Performance prediction of mercury ejector vacuum pumps as main component of the DEMO torus booster pumping system

Master Thesis 1 October 2022 – 31 March 2023 Performance prediction of mercury ejector vacuum pumps as main component of the DEMO torus booster pumping system by Clément RACINET at the KIT Department of Mechanical Engineering Examiner Prof. Dr.-Ing. Robert Stieglitz Institute for Applied Thermofluidic (IATF) Supervisor Dr.-Ing. Thomas Giegerich Institute for Technical Physics (ITEP) KIT – The Research University in the Helmholtz Association Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 2 of 94 Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 3 of 94 Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 4 of 94 DECLARATION OF OWN WORK I truthfully declare that I have written this thesis independently, that I have indicated all the aids used completely and accurately, and that I have indicated everything that has been taken from the work of others, either changed or with modifications, and that I have observed the KIT Statutes for the Safeguarding of Good Scientific Practice in the currently valid version. Ich versichere wahrheitsgemäß, die Arbeit selbstständig verfasst, alle benutzten Hilfsmittel vollständig und genau angegeben und alles kenntlich gemacht zu haben, was aus Arbeiten anderer unverändert oder mit Abänderungen entnommen wurde sowie die Satzung des KIT zur Sicherung guter wissenschaftlicher Praxis in der jeweils gültigen Fassung beachtet zu haben. I also comply that this Master Thesis is appointed in the library and is allowed to be copied. Karlsruhe, 31.03.2023 …..…..…..…..…..…..…..….. (Clément Racinet) Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 5 of 94 Acknowledgement I want to thank Dr.˗Ing. T. Giegerich for this interesting work and the assistance he gave me the last six months. I also want to thank Dr.˗Ing. C. Day who enabled this work in his task force group and Prof. Dr.˗Ing R. Stieglitz for being the supervisor of my diploma thesis. I am particularly grateful to T. Teichmann for his precious help and his knowledge that he was always ready to share with me as well as his efficiency to supervise me during these 6 months. Additionally, I want to thank A. Uihlein for the interesting discussions, for her precious advices and her daily motivation. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 6 of 94 Abbreviations Notation Meaning 1-D One-dimensional CFD Computational Fluid Dynamics DEMO Demonstration Fusion Power plant DIRL Direct Internal Recycling Loop FVM Finite Volume Method ITEP Institute for Technical Physics ITER International Thermonuclear Experimental Reactor INTL Inner Tritium Plant Loop KIT Karlsruhe Institute of Technology Kn Knudsen Number LDP Linear Diffusion Pump LRP Liquid Ring Pump Ma Mach Number MFP Metal Foil Pump NXP Nozzle X-axis Position (axial distance between nozzle and mixing chamber) OUTL Outer Tritium Plant Loop RANS equations Reynolds Averaged Navier-Stokes equations Z Atomic Number Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 7 of 94 Quantities Notation Notation Standard Unit Meaning A [m2] Area c [m s-1] Light celerity cP [J K-1 mol-1] Isobaric molar heat capacity cV [J K-1 mol-1] Isochoric molar heat capacity d [m] Diameter E [J] Energy g [m s-2] Earth gravitational constant Kn [-] Mach number L [m] Length Ma [-] Mach number M [kg mol-1] Molar mass m [kg] mass 𝐦󰇗 [kg s-1] Mass flow rate P [Pa] Pressure R [J K-1 mol-1] Perfect gas constant R* [J K-1 kg-1] Specific gas constant T [K] Temperature t [s] Time u [-] Entrainment ratio v [m s-1] Velocity x [-] Molar fraction z [-] Compressibility ratio Z [-] Atomic number γ [-] Heat capacity ratio ε [K] Lennard-Jones characteristic energy η [-] Efficiencies κ [W m−1 K−1] Heat conductivity λ [m] Free mean path μ [Pa s] Dynamic viscosity ρ [kg m-3] density σ [m] Lennard-Jones characteristic length τ [-] Diameter ratio χ [-] Pressure ratio Ψ [-] Compression ratio Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 8 of 94 Index notations Notation Meaning 0 at the inlet d In the diffuser (after the second shock wave) c At the outlet C characteristic cc At the critical point cb At the breaking point m mixed fluid p primary fluid s secondary fluid t at the nozzle throat y at the y-y section Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 9 of 94 Abstract A continuous pumping solution for a fusion reactor is currently being developed at the Karlsruhe Institute of Technology (KIT). This allows the tritium inventory to be drastically reduced compared to current discontinuous pumping systems based on cryogenic pumps. In this concept, a mercury diffusion pump is used as the high vacuum pump and a liquid ring pump, also with mercury as the operating liquid, as the backing pump. Mercury is used because of its good tritium compatibility. However, as things stand at present, the suction pressure of the backing pumps and the delivery pressure of the diffusion pumps are not compatible. A considerable pressure difference must therefore be bridged to connect these two types of pumps. Ejector pumps have been identified as a possible pumping solution for this pressure difference. These pumps can also be operated with mercury as the primary fluid to ensure compatibility with tritium. This work investigates whether and how such a pump can be used for the purpose described above. For this purpose, a new 1D model is first developed based on literature models. This model can be used to derive a pre-design for ejector pump stages quickly and cheaply. Subsequently, this design can be validated and optimised by 3D simulations using the finite volume method. Since ejectors in industry are typically used with identical operating and process media (usually steam), existing models are based on this assumption. One of the key points of the present work is therefore the extension of both 1D and 3D models to gas mixtures. The extended models are validated with literature data, in particular also with mercury as working medium. Finally, the new 1D model is used to propose a functional, multi-stage solution for the pumping system and to show potential optimisations regarding the required number of ejector and ring pump stages. Furthermore, the 3D finite volume model is used as an example to quantify the accuracy of the 1D model. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 16 of 94 2 Vacuum and tritium cycle 2.1 The DEMO fuel cycle In order to achieve optimal fusion conditions, the plasma must be maintained at high purity. So, to ensure the minimal presence of contaminants and to exhaust the fusion product helium, the torus must be continuously pumped to a fine vacuum pressure [12]. However, as previously stated, tritium is very expensive and restricted due to its hazardous nature. As a result, tritium in the exhaust gas must be entirely recycled. Without significant alterations to the fuel cycle design, scaling up from ITER to a commercial reactor would result in a tritium inventory of more than ten kilograms [13], which is seen as a show-stopper for such a machine (safety, licencing and availability problems). To solve this problem, a 3-loop fuel cycle design was developed to minimize the amount and duration of tritium inventory build-up (see Fig. 5) [13]. Metal Foil Pumps (MFP) are used to separate unburnt D-T fuel from the exhaust gas mixture [14]. It is projected that 80% of the D-T in the exhaust stream will be recycled by the Direct Internal Recycling Loop (DIRL). The remaining 20% can be recycled in a second loop called Inner Tritium Plant Loop (INTL). This second loop is slower, but will allow for an exhaustive separation of the isotopes. Finally, a third and last loop, the Outer Tritium Plant Loop (OUTL) is planned to carry out various tasks that are not part of the recycling of elements present in the tokamak. In particular, this loop will handle the production of tritium from the lithium blankets and will serve as an active air detritiation circuit to ensure that the system is perfectly sealed. 2.2 The DEMO torus pumping system At present fusion devices, cryogenic pumps are used as primary vacuum pumps in order to achieve the high vacuum required. This type of pumps works in the following way: walls cooled to a very low temperature are used to catch the particles that come into contact with them [15]. Thus, cryogenic pumps accumulate the pumped material and require a regeneration time in order to release it. This phenomenon generates a significant increase in the tritium inventory. In order to pump continuously and to limit the storage of tritium as much as possible, the idea of a non-cryogenic pumping system is followed (some examples are given in [16]). Because mercury is perfectly tritium compatible, a pumping system based on mercury pumps is currently being developed as shown in Fig. 6. The primary pumps are mercury-driven Linear Diffusion Pumps (LDPs) which can reach the high vacuum required. These are backed by also mercury-driven Liquid Ring Pumps (LRPs) which work at higher pressure and compress to ambient [17–19]. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 17 of 94 Fig. 5: Systematic illustration of interlink between residence time and tritium inventory in the three-loop fuel cycle, taken from [13]. Fig. 6: Example of a mercury pumping system for DEMO, taken from [17]. 2.3 The intermediate pumping stage (Booster Pump) As we can see in Fig. 7 one stage of a LRP as expected for DEMO can only achieve around 20 000 Pa inlet pressure. With several stages of LRP in series, an ultimate pressure of 400 Pa would be achievable [20]. On the other hand, as shown in Fig. 8, one stage of LDP can reach outlet pressures of about 1 Pa before their performance declines. Using 3 stages in a LDP, it is believed that outlet pressures of 10 Pa could be achieved [20]. Therefore, there is a gap between the highest expected outlet pressure of the LDPs and the lowest achievable inlet pressure of the LRP. In order to cover the pressure gap between the LDP and the LRP, a new type of pump should be found. Knowing that for each additional pumping stage, the number of LRP required increases drastically, reach a LRP inlet pressure of 1 000 Pa would reduce the number of LRPs by 35 %, rise to 2 500 Pa would reduce it by more than 55% and achieving 7 000 Pa would reduce this number by more than 75%. Since LRPs are linked to space needs and complex maintenance work, limiting the number of LRPs would be ideal. The purpose of this study is to determine whether this goal can be reached by mercury powered ejector pumps in order to have a complete and effective pumping system for DEMO. