Numerical Technique for Resolving the Dual Phase Lag Heat Conduction in Thin Film Metal

Abstract

In this article, a three-time level finite difference scheme is used to resolve the dual phase lag’s (DPL) heat conduction in a micro scale gold film subjected to spontaneous temperature boundary conditions without knowing the heat flux. Finite difference analog of DPL equation on applying to the intermediate grid points of the computational domain results into a system of linear, algebraic equations which can be solved using Thomas’ algorithm to finally obtain the transient temperature solution distributions in the film. The solution predicted by the DPL model is compared with that obtained by the single-phase Cattaneo–Vernotte’s model. Further, the way in which non-Fourier’s temperature distributions affected by the diffusion due to the increase in Heat Conduction Model numbers agree with the predecessor’s published results. The results by both the models revealed a finite thermal wave speed in the film contrasting the infinite speed of heat propagation as stated by the classical Fourier’s thermal model. Low spatial step and higher order finite difference schemes are recommended for better accurate numerical results of the non-Fourier’s temperature distributions occurring in the very short transient period between the instants of the suddenly applied spatial temperature gradient and the reaching of the steady state conditions.     

NON-FOURIER HEAT CONDUCTION PHENOMENA IN POROUS MATERIAL HEATED BY MICROSECOND LASER PULSE

Experiments on porous material heated by a microsecond laser pulse and the corresponding theoretical analysis are carried out. Some non-Fourier heat conduction phenomena are observed in the experimental sample. The experimental results indicate that only if the thermal disturbance is strong enough (i.e., the pulse duration is short enough and the pulse heat flux is great enough) is it possible to observe apparent non-Fourier heat conduction phenomenon in the sample, and evident non-Fourier heat conduction phenomenon can only exist in a very limited region around the thermal disturbance position. The hyperbolic heat conduction (HHC) equation and the dual-phase lag (DPL) model are employed, respectively, to describe the non-Fourier heat condution process happening in the experimental sample, and the finite-difference method (FDM) is used to solve them numerically. The numerical solutions show that both the HHC equation and the DPL model can predict the non-Fourier heat conduction phenomenon emerging in the experimental sample qualitatively. Moreover, if τ_q and τ_T are assumed to have suitable values, the theoretical result of the DPL model is more agreeable to the experimental result.     

Performance Evaluation of Two Heat Transfer Models of a Walking Beam Type Reheat Furnace

Two different heat transfer models for predicting the transient heat transfer characteristics of the slabs in a walking beam type reheat furnace are compared in this work. The prediction of heat flux on the slab surface and the temperature distribution inside the slab have been determined by considering thermal radiation in the furnace chamber and transient heat conduction in the slab. Both models have been compared for their accuracy and computational time. The furnace is modeled as an enclosure with a radiatively participating medium. In the first model, the three-dimensional (3D) transient heat conduction equation with a radiative heat flux boundary condition is solved using an in-house code. The radiative heat flux incident on the slab surface required in the boundary condition of the conduction code is calculated using the commercial software FLUENT. The second model uses entirely FLUENT along with a user-defined function, which has been developed to account for the movement of slabs. The results obtained from both models have a maximum temperature difference of 2.25%, whereas the computational time for the first model is 3 h and that for the second model is approximately 100 h.     

Inverse Transient Heat Conduction Problems of a Multilayered Functionally Graded Cylinder

Inverse transient heat conduction problems of a multilayered functionally graded (FG) cylinder are presented. The approach is based on measurement of temperature on the outer surface of the cylinder to estimate the heat flux and convection heat transfer coefficient on its inner surface. The non-Fourier heat transfer equation is employed to accurately formulate the problem. The conjugate gradient method (CGM) is used for the optimization procedure and the incremental differential quadrature method (DQM) is applied to solve the direct, sensitivity, and adjoint problems. The accuracy of the presented approach is examined by simulating the exact and noisy data through different examples. Good accuracy of the obtained results validates the presented approach.     

