modelling of a transcritical co ejector with variable geometry

Modelling of a transcritical CO2 ejector

with variable geometry

INTEGRATED MASTER OF MECHANICAL ENGINEERING

FACULTY OF ENGINEERING OF THE UNIVERSITY OF PORTO

Tomás Pinto de Freitas Teixeira da Rocha

Supervisor:

Professor Szabolcs Varga

June 2021

MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

i INTEGRATED MASTER OF MECHANICAL ENGINEERING

Abstract

Currently, energy demand is met using mostly non-renewable energy sources, which

poses serious environmental challenges. The global effort towards sustainable development

creates the need for rational use of our planet’s resources and for energy efficient processes.

Buildings account for 40% of total energy consumption in the European Union. In this

context, recent developments in efficient heat pumps have drawn great attention to this

technology for indoor space heating and cooling applications. In particular, CO2 heat pumps

are subject to extensive research because carbon dioxide is an environmentally friendly

working fluid. Its low critical temperature limits its use in subcritical cycles. However,

transcritical cycles typically have low performance due to high throttling losses. These losses

may be reduced by replacing the expansion device with an ejector, which partially recovers

the expansion work of the supercritical gas. Fixed-geometry ejectors operate with maximum

efficiency in a narrow range of operating (design) conditions and their performance is

compromised in off-design conditions. Variable geometry ejectors (VGE) may maintain

good performance under variable operating conditions. The area ratio and the nozzle exit

position are the geometric parameters with the greatest impact on its performance.

The main objective of this work is to develop a suitable mathematical model to

evaluate the performance of a transcritical CO2 VGE under variable operating conditions and

assess its benefits when compared to a fixed-geometry ejector. The developed model is based

on the CFD method using FLUENT® commercial software. It is assumed that the VGE

operates under steady-state conditions. The flow is compressible and axisymmetric. The

RNG 𝑘 − 휀 model is used to tackle turbulence. A baseline fixed-geometry ejector is designed

with an existing design tool. In this work, variable geometry is achieved by changing the area

ratio of the ejector by implementing a spindle in the primary nozzle. The VGE is first

simulated for different compression ratios using an ideal-gas approach. A real-gas approach

using the Homogeneous Equilibrium Model is also implemented to simulate the primary

nozzle of the VGE. The entrainment ratio is used as the principal performance indicator.

The results clearly show that significant improvements of ejector performance are

obtained by adjusting the area ratio, indicating that a transcritical VGE may significantly

outperform the equivalent fixed-geometry device under variable operating conditions. For

compression ratios of 1.1 and 1.2, 202% and 106% improvements in the entrainment ratio

are obtained for optimum spindle position, respectively. The ideal-gas model is a simple

approach to modelling CO2 properties, and allows for a faster numerical convergence.

However, it poorly predicts the density of the supercritical gas, resulting in a poor description

of the transcritical expansion in the primary nozzle. The HEM contemplates evaporation and

condensation effects, allowing for a more realistic prediction of the transcritical expansion

process in the primary nozzle. Simulation results show that the primary mass flow rate may

be effectively controlled using an adjustable spindle within the primary nozzle. However, the

HEM is highly sensitive to mesh design and obtaining convergence is very challenging when

compared to the ideal-gas approach. This may be explained by the dependence of fluid

properties on both pressure and enthalpy; updating the value of either pressure or enthalpy in

each cell requires that fluid properties be recalculated. Moreover, density varies significantly

in the transcritical expansion process that occurs in the primary nozzle. This property appears

in all governing equations, which may also difficult convergence.


ii INTEGRATED MASTER OF MECHANICAL ENGINEERING


iii INTEGRATED MASTER OF MECHANICAL ENGINEERING

Resumo

Atualmente, a procura energética mundial é satisfeita recorrendo, sobretudo, a fontes

não renováveis. A meta de desenvolvimento sustentável obriga ao uso racional dos recursos

do nosso planeta e à implementação de processos eficientes a nível energético. Os edifícios

representam cerca de 40% do consumo total de energia na União Europeia. Neste contexto,

desenvolvimentos recentes em bombas de calor tornam esta tecnologia cada vez mais atrativa

para aquecimento e arrefecimento de espaços interiores. Em particular, bombas de calor de

CO2 são muito atrativas pelo baixo impacto ambiental deste fluido. A sua utilização em ciclos

subcríticos é limitada pela baixa temperatura de ponto crítico. Contudo, os ciclos transcríticos

de CO2 apresentam, tipicamente, baixa performance devido às perdas no processo de

expansão. Estas podem ser minimizadas substituindo a convencional válvula de expansão

por um ejetor, permitindo recuperar parcialmente o trabalho de expansão do fluido

supercrítico. Os ejetores de geometria fixa apresentam boa performance apenas numa

reduzida gama de condições operativas (condições ótimas de funcionamento). Por outro lado,

os ejetores de geometria variável (EGV) são adequados para condições de funcionamento

variáveis. A razão de áreas e a posição do bocal primário são os parâmetros geométricos com

mais impacto na sua performance.

O principal objetivo deste trabalho é o desenvolvimento de um modelo matemático

para a análise da performance de um EGV transcrítico de CO2 em condições variáveis de

operação, e a sua comparação com um ejetor de geometria fixa. O modelo desenvolvido

recorre a técnicas de CFD e utiliza o software comercial FLUENT®. O ejetor opera em

regime estacionário e o escoamento é compressível e axissimétrico. O modelo de turbulência

utilizado é o RNG 𝑘 − 휀. A geometria de referência é estimada com base numa ferramenta

de cálculo pré-existente. Neste trabalho, a geometria variável é obtida alterando a razão de

áreas do ejetor através da implementação de uma agulha na secção convergente do bocal

primário. Numa primeira fase, o fluido de trabalho é modelado como gás ideal e o EGV é

simulado para diferentes taxas de compressão. Posteriormente, é simulado o bocal primário

do ejetor, modelando o CO2 como gás real de acordo com o Homogeneous Equilibrium

Model (HEM). A taxa de arrastamento é utilizada como principal indicador de performance.

Os resultados numéricos indicam, de forma clara, que o ajuste da razão de áreas pode

melhorar significativamente a performance de um ejetor transcrítico de CO2, indicando que

um EGV poderá operar com maior eficiência do que o equivalente ejetor de geometria fixa

quando sujeito a condições variáveis de funcionamento. Para taxas de compressão de 1.1 e

1.2, observam-se aumentos de, respetivamente, 202% e 106% na taxa de arrastamento

quando a agulha se encontra na sua posição ótima. O modelo de gás ideal modela, de forma

simples, as propriedades do CO2 e permite uma rápida convergência numérica. Contudo, é

pouco preciso na previsão da massa volúmica do fluido supercrítico, descrevendo com pouco

rigor a expansão transcrítica no bocal primário. Por outro lado, o HEM contempla a

ocorrência de condensação, permitindo uma descrição mais realista do processo de expansão.

Os resultados numéricos indicam que é possível controlar, de forma eficaz, o caudal primário

de um EGV transcrítico de CO2 recorrendo a uma agulha ajustável no bocal primário. No

entanto, o HEM é muito sensível à geometria da malha, sendo mais difícil obter uma boa

convergência numérica do que na abordagem de gás ideal. Por um lado, as propriedades do

fluido dependem da pressão e da entalpia, simultaneamente; a atualização de cada uma destas

variáveis requer que as propriedades do fluido sejam recalculadas. Por outro lado, a massa

volúmica varia significativamente durante a expansão no bocal primário. A massa volúmica

intervém em todas as equações governativas, o que poderá também dificultar a convergência.


iv INTEGRATED MASTER OF MECHANICAL ENGINEERING


v INTEGRATED MASTER OF MECHANICAL ENGINEERING

Acknowledgements

I would like to thank my family for the love and support they have always given me.

Their constant encouragement has given me the confidence to face any obstacles in my path,

and I certainly owe my success to them. Thank you to my father for never leaving my side.

I also wish to express my gratitude towards my supervisor, professor Szabolcs Varga,

for his unfailing support in every stage of this work. Working under his supervision gave me

a great sense of security for knowing I could rely on his help when facing major difficulties,

whilst allowing me to be the author of my own work. His knowledge and experience in this

area played an important role and have contributed to the quality of this work.

Finally, I am very grateful to all other members of CIENER – INEGI who helped in

numerous moments of my work. I could count on professor João Soares’, Behzad’s, and

Karla’s willingness to help in many situations, and I learnt a lot from their experience. Also,

I would like to thank André, Rui, and Akus for the great casual moments we spent at the

laboratory.

I would also like to thank my friends who have accompanied me for many years. I

can always rely on them in times of trouble and I hope our friendship lasts. Thank you to my

special lady friend for putting up with me in the stressful times of my dissertation. So much.


vi INTEGRATED MASTER OF MECHANICAL ENGINEERING


vii INTEGRATED MASTER OF MECHANICAL ENGINEERING

Table of contents

Abstract .................................................................................................................................... i

Resumo .................................................................................................................................. iii

Acknowledgements ................................................................................................................ v

Table of contents .................................................................................................................. vii

List of figures ........................................................................................................................ ix

List of tables .......................................................................................................................... xi

Nomenclature....................................................................................................................... xiii

1. Introduction ..................................................................................................................... 1

1.1. Current energy context and outlook ............................................................................ 1

Sustainable development .................................................................................. 2

Energy performance and evolution of consumption......................................... 5

Carbon footprint goals ...................................................................................... 6

Energy in buildings ........................................................................................... 8

1.2. Heat pumps .................................................................................................................. 9

Context ............................................................................................................. 9

Heat pumps and renewable energy ................................................................. 10

Restrictions on working fluids ........................................................................ 11

CO2 as working fluid in heat pumps ............................................................... 12

1.3. Objectives of the dissertation .................................................................................... 12

1.4. Structure of the dissertation ....................................................................................... 13

2. Literature review ........................................................................................................... 15

2.1. CO2 as working fluid ................................................................................................. 15

2.2. Conventional CO2 cycle ............................................................................................ 17

2.3. Modifications to the conventional CO2 cycle ............................................................ 19

2.4. The ejector expansion device .................................................................................... 21

Working principle and design......................................................................... 21

Ejector performance ....................................................................................... 24

Most relevant geometric parameters of an ejector.......................................... 27

Variable geometry ejector concept ................................................................. 29

2.5. Transcritical ejector models....................................................................................... 30

3. Development of the CFD model ................................................................................... 35

3.1. Assumptions .............................................................................................................. 35

3.2. Ejector design ............................................................................................................ 36

Effect of boundary conditions on optimal ejector design ............................... 36

Fixed geometry ejector ................................................................................... 38

Variable geometry ejector .............................................................................. 39

3.3. Mathematical model for the CFD simulations .......................................................... 42

Governing equations ....................................................................................... 42

Turbulence ...................................................................................................... 42

The finite volume method............................................................................... 45

FLUENT® ...................................................................................................... 47

3.4. Development of the numerical mesh ......................................................................... 47

Mesh geometry ............................................................................................... 47

Mesh quality evaluation ................................................................................. 48

Mesh independence of the results ................................................................... 49

4. Formulation of the energy equation for the CFD model ............................................... 51

4.1. Energy equation for CO2 as ideal-gas........................................................................ 51

Temperature-based formulation of the energy equation................................. 51


viii INTEGRATED MASTER OF MECHANICAL ENGINEERING

Simulation strategy ......................................................................................... 52

Boundary conditions ....................................................................................... 54

Mesh independence testing ............................................................................. 56

4.2. Energy equation for CO2 as real-gas ......................................................................... 57

Enthalpy-based formulation of the energy equation....................................... 57

Implementation of the HEM ........................................................................... 58

Simulation strategy ......................................................................................... 60

Boundary conditions ....................................................................................... 61

Mesh independence testing ............................................................................. 62

Influence of property table size on flow variables ......................................... 63

5. Results and discussion ................................................................................................... 65

5.1. Simulation results for the ejector flow with CO2 as ideal-gas ................................... 65

Applicability of the ideal-gas model .............................................................. 65

Ejector performance for fixed spindle position .............................................. 68

Ejector performance for fixed compression ratio ........................................... 70

Overview of the numerical results .................................................................. 72

5.2. Simulation results for the ejector flow with CO2 as real-gas .................................... 74

Applicability of the real-gas model ................................................................ 74

Overview of the numerical results .................................................................. 77

6. Conclusions and suggestions for future work ............................................................... 79

6.1. Conclusions ............................................................................................................... 79

6.2. Suggestions for future work ...................................................................................... 80

7. References ..................................................................................................................... 81

Appendix I – SpaceClaim® script for automated geometry generation .............................. 87

Appendix II – C scripts for implementation of the HEM ..................................................... 89

Appendix III – EES® script for generation of property lookup tables .............................. 101

Appendix IV – EES® script for calculation of nozzle exit pressure .................................. 107


ix INTEGRATED MASTER OF MECHANICAL ENGINEERING

List of figures

Figure 1 – Recent evolution of the energy consumption in the world and forecast for the year

2040 [2]. ................................................................................................................................. 1

Figure 2 – Overview of renewable energy sources [3]. .......................................................... 3

Figure 3 – Positive impacts of the use of renewable energy sources on the environmental,

political, social, economic, and technological fields [3]. ....................................................... 3

Figure 4 – Use, technical potential, and theoretical potential of renewable energy sources in

2004 [9]. ................................................................................................................................. 4

Figure 5 – Recent energy consumption per capita in developed and emerging countries and

forecast for the year 2050 [2]. ................................................................................................ 6

Figure 6 – Carbon emissions by sector and forecast for current policy on the reduction of the

carbon footprint [11]. .............................................................................................................. 7

Figure 7 – Recent carbon emissions per capita in developed and emerging countries and

forecast for the year 2050 [2]. ................................................................................................ 7

Figure 8 – Pressure-enthalpy diagrams for subcritical (a) and transcritical (b) refrigeration

cycles [52]............................................................................................................................. 17

Figure 9 – Effect of gas cooler pressure on the performance of a transcritical refrigerating

cycle [55]. ............................................................................................................................. 18

Figure 10 – Theoretical COP values for the transcritical carbon dioxide cycle for different

values of gas cooler pressure and exit temperature [58]. ..................................................... 18

Figure 11 – Schematic of a standard transcritical ejector cycle [45].................................... 20

Figure 12 – Pressure-enthalpy diagram of a standard transcritical ejector cycle [45]. ........ 20

Figure 13 – Schematic diagram of an ejector cross section [41]. ......................................... 22

Figure 14 – Evolution of pressure and velocity of the primary (P) and secondary (S) flows

inside an ejector. ................................................................................................................... 22

Figure 15 – Ejectors with different nozzle positions: constant area mixing (left) and constant

pressure mixing (right) ejectors [60]. ................................................................................... 23

Figure 16 – Working modes of a supersonic ejector as a function of backpressure for constant

inlet pressure [40]. ................................................................................................................ 24

Figure 17 – Effect of the expansion angle of the primary flow on the effective area of the

ejector: greater effective area for smaller expansion angle (a) and smaller effective area for

greater expansion angle (b) [89]. .......................................................................................... 26

Figure 18 – Entrainment ratio for fixed inlet conditions of the secondary flow (a) and fixed

backpressure (b) [81]. ........................................................................................................... 26

Figure 19 – Effect of area ratio on entrainment ratio and critical backpressure. ................. 27

Figure 20 – Effect of nozzle exit position on entrainment ratio for different values of primary

pressure. ................................................................................................................................ 28

Figure 21 – Schematic of a variable geometry ejector with adjustable area ratio [90]. ....... 30

Figure 22 – Performance characteristics of a variable geometry ejector. ............................ 30

Figure 23 – Classification of current two-phase carbon dioxide ejector models. ................ 32

Figure 24 – Pressure-enthalpy diagram of carbon dioxide showing saturation lines (blue and

orange), homogeneous nucleation lines (green and red), and expansion lines (pink – near-

critical expansion (1) and off-critical expansion (2)) [44]. .................................................. 33

Figure 25 – Effect of Tp, Ts, and Π on the diameter of the primary throat, the entrainment

ratio, and COP. ..................................................................................................................... 37

Figure 26 – Geometry of the ejector (dimensions in millimetres). ...................................... 39

Figure 27 – Location of secondary jet choking withing the mixing section (red) and expansion

lines of the primary flow (blue). ........................................................................................... 39

Figure 28 – Design of the VGE spindle. .............................................................................. 39


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Figure 29 – Design detail of the variable geometry ejector (primary and secondary inlets).

.............................................................................................................................................. 40

Figure 30 – Spindle position (SP) and primary nozzle exit position (NXP). ....................... 40

Figure 31 – Reynolds decomposition of instantaneous velocity. ......................................... 43

Figure 32 – First-order Central Differencing Scheme (a) and first-order Upwind Differencing

Scheme (b). ........................................................................................................................... 46

Figure 33 – Detail of the mesh near the boundary layer. ..................................................... 48

Figure 34 – Effect of mesh refinement on simulation results............................................... 49

Figure 35 – Definition of specific heat, thermal conductivity, and molecular viscosity as

ideal-gas properties. .............................................................................................................. 52

Figure 36 – Approach to the convective, diffusive, and source terms in the UDS transport

equation. ............................................................................................................................... 58

Figure 37 – Schematic of property UDFs (specific heat, density, thermal conductivity, and

viscosity). .............................................................................................................................. 58

Figure 38 – Schematic of the property (φ(p,h)) reading process for the developed model. . 59

Figure 39 – Schematic of property UDFs (Mach number, temperature, and quality). ......... 59

Figure 40 – Schematic of UDF to verify pressure and enthalpy limits. ............................... 59

Figure 41 – Detail of mesh orthogonality for mesh IG7. ..................................................... 63

Figure 42 – Static pressure and Mach number distributions along the ejector axis. ............ 65

Figure 43 – Detail of the pressure distribution for the primary flow near the primary nozzle

exit section (pressure in Pa). ................................................................................................. 66

Figure 44 – Detail of the Mach number results for the secondary flow near the inlet section

of the diffuser. ...................................................................................................................... 66

Figure 45 – Saturation lines and expansion process in the primary nozzle (1-D model and

CFD assuming ideal-gas behaviour)..................................................................................... 67

Figure 46 – Primary and secondary mass flow rates and entrainment ratio as a function of

compression ratio for a SP of 2.5 mm. ................................................................................. 68

Figure 47 – Mach number distribution for a SP of 2.5 mm and compression ratios of 1.1 (a),

1.2 (b), 1.3 (c), 1.4 (d), and 1.5 (e). ...................................................................................... 69

Figure 48 – Entrainment ratio as a function of compression ratio for SP of 1.5, 2.5 and 3.5

mm, and optimal operation line. ........................................................................................... 70

Figure 49 – Primary and secondary mass flow rates and entrainment ratio as a function of SP

for a compression ratio of 1.3. .............................................................................................. 71

Figure 50 – Mach number distribution for a compression ratio of 1.3 and SP of 5 mm (a), 4

mm (b), 2 mm (c), and 1 mm (d). ......................................................................................... 71

Figure 51 – Entrainment ratio as a function of SP for compression ratios of 1.2, 1.3 and 1.4.

.............................................................................................................................................. 72

Figure 52 – Entrainment ratio as a function of SP for different compression ratios. ........... 72

Figure 53 – Optimum SP as a function of compression ratio. .............................................. 73

Figure 54 – Entrainment ratio for a fixed-geometry ejector and a VGE with optimum SP, and

improvement on entrainment ratio. ...................................................................................... 74

Figure 55 – Saturation lines and expansion process in the primary nozzle (1-D model and

CFD assuming real-gas behaviour). ..................................................................................... 75

Figure 56 – Distribution of the first source term in the transport equation for enthalpy (source

term in W/m3). ...................................................................................................................... 76

Figure 57 – Quality and Mach number distributions along the primary nozzle axis. .......... 77

Figure 58 – Primary mass flow rate as a function of SP. ..................................................... 78

Figure 59 – Mach number distribution for SP of 7 mm (a), 6 mm (b), 5 mm (c), and 4 mm

(d). ........................................................................................................................................ 78


xi INTEGRATED MASTER OF MECHANICAL ENGINEERING

List of tables

Table 1 – GWP and ODP values for common refrigerant fluids [36] .................................. 11

Table 2 – Properties of different fluids commonly used in power and refrigeration cycles [39]

.............................................................................................................................................. 15

Table 3 – Overview of the considered non-equilibria, benefits, and challenges and limitations

of currently available carbon dioxide two-phase ejector models [44] .................................. 34

Table 4 – Operating conditions selected for dimensioning the ejector ................................ 38

Table 5 – Mass flow rate, density, cross-section area, and velocity at both inlets and the outlet

.............................................................................................................................................. 41

Table 6 – Static pressure, dynamic pressure, and respective ratio for both inlets and the outlet

.............................................................................................................................................. 41

Table 7 – Selected parameters for the definition of the turbulence model ........................... 45

Table 8 – Quality of the mesh in terms of skewness and orthogonality [113] ..................... 48

Table 9 – Selected parameters for the definition of the discretization schemes for the ideal-

gas model .............................................................................................................................. 52

Table 10 – Estimation of hydraulic diameter, Reynolds number, turbulence intensity, and

turbulence length scale for both inlets and the outlet ........................................................... 55

Table 11 – Applied boundary conditions for the ideal-gas model ....................................... 55

Table 12 – Results of the mesh sensitivity test ..................................................................... 56

Table 13 – Selected parameters for the definition of the discretization schemes for the real-

gas model .............................................................................................................................. 60

Table 14 – Applied boundary conditions for the real-gas model ......................................... 62

Table 15 – Primary mass flow rate and average Mach number at the nozzle exit section for

different meshes .................................................................................................................... 63

Table 16 – Primary mass flow rate and average Mach number at the nozzle exit section for

different interpolation schemes ............................................................................................ 64

Table 17 – 1-D model estimations and ideal-gas simulation results for pressure and density

at the primary inlet, primary nozzle throat, and primary nozzle exit section ....................... 67

Table 18 – Confirmation of stagnation conditions at the inlets and the outlet for the ideal-gas

simulations ............................................................................................................................ 68

Table 19 – 1-D model estimations and real-gas simulation results for pressure, enthalpy, and

density at the primary inlet, primary nozzle throat, and primary nozzle exit section .......... 75


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xiii INTEGRATED MASTER OF MECHANICAL ENGINEERING

Nomenclature Abbreviations

AR Area Ratio

CAM Constant-Area Mixing

CDS Centred Differencing Scheme

CFC Chlorofluorocarbon

CFD Computational Fluid Dynamics

COP Coefficient Of Performance

CPM Constant-Pressure Mixing

CRMC Constant Rate of Momentum Change

CV Control Volume

DEM Delayed Equilibrium Model

DNS Direct Numerical Simulation

GDP Gross Domestic Product

GWP Global Warming Potential

HC Hydrocarbon

HCFC Hydrochlorofluorocarbon

HEM Homogeneous Equilibrium Model

HFC Hydrofluorocarbon

HFO Hydrofluoroolefin

HRM Homogeneous Relaxation Model

NXP Nozzle eXit Position

ODP Ozone Depletion Potential

OECD Organization for Economic Co-operation and Development

QUICK Quadratic Upstream Interpolation for Convective Kinematics

RANS Reynolds-Averaged Navier Stokes

RNG Re-Normalization Group

SDG Sustainable Development Goals

SP Spindle Position

TFM Two-Fluid Model

UDS Upwind Differencing Scheme (Section 3.3.3)

User-Defined Scalar (Section 4.2.1)

VGE Variable Geometry Ejector

Symbols

𝑎 Speed of sound m.s-1

𝐶 Model constant -

𝐶𝑝 Specific heat J.kg-1.K-1

𝐷 Diameter M

ℎ Specific enthalpy J.kg-1

𝐼 Turbulence intensity -

𝑘 Thermal conductivity W.m-1.K-1

Turbulent kinetic energy (Section 3.3.2) m-2.s-2

𝑙 Turbulence length scale m

�� Mass flow rate kg.s-1

�� Universal gas constant J.mol-1.K-1

𝑀 Molecular weight kg.mol-1


xiv INTEGRATED MASTER OF MECHANICAL ENGINEERING

𝑀𝑎 Mach number -

𝑝 Pressure Pa

𝑅 Gas constant J.kg-1.K-1

𝑅𝑒 Reynolds number -

𝑠 Entropy J.kg-1.K-1

��ℎ Source term in UDS transport equation W.m-3

𝑡 Time s

𝑇 Temperature K

𝒖 Velocity vector m.s-1

𝑢∗ Shear velocity m.s-1

�� Work rate W

𝑦 Wall distance m

𝑦+ Dimensionless wall distance - Greek symbols

𝛼 Phase volume fraction -

𝛾 Polytropic coefficient -

𝛤 Diffusion coefficient in UDS transport equation kg.m-1.s-1

𝛿 Kronecker delta -

휀 Turbulent dissipation rate (Section 3.3.2) m2.s-3

Error -

𝜂 Efficiency -

𝜇 Viscosity Pa.s

𝛱 Ejector compression ratio -

𝜌 Density Kg.m-3

𝜎 Prandtl number -

𝜏 Viscous stress Pa

𝜑 Generic flow variable -

𝜔 Mass entrainment ratio -

𝛻 Divergent operator - Subscripts

𝑐𝑎 Constant-area section

𝑜 Outlet

𝑝 Primary inlet/flow

𝑝𝑡 Primary nozzle throat

𝑠 Secondary inlet/flow

𝑇 Turbulence Superscripts

Time-averaged component

′ Fluctuating component


1 INTEGRATED MASTER OF MECHANICAL ENGINEERING

1. Introduction

1.1. Current energy context and outlook

The energy dependency of modern society is undeniable, as most human activities

rely on some form of energy either directly or indirectly. Over the last decades, energy

consumption has shown a clear increasing tendency. In fact, primary energy consumption

has grown in 49% and CO2 emissions have increased by 43% from 1984 to 2004 [1]. The

coherence between the quality of life and energy consumption is rather evident. Population

growth, economic and industrial development in emerging regions, and the higher demand

for comfort levels of the population have contributed to the global increase of energy

consumption [1, 2]. Figure 1 shows recent evolution and projection of the energy

consumption in the world. In the 20th century, renewable energy sources (disregarding

hydroelectricity) had no significant impact on the global energetic mix. Over recent years,

their remarkable growth led them to a current participation of about 8%. This growth is

expected to accelerate, and forecasts suggest a penetration level of 17% on the global

energetic mix by 2040. Natural gas has also gained importance and currently represents 25%

of primary energy consumption, as environmental concerns have drawn attention to this

cleaner alternative to conventional fossil fuels (coal and oil) [3]. An expected growth of 2%

by 2040 represents an increase in annual consumption of over 400 million tonnes of oil

equivalent. The increase in energy consumption in the world is expected to be met primarily

by these two energy sources. Environmental concerns are one of the main drivers for this

transition and push towards clean energy production [3]. The use of conventional fossil fuels

such as oil and coal, which currently account for over 60% of total primary energy

consumption, will remain rather unchanged. As a result, these polluting energy sources will

most likely still account for about 50% of primary energy consumption in the near future [2].

