# Evaluating the Transient Energy Dissipation in a Centrifugal Impeller under Rotor-Stator Interaction

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods

#### 2.1. Governing Equations

_{ij}is the is the Kroneker delta, μ is dynamic viscosity. $\overline{\varphi}$ and $\varphi \prime $ are respectively the time-averaged and fluctuating component of arbitrary parameter $\varphi $. Term ρ$\overline{{u}_{i}^{\prime}{u}_{j}^{\prime}}$ is called the Reynolds stress. $\overline{{S}_{ij}}$ is the mean rate of strain tensor:

_{sta}is the static enthalpy, h

_{tot}is the total enthalpy that h

_{tot}=h

_{sta}+u

^{2}/2, λ

_{t}is the thermal conductivity. Based on these equations above, the hydraulic energy loss, dissipation and transformation to internal energy can be simulated in detail.

#### 2.2. Eddy Viscosity Turbulence Modeling

_{t}as:

_{t}based on statistics or experimental verifications.

_{k-ω}is the turbulence scale which can be expressed as:

_{ω}is the coefficient of the production term, F

_{1}is the blending function, σ

_{k}, σ

_{ω}and β

_{k}are model constants.

#### 2.3. Acoustic Analogy Method

_{A}[26]:

_{ε}is a constant equal to 0.1, ε is the eddy dissipation rate, M

_{t}is the specific turbulence kinetic energy which can be calculated by:

_{c}is the sound speed which is 340 m/s in this case. The flow-induced sound power level L

_{sp}can be calculated by:

_{ref}is the reference sound power which is 1 × 10

^{−12}W/m

^{3}in this case.

## 3. Case Description

#### 3.1. Centrifugal Air Pump Model

_{φ}can be expressed as:

#### 3.2. Flow Domain Meshing

_{stot}between impeller inlet and diffuser outlet and the y

^{+}, are checked to determine the mesh scheme for a better simulation accuracy. The specific entropy difference coefficient ΔC

_{stot}can be written as:

^{*}is the specific entropy, g is the acceleration of gravity, subscript x

_{1}and x

_{2}denotes two locations. If x

_{1}is the impeller inflow and x

_{2}is the diffuser outflow, ΔC

_{stot}, in the case that temperature changes slightly, can describe the energy loss in the flow passage from impeller inflow to diffuser outflow by considering ΔT·s* instead of $\int}T$ds [32]. Checking ΔC

_{stot}can give a better result in predicting the macro energy change in pump. Checking y

^{+}is to improve the near-wall solution especially in the viscous sub-layer and buffer layer.

_{stot}check is conducted under varying mesh node number from about 2.23 × 10

^{5}to about 3.56 × 10

^{6}based on the steady state simulation. The residual of C

_{stot}is monitored as shown in Figure 2 with the criterion of continually less than 1%. The final mesh is determined with 2,271,260 nodes and 2,073,394 elements in total. The y

^{+}value is controlled within 7.8~149.3 on both the impeller blades and diffuser blades by setting the first off-wall layer height of 0.03 mm. This y

^{+}range can be proper for applying the automatic wall treatment [33].

#### 3.3. Setup of CFD Simulation

- Velocity inlet: the velocity at inlet boundary, V
_{in}, was V_{in}= Q/A_{in}where A_{in}is the impeller inflow area; the temperature at inlet boundary, T_{in}, was 298.15 K; the pressure at inlet boundary, p_{in}, follows the Neumann condition [34]; the inlet turbulence intensity is set as medium of 5%; - Pressure outlet: the pressure at outlet boundary, p
_{out}, was 0 Pa relative to the environment pressure 1 atm; the velocity and temperature at outlet boundary, V_{out}and T_{out}, follows the Neumann condition; - No slip wall: the impeller hub, impeller shroud, impeller blade, diffuser hub, diffuser shroud and diffuser blade are all in the no slip wall type [35];
- Rotor-Stator interface: an interface is given between impeller and diffuser with conservative interface flux on mass and momentum; the mesh is connected using the general grid interface (GGI) method.

^{−5}. The transient simulation is conducted for about 0.343 s (10 impeller revolutions). The time step is 9.52 × 10

^{−5}s with the maximum iteration step number of 10 and the criterion of RMS residual less than 1 × 10

^{−6}. The advection scheme is set as high resolution.

