# Stirling Refrigerating Machine Modeling Using Schmidt and Finite Physical Dimensions Thermodynamic Models: A Comparison with Experiments

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Method

#### 2.1. Description of Experimental Installation

#### 2.2. Application of Schmidt Method with Imperfect Regeneration in the Study of a β-Type Stirling Refrigerating Machine

_{p,reg}provided by the source, as each cycle is driven by imperfect regeneration (Figure 3). Using refrigerator geometry, the volumes of compression and expansion spaces can be expressed according to the instantaneous positions of the pistons [29].

_{C}

_{0}is the swept compression volume; this is the displacer swept volume in the case of β-type Stirling machines.

_{E0}is the swept expansion volume. ${V}_{0l}$ is the overlapping volume in the case of a β-type Stirling machine and is due to the intrusion of the displacer piston into the working piston swept volume.

_{reg}, assumed to be constant on the whole length of the regenerator [28]. Therefore, in the case of the Stirling refrigerator, the regenerator efficiency is defined by:

#### 2.3. Application of Exergetic Method in the Study of β-Type Stirling Refrigerating Machine

#### 2.3.1. Exergetic Analysis Applied in the Study of the β-Type Stirling Refrigeration Machine

#### 2.3.2. Study of Heat Exchangers (Compression and Expansion Volume) and Calculation of Exergy Destroyed due to Temperature Differences

#### Cold-End Heat Exchanger Study

#### Hot-End Heat Exchanger Study

#### 2.4. Application of TDFF in the Study of the β-Type Stirling Refrigerating Machine

_{max}, V

_{max}, ${T}_{h}$, and ${T}_{l}$ as reference parameters. It is essential to consider the rotation speed as the main variable, because heat and mass transfer are dependent in a straightforward manner on speed and naturally must be expressed accordingly.

## 3. Results and Discussions

#### 3.1. Experimental Results

- ${\dot{Q}}_{l}$ is the refrigerating power of the cooling system, determined by the compensation method, (W); $\Delta {T}_{l}={T}_{l}-{T}_{wl}$, with ${T}_{l}$ representing the gas temperature measured inside the cold volume (K); and ${T}_{wl}$ is the wall temperature of the cold volume, measured with a thermocouple (K).

^{2}).

- ${c}_{w}$ is water-specific heat (J/kgK); and $\Delta {T}_{w}={T}_{w}^{e}-{T}_{w}^{i}$, with ${T}_{w}^{e}$ representing the output water temperature (K) and ${T}_{w}^{i}$ the input water temperature.

#### 3.2. Thermodynamic Analysis and Analytical Simulation Results

- (a)
- Cold-End Heat Exchanger

- (b)
- Hot-End Heat Exchanger

_{h}= −22.45 °C), after numerical simulation the 0-D model returns a value of 11.26 W (error of 44.47%), while in the FPDT model, the mechanical power required for operating the Stirling refrigerator is 19.68 W, with an error of 2.95%.

## 4. Conclusions

_{h}and T

_{wh}for the source and T

_{l}and T

_{wl}for the sink.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A | heat exchange surface, m^{2} |

