Abstract
This article presents thermal EHL calculations for line contacts using a new analytical form of the Reynolds equation for lubricants whose rheological behaviour follows a modified Carreau model proposed by Bair. The isothermal calculation process was presented in: de la Guerra (Tribol Int 82:133–141, 2015). A new parametric formula is hereby developed using the aforementioned Reynolds–Carreau equation and adding the thermal effects to the solving process. The accuracy of this formula is discussed by comparing the estimates with the experimental and numerical results available. This analytical formula provides a fast and easy calculation methodology with good accuracy within a reasonably wide range of operating conditions.
Similar content being viewed by others
Abbreviations
- a :
-
Hertzian contact half-width (m)
- c l :
-
Specific heat of the lubricant (J/kg K)
- c b :
-
Specific heat of the bodies (J/kg K)
- d 1,2 :
-
Thermally affected depth of the bodies (m)
- E′:
-
Young’s reduced modulus (Pa)
- G :
-
Shear modulus (Pa)
- \(\bar{G}\) :
-
Dimensionless material parameter
- h :
-
Film thickness profile (m)
- h N :
-
Newtonian central film thickness (m)
- h 0 :
-
Central film thickness (m)
- k l :
-
Thermal conductivity of the lubricant (W/m K)
- k b :
-
Thermal conductivity of the bodies (W/m K)
- L :
-
Contact length in flow direction (m)
- L T :
-
Thermal loading factor
- n :
-
Carreau exponent
- p :
-
Pressure (Pa)
- p m :
-
Average Hertz pressure (Pa)
- p 0 :
-
Maximum Hertz pressure (Pa)
- Q :
-
Flow rate per unit length (m2/s)
- R :
-
Reduced contact radius (m)
- \(\bar{R}\) :
-
Anuradha and Kumar factor for shear-thinning under pure rolling conditions
- \(\bar{S}\) :
-
Anuradha and Kumar factor for shear-thinning under rolling and sliding conditions
- T :
-
Temperature of the lubricant (K)
- T 0 :
-
Reference temperature of the lubricant (K)
- T 1 :
-
Temperature of the upper body (K)
- T 2 :
-
Temperature of the lower body (K)
- T b :
-
Lubricant bath temperature (K)
- u :
-
Velocity of the lubricant (m/s)
- \(\bar{U}\) :
-
Dimensionless velocity parameter
- u 1,2 :
-
Velocity of the surfaces (m/s)
- u m :
-
Average velocity of the contacting surfaces (m/s)
- Δu :
-
Sliding velocity of the contacting surfaces (m/s)
- W :
-
Normal load per unit length (N/m)
- \(\bar{W}\) :
-
Dimensionless load parameter
- x :
-
Coordinate in flow direction (m)
- z :
-
Coordinate across the film thickness (m)
- α :
-
Viscosity–pressure coefficient (Pa−1)
- β :
-
Viscosity–temperature coefficient (K−1)
- \(\dot{\gamma }\) :
-
Shear rate (s−1)
- ε :
-
Thermal expansion coefficient (K−1)
- η :
-
Viscosity (Pa s)
- κ :
-
Shear-thinning parameter
- μ :
-
Low-shear viscosity (Pa s)
- μ 0 :
-
Low-shear viscosity at ambient pressure and reference temperature (Pa s)
- ρ l :
-
Density of the lubricant (kg/m3)
- ρ b :
-
Density of the bodies (kg/m3)
- Σ :
-
Δu/u m , slide-to-roll ratio
- τ :
-
Shear stress (Pa)
- τ m :
-
Mid-plane shear stress (Pa)
- φ NN :
-
Shear-thinning factor under pure rolling conditions
- φ SRR :
-
Shear-thinning factor under rolling and sliding conditions
- φ T :
-
Thermal film thickness factor
References
Jang, J.Y., Khonsari, M.M., Bair, S.: On the elastohydrodynamic analysis of shear-thinning fluids. Proc. R. Soc. A 463, 3271–3290 (2007)
Anuradha, P., Kumar, P.: New film thickness formula for shear thinning fluids in thin film elastohydrodynamic lubrication line contacts. Proc. Inst. Mech. Eng. Part J: J. Eng. Tribol. 225, 173–179 (2011)
Carreau, P.J.: Rheological equations from molecular network theories. Trans. Soc. Rheol. 16(1), 99–127 (1972)
Bair, S.: A Reynolds–Ellis equation for line contact with shear-thinning. Tribol. Int. 39, 310–316 (2002)
Bair, S., Vergne, P., Querry, M.: A unified shear-thinning treatment of both film thickness and traction in EHD. Tribol. Lett. 18(2), 145–152 (2005)
Bair, S.: High pressure rheology for quantitative elastohydrodynamics. In: Bair, S., McCabe, C. (eds) Tribology and Interface Engineering Series. No 54 ed. Elsevier, London (2007)
