Semi-Analytical Solution for Modelling Moving Heat Sources in a Semi-Infinite Medium with Radiative and Convective Boundary Conditions

Year 2023, , 1 - 11, 01.06.2023

Antonio Metallo

https://doi.org/10.5541/ijot.1097756

Cited By: 1

Abstract

The weld quality is highly related to the thermal history of the weld and there have been many trials to monitor the quality using an infrared (IR) sensor. To obtain the real temperature of a surface based on the brightness temperature values measured by an IR camera, the emissivity value must be derived. For an accurate assessment of the emissivity, one must be aware of the melting point isotherm. The temperature profiles only depend on three factors during laser processing, specified as constants the characteristics of the material: laser beam speed (v), laser beam diameter (d), and power (P). Predicting the width of the melted zone reached during the welding process as the parameters vary is a tool for helping a quality laser processing and for determination of true temperature in laser welding using IR camera. This study describes the semi-analytical (SA) solution of the heat conduction equation for a localized moving Gaussian heat source with constant parameters on a semi-infinite medium. The solution, simple and quick to obtain, provides information on the width of the melted zone with an average error < 5 %. The outcome is assessed numerically and contrasted with FEM solutions for a Gaussian source, the latter having undergone experimental validation. With two distinct defocus values, def0 and def-6, and by varying the speed and power settings, two separate types of experiments were run. Thus, the SA solution was obtained and compared after the FEM solution had been obtained with a good approximation (max err 4.3 %, average err 2.7 %). Only in regard to the 1AL test is an error more than 5 % detected; in the other case, the average error is 3.75 %. Two more tests at the defocus values of def-4 and def-8 were conducted to confirm the model's validity as the parameters varied. Overall, the average error between the semi-analytical and the FEM solution is 4.1%. The SA solution may be used to effectively estimate the isotherms related to the melting point of aluminum (770 K). This allows to obtain a tool which helps restoring the real temperature based on the brightness values measured by the IR camera during laser welding. At the same time, this effective tool allows to investigate the importance of different processing parameters in laser manufacturing.

Keywords

: Semi-analytical solution;, laser welding, integral method, semi-infinite medium

