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Compressible high-pressure lubrication flows in thrust bearings

Published online by Cambridge University Press:  31 March 2022

S.Y. Chien Open the ORCID record for S.Y. Chien [Opens in a new window]  and
M.S. Cramer Open the ORCID record for M.S. Cramer [Opens in a new window]
S.Y. Chien
Affiliation:
Engineering Mechanics Program, Virginia Polytechnic Institute and State University, Blacksburg, VA24060, USA
M.S. Cramer*
Affiliation:
Engineering Mechanics Program, Virginia Polytechnic Institute and State University, Blacksburg, VA24060, USA
*
†Email address for correspondence: [email protected]
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Abstract

We present a detailed derivation of the Reynolds equation and its corresponding energy equation for three-dimensional, steady, laminar, compressible flows of single-phase Navier–Stokes fluids in thrust bearings. These equations are shown to be valid over most of the dense and supercritical gas regime except for the vicinity of the thermodynamic critical point. It is shown that the primary thermodynamic function governing the lubrication flow of high-pressure gases is the effective bulk modulus defined as the ratio of the bulk modulus to the shear viscosity. Numerical solutions to our Reynolds equation are obtained using a finite difference scheme for both moderate and high-speed flows. Approximate solutions to our Reynolds equation for high-speed flows are also derived through a perturbation analysis. It is found that boundary layers form on three out of four edges of the thrust pad. At the inner and outer radii of the pad, the flow is governed by a nonlinear heat equation. As the main flow leaves the pad, the flow is governed by a nonlinear relaxation equation. These three boundary layer solutions are rendered consistent by the construction of boundary layer solutions in the corner regions. A composite solution is developed which provides a single approximation and has the same accuracy as the individual approximations in their respective regions of validity.

