non-Fourier Heat Conduction Model Research Articles

Modeling heat conduction with dual-dissipative variables: A mechanism-data fusion method.

Many macroscopic non-Fourier heat conduction models have been developed in the past decades based on Chapman-Enskog, Hermite, or other small perturbation expansion methods. These macroscopic models have achieved great success in capturing non-Fourier thermal behaviors in solid materials, but most of them are limited by small Knudsen numbers and incapable of capturing highly nonequilibrium or ballistic thermal transport. In this paper, we provide a different strategy for constructing macroscopic non-Fourier heat conduction modeling, that is, using data-driven deep-learning methods combined with nonequilibrium thermodynamics instead of small perturbation expansion. We present the mechanism-data fusion method, an approach that seamlessly integrates the rigorous framework of conservation-dissipation formalism (CDF) with the flexibility of machine learning to model non-Fourier heat conduction. Leveraging the conservation-dissipation principle with dual-dissipative variables, we derive an interpretable series of partial differential equations, fine tuned through a training strategy informed by data from the phonon Boltzmann transport equation. Moreover, we also present the inner-step operation to narrow the gap from the discrete form to the continuous system. Through numerical tests, our model demonstrates excellent predictive capabilities across various heat conduction regimes, including diffusive, hydrodynamic, and ballistic regimes, and displays its robustness and precision even with discontinuous initial conditions.

Revisit nonequilibrium thermodynamics based on thermomass theory and its applications in nanosystems

Abstract The development of non-Fourier heat conduction models is encouraged by the invalidity of Fourier’s law to explain heat conduction in ultrafast or ultrasmall systems. The production of negative entropy will result from the combination of traditional nonequlibrium thermodynamics and non-Fourier heat conduction models. To resolve this paradox, extended irreversible thermodynamics (EIT) introduces a new state variable. However, real dynamics variables like force and momentum are still missing from nonequilibrium thermodynamics and EIT’s generalized force and generalized flux. Heat has both mass and energy, according to thermomass theory and Einstein’s mass-energy relation. The generalized heat conduction model containing non-Fourier effects was established by thermomass gas model. The thermomass theory reshapes the concept of the generalized force and flux, temperature, and entropy production in nonequilibrium thermodynamics and revisits the assumption for the linear regression of the fluctuations in Onsager reciprocal relation. The generalized heat conduction model based on thermomass theory has been used to study thermal conductivity, thermoelectric effect, and thermal rectification effect in nanosystems.

Nonlocal dual-phase-lag thermoelastic damping in small-sized circular cross-sectional ring resonators

The purpose of current research is to establish a theoretical size-dependent framework for estimating the amount of thermoelastic damping (TED) in miniaturized rings with circular cross section via the nonlocal dual-phase-lag (NDPL) generalized thermoelasticity theory, as one of the most exhaustive non-Fourier heat conduction models. The method used to achieve this goal is based on the definition of TED in the energy dissipation (ED) approach. To do so, after deriving heat equation in the context of NDPL model, the distribution of temperature all over the volume of the ring is specified. Then, the relations of dissipated thermal energy and strain energy in the ring are obtained. Lastly, by employing ED approach, a formula comprising the nonclassical parameters of NDPL model is rendered to determine TED value in small-scaled toroidal rings. By presenting various numerical examples, the dependence of TED on influential parameters including nonclassical thermal constants in NDPL model, ring geometry, vibrational mode number and ring material is surveyed. The Findings enlighten that the inclusion of nonlocal parameter into the model can have conspicuous impacts on TED value, especially in smaller rings or higher vibrational mode numbers. The results also reveal that among the investigated materials (i.e. silicon, copper, silver, and lead), rings made of lead and silicon exhibit the maximum and minimum TED values, respectively.