Supporting Institution

University of Salerno

References

• P. Kumar, A. N. Sinha, “Studies of temperature distribution for laser welding of dissimilar thin sheets through finite element method”, J Braz. Soc. Mech. Sci., Eng. 40, 455, 2018.
• M Miccio, R. Pierri, G. Cuccurullo, A. Metallo, P. Brachi, “Process intensification of tomato residues drying by microwave heating: experiments and simulation”, Chem. Eng. Process. Process Intensif., 156, 2020.
• I. Smurov, M. Doubenskaia, Laser-Assisted Fabrication of Materials, Springer Series in Materials Science 161, 373-422, 2013.
• T. Staudt, E. Eschner, M. Schmidt, “Temperature determination in laser welding based upon a hyperspectral imaging technique”, Manufacturing Technology, CIRP Annals, 68 (1), 225-228, 2019.
• M. Doubenskaia , M. Pavlov , S. Grigoriev , I. Smurov, “Definition of brightness temperature and restoration of true temperature in laser cladding using infrared camera”, Surface and Coatings Technology 220, 244-247, 2013.
• D. Rosenthal, The Theory of Moving Sources of Heat and its Application to Metal Treatments, Metals Handbook, 1981, 9th Ed., TranSASME, 68, pp. 3515-3528. 17 ASM, Metal Park, OH 1946.
• V. Alfieri, F. Caiazzo, V. Sergi, “Autogenous laser welding of AA2024 aluminum alloy: Process Issues and Bead Features", Procedia CIRP, 33, 2015.
• A. M. Leitner, T. Leitner, A. Schmon, K. Aziz, G. Pottlacher, “Thermophysical Properties of Liquid Aluminum”, Metall Mater Trans A 48, 3036–3045, 2017.
• Q. G. Meng, H. Y. Fang, J. G. Yang, S. D. Ji, “Analysis of temperature and stress field in Al alloy's twin wire welding”, Theoretical and Applied Fracture Mechanics, 44 (2), 178−186, 2005.
• E. Kaschnitz, W. Funk, T. Pabel, “Electrical resistivity measured by millisecond pulse-heating in comparison to thermal conductivity of the aluminum alloy Al-7Si-0.3Mg at elevated temperature”, High Temperatures-High-Pressures,-,-43 (2), 175-191, 2014.
• K. Narender, A. S. M. Rao, K. G. K. Rao, N. G. Krishna, “Temperature Dependence of Density and Thermal Expansion of Wrought Aluminum Alloys 7041, 7075 and 7095 by Gamma Ray Attenuation Method”, Journal of Modern Physics, 4 (3), 2013.
• M. Bjelić, K. Kovanda, L. Kolařík, M. N Vukićević, “Numerical modeling of two-dimensional heat-transfer and temperature-based calibration using simulated annealing optimization method: Application to gas metal arc welding”, Thermal Science, 20, 255-265, 2016.
• C. Ratti, A. S. Mujumdar, Infrared Drying, Handbook of Industrial Drying by Taylor & Francis Group, LLC, 2006.
• J. Heigel, P. Michaleris, E. Reutzel, “Thermo-mechanical model development and validation of directed energy deposition additive manufacturing of Ti-6Al-4V”, Addit. Manuf., 5, 9–19, 2015.
• Z. Fan, L. Frank, “Numerical modeling of the additive manufacturing (AM) process of titanium alloy”, Titanium Alloys—Towards Achieving Enhanced Properties for Diversified Applications, A.K.M. Ed., 3–28. 2012.
• R. Paschotta, Encyclopedia of Laser Physics and Technology, Wiley-VCH: Berlin, Germany, 2008.
• K. Suresh Kumar, T. Sparks, F. Liou, “Parameter determination and experimental validation of a wire feed Additive Manufacturing model”, International Solid Freeform Fabrication Symposium, Austin, TX, USA, 2015.
• C. Wen, I. Mudawar, “Experimental investigation of emissivity of aluminum alloys and temperature determination using multispectral radiation thermometry (MRT) algorithms”, J. Mater, Eng. Perform, 11, 551–562, 2002.
• W. Steen, J. Mazumder, Laser Material Processing, Springer: Berlin, Germany, 2010.
• AWS, Specification for Fusion Welding for Aerospace Applications, American Welding Society, Miami, Fla, USA, 2001.
• W. W. Duley, Laser welding, John Wiley and Sons Inc., New York, 1999.
• M. Doubenskaia, I. Zhirnov, V. I. Teleshevskiy, Ph. Bertrand, I. Smurov, “Determination of True Temperature in Selective Laser Melting of Metal Powder Using Infrared Camera”, Materials Science Forum,-834, 93-102,-2015.
• A. Mosavia, F. Salehib, L. Nadaie, Z. Karolyc, N. Gorjid, “Modeling the temperature distribution during laser hardening process”, Results in Physics, 16, 2020.
• J. Sundqvista, A.F.H. Kaplana, L. Shachafb, C. Kongc, “Analytical heat conduction modelling for shaped laser beams”, Journal of Materials Processing Tech., 247,48–54, 2017.
• D. Rosenthal, “Mathematical theory of heat distribution during welding and cutting”, Weld. J., 20 (5), 220–234, 1941.
• N. Rykalin, A. Uglov, A. Kokora, O. Glebov, Laser Machining and Welding, Mir Publishers, Moscow, 1978.
• H. Carslaw, J. Jaeger, Conduction of Heat in Solids, Oxford Science Publications, 1990.
• T.W. Eagar, N.S. Tsai, “Temperature fields produced by traveling distributed heat sources”, Weld. J. 62 (12), 346–355, 1983.
• J. Goldak, A. Chakravarti, M. Bibby, A double ellipsoid finite element model for welding heat sources, IIW Doc. No. 212-603-85, “A new finite element model for welding heat sources”, Metall Mater Trans B 15, 299–305, 1984.
• N.T. Nguyen, A. Otha, K. Matsuoka, N. Suzuki, Y. Maeda, “Analytic solutions for transient temperature of semi-infinite body subjected to 3-D moving heat sources”, Weld. Res. Suppl., 265–274, 1999.
• N.T. Nguyen, Y.-W. Mai, S. Simpson, A. Otha, “Analytical approximate solution for double ellipsoidal heat source in finite thick plate”, Weld. Res., 82–93, 2004.
• M. Van Elsen, M. Baelmans, P. Mercelis, J.P. Kruth, “Solutions for modelling moving heat sources in a semi-infinite medium and applications to laser material processing”, International Journal of Heat and Mass Transfer, 50 (23), 4872–4882, 2007.
• T. F. Flint, J. A. Francis, J. R. Yates, “Analytical solutions of the transient thermal field induced in finite bodies with insulating and convective boundary conditions subjected to a welding heat source”, Transactions, SMiRT-22, San Francisco, California, USA, 2013.
• R. Forslunda, A. Snisa, S. Larsson, “Analytical solution for heat conduction due to a moving Gaussian heat flux with piecewise constant parameters”, Appl. Math. Model., 66, 227-240, 2019.
• A. Metallo , "Emissivity Prediction for an IR Camera During Laser Welding of Aluminum", International Journal of Thermodynamics, 25( 4), 24-34, 2022.