# Unsteady Flow Numerical Simulations on Internal Energy Dissipation for a Low-Head Centrifugal Pump at Part-Load Operating Conditions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{QE}and the number of blades of the studied centrifugal pumps from the above-mentioned studies are summarized in Table 1. The discharge factor, specific energy factor and specific speed for the design operating condition, are defined by Equations (1)–(3).

## 2. Case Study

_{QE}) of the pump was 0.14. The assembly drawing and computing domains of the centrifugal pump of the dredge pump are shown in Figure 1. The calculation domains included the suction pipe, the centrifugal impeller, front gap, pump-out vanes, the back gap, and the volute.

## 3. Methodology

#### 3.1. Governing Equations and Numerical Setup

_{ij}, h, h

_{tot}, p, t, μ, ρ, and λ

_{t}are velocity, strain rate tensor, specific enthalpy, specific total enthalpy, pressure, time, dynamic viscosity, density, and thermal conductivity, respectively. The physical property of water was regarded as a constant in numerical simulations.

^{6}elements was utilized in the simulations, considering both the precision and the cost of calculation. The definitions of head H, power P, and efficiency η are as follows:

_{t}is the torque; A

_{in}and A

_{out}represent the inlet and outlet boundary, respectively. All of mesh elements were structural meshes, except for the impeller computing domain. The averaged y

^{+}value in each domain was less than 130.

^{−4}s, which was 1/300 of a rotating period.

#### 3.2. Experimental Setup

#### 3.3. Energy Analysis Method

_{t}is the thermal conductivity of the fluid. The items on the right side of Equation (14) describes the viscous dissipation, turbulent production, turbulent dissipation, and dissipation from heat transfer, respectively. Since the turbulent fluctuating flow parameters $\nabla \overrightarrow{c}$ and T′ could not be obtained directly from the URANS simulations, the internal energy dissipation in the turbulent fluctuating flow needed to be expressed by mean flow components. Expressions and approximations from the previous research, based on the eddy viscosity model [43], were adopted.

_{t}is the eddy viscosity; ε is the turbulence dissipation rate; and λ

_{eff}is the effective thermal conductivity.

^{+}value was larger than 50 [44], an approximation for the internal power loss near the wall, was defined as Equation (20).

_{i}is the wall shear stress vector component. The internal power loss could be expressed as:

## 4. Results

#### 4.1. Validation of Numerical Simulating Results

_{p}was defined as Equation (23), in which the U

_{2}is the rotating velocity at the impeller outlet [45]. Both, the time history and frequency domain of the monitor points pbt5 and FCA1, from the numerical and experimental results, at 1.0 × Q

_{0}and 0.8 × Q

_{0}operating conditions, are shown in Figure 7, where n represents the rotating frequency. In Figure 7a,c,e,g, it can be seen that there were three oscillations during one impeller revolution. In Figure 7b,d,f,h, the dominant frequency was the blade passing frequency as 3 × n. The pressure analysis from unsteady numerical simulation results can basically describe the periodical change of pressure fluctuations, while there exists differences in the pressure amplitudes.

#### 4.2. Internal Power Loss

#### 4.3. Local Internal Energy Dissipation in the Impeller

_{0}operating condition, see Figure 10c, the vortex with low velocity was observed near the trailing edge of the blade suction side, in one flow passage. At 0.8 × Q

_{0}and 0.7 × Q

_{0}operating conditions, see Figure 10d,e, the vortex flow existed in two flow passages. At 0.6 × Q

_{0}and 0.5 × Q

_{0}operating conditions, see Figure 10f,g, the low-velocity vortices were apparent in all the three flow passages.

_{0}, 0.8 × Q

_{0}, and 0.5 × Q

_{0}operating conditions were applied to analyze the variation of internal energy dissipation rate distribution, in the rotating impeller. The distribution of internal energy dissipation rate on the cross-section A at six different instances of the 0.8 × Q

_{0}operating condition, is shown in Figure 11. In each flow passage, the internal energy dissipation rate near the blade suction side was larger than that near the pressure side. The internal energy dissipation rate was the most significant near the blade suction side of Blade 2. The large internal energy dissipation region existed with an invariable relative position during the rotating period. However, the strength of the internal energy dissipation rate changed with the impeller rotation. The region with large internal energy dissipation rate also existed near the outlet of impeller, especially at the trailing edge of three impeller blades, as shown with the triangle marked in Figure 11b,d,f.