Figure 1 – Recent evolution of the energy consumption in the world and forecast for the year 2040 [2].



Sustainable development

Current energy technologies meet the demand using mostly non-renewable energy

sources such as fossil fuels, which cannot be sustained on a long term. Moreover, the intense

use of these energy sources poses serious environmental challenges leading to unwanted

climate changes and other biohazards. This has drawn attention to the need for efficient and

environment-friendly energy production and conversion [1]. Modern society pushes for

sustainable development, i.e., the “development that meets the needs of the present without

compromising the ability of future generations to meet their own needs” [4]. In 2015, the

United Nations established a set of 17 Sustainable Development Goals (SDG) to engage

developed and developing countries worldwide in tackling urgent problems such as poverty,

hunger, lack of access to education and healthcare, preservation of ecosystems, and climate

change [5]. The interaction between environmental and social goals is essential in

determining the viability of each. The elimination of extreme poverty means raising quality

of life for low-income populations, resulting in an increased carbon footprint [6]. This

evidences the need to evaluate possible trade-offs between different development goals. In

low-income groups, the marginal environmental impact of an increased income is greater,

which hinders low-income countries’ ability to simultaneously reach social development and

environmental preservation goals [6]. The increased environmental impact resulting from

reaching social goals worldwide may not be compatible with the sustainable use of resources

[7]. Even if extreme poverty is not completely eliminated, the effort to achieve social

development goals requires a carbon footprint reduction of about 77% in high-income

populations [6]. It is therefore clear that high-income groups have a greater ability to

effectively reduce the world’s carbon footprint. Less developed countries are more focused

on achieving social equality, so social policies may still be required to ensure basic human

rights. On the other hand, in developed countries these are mostly ensured and unquestioned.

Therefore, high-income countries may address other matters such as environmental

sustainability, also because they tend to register higher energy consumption values per capita

[6, 8].

The efficient use of our planet´s resources is essential to achieve sustainable

development. Renewable energy sources are “energy sources that are continually replenished

by nature and derived directly from the sun or other natural movements and mechanisms of

the environment” [3]. Contrary to fossil fuels, these energy sources do not rely on the

extraction of Earth’s finite reserves of natural resources. The use of renewable energy sources

directly contributes to achieving the SDG related to affordable and clean energy. Moreover,

it also contributes to the preservation of the ecosystems, as they are an eco-friendlier

alternative to fossil fuels. The advantages of renewable energy sources include reduced

carbon emissions, an important means of tackling climate change. Thus, other SDG benefit,

though indirectly, from this shift towards renewable energy sources [5]. For these reasons,

the focus on energy efficiency is expected to be complemented using renewable energy

sources [2, 3]. Figure 2 shows the variety of renewable energy sources that can be used to

produce useful energy, typically heat or electricity. Different energy sources require different

conversion technologies and are usually in different development stages. For example,

hydroelectricity is the most established renewable energy source and has been successfully

used for the production of clean and cheap electricity for many years [3]. Geothermal energy

is also a very mature technology, and its conversion potential is vast, but it is still not very

significant in a global perspective [3, 9]. In the recent past, close attention has been drawn to

wind energy which allows for predictable and safe production of electricity [3]. Solar

photovoltaic energy has had an exponential increase in recent years, as the development of



more cost-effective modules widened its range of application [3]. Bioenergy has an immense

potential but still faces many technological obstacles that affect its economic competitiveness

against conventional fossil fuels [3]. Until very recently, marine energy was rather uncharted

territory as a renewable energy source, but many research projects are currently in place and

many prototypes have already been developed and tested. This technology is close to

becoming economically viable and its potential exceeds current human needs [3]. With larger

deployment in the power system, incentivizing the use of these technologies will help reduce

the pressure on fossil fuels and the associated environmental impacts [3, 10].

Figure 2 – Overview of renewable energy sources [3].

In addition to being inexhaustible, renewable energy sources present many other

advantages when compared to conventional, fossil-fuel based energy conversion processes,

as shown in Figure 3. Apart from the environmental benefits regarding reduced carbon

emissions, the use of renewable energy impacts other fields. It promotes technological

development and innovation, not only creating diverse job opportunities but also contributing

to achieving energy self-sufficiency. Moreover, the decentralized production enabled by

renewable energy sources facilitates energy access in rural areas [3]. However, there are some

negative consequences of the use of renewable energy sources. Technologies such as biomass

energy and solar energy still face problems of cost-effectiveness. The construction of dams

for the production of hydroelectricity leads to flooding upstream and can significantly affect

local fauna and flora. Wind turbines require great amounts of land and can visually impact

landscapes [3].

Figure 3 – Positive impacts of the use of renewable energy sources on the environmental, political, social,

economic, and technological fields [3].



The potential for renewable energy production far exceeds current levels of

consumption [3, 9]. In fact, when evaluating the availability of renewable resources, it is

necessary to distinguish between theoretical potential and technical potential. Figure 4 shows

the distinction between these two concepts, and offers predictions based on energy

consumption and technological solutions available in 2004. The theoretical potential of a

resource refers to its availability, that is, the quantity or flow of the specified resource that

exists in nature. However, the exploitable resource is less than the available resource. It may

not be possible to explore a certain resource because of its location, either for social or

environmental reasons. In other cases, current technology may be the limiting factor. The

economic viability of a project is also decisive. All these limitations define the technical

potential of a renewable energy source [9]. Nonetheless, current renewable energy

consumption is far below the technical potential. On the one hand, many technologies are

still recent, which means their presence on worldwide energy systems still has a vast growing

potential [2]. On the other hand, contrarily to the fixed theoretical potential of a resource, its

technical potential evolves alongside the respective conversion technology. Innovation

tackles operational impediments and improves cost-effectiveness, thus widening the range of

application of these technologies.

Figure 4 – Use, technical potential, and theoretical potential of renewable energy sources in 2004 [9].

Padhan et al [8] explain that the consumption of renewable energy depends on

multiple factors. Per capita income, real price of oil and carbon emissions per capita have a

significant and positive impact on the consumption of renewable energy in OECD

(Organization for Economic Co-operation and Development) countries. This can be further

stimulated by conceding incentives and/or fiscal benefits to productors of clean energy.

Investment on renewable energy generation, promotion of technological innovation, and

education of the population are key factors to ensure a more significant presence of renewable

sources on the energy market. The integration of these conversion systems faces an additional

challenge as it ought to be done in useful time, considering for example the carbon footprint

goals for 2030 and 2050 [11-13]. Forecasts point towards an exponential increase of the

implementation of renewable energy technologies. More mature technologies, such as

hydropower, are expected to suffer smaller yet very significant increases. Other technologies

such as solar thermal/photovoltaic, wind power, and bio-power have a great growth potential

[3, 9].



Energy performance and evolution of consumption

The assessment of a country’s energy performance may be conducted using various

indicators. These may be physical-based indicators (energy consumed per physical unit or

output) or monetary-based indicators (energy consumed per monetary unit of output). The

most common monetary-based indicator is energy intensity and corresponds to the ratio

between energy consumption and gross domestic product (GDP) of a country. This indicator

measures the energy efficiency of a country and its ability to use energy in a productive

manner [10]. Energy intensity may also be evaluated in terms of the growth rate of both

energy consumption and GDP to contemplate a temporal dimension. This indicator is based

mostly on the energy intensity in each sector of the economy and the economic structure of

a country [10]. Energy intensity of each economic sector depends on its technological

development level and can be improved by pushing for more energy efficient processes. The

economic structure of a country defines the importance of each sector in a global perspective

and may help to identify which sector to act on in terms of energy efficiency. Many studies

analysing the evolution of energy intensity on individual countries have shown that energy

consumption has grown at a slower pace than GDP [14-16]. Additionally, studies focusing

on developed countries reported an evident decrease of energy intensity during the 20th

century [17, 18]. This has become more evident since the 1970´s, indicating a certain level

of decoupling between energy consumption and economic growth [19]. However, when

using GDP per capita to evaluate the decoupling between energy consumption and economic

growth, results are less optimistic. During the 20th century, no clear decoupling could be

detected. Economic growth was strongly dependent on energy consumption and no clear

improvement on energy efficiency was observed. A slight decoupling may have begun in the

21st century [19], perhaps as a result of energy efficiency policies and restrictions on energy

consumption in developed countries.

Despite some ambiguities in the definition of concepts such as progress and

development, a fairly evident relation exists between energy consumption and quality of life

of a population [20]. Therefore, energy consumption per capita is typically higher in

developed countries. In these countries it is a priority to monitor the energy indicators and to

promote energy saving, rational usage, energy efficient processes, and proper use of the

energy vectors. Developed countries aim to maintain energy intensity under unity, meaning

that the growth rate in GDP is accompanied by a lower growth rate in energy consumption.

In other words, energy efficiency increases. However, in developing countries restrictions on

energy consumption mean an obstacle to economic growth [6]. These restrictions are less

demanding to still allow for the countries’ development. Energy intensity is usually higher

than 1 and lower energy efficiency is expected for some time. As their development level

reaches a higher standard, it is expected that they take measures to reduce their energy

intensity indicator. Globally, energy intensity is expected to decrease because of an effort

towards higher energy efficiency. Based on current policies, the International Energy Agency

predicts an annual average improvement rate of 2.4% on energy intensity until 2030 [21].

Figure 5 shows recent energy consumption per capita in both developed and emerging

countries and a forecast for the year 2050 [2]. The Rapid scenario is based on measures

consistent with the goal of limiting global temperature rise to 2 degrees by 2100 [2]. As

referred, energy consumption per capita is typically higher in developed countries because

of higher living standards. Naturally, it is possible to achieve greater reductions in energy

consumption per capita in countries where it is currently higher, such as the United States.

These reductions are the result of the implementation of measures to ensure energy efficiency

and saving. On the contrary, in countries such as China and India the expected economic



growth requires that energy policies be alleviated [6], which allows for an increase in energy

consumption per capita and energy intensity. After this period of economic growth, it is

expected that these countries adopt stricter energy policies to reduce these indicators.

Figure 5 – Recent energy consumption per capita in developed and emerging countries and forecast for the

year 2050 [2].

Carbon footprint goals

In past years, the effort to reduce the carbon footprint has led international

organizations to act on different sectors. In 2011, the European commission defined three

distinct targets to be reached regarding energy and climate change. Member-states committed

themselves to reaching a 20% reduction of greenhouse gas emissions, a 20% share of

renewable energy sources on the European market, and a 20% overall energy efficiency

improvement by 2020 [11]. Following the path drawn by the Europe 2020 flagship, the

European Commission now proposes a series of long-term targets to be met by the year 2050.

The revised goals include a reduction of 80-95% on greenhouse gas emission compared to

1990 [11]. Figure 6 illustrates the current panorama regarding greenhouse emissions. Current

energy policies will most likely lead to a reduction of greenhouse gas emissions of 40% by

2050. This value comes short of the 80-95% target proposed by the European Commission

in 2011, resulting in the need for stronger policies and new technological solutions to be

implemented until 2050. The effectiveness of additional policies is dependent on the

development and implementation of new technological options that will allow for a cost-

effective reduction of greenhouse gas emissions [11]. Research on technologies such as low-

carbon energy sources, and carbon capture and storage solutions is fundamental to ensure the

economic viability of this generalized transition. Otherwise, continued use of other, more

polluting technologies, will worsen the current scenario. Failing to meet these targets will

lead to the need for even stricter restrictions on carbon emissions, which would harm the

overall costs of this transition [11].



Figure 6 – Carbon emissions by sector and forecast for current policy on the reduction of the carbon footprint

[11].

On a global scale, reduced emissions are expected for both developed and developing

countries until 2050 [2], as shown in Figure 7. The implementation of strict measures to limit

carbon emissions in developed countries allows for a much more significant reduction of the

carbon footprint, as they register higher emissions per capita. Because of the high living

standards of their populations, it is possible to convey efforts into tackling climate change

and other biohazards, namely through reducing carbon emissions. On the contrary, in

developing countries matters such as ensuring access to education and healthcare, and

fighting poverty are more urgent. Because reducing carbon emissions is not a priority,

developing countries are expected to achieve a less significant reduction [2, 6].

Figure 7 – Recent carbon emissions per capita in developed and emerging countries and forecast for the year

2050 [2].



The European Commission proposed in 2019 an international mechanism to mobilize

governmental efforts towards the transition into a climate neutral economy by 2050. In this

context, carbon pricing is the first measure to be adopted and/or reinforced, disincentivizing

the use of carbon-intensive processes and promoting research on energy efficient

technologies [22]. These measures, complemented by others, will motivate companies and

corporations to replace current, less efficient technologies for innovative ones. The revenue

generated from this strict carbon tax ought to be invested on green innovation. By doing so,

an opportunity is created for European companies to establish a strong presence on the energy

market. The implementation of these regulations will be accompanied with social

consequences, as specific communities will be significantly affected by these measures.

Those living in rural areas will be directly affected by higher fuel prices. Regions that are

most dependent on the production of fossil fuels will experience the disappearance of

industries and jobs [22]. The impact of these measures is uneven throughout all social strata,

as it is expected to particularly harm poorer households [23]. These must be addressed on a

national basis, as they derive from the materialization of this Green Deal specific to each

country [22].

Energy in buildings

In 2004 energy consumption in buildings already represented 37% of final energy

consumption against 28% and 32% for the industry and transport sectors, respectively [1].

Energy consumption in buildings represented 20-40% of total energy use in developed

countries by 2008 [1]. In 2016, this figure was higher than 40% in the United States and in

the European Union [24], demonstrating the increasing importance of the building sector on

the overall energy panorama and the possible impact of energy-efficient buildings on the total

energy consumption. Population growth, migration from rural regions to cities, improved

indoors comfort levels, and increase in time spent indoors have steeply raised energy

consumption levels in buildings [1, 25, 26]. Population and economic growths increase the

demand for health, education and other services and contribute to the increase of energy

consumption in commercial and public buildings. On the other hand, dwellings in developed

countries are expected to offer a high comfort level, which represents another means of

energy consumption. Increasing urbanization and economic development have also

contributed to an increasing emission of greenhouse gases [25]. In 2008, half the world’s

population lived in cities but were responsible for over two thirds of the energy consumption

worldwide [27]. Domestic energy consumption is much higher in developed countries and is

expected to continue increasing due to the installation of new appliances [1]. In this context,

great research has been conducted on the concept of zero-energy buildings. The accurate

definition of a zero-energy building varies according to different authors [28] and different

requirements are imposed on a zero-energy building. However, there is consensus that a zero-

energy building must be very energy-efficient and make use of renewable energy to satisfy

its own energy requirements. The development of zero-energy buildings requires extensive

technological innovation but may allow for a sustainable growth [24].

High comfort levels in buildings rely on the use of HVAC (Heating, Ventilation and

Air Conditioning) systems to maintain indoor air quality. These systems control ambient

parameters such as temperature and humidity to ensure optimal indoor air conditions and

comfort for the occupants. In developed countries, HVAC systems are responsible for around

one half of the energy consumption in buildings and for around one fifth of the total energy

consumption [1]. In the European Union, the main use of energy in residential buildings is

for space and water heating, followed by appliances, space cooling, lighting, and cooking. In



commercial buildings, appliances are the most intensive energy end-users, followed by space

and water heating, lighting, and space cooling [29].

1.2. Heat pumps

Context

As the need for energy efficient thermal systems rises, heat pumps regained

popularity in recent years [30]. The ability to recover thermal waste is increasingly sought as

it allows for the reduction of energy consumption, either in domestic households, service

buildings, or industries. Increased energy efficiency can be achieved with heat recovery, that

is, to recuperate otherwise waste heat and use it as an energy input in other operations. Heat

pumps are the only means of recovering thermal waste. As a result, over the past years these

systems have regained attention and been subject to many improvements [30]. One of the

reasons for this growth is associated with the technological development and cost reduction

in recent years. Heat pumps are a relevant technological solution when it comes to increasing

the efficiency of a process and/or reducing its greenhouse gas emissions. On the one hand,

this has clear economic benefits, such as reduced operating costs of HVAC systems and

industrial processes. Given the global energy context, this is a vital aspect as fossil fuel costs

continue to increase. Therefore, transitioning to more efficient technologies is evermore a

current topic. On the other hand, any decrease in energy consumption alleviates the stress on

the electrical grid and other energy sources. Depending on each country’s energetic mix, this

helps reduce the emission of polluting gases such as carbon dioxide.

The use of heat pumps on power plants allows for the recovery of waste heat and is a

cost-effective measure to reduce greenhouse gas emissions [30]. This may allow the power

sector to decrease its emissions by over 50% by 2030 and over 90% by 2050 [11]. In the

industry sector, the use of heat pumps in novel applications such as drying and desalination

processes [30] is key to potentially reducing the carbon footprint by over 30% by 2030 and

over 80% by 2050 [11]. On the residential and service sector, the role of heat pumps on

heating and cooling could help in reducing the respective greenhouse gas emissions by

around 40% by 2030 and 90% by 2050 [11].

Indoor space heating and cooling are perhaps the most typical application of heat

pumps in buildings. In the European Union alone, more than 7.5 million electrically driven

heat pumps were installed between 2010 and 2015 for heating and cooling of residential

buildings [31]. New solutions have been recently developed and implemented, such as

systems that allow for simultaneous space heating and hot water production [30]. On a larger

scale, the use of heat pumps for district heating and cooling is seen as having a tremendous

potential in terms of reducing energy consumption by recovering industrial waste heat [31,

32]. In these systems, heat pumps play an essential role in guaranteeing an efficient heat

transfer. The sheer scale of such systems indicate the ability to significantly reduce energy

consumption and carbon emission [30], especially in countries where the energetic mix is

largely dependent on fossil fuels.

As isolated units, heat pumps improve energy efficiency when compared to other

conventional solutions for heating and cooling in buildings. Because of their very high

efficiency, heat pumps meet the heating and cooling demands with a reduced energy

consumption. Overall, energy consumption is lower and carbon emissions associated to the

production of electricity are reduced. The possibility of improved energy efficiency and



reduced carbon footprint may be further explored with the use of renewable energy sources

to produce the electricity needed to drive the heat pump. The combination of renewable

electricity with an environmentally friendly working fluid may allow for the use of heat

pumps as a sustainable solution for heating and cooling in buildings. In this context, the

efficiency of a heat pump may be further improved with the use of an ejector expansion

device. This allows for the partial recovery of the expansion work which is lost in the

expansion device equipped in conventional refrigeration cycles, reducing energy

consumption, and contributing to a sustainable operation [33-35].

Heat pumps and renewable energy

The coupling of heat pumps with renewable energy sources may be achieved in two

different scales. When using a heat pump connected to the electrical grid, this is determined

by each country’s energetic mix, namely the penetration of renewable energy sources such

as wind, solar and hydroelectric. These systems are more compact as they do not need any

energy source rather than a connection to the grid. However, the environmental impact is

largely dependent on the energetic mix and these systems have a higher operation cost as it

is necessary to acquire all the electric energy for its operation. On the other hand, it is also

possible to produce the necessary electrical energy locally, for example with photovoltaic

panels. Despite the greater initial cost of the whole system, the ability to detach it from the

grid (although in most cases not completely) means that the operating cost is much lower. In

an economic perspective, the decision for either system is based on the balance between

initial and long-term costs. Governments may promote the adoption of this technology by

conceding incentives and/or other financial benefits for its purchase. Energy cost is an

important aspect, as lower energy costs difficult the economic viability of solar assisted heat

pumps, for example. Controversially, the use of heat pumps actuated by renewable energy

sources is most beneficial to reducing carbon emissions in countries where the energetic mix

relies largely on fossil fuels. As most countries still rely largely on fossil fuels for energy

production, coupling heat pumps with renewable energy sources has the potential for a great

reduction on environmental impact. Even in countries where renewable energy sources play

a significant role in the energetic mix these systems may help in reducing carbon emissions.

Although large energy production facilities have seen significant technological

improvements, heat pumps far exceed them in terms of efficiency in direct heat generation.

Current discussion related to the integration of equipment for water and space heating

in future energy systems is divided in two main perspectives. Some argue that the effort

towards zero-energy or even plus-energy buildings will remove the need for auxiliary power

from the electrical grid. By implementing solar thermal collectors, for example, buildings

become energetically independent from the grid and, in some cases, may even produce excess

heat. Absorption cycles are currently the most widespread technology for space cooling and

may be operated by the heat produced in solar thermal collectors [26]. On the other hand,

others believe that using otherwise excess heat from industrial processes, waste incineration

and power production may meet the demand for hot water and space heating. In this context,

a district heating infrastructure is required [32]. Heat pumps are a suitable and cost-effective

alternative to district heating, although their relative cost depends on the distance from the

installation to the district heating system. In Denmark the share of renewable energy sources

is one of the highest in the world, and district heating is an interesting solution for future

energy systems in large cities.



Restrictions on working fluids

As in any other equipment, the use of heat pumps leads to unwanted environmental

impacts. On the one hand, the production of the components and sub-assemblies is done by

varyingly polluting industrial processes. On the other hand, it is also necessary to equate the

(electric) energy needed to drive them. Since electricity is mostly produced in centralized

thermal powerplants, the efficiency of the process and the heat sources used determine the

level of pollution caused by particle and gas emissions into the atmosphere, infiltration of

chemicals and other harmful by-products into the soil, and thermal pollution of natural heat

sinks such as rivers and underground waterbeds.

Heat pumps operate in closed thermodynamic cycles using an adequate working fluid.

However, when using a synthetic working fluid, another aspect must be also considered.

During the operation, maintenance, and decommissioning of heat pumps, working fluid

leakage to the atmosphere is inevitable. Because of imperfect sealing between the different

components of the installation, a certain amount of working fluid is released into the

environment during the lifecycle of the equipment. Additionally, maintenance operations

usually represent losses of working fluid. It is necessary to drain the system and transfer the

fluid into temporary containers, so a small portion of the fluid is lost. Finally, when the

system completes its projected lifecycle, what to do with the working fluid remains a largely

unsolved problem. The working fluid tends to degrade during the operation of the cycle,

especially if subject to harsh operating conditions such as high temperature differences. The

environmental impact of working fluids used in refrigeration systems and heat pumps is

typically evaluated by two different indicators. The Ozone Depletion Potential (ODP) of a

fluid is defined as its ability to effectively remove ozone from the earth´s atmosphere. This

indicator is relative and calculated in comparison to a reference fluid. The Global Warming

Potential (GWP) is the ratio between the heat absorbed by a fluid in the earth´s atmosphere

and the heat absorbed by the same mass of carbon dioxide. Table 1 shows GWP and ODP

values for common refrigerant fluids.

Table 1 – GWP and ODP values for common refrigerant fluids [36]

Many working fluids for refrigeration systems and heat pumps have a high GWP. For

example, R134a has a GWP of 1300, meaning that 1 kg of this substance released into the

atmosphere has the same effect in terms of global warming as 1300 kg of carbon dioxide

[36]. Other traditional working fluids for these purposes have even higher values of GWP.

In 2014, the European Union and the European Council emitted a directive to regulate on



this matter [13]. In this document, issues such as scheduling and systems of leak checking,

recovery of working fluids after the system’s decommissioning, and training and certification

of operators are addressed. Restrictions on the use of current working fluids imposes a

reduction of 79% on equivalent carbon emissions by the year 2030. As a result, a generalized

transition towards the development and implementation of low-GWP working fluids is in

place. Environmental concerns already altered the global refrigerant market: availability of

synthetic fluids has shrunk by 63%. Additional restrictions are imposed on the GWP. Since

2018, all working fluids with a GWP higher than 150 were prohibited for use in new

equipment. On January 1st, 2020, decommissioning was imposed on refrigeration systems

with an emission equal to or greater than 40 tonnes of carbon dioxide equivalent working

with fluorinated greenhouse gases with a GWP higher than 2500 [12].

CO2 as working fluid in heat pumps

The applicability of any working fluid in heat pumps is evaluated in different aspects.

From a technical perspective, it must ensure an adequate performance for the cycle in which

it is used. The fluid properties dictate how it behaves under specific operating conditions.

These aspects limit, for example, the use of water because of its low vapor density and low

vapor pressure. Potential leakages make the toxicity and flammability of the fluid important

issues to be analyzed. In this context, ammonia is in disadvantage as it is highly toxic. These

performance-related aspects are complemented by a financial analysis since the operation of

the cycle must be economically viable. In fact, many working fluids have had limited

implementation because of their high cost of for being unsuitable for a cost-effective

operation (e.g., water).

Chlorofluorocarbon (CFC) and hydrocarbon (HC) fluids have been widely used in

the past as refrigerants, as they generally meet cost-effectiveness solutions, with minimal

problems regarding toxicity and flammability. In fact, in the past they came to replace carbon

dioxide, one of the first working fluids used in mechanical compression systems [37].

However, because of their negative impact on the environment, both CFC and HC fluids have

been subjected to strong criticism. CFC fluids have an unacceptable ODP, whereas HC have

very high GWP. Considering current legislation, these fluids are completely banned or its

use on new systems has been restricted [11-13, 26]. This created the opportunity for carbon

dioxide to reappear as a non-toxic, non-flammable, and low-environmental impact

refrigerant. Its GWP is 1 and it has a zero ODP. Its abundance in the atmosphere makes its

processing cost-effective. Because of its density and admissible working pressure, it is used

in lightweight heat pump systems. All these properties make carbon dioxide an adequate

working fluid for refrigeration systems [37, 38]. Despite all its advantages, the use of carbon

dioxide poses its own challenges. Its low critical point and high operating pressures raise

structural concerns when designing the system [26, 37]. These obstacles mean that carbon

dioxide is not yet a universal solution in heat pumps. Research on this matter has led to

improvements on the system’s mechanical components and structure [37]. Refrigeration

cycles using carbon dioxide as working fluid will be further explored in Chapter 2.

