Heat flow during the laser transformation hardening of cylindrical bodies
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I.
INTRODUCTION
THE surface hardening of steels and cast irons with highpower continuous-wave CO2 lasers has drawn a great deal of attention recently. In general, the process is carried out by moving the workpiece at a constant speed under a stationary laser beam. For flat workpieces, beam integrators or oscillators can be used to distribute uniformly the energy of the laser beam over a relatively large area on the workpiece surface, so that high coverage rates and uniform depth of surface hardening can be obtained. For cylindrical bodies, on the other hand, toric mirrors can be used to reflect an annular laser beam onto the workpiece surface in such a manner that a continuous ring-shaped band is generated around the periphery of the cylinder. This allows the entire workpiece surface to be hardened in a continuous manner, and thus avoids backtempering of already hardened material. The workpiece can be rotated in order to further ensure uniform heating of its surface. Heat flow during the laser transformation hardening of flat workpieces has been investigated in a previous study. In the present study, heat flow during the laser transformation hardening of cylindrical bodies will be investigated. Recently, Sandven2 has treated such a heat flow problem by multiplying with a geometric factor the analytical solution of Carslaw and Jaeger3 for a semi-infinite plate. Unfortunately, no sound justifications were given for the value of the geometric factor used.
II.
MATHEMATICAL MODEL
The energy balance equation, the boundary conditions, and the finite difference equation used in the heat flow simulation are introduced in the following.
(a)
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Figure 1 shows schematically the laser transformation hardening of the outer surface of a cylinder and the inner surface of a hollow cylinder. In either case, the laser beam SINDO KOU, Associate Professor, Department of Metallurgical Engineering and Materials Science, and D.K. SUN, Graduate Student, Department of Physics, are both with Carnegie-Mellon University, Pittsburgh, PA 15213. Manuscript received January 14, 1983. METALLURGICALTRANSACTIONS A
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goi,~y ~3 (b) Fig. 1 - - Schematic sketches of laser transformation hardening of cylindrical bodies: (a) surface of a cylinder; (b) inner surface of a hollow cylinder.
and the toric mirror are stationary, while the workpiece moves at a constant velocity U in its axial direction. The energy balance equation for a stationary cylindrical coordinate system (z-r) is given below: O(pH) Ot
A. Formulation of Finite Difference Equation
9 OSe, ( o -
= V" (kVT) - U O(pH)
[1]
Oz
where t is time, T is temperature, H the enthalpy, k the thermal conductivity, and p the density of the workpiece. The origin of the cylindrical coordinate coincides with the center of the laser beam on the surface of the workpiece. The L. H. S. of Eq. [1] represents the rate of enthalpy change per unit volume. The first term on the R. H. S. of the same equation corresponds to heat transfer due to conduction, while the last term corresponds to heat trans
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