller and the monitor point P

_{21}in the second-stage impeller under different conditions (Figure 15), it can be found that the fluctuation with the first harmonic of RF at the two impellers increased with the flow rate obviously. The main frequency component at the monitor point P

_{21}was the first harmonic of the RF; while, the fluctuation component at the monitor point P

_{11}was more complex compared to that at the monitor point P

_{21}, and the main fluctuation frequency was RF and its first harmonic. Under part load conditions, the main frequency was the first harmonic of the RF. As the flow rate increased, the fluctuation with RF gradually became the main fluctuating component.

_{n}was mainly caused by the blade passing through the high-speed zone twice in one rotation. As the flow rate increased, the distribution of the high-speed zone changed and merged into a single high-speed zone near the inlet direction, and the frequency of the fluctuation also changed to RF. The symmetrically-distributed high-speed zone at the leading edge of the second-stage impeller as shown in Figure 14b was due to the symmetrically-distributed structure of the inter-stage flow channel. At the intersection of the two channels, the symmetrically-distributed high-speed zones appeared near the impeller inlet and were only influenced by the flow rate of the inter-stage flow channel. This indicates that the fluctuation with the first harmonic of the RF at the leading edge of the second-stage impeller was caused by the leading edge passing the high-speed zones two times in one rotation period.

#### 3.3.2. Fluctuation at the Trailing Edge

_{n}was significantly higher than that under the other flow rates. As the flow rate increased, both the amplitude and bandwidth of the broadband decreased. Under the 1.2Q

_{n}condition, the broadband vanished at the trailing edge of the second-stage impeller, and only fluctuation with BPF and its first harmonic existed.

_{n}and 1.2Q

_{n}was analyzed. Figure 17 shows the time domain comparison of the pressure coefficient and radial velocity in one rotation period at the monitor point P

_{28}. It was found that the radial velocity reached the valley as the pressure dropped to the valley under 0.6Q

_{n}when the blade passed through the tongue and the start of the partition. Figure 18 shows the flow details when the blade rotated at the tongue position. It should be noted that the phase difference between the tongue and the start of the partition was π. Therefore, when one blade passed through the tongue, there must have been another blade passing the start of the position at the same time since there were six blades in total; see Figure 18. This is why the fluctuation with the first harmonic of the RF existed in the second-stage impeller near the trailing edge.

_{n}. Under 1.2Q

_{n}, the radial velocity distribution was smooth, but slight fluctuations were still available when the blade passed through the tongue and the start of the partition; see Figure 17. Different from the 0.6Q

_{n}condition, the impeller outlet was evenly distributed with small vortexes. It should be noted that those small vortexes near the impeller outlet remained unchanged even when the blade passed through the volute tongue and the start of the partition. This means that the impeller outflow under 1.2Q

_{n}was less affected by the volute tongue and the partition. As a result, the pressure fluctuation amplitude with 25 Hz under 1.2Q

_{n}was lower than that under 0.6Q

_{n}, as shown in Figure 12.

_{n}. A large size backflow vortex appeared in the blade flow channel. It generated a low-pressure zone between the impeller, volute tongue, and the start of the partition, which made a large pressure difference between the two sides of the tongue and the start of the partition. That is why the pressure at the monitor point P

_{28}dropped suddenly as the blade passed through the tongue and the start of the partition (Figure 17). Affected by the backflow vortex, the radial velocity of the blade was suppressed. Therefore, the fluctuation with the first harmonic of the RF was generated as the blade passed through the tongue and the start of the partition in every rotation period. However, under 1.2Q

_{n}, the flow at the impeller outlet was smooth, and the impact of the nonuniform outflow disappeared (Figure 20), which explains the pressure fluctuation performance with the first harmonic of the RF in the second-stage impeller shown in Figure 12.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Simulation domain and boundary conditions of the two-stage double-suction centrifugal pump.

**Figure 6.**Monitor points on the impeller. The positions are the same on both the first-stage impeller and second-stage impeller.

