Research Papers: Conduction

Scaling Analysis of a Moving Point Heat Source in Steady-State on a Semi-Infinite Solid

[+] Author and Article Information
Patricio F. Mendez

Chemical and Materials Engineering,
University of Alberta,
Donadeo ICE 12-332 9211 116 St,
Edmonton, AB T6G 2V4
e-mail: pmendez@ualberta.ca

Yi Lu

Chemical and Materials Engineering,
University of Alberta,
Edmonton, AB T6G 2R3, Canada
e-mail: ylu13@ualberta.ca

Ying Wang

Chemical and Materials Engineering,
University of Alberta,
Edmonton, AB T6G 2R3, Canada
e-mail: wang18@ualberta.ca

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received December 1, 2017; final manuscript received February 1, 2018; published online April 11, 2018. Assoc. Editor: Alan McGaughey.

J. Heat Transfer 140(8), 081301 (Apr 11, 2018) (9 pages) Paper No: HT-17-1725; doi: 10.1115/1.4039353 History: Received December 01, 2017; Revised February 01, 2018

This paper presents a systematic scaling analysis of the point heat source in steady-state on a semi-infinite solid. It is shown that all characteristic values related to an isotherm can be reduced to a dimensionless expression dependent only on the Rykalin number (Ry). The maximum width of an isotherm and its location are determined for the first time in explicit form for the whole range of Ry, with an error below 2% from the exact solution. The methodology employed involves normalization, dimensional analysis, asymptotic analysis, and blending techniques. The expressions developed can be calculated using a handheld calculator or a basic spreadsheet to estimate, for example, the width of a weld or the size of zone affected by the heat source in a number of processes. These expressions are also useful to verify numerical models.