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 18 of 94 Fig. 7: Mercury liquid ring pump pumping curve for one stage, taken from [21]. Fig. 8: Mercury diffusion pump pumping curve for one stage, taken from [19]. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 19 of 94 3 Ejector vacuum pumping basics 3.1 Description of the working principle Ejectors are device without moving parts that may pump, compress, or mix gases, vapours, or liquids due to the expansion of a primary fluid. The primary fluid supplies the momentum required to propel the secondary fluid. In our case, the primary fluid would be mercury vapour, while the secondary fluid is the exhaust gas mixture being pumped. A primary nozzle and a secondary nozzle are part of a traditional ejector. The latter consists of a diffuser and a receiving chamber with a constant section to ensure the mixing of the two fluids. The primary nozzle of a gas ejector is convergent-divergent (e.g. de Laval type) to produce a supersonic primary jet at its exit i.e. that its Mach Number Ma (function of the velocity v, the isentropic coefficient γ, the specific gas constant R* and the temperature T) is greater than 1: 𝑀𝑎=𝑣 √𝛾 𝑅∗𝑇 3.1-1 The basic working principle of an ejector pump is displayed in Fig. 9: the primary fluid (in our case mercury vapour) enters the primary nozzle at high pressure, and its energy is subsequently transformed into kinetic energy by expansion. Due to the Venturi effect, the static pressure of this primary fluid jet drops down, which entrains the secondary fluid (in our case the exhaust gas mixture). The entrainment ratio u is defined as the ratio of the secondary mass flow rate (𝑚󰇗𝑠) to primary mass flow rate (𝑚󰇗𝑝) (see equation 3.1-2). The two fluids mix uniformly in the mixing chamber. In the diffuser, the velocity of the mixture is reduced, converting part of its kinetic energy to static pressure. The outlet pressure of the mixture lies between the primary and secondary inlet pressures, so the ejector has provided a compression of the entrained gas [22–24]. 𝑢=𝑚󰇗𝑠 𝑚󰇗𝑝 3.1-2 Fig. 9: Schematic view and the variation of fluids pressure and velocity as a function of location along the ejector Inspired by [25] . Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 20 of 94 3.2 Discussion of operating conditions for standard vapour ejector pumps Thereafter, in order to fit in with the need for DEMO (see section 2), we will focus on vapour ejector pumps i.e. ejectors operating with gases as primary and secondary fluid. The three vapour ejector functioning modes are also depicted in Fig. 10 [22, 25, 26]: 1Critical mode: the entrainment ratio is constant and maximum as long as Pc<Pcc. In this mode, the primary fluid as well as the secondary fluid chock. The primary fluid chokes in the primary nozzle throat and the secondary fluid chokes in the mixing chamber. A shock wave to return to the subsonic regime appears in the diffuser (this is the case shown in Fig. 9). 2Subcritical mode: the entrainment ratio drops down linearly with Pc between Pcc and Pcb. In this transition phase, only the primary fluid chokes in the primary nozzle throat. However, its energy is sufficient to counter completely the outlet pressure. The system continues to pump but in a degraded way. 3Back-flow mode: The pumping action ceases (the pump will discharge through the secondary inlet), the entrainment ratio is then negative. In this case, the primary fluid has no longer enough energy to sustain against the outlet pressure. Leakage occurs and the outlet fluid can flow back through the secondary inlet. Then, two essential points can be highlighted: - the breakdown point, where 𝑢=0 and the backing pressure Pc is equal to the breakdown backing pressure Pcb. This point is reached when the primary jet can no longer sustain against the outlet pressure. - the critical point, where entrainment ratio u equals critical (and maximum) entrainment ratio and backing pressure Pc equals critical backing pressure Pcc. This point corresponds to the transition point where the mixed fluid ceases to shock in the diffuser. The boundary conditions have of course a very high influence on the working point of the pump. To perfectly constrain the pump, there are 5 degrees of freedom used as boundary conditions. Usually (also in this study), the primary and secondary inlet pressure and temperature added to the outlet pressure constitute the boundary conditions. Their effect on the pump is huge but well understood and, hence, predictable. In fact, as long as one remains in critical mode the mass flow rate of the primary fluid is linearly dependant on the primary pressure (see in section 4.2.1 below) and the secondary mass flow rate is directly proportional to the latter (u is constant in critical mode). On the other hand, the influence of the primary pressure affects the mass flow of the secondary fluid in a much more complex way. However, the influence of the secondary and primary pressures and temperature on the mass flow ratio versus outlet pressure curve always evolve in the same way (see Fig. 10), which makes it possible to adjust these parameters as best as possible according to the final needs once the pump has been dimensioned. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 21 of 94 Fig. 10: Global variation of entrainment ratio and critical point as function of the primary and secondary pressures and temperatures. Inspired by [26]. 3.3 Main geometrical parameters influencing the pumping performance Many different types of ejector designs exist, especially regarding the type of nozzle used. Some have two, three or more nozzles in series and/or in parallel. We will concentrate ourselves on the simplest version as the one shown in Fig. 11. Although the geometry of such an ejector pump is relatively simple, it still has several important parameters that determine the pump performance. Because of the number of parameters, it is important to have numerical tools with which one can perform parameter scans so as to understand their impact on the overall performance. Computational Fluid Dynamics software (CFD) is able to describe the flows in such pumps parametrically without having to manufacture and test a lot of these pumps in experiments. Despite this, each calculation made by CFD costs a lot in terms of computational time. In order to limit these costs and knowing that a lot of research on the sensitivity of these parameters has been done in the past, summarizing previous research [24, 27–32], it can be concluded that the most important parameters affecting the ejector performance are the three following ones: - The diameter ratio 𝜏=𝑑𝑚 𝑑𝑡 - The distance between primary nozzle mouth and mixing chamber (NXP) as shown on Fig. 11 - The dimensionless mixing length 𝐿𝑚 𝑑𝑚 In the present work, we focus on these parameters during the geometrical sensitivity study. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 22 of 94 Fig. 11: Main geometrical parameters in the design of a vacuum ejector. 4 Modelling of ejector pumps 4.1 State-of-the-art in ejector pump modelling In literature, two main types of simulations for ejectors are employed. The first type is a one-dimensional (1-D) simulation based on the conservation equation of the first law of thermodynamics as e.g. in [33–37], and as detailed in section 4.2. This modelling requires strong assumptions but provides results in the right order of magnitude with low computational complexity. The second type, much more frequently used in literature, is by the Finite Volume Method (FVM) as for example in [38–42] and detailed in section 4.3. This approach, if the discretization procedures are applied properly, is expected to give more precise results and is more flexible (for example by considering much better the geometrical parameters) but it has the disadvantage of being complex and much more demanding in terms of computing time and power. Both the 1-D and FVM modelling are based on the fluid mechanics equations, which means that the flow should stay in the continuum flow regime and not achieve the rarefied regime. To validate this assumption, the Knudsen Number (Kn) can be used. It is a dimensionless number defined as the ratio of the molecular mean free path length 𝜆 to a representative physical length scale LC: 𝐾𝑛=𝜆 𝐿𝐶 4.1-1 To compute the Knudsen number in our case, we use the mixing chamber diameter as characteristic length LC and the molecular mean free path length 𝜆 is taken from the literature [43]. To ensure that the pump is working on continuum flow, the Knudsen number should stay below 10-2 [44, 45]. 4.1.1 1-D modelling The basics of 1-D modelling introduced by Keenan et al. Keenan et al. introduced in [33] a one-dimensional ejector modelling method in 1950 under the sponsorship of the Elliott Company and then has taken part to a program at the Massachusetts Institute of Technology under the sponsorship of the Navy Department, Bureau of Ordnance Contract Nord 9661. The method they present is based on the assumptions of perfect and isentropic fluid. Being a pioneer, it is limited to the specific framework of pure air ejectors. They obtains very good results in this case. However, a number of their mixing equations are based on ASME formulae and therefore incorporate many experimental data. It is interesting to note that they discuss the mixing conditions of the fluids in detail and caution about the limited validity of their models. