Numerical Solution of Three-Dimensional Transient Heat Conduction Equation in Cylindrical Coordinates

Heat equation is a partial differential equation used to describe the temperature distribution in a heat-conducting body. The implementation of a numerical solution method for heat equation can vary with the geometry of the body. In this study, a three-dimensional transient heat conduction equation was solved by approximating second-order spatial derivatives by five-point central differences in cylindrical coordinates. The stability condition of the numerical method was discussed. A MATLAB code was developed to implement the numerical method. An example was provided in order to demonstrate the method. The numerical solution by the method was in a good agreement with the exact solution for the example considered. The accuracy of the five-point central difference method was compared with that of the three-point central difference method in solving the heat equation in cylindrical coordinates. The solutions obtained by the numerical method in cylindrical coordinates were displayed in the Cartesian coordinate system graphically. The method requires relatively very small time steps for a given mesh spacing to avoid computational instability. The result of this study can provide insights to use appropriate coordinates and more accurate computational methods in solving physical problems described by partial differential equations.

On the Determination of Thermal Conductivity From Frequency Domain Thermoreflectance Experiments

The Fourier and the hyperbolic heat conduction equations were solved numerically to simulate a frequency-domain thermoreflectance (FDTR) experiment. Numerical solutions enable isolation of pump and probe laser spot size effects, and use of realistic boundary conditions. The equations were solved in time domain and the phase lag between the temperature of the transducer (averaged over the probe laser spot) and the modulated pump laser signal, were computed for a modulation frequency range of 200 kHz to 200 MHz. Numerical calculations showed that extracted values of the thermal conductivity are sensitive to both the pump and probe laser spot sizes, while analytical solutions (based on Hankel transform) cannot isolate the two effects, although for the same effective (combined) spot size, the two solutions are found to be in excellent agreement. If the substrate (computational domain) is sufficiently large, the far-field boundary conditions were found to have no effect on the computed phase lag. The interface conductance between the transducer and the substrate was found to have some effect on the extracted thermal conductivity. The hyperbolic heat conduction equation yielded almost the same results as the Fourier heat conduction equation for the particular case studied. The numerically extracted thermal conductivity value (best fit) for the silicon substrate considered in this study was found to be about 82-108 W/m/K, depending on the pump and probe laser spot sizes used.

Inverse Thermoelastic Analysis of a Thick Rectangular Plate

Thermal stresses and displacement functions are obtained for a rectangular plate occupying the space R: -a < x < a, 0 < y < b, -h < z < h, with the known boundary and initial conditions. In this inverse problem the unknown surface temperature is determined on the boundary along the y-axis when the temperature at some internal point is known. The governing heat conduction equation has been solved by applying Marchi – Fasulo transform and Laplace transform techniques. The solutions are obtained in form of infinite series. The results for displacement and thermal stresses have been computed numerically and illustrated graphically for Aluminium plate. MSC 2010: 74A10,74J25, 74H99, 74D99