Hyperbolic Heat Conduction Model Research Articles

Application of hyperbolic heat conduction model in thermal lens spectroscopy

Between the techniques directed to studying the thermal properties of materials, we can cite the photothermal methods, including the thermal lens technique in this group. The technique of thermal lens spectroscopy has, in most cases, its temperature profile described by Fourier’s Law of heat conduction. In this work, the temperature distribution was successfully represented through the hyperbolic heat conduction model. However, with the application of the hyperbolic model, the differential equation that describes the problem has no analytical solution; therefore, we used two techniques for their solution. The method of lines (purely numerical method) and the generalized integral transform technique (analytical-numerical method). Both approaches have excellent rates of convergence in their solutions. They also present the new agreement proposed solution with experimental data, thus, providing a new perspective for future work in this spectroscopic technique.

Finite element method for hyperbolic heat conduction model with discontinuous coefficients in one dimension

Finite element method for hyperbolic heat conduction model with discontinuous coefficients in one dimension

A Non-Fourier Bioheat Transfer Model for Cryosurgery of Tumor Tissue with Minimum Collateral Damage

Background and objectivesIncorporation of non-Fourier heat conduction while studying heat transfer phenomena in biological materials has emerged has an important approach as it predicts better and more realistic results than Fourier based models. In this article we have proposed a non-Fourier computational model and applied the same to simulate cryosurgery of lung tumor and attempted minimization of freezing damage of healthy lung tissue using pulsed laser irradiation. MethodsA non-Fourier bioheat transfer model for phase change in biological tissues is solved via a Fourier heat conduction based solution approach. A unified model is proposed combining all variants of bioheat models: Fourier's heat conduction based Pennes’ bioheat model, hyperbolic heat conduction model and dual phase lag model. The proposed model takes into account the different thermophysical properties of frozen and unfrozen regions. In order to mimic the actual biotransport process, the blood perfusion and metabolic heat generation are switched off in the frozen region. Implicit source based enthalpy method is used to model phase change process. A new iterative enthalpy update equation is developed for capturing evolution of freezing front implicitly. Finite Volume based numerical discretization technique is used to discretize the governing PDE. The resulting discrete algebraic equation set is solved implicitly by Tri-diagonal Matrix Algorithm. The proposed model is verified with existing results from the literature. ResultsFor Fourier heat conduction, freezing time of 99.99% of tumor is 1247s, which increases to 1267s for τq= 5s (τT= 0s) and again reduces to 1255s for τq= 5s and τT= 3s. τq and τT are phase lag parameters for non-Fourier heat conduction. For τq= 5s and τT= 0.05s, the freezing damage of healthy tissue decreases by 23.76% when pulsed laser irradiation (Io = 106 W/m2) is used to warm the neighboring healthy tissue. ConclusionsSo non-Fourier bioheat transport models are better and more accurate in predicting temperature history, freezing time and freezing front propagation as compared to Fourier based models. Pulsed laser irradiation can prove to be a very efficient technique in minimizing collateral damage during cryosurgery.

Particle–Substrate Transient Thermal Evolution During Cold Spray Deposition Process: A Hybrid Heat Conduction Analysis

A hybrid analytical heat conduction model was developed to predict the transient thermal evolution at the particle–substrate interface during the deposition of cold spray process. First, three-dimensional heat conduction model based on classical diffusion approach was developed to predict the transient surface temperature of the substrate. To that end, the traveling wave solution technique was utilized in order to take into account the effect of the movement of the cold spray nozzle. The results of the analytical model were validated with the experimentally measured surface temperature which was obtained by employing a low-pressure cold spray unit to generate the impingement of a compressed air jet on a flat substrate. The analytical model was further utilized to investigate the effect of the non-dimensional characteristic velocity of the traveling heat source on the surface temperature profile of the substrate. It was found that both the maximum surface temperature and the spatial variation of surface temperature profile of the substrate decreased as the non-dimensional characteristic velocity increased. The mathematical model was further extended by developing a one-dimensional hyperbolic (non-Fourier) heat conduction model to predict the temperature rise at the particle–substrate interface during the cold spray deposition process. In order to validate the results of the hybrid model, a three-dimensional finite element model was developed in ABAQUS to simulate the thermal and dynamic behavior of a single particle upon impact. The results of the hybrid analytical model for the temperature at the particle–substrate interface were compared to the results of the numerical model, and good agreement was found. It was concluded that by coupling the classical diffusion theory and hyperbolic heat conduction approach, the proposed hybrid analytical model can be utilized to predict the transient temperature of the particle–substrate interface during the cold spray deposition process a priori before experimentation.

