# Exergy Flows inside a One Phase Ejector for Refrigeration Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Transiting Exergy in a Process with Pressure, Temperature and Velocity Variations

_{0}, Equation (1a), by its maximum value for sub-environmental conditions, Equation (1b) and by the value T

_{0}if the environmental temperature has an intermediate value, Equation (1c). Figure 1 visualises the fact that the introduction of the transiting exergy results in a shift of the reference state for exergy calculations. As a result the exergy consumed and produced in a system or its parts are represented by smaller band widths.

_{P})

_{Tin}in Equation (2) is the decrease of the specific mechanical exergy due to an isothermal pressure drop at constant temperature Tin. The term (Δe

_{T})

_{Pout}in Equation (3) is the increase of the specific thermal exergy due to an isobaric temperature drop under sub-environmental conditions at constant pressure Pout. The fluid flow rate is ($\dot{\mathrm{m}}$). The specific exergy losses (d) are also presented on the diagram in Figure 2. As illustrated in Equations (4) and (5) the values (∇E) and (ΔE) allow the computation of two important thermodynamic measures, namely exergy efficiency (ƞ

_{ex, TR}) and exergy losses (D) where (d) represents the specific exergy losses:

_{P})

_{Tin}, the segment 2′-2 is (Δe

_{T})

_{Pout}. Given that the throttling takes place under sub-environmental conditions, the lowest exergy content of the fluid is reached at point 2′ which corresponds to the lowest pressure Pout and the highest temperature Tin. This exergy value is the specific transiting exergy of the stream. If the kinetic energy of the stream cannot be neglected, the lowest velocity value should be added to the definition of transiting exergy as illustrated in Figure 1. The expressions (2) and (3) for consumed and produced exergies will be changed accordingly; the subject is important for an ejector analysis and will be discussed in the next section.

## 3. The Exergy Consumption and Production in Different Parts of a One-Phase Ejector

_{4}= 145 °C and P

_{4}= 1000 kPa, the flow rate is ${\dot{\mathrm{m}}}_{\mathrm{p}}$ = 0.19838 kg/s; the temperature and the pressure of the secondary fluid are T

_{6}= −5 °C and P

_{6}= 22.28 kPa, the flow rate is ${\dot{\mathrm{m}}}_{\mathrm{s}}$ = 0.04959 kg/s. According to information compiled by Liu and Groll [15], it has been assumed that the polytropic efficiencies are: 0.95 for the primary nozzle, 0.85 for the suction chamber and 0.78 for the diffuser.

_{3}CCl

_{2}F) was chosen. Previous studies [17,18] have found that it is suitable for ejector applications. Furthermore, R141b has a positive-slope saturated-vapor line in the temperature-entropy diagram and therefore does not require superheating. Its critical temperature and pressure are respectively 204.2 °C and 4205 kPa. Its normal boiling point is 32 °C. Its ODP (Ozon depletion potential) value is 0.1 proving that it has a negligible impact on the environment, while its global warming potential (GWP) value is 725 which is quite acceptable for the application under consideration. The profiles of pressure, temperature and velocity along the ejector are illustrated in Figure 4.

_{0}. The exergy production includes: (1) the increase of thermal exergy due to the temperature drop from T

_{0}to T

_{7p}at the sub-environmental conditions and calculated at constant outlet pressure P

_{7p}, and (2) the increase of kinetic energy due to the velocity rise from inlet to outlet.

_{6}. (ΔE) is the sum of: (1) the increase of thermal exergy due to the temperature drop in the sub-environmental region and calculated at constant outlet pressure P

_{7s}, and (2) the increase of kinetic energy due to the velocity rise from inlet to outlet.

_{7p}, and (2) the decrease of kinetic energy due to the velocity reduction. For the secondary stream it is the decrease of thermal exergy due to a temperature rise in the sub-environmental region and calculated at constant inlet pressure P

_{7s}. The produced exergy is linked to the primary and secondary streams together. For the primary it is the increase of thermal exergy due to a temperature rise in the sup-environmental region and calculated at constant outlet pressure P

_{7p}. For the secondary it is the sum of: (1) the increase of thermal exergy due to the temperature rise in the sub-environmental region and calculated at constant outlet pressure P

_{7}s, and (2) the increase of kinetic energy due to the velocity rise.

