# Study of Energy Loss Characteristics of a Shaft Tubular Pump Device Based on the Entropy Production Method

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emissions and carbon neutrality, which requires that pump devices as energy consumers must operate more efficiently. Therefore, conducting a more in-depth study on the energy loss mechanism of STPDs is of great significance.

## 2. Numerical Simulation

#### 2.1. Settings and Turbulence Model

#### 2.2. Grid Division and Scheme Selection

## 3. Model Test

## 4. Analysis Method

#### 4.1. Traditional Pressure Drop Method

_{s}in the stationary domain (inlet and outlet passage, guide vanes) and Δh

_{r}in the rotating domain (impeller) can be calculated by Equations (1) and (2), respectively. The above hydraulic loss can be converted into the energy loss by Equation (3).

_{1}and p

_{2}represent the total pressure of inlet and outlet sections, Pa; and P is the input shaft power, kW.

#### 4.2. Entropy Production Method

_{1}, u

_{2}and u

_{3}, the Φ can be expanded to Equation (6):

## 5. Results and Analysis

#### 5.1. Validation of Simulation and Comparison of Energy Analysis Methods

_{d}~1.2 Q

_{d}, the head and the shaft power curves of the STPD decrease with the increase in flow rate. The efficiency curve of the STPD increases with the increasing flow rate, reaching a maximum at 1.0 Q

_{d}and then decreasing at 1.0 Q

_{d}. The relative errors of head, shaft power and efficiency between the model test and numerical simulation are 2.49%, 2.69% and 0.62%, respectively. The maximum error of the shaft power and efficiency occurs at 0.8 Q

_{d}, and the relative errors are 4.95% and 1.80%, respectively. The relative error at low-flow-rate conditions is mainly due to the flow separation at the blades. The maximum error of the head occurs at 1.2 Q

_{d}with a relative error of 5.2%, which is caused by the low absolute value of the head at high-flow-rate conditions. Although there are still some uncertainties in numerical simulations that may cause slight deviations from experimental results, overall, the results are reliable.

_{d}~1.2 Q

_{d}. The results obtained by the entropy production method are proved to be reliable according to the traditional pressure drop method; hence, this study can be conducted based on the entropy production method.

#### 5.2. Analysis of Energy Loss in the STPD

_{d}~1.1 Q

_{d}, the loss proportion of each component from the largest to the smallest is impeller, outlet passage, guide vanes and inlet passage; however, the loss proportion of the outlet passage at 1.2 Q

_{d}is more than the impeller. According to Figure 7a, in the flow range of 0.8 Q

_{d}~1.2 Q

_{d}, the loss in the inlet passage accounts for 3.0~8.2% of the loss in the STPD, the loss in the impeller accounts for 30.9~47.7%, the loss in the guide vanes accounts for 10.2~26.2% and the loss in the outlet passage accounts for 31.5~41.4%. As shown in Figure 7b, the percentage of energy loss in each component is basically consistent with the results obtained by the traditional pressure drop method. In addition, the proportions of different types of entropy production are shown in Figure 7b. It is obvious that the DEP in each component is small and can be neglected, while the TEP is the main contributor to energy loss in STPDs, which is consistent with the conclusions of previous research [31,35].

#### 5.3. Analysis of TEP Distribution

_{d}~1.0 Q

_{d}, the TEP gradually decreases, while in the flow range of 1.0 Q

_{d}~1.2 Q

_{d}, the TEP increases, and the trend of variation of the energy loss is consistent with Figure 7. As shown in Figure 9a, it can be found that the TEP in the impeller firstly increases and then decreases along the axial direction at each condition. In IA3, the flow separation at the leading edge of the impeller leads to a surge of TEP. At 0.8 Q

_{d}, the TEP in IA3 is the largest, which is due to the decrease in the attack angle; the flow separation is more serious, as is the formation of backflow and vortex [36]. It can be found from Figure 9b that the TEP in the middle subdomains of the impeller is relatively stable, which indicates that the flow state in this region is stable. As it is influenced by the frictional resistance of the wall and the viscous resistance of the liquid, thus the flow in the axial direction near the wall is weakened and the loss increases [37]. Therefore, the TEP in IR1 and IR10 are clearly higher than that in the adjacent subdomains. Additionally, the presence of the tip leakage vortex near the shroud further promotes the TEP in IR10. This results in the TEP proportion exceeding 40% in IR10 with a volume fraction of 14%. Furthermore, the range of TLV increases as the flow velocity decreases [38], hence the TEP in IR10 is large at a small flow rate.

_{d}~1.2 Q

_{d}, the LEP in subdomains decreases along the axial direction, which indicates that the guide vanes reduce the circulation velocity of the flow out of the impeller, and finally the flow in the subdomains at the exit is stable and the TEP is lower. At 0.8 Q

_{d}, there is a vortex at the back of the guide vanes near the outlet section, therefore the TEP in GA10 increases. According to Figure 10b, along the radial direction, the TEP in GR1 and GR10 is clearly higher than that in the adjacent subdomains due to the wall effect. At 0.9 Q

_{d}~1.1 Q

_{d}, the TEP in the guide vanes is stable, which indicates that the flow is stable. At 0.8 Q

_{d}and 1.2 Q

_{d}, the distribution of TEP has no clear regularity due to the large deviation from the design flow rate, which also indicates the internal flow of guide vanes is chaotic.

