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Vacuum technology Screw spindle vacuum pumps Fig. 6: Simulation domain for a pressure-driven flow through a channel with structured surfaces a) L/h=1 b) L/h=10 Fig. 7: Mass flow rate through a channel with structured surfaces related to the respective mass flow rate through a channel with smooth surfaces as a function of the inlet pressure p 1 for different profile parameters α = β [Saz20]. Fig. 7a shows this for a length to height ratio of one, which corresponds approximately to an orifice flow. It can be seen that all four curves cause an increased throttling effect with decreasing pressure, which can be explained analogously to Fig. 3 by the fact that the proportion of gas-surface interactions increases. Furthermore, the throttling effect depends on the shape of the surface structure. A very wide profile angle causes only a slight reduction in the mass flow rate, while a profile angle α = β = 45° already causes a reduction in the mass flow rate of about 10 % in the orifice flow. If the profile angle is reduced further, the flow stagnates. This can also be seen in Fig. 7b, where the same situation is shown for a length to height ratio of ten and the molecules therefore have to pass through a significantly longer channel. Here, a reduction of about 25 % can already be achieved for smaller profile angles. It is also noticeable that the effect becomes smaller for increasing inlet pressures, so that a transfer to the vacuum pump is particularly interesting for the low-pressure side of the machine. This results in promising synergy possibilities with the previously described approach by Kösters and Eickhoff, as larger machine gaps could be realised on the low-pressure side in order to prevent overcompression during start-up. As soon as the pressure on the low-pressure side is low enough, the increased gap size would be compensated for with the help of the surface structures by reducing the mass flow rate even further than is already the case due to the rarefied gas flow with technically smooth walls. Since there is always a superposition of a pressure-driven Poiseuille flow and a shear-driven Couette flow in the gaps of vacuum pumps, the DSMC method is used to analyse the effect of such a surface structure on a pure Couette flow. The corresponding simulation domain is shown in Fig. 8. With a gap height of h = 0.3 mm, a profile depth Λ = 0.03 mm is used. Since the gap length and the gap width in vacuum pumps are much greater than the gap height, an infinitely wide and long channel is simulated for simplification, so that symmetrical boundary conditions are used in the depth direction (z) and cyclic boundary conditions in the flow direction (x) to reduce the computational effort. The latter have the property that the left and right cells are linked to each other as if the channel were continuing. Accordingly, a particle that crosses the cyclic boundary condition on the right-hand side is initialised again on the left-hand side and vice versa. The lower wall has a wall velocity of U = 10 m/s in the positive x-direction. In the reference plane shown in green, the mass flow rate is determined by calculating the sum of the particle masses in the positive x-direction minus the sum of the particle masses that cross the plane in the negative direction within a time step and then dividing by the time step: Eq. 1 Regardless of the pressure range, the mass flow rate for a pure Couette flow with technically smooth surfaces can be calculated via Eq. 2 incorporating the density ρ and the smallest cross-section area A = h b [Ple22a, Ple22b]. Fig. 8: DSMC simulation domain of a pure Couette flow through a channel with onesided surface structure. Fig. 9 shows the simulated mass flow rate of a Couette flow for different profile angles α = β in relation to the mass flow rate through a channel with smooth walls with the same gap height as a function of the pres- 50 PROCESS TECHNOLOGY & COMPONENTS 2024
Vacuum technology Screw spindle vacuum pumps Fig. 9: Mass flow rate of a pure Couette flow through a channel with surface structures related to the respective mass flow rate with technically smooth walls as a function of the pressure p for different profile angles α = β. sure p for air at T = 293 K. The error bars show the maximum statistical uncertainty of the results. It can be seen that - as with the pressure-driven flow - a reduction in the mass flow rate at same minimal gap height can also be achieved with a shear-driven flow using surface structures. The smaller the pressure, the greater the effect, whereas the effect disappears at high pressures. Here too, the greatest reduction of about 30 % can be achieved