non-Fourier Heat Transfer Research Articles

Local high temperature phenomena near the complex interfaces of thermal barrier coatings based on non-Fourier model

Thermal barrier coatings (TBCs) consist of a metal substrate (Sub), a bond coat (BC), and a ceramic top coat (TC). The ceramic coating is a porous dielectric layer with a significant relaxation time that cannot be ignored. In high-temperature environments, an irregularly shaped thermally grown oxide layer (TGO) forms between the BC and the TC. Therefore, using non-Fourier model to describe the thermal contact process of TBCs with complex interface can get more accurate results. However, it is difficult to obtain the analytical solution of this model. This paper introduces a finite volume method based on unstructured meshes to address the non-Fourier heat transfer process in the complex interfaces of multilayer TBCs. The gradient reconstruction technique is employed to calculate the cell derivatives on the surface. The method's reliability is confirmed through a comparison with a two-dimensional analytical solution. The results indicate that the thermal wave travels at a limited speed within the TC layer. After passing through the interface, the thermal wave travels very fast within the BC and Sub layers, due to the influence of thermal wave reflection and superposition, obvious local high temperature will be formed near the interface. In addition, the effects of thermal conductivity, relaxation time and interface shapes on the non-Fourier heat conduction process of TBCs are also discussed. The results of this paper are useful for the structural design and failure mechanism research in TBCs.

A hybrid shape functions based temporal finite element method to solve transient heat conduction problems

A novel temporal finite element method is presented to solve Fourier and non-Fourier heat conduction problems. Two kinds of temporal finite element models are developed using Gurtin variational principle and weighted residual technique, respectively. A kind of hybrid shape function with polynomial and trigonometric basis is stressed to give more flexible and appropriate description of temporal behavior of temperature. A recursive algorithm is developed by which the temperature solution at a specific time can be obtained only via a matrix power product with the initial condition, and a criterion of stability analysis is derived, which is can numerically be conducted when the constitution of shape function and time step size are prescribed. The proposed approach is available for Fourier and non-Fourier heat transfer problems, and can conveniently be combined with well-developed numerical algorithms for the boundary value problem, such as FE, SBFEM, etc. Various numerical examples, including those with the nonlinear thermal conductivity, radiative boundary condition, etc. are provided to illustrate the efficiency of proposed approaches, and impacts of the temporal FE model, constitution of shape functions, and step size, etc. are taken into account. Satisfactory results are achieved in comparison with the analytical and the ABAQUS based numerical solutions.

Non-Fourier computations of heat and mass transport in nanoscale solid-fluid interactions using the Galerkin finite element method

PurposeTo improve the thermal performance of base fluid, nanoparticles of three types are dispersed in the base fluid. A novel theory of non-Fourier heat transfer is used for design and development of models. The thermal performance of sample fluids is compared to determine which types of combination of nanoparticles are the best for an optimized enhancement in thermal performance of fluids. This article aims to: (i) investigate the impact of nanoparticles on thermal performance; and (ii) implement the Galerkin finite element method (GFEM) to thermal problems.Design/methodology/approachThe mathematical models are developed using novel non-Fourier heat flux theory, conservation laws of computational fluid dynamics (CFD) and no-slip thermal boundary conditions. The models are approximated using thermal boundary layer approximations, and transformed models are solved numerically using GFEM. A grid-sensitivity test is performed. The accuracy, correction and stability of solutions is ensured. The numerical method adopted for the calculations is validated with published data. Quantities of engineering interest, i.e. wall shear stress, wall mass flow rate and wall heat flux, are calculated and examined versus emerging rheological parameters and thermal relaxation time.FindingsThe thermal relaxation time measures the ability of a fluid to restore its original thermal state, called thermal equilibrium and therefore, simulations have shown that the thermal relaxation time associated with a mono nanofluid has the most substantial effect on the temperature of fluid, whereas a ternary nanofluid has the smallest thermal relaxation time. A ternary nanofluid has a wider thermal boundary thickness in comparison with base and di- and mono nanofluids. The wall heat flux (in the case of the ternary nanofluids) has the most significant value compared with the wall shear stresses for the mono and hybrid nanofluids. The wall heat and mass fluxes have the highest values for the case of non-Fourier heat and mass diffusion compared to the case of Fourier heat and mass transfer.Originality/valueAn extensive literature review reveals that no study has considered thermal and concentration memory effects on transport mechanisms in fluids of cross-rheological liquid using novel theory of heat and mass [presented by Cattaneo (Cattaneo, 1958) and Christov (Christov, 2009)] so far. Moreover, the finite element method for coupled and nonlinear CFD problems has not been implemented so far. To the best of the authors’ knowledge for the first time, the dynamics of wall heat flow rate and mass flow rate under simultaneous effects of thermal and solute relaxation times, Ohmic dissipation and first-order chemical reactions are studied.

