Moving Heat Source

Equations E5.4-12 and E5.4-13 satisfy the differential equation and the boundary and initial conditions. Therefore they form an exact solution to the problem. In the preceding solution we neglected heat convection as a result of the expansion of the melt phase due to the density decrease. The rate of melting per unit area as a function of time can be obtained from Eq. E5.4-10 

The preceding examples discuss the heat-conduction problem without melt removal in a semi-infinite solid, using different assumptions in each case regarding the thermophysical properties of the solid. These solutions form useful approximations to problems encountered in everyday engineering practice. A vast collection of analytical solutions on such problems can be found in classic texts on heat transfer in solids (10,11). Table 5.1 lists a few well-known and commonly applied solutions, and Figs. 5.5-5.8 graphically illustrate some of these and other solutions. 

Semi-infinite solid Semi-infinite solid 

Flat plate Flat plate Cylinder Cylinder 

Example 5.5 Continuous Heating of a Thin Sheet Consider a thin polymer sheet infinite in the x direction, moving at constant velocity Vq in the negative x direction (Fig. E5.5). The sheet exchanges heat with the surroundings, which is at T = T0, by convection. At x = 0, there is a plane source of heat of intensity q per unit cross-sectional area. Thus the heat source is moving relative to the sheet. It is more convenient, however, to have the coordinate system located at the source. Our objective is to calculate the axial temperature profile T(x) and the intensity of the heat source to achieve a given maximum temperature. We assume that the sheet is thin, that temperature at any x is uniform, and that the thermophysical properties are constant. 

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N. R. DesRuisseaux and R. D. Zerkle, Temperature in Semi-Infinite and Cylindrical Bodies Subjected to Moving Heat Sources and Surface Cooling, J. Heal Transfer, 92, pp. 456-464,1970. 

M. D. Bryant, Thermoelastic Solutions for Thermal Distributions Moving over Half Space Surfaces and Application to the Moving Heat Source, / Appl. Mech., 55, pp. 87-92,1988. 

For solving this problem the analytical approximate method of integral correlations proposed by Goodman (1958) and improved by Volkov et al (1988) can be employed. This method was successfully used by Fomin et al (1994) for solving more complex problem of moving heat source within the borehole in application to melting of the paraffin deposition in the annulus. 

Fomin, S.A., Wei, P. S., Chugunov V. A. 1994. Melting Paraffin Plug Between Two Co-axial Pipes by a Moving Heat Source in the Inner Pipe. ASME Journal of Heat Transfer 116 pp. 1028-1033. 

The classical starting points for heat flow analysis, originally for arc welding, are the point and line source solutions for a moving heat source, due to Rosenthal (Ref 2,3) and elegantly reassessed and extended by Myhr and Grong (Ref 4, 5). These solutions approximate the plate as being infinite in extent in two or... 

Cutting Temperature, Fig. 18 Model of shear-plane moving heat source... 

Figure 18 shows Idealized diagram of shear-plane moving heat source which was used by Show to drive the shear-plane temperature rise. Here the chip may be considered as a perfect insulator if the total heat flux through the interface is equal to the heat flux flowing into the workpiece, (1 — R )qi, where <71 (= e given by Eq. 13) is the heat flux which flows from the shear zone. The detail of theory by M.C. Shaw can be found in his book (Shaw 1984). 

Heat partitioning directly affects tool wear and surface integrity of the machined part. Therefore researchers have been aiming to predict it by different modeling and simulation techniques. One prominent approach is to utilize the moving heat source theory (Jaeger 1942) as an approximation for the stationary chip formation process. 

Heat Partitioning in Dry Milling, Fig. 2 Loewen and Shaw s moving heat source model (After Komanduri and Hou (2000))... 

The heat conduction equation for a moving heat source can be written as... 

Fig. 35. 1st row Temperature distribution calculated with ANSYS CFX . Here the moving heat source is applied within the boundary layer air/steel. 2nd row (/c/t) Temperature distribution imported from CFX to ANSYS classic. 2nd row (right) Calculated voN MiSES stress distribution during welding. 

Engineering Problem Solving A Classical Perspective 9.5 Moving Heat Source... 

Figure 11.12. Moving heat source with boundary condition of the second kind [constant thermal flux q) at the surface].

By heating one surface of a bonded sandwich structure and observing the temperature rise of the opposite face, areas of debond, which resist the transfer of heat, show as cool areas. Alternatively, if the heated face is scanned, debonds will show as hot areas. Temperature sensing is normally done with a scanning infra-red camera (e.g. AGA Thermovision). More recently, heat pulses or moving heat sources have been used (Vavilov and Taylor, 1982). Temperature sensitive paints or liquid crystals, and thermoluminescent coatings are also used. 

Grewell, D. Benatar, A. Modelling Heat Flow for a Moving Heat Source to Describe Scan Micro-Laser Welding, 60 Annual Technical Conference (ANTEC), Society of Plastics Engineers, San Francisco 5.-9. Mai 2002... 

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