Numerical simulation of thermal wave propagation and collision in thin film using finite element solution
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Numerical simulation of thermal wave propagation and collision in thin film using finite element solution R. Yuvaraj1 · D. Senthil Kumar1 Received: 6 November 2019 / Accepted: 13 January 2020 © Akadémiai Kiadó, Budapest, Hungary 2020
Abstract Numerical simulation of propagation and collision of thermal wave phenomena in the order of nanometer is carried out. The finite element procedure is developed for Cattaneo–Vernotte heat conduction equation. In this simulation, non-homogeneous boundary conditions are used. Newmark’s scheme and constant average acceleration method are used to solve second-order time derivatives. The non-dimensional results are compared with the analytical solutions of the same kind of problem solved by using superposition principle with solution structure theorems and experimental values. It is noted that this current procedure had good agreement with the results given by analytical and experimental solutions. In this study, the finite element modeling of hyperbolic equation is solved for thermal wave propagation and collision of thermal waves. The transition of hyperbolic to parabolic nature of thermal wave propagation for different layer lengths and the characteristics of nodal temperatures near to the boundary are discussed. Also this work examined the temperature variation at the center of the layer at all collisions, which shows the increase in temperature after the first collision when time is 0.5 and then propagates toward boundary with further increase in temperature. This is well above the applied boundary temperature and decrease in temperature at boundary for subsequent collisions. The transition of thermal waves from hyperbolic to parabolic nature is analyzed and is found that the transition happens for the film thickness greater than 5. This phenomenon is repeated until it reaches steady state when time is 5 for hyperbolic whereas 1.5 for parabolic mode. Keywords Finite element model · Thermal wave propagation · Thermal wave collision · Hyperbolic model · Asymmetric boundary List of symbols T Temperature (K) Tw Wall temperature (K) k Thermal conductivity (W m−1 K−1) cp Specific heat capacity (J kg−1 K−1) L Length of the layer (m) x Distance (m) l Element length (m) n Number of elements q Heat flux (W m−2) c Thermal wave speed (m s−1) t Time (s) w Weight function [K] Stiffness matrix [C] Capacitance matrix
[M] Mass matrix {F} Force vector Greek symbols 𝜏 Relaxation time (s) 𝜌 Density (kg m−3) 𝛼 Thermal diffusivity (m2 s−1) 𝜓 Approximate function 𝜃 Dimensionless temperature 𝜉 Dimensionless distance 𝜂 Dimensionless time 𝜁 Dimensionless heat flux 𝛽 Constant 𝛾 Constant
Introduction * D. Senthil Kumar [email protected] 1
Department of Mechanical Engineering, Sona College of Technology, Salem, Tamilnadu, India
The recent scenario shows that the applications of thin film require a better understanding of heat transport phenomena in the strong thermal wave environments. Thin films in the
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order of microns are developed
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