1.3. Objectives of the dissertation

This thesis is part of the research work by CIENER on the use of ejectors for various

refrigeration and heat pump cycles. The main objective of this work is to develop an adequate

mathematical model for simulation of ejector flow, allowing for the analysis of the

performance of a transcritical CO2 ejector with variable geometry under variable operating



conditions and the assessment of its benefits relative to a fixed-geometry ejector. The specific

objectives of this work are:

• Estimating ejector geometry and dimensions using an existing mathematical tool,

considering a transcritical heat pump cycle designed for the Portuguese climate,

operating on heating mode during the winter and on cooling mode during the summer,

with a heating capacity of 20 kW;

• Developing a CFD model based on the ideal-gas approach to conduct a qualitative

analysis of the benefits of a transcritical CO2 VGE when compared to the equivalent

fixed-geometry ejector, for variable operating conditions;

• Writing the necessary UDFs for the implementation of the HEM in FLUENT®,

analysing the transcritical expansion process in the primary nozzle, and assessing the

influence of the adjustable spindle on the primary mass flow rate.

1.4. Structure of the dissertation

This work consists of 7 chapters. In Chapter 1, the current energy context is explained,

and heat pump technology is addressed. The theoretical analysis of transcritical carbon

dioxide cycles, the working principle of ejectors, and the different approaches to the

modelling of CO2 properties are shown in Chapter 2. Chapter 3 defines the geometry of the

ejector and introduces the analytical models used in the numerical simulations. Chapter 4

addresses the two distinct formulations of the energy equation and their implementation in

FLUENT®. In addition, preliminary tests such as mesh independence of the results are

conducted. Lastly, simulation strategy and boundary conditions for each simulation approach

are defined. The applicability of each model is addressed in Chapter 5, which also includes

discussion of simulation results. Chapter 6 summarizes the results shown in Chapter 5,

addresses the conclusions, and presents suggestions for future work that may complement

this work. Chapter 7 lists the literature references that underlie the analyses conducted in this

work.



2. Literature review

2.1. CO2 as working fluid

Several factors must be considered for the selection of an appropriate working fluid

for a thermal cycle [39, 40]. The thermodynamic properties of the fluid are key because they

must be adjusted to the operating conditions of the cycle [41]. Under such conditions, the

fluid must be chemically stable and compatible with the construction materials used [40, 42].

Furthermore, working fluids are subject to increasingly restrictive policies regarding their

safety and environmental impact [11-13, 26, 40]. In this context, a working fluid is desirably

non-toxic, non-flammable, and has low GWP and ODP [40, 41, 43]. Apart from the technical

data, the economic aspects are also decisive. Availability, cost, and ease in processing [42]

are necessary conditions for the widespread application of a working fluid. Table 2 shows

the properties of a list of fluids commonly used in power and refrigeration cycles. The

ASHRAE (American Society of Heating, Refrigerating and Air-Conditioning Engineers)

Level for safety classifies working fluids according to their toxicity and flammability, among

other safety aspects. In this classification, A1 represents the highest level corresponding to a

non-toxic, non-flammable working fluid. If safety is a priority, only working fluids from this

class are suitable [39]. As environmental protection is at the forefront of modern society´s

concerns [11, 21, 44], the use of working fluids with high values of GWP and ODP has been

limited and/or completely banned in the European Union [12, 13, 41, 45].

Table 2 – Properties of different fluids commonly used in power and refrigeration cycles [39]

Refrigerants are usually classified by their chemical composition [41], leading to

three distinct groups: halocarbons and hydrocarbons, organic compounds, and other

refrigerants [40]. The first group includes chlorofluorocarbons (CFC),

hydrochlorofluorocarbons (HCFC), hydrofluorocarbons (HFC), hydrofluoroolefins (HFO)

and hydrocarbons (HC). Halocarbons allow for lower cooling temperatures and a better cycle

performance, but many have a significant environmental impact [40] and have had their use

limited or prohibited by recent carbon policies [12, 13, 40]. HCFCs are relatively safe, stable,

and non-toxic working fluids [40]. HFOs offer a compromise between performance, safety,

and environmental impact but face toxicity problems [40]. HCs have zero environmental



impact but are highly flammable, thus their use is limited to low capacity cycles [40]. Organic

refrigerants are compounds consisting of hydrogen and carbon, such as the R290, the R600,

and the R600a. The R290 (propane) is considered a promising substitute for other refrigerants

currently in use because of its negligible GWP and adequate thermal properties. However,

this working fluid is flammable, which requires that the charge mass of R290 in the systems

be reduced to a minimum [46]. Inorganic natural refrigerants include water, ammonia, and

carbon dioxide [40]. Water has a high heat of vaporization, is abundant and inexpensive, and

has low environmental impact [40, 45]. However, the cooling temperature of the cycle must

be kept above 0ºC to avoid freezing [40]. Not only does this limit the application range of

water as a working fluid, but it also limits the performance of the cycle [47]. Moreover, the

high specific volume of water requires large diameter pipes to reduce pressure loss in the

circuit [45, 48]. In addition, water refrigeration systems work with low operating pressures

that are usually below the ambient pressure. This facilitates the infiltration of air into the

system, which compromises its performance. For these reasons, water is not frequently used

as a working fluid in refrigeration systems [40]. Ammonia has been extensively studied as a

working fluid in refrigeration cycles for its low cost, high performance, and adequate

thermodynamic properties [40, 45]. However, because of its toxicity, it is likely to remain

restricted to industrial applications [40, 45].

Considering all these criteria for the selection of an adequate working fluid, carbon

dioxide shows great potential [38, 39, 43, 44, 49]. In fact, its use as a refrigerant was first

proposed in 1850 by Alexander Twining [50]. Refrigeration systems using carbon dioxide

and operating on a vapor compression cycle rapidly gained popularity [50]. After World War

II, development and production of synthetic working fluids suspended the use of carbon

dioxide as a refrigerant. However, given current environmental and safety concerns that limit

the use of these synthetic refrigerants, carbon dioxide is regaining attention [50]. Carbon

dioxide has a residual environmental impact when compared to other common working

fluids. With an ODP of 0 and a GWP of 1, it is one of the most environmentally friendly

substances [37-39, 43-45, 49-54]. Moreover, it is non-toxic and non-flammable [39, 43, 49-

53]. Due to its abundance on Earth´s atmosphere, the cost of extracting and processing carbon

dioxide is low [37, 39, 44, 51-53]. Considering that the carbon dioxide in thermal cycles is

extracted from the atmosphere, problems with leakage and decommissioning are less

significant when compared to others [37]. The thermodynamic properties of carbon dioxide

are also responsible for its growing popularity in recent years. It has unique critical point

properties: high critical pressure (7.38 MPa) but low critical temperature (31.1ºC) [39, 52].

The high operating pressures, alongside its high vapor density and high volumetric heating

capacity [38, 43, 52], allow for a more compact equipment [39, 44, 52]. On the other hand,

the low critical temperature means that it is possible to implement a transcritical cycle even

with low maximum temperatures [39]. The heat exchange in the supercritical region is more

efficient than when phase changing occurs [39]. In addition, transcritical carbon dioxide

cycles usually have a lower compression ratio than conventional refrigeration cycles, which

allows for a greater efficiency of the compressor [37, 52]. However, significant challenges

must be overcome for the widespread implementation of cycles using carbon dioxide [37].

The high operation pressure that allows for a compact equipment also requires careful

designing of the system components [37, 52, 53] and selection of the compressor [52, 53].

Moreover, transcritical carbon dioxide heat pumps show high irreversibility caused by

throttling losses in the expansion device [37].



2.2. Conventional CO2 cycle

Figure 8 shows pressure-enthalpy diagrams for both subcritical (Figure 8a) and

transcritical (Figure 8b) refrigeration cycles. In the first case, the cycle operates according to

a typical vapor compression cycle. The working fluid absorbs heat and evaporates at low

pressure in the evaporator, after which it is admitted in the compressor and compressed to

the condenser pressure. In the condenser, it rejects heat and condenses at high pressure [52].

An expansion device ensures the necessary pressure drop to readmit the refrigerant in the

evaporator. The transcritical cycle differs from the subcritical cycle in the heat rejection

process [37, 52, 53]. In the transcritical cycle, this occurs above the critical point [52, 53] and

exclusively through sensible cooling. Because there is no phase change, the condenser is

replaced by a gas cooler [50, 52, 55]. In the subcritical cycle the heat exchange occurs mainly

as the working fluid condenses at a constant temperature. However, in the transcritical cycle

the refrigerant loses only sensible heat, suffering a higher temperature decrease [37, 43, 53].

In the condenser of the subcritical cycle, pressure and temperature are not independent

properties because the working fluid is in the phase-change region [53]. However, in the

transcritical cycle, gas cooler pressure and temperature are independent [43, 50, 53, 55].

Figure 8 – Pressure-enthalpy diagrams for subcritical (a) and transcritical (b) refrigeration cycles [52].

As mentioned above, in subcritical operating mode it is not possible to regulate

condenser pressure and temperature separately. The performance of the subcritical cycles

decreases as the condenser pressure/temperature increases because additional compression

work is required while less heat is exchanged in the evaporator. In contrast, in transcritical

operation, it is possible to maintain either a constant gas cooler pressure or exit temperature

and control the other one. Figure 9 shows the effect of the gas cooler pressure on the

transcritical cycle for different exit temperatures. A higher gas cooler pressure

simultaneously increases the heating effect and the compression work. Depending on which

effect is stronger, this means that for each exit temperature there is an optimal gas cooler

pressure that maximizes the efficiency of the cycle [43, 55-57].



Figure 9 – Effect of gas cooler pressure on the performance of a transcritical refrigerating cycle [55].

Figure 10 shows the theoretical Coefficient Of Performance (COP) values for the

transcritical cycle using CO2, for different values of gas cooler pressure and exit temperature.

It can be seen that, for a given temperature of the working fluid at the exit of the gas cooler,

there is an optimal gas cooler pressure that maximizes the efficiency of the cycle [56]. For

this reason, controlling the gas cooler pressure is essential to ensure high performance of the

cycle [38, 55]. This optimal pressure increases with the exit temperature of the gas cooler

because the isothermal lines on the pressure-enthalpy diagram become less steep for higher

temperatures. For a fixed gas cooler pressure, the efficiency of the cycle decreases with an

increasing exit temperature of the working fluid. For a higher exit temperature, CO2 enters

the evaporator with a higher enthalpy and the cooling effect is reduced. The compression

work is unaltered because it depends only on the gas cooler pressure.

Figure 10 – Theoretical COP values for the transcritical carbon dioxide cycle for different values of gas cooler

pressure and exit temperature [58].

It is not always possible to operate a subcritical cycle because this requires a low

temperature for heat rejection, well below the critical temperature of CO2. For cooling

applications, it may be possible to run the system in subcritical mode in cold climates, if

outside temperatures are sufficiently low. However, in warmer climates this is not possible

because outside temperatures are typically above the critical temperature of CO2. In heat

pumps, the use of subcritical carbon dioxide cycles is rather limited. Considering space

heating, the minimum temperature for heat rejection is the indoor temperature, which is

typically lower than the critical temperature. However, required temperature levels could be



significantly higher depending on the heat distribution method (e.g., using radiators or floor

heating). For domestic hot water production, heat rejection temperature is limited by the

temperature of the hot water, which is normally higher than the critical temperature of CO2.

Consequently, transcritical CO2 heat pumps should be applied. When using carbon dioxide,

transcritical cycles perform worse than subcritical ones [38, 50] because of the increased

compression work. Also, the expansion process has increased energy losses, compromising

the efficiency of the cycle [37, 38, 49, 52]. To address this problem, implementing means for

expansion work recovery in the transcritical CO2 heat pump can lead to a considerable

increase in system efficiency [37, 40, 52, 53].

2.3. Modifications to the conventional CO2 cycle

Conventional vapor compression cycles are usually equipped with either a capillary

tube, a thermostatic expansion valve, or an automatic expansion valve as the expansion

device [59, 60]. Their main functions are to distribute the working fluid to the evaporator,

and to maintain a pressure difference between the condenser (subcritical cycle) or the gas

cooler (transcritical cycle) and the evaporator [37]. They restrict the refrigerant flow and

produce the necessary pressure drop through a throttling process [60]. Brown et al. [61]

performed a second-law analysis on carbon dioxide refrigeration systems for residential air-

conditioning applications and concluded that the irreversibility in the expansion device was

the main responsible for the low COP of the cycle. Similar research by Yang et al. [62] and

Robinson et al. [63] reported high exergy losses in the throttle valve. Therefore, reducing

energy losses in the expansion process could potentially lead to a significant improvement

on the efficiency of the transcritical carbon dioxide refrigeration cycle [43, 61-63]. An early

work by Kornhauser [64] pointed out the significant impact of the throttling losses on the

performance of a conventional vapor compression cycle and investigated on the use of an

ejector as a work-recovery device. The author indicated a theoretical COP improvement of

21% for an R12 system with a two-phase ejector [64]. Since then, the use of an ejector to

substitute a conventional expansion device has been extensively studied [60, 65-67]. Figure

11 shows the schematic of a standard transcritical ejector cycle. High-pressure refrigerant

leaves the gas-cooler (3) and enters on the primary or motive side into the ejector (4) [53,

65]. This high-pressure flow is used to partially compress the low-pressure (or secondary)

refrigerant coming from the evaporator (9) [64]. The primary and secondary flows mix in the

ejector (6) and leave the ejector at an intermediate pressure (7) that is lower than the primary

pressure but higher than the secondary pressure [64]. In the separator, the liquid and vapor

phases are separated. The liquid CO2 (8) is admitted to the evaporator after flowing through

an expansion device (8a), while the saturated vapor (1) is compressed before entering the gas

cooler (2) [53, 64].



Figure 11 – Schematic of a standard transcritical ejector cycle [45].

Figure 12 shows the corresponding simplified pressure-enthalpy diagram. The high-

pressure refrigerant leaving the gas cooler (3) is expanded to an intermediate pressure (7),

and not to the low pressure in the evaporator. This expansion work that would otherwise be

lost in the high stage expansion is used for partially compressing the secondary flow from

the evaporator (9) [64, 67]. The compressor suction pressure (1) is therefore higher than in a

conventional cycle [52, 53, 60, 65], thus requiring a lower energy input, and improving the

efficiency [44, 53, 67]. The cooling capacity in the evaporator is also increased because the

refrigerant enters the evaporator with lower vapor quality (8a), further improving the

performance of the cycle [44, 60, 64, 67]. The expansion process before the evaporator

(thermodynamic states 8-8a) covers a lower pressure gap, thus represents a reduced energy

loss [64].

Figure 12 – Pressure-enthalpy diagram of a standard transcritical ejector cycle [45].

Carbon dioxide has high operating pressures, and the transcritical cycle shows a large

pressure difference between heat absorption and heat rejection. As a result, there is a great

difference between an isenthalpic and an isentropic expansion in transcritical carbon dioxide

cycles. The use of an ejector changes the isenthalpic expansion that occurs in conventional

expansion devices to a theoretically isentropic process [60, 67]. Therefore, the use of a work-

recovery device such as an ejector is an interesting and low-cost solution to tackle their low

performance [38, 44, 68, 69]. Most of the reported research concerns transcritical cooling

applications. Hrnjak [70] indicated a maximum COP improvement of 44% on carbon dioxide

systems when using an isentropic ejector, compared with a 13% improvement on systems

using R134a. Later, Elbel and Lawrence [67] also showed that the COP improvement due to

the implementation of a two-phase ejector is much higher in cycles using carbon dioxide,

rather than HFC. Takeuchi et al. [71] and Ozaki et al. [72] conducted experimental work on



two-phase ejectors using carbon dioxide for an automotive system, and results showed COP

improvements of 20% in cooling applications [71, 72]. Gullo et al. [73] showed that the

ability to recover part of the expansion work of the high-pressure refrigerant allows for

energy savings up to 25% in supermarket refrigeration systems, when compared to systems

using HFC. Based on a modelling approach similar to that of Kornhauser [64], Liu et al. [74]

predicted a COP improvement between 6% and 14%. Joeng et al. [75] conducted numerical

simulations and obtained a 22% COP improvement over the conventional cycle using an

expansion valve. Elbel and Hrnjak [68] conducted experimental studies on an adjustable

ejector, and showed that it is possible to adjust the ejector to find the optimal high-side

pressure. Although an adjustable needle was shown to be less efficient, it was still possible

to increase the overall efficiency of the cycle. Simultaneous improvements on COP and

cooling capacity of 7% and 8% were observed, respectively. Research on transcritical CO2

heat pumps using an ejector have also shown promising results. Taleghani et al. [38]

investigated on the influence of the ejector geometry on its performance and reported a

maximum heating COP improvement of 12%. Jahar [56] analysed the performance of a

transcritical heat pump cycle under varying operating conditions and reported a potential

performance improvement of 14.7% when using an ejector.

Carel and Danfoss currently offer ejector solutions to improve the efficiency of

transcritical CO2 refrigeration cycles in warm climates [76, 77]. Transcritical CO2 heat

pumps using an ejector expansion device are already commercially available. These are

marketed as the Eco Cute Heat Pump and are commercialized by several Japanese

manufacturers. As of March 2015, over 4.6 million units had been sold [78].

2.4. The ejector expansion device

Working principle and design

The ejector is the main distinguishing component in a transcritical ejector cycle [41,

49]. Its principal functions are the entrainment and the compression of the low pressure

secondary flow [40]. Figure 13 shows the schematic cross section of a typical ejector. The

primary flow enters the primary nozzle where it accelerates through an isentropic expansion

process to a high velocity, low static pressure stream [40]. The low-pressure region in the

mixing chamber entrains the secondary flow into the suction chamber. The primary and

secondary flows mix in the mixing zone, and the pressure of the resulting stream increases

inside the diffuser to the desired value [40]. More details on the pressure and velocity

distributions in the ejector will be presented later in this section. Ejectors compress the

secondary flow without consuming external mechanical energy, by transferring the

mechanical energy of the primary flow to the secondary flow [79]. The expansion work of

the motive flow is not lost but partially recovered and used to compress the secondary flow

[44, 64, 67]. The fluid leaves the ejector at a pressure level between the primary inlet pressure

and the secondary inlet pressure [79]. Because of their simple configuration, ejectors are a

technologically cost-effective and safe solution, and their integration leads to simple system

layouts [41]. The absence of moving parts provides them with great reliability, and little

needs for maintenance [41, 49, 80].



Figure 13 – Schematic diagram of an ejector cross section [41].

Figure 14 shows the idealized evolution of velocity and static pressure for both the

primary and secondary flows along the ejector axis. The primary flow enters the ejector at

subsonic speed [41]. The upstream pressure in the motive nozzle forces the flow to accelerate

along the converging section to sonic conditions (Ma=1) at the throat [44, 81]. This

acceleration is accompanied by a decrease in pressure [41]. The flow is further accelerated

in the diverging section and transitions into the supersonic regime (Ma>1) [26, 41, 81, 82] as

primary flow pressure continues to decrease [41, 65]. The motive flow exits the primary

nozzle, creating a low-pressure zone that entrains the secondary flow [26, 41, 44, 49, 66, 81,

82]. The expansion of the primary flow forms a converging duct for the secondary flow,

forcing it to accelerate [83]. The secondary flow reaches sonic condition, after which mixing

with the primary flow begins [26, 41, 83] with exchange of mass, momentum, and heat [44].

Mixing occurs in the mixing section and is complete as the flow enters the constant-area

section. Due to the higher pressure downstream, the flow suffers a normal shock wave in the

constant-area section or in the diffuser. The flow becomes subsonic [26, 41] and pressure

increases [82]. In the diffuser, the additional deceleration of the stream leads to a further

increase in pressure [26, 41, 44, 65, 66, 82].

Figure 14 – Evolution of pressure and velocity of the primary (P) and secondary (S) flows inside an ejector.

Munday and Bagster [83] considered that mixing does not occur immediately after

the primary flow leaves the nozzle exit. As the primary flow exits the motive nozzle, it begins

to interact with the secondary flow and a shear mixing layer develops between the two flows

[84]. This shear mixing layer consists of large-scale vortices whose stretching and interaction



with the primary and secondary flows promotes the entrainment and mixing of the flows [84].

Within a short distance of the nozzle exit, the primary flow is supersonic, but the secondary

flow is subsonic [84]. Despite being thin, the shear layer acts as a barrier between the primary

and secondary flows, and no significant mixing occurs [84]. The mixing layer intrudes into

the secondary flow and accelerates it as it moves along the converging duct created by the

expanding primary flow [84]. At some point of this converging duct, referred to as

“hypothetical throat”, the secondary flow reaches sonic condition. Simultaneously, the

primary flow is decelerated [84]. The mixing layer thickens, starting the mixing effect [84].

Ejectors can be classified according to nozzle position, nozzle design, and number of

phases of the working fluid [40]. In terms of nozzle position, the most common ejector

configurations are designated as Constant Area Mixing (CAM) and Constant Pressure

Mixing (CPM) [40], as shown in Figure 15. The nozzle exit in CAM ejectors is placed in the

constant-area section, where the primary and secondary flows are mixed [40, 41, 85]. In CPM

ejectors the nozzle exit is placed in the suction section and the fluids are mixed in the mixing

section at constant pressure [40, 41, 85]. Despite providing inferior mass flow rates than

CAM ejectors, CPM ejectors have been reported to have better performance [41, 85, 86].

CPM ejectors can operate under higher backpressures and are the most common ejector

design [40]. In an attempt to combine the positive aspects of both CAM and CPM ejectors,

and reduce the irreversibility caused by the shockwaves, Eames [87] proposed a novel nozzle

configuration called the Constant Rate of Momentum-Change (CRMC) ejector. In this

design, a variable area section rather than a constant-area section is used to provide optimal

flow area and increase ejector efficiency [40].

Figure 15 – Ejectors with different nozzle positions: constant area mixing (left) and constant pressure mixing

(right) ejectors [60].

Nozzle design affects the flow inside an ejector. If the nozzle has a convergent shape,

the ejector works in a subsonic regime and the primary flow is, at most, sonic at the exit

section of the suction chamber [40]. Subsonic ejectors are used to provide a small

compression of the secondary flow, but they must also provide a small pressure loss for the

primary flow [40]. On the other hand, supersonic ejectors are used when a high pressure

difference is needed [40]. The primary flow reaches the supersonic regime, thus creating a

low-pressure zone at the nozzle exit that entrains a high mass flow rate of secondary flow

[40]. Figure 16 shows the three working modes of a supersonic ejector as a function of

backpressure for constant inlet pressures. In critical mode, the entrainment ratio is constant

because both the primary and secondary flows are choked [40, 49]. Therefore, variations on

backpressure (downstream conditions) do not influence the mass flow rate inside the primary

nozzle throat and in the hypothetical throat [81, 88]. This latter is the cross section where the

secondary stream reaches the speed of sound. Critical backpressure marks the transition

between critical and subcritical modes and is the maximum backpressure that permits double

choking [49]. Critical backpressure (𝑝𝑐𝑟𝑖𝑡) may be seen as a performance indicator, as it

limits the on-design operation range of the ejector [80]. When operating at the critical point,

the ejector is at its optimal performance [38, 49] and the shockwave occurs at the inlet section



of the constant-area section [49]. As backpressure further increases, the ejector transitions

into subcritical mode and the shock on the primary jet occurs before the secondary flow could

reach choking [49, 88], interfering with the mixing process [85]. Thus, only the primary flow

is choked, and the entrainment ratio depends on the difference between the secondary inlet

pressure and backpressure [40, 49]. For very high values of the backpressure (above 𝑝𝑚𝑎𝑥),

the ejector malfunctions and the secondary flow is reversed (backflow) [40, 49, 85].

Figure 16 – Working modes of a supersonic ejector as a function of backpressure for constant inlet pressure

[40].

Ejectors may also be classified according to the number of phases of the working

fluid inside the device. They can either be single phase, if both the primary and the secondary

flows are gases or liquids, or two-phase [40]. Research on single phase ejectors is vast [40].

Within two-phase ejectors, there is further distinction between condensing ejectors

(condensation of the primary flow occurs) and two-phase ejectors (the flow at the outlet is

two-phase) [40]. Two-phase ejectors are an interesting solution for the low performance of

transcritical carbon dioxide cycles [44]. They are low-cost devices with long self-life, and

suitable for two-phase flows [44]. The absence of moving parts provides them with great

reliability [41, 44]. For these reasons, Elbel and Lawrence [67] suggested the use of ejector

expansion devices in HVAC units. However, because of the unique critical point properties

of carbon dioxide (low critical temperature and high critical pressure), they require a robust

construction and have a higher initial cost when applied in transcritical systems [80].

Moreover, the understanding of the complex flow inside a two-phase ejector is still limited

[40].

Ejector performance

The assessment of a two-phase ejector performance is typically based on the

following indicators: the mass entrainment ratio, the compression ratio, the pressure lift, and

the ejector isentropic efficiency [44]. The mass entrainment ratio (𝜔) is the ratio of the

secondary mass flow rate (��𝑠) to the motive mass flow rate (��𝑝) [26, 40, 41, 44, 49, 56, 60,

85] as:

𝜔 =��𝑠

��𝑝 (1)

The higher the mass flow rate, the better the efficiency of the cycle. The compression ratio

(𝛱) is defined as the ratio of the ejector outlet pressure (𝑝𝑜) to the secondary inlet pressure

(𝑝𝑠) [40, 44, 49, 56, 60]. The pressure lift (𝑝𝑙𝑖𝑓𝑡) is the difference between the outlet and the



secondary inlet pressures [44]. These indicators limit the maximum temperature for heat

rejection [41, 85], therefore defining the operative range of the cycle [40]. The compression

ratio and the pressure lift are defined according to the following equations:

𝛱 =𝑝𝑜

𝑝𝑠 (2)

𝑝𝑙𝑖𝑓𝑡 = 𝑝𝑜 − 𝑝𝑠 (3)

Finally, the ejector isentropic efficiency (𝜂𝑒𝑗𝑒𝑐𝑡𝑜𝑟) is the ratio between the work rate

recovered in the ejector (��𝑟) and the maximum work rate recovery potential for an isentropic

expansion (��𝑟,𝑚𝑎𝑥) [44]. The ejector isentropic efficiency is defined as:

𝜂𝑒𝑗𝑒𝑐𝑡𝑜𝑟 =��𝑟

��𝑟,𝑚𝑎𝑥= 𝜔

ℎ(𝑃7,𝑠5)−ℎ5

ℎ3−ℎ(𝑃7,𝑠3) (4)

where the thermodynamic states refer to Figure 12. Ejector efficiency is normally inferior to

0.2 when the working fluid is either R404 or R134a [44]. However, the use of carbon dioxide

allows for efficiencies ranging between 0.2 and 0.4 [44]. Apart from this higher ejector

efficiency, the thermodynamic properties of carbon dioxide favour its use in expansion work-

recovery devices [44]. However, the flow inside a carbon dioxide ejector is very complex

and highly dependent on its geometry [44]. The geometry of these ejectors should be

specifically designed for the desired range of application, which is not a simple task. There

are no guidelines available in literature that could be easily followed. For optimal ejector

performance, the above-mentioned parameters should be as high as possible [60, 85].

