Theoretical Analysis and Experimental Evidence of Non-Fourier Heat Conduction Behavior

This paper consists of two parts. (1) For a hollow sphere with sudden temperature changes on its inner and outer surfaces, the hyperbolic heat conduction equation is employed to describe this extreme thermal case and an analytical expression of its temperature distribution is obtained. According to the expression, the non-Fourier heat conduction behavior that will appear in the hollow sphere is studied and some qualitative conditions that will result in distinct non-Fourier behavior in the medium is ultimately attained. (2) A novel experiment to observe non-Fourier heat conduction behavior in porous material (mainly ordinary duplicating paper) heated by a microsecond laser pulse is presented. The conditions for observing distinct non-Fourier heat conduction behavior in the experimental sample agree well with the theoretical results qualitatively.

Two exact solutions of the DPL non-Fourier heat conduction equation with special conditions

This paper presents two exact explicit solutions for the three dimensional dual-phase lag （DLP） heat conduction equation, during the derivation of which the method of trial and error and the authors＇ previous experiences are utilized. To the authors＇ knowledge, most solutions of 2D or 3D DPL models available in the literature are obtained by numerical methods, and there are few exact solutions up to now. The exact solutions in this paper can be used as benchmarks to validate numerical solutions and to develop numerical schemes, grid generation methods and so forth. In addition, they are of theoretical significance since they correspond to physically possible situations. The main goal of this paper is to obtain some possible exact explicit solutions of the dual-phase lag heat conduction equation as the benchmark solutions for computational heat transfer, rather than specific solutions for some given initial and boundary conditions. Therefore, the initial and boundary conditions are indeterminate before derivation and can be deduced from the solutions afterwards. Actually, all solutions given in this paper can be easily proven by substituting them into the governing equation.

Numerical Analysis of Non-Fourier Heat Transfer in a Solid Cylinder with Dual-Phase-Lag Phenomenon

In this study,transient non-Fourier heat transfer in a solid cylinder is analytically solved based on dual-phase-lag for constant axial heat flux condition.Governing equations for the model are expressed in two-dimensional cylindrical coordinates;the equations are nondimensionalized and exact solution for the equations is presented by using the separation of variable method.Results showed that the dual-phase-lag model requires less time to meet the steady temperature compared with single-phase-lag model.On the contrary,thermal wave diffusion speed for the dual-phase-lag model is greater than the single-phase-lag model.Also the effect of relaxation time in dual-phase-lag model has been taken on consideration.

Time fractional dual-phase-lag heat conduction equation

We build a fractional dual-phase-lag model and the corresponding bioheat transfer equation, which we use to interpret the experiment results for processed meat that have been explained by applying the hyperbolic conduction. Analytical solutions expressed by H-functions are obtained by using the Laplace and Fourier transforms method. The inverse fractional dual-phase-lag heat conduction problem for the simultaneous estimation of two relaxation times and orders of fractionality is solved by applying the nonlinear least-square method. The estimated model parameters are given. Finally, the measured and the calculated temperatures versus time are compared and discussed. Some numerical examples are also given and discussed.

Fractional Cattaneo heat equation in a semi-infinite medium

To better describe the phenomenon of non-Fourier heat conduction, the fractional Cattaneo heat equation is introduced from the generalized Cattaneo model with two fractional derivatives of different orders. The anomalous heat conduction under the Neumann boundary condition in a semi-infinity medium is investigated. Exact solutions are obtained in series form of the H-function by using the Laplace transform method. Finally, numerical examples are presented graphically when different kinds of surface temperature gradient are given. The effects of fractional parameters are also discussed.

Model of Fractional Heat Conduction in a Thermoelastic Thin Slim Strip under Thermal Shock and Temperature-Dependent Thermal Conductivity

The present paper paper,we estimate the theory of thermoelasticity a thin slim strip under the variable thermal conductivity in the fractional-order form is solved.Thermal stress theory considering the equation of heat conduction based on the time-fractional derivative of Caputo of order
is applied to obtain a solution.We assumed that the strip surface is to be free from traction and impacted by a thermal shock.The transform of Laplace(LT)and numerical inversion techniques of Laplace were considered for solving the governing basic equations.The inverse of the LT was applied in a numerical manner considering the Fourier expansion technique.The numerical results for the physical variables were calculated numerically and displayed via graphs.The parameter of fractional order effect and variation of thermal conductivity on the displacement,stress,and temperature were investigated and compared with the results of previous studies.The results indicated the strong effect of the external parameters,especially the timefractional derivative parameter on a thermoelastic thin slim strip phenomenon.

Jeffrey fluid flow due to curved stretching surface with Cattaneo-Christov heat flux

The two-dimensional(2D) motion of the Jeffrey fluid by the curved stretching sheet coiled in a circle is investigated. The non-Fourier heat flux model is used for the heat transfer analysis. Feasible similarity variables are used to transform the highly nonlinear ordinary equations to partial differential equations(PDEs). The homotopy technique is used for the convergence of the velocity and temperature equations. The effects of the involved parameters on the physical properties of the fluid are described graphically.The results show that the curvature parameter is an increasing function of velocity and temperature, and the temperature is a decreasing function of the thermal relaxation time.Besides, the Deborah number has a reverse effect on the pressure and surface drag force.