A thermodynamic analysis of an enhanced theory of heat conduction model: Extended influence of finite strain and heat flux

An Eulerian form of thermodynamically consistent generalized thermoelastic model capable of accounting for thermal signals is proposed. The heat flux is assumed to consist of both energetic and dissipative components. The constitutive relations for the stress, entropy and the heat fluxes are derived in the spatial coordinate system. Later on, a domain of dependence inequality for the proposed hyperbolic type heat conduction model is proved. The linearised form of this improved thermoelasticity theory of finite deformation is employed to study the thermoelastic interactions due a continuous source of heat in isotropic elastic solids. This newly developed model is also applied in a thermoelastic saturated porous medium. In order to obtain the exact analytical expressions of the field functions, appropriate integral transformations are employed in a convenient way.

Extensional and flexural modes of Rayleigh–Lamb wave in an orthotropic thermoelastic layer lying over a viscoelastic half-space

In this article the characteristics of the extensional and flexural modes, propagating in a thermoelastic orthotropic layer lying over a viscoelastic half-space, are analyzed. The complete analysis is carried out in the framework of a thermodynamically consistent hyperbolic type heat conduction model without energy dissipation. The normal-mode-analysis is adopted and a general form of dispersive equation is derived for an anisotropic thermoelastic layered medium. A prominent distinction with the isotropic elastic solids is observed in the symmetric as well as anti-symmetric modes of dispersion curves. In turn, such deformation reshapes the wave propagation while the deformation stiffening changes significantly the phase velocities of the wave till the acoustic radiation stresses are balanced by elastic stresses in the current configuration of the hyperelastic medium.

Convergence of finite element methods for hyperbolic heat conduction model with an interface

The paper concerns numerical study of non-Fourier bio heat transfer model in multi-layered media. Specifically, we employ the Maxwell–Cattaneo equation on the physical media with discontinuous coefficients. A fitted finite element method is proposed and analyzed for a hyperbolic heat conduction model with discontinuous coefficients. Typical semidiscrete and fully discrete schemes are presented for a fitted finite element discretization with straight interface triangles. The fully discrete space–time finite element discretizations are based on second order in time Newmark scheme. Optimal a priori error estimates for both semidiscrete and fully discrete schemes are proved in L∞(L2) norm. Numerical experiments are reported for several test cases to confirm our theoretical convergence rate. Finite element algorithm presented here can be used to solve a wide variety of hyperbolic heat conduction models for non-homogeneous inner structures.

Dual-phase-lag heat conduction in the composites by introducing a new application of DQM

The first application of the differential quadrature method (DQM) in solving the nonlinear dual-phase-lag (DPL) heat conduction equation is demonstrated here. To show the effect of DPL parameters, the temperature response of the medium was obtained from Fourier’s low, hyperbolic heat conduction and hyperbolic type DPL heat conduction model were compared. Furthermore, the transient temperature and resultant heat flux distributions have been found for various types of dynamic thermal loading. We show whether thermal waves exist or not in hyperbolic type DPL heat conduction by considering the time lag parameter in the microstructural interactions of fast transient heat conduction. Also, overshooting which is one of the hyperbolic heat conduction is investigated here. The numerical solution at each time level depends on the solutions at its previous levels. This means the temperature and heat flux obtained at the time nth are the initial conditions for the time (n + 1)th. After demonstrating the convergence and accuracy of the method, the effects of different parameters on the temperature and heat flux distribution of the medium are studied.

Thermoelastic Response of a Rectangular Plate under Hyperbolic Heat Conduction Model in Differential Transform Domain

In the present work, a semi-analytical solution is presented for the thermoelastic response of a finite plate of rectangular geometry considering hyperbolic heat conduction model. The solution of thermoelastic displacement, thermal stresses and temperature are obtained using differential transform method under hyperbolic, non-Fourier heat conduction theory. For special case, thermal stresses and displacement functions are determined numerically and plotted graphically to analyze the effect of the thermal relaxation time.

Investigation of the thermal-elastic problem in cracked semi-infinite FGM under thermal shock using hyperbolic heat conduction theory

In this paper, a thermoelastic analytical model is established for a functionally graded half-plane containing a crack under a thermal shock in the framework of hyperbolic heat conduction theory. The moduli of functionally graded materials (FGMs) are assumed to vary exponentially with the coordinates. By employing the Fourier transform and Laplace transform, coupled with singular integral equations, the governing partial differential equations under mixed, thermo-mechanical boundary conditions are solved numerically. For both the temperature distribution and transient stress intensity factors (SIFs) in FGMs, the results of hyperbolic heat conduction model are significantly different than those of Fourier’s Law, which should be considered carefully in designing FGMs.