c | specific heat, Jkg^{−1}K^{−1} |

C | stroke of the piston, m |

D | diameter of the piston, m |

Ex | Exergy, J |

${\dot{E}}_{x}$ | exergy flow rate, W |

h | convective heat transfer coefficient, Wm^{−2}K^{−1} |

I | current, A |

k | losses factor in regenerator, |

K | heat exchanger conductance, WK^{−1} |

m | mass, kg |

n | engine rotation speed, rot·s^{−1} |

p | pressure, Pa |

Q | heat, J |

$\dot{Q}$ | heat transfer rate, W |

R | gas constant, Jkg^{−1}K^{−1} |

S | entropy, JK^{−1} |

s | specific entropy, Jkg^{−1}K^{−1} |

T | temperature, K |

U | voltage, V |

V | volume, m^{3} |

W | work, J |

$\dot{W}$ | mechanical power, W |

Greek symbols | |

γ | adiabatic exponent, - |

φ | rotation angle, ° |

φ_{0} | phase lag angle, ° |

ε | volumetric compression ratio (V_{max}/V_{min}), - |

η | efficiency, - |

$\Pi $ | entropy increase, JK^{−1} |

$\dot{\Pi}$ | rate of entropy increase, WK^{−1} |

$\xi $ | dissipation coefficient, - |

Subscripts | |

C | compression |

ε | depending on ε |

ex | exergetic |

E | expansion |

d | displacer |

h | hot on working gas side |

l | low on working gas side |

m | dead |

max | maximum |

min | minimum |

p | piston |

rev | reversible |

reg | regenerator |

v | constant volume (specific heat) |

wl | wall on source side |

wh | wall on sink side |

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**Figure 7.**Exo-irreversible reversed Stirling cycle. (

**A**) Logp-LogV diagram in the range limit of ${p}_{\mathrm{max}}$, ${V}_{\mathrm{max}}$, ${T}_{l}$ and ${T}_{l}$; (

**B**) energy balance scheme.

n (rot/s) | T_{l}(K) | T_{h}(K) | Δ
$\mathit{T}$ (K) | ${\dot{\mathit{Q}}}_{\mathit{l}}$ (W) | ${\dot{\mathit{Q}}}_{\mathit{h}}$ (W) | ${\dot{\mathit{W}}}_{\mathbf{exp}}$ (W) | COP_{exp}(–) |
---|---|---|---|---|---|---|---|

2.85 | 249 | 348.69 | 99.69 | 12.35 | 32.57 | 20.22 | 0.61 |

3.04 | 249.5 | 343.99 | 94.49 | 13.40 | 33.56 | 20.16 | 0.66 |

3.31 | 250.3 | 338.03 | 87.73 | 14.70 | 34.54 | 19.84 | 0.74 |

3.47 | 250.4 | 332.72 | 82.32 | 16.50 | 35.53 | 19.03 | 0.87 |

3.60 | 249 | 330.34 | 81.34 | 17.94 | 37.51 | 19.57 | 0.92 |

3.86 | 250.7 | 329.10 | 78.40 | 19.20 | 39.48 | 20.28 | 0.95 |

${\mathit{A}}_{\mathit{h}}$ (m ^{2})
| ${\mathit{A}}_{\mathit{l}}$ (m ^{2})
| ${\mathit{V}}_{\mathbf{min}}\cdot {10}^{-4}$ (m ^{3})
| ${\mathit{V}}_{\mathbf{max}}\cdot {10}^{-4}$ (m ^{3})
| ${\mathit{D}}_{\mathit{p}}={\mathit{D}}_{\mathit{d}}$ (m) | ${\mathit{C}}_{\mathit{p}}={\mathit{C}}_{\mathit{d}}$ (m) | ${\mathit{\phi}}_{0}$ (°) |
---|---|---|---|---|---|---|

0.01885 | 0.03717 | 1.906 | 3.278 | 0.06 | 0.0484 | 110 |

p_{min} = 70,000 Pa | p_{max} = 211,600 Pa | ||||
---|---|---|---|---|---|

n (rot/s) | T_{l}(K) | T_{wl}(K) | T_{h}(K) | T_{wh}(K) | T_{0}(K) |

3.86 | 250.7 | 268.5 | 329.1 | 295 | 293 |

Experiment | 0-D Model | 0-D Error (%) | FPDT Model | FPDT Error (%) | |
---|---|---|---|---|---|

