# Numerical Delineation of 3D Unsteady Flow Fields in Side Channel Pumps for Engineering Processes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{n}), 1200 Hz (48 × f

_{n}), and 1800 Hz (72 × f

_{n}). For this reason, exchanged flow times between the impeller and side channel is mainly responsible for the pressure fluctuation which subsequently affects the noise and vibration generation in the side channel pump. Hence, the results could be used as a reference for Noise-Vibration-Harshness (NVH) study in turbomachinery especially modifying the side channel pump in order to improve the operational reliabilities for many engineering processes.

## 1. Introduction

_{s}) with a magnitude range of 5 to 20 [12]. Due to its unique flow characteristics, it has attracted a lot of research attention after Siemen and Hinsch [13] first designed it. The momentum exchange theory used by Engels [14], Iverson [15] and Grabow [16] has been considered to be more efficient in predicting the flow behavior inside the side channel pump. Badami [17] analyzed the flow pattern in a regenerative pump using the theoretical and experimental methods. The effects of blade angle on the hydraulic performance characteristics of a regenerative pump were also studied by Choi et al. [18]. Maity et al. [19] investigated the inner flow behavior of a regenerative pump using CFD for performance prediction. Zhang et al. [20] reported that axial and radial clearances of a side channel pump greatly affect the magnitude of flow losses as well as the pump efficiency. Fleder and Bӧhle [21] conducted numerical and experimental studies on different geometrical conditions such as blade length, blade width and side channel height to analyze the internal flow characteristics and its influence on the hydraulic performance of a side channel pump. Recently, Zhang et al. [22,23] put forward that the greater blade suction angle, the higher the head performance. Even though, the blade suction angle had a minimal effect on the efficiency, the blades profiled with suction angle 30° produced optimal performance compared to 10° and 20°. Again, Zhang et al. [23] in 2018 studied the transient flow characterization of side channel pump under different time steps.

## 2. Pump Geometry

_{s}= nQ

^{0.5}/H

^{0.75}) of the pump models is 8.6. The models composed of 24 isosceles-triangular shaped impeller blades designed with suction angle 30° and circular side channel as shown in Figure 1. The rotating speed of the shaft is designed to operate at 1500 r/min and the radial and axial gaps are set to 0.2 mm. Table 1 summarizes the main geometrical parameters of the pump models.

## 3. Numerical Calculations (CFD) Setup

#### 3.1. Governing Equations

_{1}and F

_{2}.

_{1}and F

_{2}are:

_{t}is calculated using:

_{k}

_{1}= 1.176, σ

_{ω}

_{1}= 2.0, β

_{1}= 0.075, α

_{1}= 5/9, β′ = 0.09 [30]. Meanwhile, the k-ε equation for the free-stream flows is solved using the following coefficients: σ

_{k}

_{2}= 1.0, σ

_{ω}

_{2}=1/0.856, β

_{2}= 0.828 [30].

#### 3.2. Mesh Independence Analysis

_{BEP}. Figure 2 shows the mesh independence analysis carried out on pump case. The head is calculated using Equation (11). The mesh C which consist 4.5 million is selected for further numerical calculations after following the procedures outlined by Celik et al. [31]. For consistency with the mesh independence analysis, cases 1 and 3 are then gridded with the same blocking method with an approximated grid number of pump cases 1 and 3.

#### 3.3. Boundary Conditions Settings

^{−4}s corresponding to one degree. The chosen time step is small enough to acquire the necessary time resolution. A maximum of 10 iterations are carried out at each time step and the convergence criterion at maximum residual is limited to 1.0 × 10

^{−6}. Time discretization is carried out with the second order backward Euler scheme. The impeller is set to rotate for six complete revolutions representing a total time of 2.4 × 10

^{−1}s to predict and evaluate the unsteady flow fields at various monitoring points at different operating conditions.

#### 3.4. Monitoring Points

## 4. Results and Discussion

#### 4.1. Hydraulic Performance Characteristics

^{3}h

^{−}

^{1}) and its maximum measuring error is ±0.5%. The shaft power of the tested pump is driven by variable-speed electric AC motor and controlled by a frequency drive of 0 to 50 Hz. The readings (20 sample data) are taken at constant speed for each valve opening at different flow rates.

