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

2016 ◽  
Vol 16 (4) ◽  
pp. 85-90 ◽  
Author(s):  
B. Mochnacki ◽  
M. Ciesielski
Keyword(s):  
Mushy Zone ◽  
Metal Film ◽  
Control Volume ◽  
Extreme Temperature ◽  
Internal Heat ◽  
Melting Process ◽  
Thin Metal Film ◽  
Heating Process ◽  
Phase Lag ◽  
Dual Phase

Abstract 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.

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

2016 ◽  
Vol 16 (1) ◽  
pp. 41-44 ◽  
Author(s):  
E. Majchrzak ◽  
B. Mochnacki
Keyword(s):  
Laser Pulse ◽  
Metal Film ◽  
Initial Conditions ◽  
Control Volume ◽  
Extreme Temperature ◽  
Laser Action ◽  
Internal Heat ◽  
Internal Heat Source ◽  
Thin Metal Film ◽  
Phase Lag

Abstract 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.

scholarly journals Second-Order Dual Phase Lag Equation. Modeling of Melting and Resolidification of Thin Metal Film Subjected to a Laser Pulse

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 999 ◽  
Author(s):  
Ewa Majchrzak ◽  
Bohdan Mochnacki
Keyword(s):  
Laser Pulse ◽  
Latent Heat ◽  
Metal Film ◽  
Thin Metal Film ◽  
Second Order ◽  
Phase Changes ◽  
Thin Metal ◽  
Phase Lag ◽  
Dual Phase ◽  
Numerical Computations

The process of partial melting and resolidification of a thin metal film subjected to a high-power laser beam is considered. The mathematical model of the process is based on the second-order dual phase lag equation (DPLE). Until now, this equation has not been used for the modeling of phase changes associated with heating and cooling of thin metal films and the considerations regarding this issue are the most important part of the article. In the basic energy equation, the internal heat sources associated with the laser action and the evolution of phase change latent heat are taken into account. Thermal processes in the domain of pure metal (chromium) are analyzed and it is assumed that the evolution of latent heat occurs at a certain interval of temperature to which the solidification point was conventionally extended. This approach allows one to introduce the continuous function corresponding to the volumetric fraction of solid or liquid state at the neighborhood of the point considered, which significantly simplifies the phase changes modeling. At the stage of numerical computations, the authorial program based on the implicit scheme of the finite difference method (FDM) was used. In the final part of the paper, the examples of numerical computations (including the results of simulations for different laser intensities and different characteristic times of laser pulse) are presented and the conclusions are formulated.

Modeling of Laser-Soft Tissue Interactions Using the Dual-Phase Lag Equation: Sensitivity Analysis with Respect to Selected Tissue Parameters

2017 ◽  
Vol 379 ◽  
pp. 108-123
Author(s):  
Ewa Majchrzak ◽  
Marek Jasiński ◽  
Łukasz Turchan
Keyword(s):  
Sensitivity Analysis ◽  
Soft Tissue ◽  
Laser Irradiation ◽  
Soft Tissues ◽  
Initial Conditions ◽  
Internal Heat ◽  
Bioheat Transfer ◽  
Phase Lag ◽  
Dual Phase ◽  
Coupled Equations

Thermal processes occurring in soft tissues are subjected to laser irradiation are analyzed. The transient bioheat transfer is described by the generalized dual-phase lag model. This model consists of two coupled equations concerning the tissue and blood temperatures supplemented by the appropriate boundary and initial conditions. The efficiency of the internal heat source connected to the laser irradiation results from the solution of the diffusion equation. This approach is acceptable when the scattering dominates over the absorption for wavelengths between 650 and 1300 nm, and just such a situation occurs in the case of soft tissues. Sensitivity analysis with respect to the parameters occurring in the mathematical model is done using the direct approach (differentiation of the basic equations and the boundary-initial conditions with respect to the parameter considered), especially the absorption coefficient and scattering coefficient of the soft tissue are considered. At the stage of numerical modeling the basic problem and additional problems connected with the sensitivity functions are solved using the finite difference method. In the final part the conclusions and examples of computations are presented.

