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.

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.

Analytical solution for the time-fractional heat conduction equation in spherical coordinate system by the method of variable separation

In this paper,using the fractional Fourier law,we obtain the fractional heat conduction equation with a time-fractional derivative in the spherical coordinate system.The method of variable separation is used to solve the timefractional heat conduction equation.The Caputo fractional derivative of the order 0 〈 α≤ 1 is used.The solution is presented in terms of the Mittag-Leffler functions.Numerical results are illustrated graphically for various values of fractional derivative.

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.

Effect of thermal conductivity on heat transfer for a power-law non-Newtonian fluid over a continuous stretched surface with various injection parameters

The two-dimensional non-Newtonian steady flow on a power-law stretched surface with suction or injection is studied. Thermal conductivity is assumed to vary as a linear function of temperature. The transformed governing equations in the present study are solved numerically using the Runge-Kutta method. Through a comparison, results for a special case of the problem show excellent agreement with those in a previous work. Two cases are considered, one corresponding to a cooled surface temperature and the other to a uniform surface temperature. Numerical results show that the thermal conductivity variation parameter, the injection parameter, and the power-law index have significant influences on the temperature profiles and the Nusselt number.

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.

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.

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.

Statistical Analysis of Subsurface Diffusion of Solar Energy with Implications for Urban Heat Stress

Analysis of hourly underground temperature measurements at a medium-size (by population) US city as a function of depth and extending over 5+ years revealed a positive trend exceeding the rate of regional and global warming by an order of magnitude. Measurements at depths greater than ~2 m are unaffected by daily fluctuations and sense only seasonal variability. A comparable trend also emerged from the surface temperature record of the largest US city (New York). Power spectral analysis of deep and shallow subsurface temperature records showed respectively two kinds of power-law behavior: 1) a quasi-continuum of power amplitudes indicative of Brownian noise, superposed (in the shallow record) by 2) a discrete spectrum of diurnal harmonics attributable to the unequal heat flux between daylight and darkness. Spectral amplitudes of the deepest temperature time series (2.4 m) conformed to a log-hyperbolic distribution. Upon removal of seasonal variability from the temperature record, the resulting spectral amplitudes followed a log-exponential distribution. Dynamical analysis showed that relative amplitudes and phases of temperature records at different depths were in excellent accord with a 1-dimensional heat diffusion model.

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.

一维热传导方程推导及定解条件小结

以均匀且各向同性的细棒为例,用两种方法推导细棒在传热过程中温度所满足的偏微分方程,且对一些常见的定解条件进行总结说明.

Thermal vibration phenomenon of single phase lagging heat conduction and its thermodynamic basis

In the present work it is shown that the single phase lagging heat conduction not only avoids the infinite heat propagation speed assumed by the conventional Fourier law, but also complies with Galilean principle of relativity. Therefore it is more advantageous than the Cattaneo-Vernotte model. Based on the single-phase-lagging heat conduction model, the condition for the occurrence of thermal vibration of heat conduction is established. In order to resolve the contradiction that the thermal vibration violates the second law of thermodynamics, the extended irreversible thermodynamics is improved and a generalized entropy definition is introduced. In the framework of the newly-developed extended irreversible thermodynamics the thermal vibration phenomena are consistent with the second law of thermodynamics.

大跨度斜拉桥钢-混组合梁一维日照温度场高效数值模型构建及试验验证

由于内部材料的分层,钢-混组合梁整体竖向温度梯度分布不均匀,温度场求解困难且模拟精度和效率不理想。该文以傅里叶热传导定律为理论基础,推导了钢-混组合梁的一维日照温度场数值计算通用模型,采用逐次超松弛迭代法进行求解;通过一个钢-混组合梁数值案例,对比分析了该方法与COMSOL的计算精度和效率;基于某大跨度斜拉桥的温度健康监测数据,验证了该方法对于日温度时程和全截面竖向温度梯度仿真的精度,并将竖向温度梯度的仿真结果与我国桥梁规范进行了对比。结果表明:该方法计算精度与物理场仿真软件保持一致,计算效率提升了78.3%,并能准确模拟大跨度斜拉桥钢-混组合梁的竖向温度分布,为大跨度斜拉桥钢-混组合梁温度效应的计算提供了强有力工具。

基于有限元分析方法对高温作业专用服装设计的研究

本文针对的主要问题是在时间和厚度等可变因素的共同影响下,运用有限元分析方法,设计出合理的一维热传导模型,并应用到实际的高温防护服设计中。考虑到三维模型较为复杂,本文采用降低模型维度的方法进行分析,由简入繁。根据题目中给定的基本数据,利用Fourier定律、能量守恒定律、合理的数学建模知识以及使用Matlab计算软件,解决了题目中的两个问题。

瞬态非傅里叶导热效应判据的探讨

讨论了通用傅里叶导热定律的数学模型 ,推导了半无限大物体在第一类边界条件下的基于非傅里叶导热定律的双曲线型偏微分方程的解析解。通过分析非傅里叶导热定律在瞬态条件下温度分布的变化过程 ,提出了非傅里叶导热效应的瞬态判据以及非傅里叶导热的作用范围 ,对研究工程中瞬态导热问题 (如激光处理 ,电子束处理等 )

强瞬态非傅立叶导热效应判据与激光冲击硬化应用的探讨

为了更好地反映激光冲击硬化的强瞬态特征及建立非傅立叶导热效应作用范围的判据 ,本文讨论了通用傅立叶导热定律的数学模型 ,推导了半无限大物体在第一类边界条件下基于非傅立叶导热定律的双曲线型偏微分方程的解析解。通过分析非傅立叶导热定律在瞬态条件下温度分布的变化过程 ,提出了非傅立叶导热效应的强瞬态判据。并就强瞬态导热现象对激光热处理的影响进行了讨论。指出按照非傅立叶导热定律计算能更好反映激光作用的热冲击特征。

热质

热是介质分子无规运动的能量。按爱因斯坦的质能关系式可求得热能所当量的质量,称为热质。从参量热流密度中可导出热质速度,建立了声子气(热质气)的状态方程后可求得热质运动的驱动力。鉴于热量在介质中传递本质上是热质在介质中的运动,所以基于热质守恒与热质动量守恒的热质运动方程,能普适地描述热量传递过程,因此它就是普适导热定律。当热流密度不是很大时,普适导热定律退化为傅里叶导热定律;当瞬态热流很大但忽略空间惯性力时,普适导热定律退化为CV(Cattaneo-Vernotte,CV)模型;而当稳态热流密度很大时,普适导热定律退化为有空间惯性力项的稳态导热定律,它表明稳态导热情况下也会出现非傅里叶导热。