Skip to main content
SpringerLink
Log in

Modeling of spray-formed materials: Geometrical considerations

  • Published:
Metallurgical and Materials Transactions A Aims and scope Submit manuscript

Abstract

In this article, a mathematical model is formulated to predict the evolution and final geometry of an axisymmetric billet (i.e., round) obtained using an off-axis spray arrangement. The model is formulated by calculating the shape change of a profile curve of a billet surface, based on an axisymmetric surface. On the basis of this model, a methodology to determine the “shadowing effect” coefficient is presented. The modeling results suggest that there are three distinct regions in a spray-formed billet: a base transition region, a uniform diameter region, and an upper transition region. The effects of several important processing parameters, such as the withdrawal velocity of substrate, maximum deposition rate, spray distribution coefficient, initial eccentric distance, and rotational velocity of substrate, on the shape factors (e.g., the diameter size of the uniform region and the geometry of the transition regions) are investigated. The mechanisms responsible for the formation of the three distinct regions are discussed. Finally, the model is then implemented and a methodology is formulated to establish optimal processing parameters during spray forming, paying particular attention to deposition efficiency.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

\(\dot H\) :

deposition rate per unit area (mm3/mm2 · s)

Δh :

normal growth distance of a point during a time interval (mm)

Δt :

deposition time interval (s)

d s :

reference distance, i.e., distance between the atomizer and a reference plane perpendicular to the spray axis (mm)

d p :

distance between the atomizer and any plane perpendicular to the spray axis (mm)

d r :

the distance between the atomizer and the rotation axis (mm)

a s :

maximum deposition rate at reference distance (mm/s)

b s :

spray distribution coefficient at reference distance (mm−2)

a :

maximum deposition rate (mm/s)

b :

spray distribution coefficient (mm−2)

s :

the shortest distance between a deposited point and the spray axis (mm)

ξ :

shadowing effect coefficient

x p, yp, zp :

Cartesian coordinates of a point P in billet surface

x 1 y 1, z 1 and x 2, y 2, z 2 :

Cartesian coordinates of any two neighboring points in a profile curve

a t :

position vector of the atomizer

p :

position vectors of point P in billet surface

p′ :

position vector of a point P′ very near P in the element curve

e n :

unit normal vector of a point P

e s :

unit vector of spray axis

e f :

unit vector of flight direction from the atomizer to a point P

n :

tangential vector of a point P

m :

normal vector of a plane determined by Y-axis and a profile curve

‖V‖:

Euclidean norm of a vector V

β :

angle between the positive direction of X-axis and the plane determined by a profile curve and Y-axis (°)

ω :

rotational velocity of substrate (°/s)

ν :

withdrawal velocity of substrate (mm/s)

ν cri :

critical withdrawal velocity of substrate to spray form round billets (mm/s)

l e :

initial eccentric distance of the spray axis (mm)

l em :

maximum initial eccentric distance to spray form billets (mm)

φ :

inclined angle of the spray axis to the rotation axis (°)

Y :

deposition efficiency in stable growth stage

D :

diameter of uniform region (mm)

R p :

required billet diameter in industrial production (mm)

O :

the center of substrate

V :

vertex of a round billet

T :

intersection point between the spray axis and the profile curve in XOY plane (X>0)

A 0 :

initial position of the atomizer

A t :

atomizer position during spray forming

S 0 :

intersection point between the initial spray axis and the substrate

S :

intersection point between the spray axis and billet surface at any moment

P, Q, P 1, P 2, P j, Pn :

points in billet surface

Σ :

a profile curve

References

  1. E.J. Lavernia and Y. Wu: Spray Atomization and Deposition, John Wiley & Sons, Inc., New York, NY, 1996.

    Google Scholar 

  2. P.S. Grant: Progr. Mater. Sci., 1995, vol. 39, pp. 497–545.

    Article  CAS  Google Scholar 

  3. P. Mathur, D. Apelian, and A. Lawley: Acta Metall., 1989, vol. 37, pp. 429–43.

    Article  CAS  Google Scholar 

  4. P.S. Grant, W.T. Kim, B.P. Bewlay, and B. Cantor: Scripta Metall., 1989, vol. 23, pp. 1651–56.

    Article  CAS  Google Scholar 

  5. A.G. Leatham and A. Lawley: Int. J. Powder Metall., 1993, vol. 29, p. 321.

    CAS  Google Scholar 

  6. Z.H. Lee, H. Hu, E.J. Lavernia, and D.R. White: Metall. Mater. Trans. A, 2000, vol. 31A, pp. 723–33.

    Google Scholar 

  7. G. Hartmann, P.N. Hansen, and P.R. Sahm: Modeling and Control of Casting and Welding Processes IV, TMS, Warrendale, PA, 1988, pp. 915–21.

    Google Scholar 

  8. B.W. Oh and Z.H. Lee: Proceeding of Modeling of Casting and Solidification Process, Yonsei University, Seoul, Korea, 1991, pp. 13–22.

    Google Scholar 

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

    CAS  Google Scholar 

  10. C.Y. Tsao and N.J. Grant: Int. J. Powder Metall., 1994, vol. 30, pp. 323–33.

    CAS  Google Scholar 

  11. Men G. Chu: Proc. 3rd Int. Conf. on Spray Forming, Wales, United Kingdom, Osprey Metals Ltd., Neath, West Glamorgan, 1997, pp. 27–35.

    Google Scholar 

  12. R. Rebis, C. Madden, T. Zappia, and C. Cai: Proc. 3rd Int. Conf. on Spray Forming, Wales, United Kingdom, Osprey Metals Ltd., Neath, West Glamorgan, 1996, pp. 281–85.

