# Analysis of an Elasto-Hydrodynamic Seal by Using the Reynolds Equation
## 1 Introduction  
Power generation has been, and still is, at the very core of economic, environmental, societal, and ethical discussions in engineering. Triggered mainly by environmental concerns, there have been unprecedented investments and R&D efforts in the power generation field recently. These efforts focused not only on innovative renewable energy generation but also on efficiency improvements in the current power plant technologies. In this context, supercritical carbon dioxide (sCO\({}_{2}\)) power generation technology has received increasing attention in the last decade. sCO\({}_{2}\) power technology is superior to the traditional water-based, air-breathing, direct-fired, open Brayton cycles or indirect-fired, closed Rankine cycles in terms of efficiency and equipment footprint. They have promising potential in nuclear power production, fossil fuel power plants, concentrated solar power, geothermal power, and ship propulsion. To unlock the potential of sCO\({}_{2}\) power cycles, the technology readiness must be demonstrated on the scale of 10-600 MWe and at the sCO\({}_{2}\) temperatures and pressures of 350-700 \({}^{\circ}\)C and 20-35 MPa for nuclear industries [1; 2; 3; 4]. The lack of suitableshaft seals to withstand sCO\({}_{2}\) operating conditions is one of the main challenges at the component level. So far, conventional seals all suffer from the incapability of handling sCO\({}_{2}\) pressures and temperatures in one way or another. For example, labyrinth, brush, and compliant seals suffer from high leakage rates, bristle wear, and scalability issues, respectively [5; 6; 7; 8; 9; 10]. There is an ongoing worldwide effort to develop effective sealing technologies for sCO\({}_{2}\) turbomachinery [11; 12]. [PERSON] et al. ran experimental and CFD analyses on pressure distribution on conventional dry gas seals at different film thickness values using air as a working fluid. They then compared these data and proposed its prospectus in sCO\({}_{2}\) power cycles [13]. [PERSON] et al. developed dry gas seals and ran an experimental analysis on them at different pressures for both controlled and uncontrolled temperatures. The isothermal expansion due to extremely low leakage had been confirmed by successful experimental test results. Furthermore, solid predictive capability with CO\({}_{2}\) was also demonstrated [14]. In addition, [PERSON] proposed the concept of a dry gas seal which is a slight modification and feature enhancement of the current available dry gas seals. However, no scientifically sound data have yet been shown to support their concept [15]. [PERSON] et al. discussed various design challenges for large-diameter seals operating at pressures higher than 6.9 MPa. They also discussed the operating conditions expected for the shaft end seal. As a solution, they proposed a face seal concept that can be applicable at a utility-scale for sCO\({}_{2}\) power cycles [16]. However, these concepts are all still in the development phase. To the authors' knowledge, there is no known sealing solution that can be applied to the sCO\({}_{2}\) turbomachinery. As a potential solution, we propose a patented Elasto-Hydrodynamic (EHD) high-pressure, high-temperature, and scalable shaft end seal for sCO\({}_{2}\) cycles. Under the EHD mechanism, the higher the pressure, the tighter the sealing becomes, while still sustaining a continuous sCO\({}_{2}\) film. The EHD seal has a very simple, sleeve-like structure, wrapping on the rotor with tight clearances, with typically less than 50 \(\upmu\)m initial clearances. Because of its unique design, the pressure differential across the seal forces the seal to bend downward, forming a throat closer to the root of the seal. We hypothesize that the throat achieves the physically possible minimum clearance, acting as an effective flow restrictor.