Increasing the entrainment ratio results in a lower mass flow in the compressor for a given

cooling capacity [60]. A higher pressure ratio results in a lower compression ratio in the

compressor, therefore a lower compression work [60]. The efficiency of the ejector increases

with both the entrainment ratio and the compression ratio. There are limitations to the

increase of the entrainment ratio [60]. An excessive entrainment ratio means a low mass flow

rate of the primary flow, the primary flow loses its ability to compress the secondary flow

[60] and the critical backpressure reduces.

The expansion angle of the primary flow dictates the area of the hypothetical throat

of the secondary flow, also known as the effective area of the ejector. Figure 17 shows the

effect of the expansion angle of the primary flow on the effective area. For a given geometry,

the effective area varies inversely with the expansion angle of the primary flow. To ensure

critical operation, the effective area must be equal to or smaller than the critical area of the

secondary flow for it to reach sonic condition. However, if the effective area is much smaller

than the critical area of the secondary flow, the entrainment ratio decreases. For any set of

operating conditions, the effective area should be as close to the critical area as possible to

maximize the entrainment ratio.



Figure 17 – Effect of the expansion angle of the primary flow on the effective area of the ejector: greater

effective area for smaller expansion angle (a) and smaller effective area for greater expansion angle (b) [89].

The performance of a fixed geometry ejector is a function of the inlet (primary and

secondary) and outlet conditions [81]. Figure 18a shows the effect of both the primary inlet

pressure and backpressure on the entrainment ratio for fixed secondary inlet conditions.

Decreasing primary pressure results in a smaller primary jet area in the mixing section and

therefore a larger effective area. Given that the secondary flow chokes, its mass flow rate is

higher and so is the entrainment ratio [81]. However, a lower backpressure is required for the

secondary flow to choke because of the larger effective area, leading to a reduced critical

backpressure [81]. In contrast, a higher pressure of the primary flow compromises the

entrainment effect. The primary flow reaches the nozzle exit plain in a highly under-

expanded state and therefore the effective area becomes smaller and the entrainment ratio

decreases [81]. The smaller effective area allows the secondary flow to reach sonic condition

for higher backpressures [81]. Figure 18b shows the effect of both secondary flow pressure

and primary flow pressure on ejector performance for a fixed backpressure. For a fixed

secondary pressure, there is an optimal value of primary pressure (𝑝𝑝) that leads to the

maximum entrainment ratio [81, 90]. As 𝑝𝑝 increases, the area of the primary jet core also

increases, resulting in a smaller effective area. Initially, the entrainment ratio increases but

the secondary flow is not choked. When the effective area reaches the critical area of the

secondary flow, the entrainment ratio is maximum. A further increase of the primary pressure

results in an effective area smaller than the critical area, which compromises the entrainment

ratio as the entrained mass flow rate decreases. A higher secondary flow pressure ensures

sonic condition in a larger effective area, resulting in a lower optimal primary pressure [81].

Figure 18 – Entrainment ratio for fixed inlet conditions of the secondary flow (a) and fixed backpressure (b)

[81].



Most relevant geometric parameters of an ejector

Ejector geometry is described by flow path diameters and converging and diverging

half angles, and they should be given a proper design. The main geometric parameters that

affect the performance of a transcritical carbon dioxide ejector are the area ratio, the primary

nozzle exit diameter, the nozzle exit position, the convergence angle of the mixing section,

the length of the constant-area section, and the divergence angle of the diffuser [91]. The area

ratio (𝐴𝑅) of an ejector is the ratio between the area of the constant-area section (𝐴𝑐𝑎) and

the area of the primary nozzle throat (𝐴𝑝𝑡) according to the following formula:

𝐴𝑅 =𝐴𝑐𝑎

𝐴𝑝𝑡 (5)

Figure 19 shows the effect of the area ratio on the performance of the ejector. For a given

backpressure (𝑝𝑜∗) and when the secondary flow still reaches the sonic condition, the

entrainment ratio increases with an increase in the area ratio because of the larger

hypothetical throat area. However, a lower critical backpressure is observed [92]. Therefore,

the area ratio has a dual effect on ejector performance. On the one hand, a high area ratio

allows for a higher entrainment ratio but requires a higher primary flow pressure to ensure

double choking [90, 92]. On the other hand, an ejector with a low area ratio may operate on

a wider range of primary inlet pressures but with a lower entrainment ratio [90, 92].

Consequently, there is an optimal value of area ratio that maximizes the entrainment ratio

depending on the operating conditions [90, 92]. Using numerical analysis, Wang et al. [90]

observed a linear dependence of the optimal area ratio on the primary pressure. The effective

area for the secondary stream reduces with primary inlet pressure because of the increased

expansion of the primary jet as it leaves the nozzle section. A linear relationship between

optimal area ratio and primary pressure was also reported by Yan et al. [92] based on

experimental data.

Figure 19 – Effect of area ratio on entrainment ratio and critical backpressure.

The geometry of the primary nozzle has a significant impact on the ejector flow

because it induces the supersonic jet downstream to the nozzle exit plane. Therefore,

optimizing the nozzle shape may lead to significant improvements on ejector performance.

Fu et al. [93] analysed the influence of primary nozzle exit diameter and divergence angle on

ejector performance. The expansion of the primary flow in the divergent section of the

primary nozzle depends on the area ratio between the exit and the throat sections. For a small

exit diameter, the primary flow is not fully expanded and its velocity at the exit section is too

low. Consequently, the pressure at the exit section is not sufficiently low to draw the

secondary flow into the suction chamber, and backflow or very low entrainment occurs. For

an adequate expansion of the primary flow, a minimum diameter for the primary nozzle exit



is required. In this case, the ejector operates in critical mode, providing a high, constant

entrainment ratio. Further increasing the primary nozzle exit diameter means that the

expanded primary flow occupies a higher section area in the suction chamber. As a result,

the effective area for the secondary flow is reduced, and its mass flow rate decreases. The

divergence angle of the primary nozzle also affects ejector performance. Increasing the

divergence angle results in a shorter divergent section, reducing pressure loss due to friction

between the primary flow and the nozzle walls. However, large angles may cause boundary

layer separation, which reduces the isentropic efficiency of the ejector. Because of the small

lengths of the diverging section, the influence of this geometric parameter is rather

insignificant, and an ejector may operate with acceptable performance on a wide range of

divergence angles.

The nozzle exit position (NXP) refers to how the primary nozzle exit is positioned

relative to the inlet of the mixing chamber. For a null NXP, the nozzle exit section is perfectly

aligned with the inlet of the mixing chamber. As the NXP decreases, the motive nozzle moves

backwards and away from the constant-area section. Contrarily, a positive NXP means that

the exit section of the primary nozzle is positioned within the mixing chamber and closer to

the constant-area section. Figure 20 shows the effect of the NXP on the entrainment ratio. As

the NXP increases, the entrainment ratio also increases until reaching the optimal (maximum)

value [48, 90, 94]. Pianthong et al. [86] and Chunnanond and Aphornratana [89] related the

increase in the NXP to a stronger compression effect on the expanded primary flow, leading

to a smaller expansion angle and a larger hypothetical throat of the secondary flow. Further

increasing the NXP leads to a lower entrainment ratio and a worse ejector performance [48,

94] because of a loss of momentum in the primary flow [86]. The optimal NXP decreases

linearly with the increase in primary flow pressure. The maximum entrainment ratio obtained

with the optimal NXP decreases with the increase in primary flow pressure [90]. Chen et al.

[48] also observed that there is an optimal NXP value that maximizes critical backpressure

and therefore broadens the application range of the ejector. However, the influence of NXP

on critical backpressure is slim [48].

Figure 20 – Effect of nozzle exit position on entrainment ratio for different values of primary pressure.

The convergence angle of the mixing section affects the effective area of the ejector

and the secondary flow. Increasing the angle places the ejector walls further away from the

expanded primary flow, resulting in a larger effective area. Initially, this increases the

entrainment mass flow rate. However, an excessive convergence angle leads to a low speed,

high pressure secondary flow. The resulting adverse pressure gradients cause separation of

the boundary layer and backflow [95]. Therefore, there is an optimal value for this parameter



that maximizes the entrainment ratio. This value is dependent on ejector type, working fluid

and operating conditions.

The length of the constant-area section of the ejector has a significant impact on the

critical backpressure, and therefore influences the entrainment ratio. For a given geometry,

increasing this length initially leads to an increase in the entrainment ratio until its maximum

value [48, 95]. In this range, the mixing process has less energy losses and the normal shocks

in the diffuser eventually weaken [95]. If the length of the constant-area section is increased

beyond its optimal value, the entrainment ratio decreases [48, 95] because the secondary flow

is not choked.

For low divergence angles of the diffuser, the pressure drop due to friction with the

ejector walls increases. On the other hand, a high divergence angle creates a strong adverse

pressure gradient in the diffuser, promoting boundary layer separation and compromising

ejector efficiency. An optimal full angle of 5º has been reported for transcritical carbon

dioxide ejectors [68, 96].

Variable geometry ejector concept

Ejectors are designed to operate in critical mode, as this allows for higher entrainment

ratio and efficiency. Moreover, the backpressure should be as high as possible to maximize

the compression ratio within critical operation. This leads to a higher suction pressure in the

compressor, reducing the necessary compression work and improving the efficiency of the

cycle. The design of a fixed geometry ejector is optimized for a specific set of operating

conditions, according to the properties of the working fluid and desired cycle capacity [80].

However, it is not always possible to ensure critical mode operation, and an ejector may

operate in off-design conditions. Off-design operation of a fixed geometry ejector occurs

when at least one of the inlet conditions (of the primary or secondary flows) or the

backpressure is altered [81]. For a fixed geometry ejector, restoring critical operation requires

that at least one of the boundary conditions be altered [81]. When the ejector operates at off-

design conditions, for example, in subcritical mode, its entrainment ratio decreases

significantly [81]. For this reason, a fixed geometry ejector is expected to operate with

maximum efficiency only under a narrow range of operating conditions [80]. Performance is

compromised as operating conditions deviate from design values, which is considered to be

one of the main drawbacks of ejector refrigeration systems [90].

A possible approach to this problem is the use of multi-ejectors [97]. In this case, at

any given point only a set of the available ejectors are turned on, allowing for a step

adjustment of the high-side pressure while maintaining efficient operation [97]. However,

for each combination of active ejectors, they shall still operate under optimal conditions [97].

Another solution is to use a variable geometry ejector (VGE). A VGE can adapt to variable

operating conditions [26], which is not possible with a fixed geometry ejector. The benefits

of using a variable geometry ejector are increasingly significant as the operating conditions

deviate from the design conditions [26]. Yan et al. [48, 92] investigated on the effect of

different geometric parameters on the performance of an ejector and showed that the area

ratio and the nozzle exit position play the most determinant roles. For this reason, the

optimization of the geometry should focus on these two design parameters. Adjusting the AR

and the NXP allows for higher entrainment ratios on a wide range of operating conditions

[90, 98]. However, a variable AR has been shown to have a more significant effect than a

variable NXP [90]. Figure 21 shows the concept of a VGE with adjustable area ratio. In this



configuration, an adjustable needle inside the converging section of the primary nozzle is

used to regulate the effective area of the primary flow, therefore altering the area ratio of the

ejector. As the spindle moves upstream, the cross-section area of the primary nozzle throat

increases, and the area ratio decreases.

Figure 21 – Schematic of a variable geometry ejector with adjustable area ratio [90].

Figure 22 depicts the performance characteristics of a variable geometry ejector.

Suppose an ejector operates on its critical point with a backpressure 𝑝𝑜1 and an entrainment

ratio 𝜔1. If the backpressure increases (𝑝𝑜2), a fixed geometry ejector operates on subcritical

mode and the entrainment ratio is reduced to 𝜔∗. However, with a VGE it is possible to

reduce the area ratio and adjust its performance curve to obtain a higher entrainment ratio

(𝜔2). Considering a fixed geometry ejector working on its critical point (𝑝𝑜2, 𝜔2), the

entrainment ratio remains constant if the backpressure decreases (𝑝𝑜1). In contrast, the area

ratio of a VGE may be increased to maximize the entrainment ratio (𝜔1). A numerical model

developed by Varga et al. [98] showed that the adjustable spindle configuration allowed for

a significant performance improvement of a R600a ejector. Wang et al. [90] also reported a

significant effect of area ratio on the entrainment ratio.

Figure 22 – Performance characteristics of a variable geometry ejector.

2.5. Transcritical ejector models

Experimental work is the most reliable approach for geometry optimization and

remains the most common methodology [44]. However, this leads to an ejector geometry that

is only optimal for the specific conditions of the study [67]. This urges the need for

generalized studies on ejector geometry, directing the attention towards numerical

simulations. Experimentally validated numerical modelling is a versatile tool offering a

systematic approach to the study of ejector geometry. Ejector flow simulations may rely on

zero-dimensional (0-D), one-dimensional (1-D), two-dimensional (2-D), or three-

dimensional (3-D) approaches. All these models are based on equations for the conservation

of mass, energy, and momentum. Heat transfer between the flow and the ejector walls is



usually neglected [81]. To predict ejector performance, most models attempt to determine

global performance indicators such as the entrainment ratio. It is necessary to define the

adequate boundary conditions, i.e., temperature and pressure at the primary and secondary

nozzles, and at the outlet [44].

The simpler models are known as thermodynamic relation models or 0-D models, and

do not tackle the flow inside the ejector, but the global thermodynamic processes that take

place as the refrigerant flows from the inlet to the outlet. Because of their simplicity, the error

margin could be very substantial (10-15% error in predicting the motive mass flow rate [49]).

Moreover, their application range is limited to fixed geometry and on-design conditions and

the goodness of the empirical constants they rely on [44]. For a more detailed solution, 1-D

models are used to simulate the flow in one spatial direction. These models, complemented

by experimental data, are now capable of providing better predictions of two-phase ejector

flow [44]. Both 0-D and 1-D simulations have the advantage of requiring little computational

effort. The prediction error with these models is largely dependent of the validity of the

empirical constants applied [91]. However, these constants may vary from case to case,

meaning that considerable effort is needed to experimentally validate these models. In

addition, these models provide little insight into the complex flow inside the ejector, namely

the occurrence of oblique and normal shock waves, and shear mixing, needed for geometry

optimization [91].

Despite the advances on thermodynamic models and their ability to provide useful

results with little calculation effort, they are fundamentally uncapable of correctly

reproducing flow physics. For optimization purposes, the description of shock waves,

boundary layers, and mixing is necessary [82]. The challenge of modelling a two-phase

ejector flow also results of their sensitivity to the boundary conditions. Smolka et al. [99]

reported significant variations of the motive mass flow rate predicted by the model as a result

of changing the boundary conditions within the limits of the experimental uncertainty.

Therefore, improved models and high precision experimental data are required to produce

accurate predictions of ejector performance [44]. Current research on two-phase ejector flow

involves numerical simulation using Computational Fluid Dynamics (CFD) techniques.

These describe the flow inside the ejector more accurately, either in 2-D or 3-D, and therefore

have the potential to provide more fundamental predictions than the simpler models.

Moreover, these models depend less on experimentally determined constants and therefore

their validity is more general [44]. The 3-D models are used to obtain a detailed description

of local flow inside an ejector. However, the ejector has a strong axial symmetry, and so its

geometry may be modelled in only two dimensions. Depending on the design, local

asymmetry may occur but this usually has little effect when transferring the real 3-D

geometry into a simplified 2-D model [86].

CFD techniques may use the real properties of CO2, which should provide

significantly more accurate results [86]. Figure 23 shows the classification of current carbon

dioxide ejector models based on a CFD approach [44]. Because of the transcritical nature of

these devices, all developed approaches apply multiphase models. Within these, two fluid

models consider two distinct fluids, and one set of equations is solved for each. On the other

hand, pseudo-fluid models solve a single set of equations by averaging the properties of the

wet vapor.



Figure 23 – Classification of current two-phase carbon dioxide ejector models.

Mathematical models derived from the pseudo-fluid approach may be further

classified according to whether equilibrium of phases is considered in the flow. A common

assumption is that the flow is homogeneous, meaning that the different phases have identical

velocity and pressure fields. This simplifies the model as it allows for the treatment of the

different phases as a single pseudo-fluid with thermodynamic properties obtained by

averaging the properties of the individual phases according to their fractions. One of the

benefits of the homogeneous flow approach is the ability to define the thermodynamic state,

and thus the thermodynamic properties in the phase change region, through just the pressure

and the enthalpy. Using density-energy formulations necessarily leads to the conservation of

both mass and energy, thus producing more accurate thermodynamic relations [44]. The

Homogeneous Equilibrium Model (HEM) determines the phase fractions considering

thermodynamic equilibrium between the different phases. This approach accurately predicted

motive and suction mass flow rates in a transcritical carbon dioxide heat pump, for operation

near the critical point. The average deviation from experimental data was 5.6% for motive

mass flow rate and 10.1% for suction mass flow rate [99].

However, assuming thermodynamic equilibrium limits the application range of the

HEM approach [100]. Because of the rapid depressurization of the motive flow, the saturation

temperature drops below the liquid temperature and the liquid becomes superheated. This

forces the liquid to evaporate until equilibrium is reached. The upper limit of superheating is

imposed by the homogeneous nucleation line; therefore, any perturbation causes instant

phase change when the liquid is further superheated. This phenomenon is demonstrated in

Figure 24. For a near-critical expansion (1), phase change occurs rapidly, and non-

equilibrium may be neglected. However, as the expansion of the motive flow occurs further

away from the critical point (off-critical expansion, 2), the homogeneous nucleation line and

the saturation line diverge, allowing for a greater level of superheating. The resulting non-

equilibrium becomes increasingly significant [44]. To model the non-equilibrium phase

change, the Homogeneous Relaxation Model (HRM) treats the phase change as a relaxation

process towards the equilibrium vapor quality, rather than as an instantaneous process. By

introducing a relaxation time, the onset of phase-change is delayed. However, the advantage

of this model is dependent on a correct estimation of the relaxation time [44]. Moreover,

assuming a constant relaxation time limits the accuracy of the HRM in some operating

conditions [101]. The HRM shows better accuracy than the HEM in off-critical conditions,

but is outperformed in supercritical conditions [101]. In this context, a variable relaxation

time was proposed by Haida et al. [102]. The resulting HRM performed well in both near-

critical and off-critical operating conditions.



Figure 24 – Pressure-enthalpy diagram of carbon dioxide showing saturation lines (blue and orange),

homogeneous nucleation lines (green and red), and expansion lines (pink – near-critical expansion (1) and off-critical expansion (2)) [44].

Mixture models treat the phase change mechanism by explicitly including terms for

condensation and evaporation in the governing equations, thus the estimation of the phase

fraction is more accurate. Giacomelli et al. [103] reported a significantly better performance

of the mixture model when compared to the HEM. The HEM predicted a flow pattern with

evident discontinuities in density, whereas the mixture model produces a smooth evolution

of density. However, the mixture model presented numerical instabilities and slow

convergence, impeding its use for geometry optimization.

The HEM is a more restrictive approach to two-phase flow modelling as it only allows

for saturated conditions of vapor and liquid. On the other hand, the mixture model includes

both saturated and meta-stable conditions of vapor and liquid. An intermediate approach is

the Delayed Equilibrium Model (DEM), which considers only saturated vapor but models

the liquid phase as a combination of saturated and meta-stable liquid. As discussed by

Bartosiewicz and Seynhaeve [104], it is assumed that only a fraction of the liquid is

superheated, while the remaining liquid is in saturated conditions. The DEM has shown lower

accuracy than the HEM in predicting the pressure field in a converging-diverging nozzle

[105].

All previously discussed models assume homogeneous flow. The drift flux model

considers different velocities of the liquid and vapor phases (momentum non-equilibrium).

This model reveals pressure waves at the primary nozzle exit, which are smoothed by the

homogeneous flow models. However, the drift flux model was found to have little influence

on the ejector performance [106]. Research on this model is still limited [44].

It is possible to analyse thermal non-equilibrium in a two-phase ejector flow. If the

two phases are at different temperatures, heat transfer between them should be considered.

Pressure non-equilibrium is also possible but it is often neglected because of its short time

scale [44]. The two-fluid model (TFM) considers the vapor and liquid phases as separate

fluids. From this approach results twice as many governing equations which need to be solved

simultaneously. Phase change and phase slip are directly modelled, and non-equilibria

between the phases is explicitly captured. Despite requiring less sub-modelling, the TFM still

requires modelling of the interaction between phases. Moreover, its complexity requires

more accurate experimental data for validation, which is not yet available [44].



Table 3 summarizes the most relevant characteristics of CFD approaches found in the

literature to simulate transcritical ejector flow with their benefits, challenges, and limitations.

Table 3 – Overview of the considered non-equilibria, benefits, and challenges and limitations of currently

available carbon dioxide two-phase ejector models [44]

Model Non-

equilibrium Benefits Challenges/limitations

HEM None

• Simplicity and stability

• Accurate at supercritical

conditions

• Extensively tested in

literature

• Does not consider meta-

stability

HRM Chemical

• Considers meta-stability

• Extended with variable

relaxation time for

subcritical conditions

• Empirically based parameters

for relaxation time

• Requires tuning of

parameters

Mixture Chemical

• Considers meta-stability

• Can more accurately

evaluate the phase

fractions by mass transfer

modelling

• Highly accurate results

for motive flows

• Increased complexity

• Requires tuning of model

parameters

• Less profound literature

database on carbon dioxide

ejectors

• Not yet tested for low motive

pressures



3. Development of the CFD model 3.1. Assumptions

As referred in Chapter 2, CFD techniques may be used to model ejector flow in two

or three dimensions. 3-D modelling provides a deeper insight into flow physics, namely the

occurrence of shockwaves, the development of the shear mixing layer, and turbulence.

However, this level of detail comes at a considerably higher computational cost. The

governing equations become more complex as they consider mass and energy transfer in an

additional direction. Moreover, the computational domain requires a larger number of

elements for spatial discretization, further extending computing time. A time-effective

solution is to take advantage of the rotational symmetry of the ejector, and to model the flow

as axisymmetric. Using this 2-D approach has been shown to lead to similar simulation errors

when compared to a full, 3-D simulation [86]. However, the axial symmetry of the ejector is

not perfect. In fact, the secondary nozzle inlet is typically placed on the side of the nozzle

section, breaking this symmetry. Bartosiewicz et al. [99] conducted a 3-D simulation and

reported some asymmetries for the local pressure distribution in the mixing section and in

the diffuser. Mazzelli et al. [107] also reported that 3-D modelling was necessary for off-

design operation, while 2-D models provided accurate results for on-design operation. After

weighing the benefits and limitations of each approach, a 2-D axisymmetric approach was

selected. A substantial number of simulations is necessary to analyse the influence of ejector

geometry on its performance, therefore a short computational time is an essential constraint.

Pressure losses are another relevant aspect to analyse in the heat pump cycle. In fact,

due to friction between the fluid and the pipe walls, the static pressure drops as the fluid

travels between the different system components. Therefore, the primary flow enters the

ejector at a lower pressure than it leaves the gas cooler. Similarly, the inlet pressure of the

secondary flow is lower than the evaporator pressure. Lastly, there is a pressure drop as the

CO2 leaves the ejector and flows into the separator. These pressure losses depend on the

length and diameter of the pipes and fittings. They are usually negligible when compared to

the static pressure in the cycle. Moreover, the inlet and outlet velocities are typically small,

and so the refrigerant carries a low level of kinetic energy. Therefore, the difference between

the static and stagnation pressures is negligible. It is reasonable to assume that the stagnation

pressure of the primary flow equals the gas cooler pressure. Accordingly, the stagnation

pressure of the secondary flow is assumed equal to the evaporator pressure; and finally, the

stagnation pressure of the outgoing flow is equal to the pressure in the separator.

The physical properties of a fluid are typically divided into thermophysical and

transport properties. The thermophysical properties include density, specific heat, vapor

volume fraction, and temperature. The transport properties include kinematic viscosity and

thermal conductivity. Two independent thermodynamic properties are necessary to

characterize the thermodynamic state of the CO2, and therefore its properties. For example,

pressure and temperature define the thermodynamic state of a substance in the liquid or vapor

states. In the phase-change region, enthalpy should be used instead of temperature. There are

several approaches to obtaining the properties of the fluid along the flow path. One possibility

is to use property tables and interpolation when the known independent properties do not

match the tabulated values. A second approach is based on the use of equations of state.

These have the advantage of requiring less calculation time, while maintaining considerable

precision. Equations of state with varying levels of precision have already been developed

for carbon dioxide. A third approach is to model the properties of the fluid based on



theoretical assumptions. Modelling the fluid as an ideal-gas, for example, significantly

reduces the calculation time necessary to obtain the desired properties. However, considering

ideal properties for the CO2 flow in ejectors provides only a vague approach since it does not

capture the phase-change process during the transcritical expansion. In the first section of

this work, the CO2 is modelled as an ideal-gas to study the effectiveness of an adjustable

spindle on the optimization of ejector geometry for varying operating conditions. In a

subsequent analysis, the primary nozzle is simulated for different spindle positions. In this

context, the real-gas properties of carbon dioxide are interpolated from property libraries

extracted from EES®.

