Dual Phase Lag Equation Research Articles

Numerical simulation of heat transfer properties of skin tissue acted on repetitive laser irradiation

Pulsed laser has been widely used in the clinical treatment of various diseases and injuries due to its unique advantages in the process of interaction with skin tissue. In order to ensure the therapeutic effect and patient safety, it is necessary to accurately monitor the time-space response of the whole-field temperature of the target tissue irradiated by pulsed laser. A research strategy to explore the thermal response and incidental thermal damage of skin tissue induced by repeated pulses laser has been proposed in present work. A theoretical model involved variable physical properties was first proposed, in which the dual-phase lag (DPL) equation and Henrique burn equation were employed to evaluate the tissue temperature and thermal damage induced by repeated pulse laser irradiation. With the help of finite difference method, the governing equation with variable physical properties was then numerical solved to obtain the time-space distribution of tissue temperature, and the evolution process of the thermal damage was further calculated. The effects of laser input parameters and variable physical parameters on the laser penetration depth, temperature distribution and burn degree of the irradiated tissue were analyzed.

Sensitivity of a Process for Heating Thin Metal Film Described by the Dual-Phase Lag Equation with Temperature-Dependent Thermophysical Parameters to Perturbations of Lag Times

In the paper, an equation with two delay times (dual-phase lag Equation (DPLE)) in a version that takes into account the dependence of thermophysical parameters (volumetric specific heat and thermal conductivity) on temperature is considered. In particular, an analysis of the sensitivity of transient temperature field in relation to disturbances in delay times (the relaxation and thermalization times) is performed. The sensitivity model concerns the process of heating an ultrathin metal layer with a laser beam. First, the equation with two delay times in the case of temperature-dependent thermophysical parameters is presented. Next, the sensitivity equations with respect to delay times are derived using the direct method. The algorithms for solving the basic and sensitivity tasks are also briefly presented. At the stage of computations, an authorial program based on the implicit scheme of a finite-difference method is developed. In the final part of the paper, examples of numerical solutions (for layers made from gold and nickel) are presented. The research conducted here shows that disturbances in the temperature field are clearly visible and depend, on the one hand, on the thermophysical parameters of the material, and on the other hand, on the intensity of heating with an external heat source.

Numerical modelling of the laser high-temperature hyperthermia using the dual-phase lag equation

Numerical modelling of the laser high-temperature hyperthermia using the dual-phase lag equation

Modelling of thin metal film heating using the dual-phase lag equation with temperature-dependent parameters

The paper presents a mathematical description and numerical algorithm simulating the thermal processes occurring in the metal microdomain subjected to the ultrashort laser pulse. The model of these processes is based on an equation with two delay times (the dual-phase lag equation – DPLE) supplemented by the appropriate boundary and initial conditions. The energy equation is formulated in the version assuming the variability of thermophysical parameters with temperature (volumetric specific heat and thermal conductivity are temperature-dependent). Taking into account the geometric properties of the laser beam, the problem is treated as an axisymmetric task, while the thermal impact of the laser is taken into account by the introducing into DPLE a component related to the internal heat sources in the domain considered. At the stage of numerical modeling the implicit scheme of the finite difference method is used. In the program simulating the heating/cooling processes occurring in the metal microdomain, the possibility of the melting and resolidification effects are also considered. In the final part of the work, the numerous examples of numerical computations and the resulting conclusions are presented. It turned out, among others, that especially at high temperatures, the consideration of parameters variability (gold, nickel) causes that the results of numerical simulations differ visibly from classical solutions for the constant values of parameters.

Numerical analysis of biological tissue heating using the dual-phase lag equation with temperature – dependent parameters

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

Sensitivity analysis of temperature in heated soft tissues with respect to time delays

Axisymmetric tissue region heated by an external heat flux is considered. The mathematical model is based on the dual-phase lag equation supplemented by appropriate boundary and initial conditions. This equation, in relation to the Pennes’ equation, has two additional parameters, namely the relaxation time and the thermalization time. The aim of this research is to estimate the temperature changes due to changes of these parameters. To achieve this, sensitivity analysis methods are used. The basic problem and additional ones related to the sensitivity functions are solved using the implicit scheme of the finite difference method. The performed computations show that the temperature changes caused by changes in the relaxation and thermalization times are larger for higher values of the external heat flux and shorter times of its action.

Numerical modeling of laser photothermal heating in organic and inorganic structures

In this paper, we discuss flaws with static heating models as well as flaws with diffusive transport models such as the Fourier-Biot equation, the Cattaneo-Vernotte thermal wave model, and the hyperbolic heat equation and how the Dual-Phase-Lag (DPL) heat transfer model corrects for these flaws. We present novel numerical solutions and coding methodology in detail for solutions of the DPL heat equations, which take into account nanoscale heating effects such as phonon and electron interactions, for both organic and inorganic media for the laser heating of various structures and boundary conditions. We then use the models presented to validate our results against the experimental and theoretical results of various others for time scales ranging from 100 fs to 100 s and from spatial scales all the way from macroscale down to nanometers in both organic and inorganic media and systems of different kinds to show the validity of not only the DPL equation but also our solution methodology to a wide variety of problems.

