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Mathematical modeling of the isothermal impingement of liquid droplets in spraying processes

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Abstract

A mathematical representation has been developed and computed results are presented describing the spreading of droplets impacting onto a solid substrate. Problems of this type are of major practical interest in plasma spraying (PS) and in spray forming (SF) operations. While the present study was confined to the fluid flow aspects of the process, information has been generated on both the final splat dimensions and on the time required to complete the spreading process. Through this treatment, it is possible to relate these quantities (the splat size and the spreading time) to the operating conditions,i.e., droplet size and droplet velocity, and material properties. The theoretical predictions were found to be in good agreement with both Madejski’s asymptotic solution[17] and with available experimental results. For typical SF conditions (droplet sizes in the 100-µm range and droplet velocities in the 100 m/s range), the spreading times were of the order of microseconds,i.e., significantly shorter than the estimated solidification time.

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References

  1. F.T. Dodge:J. Colloid Interface Sci., 1988, vol. 121 (1), pp. 154–60.

    Article  Google Scholar 

  2. H.P. Greenspan:J. Fluid Mech., 1978, vol. 84, p. 125.

    Article  Google Scholar 

  3. G.I. Shapiro:J. Appl. Mech. Tech. Phys. (Russian), 1983, vol. 24, p. 321.

    Article  Google Scholar 

  4. P. Savic and G.T. Boult:Proc. Heat Transfer Fluid Mechanics Institute, Stanford University Press, Stanford, CA, 1957, pp. 43–83.

    Google Scholar 

  5. L.M. Hocking:Q. J. Mech. Appl. Math., 1983, vol. 36, part I.

  6. F.H. Harlow and J.P. Shannon:J. Appl. Phys., 1967, vol. 38, p. 10.

    Article  Google Scholar 

  7. F.H. Harlow and J.E. Welch:Phys. Fluids, 1965, vol. 8, p. 12.

    Article  Google Scholar 

  8. A.M. Worthington:A Study of Flashes, The MacMillan Company, New York, NY, 1963.

    Google Scholar 

  9. H.E. Edgerton and J.R. Killian, Jr.:Flash, Charles T. Branford, Boston, MA, 1954.

    Google Scholar 

  10. J.E. Field, M.B. Lesser, and P.N.H. Davies:Proc. 5th Int. Conf. on Erosion by Solid and Liquid Impact, University of Cambridge, Cambridge, United Kingdom, 1976.

    Google Scholar 

  11. M.B. Lesser:Proc. 5th Int. Conf. on Erosion by Solid and Liquid Impact, University of Cambridge, Cambridge, United Kingdom, 1976.

    Google Scholar 

  12. F.P. Bowden and J.E. Field:Proc. R. Soc, 1964, vol. A282, p. 331.

    Article  Google Scholar 

  13. F.J. Heymann:J. Appl. Phys., 1969, vol. 40, p. 13.

    Article  Google Scholar 

  14. J.M. Houben:2nd National Conf. on Thermal Spraying, ASM, Long Beach, CA, 1984.

    Google Scholar 

  15. H. Jones:J. Phys. D: Appl. Phys., 1971, vol. 4, pp. 1657–60.

    Article  Google Scholar 

  16. CA. Williams and H. Jones:Mater. Sci. Eng., 1975, vol. 19, pp. 293–97.

    Article  Google Scholar 

  17. J. Madejski:Int. J. Heat Mass Transfer, 1976, vol. 19, pp. 1009–13.

    Article  Google Scholar 

  18. R. McPherson:J. Mater. Sci., 1980, vol. 15, pp. 3141–49.

    Article  Google Scholar 

  19. D. Apelian and M. Flemings: inAdvanced High Temperature Alloys: Processing and Properties, S.M. Allen, R.M. Pelloux, and R. Widmer, eds., ASM, Metals Park, OH, 1986.

    Google Scholar 

  20. D. Apelian, G. Gillen, and A. Leatham:Conf. Proc, ASM Meeting, Orlando, FL, Oct. 1986, pp. 107–25.

    Google Scholar 

  21. E. Garrity, D. Wei, and D. Apelian: inModeling and Control of Casting and Welding Processes IV, A. Giamei, ed., TMS-AIME, Warrendale, PA, 1988, p. 593.

