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

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 푘 − 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.

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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 푘 − 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.

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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.

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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

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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

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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

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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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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

푎 Speed of sound m.s-1 퐶 Model constant - -1 -1 퐶푝 Specific heat J.kg .K 퐷 Diameter M ℎ Specific enthalpy J.kg-1 퐼 Turbulence intensity - 푘 Thermal conductivity W.m-1.K-1 Turbulent kinetic energy (Section 3.3.2) m-2.s-2 푙 Turbulence length scale m 푚̇ Mass flow rate kg.s-1 푀̅ Universal gas constant J.mol-1.K-1 푀 Molecular weight kg.mol-1

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푀푎 Mach number - 푝 Pressure Pa 푅 Gas constant J.kg-1.K-1 푅푒 Reynolds number - 푠 Entropy J.kg-1.K-1 -3 푆ℎ̇ Source term in UDS transport equation W.m 푡 Time s 푇 Temperature K 풖 Velocity vector m.s-1 푢∗ Shear velocity m.s-1 푊̇ Work rate W 푦 Wall distance m 푦+ Dimensionless wall distance -

Greek symbols

훼 Phase volume fraction - 훾 Polytropic coefficient - 훤 Diffusion coefficient in UDS transport equation kg.m-1.s-1 훿 Kronecker delta - 휀 Turbulent dissipation rate (Section 3.3.2) m2.s-3 Error - 휂 Efficiency - 휇 Viscosity Pa.s 훱 Ejector compression ratio - 𝜌 Density Kg.m-3 𝜎 Prandtl number - 휏 Viscous stress Pa 휑 Generic flow variable - 휔 Mass entrainment ratio - 훻 Divergent operator -

Subscripts

푐푎 Constant-area section 표 Outlet 푝 Primary inlet/flow 푝푡 Primary nozzle throat 푠 Secondary inlet/flow 푇 Turbulence

Superscripts

̅ Time-averaged component ′ Fluctuating component

xiv INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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].

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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

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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].

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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].

4 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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

5 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY 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].

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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].

7 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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

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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

9 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY 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.

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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

11 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY 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

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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.

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14 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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

15 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY 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].

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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].

17 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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

18 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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].

19 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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

20 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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].

21 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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

22 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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 (푝푐푟푖푡) 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

23 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY 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 푝푚푎푥), 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 (휔) is the ratio of the secondary mass flow rate (푚̇ 푠) to the motive mass flow rate (푚̇ 푝) [26, 40, 41, 44, 49, 56, 60, 85] as:

푚̇ 푠 휔 = (1) 푚̇ 푝

The higher the mass flow rate, the better the efficiency of the cycle. The compression ratio (훱) is defined as the ratio of the ejector outlet pressure (푝표) to the secondary inlet pressure (푝푠) [40, 44, 49, 56, 60]. The pressure lift (푝푙푖푓푡) is the difference between the outlet and the

24 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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:

푝표 훱 = (2) 푝푠 푝푙푖푓푡 = 푝표 − 푝푠 (3)

Finally, the ejector isentropic efficiency (휂푒푗푒푐푡표푟) is the ratio between the work rate recovered in the ejector (푊̇푟) and the maximum work rate recovery potential for an isentropic expansion (푊̇푟,푚푎푥) [44]. The ejector isentropic efficiency is defined as:

푊̇ 푟 ℎ(푃7,푠5)−ℎ5 휂푒푗푒푐푡표푟 = = 휔 (4) 푊̇ 푟,푚푎푥 ℎ3−ℎ(푃7,푠3) 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.

25 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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 (푝푝) that leads to the maximum entrainment ratio [81, 90]. As 푝푝 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].

26 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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 (퐴푅) of an ejector is the ratio between the area of the constant-area section (퐴푐푎) and the area of the primary nozzle throat (퐴푝푡) according to the following formula:

퐴푐푎 퐴푅 = (5) 퐴푝푡

Figure 19 shows the effect of the area ratio on the performance of the ejector. For a given ∗ backpressure (푝표) 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

27 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY 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

28 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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

29 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY 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 푝표1 and an entrainment ratio 휔1. If the backpressure increases (푝표2), a fixed geometry ejector operates on subcritical mode and the entrainment ratio is reduced to 휔∗. However, with a VGE it is possible to reduce the area ratio and adjust its performance curve to obtain a higher entrainment ratio (휔2). Considering a fixed geometry ejector working on its critical point (푝표2, 휔2), the entrainment ratio remains constant if the backpressure decreases (푝표1). In contrast, the area ratio of a VGE may be increased to maximize the entrainment ratio (휔1). 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

30 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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.