scholarly journals The elastic wave motions for a photothermal medium of a dual-phase-lag model with an internal heat source and gravitational field

2016 ◽  
Vol 94 (4) ◽  
pp. 400-409 ◽  
Author(s):  
Kh. Lotfy
Keyword(s):  
Heat Source ◽  
Thermal Loading ◽  
Harmonic Wave ◽  
Internal Heat ◽  
Internal Heat Source ◽  
Transfer Model ◽  
Phase Lag ◽  
Dual Phase ◽  
Horizontal Distance ◽  
Lag Model

In this work, the dual-phase-lag (DPL) heat transfer model is introduced to study the problem of an isotropic generalized thermoelastic medium with an internal heat source that is moving with a constant speed. Thermal loading at the free surface of a semi-infinite semiconconducting medium coupled plasma waves with the effect of mechanical force during a photothermal process to study the effect of a gravity field. Harmonic wave analysis is used to obtain exact expressions for the considered variables, also the carrier density coefficients were obtained analytically. The variations of the considered variables through the horizontal distance are illustrated graphically under the effects of several parameters based on the DPL model. The results are discussed and depicted graphically.

scholarly journals First and second order dual phase lag equation. Numerical solution using the explicit and implicit schemes of the finite difference method

2018 ◽  
Vol 240 ◽  
pp. 05018 ◽  
Author(s):  
Ewa Majchrzak ◽  
Bohdan Mochnacki
Keyword(s):  
Thin Metal Film ◽  
Phase Lag ◽  
Dual Phase ◽  
Mathematical Form ◽  
Fourier Law ◽  
Delay Times ◽  
Left And Right ◽  
Explicit And Implicit Schemes ◽  
Theoretical Considerations ◽  
Generalized Fourier Law

In the paper the different variants of the dual phase lag equation (DPLE) are considered. As one knows, the mathematical form of DPLE results from the generalization of the Fourier law in which two delay times are introduced, namely the relaxation time τq and the thermalization one τT. Depending on the order of development of the left and right hand sides of the generalized Fourier law into the Taylor series one can obtain the different forms of the DPLE. It is also possible to consider the others forms of equation discussed resulting from the introduction of the new variable or variables (substitution). In the paper a thin metal film subjected to a laser pulse is considered (the 1D problem). Theoretical considerations are illustrated by the examples of numerical computations. The discussion of the results obtained is also presented.

Numerical solutions of the second-order dual-phase-lag equation using the explicit and implicit schemes of the finite difference method

2019 ◽  
Vol 30 (4) ◽  
pp. 2099-2120
Author(s):  
Ewa Majchrzak ◽  
Bohdan Mochnacki
Keyword(s):  
Finite Difference Method ◽  
Laser Pulse ◽  
Metal Film ◽  
Numerical Solutions ◽  
Second Order ◽  
Phase Lag ◽  
Dual Phase ◽  
Content Type ◽  
The Finite Difference Method ◽  
Lag Times