Nonlinear Thermal Buoyancy on Ferromagnetic Liquid Stream Over a Radiated Elastic Surface with Non Fourier Heat Flux

The current article discusses the heat transfer characteristics of ferromagnetic liquid over an elastic surface with the thermal radiation and non-Fourier heat flux.In most of the existing studies,the heat flux is considered as constant,but whereas we incorporated the non-Fourier flux to get the exact performance of the flow.Also,we excluded the PWT and PHF cases to control the boundary layer of the flow.The governing equations related to our contemplate are changed into non-linear ordinary differential equations(ODE’s)by utilizing appropriate similarity changes,which are at the point enlightened by Runge–Kutta based shooting approach.The equations are broken down concerning boundary conditions and to be explained prescribed wall temperature(PWT)and prescribed heat flux(PHF)cases.The impacts of diverse non-dimensional physical parameters on velocity and temperature profiles are laid out graphically.Also,the assortment of skin friction and local Nusselt number for both PWT and PHF cases for various assessments of non-dimensional parameters have been sorted out.Towards the wrap-up of the examination,we suspect that the friction factor coefficient is higher in the PWT case compared to the PHF case.This result helps to conclude that the flux conditions are useful for cooling applications.

Vibration of Nano Beam Induced by Ramp Type Heating

The non-Fourier effect in heat conduction and the coupling effect between temperature and strain rate, became the most significant effects in the nano-scale beam. In the present study, a generalized solution for the generalized thermoelastic vibration of a bounded nano-beam resonator induced by ramp type of heating is developed and the solutions take into account the above two effects. The Laplace transforms and direct method are used to determine the lateral vibration, the temperature, the displacement, the stress and the energy of the beam. The effects of the relaxation time and the ramping time parameters have been studied with some comparisons.

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

Dual-phase lag (DPL) model is used to describe the non-Fourier heat conduction in a finite medium where the boundary at x=0 is heated by a rectangular pulsed energy source and the other boundary is tightly contactal with another medium and satisfies the continuous boundary condition. Numerical solution of thes kind of non-Fourier heat conduction is presented in this paper. The results are compared with those predicted by the hyperbolic heat conduction (HHC) equation.

Nonlinear analysis of a non-Fourier heat conduction problem in a living tissue heated by laser source

The effect of laser, as a heat source, on a one-dimensional finite living tissue was stud- ied in this paper. The dual phase lagging （DPL） non-Fourier heat conduction model was used for thermal analysis. The thermal conductivity was assumed temperature- dependent, resulting in a nonlinear equation. The obtained equations were solved using the approximate-analytical Adomian decomposition method （ADM）. It was concluded that the nonlinear analysis was important in nomFourier heat conduction problems. Moreover, a good agreement between the present nonlinear model and experimental result was obtained.

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.

Non-Fourier heat conduction analysis of a 2-D plate with inner cracks at arbitrary direction angles

This paper investigates the problem of a 2-D plate containing internal cracks at arbitrary angles to the heating surface under thermal shock.The finite difference method is applied to solve the temperature distribution and a difference scheme of the oblique crack is developed.For the first time,a non-Fourier heat transfer analysis method of a 2-D plate with inner cracks at arbitrary direction angles is established.Specifically,for a strip with an inner crack paralleling to the surface,numerical results are compared with the analytical solutions by rotating the boundary,and perfect agreement is obtained.Numerical experiments are further presented to investigate the influence of crack orientation angle and multi-crack distribution on non-Fourier heat conduction.

Non-Fourier Heat Conduction Effects During High-Energy Beam Metalworking

Non-Fourier heat conduction induced by ultrafast heating of metals with a high-energy density beam was analyzed. The non-Fourier effects during high heat flux heating were illustrated by comparing the transient temperature response to different heat flux and material relaxation times. Based on the hyperbolic heat conduction equation for the non-Fourier heat conduction law, the equation was solved using a hybrid method combining an analytical solution and numerical inversion of the Laplace transforms for a semi- infinite body with the heat flux boundary. Analysis of the temperature response and distribution led to a crite- rion for the applicability of the non-Fourier heat conduction law. The results show that at a relatively large heat flux, such as greater than 108 W/cm2, the heat-affected zone in the metal material experiences a strong thermal shock as the non-Fourier effects cause a large step increase in the surface temperature. The results provide a method for analyzing transient heat conduction problems using a high-energy density beam, such as electron beam deep penetration welding.

THERMOELASTIC DAMPING IN A MICRO-BEAM RESONATOR TUNABLE WITH PIEZOELECTRIC LAYERS

In this paper,thermoelastic damping （TED） in a micro-beam resonator with a pair of piezoelectric layers bonded on its upper and lower surfaces is investigated.Equation of motion is derived and the thermoelasticity equation is governed using two dimensional non-Fourier heat conduction model based on continuum theory frame.Applying Galerkin discretization method and complex-frequency approach to solve the equations of coupled thermoelasticity,we study TED of a clamped-clamped micro-beam resonator.The presented results demonstrate that thickness of the piezoelectric layers and application of DC voltage to them can affect the TED ratio and the critical thickness value of the resonator.

Study on thermal wave based on the thermal mass theory

The conservation equations for heat conduction are established based on the concept of thermal mass.We obtain a general heat conduction law which takes into account the spatial and temporal inertia of thermal mass.The general law introduces a damped thermal wave equation.It reduces to the well-known CV model when the spatial inertia of heat flux and temperature and the temporal inertia of temperature are neglected,which indicates that the CV model only considers the temporal inertia of heat flux.Numerical simulations on the propagation and superposition of thermal waves show that for small thermal perturbation the CV model agrees with the thermal wave equation based on the thermal mass theory.For larger thermal perturbation,however,the physically impossible phenomenon pre-dicted by CV model,i.e.the negative temperature induced by the thermal wave superposition,is eliminated by the general heat conduction law,which demonstrates that the present heat conduction law based on the thermal mass theory is more reasonable.