_{0}operating condition is shown in Figure 12. In all flow passages, there were regions with large internal energy dissipation rate. The strength of the internal energy dissipation rate in each passage, varied with time. It is similar to the flow characteristic at the 0.8 × Q

_{0}operating condition. Taking the flow structure between Blade 1 and Blade 3 as an example, when the impeller rotated from 0° to 60°, the internal energy dissipation strengthened. At the relative position of 120°, the flow separation and detachment developed with a large internal energy dissipation. With the impeller rotating, the detached flow propagated downstream to the trailing edge of the blade suction side and the internal energy dissipation rate decreased. Vortex flow with high internal energy dissipation rate occupied a large area of the flow passage, causing flow blockage. Compared with the 0.8 × Q

_{0}operating condition, the internal energy dissipation near the impeller outlet was not significant.

#### 4.4. Pressure Fluctuations in the Impeller

_{0}, 0.8 × Q

_{0}, and 0.5 × Q

_{0}operating conditions are listed in Table 3.

_{0}and 0.8 × Q

_{0}operating conditions, and the amplitudes decreased with a smaller discharge. In addition, a low-frequency like 0.5 × n was found with a relatively small amplitude at the 0.8 × Q

_{0}operating condition.

_{0}operating condition, the dominant frequency was 3 × n with the maximum amplitudes at the blade leading edge. The low-frequency component was more significant with a frequency of 0.7 × n. Its amplitude reached 40% of the amplitude of 3 × n, at monitor point P11 and was 38% of the amplitude of 3 × n at the monitor point S13.

_{0}operating condition. Time history of the pressure signal and pressure components of frequencies, like 3 × n and 0.7 × n, at 0.5 × Q

_{0}operating condition is shown in Figure 13. It was found that the pressure signal was mainly affected by the two pressure components of 3 × n and 0.7 × n. The pressure signal in monitor point P11 was more complex than the superposition of the two waves.

## 5. Discussion

_{0}operating condition, the internal energy dissipation rate of the vortex inside one flow passage changed with the rotating impeller. The vortex flow was related to the internal energy dissipation in the impeller and had an unsteady feature. This flow structure brought about a pressure fluctuation with a low-frequency, which is 50% of the rotating frequency. The amplitude was not obvious because the separating vortex flow only existed in one impeller passage and occupied a small part of the passage.

_{0}operating condition, the vortex detachment and propagation generated the internal energy dissipation and enhanced the unsteady characteristic. The periodical change of internal energy dissipation strength showed that the vortex structure was a form of rotating stall. The periodical separation and detachment made the low-frequency components more complicated, with higher amplitudes. The characteristic frequency was 70% of the rotating frequency, which corresponded with the feature of the rotating stall phenomenon, inside the impeller [49,50,51,52].

_{0}operating condition is shown in Figure 14. The reverse flow developed at the suction pipe outlet, which had an influence on the flow and the pressure fluctuations, downstream of the impeller. Therefore, the pressure fluctuation at the blade leading edge contained more complex frequency components.

## 6. Conclusions

_{0}operating condition, the impeller experienced an unsteady rotating stall. The pressure fluctuation analysis made the unsteady characteristic of these flow phenomena apparent. At the 0.5 × Q

_{0}operating condition, there was an obvious low-frequency of pressure fluctuation, like 0.7 × n in the impeller, which was related to the rotating stall phenomenon. The reverse flow near the interface between the suction pipe and the impeller had an influence on the flow patterns, downstream. The cause for the complicated pressure components at the blade leading edge was explained as the reverse flow at the pipe outlet.

_{0}operating condition, the internal energy dissipation due to the rotor–stator interaction and pressure amplitudes of rotating frequency were not obvious. The blade passing frequency was the dominant frequency at the impeller inlet, due to the flow interaction between the pipe and the impeller.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | Area (m^{2}) |

A_{in} | Inlet boundary area (m^{2}) |

A_{out} | Outlet boundary area (m^{2}) |

c_{p} | Pressure normalization (-) |

$\overrightarrow{C}$ | Flow velocity (m·s^{−}^{1}) |

D | Impeller diameter (m) |

$\stackrel{\overline{\xaf}}{D}$ | Strain rate tensor (s^{−}^{1}) |

e | Specific internal energy (J·kg^{−1}) |

${E}_{\mathit{nD}}^{0}$ | Specific energy factor at design operating condition (-) |

