Transient heat conduction in spheres for Fo < 0.3 and finite Bi

Using Laplace transforms, an analytical solution was obtained for transient heat conduction in spheres exposed to surroundings at a uniform temperature and finite Bi numbers. The solution is explicit, and valid during early transients, for Fourier numbers Fo < 0.3     

Transient hyperbolic heat conduction in thick-walled FGM cylinders and spheres with exponentially-varying properties

This paper focuses on non-Fourier hyperbolic heat conduction analysis for heterogeneous hollow cylinders and spheres made of functionally graded material (FGM). All the material properties vary exponentially across the thickness, except for the thermal relaxation parameter which is taken to be constant. The cylinder and sphere are considered to be cylindrically and spherically symmetric, respectively, leading to one-dimensional heat conduction problems. The problems are solved analytically in the Laplace domain, and the results obtained are transformed to the real-time space using the modified Durbin’s numerical inversion method. The transient responses of temperature and heat flux are investigated for different inhomogeneity parameters and relative temperature change values. The comparisons of temperature distribution and heat flux between various time and material properties are presented in the form of graphs.     

Steady state Joule heating with temperature dependent conductivities

A general solution is given to the nonlinear steady state heat conduction equation for the case in which a metal is heated by electrical conduction currents. The solution is valid for any temperature variation in the thermal and electrical conductivities and is illustrated by application to a typical Joule heating situation in one dimension. Comparison of the case of temperature dependent conductivities for a metal with the solution which assumes both of these conductivities to remain constant with varying temperature reveals some interesting differences between the two cases.     

Transient conjugate mixed convection from a sphere in a porous medium saturated with cold pure or saline water

In this paper, a numerical investigation of the transient conjugate mixed convection flow about a sphere embedded in a porous medium saturated with pure or saline water is carried out. The effect of density extremum is considered by using the nonlinear dependence of density on the temperature. The salinity effects are considered by assuming uniform saline concentration over the domain considered. The direction of the natural convection is changed either to aiding or to opposing the upcoming flow direction simulating the sphere is either hot or cold relative to the surrounding temperature. Results show that the initial temperature differences as well as the saline concentration alter the transient heat transfer rate in conceivable degree. It was found that the heat capacity ratio between the sphere and the surrounding media has more significant effect on the calculated heat transfer rate than the thermal conductivity ratio. The study is performed by using six nondimensional parameters and results are discussed in detail. Received on 10 November 1997     

Experimental study of steady and transient natural convection heat transfer of mercury

An experimental investigation of transient and steady-state natural convection in a narrow vertical rectangular channel following a step-change in uniform wall-heat-flux is presented. The construction and instrumentation for two test sections are described. These test sections formed a rectangular channel 15.2×2.54×25.4 cm and consisted of: 1) both 15.2×25.4 cm faces heated uniformly by constant radiant heat flux with mercury as the fluid, and 2) the same boundary conditions as 1 but lead was used to thermally model the mercury. Initially the fluid was stagnant and at a uniform temperature. The transient was initiated by suddenly increasing the wall-heat-flux from zero to some constant, preselected value using radiant heating. Temperature-time histories were measured during the transient and steady-state regimes at several locations on the wall and in the fluid. Transient and steady-state heat transfer results are reported. The results show that when the wall-heat-flux on both faces is sufficiently large, the primary mechanism for energy transport in the fluid is molecular conduction. For lower values of imposed heat flux, natural convection, as well as conduction, contributed to the energy transfer.     

Stochastic thermoelastic problem of a functionally graded plate under random temperature load

This study attempts to derive the statistics of temperature and thermal stress in functionally graded material (FGM) plates exposed to random external temperatures. The thermomechanical properties of the FGM plates are assumed to vary arbitrarily only in the plate thickness direction. The external temperatures are expressed as random functions with respect to time. The transient temperature field in the FGM plate is determined by solving a nonhomogeneous heat conduction problem for a multilayered plate with linear nonhomogeneous thermal conductivity and different homogeneous heat capacity in each layer. The autocorrelations and power spectrum densities (PSDs) of temperature and thermal stress are derived analytically. These statistics for FGM plates composed of partially stabilised zirconia (PSZ) and austenitic stainless steel (SUS304) are computed under the condition that the fluctuation in the external temperature can be considered as white noise or a stationary Markov process.     

Ground thermal response to heat conduction in a power transmission tower foundation

An analytical formulation is developed to predict transient heat conduction in a semi-infinite medium with a vertical finite line heat source, which represents a buried tower of a power transmission line foundation. Unlike past studies with a constant line heat source, the current model develops a time-dependent variable heating strength, as well as a time varying surface temperature of the ground. An approximate VHS model (variable heating strength) is developed for sinusoidal variations of the line source strength and surface temperature, in order to simulate seasonal variations of ground temperatures. The VHS model reduces computational time and exhibits good accuracy, when compared against a full exact solution. The model is applied to heat conduction in a tower foundation, with time-varying ground surface temperatures. Effects of ground thermal conductivity and diffusivity, as well as variations of the line source strength, are investigated in this article.     

