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Publisher Summary This chapter explores the different processes through which heat is transferred in a furnace, namely, conduction, convection, radiation, and electrical heating. The primary objective of a furnace is to transfer thermal energy to the product. In a solid, the flow of heat by conduction is the result of the transfer of vibrational energy from one molecule to the next, and in fluids it occurs in addition as a result of the transfer of kinetic energy. Under transient conduction conditions, the temperature gradient through the system varies with time. Heat transfer by convection is attributable to macroscopic motion of a fluid and is thus confined to liquids and gases. Convective heat transfer is in reality the conduction of heat through a flowing fluid to a fixed surface, whereby the conductivity is defined by a convective heat transfer coefficient. All materials radiate thermal energy in the form of electromagnetic waves. When radiation falls on a surface it may be reflected, transmitted, or absorbed. The fraction of energy that is absorbed is manifested as heat. The physical laws governing thermal radiation are well established but are insufficient to describe technical processes quantitatively. Heat can be transferred by other parts of the electromagnetic spectrum, particularly in the long wave region, in and beyond the infrared. The primary source of energy for heating in this region is electricity. If there is a temperature difference between two parts of a system, then heat will be transferred by one or more of three methods.

In recent years considerable interest has been aroused to the problem of high-temperature heat conduction of solid-state body subject to strong high-frequency electronic and laser beams [1].

This paper deals with the time differential dual-phase-lag heat transfer models aiming, at first, to identify the eventually restrictions that make them thermodynamically consistent. At a first glance it can be observed that the capability of a time differential dual-phase-lag model of heat conduction to describe real phenomena depends on the properties of the differential operators involved in the related constitutive equation. In fact, the constitutive equation is viewed as an ordinary differential equation in terms of the heat flux components (or in terms of the temperature gradient) and it results that, for approximation orders greater than or equal to five, the corresponding characteristic equation has at least a complex root having a positive real part. That leads to a heat flux component (or temperature gradient) that grows to infinity when the time tends to infinity and so there occur some instabilities. Instead, when the approximation orders are lower than or equal to four, this is not the case and there is the need to study the compatibility with the Second Law of Thermodynamics. To this aim the related constitutive equation is reformulated within the system of the fading memory theory, and thus the heat flux vector is written in terms of the history of the temperature gradient and on this basis the compatibility of the model with the thermodynamical principles is analyzed.

Development of constitutive equations of Oldroyd fluids with hyperbolic heat conduction, directly from the free energy and dissipation functions, is the focus of this work. Two models from hyperbolic thermoelasticity of solids – theory with one thermal relaxation time and theory with two thermal relaxation times – provide a stepping-stone for this task. The setting for these models is offered, respectively, by the primitive thermodynamics of Edelen and the thermodynamic orthogonality of Ziegler, both cases involving an internal parameter: the inelastic strain. The identified free energy and dissipation functions show that, in the first case, the hyperbolic (telegraph-like) heat conduction is governed by the Maxwell-Cattaneo model involving one thermal relaxation time, whereas, in the second case, by the Fourier law and two thermal relaxation times in the constitutive laws for stress and entropy. In both cases, the thermal fields vanish from the mechanical equations as the Oldroyd fluid becomes incompressible, whereas the mechanical fields are present in the telegraph-type equations for temperature.

In this work, we start with several concerns regarding Galilean variance, pointed out by Christov and Jordan (2005), of the Cattaneo-Vernotte heat conduction in moving media. We then describe a generalized heat transport model in moving media and its underlying theory emanating from the Boltzmann Transport Equation, which achieves the Galilean invariance in all the inertial frameworks. The resulting model recovers heat transport characteristics of different scales with respect to both space (from ballistic to diffusive limits) and time (from finite to infinite heat propagation speeds).