Purpose The purpose of this paper is the application of the finite difference method (FDM) for numerical modeling of the microscale heat transfer processes occurring in the domain of thin metal film subjected to a laser pulse. The problem discussed is described by the different variants of the second-order dual-phase-lag equation (DPLE). The laser action is taken into account by the introduction of internal volumetric heat source to the governing equation. The capacity of the source is dependent on the geometrical co-ordinates and duration of the laser beam. The modified forms of DPLE presented in the paper, resulting from the certain substitutions introduced to the basic equation. Design/methodology/approach At the stage of numerical computations, the different variants of the FDM are applied. Both the explicit and implicit FDM schemes are used. The formula determining the capacity of the internal heat source suggests the formulation of the task discussed using the cylindrical co-ordinate system. The in-house programs realizing the numerical computations concern the axially-symmetrical tasks. In this paper, the metal films made of the nickel and gold are considered. Findings The algorithms presented make possible to analyze the heating/cooling processes occurring in the domain of metal film having a thickness Z for the different laser parameters (laser intensity, characteristic time of laser pulse and laser beam radius) and the different materials (optical penetration depth, reflectivity of irradiated surface, lag times, thermal conductivity and volumetric specific heat). Research limitations/implications Not for all metals, one can find information on lag times. In the literature, analytical formulas can be found to calculate these values, but they are strongly approximated. It should be pointed out that there are some limitations concerning the delay times of material considered, which assure the physical correctness of the second-order DPLE. Originality/value The FDM algorithm concerns the three-dimensional cylindrical domain while a large majority of the second-order DPLE numerical solutions have been obtained for the one-dimensional tasks. Both the implicit and explicit numerical schemes are proposed and the testing computations confirm the correctness and effectiveness of the algorithms presented.

scholarly journals Numerical Simulation of Thermal Processes in a Domain of Thin Metal Film Subjected to an Ultrashort Laser Pulse

Materials ◽  
2018 ◽  
Vol 11 (11) ◽  
pp. 2116 ◽  
Author(s):  
Ewa Majchrzak ◽  
Bohdan Mochnacki
Keyword(s):  
Laser Pulse ◽  
Metal Film ◽  
Ultrashort Laser Pulse ◽  
Initial Conditions ◽  
Time Derivative ◽  
Thin Metal Film ◽  
Thin Metal ◽  
Thermal Processes ◽  
Phase Lag ◽  
Ultrashort Laser

A thin metal film subjected to an ultrashort laser pulse is considered. With a sufficiently high laser intensity the process of the film heating may cause metal melting and even ablation. In this work, the numerical model of the melting and resolidification processes is presented. The mathematical model is based on the dual phase lag equation in which two positive constants appear, this means the relaxation and thermalization times. The considered equation contains a second-order time derivative and higher order mixed derivative in both time and space and should be supplemented by the appropriate boundary and initial conditions. The model of the melting and resolidification is presented in two versions. The first can be called ‘the introduction of the artificial mushy zone sub-domain’, while the second ‘the two forms of the basic energy equation’. At the stage of numerical computations, the implicit scheme of the finite difference method is used. The numerical algorithm is tested for the two proposed models which are applied to the computations concerning the thermal processes occurring in the cylindrical micro-domain (chromium, gold) subjected to an ultrashort laser pulse.

Thermal Dispersion in Finite Medium Under Periodic Surface Disturbance Using Dual-Phase-Lag Model

2015 ◽  
Vol 138 (3) ◽  
Author(s):  
Tung T. Lam ◽  
Ed Fong
Keyword(s):  
Heat Flux ◽  
Solution Structure ◽  
Superposition Principle ◽  
Internal Heat ◽  
Transient Heat Conduction ◽  
Fourier Series Expansion ◽  
Hyperbolic Model ◽  
Phase Lag ◽  
Dual Phase ◽  
Lag Model

Transient heat conduction in finite thin films subjected to time-varying surface heat flux incidences at both boundaries and internal heat generation is investigated via the dual-phase-lag (DPL) hyperbolic model. Analytical solution of the temperature profiles inside the solid is derived by using the superposition principle and the method of Fourier series expansion in conjunction with the solution structure theorems. For comparison purposes, the classical diffusion, Cattaneo–Vernotte (C–V) model, and simplified thermomass (TM) models are deduced from the generalized DPL model. This is made possible by adjusting the temperature and heat flux relaxation parameters, and offers the opportunity to examine various interconnected non-Fourier conduction heat transfer characteristics including wave and diffusion effects as well as their interrelationship. Details of this process are examined and results are explored in this study.

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