Numerical Analysis of Non-Fourier Model-Based Bio-Heat Transfer in the Laser-irradiated Axisymmetric Living Tissue.

The current work is related to the numerical investigation of non-Fourier heat transfer inside the short-pulsed laser-irradiated axisymmetric soft tissue phantom. It utilizes the modified discrete ordinate method to solve the transient radiative transfer equation (TRTE) for determining the intensity field. The laser energy absorbed by the soft tissue phantom behaves like a source in the Fourier/non-Fourier heat conduction model based-bio-heat transfer equation (BHTE), which is solved by employing the finite volume method (FVM) to determine the temperature distribution. Despite the prevalent use of non-Fourier BHTE for this purpose, a second law analysis is considered crucial to detect any potential anomalies. Equilibrium entropy production rates (EPR) are initially computed based on classical irreversible thermodynamics (CIT), which may yield negative values, possibly contravening the second law. Consequently, the EPR based on CIT is adjusted using the extended irreversible thermodynamics (EIT) hypothesis to ensure positivity. After that, the current research findings are compared with the results from the literature, and found good agreement between them. Then, the independent study is performed to select the optimum grid size, control angle size, and time step. A comparative analysis of results between the traditional Fourier and non-Fourier models has been performed. The impact of different parameters on the temperature fields and EPRs, is discussed. The effect of the optical properties of the inhomogeneity on the temperature distribution has been investigated. This study may help to enhance the effectiveness of the laser-based photo-thermal therapy.

From clusters of moving molecules to continua: Material elements as open systems

We discuss the discrete-to-continuum transition in the description of matter, starting from a cluster of flowing molecules with equal mass and ending up with a non-simple fluid. We account for the fluctuations beyond a local affine approximation of the velocity distribution of molecules within a space window adopted to compute some prominent statistics. The resulting continuum picture accounts for local mass variation in each space window corresponding to a point in the continuum scale. From a statistical viewpoint, every material element is thus considered as a grand-canonical ensemble. So-called C-derivatives account for macroscopic-to-mesoscopic relative motion. When considered for second-rank tensors, they extend Truesdell’s derivative and reduce to Oldroyd’s one when the macroscopic-to-mesoscopic relative motion vanishes. Fluctuations beyond an affine component best approximating the kinetic energy are summarized into a second-rank symmetric tensor whose time variation enters a balance equation governing transfer from velocity fluctuations to heat. Eventually, we discuss essential elements of thermodynamics in the present setting. What emerges is the possibility of a non-Fourier type heat transfer. The results address computational schemes for field representations of sparse phase dynamics, such as granular materials, and the one of bodies with transport of scattered molecules, such as pollutants in fluids or proteins in biological tissues.

Nonlinear thermal analysis of two-dimensional materials with memory

A nonlinear hyperbolic heat transport equation has been proposed based on the Cattaneo model without mechanical effects. We analyze the two-dimensional Maxwell-Cattaneo-Vernotte heat equation in a medium subjected to homogeneous and non-homogeneous boundary conditions and with thermal conductivity and relaxation time linearly dependent on temperature. Since these nonlinearities are essential from an experimental point of view, it is necessary to establish an effective and reliable way to solve the system of partial differential equations and study the behavior of temperature evolution. A numerical scheme of finite differences for the solution of the two-dimensional non-Fourier heat transfer equation is introduced and studied. We also investigate the attributes of the numerical method from the aspects of stability, dissipation and dispersive errors.

Thermal analysis of magnetohydrodynamics (MHD) Casson fluid with suspended iron (II, III) oxide-aluminum oxide-titanium dioxide ternary-hybrid nanostructures

This study is carried out to enhance and analyze the thermal performance of non-Newtonian Casson fluid by immersing Ternary hybrid nanoparticles Fe3O4-Al2O3-TiO2 uniformly. To model the behaviour of such complex phenomena mathematically, a system of complex transport differential equations is developed by utilizing a non-Fourier heat transfer model for energy transport. The non-dimensional system of transport equations involving physical parameters is analyzed numerically by employing the Galerkin finite element methodology (G-FEM). The simulations predicted promising enhanced heat transfer characteristics in Ternary hybrid Casson fluid. The Casson fluid parameter has shown remarkable behavior on temperature distribution. Ternary hybrid nanoparticles have increased the overall thermal conductivity of the fluid and consequently, the temperature distribution is significantly raised. Hence ternary hybridity of nano-structures in Casson fluid is shown as an effective technique in improving the thermal performance of complex fluid.