## 4. Numerical-Experimental Verification Study

_{i}

_{2}is chosen for verification by dividing into radial component C

_{r}and tangential component C

_{t}. Figure 4 shows the comparison between experimental and numerical data by plotting curves. Figure 5 shows the comparison between experimental and numerical data on contours. Parameter G

_{z}is the relative impeller blade channel position where 0~2 means two channels. Parameter S

_{p}is the spanwise position where 0 is at hub and 1 is at shroud.

## 5. Transient Flow Field Analysis at Lower-Load

#### 5.1. Velocity Fields

_{vrel}is defined as:

_{rel}is the relative velocity.

_{vrel}region can be mainly found on the suction side of impeller blade and also found near the impeller blade trailing-edge on the pressure side. There are also three mainly low C

_{vrel}regions that located at the diffuser outlet, in the impeller trailing-edge wake and near the convex side of diffuser blade. Figure 7 shows the instantaneous flow regime using relative velocity vectors with enlarged views around leading-edges and trailing-edges. A stall vortex flow can be observed in the diffuser outlet low C

_{vrel}region. Flow separations are also found on the convex side of the diffuser blade near leading-edge and in the impeller blade’s trailing-edge wake. On the contrary, the flow regime is well-behaved near the impeller leading-edge especially on the blade suction side. Therefore, the low C

_{vrel}region is related to the local undesirable flow pattern like separation, wake and vortex. The high C

_{vrel}region is because of the smooth local-flow.

#### 5.2. Energy Dissipation Analysis

_{stot}

^{*}can be defined as:

_{stot}

^{*}contour in the impeller and diffuser is shown in Figure 8. The lowest C

_{stot}

^{*}region is near the impeller blade leading-edge on the suction side. Three mainly high C

_{stot}

^{*}regions can be found at the diffuser outlet, in the impeller trailing-edge wake and near the convex side of diffuser blade. The high C

_{stot}

^{*}regions overlap the low C

_{vrel}regions as shown in the relative velocity contour map. The low C

_{stot}

^{*}regions also overlap the well-behaved flow regions. It revealed that the energy dissipation (transformed to internal energy) is caused by the local undesirable flow regime.

_{stot}

^{*}on the diffuser blade leading-edge. On Figure 6, high C

_{stot}

^{*}regions can be obviously found on the leading-edge of 4 of the 12 diffuser blades. The high C

_{stot}

^{*}regions on the convex side of the diffuser blade also only occur in some specific channels. These two high C

_{stot}

^{*}regions seem to be random or rotationally periodic. To understand this phenomenon and its transient change, the internal flow observation is conducted in the region shown in Figure 9 within one impeller revolution. Monitoring points P

_{1}to P

_{4}are also set in the typical high and low C

_{stot}

^{*}regions as indicated in Figure 9.

_{stot}

^{*}contour pulsation in one impeller revolution by plotting sub-maps for each 1/18 revolution. The high C

_{stot}

^{*}region consistently exists on the diffuser blade near outlet which is the stall vortex flow site. However, the high C

_{stot}

^{*}region in the diffuser trailing-edge wake changed periodically. The high C

_{stot}

^{*}regions on the diffuser blade’s convex side also periodically generates and disappears. The high C

_{stot}

^{*}regions form a street from diffuser blade leading-edge to trailing-edge. Another high C

_{stot}

^{*}region occurs periodically on the diffuser leading-edge. Obviously, it is caused by the rotor-stator interaction. When impeller blade trailing-edge wake passes by the diffuser blade leading-edge, high C

_{stot}

^{*}generates. The high C

_{stot}

^{*}region in the impeller blade trailing-edge wake is relatively stable which consistently exists during impeller rotation.

_{1}~P

_{4}shown in Figure 7, the transient C

_{stot}

^{*}pulsation is analyzed in frequency in Figure 11. According to the rotating speed of 1750 r/min, the impeller frequency f

_{imp}is 29.167 Hz, the impeller blade frequency f

_{ib}=Z

_{i}·f

_{imp}is 204.167 Hz. Figure 11a is on P

_{1}which located on diffuser blade leading-edge and is near the impeller blade trailing-edge. The frequency on P

