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The analytical solution derived by Rosenthal for three‐dimensional heat flow in a semi‐infinite workpiece during welding, Figure 2.18, is as follows [25]:

(2.9)

where R is the radial distance from the origin, namely, (x ² + y ² + z ² ) ^{1/ 2}. Equation (2.9) can be used to calculate the steady‐state temperature T(x, y, z), with respect to the moving heat source, at any location in the workpiece (x, y, z), for instance, at x = 1 cm, y = 4 cm, and z = 0 cm, as shown in Figure 2.18. The temperatures at other locations along y = 4 cm can also be calculated, and the temperature distribution along y = 4 cm can thus be determined.

Figure 2.18 Three‐dimensional heat flow during welding of semi‐infinite workpiece.