Heat transfer analysis in a longitudinal porous trapezoidal fin by non-Fourier heat conduction model: An application of artificial neural network with Levenberg–Marquardt approach

Thermal critical issues commonly occur in advanced electrical devices as a response to excessive heat generation or a loss in efficient surface area for heat exclusion. This issue can be addressed by utilizing the extended surface to improve the heat transfer performance of the devices. Thus, the present analysis is devoted to scrutinizing the non-Fourier unsteady heat transference of a trapezoidal porous fin. Levenberg–Marquardt technique of backpropagation artificial neural network (LMT-BANN) is employed here to analyze the thermal variation in the fin. The developed governing equation is the hyperbolic heat conduction equation (HHCE), which is transformed into a dimensionless partial differential equation (PDE) using dimensionless variables. LMT-BANN is employed on thermal numerical data and is developed to trace numerical approximation of fin problem using a methodology that includes testing, training, and validation. A graphical visualization of the consequences of thermal variables on the temperature field is presented. The notable evidence of this study reveals that as the magnitude of the convection factor increases, thermal dissipation through the fin gradually decreases. Further, the LMT-BANN technique has been determined to be an effective, reliable, and rapidly convergent stochastic computational solver that can be used effectively for examining the thermal model.

Comparative analysis of thermal damage to laser-irradiated breast tumor based on Fourier conduction and non-Fourier heat conduction models: A numerical study

In recent decades, laser technology has been widely used in surgeries and thermal therapies. To ensure patient safety and enhance therapeutic effectiveness in laser applications, it is crucial to accurately analyze burn injuries during the treatment process. In this study, a bioheat transfer model that includes the effects of metabolic heat generation, capillary blood vessels, and relaxation times of heat flux and temperature gradient was developed to study the non-Fourier heat conduction behavior in 2-D spherical breast tumor during the laser hyperthermia process. By employing different heat conduction models and the Arrhenius burn integration, the temperature distribution and laser-induced thermal damage to tumors were obtained. The progression of burn injury in time and the effect of various parameters on it were investigated. It was observed that the DPL model provides more reliable results in comparison with the CV model in predicting thermal damage to bio-tissue when non-Fourier effects are taken into consideration. The results showed that increasing heat flux phase lag leads to severe thermal damage, earlier onset of empyrosis, and wider and deeper burn injury. However, a greater phase lag of temperature gradient results in lower and superficial thermal damage, milder burn, later onset of empyrosis, and smaller burned area. The effect of laser power density on the onset time of different degrees of burn and their sensitivity to input power were investigated, and the results were represented graphically. Finally, investigation of wave propagation in various tissues revealed that the wave propagation speed is not constant and reduces as time increases.

Correction to: A numerical study on non-Fourier heat conduction model of phase change problem with variable internal heat generation

Correction to: A numerical study on non-Fourier heat conduction model of phase change problem with variable internal heat generation

Temperature response in skin tissue during hyperthermia based on three-phase-lag bioheat model using RBF meshfree method

Various non-Fourier heat conduction models have been proposed to overcome the paradox of the infinite propagation speed of heat signals in Fourier’s law. The three-phase-lag (TPL) heat model is a non-Fourier model based on the linearized theory of coupled thermoelasticity that combines heat flux, temperature gradient, and thermal displacement gradient. The present paper proposes a novel method with meshfree and spectral nature to solve the two-dimensional three-phase lag (TPL) bioheat model for tissue heating during hyperthermia. In the space domain, radial basis functions (RBFs) and, in the time direction, shifted Chebyshev polynomials are employed to solve the model. The use of Chebyshev polynomials in the time direction enables one to get the solution in the whole time domain simultaneously with fewer nodes. Further, there is no need for background meshes in space due to the meshless nature of RBFs, and the proposed method is equally applicable to irregular domains. The proposed method is validated with the existing semi-analytical solution of the TPL model in one spatial dimension and found to be in good agreement. The temperature profile and the impact of various parameters, such as phase lag times, blood perfusion rate, and heat source parameters on heat transfer in tissue, have also been discussed.