_{d}. (ΔE) is the increase of kinetic energy due to the velocity rise from the inlet to the outlet.

_{0}and calculated at constant inlet pressure P

_{6}. The produced exergy (ΔE) is linked to the secondary stream only. The exergy production is the increase of thermo-mechanical exergy due to the pressure rise from the inlet to the outlet and to the temperature rise from T

_{0}to T

_{1}.

## 4. Analysis of Numerical Results

_{ex, GR}) and exergy efficiencies taking into account the transition streams (ƞ

_{ex, TR}) are calculated according to Equations (1), (4), (5) and are presented in the table as well.

_{ex, TR}), may be used to complement this analysis.

_{m-d}is larger than D

_{7p-m+7s-m}, however ƞ

_{m-d}is higher than ƞ

_{7p-m+7s-m}. It means that the transformation of thermal and kinetic exergies into thermal and mechanical exergies, taking place within the zone of shock, is thermodynamically more efficient than the transformation of thermal and kinetic exergies into kinetic exergy of the secondary stream, taking place within the zone of mixing. Thus, particular attention should be paid to the improvement of the mixing process. The analysis of expression (12) for (∇E) reveals that the most important factor influencing the irreversible losses in the zone of mixing is the decrease of thermal exergy due to the temperature rise in the sub-environmental region (from 257 K to 289 K for the secondary stream and from 284 K to 289 K for the primary stream). As a result the cold created in the evaporator of a refrigeration system, in the diverging part of the nozzle (thr-7p) and in the suction section of the entrained stream (6-7s) is completely destroyed in the zone of mixing. To our knowledge this type of result has never been published in the scientific literature. The engineering proposals regarding the reduction of (∇E) in this zone and the consequent reduction of exergy losses will be discussed in future publications.

_{ex, GR}) with the efficiency (ƞ

_{ex, TR}) taking into account the transiting exergy flow. First of all let’s notice that the value of (ƞ

_{ex, GR}) for the suction section of the entrained stream (6–7s) is higher than 100%, and thus does not have any physical meaning. The explanation of this unacceptable result is the following; because of the vacuum conditions the exergies E

_{7s}and E

_{6}are negative, where E

_{7s}is lower than E

_{6}by the value of exergy losses. It means that absolute value of E

_{7s}is higher than E

_{6}. As a result the ratio E

_{7s}/E

_{6}is higher than 1. Contrary to (ƞ

_{ex, GR}), the value of (ƞ

_{ex, TR}) is lower than 100%, and it reflects in this way the exergy losses occurring within the suction section of the entrained stream.

_{ex, GR}) the section (m-d) is less efficient and according to (ƞ

_{ex, TR}) it is more efficient. The main reason of this contradiction is the fact that exergy flows at the entries and exists of each zone present the summation of positive terms (thermal and kinetic exergies) and negative ones (mechanical exergies under vacuum conditions). As a result their ratio (ƞ

_{ex, GR}) does reflect the mutual transformation of one form of exergy to another. On the contrary, given that the transiting part of mechanical exergy is negative, the application of (ƞ

_{ex, TR}) reflects this mutual transformation. Thus by transforming the kinetic exergy to mechanical one, the zone of the shock is more efficient than the mixing zone. The third observation concerns the constant area section (d-8). According to (ƞ

_{ex, TR}), it is the less efficient part of the ejector. According to (ƞ

_{ex, GR}), it is one of the most efficient. The main reason of this discrepancy is the presence of an important transiting exergy flow; this fact is ignored when the Grassman exergy efficiency is applied.

