# Numerical Simulation of the Heat Transfer and Flow Characteristics of Pulse Tube Refrigerators

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Numerical Model

#### 3.1. Geometric Model

#### 3.2. Initial and Boundary Conditions

#### 3.3. Solving Method

^{−6}, and the remaining items are 1 × 10

^{−3}. Standard k-ε turbulent model is chosen to solve the turbulence and the PISO method is used to solve pressure-velocity coupling. The PRESTO! method is adopted for the pressure term, and each cycle has 40 time steps; thus, the accuracy of the model can be guaranteed.

#### 3.4. Model Validation

## 4. Results and Discussion

#### 4.1. Effect of the Conical Tube on the Cooling Rate

#### 4.2. Temperature Variation in the Pulse Tube

#### 4.3. Variation of Velocity in the Pulse Tube

#### 4.4. Multi-Dimensional Flow Analysis in Pulsed Tubes

^{−1}(velocity field of spin). The vorticity difference of the other three models is very small (below 700 S

^{−1}). For the Original Model, the vorticity near the wall is much greater when the flow is changing direction than that when the flow is steady. This is also the time when vortices are likely to be generated. The Original Model has a very large vorticity gradient at the boundary close to the wall. A very large vorticity gradient can lead to strong vortices forming easily near the wall, which are generated near the inlet and developed with the fluid flow, disrupting the smooth linear flow in the pulse tube and leading to a mixture of hot and cold fluids, resulting in energy loss. Models 1, 2, and 3 do not generate strong vortices due to the small vorticity at the boundary and the relatively small radial and axial velocities, which ensure a smoother temperature gradient in the pulse tube.

#### 4.5. Effect of the Conical Tube on the Cooling Capacity and COP

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A_{0} | Maximum displacement | t | Time |

CD | Taper pulse tube | $\stackrel{=}{t}$ | Stress tensor |

CHX | Cold Heat Exchanger | $\mathrm{u},\overrightarrow{v}$ | Velocity |

COP | Coefficient of Performance | v | Viscosity |

Ef | Fluid total energy | w | Angular frequency |

Es | Solid total energy | WHX1 | Aftercooler |

F | Body force | WHX2 | Hot heat exchanger |

f | Frequency | ρ_{f}, ρ | Fluid density |

kf, keff | Thermal conductivity | ρ_{s} | Solid density |

P | Pressure | Φ | Porosity |

S | Source term | ω | Vorticity |

T | Temperature |

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**Figure 1.**Four types of pulse tube structures. (

**a**) Original Model. (

**b**) Model 1. (

**c**) Model 2. (

**d**) Model 3.

Components | Number | Radius (mm) | Length (mm) | Material | Boundary Conditions |
---|---|---|---|---|---|

Compressor | A | 9.54 | 7.5 | Steel | Adiabatic |

Transfer line | B | 1.55 | 101 | Steel | Adiabatic |

WHX1 | C | 4 | 20 | Copper | 300 K |

Regenerator | D | 4 | 58 | Steel | Adiabatic |

CHX | E | 3 | 5.7 | Copper | Adiabatic |

Pulse tube | F | 2.5 | 60 | Steel | Adiabatic |

WHX2 | G | 4 | 10 | Copper | 300 K |

Inertance Tube | H | 0.425 | 684 | Steel | Adiabatic |

gas reservoir | I | 13 | 130 | Steel | Adiabatic |

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**MDPI and ACS Style**

Meng, Y.; Cui, Z.; Shao, W.; Ji, W.
Numerical Simulation of the Heat Transfer and Flow Characteristics of Pulse Tube Refrigerators. *Energies* **2023**, *16*, 1906.
https://doi.org/10.3390/en16041906

**AMA Style**

Meng Y, Cui Z, Shao W, Ji W.
Numerical Simulation of the Heat Transfer and Flow Characteristics of Pulse Tube Refrigerators. *Energies*. 2023; 16(4):1906.
https://doi.org/10.3390/en16041906

**Chicago/Turabian Style**

Meng, Yuan, Zheng Cui, Wei Shao, and Wanxiang Ji.
2023. "Numerical Simulation of the Heat Transfer and Flow Characteristics of Pulse Tube Refrigerators" *Energies* 16, no. 4: 1906.
https://doi.org/10.3390/en16041906