In the present study, the dual-phase-lag heat conduction model was employed to describe the thermal behavior of a square plate which is heated by a moving temporally non-Gaussian heat source. The Green's function approach was utilized to develop a general solution for the non-Fourier heat conduction equation in a two-dimensional finite medium. The temperature responses of the medium obtained from Fourier's law, hyperbolic heat conduction model and dual-phase-lag heat conduction model were compared to show the influences of the phase lag parameters. In addition, the effect of scanning speed of heat source on the temperature distribution was discussed. The vector diagrams of heat flux were illustrated to show its propagation.

In this paper, a new and effective radial integration boundary element method (RIBEM) is presented to solve nonlinear heat conduction with temperature dependent thermal conductivity of materials. Boundary-domain integral equation is formulated for nonlinear heat conduction by utilizing the fundamental solutions for the corresponding linear heat conduction, which contains a domain-integral due to the temperature dependence of the thermal conductivity of the materials. Two different approaches based on the radial basis functions are implemented to approximate the unknowns appearing in domain integrals. The arising domain-integral is converted into the equivalent boundary integrals using the radial integration method (RIM), resulting in a pure boundary element analysis algorithm. Newton−Raphson iterative method is applied to solve the final system of algebraic equations after the discretization. Numerical examples are presented to demonstrate the accuracy and the efficiency of the present method.

A number of contributions have been made to model thermal contact problems involving perfect/imperfect thermal interfaces. The Fourier-, Cattaneo-, and Jeffreys -types of thermal flux modes have been exploited to govern the heat transfer processes of adjacent regions. However, most of the existing studies consider using only one type of these thermal flux models to describe the thermal physics of the entire system such that the mathematical model may fail to precisely describe the physics within the multi-domain system, especially, when the mean free paths of different adjacent domains span in a large range with respect to the characteristic dimension of the material. To circumvent this issue, a generalized heat conduction model, termed C- and F-processes heat conduction model, with the consideration of perfect/imperfect thermal interface is proposed and formulated under the expression of a first-order time dependent differential-algebraic equation in which the interface conditions are treated as algebraic equations. The unified time integration of GSSSS-1 is extended to solve the resulting differential-algebraic system with Index 2 constraints. The numerical results illustrate that the proposed framework has a better capacity of describing the thermal contacts with/without the thermal resistance of Fourier-Fourier, Fourier-Cattaneo, and other general cases.

© 2017, Springer-Verlag GmbH Austria. We study the heat conduction with a general form of a constitutive equation containing fractional derivatives of real and complex order. Using the entropy inequality in a weak form, we derive sufficient conditions on the coefficients of a constitutive equation that guarantee that the second law of thermodynamics is satisfied. This equation, in special cases, reduces to known ones. Moreover, we present a solution of a temperature distribution problem in a semi-infinite rod with the proposed constitutive equation.

This article proposes a new nanoscale heat transfer model based on the Caputo type fractional dual-phase-lagging (DPL) heat conduction equation with the temperature-jump boundary condition. The model is proved to be well-posed. A finite difference scheme based on the L1 approximation for the Caputo derivative is then presented for solving the fractional DPL model. Unconditional stability and convergence of the scheme are proved by using the discrete energy method. Three numerical examples are given to verify the accuracy of the scheme. Results show the convergence order to be $$O(\tau ^{2-\alpha }+h^2)$$ , which coincides with the theoretical analysis. A simple nanoscale semiconductor silicon device is illustrated to show the applicability of the model. It is seen from the numerical result that when $$\alpha =1$$ , the fractional DPL reduces to the conventional DPL and the obtained peak temperature is almost identical to those obtained in the literature. However, when $$\alpha

A mathematical theory of constructing integral transforms and formulas of their inversion for the generalized equation of nonstationary heat conduction in an infinite region, bounded from the inside by a plane, a cylindrical, or a spherical surface, and the Green′s function method have been developed. Integral relations for analytical solutions of boundary-value problems in the indicated region are proposed. Illustrative examples are considered.

