# An Object-Oriented R744 Two-Phase Ejector Reduced-Order Model for Dynamic Simulations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The R744 Two-Phase Mathematical Approaches

#### 2.1. 0D Model Using the Bernoulli Equation

#### 2.2. 1D Homogeneous Equilibrium Model

- A negligible pressure drop in the gas cooler, evaporator and all connections;
- There is no heat loss to the environment from the system, except via heat rejection in the gas cooler;
- The liquid and vapour streams outflowing from the separator are saturated;
- The flows across expansion devices are isenthalpic;
- The compressor has a given isentropic efficiency;
- The evaporator has a given superheat degree, and the gas cooler has a given outlet temperature;
- The flow in the ejector is considered to be a 1D homogeneous equilibrium flow;
- The motive and suction streams enter the constant area mixing zone with the same pressure, and there is no mixing between them before the inlet of the constant area mixing;
- The expansion efficiencies of the motive and suction streams as well as the efficiency of the ejector diffuser are given constants.

#### 2.3. A Hybrid Reduced-Order Model

**U**utilising a linear combination of the snapshots. Moreover, the POD model requires an additional interpolation procedure to evaluate the ejector behaviour continuously for different operating conditions. The radial basis interpolation functions were applied for the investigated ROM model. In this study, the thin plate spline radial function with a smoothness factor was employed. The implementation of RBF into the POD model reduces the dimensionality of the ROM to the number of unknown parameters defined as the boundary conditions of the CO${}_{2}$ two-phase ejector, listed below:

- Motive nozzle pressure,
- Motive nozzle specific enthalpy,
- Suction nozzle pressure,
- Suction nozzle specific enthalpy,
- Outlet pressure.

## 3. Object-Oriented Modelling of the R744 Transcritical System

## 4. Comparison of the Investigated R744 Two-Phase Ejector Numerical Models

## 5. The System Energy Performance Comparison Using Different Two-Phase Ejector Models

## 6. Dynamic Simulations of the R744 Two-Phase Ejector Integrated With the Refrigeration System

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$eff$ | effective cross-section area | ||

a | flow rate ratio, m${}^{2}\xb7$kg${}^{-1}\xb7$s${}^{-1}$ | $ev$ | evaporator |

h | specific enthalpy, kJ· kg${}^{-1}$ | $exp$ | experimental data |

$fv$ | flash valve | ||

${k}_{v}$ | valve flow coefficient, m${}^{3}\xb7$ s${}^{-1}$ | $lr$ | liquid receiver |

$\dot{m}$ | mass flow rate, kg· s${}^{-1}$ | $max$ | maximal value |

p | pressure, Pa | $mix$ | mixing section |

s | specific entropy, kJ· kg${}^{-1}\xb7$ K${}^{-1}$ | $MN$ | motive nozzle |

T | temperature, K | $MNm$ | motive stream at the inlet of CAMS |

u | stream velocity, m·s${}^{-1}$ | $OUT$ | outlet |

$\mathit{U}$ | snapshot basis matrix, - | $rec$ | recovered ejector expansion work rate |

$\nu $ | specific volume, m${}^{3}\xb7$kg${}^{-1}$ | $SN$ | suction nozzle |

$\dot{W}$ | work rate, W | $SNm$ | suction stream at the inlet of CAMS |

x | vapour quality, − | v | valve |

Greek Symbols | Abbreviations | ||

$\chi $ | mass entrainment ratio, − | $CAMS$ | Constant Area Mixing Section |

$\delta $ | relative difference, % | $COP$ | Coefficient of Performance |

$\eta $ | ejector efficiency, % | $GWP$ | Global Warming Potential135 |

$\mathsf{\Pi}$ | pressure ratio, − | $HEM$ | Homogeneous Equilibrium Model |

$\rho $ | density, kg· m${}^{-3}$ | $HPV$ | High Pressure Valve |

$\tau $ | time, s | $HRM$ | Homogeneous Relaxation Model |

Subscript | $IHX$ | Internal Heat Exchanger | |

$amb$ | ambient | $MFR$ | Mass Flow Rate |

$bm$ | beginning of the mixing section | $ODP$ | Ozone Depletion Potentia |

$comp$ | compressor | $ROM$ | Reduced-Order Model |

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**Figure 2.**The operational envelope of the hybrid reduced-order model (ROM): (

