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.     

Approximate analysis for transient heat conduction with radiation in a slab

In an attempt to minimize the numerical computations associated with the solution of transient heat conduction with radiation in a slab, a perturbation type of analysis is being applied to the temperature field and radiation heat flux simultaneously. The resulting partial differential equations for the perturbation functions for the temperature are solved in explicit forms by use of the energy integral methods, while the radiation heat flux is determined by an appropriate scheme of approximating the temperature distribution in the slab. Included in the analysis are the effects of the parameters: the optical thickness, the ratio of conduction transport to radiation and the wall emissivity. It is found that, in a wide range of these governing parameters, the results compare very favorably with those obtained by the numerical solution of the formulated integro-differential equation. With the present analysis, the temperature, conduction and radiation heat fluxes can be predicted without resorting to lengthy numerical analysis.     

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.     

Non-Fourier heat conduction in an exponentially graded slab

The present article investigates one-dimensional non-Fourier heat conduction in a functionally graded material by using the differential transformation method. The studied geometry is a finite functionally graded slab, which is initially at a uniform temperature and suddenly experiences a temperature rise at one side, while the other side is kept insulated. A general non-Fourier heat transfer equation related to the functionally graded slab is derived. The problem is solved in the Laplace domain analytically, and the final results in the time domain are obtained by using numerical inversion of the Laplace transform. The obtained results are compared with the exact solution to verify the accuracy of the proposed method, which shows excellent agreement.     

Non-fourier heat conduction in a plane slab with prescribed boundary heat flux

The non-stationary heat conduction in an infinitely wide plane slab with a prescribed boundary heat flux is studied. An arbitrary time dependent boundary heat flux is considered and a non-vanishing thermal relaxation time is assumed. The temperature and the heat flux density distributions are determined analytically by employing Cattaneo-Vernotte's constitutive equation for the heat flux density. It is proved that the temperature and the heat flux density distributions can be incompatible with the hypothesis of local thermodynamic equilibrium. A comparison with the solution which would be obtained by means of Fourier's law is performed by considering the limit of a vanishing thermal relaxation time.     

Thermal and mechanical analysis of material response to non-steady ramp and steady shock wave loading

Ramp wave experiments on the Sandia Z accelerator provide a new approach to study the rapid compression response of materials at pressures, temperatures and stress or strain rates not attainable in conventional shock experiments. Due to its shockless nature, the ramp wave experiment is often termed as an isentropic (or quasi-isentropic) compression experiment (ICE). However, in reality there is always some entropy produced when materials are subjected to large amplitude compression even under shockless loading. The entropy production mechanisms that cause deformation to deviate from the isentropic process can be attributed to mechanical and thermal dissipations. The former is due to inelasticity associated with various deformation mechanisms and the rate effect that is inherent in all the deformation processes and the latter is due to irreversible heat conduction. The main purpose of the current study is to gain insights into the effects of ramp and shock loading on the entropy production and thermomechanical responses of materials. Another purpose is to investigate the role of heat conduction in the material response to both the non-steady ramp wave and steady shock.Numerical simulations are used to address the aforementioned research objectives. The thermomechanical response associated with a steady shock wave is investigated first by solving a set of nonlinear ordinary differential equations. Using the steady wave solutions as the reference, the material responses under non-steady ramp waves are then studied with numerical wave propagation simulation. It is demonstrated that the material response to ramp and shock loading is essentially a manifestation of the interaction between the time scale associated with the loading and the intrinsic time scales associated with mechanical deformation and heat transfer. At lower loading rates as encountered in ramp loading, the loading path is closer to an isentrope and results in lower entropy production. The reasonable ramp rate to obtain a quasi-isentropic state depends on the intrinsic time scales of the dissipation mechanisms which are strongly material dependent. Thus shockless loading does not necessarily produce an isentropic response. Between two equilibrium states, heat conduction was shown to have significant effect on the temperature history but it contributes little to the overall temperature change if the specific heat remains constant. It also affects the history of entropy, but only the irreversible part of heat conduction contributes to the net entropy change. The various types of thermomechanical responses of materials would manifest themselves more significantly in terms of the thermal history than the mechanical history. Thus temperature measurement appears to be an important experimental tool in distinguishing the various mechanisms for the thermomechancial responses of the materials.     