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Rosenthal, D. , 1935, “Etude Théorique Du Régime Thermique Pendant La Soudure à L'Arc,” Comptes Rendus (2eme Congres National Des Sciences), pp. 1277–1292.
Rosenthal, D. , 1946, “The Theory of Moving Sources of Heat and Its Application to Metal Treatments,” Trans. ASME, 68, pp. 849–866.
Rykalin, N. N. , 1951, Calculation of Heat Flow in Welding, Mashgis, Moscow, Russia.
Li, W. B. , Easterling, K. E. , and Ashby, M. F. , 1986, “Laser Transformation Hardening of Steel-II. Hypereutectoid Steels,” Acta Metall., 34(8), pp. 1533–1543. [CrossRef]
Komanduri, R. , and Hou, Z. B. , 2001, “Thermal Analysis of the Laser Surface Transformation Hardening Process,” Int. J. Heat Mass Transfer, 44(15), pp. 2845–2862. [CrossRef]
Hill, J. W. , Lee, M. J. , and Spalding, I. J. , 1974, “Surface Treatments by Laser,” Opt. Laser Technol., 6(6), pp. 276–278. [CrossRef]
Hou, Z. B. , and Komanduri, R. , 2000, “General Solutions for Stationary/Moving Plane Heat Source Problems in Manufacturing and Tribology,” Int. J. Heat Mass Transfer, 43(10), pp. 1679–1698. [CrossRef]
Jaeger, J. C. , 1942, “Moving Sources of Heat and the Temperature of Sliding Contacts,” Proc. R. Soc. New South Wales, 76, pp. 203–224.
Bulsara, V. H. , Ahn, Y. , Chandrasekar, S. , and Farris, T. N. , 1997, “Polishing and Lapping Temperatures,” ASME J. Tribol., 119(1), pp. 163–170. [CrossRef]
Malkin, S. , 1974, “Thermal Aspects of Grinding: Part 2 – Surface Temperatures and Workpiece Burn,” ASME J. Eng. Ind., 96(4), pp. 1184–1191. [CrossRef]
Komanduri, R. , and Hou, Z. B. , 2009, “Unified Approach and Interactive Program for Thermal Analysis of Various Manufacturing Processes With Application to Machining,” Mach. Sci. Technol., 13(2), pp. 143–176. [CrossRef]
Dutt, R. P. , and Brewer, R. C. , 1965, “On the Theoretical Determination of the Temperature Field in Orthogonal Machining,” Int. J. Prod. Res., 4(2), pp. 91–114. [CrossRef]
Kolonits, F. , 2016, “Analysis of the Temperature of the Rail/Wheel Contact Surface Using a Half-Space Model and a Moving Heat Source,” Proc. Inst. Mech. Eng., Part F, 230(2), pp. 502–509. [CrossRef]
Knothe, K. , and Liebelt, S. , 1995, “Determination of Temperatures for Sliding Contact With Applications for Wheel-Rail Systems,” Wear, 189(1–2), pp. 91–99. [CrossRef]
Wei, P. S. , and Giedt, W. H. , 1985, “Surface Tension Gradient-Driven Flow Around an Electron Beam Welding Cavity,” Weld. J., 64(9), pp. s251–s259. https://app.aws.org/wj/supplement/WJ_1985_09_s251.pdf
Friedman, E. , 1975, “Thermomechanical Analysis of the Welding Process Using the Finite Element Method,” ASME J. Pressure Vessel Technol., 97(3), pp. 206–213. [CrossRef]
Goldak, J. , Chakravarti, A. , and Bibby, M. , 1984, “A New Finite Element Model for Welding Heat Sources,” Metall. Trans. B, 15(2), pp. 299–305. [CrossRef]
Rohsenow, W. M. , Hartnett, J. P. , and Cho, Y. I. , 1998, Handbook of Heat Transfer, 3rd ed., McGraw-Hill, New York.
Eagar, T. W. , and Tsai, N. S. , 1983, “Temperature Fields Produced by Traveling Distributed Heat Sources,” Weld. J., 62(12), pp. 346–355. http://files.aws.org/wj/supplement/WJ_1983_12_s346.pdf
Van Elsen, M. , Baelmans, M. , Mercelis, P. , and Kruth, J.-P. , 2007, “Solutions for Modelling Moving Heat Sources in a Semi-Infinite Medium and Applications to Laser Material Processing,” Int. J. Heat Mass Transfer, 50(23–24), pp. 4872–4882. [CrossRef]
Winczek, J. , 2010, “Analytical Solution to Transient Temperature Field in a Half-Infinite Body Caused by Moving Volumetric Heat Source,” Int. J. Heat Mass Transfer, 53(25–26), pp. 5774–5781. [CrossRef]
Gajapathi, S. S. , Mitra, S. K. , and Mendez, P. F. , 2011, “Controlling Heat Transfer in Micro Electron Beam Welding Using Volumetric Heating,” Int. J. Heat Mass Transfer, 54(25–26), pp. 5545–5553. [CrossRef]
Mendez, P. F. , Tello, K. E. , and Gajapathi, S. S. , 2012, “Generalization and Communication of Welding Simulations and Experiments Using Scaling Analysis,” Ninth International Conference on Trends in Welding Research, Chicago, IL, June 4–8, pp. 249–258.
Mendez, P. F. , 2011, “Synthesis and Generalisation of Welding Fundamentals to Design New Welding Technologies: Status, Challenges and a Promising Approach,” Sci. Technol. Weld. Joining, 16(4), pp. 348–356. [CrossRef]
Muzychka, Y. S. , and Yovanovich, M. M. , 2001, “Thermal Resistance Models for Non-Circular Moving Heat Sources on a Half Space,” ASME J. Heat Transfer, 123(4), pp. 624–632. [CrossRef]
Incropera, F. P. , and DeWitt, D. P. , 1985, Fundamentals of Heat and Mass Transfer, 2nd ed., Wiley, New York.