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 23 of 94 1-D simulation by Huang et al. Based on Keenan's et al. article [33], Huang et al. [34] rework and simplify Keenan's model so that it no longer refers to ASME equations. They use the same assumptions as those introduced by Keenan, i.e. those of a perfect isentropic fluid. They introduce 4 efficiency parameters and based on experiments, give experimental values of these efficiencies as a function of the fluid and the diameter ratio τ. They always work in a pure case but extend to other fluids such as steam or the refrigerant R141b. Amelioration and first mixture modelling by He et al. He et al. further develop the 1-D model by taking into account the entropy variation [35]. However, this requires a very detailed knowledge of the fluid used. They also address the case of modelling the mixture but only in the two-phase case (presence of the liquid and gas phase of the same fluid) but not in the case of two different fluids. Moreover, this modelling is mainly based on experimental data. They also present numerous experimental cases involving many fluids. The Kumar and Ooi modelling Kumar and Ooi present in [36] a more theoretical and less experimental model of an ejector based on the He et al. article [35]. They greatly simplify the equations used by He et al. and focus more specifically on Fanno flows. Fanno flow is adiabatic flow in a duct of constant area where the effect of friction is taken into account. It is noted that the equations they arrive at are very similar to those of Keenan et al. and Huang et al. [33]. Their case is however reduced the case of using R141b as the primary and secondary fluid. Li et al. model extension Based on the previous work, Li et al. present an algorithm to calculate the performance of the ejector not only in the critical case but also in the subcritical case. However, they model it only in a linear way. This allows the backing pressure to be calculated [37]. They also study in depth the efficiency parameters as introduced by Huang et al. for new fluids [33]. However, they are limited to the case where the primary and secondary fluids are the same. Conclusion of the state-of-the-art for 1-D modelling Thus, while there are several high-performance 1-D models, they are all mono-fluidic. Indeed, studying ejectors in the thermal context (especially for heat pumps), the models developed are very well adapted to this need, based largely on a perfect and thorough knowledge of the fluid used, which is not our case. An extension of these models based on stronger assumptions but requiring less knowledge of the fluid will be more suitable in our case. Furthermore, it will be necessary to extend our model to the specific case where the primary and secondary fluids are not the same. 4.1.2 FVM modelling There is a very large amount of data on FVM simulation of ejector pumps. We will therefore present here only three very complete articles that are perfectly representative of the literature on the FVM simulation of ejectors. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 24 of 94 Sriveerakul et al. Sriveerakul et al. present a paper in two parts on pure steam ejectors [38, 39]. They compare FVM simulations they have performed with the corresponding experiments also performed by them. Their experimental protocol has a huge positive point, which is that they are able to take a pressure reading at several points of the ejector as shown in Fig. 12. Thus, a local comparison of the values is possible and one is much more precise on the analysis of possible deviations. Thus, it can be noticed that the largest areas of difference appear in the mixing chamber. Moreover, their simulation also extends to subcritical mode. Fig. 12: Test bench with multiple pressure points. Taken from [38]. Croquer et al. Croquer et al. published a two-part article simulating the case of a single-fluid ejector operating on R134a [40, 41]. This article presents a complete and detailed study of the solvers used and their influence on the results. They also study in an exhaustive way some hypotheses such as the ideal gas and several options of solvers frequently used (as we will detail in section 4.3). Sicuro et al. Sicuro et al. have published an article presenting an FVM model of an air ejector in the aeronautical context [42]. They have studied in depth the different solvers and the influence of these on the results. They also took a special interest in the throttling of the secondary fluid (as defined in section 4.2), which is a key point of this study. Then, they performed a sensitivity study and an optimization of the ejector geometry. Conclusion of the state-of-the-art for FVM modelling There are plenty of sources that perform FVM simulations. These allow enormous flexibility since, in addition to being able to take into account the entire geometry and weaker thermodynamic assumptions; they extend perfectly to both the critical and sub-critical regime. Moreover, these simulations allow highlighting complex phenomena such as the throttling of the secondary fluid or the type of shock wave that appears in the diffuser. They also allow a geometrical optimisation of the pump. However, this type of simulation has a significant disadvantage in terms of time and complexity of the calculations. If the FVM modelling is a tool that will be of great help to refine our results, it Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 25 of 94 remains very expensive and it seems unreasonable to carry out the whole of our work based only on this modelling. 4.1.3 Our modelling approach We can therefore note that despite a rich literature, there is no source that explicitly deals with cases where the primary and secondary fluids are very different. For modelling mercury ejector pumps we take the following two-stage approach: first, we start by implementing a one-dimensional model based on the Li et al. article [37] which we will adapt to the case of fluid mixtures with very light secondary fluids (cf. section 4.2). That will allows us to produce a rough pre-design with minimal computational effort. This pre-design will subsequently be optimized using detailed FVM simulations in mixture case (cf. section 4.3) to produce the final design. 4.2 1-D modelling approach 4.2.1 Description of the initial model The initial 1-D model is mainly based on literature [36, 37, 46], reflecting the standard operation cases of ejector pumps in industry. While these models are based on fundamental thermodynamic relations, they are using several efficiency parameters to fit to experimental data. However, because there is currently a lack of such data for mercury ejector pumps, our model does not initially rely on any efficiency factor. The models in the literature are in close agreement. In order to better understand the notations used, the notations are explained in Fig. 11 and Fig. 13. Fig. 13: Graphic representations of the variables involved in the 1-D model. The colours represent the different fluids; the arrows represent the speed of the fluid. The initial 1-D model is using the following hypotheses: 1The working fluid is an ideal gas with constant heat capacity 𝑐𝑝 and heat capacity ratio 𝛾 2The flow inside the ejector is one dimensional, steady and isentropic with the exception of shock waves Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 32 of 94 4.2.3 Mixing law model The literature 1-D model has to be adapted to work in case that primary and secondary fluid are not the same. By assuming the hypotheses of 𝑛 perfect gases of molar isobaric heat capacity 𝑐𝑃𝑖 and heat capacity ratio 𝛾𝑖 mixing in an isentropic way, we can demonstrate (complete demonstration available in Appendix 2) that the molar isobaric heat capacity 𝑐𝑃 and the heat capacity ratio 𝛾 of the mixture are: 𝑐𝑃=(∑𝑥𝑖 𝑅𝑖∗ 𝑛 𝑖)−1+∑𝑥𝑖 𝑐𝑃𝑖 𝛾𝑖 𝑛 𝑖=1 4.2-18 𝛾=1+ 1 ∑∑ 𝑥𝑖𝑥𝑗 𝛾𝑖−1𝑀𝑗 𝑀𝑖 4.2-19 with 𝑥𝑖 the molar fraction of the species 𝑖, 𝑀𝑖 the molar mass of the species 𝑖, 𝑅𝑖∗=𝑅 𝑀𝑖 the specific perfect gas constant of the species 𝑖 and 𝛾𝑖 the heat capacity ratio of the species 𝑖. Benchmarking the new iterative mixture 1-D model Since there is no literature on multispecies ejector simulation, we have to compare to experimental data (the latter being also rare). In this context, the paper of Reddick et al. [49] is the only suitable reference with sufficient geometric references to be able to run a simulation and have a point of comparison. It presents parametric experiments for pumping CO2 with a steam ejector. Based on [49] we can benchmark the improved 1-D-model for two experimental cases: 1Primary fluid: steam, secondary fluid: steam (i.e. single species case) 2Primary fluid: steam, secondary fluid: steam + CO2 mixture The experiments covered 4 nozzle diameters, different CO2 mass fractions, and varied inlet pressure. The results of the entrainment ratio agreement with the experiment are shown on Fig. 19. The complete curves are available in Appendix 3. We can see that the results for the entrainment ratio are in very good agreement with the experimental measurements. This also verifies that the newly developed mixture model works correctly in the edge case of similar species. The values of the critical pressures are less accurate but the pump should work at the plateau point so it will not be a huge problem for us but remain a point of future improvement of this 1-D model. Fig. 19: Results of the entrainment ratio computed with the iterative model for the pure steam (blue) and a mixing of steam and CO2 (orange). The experimental reference comes from [49]. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 33 of 94 4.2.4 Conclusion and discussion and limitation of the 1-D model In view of the results obtained we can conclude that the new improved one-dimensional model works correctly. A possible evolution of this model would be to justify in a more rigorous way the choice of the operating point in the case where the second fluid does not choke. Furthermore, a new model where we would take into account a progressive mixing between the primary and secondary fluid could be much more accurate but it would also be much more complex to implement [50, 51]. For our problem, the developed model is felt to be sufficient to give an initial indication, which we will refine subsequently by detailed simulations using the finite volume method, as is described in the next section. Since it is assumed that the velocity of the secondary fluid is constant at the thermodynamic throat created by the primary fluid and the mixing chamber (by definition of 1-D modelling) and that the surface area of the mixing chamber is calculated by subtracting the surface area of the mixing chamber from the surface area of the primary flow (see equation 4.2-5), it follows that the diameter of the mixing chamber is not an output data but an input data. Indeed, by increasing the diameter of the mixing chamber without changing the diameter of the primary nozzle throat (which allows to calculate the surface of the thermodynamic throat), it is possible to increase the secondary mass flow arbitrarily, which represents a flagrant violation of the energy conservation. In order to avoid this deviation, it was decided to always take a diameter of the mixing chamber proportional to the diameter of the primary nozzle. In order to determine the value of this coefficient, a study of the literature was performed and is shown in Fig. 20. In view of the results, it was decided to use a diameter ratio (defined as mixing chamber diameter over nozzle throat diameter) of τ=3.7. Fig. 20: Evaluation of the diameter ratio in view of the values in the literature. All detailed values and references are given in Appendix 4. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 34 of 94 4.3 FVM modelling approach Concept of FVM method In the case of complex systems, it is possible to solve the equations of fluid mechanics numerically. This field is called Computational Fluid Dynamics (CFD). The finite volume method (FVM) is a branch of the CFD. It is a method for generating and solving partial differential equations numerically by turning it into discrete algebraic equations over differential finite volumes. The FVM is regarded as a strictly conservative approximation since the flux entering and exiting the volumes are equal. The FVM is advantageous in CFD because of its conservatism and the straightforward, non-intrusive implementation of the boundary conditions. In our case, we will use the ANSYS Fluent® software. The Navier-Stokes equations The Navier-Stokes equations represent the equations of the three great principles of physical conservation which are the conservation of mass, the conservation of momentum (Newton's second law) and the conservation of energy. Using the Eulerian notation (the one adapted to the FVM), the Navier-Stokes equation system reads: Continuity equation (conservation of mass) [52, 53]: 𝜕𝜌 𝜕𝑡+∇ 󰇍 󰇍 ∙(𝜌𝑉 󰇍 )=0 4.3-1 Momentum conservation equation[52, 53]: 𝜕(𝜌 𝑉 󰇍 ) 𝜕𝑡 +∇ 󰇍 󰇍 ∙(𝜌 𝑉 󰇍 󰇍 󰇍 𝑉 󰇍 )=−∇ 󰇍 󰇍 𝑃+ ∇ 󰇍 󰇍 Σ+𝜌𝑔 +𝑓 4.3-2 Energy conservation equation [52, 53]: 𝜕(𝜌 E) 𝜕𝑡 +∇ 󰇍 󰇍 ∙(𝜌 E 𝑉 󰇍 )=−∇ 󰇍 󰇍 (𝑃 𝑉 󰇍 )+ ∇ 󰇍 󰇍 (Σ 𝑉 󰇍 )−∇ 󰇍 󰇍 (κ ∇T)+𝜌𝑔 𝑉 󰇍 +𝑞 4.3-3 In these equations, - 𝑡 represent the time - ρ is the volume mass of the fluid - 𝑉 󰇍 is the Eulerian speed of the fluid - 𝑃 is the pressure tensor - Σ is the viscous stress tensor - κ is the thermal conductivity - T is the temperature - 𝑔 is gravity acceleration constant - 𝐸 is the total mass energy - 𝑓 External forces - 𝑞 External sources and sinks of energy Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 35 of 94 Turbulence modelling There are many different ways to simulate turbulences from the Navier-Stokes equations. In order to save a lot of time and power computation and considering that we are working on steady state, we concentrate on the Reynold Average Navier-Stokes equations (RANS equations). These equations are the time averaged version of the fluid motion equations and makes it possible to describe the intrinsically unstable turbulent flow in an averaged steady state [54]. In agreement with the literature [24, 27–32], it has been determined that the two most suitable systems of equations are the k-ε realizable model and k-ω SST model [55]. In order to improve the speed of calculation, the models presented can neglect certain parts of the equations. These can then be taken into consideration as an option at the user's discretion. In the following section, we will focus on 5 different options: 1Production Kato Launder The Kato-Launder modification is an ad-hoc modification of the turbulent production term in the k equation. The main purpose of the modification is to reduce the tendency that many two-equation models have to over-predict the turbulent production in regions with large normal strain, i.e. regions with strong acceleration or deceleration [56]. 2Viscous heating Viscous heating represents the effect of an irreversible process by means of which the work done by a fluid on adjacent layers due to the action of shear forces is transformed into heat [57]. 3Low Reynolds corrections In contrast with the high Reynolds number turbulence models, there is no need of the wall functions to bridge between the core and the near wall regions in the low Reynolds corrections k-ε models. The low Reynolds corrections models use damping functions instead to account for the near wall effects. The damping functions [55] bring about the opportunity of modelling of a broader range of the flow fields [58]. The limitations imposed by the wall functions are described by Wilcox [59]. 4Compressibility effects For high-Mach-number flows, compressibility affects turbulence through so-called "dilatation dissipation'', which is normally neglected in the modelling of incompressible flows [59]. Neglecting the dilatation dissipation fails to predict the observed decrease in spreading rate with increasing Mach number for compressible mixing and other free shear layers. To account for these effects in the k-ε models in ANSYS Fluent®, a dilatation dissipation term is included in the k equation [55]. This term is modelled according to a proposal by Sarkar [60]. 5Production limiter A disadvantage of standard two-equation turbulence models is the excessive generation of the turbulence energy in the vicinity of stagnation points. In order to avoid the build-up of turbulent kinetic energy in the stagnation regions, the production term in the turbulence equations can be limited [55]. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 36 of 94 k-ω SST model The different options of the k-ω SST model have been studied on the case of [39]. The results are listed in Tab. 1. It seems that for our case, the options of viscous heating, compressibility effects and Kato Launder production should be retained. Tab. 1: Options study on the k-ω SST model based on the entrainment ratio. References comes from [39]. case studied Production Kato Launder Viscous heating Low Reynolds corrections Compressibility effects Error 1 no yes no no 30.73% 1 yes no no yes 18.30% 1 no no no yes 18.29% 1 no no no no 18.29% 1 no no yes no 34.08% 1 yes yes no yes 12.50% 2 yes yes no yes 3.15% 3 yes yes no yes 13.68% ∗error=|secondary mass flow rate for without options − secondary mass flow rate for with options| |secondary mass flow rate for without options| k-ε realizable model The different options of the k-ε realizable model have been studied on the case of [39]. The results are shown in the Tab. 2. It seems that the case with the options of viscous heating, compressibility effects and limiter production should be retained. Tab. 2: Options study on the k-ε realizable model based on the entrainment ratio. References comes from [39]. Case studied Viscous heating Compressibility effects Production limiter Error 2 no no no 16.17% 2 no no yes 26.29% 2 no yes no 3.27% 2 no yes yes 5.65% 2 yes no no 14.12% 2 yes no yes 12.58% 2 yes yes no 3.55% 2 yes yes yes 3.33% ∗error=|secondary mass flow rate for without options − secondary mass flow rate for with options| |secondary mass flow rate for without options| Based on the literature survey and the study regarding the solver options summarized above we can conclude that the choice of the turbulence model and its options has a relatively small influence on the final solution. As we found that the k-ω SST model is converging a bit faster than the k-ε realizable model, we chose the k-ω SST model for all simulations in the present work. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 37 of 94 Thermo-physical properties Equation of state If the velocity, pressure and temperature are calculated from the conservation equations above, the system is still missing equations to make it complete. In particular, an equation of state is missing. In the literature, the compressible ideal gas law is commonly used [38–42, 61, 62] in order to compute the density. This is also the case for our FVM model: 𝜌=𝑀𝑃 𝑅𝑇 4.3-4 In some applications in literature, the applicability of the ideal gas law is questionable. One simplest way to check the validity of the ideal gas law is to check the compressibility ratio z defined as 𝑧= 𝑃𝑉 𝑛𝑅𝑇 4.3-5 with P the pressure, V the volume, n the mole number, R the ideal gas constant, T the temperature and M the molar mass. Thus, the compressibility ratio gives the deviation of the ideal gas law. The article from Croquer et al. [40] studies the influence of such an assumption on the performances of an ejector pump. They use R134a as working fluid at 3188 kPa and 89.13°C. By comparing it to the Redlich-Kwong-Soave equation of state and to an experimental database, they show a constant difference in the entrainment ratio of 12% and an unnoticeable difference on the Mach number and on the pressure. At this pressure and temperature, the compressibility ratio for R134a is smaller than 0.55 [63]. We will show later that we are always well above this value, thus justifying the use of this equation. Transport properties Thus, this equation system is complete but is dependent on fluid transport properties such as viscosity, thermal conductivity and mass diffusion coefficients. While the values of the latter are available in the ANSYS Fluent® database for very well-known fluids such as steam, they are not available for fluids such as mercury vapour or helium. In order to obtain a value for the latter, we use the kinetic theory based on Lennard-Jones potentials (experimentally determined) and degrees of freedom of the specie. Thus, the molar specific heat capacity cp is computed as function of the degrees of freedom f and of the molar mass M as depicted in equation 4.3-9 [55]. 𝑐𝑝=1 2∙𝑅 𝑀(𝑓+2) 4.3-6 To compute the diffusion coefficient, Fluent uses a modified form of the Chapman-Enskog formula (detailed in [64]) [55]. For the viscosity 𝜇, ANSYS Fluent® uses the equation 4.3-7 [55] dependant on the characteristic length 𝜎 and the energy parameter 𝜖/𝑘𝐵. The 𝛺𝜇 is the double integral molecular distribution function as defined by Chapman-Enskog [64]. 