Abstract
Heat equations can estimate the thermal distribution and phase transformation in real-time based on the operating conditions and material properties. Such wonderful features have enabled heat equations in various fields, including laser and electron beam processing. The integral transform technique (ITT) is a powerful general-purpose semi-analytical/numerical method that transforms partial differential equations into a coupled system of ordinary differential equations. Under this category, Fourier and non-Fourier heat equations can be implemented on both equilibrium and non-equilibrium thermo-dynamical processes, including a wide range of processes such as the Two-Temperature Model, ultra-fast laser irradiation, and biological processes. This review article focuses on heat equation models, including Fourier and non-Fourier heat equations. A comparison between Fourier and non-Fourier heat equations and their generalized solutions have been discussed. Various components of heat equations and their implementation in multiple processes have been illustrated. Besides, literature has been collected based on ITT implementation in various materials. Furthermore, a future outlook has been provided for Fourier and non-Fourier heat equations. It was found that the Fourier heat equation is simple to use but involves infinite speed heat propagation in comparison to the non-Fourier heat equation and can be linked with the Two-Temperature Model in a natural way. On the other hand, the non-Fourier heat equation is complex and involves various unknowns compared to the Fourier heat equation. Fourier and Non-Fourier heat equations have proved their reliability in the case of laser–metallic materials, electron beam–biological and –inorganic materials, laser–semiconducting materials, and laser–graphene material interactions. It has been identified that the material properties, electron–phonon relaxation time, and Eigen Values play an essential role in defining the precise results of Fourier and non-Fourier heat equations. In the case of laser–graphene interaction, a restriction has been identified from ITT. When computations are carried out for attosecond pulse durations, the laser wavelength approaches the nucleus-first electron separation distance, resulting in meaningless results.
Highlights
Various engineering problems can be modeled using partial differential equations (PDEs) with initial and boundary conditions
All methods reduce the PDEs to a set of ordinary differential equations (ODEs) that can be solved via well-known techniques
In the non-Fourier heat equation and Fourier equation, the heat transfer coefficient is the only parameter that provides heat output from the system, following the principle of the first law of thermodynamics. It is why the heat transfer coefficient has been considered in the non-Fourier heat equation, as mentioned in Appendix A
Summary
Various engineering problems can be modeled using partial differential equations (PDEs) with initial and boundary conditions For this purpose, numerical approaches, including finite element, finite difference, boundary element, and spectral techniques, are usually applied [1]. To the best of the authors’ knowledge, this is the first study that compiles ITT implementation on Fourier and non-Fourier heat equations along with the user-defined codes in the field of experimental physics. ITT is illustrated as a reliable tool for solving heat equations engineering problems. This technique converts non-linear PDEs to a coupled non-linear ODEs so that they can be solved numerically. One will be able to implement ITT, for Fourier and non-Fourier, in various engineering applications
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