# Isobaric Expansion Engine Compressors: Thermodynamic Analysis of the Simplest Direct Vapor-Driven Compressors

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## Abstract

**:**

## 1. Introduction

## 2. Basic Schemes of Vapor-Driven Compressors

## 3. Relations between the Pressures and Piston Areas

## 4. Efficiency of Driving Vapor Use

- The process in the compressor is either adiabatic or isothermal.
- The process in the driver is adiabatic.
- The driver only performs useful work on the compression.
- The minimum volumes of the compression and driving cylinder are zero (no dead volume).
- The temperature and pressure of the fluids in the driver and compressor are uniform.
- Mechanical friction between moving and stationary parts in contact (such as piston and cylinder, piston rod and stuffing box) is negligible.
- The inertia of the pistons, piston rods and the fluids is negligible; this is justified for IE engines operating at low frequencies.
- The cross-sectional area of the piston rods is much smaller than the area of the pistons.
- The heat capacities of ideal gases are constant.

#### 4.1. Ideal Gases

#### 4.2. Real Gases

## 5. IE Engine Compressor Efficiency

_{heater}, do not depend on the compression process, whereas the regenerated heat is determined by the temperature of working fluid (driving vapor) discharged from the driver to the recuperator, which depends on its initial value, i.e., on the temperature of the driving vapor at the end of the compression stroke.

## 6. Validation of the Engine Compressor Concept

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$A$ | Cross-sectional area (m^{2}) | Greek Letters | |

${c}_{p}$ | Heat capacity at constant pressure (J/kg K) | $\alpha $ | Relative vapor use efficiency |

${c}_{v}$ | Heat capacity at constant volume (J/kg K) | $\beta $ | Isobaric expansion coefficient |

$h$ | Specific enthalpy (J/kg) | $\gamma $ | Heat capacity ratio |

IE | Isobaric expansion | $\eta $ | Thermal efficiency |

$k$ | Fraction of the feed pump work | $\mu $ | Molar mass (kg/kmol) |

$m$ | Mass of the driving vapor (kg) | $\tau $ | Dimensionless temperature of the driving vapor at the end of the compression stroke |

$\dot{m}$ | Mass rate of flow (kg/s) | ||

$P$ | Pressure (bar) | Subscripts | |

$Q$ | Heat (J) | a | Ambient |

$r$ | Pressure ratio | c | Compressor |

$R$ | Specific gas constant (the molar gas constant divided by the molar mass) (J/K kg) | d | Driver |

$t$ | Time | e | End of the compression stroke |

$T$ | Temperature (°C, K) | fp | Feed pump |

$u$ | Specific internal energy (J/kg) | H | High |

$v$ | Specific volume (m^{3}/kg) | L | Low |

$V$ | Volume (m^{3}) | p | Ideal pump |

$W$ | Work (J) | r | Receiver |

$w$ | Specific work (J/kg) | R | Recuperator |

$z$ | Work ratio |

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**Figure 4.**Dimensionless temperature of the driving gas at the end of the compression stroke (

**a**) and relative efficiency (

**b**) for different heat capacity ratios as a function of the relative work; $r$ = 3.

**Figure 5.**Dimensionless temperature of the driving gas at the end of the compression stroke (

**a**) and relative efficiency (

**b**) for different heat capacity ratios as a function of the driver pressure ratio; $z$ = 0.5.

**Figure 6.**Dimensionless temperature of the driving gas at the end of the compression stroke (

**a**) and efficiency of the driving gas use (

**b**) as a function of the compressor pressure ratio and different $\gamma ;$ ${\gamma}_{c}$ = 1.4, $r$ = 3; solid lines—adiabatic compression, dashed lines—isothermal compression.

**Figure 7.**Dimensionless temperature of the driving gas at the end of the compression stroke (

**a**) and efficiency of the driving gas use (

**b**) as a function of the driver pressure ratio and different $\gamma ;$ ${\gamma}_{c}$ = 1.4, ${r}_{c}$ = 5; solid lines—adiabatic compression, dashed lines—isothermal compression.

**Figure 8.**Temperature at the end of the compression stroke (a) and relative efficiency (b) for R134a and ammonia as a function of the relative work.

**Figure 9.**Dimensionless temperatures of the driving R134a (

**left**) and ammonia (

**right**) at the end of compression stroke as a function of the compressor pressure ratio; solid lines—adiabatic compression, dashed lines—isothermal compression.

**Figure 10.**Efficiency of the driving R134a (

**left**) and ammonia (

**right**) use as a function of the compressor pressure ratio; solid lines—adiabatic compression, dashed lines—isothermal compression.

Fluid | ${\mathit{P}}_{\mathit{H}}\left(\mathbf{b}\mathbf{a}\mathbf{r}\right)$ | ${\mathit{P}}_{\mathit{L}}$$\left(\mathbf{b}\mathbf{a}\mathbf{r}\right)$ | ${\mathit{T}}_{\mathit{H}}\left(\xb0\mathbf{C}\right)$ | $\mathit{r}$ |
---|---|---|---|---|

R134a | 20 and 30 | 7.7 | 90 | 2.6 and 3.9 |

Ammonia | 20 and 30 | 10.0 | 70 | 2 and 3 |

${\mathit{P}}_{\mathit{H}}\left(\mathbf{b}\mathbf{a}\mathbf{r}\right)$ | R134a | Ammonia | ||||
---|---|---|---|---|---|---|

${\mathit{T}}_{\mathit{H}}\left(\xb0\mathbf{C}\right)$ | $\mathit{r}$ | $\mathit{\gamma}$ | ${\mathit{T}}_{\mathit{H}}\left(\xb0\mathbf{C}\right)$ | $\mathit{r}$ | $\mathit{\gamma}$ | |

20 | 90 | 2.6 | 1.30 | 70 | 2 | 1.49 |

30 | 90 | 3.9 | 1.91 | 70 | 3 | 1.69 |

Fluid | ${\mathit{P}}_{\mathit{c}\mathit{L}}\left(\mathbf{b}\mathbf{a}\mathbf{r}\right)$ | ${\mathit{P}}_{\mathit{c}\mathit{H}}\left(\mathbf{b}\mathbf{a}\mathbf{r}\right)$ | ${\mathit{T}}_{\mathit{c}\mathit{L}}\left(\xb0\mathbf{C}\right)$ | ${\mathit{r}}_{\mathit{c}}$ |
---|---|---|---|---|

R134a | 1 | 1–10 | 20 | 1–10 |

Ammonia | 1 | 1–10 | 0 | 1–10 |

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**MDPI and ACS Style**

Kronberg, A.; Glushenkov, M.; Roosjen, S.; Kersten, S.
Isobaric Expansion Engine Compressors: Thermodynamic Analysis of the Simplest Direct Vapor-Driven Compressors. *Energies* **2022**, *15*, 5028.
https://doi.org/10.3390/en15145028

**AMA Style**

Kronberg A, Glushenkov M, Roosjen S, Kersten S.
Isobaric Expansion Engine Compressors: Thermodynamic Analysis of the Simplest Direct Vapor-Driven Compressors. *Energies*. 2022; 15(14):5028.
https://doi.org/10.3390/en15145028

**Chicago/Turabian Style**

Kronberg, Alexander, Maxim Glushenkov, Sander Roosjen, and Sascha Kersten.
2022. "Isobaric Expansion Engine Compressors: Thermodynamic Analysis of the Simplest Direct Vapor-Driven Compressors" *Energies* 15, no. 14: 5028.
https://doi.org/10.3390/en15145028