**a**) the motive nozzle conditions; (

**b**) the suction nozzle conditions.

**Figure 4.**The computational procedure flowchart of the R744 transcritical system at a single time step.

**Figure 5.**Selected operating conditions: (

**a**) the motive nozzle conditions on the R744 pressure-specific diagram; (

**b**) the pressure ratio in terms of the suction nozzle pressure.

**Figure 6.**Relative difference between numerical result for each the R744 two-phase ejector mathematical model and measurements: (

**a**) the motive nozzle mass flow rate (MFR); (

**b**) the mass entrainment ratio together with the ejector efficiency given from the experimental data.

**Figure 7.**The energy performance of the R744 system based on different two-phase ejector models under the operating conditions given by Banasiak et al. [33]: (

**a**) coefficient of performance (COP); (

**b**) COP accuracy.

**Figure 8.**The relationship between the ambient conditions (

**left**) and the gas cooler outlet pressure/motive nozzle pressure (

**right**) during the summer campaign in different regions: (

**a**) Mediterranean zone; (

**b**) South America zone; (

**c**) South Asia zone.

**Figure 9.**The ejector efficiency at different pressure ratios during the summer campaign in the Mediterranean region.

**Figure 10.**The ejector efficiency at different pressure ratios during the summer campaign in the South America region.

**Figure 11.**The ejector efficiency at different pressure ratios during the summer campaign in the South Asia region.

**Figure 12.**The coefficient of performance for 24 h of the summer campaign in different hot climate zones.

**Table 1.**The main geometry parameters of the R744 two-phase ejector installed in the multi-ejector module [33].

Parameter Name | Unit | Dimension |
---|---|---|

Motive nozzle inlet diameter | mm | 3.80 |

Motive nozzle throat diameter | mm | 1.41 |

Motive nozzle outlet diameter | mm | 1.58 |

Motive nozzle converging angle | ${}^{\circ}$ | 30.00 |

Motive nozzle diverging angle | ${}^{\circ}$ | 2.00 |

Diffuser outlet diameter | mm | 8.40 |

Diffuser angle | ${}^{\circ}$ | 5.00 |

**Table 2.**The set of operating conditions adapted from [26].

ID | Motive Nozzle Inlet | Suction Nozzle Inlet | Outlet | ||
---|---|---|---|---|---|

p_{MN} | T_{MN} | p_{SN} | T_{SN} | p_{OUT} | |

bar | K | bar | K | bar | |

#1 | 94.46 | 308.43 | 27.21 | 275.75 | 32.85 |

#2 | 86.04 | 304.48 | 27.32 | 273.61 | 32.90 |

#3 | 91.91 | 304.13 | 31.41 | 278.43 | 38.24 |

#4 | 87.86 | 301.55 | 31.55 | 278.66 | 38.29 |

#5 | 80.62 | 299.40 | 31.58 | 278.49 | 38.48 |

#6 | 78.45 | 301.71 | 31.72 | 278.86 | 38.28 |

#7 | 76.56 | 301.49 | 27.33 | 274.01 | 32.87 |

#8 | 75.79 | 301.22 | 28.17 | 275.73 | 36.80 |

#9 | 66.51 | 295.56 | 28.21 | 275.36 | 34.85 |

#10 | 66.62 | 295.53 | 27.87 | 274.93 | 32.88 |

#11 | 61.79 | 293.42 | 29.93 | 276.73 | 33.87 |

#12 | 59.27 | 291.58 | 29.14 | 277.44 | 34.83 |

#13 | 58.41 | 283.15 | 27.82 | 277.71 | 34.83 |

#14 | 53.93 | 279.48 | 27.30 | 278.85 | 34.23 |

ID | Experimental Data | 0D Model | 1D Model | Hybrid ROM | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\dot{\mathit{m}}}_{\mathit{MN}$ | $\dot{\mathit{m}}}_{\mathit{SN}$ | $\mathit{\chi}$ | $\mathit{\delta}}_{{\dot{\mathit{m}}}_{\mathit{MN}}$ | $\mathit{\delta}}_{{\dot{\mathit{m}}}_{\mathit{SN}}$ | $\mathit{\delta}}_{\mathit{\chi}$ | $\mathit{\delta}}_{{\dot{\mathit{m}}}_{\mathit{MN}}$ | $\mathit{\delta}}_{{\dot{\mathit{m}}}_{\mathit{SN}}$ | $\mathit{\delta}}_{\mathit{\chi}$ | $\mathit{\delta}}_{{\dot{\mathit{m}}}_{\mathit{MN}}$ | $\mathit{\delta}}_{{\dot{\mathit{m}}}_{\mathit{SN}}$ | $\mathit{\delta}}_{\mathit{\chi}$ | |

kg·s$}^{-\mathbf{1}$ | kg·s$}^{-\mathbf{1}$ | - | % | % | % | % | % | % | % | % | % | |