Thermodynamic stability analysis of non-Fourier model-based bio-heat transfer in laser-irradiated cylindrical-shaped multilayered biological tissue

The present research focuses on examining the thermic response of living tissue in the form of a triple-layered cylindrical structure when subjected to laser light and the compatibility analysis of non-Fourier heat transfer with thermodynamics second law. The temperature field in the triple-layered cylindrical living tissue subjected to laser light is determined by numerically solving the transient radiative transfer equation in conjunction with the dual phase lag (DPL) based bioheat equation. Once the temperature field is known, the equilibrium and nonequilibrium entropy production rate (EPR) is calculated based on the hypothesis of classical irreversible thermodynamics and extended irreversible thermodynamics, respectively. The present results are verified against the data from the literature and found a good match between them. A comparative analysis of the Fourier and non-Fourier models is accomplished. The equilibrium and nonequilibrium EPR values for the Fourier model are found to be positive. While the equilibrium EPR is negative for non-Fourier heat conduction and does not satisfy the thermodynamics second law, nonequilibrium EPR is always a positive value for Fourier, DPL, and hyperbolic models and satisfies thermodynamics second law. It has been investigated how thermal relaxation times affect the temperature field and EPRs in tissue are subjected to laser light.     

Numerical Simulation of Thermal Damage to Living Biological Tissues Induced by Laser Irradiation Based on a Generalized Dual Phase Lag Model

A generalized dual phase lag (DPL) bioheat model based on the nonequilibrium heat transfer in living biological tissues is applied to investigate thermal damage induced by laser irradiation. Comparisons of the temperature responses and thermal damages between the generalized and classical DPL bioheat model, derived from the constitutive DPL model and Pennes bioheat equation, are carried out in this study. It is shown that the generalized DPL model could predict significantly different temperature and thermal damage from the classical DPL model and Pennes bioheat conduction model. The generalized DPL equation can reduce to the classical Pennes heat conduction equation only when the phase lag times of temperature gradient (τ _T ) and heat flux vector (τ _q ) are both zero. The effects of laser parameters such as laser exposure time, laser irradiance, and coupling factor on the thermal damage are also studied.     

阶跃热流加热下两接触平板间的非Fourier传热研究

建立两接触平板在阶跃热流加热下的非Fourier导热模型,研究了两平板间的接触热阻及平板的导热系数、导温系数、松弛时间对非Fourier传热过程的影响,指出各平板内热波的传递具有反射性、波幅衰减性和瞬时性的特点。     

Notable physical anomalies manifested in non-Fourier heat conduction under the dual-phase-lag model

In this study, a number of notable physical anomalies concerning non-Fourier heat conduction under the dual-phase-lag (DPL) model are observed and investigated. It is found that, during the transient heat transfer process, the over-diffusion mode predicts a “hyper-active” to “under-active” transition in thermal behavior. The main cause behind it lies in the time-varying effect of τ_T (the phase lag of the temperature gradient) on the thermal response. Also, change of polarity in reflected thermal waves can be observed when a constant-temperature boundary is involved, which hints that a heating process may be followed by a spontaneous cooling effect. A fairly strong connection is present between the τ_T-induced dispersive effect and an unusual thermal accumulation phenomenon in an on–off periodic heating process. Furthermore, a paradox involving a moving medium is detected in the DPL model, which can be solved by replacing the temporal partial derivatives in the DPL equation with the material derivatives. During the process of analysis, a high-order characteristics-based TVD scheme is relied on to provide accurate and reliable numerical simulations to the DPL heat conduction equation under various initial-boundary conditions.     