for profile angles α = β = 30°. Conclusion and outlook The theoretical investigations suggest that a microscopic surface structure in the low-pressure range can achieve a significant reduction of up to 30 % in the gap mass flow rates in vacuum pumps without jeopardising operational safety. On the one hand, this can be used to ensure that the machine has a significantly better suction speed at lower suction pressure ranges with the same gap height, as the measurement results from Dreifert and Müller show with regard to the change in gap height. On the other hand, the idea of Kösters and Eickhoff could be pursued, with which the gap height on the low-pressure side of the machine is increased in order to reduce over-compression at high suction pressures. As the surface structure produces a significantly greater throttling effect, particularly at low suction pressures, but has hardly any effect at high suction pressures, over-compression can be reduced without the machine deteriorating at low suction pressures. Due to the great potential for improvement, the applicability of surface structures in rarefied gas flows is being investigated in more detail in a current cooperative research project between the Chair of Fluidics and the Institute of Machining Technology at TU Dortmund University. In the course of the project, the shape of the structure is being optimised in order to achieve the greatest possible throttling effect on the one hand and to enable efficient production on the other hand. A central challenge in production is the small profile depth of the surface structure - this is referred to as micro-machining. The dimensions of the burrs can be of the same order of magnitude as the profile depth. For this reason, a special tool is developed with the aid of a finite element chip formation simulation, whereby various geometric adjustments to the tool can be simulatively investigated to minimise burr formation. The most promising tool variants are then manufactured and used to prepare samples with the identified surface structures. On the one hand, these are analysed metrologically, which enables the chip formation simulation to be validated, and on the other hand they are used on a vacuum test rig in which the throttling effect can be investigated. Acknowledgements Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation). Gefördert durch die Deutsche Forschungsgemeinschaft (DFG) – Projektnummer 513663608. Bibliography [Bir94] Bird, G. A.: Molecular gas dynamics and the direct simulation of gas flows (Clarendon Press, Oxford, 1994). [Brü21] Brümmer, B.; Pleskun, H.: Verfahren und Vorrichtung zur Beeinflussung verdünnter Gasströmungen mit Hilfe von Rauheiten aufweisenden Oberflächen, insbesondere an Vakuumpumpen, MEMS, Patent, DE 102021002290, 2021. [Dre14] Dreifert, T.; Müller, R.: Screw Vacuum pumps - The state of the art: International Conference on Screw machines 2014: VDI-Berichte 2228, pp. 29-42 (VDI-Verlag, 2014). [Jou18] Jousten, K.: Wutz - Handbuch der Vakuumtechnik, Vol. 12 (Vieweg+Teubner, 2018). Symbols and abbreviations symbol unit explanation b m gap width h m gap height L m gap length . m kg⁄s mass flow rate m kg mass p Pa pressure t s time T K temperature U m⁄s wall velocity α ° profile angle β ° profile angle Λ m profile depth ρ kg⁄m 3 density index or abbreviation eff suction Wiesbaden, [Kös06] Kösters, H.; Eickhoff, J.: Trockene Schraubenvakuumpumpe mit hoher innerer Verdichtung, Schraubenmaschinen 2006: VDI- Berichte 1932, pp. 423-428 (VDI- Verlag, 2006). 1 inlet 2 outlet explanation effective value suction value + positive direction - negative direction The Authors: Sven Brock 1 , Heiko Pleskun 1 , Gero Polus 2 , Jannis Saelzer 2 , Prof. Dr.-Ing. Prof. h.c. Dirk Biermann 2 , Prof. Dr.-Ing. Andreas Brümmer 1 1 Chair of Fluidics, TU Dortmund University, 44227 Dortmund, Germany https://ft.mb.tu-dortmund.de/ 2 Institute of Machining Technology, TU Dortmund University, 44227 Dortmund, Germany https://isf.mb.tu-dortmund.de/ [Moe23] Moesch, T. W. et al.: Thermodynamic analysis of a conical screw spindle compressor for R718: ICR2023 - 26th International Congress of Refrigeration, p. 012016, 2023. [Ple22a] Pleskun, H.; Bode, T., Brümmer, B.: Couette flow in a rectangular channel in the whole range of the gas rarefaction, Physics of Fluids, Vol. 34, p. 032004, 2022. [Ple22b] Pleskun, H.; Brümmer, B.: Gas-surface interactions of a Couette-Poiseuille flow in a rectangular channel, Physics of Fluids, Vol. 34, p. 082009, 2022. [Saz20] Sazhin, O.: Rarefied gas flow through a rough channel into a vacuum: Microfluid Nanofluid, Vol. 24, 2020. PROCESS TECHNOLOGY & COMPONENTS 2024 51
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