#1 | 0.084 | 0.035 | 0.417 | 27.0 | 5.8 | $-16.7$ | 27.0 | 13.1 | $-11.0$ | 0.9 | 1.0 | 1.0 |

#2 | 0.079 | 0.032 | 0.409 | 20.3 | $-12.5$ | $-27.3$ | 20.3 | 0.7 | $-16.3$ | 0.8 | 0.5 | 0.7 |

#3 | 0.095 | 0.033 | 0.344 | 22.6 | $-15.3$ | $-30.9$ | 22.6 | $-0.7$ | $-19.0$ | 0.5 | 0.6 | 0.6 |

#4 | 0.097 | 0.032 | 0.326 | 18.1 | $-23.0$ | $-34.8$ | 18.1 | $-6.3$ | $-20.7$ | 0.4 | 0.2 | 0.3 |

#5 | 0.090 | 0.025 | 0.278 | 11.7 | $-29.0$ | $-36.4$ | 11.7 | $-0.6$ | $-11.0$ | 0.1 | 0.8 | 0.5 |

#6 | 0.073 | 0.026 | 0.349 | 9.9 | $-35.9$ | $-41.6$ | 9.9 | $-13.7$ | $-21.5$ | 0.5 | 0.1 | 0.3 |

#7 | 0.067 | 0.028 | 0.411 | 8.6 | $-33.1$ | $-38.4$ | 8.6 | $-13.5$ | $-20.4$ | 0.1 | 0.5 | 0.3 |

#8 | 0.067 | 0.011 | 0.166 | 6.6 | $-8.3$ | $-14.0$ | 6.6 | 44.2 | 35.3 | 1.0 | 0.9 | 1.0 |

#9 | 0.072 | 0.014 | 0.192 | $-11.9$ | $-37.1$ | $-28.6$ | $-11.9$ | 19.7 | 35.8 | 0.5 | 0.8 | 0.7 |

#10 | 0.072 | 0.022 | 0.304 | $-10.9$ | $-42.7$ | $-35.7$ | $-10.9$ | $-21.1$ | $-11.4$ | 0.6 | 0.5 | 0.6 |

#11 | 0.072 | 0.019 | 0.259 | $-26.8$ | $-40.0$ | $-18.0$ | $-26.8$ | $-32.4$ | $-7.6$ | 0.2 | 0.4 | 0.3 |

#12 | 0.076 | 0.009 | 0.116 | $-31.3$ | $-26.5$ | 7.1 | $-31.3$ | 2.6 | 49.4 | 0.8 | 0.1 | 0.5 |

#13 | 0.103 | 0.007 | 0.064 | $-8.9$ | $-11.1$ | $-2.5$ | $-8.9$ | 36.4 | 49.7 | 0.1 | 0.5 | 0.3 |

#14 | 0.100 | 0.003 | 0.031 | $-7.4$ | 36.5 | 47.4 | $-7.4$ | 59.4 | 72.1 | 0.5 | 0.1 | 0.3 |

**Table 4.**Operating conditions of the R744 vapour compression system adapted from [33].

Parameter | Test A | Test B |
---|---|---|

Evaporation temperature, ${}^{\circ}$C | $-5$ | 8 |

Evaporator outlet superheat, K | 10 | 10 |

Gas cooler outlet temperature, ${}^{\circ}$C | 25 | 30 |

Liquid separator pressure, bar | 34 | 35 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Haida, M.; Fingas, R.; Szwajnoch, W.; Smolka, J.; Palacz, M.; Bodys, J.; Nowak, A.J. An Object-Oriented R744 Two-Phase Ejector Reduced-Order Model for Dynamic Simulations. *Energies* **2019**, *12*, 1282.
https://doi.org/10.3390/en12071282

**AMA Style**

Haida M, Fingas R, Szwajnoch W, Smolka J, Palacz M, Bodys J, Nowak AJ. An Object-Oriented R744 Two-Phase Ejector Reduced-Order Model for Dynamic Simulations. *Energies*. 2019; 12(7):1282.
https://doi.org/10.3390/en12071282

**Chicago/Turabian Style**

Haida, Michal, Rafal Fingas, Wojciech Szwajnoch, Jacek Smolka, Michal Palacz, Jakub Bodys, and Andrzej J. Nowak. 2019. "An Object-Oriented R744 Two-Phase Ejector Reduced-Order Model for Dynamic Simulations" *Energies* 12, no. 7: 1282.
https://doi.org/10.3390/en12071282