_{1}is mainly dominated by f

_{ib}. The 2-times to 7-times f

_{ib}frequencies are also strong. The C

_{stot}

^{*}on P

_{1}seems to be influenced by the rotor-stator interaction. Figure 11b is on P

_{2}which located on diffuser blade’s convex side. The frequency that f

_{ib}also dominates and with also 2 to 4 times f

_{ib}peaks. The C

_{stot}

^{*}on P

_{2}are also under the rotor-stator interaction. Figure 11c is on P

_{3}which is in the stall vortex on diffuser blade trailing-edge near outlet. The dominate frequency is a very-low stall frequency f

_{TEv}which is about 1/5 f

_{imp}. It is mainly and strongly influenced by the stable stalled vortex structure which is similar as in the rotating stall cases [36,37]. Figure 11d is on P

_{4}which is in the diffuser blade trailing-edge wake. The frequency on P

_{4}becomes complex with both the stall frequency f

_{TEv}and the 1~4 times of impeller blade frequency f

_{ib}. It shows that the C

_{stot}

^{*}in diffuser blade trailing-edge wake is influenced by both the impeller incoming flow and the trailing-edge stall vortex flow.

#### 5.3. Turbulence Kinetic Energy Fields

_{k}is defined for analysis:

_{k}region mainly occurs on the convex side of diffuser blade, in the diffuser blade trailing-edge wake and in the impeller blade trailing-edge wake. These three regions overlap the high C

_{stot}

^{*}regions in Figure 8. However, the stall vortex flow site on diffuser blade near outlet is the low C

_{k}region. It does not accord with the local high C

_{stot}

^{*}characteristic.

_{k}values at diffuser inlet.

#### 5.4. Flow Induced Noise

_{sp}is analyzed as the instantaneous contour in Figure 13. The high L

_{sp}regions completely accord with the high C

_{k}regions and are obviously stronger than the surrounded low L

_{sp}sites. To understand the L

_{sp}pulsation during impeller rotation, the L

_{sp}contour is monitored and analyzed within one impeller revolution.

_{sp}fields. The high L

_{sp}regions on diffuser blade are no more a street but are triggered from leading-edge and separate into mid-channel. The diffuser leading-edge high and low L

_{sp}regions are pulsating due to rotor-stator interaction. The stall vortex region on the diffuser blade near outlet is always low in L

_{sp}intensity. The high L

_{sp}regions in the impeller blade trailing-edge wake and in the diffuser blade trailing-edge wake always exist and are relatively stable.

_{1}~P

_{4}shown in Figure 7, the transient L

_{sp}pulsation is analyzed in frequency in Figure 15. Figure 15a is on P

_{1}. The frequency on P

_{1}is also mainly dominated by f

_{ib}which is the same as the C

_{stot}

^{*}pulsation. The 2-times and 3-times f

_{ib}frequencies are also strong, which indicate the rotor-stator interaction effect on L

_{sp}on P

_{1}. Figure 15b is on P

_{2}. The frequency f

_{ib}is the only dominate frequency greater than other values. The flow-induced noise field on P

_{2}is also under the rotor-stator interaction. Figure 15c is on P

_{3}. The stall frequency f

_{TEv}≈1/5 f

_{imp}is the strongest frequency. The 2 and 3 times f

_{imp}are also obvious as peaks. Both the stall frequency and impeller frequency can be found on the L

_{sp}on P

_{3}but the stall frequency dominates. Figure 15d is on P

_{4}. The L

_{sp}pulsation frequency on P

_{4}is not the same as the C

_{stot}

^{*}pulsation frequency. Due to the strong trailing-edge wake, the stall frequency of L

_{sp}caused by trailing-edge stall vortex does not strongly impact the local flow. Thus, frequency f

_{TEv}≈1/5 f

_{imp}is not found on Figure 15d but the f

_{ib}frequency dominates. The flow induced-noise in the trailing-edge wake is mainly influenced by impeller incoming flow.