Non-Fourier thermal wave in 2D cellular metamaterials: From transient heat propagation to harmonic band gaps

Tailoring the nano/microarchitecture of advanced materials is a new root for manipulating their behavior to attain unparalleled multifunctional properties, including guided transitional waves. Elastic , acoustic, and electromagnetic wave responses in architected materials have extensively been studied, while only few studies have been allocated to the thermal wave propagation in these engineered materials. In this article, we explore the propagation of thermal waves in cellular metamaterials as a category of rationally-designed architected materials. First, the propagation of an instantaneous heat excitation through a two-dimensional (2D) cellular medium is investigated in steady-state and transient modes by Fourier and non-Fourier (Cattaneo-Vernotte and dual-phase-lag) heat conduction theories. The effects of relative density and microarchitecture of the cellular domain on the heat propagation characteristics are investigated, connecting these traits to the effective thermal conductivity of the cellular domain using an asymptotic homogenization method. Second, by implementing the Bloch theorem for non-Fourier heat conduction models, complex dispersion curves are obtained for five alternative cellular architectures with Archimedean tilings. The cellular microstructure, on the one hand, affects the transient temperature distribution resulting from an instantaneous source and temporarily creates domains with temperatures as high as two times the applied temperature difference in the external boundaries, while on the other hand, blocks harmonic thermal excitations in a range of frequencies (thermal band gaps) from passing the material. Moreover, the relative density of cellular architectures significantly affects the characteristics of thermal band gaps; for example, the first thermal band gap width may be reduced by 80% when relative density of cellular architectures changes from 0.2 to 0.6. Our findings can pave a path towards developing architected multifunctional metamaterials with programmable thermal isolation/guiding properties.

Novel Non-Fourier Ice Heat Conduction Model Considering Latent Heat Via Specific Heat Capacity Functionalization and Its Potential Application

Abstract Proton exchange membrane fuel cells are widely used in the automotive and aviation fields due to their high efficiency and environmental friendliness. However, the long starting time of fuel cell vehicles at low temperatures restricts large-scale commercialization. In this work, for the problem of rapid ice melting during a cold start, it is found that when Fourier’s law is adopted, the error is as much as three times higher compared with the non-Fourier heat conduction law, and for ice, the influence of latent heat cannot be ignored, so a novel non-Fourier ice heat conduction model considering latent heat via specific heat capacity functionalization is established. The results demonstrate that the temperature curve first remains unchanged with time and then changes suddenly after the arrival of the heat wave. When the temperature rises to the phase change range, the temperature hardly changes before the completion of the phase change, and then finally rises slowly. Changing the thermal relaxation time may significantly affect the temperature response. The research conclusions of this paper have scientific guiding significance for the materials and structures working in extreme thermal environments such as low temperatures and ultra-high temperature change rates, as well as the design of fuel cell vehicles.

Significance of non-Fourier heat conduction in the thermal analysis of a wet semi-spherical fin with internal heat generation

The transient temperature distribution of a wet semi-spherical porous fin with convection, internal heat generation, and radiation is researched using a non-Fourier heat conduction (NFHC) model. A mathematical model for transient heat transfer in a dehumidifying environment is developed from the non-Fourier law and the corresponding governing equation is formulated. This equation is subjected to nondimensionalization using suitable dimensionless terms and is further solved by employing the finite difference method (FDM). A substantial deviation in thermal response by considering non-Fourier law under unsteady heat transmission has been analyzed graphically with the influence of various nondimensional parameters on the dry and wetted fins surface. The examination reveals the notable difference in temperature variation between the dry and wetted fins surfaces. Some important key results of the research analysis include that the transient thermal distribution through the fin decrease as the magnitude of the convective, wet porous and radiation-conduction parameters develop. For higher generating numbers, thermal distribution will be higher. The temperature in the hyperbolic model increases after some time in the existence of relaxation time and the temperature in a non-Fourier model flow in the form of a thermal wave.