31 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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.

32 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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].

33 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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] Non- Model Benefits Challenges/limitations equilibrium • Simplicity and stability • Accurate at supercritical • Does not consider meta- HEM None conditions stability • Extensively tested in literature • Considers meta-stability • Empirically based parameters • Extended with variable for relaxation time HRM Chemical relaxation time for • Requires tuning of subcritical conditions parameters • Increased complexity • Considers meta-stability • Requires tuning of model • Can more accurately parameters evaluate the phase • Less profound literature Mixture Chemical fractions by mass transfer database on carbon dioxide modelling ejectors • Highly accurate results • Not yet tested for low motive for motive flows pressures

34 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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

35 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY 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 (푇푝) 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 푇푝 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 (푇푠) 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.

36 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

Figure 25 summarizes the effect of 푇푝, 푇푠, and 훱 on the diameter of the primary nozzle throat (퐷푝푡), the entrainment ratio, and COP. An increase in 푇푝 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 푇푠 results in a higher evaporation pressure and a higher inlet pressure in the compressor. A higher primary mass flow rate (higher 퐷푝푡) 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 푇푠, 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 푇푠. However, further increasing 푇푠 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 푇푠. An increase in 훱 for fixed evaporator and gas cooler pressures has a similar effect to an increase in 푇푠. A higher primary mass flow rate is necessary due to the lower enthalpy of the fluid after compression, and so 퐷푝푡 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 훱, 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

37 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY 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 푻 [ºC] 43 푻 [ºC] 8 풑 풔 휫 [-] 1.3 20 kW 푷풑 [MPa] 9.30 푷풔 [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.

38 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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

39 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY 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).

40 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

Assuming stagnation conditions at both inlets and at the outlet is translated into the following mathematical formulation:

1 푝 = 푝 + 𝜌푢2 ≈ 푝 (6) 0 2

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 (푚̇ ) and density (𝜌) calculated by the 1-D model used to estimate the geometry of the ejector. The cross-section area (퐴) and velocity (푢) 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 풌품 풎̇ [ ] 0.2485 0.05958 0.3081 풔 풌품 흆 [ ] 435.3 100.7 187.1 풎ퟑ 푨 [풎풎ퟐ] 520.6 53.37 205.9 풎 풖 [ ] 1.097 11.10 8.000 풔

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 Primary inlet Secondary inlet Outlet 풑 [푷풂] 9.300x106 3.770x106 4.901x106 ퟏ 흆풖ퟐ[푷풂] 261.7 6188 5986 ퟐ ퟏ 흆풖ퟐ ퟐ [%] < 0.01 0.09 0.02 풑

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.

41 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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:

훻. (𝜌풖) = 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:

훻. (𝜌풖풖) = −훻푝 + 훻. 흉 (8)

In this work, all external forces were neglected. 흉 is the viscous shear stress tensor and refers to the viscous stresses acting on the fluid element. Each component 휏푖푗 is defined as follows for steady-state flows:

휕푢푖 휕푢푗 2 휕푢푘 휏푖푗 = 휇 [( + ) − 훿푖푗 ] (9) 휕푥푗 휕푥푖 3 휕푥푘 where 휇 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 (푅푒) 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:

𝜌푢퐷 푅푒 = (10) 퐷 휇

In equation 10, 퐷 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

42 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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 (푢) in its time-averaged (푢̅) and fluctuating (푢′) 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:

훻. (𝜌풖̅풖̅) = −훻푝̅ + 훻. (흉̅ + 흉푻) (11)

These equations govern the transport of the averaged flow quantities (풖̅), 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 (흉푻), account for the dissipative nature of the fluctuating velocities and are defined as:

̅̅´̅̅̅´ 휏푇푖푗 = −𝜌푢푖푢푗 (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]:

휕푢̅푖 휕푢̅푗 2 휕푢̅̅푘̅ 2 ̅̅´̅̅̅´ 휏푇푖푗 = −𝜌푢푖푢푗 = 휇푇 [( + ) − 훿푖푗 ] − 훿푖푗𝜌푘 (13) 휕푥푗 휕푥푖 3 휕푥푘 3

