Heat and Mass Transfer

, Volume 45, Issue 10, pp 1247–1251 | Cite as

Cut-off cooling velocity profiling inside a keyhole model using the Boubaker polynomials expansion scheme

  • S. Amir Hossein A. E. TabatabaeiEmail author
  • Tinggang Zhao
  • O. Bamidele Awojoyogbe
  • Folorunsho O. Moses


The time dependent heating and cooling velocities are investigated in this paper. The temperature profile is found by using a keyhole approximation for the melted zone and solving the heat transfer equation. A polynomial expansion has been deployed to determine the cooling velocity during welding cut-off stage. The maximum cooling velocity has been estimated to be V max ≈ 83°C s−1.


Welding Laser Welding Cooling Velocity Isothermal Expansion Welding Keyhole 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols


Thermal diffusivity (m2 s−1)


Keyhole height (m)


Thermal conductivity (Wm−1 K−1)


Prefixed integer


Fluid pressure at mean temperature (Pa)


Power per unit volume (Wm−3)


Absolute temperature (K)


Maximum absolute temperature (K)


Room absolute temperature (K)

Greek letters


Boubaker polynomials minimal positive roots (dimensionless)

\( \varpi \)

Constant (dimensionless)


Density (Kg m−3)


Heat capacity ratio (dimensionless)


Real coefficients (dimensionless)



The authors would like to acknowledge help and assistance from Associate Professor Karem Boubaker from University of Tunisia.