Dual-phase-lag heat conduction in FG carbon nanotube reinforced polymer composites

The numerical simulation of non-Fourier, dual-phase-lag (DPL) heat conduction in carbon nanotube (CNT) reinforced composites has been performed by developing the differential quadrature method (DQM) application. Although using the non-Fourier heat conduction has become popular, most of the simulations are conducted over simple geometries due to numerical restrictions. DQM copes with this problem easily even with complicated boundary conditions. The proposed method discretizes the time domain in the form of blocks. Each block contains several time levels, and each block needs to solve separately by using the output of one block as an initial condition of the next block. The effect of volume fraction of CNT on the temperature and heat flux profile is investigated, and all parameters are considered temperature dependent. To show the influence of time lag parameters, the Fourier, hyperbolic heat conduction and DPL heat conduction models were compared. Furthermore, various types of dynamic thermal loading have been examined to obtain the transient temperature distributions. We show the presence of heat wave in DPL heat conduction by considering the time lag parameter in the microstructural interactions of fast transient heat conduction. It is found that considering the mechanical properties of the CNTs dependent to the temperature is crucial to obtain accurate results.

Three mathematical representations and an improved ADI method for hyperbolic heat conduction

Hyperbolic heat conduction models have been proposed to characterize the breakdown of Fourier’s law, i.e. thermal waves. In this paper, three mathematical representations for hyperbolic heat conduction, namely temperature representation, hybrid representation and heat flux representation, and their corresponding characteristics are first analyzed. The hybrid representation is demonstrated to be preferable for numerical calculations and contains sufficient heat transport information. Specifically designed for solving transient heat conduction problems in the hybrid representation, an improved alternative direction implicit (ADI) method based on staggered grids is then developed. This algorithm focuses on the entire hyperbolic equation set instead of one single hyperbolic equation, and it adopts chasing method rather than iteration, which enables to significantly save computing time and storage space. Characteristics analyses on the definite conditions show that for each side only one boundary condition is necessary for Cattaneo-Vernotte (CV) type hyperbolic heat conduction. The advantages of the hybrid representation are also demonstrated by numerical simulations. Besides, the initial heat flux, which implies the initial phonon momentum in dielectrics, has an important influence on the propagation patterns of thermal waves, changing the way of energy conveying. The mechanism of phonon momentum conservation leads to the vector characteristics of thermal waves, and causes the direction preference in hyperbolic heat conduction.

Heating Effects Induced in a Finite Silver Selenide Slab by a Modelled Laser Internal Source Using the Hyperbolic Heat Conduction (HHCE) Model in Dimensionless Domain

Lasers of high power densities are useful for a variety of material processing techniques. Laser heating of a finite homogeneous Silver Selenide slab is studied according to the hyperbolic heat conduction model. Laplace Integral transform technique is used to get the solution. This material suffers phase transition from semiconductor to metallic phase at 403 K. It has vital technological applications. The obtained temperature field makes it possible to determine the time required to initiate phase transition or melting. The functional dependence of the obtained functions is revealed. Different laser power densities are considered as illustrative examples.

A model of femtosecond laser ablation of metal based on dual-phase-lag model

Femtosecond laser ablation possesses a variety of applications due to its better control, high power density, smaller heat-affected zone, minimal collateral material damage, lower ablation thresholds, and excellent mechanical properties. The non-Fourier effect in heat conduction becomes significant when the heating time becomes extremely small. In order to analyze the femtosecond laser ablation process, a hyperbolic heat conduction model is established based on the dual-phase-lag model. Taken into account in the model are the effect of heat source, laser heating of the target, the evaporation and phase explosion of the target material, the formation and expansion of the plasma plume, and interaction of the plasma plume with the incoming laser. Temperature-dependent optical and thermophysical properties are also considered in the model due to the fact that the properties of the target will change over a wide range in the femtosecond laser ablation process. The effects of the plasma shielding, the ratio of the two delay times, and laser fluence are discussed and the effectiveness of the model is verified by comparing the simulation results with the experimental results. The results show that the plasma shielding has a great influence on the femtosecond laser ablation process, especially when the laser fluence is high. The ratio between the two delay times (the ratio <i>B</i>) has a great influence on the temperature characteristic and ablation characteristic in the femtosecond laser ablation process. The augment of the ratio <i>B</i> will increase the degree of thermal diffusion, which will lower down the surface temperature and accelerate the ablation rate after the ablation has begun. The ablation mechanism of femtosecond laser ablation is dominated by phase explosion. The heat affected zone of femtosecond laser ablation is small, and the heat affected zone is less affected by laser fluence. The comparison between the simulation results and the experimental results in the literature shows that the model based on the dual-phase-lag model can effectively simulate the femtosecond laser ablation process.