This approach introduces the concept of turbulent viscosity (휇푇). The molecular viscosity (휇) 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. 푘 − 휀 turbulence models define it as follows [112]:

43 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

푘2 휇 = 𝜌퐶 (14) 푇 휇 휀 where 퐶휇 is a constant. The 푘 − 휀 models are two-equation models because they involve the numerical solution of two transport equations, one for the turbulent kinetic energy (푘) and another one for the dissipation rate of turbulent kinetic energy (휀). In the standard 푘 − 휀 model, these are defined as follows [113]:

휇푇 훻. (𝜌풖푘) = 훻. [(휇 + ) 훻푘] + 퐺푘 + 퐺푏 − 𝜌휀 − 푌푀 (15) 𝜎푘 2 휇푇 휀 휀 (16) 훻. (𝜌풖휀) = 훻. [(휇 + ) 훻휀] + 퐶휀1 (퐺푘 + 퐶휀3퐺푏) − 퐶휀2𝜌 𝜎휀 푘 푘

𝜎푘 and 𝜎휀 are the turbulent Prandtl numbers for 푘 and 휀, respectively. 퐺푘 and 퐺푏 are the generation of 푘 due to the mean velocity gradients and buoyancy, respectively. 푌푀 is the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. 퐶휀1, 퐶휀2 and 퐶휀3 are constants depending on the variant of the 푘 − 휀 model used. The RNG 푘 − 휀 model is derived from the instantaneous Navier-Stokes equations using renormalization group (RNG) techniques. It is based on the standard 푘 − 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 푘 − 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 푘 − 휀 model more reliable, and extend its applicability to a wider range of flows, when compared to the standard 푘 − 휀 model [113]. The Realizable 푘 − 휀 model is a relatively recent improvement of the standard 푘 − 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 푘 − 휀 model is not realizable). The Realizable 푘 − 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 푘 − 휀 and the Realizable 푘 − 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 푘 − 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

44 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

elements in the mesh is reduced and simulations converge more quickly. The applicability of wall functions is assessed by the dimensionless wall distance (푦+), defined as follows:

𝜌푢∗푦 푦+ = (17) 휇

푦 is the distance of the cell node to the wall. 푢∗ is known as shear velocity and is defined as follows:

휏푤 푢∗ = √ (18) 𝜌

+ 휏푤 is the shear stress near the wall. 푦 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 푦+, 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 푦+ 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 푦+ 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 푘 − 휀 퐶휇 0.0845

퐶휀1 1.42

퐶휀2 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

45 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY 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 (휑) varies linearly between the neighboring cells. 휑푤 designates the value of 휑 on the left surface of control volume P and is calculated through linear interpolation between the value of 휑 on the center of control volumes W (휑푊) and P (휑푃). Similarly, the value of 휑 on the right surface of control volume P (휑푒) is linearly interpolated between the cell centre values for control volumes P and E (휑퐸). 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 휑푤 on control volume P is set equal to the cell-center value of the upstream cell (휑푊). The cell- center immediately upstream to 휑푒 refers to control volume P, therefore 휑푒 = 휑푃.

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

46 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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

47 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY gradients that occur near the ejector walls, and provides a better perception of the 푦+ 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

48 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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 휑 evolves asymptotically, as shown in Figure 34. As the number of elements in the mesh (푁) increases from 푁1 to 푁2, the results tend to vary (휀휑). 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.

49 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

50 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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:

풖2 훻. [𝜌풖 (퐶 푇 + )] = 훻. [푘훻푇 + 휏. 풖] (19) 푝 2 where 퐶푝 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:

풖2 훻. [𝜌풖 (퐶 푇 + )] = 훻. [(푘 + 푘 )훻푇 + (휏 + 휏 ). 풖] (20) 푝 2 푇 푇

Diffusive heat transfer is increased through a turbulent thermal conductivity (푘푇). Additionally, turbulent stresses (휏푇) 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 푘 − 휀 turbulence models, this effect is accounted for with the term 푌푀 for dilatation dissipation in the transport equation for 푘 (see equation 15), as follows [125]:

2 푌푀 = 2𝜌휀푀푎푇 (21)

In equation 21, 푀푎푇 is the turbulent Mach number, defined as:

푘 푀푎 = √ (22) 푇 푎2 where 푎 is the speed of sound, defined as:

푎 = √훾푅푇 (23)

The constant 훾 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:

51 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

푝 + 푝표푝 𝜌 = 푀̅ (24) 푀 푇 where 푝표푝 is the operating pressure, 푀̅ is the universal gas constant, and 푀 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 푇 , the software interpolates each property (퐶푃, 푘 and 휇 ).