MODELING THE THERMAL RESPONSE OF LASER-IRRADIATED BIOLOGICAL SAMPLES THROUGH GENERALIZED NON-FOURIER HEAT CONDUCTION MODELS: A REVIEW

MODELING THE THERMAL RESPONSE OF LASER-IRRADIATED BIOLOGICAL SAMPLES THROUGH GENERALIZED NON-FOURIER HEAT CONDUCTION MODELS: A REVIEW

Estimating Relaxation Time and Fractionality Order Parameters in Fractional Non-Fourier Heat Conduction Using Conjugate Gradient Inverse Approach in Single and Three-Layer Skin Tissues

In this work, the relaxation parameter (τ) and fractionality order (α) in the fractional single phase lag (FSPL) non-Fourier heat conduction model are estimated by employing the conjugate gradient inverse method (CGIM). Two different physics of skin tissue are chosen as the studied cases; single and three-layer skin tissues. Single-layer skin is exposed to laser radiation having the constant heat flux of Qin. However, a heat pulse with constant temperature is imposed on the three-layer skin. The required inputs for the inverse problem in the fractional diffusion equation are chosen from the outcomes of the dual phase lag (DPL) theory. The governing equations are solved numerically by utilizing implicit approaches. The results of this study showed the efficiency of the CGIM to estimate the unknown parameters in the FSPL model. In fact, obtained numerical results of the CGIM are in excellent compatibility with the FSPL model.

Non-Fourier Heat Conduction and Thermal-Stress Analysis of a Spherical Ice Particle Subjected to Thermal Shock in PEM Fuel Cell at Quick Cold Start-Up

Quick start at low temperature is one of the key bottleneck technologies that restrict the large-scale commercialization of proton exchange membrane (PEM) fuel cell vehicles. Ice and devices in battery systems based on thermal deicing are inevitably subjected to thermal shock. In order to study this problem, a non-Fourier heat conduction model is established to study the temperature response of a spherical ice particle subjected to thermal shock with different boundary conditions on the surface. Furthermore, distribution of thermal stress in the particle is obtained using the calculated temperature field, and the effect of thermal relaxation time and boundary conditions on the temperature response as well as the thermal stress field are also analyzed. The results, which are significantly different from that obtained using Fourier law of heat conduction, show that the mechanical stresses and the serious expansion of the ice particle may lead to the severe deformation of the devices connected to the ice during the cold start-up, imposing a great challenge in controlling structural reliability of PEM fuel cells. The numerical results are expected to provide a scientific theoretical basis for PEM fuel cell design and low-temperature start-up control.

A numerical study on non-Fourier heat conduction model of phase change problem with variable internal heat generation

A numerical study on non-Fourier heat conduction model of phase change problem with variable internal heat generation

Thermodynamic modeling of viscoelastic thin rotating microbeam based on non-Fourier heat conduction

In this investigation, the system of equations governing the thermoelastic behavior of a rotating viscoelastic microbeam has been derived based on the non-Fourier heat conduction model. The viscoelastic properties of the material are incorporated in terms of the Kelvin-Voigt scheme. This considered microbeam is axially compressed and rotated at a uniform angular velocity in addition to exposing it to a femtosecond laser pulse heat source. The proposed model can be reduced to the thermoelastic beam theory when the viscous and rotation parameters are neglected. To obtain an analytical solution for the studied fields, the Laplace transform technique is applied. The effect of the viscous damping coefficient on the microbeam structure has been studied. Also, the effect of the axial load, laser pulse duration and angular velocity on the thermal and elastic waves of the rotating microbeam has been presented graphically and analyzed in detail.

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.