Size effects are discussed for several non-linear heat conduction models with flux-limited behaviors, including the phonon hydrodynamic, Lagrange multiplier, hierarchy moment, nonlinear phonon hydrodynamic, tempered diffusion, thermon gas and generalized nonlinear models. For the phonon hydrodynamic, Lagrange multiplier and tempered diffusion models, heat flux will not exist in problems with sufficiently small scale. The existence of heat flux needs the sizes of heat conduction larger than their corresponding critical sizes, which are determined by the physical properties and boundary temperatures. The critical sizes can be regarded as the theoretical limits of the applicable ranges for these non-linear heat conduction models with flux-limited behaviors. For sufficiently small scale heat conduction, the phonon hydrodynamic and Lagrange multiplier models can also predict the theoretical possibility of violating the second law and multiplicity. Comparisons are also made between these non-Fourier models and non-linear Fourier heat conduction in the type of fast diffusion, which can also predict flux-limited behaviors.

This paper presents an exact analytical solution for transient anisotropic heat conduction in a truncated composite conical shell for the first time. The fibers of composite conical shell are winded around the body in any arbitrary direction. Heat convection between the ambient fluid flow and composite conical shell is modeled and the exact solution of heat conduction equation is obtained by combining two popular analytical methods. First, by means of an integral transform in angular direction, the heat conduction equation is transformed and after that the transferred heat equation is solved using separation of variables method. Eventually, an inverse transformation is used in order to achieve the final exact solution of heat conduction equation. Validation of the present analytical solution is performed by comparing the analytical results with the solution of second order finite different method. Besides, the capability of present solution in solving the industrial cases is tested. The present solution is applicable in some industrial cases such as cooling pin fins and aeronautical instruments.

Background In this study, a quasi-analytical solution for longitudinal fin and pin heat conduction problems is investigated. Methods The differential transform method, which is based on the Taylor series expansion, is adapted for the development of the solution. The proposed differential transform solution uses a set of mathematical operations to transform the heat conduction equation together with the fin profile in order to yield a closeform series of homogeneous extended surface heat diffusion equation. Results and conclusions The application of the proposed differential transform method solution to longitudinal fins of rectangular and triangular profiles and pins of cylindrical and conical profiles heat conduction problems showed an excellent agreement on both fin temperature and efficiencies when compared to exact results. Therefore, the proposed differential transform method can be useful for optimal design of practical extended surfaces with suitable profile for temperature response.

In polar coordinates, a discrete analog of the conjugate-operator model of a heat conduction problem is formulated to hold the structure of the original model. The difference scheme converges with second-order accuracy in the case of discontinuous parameters of the medium in the Fourier law and irregular grids. An efficient algorithm for solving the discrete conjugate-operator model when heat conduction tensor is a unit operator is proposed.

Variants of the differential equation of heat conduction in a solid body, which follow from the Fourier and Cattaneo–Vernotte hypotheses and the Lykov equation, are considered. A boundary value problem describing temperature fields in a body (cylinder) upon cyclic heat transfer with cold and hot media is formulated. An analytical solution to the boundary value problem with a hyperbolic differential equation of heat conduction with allowance for thermal relaxation and temperature damping with cyclic boundary conditions of the third kind is given. The thermal transient processes calculated by the classical heat conductance equation and hyperbolic equation of heat conduction on the axis of the cylinder at different values of factors such as the ratio of the thermal damping time to the thermal relaxation time, the duration of cyclic periods, the Fourier relaxation number, and the Biot number are compared. A conclusion is made that the theory of regenerative air heater should be improved by taking into account thermal relaxation and thermal damping in the nozzle and measurements of the thermal relaxation and thermal damping times of the corresponding materials.