    Google Scholar 

  22. A.R.E. Singer and R.W. Evans:Met. Technol., 1983, vol. 10, pp. 61–68.

    Article  Google Scholar 

  23. P. Mathur, D. Apelian, and A. Lawley:Acta Metall., 1989, vol. 37, p. 429.

    Article  Google Scholar 

  24. P. Mathur, S. Annavarapu, D. Apelian, and A. Lawley:J. Mater., 1989, vol. 41 (10), pp. 23–28.

    Google Scholar 

  25. S. Annavarapu, D. Apelian, and A. Lawley:Metall. Trans. A, 1990, vol. 21A, pp. 3237–56.

    Article  Google Scholar 

  26. S. Annavarapu, D. Apelian, and A. Lawley:Metall. Trans. A, 1988, vol. 19A, pp. 3077–86.

    Article  Google Scholar 

  27. G. Hertman, P.N. Hansen, and P.R. Sahm: inModeling of Casting and Welding Processes IV, A.F. Giamei and G.J. Abbaschian, eds., TMS, Warrendale, PA, 1988, pp. 915–20.

    Google Scholar 

  28. E. Gutierrez-Miravete, G. Trapaga, and J. Szekely: inCasting of Near Neat Shape Products, Y. Sahai, J.E. Battles, R.S. Carbonara, and C.E. Mobley, eds., TMS, Warrendale, PA, 1988, pp. 133–51.

    Google Scholar 

  29. G. Trapaga: Sc.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1989.

  30. N. El-Kaddah, J. McKelliget, and J. Szekely:Metall. Trans. B, 1984, vol. 15B, pp. 59–70.

    Article  Google Scholar 

  31. J.A. Lewis and W.H. Gauvin:AIChE J., 1973, vol. 19 (5), pp. 983–90.

    Article  Google Scholar 

  32. M.I. Boulos and H. Gauvin:CJChE J., 1974, vol. 52, p. 355.

    Google Scholar 

  33. E. Pfender:Pure Appl. Chem., 1985, vol. 57, p. 1179.

    Article  Google Scholar 

  34. D. Apelian, D. Wein, and M. Paliwal:Thin Solid Films, 1984, vol. 118, pp. 395–407.

    Article  Google Scholar 

  35. E. Bourdin, P. Fauchais, and M. Boulos:Int. J. Heat Mass Transfer, 1983, vol. 26, p. 567.

    Article  Google Scholar 

  36. P. Fauchais, M. Vardelle, A. Vardelle, and J.F. Coudert:Metall. Trans. B, 1989, vol. 20B, pp. 263–76.

    Article  Google Scholar 

  37. M. Vardelle, A. Vardelle, P. Fauchis, and M.I. Boulos:AIChE J., 1983, vol. 29 (2), pp. 236–43.

    Article  Google Scholar 

  38. Y.P. Chyou and E. Pfender.Plasma Chem. Plasma Process, 1989, vol. 9, pp. 45–71.

    Article  Google Scholar 

  39. J.M. Floryan and H. Rasmussen:Appl. Mech. Rev., 1989, vol. 42 (12), pp. 323–41.

    Article  Google Scholar 

  40. FLOW-3D, Report No. FSI-88-00-l, Flow Science Inc., Los Alamos, NM, 1988, vols. 1-4.

  41. B.J. Daly:J. Comput. Phys., 1969, vol. 4, pp. 97–117.

    Article  Google Scholar 

  42. J.E. Welch, F.H. Harlow, J.P. Shannon, and B.J. Daly: Los Alamos Scientific Laboratory Report LA-3425, Los Alamos, NM, 1975.

  43. C.W. Hirt, B.D. Nichols, and N.C. Romero: Los Alamos Scientific Laboratory Report No. LA-5852, Los Alamos, NM, 1975.

  44. C.W. Hirt and B.D. Nichols:J. Comp. Phys., 1981, vol. 39, p. 201.

    Article  Google Scholar 

  45. P. Zoltowski:Rev. Int. Hautes. Temper. Refract., 1968, vol. 5, p. 253.

    Google Scholar 

  46. D. Backman, E. Russell, D. Wei, and Y. Pang: inIntelligent Processing of Materials, H. Wadley and W. Eckgart, Jr., eds., TMS, Warrendale, PA, 1990.

    Google Scholar 

  47. D. Backman:J. Met., 1990, vol. 42, pp. 17–20, 915.

    Google Scholar 

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Authors and Affiliations

  1. Department of Materials Science and Engineering, Massachusetts Institute of Technology, 02139, Cambridge, MA

    Julian Szekely (Professor of Materials Engineering)

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  1. Gerardo Trapaga
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  2. Julian Szekely
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formerly Graduate Student, Postdoctoral Associate.

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Trapaga, G., Szekely, J. Mathematical modeling of the isothermal impingement of liquid droplets in spraying processes. Metall Trans B 22, 901–914 (1991). https://doi.org/10.1007/BF02651166

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