Role of 3D Thermal Diffusivity in the Numerical Resolution of the Heat Equation

Authors : Ismail Abbas

Volume/Issue : Volume 6 - 2021, Issue 7 - July

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Abstract : The ad hoc one-dimensional definition of the scalar thermal diffusion coefficient D defined as K / Roh C is short and inadequate to deal with the resolution of the 2D and 3D thermal diffusion equation. We have alternatively applied the chains of matrix B to the solution of the 2D and 3D heat diffusion equation for stationary solutions and time-dependent transient solutions. The role of 3D thermal diffusivity in the numerical resolution of the heat equation is carefully studied throgh the repeated variation of the main diagonal entry of matrix B,RO in the interval [0,1]. It is obvious that thermal diffusivity is related to RO, one of them produces the other. The chains of the matrix B using the 3D diffusion coefficient combine D, dt and the Laplace operator in an inseparable block and define a new technique to solve the diffusion of heat in different situations.In this article, we have applied the B chains to solve five different examples of heat diffusion in 2D and 3D geometries for both time-dependent and stationary conditions and the presented digital solutions are surprisingly precise, fast and stable

The ad hoc one-dimensional definition of the scalar thermal diffusion coefficient D defined as K / Roh C is short and inadequate to deal with the resolution of the 2D and 3D thermal diffusion equation. We have alternatively applied the chains of matrix B to the solution of the 2D and 3D heat diffusion equation for stationary solutions and time-dependent transient solutions. The role of 3D thermal diffusivity in the numerical resolution of the heat equation is carefully studied throgh the repeated variation of the main diagonal entry of matrix B,RO in the interval [0,1]. It is obvious that thermal diffusivity is related to RO, one of them produces the other. The chains of the matrix B using the 3D diffusion coefficient combine D, dt and the Laplace operator in an inseparable block and define a new technique to solve the diffusion of heat in different situations.In this article, we have applied the B chains to solve five different examples of heat diffusion in 2D and 3D geometries for both time-dependent and stationary conditions and the presented digital solutions are surprisingly precise, fast and stable