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。     

钢-混凝土组合梁收缩徐变效应的随机分析

钢-混凝土组合梁是由混凝土板和钢梁通过剪力键连接而成的组合结构。由于混凝土的收缩徐变,将引起结构内力和应力重分布。混凝土收缩徐变具有离散性大的特点,进而导致结构长期响应表现出随机性。本文综合考虑徐变模型、收缩模型、混凝土抗压强度、混凝土弹性模量、环境湿度、钢材弹性模量、荷载以及剪力键刚度的随机性对钢-混凝土组合梁结构响应的影响。利用拉丁超立方抽样技术和基于响应面方法的蒙特卡洛抽样,研究了钢-混凝土组合梁挠度和应力时变效应的概率问题。     

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.     

Analytical solutions for the time-dependent behaviour of composite beams with partial interaction

This paper presents a generic modelling for the time-dependent analysis of composite steel–concrete beams with partial shear interaction that occurs due to the deformation of the shear connection. The time effects considered in this modelling are those that arise from shrinkage and creep deformations of the concrete slab, and these effects are modelled using algebraic representations such as those of the age-adjusted effective modulus method (AEMM) and the mean stress method (MS), which are viscoelastic models for concrete deformation that can be stated algebraically. The generic model lends itself to closed form solutions for the analysis of composite beams subjected to a generic applied loading under a variety of end conditions. In this paper, the generic model is applied for the time-dependent analysis of composite beams that are simply supported and encastré, and to a propped cantilever, that are subjected to uniformly distributed loading and shrinkage deformations. Various representations of the structural behaviour of these beams are given in closed form which can also be used to benchmark available modelling techniques, i.e. finite element and finite difference formulations, which require a spatial discretisation to be specified as well as the time discretisation to perform a time analysis.     

Hamiltonian description of the heat conduction

It is shown that the linear boundary value problems of the heat conduction in a homogeneous slab can be mapped on the initial value problem for a Hamiltonian motion whose phase-space trajectories are subject to an additional restriction, the “arrival condition”. The physical consequences of this formal analogy for the macroscopic heat conduction are discussed in detail. Received on 27 April 1998     

Transient heat flow in a composite slab-constant flux, zero flux boundary conditions

Summary The time-dependent, one-dimensional equation of heat conduction is solved for a slab of two layers in perfect thermal contact. At one boundary there is a constant heat flux into the slab, and at the other boundary there is a zero flux. The solution for the temperature distribution is obtained with the aid of the Laplace transformation.This work was supported by the U.S. Naval Weapons Evaluation Facility, Albuquerque, N. M., U.S.A.     

An improved lumped analysis for transient heat conduction by using the polynomial approximation method

In this study, unsteady state one-dimensional heat conduction is analyzed using a polynomial approximation method. As a classical lumped model is only applicable for use with Biot numbers of less than 0.1, and additionally, it cannot be used for high-temperature gradients within the region, an improved lumped model is implemented for a typical long slab, long cylinder and sphere. It has been shown that in comparison to a finite difference solution, the improved model is able to calculate average temperature as a function of time for higher value of Biot numbers. The comparison also shows that the presented model has better accuracy when compared with others recently developed models. The simplified relations obtained in this study can be used for engineering calculations in many conditions.     