_{BEP}are potted on the y and x axes, respectively. The efficiency curve increases gradually and decreases after the best efficiency, Q

_{BEP}. The Q

_{BEP}marks the maximum efficiency and highlighted in red. The value of the Q

_{BEP}is 10 m

^{3}h

^{−1}. The CFD efficiency is higher than the experimental measurements at all operating conditions. This is because mechanical losses were not taken into consideration during the numerical calculations. The Q

_{BEP}of the CFD efficiency curve is 35.9% whereas that of the experimental efficiency is 34.2%. The largest deviation on the efficiency curves occurs at the Q

_{BEP}and is about 4.90%. Figure 6b shows that higher head values are concurrent with lower flows. The rise in pressure is due to the high circulatory velocity the fluid attains at low flows [32]. Largely, the experimental heads are higher than the CFD heads. The best agreement between the curves occurred at part-load conditions from 0.4 Q

_{BEP}to 0.8 Q

_{BEP}. At the Q

_{BEP}, the deviation between the experimental and CFD head coefficients is 5.27%. This shows that there is a good agreement between the CFD and experimental results thus the CFD results can be used for further analyses. However, the hydraulic performance comparison of the three pump cases has been conducted in details by Pei et al. [24].

#### 4.2. Pressure Fluctuation Intensity

#### 4.2.1. Definition of Pressure Fluctuation Intensity

_{0}represents the initial time for the transient calculations and $\Delta t$ is the time-step for the transient calculation, j is the number of time-steps. It is worth noticing that, ${C}_{p}^{*}$ and ${C}_{w}^{*}$ defined in Equation (21) are independent of instantaneous rotating time and position.

#### 4.2.2. Impeller Passage

#### 4.2.3. Side Channel Passage

#### 4.3. Time Domain History of Pressure Fluctuation Intensity at Monitoring Points

#### 4.3.1. Inflow Region

_{p}of the inner radius (D3P3) and outer radius (D3P1) inside the impeller passage for all three cases at the inflow region. The pressure coefficient, Cp is plotted on the y and the timestep of the impeller rotation is plotted on the x-axis. The timestep is measured in degree. According to Figure 11, negative C

_{p}value refers to a pressure that is lower than the reference point, which is selected at the inflow. The diameter line, D3 is located 30° from the relative position, ϕ. The pressure amplitudes of all pump cases depict a gentle rise from the inflow to outflow except in the interrupter region where significant amount of pressure energy is lost because there is no exchange of flow. All three pump cases operate with a 24 blade impeller but in case 1 during the flow exchange between the impeller and side channel at both D3P1 and D3P3, 21 regular fluctuations of the pressure appeared in a complete circulatory cycle of the impeller. During the complete circulatory cycle, one blade stays in the interrupter and the other two at the inflow and outflow regions respectively. Case 2 reveals 20 regular pressure fluctuations with two blades in the interrupter while one blade is at inflow region and another in the outflow region in complete circulatory cycle. Pump case 3 on the other hand shows 19 regular pressure fluctuations because it has the largest wrapping angle. Three blades stay in the interrupter in a complete circulatory cycle while the other two are close to the inflow and outflow regions. So, the number of circulations can easily be found as a function of the number of impeller blades.

_{p}= −1.43 and −1.18 at D3P1 and D3P3 respectively. Even though, D3P1 at the inflow region is expected to attain higher static pressure, D3P3 for all three pump cases record higher pressure fluctuation intensity than D3P1. This is due to the fact that at the inflow region, the incident fluid from the inlet pipe begins to experience the centrifugal force imparted by the blades in the impeller flow passage and also the high-pressurized fluid in the blade passage interfered with the low-pressurized flow from the inlet pipe. This phenomenon makes the inflow region highly susceptible to vortex and generation of unwanted noise for all pump cases. The flow exchange between the impeller and side channel is mainly responsible for the pressure fluctuations generated in side channel pumps. The fluctuation times are dependent on the exchange times thus the smaller the wrappings angle the higher fluctuation frequency.