𝜇=2.67 10−6∙√𝑀 𝑇 𝜎2𝛺𝜇(𝑇 𝜖/𝑘𝐵) 4.3-7 Then, to compute the thermal conductivity κ, ANSYS Fluent® uses the following equation [55]: 𝜅=15 4∙𝑅 𝑀𝜇 [4 15𝑐𝑝𝑀 𝑅+1 3] 4.3-8 Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 38 of 94 In order to verify the hypothesis made on the basis of kinetic theory, it was decided to compare with experimental data from the literature. Indeed, the article by Vukalovich et al. [65] is a proven reference and corroborated by other independent studies as [66–68] on mercury vapour data. The comparison was carried out on the μ dynamic viscosity. The results presented in Fig. 21 show a very strong corroboration between the value of the dynamic viscosity obtained during our simulation with ANSYS Fluent® and the results provided by the literature. We can thus validate the use of kinetic theory. Fig. 21: Results of the dynamic viscosity μ with the kinetic theory compared to the empirical values from [65]. 4.3.1 Description of the model implementation Axisymmetric and boundary conditions In order to simplify the computations, we compare the pressure ratio as define in equation 4.3-9 in axisymmetric 2D model and 3D model. The pressure ratio 𝜒 characterises the compression achieved (pressure change between secondary fluid suction and discharge pressure) in relation to the maximum compression ever achieved (pressure between primary and secondary fluid). It should not be confused with the compression ratio 𝛹 as defined by equation 4.3-10. The latter is the ratio of critical backing pressure on secondary inlet pressure. The Fig. 22 shows that the difference between 3D and axisymmetric 2D cases are very small. In consequence and inspecting the geometry of an ejector pump, we assume that the simulation can be reduced to a two-dimensional axisymmetric simulation. We can then apply the following boundary conditions: Primary total pressure and temperature at the primary inlet, total secondary pressure and Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 39 of 94 temperature at the secondary inlet and total backing pressure at the outlet. The wall is considered as perfectly adiabatic with a no-slip condition. Fig. 22: Results of the pressure ratio 𝜒 as unction of the entrainment ratio for 2-D and 3-D FVM simulations. The experimental data comes from [69]. Choice of the mesh It was decided to have a minimum of 20 elements in the nozzle throat diameter (equivalent to the smallest dimension). In order to ensure a good overall quality of the mesh, we also used three indicators: 1The aspect ratio The aspect ratio is a measure of the stretching of a cell. It is computed as the ratio of the maximum value to the minimum value of any of the following distances: the distances between the cell centroid and face centroids, and the distances between the cell centroid and nodes (see Fig. 23 ). Based on the software recommendations, we use a maximum aspect ratio of 18 [55]. 𝜒 = 𝑃𝑐−𝑃𝑠0 𝑃𝑃0−𝑃𝑠0 4.3-9 𝛹 =𝑃𝑐𝑐 𝑃𝑠0 4.3-10 Fig. 23: Definition of the aspect ratio for 2D quadrilateral element. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 40 of 94 2The orthogonality Orthogonality is a measure of the quality of a mesh, and is calculated as shown on Fig. 24 from the dot products of each vector 𝑟𝑓𝑖 󰇍 󰇍 󰇍 pointing from the centroid of a cell toward the center of each of its faces, and the corresponding face area vector 𝐴𝑖 󰇍 󰇍 󰇍 as Based on the software recommendations, we use a minimum orthogonality above 0.1 [55]. 3The skewness The skewness is a non-dimensional parameter calculated using the normalized angle deviation method (see Fig. 25), and is defined by where qmax is the largest angle in the face or cell, qmin is the smallest angle in the face or cell and qe is the angle for an equiangular face or cell (e.g., 60° for a triangle and 90° for a square). Based on the software recommendations, we use a maximum skewness below 0.95 with an average value less than 0.33 [55]. Fig. 24: Definition of the orthogonality for 2D quadrilateral element. Fig. 25: Definition of the skewness for 2D quadrilateral element. In order to prevent problems in the walls, a gradual refinement of the walls was carried out (see Fig. 26). To ensure the validity of such a method, a mesh sensitivity study was performed (see Tab. 3). An element size of 0.25 mm corresponds to 9 elements in the nozzle throat radius and 0.2mm correspond to 11 elements. We can see that a huge variation of the number of integration points (points where the different variables of the simulation are computed) by factor of 1.8 obtained by increasing the number of element and by a factor of 4 obtain by increasing the order of each element (see Fig. 27) doesn’t change significantly the global results. Indeed, the variation of the secondary mass flow rate between the coarsest and finest mesh sizes is well below 1%. Thus, we can be confident on the mesh quality. orthogonality= 𝑚𝑖𝑛 𝑖[𝐴𝑖 󰇍 󰇍 󰇍 ∙𝑟𝑓𝑖 󰇍 󰇍 󰇍 |𝐴𝑖 󰇍 󰇍 󰇍 ||𝑟𝑓𝑖 󰇍 󰇍 󰇍 |] 4.3-11 skewness= 𝑚𝑎𝑥[𝑞𝑚𝑎𝑥−𝑞𝑒 180−𝑞𝑒,𝑞𝑒−𝑞𝑚𝑖𝑛 𝑞𝑒] 4.3-12 Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 41 of 94 Fig. 26: Typical mesh pattern. Fig. 27: 2D quadrilateral mesh parameter. Linear element on right (with one integration point) and quadratic element on the left (with four integration points). Tab. 3: Results of the mesh convergence element type element size [mm] mesh nodes mesh elements sec. mass flow [kg s-1] sec. error* Throughput [Pa m3 s-1] Throughput [Torr L s-1] linear 0.25 133102 130792 1.77E-04 8.37E-03 15.2676115 114.507086 linear 0.2 248189 245192 1.76E-04 5.69E-07 15.1408825 113.556619 quadratic 0.2 248189 245192 1.76E-04 0 15.1408739 113.556554 element type element size [mm] mesh nodes mesh elements Primary mass flow [kg s-1] primary error* outlet mass flow rate [kg s-1] outlet error* mass conservatio n [kg s-1] linear 0.25 133102 130792 0.0042812 2.34E-06 -0.004458 3.31E-04 2.00E-10 linear 0.2 248189 245192 0.0042812 0.00 -0.004457 1.35E-06 -1.24E-08 quadratic 0.2 248189 245192 0.0042812 0.00 -0.004457 0.00 -6.50E-09 ∗error=|value for quadratic 0.2mm element − value for the simulated case| |value for quadratic 0.2mm element| Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 48 of 94 5 DEMO ejector pump development 5.1 Summary of requirements In order to guarantee a homogeneous vacuum, the tokamak is divided into 10 pumping sections, each of which representing 45 Pa m3 s-1 to be pumped. As defined in section 0, the requirement for this study is to bridge a pressure gap from 10 Pa to a pressure between 400 Pa and 7 000 Pa. As shown in Fig. 36, we separate the problem in two parts. The first stage of ejector pump will be directly integrated into the pumping ports after the LDPs in the tokamak building. For those ejectors the goal is to pump 9 Pa m3 s-1 per pumping port at an inlet pressure of approx. 10 Pa and against a backing pressure between 50 to 100 Pa. These ejectors should remain below 2 meters long to fit in the available space under the tokamak. For the second stage of ejectors, it will be organized in packs to fit in gloveboxes. A CAD drawing of a typical example of such a pack measuring 660x548x710 mm can be found in Appendix 5. The target working pressure range of such an ejector pack is an inlet pressure of 50 Pa to 100 Pa and an outlet pressure of 400 Pa to 7 000 Pa. The total throughput to achieve is around 360 Pa m3 s-1 for the DIRL and 90 Pa m3 s-1 for the INTL. With this information, we can roughly determine the number of ejector pumps and their primary pressures. Then, a discussion could be held to determine whether it is preferable to use LRPs or ejector pumps and if the ejector pumps could be the solution to bridge the pressure gap between the LDPs and the LRPs. In order to have realistic results and taking into account the lack of literature on tritium or deuterium, helium was the gas species used for the dimensioning of this vacuum system. Fig. 36: Global organisation of the DEMO pumping system including the ejectors (preliminary concept). Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 49 of 94 5.2 Sizing of the INTL first stage ejector pump In view of the equations used, the assumptions made in the 1-D modelling, and the complexity of the FVM calculations, it is believed that the first compression stage is the most difficult to design, especially considering the size limitation (the length of these ejectors should not exceed 2 m as mentioned above). Indeed, as far as 1-D modelling is concerned, the assumptions of constant velocity, as a direct consequence of the one-dimensional aspect of the model, becomes less and less credible at low pressure. We are also limited in the number of possible compression stages (the best being to realize only one compression stage). In view of these constraints, and knowing that we work at low pressure (10 Pa as inlet pressure), a minimum size of mixing chamber diameter is given by a limit Knudsen number of 10-2 at a value of 175 mm. By upscaling the article pump [70], this diameter gives us an overall pump length of about 2m. According to Fig. 20, the primary nozzle throat diameter is then fixed to 50 mm. 1-D approach for the choice of the primary pressure The aim is to pump 9 Pa m3 s-1 with a compression factor of at least 5 (in order to be in the pressure range above 50 Pa as define previously). Note that on the Fig. 37 this minimum compression factor is guaranteed. The aim is then to maximise the compression factor while minimising the number of pumps (i.e. maximising the throughput). Thus, placing ourselves at the upper limit of 9 Pa m3 s-1 allows us to use only one ejector pump per tokamak section and to maximise the critical backing pressure. From the results of the 1-D simulation shown in Fig. 37, it appears that a primary pressure of 1350 Pa is the maximum pressure one case use to guarantee a throughput of 9 Pa m3 s-1. At this inlet pressure, the achievable backing pressure is around 160 Pa, which in the goal range (more than 3 times higher than the minimum required). However, as mentioned earlier, the pressure range in which this pump operates is at the limits of our hypotheses so that a large overestimation of the flow and pressure values can be presumed. Fig. 37: Results of the 1-D model for the secondary throughput and the critical backing pressure as function of the primary pressure. Primary nozzle throat of 50 mm, secondary inlet pressure of 10 Pa. The black solid line is the maximum throughput required by our system. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 50 of 94 Comparison with the FVM approach In order to improve the predictions made by the 1-D model, it was decided to carry out FVM simulations to determine efficiency factors on backing pressure and throughput. Since it is believed that the efficiency is worst at low pressure (which will be validated later), we decided to perform our simulations at low pressure in order to remain conservative in determining these efficiencies. A minimum pressure of 500 Pa, the approximate mercury operating pressure of the diffusion pumps, was chosen as this appears in Fig. 37 to be a suitable operating point. It seems unnecessary to go below this pressure, as the backing pressure is already in the desired low-pressure range. A first simulation with a backing pressure of 10.5 Pa was performed in order to compare the mass flow rates in critical regimes (a compression factor very close to 1, ensures a critical regime as depicted in Fig. 10). Summaries of the results can be found in Tab. 6 and the detailed results in the Appendix 6 and Appendix 8. Although for the primary flow rate, the results are very close to the those one can compute from the 1-D equation 4.2-1 (0.09% relative error), the secondary mass flow rate is well below the estimate from the 1-D simulation. An efficiency of our 1-D model on the secondary mass flow rate (which correspond to the secondary mass flow rate computed by FVM over the secondary mass flow rate computed by 1-D model) can be calculated at about ηms=0.32. A new simulation was performed at 20 Pa backing pressure (compression ratio of 2). At this pressure, it can be seen from the decrease of the secondary mass flow rate that we are in the vicinity of the critical point. Thus, a second 1-D model efficiency on the value of this critical pressure (critical backing pressure computed by FVM over the critical backing pressure computed by 1-D model) of about ηPcb=0.28 can be calculated. Tab. 6: Secondary mass flow rates and throughput for primary inlet pressure of 500 Pa and secondary inlet pressure of 10 Pa. Secondary mass flow rates [kg s-1] Throughput [Pa m3 s-1] FVM simulation with 10.5 Pa backing pressure 2.673E-5 16.1 FVM simulation with 20 Pa backing pressure 2.585E-5 15.6 1-D simulation 8.2443E-5 49.7 1-D simulation integrating the efficiencies One can think (and it will be demonstrated later) that the predictive efficiencies on the secondary fluid of the 1-D model are increasing (or in the worst-case stagnating) with the increase of the different pressures. Thus, integrating the previously calculated efficiencies and performing a new 1-D simulation, we obtain the results shown in Fig. 38. It can be seen that a primary pressure of 1500 Pa would allow us to achieve our minimum objective of a 50 Pa backing pressure in one stage. Achieving this backing pressure in one stage limits the throughput to 1.5 Pa m3 s-1, which means 6 ejector pumps i.e. 2 ejectors per diffusion pumps which, using a positioning as shown in Fig. 39, would appear not to be an insurmountable problem. The required power per ejector can then be estimated from the mercury mass flow rate. The mass flow rate follows directly from the choked flow condition in the nozzle throat (cf. equation 4.2-1) and the stagnation conditions. Assuming a constant vaporization enthalpy of 310 kJ kg-1 [68] this yields a power of around 1.67 kW per ejector (so a total of around 10 kW for the 6 ejectors and 100kW for the all tokamak). Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 51 of 94 Fig. 38: Results of the 1-D model including the efficiencies for the throughput and the critical backing pressure as function of the primary pressure. Primary nozzle throat of 50 mm, secondary inlet pressure of 10 Pa. Fig. 39: Positioning of ejector pumps (green) to diffusion pumps (purple). Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 52 of 94 FVM validation of the INTL first stage ejector pump As expected, the prediction error decreases with increasing primary pressure. Thus, in order to validate our previously defined operating point at 1500 Pa, we calculated using the FVM method the ejector with a primary pressure of 1500 Pa and found a throughput of 1.6 Pa m3 s-1 (complete results in Appendix 9 and Appendix 10). This value is higher than the predictions of the 1-D model with efficiency factors that predicted a flow rate of 1.48 Pa m3 s-1. This validates our earlier assumption that efficiencies increase with pressure and confirms that the realised sizing is conservative. Conclusion on INTL first stage ejector pump In view of the results presented above, we can guarantee a minimum compression ratio of 5 i.e. to a minimum backing pressure of 50 Pa which is in the target range. The achievement of this objective in its lower part, is certainly a disadvantage with respect to the final objective but allows in the case where the MFPs only discharge at 50 Pa in the DIRL to be able to use the same solutions (detailed in the following section) as in the INTL. Thus, the work presented will cover all possible pressure ranges. 5.3 Sizing of the 6-pack ejector pump Using the ejector proportions determined above, and based on the desired final size in order to fit in a commercial glovebox, we can determine a primary nozzle diameter of approximately 10 mm. In order to allow for semi-series production, we decided to use the same size for all ejectors. Using the 1-D model with the efficiencies computed for the first stage (see above), we are able to predict (in a very conservative way) the rough number of pumps needed. A typical example of such an analysis is shown on Tab. 7. The power are computed in the same way as above First of all, let us recall that the problem is a multifactor optimisation problem. For the results presented, and remembering that the aim is to demonstrate the possibility of using ejectors, only the pressures and throughput (in order to obtain the number of pumps needed) have been studied. Typically, the heating power for the primary fluid is only an informative output that has not been considered in the optimisation here. With this in mind, it can first be noted that using ejectors may be a viable solution. Furthermore, we must remember that it is likely that the efficiencies used to dimension this problem were calculated under very low-pressure conditions and that these may increase with primary and secondary pressure. A FVM simulation based on the first stage has been made to confirm it. Their results are available in Appendix 11. It appears that a throughput of 3.74 Pa m3 s-1 instead of 3.3 Pa m3 s-1 is found, reflecting thus a better efficiency. This confirms once again that the results found are conservative. Indeed, it can be seen that with a total of 53 packs, the minimum target pressure can be reached. By adding only two packs, the pressure of 900 Pa could be reached, which would allow the number of LRPs to be reduced significantly. The last stage presented in this table is purely informative. It can be believed regarding the particularly impressive performances that we are at the limits of the predictions of our model. It should be noted, however, that the power increases with each stage and that the power required for the last stages is significant and therefore calls for discussion with regard to the other solutions and the overall objectives. In addition, it was assumed that between each stage, the primary fluid was completely liquefied and therefore the secondary fluid was cooled to 20°C. The liquefaction of the secondary fluid has not been studied here and may represent a problem with regard to the powers involved. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 53 of 94 Tab. 7: Possible realization of a pumping system based on ejectors modelled by the 1-D simulation. Property Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Inlet pressure [Pa] 50 100 300 900 1700 3950 6500 9880 14800 22000 Primary nozzle diameter [mm] 10 10 10 10 10 Primary pressure [Pa] 2500 8250 24000 35000 90000 Primary temperature [K] 475 512 550 565 607 Primary mass flow rate [kg s-1] 0.00101 0.00322 0.00905 0.01303 0.03232 Power per ejector [kW] 0.31 1.00 2.81 4.04 10.02 Backing pressure [Pa] 100 300 900 1700 3950 6500 9880 14800 22000 34000 Throughput [Pa m3 s-1] 3.3 3.7 12 70 106 390 740 1215 1925 2975 Number ejector DIRL 110 98 30 6 4 0.9230769 0.4864864 0.2962963 0.1870129 0.121008 Number ejector INTL 28 25 8 2 1 0.2307692 0.1216216 0.0740740 0.0467532 0.030252 Number pack DIRL 19 17 5 1 1 Number pack INTL 5 5 2 1 1 Power of a pack [kW] 1.89 6.00 16.84 24.23 60.12 Power the stage [kW] 45.27 131.91 117.85 48.46 120.25 Cumulative number pack 24 46 53 55 57 Cumulative power [kW] 45.27 177.18 295.03 343.49 463.74 Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 54 of 94 5.4 Sensitivity study 5.4.1 Pressure sensitivity Primary pressure sensitivity As shown in Fig. 40 and Fig. 41, the pump parameters are strongly dependant on the primary pressure. Thus, the choice of the primary pressure is very important. Indeed, it is a compromise that must be made between the maximum critical pressure against which one can pump and the flow rate that will be pumped. The more gas you pump, the lower the pressure at which you can pump it. This choice is all the more complex as both curves show a very rapid growth (respectively decrease). It is therefore an optimisation problem based on the objectives to be defined. Fig. 40: Sensitivity of the primary pressure in with a critical secondary mass flow rate point of view. Complete results are in Appendix 12. Fig. 41: Sensitivity of the primary pressure with a critical pressure point of view. Complete results are in Appendix 12. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 55 of 94 Secondary pressure sensitivity Fig. 42 shows the sensitivity of the secondary mass flow rate to the secondary pressure. In contrast to the primary pressure, a nearly linear variation can be seen here as predicted in section 3.2. Thus, an increase in secondary pressure will always increase the performance of the pumping system. A design that minimises this pressure, as has been done here, is therefore very conservative and guarantees minimal pump operation. Fig. 42: Sensitivity of the secondary pressure. Complete data in Appendix 14. 