Thermal characteristics of longitudinal fin with Fourier and non-Fourier heat transfer by Fourier sine transforms

The quest for high-performance of heat transfer components on the basis of accommodating shapes, smaller weights, lower costs and little volume has significantly diverted the industries for the enhancement of heat dissipation with variable thermal properties of fins. This manuscript proposes the fractional modeling of Fourier and non-Fourier heat transfer of longitudinal fin via non-singular fractional approach. The configuration of longitudinal fin in terms of one dimension is developed for the mathematical model of parabolic and hyperbolic heat transfer equations. By considering the Fourier and non-Fourier heat transfer from longitudinal fin, the mathematical techniques of Fourier sine and Laplace transforms have been invoked. An analytic approach is tackled for handling the governing equation through special functions for the fractionalized parabolic and hyperbolic heat transfer equations in longitudinal fin. For the sake of comparative analysis of parabolic verses hyperbolic heat conduction of fin temperature, we depicted the distinct graphical illustrations; for instance, 2-dimensional graph, bar chart, contour graphs, heat graph, 3-dimensional graphs and column graphs on for the variants of different rheological impacts of longitudinal fin.

Thermo-vibrational analyses of skin tissue subjected to laser heating source in thermal therapy

Laser-induced thermal therapy, due to its applications in various clinical treatments, has become an efficient alternative, especially for skin ablation. In this work, the two-dimensional thermomechanical response of skin tissue subjected to different types of thermal loading is investigated. Considering the thermoelastic coupling term, the two-dimensional differential equation of heat conduction in the skin tissue based on the Cattaneo–Vernotte heat conduction law is presented. The two-dimensional differential equation of the tissue displacement coupled with the two-dimensional hyperbolic heat conduction equation in the tissue is solved simultaneously to analyze the thermal and mechanical response of the skin tissue. The existence of mixed complicated boundary conditions makes the problem so complex and intricate. The Galerkin-based reduced-order model has been utilized to solve the two-sided coupled differential equations of vibration and heat transfer in the tissue with accompanying complicated boundary conditions. The effect of various types of heating sources such as thermal shock, single and repetitive pulses, repeating sequence stairs, ramp-type, and harmonic-type heating, on the thermomechanical response of the tissue is investigated. The temperature distribution in the tissue along depth and radial direction is also presented. The transient temperature and displacement response of tissue considering different relaxation times are studied, and the results are discussed in detail.

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.

Two-Dimensional Vibrating And Thermal Response of Skin Tissue To Different Types of Heat Sources Considering The Hyperbolic Heat Transfer Equation

Laser-induced thermal therapy, due to its applications in various clinical treatments, has become an efficient alternative, especially for skin ablation. In this work, the two-dimensional thermomechanical response of skin tissue subjected to different types of thermal loading is investigated. Considering the thermoelastic coupling term, the two-dimensional differential equation of heat conduction in the skin tissue based on the Cattaneo–Vernotte heat conduction law is presented. The two-dimensional differential equation of the tissue displacement coupled with the two-dimensional hyperbolic heat conduction equation of tissue is solved simultaneously to analyze the thermal and mechanical response of the skin tissue. The existence of mixed complicated boundary conditions makes the problem so complex and intricate. The Galerkin-based reduced-order model has been utilized to solve the two-sided coupled differential equations of skin displacement and heat transfer with accompanying complicated boundary conditions. The effect of various types of heating sources such as thermal shock, single and repetitive pulses, repeating sequence stairs, ramp-type, and harmonic-type heating, on the thermomechanical response of the tissue is investigated. The temperature distribution in the tissue along the depth and radial direction is also presented. The transient temperature and displacement response of tissue considering different relaxation times are studied, and the results are discussed in detail.

Reduced Differential Transform Method for Thermoelastic Problem in Hyperbolic Heat Conduction Domain

In the present study, we have applied the reduced differential transform method to solve the thermoelastic problem which reduces the computational efforts. In the study, the temperature distribution in a two-dimensional rectangular plate follows the hyperbolic law of heat conduction. We have obtained the generalized solution for thermoelastic field and temperature field by considering non-homogeneous boundary conditions in the x and y direction. Using this method one can obtain a solution in series form. The special case is considered to show the effectiveness of the present method. And also, the results are shown numerically and graphically. The study shows that this method provides an analytical approximate solution in very easy steps and requires little computational work.

A control volume finite element method for the thermoelastic problem in functional graded material with one relaxation time

In this paper, a new direct vertex-centred finite volume method (CV-FEM) has been developed for the thermoelastic problem in functional graded material (FGM) based on Lord-Shulman theory. The heat conduction equation in Lord-Shulman theory is modified by considering the product term of spatial gradient of relaxation time and the heat flux rate, and it makes the present method more accurate to capture characteristics of a thermoelastic wave in inhomogeneous and FGM compared with previous methods. Some benchmark examples are used to demonstrate the capability of the present method for hyperbolic heat conduction and thermoelastic coupled problems. The effects of the ‘product-term’ on the wave propagation are studied by a heat conduction problem in inhomogeneous material and a thermoelastic problem in FGM. The FGM results show that its effect on the thermoelastic response is significant even for a linear variation of material properties.

Convergence analysis of a variational quasi-reversibility approach for an inverse hyperbolic heat conduction problem

We study a time-reversed hyperbolic heat conduction problem based upon the Maxwell–Cattaneo model of non-Fourier heat law. This heat and mass diffusion problem is a hyperbolic type equation for thermodynamics systems with thermal memory or with finite time-delayed heat flux, where the Fourier or Fick law is proven to be unsuccessful with experimental data. In this work, we show that our recent variational quasi-reversibility method for the classical time-reversed heat conduction problem, which obeys the Fourier or Fick law, can be adapted to cope with this hyperbolic scenario. We establish a generic regularization scheme in the sense that we perturb both spatial operators involved in the PDE. Driven by a Carleman weight function, we exploit the natural energy method to prove the well-posedness of this regularized scheme. Moreover, we prove the Hölder rate of convergence in the mixed {L^{2}}–{H^{1}} spaces.