#### 4.3.2. 180° Offset Region

#### 4.3.3. Outflow Region

#### 4.3.4. Interrupter Region

#### 4.4. Frequency Domain History at Monitoring Points

#### 4.4.1. Impeller Passage

_{n}and the blade passing frequency (BPF). The shaft frequency occurred at 25 Hz while the blade passing frequencies occurred at multiples of the shaft frequency:

_{n}), 1200 Hz (48 × f

_{n}), 1800 Hz (72 × f

_{n}), and continued in the same manner. Here, f

_{n}is 25 Hz, the shaft rotating frequency. It is important to note that the excitation of these harmonics is mainly associated with the flow exchange between the impeller and side channel. Likewise, the flow exchange times greatly depends on the size of the wrapping angle and blade numbers. Meanwhile, case 1 marks the highest pressure coefficient amplitude on the blade passing frequency harmonic at all monitoring points in the impeller flow passage. It is therefore obvious that the pressure coefficient amplitudes decreases as the wrapping angle increases from 15° to 45°.

_{n}at inflow region of case 1 records the highest pressure fluctuation, C

_{p}= 1.06 and 1.11 at D3P1 and D3P3, respectively. Meanwhile, case 3 reaches the lowest C

_{p}at f

_{n}and 1st excitation frequency of 600 Hz. In between the shaft and first excitation frequencies are associated noise accompanying flow. These emanating sub-frequencies are from undesirable flow patterns which are mainly caused by the high-pressurized fluid in the impeller passage from outflow to inflow which subsequently tends to mix with the low-pressurized fluid from the suction pipe.

#### 4.4.2. Side Channel Passage

_{n}), 1200 Hz (48 × f

_{n}), 1800 Hz (72 × f

_{n}), and continued in the same manner. The pressure fluctuation harmonics of the side channel passage appear at multiples of the shaft frequency and the multiples are related to the blade numbers. Just like the impeller passage, the pressure coefficient amplitudes reduce as the frequency increases. The excited harmonics generated in the side channel also confirm that the pressure fluctuation in the side channel pump is mainly affected by the exchange flow. Predominantly, the pressure coefficient amplitudes at frequencies between 0–200 Hz register relatively low magnitude for all monitoring points especially at the inflow region in Figure 20a. The reason is that the flow receives minimal centrifugal force at the inflow region.

#### 4.4.3. Impeller and Side Channel Passages

_{n}), 1200 Hz (48 × f

_{n}), 1800 Hz (72 × f

_{n}) and so on. It still predicts that the flow exchange between the impeller and the side channel is the main reason causing pressure fluctuations both in the impeller and side channel passages. The pressure coefficient amplitude of the selected monitoring point declines gradually with the increasing excited frequency.

#### 4.5. Relative Velocity Fluctuation Intensity

#### 4.5.1. Definition of Relative Velocity Fluctuation Intensity

#### 4.5.2. Impeller Passage

#### 4.5.3. Side Channel Passage

#### 4.6. Turbulent Kinetic Energy Intensity (TKE)

## 5. Conclusions

_{n}), 1200 Hz (48 × f

_{n}), 1800 Hz (72 × f

_{n}), and continued in the same manner. From the time and frequency data, it can be confirmed that the pressure fluctuation intensity is mainly associated with the flow exchange times between the moving impeller and static side channel. The flow exchange is also related to the size of the wrapping angle and the blade number.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