Numerical study into the fin performance subjected to different periodic base temperatures employing Fourier and non-Fourier heat conduction models

Abstract In this study, the instantaneous and time-averaged fin efficiencies of a convective fin subjected to sinusoidal, triangular and square periodic base temperatures is assessed with Fourier and non-Fourier models using Spectral-Finite volume approach. Validation of the used approach is performed by comparison of current results with those of an available work and good agreement is observed. It is shown that the maximum instantaneous fin efficiencies of sinusoidal and triangular stimulations depend on the model used for heat transfer and are more sensitive to the change in the relaxation time than the square disturbance. For Fourier model, the time-averaged fin efficiency changes as the disturbance frequency varies while for non-Fourier behavior, dependency of time-averaged fin efficiency to the disturbance frequency is observed within a specific range of disturbance frequency. Square disturbance along with non-Fourier model predicts a higher value for the time-averaged fin efficiency than the other disturbances especially at high relaxation times, disturbance frequencies and amplitudes. In non-Fourier fin model, time-averaged fin efficiency is independent of relaxation time at very low disturbance frequencies. Fourier and non-Fourier models result in a same time-averaged fin efficiency at low disturbance amplitudes whereas non-Fourier model leads to higher time-averaged fin efficiency at high disturbance amplitudes.

Numerical investigation on plume expansion, components evolution and ionization of polytetrafluoroethylene (PTFE) irradiated by nanosecond laser

To investigate the plume expansion, components evolution and ionization in the laser ablation pulsed plasma thruster, which is difficult to obtain from experiments, the non-Fourier heat conduction model, fluid dynamics model and thermochemical model are established. The non-Fourier heat conduction model is established to calculate the propellant ablation, taking into account temperature dependent material properties, phase transition, depolymerization of polytetrafluoroethylene propellant, the reflection of the laser beam at the surface of the propellant, and non-Fourier effect. The fluid dynamics model is established to calculate the plume expansion, taking into account plume absorption and shielding. The thermochemical model is established to calculate the components evolution and ionization, taking into account nine chemical and ionization reactions with 12 components. By coupling the three models, the thermochemical characteristics, plume expansion and ionization of polytetrafluoroethylene irradiated by a nanosecond laser are numercially calculated.

Non-Fourier Heat Conduction of a Functionally Graded Cylinder Containing a Cylindrical Crack

The present work investigates the problem of a cylindrical crack in a functionally graded cylinder under thermal impact by using the non-Fourier heat conduction model. The theoretical derivation is performed by methods of Fourier integral transform, Laplace transform, and Cauchy singular integral equation. The concept of heat flux intensity factor is introduced to investigate the heat concentration degree around the crack tip quantitatively. The temperature field and the heat flux intensity factor in the time domain are obtained by transforming the corresponding quantities from the Laplace domain numerically. The effects of heat conduction model, functionally graded parameter, and thermal resistance of crack on the temperature distribution and heat flux intensity factor are studied. This work is beneficial for the thermal design of functionally graded cylinder containing a cylindrical crack.

Thermoelastic damping in micro and nano-mechanical resonators utilizing entropy generation approach and heat conduction model with a single delay term

Analysis of thermoelastic damping (TED) of micro and nano-beam resonators plays a very important role in designing the resonator with high-quality factors. TED of micro and nano-beam resonators has therefore been an interesting and challenging area of research in recent years. Prediction of TED is more challenging under the effect of non-Fourier heat conduction. In the past, several works have been dedicated to TED modeling with the influence of non-Fourier heat conduction in view of the fact that non-Fourier heat conduction model is more relevant for analysis of TED for small scale devices. The present work aims to investigate TED in micro and nano-mechanical resonators utilizing non-Fourier heat conduction model with a single delay term introduced by Quintanilla in 2011. We derive an explicit formula of the quality factor for TED based on the entropy generation approach. With the help of numerical results, we present the influence of TED in the context of normalized frequency as well as beam thickness. Further, the effects of the time delay parameter and the material constants on TED have been discussed in detail. The results of the present model are compared to those obtained for GN-III model. It has been observed that the current model offers a high-quality factor as compared to GN-III model.