_{ex, TR}) of the overall ejector, calculated according to Equations (20), (21) and (4) is low and equals 18.3%, contrary to the “optimistic” value 37.5% of (ƞ

_{ex, GR}). The main reason of this difference is the presence of an important transiting exergy flow of 3.2 kW.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

Symbols | Greek Symbols | ||

A | Area, cm^{2} | η | Efficiency, % |

Co | Condenser | ∇ | Thermodynamic metric: consumption |

D | Destroyed exergy, kW | Δ | Thermodynamic metric: production |

d | Specific exergy losses, kJ/kg | Subscripts | |

E | Exergy, kW | 0 | Dead state |

e | Specific exergy, kJ/kg | 4, thr, | States in one phase ejector |

Ev | Evaporator | d | Downstream |

Ge | Generator | in | Inlet |

h | Specific enthalpy, kJ/kg | M, m | Mixing |

L | Ejector part‘s length, cm | max | Maximal |

M | Mach number | min | Minimal |

$\dot{\mathrm{m}}$ | Mass flowrate, kg/s | out | Outlet |

P | Pressure, kPa | p | Primary |

T | Temperature, °C, K | s | Secondary |

V | Velocity, m/s | thr | Throat |

X | Ejector part’s length, cm | tr | Transiting |

## References

- Chen, X.; Omer, S.; Worall, M.; Riffat, S. Recent developments in ejector refrigeration technologies. Renew. Sustain. Energy Rev.
**2013**, 19, 629–651. [Google Scholar] [CrossRef] - Arbel, A.; Shklyar, A.; Hershgal, D.; Barak, M.; Sokolov, M. Ejector irreversibility characteristics. J. Fluid Eng.
**2003**, 125, 121–129. [Google Scholar] [CrossRef] - Al-Najem, N.M.; Darwish, M.A.; Youssef, F.A. Thermovapor compression desalters: Energy and availability—Analysis of single- and multi-effect systems. Desalination
**1997**, 110, 223–238. [Google Scholar] [CrossRef] - McGovern, R.K.; Narayan, G.P.; Lienhard, V.J.H. Analysis of reversible ejectors and definition of an ejector efficiency. Int. J. Therm. Sci.
**2012**, 54, 153–166. [Google Scholar] [CrossRef] - Brodyansky, V.M.; Sorin, M.; LeGoff, P. The Efficiency of Industrial Processes: Exergy Analysis and Optimization; Elsevier Science B. V: Amsterdam, The Netherlands, 1994. [Google Scholar]
- Kotas, T.J. The Exergy Method of Thermal Plant Analysis, 2th ed.; Krieger Publishing: Malabar, FL, USA, 1995. [Google Scholar]
- Marmolejo-Correa, D.; Gundersen, T. A comparison of exergy efficiency definitions with focus on low temperature processes. Energy
**2012**, 44, 477–489. [Google Scholar] [CrossRef] - Zanchini, E. A more general exergy function and its application to the definition of exergy efficiency. Energy
**2015**, 87, 352–360. [Google Scholar] [CrossRef] - Lazzaretto, A.; Tsatsaronis, G. SPECO: A systematic and general methodology for calculating efficiencies and costs in thermal systems. Energy
**2006**, 31, 1257–1289. [Google Scholar] [CrossRef] - Lior, N.; Zhang, N. Energy, exergy, and Second Law performance criteria. Energy
**2007**, 32, 281–296. [Google Scholar] [CrossRef] - Keenan, H.; Neumann, E.P.; Lustwerk, F. An investigation of ejector design by analysis and experiment. J. Appl. Mech. Trans. ASME
**1950**, 72, 299–309. [Google Scholar] - Sun, D.W.; Eames, I.W. Performance characteristics of HCFC-123 ejector refrigeration cycles. Int. J. Energy Res.
**1996**, 20, 871–885. [Google Scholar] [CrossRef] - Sun, D.W. Experimental investigation of the performance characteristics of a steam jet refrigeration system. Energy Sources
**1997**, 19, 349–367. [Google Scholar] [CrossRef] - Khennich, M.; Galanis, N.; Sorin, M. Effects of design conditions and irreversibilities on the dimensions of ejectors in refrigeration systems. Appl. Energ.
**2016**. submitted. [Google Scholar] - Liu, F.; Groll, E.A. Study of ejector efficiencies in refrigeration cycles. Appl. Therm. Eng.
**2013**, 52, 360–370. [Google Scholar] [CrossRef] - Khennich, M.; Sorin, M.; Galanis, N. Effects of condenser pressure on the size and operation conditions of ejector refrigeration systems. In Proceedings of the International Conference on Innovative Technologies (INTECH), Dubrovnik, Croatia, 9–11 September 2015.
- Huang, B.J.; Chang, J.M. Empirical correlation for ejector design. Int. J. Refrig.
**1999**, 22, 379–388. [Google Scholar] [CrossRef] - Dorantes, R.; Lallemand, A. Prediction of performance of a jet cooling system operating with pure refrigerants or non-azeotropic mixtures. Int. J. Refrig.
**1995**, 18, 21–30. [Google Scholar]