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 Type Parameter Selected 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

52 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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

53 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY 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 (퐼) 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]:

1 − 퐼 = 0.16푅푒 8 (25) 퐷퐻

In equation 25, 푅푒퐷퐻 is the flow Reynolds number, calculated with the hydraulic diameter (퐷퐻) of the flow section. The turbulence length scale (푙) 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]:

푙 = 0.07퐷퐻 (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.

54 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

Table 10 – Estimation of hydraulic diameter, Reynolds number, turbulence intensity, and turbulence length scale for both inlets and the outlet Primary inlet Secondary inlet Outlet 푫푯 [풎풎] 17.6 6.09 16.2

3 3 3 푹풆푫푯 [−] 2.68x10 4.03x10 7.97x10

푰 [%] 3.35% 3.22% 2.93%

풍 [풎풎] 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 Turbulence Intensity (%) 5 Turbulence Length Scale (mm) 0.430 Outlet Type Pressure-outlet Pressure (MPa) 4.147 / 4.524 / 4.901 / 5.278 / 5.655 Turbulence Intensity (%) 10 Turbulence Length Scale (mm) 1.13 Walls Momentum Type Stationary wall Shear condition No slip Roughness models Standard Roughness height 0 Temperature Boundary condition Specified flux Boundary value 0

55 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

Mesh independence testing

The validity of the selected wall functions requires that the 푦+ 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 푦+ 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 푦+ 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 푦+ 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 푦+ 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 (휀휔) 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 Number of 풎̇ 휺 풎̇ 휺 흎 휺 Mesh 풑 풎̇ 풔 풔 풎̇ 푷 흎 elements [kg/s] [%] [kg/s] [%] [-] [%] 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).

56 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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:

풖2 훻. [𝜌풖 (ℎ + )] = 훻. [(훤훻ℎ) + 흉. 푢] (27) 2 with,

푘 훤 = ( ) 휕ℎ (28) 휕푇 푝

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:

̇ ̇ ̇ 훻. (𝜌풖ℎ) = 훻. (훤ℎ,푒푓푓훻ℎ) + 푆ℎ1 + 푆ℎ2 + 푆ℎ3 (29)

The effective diffusion coefficient (훤ℎ,푒푓푓) is the sum of the laminar diffusion coefficient in equation 28 and the turbulent diffusion coefficient (훤푇):

훤ℎ,푒푓푓 = 훤 + 훤푇 (30)

The turbulent diffusion coefficient is determined from:

휇푇 훤푇 = (31) 𝜎푇

̇ ̇ ̇ In equation 31, 𝜎푇 is the turbulent Prandtl number. The source terms 푆ℎ1, 푆ℎ2 and 푆ℎ3 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:

̇ 푆ℎ1 = 풖. 훻푝 (32) 휕푢 2 휕푣 2 휕푤 2 휕푢 휕푣 2 푆̇ = (휇 + 휇 ) {2 [( ) + ( ) + ( ) ] + ( + ) ℎ2 푇 휕푥 휕푦 휕푧 휕푦 휕푥 (33) 휕푢 휕푤 2 휕푣 휕푤 2 2 2 + ( + ) + ( + ) − (훻. 풖)2} − 𝜌푘훻. 풖 휕푧 휕푥 휕푧 휕푦 3 3 ̇ 푆ℎ3 = −𝜌풖. 훻푘 (34)

57 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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]:

휕𝜌휑 + 훻. (𝜌풖휑) = 훻. (훤훻휑) + 푆̇ (35) 휕푡 휑

In equation 35, 휑 is an arbitrary scalar, 훤 is the diffusion coefficient, and 푆휑̇ 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).

58 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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.

59 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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 Type Parameter Selected 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

60 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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

61 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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 Turbulence Intensity (%) 5 Turbulence Length Scale (mm) 1.23 Primary nozzle exit section Type Pressure-outlet Pressure (MPa) Defined for each SP Turbulence Intensity (%) 5 Turbulence Length Scale (mm) 0.220 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 푦+ 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 (푀푎푁푋) was assessed as a relevant output variable. This variable is affected by both the velocity and enthalpy fields. 푀푎푁푋 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.