  1. 1.
    Singh RK, Narayan J (1990) Pulsed-laser evaporation technique for deposition of thin films: physics and theoretical model. Phys Rev B 41:8843–8859CrossRefGoogle Scholar
  2. 2.
    Anisimov SI, Luk’yanchuk BS, Luches A (1996) An analytical model for three-dimensional laser plume expansion into vacuum in hydrodynamic regime. Appl Surf Sci 96–98:24–32CrossRefGoogle Scholar
  3. 3.
    Koopman DW (1971) Langmuir probe and microwave measurements of streaming plasmas generated by focused laser pulses. Phys Fluids 14:1707–1716CrossRefGoogle Scholar
  4. 4.
    Toftmann B, Schou J, Hansen TN, Lunney JG (2000) Angular distribution of electron temperature and density in a laser-ablation plume. Phys Rev Lett 84:3998–4001CrossRefGoogle Scholar
  5. 5.
    Weaver I, Martin GW, Graham WG, Morrow T, Lewis CLS (1999) The langmuir probe as a diagnostic of the electron component within low temperature laser ablated plasma plumes. Rev Sci Instrum 70:1801–1805CrossRefGoogle Scholar
  6. 6.
    Doggett B, Budtz-Joergensen C, Lunney JG, Sheerin P, Turner MM (2005) Behaviour of a planar langmuir probe in a laser ablation plasma. Appl Surf Sci 247:134–138CrossRefGoogle Scholar
  7. 7.
    Chaouachi A, Boubaker K, Amlouk M, Bouzouita H (2007) Enhancement of pyrolysis spray disposal performance using thermal time-response to precursor uniform deposition. Eur Phys J Appl Phys 37:105–109CrossRefGoogle Scholar
  8. 8.
    Ghanouchi J, Labiadh H, Boubaker K (2008) An attempt to solve the heat transfer equation in a model of pyrolysis spray using 4q-order Boubaker polynomials. Int J Heat Tech 26:49–52Google Scholar
  9. 9.
    Awojoyogbe OB, Boubaker K (2009) A solution to Bloch NMR flow equations for the analysis of homodynamic functions of blood flow system using m-Boubaker polynomials. Curr Appl Phys 9:278–283CrossRefGoogle Scholar
  10. 10.
    Boubaker K (2007) On modified Boubaker polynomials: some differential and analytical properties of the new polynomials issued from an attempt for solving bi-varied heat equation. Trends Appl Sci Res 2:540–544 (by Academic Journals ‘aj’ New York)Google Scholar
  11. 11.
    Labiadh H (2007) A Sturm-Liouville shaped characteristic differential equation as a guide to establish a quasi-polynomial expression to the Boubaker polynomials. J Differ Equ Control Processes 2:117–133MathSciNetGoogle Scholar
  12. 12.
    Gallusser R, Dressler K (1971) Application of the coulomb approximation to the Rydberg transitions of the NO molecule. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 22:792–794Google Scholar
  13. 13.
    Armstrong BH, Purdum KL (1966) Extended use of the coulomb approximation: mean powers, a sum rule, and improved transition integrals. Phys Rev 150:51–58CrossRefGoogle Scholar
  14. 14.
    Paul A, Debroy T (1988) Free surface flow and heat transfer in conduction mode laser welding. Metall Mater Trans B 19:851–858Google Scholar
  15. 15.
    Andreassen E, Myhre OJ, Oldervoll F, Hinrichsen EL, Grøstad K, Braathen MD (1995) Nonuniform cooling in multifilament melt spinning of polypropylene fibers: cooling air speed limits and fiber-to-fiber variations. J Appl Polym Sci 58:1619–1632CrossRefGoogle Scholar
  16. 16.
    Belcher SL (2007) Practical guide to injection blow molding, ISBN 0824757912, 9780824757915, CRC Press, Boca RatonGoogle Scholar
  17. 17.
    Santos CAC, Quaresma JNN, Lima JA (2001) Convective heat transfer in ducts: the integral transform approach: the integral transform approach, ISBN 8587922238, 9788587922236, E-papers Servicos Editoriais LtdaGoogle Scholar
  18. 18.
    Mughal MP, Fawad H, Mufti R (2006) Finite element prediction of thermal stresses and deformations in layered manufacturing of metallic parts. Acta Mech 183:61–79zbMATHCrossRefGoogle Scholar
  19. 19.
    Dowden J, Postacioglu N, Davis M, Kapadia P (1987) A keyhole model in penetration welding with a laser. J Phys D Appl Phys 20:36–44CrossRefGoogle Scholar
  20. 20.
    Semak VV, Bragg WD, Damkroger B, Kempka S (1999) Transient model for the keyhole during laser welding. J Phys D Appl Phys 32:61–64CrossRefGoogle Scholar
  21. 21.
    Ki H, Mazumder J, Mohanty PS (2002) Modeling of laser keyhole welding: Part II. simulation of keyhole evolution, velocity, temperature profile, and experimental verification. Metall Mater Trans A 33:1831–1842CrossRefGoogle Scholar
  22. 22.
    Rai R, Kelly SM, Martukanitz RP, DebRoy T (2008) A convective heat-transfer model for partial and full penetration keyhole mode laser welding of a structural steel. Metall Mater Trans A 39:98–112CrossRefGoogle Scholar