A peridynamic model for coupled thermo-mechanical-oxygenic analysis of C/C composites with SiC coating

The oxidation induced by coating cracking is a general failure mode of high-temperature composites such as C/C and C/SiC. However, this failure mode has not been fully investigated due to the lack of a method suitable for both crack simulation and oxidation interface capture. In this paper, a coupled thermo-mechanical-oxygenic peridynamic model is proposed for SiC-coated C/C composites in an oxidizing environment. The ordinary state-based peridynamics is extended to plane strain thermoelastic problems for orthotropic media to calculate the crack distribution. Cracks in the coating are modeled as oxygen diffusion channels and the peridynamic oxidation equation is developed to simulate the oxidation of the C/C substrate. Furthermore, the peridynamic non-Fourier heat transfer equation is employed to consider the finite thermal propagation speed and the model is validated by experimental results. Numerical examples are presented to investigate the oxidation failure after thermal shock, non-Fourier thermal shock fracture, and stress oxidation failure of SiC-coated C/C composites.

Numerical Investigation of the Electrothermal Properties of SOI FinFET Transistor

This paper investigates the non-Fourier transient heat transfer in an SOI FinFET transistor. The calibrated drift-diffusion (D-D) model in conjunction with the ballistic diffusive (BDE) model is used as an electrothermal model to predict phonon and electron transports in the quasi-ballistic regime. The finite element method has been employed to generate the numerical results. The proposed mathematical formulation was found to capture the transfer characteristics and the temporal temperature as given by TCAD simulation and experimental data. On the other hand, we have demonstrated that after 100 ns, the 14 nm Bulk FinFET supports better temperature distribution than the 14 nm SOI FinFET.

Transient Temperature Distribution in a Half-Space Due to Local Surface Heating Via Non-Fourier Fractional Dual-Phase-Lag Model

Abstract The non-Fourier heat transfer in a half-space is analyzed under sudden heating or cooling on a local surface. The non-Fourier heat transfer effect is described by the time-fractional dual-phase-lag (DPL) model, where the fractional derivative without singular kernel is used. An axisymmetric mixed initial-boundary value problem is solved by the use of the Hankel and Laplace transforms. Two typical cases of sudden temperature rising on a circular zone of the surface or an instantaneous surface heat source are analyzed. For sudden temperature rises, the heat flux and temperature gradient exhibit an inverse square-root singularity near the boundary of the heating zone and their dynamic intensity factors are computed numerically in the time domain. For the instantaneous surface point heat source, an exact solution of the transient temperature at any position in the Laplace domain is obtained. The effects of the fractional order and relaxation time on the temperature distribution and heat flux response are elucidated. The singular behavior of the transient thermal response and the non-Fourier effect of heat transfer are shown.

A Semi-Analytical Description of Radiation Coupled with Thermal Wave Conductive Transfer in Inhomogeneous Solid Cylinder

This study presents the analytical layered solution for thermal radiations coupled with non-Fourier conductive heat transfer, formulated from the Cattaneo-Vernotte flux model in inhomogeneous solid cylinder. This solution is built by combining the spherical harmonics for volumetric thermal radiations with the D1Q3 scheme of lattice Boltzmann method (LBM) for hyperbolic energy equation where the gradient of static particle distribution function was discretized at implicit time. The accuracy of the proposed model for dealing with radiation/conduction problems is investigated by considering a semitransparent radiative transfer in a cylinder with temperature dependent thermal conductivity and space dependent scattering albedo. The effects of different parameters, such as scattering albedo, refractive index, thermal conductivity, emissivity, optical thickness, and the conduction-radiation parameter on both radiations and temperature distributions for steady and transient states are studied. Results of the present work are benchmarked against those available in the literature with accuracy greater than 98.9% for a large interval of parameter sets and therefore, excellent agreement has been obtained. It also establishes from this study that the proposed layered approach is an efficient and accurate method for radiative analysis in inhomogeneous media while the D1Q3 scheme is suitable to accommodate thermal wave front in non-Fourier analysis.

