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.

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.

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).

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.

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.

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 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.

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 thermal processes proceeding in the casting-mould system are analysed. Solidification process can be described in the different ways. One of them consists in the application of the Fourier-Kirchhoff equation in which the source function controlling the phase change is introduced. In this paper, two approaches to the modelling of pure metals solidification, i.e. the solidification proceeding at the constant temperature and in the artificially introduced interval of temperature are considered. Numerical model used at the stage of computations is constructed using the Control Volume Method in which the principle of conservation of energy (taking also into account the release of latent heat of solidification) is applied to each control volume ensuring the correctness of the model. The use of Voronoi/Thiessen tessellation to the discretization of the casting-mould system can accurately reproduce the shape of each sub-domain. In the final part of the paper, the examples of computations are shown.

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.

Solidification and cooling processes proceeding in the metal domain can be described in different ways. One of them consists in the application of the Fourier-type equation in which the additional term (source function) controlling the solidification process is introduced. In this paper this type of energy equation is used, but for the phase change modeling the equation discussed is in some way transformed. Such a modification is possible if one considers the pure metal for which the solidification point is a constant value. The numerical model used at the stage of computations is based on the Control Volume Method. In the final part of the paper, examples of computations are shown.

A cylindrical skin tissue domain subjected to an external heat flux is considered. Thermal processes in the domain considered are described by the Cattaneo-Vernotte equation supplemented by the appropriate boundary and initial conditions. The aim of considerations is the identification of external heat flux and relaxation time on the basis of ‘measured’ heating/cooling curves at the set of selected points located on the surface of the skin. The direct problem is solved using the implicit scheme of the Finite Difference Method (FDM), while at the stage of the inverse problem solution, the evolutionary algorithm is applied. In the final part of the paper the examples of computations are presented.

In the paper the soft tissue freezing process is considered. The tissue sub-domain is subjected to the action of cylindrical cryoprobe. Thermal processes proceeding in the domain considered are described using the dual-phase lag equation (DPLE) supplemented by the appropriate boundary and initial conditions. DPLE results from the generalization of the Fourier law in which two lag times are introduced (relaxation and thermalization times). The aim of research is the identification of these parameters on the basis of measured cooling curves at the set of points selected from the tissue domain. To solve the problem the evolutionary algorithms are used. The paper contains the mathematical model of the tissue freezing process, the very short information concerning the numerical solution of the basic problem, the description of the inverse problem solution and the results of computations.

In the paper, the thermal processes occurring in the axially-symmetrical domain of multi-layered thin metal film subjected to a strong laser pulse are considered. The problem is described by the system of dual-phase lag equations (DPLE) supplemented by the appropriate boundary and initial conditions (in the form corresponding to the model considered). The possibility of domain melting is also taken into account. At the stage of numerical computations, the implicit scheme of the finite difference method is used. The problems connected with the modeling of the thermal contact conditions between sub-domains and finally the simulations of the melting and resolidification processes are discussed in detail.The continuity condition given on the contact surface is considered in the form in which the lag times are taken into account. The results confirm the differences between often presented solutions, using the dual-phase lag model supplemented by the 'macroscopic' type of these conditions. The model of melting and resolidification results from the modified form of the DPLE for which the derivative of temperature with respect to time is equal to zero. It results from the fact that the melting of pure metals proceeds at the constant temperature.

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.

The numerical modeling of thermal processes in domain of biological tissue (the male thigh) secured by multilayered protective clothing being in the thermal contact with the environment is discussed. The thigh is treated as the nonhomogeneous domain in which the sub-domains of skin tissue, fat, muscle, bone and blood vessels are distinguished. Between the protective clothing and skin tissue the air gap is taken into account. The heat transfer is described by the system of hyperbolic Cattaneo-Vernotte equations (for the tissue sub-domains) and parabolic Fourier equations (for the remaining sub-domains). The process of external heating is determined by the appropriate boundary condition and the internal heat source (in the fabric sub-domain) related to the absorption of incident thermal radiation. The mathematical model is solved numerically using the control volume method, while the considered sub-domains (the 2 D problem) are covered by the Voronoi meshes. In the final part of the article, the example of computations is presented.

Thermal processes occuring in the solid bodies are, as a rule, described by the well-known Fourier equation (or the system of these equations) supplemented by the appropriate boundary and initial conditions. Such a mathematical model is sufficiently exact to describe the heat transfer processes in the macro scale for the typical materials. It turned out that the energy equation based on the Fourier law has the limitations and it should not be used in the case of the microscale heat transfer and also in the case of materials with a special inner structure (e.g. biological tissue). The better approximation of the real thermal processes assure the modifications of the energy equation, in particular the models in which the so-called lag times are introduced. The article presented is devoted to the numerical aspects of solving these types of equations (in the scope of the microscale heat transfer). The results published by the other authors can be found in the references posted in the works cited below.