PurposeIn this paper, the mathematical and numerical models of heat transfer processes based on the dual-phase lag equation (DPLE; the energy equation with two delay times) are considered. This type of equation is, as a rule, applied for mathematical description of thermal processes taking place in the microdomains (microscale heat transfer) and also for the modeling of problems associated with the heat transfer in biological tissue domains, which follows from the specific tissue structure. The purpose of this paper is to show the correct form of boundary conditions supplementing the DPL equation (equations). The solutions of different DPLE variants discussed in the literature are obtained using the classical form of boundary conditions (as in the Fourier equation), which is not completely correct. In this paper, the proper mathematical form of the Neumann, Robin and continuity conditions is presented.Design/methodology/approachThe second part of the paper is devoted to the numerical aspects of solving the problems basing on the mathematical description formulated in this way. At the stage of computations, the finite difference method in an implicit scheme is applied (1D and axially-symmetrical problems are considered). One of the examples was also solved using the generalized boundary element method. For numerical computations, an authorial computer program was developed, which performs simulations related to the modeling of thermal processes based on DPL as well as on the Cattaneo-Vernotte and Fourier equations.FindingsThe results of numerous simulations concerning both microscale heat transfer and bioheat transfer are shown, including a comparison of solutions using the classical approach to boundary-initial conditions and the approach presented in this paper.Research limitations/implicationsDelay times values are not known for all materials, whereas the values presented in the literature sometimes differ from each other (especially in the case of biological tissue). In some works, it is emphasized that there are some limitations concerning the delay times of material considered, which assure the physical correctness of DPLE.Originality/valueThe correct formulations of boundary and initial conditions supplementing the dual-phase lag model are presented.

Abstract The heating of thin metal film subjected to the ultrafast laser pulse is analyzed. The thermal processes occurring in the domain considered are described using the hyperbolic two-temperature model supplemented by the procedure of melting and resolidification modelling. The problem is solved by means of the implicit scheme of the finite difference method. The examples of computations for different laser intensities and characteristic times of laser pulse are presented.

The heating of a thin metal film subjected to the ultra-short laser pulse is presented. Mathematical description of this process is based on the system of equations describing the electron and lattice temperatures and dependences between intensity of heat fluxes and temperature gradients supplemented by appropriate boundary and initial conditions. In this approach, a system of four equations needs to be solved. In this paper, another method of solution of the above formulated problem is proposed. Using appropriate mathematical manipulations, instead of four equations, two equations describing the lattice and electron temperature distributions are obtained. This system of two equations is solved using an implicit scheme of the finite difference method. The results obtained using both approaches were compared. They were almost identical, which confirms the correctness of the proposed method.

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.

In the paper, thermal processes occurring in a soft tissue subjected to laser irradiation are analyzed. The bioheat transfer in an axisymmetric domain is described by a dual-phase lag equation, which takes into account temperature-dependent thermophysical parameters of the tissue. The source term in this equation is related to laser irradiation, and is determined by solving the optical diffusion equation. It is assumed that the optical parameters depend on the Arrhenius integral, which is a measure of the degree of tissue destruction. In the model, the process of evaporation of water contained in the tissue is also considered.

The thermal processes proceeding within a perfused tissue in the presence of a vessel are considered. The Pennes bio-heat transfer equation determines the steady state temperature field in tissue sub-domain, while the ordinary differential equation resulting from the energy balance describes the change of blood temperature along the vessel. The coupling of above equations results from the boundary condition given on the blood vessel wall. The problem is solved using the combined numerical algorithm, in particular the boundary element method (for the tissue sub-domain) and the finite differences method (for blood vessel sub-domain).

In the paper the numerical analysis of thermal processes proceeding in the domain of biological tissue subjected to an external heat source is presented. Heat transfer in the skin tissue was assumed to be transient and two-dimensional. The bioheat transfer in the domain considered is described by the system of Pennes equations determining the temperature field in successive skin layers. Between the layers the ideal contact is assumed. On the selected part of skin surface the Neumann condition determining the value of external heat source is given, on the conventionally assumed internal surface of the tissue the no-flux condition is accepted. For time t =0 the initial distribution of temperature is known. The degree of the skin burn can be predicted on the basis of the so-called Henriques integrals and the main subject of the paper is the sensitivity analysis of these integrals with respect to the skin parameters. On the stage of numerical computations the boundary element method has been used. In the final part of the paper the results obtained are shown.