Özisik, M. N. , 1993, Heat Conduction, 2nd ed., Wiley, New York.
Seyffarth, P. , Meyer, B. , and Scharff, A. , 1992, Grosser Atlas Schweiss-ZTU-Schaubilder, Fachbuchreihe Schweisstechnik. Deutscher Verlag für Schweisstechnik, Düsseldorf, Germany.
Wood, G. , Islam, S. A. , and Mendez, P. F. , 2014, “Calibrated Expressions for Welding and Their Application to Isotherm Width in a Thick Plate,” Soldagem Inspeção, 19(3), pp. 212–220. [CrossRef]
Wilson, H. A. , 1904, “On Convection of Heat,” Proc. Cambridge Philos. Soc., 12, pp. 406–423.
Roberts, O. F. T. , 1923, “The Theoretical Scattering of Smoke in a Turbulent Atmosphere,” Proc. R. Soc. A, 104(728), pp. 640–654. [CrossRef]
Christensen, N. , Davies, V. , de, L. , and Gjermundsen, K. , 1965, “Distribution of Temperatures in Arc Welding,” British Weld. J., 12(2), pp. 54–75.
Mendez, P. F. , 2010, “Characteristic Values in the Scaling of Differential Equations in Engineering,” ASME J. Appl. Mech., 77(6), p. 061017. [CrossRef]
Dantzig, J. A. , and Tucker, C. L. , 2001, Modeling in Materials Processing, Cambridge University Press, Cambridge, UK. [CrossRef]
Buckingham, E. , 1914, “On Physically Similar Systems; Illustrations of the Use of Dimensional Equations,” Phys. Rev., 4(4), pp. 345–376. [CrossRef]
Washio, T. , and Motoda, H. , 1999, “Extension of Dimensional Analysis for Scale-Types and Its Application to Discovery of Admissible Models of Complex Processes,” International Workshop on Similarity Method, pp. 129–147.
Myhr, O. R. , and Grong, Ø. , 1990, “Dimensionless Maps for Heat Flow Analyses in Fusion Welding,” Acta Metall. Et Mater., 38(3), pp. 449–460. [CrossRef]
Grong, Ø. , 1994, Metallurgical Modelling of Welding, 1st ed., Institute of Materials, Cambridge, UK.
Fuerschbach, P. W. , and Eisler, G. R. , 2002, “Determination of Material Properties for Welding Models by Means of Arc Weld Experiments,” Sixth International Trends in Welding Research, Pine Mountain, Georgia, Apr. 15–19. http://smartweld.sourceforge.net/Pdf_docs/Trends02.pdf
Churchill, S. W. , and Usagi, R. , 1972, “A General Expression for the Correlation of Rates of Transfer and Other Phenomena,” AIChE J., 18(6), pp. 1121–1128. [CrossRef]
Mendez, P. F. , and Eagar, T. W. , 2012, “Order of Magnitude Scaling: A Systematic Approach to Approximation and Asymptotic Scaling of Equations in Engineering,” ASME J. Appl. Mech., 80(1), p. 011009. [CrossRef]
Mendez, P. F. , and Ordóñez, F. , 2005, “Scaling Laws From Statistical Data and Dimensional Analysis,” ASME J. Appl. Mech., 72(5), pp. 648–657. [CrossRef]
Goldak, J. , Asadi, M. , and Alena, R. G. , 2010, “Why Power Per Unit Length of Weld Does Not Characterize a Weld?,” Comput. Mater. Sci., 48(2), pp. 390–401. [CrossRef]
The James F. Lincoln Arc Welding Foundation, 2000, The Procedure Handbook of Arc Welding, 14th ed., The James F. Lincoln Arc Welding Foundation, Cleveland, OH.
Fuerschbach, P. W. , 1995, “A Dimensionless Parameter Model for Arc Welding Processes,” Fourth International Conference on Trends in Welding Research, Gatlinburg, TN, June 5–8, pp. 493–497.
Inagaki, M. , Nakamura, H. , and Okada, A. , 1965, “Studies of Cooling Processes in the Cases of Welding With Coated Electrode and Submerged Arc Welding,” J. Jpn. Weld. Soc., 34(10), pp. 1064–1075. [CrossRef]


Grahic Jump Location
Fig. 1

Point heat source moving with constant velocity on a semi-infinite solid

Grahic Jump Location
Fig. 2

Characteristic values ymax and xmax for a point heat source moving with constant velocity on a semi-infinite solid

Grahic Jump Location
Fig. 4

Blending error for isotherm width ymax as a function of Ry for exponents n at or near the optimal value

Grahic Jump Location
Fig. 3

Dimensionless isotherm width ymax* as a function of Ry

Grahic Jump Location
Fig. 7

Dimensionless location xmax of maximum isotherm width as a function of Ry

Grahic Jump Location
Fig. 5

Maximum blending error for isotherm width ymax as a function of blending parameter n

Grahic Jump Location
Fig. 6

Correction factors for maximum isotherm width ymax

Grahic Jump Location
Fig. 11

Weld bead corresponding to Ry = 20.84. Scale is in mm.

Grahic Jump Location
Fig. 8

Error of blending for location xmax of maximum isotherm width as a function of Ry for blending parameter n at or near the optimal value

Grahic Jump Location
Fig. 9

Maximum blending error for location xmax of maximum isotherm width as a function of blending parameter n

Grahic Jump Location
Fig. 10

Correction factors for location xmax of maximum isotherm width



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