62 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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

Mesh Number of elements 풎̇ 풑 [kg/s] 휺풎̇ 풑 [%] 푴풂푵푿 [-] 휺푴풂푵푿 [-] 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.

63 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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 풎̇ [kg/s] 휺풎̇ [%] 푴풂 [kg/s] 휺풎̇ [%] Pressure Enthalpy Total 풑 풑 푵푿 푴풂푵푿 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

64 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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 푝 and 푀푎, 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.

65 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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.

66 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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 풑 [MPa] 흆 [kg/m3] Location 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.

67 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

Table 18 – Confirmation of stagnation conditions at the inlets and the outlet for the ideal-gas simulations Primary inlet Secondary inlet Outlet

풖 [m/s] 2.26 24.4 15.1

풑 [Pa] 9.30x106 3.75 x106 4.90 x106

ퟏ 흆풖ퟐ [Pa] 411.38 20715 9702.6 ퟐ ퟏ 흆풖ퟐ ퟐ [%] <0.01 0.55 0.20 풑

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 훱. 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

68 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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 (푀푎<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.

69 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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.

70 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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

71 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY 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.

72 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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.

73 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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.

74 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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 풑 [MPa] 풉 [kJ/kg] 흆 [kg/m3] Location 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 (푆ℎ1) 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

75 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY 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 (푆ℎ2) and third (푆ℎ3) 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.

76 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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.

77 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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).

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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.

79 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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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7. References

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74. J. P. Liu, L.P.C., Z. J. Chen, Thermodynamic analysis on transcritical R744 vapor- compression/ejection hybrid refrigeration cycle, in 5th IIR-Gustav Lorentzen Conference on Natural Working Fluids. 2002: Guangzhou, China. p. 184-188. 75. Jeong, J., Saito, K., Kawai, S., Yoshikawa, C., Hattori, K., Efficiency enhancement of vapor compression refrigerator using natural working fluids with two-phase flow ejector, in 6th IIR- Gustav Lorentzen Conference on Natural Working Fluids. 2004: Glasgow, UK. 76. Carel, EmJ - Electronic Modulating Ejector. https://www.carel.com/emj-electronic- modulating-ejector. 77. Danfoss, Multi Ejector Solution. https://www.danfoss.com/en/products/valves/dcs/electric- expansion-valves/multi-ejector-solution/#tab-overview. 78. Saikawa, M., An Inside Story Behind the Advent of “Eco Cute” CO2 Heat Pump Water Heater for Residential Use, in Heat Pumping Technologies. 2015: https://heatpumpingtechnologies.org/publications/an-inside-story-behind-the-advent-of-eco- cute-co2-heat-pump-water-heater-for-residential-use/. 79. Bartosiewicz, Y., et al., Numerical and experimental investigations on supersonic ejectors. International Journal of Heat and Fluid Flow, 2005. 26(1): p. 56-70. 80. Pereira, P.R., et al., Experimental results with a variable geometry ejector using R600a as working fluid. International Journal of Refrigeration, 2014. 46: p. 77-85. 81. Galanis, N. and M. Sorin, Ejector design and performance prediction. International Journal of Thermal Sciences, 2016. 104: p. 315-329. 82. He, S., Y. Li, and R.Z. Wang, Progress of mathematical modeling on ejectors. Renewable and Sustainable Energy Reviews, 2009. 13(8): p. 1760-1780. 83. Munday, J.T. and D.F. Bagster, A New Ejector Theory Applied to Steam Jet Refrigeration. Industrial & Engineering Chemistry Process Design and Development, 1977. 16(4): p. 442- 449. 84. Chou, S.K., P.R. Yang, and C. Yap, Maximum mass flow ratio due to secondary flow choking in an ejector refrigeration system. International Journal of Refrigeration, 2001. 24(6): p. 486- 499. 85. Chunnanond, K. and S. Aphornratana, Ejectors: applications in refrigeration technology. Renewable and Sustainable Energy Reviews, 2004. 8(2): p. 129-155. 86. Pianthong, K., et al., Investigation and improvement of ejector refrigeration system using computational fluid dynamics technique. Energy Conversion and Management, 2007. 48(9): p. 2556-2564. 87. Eames, I.W., A new prescription for the design of supersonic jet-pumps: the constant rate of momentum change method. Applied Thermal Engineering, 2002. 22(2): p. 121-131. 88. Sun, D.-W., Variable geometry ejectors and their applications in ejector refrigeration systems. Energy, 1996. 21(10): p. 919-929. 89. Chunnanond, K. and S. Aphornratana, An experimental investigation of a steam ejector refrigerator: the analysis of the pressure profile along the ejector. Applied Thermal Engineering, 2004. 24(2): p. 311-322. 90. Wang, L., et al., Auto-tuning ejector for refrigeration system. Energy, 2018. 161: p. 536-543. 91. He, Y., et al., Synergistic effect of geometric parameters on CO2 ejector based on local exergy destruction analysis. Applied Thermal Engineering, 2021. 184: p. 116256. 92. Jia, Y. and C. Wenjian, Area ratio effects to the performance of air-cooled ejector refrigeration cycle with R134a refrigerant. Energy Conversion and Management, 2012. 53(1): p. 240-246. 93. Fu, W., et al., Numerical study for the influences of primary nozzle on steam ejector performance. Applied Thermal Engineering, 2016. 106: p. 1148-1156. 94. Zhu, Y., et al., Numerical investigation of geometry parameters for design of high performance ejectors. Applied Thermal Engineering, 2009. 29(5): p. 898-905. 95. Wu, H., et al., Numerical investigation of the influences of mixing chamber geometries on steam ejector performance. Desalination, 2014. 353: p. 15-20. 96. Banasiak, K., A. Hafner, and T. Andresen, Experimental and numerical investigation of the influence of the two-phase ejector geometry on the performance of the R744 heat pump. International Journal of Refrigeration, 2012. 35(6): p. 1617-1625. 97. Smolka, J., et al., Performance comparison of fixed- and controllable-geometry ejectors in a CO2 refrigeration system. International Journal of Refrigeration, 2016. 65: p. 172-182. 98. Varga, S., P.M.S. Lebre, and A.C. Oliveira, CFD study of a variable area ratio ejector using R600a and R152a refrigerants. International Journal of Refrigeration, 2013. 36(1): p. 157- 165.