  23. 23.
    Al-Kazzaz H, Medraj M, Caoand X, Jahazi M (2008) Nd:YAG laser welding of aerospace grade ZE41A magnesium alloy: modeling and experimental investigations. Mater Chem Phys 109:61–76Google Scholar
  24. 24.
    Kaplan A (1994) A model of deep penetration laser welding based on calculation of the keyhole profile. J Phys D Appl Phys 27(180):5–1814Google Scholar
  25. 25.
    Lampa C, Kaplan AFH, Powell J, Magnusson C (1997) An analytical thermodynamic model of laser welding. J Phys D Appl Phys 30(9):1293–1299CrossRefGoogle Scholar
  26. 26.
    Jin X, Li L, Zhang Y (2002) A study on Fresnel absorption and reflections in the keyhole in deep penetration laser welding. J Phys D Appl Phys 35:2304–2310CrossRefGoogle Scholar
  27. 27.
    Solana GNegro (1997) A study of the effect of multiple reflections on the shape of the keyhole of the keyhole in the laser processing of materials. J Phys D Appl Phys 30:3216–3222CrossRefGoogle Scholar
  28. 28.
    Wu CS, Wang HG, Zhang YM (2006) A new heat source model for keyhole plasma arc welding in FEM analysis of the temperature profile. Weld J 85:284–289Google Scholar
  29. 29.
    Yamamoto N, Genma K (2007) On error estimation of finite element approximations to the elliptic equations in nonconvex polygon domains. J Comput Appl Math 199:286–296Google Scholar
  30. 30.
    Tabata M (2007) Discrepancy between theory and real computation on the stability of some finite element schemes. J Comput Appl Math 199:424–431zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Lamba H, Seaman T (2006) Mean-square stability properties of an adaptive time-stepping SDE solver. J Comput Appl Math 194:245–254zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Chantasiriwan S (2000) Inverse determination of steady-state heat transfer coefficient. Int Comm Heat Mass Transf 27(8):1155–1164CrossRefGoogle Scholar
  33. 33.
    Erdogdu F (2005) Mathematical approaches for use of analytical solutions in experimental determination of heat and mass transfer parameters. J Food Eng 68:233–238CrossRefGoogle Scholar
  34. 34.
    Kusiak A, Battaaglia JL, Marchal R (2005) Influence of CrN coating in wood machining from heat flux estimation in the tool. Int J Therm Sci 44:289–301CrossRefGoogle Scholar
  35. 35.
    Cohen K, Siegel S, McLaughlin (2006) A heuristic approach to effective sensor placement for modelling o a cylinder wake. Comput Fluids 35:103–120zbMATHCrossRefGoogle Scholar
  36. 36.
    Chen CK, Wu LW, Yang YT (2006) Application of the inverse method to the estimation of heat flux and temperature on the external surface in laminar pipe flow. Appl Therm Eng 26:1714–1724CrossRefGoogle Scholar
  37. 37.
    Benjamin SF, Roberts CA (2002) Measuring flow velocity at elevated temperature with a hot wire anemometer calibrated in cold flow. Int J Heat Mass Transf 45:703–706CrossRefGoogle Scholar
  38. 38.
    Uselton S, Ahrens J, Bethel W, Treinish L (1998) Multi-source data analysis challenges. In: Proceedings of IEEE Vis 98(VIZ98)Google Scholar
  39. 39.
    Emery AF, Nenarokomov AV, Fadale TD (2000) Uncertainties in parameter estimation: the optimal experiment design. Int J Heat Mass Transf 43:3331–3339zbMATHCrossRefGoogle Scholar
  40. 40.
    Refsgaard JC, Henriksen HJ (2004) Modelling guidelines terminology and guiding principles. Adv Water Resour 27:71–82CrossRefGoogle Scholar
  41. 41.
    Lauwagie T, Sol H, Heylen W (2006) Handling uncertainties in mixed numerical-experimental techniques for vibration based material identification. J Sound Vib 291:723–739CrossRefGoogle Scholar
  42. 42.
    Ramroth WT, Krysl P, Asaro RJ (2006) Sensitivity and uncertainty analysis for FE thermal model of FRP panel exposed to fire. Compos Part A 37:1082–1091CrossRefGoogle Scholar
  43. 43.
    Hsu PT (2006) Estimating the boundary condition in a 3D inverse hyperbolic heat conduction problem. Appl Math Comput 177:453–464zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • S. Amir Hossein A. E. Tabatabaei
    • 1
    Email author
  • Tinggang Zhao
    • 2
  • O. Bamidele Awojoyogbe
    • 3
  • Folorunsho O. Moses
    • 4
  1. 1.Sadra Institute of Higher EducationTehranIran
  2. 2.College of MathematicsLanzhou City UniversityLanzhouPeople’s Republic of China
  3. 3.Department of PhysicsFederal University of TechnologyMinnaNigeria
  4. 4.Department of PhysicsFederal University of TechnologyAkureNigeria

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