As described in the previous chapter, there is a multitude of models to describe the

properties of a two-phase flow. These vary in precision, and more accurate models usually

require higher computational effort and lead to longer simulation times. For this reason, they

are not suitable approaches for geometry optimization, in which case many simulations are

required for tuning of the different geometric parameters. The HEM is the simplest two-phase

model and should allow for rapid numerical convergence [44]. Although it loses accuracy in

off-critical conditions, it is accurate for modelling ejector operation near the critical point.

Giacomelli et al. [108, 109] concluded that the HEM is an efficient approach to modelling

the two-phase flow in carbon dioxide ejectors, producing reasonable results. However,

increased accuracy in describing flow physics requires that meta-stability be considered.

More complex models, such as the HRM, maintain reasonable accuracy even in off-critical

operating conditions, because they account for the effects of meta-stability. However, the

computational effort is significantly increased, leading to longer simulation times.

Considering the benefits and limitations of each model, the HEM was preferred in this work.

3.2. Ejector design

Effect of boundary conditions on optimal ejector design

The first step in studying the optimal geometry of a VGE is establishing a range of

realistic operating conditions, which means defining the boundary conditions at both inlets

and at the outlet. A parametric analysis was carried out regarding the operating conditions

and their impact on ejector dimensions and cycle performance. The 1-D EES® model

developed by Marques [110] was used. The temperature of the primary flow (𝑇𝑝) was varied

from 35ºC to 43ºC with a 2ºC step. For the Portuguese climate, this temperature range is

compatible with floor heating applications and allows for an efficient heat rejection if the

heat pump operates as an air conditioning system. Therefore, it is possible to simulate a heat

pump system operating in heating mode during the winter and in cooling mode during the

summer. Sawalha [58] studied the effect of gas cooler pressure on the cycle COP for different

gas cooler exit temperatures (see Figure 10). Based on this work and considering 𝑇𝑝 ranging

from 35ºC to 43ºC, the gas cooler pressure was set to 9.3 MPa. It was also necessary to define

the inlet conditions for the secondary flow coming from the evaporator. The temperature of

the secondary flow (𝑇𝑠) was varied from 0ºC to 8ºC with a 2ºC step. A 5ºC superheating was

considered, resulting in evaporation temperatures between -5ºC and 3ºC. This temperature

range is compatible with a heating application even during the winter and allows for the heat

pump to operate as an air conditioning system during the summer. Pressure and temperature

are dependent in the evaporator. The outlet pressure was defined via the compression ratio

of the ejector. Values of 1.1, 1.2, and 1.3 were analysed.



Figure 25 summarizes the effect of 𝑇𝑝, 𝑇𝑠, and 𝛱 on the diameter of the primary nozzle

throat (𝐷𝑝𝑡), the entrainment ratio, and COP. An increase in 𝑇𝑝 means a lower temperature

glide in the gas cooler, therefore a higher primary mass flow rate is necessary to ensure the

heating effect. Accordingly, a higher primary throat diameter is required to assure a larger

mass flow rate and the compression work increases. The primary flow enters the ejector with

a higher enthalpy because of its higher temperature, also leading to a higher enthalpy at the

outlet. The higher vapor quality means that less fluid is directed to the evaporator, i.e., the

secondary mass flow rate decreases. As a result, the entrainment ratio decreases, and so does

the performance (COP) of the heat pump cycle. Increasing 𝑇𝑠 results in a higher evaporation

pressure and a higher inlet pressure in the compressor. A higher primary mass flow rate

(higher 𝐷𝑝𝑡) is necessary because the refrigerant leaves the compressor with a lower enthalpy.

The primary mass flow rate increases, but because the saturated liquid at the ejector outlet

enters the evaporator with a higher enthalpy, a higher secondary mass flow rate is necessary

to ensure heat absorption. For a small increase in 𝑇𝑠, a higher secondary mass flow rate is

also necessary because the CO2 enters the evaporator with a lower enthalpy. Therefore, the

entrainment ratio increases. The effect of the reduced pressure lift in the compressor

supplants the higher mass flow rate and the compression work decreases. Consequently, the

performance of the cycle is boosted and the COP increases with 𝑇𝑠. However, further

increasing 𝑇𝑠 means that the increase in primary mass flow rate becomes more significant

than the decrease in compression ratio, and the compression work increases. The amount of

heat absorbed in the evaporator decreases, leading to a lower secondary mass flow rate. The

entrainment ratio decreases rapidly, along with COP. The maximum values of entrainment

ratio and COP are functions of 𝑇𝑠. An increase in 𝛱 for fixed evaporator and gas cooler

pressures has a similar effect to an increase in 𝑇𝑠. A higher primary mass flow rate is

necessary due to the lower enthalpy of the fluid after compression, and so 𝐷𝑝𝑡 increases. On

the other hand, a higher fraction of the pressure lift between the evaporator and the gas cooler

is ensured by the ejector rather than by the compressor. The optimal values of 𝛱, maximizing

the entrainment ratio and COP of the cycle, inter-depend on the other operating conditions.

Figure 25 – Effect of Tp, Ts, and Π on the diameter of the primary throat, the entrainment ratio, and COP.

Another criterion for the dimensioning of the VGE was set according to construction

constraints. The high volumetric heating/cooling capacity of CO2 results in a very compact

system [39, 44, 52], which means that the dimensions of the ejector are very small. Precise

machining of small dimensions is a complex and expensive task because of the tight

dimensional tolerances, thus excessively small diameters and lengths should be avoided.

Moreover, it is necessary to ensure that the VGE operates well on the range of operating

conditions considered. For varying inlet and outlet conditions, the area ratio of the ejector

may be regulated but it has a minimum value. This lower limit is defined by fixed geometric

parameters, such as the diameters of the primary nozzle throat and the constant-area section.

For a given ejector geometry, it is not possible to decrease the area ratio past this threshold.

Therefore, the diameter of the primary throat must be equal to the value determined from its



design operating conditions. For different operating conditions, i.e., requiring a higher area

ratio, the adjustable spindle inside the primary nozzle is used to adjust (decrease) the area of

the primary nozzle throat.

The dimension of the ejector depends on the primary and secondary mass flow rates,

which are influenced by the operating conditions and capacity of the cycle. As the heating

effect increases, the primary mass flow rate also increases to ensure the necessary heat

rejection in the gas cooler. Additional heat absorption is required in the evaporator to

maintain steady-state operation. Therefore, the required cross-section area for the different

ejector sections increases linearly with the heating capacity of the cycle. To address the

challenges of manufacturing a very small ejector, it is possible to simply increase the cycle

capacity. However, this also increases the cost of the heat pump, as well as the dimension of

its other components, therefore limiting the applicability of the system. Considering the

compromise between heating capacity and production constraints, the heating nominal target

power was set to 20 kW.

The benefit of integrating an ejector into a conventional transcritical CO2 cycle lays

in the possibility of improving its performance. Therefore, it was also necessary to analyse

the COP of the transcritical ejector cycle for the different operating conditions. Considering

typical values of transcritical CO2 heat pumps, the scope of the analysis was limited to values

of COP equal to or higher than 2. Moreover, to justify the increased cost and complexity of

a cycle equipped with an ejector, a significant COP improvement should be verified when

compared to a conventional transcritical cycle.

Fixed geometry ejector

Table 4 shows the operating conditions selected for the design of the ejector, based

on the criteria established in Section 3.2.1.

Table 4 – Operating conditions selected for dimensioning the ejector

Primary inlet Secondary inlet Outlet Heating capacity

𝑻𝒑 [ºC] 43 𝑻𝒔 [ºC] 8 𝜫 [-] 1.3 20 kW

𝑷𝒑 [MPa] 9.30 𝑷𝒔 [MPa] 3.77

A number of geometric factors were defined according to the work by Banasiak and Hafner

[111]. Because of the different mass flow rates and therefore overall dimension of the ejector,

these factors were the divergence/convergence angles and relative lengths, which should

allow for an adequate scaling of the geometry. The divergence angle of the motive nozzle

was set to 1º. The authors propose a value of 21º for the convergence angle of the mixing

section, similarly to Taleghani et al. [49] who propose a value of 20º. However, to reduce the

risk of boundary layer separation, under variable operating conditions, in the transition zone

to the constant-area section, this value was reduced to 15º. The length of the constant-area

section was defined as 8 times the respective diameter. This is in agreement with Taleghani

et al. [49], who analysed a constant-area section length of 7.344 times its diameter. The

divergence angle of the diffuser was set to its optimal value of 5º, as previously referred [68,

96]. To define the diffuser exit section, an outlet velocity of 8 m/s was considered. This value

is based on experimental results by Smolka et al. [99]. Using the previously defined operating

conditions as input, this model was employed as a first approach to the geometry of the VGE.

Figure 26 shows the final geometry selected for the VGE, based on the operating conditions

defined in Table 4. All dimensions in the figure are indicated in millimetres.



Figure 26 – Geometry of the ejector (dimensions in millimetres).

The mathematical model used also predicts the location where the secondary stream reaches

sonic speed, as signalled in Figure 27 (red dashed line). This figure also depicts the expansion

lines of the primary flow (blue) and the diameter of the expanded primary jet at the double

choking section.

Figure 27 – Location of secondary jet choking withing the mixing section (red) and expansion lines of the

primary flow (blue).

Variable geometry ejector

In the previous section, the mathematical model by Marques [110] was used to

estimate the optimal geometry of a fixed geometry ejector, given a set of operating

conditions. This serves as the basis for the definition of the geometry of the VGE, although

some geometry modifications are necessary. A spindle was introduced through the primary

inlet to adjust the free cross-section area in the primary throat and, therefore, the area ratio

of the ejector. The dimensions of the spindle (in millimetres) are shown in Figure 28.

Figure 28 – Design of the VGE spindle.

To ensure that the primary flow is choked near the primary nozzle throat as opposed to the

convergent section of the primary nozzle, the convergence angle of the primary inlet must be

greater than the angle of the spindle tip. The latter was calculated to be 9.5⁰, so the former

was set at 15º (equal to the convergence angle of the mixing section). To reach the maximum



primary mass flow rate for any given set of boundary conditions, the spindle must be

positioned outside the primary throat to maximize its effective area. The spindle has a total

length of 40 mm, and a length of 46 mm was defined for the primary nozzle convergent

section, allowing for the spindle tip to be positioned upstream without reducing its effective

area. The secondary inlet was enlarged to ensure a sufficiently small inlet velocity and its

convergence angle was also set at 15º. The inner diameter of the secondary inlet was

increased by 1 mm to provide a more realistic ejector geometry and to prevent mesh cracking.

Mesh cracking may occur when the distance between two non-adjacent cell centres is lower

than the solver’s numerical precision. In this work, this problem arises near in the exit section

of the primary nozzle, since the nozzle wall is very thin. To allow for an adjustable NXP, the

divergent section of the primary nozzle was moved downstream, creating a short, constant-

area section that precedes the diverging section of the nozzle. After performing a set of

preliminary simulations, a recirculation zone was observed at the inlet of the primary nozzle

throat. In this region, the flow detaches from the wall as the ejector geometry transitions from

the primary convergent to the primary throat. This recirculation zone reduces the effective

area of the primary flow, resulting in a lower primary mass flow rate. Moreover, numerical

convergence is slower since the presence of the recirculation zone induces local flow

instability. For a faster convergence, a rounded inlet in the primary throat was defined for all

subsequent meshes. The final geometry of the ejector is shown in Figure 29.

Figure 29 – Design detail of the variable geometry ejector (primary and secondary inlets).

Figure 30 shows how the spindle position (SP) and the primary nozzle exit position (NXP)

are measured. A SP=0 corresponds to the case when the primary flow is completely blocked

by the spindle. As the SP increases, the spindle moves upstream and the restriction it imposes

on the primary flow decreases. Under constant operating conditions, the primary mass flow

rate is expected to increase until the effective area of the primary flow reaches its maximum.

Beyond this point, the mass flow rate remains constant. A NXP=0 corresponds to having the

primary nozzle exit perfectly aligned with the inlet of the mixing chamber. As the NXP

increases, the nozzle exit is shifted downstream; inversely, the nozzle moves upstream as the

NXP decreases.

Figure 30 – Spindle position (SP) and primary nozzle exit position (NXP).



Assuming stagnation conditions at both inlets and at the outlet is translated into the

following mathematical formulation:

𝑝0 = 𝑝 +1

2𝜌𝑢2 ≈ 𝑝 (6)

Given the estimated ejector geometry, it is then necessary to verify that the refrigerant has

negligible velocity at the inlets and at the outlet, otherwise it would carry substantial kinetic

energy and the assumption of stagnation conditions would be invalid. Table 5 shows, for both

inlets and the outlet, the values of mass flow rate (��) and density (𝜌) calculated by the 1-D

model used to estimate the geometry of the ejector. The cross-section area (𝐴) and velocity

(𝑢) are calculated based on the geometry obtained with EES® and the considerations

regarding stagnation conditions.

Table 5 – Mass flow rate, density, cross-section area, and velocity at both inlets and the outlet

Primary inlet Secondary inlet Outlet

�� [𝒌𝒈

𝒔] 0.2485 0.05958 0.3081

𝝆 [𝒌𝒈

𝒎𝟑] 435.3 100.7 187.1

𝑨 [𝒎𝒎𝟐] 520.6 53.37 205.9

𝒖 [𝒎

𝒔] 1.097 11.10 8.000

It was established that the dynamic pressure at the inlets and at the outlet should not exceed

1% of the respective static pressure. Table 6 shows the static and dynamic pressures for both

inlets and the outlet, as well as the respective ratio.

Table 6 – Static pressure, dynamic pressure, and respective ratio for both inlets and the outlet


𝒑 [𝑷𝒂] 9.300x106 3.770x106 4.901x106

𝟏

𝟐𝝆𝒖𝟐[𝑷𝒂] 261.7 6188 5986

𝟏𝟐 𝝆𝒖𝟐

𝒑 [%] < 0.01 0.09 0.02

The dynamic pressure of the flow at the inlets and at the outlet is negligible when compared

to the respective static pressure, and so the assumption of stagnation conditions is verified at

all boundaries. This verification is based on values of density and mass flow rates calculated

with the 1-D model, thus the following numerical simulations may produce different results.

However, given such low ratios between dynamic and static pressure, possible fluctuations

of these values should not exceed the established criterion.



3.3. Mathematical model for the CFD simulations

Governing equations

The governing equations of fluid flow, continuity and energy are the mathematical

formulation of the conservation laws in fluid dynamics. When applied to a fluid continuum,

these equations describe the change of a given flow property. Under steady-state conditions,

the law of conservation of mass states that the mass of a fluid element is constant so that the

sum of mass flow rates entering and leaving the element is null, therefore:

𝛻. (𝜌𝒖) = 0 (7)

The law of conservation of momentum, or Newton’s second law of motion, states that, for a

fluid particle, the rate of change of linear momentum is equal to the sum of the external forces

acting on the particle, according to:

𝛻. (𝜌𝒖𝒖) = −𝛻𝑝 + 𝛻. 𝝉 (8)

In this work, all external forces were neglected. 𝝉 is the viscous shear stress tensor and refers

to the viscous stresses acting on the fluid element. Each component 𝜏𝑖𝑗 is defined as follows

for steady-state flows:

𝜏𝑖𝑗 = 𝜇 [(𝜕𝑢𝑖

𝜕𝑥𝑗+

𝜕𝑢𝑗

𝜕𝑥𝑖) −

2

3𝛿𝑖𝑗

𝜕𝑢𝑘

𝜕𝑥𝑘 ] (9)

where 𝜇 is the dynamic viscosity of the fluid.

The law of conservation of energy, or 1st law of thermodynamics, states that, for a

fluid particle, the rate of change of energy is equal to the sum of the heat addition and the

work done on the particle. This equation may be formulated using temperature (see Section

4.1) or specific enthalpy (see Section 4.2) as the independent flow variable.

Turbulence

Reynolds number (𝑅𝑒) is a non-dimensional parameter that evaluates the ratio

between inertial and viscous effects in a flow. It is based on a characteristic dimension of the

flow and is calculated as follows for flows within circular ducts:

𝑅𝑒𝐷 =𝜌𝑢𝐷

𝜇 (10)

In equation 10, 𝐷 is the diameter of the duct. For a low Reynolds number, viscous friction

prevents turbulent movement, and the flow is laminar. However, when a fluid particle carries

excessive kinetic energy (high Reynolds number), the dampening effect of the fluid’s

viscosity is insufficient, and the flow becomes turbulent [112]. Turbulent flows are

characterized by a fluctuating velocity field and intensive mixing of both momentum and

energy, causing the transported properties to fluctuate as well.

A possible approach to modelling turbulence is to directly solve these fluctuations,

which requires high spatial and time resolutions. This is known as Direct Numerical



Simulation (DNS) and is used to establish reference results for standard flows. Although it

produces accurate results, DNS is very computationally expensive, therefore it is inadequate

for many practical engineering applications. Another solution for modelling turbulence is to

remove the need for solving the small turbulent scales, for example, through Reynolds

decomposition, resulting in a simpler equation system. This approach is less accurate than

DNS but may still provide satisfactory results in reasonable time. Reynolds decomposition

consists of separating the instantaneous velocity (𝑢) in its time-averaged (��) and fluctuating

(𝑢′) components, as shown in Figure 31.

Figure 31 – Reynolds decomposition of instantaneous velocity.

Introducing this formulation in the momentum equations leads to the Reynolds-averaged

Navier Stokes (RANS) equations, which are written as follows:

𝛻. (𝜌��) = −𝛻�� + 𝛻. (�� + 𝝉𝑻) (11)

These equations govern the transport of the averaged flow quantities (��), removing the need

to solve the small scales of turbulence. Instead, all scales of turbulence are modelled, which

significantly reduces the computational effort necessary to conduct the simulation. RANS

equations have a similar form as the instantaneous Navier-Stokes equations, differing in the

fact that the velocities and other field variables now represent time-averaged values. The

additional terms in equation 11, called Reynolds stresses (𝝉𝑻), account for the dissipative

nature of the fluctuating velocities and are defined as:

𝜏𝑇𝑖𝑗= −𝜌𝑢𝑖

´𝑢𝑗´ (12)

These new terms must be modelled to provide closure for the RANS equations [113]. The

classical closure for RANS is based on the Boussinesq hypothesis, which states that turbulent

diffusion of turbulent transport properties is proportional to the gradient of the transported

properties. The Boussinesq hypothesis for turbulence is written as [112]:

𝜏𝑇𝑖𝑗= −𝜌𝑢𝑖

´𝑢𝑗´ = 𝜇𝑇 [(

𝜕𝑢��

𝜕𝑥𝑗+

𝜕𝑢��

𝜕𝑥𝑖) −

2

3𝛿𝑖𝑗

𝜕𝑢𝑘

𝜕𝑥𝑘] −

2

3𝛿𝑖𝑗𝜌𝑘 (13)

This approach introduces the concept of turbulent viscosity (𝜇𝑇). The molecular viscosity (𝜇)

is a property of the fluid and determines the molecular dissipation of energy in a flow.

Analogously, the turbulent viscosity is responsible for turbulent dissipation on small scales.

However, this does not depend only on the properties of the fluid, but also on the

characteristics of the flow. There are several different models to determine the turbulent

viscosity. 𝑘 − 휀 turbulence models define it as follows [112]:



𝜇𝑇 = 𝜌𝐶𝜇

𝑘2

휀 (14)

where 𝐶𝜇 is a constant. The 𝑘 − 휀 models are two-equation models because they involve the

numerical solution of two transport equations, one for the turbulent kinetic energy (𝑘) and

another one for the dissipation rate of turbulent kinetic energy (휀). In the standard 𝑘 − 휀

model, these are defined as follows [113]:

𝛻. (𝜌𝒖𝑘) = 𝛻. [(𝜇 +𝜇𝑇

𝜎𝑘) 𝛻𝑘] + 𝐺𝑘 + 𝐺𝑏 − 𝜌휀 − 𝑌𝑀 (15)

𝛻. (𝜌𝒖휀) = 𝛻. [(𝜇 +𝜇𝑇

𝜎𝜀) 𝛻휀] + 𝐶𝜀1

휀

𝑘(𝐺𝑘 + 𝐶𝜀3

𝐺𝑏) − 𝐶𝜀2𝜌

휀2

𝑘 (16)

𝜎𝑘 and 𝜎𝜀 are the turbulent Prandtl numbers for 𝑘 and 휀, respectively. 𝐺𝑘 and 𝐺𝑏 are the

generation of 𝑘 due to the mean velocity gradients and buoyancy, respectively. 𝑌𝑀 is the

contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation

rate. 𝐶𝜀1, 𝐶𝜀2

and 𝐶𝜀3 are constants depending on the variant of the 𝑘 − 휀 model used. The

RNG 𝑘 − 휀 model is derived from the instantaneous Navier-Stokes equations using

renormalization group (RNG) techniques. It is based on the standard 𝑘 − 휀 model and

presents significant improvements. Firstly, the transport equation for 휀 has an additional term

which improves the accuracy for high local strain variations. Secondly, the effect of swirl on

turbulence is accounted for, which improves the accuracy for swirling flows. Moreover, an

analytical formula is used to determine turbulent Prandtl numbers, contrary to the user-

defined, constant values in the standard 𝑘 − 휀 model. Lastly, an analytically-derived

differential formula is used to determine effective viscosity, which accounts for low-

Reynolds-number effects. These improvements make the RNG 𝑘 − 휀 model more reliable,

and extend its applicability to a wider range of flows, when compared to the standard 𝑘 − 휀

model [113]. The Realizable 𝑘 − 휀 model is a relatively recent improvement of the standard

𝑘 − 휀 model. It contains a new formulation for the turbulent viscosity, and a new transport

equation for 휀 derived from an exact equation for the transport of the mean-square vorticity

fluctuation. The term “Realizable” means that this model satisfies certain mathematical

restrictions on the Reynolds stresses, resulting of the physics of turbulent flows (the RNG

𝑘 − 휀 model is not realizable). The Realizable 𝑘 − 휀 model is more accurate in modelling

planar and round jets, and may also outperform other models for flows involving rotation,

boundary layers under strong adverse pressure gradients, separation, and recirculation [113].

Both the RNG 𝑘 − 휀 and the Realizable 𝑘 − 휀 models have been reported to perform well for

ejector flows [99, 114-119].

Turbulence model predictions are affected by the presence of walls, near which the

effect of viscosity results in strong pressure and velocity gradients. An accurate description

of the near-wall flow is key to accurately simulating wall bounded turbulent flows. However,

many turbulence models (such as 𝑘 − 휀 models) are only valid for the modelling of fully

developed turbulence. Consequently, they are limited in describing the flow near walls,

where the local velocity is lower. A solution to this problem is to use wall functions. These

empirically derived equations are used to model the region of the flow adjacent to the walls

and require that the centre of the first numerically discretized control volume (CV) be placed

within the log-law region of the velocity profile to ensure accurate results. When using this

approach, it is not necessary to resolve the boundary layer, which would require a very high

number of CVs near the wall to adequately capture the strong gradients. The number of



elements in the mesh is reduced and simulations converge more quickly. The applicability of

wall functions is assessed by the dimensionless wall distance (𝑦+), defined as follows:

𝑦+ =𝜌𝑢∗𝑦

𝜇 (17)

𝑦 is the distance of the cell node to the wall. 𝑢∗ is known as shear velocity and is defined as

follows:

𝑢∗ = √𝜏𝑤

𝜌 (18)

𝜏𝑤 is the shear stress near the wall. 𝑦+ is usually calculated for the cells adjacent to the walls

of the calculation domain and may be interpreted as a local Reynolds number, therefore

defining the ratio between viscous and turbulent effects. For low values of 𝑦+, the flow within

the cell is laminar. In this case, a coarser mesh should be used to ensure that the centre of the

first cell is at a sufficient distance from the wall to fall within the log-law region. A high

value of 𝑦+ indicates a fully turbulent flow within the wall-adjacent cell, thus the wall is not

properly resolved. A finer mesh is necessary to better resolve the wall region and produce

more accurate results. Literature [120] recommends a value of 𝑦+ between 30 and 300. In

this work, the wall functions proposed by Launder and Spalding [121] were used. In

conjunction with these standard wall functions, scalable wall functions were set to force the

usage of the log law. Table 7 shows the selected parameters for the definition of the

turbulence model.

Table 7 – Selected parameters for the definition of the turbulence model

Type Parameter Selected

Turbulence Model RNG 𝑘 − 휀

𝐶𝜇 0.0845

𝐶𝜀1 1.42

𝐶𝜀2 1.68

Wall Prandtl number 0.85

Near-wall treatment Scalable wall functions

The finite volume method

Deriving an analytical solution of the governing equations of a flow (momentum,

continuity, and energy) is only possible for some simple, theoretical cases. These include

simple geometries and laminar flows, for which velocity and temperature profiles may be

determined. However, the flow geometries to be studied are more complex for practical

engineering applications. Moreover, the flow itself may show complex physics, such as

shockwaves, boundary layer separation, recirculation, and turbulence. As a result, it is either

impractical or impossible to reach an analytical solution of the governing equations. A

different approach is to divide the fluid continuum into small fluid elements, and to discretize

the governing equations for each element to calculate the desired flow properties using

numerical approximation. The discretization of the differential equations leads to a set of

simple, algebraic equations, which may be solved numerically using a variety of solving

methods. This procedure does not produce an exact solution, but it may lead to sufficiently

accurate results. The discretization of the fluid continuum, and consequently of the governing



equations, may be achieved through three different techniques. The simplest is the finite

difference method, but its usability is rather limited when dealing with complex geometries.

Moreover, it is not necessarily conservative, that is, it is not guaranteed that it respects the

laws of conservation of mass, momentum, and energy. The other two approaches, finite

element and finite volume techniques, are suitable for complex flow geometries, and they are

intrinsically conservative. However, the finite element technique is more computationally

demanding, which compromises its popularity for flow simulations [122]. Most commercial

software packages for flow simulation rely on the finite volume approach. Here, the changes

of mass, momentum, and energy are intuitively accounted for as fluid crosses the boundaries

of the discrete computational volumes within the flow domain. The discretized governing

equations are written for each control volume within the calculation domain. They are

integrated using Gauss’s divergence theorem according to which the volume integrals are

expressed in terms of fluxes across the surfaces of each control volume. The surface flux is

calculated as the derivative of the variable on that surface, which itself is approximated by

the values in the centre of the neighbouring control volumes. The resulting linearized

equations are used to build a set of equations for the unknown flow variables. For more

detailed information on the finite volume method, the reader is referred to [120].