_{d}~1.1 Q

_{d}, the distribution characteristics of TEP along the axial direction are consistent. The TEP increases first, with the maximum in the OA3 and OA4, and then decreases gradually. The volume of OA1~OA4 accounts for 30%; however, the TEP accounts for more than 65%. In the rear part, the energy loss is very small because the velocity and circulation have decreased due to the completion of the flow diffusion. At 0.8 Q

_{d}, the TEP value in the OA1 is the largest, which is due to the effect of the velocity circulation and guide vanes’ wake vortex (GWV). At 1.2 Q

_{d}, due to the larger flow velocity, the TEP values in OA2~OA10 are larger than those at other flow-rate conditions.

#### 5.4. Visualization Analysis

_{d}. The ILV is the flow separation caused by the large attack angle at a small flow rate. There is also a high TEPR region in the impeller passage, which is caused by the hub vortex of the impeller (IHV). At 1.0 Q

_{d}and 1.2 Q

_{d}, the attack angle decreases and the ILV disappears, hence the TEPR is significantly lower. In the middle passage (span = 0.5), the flow moves along the blade airfoil, hence the TEPR is obviously lower than that near the hub and shroud. Near the shroud region (span = 0.97), the TLV leads to the high TEPR region, and as the flow rate increases; the head and the pressure difference between the pressure side (PS) and suction side (SS) decrease, which leads to a reduction of the area of the TLV and high TEPR region.

_{d}, because of the misfit of flow direction and the guide vanes placement angle, there is a guide vanes separation vortex (GSV) in the SS side near the outlet, and the TEPR is high in this region. At 1.0 Q

_{d}, the flow in the guide vanes is stable without flow separation, hence the TEPR is low. At 1.2 Q

_{d}, the location of GSV shifts to the PS of the guide vanes near the inlet and results in a high TEPR region.

_{d}, the presence of GHV near the hub leads to a small high TEPR region. Along the axial direction, GSV appears in the SS of the guide vanes and gradually increases, hence the area of the high TEPR region also increases. When the flow rate is 1.0 Q

_{d}and the TEPR is small in each section of the guide vanes, there is only a small high TEPR region that appears near the hub caused by GHV and gradually increases with the axial direction. At 1.2 Q

_{d}, in each monitoring section, there are high TEPR regions near the hub and the PS of the guide vanes due to the influence of GSV and GHV. Along the axial direction, the high TEPR region gradually decreases because of the decrease in the GSV range.

## 6. Conclusions

- The energy loss in each part of the STPD is analyzed, and the results show that the prediction of the energy loss by the above two methods is consistent. The TEP is the primary cause of energy loss, while the EPW is the secondary source. TEP accounts for 65.40~77.88% of the energy loss in the STPD, whereas EPW accounts for 21.54~33.63%.
- At 0.8 Q
_{d}~1.1 Q_{d}, the component with the largest energy loss in an STPD is the impeller. Due to the wall effect and TLV, the energy loss in the region near the impeller shroud accounts for more than 40% of the total loss in the impeller. Therefore, it is necessary to consider the influence of the tip clearance of the impeller to reduce losses. - At 1.2 Q
_{d}, the component with the largest energy loss in an STPD is the outlet passage. The energy loss in the outlet passage is mainly concentrated in the entrance part due to the wake flow of the guide vanes. Therefore, the adoption of appropriate measures at the entrance part can be considered to eliminate the influence of the wake flow and reduce loss. - The visualization of the TEPR of the main components of the STPD clearly identify the vortex regions, which are accurately verified by the flow field structure. The main vortex structures in the impeller are IWV, IHV and TLV, while the main vortex structures in the guide vanes are GWV, GHV and GSV. It is clearly found that the TEPR is significantly higher in the vortex region. The distribution of TEPR in the outlet passage is clearly influenced by the guide vanes, and the influence gradually decreases along the axial direction.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature and Abbreviations

$\u2206{h}_{s}$: | Total pressure loss in the stationary domain, m |

$\u2206{h}_{r}$: | Total pressure loss in the stationary domain, m |

${p}_{1}$: | Total pressure of inlet section, Pa |

${p}_{2}$: | Total pressure of outlet section, Pa |

ρ: | Density of fluid, kg/m^{3} |

Q: | Flow rate, m^{3}/s |

$\dot{Q}$: | Energy conversion rate |

${u}_{i}$: | Velocity with different directions in Cartesian coordinates, m/s |

${\delta}_{ij}$: | Crowe dick symbol |

STPD: | Shaft tubular pump device |

EPW: | Entropy production in the wall region |

TEP: | Turbulent entropy production |

LEP: | Local entropy production |

LEPR: | Local entropy production rate |

TEPR: | Turbulent entropy production rate |

LEP: | Local entropy production |

GCI: | Grid convergence index |

TLV: | Tip leakage vortex |

IWV: | Impeller wake vortex |

ILV: | Vortex at leading edge of impeller |

IHV: | Impeller hub vortex |

GWV: | Guide vanes wake vortex |

GHV: | Guide vanes hub vortex |

GSV: | Guide vanes separation vortex |

PS: | Pressure side |

SS: | Suction side |

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**Figure 7.**Energy loss in each part of an STPD at different flow-rate conditions. (