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99. Smolka, J., et al., A computational model of a transcritical R744 ejector based on a homogeneous real fluid approach. Applied Mathematical Modelling, 2013. 37(3): p. 1208- 1224. 100. Palacz, M., et al., Application range of the HEM approach for CO2 expansion inside two- phase ejectors for supermarket refrigeration systems. International Journal of Refrigeration, 2015. 59: p. 251-258. 101. Palacz, M., et al., HEM and HRM accuracy comparison for the simulation of CO2 expansion in two-phase ejectors for supermarket refrigeration systems. Applied Thermal Engineering, 2017. 115: p. 160-169. 102. Haida, M., et al., Modified homogeneous relaxation model for the R744 trans-critical flow in a two-phase ejector. International Journal of Refrigeration, 2018. 85: p. 314-333. 103. Giacomelli, F., et al., Experimental and computational analysis of a R744 flashing ejector. International Journal of Refrigeration, 2019. 107: p. 326-343. 104. Bartosiewicz, Y. and J.-M. Seynhaeve, Delayed Equilibrium Model (DEM) of Flashing Choked Flows Relevant to LOCA and Implementation in System Codes. Vol. 2. 2014. 105. Angielczyk, W., et al., Prediction of critical mass rate of flashing carbon dioxide flow in convergent-divergent nozzle. Chemical Engineering and Processing - Process Intensification, 2019. 143: p. 107599. 106. Yazdani, M., A.A. Alahyari, and T.D. Radcliff, Numerical modeling of two-phase supersonic ejectors for work-recovery applications. International Journal of Heat and Mass Transfer, 2012. 55(21): p. 5744-5753. 107. Mazzelli, F., et al., Computational and experimental analysis of supersonic air ejector: Turbulence modeling and assessment of 3D effects. International Journal of Heat and Fluid Flow, 2015. 56: p. 305-316. 108. Giacomelli, F., F. Mazzelli, and A. Milazzo, A novel CFD approach for the computation of R744 flashing nozzles in compressible and metastable conditions. Energy, 2018. 162: p. 1092-1105. 109. Giacomelli, F., F. Mazzelli, and A. Milazzo, Evaporation in supersonic CO2 ejectors: analysis of theoretical and numerical models. 2016. 110. Marques, J.S., Desenvolvimento de um modelo de dimensionamento de ejetores transcríticos com fluido de trabalho CO2 para sistemas de refrigeração, in Departamento de Engenharia Mecânica. 2018, Faculdade de Engenharia da Universidade do Porto. 111. Banasiak, K. and A. Hafner, 1D Computational model of a two-phase R744 ejector for expansion work recovery. International Journal of Thermal Sciences, 2011. 50(11): p. 2235- 2247. 112. Sadrehaghighi, I., Turbulence Modeling. 2021. 113. ANSYS FLUENT 12.0 Theory Guide. 2009. 114. Varga, S., A.C. Oliveira, and B. Diaconu, Numerical assessment of steam ejector efficiencies using CFD. International Journal of Refrigeration, 2009. 32(6): p. 1203-1211. 115. Sriveerakul, T., S. Aphornratana, and K. Chunnanond, Performance prediction of steam ejector using computational fluid dynamics: Part 1. Validation of the CFD results. International Journal of Thermal Sciences, 2007. 46(8): p. 812-822. 116. Sriveerakul, T., S. Aphornratana, and K. Chunnanond, Performance prediction of steam ejector using computational fluid dynamics: Part 2. Flow structure of a steam ejector influenced by operating pressures and geometries. International Journal of Thermal Sciences, 2007. 46(8): p. 823-833. 117. Rusly, E., et al., CFD analysis of ejector in a combined ejector cooling system. International Journal of Refrigeration, 2005. 28(7): p. 1092-1101. 118. Marynowski, T., P. Desevaux, and Y. Mercadier, Experimental and Numerical Visualizations of Condensation Process in a Supersonic Ejector. J. Visualization, 2009. 12: p. 251-258. 119. Bartosiewicz, Y., Z. Aidoun, and Y. Mercadier, Numerical assessment of ejector operation for refrigeration applications based on CFD. Applied Thermal Engineering, 2006. 26(5): p. 604- 612. 120. J. H. Ferziger, M.P., Computational Methods for Fluid Dynamics. 4th ed. 2002: Springer. 121. Launder, B.E. and D.B. Spalding, The numerical computation of turbulent flows. Computer Methods in Applied Mechanics and Engineering, 1974. 3(2): p. 269-289. 122. Norton, T. and D.-W. Sun, Computational fluid dynamics (CFD) – an effective and efficient design and analysis tool for the food industry: A review. Trends in Food Science & Technology, 2006. 17(11): p. 600-620.

