# Isobaric Expansion Engines–Compressors: Thermodynamic Analysis of Multistage Vapor Driven Compressors

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## Abstract

**:**

## 1. Introduction

## 2. Process Schemes and Assumptions

- The processes occurring in the compressor and driver are assumed to be adiabatic and reversible.
- The driver only performs useful work on the compression process.
- The minimum volumes of the compression and driving cylinders are negligible, indicating no clearance.
- The temperature and pressure of the fluids in the driver and compressor are assumed to be uniform.
- Mechanical friction between moving and stationary parts in contact, such as the piston and cylinder or the piston rod and stuffing box, is considered to be negligible.
- The inertia of the pistons, piston rods, and fluids is assumed to be negligible.
- The cross-sectional area of the piston rods is much smaller compared to the area of the pistons.
- The compressed and driving fluids are treated as ideal gases with constant heat capacities.
- Intercooling is assumed to be perfect, meaning that the compressed gas is cooled to the temperature of the intake gas.
- The pressure ratio is the same in all stages, which is the condition for achieving minimum work for compression with perfect intercooling [32].

## 3. Efficiency of Driving Gas Utilization

## 4. Dimensions of the Compressors and Drivers

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

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

${c}_{p}$ | Heat capacity at constant pressure (J/kg K) |

${c}_{v}$ | Heat capacity at constant volume (J/kg K) |

$H$ | Specific enthalpy (J/kg) |

IE | Isobaric expansion |

$k$ | Constant, 1 or ${\gamma}_{c}$ |

$m$ | Mass of the driving fluid (kg) |

$n$ | Number of compression stages |

$P$ | Pressure (bar) |

$Q$ | Heat (J) |

$r$ | Pressure ratio |

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

$T$ | Temperature (K) |

$U$ | Specific internal energy (J/kg) |

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

$V$ | Volume (m^{3}) |

$W$ | Work (J) |

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

Greek letters | |

$\alpha $ | Relative vapor use efficiency |

$\beta $ | Dimensionless mass |

$\gamma $ | Heat capacity ratio, ${c}_{p}/{c}_{v}$ |

$\mu $ | Molar mass (kg/kmol) |

$\nu $ | Dimensionless volume |

$\tau $ | Dimensionless temperature |

$\omega $ | Dimensionless work |

Subscripts | |

0 | At the inlet of the driver |

c | Compressor |

d | Driver |

H | High |

1, 2, … i | Stage number |

L | Low |

n | Number of stages |

ss | Single-stage |

t | Total |

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**Figure 2.**Simplified scheme of a two-stage vapor-driven compressor; modifications that are needed for matching the arbitrary pressures in the compressors and drivers are not shown.

**Figure 3.**Efficiency of the driving gas use for one-, two-, and three-stage compression depending on the compressor pressure ratio at ${r}_{d}$ = 3, ${\gamma}_{c}$ = 1.4 (

**a**), and driver pressure ratio at ${r}_{c}$ = 10, ${\gamma}_{c}$ = 1.4 (

**b**).

**Figure 4.**Efficiency of the driving gas used for two-stage compression as a function of the compressor pressure ratio at ${r}_{d}$ = 3 and at different ratios of heat capacities of the compressed gas (

**a**) and driving gas (

**b**).

**Figure 5.**Relative amount of the consumed driving gas as a function of the pressure ratio in the compressors at ${r}_{d}$ = 3 (

**a**) and drivers at ${r}_{c}$ = 10 (

**b**) in two- and three-stage compressions; ${\gamma}_{c}$ = ${\gamma}_{d}$ = 1.4; solid lines—without intercooling, dashed lines—with intercooling.

**Figure 6.**Relative work in two-, three-, and infinite-stage compressions as a function of the pressure ratio in the compressor.

**Figure 7.**Total volume of the compressors (

**a**) and drivers (

**b**) for $n$-stage compression with and without intercooling relative to their values for single-stage compressions; solid lines—without intercooling, dashed lines—with intercooling; ${\gamma}_{c}$ = 1.4.

**Figure 8.**Ratio of the total volumes of the two-stage and single-stage units (drivers and compressors) depending on the compressor pressure ratio at ${r}_{d}$ = 3, ${\gamma}_{c}$ = 1.4 (

**a**), and driver pressure ratio at ${r}_{c}$ = 10, ${\gamma}_{c}$ = 1.4 (

**b**).

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**MDPI and ACS Style**

Kronberg, A.; Glushenkov, M.; Roosjen, S.; Kersten, S.
Isobaric Expansion Engines–Compressors: Thermodynamic Analysis of Multistage Vapor Driven Compressors. *Energies* **2023**, *16*, 6791.
https://doi.org/10.3390/en16196791

**AMA Style**

Kronberg A, Glushenkov M, Roosjen S, Kersten S.
Isobaric Expansion Engines–Compressors: Thermodynamic Analysis of Multistage Vapor Driven Compressors. *Energies*. 2023; 16(19):6791.
https://doi.org/10.3390/en16196791

**Chicago/Turabian Style**

Kronberg, Alexander, Maxim Glushenkov, Sander Roosjen, and Sascha Kersten.
2023. "Isobaric Expansion Engines–Compressors: Thermodynamic Analysis of Multistage Vapor Driven Compressors" *Energies* 16, no. 19: 6791.
https://doi.org/10.3390/en16196791