**a**) Traditional pressure drop method; and (

**b**) entropy production method.

**Figure 8.**The division diagram. (

**a**) The subdomains of the impeller; and (

**b**) the subdomains of the guide vanes.

**Figure 13.**The distribution of TEPR in the impeller: (

**a**) 0.8 Q

_{d}; (

**b**) 1.0 Q

_{d}; and (

**c**) 1.2 Q

_{d}.

**Figure 14.**The flow characteristics in the impeller: (

**a**) 0.8 Q

_{d}; (

**b**) 1.0 Q

_{d}; and (

**c**) 1.2 Q

_{d}.

**Figure 15.**The distribution of TEPR in the guide vanes: (

**a**) 0.8 Q

_{d}; (

**b**) 1.0 Q

_{d}; and (

**c**) 1.2 Q

_{d}.

**Figure 16.**The flow characteristics in the guide vanes: (

**a**) 0.8 Q

_{d}; (

**b**) 1.0 Q

_{d}; and (

**c**) 1.2 Q

_{d}.

**Figure 17.**Matching relationship between the flow direction and the guide vanes placement angle: (

**a**) 0.8 Q

_{d}; (

**b**) 1.0 Q

_{d}; and (

**c**) 1.2 Q

_{d}.

**Figure 19.**The distribution of TEPR and flow characteristics in the monitoring sections of the impeller: (

**a**) 0.8 Q

_{d}; (

**b**) 1.0 Q

_{d}; and (

**c**) 1.2 Q

_{d}.

**Figure 20.**The distribution of TEPR and flow characteristics in the monitoring sections of the guide vanes: (

**a**) 0.8 Q

_{d}; (

**b**) 1.0 Q

_{d}; and (

**c**) 1.2 Q

_{d}.

**Figure 21.**The distribution of TERP in outlet passage: (

**a**) horizontal section; and (

**b**) vertical section.

**Figure 23.**The distribution of TEPR in the monitoring sections of the outlet passage: (

**a**) 0.8 Q

_{d}; (

**b**) 1.0 Q

_{d}; and (

**c**) 1.2 Q

_{d}.

Parameters | Values |
---|---|

Diameter of pump | 300 mm |

Impeller blade number | 3 |

Guide vanes blade number | 6 |

Tip clearance | 0.2 |

Rotational speed | 1450 r/min |

Parameters | Values |
---|---|

Inlet | Total pressure (1 atm) |

Outlet | Mass flow rate |

Wall | No-slip |

Convergence accuracy | 1 × 10^{−4} |

Time step | 3.45 × 10^{−4} (s) |

Total time | 0.828 (s) |

Scheme | Number of Grid/10^{6} | Efficiency/% |
---|---|---|

1 | 1.7 | 81.87 |

2 | 3.9 | 81.64 |

3 | 7.1 | 81.49 |

4 | 8.6 | 81.40 |

5 | 8.8 | 81.39 |

Parameters | Values |
---|---|

N_{1}, N_{2}, N_{3} | 8.6 M, 3.9 M, 1.7 M |

r_{21}, r_{32} | 1.302, 1.319 |

Φ_{1}, Φ_{2}, Φ_{3} | 81.40%, 81.64%, 81.87% |

p | 2.28 |

${\Phi}_{ext}^{21}$,${\Phi}_{ext}^{32}$ | 1.22, 1.21 |

${e}_{a}^{21}$, ${e}_{a}^{32}$ | 0.29%, 0.17% |

GC1_{21} | 0.44% |

GCI_{32} | 0.24% |

Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Efficiency/% | 82.12 | 82.18 | 82.04 | 82.26 | 82.09 | 82.15 | 82.27 | 82.20 | 82.19 | 82.22 |

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## Share and Cite

**MDPI and ACS Style**

Ji, D.; Lu, W.; Xu, B.; Xu, L.; Lu, L.
Study of Energy Loss Characteristics of a Shaft Tubular Pump Device Based on the Entropy Production Method. *Entropy* **2023**, *25*, 995.
https://doi.org/10.3390/e25070995

**AMA Style**

Ji D, Lu W, Xu B, Xu L, Lu L.
Study of Energy Loss Characteristics of a Shaft Tubular Pump Device Based on the Entropy Production Method. *Entropy*. 2023; 25(7):995.
https://doi.org/10.3390/e25070995

**Chicago/Turabian Style**

Ji, Dongtao, Weigang Lu, Bo Xu, Lei Xu, and Linguang Lu.
2023. "Study of Energy Loss Characteristics of a Shaft Tubular Pump Device Based on the Entropy Production Method" *Entropy* 25, no. 7: 995.
https://doi.org/10.3390/e25070995