A study on the non-Fourier effects in spherical media due to sudden temperature changes on the surfaces

The non-Fourier effects on the dynamic thermal behavior of spherical media, including solid, hollow and bi-layered composite spheres, due to sudden temperature changes on the surfaces are investigated. The temperature and heat flux histories in the spherical media are predicted by an analytical–numerical technique. The speed of heat propagation is finite, as revealed in the temperature and heat flux calculated by using the hyperbolic heat-conduction equation. Effects of different parameters such as the relaxation time, the imposed temperature ratio on the inner and outer layers of the hollow sphere, the thermal diffusivity ratio, and the relaxation time ratio of the composite sphere are studied and presented.     

基于状态空间理论的功能梯度圆球瞬态热传导分析1)

基于状态空间理论研究功能梯度圆球的球对称瞬态热传导问题。根据热传导方程和热流密度的定义,取温度场和热流密度为系统的状态向量,通过将圆球分层和在时域内应用差分格式对控制方程进行离散,建立了系统的状态方程,给出了功能梯度圆球瞬态热传导问题的半解析解。算例分析表明：本文解不但结果正确、计算效率高,而且适用于材料参数沿径向任意梯度变化的圆球瞬态热传导分析。     

Transient thermal cracking associated with non-classical heat conduction in cylindrical coordinate system

This paper studies the fracture behavior of a thermoelastic cylinder subjected to a sudden temperature change on its outer surface within the framework of non-classical heat conduction.The heat conduction equation is solved by separation of variable technique.Closed form solution for the temperature field and the associated thermal stress are established.The critical parameter governing the level of the transient thermal stress is identified.Exact expression for the transient stress intensity factor is obtained for a crack in the cylinder.The difference between the non-classical solutions and the classical solution are discussed.It is found that the traditional classical heat conduction considerably underestimates the transient thermal stress and thermal stress intensity factor.     

Finite Péclet number forced convection past a sphere in a porous medium using a thermal nonequilibrium model

We study the finite-Péclet number forced convective heat transfer from a uniform temperature sphere placed in otherwise uniform fluid stream within a porous medium. A numerical study is undertaken to determine how the lack of local thermal equilibrium between the phases affects temperature fields of the two phases and the respective rates of heat transfer from the sphere. On the upstream side of the sphere the temperature field extends further from the sphere in the solid phase than it does for the fluid phase, but the opposite is true on the downstream side.     

一种考虑瞬态效应的散热结构导热路径设计的拓扑优化模型

实际工程中,热载荷多数具有短时和周期性特点,瞬态效应显著。目前的散热结构导热路径设计多基于稳态热传导模型,未考虑瞬态效应。本文提出了一种以区域温度控制函数作为设计目标的瞬态热传导问题的拓扑优化模型,能够实现在整个时间历程上特定区域内最高温度最小。使用伴随变量法,推导了目标函数关于设计变量的敏度计算格式。算例表明,基于本文优化模型获得的散热路径设计与基于稳态热传导模型的结果有明显差别,具有更优的散热性能。因此,时变热荷载下的散热结构构型设计需要考虑瞬态响应的影响。     

The effect of local thermal non-equilibrium on conduction in porous channels with a uniform heat source

We examine the effect of local thermal non-equilibrium on the steady state heat conduction in a porous layer in the presence of internal heat generation. A uniform source of heat is present in either the fluid or the solid phase. A two-temperature model is assumed and analytical solutions are presented for the resulting steady-state temperature profiles in a uniform porous slab. Attention is then focussed on deriving simple conditions which guarantee local thermal equilibrium.     

非傅立叶导热的最新研究进展

对迄今为止有关非傅立叶导热的研究成果进行了全面的综述,其中包括作者在该领域的最新研究进展:空心球体介质双曲线非傅立叶导热模型的分析求解,室温条件下多孔材料内非傅立叶导热的实验结果及数值模拟,非傅立叶导热的“瞬时薄层”模型,非傅立叶导热和非费克质量传递的耦合分析,非傅立叶导热的分子动力学模拟等．文中还对下一步的研究工作进行了展望.      

Similarity between unsteady-state conduction in a planar slab for short times and steady-state conduction in a uniform,straight fin

The principal goal of this study is two-fold. First, to elucidate an analogy between unsteady-state conduction in a planar slab for short times and steady-state conduction in a straight fin of uniform cross section. Second, to present approximate analytical solutions of the transient heat conduction equation for short times in a plane having a uniform initial temperature and subjected to a uniform surface temperature (Dirichlet boundary condition). Use was made of a hybrid computational method, theTransversal Method Of Lines (TMOL) to bypass the classical solution techniques for partial differential equations and exploit the physical analogy with the steady-state, heat conduction in a straight fin. The resulting quasi-steady, approximate analytical solution is very easy to employ and is suitable for obtaining quality short-time temperature distributions in the slab.     