Fluid | Primary Fluid | Secondary Fluid | Mixed Fluid | ||||||
---|---|---|---|---|---|---|---|---|---|

States | 4 | thr | 7p | 6 | 7s | m = u | d | 8 | 1 |

T (°C) | 145.0 | 125.7 | 11.2 | −5.0 | −15.8 | 16.2 | 102.9 | 102.6 | 106.0 |

T (K) | 418.2 | 398.9 | 284.4 | 268.2 | 257.4 | 289.4 | 376.0 | 375.7 | 379.2 |

P (kPa) | 1000.0 | 603.84 | 13.06 | 22.28 | 13.06 | 13.06 | 86.57 | 83.21 | 90.84 |

V (m/s) | 0.0 | 156.2 | 440.2 | 0.0 | 129.2 | 378.0 | 73.3 | 76.2 | 0.0 |

M (–) | 0.0 | 0.972 | 2.956 | 0.0 | 0.910 | 2.517 | 0.435 | 0.453 | 0.0 |

ṁ (kg/s) | 0.19838 | 0.19838 | 0.19838 | 0.04959 | 0.04959 | 0.24797 | 0.24797 | 0.24797 | 0.24797 |

Section | Exergy Consumed (∇E, kW) | Exergy Produced (ΔE, kW) | Exergy Losses (D, kW) | Transiting Exergy (E^{tr}, kW) | Exergy Efficiency (ƞ_{ex, TR}) | Exergy Efficiency (ƞ_{ex, GR}) |
---|---|---|---|---|---|---|

4-thr | 2.515 | 2.421 | 0.094 | 11.906 | 96.3% | 99.3% |

thr-7p | 17.562 | 16.805 | 0.757 | −3.236 | 95.7% | 94.7% |

6-7s | 0.539 | 0.457 | 0.082 | −1.388 | 84.8% | 109.7% |

7p-m + 7s-m | 5.121 | 3.130 | 1.991 | 7.517 | 61.1% | 84.2% |

m-d | 17.051 | 11.764 | 5.287 | −6.404 | 69.0% | 50.3% |

d-8 | 0.211 | 0.055 | 0.156 | 5.149 | 26.1% | 97.1% |

8-1 | 0.721 | 0.609 | 0.112 | 4.483 | 84.5% | 97.9% |

total ejector | 10.374 | 1.894 | 8.480 | 3.199 | 18.3% | 37.5% |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Khennich, M.; Sorin, M.; Galanis, N. Exergy Flows inside a One Phase Ejector for Refrigeration Systems. *Energies* **2016**, *9*, 212.
https://doi.org/10.3390/en9030212

**AMA Style**

Khennich M, Sorin M, Galanis N. Exergy Flows inside a One Phase Ejector for Refrigeration Systems. *Energies*. 2016; 9(3):212.
https://doi.org/10.3390/en9030212

**Chicago/Turabian Style**

Khennich, Mohammed, Mikhail Sorin, and Nicolas Galanis. 2016. "Exergy Flows inside a One Phase Ejector for Refrigeration Systems" *Energies* 9, no. 3: 212.
https://doi.org/10.3390/en9030212