A mathematical model of heat conduction in a prolate spheroidal coordinate system with applications to the theory of welding

Abstract

The steady-state heat conduction equation is solved in a prolate spheroidal coordinate system. Some simple solutions are derived in the limiting cases of both low and high Peclet numbers. The analysis was carried out in such a way as to avoid the physical details of conditions inside 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 experimentally, as is the precise detailed shape of the pool; the solutions obtained can be verified experimentally. The high-Peclet-number approximation is likely to be particularly useful in the case of laser welding where large translation speeds of the weld piece are of interest. The solution of the problem is given in the form of a series as well as in an asymptotic form. The asymptotic method of solution presented can be adapted to any smooth shape of weld pool with only minor alterations, since the method involves integration in the tangent plane to the weld 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. These are prolate spheroidal weld pool geometries with various aspect ratios and cylindrical weld pool geometries with elliptical or circular cross sections. The results of these comparisons were found to be satisfactory.