Analytical solution of heat-transfer in central part of tibia bone tissue using non-Fourier heat equation with a laser heat source

Hyperbolic heat transfer equation considering a laser heat source is analytically investigated using Laplace transform method to study the temperature distribution in the central part of the tibia bone tissue. The target zone is the compact layer of tibia, and the laser irradiation in four near-surface points during 2 s is taken into account. Holmium–Yttrium–Aluminum-Garnet (Ho-YAG) laser is utilized in the simulations. Results indicate that there is a trilateral relationship among the temperature changes of various points in the tissue, the power used and the wavelength. While neglecting the effects of blood circulation upon laser interaction with the bone surface, the results reveal that the temperature of the points adjacent to the bone surface dramatically increases up to 85 °C. Meanwhile, the heat source is provided based on the Beer-Lambert law, and finally it demonstrates that the penetration depth exponentially reduces the intensity and causing considerable decrease in temperature. For the first time, the non-Fourier heat transfer equation was solved using the Laplace transform method for the tibia bone, and its results were compared with the numerical solution results using the Leap- Frog method. Knowing the temperature changes caused by laser inside the tissue is very crucial in Laser tissue welding, tissue ablation, photocoagulation and photo-thermal denaturation, angioplasty, treatment of granulation tissues, change and deformation of cartilage.

Fractional Dual-Phase-Lag Non-Fourier Heat Transfer in a Bimaterial with a Circular Interface Insulator

The transient temperature response of a bimaterial with a circular insulated interface region is studied under sudden heating or cooling. The time-fractional dual-phase-lag heat conduction model is adopted to simulate the non-Fourier effect. The problem is reduced to an initial-boundary value problem. The Laplace transform is applied to convert the problem to a mixed boundary value problem, and then the Hankel transform reduces it to a Fredholm integral equation. Special situations for asymptotic thermal behavior near the insulated circular edge and for the steady-state cases are discussed, respectively. The dynamic intensity factors of heat flux and temperature gradient near the insulated circular edge are computed numerically through Stehfest’s Laplace inversion transform technique. The influences of fractional order and relaxation times on the instantaneous temperature change are analyzed. The exact solution of temperature fields for the steady-state case is derived and displayed graphically. The wave-like diffusion behavior of the fractional dual-phase-lag model is interpreted.

Investigation of variable thermal relaxation time in non-Fourier heat transfer flow with nonlinear thermal stratification

Energy utilization is a significant aspect of the manufacturing industrial processes, and inefficient transferring of heat is one of the disadvantages which can affect the quality measures of end product. Therefore, by controlling the heat transferring rate, the efficiency of manufacturing processes can be improved. Thus, the variable thermal relaxation time factor in the theory of Cattaneo-Christov is main concern of this analysis. The non-isothermal flow by means of a porous medium is modeled and analyzed for a viscous fluid. The fluid deformation is discussed under stretching mechanism. A study is also executed to interrogate the heat transfer features under the heat flux model of Cattaneo-Christov. Thermal relaxation time is also considered which is determined by temperature. The aspect of thermal stratification is also implemented to see the modifications in temperature. The flow and energy equations are reduced by appropriate variables. The reduced constitutive flow equations are then computed using an analytical convergent technique. The significant impacts of the numerous parameters on the flow field and heat transport features are evaluated in detail through various plots. Skin friction coefficient is evaluated mathematically. Results found that the temperature significantly decreases with increased stratification effects. For larger thermal relaxation time parameter, the temperature function shows increasing behavior.

Isogeometric dual reciprocity BEM for solving non-Fourier transient heat transfer problems in FGMs with uncertainty analysis

In this paper, the precise integration multi-patch isogeometric dual reciprocity boundary element method (IG-DRBEM) of the non-Fourier transient heat transfer problem in functionally graded materials (FGMs) is proposed, and the stochastic analysis of multidimensional material uncertainty by hyperbolic truncated adaptive sparse polynomial chaos expansion (PCE) is established based on the determine theory. Up to now, most research results of the isogeometric analysis boundary element method (IGABEM) in transient heat transfer problems are based on Fourier's law. The main reason is the higher-order derivative term with respect to time in the non-Fourier heat transfer control equation leads to additional domain integrals in the boundary-domain integral equation. The dual reciprocity method (DRM) can transform the domain integral to the boundary conveniently and maintain the dimension reduction advantage of the BEM. The introduction of the precise integration method (PIM) can improve the stability and accuracy of the time-domain solution. In the random analysis based on IG-DRBEM, a sparse PCE strategy with hyperbolic truncation is constructed to overcome the dimensional disaster difficulty of traditional PCE. Through the proposed adaptive iterative process, a relatively small number of PCE terms are retained compared with the full PCE on the premise of meeting the error estimation requirements. The presented method is verified to have good accuracy and high efficiency by the complex model similar to actual engineering, which further expands the application extent of IG-DRBEM.