## 6. Conclusions

- (1)
- The temperature and static entropy patterns can be successfully found by applying the total energy governing equations. The flow energy which transferred to internal energy can be quantitatively known. The flow-induced noise, which might be another energy dissipation source, can be also predicted based on the turbulence flow modeling. It is found strongly relative to the turbulence kinetic energy and dissipation rate.
- (2)
- The high static entropy sites are related to the local low velocity regions. According to the vectors of relative velocity, these low velocity regions have undesirable flow regime. The main high static entropy sites locate in the impeller trailing-edge wake, on the diffuser blade leading-edge and convex side, in the diffuser blade trailing-edge stall vortex near outlet and in the diffuser blade trailing-edge wake. The high noise sites overlap some of the high static entropy sites. These overlapped noisy sites are also due to the local undesirable flow regime. There is an exception as stall vortex near diffuser blade trailing-edge. The flow-induced noise is constantly very low because of the local low turbulence kinetic energy. Accordingly, different flow structures have different energy dissipation mechanisms. In this case, rotor-stator interaction affects both the internal energy and the flow-induced noise. The stalled vortex flow mainly causes the internal energy variation but weak in producing noise.
- (3)
- These high energy dissipation regions perform differently during rotor-stator interaction. Based on the frequency analysis, the transient characteristics of energy dissipation during impeller rotation can be clarified in detail. The flow regime on the diffuser blade leading-edge, on the diffuser blade’s convex side and in the diffuser blade trailing-edge wake are influenced mainly by the impeller frequency or the impeller blade frequency. It shows that the rotor-stator interaction affects the flow regime, energy dissipation and flow-induced noise from diffuser inlet to outlet. The diffuser blade trailing-edge stall vortex is not strongly influenced by rotor-stator interaction. It keeps the stall frequency that is about 1/5 impeller frequency. This stall frequency also affects the diffuser trailing-edge wake region.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Latin Letters | |||

A_{in} | impeller inflow area | Q | flow rate |

c_{h} | heat capacity | R_{cn} | Cone radius |

C_{k} | turbulence kinetic energy coefficient | R_{d1} | Diffuser blade inlet radius |

C_{r} | radial component of velocity | R_{d2} | Diffuser blade outlet radius |

C_{stot}^{*} | energy dissipation coefficient | R_{hub} | Hub arc radius |

C_{t} | tangential component of velocity | R_{i1} | Impeller blade inlet radius |

C_{vrel} | relative velocity coefficient | R_{i2} | Impeller blade outlet radius |

C_{φ} | flow rate coefficient | R_{in} | Impeller inflow radius |

C_{ω} | production term coefficient in SST model | R_{shr} | Shroud arc radius |

F_{1} | blending function in SST model | s^{*} | specific entropy |

f_{ib} | impeller blade frequency | S_{ij} | strain tensor |

f_{imp} | impeller frequency | S_{p} | spanwise position |

f_{TEv} | diffuser trailing-edge stall frequency | t | time |

g | acceleration of gravity | T | temperature |

G_{z} | relative impeller blade channel position | T_{in} | temperature at inlet boundary |

h_{b} | Height of impeller/diffuser blade | T_{out} | temperature at outlet boundary |

h_{cn} | Height of cone | u | velocity |

h_{hub} | Height over cone on hub | U_{i2} | Rotational linear speed at impeller outlet |

h_{sta} | static enthalpy | V_{c} | sound speed |

h_{tot} | total enthalpy | V_{in} | velocity at inlet boundary |

k | turbulence kinetic energy | V_{out} | velocity at outlet boundary |

l_{k-ω} | turbulence scale in SST model | V_{rel} | relative velocity |

L_{sp} | sound power level | W_{A} | sound power |

M_{t} | specific turbulence kinetic energy | W_{ref} | reference sound power |

n | Rotational speed | x | coodinate component |

P | turbulence production term in SST model | y^{+} | dimensionless height off-wall |

P_{1}, P_{2}, P_{3}, P_{4} | monitoring point 1 to 4 | Z_{d} | Diffuser blade number |

p_{in} | pressure at inlet boundary | Z_{i} | Impeller blade number |

p_{out} | pressure at outlet boundary | ||

Greek Letters | |||

α_{ε} | model constant in acoustic analogy | μ_{t} | eddy viscosity |

β_{k} | model constant in SST model | ρ | density |

ΔC_{stot} | specific entropy difference coefficient | σ_{k} | model constant in SST model |

δ_{ij} | Kroneker delta | σ_{ω} | model constant in SST model |

ε | eddy dissipation rate | ϕ | denotation for arbitrary parameter |

λ_{t} | thermal conductivity | in governing equations | |

μ | dynamic viscosity | ω | specific turbulence dissipation rate |

Acronyms | |||

CFD | computational fluid dynamics | LNR | low noise region |

EXP | experiment | LVR | low C_{vrel} region |

GGI | general grid interface | RANS | Reynolds-averaged Navier-Stokes |

HKR | high k region | REV | impeller revolution |

HNR | high noise region | RMS | root mean square |

HVR | high C_{vrel} region | SS | blade suction side |

LE | blade leading-edge | SST | shear stress transport |

LKR | low k region | TE | blade trailing-edge |

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**Figure 2.**Mesh scheme determination and diagram (

**a**) mesh scheme determination by checking the residual of ΔC

_{stot}; (

**b**) diagram of the mesh scheme.

**Figure 6.**Instantaneous contour of relative velocity coefficient C

_{vrel}with indications of the local low and high regions. LVR: low C

_{vrel}region; HVR: high C

_{vrel}region; SS: blade suction side.