Different methods exist to approximate the surface values of the unknown flow

variables based on the values calculated for the centre of neighbouring control volumes

(nodal values). Figure 32 shows the difference between a first-order Central Differencing

Scheme (CDS) and a first-order Upwind Differencing Scheme (UDS). A first-order CDS

assumes that any flow variable (𝜑) varies linearly between the neighboring cells. 𝜑𝑤

designates the value of 𝜑 on the left surface of control volume P and is calculated through

linear interpolation between the value of 𝜑 on the center of control volumes W (𝜑𝑊) and P

(𝜑𝑃). Similarly, the value of 𝜑 on the right surface of control volume P (𝜑𝑒) is linearly

interpolated between the cell centre values for control volumes P and E (𝜑𝐸). This approach

does not contemplate the direction of the flow, thus is more adequate when the flow is

governed by diffusion phenomena. However, as the velocity of the flow increases, the

convective terms become increasingly significant. This affects numerical stability and

imposes an upper limit on the dimension of the control volumes. A solution to this problem

is using an UDS, which favours the upstream conditions to determine the surface value on a

given control volume. For the first-order UDS, the cell-centre value of any flow variable is

assumed to represent the average for that cell and to be constant throughout the entire cell.

Therefore, the surface values are identical to the nodal values for each cell and the surface

value of a cell is set as the cell-centre value of the upstream cell. The surface value 𝜑𝑤 on

control volume P is set equal to the cell-center value of the upstream cell (𝜑𝑊). The cell-

center immediately upstream to 𝜑𝑒 refers to control volume P, therefore 𝜑𝑒 = 𝜑𝑃.

Figure 32 – First-order Central Differencing Scheme (a) and first-order Upwind Differencing Scheme (b).



In a second-order upwind scheme, the surface value is calculated using a multidimensional

linear reconstruction approach. The cell-centre value and gradient of the field variable at the

upstream cell are used to determine the surface value [123]. QUICK (Quadratic Upstream

Interpolation for Convective Kinematics) schemes are based on a weighted average of

second-order upwind and central interpolations of the variable. Second-order UDS and

QUICK schemes are typically more accurate on structured meshes aligned with the flow

direction [124].

Pressure-coupled solver schemes are preferred because they have been reported to

perform well in terms of computational time and numerical stability for the simulation of

ejector flow [99]. These schemes simultaneously solve the continuity and momentum

equations, after which they tackle the energy and turbulence equations. Lastly, the

thermodynamic properties of the fluid are determined. In this work, the flow was simulated

as pseudo-transient, i.e., the flow adjusts itself to the boundary conditions over time. The

pseudo-transient formulation treats the model as if it were transient, therefore limiting the

rate of change of any flow variable to that which would occur in a transient flow during the

defined time step. An artificial, transient term is added to each equation, allowing for the

solution to march forward in time. The solver finds an instantaneous solution in each time

step, as opposed to aiming directly towards the final solution. As the simulation converges,

the transient oscillations fade and the solution to the steady-state flow is obtained.

FLUENT®

FLUENT® is a commercial CFD package by ANSYS™ for the simulation of a wide

range of compressible or incompressible, laminar or turbulent flows. Its applications include

laminar non-Newtonian flows, heat transfer in turbomachinery and automotive engine

components, pulverized coal combustion in utility boilers, and flows through compressors,

pumps, and fans. Another interesting application of this software is the modelling of multi-

phase flows, including gas-liquid, gas-solid, liquid-solid, and gas-liquid-solid flows. A wide

range of turbulence models are available, some of which contemplate the effects of

compressibility and buoyancy on the turbulent flow. Extended wall functions and zonal

models may be applied to ensure satisfactory near-wall accuracy. Overall, FLUENT® is a

versatile tool for the simulation of complex flows and geometries.

3.4. Development of the numerical mesh

Mesh geometry

The baseline mesh was built considering 50 cells along the vertical direction in the

primary nozzle, and 30 in the secondary nozzle (not including the boundary layer). Along the

ejector axis, an approximate element size of 0.2 mm was defined. The mesh density was

varied according to the expected flow profile, e.g., as the fluid accelerates towards the

primary nozzle throat, the element size was reduced to adequately capture the velocity

gradients. Inversely, as the flow decelerates in the diffuser, the size of the elements was

progressively increased towards the outlet. This allows for a significant reduction of the total

number of mesh elements without compromising the adequate reproduction of flow physics.

The geometry of the mesh was then defined for the boundary layer. A thickness of

0.05 mm was proposed for the initial mesh, and the boundary layer was divided into 6

separate elements along its thickness. The size of these elements was set to increase with wall

distance. This high mesh density is essential to capture the strong pressure and velocity



gradients that occur near the ejector walls, and provides a better perception of the 𝑦+ value

(see Section 3.4.2.). The mesh is significantly finer inside the boundary layer, therefore an

abrupt change in size should be avoided at the boundary layer interface. For this purpose, a

variable cell size was also defined outside the boundary layer to ensure a smooth transition.

This is shown in detail in Figure 33.

Figure 33 – Detail of the mesh near the boundary layer.

The flow is also limited by the spindle, which requires an additional boundary layer. In the

secondary inlet, the calculation domain is limited on both sides by the ejector wall, and so

two separate layers were introduced. The definition of the boundary layers caused a slight

distortion of the mesh. However, good orthogonality and skewness were still verified (see

Section 3.4.2). The proposed baseline mesh has a total of 75854 elements.

Mesh quality evaluation

For the purpose of assessing the geometric quality of the mesh, several indicators may

be calculated for each control volume, allowing for the identification of the regions in need

of improvement. The most common indicators used to evaluate the quality of the mesh are

skewness and orthogonality. Skewness measures the deformation of each control volume by

comparing its shape to that of a quadrilateral element of the same area. A perfectly

symmetrical element has a skewness value of 0, while a heavily distorted element has a

skewness value closer to 1. Orthogonality assesses the proximity of the mesh geometry to a

purely orthogonal mesh. The ideal value of orthogonality is 1, while a lower value indicates

poorer mesh quality. Table 9 shows the mesh quality requirements typically employed when

using FLUENT®.

Table 8 – Quality of the mesh in terms of skewness and orthogonality [113]

Skewness Orthogonality

0 - 0.25 0.95 - 1 Excellent

0.25 - 0.5 0.7 - 0.95 Very good

0.5 - 0.8 0.2 - 0.69 Good

0.8 - 0.94 0.15 - 0.2 Acceptable

0.95 - 0.97 0.001 - 0.14 Bad

0.98 - 1 0 - 0.001 Unacceptable

The baseline mesh has a maximum skewness of 0.22, which means that the overall skewness

quality of the mesh is excellent. The minimum orthogonal quality of the mesh is 0.85,

evidencing that it could be improved. However, the quality of the mesh is lower than 0.95

(excellent) only for a small number of CVs, namely in the boundary layer in the converging



section of the primary nozzle. Globally, the baseline mesh fulfilled the necessary quality

requirements. Nonetheless, the quality of the mesh is expected to improve with further

refinement for mesh independence testing (see Section 3.4.3). Smaller elements may adapt

to the geometry with less deformation, thus ensuring even better skewness and orthogonality.

Mesh independence of the results

Mesh refinement plays a key role in determining the accuracy of the results. An

essential step for the validation of the simulation procedure and results is obtaining mesh

independence, i.e., designing a mesh with a refinement such that the results are not depending

on the mesh itself. Coarse meshes are less capable of adequately reproducing the physics of

the flow, for example, velocity and pressure gradients. However, a smaller number of

elements leads to faster simulations. On the other hand, a smaller distance between cell

centres allows for a higher precision of the discretization schemes, i.e., the discretization

error is reduced. Consequently, the results become more accurate, but finer meshes show

lower convergence rates and require longer simulation times. As a result, there is a

compromise between the accuracy of the results and the simulation time necessary to obtain

them.

As mesh refinement increases, any evaluation parameter 𝜑 evolves asymptotically,

as shown in Figure 34. As the number of elements in the mesh (𝑁) increases from 𝑁1 to 𝑁2,

the results tend to vary (휀𝜑). For consecutive mesh refinements, the variation of the

evaluation parameter tends to decrease.

Figure 34 – Effect of mesh refinement on simulation results.

Mesh independence was verified separately for the modelling of CO2 as an ideal-gas and as

a real-gas. Further along this work, different SP are simulated, requiring different mesh

geometries. However, these do not differ significantly, as the same mesh density was

maintained during all simulations. The influence of the mesh on the results is predominantly

related to the general dimension of the mesh elements, rather than with the exact geometry

of each element. Therefore, it was expected that mesh independence applied for these small

modifications.



4. Formulation of the energy equation for the CFD model 4.1. Energy equation for CO2 as ideal-gas

Temperature-based formulation of the energy equation

The energy equation may be formulated using temperature as the independent

variable. In this formulation, the energy equation serves as a transport equation for

temperature. For a steady-state, laminar flow, it may be written as follows:

𝛻. [𝜌𝒖 (𝐶𝑝𝑇 +𝒖2

2)] = 𝛻. [𝑘𝛻𝑇 + 𝜏. 𝒖] (19)

where 𝐶𝑝 is the specific heat of the fluid. As mentioned in Section 3.3.2, turbulence is a

highly dissipative phenomenon that needs to be contemplated in the energy equation. The

energy equation may be derived for turbulent flows, taking the following form:

𝛻. [𝜌𝒖 (𝐶𝑝𝑇 +𝒖2

2)] = 𝛻. [(𝑘 + 𝑘𝑇)𝛻𝑇 + (𝜏 + 𝜏𝑇). 𝒖] (20)

Diffusive heat transfer is increased through a turbulent thermal conductivity (𝑘𝑇).

Additionally, turbulent stresses (𝜏𝑇) add to laminar stresses and therefore contribute to

viscous dissipation.

For high Mach number flows, compressibility affects turbulence through dilatation

dissipation. In this case, this effect should be considered, otherwise the expected

compressible mixing is poorly captured. In the 𝑘 − 휀 turbulence models, this effect is

accounted for with the term 𝑌𝑀 for dilatation dissipation in the transport equation for 𝑘 (see

equation 15), as follows [125]:

𝑌𝑀 = 2𝜌휀𝑀𝑎𝑇2 (21)

In equation 21, 𝑀𝑎𝑇 is the turbulent Mach number, defined as:

𝑀𝑎𝑇 = √𝑘

𝑎2 (22)

where 𝑎 is the speed of sound, defined as:

𝑎 = √𝛾𝑅𝑇 (23)

The constant 𝛾 is the polytropic coefficient and R is the specific gas constant.

When simulating CO2 as ideal-gas, pressure and temperature may be used as

independent properties to uniquely identify its thermodynamic state. Density, for example,

is a function of both these properties and its calculation is handled internally by FLUENT®,

as follows:



𝜌 =𝑝 + 𝑝𝑜𝑝

��𝑀 𝑇

(24)

where 𝑝𝑜𝑝 is the operating pressure, �� is the universal gas constant, and 𝑀 is the molecular

weight of the fluid (44.01 kg/kmol for CO2). Other properties, such as specific heat, thermal

conductivity, and molecular viscosity, are functions of temperature. To address this matter,

EES® was used to calculate these properties across the expected temperature range and these

values were manually introduced in FLUENT®. For each cell, the solver uses the local

temperature to determine the local properties, as shown in Figure 35. In a cell with

temperature 𝑇∗, the software interpolates each property (𝐶𝑃∗, 𝑘∗ and 𝜇∗).

Figure 35 – Definition of specific heat, thermal conductivity, and molecular viscosity as ideal-gas properties.

Specific heat, thermal conductivity, and molecular viscosity were defined on a temperature

range from 156.15 K (-117ºC) to 316.15 K (43ºC), with a 20ºC step. In some simulations,

temperatures under this lower limit were observed, meaning that fluid properties are not

adequately determined in some cells. However, these very low temperatures are not realistic

and would not be replicated in an experimental installation. Although the ejector is modelled

as adiabatic, some heat exchange should occur through the ejector walls. Moreover, the

metallic structure favours axial heat conduction along the ejector wall. For both these reasons,

it is not expected that the temperature of the supersonic jet drops below -20/-30ºC.

Simulation strategy

Smolka et al. [99] reported that using a first-order discretization of the governing

equations results in a poor prediction of the flow physics, namely in the constant-area section.

When using this discretization scheme, a shockwave does not occur in the ejector. When

compared to higher order discretization schemes (second-order and QUICK), performance

indicators such as the entrainment ratio vary significantly. For this reason, second-order

discretization schemes were selected for every flow variable, as shown in Table 9.

Table 9 – Selected parameters for the definition of the discretization schemes for the ideal-gas model


Discretization Scheme

Gradient Least Squares Cell Based

Pressure Second-order

Density Second-order upwind

Momentum Second-order upwind

Turbulence kinetic Energy Second-order upwind

Turbulence Dissipation Rate Second-order upwind

Energy (modified) Second-order upwind



To analyse the impact of spindle position on ejector performance, different SP were

simulated. SP ranging from 1 to 7 mm were simulated with a 0.5 mm step, for a total of 13

different ejector geometries. Since the ejector is not designed to work with a very restricted

primary flow, the minimum SP simulated was 1 mm. For a SP under this value, the primary

mass flow rate is very low, and the entrainment of the secondary flow is insufficient and not

compatible with adequate ejector operation. A very restricted primary flow would require a

low compression ratio to ensure critical operation, resulting in poor ejector performance. On

the other hand, as the spindle is moved upstream and away from the primary throat, it is

expected that the variation of the entrainment ratio becomes less significant. In fact, as SP

increases, the effective area of the primary flow is decreasingly reduced. For a sufficiently

high SP, the spindle does not affect the section area on the primary throat and the primary

mass flow rate becomes constant. For this reason, a maximum SP of 7 mm was simulated.

For each SP, the outlet pressure was varied according to different compression ratios.

Compression ratios of 1.1, 1.2, 1.3, 1.4, and 1.5 were simulated. Since the goal of the ejector

is to provide a significant pressure lift to the secondary flow, it is not expected to operate

under very low compression ratios. For this reason, a minimum compression ratio of 1.1 was

established. Moreover, a high compression ratio typically results in a low secondary mass

flow rate, which compromises ejector performance. Consequently, the maximum

compression ratio to simulate was set at 1.5. A total of 65 simulations were conducted.

An automated process of geometry generation was implemented to expedite the

creation of different ejector geometries, e.g., the effect or a round surface in the primary inlet

to avoid recirculation. Using FLUENT® script capabilities, the generation of a first geometry

was recorded. To alter specific geometric parameters, the coordinates of the associated points

were manually altered within the script. When running the tailored scripts, FLUENT®

automatically generated the necessary ejector geometries. An overview of these scripts is

shown in Appendix I. However, extending this automated process to the generation of the

different meshes is not effective. For this reason, the meshes used to simulate the different

SP were adapted from previous ones, rather than being created from a new ejector geometry.

The objective of this work is not the detailed characterization of ejector flow, rather

it aims to analyse the effect of ejector geometry on global performance indicators such as the

entrainment ratio. Therefore, it is not necessary to reach a very low relative residual in each

governing equation, and so a maximum relative residual of 10-5 was established for all flow

variables. The mass flow rate error (mass imbalance between the inlets and the outlet) was

set as an external report. Its value should not exceed 10-6, a very strict criterion given that the

mass flow rates in the conducted simulations range from 10-2 to 100. The individual mass

flow rates (primary, secondary and outlet) were introduced as external reports to calculate

the entrainment ratio. To verify final convergence, it was established that all these mass flow

rates should be stable, i.e., show a constant value for a minimum of 100 iterations.

A simulation was initialized for each value of the compression ratio using the mesh

corresponding to the maximum SP. In each case, the pressure was initialized at a value

slightly lower than the outlet pressure throughout the whole calculation domain. To ensure

that the velocity field is adequately initialized, the initial axial and radial velocities were set

at 10 m/s and 0 m/s, respectively. Turbulent kinetic energy was set at 1 m2/s2 and turbulent

dissipation rate was set at 1 m2/s3. Temperature was initialized at 300 K, an intermediate

value between the primary and secondary inlets. Boundary conditions were set to the desired

values at both inlets and the outlet. The first simulation was conducted with first-order

discretization schemes for all flow variables to ensure numerical stability. After convergence



was obtained, these were individually switched to second-order schemes. Final convergence

was obtained with second-order schemes in all cases. In the following simulations, the

spindle was positioned further downstream, leading to a stronger restriction of the primary

flow. The results from the previous simulations were interpolated into the new mesh

geometries, allowing for the following simulations to run with second-order schemes from

the start. For an adequate compromise between numerical stability and convergence rate, the

under-relaxation factors were set as follows: 0.3 for pressure, 0.6 for density, and 0.8 for

turbulent kinetic energy, turbulent dissipation rate, turbulent viscosity, and energy. As for

momentum, most simulations ran with an under-relaxation factor of 0.2. However, in some

cases, this resulted in oscillating residuals and a slower convergence. This problem was

addressed by reducing the momentum under-relaxation factor to 0.1, providing a more stable

convergence. The pseudo-time step was set at 5x10-5 s to ensure numerical stability and a

steady convergence.

Boundary conditions

Solving the flow inside the calculation domain requires imposing adequate boundary

conditions for each variable. Pressure and temperature were set at the inlets for the ideal-gas

model, according to the initial design conditions [110]. Outlet pressure was selected

according to the defined compression pressure ratios, as referred in Section 4.1.2. The energy

equation is intrinsically conservative, therefore the energy balance between the inlets and the

outlet is ensured.

Turbulence properties also need to be defined at the inlets and the outlet. Turbulence

intensity (𝐼) compares the instantaneous velocity fluctuations with the mean flow velocity.

For a fully-developed duct flow, it can be estimated from the following formula [126]:

𝐼 = 0.16𝑅𝑒𝐷𝐻

−18 (25)

In equation 25, 𝑅𝑒𝐷𝐻 is the flow Reynolds number, calculated with the hydraulic diameter

(𝐷𝐻) of the flow section. The turbulence length scale (𝑙) is a physical quantity related to the

dimension of the large eddies that contain the energy in turbulent flows. In duct flows, the

turbulence length scale is limited by the dimension of the duct because the eddies cannot be

larger than the duct itself. It is possible to estimate the turbulence length scale with the

following correlation [126]:

𝑙 = 0.07𝐷𝐻 (26)

Based on the mass flow rates estimated with the 1-D model and the ejector geometry defined

in Section 3.2, both turbulence intensity and turbulent length scale were estimated, as shown

in Table 10. The values calculated for turbulence intensity are in the same range as the values

typically used by the research group at CIENER - INEGI. Based on previous experience,

turbulence intensity was set at 5% for both inlets and 10% at the outlet. The values of

turbulent length scale were set according to Table 10.



Table 10 – Estimation of hydraulic diameter, Reynolds number, turbulence intensity, and turbulence length scale for both inlets and the outlet


𝑫𝑯 [𝒎𝒎] 17.6 6.09 16.2

𝑹𝒆𝑫𝑯 [−] 2.68x103 4.03x103 7.97x103

𝑰 [%] 3.35% 3.22% 2.93%

𝒍 [𝒎𝒎] 1.23 0.430 1.13

The flow is physically limited by the walls. Therefore, velocity in the direction

perpendicular to the ejector wall is null. This is directly contemplated on the governing

equations that are explicitly written in 2-D. Finally, there is no heat transfer between the

working fluid and the ejector walls, which is assured by imposing a zero-gradient boundary

condition on temperature. Table 11 shows the parameters selected for the definition of the

boundary conditions.

Table 11 – Applied boundary conditions for the ideal-gas model

Boundary Conditions Parameter Selected

Primary Inlet

Type Pressure-inlet

Pressure (MPa) 9.30

Temperature (ºC) 43

Turbulence Intensity (%) 5

Turbulence Length Scale (mm) 1.23

Secondary Inlet

Type Pressure-inlet

Pressure (MPa) 3.77

Temperature (ºC) 8



Outlet

Type Pressure-outlet

Pressure (MPa) 4.147 / 4.524 / 4.901 / 5.278 / 5.655



Walls

Momentum

Type Stationary wall

Shear condition No slip

Roughness models Standard

Roughness height 0

Temperature

Boundary condition Specified flux

Boundary value 0



Mesh independence testing

The validity of the selected wall functions requires that the 𝑦+ criterion be met.

Therefore, a preliminary simulation was performed using the baseline numerical mesh and

under the operating conditions presented in Table 11 (compression ratio of 1.3). The

maximum 𝑦+ value obtained was 126, well under the upper limit recommended in literature

[120]. The minimum observed value was of 10, slightly lower than the lower limit suggested

by the same author. However, the mesh was accepted because a low value of 𝑦+ means that

the velocity gradient near the wall is well captured by the first mesh elements. This also

means that the mesh could be coarsened, if needed, without violating the 𝑦+ criterion.

However, it was shown that this boundary layer geometry provided a smooth convergence.

Moreover, the number of elements inside the boundary layer is negligible when compared to

the whole ejector; therefore, coarsening the boundary layer should not lead to a significant

improvement on the convergence rate of the simulation. This validation was only addressed

once, as it is not expected that 𝑦+ varies significantly across the following simulations.

Mesh independence is usually assessed according to some predefined criteria. In this

section, it was considered that the primary and secondary mass flow rates, as well as the

entrainment ratio (as being one of the main performance indicators), should not vary more

than 0.5% between two meshes with different levels or refinement. This is in agreement with

the methodology employed by Li et al. [127]. To verify mesh independence of the results,

the baseline mesh (mesh IG1) was simulated under the operating conditions defined in Table

11 (compression ratio of 1.3). To ensure that the spindle would not affect the primary mass

flow rate results, it was positioned outside the primary nozzle throat. This way, the primary

mass flow rate was expected to be at its maximum under the given primary inlet conditions,

corresponding to a fixed geometry device. After the simulation, a coarser mesh (mesh IG2)

was created and simulated under identical boundary conditions. The entrainment ratio varied

more than 0.5% between meshes IG1 and IG2; therefore, mesh IG2 was discarded since the

mesh resolution significantly affects the results. Mesh IG1 was further refined, resulting in

mesh IG3. Between meshes IG1 and IG3, both the primary and secondary mass flow rates

suffered only a slight variation, resulting in a variation of the entrainment ratio (휀𝜔) lower

than 0.5%. Table 12 shows the primary and secondary mass flow rates together with the

entrainment ratio, indicating their relative differences.

Table 12 – Results of the mesh sensitivity test

Mesh Number of

elements

��𝒑

[kg/s]

𝜺��𝒔

[%]

��𝒔

[kg/s]

𝜺��𝑷

[%]

𝝎

[-]

𝜺𝝎

[%]

IG2 61208 0.173 -0.71 0.0870 0.12 0.504 0.84

IG1 75854 0.174 - 0.0868 - 0.500 -

IG3 92024 0.174 -0.0037 0.0869 0.087 0.500 0.034

Mesh IG1 is used in the following simulations, for it provides results that are mesh

independent while allowing for a shorter simulation time (due to its lower number of

elements).



4.2. Energy equation for CO2 as real-gas

Enthalpy-based formulation of the energy equation

The energy equation may be generalized when using the specific enthalpy as

independent variable. This approach eliminates the need for an additional equation for the

phase marker (phase volume or mass fraction) in problems with phase change [99]. The

thermodynamic state of the refrigerant is fully defined by the two independent phase

variables pressure and enthalpy, even in the phase change region. In a steady-state, laminar

flow, the energy equation may be written as follows:

𝛻. [𝜌𝒖 (ℎ +𝒖2

2)] = 𝛻. [(𝛤𝛻ℎ) + 𝝉. 𝑢] (27)

with,

𝛤 = (𝑘

𝜕ℎ𝜕𝑇

)

𝑝

(28)

Implementing the enthalpy-based formulation of the energy equation for a turbulent flow

requires significant modifications. Based on the work by Smolka et al. [99], the energy

equation under steady-state conditions takes the following form:

𝛻. (𝜌𝒖ℎ) = 𝛻. (𝛤ℎ,𝑒𝑓𝑓𝛻ℎ) + ��ℎ1+ ��ℎ2

+ ��ℎ3 (29)

The effective diffusion coefficient (𝛤ℎ,𝑒𝑓𝑓) is the sum of the laminar diffusion coefficient in

equation 28 and the turbulent diffusion coefficient (𝛤𝑇):

𝛤ℎ,𝑒𝑓𝑓 = 𝛤 + 𝛤𝑇 (30)

The turbulent diffusion coefficient is determined from:

𝛤𝑇 =𝜇𝑇

𝜎𝑇 (31)

In equation 31, 𝜎𝑇 is the turbulent Prandtl number. The source terms ��ℎ1, ��ℎ2

and ��ℎ3 describe

the mechanical energy, the irreversible dissipation of the kinetic energy variations, and the

dissipation of the turbulent kinetic energy, respectively. These terms are defined as:

��ℎ1= 𝒖. 𝛻𝑝 (32)

��ℎ2= (𝜇 + 𝜇𝑇) {2 [(

𝜕𝑢

𝜕𝑥)

2

+ (𝜕𝑣

𝜕𝑦)

2

+ (𝜕𝑤

𝜕𝑧)

2

] + (𝜕𝑢

𝜕𝑦+

𝜕𝑣

𝜕𝑥)

2

+ (𝜕𝑢

𝜕𝑧+

𝜕𝑤

𝜕𝑥)

2

+ (𝜕𝑣

𝜕𝑧+

𝜕𝑤

𝜕𝑦)

2

−2

3(𝛻. 𝒖)2} −

2

3𝜌𝑘𝛻. 𝒖

(33)

��ℎ3= −𝜌𝒖. 𝛻𝑘 (34)



In FLUENT®, the built-in energy equation is temperature-based, thus not suitable for

solving transcritical flow problems. Therefore, a new scalar transport equation was

implemented for the enthalpy-based energy equation. For an arbitrary User-Defined Scalar

(UDS), FLUENT® solves a generic transport equation in the form of [113]:

𝜕𝜌𝜑

𝜕𝑡+ 𝛻. (𝜌𝒖𝜑) = 𝛻. (𝛤𝛻𝜑) + ��𝜑 (35)

In equation 35, 𝜑 is an arbitrary scalar, 𝛤 is the diffusion coefficient, and ��𝜑 is the scalar

source term. The enthalpy-based formulation of the energy equation in equation 29 can be

adapted to equation 35. For a steady-state flow, the time derivative on the left-hand side is

neglected. The convective terms of the UDS transport equation are based on the mass flow

rates per unit volume in the different spatial directions. The diffusive term on the right-hand

side is defined according to equation 30. For this purpose, a User-Defined Function (UDF)

was written (DEFINE_DIFFUSIVITY). The source term in equation 35 is the sum of the

individual source terms in equations 32-34. For their implementation, three UDFs

(DEFINE_SOURCE) were written and compiled to FLUENT®. All UDFs written for the

implementation of the HEM are shown in Appendix II. Figure 36 shows how the convective,

diffusive, and source terms of the UDS transport equation were addressed.