g | Gravitational acceleration (m·s^{−}^{2}) |

h | Specific enthalpy (J·kg^{−1}) |

h_{tot} | Specific total enthalpy (J·kg^{−1}) |

H | Head (m) |

H_{0} | Rated head (m) |

$\stackrel{\overline{\xaf}}{I}$ | Unit tensor (-) |

n | Rotating speed (s^{−}^{1}) |

N_{QE} | Specific speed (-) |

p | Pressure (Pa) |

P | Power (W) |

$\stackrel{\overline{\xaf}}{P}$ | Total stress tensor (Pa) |

P_{loss,A} | Internal power loss near wall (W) |

P_{loss,h} | Hydraulic power loss (W) |

P_{loss,i} | Internal power loss (W) |

$\overrightarrow{q}$ | Heat flux (W·m^{−}^{2}) |

Q | Discharge (m^{3}·s^{−}^{1}) |

Q_{0} | Design discharge (m^{3}·s^{−}^{1}) |

${Q}_{\mathit{nD}}^{0}$ | Discharge factor at design operating condition (-) |

R | Radial coordinate (m) |

t | Time (s) |

T | Temperature (K) |

T_{t} | Torque (N·m) |

U_{2} | Circumferential velocity at the impeller outlet (m·s^{−}^{1}) |

V | Volume (m^{3}) |

y+ | Dimensionless wall distance (-) |

Z | Height coordinate (m) |

$\epsilon $ | Turbulence dissipation rate (m^{2}·s^{−}^{3}) |

${\lambda}_{\mathit{eff}}$ | Effective thermal conductivity (W·K^{−1}·m^{−}^{1}) |

${\lambda}_{t}$ | Thermal conductivity (W·K^{−1}·m^{−}^{1}) |

$\eta $ | Efficiency (-) |

$\mu $ | Dynamic viscosity (kg·m^{−1}·s^{−}^{1}) |

${\mu}_{t}$ | Eddy viscosity (kg·m^{−1}·s^{−}^{1}) |

$\rho $ | Density (kg·m^{−3}) |

$\tau $ | Wall shear stress (Pa) |

$\theta $ | Angular coordinate (°) |

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**Figure 1.**Assembly drawing and computing domains of the centrifugal pump. (

**a**) Cross-sectional assembly drawing of the centrifugal pump; (

**b**) computing domains; and (

**c**) meridional view of the computing domains.

**Figure 3.**Monitor point arrangements in the numerical simulations. (

**a**) Impeller; (

**b**) volute; and (

**c**) front chamber.

**Figure 5.**The influence of clocking positions on the head and power at different operating conditions—(

**a**) head; and (

**b**) power.

**Figure 6.**Comparison of the head and efficiency, between the numerical simulations and experimental results—(

**a**) head; and (

**b**) efficiency.

**Figure 7.**Time history and frequency domain of the monitor points pbt5 and FCA1 from the numerical and experimental results (

**a**) Time history (pbt5, 1.0 × Q

_{0}); (

**b**) frequency domain (pbt5, 1.0 × Q

_{0}); (

**c**) time history (FCA1, 1.0 × Q

_{0}); (

**d**) frequency domain (FCA1, 1.0 × Q

_{0}); (

**e**) time history (pbt5, 0.8 × Q

_{0}); (

**f**) frequency domain (pbt5, 0.8 × Q

_{0}); (

**g**) time history (FCA1, 0.8 × Q

_{0}); and (

**h**) frequency domain (FCA1, 0.8 × Q

_{0}).

**Figure 9.**Influence of the discharge on the internal power loss in each component for the centrifugal pump.

**Figure 10.**Relative flow velocity vector on the cross-section A at the design and part-load operating conditions, (

**a**) cross-section A; (

**b**) 1.0 × Q

_{0}; (

**c**) 0.9 × Q

_{0}; (

**d**) 0.8 × Q

_{0}; (

**e**) 0.7 × Q

_{0}; (

**f**) 0.6 × Q

_{0}; and (

**g**) 0.5 × Q

_{0}.

**Figure 11.**Internal energy dissipation rate on the cross-section A at six different moments at 0.8 × Q

_{0}operating condition (

**a**) t = 0.84 s (0°); (

**b**) t = 0.85 s (60°); (

**c**) t = 0.86 s (120°); (

**d**) t = 0.87 s (180°); (

**e**) t = 0.88 s (240°); and (

**f**) t = 0.89 s (300°).

**Figure 12.**Internal energy dissipation rate on the cross-section A at six different moments at 0.5 × Q

_{0}operating condition. (

**a**) t = 0.84 s (0°); (

**b**) t = 0.85 s (60°); (

**c**) t = 0.86 s (120°); (

**d**) t = 0.87 s (180°); (

**e**) t = 0.88 s (240°); (

**f**) t = 0.89 s (300°).

**Figure 13.**Time history of pressure signal and pressure components of frequencies as 3 × n and 0.7 × n at 0.5 × Q

_{0}operating condition. (

**a**) Monitor point P11; and (

**b**) monitor Point S13.

References | N_{QE} | Number of Blades | References | N_{QE} | Number of Blades |
---|---|---|---|---|---|