**Figure 7.**Instantaneous vectors of relative velocity with enlarged views. LE: leading-edge; TE: trailing-edge.

**Figure 8.**Instantaneous contour of local dissipation coefficient C

_{stot}

^{*}with indications of low and high regions.

**Figure 9.**Observation region and monitoring points for the transient pulsation of internal flow characteristics.

**Figure 11.**C

_{stot}

^{*}pulsation on P

_{1}, P

_{2}, P

_{3}and P

_{4}within one impeller revolution. (

**a**) on P

_{1}; (

**b**) on P

_{2}; (

**c**) on P

_{3}; (

**d**) on P

_{4}.

**Figure 12.**Instantaneous contour of turbulence kinetic energy coefficient C

_{k}with the indications of local low and high regions. HKR: high k region; LKR: low k region; LE: leading-edge; TE: trailing-edge.

**Figure 13.**Instantaneous contour of flow-induced noise level L

_{sp}with the indications of local low and high regions. HNR: high noise region; LNR: low noise region; LE: leading-edge; TE: trailing-edge.

**Figure 15.**L

_{sp}pulsation on P

_{1}, P

_{2}, P

_{3}and P

_{4}within one impeller revolution. (

**a**) on P

_{1}; (

**b**) on P

_{2}; (

**c**) on P

_{3}; (

**d**) on P

_{4}.

Parameter | Value and (Unit) |
---|---|

Impeller blade inlet radius R_{i}_{1} | 0.120 (m) |

Impeller blade outlet radius R_{i}_{2} | 0.210 (m) |

Diffuser blade inlet radius R_{d}_{1} | 0.222 (m) |

Diffuser blade outlet radius R_{d}_{2} | 0.332 (m) |

Hub arc radius R_{hub} | 0.100 (m) |

Shroud arc radius R_{shr} | 0.025 (m) |

Impeller inflow radius R_{in} | 0.092 (m) |

Cone radius R_{cn} | 0.009 (m) |

Height of cone h_{cn} | 0.026 (m) |

Height over cone on hub h_{hub} | 0.087 (m) |

Height of impeller/diffuser blade h_{b} | 0.020 (m) |

Impeller blade number Z_{i} | 7 (-) |

Diffuser blade number Z_{d} | 12 (-) |

Parameter | Value and (Unit) |
---|---|

Rotational speed n | 2000 (r/min) |

Rotational linear speed at impeller outlet U_{i}_{2} | 43.98 (m/s) |

Flow rate coefficient C_{φ} | 0.048 |

Fluid medium dynamic viscosity μ | 1.83 × 10^{−5} (kg/m·s) |

Fluid medium density ρ | 1.2 (kg/m^{3)} |

Fluid medium thermal conductivity λ_{t} | 0.0261 (W/m·K) |

Fluid specific heat capacity c_{h} | 1004 (J/kg·K) |

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## Share and Cite

**MDPI and ACS Style**

Tao, R.; Zhao, X.; Wang, Z.
Evaluating the Transient Energy Dissipation in a Centrifugal Impeller under Rotor-Stator Interaction. *Entropy* **2019**, *21*, 271.
https://doi.org/10.3390/e21030271

**AMA Style**

Tao R, Zhao X, Wang Z.
Evaluating the Transient Energy Dissipation in a Centrifugal Impeller under Rotor-Stator Interaction. *Entropy*. 2019; 21(3):271.
https://doi.org/10.3390/e21030271

**Chicago/Turabian Style**

Tao, Ran, Xiaoran Zhao, and Zhengwei Wang.
2019. "Evaluating the Transient Energy Dissipation in a Centrifugal Impeller under Rotor-Stator Interaction" *Entropy* 21, no. 3: 271.
https://doi.org/10.3390/e21030271