DPL Model Analysis of Non—Fourier Heat Conduction Restricted by Continuous Boundary Interface

IntroductionAs widely known, the hahonal Fourier law isbased on a large quantity of regular heat transfer (i.e. thethermal bine scale is comparatively lOng and the heatflux density is comparatively small) experiments and it'sjust a phenomenological descriphon of regular thermalProcesses. The Fourier law itself mpes an infinitespeed of Propagation of thermal distUrbance, indicatingthat a local change in tempera~ causes aninstantaneous per'tUrbation in the temperatore at eachPOint in the medi…     

抗事故包装箱结构中木材层的非傅立叶热传导

建立了抗事故包装箱结构中防火缓冲层材料的一维非傅立叶导热模型,模型考虑了材料的热解作用。通过数值模拟得到了有意义的结果:对一定条件下的薄木板内导热,存在一个发生非傅立叶导热效应的临界边界温度值;给定条件下的临界边界温度值为984K。给出了防火缓冲层材料的选取原则,这对抗事故包装箱结构设计具有一定的指导意义。     

Application of CESE method to simulate non-Fourier heat conduction in finite medium with pulse surface heating

This study employs the space–time conservation element and solution element (CESE) method to simulate the temperature and heat flux distributions in a finite medium subject to various non-Fourier heat conduction models. The simulations consider three specific cases, namely a single phase lag (SPL) thermal wave model with a pulsed temperature condition, a SPL model with a surface heat flux input, and a dual phase lag (DPL) thermal wave model with an initial deposition of thermal energy. In every case, the thermal waves are simulated with respect to time as the thermal wave propagates through the medium with a constant velocity. In general, the simulation results are found to be in good agreement with the exact analytical solutions. Furthermore, it is shown that the CESE method yields low numerical dissipation and dispersion errors and accurately models the propagation of the wave form even in its discontinuous portions. Significantly, compared to traditional numerical schemes, the CESE method provides the ability to model the behavior of the SPL thermal wave following its reflection from the boundary surface. Further, a numerical analysis is performed to establish the CESE time step and mesh size parameters required to ensure stable solutions of the SPL and DPL thermal wave models, respectively.     

A new numerical method to simulate the non-Fourier heat conduction in a single-phase medium

Many non-equilibrium heat conduction processes can be described by the macroscopic dual-phase lag model (DPL model). In this paper, a numerical method, which combines the dual reciprocity boundary element method (DRBEM) with Laplace transforms, is constructed to solve such mathematical equation. It is used to simulate the non-Fourier phenomenon of heat conduction in a single-phase medium, then numerically predict the differences between the thermal diffusion, the thermal wave and the non-Fourier heat conduction under different boundary conditions including pulse for one- and two-dimensional problems. In order to check this numerical method's reliability, the numerical solutions are still compared with two known analytical solutions.     

Treatment of the interfacial temperature jump condition with non-Fourier heat conduction effects

Transient energy transport with non-Fourier heat conduction effects in a two-layer structure of dissimilar materials subject to ultra-fast laser heating is studied using the proper interfacial temperature jump condition. The solution obtained is compared with solutions available in literature that use diffusion-type interfacial conditions in conjunction with non-Fourier heat conduction effects. The dual-phase lag heat conduction model is used in this work as it includes both the temporal and spatial non-Fourier effects. It is found that the diffusion-type interfacial temperature jump condition with non-Fourier heat conduction models can lead to discrepancies and erroneous trends in theoretical predictions. Moreover, a comparison between the functional forms of the two solutions obtained utilizing both interfacial conditions shows that implementing the proper interfacial temperature jump condition does not add any complexity to the solution obtained. This study – implementing the proper interfacial temperature jump condition – is further extended to show the strong effects of the thermal contact conductance and the surface layer thickness on the transient thermal response of a two-layer material in a semi-infinite domain subject to ultra-fast laser heating processes in terms of the reflectivity change of the surface layer, the temperature distribution in the two-layer structure as well as the temporal variation of the interfacial temperature difference.     