De la Guerra, E., Echávarri, J., Chacón, E., Lafont, P., Díaz, A., Munoz-Guijosa, J.M., Muñoz, J.L.: New Reynolds equation for line contact based on the Carreau model modification by Bair. Tribol. Int. 55, 141–147 (2012)
Bair, S., Khonsari, M.M.: Reynolds equation for common generalized Newtonian model and an approximate Reynolds–Carreau equation. Proc. Inst. Mech. Eng. Part J: J. Eng. Tribol. 220(4), 365–374 (2006)
Grubin, A.N.: Fundamentals of the hydrodynamic theory of lubrication of heavily loaded cylindrical surfaces. Book No. 30 (1949), Central Scientific Research Institute for Technology and Mechanical Engineering, Moscow (DSIR Translation)
De la Guerra, E., Echávarri, J., Sánchez, A., Chacón, E.: Film thickness predictions for line contact using a new Reynolds–Carreau equation. Tribol. Int. 82, 133–141 (2015)
Habchi, W., Vergne, P., Bair, S., Andersson, O., Eyheramendy, D., Morales-Espejel, G.E.: Influence of pressure and temperature dependence of thermal properties of a lubricant on the behavior of circular TEHD contacts. Tribol. Int. 43, 1842–1850 (2010)
Echávarri, J., Lafont, P., Chacón, E., de la Guerra, E., Díaz, A., Munoz-Guijosa, J.M., Muñoz, J.L.: Analytical model for predicting the friction coefficient in point contacts with thermal elastohydrodynamic lubrication. Proc. Inst. Mech. Eng. Part J: J. Eng. Tribol. 225, 181–191 (2011)
Anuradha, P., Kumar, P.: New minimum film thickness formula for EHL rolling/sliding line contacts considering shear thinning behaviour. Proc. Inst. Mech. Eng. Part J: J Eng. Tribol. 227(3), 187–198 (2012)
Carli, M., Sharif, K.J., Ciulli, E., Evans, H.P., Snidle, R.W.: Thermal point contact EHL analysis of rolling/sliding contacts with experimental comparison showing anomalous film shapes. Tribol. Int. 42(4), 517–525 (2009)
Spikes, H.A., Anghel, V., Glovnea, R.: Measurement of the rheology of lubricant films within elastohydrodynamic contacts. Tribol. Lett. 17, 593–605 (2004)
Habchi, W., Eyheramendy, D., Bair, S., Vergne, P., Morales-Espejel, G.: Thermal elastohydrodynamic lubrication of point contacts using a Newtonian/generalized Newtonian lubricant. Tribol. Lett. 30, 41–52 (2008)
Raisin, J., Fillot, N., Dureisseix, D., Vergne, P., Lacour, V.: characteristic times in transient thermal elastohydrodynamic line contacts. Tribol. Int. 82, 472–483 (2015)
Stachowiak, G.W., Batchelor, A.W.: Engineering Tribology. Elsevier, Oxford (2005)
Wilson, W.R.D.: A Framework for thermohydrodynamic lubrication analysis. J. Tribol. 120(2), 399–405 (1998)
De la Guerra, E., Echávarri, J., Chacón, E., Del Río, B.: A thermal resistances-based approach for thermal-elastohydrodynamic calculations in point contacts. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. (2017). https://doi.org/10.1177/0954406217713231
Hsiao, H.S., Hamrock, B.J.: A complete solution for thermal-elastohydrodynamic lubrication of line contacts using circular non-Newtonian fluid model. J. TWM. ASME paper 91-Trib-24 (1992)
Kumar, P., Anuradha, P., Khonsari, M.M.: Some important aspects of thermal elastohydrodynamic lubrication. Proc. Inst. Mech. Eng. Part C: J Mech. Eng. Sci. 224, 2588–2598 (2010)
Abadie, J., Carpentier, J.: Generalization of the Wolfe reduced gradient method to the case of nonlinear constraints. In: Fletcher, R. (ed.) Optimization. Academic, New York (1969)
Höhn, B.R., Michaelis, K., Mann, U.: Measurement of oil film thickness in elastohydrodynamic contacts influence of various base oils and Vl-lmprovers. Tribol. Ser. 31, 225–234 (1996)
Acknowledgements
This work was carried out as a part of the Research Project DPI2013-48348-C2-2-R, financed by the Spanish Ministry of Economy and Competitiveness. We would also like to thank the Lubricants Laboratory of Repsol.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
de la Guerra Ochoa, E., Echávarri Otero, J., Sánchez López, A. et al. Film Thickness Formula for Thermal EHL Line Contact Considering a New Reynolds–Carreau Equation. Tribol Lett 66, 31 (2018). https://doi.org/10.1007/s11249-018-0981-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11249-018-0981-6