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123. Barth, T. and D. Jespersen. The design and application of upwind schemes on unstructured meshes. 1989. 124. Leonard, B. and S. Mokhtari, ULTRA-SHARP Nonoscillatory Convection Schemes for High- Speed Steady Multidimensional Flow. 1990. 125. Sarkar, S. and L. Balakrishnan, Application of a Reynolds-Stress Turbulence Model to the Compressible Shear Layer., in ICASE Report 90-18. 1990, NASA. 126. ANSYS FLUENT 12.0 User Guide. 2009. 127. Li, S., et al., Effects of working conditions on the performance of an ammonia ejector used in an ocean thermal energy conversion system. The Canadian Journal of Chemical Engineering. n/a(n/a). 128. Lund, H. and T. Flåtten, Equilibrium conditions and sound velocities in two-phase flows. 2010.

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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 (푥=푦=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.

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88 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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.

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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.

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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.

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Figure II.5 – DEFINE_PROPERTY UDF for the interpolation of density.

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Figure II.6 – DEFINE_PROPERTY UDF for the interpolation of thermal conductivity.

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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

94 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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.

95 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

Figure II.9 – DEFINE_ON_DEMAND UDF for the calculation of Mach number.

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Figure II.10 – DEFINE_ON_DEMAND UDF for the calculation of temperature.

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Figure II.11 – DEFINE_ON_DEMAND UDF for the calculation of quality.

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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,

99 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY 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.

100 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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.

101 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

102 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

103 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

104 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

Figure III.1 – EES(R) script for generation of property lookup tables.

105 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

106 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

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.

107 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

108 INTEGRATED MASTER OF MECHANICAL ENGINEERING MODELLING OF A TRANSCRITICAL CO2 EJECTOR WITH VARIABLE GEOMETRY

Figure IV.1 – EES® script for calculation of nozzle exit pressure.

109 INTEGRATED MASTER OF MECHANICAL ENGINEERING