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.

The boundary element method is applied for numerical simulation of the freezing process proceeding in biological tissue under the influence of cylindrical cryoprobe. From the mathematical point of view the problem discussed belongs to the group of moving boundaries ones for which the mushy zone sub-region (intermediate phase) is considered. In this paper the mathematical model of the process is formulated using the fixed domain approach and a parameter called the substitute thermal capacity determines the evolution of latent heat. On a stage of numerical computations the generalized variant of the alternating phase truncation method (APTM) is applied and the basic mathematical model is rebuilt by the introduction of the enthalpy function. The boundary element method together with APTM leads to the simple and effective numerical algorithm because the difficulties connected with the non-linear problem modelling can be omitted. In the final part of the paper the results of computations are shown.

Application of the boundary element method for approximate solution of non-steady and nonlinear thermal diffusion problems is not possible in a direct way. The fundamental solutions (being a basis of the BEM algorithm) are known only for linear problems-in particular the linear form of the Fourier equation is required. On the other hand, the numerous advantages of the boundary element method are a sufficient justification for the examinations concerning the adaptation of the method in this direction. In the paper, the numerical procedures "linearizing" the typical mathematical model of heat conduction process will be discussed. Combining the basic BEM algorithm for linear Fourier equation with procedures correcting the temporary solutions for successive values of time, we obtain a simple tool which allows us to solve a large class of the practical problems concerning the heat conduction processes. In this paper we will discuss in turn the algorithms called the temperature field correction method (TFCM), the alternating phase truncation method (APTM) and the artificial heat source method (AHSM). In the final part of the paper, some examples of numerical solutions will be presented.

The mathematical model of solidification process can be formulated using the macro or micro-macro approach. In this paper the second generation model (micro-macro one) is considered. The driving force of crystallization is the local and temporary undercooling below solidification point Tcr . The nucleation and nuclei growth are proportional to the second power of undercooling. Formulas determining the phenomena previously mentioned contain coefficients called the nucleation coefficient and nuclei growth one. These coefficients are assumed to be interval values. For above assumptions the problem has been solved bymeans of interval boundary element method. In the final part of the paper the results of computations are shown.

In the paper the problems connected with numerical modelling of bio-heat transfer processes are discussed. The mathematical model of phenomena discussed bases on the Pennes equation, at the same time the steady and transient tasks are considered. The basic equation is supplemented by the adequate geometrical, physical, boundary and (in the case of transient heat transfer) initial conditions. In the first part of' the paper the examples of direct solutions are discussed. Next the possibilities of sensitivity analysis applications in the domain of bio-heat transfer are presented. In the final part the selected solutions of inverse problems are shown. On the stage of numerical simulations both in the case of direct and inverse problems, as a rule, the different variants of the boundary element method have been used.

The dual-phase lag equation is formulated for the case when the thermophysical parameters occurring in this equation are temperature-dependent. The axial-symmetrical domain of biological tissue heated by an external heat source is considered. The problem is solved using the implicit scheme of the finite difference method. At the stage of numeri-cal computations, the analytical relationships taken from the literature describing changes in parameters are taken into account.

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.

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.

The paper concerns the problems related to applying the complex variable step method for the sensitivity analysis of the steady temperature field in the solid body domain due to the perturbations of the geometrical and physical parameters. The optimization problem using the approach proposed is also discussed. At the stage of numerical modelling, the boundary element method is used. The first part of the paper is devoted to the shape sensitivity. The results obtained are compared with the solution resulting from the implicit approach of sensitivity analysis. In the second part, the practical problem concerning optimizing the geometry of continuous casting mould cross-section is considered. The project variable vector contains the cooling pipes' radius and the volume flow rate of the cooling water. The numerical results and the conclusions are presented in the final part of the paper.

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.

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.

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.

In the paper a description of heat transfer in a one-dimensional metal films is considered. The Boltzmann transport equation and a two-temperature model supplemented by appropriate boundary and initial conditions are applied to analyze the thermal process in a heated thin metal film. The problem considered is solved by the lattice Boltzmann method and the finite difference method respectively. The internal heat source is given in two different ways. In the first way, the internal heat source is considered a constant value, while in the second way an exponential function which simulates irradiation using a laser pulse is used. In the final part of the paper numerical examples of comparison of two methods and conclusions are presented.