This paper presents a pure analytical solution of one-dimensional hyperbolic heat conduction equation in a homogeneous finite medium under series of time pulsed heat source which is exponentially distributed and acts symmetrically on both sides. The solution is obtained without any numerical procedures, using the Eigenvalue function. The problem is solved under two types of step and exponential time pulse series functions, which are used in simulation of laser interaction of tissues, and the closed-form solutions are introduced. The ability of the solution to estimate the effect of pulse duration and intensity is investigated. The results can be applied as a verification branch for other numerical solutions such as pulse laser interaction phenomenon.

In this paper, the problem of Non-Fourier heat conduction in a one-dimensional finite body was studied. Non-Fourier heat conduction model of thermal wave was used for thermal analysis of the problem. Temperature-dependent heat conductivity was assumed and a nonlinear equation was obtained. To solve the equations, semi-analytical methods of Reduced Differential Transform Method (RDTM) and Homotopy Perturbation Method (HPM) were applied. It was concluded that nonlinear analysis of Non-Fourier heat conduction problems is highly important and higher accuracy of RDTM with respect to HPM was specified.

Approximate analysis is a major application of variational principles for heat conduction. Recently, O’Toole’s variational principle for Fourier’s law has been extended to non-Fourier heat conduction models, which are applied to approximate analyses based on the Rayleigh–Ritz method. Suitable trial functions satisfying boundary conditions are sought, and then substituted into the variational principles to obtain the undetermined coefficients. From the inverse Laplace transforms, the approximate solutions are obtained. Examples are provided for 1D problems for different heat conduction models. The largest calculation errors are one or two orders of magnitude smaller than the equilibrium temperature, which will tend to be zero.

An analytical discrete-variable model has been developed to describe heat conduction in nano-sized systems. The model assumes that the system consists of a homogeneous array of cells with characteristic size h; each cell interacts with the nearest neighbors in discrete time step τ and all the cells compute their new state simultaneously. In the continuum limit h → 0 and τ → 0 , the model reduces to classical heat diffusion equation of parabolic type or heat conduction equation of hyperbolic type, depending on the choice of scaling invariant. The model is applied to heat conduction in nano-films with emphasis on the transition from the diffusive to ballistic heat transport, which occurs with decreasing film thickness. This model provides a simple method for predicting in a self consistent manner the effective cross-plane thermal conductivities, the temperature jump at the boundaries, the heat flux across the film, and the temperature gradient within the film as functions of the film thickness. The results are in good agreement with molecular dynamic and Monte Carlo simulations.

Vortex is one of the most important characteristics for flow field, but for heat conduction, vortex problems have not been studied. In this paper, heat vortex is mainly investigated from the viewpoint of heat flux rotation. The heat flux rotation is calculated and discussed for Fourier’s law and typical non-Fourier heat conduction models, including the Cattaneo-Vernotte (CV) model, Jeffrey model, Guyer-Krumhansl (GK) model and dual-phase-lagging (DPL) model. It is found that the change rule of heat flux rotation is only determined by the heat conduction law, and the change rule can also reflect certain relaxation process in non-Fourier heat conduction. These conclusions can provide a perspective for proving non-Fourier heat conduction, which is more rigorous than heat wave phenomena or finite propagation velocity. The heat flux rotation can also shows particular characteristics of non-Fourier heat conduction, which is quite different from mechanical phenomena. Different from the heat flux rotation, the entropy flux rotation for non-Fourier heat conduction does not change in an established rule.

In this paper, a new and simple boundary-domain integral equation is presented for solving transient nonlinear heat conduction problems with temperature-dependent conductivity of materials. The boundary-domain integral equation is formulated for transient nonlinear heat conduction problems by using the fundamental solution for the corresponding steady linear heat conduction problems, which results in the appearance of domain integrals. The arising domain integrals are converted into equivalent boundary integrals using the radial integration method (RIM) by expressing the temperature as a series of radial basis functions. This treatment results in a pure boundary element algorithm and requires no internal cells to evaluate the domain integrals. Based on the finite difference technique, an implicit time marching solution scheme is developed for solving the time-dependent system of equations. To solve the final system of algebraic equations, the Newton-Raphson iterative method is applied. Numerical examples are presented to demonstrate the accuracy and efficiency of the present method.

