A MATHEMATICAL-MODEL OF HEAT-CONDUCTION IN A PROLATE SPHEROIDAL COORDINATE SYSTEM WITH APPLICATIONS TO THE THEORY OF WELDING

The steady-state heat conduction equation is solved in a prolate spher oidal coordinate system. Some simple solutions are derived in the limi ting cases of both low and high Peclet numbers. The analysis was carri ed out in such a way as to avoid the physical details of conditions in side the weld pool, so that the solutions are restricted to the solid region outside the weld pool. This procedure was specifically adopted because these conditions are difficult to gain access to experimentall y, as is the precise detailed shape of the pool; the solutions obtaine d can be verified experimentally. The high-Peclet-number approximation is likely to be particularly useful in the case of laser welding-wher e large translation speeds of the weld piece are of interest. The solu tion of the problem is given in the form of a series as well as in an asymptotic form. The asymptotic method of solution presented here can be adapted to any smooth shape of weld pool with only minor alteration s, since the method involves integration in the tangent plane to the w eld pool and the results of such an integration are independent of the global form of the weld pool. The asymptotic result is compared with the exact solution in a number of special geometric configurations. Th ese are prolate spheroidal weld pool geometries with various aspect ra tios and cylindrical weld pool geometries with elliptical or circular cross sections. The results of these comparisons were found to be sati sfactory.