Hyperbolic heat conduction based weight function method for thermal fracture of functionally graded hollow cylinders

This article introduces a weight function method for fracture analysis of a circumferentially cracked functionally graded hollow cylinder subjected to thermal loads. The non-Fourier hyperbolic heat conduction model is used to determine the transient temperature distribution in the functionally graded cylinder. Solutions for transient temperature and stress distributions in the uncracked cylinder are derived by converting the governing differential equations into Fredholm integral equations in the Laplace domain. A numerical Laplace inversion technique is used to calculate the wave-like temperature and stress solutions in the time domain. These solutions are utilized to determine stresses acting on the faces of the circumferential crack in the local perturbation problem. A weight function technique is developed to compute the corresponding mode I thermal stress intensity factors. Comparisons to the results generated by finite difference and finite element methods demonstrate the high level of accuracy attained by the application of the developed procedures. Further parametric analyses are presented to illustrate the influences of dimensionless time, crack depth to thickness ratio, power law index, and thermal relaxation time upon the stress intensity factors.

An enhanced explicit technique for the solution of non-Fourier heat transfer problems

In this work, an enhanced explicit technique is proposed to analyze hyperbolic heat conduction models. As usual, the explicit approach allows the solution of the problem to be carried out without dealing with any system of equations, featuring a very efficient methodology. In addition, the proposed technique enables algorithmic dissipation, allowing the influence of spurious high modes to be properly eliminated, without introducing significant period elongation and amplitude decay errors into the analysis. As an explicit approach, the technique is conditionally stable; however, it exhibits high stability limits (its critical time-step is around 1.8 times that of the Central Difference Method), emphasizing its effectiveness. The technique is very accurate, truly self-starting and extremely direct to implement. At the end of the manuscript, numerical results are presented, illustrating the good performance of the discussed technique.

Pulsed Laser Heating of a Finite Silver Selenide Slab Using (HHCE) Model

Heat induced in a finite Silver Selenide slab by an external pulsed laser source is studied in dimensionless scale according to hyperbolic heat conduction model (HHCE) using Laplace integral transform technique. The temperature profile, the critical time required to initiate phase transition and that to initiate damage at the front surface are obtained for different pulses and are illustrated graphically.

Analytical solution of dual-phase-lag heat conduction in a finite medium subjected to a moving heat source

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.

Transient thermal stress intensity factors for a circumferential crack in a hollow cylinder based on generalized fractional heat conduction

A generalized fractional heat conduction theory is applied to investigate the transient thermal fracture problem of a hollow cylinder with an embedded or surface circumferential crack. Integral transform technique is used to solve an associated initial-boundary value problem. Explicit temperature field and thermal stresses are given in the Laplace transform domain for a circumferentially cracked hollow cylinder subjected to thermal shock at the inner surface and with an insulated outer surface. Numerical results in the time domain are obtained by using numerical inversion of the Laplace transform. Transient thermal stresses induced by a crack are determined and thermal stress intensity factors at the crack front are calculated. The effects of fractional order, phase lag of heat flux on the transient temperature field, thermal stresses and thermal stress intensity factors are illustrated graphically for internal and surface cracks. The obtained results based on non-Fourier law of fractional heat conduction are compared with those using the classical Fourier law and hyperbolic heat conduction models, respectively.

Finite integral transform-based analytical solutions of dual phase lag bio-heat transfer equation

• Analytical solution of the non-Fourier bio heat transfer equation. • Finite integral transform. • Time independent/dependent boundary conditions. • Coupling of the transient RTE and non-Fourier heat conduction models. • Thermal analysis of laser-irradiated biological tissue phantoms. A finite integral transform (FIT)-based analytical solution to the dual phase lag (DPL) bio-heat transfer equation has been developed. One of the potential applications of this analytical approach is in the field of photo-thermal therapy, wherein the interest lies in determining the thermal response of laser-irradiated biological samples. In order to demonstrate the applicability of the generalized analytical solutions, three problems have been formulated: (1) time independent boundary conditions (constant surface temperature heating), (2) time dependent boundary conditions (medium subjected to sinusoidal surface heating), and (3) biological tissue phantoms subjected to short-pulse laser irradiation. In the context of the case study involving biological tissue phantoms, the FIT-based analytical solutions of Fourier, as well as non-Fourier, heat conduction equations have been coupled with a numerical solution of the transient form of the radiative transfer equation (RTE) to determine the resultant temperature distribution. Performance of the FIT-based approach has been assessed by comparing the results of the present study with those reported in the literature. A comparison of DPL-based analytical solutions with those obtained using the conventional Fourier and hyperbolic heat conduction models has been presented. The relative influence of relaxation times associated with the temperature gradients ( τ T ) and heat flux ( τ q ) on the resultant thermal profiles has also been discussed. To the best of the knowledge of the authors, the present study is the first successful attempt at developing complete FIT-based analytical solution(s) of non-Fourier heat conduction equation(s), which have subsequently been coupled with numerical solutions of the transient form of the RTE. The work finds its importance in a range of areas such as material processing, photo-thermal therapy, etc.