The modeling of heat conduction is considered by letting the time derivative, in the Cattaneo–Maxwell equation, be replaced by a derivative of fractional order. The purpose of this new approach is to overcome some drawbacks of the Cattaneo–Maxwell equation, for instance possible fluctuations which violate the non-negativity of the absolute temperature. Consistency with thermodynamics is shown to hold for a suitable free energy potential, that is in fact a functional of the summed history of the heat flux, subject to a suitable restriction on the set of admissible histories. Compatibility with wave propagation at a finite speed is investigated in connection with temperature-rate waves. It follows that though, as expected, this is the case for the Cattaneo–Maxwell equation, the model involving the fractional derivative does not allow the propagation at a finite speed. Nevertheless, this new model provides a good description of wave-like profiles in thermal propagation phenomena, whereas Fourier’s law does not.

Heat transport may behave as wave propagation when the time scale of processes decreases to be comparable to or smaller than the relaxation time of heat carriers. In this work, a generalized heat transport equation including nonlinear, nonlocal and relaxation terms is proposed, which sums up the Cattaneo–Vernotte, dual-phase-lag and phonon hydrodynamic models as special cases. In the frame of this equation, the heat wave propagations are investigated systematically in nonequilibrium steady states, which were usually studied around equilibrium states. The phase (or front) speed of heat waves is obtained through a perturbation solution to the heat differential equation, and found to be intimately related to the nonlinear and nonlocal terms. Thus, potential heat wave experiments in nonequilibrium states are devised to measure the coefficients in the generalized equation, which may throw light on understanding the physical mechanisms and macroscopic modeling of nanoscale heat transport.

The least action principle is used in various disciplines including linear transport processes. However, non-equilibrium thermodynamic analyses of linear transport processes involve the dot product of the thermodynamic flux and the thermodynamic force which must be the entropy production rate in the linear phenomenological law for such processes; thus, Fourier’s heat conduction law cannot be derived based on the variation of the entropy production rate. A generalized linear phenomenological law for various types of linear transport processes, including heat conduction, mass diffusion, electric conduction and fluid flow in porous medium is introduced here where the dot product of a generalized flux and a generalized force is taken as the action to give the generalized linear phenomenological law. For heat conduction, the entransy dissipation rate, which is the dot product of the heat flux and the negative of the temperature gradient, is taken as the action, and the variation of the entransy dissipation rate then leads to Fourier’s heat conduction law with constant thermal conductivity. Hence, the action of heat conduction process is the entransy dissipation rate rather than the entropy production rate. A nonlinear constitutive relation for the heat conduction with temperature dependent thermal conductivity is then converted to a linear problem by introducing a generalized temperature, which gives a least generalized entransy dissipation principle for nonlinear heat conduction processes. Finally, the least entransy dissipation principle is applied to optimize a one-dimensional heat conduction problem without heat-work conversion as an example where the minimum entropy generation principle is not applicable.