5.4.2 Geometrical Sensitivity The diameter ratio sensitivity As predicted in section 3.3, Fig. 43 shows a significant sensitivity of the secondary mass flow to the diameter ratio. It is important to note that the maximum does not appear to be reached for the value 3.7 as previously assumed. Thus, a modification of this parameter on the DEMO ejectors could allow a significant improvement of the pumping capacities. Further study would be required to confirm or deny this. Fig. 43: Sensitivity of the diameter ratio. Complete results are in Appendix 16. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 56 of 94 NXP sensitivity We notice on the Fig. 44 curve that the variation of the throughput is highly dependent on the value of the NXP. It can be seen that there is a maximum. In view of the strong variation around this maximum, we can say that an optimisation of this parameter is essential when designing such an ejector. It should also be kept in mind that such a maximum is probably also dependent on the thermodynamic boundary conditions. A study in this sense should therefore be considered. Fig. 44: NXP sensitivity. Complete results are in Appendix 13. The dimensionless mixing length sensitivity The dimensionless mixing length sensitivity is shown in Fig. 45. Although it shows a maximum, it is much less sensitive than that of the NXP. However, as with the NXP, a study to confirm or deny the interdependence of this maximum with the boundary conditions would allow a much more reliable optimisation of this type of pump. Fig. 45: Dimensionless mixing length sensitivity. Complete results are in Appendix 15. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 57 of 94 5.5 Discussion of potential performance improvements 5.5.1 Reducing the power consumption of the heater In view of the power required to operate these pumps (especially in the highest-pressure range), and in order to optimise the overall efficiency of the system, it may be worthwhile to extract the heat from the pump outlet fluid, which is at a relatively high temperature and which we wish to liquefy in order to preheat the liquid mercury that will be used as the primary fluid for the pump. This can be done by a heat exchanger or a heat pump. Indeed, considering a coefficient of performance of about 3 (which is in the low range of what is commercially available), this would mean that for an electrical power consumption of 1kW, we would cool the output fluid by about 2kW and heat the input fluid by 3kW. Moreover, rather than heating the primary fluid only with an electric heater, it may be interesting to use the cooling fluid of the blankets to exploit some of its energy without the efficiency of an electric turbine. While such a solution is interesting, it cannot be unique and independent, as the system would then not be able to function for a start-up when the tokamak cooling fluid is cold. A diagram summarising all the energy transfers mentioned here has been drawn up (see Fig. 46). Finally, a possible solution to reduce the heating power and increase the overall performance of some pumps would be not to liquefy the mercury between two compression stages. This would increase the temperature of the absorbed fluid and thus increase the performance of the pump. However, this solution has significant disadvantages, like the high dilution of helium, deuterium and tritium in the mercury, and the significant increase in the amount of mercury required. Fig. 46: Conceptual diagram of the different ways of condensing and vaporizing mercury. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 64 of 94 [62] L. Macia, R. Castilla, P. J. Gamez-Montero, S. Camacho, and E. Codina, “Numerical Simulation of a Supersonic Ejector for Vacuum Generation with Explicit and Implicit Solver in Openfoam,” Energies, vol. 12, no. 18, Art. no. 18, Jan. 2019, doi: 10.3390/en12183553. [63] J. Yang, D. Fang, X. Meng, and J. Wu, “Experimental measurements and correlations of the vapor phase pvTx behavior of binary mixtures of 1,1,1,2-tetrafluoroethane (R134a) + 2,3,3,3tetrafluoroprop-1-ene (R1234yf),” Int. J. Refrig., vol. 132, pp. 263–275, Dec. 2021, doi: 10.1016/j.ijrefrig.2021.09.018. [64] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases. Cambridge University Press, 1990. [65] M. P. Vukalovich, A. I. Ivanov, and L. R. Fokin, “Yak ovlev, AT, Teplofizicheskie svoistva rtuti (Thermal and Physical Properties of Mercury), Moscow: Izd,” Stan Dartov, vol. 4, no. 3.5, pp. 3– 0, 1971. [66] M. L. Huber, A. Laesecke, and D. G. Friend, “The vapor pressure of mercury”. [67] G. J. F. Holman and C. A. ten Seldam, “A Critical Evaluation of the Thermophysical Properties of Mercury,” J. Phys. Chem. Ref. Data, vol. 23, no. 5, pp. 807–827, Sep. 1994, doi: 10.1063/1.555952. [68] L. R. Fokin, V. N. Popov, and S. P. Naurzakov, “Equation of state and thermodynamic properties of saturated and superheated mercury vapors up to 1650 K and 125 MPa,” High Temp., vol. 49, no. 6, pp. 832–840, Dec. 2011, doi: 10.1134/S0018151X11050075. [69] K. Aldas and R. Yapıcı, “Investigation of Effects of Scale and Surface Roughness on Efficiency of Water Jet Pumps Using CFD,” Eng. Appl. Comput. Fluid Mech., vol. 8, pp. 14–25, Mar. 2014, doi: 10.1080/19942060.2014.11015494. [70] B. D. Power and N. T. M. Dennis, “An Unusual Mercury Ejector Pump,” in Fundamental Problems in Vacuum Techniques Ultra-High Vacuum, E. Thomas, Ed., Pergamon, 1960, pp. 197–201. doi: 10.1016/B978-1-4832-8302-9.50040-5. [71] J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, The Molecular Theory of Gases and Liquids. Wiley, 1954. [72] J. Zhang et al., “Research on performance of multi-nozzle ejector for aeroengine air system,” Therm. Sci. Eng. Prog., vol. 38, p. 101651, Feb. 2023, doi: 10.1016/j.tsep.2023.101651. [73] J. Han, J. Feng, T. Hou, and X. Peng, “Performance investigation of a multi-nozzle ejector for proton exchange membrane fuel cell system,” Int. J. Energy Res., vol. 45, no. 2, pp. 3031–3048, 2021, doi: 10.1002/er.5996. [74] Y. Shan, J. Zhang, and X. Ren, “Numerical modeling on pumping performance of piccolo-tube multi-nozzles supersonic ejector in an oil radiator passage,” Energy, vol. 158, pp. 216–227, Sep. 2018, doi: 10.1016/j.energy.2018.06.001. [75] M. A, T. S, R. Divvela, and R. Ethirajan, “Nozzle Geometry Effect on Twin Jet Flow Characteristics,” Int. J. Turbo Jet-Engines, vol. 39, no. 4, pp. 525–532, Dec. 2022, doi: 10.1515/tjj-2019-0049. [76] J. Bodys et al., “Performance of fixed geometry ejectors with a swirl motion installed in a multiejector module of a CO2 refrigeration system,” Energy, vol. 117, pp. 620–631, Dec. 2016, doi: 10.1016/j.energy.2016.07.037. [77] J. Parveen Banu and A. Mani, “Numerical studies on ejector with swirl generator,” Int. J. Therm. Sci., vol. 137, pp. 589–600, Mar. 2019, doi: 10.1016/j.ijthermalsci.2018.11.033. [78] “Flow visualization of the jet exiting a chevron nozzle,” Aeronaut. Aerosp. Open Access J., vol. Volume 2, no. Issue 2, Mar. 2018, doi: 10.15406/aaoaj.2018.02.00031. [79] J. R. von Mayer, Die Mechanik der Wärme: in gesammelten Schriften. J.G. Cotta, 1893. [80] J. García del Valle, J. M. Saíz Jabardo, F. Castro Ruiz, and J. F. San José Alonso, “An experimental investigation of a R-134a ejector refrigeration system,” Int. J. Refrig., vol. 46, pp. 105–113, Oct. 2014, doi: 10.1016/j.ijrefrig.2014.05.028. [81] B. Kitrattana, S. Aphornratana, and T. Thongtip, “One dimensional steam ejector model based on real fluid property,” Therm. Sci. Eng. Prog., vol. 25, p. 101016, Oct. 2021, doi: 10.1016/j.tsep.2021.101016. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 65 of 94 [82] A. B. Little, Y. Bartosiewicz, and S. Garimella, “Visualization and Validation of Ejector Flow Field With Computational and First-Principles Analysis,” J. Fluids Eng., vol. 137, no. 5, May 2015, doi: 10.1115/1.4029534. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 66 of 94 Appendix Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 67 of 94 Appendix 1: Iterative 1-D model results for the experiment from [34] Tab. A1-1: Iterative 1-D model results for the experiment from [34]. Pp0 [Pa] Tp0 [K] Ps0 [Pa] Ts0 [K] dm [mm] dt [mm] Entrainment ratio 1-D Experimental entrainment ratio Error 604000 368.15 40000 281.15 9.2 2.82 0.47858721 0.4377 9.34% 604000 368.15 40000 281.15 8.84 2.82 0.41816401 0.3937 6.21% 604000 368.15 40000 281.15 8.1 2.64 0.3878042 0.3457 12.18% 604000 368.15 40000 281.15 8.54 2.82 0.370049 0.3505 5.58% 604000 368.15 40000 281.15 7.6 2.64 0.30595365 0.2814 8.73% 604000 368.15 40000 281.15 8.1 2.82 0.30329935 0.2902 4.51% 604000 368.15 40000 281.15 7.6 2.82 0.23313888 0.2273 2.57% 604000 368.15 40000 281.15 7.34 2.64 0.26610372 0.2552 4.27% 604000 368.15 40000 281.15 7.34 2.82 0.19918532 0.2053 -2.98% 604000 368.15 40000 281.15 6.7 2.64 0.17632829 0.1859 -5.15% 538000 363.15 40000 281.15 8.1 2.64 0.45619898 0.4446 2.61% 538000 363.15 40000 281.15 7.6 2.64 0.3639998 0.3488 4.36% 538000 363.15 40000 281.15 7.34 2.64 0.31897637 0.304 4.93% 538000 363.15 40000 281.15 6.98 2.64 0.26007443 0.2718 -4.31% 538000 363.15 40000 281.15 6.7 2.64 0.21712903 0.2246 -3.33% 465000 357.15 40000 281.15 8.1 2.64 0.55555735 0.5387 3.13% 465000 357.15 40000 281.15 7.6 2.64 0.44859966 0.4241 5.78% 465000 357.15 40000 281.15 7.34 2.64 0.39625538 0.3883 2.05% 465000 357.15 40000 281.15 6.98 2.64 0.32756407 0.3117 5.09% 465000 357.15 40000 281.15 6.7 2.64 0.27730281 0.288 -3.71% 400000 351.15 40000 281.15 8.1 2.64 0.67559465 0.6227 8.49% 400000 351.15 40000 281.15 7.6 2.64 0.55121386 0.4889 12.75% 400000 351.15 40000 281.15 7.34 2.64 0.49019131 0.4393 11.58% 400000 351.15 40000 281.15 6.98 2.64 0.40995938 0.3922 4.53% 400000 351.15 40000 281.15 6.7 2.64 0.35105417 0.3257 7.78% 604000 368.15 47000 285.15 8.84 2.82 0.52256896 0.4989 4.74% 604000 368.15 47000 285.15 8.54 2.82 0.46577427 0.4048 15.06% 604000 368.15 47000 285.15 8.1 2.64 0.48675399 0.4541 7.19% 604000 368.15 47000 285.15 7.34 2.64 0.34260374 0.3503 -2.20% 604000 368.15 47000 285.15 7.6 2.82 0.30335596 0.304 -0.21% 604000 368.15 47000 285.15 6.7 2.64 0.23539247 0.235 0.17% 538000 363.15 47000 285.15 8.1 2.64 0.56822665 0.5422 4.80% 538000 363.15 47000 285.15 7.34 2.64 0.40606339 0.4034 0.66% 538000 363.15 47000 