Subcritical crack growth of edge and center cracks in façade rock panels subject to periodic surface temperature variations

This paper examines subcritical cracking in a rock panel or slab containing either a pre-existing edge or a center crack perpendicular to the panel surface. The panel is subject to periodic surface temperature variation on one side of the panel while the other is kept at a constant temperature. The thermally induced stress intensity factors are determined using superposition technique by employing the fundamental point load solution for an edge crack or a center crack in a slab of finite thickness. Rock panel is modeled as a long elastic strip with either a free or a fully constrained lateral end condition. The temperature variations versus time at various depths of the rock panel appear roughly as a sinusoidal function. The lateral thermal stress for the free end case is larger than the constrained end case; whereas stress intensity factors for both edge and center cracks in the constrained end slab are 1000 times larger than that of free end case. Subcritical crack propagation in rock panels on façade is then estimated as a function of time. This subcritical crack propagation continues until a critical crack size is attained and the rock panel will fail under wind load. This new theoretical framework provides a new paradigm to examine the mechanisms of time-dependent cracking in rock panels on façade of buildings.     

Network simulation method for solving phase-change heat transfer problems with variable thermal properties

 The transient heat conduction equation in a finite slab undergoing phase change (two-phase problem of melting and solidification), with isothermal, adiabatic or convective boundary conduction is studied by the network simulation method; solid phase conductivity and specific heat are assumed to be dependent on temperature. Ablation, as a particular case, is also analysed. A network model is established for a cell and boundary conditions are added to complete the whole network model. No restrictions exist, as to the kinds of linear and non-linear boundary conditions, Stefan number values or the initial conditions (when hypotheses concern of the Stefan problem, numerical and exact solutions are compared for a large interval of Stefan numbers; simulation values show good agreement). Movement of the solid–liquid boundary and thermal fields are determined in all cases. Received on 10 May 2000 / Published online: 29 November 2001     

A steady conjugate heat transfer problem with conduction and free convection

Summary A steady conjugate heat transfer problem dealing with conduction in a heat-generating slab and free convection in the surrounding fluid is studied analytically. Free convection is analyzed by a Görtler-type series solution to the boundary-layer equations for non-uniform surface-temperature variations, while conduction is treated by the standard technique of Fourier transforms. Interfacial temperature and heat flux variations from both solutions in series forms are then formally matched to yield algebraic relations for the coefficients in the series. These coefficients can then be simply evaluated in a given problem in terms of three physical parameters. A numerical example is shown.     

An inverse problem to estimate temperature dependent heat capacity under convection processes

Solutions of the heat capacity versus temperature in a one-dimensional slab have been studied for different types of dependency (lineal, sinusoidal, piece-wise and rectangular) under boundary conditions of natural and forced convection on both sides of the slab. The input data of this inverse problem are the temperature history ("measurements") at a particular location within the slab, obtained by adding a specified random error to the set of temperatures which are the solution of the direct problem. No prior information is used as regards the temperature-dependent functional forms of the unknown heat capacity. In all cases, a piece-wise function is used to approach the solution. Using a programming routine that minimises a classical predefined functional, successive stretches of this piece-wise function are obtained step by step by (i) fixing its length and (ii) increasing or decreasing its slope. The Network Simulation Method is used to solve both the direct and inverse problems. No mathematical manipulations of the finite-difference differential equations are required by the programmer, since they are contained in the computer code used in the method. The basic network for the inverse problem, which is basically the same as for the direct problem, is easy to design and has very few devices. Several examples are shown to prove the accuracy and effectiveness of the proposed method.     

Periodic heat conduction with relaxation time in cylindrical geometry

Steady-periodic heat conduction with relaxation time in an infinitely long hollow cylinder is considered. Four boundary value problems, with boundary conditions of the first and of the second kind, are solved analytically. The solution for a solid cylinder with a sinusoidally varying surface temperature is obtained as a special case of a solution found for the hollow cylinder. The effects of the relaxation time on the steady-periodic temperature field are analysed, in details, for a solid cylinder with a sinusoidally varying surface temperature and for a hollow cylinder with a sinusoidally varying heat flux at the inner surface and with a constant temperature at the outer surface. The results show that thermal resonances may occur and suggest that accurate measurements of the relaxation time could be obtained by means of experiments on steady-periodic heat conduction in cylindrical geometry. Received on 15 April 1997