The modeling tasks of thermal effect of power impulse action on a surface of the plate of VК10 (КS) alloy were considered and solved. As modeling tasks for homogeneous equations of parabolic and hyperbolic heat conductions, a wave equation in a cylindrical solid of final sizes with boundary conditions of III kind were chosen. The action of power impulse from an exterior radiant was modeled by sudden appearance of initial high temperature, which spreads on a plate body under the laws expressed by various heat conduction equations, on one of the surface ends of a cylinder. Approaches of temperature fields were received in the form of a series segment of functions from eigenvalues of tasks, gradients of fields were defined. Simultaneous presence in the equation of heat conductivity of private derivatives on time of the first and second usages (the hyperbolic equation), statement of the task for it with boundary conditions of III kind and the entry condition at a cylinder end surface provides two ways (modes) of the problem’s decision, both of diffusion type. For the value of the relaxation time of the heat flux of 10–11 s, the complete cooling of the cylindrical sample (tungsten carbide) in the first mode is minutes, in the second – 10–10 s. It can be concluded that the modes for solving the problem for the hyperbolic heat equation do not correspond to the actual pattern of heat propagation. However, the linear combination of these modes as a solution of the problem preserves the possibility of obtaining a diffusion dynamics adequate to the actual process. Gradients of the temperature field in the solutions of the problems for the parabolic heat conduction equation and the wave equation are in the same order of values. The temperature field of the moving thermal wave for several of its first reflections in experimental samples should be taken into account when evaluating phase transformations and temperature stresses. The results of the theoretical analysis are compared with changes in the microstructure of the near-surface layer of a plate of alloy VK10 (KS), subjected to electric explosive loading by plasma of a titanium foil.

In this paper, a new and simple boundary-domain integral equation is presented to solve nonlinear heat conduction problems with temperature-dependent conductivity of materials. The boundary-domain integral equation is formulated for nonlinear heat conduction problems by using the fundamental solutions for the corresponding linear heat conduction problems, which results in the appearance of a domain integral due to the variation of the heat conductivity with temperature. The arising domain integral is converted into an equivalent boundary integral using the radial integration method (RIM) by expressing the temperature as a series of basis functions. This treatment results in a pure boundary element algorithm and requires no internal cells to evaluate the domain integral. To solve the final system of algebraic equations formed by discretizing the boundary of the problem into boundary elements, the Newton–Raphson iterative method is applied. Numerical examples are presented to demonstrate the accuracy and efficiency of the present method.

The current work aims to investigate the features of an improved heat conduction relation and chemical processes on the unsteady second grade fluid over a stretching surface. The features of unsteady Cattaneo-Christov heat relation is incorporated in the energy equation. The unsteady Cattaneo-Christov heat flux relation is the generalization of Fourier’s laws in which the time space upper-convected derivative is utilized to describe the heat conduction phenomena. Additionally, the mass transfer phenomena is conducted by utilizing heterogeneous-heterogeneous processes. Heterogeneous-homogenous phenomena’s are taking place on the surface of the wall and ambient fluid, respectively. Moreover, these processes are given by isothermal cubic auto-catalator and first order kinetics. The self-similarity transformation is used to transfer the governing PDEs into the ODEs. The resulting problem under consideration is solved analytically by using HAM. The effect of non-dimensional pertinent parameters on the temperature and concentration distribution are deliberated by using graphs and tables. Results show that the temperature profile diminishes for augmented values of the thermal relaxation parameter. In addition to this, it has come to the observation that the liquid temperature is higher for classical Fourier’s law when compared to non- Fourier’s law. Keywords: Second grade fluid, Cattaneo-Christov heat flux model, Heterogeneous-homogenous processes

We obtain solutions for differential equations, describing a broad range of physical problems by the operational method with recourse to inverse differential operators, integral transforms and operational exponent. Generalized families of orthogonal polynomials and special functions are also employed with recourse to their operational definitions. The evolutional type problems for heat transfer in various heat conduction models are studied. Exact analytical solutions for Guyer-Krumhansl hyperbolic heat equation are obtained and compared with those of Fourier and Cattaneo equations. Modelling heat pulse propagation from a laser source is performed in the framework of Fourier, Cattaneo and Guyer-Krumhansl heat transfer models. Compliance of obtained solutions with the maximum principle is studied.

