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Integro-differential form of the first-order dual phase lag heat transfer equation and its numerical solution using the Control Volume Method

Ciesielski M., Mochnacki B., Majchrzak E.

Archives of Mechanics

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2020

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Vol. 72, nr 5

415--444

EN

The start point of the dual phase lag equation (DPLE) formulation is the generalized Fourier law in which two positive constants (the relaxation and thermalization times) appear. This type of equation can be used (among others) to describe the heat conduction processes proceeding in micro-scale. Depending on the number of components in the development of the generalized Fourier law into a power series, one can obtain both the first-order DPLE and the second-order one. In this paper the first-order dual phase lag equation is considered. The primary objective of this research is the transformation of DPLE differential form to the integro-differential one supplemented by the appropriate boundary-initial conditions. The obtained form of the differential equation is much simpler and more convenient at the stage of numerical computations – the numerical algorithm based on the three-time-level scheme reduces to the two-time-level one. To find the numerical solution, the Control Volume Method is used (the heating of thin metal film subjected to a laser beam is considered). The choice of the numerical method was not accidental. The method has a simple physical interpretation ensuring the preservation of the local and global energy balances. To our knowledge, it has not been used so far in this type of tasks. In the final part of the paper the examples of numerical simulations are presented and the conclusions are formulated.

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Application of the Alternating Direction Implicit Method for numerical solution of the dual phase lag equation

Ciesielski M.

Journal of Theoretical and Applied Mechanics

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2017

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Vol. 55 nr 3

839—852

EN

The problem discussed in the paper concerns the numerical modeling of thermal processes proceeding in micro-scale described using the Dual Phase Lag Model (DPLM) in which the relaxation and thermalization time appear. The cylindrical domain of a thin metal film subjected to a strong laser pulse beam is considered. The laser action is taken into account by the introduction of an internal heat source in the energy equation. At the stage of numerical modeling, the Control Volume Method is used and adapted to resolve the hyperbolic partial differential equation. In particular, the Alternating Direction Implicit (ADI) method for DPLM is presented and discussed. The examples of computations are also presented.

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Micro-scale heat transfer : algorithm basing on the Control Volume Method and the identification of relaxation and thermalization times using the search method

Mochnacki B., Ciesielski M.

Computer Methods in Materials Science

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2015

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Vol. 15, No.2

353--361

EN

The thermal processes proceeding in micro-domains can be described, among others, using the dual phase lag model (DPLM). According to the newest opinions the DPLM constitutes a very good description of the real heat transfer processes proceeding in the micro-scale, in particular on account of extremely short duration, extreme temperature gradients and the very small geometrical dimensions of domain considered. The base of DPLM formulation is a generalized form of Fourier law in which two times τq, τγ appear (the relaxation time and thermalization one, respectively). The numerical solution of the problem discussed bases on the author’s version of the Control Volume Method adapted to resolve the hyperbolic partial differential equations. The example illustrating the method application concerns the estimation of τq and τγusing the algorithm basing on the search method and the thin metal film subjected to the laser pulse is considered.

PL

Procesy cieplne zachodzące w mikro-obszarach mogą być opisane między innymi za pomocą modelu matematycznego z dwoma czasami opóźnień (DPLM). Według najnowszych opinii, model DPLM stanowi bardzo dobry opis rzeczywistych procesów przepływu ciepła w mikroskali, w szczególności ze względu na ekstremalnie krótki czas ich trwania, ekstremalne gradienty temperatury i bardzo małe wymiary geometryczne rozważanego obszaru . Podstawą formułowania DPLM jest uogólnienie prawa Fouriera, w którym występują dwa czasy opóźnień τq i τγ (odpowiednio-czas relaksacji i termalizacji). Numeryczne rozwiązanie omówionego zagadnienia opiera się na autorskiej wersji Metody Bilansów Elementarnych dostosowanej do rozwiązywania hiperbolicznych równań różniczkowych cząstkowych. Przykład ilustrujący zastosowanie metody dotyczy oszacowania czasów τq i τγ za pomocą algorytmu opartego na metodzie przeszukiwania, oraz rozpatrywana jest cienka folia metalowa poddawana działaniu impulsu laserowego.

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Numerical analysis of short-pulse laser interactions with thin metal film

Majchrzak E., Poteralska J.

Archives of Foundry Engineering

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2010

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Vol. 10, iss. 4

123--128

EN

Thin metal film subjected to a short-pulse laser heating is considered. The hyperbolic two-temperature model describing the temporal and spatial evolution of the lattice and electrons temperatures is discussed. At the stage of numerical computations the finite difference method is used. In the final part of the paper the examples of computations are shown.