Temperature and Motion Analysis of a Rotational Maxwell Fluid with non-Constant Thermal Conductivity and non-Fourier Heat Transfer Model, Using Homotopy Perturbation Method

Temperature and Motion Analysis of a Rotational Maxwell Fluid with non-Constant Thermal Conductivity and non-Fourier Heat Transfer Model, Using Homotopy Perturbation Method

Non-Fourier heat transfer in a moving longitudinal radiative-convective dovetail fin

The purpose of this research is to estimate the non-Fourier temperature distribution in dovetail fin. The Cattaneo-Vernotte heat model is utilized to estimate heat conduction behavioral patterns in dovetail fin. In addition, the variation in the temperature profile has been inspected for both non-Fourier and Fourier models. The governing equation involves the linear variance of thermal conductivity, the power-law dependence of the heat transfer coefficient, and the constant surface emissivity. This equation is transmuted by using dimensionless variables, yielding a dimensionless partial differential equation (PDE). To solve the obtained PDE, a numerical technique called the finite difference method (FDM) is employed. The graphical illustration is provided to explain the upshot of thermal parameters on the temperature gradient. The significant findings of this investigation exhibit that as the scale of the convection and radiation variables rises, the thermal response in the fin gradually decreases. Also, the temperature field enhances for Peclet number and ambient temperature parameter. Moreover, the temperature drops from the fin's base to its tip for the Fourier effect. The temperature drops quickly at a point on the fin in comparison to the surrounding value, and this minimal temperature is preserved throughout the rest of the fin.

A comparative numerical study of a semi-infinite heat conductor subject to double strip heating under non-Fourier models

The present work considers a two-dimensional (2D) heat conduction problem in the semi-infinite domain based on the classical Fourier model and other non-Fourier models, e.g., the Maxwell–Cattaneo–Vernotte (MCV) equation, parabolic, hyperbolic, and modified hyperbolic dual-phase-lag (DPL) equations. Using the integral transform technique, Laplace, and Fourier transforms, we provide a solution of the problem (Green’s function) in Laplace domain. The thermal double-strip problem, allowing the wave interference within the heat conductor, is considered. A numerical technique, based on the Durbin series for inverting Laplace transform and the trapezoidal rule for calculating an integral form of the solution in the double-strip case, is adopted to recover the solution in the physical domain. Finally, discussions for different non-Fourier heat transfer situations are presented. We compare among the speeds of hyperbolic heat transfer models and shed light on the concepts of flux-precedence and temperature-gradient-precedence, hallmarks of the lagging response idea. Otherwise, we emphasize the existence of a relationship between the waves speed and the time instant of interference onset, underlying the five employed heat transfer models.

A peridynamic model for non-Fourier heat transfer in orthotropic plate with uninsulated cracks

In this paper, the transient temperature response of a cracked plate under thermal shock is investigated. To eliminate the problems caused by the assumption of infinite heat propagation speed, the non-Fourier heat transfer theory is adopted. A peridynamic model is developed to consider the non-Fourier effect, the orthotropy of thermal conductivity, and the crack thermal resistance. This model avoids the spatial derivative and is efficient to analyze the problems with discontinuities. Based on the Kapitza thermal resistance model, a thermal resistance bond is proposed to deal with the uninsulated crack. The explicit and implicit discrete schemes of peridynamic formulation are presented to solve the temperature field. The model is verified by the analytical solution and excellent agreements are obtained. For the non-Fourier phenomenon observed in bismuth, numerical examples are presented for analyzing the effects of crack thermal resistance, crack orientation angle, and multi-crack distribution on non-Fourier heat transfer.

Non-Fourier heat and mass transfer enhancement in magnetohydrodynamic ionized fluid

Ethylene glycol has the characteristics of plasma and off course composed of ionized particles and it is also revealed by the literature that its rheological behavior is the best characterized by the power-law rheological model. Further, nanoparticles and are antioxidant, antifriction, and antiaging, therefore, their simultaneous suspension in ethylene glycol makes the mixture an efficient coolant for engines. Here, in this study, the objective is to investigate the thermal performance of liquid mixture (the mixture of ethylene glycol and nanoparticles, and ) theoretically. For this conservation laws and correlations among the thermophysical properties are simultaneously used for the modeling of heat transfer in ethylene glycol containing nanoparticles ( and ). Models are transformed into their dimensionless form using the approach of symmetry analysis. The dimensionless models are solved numerically by the finite element method (FEM). Simulations are visualized and data is recorded in the form of graphs and numerical values. Comparative analysis has revealed that the thermal efficiency of ethylene glycol is much improved due to the simultaneous dispersion of and . It is also found that ethylene glycol with a single type of nanoparticles () is a less efficient coolant than the efficiency of ethylene glycol with hybrid nanoparticles.