It is well known that the continuum model of Fourier's law of heat conduction violates the relativity theory, admits an instantaneous thermal response, and assumes a quasi-equilibrium thermodynamic condition. Transient heat transport, however, is a non-equilibrium phenomenon with a finite thermal wave speed for applications involving very low temperatures, extremely high temperature gradients, and ballistic heat transfers. Hyperbolic and phase-lag heat conduction models have enabled detection of the finite thermal wave speed in heat transport. To accommodate effects of thermomass and size-dependency of thermophysical properties on nano/microscale heat transport and to remove the theoretical singularity of temperature gradients across the thermal wavefront, a nonlocal, fractional-order, three-phase-lag heat conduction is introduced. The model is capable of simulating heat conduction phenomena in multiple spatio-temporal scales. To confirm the existence of thermal waves in nano/microscale heat transport, a molecular dynamics simulation is implemented for the heat transfer within a nanoscale copper slab. Correlating thermal responses in continuum and atomistic scales sheds light on the effect of length scale, fractional order, and phase-lags in multiscale heat transport. The multiscale simulation is of practical importance for microelectromechanical system design, photothermal techniques, and ultrafast laser-assisted processing of advanced materials.

This paper analyzes a hyperbolic heat conduction problem in processed meat with the non-homogenous initial temperature. This problem is related to an experimental study for the exploration of thermal wave behavior in biological tissue. Because the fundamental solution of the hyperbolic heat conduction model is difficult to be obtained, a modified numerical scheme is extended to solve the problem. The present results deviate from that in the literature and depict that the reliability of the experimentally measured properties presented in the literature is doubtful.

We investigate the generalized partial difference operator and propose a model of it in discrete heat equation in this paper. The diffusion of heat is studied by the application of Newton’s law of cooling in dimensions up to three and several solutions are postulated for the same. Through numerical simulations using MATLAB, solutions are validated and applications are derived.

The classical variational principle does not exist for parabolic and hyperbolic heat conduction equations, which has led to the demand for special variational methods for heat conduction. O’Toole (1967) first used Laplace transforms for the variational principle only for Fourier’s law with the first type of boundary condition. In this paper, the Laplace transform strategy is extended to other parabolic and hyperbolic heat conduction models and other types of boundary conditions. Generalized variational principles are given for heat conduction models including Fourier’s law, the Cattaneo–Vernotte (CV) model, the Jeffrey model, the two-temperature model and the Guyer–Krumhansl (GK) model, based on Laplace transforms. The Laplace transform method transforms the heat conduction equations of these models into linear variational equations whose variational principles are already known. For the three standard types of boundary conditions, these generalized variational principles are strictly equivalent to the heat conduction equations for these models. The Laplace transform method has stronger convergence in infinite temporal domain problems. In physics, the Laplace transform method is understood as replacing the time dimension with the frequency of the temperature change and the rate of the entropy change.

Virtual boundary meshfree Galerkin method (VBMGM) is developed to solve the multi-domain variable coefficient heat conduction problem with heat source. Both the virtual source function on the virtual boundary and heat source within each subdomain are approximate by the radial basis function interpolation. The equation of the proposed technique is derived by one of the weighted residual methods, namely the Galerkin method, for multi-domain variable coefficient heat conduction problem with heat source. Therefore, VBMGM simultaneously possesses the merits of Galerkin method, boundary element method and meshfree method. And the numerical meanings of the weighted values are definite, such as the partial derivative heat source expressions, the temperature, the heat flux and the continuity conditions, in VBMGM for multi-domain variable coefficient heat conduction problem with heat source. The specific and detailed equation of VBMGM is obtained for multi-domain variable coefficient heat conduction problem with heat source. The accuracy and the stability of the proposed technique are validated by two numerical examples for multi-domain variable coefficient heat conduction problem with heat source.

In this work, an efficient numerical method with a high accuracy is proposed for solving the heat conduction problems. In this method, the governing equation of heat conduction in the partial differential equation form is firstly integrated over the small volume around each node point. In the resulting integrals the spatial derivatives of the unknown temperature and heat flux disappear. Then the numerical quadrature is employed to discretize the integrals. Numerical results show that when the same amount of the computer memory and CPU-time is consumed the proposed method can achieve a high accuracy in comparison with the finite volume method (FVM). Furthermore, the proposed method is more accurate than the finite element method (FEM) and boundary element method (BEM) for multi-dimensional heat conduction problems.