3 | 0.17 | 1 | 21 | 0.09 | 6 |

8 | 0.09 | 4 | 22 | 0.20 | 5 |

9 | 0.03 | 7 | 23 | 0.02 | 7 |

10 | 0.17 | 6 | 25 | 0.04 | 4 |

11 | 0.13 | 5 | 26 | 0.06 | 5 |

12 | 0.07 | 7 | 27 | 0.08 | 7 |

16 | 0.08 | 6 | 28 | 0.04 | 6 |

18 | 0.07 | 6 | - | - | - |

Monitor Point | Z (m) | R (m) | θ (°) | Monitor Point | Z (m) | R (m) | θ (°) |
---|---|---|---|---|---|---|---|

S11 | 0.053 | 0.060 | 171.8 | pbt1 | 0.000 | 0.374 | 36.1 |

S12 | 0.047 | 0.089 | 131.7 | pbt2 | 0.000 | 0.292 | 0.0 |

S13 | 0.035 | 0.100 | 92.7 | pbt3 | 0.000 | 0.277 | −90.0 |

S14 | 0.035 | 0.119 | 58.5 | pbt4 | 0.000 | 0.260 | 180.0 |

S15 | 0.035 | 0.137 | 30.0 | pbt5 | 0.000 | 0.242 | 90.0 |

S16 | 0.035 | 0.153 | −3.9 | pbt6 | 0.000 | 0.236 | 66.3 |

P11 | 0.053 | 0.084 | 58.2 | pbt7 | 0.000 | 0.270 | 52.8 |

P12 | 0.037 | 0.098 | 20.8 | FCA1 | 0.055 | 0.155 | 90.0 |

P13 | 0.036 | 0.112 | −13.4 | FCA2 | 0.055 | 0.155 | 180.0 |

P14 | 0.036 | 0.127 | −44.0 | FCA3 | 0.055 | 0.155 | 270.0 |

P15 | 0.036 | 0.142 | −71.1 | FCA4 | 0.055 | 0.155 | 0.0 |

P16 | 0.036 | 0.154 | −93.7 | - | - | - | - |

**Table 3.**Characteristic frequencies and amplitudes of pressure fluctuations at 1.0 × Q

_{0}, 0.8 × Q

_{0}, and 0.5 × Q

_{0}operating conditions.

Discharge | Position | Characteristic Frequency | Maximum |c_{p}| | Monitor Point with Maximum |c_{p}| |
---|---|---|---|---|

1.0 × Q_{0} | Pressure Side | n | 0.079 | P15 |

1.0 × Q_{0} | Suction Side | n | 0.065 | S16 |

0.8 × Q_{0} | Pressure Side | n | 0.050 | P15 |

0.8 × Q_{0} | Pressure Side | 0.5 × n | 0.011 | P16 |

0.8 × Q_{0} | Suction Side | n | 0.043 | S16 |

0.8 × Q_{0} | Suction Side | 0.5 × n | 0.008 | S14 |

0.5 × Q_{0} | Pressure Side | 3 × n | 0.030 | P11 |

0.5 × Q_{0} | Pressure Side | 0.7 × n | 0.012 | P11 |

0.5 × Q_{0} | Suction Side | 3 × n | 0.034 | S11 |

0.5 × Q_{0} | Suction Side | 0.7 × n | 0.011 | S13 |

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**MDPI and ACS Style**

Zhao, X.; Luo, Y.; Wang, Z.; Xiao, Y.; Avellan, F.
Unsteady Flow Numerical Simulations on Internal Energy Dissipation for a Low-Head Centrifugal Pump at Part-Load Operating Conditions. *Energies* **2019**, *12*, 2013.
https://doi.org/10.3390/en12102013

**AMA Style**

Zhao X, Luo Y, Wang Z, Xiao Y, Avellan F.
Unsteady Flow Numerical Simulations on Internal Energy Dissipation for a Low-Head Centrifugal Pump at Part-Load Operating Conditions. *Energies*. 2019; 12(10):2013.
https://doi.org/10.3390/en12102013

**Chicago/Turabian Style**

Zhao, Xiaoran, Yongyao Luo, Zhengwei Wang, Yexiang Xiao, and François Avellan.
2019. "Unsteady Flow Numerical Simulations on Internal Energy Dissipation for a Low-Head Centrifugal Pump at Part-Load Operating Conditions" *Energies* 12, no. 10: 2013.
https://doi.org/10.3390/en12102013