    Google Scholar 

  13. A.G. Leatham: Proc. 2nd Int. Conf. on Spray Forming, Swansea, United Kingdom, Osprey Metals Ltd., Neath, West Glamorgan, 1994, pp. 129–40.

    Google Scholar 

  14. I.A. Frigaard: SIAM J. Appl. Math., 1995, vol. 55, pp. 1161–1203.

    Article  Google Scholar 

  15. H.K. Seok, D.H. Yeo, K.H. Oh, H.Y. Ra, D.S. Shin, and H.I. Lee: Proc. 3rd Int. Conf. on Spray Forming, Wales, United Kingdom, 1996, pp. 287–95.

  16. H.K. Seok, D.H. Yeo, K.H. Oh, H.I. Lee, and H.Y. Ra: Metall. Mater. Trans. B, 1998, vol. 29B, pp. 699–708.

    Google Scholar 

  17. P. Mathur, D. Apelian, and A. Lawley: Powder Metall., 1991, vol. 34, p. 109.

    CAS  Google Scholar 

  18. N. Muhamad, J.O. Medwell, and D.T. Gethin: Powder Metall., 1995, vol. 38, pp. 214–20.

    CAS  Google Scholar 

  19. I.A. Frigaard and O. Scherzer: SIAM J. Appl. Math., 1997, vol. 57, pp. 649–82.

    Article  Google Scholar 

  20. I.A. Frigaard: J. Mater. Proc. Manuf. Sci., 1994, vol. 3, pp. 173–92.

    CAS  Google Scholar 

  21. I.A. Frigaard: J. Mater. Proc. Manuf. Sci., 1995, vol. 3, pp. 257–75.

    CAS  Google Scholar 

  22. S. Kang, D.H. Chang, E.S. Lee, and S. Ahn: Mater. Sci. Eng. A, vol. 260A, 1999, pp. 161–69.

    Google Scholar 

  23. C. Kramer, V. Uhlenwinkel, and K. Bauckhage: Proc. 3rd Int. Conf. on Spray Forming, Wales, United Kingdom, 1996, pp. 273–80.

  24. P. Mathur, S. Annavarapu, D. Apelian, and A. Lawley: Mater. Sci. Eng. A, vol. 142A, 1991, pp. 261–76.

    Google Scholar 

  25. S. Elgobashi, T. Abu-arab, M. Rizk, and A. Mostafa: Int. J. Multiphase Flow, 1984, vol. 10, pp. 697–710.

    Article  Google Scholar 

  26. C.-Y.A. Tsao and N.J. Grant: Powder Metall. Int., 1994, vol. 30, pp. 323–33.

    CAS  Google Scholar 

  27. U. Fritsching, H. Zhang, and K. Bauckhage: Steel Res., 1994, vol. 65, p. 273.

    CAS  Google Scholar 

  28. Q. Xu, V.V. Gupta, and E.J. Lavernia: Metall. Mater. Trans. B, 1999, vol. 30B, pp. 527–39.

    Article  CAS  Google Scholar 

  29. D.M.Y. Sommerville: Analytical Geometry of Three Dimensions, The Syndics of Cambridge University Press, London, 1959.

    Google Scholar 

  30. Aderson, R.S. Figliola, and H. Morton: Mater. Sci. Eng. A, 1991, vol. 148A, p. 101.

    Google Scholar 

  31. J.C. Baram, M.K. Veistinen, E.J. Lavernia, M. Abinante, and N.J. Grant: J. Mater. Sci., 1988, vol. 23, p. 2457.

    Article  CAS  Google Scholar 

  32. M.K. Veistinen, E.J. Lavernia, M. Abinante, and N.J. Grant: Mater. Lett., 1987, vol. 5, p. 373.

    Article  Google Scholar 

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

    CAS  Google Scholar 

  34. W.D. Cai, J. Smugeresky, and E.J. Lavernia: Mater. Sci. Eng. A, 1998, vol. 241, pp. 60–71.

    Google Scholar 

  35. W.D. Cai and E.J. Lavernia: Metall. Mater. Trans. B, 1998, vol. 29B, pp. 1085–96.

    Google Scholar 

  36. W.D. Cai and E.J. Lavernia: Metall. Mater. Trans. B, 1998, vol. 29B, pp. 1097–1106.

    Google Scholar 

  37. W.D. Cai and E.J. Lavernia: Mater. Sci. Eng. A, 1997, vols. 226–228, pp. 8–12.

    Google Scholar 

  38. S. Annavarapu and R. Doherty: Int. J. Powder Metall., 1993, vol. 29, pp. 331–43.

    CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemical and Biochemical Engineering and Materials Science, the University of California Irvine, 92697-2575, Irvine, CA

    Y. J. Lin (Graduate Student) & E. J. Lavernia (Professor)

  2. Department of Mechanical and Aerospace Engineering, the University of California Irvine, 92697-2575, Irvine, CA

    J. E. Bobrow (Professor)

  3. the Materials Systems Reliability Department, Ford Research Laboratory, 48121-2053, Dearborn, MI

    D. R. White (Principal Engineering Specialist)

Authors
  1. Y. J. Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. J. Lavernia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. E. Bobrow
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. D. R. White
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lin, Y.J., Lavernia, E.J., Bobrow, J.E. et al. Modeling of spray-formed materials: Geometrical considerations. Metall Mater Trans A 31, 2917–2929 (2000). https://doi.org/10.1007/BF02830347

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02830347

Keywords

Search

Navigation