# Control-Volume-Based Exergy Method of Truncated Busemann Inlets in Off-Design Conditions

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Truncated and Corrected Busemann Inlet Design Methods

#### 2.2. Exergy-Based Approach

#### 2.3. Numerical Methods and Boundary Conditions

#### 2.4. Performance Indicators

## 3. Validations

#### 3.1. Grid Dependency Validation

^{+}values below 1. Refinement on the edge was performed to smooth transitions around corners. The analysis of the exit Mach number and exit static pressure varies with different numbers of grids and was conducted to determine the suitable number of grids, as presented in Figure 4. It can be observed that the differences in the exit Mach number and static pressure between 3 million grids and 12 million grids are 0.016% and 0.034%, respectively. As the difference is much smaller than the errors of the numerical process, the number of grids chosen for the numerical simulations is in the middle range of about seven million.

#### 3.2. Comparison of Design Data and Numerical Models

#### 3.3. Exergy Analysis of Inviscid Flow in Busemann Inlets

^{−1}) in the designed Busemann inlet was obtained by the formula, as presented in Table 2.

^{−1}, 12.12%) and some was turned into thermal exergy (4972.74 J∙s

^{−1}, 13.54%), while the remaining main energy was still reserved in the high Mach number airflow as a form of kinetic exergy (25,274.78 J∙s

^{−1}, 68.81%). A very small amount of exergy turned into transverse kinetic exergy, as listed in Table 3. Thus, the difference between the exergy inflow and exergy outflow is 42.93 J∙s

^{−1}. This value is consistent with the anergy calculated from the Gouy–Stodola theorem with an error of 0.21%. That is, the data listed in Table 2 and Table 3 proved that the exergy method adopted and the calculation process are completely effective and correct. To be noted, the anergy produced by shock waves accounts for only 0.12% of the total incoming exergy.

## 4. Results and Discussion

#### 4.1. Flow Field Exergy Loss Analysis

#### 4.2. Exergy Distribution Quantitative Analysis

#### 4.3. Total Performance Parameter Analysis

^{−1}to 877.28 J∙s

^{−1}and viscous rises from 479.7 J∙s

^{−1}to 744.17 J∙s

^{−1}as the Mach number increases from 4.5 to 6. An intersection point was observed between these two curves when the Mach number was between 5 and 5.5. The thermal anergy is lower than the viscous anergy when the Mach number is below the intersection point. However, after the intersection point, the thermal anergy rapidly increases and is higher than the viscous anergy. It is probable that the convective heat transfer coefficient increases significantly and heat mixing in the inlet becomes more intense when the Mach number is above the design point, whereas the viscosity coefficient varies less with the airflow speed and temperature. As the shock is almost negligible, it implies that the optimization of the minimum total exergy destroyed can be made to find the particular Mach number when the sum of thermal anergy and viscous anergy is minimum.

## 5. Conclusions

- (1)
- Compared to other traditional performance indicators such as the total pressure recovery coefficient, only providing an overall performance value, the exergy method can interpret the amount and the evolution process of each amount of exergy destroyed in the inlet. In the Busemann inlet, the exergy destroyed can be decomposed into shock wave anergy, viscous anergy and thermal anergy. Shock wave anergy accounts for less than 4% of the total exergy destroyed, while thermal anergy and viscous anergy have a roughly equivalent magnitude and contribute to almost all the remaining anergy. The vast majority of the inflow exergy is converted into boundary pressure work and thermal exergy.
- (2)
- The total pressure recovery coefficient, total anergy and static temperature ratio of the Busemann inlets increase nonlinearly with the Mach number without any deviation due to the influence of the design point. However, the Busemann inlet on-design has the maximum pressure uniformity at the exit.
- (3)
- The exergy efficiency of the Busemann inlets is higher than the total pressure recovery coefficient since some of the thermal exergy is treated as a loss in the calculation of the total pressure recovery coefficient, but further enters the combustion chamber and is converted into useful work. An intersection point was found between the curves of the viscous anergy and thermal anergy when the Mach number was between 5.0 and 5.5. It implies that the minimum total exergy destroyed at a particular Mach number can be observed if the optimization of the total anergy in the inlet was carried out.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 15.**Total pressure recovery coefficient and exergy destroyed efficiency versus flight Mach number.

**Figure 16.**Contours of total pressure recovery coefficient and entropy production at the cross section of outlets.

Parameter | Symbol | Value | Unit |
---|---|---|---|

Freestream Mach number | $M{a}_{\infty}$ | 5 | _{−} |

Throat Mach number | $M{a}_{throat}$ | 3 | _{−} |

Shape of exit | _{−} | circle | _{−} |

Radius of exit | ${r}_{0}$ | 10 | cm |

Ratio of specific heat | $\gamma $ | 1.4 | _{−} |

Freestream pressure | ${p}_{\infty}$ | 1170 | Pa |

Freestream temperature | ${T}_{\infty}$ | 226.65 | K |

Entropy Production Rate (J∙K ^{−1}) | Anergy Calculated from Entropy (J∙s ^{−1}) | Anergy Calculated from Balance Equations (J∙s ^{−1}) | Errors |
---|---|---|---|

0.189 | 42.84 | 42.93 | 0.21% |

Parameters | Inflow | Outflow | Difference Value |
---|---|---|---|

Streamwise kinetic exergy deposition rate, $\dot{{\mathrm{E}}_{u}}$ (J∙s^{−1}) | 34,742.70 | 25,274.78 | −9467.92 |

Transverse kinetic exergy deposition rate, $\dot{{\mathrm{E}}_{vw}}$ (J∙s^{−1}) | 0.0 | 2.42 | 2.42 |

Boundary pressure work rate, $\dot{{\mathrm{E}}_{p}}$ (J∙s^{−1}) | 0.01 | 4449.84 | 4449.83 |

Rate of thermal exergy, $\dot{{\mathrm{E}}_{th}}$ (J∙s^{−1}) | 1986.71 | 6959.45 | 4972.74 |

Total (J∙s^{−1}) | 36,729.42 | 36,686.49 | 42.93 |

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**MDPI and ACS Style**

Zhu, M.; Zhou, S.; Liu, Y.; Li, Z.; Chen, Z.
Control-Volume-Based Exergy Method of Truncated Busemann Inlets in Off-Design Conditions. *Processes* **2024**, *12*, 535.
https://doi.org/10.3390/pr12030535

**AMA Style**

Zhu M, Zhou S, Liu Y, Li Z, Chen Z.
Control-Volume-Based Exergy Method of Truncated Busemann Inlets in Off-Design Conditions. *Processes*. 2024; 12(3):535.
https://doi.org/10.3390/pr12030535

**Chicago/Turabian Style**

Zhu, Meijun, Shuai Zhou, Yang Liu, Zhehong Li, and Ziyun Chen.
2024. "Control-Volume-Based Exergy Method of Truncated Busemann Inlets in Off-Design Conditions" *Processes* 12, no. 3: 535.
https://doi.org/10.3390/pr12030535