**Figure 7.**Grid convergence analysis. (

**a**) Fine grid convergence index; (

**b**) extrapolated value compared with the result of the fine grid.

**Figure 9.**Simulated pump performance compared with experimental result [24].

**Figure 10.**Pressure coefficient distribution with three different time steps at Monitor Point F

_{1}: (

**a**) time domain distribution; (

**b**) frequency domain distribution.

**Figure 11.**Pressure coefficient distributions in the frequency domain at different monitor points under four typical flow rates of the two-stage double-suction centrifugal pump.

**Figure 12.**The pressure fluctuation amplitude distribution at the Blade Passing Frequency (BPF) (75 Hz), the first harmonic of BPF (150 Hz), and the first harmonic of Rotation Frequency (RF) (25 Hz) along the flow direction in each part of the two-stage double-suction centrifugal pump under four typical flow rates.

**Figure 13.**Comparison of the time domain pressure distribution, frequency domain characteristic, and the coherence of Monitor Point I

_{1}at the suction chamber and the monitor point P

_{11}at the leading edge of the first-stage impeller under four typical flow rates.

**Figure 14.**Velocity distributions at the first-stage impeller and the second-stage impeller inlet under four typical flow rates: (

**a**) velocity distribution at the first-stage impeller inlet; (

**b**) velocity distribution at the second-stage impeller inlet.

**Figure 15.**Pressure fluctuation distribution in the frequency domain at Monitor Point P

_{11}in the first-stage impeller and Monitor Point P

_{21}in the second-stage impeller under four typical flow rates.

**Figure 16.**Pressure fluctuation at the trailing edge of both the first-stage impeller (P

_{17}~P

_{19}) and the second-stage impeller (P

_{27}~P

_{29}) under four typical flow rates. The red dot indicates the pressure amplitude at 25 Hz.

**Figure 17.**Comparison of the pressure distribution and radial velocity at the trailing edge of the second-stage impeller (Monitor Point P

_{28}) under 0.6Q

_{n}and 1.2Q

_{n}. The red dashed line indicates the moment when the blade passes through the tongue and the start of the partition.

**Figure 18.**Isosurface of the Q distribution (colored by pressure) in the second-stage impeller and the volute at the moment indicated by the red lines in Figure 17. The black dashed line indicates the position of the impeller outlet. (

**a**) 0.6Q

_{n}; (

**b**) 1.2Q

_{n}.

**Figure 19.**Velocity vector and pressure distribution near the volute tongue and the start of the partition in the second-stage impeller under the 0.6Q

_{n}condition.

**Figure 20.**Velocity vector and pressure distribution near the volute tongue and the start of the partition in the second-stage impeller under the 1.2Q

_{n}condition.

y Plus | Suction Chamber | First-Stage Impeller | Inter-Stage Flow Channel | Second-Stage Impeller | Volute |
---|---|---|---|---|---|

Averaged | 21.16 | 38.74 | 55.99 | 49.70 | 73.15 |

Maximum | 57.00 | 110.34 | 221.79 | 111.31 | 162.53 |

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**MDPI and ACS Style**

Wei, Z.; Yang, W.; Xiao, R.
Pressure Fluctuation and Flow Characteristics in a Two-Stage Double-Suction Centrifugal Pump. *Symmetry* **2019**, *11*, 65.
https://doi.org/10.3390/sym11010065

**AMA Style**

Wei Z, Yang W, Xiao R.
Pressure Fluctuation and Flow Characteristics in a Two-Stage Double-Suction Centrifugal Pump. *Symmetry*. 2019; 11(1):65.
https://doi.org/10.3390/sym11010065

**Chicago/Turabian Style**

Wei, Zhicong, Wei Yang, and Ruofu Xiao.
2019. "Pressure Fluctuation and Flow Characteristics in a Two-Stage Double-Suction Centrifugal Pump" *Symmetry* 11, no. 1: 65.
https://doi.org/10.3390/sym11010065