JFM classification

Compressible Flows: Compressible Flows Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 939 , 25 May 2022 , A38
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Almqvist, A., Burtseva, E., Perez-Rafols, F. & Wall, P. 2019 New insights on lubrication theory for compressible fluids. Inl J. Engng Sci. 145, 103170.CrossRefGoogle Scholar
Bell, I.H., Wronski, J., Quoilin, S. & Lemort, V. 2014 Pure and pseudo-pure fluid thermophysical property evaluation and the open-source thermophysical property library CoolProp. Ind. Engng Chem. Res. 53 (6), 2498–2508.CrossRefGoogle ScholarPubMed
Chien, S.Y. 2019 Compressible lubrication theory in pressurized gases. PhD thesis, Virginia Polytechnic Institute and State University.Google Scholar
Chien, S.Y. & Cramer, M.S. 2019 a Load and loss for high speed lubrication flows of pressurized gases between non-concentric cylinders. J. Fluid Mech. 867, 1–25.CrossRefGoogle Scholar
Chien, S.Y. & Cramer, M.S. 2019 b Pressure, temperature, and heat flux in high speed lubrication flows of pressurized gases. Tribol. Intl 129, 468–475.CrossRefGoogle Scholar
Chien, S.Y. & Cramer, M.S. 2019 c Virial approximation for load and loss in high-speed journal bearings using pressurized gases. Fluids 4 (1), 27.CrossRefGoogle Scholar
Chien, S.Y., Cramer, M.S. & Untaroiu, A. 2017 Compressible Reynolds equation for high-pressure gases. Phys. Fluids 29 (11), 116101.CrossRefGoogle Scholar
Chung, T.H., Ajlan, M., Lee, L.L. & Starling, K.E. 1988 Generalized multiparameter correlation for nonpolar and polar fluid transport properties. Ind. Engng Chem. Res. 27 (4), 671–679.CrossRefGoogle Scholar
Chung, T.H., Lee, L.L. & Starling, K.E. 1984 Applications of kinetic gas theories and multiparameter correlation for prediction of dilute gas viscosity and thermal conductivity. Ind. Engng Chem. Fundam. 23 (1), 8–13.CrossRefGoogle Scholar
Ciuperca, I.S., Feireisl, E., Jai, M. & Petrov, A. 2018 A rigorous derivation of the stationary compressible Reynolds equation via the Navier–Stokes equations. Math. Models Meth. Appl. Sci. 28, 697–732.CrossRefGoogle Scholar
Conboy, T.M. 2013 Real-gas effects in foil thrust bearings operating in the turbulent regime. J. Tribol. 135 (3), 031703.CrossRefGoogle Scholar
Conboy, T.M., Wright, S.A., Pasch, J., Fleming, D., Rochau, G. & Fuller, R. 2012 Performance characteristics of an operating supercritical $\mathrm {CO}_2$ Brayton cycle. Trans. ASME: J. Engng Gas Turbines Power 134 (11), 111703.Google Scholar
Crespi, F., Gavagnin, G., Sánchez, D. & Martínez, G.S. 2017 Supercritical carbon dioxide cycles for power generation: a review. Appl. Energy 195, 152–183.CrossRefGoogle Scholar
DellaCorte, C., Radil, K.C., Bruckner, R.J. & Howard, S.A. 2008 Design, fabrication, and performance of open source generation I and II compliant hydrodynamic gas foil bearings. Tribol. Trans. 51 (3), 254–264.CrossRefGoogle Scholar
Dostal, V., Driscoll, M.J. & Hejzlar, P. 2004 A supercritical carbon dioxide cycle for next generation nuclear reactors. Tech. Rep. MIT-ANP-TR-100.Google Scholar
Dousti, S. & Allaire, P. 2016 A compressible hydrodynamic analysis of journal bearings lubricated with supercritical carbon dioxide. In Proceedings of Supercritical CO2 Power Cycle Symposium. San Antonio, TX.Google Scholar
Dupuy, F., Bou-Said, B., Garcia, M., Grau, G., Rocchi, J., Crespo, M. & Tichy, J. 2016 Tribological study of a slider bearing in the supersonic regime. ASME J. Tribol. 138, 041702.CrossRefGoogle Scholar
Dupuy, F., Bou-Said, B. & Tichy, J. 2015 High-speed subsonic compressible lubrication. ASME J. Tribol. 137, 041702.CrossRefGoogle Scholar
van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. Parabolic.Google Scholar
Gross, W.A., Matsch, L.A., Castelli, V., Eshel, A., Vohr, J.H. & Wildmann, M. 1980 Fluid Film Lubrication. John Wiley and Sons, Inc.Google Scholar
Guenat, E. & Schiffmann, J. 2018 Real-gas effects on aerodynamic bearings. Tribol. Intl 120, 358–368.CrossRefGoogle Scholar
Hamrock, B.J., Schmidt, S.R. & Jacobson, B.O. 2004 Fundamentals of Fluid Film Lubrication. CRC.CrossRefGoogle Scholar
Heshmat, H., Walton, J.F. & Cordova, J.L. 2018 Technology readiness of 5th and 6th generation compliant foil bearing for 10 MWE s-CO$_{2}$ turbomachiery systems. In Proceeding of the 6th International Supercritical CO2 Power Cycles Symposium. Pittsburg, PA.Google Scholar
Kim, D. 2016 Design space of foil bearings for closed-loop supercritical CO$_2$ power cycles based on three-dimensional thermohydrodynamic analyses. Trans. ASME: J. Engng Gas Turbines Power 138 (3), 032504.Google Scholar
Lemmon, E.W., Huber, M.L. & McLinden, M.O. 2002 NIST reference fluid thermodynamic and transport properties–REFPROP. NIST standard Reference Database 23, v7.Google Scholar
Marusic-Paloka, E. & Starcevic, M. 2010 Derivation of Reynolds equation for gas lubrication via asymptotic analysis of the compressible Navier–Stokes system. Nonlinear Anal. Real World Applics. 11, 4565–4571.CrossRefGoogle Scholar
Nayfeh, A.H. 1981 Introduction to Perturbation Methods. Wiley Interscience.Google Scholar
Peng, Z.C. & Khonsari, M.M. 2004 On the limiting load-carrying capacity of foil bearings. J. Tribol. 126 (4), 817–818.CrossRefGoogle Scholar
Pinkus, O. & Sternlicht, B. 1961 Theory of Hydrodynamic Lubrication. McGraw-Hill.Google Scholar
Qin, K. 2017 Development and application of multiphysics simulation tools for foil thrust bearings operating with carbon dioxide. PhD thesis, University of Queensland.Google Scholar
Reid, R.C., Prausnitz, J.M. & Poling, B.E. 1987 The Properties of Gases and Liquids. McGraw-Hill.Google Scholar
Reynolds, O. 1886 On the theory of lubrication and its application to Mr. Beauchamp Tower's experiments, including an experimental determination of the viscosity of olive oil. Proc. R. Soc. Lond. 40 (242–245), 191–203.Google Scholar
Szeri, A.Z. 2010 Fluid Film Lubrication. Cambridge University Press.CrossRefGoogle Scholar
Wright, S.A., Radel, R.F., Vernon, M.E., Robert, G.E. & Pickard, P.S. 2010 Operation and analysis of a supercritical $\mathrm {CO}_2$ Brayton cycle. Sandia Rep. No. SAND2010-0171.CrossRefGoogle Scholar
Zagarola, M.V. & McCormick, J.A. 2006 High-capacity turbo-brayton cryocoolers for space applications. Cryogenics 46, 169–175.CrossRefGoogle Scholar