The description of heat conduction in microstructured solids is presented in the framework of the dual internal variable approach. One of the internal variables is identified with microtemperature, i.e. the fluctuation of macroscopic temperature due to the inhomogeneity of the body. It is shown that the microstructure influence may result in a hyperbolic heat propagation for the microtemperature. The macroscale heat conduction is described by a parabolic equation which is coupled with the hyperbolic equation for the microtemperature.

This work investigates the performance of P1 method, FVM and SP3 method for 2D combined conduction and radiation heat transfer problem. Results based on the Monte Carlo method coupled with the energy equation are used as the benchmark solutions. Effects of the conduction-radiation parameter and optical thickness are considered. Performance analyses in term of the accuracy of heat flux and temperature predictions and of computing time are presented and analyzed.

In this study, an inverse analysis is presented to determine the time-dependent surface heat flux of a living tissue in a one-dimensional non-Fourier bio-heat conduction problem by using temperature responses measured within the medium. The properties of tissue are assumed to be function of temperature. Therefore, this problem is classified as a nonlinear inverse problem of hyperbolic bio-heat conduction. The inverse problem is solved through the minimization of an objective function by the conjugate gradient method. Three examples are presented to show that the proposed method can be implemented to estimate the unknown surface heat flux. In addition, the effects of measurement error, measurement location, initial guess, nonlinearity and thermal relation time on the precision of the estimation are investigated. The results show that the proposed method is an accurate and robust method to determine inversely the surface heat flux in hyperbolic bio-heat conduction with temperature-dependent properties.

In this paper, a consistent MPS (Moving Particle System Semi-implicit/Moving Particle System Simulation) method is established emanating from Taylor Expansion and Gauss’s Theorem. The proposed MPS method preserves the consistency between the Gradient and Laplacian approximations, and also has the ability to recover the conventional MPS method which lacks consistency. In addition, the proposed method preserves the first order accuracy in space whereas the traditional MPS method loses the accuracy in the case with arbitrary and non-uniform particle discretization. The applicability of the proposed method for the analysis of first- and second-order transient hyperbolic and parabolic systems is illustrated via a recently developed C–F model (Cattaneo–Fourier model) of heat conduction that has been proven to govern thermal transport in different scales, and also capture the various heat conduction physical phenomena. The approaches of imposing different boundary conditions in the heat conduction problems are investigated comprehensively as well. In addition, a unified GS4 i-Integration framework is exploited in the numerical simulations involving first- and second-order systems in time.

Time-nonlocal generalization of the classical Fourier law with the “long-tail” power kernel can be interpreted in terms of fractional calculus (theory of integrals and derivatives of noninteger order) and leads to the time-fractional heat conduction equation with the Caputo derivative. Fractional heat conduction equation with the harmonic source term under zero initial conditions is studied. Different formulations of the problem for the standard parabolic heat conduction equation and for the hyperbolic wave equation appearing in thermoelasticity without energy dissipation are discussed. The integral transform technique is used. The corresponding thermal stresses are found using the displacement potential.

The paper presents a mathematical model of a heat transfer process for non-contacting face seals. For a slewing ring (rotor), a time fractional Fourier law obtained from fractional calculus was applied. Heat exchange in the stator was described using Laplace’s equation. Heat flux generated in the medium layer separating the rings is discharged first to a working ring and the to a surrounding medium. Distribution of temperature fields in sealing rings was determined with the use of analytical methods, using integral transformation method. The fractional heat conduction equation (for the rotor) was considered in the light of the case 0 α ⩽ 2 with the special attention paid to α = 1 for the classical heat conduction equation. The proposed way of solving is very broad and involves many theoretical and practical problems, which were taken into account and described in the work. Finally, numerical results are presented graphically for various values of both operational and geometric parameters and order of fractional derivative.