${\dot{Q}}_{l}(\mathrm{W})$ | 19.21 | 29.17 | 51.92 | 17.79 | 7.39 |

$\left|{\dot{Q}}_{h}\right|(\mathrm{W})$ | 39.47 | 40.22 | 1.90 | 37.48 | 5.04 |

${\dot{W}}_{l}(\mathrm{W})$ | 20.28 | 11.26 | 44.47 | 19.68 | 2.95 |

Experiment | 0-D Model | 0-D Error (%) | FPDT Model | FPDT Error (%) | |
---|---|---|---|---|---|

$\left|\dot{E}{x}_{{Q}_{l}}^{{T}_{l}}\right|(\mathrm{W})$ | 3.24 | 4.92 | 51.85 | 3 | 7.41 |

$\dot{E}{x}_{{Q}_{wl}}^{{T}_{wl}}(\mathrm{W})$ | 1.75 | 2.66 | 52 | 1.62 | 7.43 |

$\dot{E}{x}_{l}^{D}(\mathrm{W})$ | 1.49 | 2.26 | 51.67 | 1.38 | 7.38 |

${\eta}_{e{x}_{l}}(\%)$ | 54.01 | 54.06 | 0.09 | 54.08 | 0.13 |

${\zeta}_{l}(\%)$ | 45.98 | 45.93 | 0.10 | 46 | 0.15 |

Experiment | 0-D Model | 0-D Error (%) | FPDT Model | FPDT Error (%) | |
---|---|---|---|---|---|

$\left|\dot{E}{x}_{{Q}_{h}}^{{T}_{h}}\right|(\mathrm{W})$ | 4.33 | 4.41 | 1.85 | 4.11 | 5.08 |

$\dot{E}{x}_{{Q}_{wh}}^{{T}_{wh}}(\mathrm{W})$ | 0.27 | 0.27 | 0 | 0.25 | 7.40 |

$\dot{E}{x}_{h}^{D}(\mathrm{W})$ | 4.06 | 4.13 | 1.70 | 3.86 | 4.92 |

${\eta}_{e{x}_{h}}(\%)$ | 6.23 | 6.12 | 1.76 | 6.18 | 0.80 |

${\zeta}_{h}(\%)$ | 93.76 | 93.65 | 0.11 | 93.91 | 0.16 |

Experiment | 0-D Model | FPDT Model | |
---|---|---|---|

${\eta}_{EX}(\%)$ | 8.63 | 23.65 | 8.23 |

n = 3.86 (rot/min) | |||||
---|---|---|---|---|---|

$CO{P}_{\mathrm{exp}}$ | ${\dot{W}}_{\mathrm{exp}}$ | $CO{P}_{0-D}$ | ${\dot{W}}_{0-D}$ | $CO{P}_{FPDT}$ | ${\dot{W}}_{FPDT}$ |

(–) | (W) | (–) | (W) | (–) | (W) |

0.947 | 20.280 | 2.570 | 11.260 | 0.905 | 19.68 |

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**MDPI and ACS Style**

Dobre, C.; Grosu, L.; Dobrovicescu, A.; Chişiu, G.; Constantin, M.
Stirling Refrigerating Machine Modeling Using Schmidt and Finite Physical Dimensions Thermodynamic Models: A Comparison with Experiments. *Entropy* **2021**, *23*, 368.
https://doi.org/10.3390/e23030368

**AMA Style**

Dobre C, Grosu L, Dobrovicescu A, Chişiu G, Constantin M.
Stirling Refrigerating Machine Modeling Using Schmidt and Finite Physical Dimensions Thermodynamic Models: A Comparison with Experiments. *Entropy*. 2021; 23(3):368.
https://doi.org/10.3390/e23030368

**Chicago/Turabian Style**

Dobre, Cătălina, Lavinia Grosu, Alexandru Dobrovicescu, Georgiana Chişiu, and Mihaela Constantin.
2021. "Stirling Refrigerating Machine Modeling Using Schmidt and Finite Physical Dimensions Thermodynamic Models: A Comparison with Experiments" *Entropy* 23, no. 3: 368.
https://doi.org/10.3390/e23030368