Transient hygrothermoelastic response in a porous cylinder subjected to ramp-type heat-moisture loading

Abstract

The classical theory of heat conduction (Fourier theory) predicts an infinite speed for thermal disturbance propagation, which is physically unrealistic. By extending the classical Fourier heat conduction and Fick’s diffusion, this article develops hyperbolic diffusion/heat conduction laws with phase lags of heat/moisture flux to simulate coupled heat-moisture diffusion-propagation behavior with the Defour and Soret effects. A porous cylinder subjected to a ramp-type heating and humidifying at the surface is studied. The Laplace transform is used to obtain a closed-form solution of the temperature, moisture, displacements and stresses in the cylinder. Numerical results are calculated via the inversion of the Laplace transform. Obtained results show that the thermal/moisture relaxation time or phase lag plays a significant role in affecting transient hygrothermoelastic field. For a non-vanishing phase lag, non-Fourier and non-Fickian effects exist and hygrothermal waves have finite propagation speeds. The influences of the phase lag of heat/moisture flux and ramp-type time parameter on the transient response of hygrothermoelastic field are presented graphically. A comparison of the numerical results based on the classical model and the present one is made. The non-Fourier heat conduction and non-Fickian diffusion can effectively avoid the shortcomings induced by the classical Fourier and Fick laws.     

Effect of boundary phonon scattering on Dual-Phase-Lag model to simulate micro- and nano-scale heat conduction

In this paper, one-dimensional heat conduction within a thin slab for Knudsen numbers more than 0.1 is implemented using the Dual-Phase-Lag (DPL) model including phonon scattering boundary condition. The Dual-Phase-Lag equation is solved with a stable and convergent finite difference scheme. Also the Laplace transformation technique is employed to solve DPL equation analytically. The results show that in the smaller values of Knudsen number, the results of the DPL model lay very close to the solution of the Boltzmann equation. Also, it is shown that moving towards the steady state, the DPL model reduces to the Cattaneo and Vernotte (CV) model and has results more accurate than the Ballistic-Diffusive Equations (BDE). It is also shown that the temperature distribution is closer to the results of Boltzmann equation relative to the heat flux distribution. Due to the simplicity of derivation of the DPL model formulation and its possibility for developing to higher dimensions, using the DPL model with new boundary condition is recommended to simulate nano- and micro-scale heat conduction. To investigate the accuracy of the DPL model, its results are compared with the results obtained from BDE model, and Boltzmann equation.     

An axisymmetric bioheat transfer analysis through a unified model

A unified model is developed for the analysis of heat transfer (radiation and non-Fourier conduction) in an axisymmetric participating medium. The proposed model includes three different variants of hyperbolic–parabolic heat conduction models, that is, the single phase lag model, dual phase lag model, and the Fourier (no phase lag) model. The radiating-conducting medium is radiatively absorbing, emitting, and isotropically scattering. Significance of all the above mentioned models on the heat transfer characteristics is investigated in a two-dimensional axisymmetric geometry. The equation of transfer and the coupled non-Fourier conduction-radiation equation are solved via finite volume method. A fully implicit scheme is used to resolve the transient terms in the energy equation. For spatial resolution of radiation information, the STEP scheme is applied. Tri-diagonal-matrix-algorithm is used to solve the resulting set of linear discrete equations. Effects of two important influencing parameters: the scattering albedo and the radiation- conduction parameter are studied on the temporal evolution of temperature field in the radiatively participating medium. The non-Fourier effect of heat transport captured well with the proposed unified model. A good agreement can be found between the proposed model predictions and those available in the literature. It is also found that when the phase lag of the temperature gradient and the heat flux are the same, it reduces to conventional Fourier conduction-radiation and the wave behavior diminishes. However, the reduction to this Fourier model fails in the presence of constant blood perfusion and metabolic heat generation.     