g | Acceleration due to gravity |

Ω | Angular speed |

Γ | Auxillary variables in turbulence model |

s | Axial and radial clearance width |

Z | Blade number |

θ | Blade suction angle |

F_{1} | Blending or auxillary functions in turbulence model |

U_{2} | Circumferential velocity of impeller outlet |

C_{p} | Coefficient of pressure fluctuation intensity |

C_{w} | Coefficient of velocity fluctuation intensity |

x, y, z | Coordinates in stationary frame |

ρ | Density |

D | Diameter of impeller |

Δ | Difference |

ϵ | Dissipation of kinetic energy of turbulence |

μ | Dynamic viscosity |

η | Efficiency |

Q | Flow rate |

h | Height of the side height |

ψ | Head coefficient |

H | Head |

h | Height of the side height |

y^{+} | Non-dimensional wall distance |

P | Pressure |

k | Kinetic energy of turbulence |

δ_{ij} | Kronecker’s delta |

n | Rotational speed |

$-\overline{\rho {u}_{i}^{\prime}{u}_{j}^{\prime}}$ | Reynolds-stress tensor |

w | Relative velocity |

ω | Specific dissipation of turbulence kinetic energy |

N | Sample number |

f_{o} | shaft frequency |

n_{s} | specific speed |

t | Time |

M | Torque |

β*, γ | Turbulence –model coefficients |

μ_{T} | Turbulent viscosity |

σ_{k}, σ_{ω} | Turbulence-model coefficients |

u_{i} | Velocity components (u, v, w) in Cartesian directions: x, y, z |

φ | Wrapping angle |

## Subscripts

0 | Origin |

x_{i} | Cartesian coordinates: x, y, z |

i, j | Components in different directions |

## Abbreviations

BPF | Blade passing frequency |

3-D | Three dimensional |

CFD | Computational fluid dynamics |

SST | Shear stress transport |

RANS | Reynolds-averaged Navier-Stokes |

BEP | Best efficiency point |

Ave | Average |

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**Figure 11.**Time domain history of pressure fluctuation coefficient in impeller passage at inflow region for all pump cases.

**Figure 12.**Time domain history of pressure fluctuation coefficient in side channel passage at inflow region for all pump cases.

**Figure 13.**Time domain history of pressure fluctuation in impeller passage at 180° offset region for all pump cases.

**Figure 14.**Time domain history of pressure fluctuation in side channel at 180

^{º}offset region for all pump cases.

**Figure 15.**Time domain history of pressure fluctuation coefficient in impeller passage at outflow region for all pump cases.

**Figure 16.**Time domain history of pressure fluctuation coefficient in side channel passage at outflow region for all pump cases.

**Figure 17.**Time domain history of pressure fluctuation coefficient in impeller passage at interrupter region for all pump cases.

**Figure 18.**Frequency domain history of pressure fluctuation in impeller passage for three pump cases.

**Figure 19.**Detailed frequency domain history of pressure fluctuation in impeller passage for all pump cases below 600 Hz.

**Figure 20.**Frequency domain history of pressure fluctuation coefficient in side channel passage for three pump cases.

**Figure 21.**Frequency domain history of pressure fluctuation of the impeller and side channel passages for three pump cases.

Domain | Parameters | Value |
---|---|---|

Impeller | Outer diameter, D_{2} (mm) | 150 |

Inner diameter, D_{1} (mm) | 80 | |

Blade width, w (mm) | 15 | |

Blade thickness, b (mm) | 2 | |

Suction angle, θ (°) | 30 | |

Radial clearance, σ (mm) | 0.2 | |

Axial clearance, s (mm) | 0.2 | |

Side Channel | Wrapping angle, φ (°) | 15, 30, 45 |

Diameter, t (mm) | 35.2 |

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**MDPI and ACS Style**

Zhang, F.; Chen, K.; Appiah, D.; Hu, B.; Yuan, S.; Asomani, S.N.
Numerical Delineation of 3D Unsteady Flow Fields in Side Channel Pumps for Engineering Processes. *Energies* **2019**, *12*, 1287.
https://doi.org/10.3390/en12071287

**AMA Style**

Zhang F, Chen K, Appiah D, Hu B, Yuan S, Asomani SN.
Numerical Delineation of 3D Unsteady Flow Fields in Side Channel Pumps for Engineering Processes. *Energies*. 2019; 12(7):1287.
https://doi.org/10.3390/en12071287

**Chicago/Turabian Style**

Zhang, Fan, Ke Chen, Desmond Appiah, Bo Hu, Shouqi Yuan, and Stephen Ntiri Asomani.
2019. "Numerical Delineation of 3D Unsteady Flow Fields in Side Channel Pumps for Engineering Processes" *Energies* 12, no. 7: 1287.
https://doi.org/10.3390/en12071287