Temperature and thermal stresses in material of a pad during braking

The transient temperature field and corresponding quasi-static thermal stresses are analysed in a system consisting of a semi-space and a strip. The strip is heated on its outer surface by a heat flux with the intensity equal to the specific power of friction during braking with a uniform retardation. The evolution and distribution in depth from a surface of friction for temperatures and thermal stresses were investigated for the metal-ceramic FMK-11 material of the strip.     

Transient heat conduction in two-dimensional functionally graded hollow cylinder with finite length

In this paper a thick hollow cylinder with finite length made of two dimensional functionally graded material (2D-FGM) subjected to transient thermal boundary conditions is considered. The volume fraction distribution of materials, geometry and thermal boundary conditions are assumed to be axisymmetric but not uniform along the axial direction. The finite element method with graded material properties within each element is used to model the structure and the Crank–Nicolson finite difference method is implemented to solve time dependent equations of the heat transfer problem. Two-dimensional heat conduction in the cylinder is considered and variation of temperature with time as well as temperature distribution through the cylinder are investigated. Effects of variation of material distribution in two radial and axial directions on the temperature distribution and time response are studied. The achieved results show that using two-dimensional FGM leads to a more flexible design so that transient temperature, maximum amplitude and uniformity of temperature distributions can be modified to achieve required specifications by selecting a suitable material distribution profile in two directions.     

Heat wave phenomena in solids subjected to time dependent surface heat flux

Hyperbolic heat conduction in a plane slab, infinitely long solid cylinder and solid sphere with a time dependent boundary heat flux is analytically studied. The solution is based on the separation of variables method and Duhamel’s principle. The temperature distribution, the propagation and reflection of the temperature wave and the effect of geometry on the shape of the wave front are studied for the case of a rectangular pulsed boundary heat flux. Comparisons with the solution obtained for Fourier heat conduction are performed by considering the limit of a vanishing thermal relaxation time.     

Effect of wall heat transfer on shock-tube test temperature at long times

When performing chemical kinetics experiments behind reflected shock waves at conditions of lower temperature (<1,000 K), longer test times on the order of 10–20 ms may be required. The integrity of the test temperature during such experiments may be in question, because heat loss to the tube walls may play a larger role than is generally seen in shock-tube kinetics experiments that are over within a millisecond or two. A series of detailed calculations was performed to estimate the effect of longer test times on the temperature uniformity of the post-shock test gas. Assuming the main mode of heat transfer is conduction between the high-temperature gas and the colder shock-tube walls, a comprehensive set of calculations covering a range of conditions including test temperatures between 800 and 1,800 K, pressures between 1 and 50 atm, driven-tube inner diameters between 3 and 16.2 cm, and test gases of N₂ and Ar was performed. Based on the results, heat loss to the tube walls does not significantly reduce the area-averaged temperature behind the reflected shock wave for test conditions that are likely to be used in shock-tube studies for test times up to 20 ms (and higher), provided the shock-tube inner diameter is sufficiently large (>8cm). Smaller diameters on the order of 3 cm or less can experience significant temperature loss near the reflected-shock region. Although the area-averaged gas temperature decreases due to the heat loss, the main core region remains spatially uniform so that the zone of temperature change is limited to only the thermal layer adjacent to the walls. Although the heat conduction model assumes the gas and wall to behave as solid bodies, resulting in a core gas temperature that remains constant at the initial temperature, a two-zone gas model that accounts for density loss from the core to the colder thermal layer indicates that the core temperature and gas pressure both decrease slightly with time. A full CFD solution of the shock-tube flow field and heat transfer at long test times was also performed for one typical condition (800 K, 1 atm, Ar), the results of which indicate that the simpler analytical conduction model is realistic but somewhat conservative in that it over predicts the mean temperature loss by a few Kelvins. This paper presents the first comprehensive study on the effects of long test times on the average test gas temperature behind the reflected shock wave for conditions representative of chemical kinetics experiments.     

Transient heat conduction from a vertical rod buried in a semi-infinite medium with variable heating strength

In this paper, a variable heating strength model (VHS model) is developed to predict transient heat conduction from a vertical rod buried in a semi-infinite medium. Unlike past studies, the current VHS model permits a VHS along the rod. Both axial heat conduction through the rod and lateral heat conduction to the surrounding ground are modeled. A derived distribution of axial heating strength is then applied to a finite line heat source model to predict transient temperature changes in the surrounding medium. The predicted results show how the rod’s radius and ground’s thermal conductivity affect the vertical variation of heating strength and temperature response. Additional simulations predict the long-term temperature increase in the ground, due to a power transmission tower installed in a region of initially frozen ground.