A heat transfer model for the analysis of transient heating of the slab in a direct-fired walking beam type reheating furnace

A mathematical heat transfer model for the prediction of heat flux on the slab surface and temperature distribution in the slab has been developed by considering the thermal radiation in the furnace chamber and transient heat conduction governing equations in the slab, respectively. The furnace is modeled as radiating medium with spatially varying temperature and constant absorption coefficient. The steel slabs are moved on the next fixed beam by the walking beam after being heated up through the non-firing, charging, preheating, heating, and soaking zones in the furnace. Radiative heat flux calculated from the radiative heat exchange within the furnace modeled using the FVM by considering the effect of furnace wall, slab, and combustion gases is introduced as the boundary condition of the transient conduction equation of the slab. Heat transfer characteristics and temperature behavior of the slab is investigated by changing such parameters as absorption coefficient and emissivity of the slab. Comparison with the experimental work show that the present heat transfer model works well for the prediction of thermal behavior of the slab in the reheating furnace.     

Estimation of the transient surface temperature and heat flux of a steel slab using an inverse method

In the steel industry it is of great importance to be able to control the surface temperature and heating- or cooling rates during heat treatment processes. An experiment was performed in which a steel slab was heated up to 1250 °C in a fuel fired test furnace. The transient surface temperature and heat flux of a steel slab is calculated using a model for inverse heat conduction. That is, the time dependent local surface temperature and heat flux of a slab is calculated on the basis of temperature measurements in selected points of its interior by using a model of inverse heat conduction. Time- and temperature histories were measured at three points inside a steel slab. Measured temperature histories at the two lower locations of the slab were used as input to calculate the temperature at the position of the third location. A comparison of the experimentally measured and the calculated temperature histories was made to verify the model. The results showed very good agreement and suggest that this model can be applied to similar applications in the Steel industry or in other areas where the target of investigation for some reason is inaccessible to direct measurements.     

Inverse Hyperbolic Heat Conduction in Fins with Arbitrary Profiles

This study aims to estimate unknown base temperature distribution in different non-Fourier fins. The Cattaneo–Vernotte (CV) heat model is used to predict the heat conduction behavior in these fins. This inverse problem is solved by the function-estimation version of the Adjoint conjugate gradient method (ACGM) based on boundary temperature measurements. The ACGM includes direct, sensitivity, and adjoint problems. For each of these problems, a one-dimensional general formulation of the non-Fourier model for longitudinal fins with arbitrary profile is driven and solved by an implicit finite difference method. In this study, three different profiles are considered: triangular, convex parabolic, and concave parabolic. For each of them, two different base temperature distributions are estimated using an inverse method. Moreover, the effects of sensor positions at the fin tip and a specific place in-between are considered on the base temperature estimation. A close agreement between the exact values and the estimated results is found, confirming the validity and accuracy of the proposed method. The results show that the ACGM is an accurate and stable method to determine the thermal boundary conditions in different non-Fourier fin problems.     

Numerical Simulation of Dual-Phase-Lag Model and Inverse Fractional Single-Phase-Lag Problem for the Non-Fourier Heat Conduction in a Straight Fin

In recent years,many studies have been done on heat transfer in the fin under unsteady boundary conditions using Fourier and non-Fourier models.In this paper,unsteady non-Fourier heat transfer in a straight fin having an internal heat source under periodic temperature at the base was investigated by solving numerically Dual-Phase-Lag and Fractional Single-Phase-Lag models.In this way,the governing equations of these models were presented for heat conduction analysis in the fin,and their results of the temperature distribution were validated using the theoretical results of Single and Dual-Phase-Lag models.After that,for the first time the order of fractional derivation and heat flux relaxation time of the fractional model were obtained for the straight fin problem under periodic temperature at the base using Levenberg-Marquardt parameter estimation method.To solve the inverse fractional heat conduction problem,the numerical results of Dual-Phase-Lag model were used as the inputs.The results obtained from Fractional Single-Phase-Lag model could predict the fin temperature distribution at unsteady boundary condition at the base as well as the Dual-Phase-Lag model could.