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# Geometrical Effects on the Temperature Distribution in a Half-Space Due to a Moving Heat Source

[+] Author and Article Information
Mohsen Akbari1

Mechatronic Systems Engineering, School of Engineering Science, Simon Fraser University, Surrey, BC, V3T 0A3, Canadamohsen_akbari@sfu.ca

David Sinton

Department of Mechanical Engineering, University of Victoria, Victoria, BC, V8W 2Y2, Canada

Majid Bahrami

Mechatronic Systems Engineering, School of Engineering Science, Simon Fraser University, Surrey, BC, V3T 0A3, Canada

1

Corresponding author.

J. Heat Transfer 133(6), 064502 (Mar 02, 2011) (10 pages) doi:10.1115/1.4003155 History: Received February 26, 2010; Revised November 24, 2010; Published March 02, 2011; Online March 02, 2011

## Abstract

Fundamental problem of heat transfer within a half-space due to a moving heat source of hyperelliptical geometry is studied in this work. The considered hyperelliptical geometry family covers a wide range of heat source shapes, including star-shaped, rhombic, elliptical, rectangular with round corners, rectangular, circular, and square. The effects of the heat source speed, aspect ratio, corners, and orientation are investigated using the general solution of a moving point source on a half-space and superposition. Selecting the square root of the heat source area as the characteristics length scale, it is shown that the maximum temperature within the half-space is a function of the heat source speed (Peclet number) and its aspect ratio. It is observed that the details of the exact heat source shape have negligible effect on the maximum temperature within the half-space. New general compact relationships are introduced that can predict the maximum temperature within the half-space with reasonable accuracy. The validity of the suggested relationships is examined by available experimental and numerical data for the grinding process, for medium Peclet numbers. For ultrafast heat sources, an independent experimental study is performed using a commercial $CO2$ laser system. The measured depth of the engraved grooves is successfully predicted by the proposed relationships.

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## Figures

Figure 1

Schematic of a symmetrical moving plane heat source with arbitrary shape

Figure 2

Hyperellipse as a general symmetric geometry covers an array of source geometries: (a) star shape (n<1), (b) rhombic (n=1), (c) elliptical (n=2), (d) rectangular with round corners (n=4), and (e) rectangular (n→∞)

Figure 3

Schematic diagram of the laser beam, the engraved groove, and cross-sectional image of the wooden substrate (red oak) following laser exposure at the typical relative beam speed of U∗=U/Umax=0.6 and relative power of P∗=P/Pmax.=0.4. Image is taken by an inverted microscope (Unitron, Commack, NY) with a 10× magnification.

Figure 4

Effect of heat source shape on the quasi-steady maximum temperature for two typical aspect ratios, 0.1 and 0.5, and Peclet number of Pe=10

Figure 5

Effect of the aspect ratio on the maximum temperature of a plane moving heat source at various source speeds (Peclet number)

Figure 6

Maximum temperature as a function of Peclet number for aspect ratio of εm=1, a typical value. Note that two asymptotes for stationary, Pe→0, and fast moving heat source, Pe→∞, can be recognized in the present plot.

Figure 7

Variation in θmax∗/θmax,0∗ as a function of dimensionless depth for elliptical (n→2) and rectangular (n→∞) heat sources with Pe=10 and various aspect ratios

Figure 8

Maximum dimensionless temperature in a half-space. Rectangular heat source with Pe=2.86 and aspect ratio of εm=4.39; experimental data from Ref. 25; see Table 4 for the experiment parameters.

Figure 9

Effect of laser beam speed on the depth of engraved channels using single pass ablation. Square (◻) and delta (△) symbols are the present experimental measurements for percentage powers of 20% and 40%, respectively. Two limits of the reported ignition temperature are shown as the dashed line for 500°C and solid line for 420°C.

Figure 10

Effect of laser power on the depth of engraved channels using single pass ablation at maximum beam speed. Square (◻) symbols are the present experimental measurements maximum beam speed. Two limits of the reported ignition temperature are shown as the dashed line for 500°C and solid line for 420°C.

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