285.15 6.7 2.64 0.28490118 0.2946 -3.29% 465000 357.15 47000 285.15 8.1 2.64 0.68625073 0.635 8.07% 465000 357.15 47000 285.15 7.34 2.64 0.4984768 0.479 4.07% 465000 357.15 47000 285.15 6.7 2.64 0.35749649 0.3398 5.21% 400000 351.15 47000 285.15 8.1 2.64 0.82850572 0.7412 11.78% 400000 351.15 47000 285.15 7.34 2.64 0.61040023 0.6132 -0.46% ∗error=Entrainment ratio 1-D − Experimental entrainment ratio Experimental entrainment ratio Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 68 of 94 Appendix 2: Demonstration of the mixing gas law The heat capacity ratio is defined as: 𝛾=𝑐𝑃 𝑐𝑉 A2-1 And the relation of Mayer which holds for ideal gases states [79]: 𝑐𝑃=𝑐𝑉+𝑅∗ A2-2 With 𝑅∗ the specific gas law constant: 𝑅∗=𝑅 𝑀 A2-3 With 𝑅 the universal molar gas constant and 𝑀 the molar mass. For an ideal mixture, we define the mixture molar mass and the mixture isochoric specific heat: 𝑀=∑𝑥𝑖∙𝑀𝑖 𝑛 𝑖=1 A2-4 𝑐𝑉=∑𝑥𝑖∙𝑐𝑉𝑖 𝑛 𝑖=1 A2-5 It comes then: 𝑅∗=𝑅 𝑀=𝑅 ∑𝑥𝑖∙𝑀𝑖= 1 ∑𝑥𝑖 𝑅𝑖∗ A2-6 And from the relation of Mayer it comes: 𝑐𝑉 𝑅∗= 1 𝛾−1 𝑎𝑛𝑑 𝑐𝑃 𝑅∗= 𝛾 𝛾−1 A2-7 And for two species 𝑖 and 𝑗: 𝑐𝑉𝑖 𝑅𝑗∗=𝑐𝑉𝑖 𝑀𝑗 𝑅=𝑐𝑉𝑖𝑀𝑖 𝑀𝑗 𝑅 𝑀𝑖=𝑐𝑉𝑖 𝑅𝑖∗∙𝑀𝑗 𝑀𝑖=1 𝛾𝑖−1 𝑀𝑗 𝑀𝑖 A2-8 So, for a mixture of n gases the Mayer relation gives us: 𝑐𝑃=𝑐𝑉+𝑅∗= A2-9 So, by definition it comes: 𝛾 =𝑐𝑃 𝑐𝑉=𝑐𝑉+𝑅∗ 𝑐𝑉=1+𝑅∗ 𝑐𝑉 A2-10 =1+ 1 ∑𝑥𝑖 𝑅𝑖∗ ∑𝑥𝑖∙𝑐𝑉𝑖 A2-11 =1+ 1 (∑𝑥𝑖 𝑅𝑖∗)(∑𝑥𝑗∙𝑐𝑉𝑗) A2-12 =1+ 1 ∑∑𝑥𝑖𝑥𝑗𝑐𝑉𝑖 𝑅𝑗∗ A2-13 =1+ 1 ∑∑𝑥𝑖𝑥𝑗 𝛾−1𝑀𝑗 𝑀𝑖 A2-14 Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 69 of 94 Appendix 3: Iterative 1-D model results with mixing law for the experiment from [49] Nozzle Diameter: 4.03 mm Nozzle Diameter: 4.03 mm CO2 mass fraction: 0 % CO2 mass fraction: 21% Nozzle Diameter: 4.23 mm Nozzle Diameter: 4.23 mm CO2 mass fraction: 0 % CO2 mass fraction: 23 % Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 70 of 94 Nozzle Diameter: 4.59 mm Nozzle Diameter: 4.59 mm CO2 mass fraction: 0 % CO2 mass fraction: 27 % Nozzle Diameter: 5.09 mm Nozzle Diameter: 5.09 mm CO2 mass fraction: 0 % CO2 mass fraction: 30 % Fig. A3-1: Comparison of the experimental data from [49] with the improved 1-D model developed in this thesis. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 71 of 94 Appendix 4: Different diameter ratios from the literature. Tab. A4-1: Different diameter ratios from the literature. gas dt [mm] dm [mm] τ Reference mercury/air 4.54 11 2.42290749 [70] CO2 2 4 2 [28] R134a 2 4.8 2.4 [40, 41] R134a 1 2.4 2.4 [80] R134a 1 2.88 2.88 [80] R134a 1 1.92 1.92 [80] R141b 2.64 6.7 2.53787879 [34] R141b 2.64 6.98 2.64393939 [34] R141b 2.64 7.34 2.78030303 [34] R141b 2.64 7.6 2.87878788 [34] R141b 2.64 8.1 3.06818182 [34] R141b 2.82 7.34 2.60283688 [34] R141b 2.82 7.6 2.69503546 [34] R141b 2.82 8.1 2.87234043 [34] R141b 2.82 8.54 3.02836879 [34] R141b 2.82 8.84 3.13475177 [34] R141b 2.82 9.2 3.26241135 [34] steam 2 19 9.5 [81] steam 2.3 19 8.26086957 [81] steam 1.5 13.4 8.93333333 [81] steam 1.7 13.4 7.88235294 [81] steam 2 19 9.5 [81] steam 1.5 13.4 8.93333333 [81] steam 2.63 15.8 6.00760456 [61] steam 2 19 9.5 [38, 39] steam 1.75 19 10.8571429 [38, 39] steam 1.5 19 12.6666667 [38, 39] steam 4.03 9.5 2.3573201 [49] steam 4.23 9.5 2.24586288 [49] steam 4.59 9.5 2.06971678 [49] steam 5.09 9.5 1.86640472 [49] air/air 6.07 27 4.44810544 [82] air/air 2.54 10.16 4 [33] air/air 2.54 15.24 6 [33] air/air 2.54 25.4 10 [33] air/air 5.08 10.16 2 [33] air/air 5.08 15.24 3 [33] air/air 5.08 25.4 5 [33] Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 72 of 94 Appendix 5: Typical example of a 6-ejector pack Fig. A5-1: Typical example of a 6-ejector pack. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 73 of 94 Fig. A5-2: detail view of the primary nozzle part. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 80 of 94 Fig. A8-3: Results for the molar fraction of helium. Fig. A8-4: Results for the Mach number. Fig. A8-5: Results for the density. Fig. A8-6: Results for the velocity. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 81 of 94 Appendix 9: FVM Results for Pp0=1500 Pa; Ps0=10 Pa Pc=10.5 Pa Tab. A9-1: geometrical parameters. Nozzle diameter: 50 mm Diameter ratio: 3.7 Dimensionless mixing length: 2 NXP: 20 mm Tab. A9-2: mass flow rate results [kg s-1]. Primary inlet 0.01420964 Secondary inlet 2.96E-06 Outlet -0.0142126 Mass conservation 4.42E-09 Fig. A9-1: Results for the static pressure. Fig. A9-2: Results for the static temperature. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 82 of 94 Fig. A9-3: Results for the molar fraction of helium. Fig. A9-4: Results for the Mach number. Fig. A9-5: Results for the density. Fig. A9-6: Results for the velocity. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 83 of 94 Appendix 10: FVM Results for Pp0=500 Pa; Ps0=10 Pa Pc=50 Pa Tab. A10-1: geometrical parameters. Nozzle diameter: 50 mm Diameter ratio: 3.7 Dimensionless mixing length: 2 NXP: 20 mm Tab. A10-2: mass flow rate results [kg s-1]. Primary inlet 0.01420964 Secondary inlet 3.01E-06 Outlet -0.01421263 Mass conservation 9.20E-09 Fig. A10-1: Results for the static pressure. Fig. A10-2: Results for the static temperature. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 84 of 94 Fig. A10-3: Results for the molar fraction of helium. Fig. A10-4: Results for the Mach number. Fig. A10-5: Results for the density. Fig. A10-6: Results for the velocity. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 85 of 94 Appendix 11: Simulation of the first stage (Pp0=2500 Pa; Ps0=50 Pa Pc=100 Pa) of the pack ejectors Tab. A11-1: geometrical parameters. Nozzle diameter: 10 mm Diameter ratio: 3.7 Dimensionless mixing length: 2 NXP: 2 mm Tab. A11-2: mass flow rate results [kg s-1]. Primary inlet 0.000990841 Secondary inlet 6.11E-06 Outlet -0.000990841 Mass conservation 5.20E-10 Fig. 48: Results for the static pressure. Fig. A11-2: Results for the static temperature. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 86 of 94 Fig. A11-3: Results for the molar fraction of helium. Fig. A11-4: Results for the Mach number. Fig. A11-5: Results for the density. Fig. A11-6: Results for the velocity. Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 87 of 94 Appendix 12: Primary pressure sensitivity results The Geometry used is the 50mm primary throat ejector. The global results are given in the followings tables Tab. A12-1: Primary pressure sensitivity results. Pp0 [Pa] Ps0 [Pa] Pcb [Pa] mp [kg s-1] ms [kg s-1] out [kg s-1] mass conservation [kg s-1] entrainment ratio 500 10 15 0.004761 2.67E-05 -0.00478837 -5.02E-11 5.61E-03 500 10 10.5 0.004761 2.67E-05 -0.0047884 -7.95E-10 5.62E-03 500 10 20 0.004761 2.59E-05 -0.00478752 -5.90E-11 5.43E-03 500 10 25 0.004761 1.77E-05 -0.00477933 -8.81E-11 3.71E-03 500 10 22 0.004761 2.43E-05 -0.00478594 6.58E-09 5.10E-03 1500 10 10.5 0.014209 2.96E-06 -0.0142126 4.42E-09 2.08E-04 1500 10 50 0.014209 3.01E-06 -0.01421263 9.20E-09 2.12E-04 1500 10 52 0.014209 1.38E-06 -0.01421102 -1.15E-09 9.71E-05 1500 10 55 0.014209 1.21E-06 -0.01421085 -1.84E-09 8.50E-05 1000 10 10.5 0.014382 3.37E-06 -0.01438564 9.52E-08 2.34E-04 1000 10 40 0.014382 3.36E-06 -0.01438552 1.98E-07 2.33E-04 1000 10 45 0.014382 3.30E-06 -0.01438585 -1.91E-07 2.29E-04 1000 10 50 0.009443 1.54E-06 -0.00944492 3.67E-08 1.63E-04 Appendix 13: NXP sensitivity results Tab. A13-1: NXP sensitivity results. NXP variation sec mass flow Throughput [Pa m3 s-1] primary mass flow NXP [mm] Throughput variation 1 6.70E-05 5.77193175 0.00270333 26.3055216 1 1.5 2.59E-04 22.3458971 0.00270721 35.7658764 3.87147632 1.75 2.75E-04 23.6608013 0.00271048 39.9092916 4.0992864 2 2.62E-04 2.25E+01 0.00271007 43.7708387 3.90280805 3 1.81E-04 15.5475333 0.00270469 57.2346782 2.69364469 4 1.03E-04 8.88888489 0.00270532 68.6064382 1.54001906 Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 88 of 94 Appendix 14: Secondary inlet pressure sensitivity results Tab. A14-1: Secondary pressure sensitivity results. Pp0 [Pa] Ps0 [Pa] Pc [Pa] mp [kg s-1] ms [kg s-1] out [kg s-1] mass conservation [kg s-1] entrainment ratio 1000 10 10.5 0.01438236 3.37E-06 -0.01438564 9.52E-08 2.34E-04 1000 10 40 0.01438236 3.36E-06 -0.01438552 1.98E-07 2.33E-04 1000 10 45 0.01438236 3.30E-06 -0.01438585 -1.91E-07 2.29E-04 1000 10 50 0.00944341 1.54E-06 -0.00944492 3.67E-08 1.63E-04 1000 15 16 0.00944362 3.55E-05 -0.00947904 9.86E-08 3.76E-03 1000 15 40 0.00944362 3.55E-05 -0.009479 1.07E-07 3.76E-03 1000 15 45 0.00944335 2.94E-05 -0.00947288 -1.13E-07 3.12E-03 1000 20 22 0.00944479 5.35E-05 -0.00949856 -2.55E-07 5.67E-03 1000 20 40 0.00944345 5.32E-05 -0.0094967 -1.92E-08 5.64E-03 1000 20 45 0.0094436 3.12E-05 -0.00947485 -3.08E-08 3.31E-03 Appendix 15: Sensitivity of the dimensionless mixing length Tab. A15-1: Sensitivity of the dimensionless mixing length. Pp0 [Pa] Ps0 [Pa] Pc [Pa] length ratio mp [kg/s] ms [kg/s] out [kg/s] mass conservation [kg/s] 1000 15 35 2 0.00944362 3.55E-05 -0.009479 1.07E-07 1000 15 35 1 0.0094369 3.66E-05 -0.00947348 3.82E-08 1000 15 35 3 0.00941377 1.77E-05 -0.00943147 4.90E-09 1000 15 35 0 0.00946435 3.10E-05 -0.00949548 -1.20E-07 Appendix 16: Sensitivity of the diameter ratio Tab. A16-1: Sensitivity of the diameter ratio. Pp0 [Pa] Ps0 [Pa] Pc [Pa] diameter ratio mp [kg/s] ms [kg/s] out [kg/s] 1000 15 35 3 0.00949602 9.34E-06 -0.00950533 1000 15 35 3.3 0.00946582 2.59E-05 -0.00949204 1000 15 35 3.5 0.00944362 3.55E-05 -0.009479 1000 15 35 3.7 0.00941464 2.05E-05 -0.00943513 1000 15 35 3.9 0.00942638 4.82E-05 -0.00947462 1000 15 35 4.5 0.0093631 2.98E-05 -0.00939307 Master Thesis Date Page Performance prediction of mercury ejector vacuum pumps for the DEMO booster pumping system 31 March 2023 89 of 94 Appendix 17: Cooling ejector walls simulations Tab. A17-1: Cooling ejector walls simulations. Case Pp0 [Pa] Ps0 [Pa] Pc [Pa] mp [kg/s] ms [kg/s] out [kg s-1] mass conservation [kg s-1] variation reference 1000 20 40 0.009443 5.32E-05 -0.00949 -1.92E-08 diffuser cooled 1000 20 40 0.009443 5.98E-05 -0.009503 2.32E-08 12.40% diffuser and mixing chamber cooled 1000 20 40 0.009443 5.68E-05 -0.00950 8.87E-10 6.68% mixing chamber cooled 1000 20 40 0.009443 5.62E-05 -0.00949 1.71E-08 5.66% Appendix 18: Cooling ejector diffuser wall Tab. A18-1: geometrical parameters. Nozzle diameter: 50 mm Diameter ratio: 3.7 Dimensionless mixing length: 2 NXP: 20 mm Fig. A18-1: Results for the static pressure. Fig. A18-2: Results for the static temperature.