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Film Thickness Formula for Thermal EHL Line Contact Considering a New Reynolds–Carreau Equation

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.

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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

  1. Jang, J.Y., Khonsari, M.M., Bair, S.: On the elastohydrodynamic analysis of shear-thinning fluids. Proc. R. Soc. A 463, 3271–3290 (2007)

    Article  Google Scholar 

  2. 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)

    Article  Google Scholar 

  3. Carreau, P.J.: Rheological equations from molecular network theories. Trans. Soc. Rheol. 16(1), 99–127 (1972)

    Article  Google Scholar 

  4. Bair, S.: A Reynolds–Ellis equation for line contact with shear-thinning. Tribol. Int. 39, 310–316 (2002)

    Article  Google Scholar 

  5. 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)

    Article  Google Scholar 

  6. 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)

  7. 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)

    Article  Google Scholar 

  8. 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)

    Article  Google Scholar 

  9. 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)

  10. 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)

    Article  Google Scholar 

  11. 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)

    Article  Google Scholar 

  12. 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)

    Article  Google Scholar 

  13. 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)

    Article  Google Scholar 

  14. 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)

    Article  Google Scholar 

  15. Spikes, H.A., Anghel, V., Glovnea, R.: Measurement of the rheology of lubricant films within elastohydrodynamic contacts. Tribol. Lett. 17, 593–605 (2004)

    Article  Google Scholar 

  16. 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)

    Article  Google Scholar 

  17. Raisin, J., Fillot, N., Dureisseix, D., Vergne, P., Lacour, V.: characteristic times in transient thermal elastohydrodynamic line contacts. Tribol. Int. 82, 472–483 (2015)

    Article  Google Scholar 

  18. Stachowiak, G.W., Batchelor, A.W.: Engineering Tribology. Elsevier, Oxford (2005)

    Google Scholar 

  19. Wilson, W.R.D.: A Framework for thermohydrodynamic lubrication analysis. J. Tribol. 120(2), 399–405 (1998)

    Article  Google Scholar 

  20. 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

    Google Scholar 

  21. 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)

  22. 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)

    Article  Google Scholar 

  23. 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)

    Google Scholar 

  24. 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)

    Article  Google Scholar 

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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.

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Correspondence to Eduardo de la Guerra Ochoa.

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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

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  • DOI: https://doi.org/10.1007/s11249-018-0981-6

Keywords

  • Reynolds–Carreau equation
  • Shear-thinning
  • Thermal effects
  • Film thickness