Figure 36 – Approach to the convective, diffusive, and source terms in the UDS transport equation.

Implementation of the HEM

In FLUENT®, the specific enthalpy cannot be directly used to define the

thermodynamic state of the fluid. To solve this problem, fluid properties were determined by

specifically written UDFs (DEFINE_PROPERTY). This procedure was adopted for density,

viscosity, and thermal conductivity, as these are necessary for solving the governing

equations. The UDFs use the values of pressure and enthalpy as independent variables in

order to calculate the desired property by interpolation. For the specific heat, a dedicated

UDF (DEFINE_ADJUST) was written to update this property after each iteration. The

specific heat is initialized using an additional UDF (DEFINE_INIT).

Figure 37 – Schematic of property UDFs (specific heat, density, thermal conductivity, and viscosity).



Real-gas properties of CO2 were calculated from a predefined table generated by

EES® as a function of pressure and enthalpy. The EES® script is shown in Appendix III. A

UDF (DEFINE_EXECUTE_ON_LOADING) was written so that FLUENT® reads and stores

this information in lookup tables when the UDF library is loaded onto the solver, as shown

in Figure 38. After that, the property tables are available for the solver.

Figure 38 – Schematic of the property (φ(p,h)) reading process for the developed model.

The Mach number distribution is an important result for supersonic compressible

flows. The speed of sound was also obtained by EES® as a function of pressure and enthalpy.

The Mach number distribution in the ejector is obtained by executing a UDF

(DEFINE_ON_DEMAND) after the solution for the other field variables is obtained. This

approach was preferred over another property UDF in order to reduce computational time.

As the speed of sound and Mach number are not part of the governing equations, it is not

necessary to update them after each iteration, which would lead to a slower numerical

convergence. An equivalent approach was used to determine the temperature field and the

quality in each cell of the calculation domain.

Figure 39 – Schematic of property UDFs (Mach number, temperature, and quality).

Interpolation is a valid method to determine fluid properties when the independent

variables (pressure and enthalpy) are within the extreme values in the property tables.

Although it was ensured that the scripts still return a property value when pressure and/or

enthalpy exceed the table limits, this should be avoided because fluid properties are not

correctly determined, which compromises the accuracy of the results. Therefore, an

additional UDF was created to evaluate whether the values of pressure and enthalpy in each

cell were within range, as shown in Figure 40. If the simulated pressure and/or enthalpy field

value exceed the limits, the property tables should be redefined.

Figure 40 – Schematic of UDF to verify pressure and enthalpy limits.



Simulation strategy

As previously referred in Section 4.1.2, second-order discretization schemes should

be used for the simulation of a transcritical CO2 ejector. However, convergence could not be

achieved within the predefined limit with the developed model when using a second-order

upwind scheme for the momentum equations. For this reason, these equations were

discretized using a first-order scheme, as shown in Table 13.

Table 13 – Selected parameters for the definition of the discretization schemes for the real-gas model


Discretization Scheme

Gradient Least Squares Cell Based

Pressure Second-order

Momentum First-order upwind

Turbulence kinetic Energy Second-order upwind

Turbulence Dissipation Rate Second-order upwind

Energy (modified) Second-order upwind

The HEM was not implemented for the simulation of the whole ejector, as this posed

serious convergence difficulties that could not be overcome during the timeline of this work.

In this context, the HEM was used to simulate the flow within the primary nozzle in order to

assess the impact of applying a spindle in the primary nozzle. The geometry of the nozzle

was the same as in the full ejector simulated with the ideal-gas model, including the

adjustable spindle. By adjusting the position of the spindle, the variation of the primary mass

flow rate could be analysed. As before, simulations were carried out for SP ranging from 1

to 7 mm, with a 0.5 mm step. The pressure boundary condition at the nozzle exit section was

defined according to the SP (see section 4.2.3). Since the ejector always operates with choked

primary flow, the primary mass flow rate is a function of SP for fixed primary inlet

conditions.

As referred, the geometry of the primary nozzle to simulate with the HEM was

identical to the full ejector simulated with the ideal-gas model. To generate such geometry,

a new SpaceClaim® script was written (see Appendix I). This ensures identical nozzle

geometries while expediting the process of generating the geometry.

The same convergence criteria were adopted as in the ideal-gas simulations. A

maximum relative residual of 10-5 was admitted for all flow variables. The permissible mass

imbalance between the inlet and the nozzle exit section was set to 10-6. During the iterations,

the (primary) mass flow rate was also introduced as an external report. It was established that

it should remain constant up to the 4th significant figure for a minimum of 100 iterations for

final convergence to be verified.

A number of simulation strategies were applied to obtain good numerical

convergence with the real-gas model. Most of these strategies were established using a

trail/error approach. The first simulation was performed with an open spindle, leading to a

maximum mass flow rate for the given inlet conditions. Pressure was initialized at a value

slightly higher than the outlet pressure throughout the flow domain. The velocity field was

initialized with axial and radial velocities of 10 m/s and 0 m/s, respectively. Turbulent kinetic

energy was set at 1 m2/s2 and turbulent dissipation rate was set at 1 m2/s3. Enthalpy was

initialized at -150000 J/kg, an intermediate value between the inlet and the estimated outlet



values. Boundary conditions were set to the desired values at both inlets and the outlet.

Initially, the calculation was carried out using first-order discretization schemes for all flow

variables until obtaining numerical stability. Later, they were switched to second-order

schemes for better accuracy (except for the momentum equations). For latter simulations with

different spindle positions, solution initialization was carried out by interpolating the results

from the open spindle case. This approach led to shorter simulation times and faster

convergence. Numerical simulation of transcritical CO2 ejectors is a relatively new research

topic. There is no sufficiently detailed information in the open literature on how to obtain

convergence. The under-relaxation factors were first set according to the work by Smolka et

al. [99]: 0.1 for pressure, momentum, and density; 0.6 for turbulent kinetic energy, turbulent

dissipation rate, and turbulent viscosity. After initial convergence, the relaxation factors were

increased according to: 0.3 for pressure, 0.2 for momentum, 0.7 for density, and 1 for

turbulent kinetic energy, turbulent dissipation rate, and turbulent viscosity. The under-

relaxation factor for the modified energy equation was maintained at 0.75 (default option in

FLUENT® for pseudo-transient simulations). The pseudo-time step was reduced to 5x10-6 s.

As referred in Section 4.2.2, the properties of CO2 as a real fluid are calculated from

lookup tables, as a function of static pressure and specific enthalpy, which are loaded onto

the solver. These property tables were defined for a range of operating conditions that were

the outputs of the 1-D model (see Section 3.2.3). The minimum pressure is expected at the

nozzle exit section, while the maximum pressure corresponds to the primary inlet pressure

(9.3 MPa). For this reason, the minimum and maximum pressure values in the property tables

were set at 1 and 10 MPa, respectively. A total of 46 interpolation points on pressure resulted

in pressure intervals of 200 kPa. As for enthalpy, the 1-D model estimates a minimum

enthalpy of -161 kJ/kg at the primary nozzle exit section and a maximum enthalpy of -146

kJ/kg at the primary inlet. Considering these values, the minimum and maximum enthalpy

values were set at -200 and -100 kJ/kg, respectively, to cover different conditions that could

be expected in the ejector flow field. Using 51 interpolation points allowed for the real-gas

properties to be determined for enthalpy intervals of 2 kJ/kg. The effect of the interpolation

scheme on the results is further analysed in Section 4.2.6.

Boundary conditions

At the primary inlet, pressure and enthalpy values were set according to the ejector

design conditions. For turbulence, boundary conditions were set according to the calculations

shown in Section 4.1.3. As the spindle is moved downstream, the effective area of the primary

flow decreases; therefore, maintaining a constant outlet pressure would result in an over-

expanded primary jet in the nozzle diverging section. To address this matter, the 1-D model

was adapted to calculate the necessary nozzle exit pressure to ensure an adequate expansion

of the primary jet – this script is shown in Appendix IV. For this reason, the calculated nozzle

exit pressure values could be considered with some degree of uncertainty. After a set of

preliminary simulations for each SP, the pressure boundary condition at the outlet of the

calculation domain was adjusted to allow for an adequate expansion of the primary jet.

Generally, the numerical simulations were expected to produce somewhat different

mass flow rates than the 1-D model; thus, setting the outlet enthalpy according to the result

given by the 1-D model could violate the 1st Law of Thermodynamics. This problem was

addressed by imposing a zero-gradient boundary condition, instead of a specified value. The

solution to the enthalpy-based energy equation ensures conservation of energy, as is the case

for the conventional, temperature-based formulation. Turbulence intensity was set at 5% at

the nozzle exit section and the turbulence length scale was calculated according to equation



26 for the respective diameter. Boundary conditions at the walls were set identical to the

ideal-gas simulations. Table 14 summarizes the applied boundary conditions.

Table 14 – Applied boundary conditions for the real-gas model

Boundary Conditions Parameter Selected

Primary Inlet

Type Pressure-inlet

Pressure (MPa) 9.30

Enthalpy (J/kg) -146191



Primary nozzle exit section

Type Pressure-outlet

Pressure (MPa) Defined for each SP



Walls

Momentum

Type Stationary wall

Shear condition No slip

Roughness models Standard

Roughness height 0

Enthalpy

Boundary condition Specified flux

Boundary value 0

Mesh independence testing

The baseline mesh resulted in 𝑦+ values ranging from 4 to 149. These are in

agreement with literature recommendations [120]; therefore, the geometry of the boundary

layer was accepted. Once again, these values are not expected to vary significantly when

simulating different models with different spindle positions, thus the quality of the boundary

layer was only verified for the baseline mesh.

Besides the primary mass flow rate, the area-averaged flow Mach number at the

nozzle exit section (𝑀𝑎𝑁𝑋) was assessed as a relevant output variable. This variable is

affected by both the velocity and enthalpy fields. 𝑀𝑎𝑁𝑋 is later compared to the 1-D model

estimation under the same operating conditions. The nozzle geometry and the baseline mesh

(mesh RG1) were directly adapted from the full ejector mesh which was previously verified

for mesh independence (see Section 4.1.4) using the boundary conditions defined in Table

15. After the simulation, a coarser version of this mesh (mesh RG2) was generated, and the

ejector was simulated with the new mesh geometry. Only a small variation was observed for

both the primary mass flow rate and the exit Mach number. For this reason, the ejector was

simulated with consecutively coarser meshes, as shown in Table 15.



Table 15 – Primary mass flow rate and average Mach number at the nozzle exit section for different meshes

Mesh Number of elements ��𝒑 [kg/s] 𝜺��𝒑 [%] 𝑴𝒂𝑵𝑿 [-] 𝜺𝑴𝒂𝑵𝑿

[-]

RG1 14694 0.26262 - -

RG2 12141 0.26260 0.00066 1.2770 0.0352

RG3 9984 0.26258 0.00704 1.2766 0.0362

RG4 8304 0.26256 0.00703 1.2760 0.0442

RG5 6776 0.26256 0.000788 1.2754 0.0475

RG6 5658 0.26254 0.00789 1.2746 0.0632

RG7 4712 0.26253 0.00200 1.2738 0.0595

RG8 3922 0.26252 0.00306 1.2729 0.0761

RG9 3219 0.26257 -0.0177 1.2709 0.152

Although meshes RG8 and RG9 verified mesh independence of the results, the low number

of CVs hindered convergence. In fact, such coarse meshes lack in quality, namely in terms

of orthogonality. In the convergent section of the primary nozzle, mesh RG7 showed a

minimum orthogonal quality of 0.532 (see Figure 41), an acceptable value according to Table

8. Meshes RG8 and RG9, however, show a minimum orthogonality of 0.415 and 0.188,

respectively, because of the larger mesh elements. For this reason, mesh RG7 was selected

as a compromise between mesh size and quality.

Figure 41 – Detail of mesh orthogonality for mesh IG7.

When modelling CO2 as real-gas, mesh independence of the results was obtained for

a significantly smaller number of mesh elements in the primary nozzle when compared to the

ideal-gas simulations. The flow inside the primary nozzle is characterised by the Ma=1

threshold in primary nozzle throat. Therefore, it may occur that the physics of the flow is

more dependent on this constraint than on the numerical mesh, allowing for the use of a

relatively coarse mesh without influencing the mass flow rate. On the other hand, the use of

first-order discretization schemes may lack precision in describing the physics of the flow in

the primary nozzle. Second-order schemes may be necessary to accurately model the flow,

potentially leading to the need for a more refined mesh.

Influence of property table size on flow variables

To assess the impact of the physical property interpolation scheme, mesh RG7 was

used with different number of interpolation points for the property tables. First, the original

lookup tables with 46 points for pressure and 51 points for the specific enthalpy were tested.



Later, two other sets of interpolation tables were built with different numbers of data points,

as shown in Table 16. The results for the primary mass flow rate and average Mach number

at the nozzle exit section are also indicated. It may be noted that the size of the lookup tables

had no significant impact on the simulated mass flow rate and Mach number. Additionally,

the density along the nozzle axis was compared for the three different table sizes. From these

values, the maximum and mean relative errors were determined. Both these indicators are

kept below 0.5%. The results indicate that the size of the lookup property tables had little

influence on the results, therefore all latter simulations were carried out using the original

46x51 data points.

Table 16 – Primary mass flow rate and average Mach number at the nozzle exit section for different

interpolation schemes

Number of interpolation points ��𝒑 [kg/s] 𝜺��𝒑

[%] 𝑴𝒂𝑵𝑿 [kg/s] 𝜺��𝑴𝒂𝑵𝑿 [%]

Pressure Enthalpy Total

24 26 624 0.26267 0.052 1.2735 -0.028

45 51 2346 0.26253 - 1.2738 -

91 101 9101 0.26251 <0.01 1.2735 -0.026



5. Results and discussion

5.1. Simulation results for the ejector flow with CO2 as ideal-gas

Applicability of the ideal-gas model

In this section, the CFD results are shown for the ejector operating with a compression

ratio of 1.3 and the adjustable spindle fully open, which means the spindle does not restrict

the primary flow. These operating conditions correspond to the design values used to obtain

the initial dimensions of the ejector using the mathematical model by Marques [110].

Figure 42 shows the static pressure and Mach number distributions along the ejector

axis. After entering the ejector through the primary inlet (1), the CO2 is accelerated due to

the reduction of the cross-section area along the convergent section of the primary nozzle,

leading to a decrease in static pressure. The primary flow reaches sonic speed in the primary

nozzle throat and continues to accelerate along the divergent section of the primary nozzle.

This is accompanied by a decrease in pressure, as expected. The first shockwave, indicated

by a sudden change in 𝑝 and 𝑀𝑎, occurs at the exit section of the primary nozzle, as the

primary and secondary streams begin to interact (2). This indicates that the primary jet is

over-expanded at the nozzle exit section, i.e., the static pressure of the primary stream at the

exit section of the primary nozzle is lower than the static pressure in the mixing chamber. A

rapid increase in pressure is accompanied by a decrease in velocity and Mach number.

Further downstream the mixing chamber, the flow recovers from this first shock wave and

suffers a series of additional shockwaves (shock-train) at supersonic speed. Apart from these

fluctuations, pressure and velocity remain virtually constant throughout the constant-area

section, where mixing of the primary and secondary flows takes place. At the inlet section of

the diffuser (3), the expansion of the supersonic mixed flow causes an increase in the Mach

number and a decrease in pressure. After this, a final shockwave occurs, and the flow velocity

drops below the Ma=1 threshold. The subsonic flow further decelerates in the diffuser,

allowing for the static pressure to increase.

Figure 42 – Static pressure and Mach number distributions along the ejector axis.



As previously mentioned, the pressure and Mach number fields suggest that the

primary jet is slightly over-expanded as it leaves the primary nozzle. Figure 43 shows the

pressure distribution for the primary flow near the primary nozzle exit section. It may be

noted that a pressure peak occurs near the ejector axis immediately after the primary stream

leaves the primary nozzle, as identified in the figure. This indicates that the nozzle exit

section is slightly over-sized, leading to a sudden pressure drop of the supersonic primary

flow in the divergent section of the nozzle. This is followed by a slight expansion of the

primary jet downstream (see Figure 42) with the corresponding increase in Mach number and

decrease in static pressure.

Figure 43 – Detail of the pressure distribution for the primary flow near the primary nozzle exit section

(pressure in Pa).

As shown in Figure 27, it is expected that the secondary stream reaches sonic speed

in the mixing chamber, more specifically at the hypothetical throat. However, the numerical

results show that the secondary stream reaches Ma=1 at the inlet section of the diffuser.

Figure 44 shows a local detail of the Mach number results for the secondary stream. Because

of the diverging geometry, the fluid starts to expand in the diffuser, which is followed by a

strong oblique shock wave. This is in disagreement with the assumed location of the

hypothetical throat in the mathematical model by Marques [110]. Moreover, a slight

recirculation zone is observed near the wall of the diffuser inlet. However, this should not

affect the primary nor the secondary mass flow rates, as both streams reach sonic speed

upstream. In Figure 44, the inner diameter of the hypothetical throat can be measured, leading

to an estimated effective area of 4.8 mm2. This is in reasonable agreement with the 5.2 mm2

estimated by the 1-D model.

Figure 44 – Detail of the Mach number results for the secondary flow near the inlet section of the diffuser.

One of the main limitations of the ideal-gas model is its inability to accurately predict

the condensation and evaporation processes that occur inside the ejector. Figure 45 depicts

the expansion of the primary flow along the primary nozzle in terms of static pressure and

specific volume. The primary flow enters the primary inlet at a higher pressure; therefore, its

specific volume is lower. As the flow accelerates, the pressure decreases and its specific

volume increases. In this figure, the saturation lines are also shown for CO2. The expansion

process predicted with the 1-D model is also included.



Figure 45 – Saturation lines and expansion process in the primary nozzle (1-D model and CFD assuming ideal-

gas behaviour).

Figure 45 indicates a clear difference between the results estimated with the 1-D model and

the results obtained by CFD. The 1-D model predicts that the supercritical gas expands into

the wet vapor region, whereas the ideal-gas simulations show that the fluid remains in the

superheated region. Table 17 compares the pressure and density results for the primary inlet,

the primary nozzle throat, and the exit section of the primary nozzle. It may be seen from the

table that the ideal-gas model predicts a stronger expansion process along the primary nozzle.

In fact, the simulations register significantly lower values of static pressure both at the

primary nozzle throat and at the primary nozzle exit section. This phenomenon is partly

responsible for the lower specific volumes obtained with the CFD model. Additionally, the

CFD model underpredicts the density of CO2 even for an equivalent thermodynamic state. In

fact, for the primary inlet state, the 1-D model estimates a density of 435.3 kg/m3, while the

ideal-gas model predicts a density of 155.7 kg/m3. Both factors contribute to deviation of the

expansion line from the two-phase region, as shown in Figure 45. In addition, the entrainment

ratio obtained in the ideal-gas simulations (0.174 kg/s) significantly differs from the 1-D

model estimations (0.249 kg/s). This 30.1% difference may result from the poor estimation

of the density of the supercritical gas by the ideal-gas model.

Table 17 – 1-D model estimations and ideal-gas simulation results for pressure and density at the primary inlet,

primary nozzle throat, and primary nozzle exit section

Location 𝒑 [MPa] 𝝆 [kg/m3]

1-D Ideal-gas 1-D Ideal-gas

Primary inlet 9.3 435.3 155.7

Primary nozzle throat 5.8 5.0 273.3 96.93

Primary nozzle exit section 4.2 3.1 186.2 67.11

Stagnation conditions were assumed at both inlets and the outlet. To verify this

assumption, average flow velocities at each ejector boundary, together with the respective

averaged static and dynamic pressures, are shown in Table 18. The average flow velocities

at the boundaries calculated by the CFD model are significantly higher than estimated by the

1-D model. However, the inlets and the outlet were fairly oversized in terms of ensuring

stagnation conditions. Therefore, neglecting the dynamic pressure at the boundaries still

represents an error lower than 1%. Although the mass flow rates vary according to the SP

and the compression ratio, these variations should not violate the previous criterion.



Table 18 – Confirmation of stagnation conditions at the inlets and the outlet for the ideal-gas simulations


𝒖 [m/s] 2.26 24.4 15.1

𝒑 [Pa] 9.30x106 3.75 x106 4.90 x106

𝟏

𝟐𝝆𝒖𝟐 [Pa] 411.38 20715 9702.6

𝟏

𝟐𝝆𝒖𝟐

𝒑 [%] <0.01 0.55 0.20

Ejector performance for fixed spindle position

In this section, the effect of the compression ratio on ejector performance is analysed

for a fixed SP. Figure 46 shows the primary and secondary mass flow rates and the

entrainment ratio as a function of the compression ratio, for a SP of 2.5 mm. As expected,

the primary mass flow rate is constant. The primary stream reaches sonic speed in the primary

nozzle throat; therefore, the mass flow rate of the choked primary flow is independent of 𝛱.

It is also clear that the entrainment ratio remains constant for low compression ratios because

the ejector is operating in critical mode (although not necessarily at the critical point). When

the ejector operates in critical mode, the secondary mass flow rate depends only on the

effective area of the hypothetical throat. For a fixed SP, this is constant because the primary

mass flow rate is also constant. Therefore, decreasing the backpressure has no effect on the

entrainment ratio.

Figure 46 – Primary and secondary mass flow rates and entrainment ratio as a function of compression ratio

for a SP of 2.5 mm.

Figure 47 shows the Mach number distribution for a SP of 2.5 mm and compression

ratios from 1.1 to 1.5. In Figures 47a and 47b, the secondary flow reaches sonic speed in the

constant-area section. When using a SP of 2.5 mm, backpressure may be increased to a

compression ratio of 1.2 without compromising ejector performance. A small decrease in the

entrainment ratio is observed when the backpressure is increased to 1.3. In this case, the

ejector is not operating in critical mode because the secondary stream does not reach sonic

condition. Thus, the secondary mass flow rate is inversely proportional to the compression



ratio. For this reason, increasing the backpressure does have a negative effect on the

entrainment ratio. It may be concluded that the critical compression ratio for the ejector

operating with a SP of 2.5 mm is between 1.2 and 1.3, corresponding to a critical

backpressure between 4.524 and 4.901 MPa. In Figures 47c and 47d, it may be noted that the

secondary stream remains subsonic (𝑀𝑎<1) throughout the whole ejector. In this case, the

compression ratio impacts the entrainment ratio. For a fixed secondary pressure, a higher

backpressure results in lower secondary mass flow rate and entrainment ratio. It may also be

noted that the effective area of the secondary stream is significantly reduced as the

compression ratio increases from 1.3 to 1.4. Additionally, a recirculation zone is visible at

the inlet of the constant-area section in Figure 47d. This significantly reduces the cross-

section area of the secondary flow in the mixing chamber and contributes to the reduction of

the secondary mass flow rate. Further increasing the backpressure leads to ejector

malfunction, as shown in Figure 47e. The secondary flow is reversed and the primary jet

flows outwards through the secondary inlet.

Figure 47 – Mach number distribution for a SP of 2.5 mm and compression ratios of 1.1 (a), 1.2 (b), 1.3 (c), 1.4

(d), and 1.5 (e).

The performance curve of the ejector may be constructed by indicating the

entrainment ratio, as a function of the compression ratio, for each SP. Figure 48 illustrates

such a curve for three different SP: 1.5, 2.5 and 3.5 mm. The optimal operation line is also

shown. When the ejector operates with a backpressure under its critical value (left to the

critical point), the spindle may be moved downstream to further restrict the primary flow.

Not only does the primary mass flow rate decrease, but also the choked secondary mass flow

rate increases because of a higher effective area at the hypothetical throat. Consequently, a

significantly higher entrainment ratio is achieved. In contrast, if the ejector operates in

subcritical mode due to a high backpressure, the spindle should be shifted upstream as to

increase the primary nozzle throat area and obtain a higher primary mass flow rate. The

stronger primary jet may now transfer momentum to the secondary stream more efficiently,

increasing the secondary mass flow rate. Globally, the entrainment ratio increases.



Figure 48 – Entrainment ratio as a function of compression ratio for SP of 1.5, 2.5 and 3.5 mm, and optimal

operation line.

Ejector performance for fixed compression ratio

In this section, the effect of SP on ejector performance is analysed for a fixed

compression ratio. Figure 49 shows the primary and secondary mass flow rates and the

entrainment ratio as a function of SP, for a compression ratio of 1.3. As the SP decreases, the

spindle moves downstream and restricts the primary flow. Therefore, the primary mass flow

rate decreases with a decreasing SP, as expected. For a SP>3 mm, the secondary stream is

choked; therefore, the secondary mass flow rate is only limited by the effective area at the

hypothetical throat. A lower primary mass flow rate means a lower cross-section area of the

expanded primary jet, resulting in a higher area for the secondary stream to be entrained.

Consequently, the secondary mass flow rate and the entrainment ratio increase with a

decrease in the SP. Given that the secondary flow remains choked, the entrainment ratio

reaches a maximum value. For a compression ratio of 1.3, the optimum SP is 2.5 mm because

it maximizes the entrainment ratio. As the SP continues to decrease, so does the primary mass

flow rate. Consequently, the primary jet’s ability to transfer momentum to the secondary

stream, accelerating and entraining it into the mixing chamber, is compromised. In this case,

the secondary mass flow rate drops rapidly because the ejector operates in subcritical mode,

and the entrainment ratio decreases. For a SP<2 mm, the secondary flow is reversed, and the

ejector operates in backflow mode.



