Phase-change heat transfer in laser transformation hardening by moving Gaussian rectangular heat source

Abstract

An analytical solution is developed for a quasi-steady phase-change heat transfer problem in which a plane slab is heated by a Gaussian rectangular heat source moving at a constant speed over the upper surface. The physical model is set up to simulate a laser transformation hardening process. In the mathematical formulation, nonhomogeneous terms arise from the Gaussian heat flux and the heat absorption and release at the moving phase-change interfaces. After conversion to dimensionless variables, a source-and-sink method in conjunction with the superposition technique and the method of undetermined parameters is applied to derive the body temperature of the workpiece. An iteration method based on the concept of perturbation methods is employed to obtain the numerical results. The high convergence rate of the numerical solutions reveals the efficacy of the method used. The theoretical predictions are in reasonable agreement with published experimental results. The effects of the heat flux distribution and the Peclet number on the penetration depth and the material cooling rate are discussed in a parametric study. Nomenclature

Non-fourier temperature field in a solid homogeneous finite hollow cylinder

“…Two cases of boundary conditions on lateral surface of cylinder including constant and Gaussian distributed heat flux with the same integrated power, are modeled. The Gaussian distribution was used for modeling irradiative or laser heating [23,24]. Numerical evaluation of the analytical solution is done in which the use is made of special functions (Bessel Functions) provided by Matlab.…”

“…Finally, the proposed solution is applied to some special cases of constant surface heat flux and the Gaussian distribution surface heat flux with the same total energy applied to the lateral surface of the cylinder. The Gaussian distribution surface heat flux was used for modeling of laser or irradiation heating in literatures [23,24]. These thermal boundary conditions could be imposed by the irradiation generated by halogen lamps and can be controlled by using ellipsoidal mirror furnaces [25].…”

Non-Fourier heat conduction in a finite hollow cylinder with periodic surface heat flux

“…where A n and B n are constants, and they are determined from the non-homogeneous boundary conditions (11).…”

“…The Gaussian distribution was used for modeling the laser heating before [10][11][12]. The Gaussian function adopted here is as follows: where q m is maximum flux in Gaussian profile, l is the cylinder length, and σ is the factor modeling the heat flux distribution and gets the values of (0.2) in our modeling.…”

“…A shockwave is produced by this boundary condition through the medium. The Gaussian distribution surface heat flux was used for modeling of laser or irradiation heating in literatures [11,12]. The behavior of the solution is studied under these conditions.…”