Fractional Dual-Phase Lag Equation—Fundamental Solution of the Cauchy Problem

In the paper, a fundamental solution of the fractional dual-phase-lagging heat conduction problem is obtained. The considerations concern the 1D Cauchy problem in a whole-space domain. A solution of the initial-boundary problem is determined by using the Fourier–Laplace transform technique. The final form of solution is given in a form of a series. One of the properties of the derived fundamental solution of the considered problem with the initial condition expressed be the Dirac delta function is that it is symmetrical. The effect of the time-fractional order of the Caputo derivatives and the phase-lag parameters on the temperature distribution is investigated numerically by using the method which is based on the Fourier-series quadrature-type approximation to the Bromwich contour integral.

Dual-phase lag model of heat transfer between blood vessel and biological tissue.

A single blood vessel surrounded by the biological tissue with a tumor is considered. The influence of the heating technique (e.g. ultrasound, microwave, etc.) is described by setting a fixed temperature for the tumor which is higher than the blood and tissue temperature. The temperature distribution for the blood sub-domain is described by the energy equation written in the dual-phase lag convention, the temperature distribution in the biological tissue with a tumor is described also by the dual-phase lag equation. The boundary condition on the contact surface between blood vessel and biological tissue and the Neumann condition are also formulated using the extended Fourier law. So far in the literature, the temperature distribution in a blood vessel has been described by the classical energy equation. It is not clear whether the Fourier's law applies to highly heated tissues in which a significant thermal blood vessel is distinguished, therefore, taking into account the heterogeneous inner structure of the blood, the dual-phase lag equation is proposed for this sub-domain. The problem is solved by means of the implicit scheme of the finite difference method. The computations were performed for various values of delay times, which were taken from the available literature, and the influence of these values on the obtained temperature distributions was discussed.

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

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.

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

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.

Semi-analytical solutions for different non-linear models of dual phase lag equation in living tissues

In this study, the heat transfer in a multilayer heated living tissue is investigated. A non-linear dual phase lag (DPL) heat conduction model is presented assuming temperature-dependent blood perfusion rate, heat conductivity and metabolic heat generation. Semi -analytical solutions for the non-linear model are obtained employing the Galerkin weighted residuals method. The effects of the variable thermophysical properties on the temperature distribution within the tissue are investigated through comparison of the results with the corresponding results for the constant thermophysical properties. The results show that the temperature dependence of the blood perfusion rate has a significant effect on the temperature distribution in the different layers of the tissue. As far as the tumor region is considered, 0.43 °C increase in its temperature is observed in comparison with that obtained assuming the constant blood perfusion rate at t = 1800 s. Moreover, increasing the temperature dependence of the thermal conductivity leads to the reduction in the temperature of the tissue except in the muscle layers. Considering the temperature-dependent metabolic heat generation increases the temperature of all of the layers compared to the corresponding temperatures obtained assuming the constant metabolic heat generation.

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

PurposeThe 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/approachAt 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.FindingsThe 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/implicationsNot 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/valueThe 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.

Numerical model of biological tissue heating using the models of bio–heat transfer with delays

The numerical model of thermal processes in domain of biological tissue subjected to an external heat source is discussed. The model presented is based on the second order dual–phase–lag equation (DPLE) in which the relaxation time and thermalization time thermalization time (τq and τT) are tak n into account. In this paper the homogeneous, cylindrical skin tissue domain is considered. The most important aim of the research is to compare the results obtained using the classical model (the first-orderDPLE) with the numerical solution resulting from the higher order form of this equation. At the stage of numerical computations the Finite Difference Method (FDM) is applied. In the final part of the paper the examples of computations are shown.

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

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.

Transient Analysis for Electrothermal Properties in Nanoscale Transistors

Due to the significance of electron and heat transfer in designing the nanoscale semiconductor devices, the transient analysis of electrothermal properties has attracted extensive attention. In this paper, the density-gradient and dual-phase-lag (DPL) equations are first combined to predict the electron and heat transport in nanoscale transistors. The DPL equation is solved with the consideration of the temperature jump boundary condition that dealing with phonon–wall collisions. We have shown that the temporal and spatial distributions of related physical variables can be obtained by self-consistently solving these equations. Furthermore, the spectral element time-domain method is used to discretize these equations. Numerical results of electrothermal properties for both 2-D and 3-D field-effect transistors have been demonstrated to show the robustness and universality of the proposed model. Therefore, the model we proposed can be used with the temporal and spatial distributions, which could be helpful for evaluating the electrothermal performance and computational designing of nanoscale transistors.

Identification of Laser Intensity Assuring the Destruction of Target Region of Biological Tissue Using the Gradient Method and Generalized Dual-Phase Lag Equation

Temperature distribution in biological tissue is described by dual-phase lag model supplemented by appropriate boundary and initial conditions. Laser–tissue interactions are taken into account in the source function occurring in this model. The problem is solved using the finite difference method. Next, the Arrhenius integral which is a measure of tissue destruction is calculated. The inverse problem formulated here is to estimate the laser intensity assuring the destruction of this tissue region for which the Arrhenius integral exceeds the critical value.

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

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.

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

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 τqand 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.

Thermodynamical consistency of the dual-phase-lag heat conduction equation

Dual phase lag equation for heat conduction is analyzed from the point of view of non-equilibrium thermodynamics. Its first order Taylor series expansion is consistent with the second law as long as the two relaxation times are not negative.