Figure 49 – Primary and secondary mass flow rates and entrainment ratio as a function of SP for a

compression ratio of 1.3.

Figure 50 shows the Mach number distribution for a compression ratio of 1.3 and SP

of 5 mm (a), 4 mm (b), 2 mm (c), and 1 mm (d), respectively. For a SP higher than its

optimum value (2.5 mm), the secondary stream reaches sonic speed (Figures 50a and 50b).

However, the hypothetical throat is larger in the second case on account of a smaller primary

jet expansion, resulting in higher secondary mass flow rate and entrainment ratio. By

decreasing the SP, it is therefore possible to improve the ejector performance. However,

moving the spindle downstream past its optimal position decreases the entrainment ratio. In

Figure 50c, the primary flow is no longer capable of accelerating the secondary stream to

sonic speed and the ejector operates in subcritical mode. By further restricting the primary

flow, the primary jet is unable to overcome the adverse pressure lift between the inlet and the

outlet, and the secondary flow is reversed (Figure 50d).

Figure 50 – Mach number distribution for a compression ratio of 1.3 and SP of 5 mm (a), 4 mm (b), 2 mm (c),

and 1 mm (d).

For each compression ratio, there is an optimum SP value that maximizes the

entrainment ratio. With this SP, the VGE operates with the lowest primary mass flow rate



that still allows for the secondary stream to reach sonic speed. Figure 51 shows the

entrainment ratio as a function of SP for compression ratios of 1.2, 1.3 and 1.4. As the

compression ratio increases, a smaller effective area at the hypothetical throat is necessary to

ensure the secondary stream is choked. This may be achieved by moving the spindle upstream

to increase the primary mass flow rate. The transfer of momentum is stronger and the

secondary stream achieves a higher velocity. Additionally, the expanded primary jet occupies

a larger cross-section area at the hypothetical throat. Consequently, the secondary stream

eventually reaches sonic condition for a sufficiently small effective area. In these conditions,

decreasing the backpressure has no impact on the entrainment ratio because both the primary

and secondary mass flow rates remain constant. The optimum SP value increases with the

compression ratio because a smaller effective area at the hypothetical throat is required for

the secondary stream to be choked.

Figure 51 – Entrainment ratio as a function of SP for compression ratios of 1.2, 1.3 and 1.4.

Overview of the numerical results

In the previous sections, ejector performance was analysed for both fixed SP and

fixed compression ratio. Figure 52 summarizes the results of all CFD simulations using the

ideal-gas model approach and shows the entrainment ratio as a function of SP for the different

compression ratios. From this figure, the analysis drawn in the previous sections is repeated:

there is an optimum SP that maximizes the entrainment ratio for each compression ratio.

Figure 52 – Entrainment ratio as a function of SP for different compression ratios.



Figure 53 shows the optimum SP as a function of the compression ratio. For a higher

backpressure, a lower effective area at the hypothetical throat is required for the ejector to

operate in critical mode (optimum performance). Consequently, the spindle must be moved

upstream to allow for a higher primary mass flow rate, resulting in a larger cross-section area

of the expanded primary jet. For a compression ratio of 1.1, an uncertainty remains whether

the optimum SP is 1 mm. In fact, an additional simulation would be necessary to verify

whether a SP of 0.5 mm would lead to a higher entrainment ratio depending on the obtained

ejector operating regime. However, such a small SP would lead to a very reduced primary

mass flow rate.

Figure 53 – Optimum SP as a function of compression ratio.

Figure 54 quantifies the improvement on ejector performance by comparing the

entrainment ratio obtained with a fixed-geometry ejector and a VGE, for each compression

ratio. The geometry of the ejector used in the previous simulations was calculated based on

a compression ratio of 1.3. Under these operating conditions, the ejector should show optimal

performance with the spindle fully retracted. However, a 55.9% improvement on the

entrainment ratio was obtained when adjusting the spindle to its optimum position (SP=2.5

mm). This discrepancy may be due to the simplicity of any 1-D model and its inability to

predict complex flow physics, such as shock waves and mixture layers. Moreover, the ejector

geometry was calculated based on real-gas properties of CO2, whereas the CFD simulations

were based on an ideal-gas approach. As mentioned in Section 5.1.1, the ideal-gas model

poorly predicts the real-gas properties of CO2 in its transcritical expansion within the primary

nozzle. For a higher compression ratio, no tangible performance improvement was expected.

In fact, these off-design conditions require a smaller hypothetical throat, i.e., a larger cross-

section area of the expanded primary jet; however, the primary mass flow rate cannot be

increased past its maximum value, corresponding to having the spindle fully retracted.

Nonetheless, a 24.1% increase on the entrainment ratio was obtained for a compression ratio

of 1.4 by adjusting the spindle. For a compression ratio of 1.5, only a slight improvement

was verified (0.47%), indicating that this compression ratio is closer to the real design

conditions of the ideal-gas ejector. A lower compression ratio allows for critical operation

for a larger hypothetical throat. Consequently, the spindle may be moved further downstream,

reducing the primary mass flow rate, and maximizing the entrainment ratio. As a result, a

more significant performance improvement may be achieved: 202% and 106% for

compression ratios of 1.1 and 1.2, respectively. Globally, simulation results indicate that a

VGE may allow for significant performance improvements for operation under a variable

backpressure.



Figure 54 – Entrainment ratio for a fixed-geometry ejector and a VGE with optimum SP, and improvement on

entrainment ratio.

5.2. Simulation results for the ejector flow with CO2 as real-gas

Applicability of the real-gas model

In this section, simulation results for the fully open spindle position case are analysed

and compared with the results for the ideal-gas simulations and the 1-D model. A primary

mass flow rate of 0.2625 kg/s is observed for a fully open spindle. This value is in agreement

with the 1-D model estimation of 0.2485 kg/s, showing only an error of 5.3%.

As mentioned in Section 5.1.1, one of the limitations of the ideal-gas model was the

poor description of the transcritical expansion process. In fact, the ideal-gas model

significantly underestimates the density of the supercritical CO2, resulting in an expansion

curve placed outside the wet vapor region (see Figure 45). In contrast, simulations conducted

with the real-gas model predict a transcritical expansion that takes the supercritical gas to the

wet vapor domain, as shown in Figure 55. Looking at the figure, one may note that the CFD

approach predicts an expansion curve that intersects the saturation lines near the critical

point, in agreement with the estimations of the 1-D model. Moreover, the near-critical

expansion is compatible with the application range of the HEM.



Figure 55 – Saturation lines and expansion process in the primary nozzle (1-D model and CFD assuming real-

gas behaviour).

Table 19 compares the pressure, enthalpy, and density results for the inlet, the nozzle throat,

and the exit section of the primary nozzle. For the primary inlet, pressure and enthalpy were

set as boundary conditions, hereby defining the density of the supercritical gas. At the

primary nozzle throat, the pressure predicted by the CFD simulations is 10.3% lower than

the estimations of the 1-D model. Density is also 12.5% lower in the simulation results. One

reason for this may be the fact that the 1-D model accounts for irreversibilities in the primary

nozzle convergent section through an empirical constant (isentropic efficiency). This simple

approach lacks in accuracy and the empirical coefficient needs to be experimentally verified

or adjusted by a more fundamental simulation method such as CFD (the effect of the

irreversibilities in the expansion process is addressed later in this section). At the exit section

of the primary nozzle, pressure is set as boundary condition and only a slight difference of

0.87% in enthalpy is registered between the 1-D model and numerical simulations. As a

result, it was considered that density is accurately predicted by the developed CFD model.

Moreover, the real-gas simulations indicate an average Mach number of 1.274 for the nozzle

exit section (see Section 4.2.5). This is in reasonable agreement with the value of 1.224

estimated by the 1-D model, showing a relative error of only 3.9%.

Table 19 – 1-D model estimations and real-gas simulation results for pressure, enthalpy, and density at the

primary inlet, primary nozzle throat, and primary nozzle exit section

Location 𝒑 [MPa] 𝒉 [kJ/kg] 𝝆 [kg/m3]

1-D Real-gas 1-D Real-gas 1-D Real-gas

Primary inlet 9.3 -146.2 435.3

Primary nozzle throat 5.8 5.2 -154.8 -157.4 273.3 239.1

Primary nozzle exit section 4.2 -161.1 -162.5 186.2 190.8

By analysing the numerical data in the transport equation for enthalpy (see equation

29), the order of magnitude of the variables is 102 for density, 102 for velocity, 105 for

enthalpy, and 10-3 for the dimensions of the ejector. This results in an order of magnitude of

1012 for the convective terms in the UDS equation. The order of magnitude of the source

terms should be in the same range for them to have an impact on the flow properties. The

distribution of the first source term (��ℎ1) in the transport equation for enthalpy is shown in

Figure 56 (the units are W/m3). This source term describes the flow’s mechanical energy and

ensures that the acceleration of the flow along the primary nozzle is coupled with a decrease

in enthalpy. This source term is intrinsically connected to the physics of the flow and



neglecting it leads to an unrealistic description of the physics of a supersonic flow. It greatly

influences the flow in the primary nozzle throat because the flow reaches supersonic speed

in this section. The second (��ℎ2) and third (��ℎ3

) source terms in equation 29 are dependent

on velocity and turbulent kinetic energy gradients, which are only significant near the wall

of the primary nozzle throat. The volume-averaged values for these source terms have an

order of magnitude of 106. Consequently, irreversibilities have little impact on the primary

nozzle and the transcritical expansion shown in Figure 55 closely follows an isentropic

process. In fact, neglecting these terms results in a variation of <0.05% on both the

entrainment ratio and the average Mach number at the nozzle exit section. Nonetheless, these

source terms may be relevant for the description of the complex flow physics of a complete

ejector flow including the turbulent mixing layer and shockwaves.

Figure 56 – Distribution of the first source term in the transport equation for enthalpy (source term in W/m3).

Figure 57 shows the axial distribution of the quality and Mach number in the primary

nozzle for the transcritical expansion. The fluid enters the nozzle in supercritical state (as

indicated by a quality of 1) with low velocity. The Mach number increases gradually as the

fluid enters the converging section of the primary nozzle. The resulting pressure decrease

eventually takes the supercritical gas into the wet vapour region. For an axial position of

about 7.5 mm, fluid quality drops below unity and condensation starts. The rapid decrease in

quality indicates that most of the condensation occurs in this section. This is in agreement

with the expansion process shown in Figure 55. Simultaneously, the speed of sound decreases

rapidly, resulting in a sudden increase in flow Mach number. The flow reaches sonic speed

(Ma=1) for an axial position of about 10 mm, i.e., in the nozzle throat. The flow further

expands in the diverging section of the nozzle and additional condensation occurs, as

indicated by the decreasing quality. At the nozzle exit section, the flow registers a Mach

number of 1.30 and a quality of 0.61.



Figure 57 – Quality and Mach number distributions along the primary nozzle axis.

In order to validate the geometry of the primary nozzle, the assumption of stagnation

conditions was evaluated. The average flow velocity at the primary inlet is 1.24 m/s, resulting

in an average dynamic pressure of 341 Pa. For a static pressure boundary condition of 9.3

MPa, this represents less than 1% of total flow pressure. In the exit section of the primary

nozzle, stagnation conditions are not verified due to the expansion of the primary jet. With

an average flow velocity of 178 m/s and an average density of 191 kg/m3, the dynamic

pressure at the nozzle exit section is 3.021 MPa. For a static pressure boundary condition of

4.233 MPa, this represents over 41% of total flow pressure. However, the outlet of the

calculation domain does not correspond to the outlet of a functioning ejector; therefore, no

limitation is imposed on the local velocity and dynamic pressure.

Overview of the numerical results

The primary mass flow rate is shown in Figure 58 as a function of SP. When the

spindle tip is further upstream from the primary nozzle throat (SP>6 mm), the mass flow rate

remains nearly constant. The primary flow reaches sonic speed in the primary nozzle throat,

where the cross-section area is minimum. In this case, the spindle does not restrict the primary

flow. A small decrease in the primary mass flow rate (from 0.2624 to 0.2615 kg/s) is visible

as SP moves from 7 to 6 mm. Moving the spindle downstream leads to a higher pressure loss

along the converging section of the primary nozzle. In addition to the flow interacting with

the spindle across a higher surface area, the average hydraulic diameter of the converging

section of the nozzle is reduced. Both these factors promote pressure loss, resulting in a lower

pressure upstream to the cross section where choking occurs and thus in a somewhat lower

mass flow rate. As the spindle moves from SP=7 mm to SP=6 mm, the average static pressure

at the throat is reduced from 5.18 MPa to 5.15 MPa. This effect is, however, negligible since

it produces a very small variation of the mass flow rate. As SP continues to decrease (SP<6

mm), the mass flow rate starts to drop rapidly. This is because the spindle restricts the primary

flow and forces it to reach sonic speed upstream to the nozzle throat. For a SP of 1 mm, the

primary mass flow rate is about 50% of the expected mass flow rate under design conditions.

A SP smaller than 1 mm would require a very low ejector compression ratio to ensure critical

operation, thus it was not considered in this work.



Figure 58 – Primary mass flow rate as a function of SP.

An additional aspect to analyse is the adequacy of spindle geometry for an accurate control

of the primary mass flow rate. Ideally, spindle geometry should allow for a linear variation

of the primary mass flow rate with SP, which is not verified by the simulation results. In fact,

given that the choking section has an annular geometry, its cross-section area is a quadratic

function of SP. A fixed variation of the SP has an increasing impact on the mass flow rate as

the spindle is positioned further downstream.

Figure 59 shows the Mach number distribution of the primary flow within the primary

nozzle for SP of 7 mm (a), 6 mm (b), 5 mm (c), and 4 mm (d). When the spindle is positioned

upstream it does not affect the primary flow, which reaches sonic condition in the primary

nozzle throat, as shown in Figures 59a and 59b. The constant cross-section area results in a

mass flow rate that is nearly independent from the SP. When the spindle moves further

downstream, it begins to affect the primary flow, as shown in Figure 59c. As a result, the

flow reaches Ma=1 upstream from the primary nozzle throat. The effective throat area is

determined by the SP, allowing for a significant adjustment of the primary mass flow rate.

Moving the spindle further downstream (see Figure 59d) means that the primary mass flow

rate continues to decrease, as the effective area is limited by the outer diameter of the spindle.

Figure 59 – Mach number distribution for SP of 7 mm (a), 6 mm (b), 5 mm (c), and 4 mm (d).



6. Conclusions and suggestions for future work

6.1. Conclusions

Decarbonization in all sectors of human activity, including the building sector, is

essential to ensure sustainable development and secure life for the future generations and our

planet in general. Cooling and heating solutions will require efficient systems driven by

renewable energy and using environment-friendly working fluids. Transcritical CO2 heat

pumps driven by renewable electricity satisfy these conditions, however there is a need for

improving their efficiency. Using an ejector for partial recovery of the expansion work is a

promising option as long as they can respond to variable operating conditions with high

performance. The present work focuses on the assessment of a variable geometry ejector

concept using a numerical approach. The primary objective was to develop an adequate

mathematical model, and to analyse the performance of a transcritical CO2 ejector with

variable geometry under variable operating conditions and assess its benefits relative to a

fixed-geometry ejector. In addition, the Homogeneous Equilibrium Model is implemented

for the modelling of CO2 as real-gas.

Ejector geometry was estimated for a heat pump cycle designed for the Portuguese

climate, with a heating capacity of 20 kW, and capable of operating on heating mode in cold

weather and on cooling mode in hot weather. The most suitable design conditions were found

to be as follows: primary pressure and temperature of 9.3 MPa and 43ºC, respectively;

secondary pressure and temperature of 3.77 MPa and 8ºC, respectively; outlet pressure of

4.90 MPa. The diameters of the primary nozzle throat and the constant-area section were

estimated at 2.972 mm and 4.636 mm, respectively. The area ratio was adjusted with SP

ranging from 1 to 7 mm. Two different CFD models were developed: a conventional ideal-

gas model and a real-gas model using the HEM approach. For the ideal-gas model, mesh

sensitivity results indicated that a structured numerical grid with 75854 CVs produced mesh

independent results for the primary and secondary mass flow rates and the entrainment ratio.

For the real-gas approach, mesh independence of the results (primary mass flow rate and

average Mach number at the primary nozzle exit section) was verified for a structured grid

with 4712 elements in the primary nozzle.

The simulation results with the ideal-gas model indicate that a VGE may maintain

good performance for a wider range of operating conditions than a fixed-geometry ejector.

As the compression ratio decreases from its design value, the spindle may be shifted

downstream to restrict the primary flow and entrain a higher secondary mass flow rate,

significantly improving the entrainment ratio. For compression ratios of 1.1 and 1.2, 202%

and 106% improvements on the entrainment ratio were obtained for optimum spindle

position, respectively. However, the ideal-gas model poorly predicts the density of the

supercritical fluid in the primary nozzle, resulting in an expansion process that significantly

differs from the 1-D model estimations. Moreover, ejector geometry was estimated based on

real-gas properties of CO2, resulting in a geometry that is not optimal when simulating carbon

dioxide as ideal-gas. In fact, adjusting the spindle allowed for a significant performance

improvement even for design conditions. In addition, recirculation zones and shockwaves in

the primary nozzle were observed. These flow phenomena should be avoided as they

compromise ejector efficiency and indicate that the ejector is not operating under optimum

conditions.



HEM presented convergence difficulties, and first-order upwind schemes were

necessary for the discretization of the momentum equations to allow for a stable calculation.

Moreover, this model proved to be very sensitive to mesh geometry and many steps of mesh

optimization were required. When simulating CO2 as real-gas, obtaining convergence as

significantly more challenging than when using an ideal-gas model. In this work, only the

primary nozzle of the ejector was simulated. This allowed for an insightful analysis of the

HEM and the behaviour of CO2 in the transcritical expansion. Despite being the simplest

approach to the modelling of real-gas properties, the HEM predicts a transcritical expansion

into the wet vapor domain. For operation with a retracted spindle, reasonable agreement

between the 1-D model and simulation results was observed for the static pressure, enthalpy,

and density values at the primary nozzle. The irreversibilities of the transcritical expansion

have little impact on the flow within the primary nozzle, and may therefore be neglected;

however, simulation of the complete ejector may require that these terms be included in the

transport equation for enthalpy for an accurate description of the flow.

6.2. Suggestions for future work

A SpaceClaim® script was implemented to automate the process of generating the

ejector geometry. However, this approach was not adopted for mesh generation due to

internal software limitations, particularly the numbering of the different mesh blocks within

the software. It is recommended that the automation process be extended to the generation of

the numerical mesh in order to reduce the time needed for mesh optimization and sensitivity

analysis.

In this work, the first steps were taken for the implementation of the HEM into

FLUENT® for the simulation of the complex flow in a variable geometry ejector. However,

convergence difficulties limited the scope of the analysis to the simulation of the primary

nozzle and the use of first-order upwind schemes for the discretization of the momentum

equations. In future work, this model may be used for the simulation of a complete ejector.

Firstly, the geometry of the primary nozzle should be assessed for an adequate expansion of

the primary jet. Additionally, the use of second-order discretization schemes for the equations

of momentum was reported to lead to a more accurate description of flow physics.

Consequently, the simulation of a complete ejector may benefit from this approach.

In addition to the area ratio, the position of the nozzle exit plane is also known to have

an impact on ejector performance; thus, the CFD simulations of the transcritical VGE may

be extended with the analysis of the effect of the NXP on the entrainment ratio under variable

operating conditions. In addition, the impact of geometry ejector on the fluid quality at the

outlet is an interesting research topic. In this context, ejector geometry may be optimized to

ensure that the liquid and vapor fractions at the outlet match the entrainment ratio, allowing

for the performance of the cycle to be controlled by the ejector.



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Appendix I – SpaceClaim® script for automated geometry generation

The automated generation of the different geometries used in this work starts by

defining the sketch plane and creating a new sketch, as shown in Figure I.1. After these steps,

the geometry is generated by drawing consecutive lines that define the general contour of the

calculation domain. The first line starts at the origin of the referential (𝑥=𝑦=0) and XX and

YY designate the coordinates of the end point. For each of the following lines, the starting

point coincides with the finish point of the previous line. The finish point of the last line is

set as the origin to create a closed domain, after which a command is issued to end the sketch.

Figure I.1 – SpaceClaim® script for automated geometry generation.



Appendix II – C scripts for implementation of the HEM

As mentioned in Section 4.2.1, the implementation of a UDS transport equation for

enthalpy requires the definition of the diffusion coefficient and the source terms. The UDFs

written for this purpose are shown in Figures II.1 and II.2, respectively.

Figure II.1 – DEFINE_DIFFUSIVITY UDF for the definition of the diffusion coefficient in the UDS transport

equation.

Figure II.2 – DEFINE_SOURCE UDFs for the definition of the source terms in the UDS transport equation.



A dedicated UDF was written to compile the property tables built in EES® into

FLUENT®. Several distinct data structures are created: an ordered list containing the

pressure values, as well as the enthalpy of saturated liquid and saturated vapor at that same

pressure; another ordered list containing the enthalpy values; a matrix for each property,

organized by both pressure (columns) and enthalpy (rows). This is shown in Figure II.3.



Figure II.3 – EXECUTE_ON_LOADING UDFs for the compilation of the property lookup tables.

The initialization of specific heat required an additional UDF, as shown in Figure II.4.

Figure II.4 – DEFINE_INIT UDF for the initialization of specific heat.

For density, thermal conductivity, and molecular viscosity, three

DEFINE_PROPERTY UDFs were written so that the solver could determine the necessary

fluid properties. In each of these UDFs, the algorithm determines the position of the CV’s

pressure and enthalpy values within the ordered lists created upon compiling the property

lookup tables. The corresponding fluid property is determined through bilinear interpolation,

as shown in Figures II.5, II.6, and II.7 for density, thermal conductivity, and molecular

viscosity, respectively.



Figure II.5 – DEFINE_PROPERTY UDF for the interpolation of density.



Figure II.6 – DEFINE_PROPERTY UDF for the interpolation of thermal conductivity.



Figure II.7 – DEFINE_PROPERTY UDF for the interpolation of molecular viscosity.

A similar approach was adopted for the interpolation of specific heat. However,

because the solver only allows for the implementation of a temperature-dependent specific

heat, this property was stored as a User-Defined Memory and a DEFINE_ADJUST UDF was

written for its calculation, as shown in Figure II.8



Figure II.8 – DEFINE_ADJUST UDF for the interpolation of specific heat.

Mach number, temperature, and quality distributions are calculated only for the

converged simulation using DEFINE_ON_DEMAND UDF, as shown in Figures II.9, II.10,

and II.11, respectively. These UDFs have a similar structure to the DEFINE_PROPERTY

and DEFINE_ADJUST UDFs used to determine fluid properties.



Figure II.9 – DEFINE_ON_DEMAND UDF for the calculation of Mach number.



Figure II.10 – DEFINE_ON_DEMAND UDF for the calculation of temperature.



Figure II.11 – DEFINE_ON_DEMAND UDF for the calculation of quality.



A final UDF was written to determine whether fluid properties for the converged

simulation were adequately calculated. This UDF compares the values of pressure and

enthalpy in each cell to the lower and upper limits on the property tables, and returns this

information to the user, as shown in Figure II.12.

Figure II.12 – DEFINE_ON_DEMAND UDF to check pressure and enthalpy limits.

To enable certain macros and calculation capabilities of the solver, it was necessary

to refer to the header files in which they are defined. For example, the udf.h header file defines

all the macros used in the UDFs, and therefore must be included in every script. The mem.h

header file was included to allow the solver to access material property macros in each cell,



returning values of pressure, velocity, thermal conductivity, and turbulent viscosity, as well

as their respective gradients. The metric.h header file terms allows the solver to access the

coordinates of the cell centroid. The external.h header file was created to allow for the

declaration of global variables, i.e., variables that are to be accessed by different scripts and

UDFs, as shown in Figure II.13. With this data structure, it is possible to build different

scripts for reading and interpolating property values, as they are retained as global variables.

Figure II.13 – Header file for declaration of global variables.



Appendix III – EES® script for generation of property lookup tables

The lookup tables for the real-gas properties of CO2 were built in EES®. For this

purpose, the script shown in Figure III.1 was written. The user defines the desired number of

interpolation points on pressure and enthalpy, as well as the respective lower and upper

limits. The script first calculates and writes an array containing an ordered list of the pressure

and enthalpy values, so that these may be compiled as a separate file into FLUENT®. For

each value of pressure, the enthalpy for saturated liquid and saturated vapor is also written

so that fluid quality may be calculated a posteriori. For each combination of pressure and

enthalpy, density and temperature are directly returned by the software. For the remaining

properties, this procedure is not valid in the wet vapor region. To address this matter,

auxiliary functions were written to determine whether the fluid is in wet vapor state, in which

case the desired property is obtained by a linear interpolation between the respective value

in the saturated liquid and saturated vapor states, based on the quality. This approach is used

to determine viscosity, thermal conductivity, and specific heat. Speed of sound, however,

requires yet another approach. The model used to estimate the general dimensions of the

ejector adopted the mathematical formulation developed by Lund and Flåtten [128]. To

maintain coherency, this model was also employed to determine the speed of sound in the

two-phase region.



Figure III.1 – EES(R) script for generation of property lookup tables.



Appendix IV – EES® script for calculation of nozzle exit pressure

When simulating the primary nozzle of the ejector with the HEM, moving the

adjustable downstream restricts the primary flow and leads to a smaller mass flow rate.

Consequently, the area ratio between the nozzle exit section and the effective throat of the

primary flow increases. Maintaining a constant pressure at the outlet of the calculation

domain would result in an over-expanded jet in the diverging section of the primary nozzle.

The EES® model used to estimate ejector dimensions was adapted to calculate the necessary

pressure value to ensure an adequate expansion of the primary jet. The physical dimensions

of the ejector are not calculated but fed as input to this model. For each SP, the mass flow

rate is calculated assuming Ma=1 at the throat.



Figure IV.1 – EES® script for calculation of nozzle exit pressure.

modelling of a transcritical co ejector with variable geometry

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