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1

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.

2

Implicit scheme of the finite difference method for the second-order dual phase lag equation

Majchrzak E., Mochnacki B.

Journal of Theoretical and Applied Mechanics

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2018

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Vol. 56 nr 2

393--402

EN

The second-order dual phase lag equation (DPLE) as a mathematical model of the microscale heat transfer is considered. It is known that the starting point determining the final form of this equation is the generalized Fourier law in which two positive constants (the relaxation and thermalization times) appear. Depending on the order of the generalized Fourier law expansion into the Taylor series, different forms of the DPLE can be obtained. As an example of the problem described by the second-order DPLE equation, thermal processes proceeding in the domain of a thin metal film subjected to a laser pulse are considered. The numerical algorithm is based on an implicit scheme of the finite difference method. At the stage of numerical modeling, the first, second and mixed order of the dual phase lag equation are considered. In the final part of the paper, examples of different solutions are presented and conclusions are formulated.

3

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.

4

Modeling of Melting and Resolidification in Domain of Metal Film Subjected to a Laser Pulse

Majchrzak E., Mochnacki B.

Archives of Foundry Engineering

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2016

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Vol. 16, iss. 1

41--44

EN

Thermal processes in domain of thin metal film subjected to a strong laser pulse are discussed. The heating of domain considered causes the melting and next (after the end of beam impact) the resolidification of metal superficial layer. The laser action (a time dependent bell-type function) is taken into account by the introduction of internal heat source in the energy equation describing the heat transfer in domain of metal film. Taking into account the extremely short duration, extreme temperature gradients and very small geometrical dimensions of the domain considered, the mathematical model of the process is based on the dual phase lag equation supplemented by the suitable boundary-initial conditions. To model the phase transitions the artificial mushy zone is introduced. At the stage of numerical modeling the Control Volume Method is used. The examples of computations are also presented.

5

Dual Phase Lag Model of Melting Process in Domain of Metal Film Subjected to an External Heat Flux

Mochnacki B., Ciesielski M.

Archives of Foundry Engineering

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2016

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Vol. 16, iss. 4

85--90

EN

Heating process in the domain of thin metal film subjected to a strong laser pulse are discussed. The mathematical model of the process considered is based on the dual-phase-lag equation (DPLE) which results from the generalized form of the Fourier law. This approach is, first of all, used in the case of micro-scale heat transfer problems (the extremely short duration, extreme temperature gradients and very small geometrical dimensions of the domain considered). The external heating (a laser action) is substituted by the introduction of internal heat source to the DPLE. To model the melting process in domain of pure metal (chromium) the approach basing on the artificial mushy zone introduction is used and the main goal of investigation is the verification of influence of the artificial mushy zone ‘width’ on the results of melting modeling. At the stage of numerical modeling the author’s version of the Control Volume Method is used. In the final part of the paper the examples of computations and conclusions are presented.

6

Numerical analysis of artificial hyperthermia treatment

Turchan Ł., Majchrzak E.

Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska

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2014

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nr 1

48--53

EN

This paper presents numerical modelling of artificial hyperthermia treatment. Presented model takes into account not only the temperature distributions but also the thermal dose parameter. Obtaining of temperature distributions takes advantage of the generalized dual phase lag equation. For computer calculations the parallelized algorithm was prepared.

PL

Artykuł dotyczy numerycznego modelowania zabiegu sztucznej hipertermii. Analiza skuteczności zabiegu jest rozpatrywana nie tylko na podstawie czasoprzestrzennych rozkładów temperatury, ale także w oparciu o parametr dawki termicznej. Do modelowania przepływu ciepła w rozpatrywanym obszarze wykorzystano uogólnione równanie z dwoma czasami opóźnień. Na potrzeby obliczeń numerycznych napisano autorski program oparty o obliczenia równoległe.

7

Estimation of relaxation and thermalization times in microscale heat transfer model

Mochnacki B., Paruch M.

Journal of Theoretical and Applied Mechanics

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2013

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Vol. 51 nr 4

837--845

EN

The energy equation corresponding to the dual phase lag model (DPLM) results from the generalized form of the Fourier law, in which the two ‘delay times’ (relaxation and thermalization time) are introduced. The DPLM should be used in the case of microscale heat transfer analysis, in particular when thermal processes are characterized by extremely short duration (e.g. ultrafast laser pulse), considerable temperature gradients and very small dimensions (e.g. thin metal film). In this paper, the problem of relaxation and thermalization time identification is discussed, at the same time the heat transfer processes proceeding in the domain of a thin metal film subjected to a laser beam are analyzed. The solution presented bases on the application of evolutionary algorithms. The additional information concerning the transient temperature distribution on a metal film surface is assumed to be known. At the stage of numerical realization, the finite difference method (FDM) is used. In the final part of the paper, an example of computations is presented.

8

Application of evolutionary algorithms for identification of dual phase lag model parameters

Mochnacki B., Paruch M.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2011

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Vol. 10, nr 1

189-198

EN

The dual phase lag model (DPLM) based on the generalized form of Fourier law, in particular the introduction of two 'delay times' (relaxation time τq and thermalization time τT) leads to the considered form of energy equation. This equation should be applied in the case of microscale heat transfer modeling. In particular, DPLM constitutes a good approximation of thermal processes which are characterized by extremely short duration (e.g. ultrafast laser pulse), extreme temperature gradients and geometrical features of the domain considered (e.g. thin metal film). In this paper, the identification problem of two of the above mentioned positive constants τq, τT is discussed and the thermal processes proceeding in the domain of thin metal film subjected to a laser beam are analyzed. At the stage of computations connected with the identification problem solution, evolutionary algorithms are used. To solve the problem, additional information concerning the transient temperature distribution on a metal film surface is assumed to be known.

9

Estimation of dual phase lag model parameters using the evolutionary algorithms

Mochnacki B., Paruch M.

Archives of Foundry Engineering

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2011

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Vol. 11, iss. 3 spec.

277--281

EN

Generalization of Fourier law, in particular the introduction of two ‘delay times’ (relaxation time τq and thermalization time τT) leads to the new form of energy equation called the dual-phase-lag model (DPLM). This equation should be applied in a case of microscale heat transfer modeling. In particular, DPLM constitutes a good approximation of thermal processes which are characterized by extremely short duration (e.g. ultrafast laser pulse), extreme temperature gradients and geometrical features of domain considered (e.g. thin metal film). The aim of considerations presented in this paper is the identification of two above mentioned positive constants τq, τT. They correspond to the relaxation time, which is the mean time for electrons to change their energy states and the thermalization time, which is the mean time required for electrons and lattice to reach equilibrium. In this paper the DPLM equation is applied for analysis of thermal processes proceeding in a thin metal film subjected to a laser beam. At the stage of computations connected with the identification problem solution the evolutionary algorithms are used. To solve the problem the additional information concerning the transient temperature distribution on a metal film surface is assumed to be known.