Time-fractional radial heat conduction in a cylinder and associated thermal stresses

The theory of thermal stresses based on the heat conduction equation with the Caputo time-fractional derivative of order 0 < α ≤ 2 is used to investigate axisymmetic thermal stresses in a cylinder. The solution is obtained applying the Laplace and finite Hankel integral transforms. The Dirichlet and two types of Neumann problems with the prescribed boundary value of the temperature, the normal derivative of the temperature, and the heat flux are considered. Numerical results are illustrated graphically.     

On the thermal rectification factor in steady heat conduction

Thermal rectification in heat conduction problems has been extensively studied in planar slabs. Here we consider the rectification problem in planar, cylindrical and spherical geometries involving two layers one of which has a temperature variable heat conductivity. The rectification factor is analytically calculated. It is shown that a maximum theoretical value of 1.618 is obtained.     

Inverse transient heat conduction problems and identification of thermal parameters

This work deals with the estimation of polymers properties. An inverse analysis based on finite element method is applied to identify simultaneously the constants thermal conductivity and heat capacity per unit volume. The inverse method algorithm constructed is validated from simulated transient temperature recording taken at several locations on the surface of the solid. Transient temperature measures taped with infrared camera on polymers were used for identifying the thermal properties. The results show an excellent agreement between manufacturer and identified values.     

The thermal response of a metal slab to a class of radially dependent heat inputs

Summary The temperature fields are deduced for an infinite metal slab of finite thickness produced by a heat flux input which depends on the radial coordinate. The formal solution is obtained by means of transform techniques.Several representations of the Green's function are obtained as well as two approximate representations. One of these approximations converges well for time large the other for time small.A small time asymptotic expansion is given for input flux densities which satisfy certain smoothness requirements.Two inputs have been considered in detail. One is the uniform spot and the other is the Gaussian. An extension of the results obtained by Oosterkamp³) has been obtained along with an estimate of the validity.The work reported in this paper was supported in part by Contract AF 30(602)-2305 between The Ohio State University Research Foundation and Rome Air Development Center, Griffiss Air Force Base, New York.     

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.     

Non-Fourier heat conduction by axisymmetric thermal waves in an infinite solid medium

The non-stationary heat conduction in an infinite solid medium internally bounded by an infinitely long cylindrical surface is considered. A uniform and time- dependent temperature is prescribed on the boundary surface. An analytical solution of the hyperbolic heat conduction equation is obtained. The solution describes the wave nature of the temperature field in the geometry under consideration. A detailed analysis of the cases in which the temperature imposed on the boundary surface behaves as a square pulse or as an exponentially decaying pulse is provided. The evolution of the temperature field in the case of hyperbolic heat conduction is compared with that obtained by solving Fourier's equation. Received on 28 January 1998     

A generalized thermal boundary condition for the hyperbolic heat conduction model

 A generalized thermal boundary condition is derived for the hyperbolic heat conduction equation to include all thermal effects of a thin layer, whether solid-skin or fluid film, moving or stationary, in perfect or imperfect thermal contact with an adjacent domain. The thin layer thermal effects include, among others, thermal capacity of the layer, thermal diffusion, enthalpy flow, viscous dissipation within the layer and convective losses from the layer. Six different kinds of thermal boundary conditions can be obtained as special cases of the generalized boundary condition. The importance of the generalized boundary condition is demonstrated comprehensively in an example. The effects of different geometrical and thermophysical properties on the validity of the generalized thermal boundary condition are investigated. Received on 23 May 2001 / Published online: 29 November 2001     

Effect of thermal losses on the microscopic hyperbolic heat conduction model

The effects of radiative losses on the thermal behavior of thin metal films, as described by the microscopic two-step hyperbolic heat conduction model, are investigated. Different criteria, which determine the ranges within which thermal radiative losses are significant, are derived. It is found that radiative losses from the electron gas are significant in thin films having [(C_R e_e^4/₃ T_￥ ⁴ )/(k_e^1/₃ L^2/₃ G)] 3 4.6 ×10⁷{{C_R \epsilon _e^{{4 \over 3}} T_\infty ^4 } \over {k_e^{{1 \over 3}} L^{{2 \over 3}} G}}\geq 4.6 \times 10^7 for /_o > 4 and F_F < 1 and [(C_R e_e^3/₂ T_￥ ^9/₂)/(k_e^1/₂ L^1/₂ G)] 3 7.4 ×10¹⁰{{C_R \epsilon _e^{{3 \over 2}} T_\infty ^{{9 \over 2}}} \over {k_e^{{1 \over 2}} L^{{1 \over 2}} G}}\geq 7.4 \times 10^{10} for /_o < 4 and F_F > 1.     

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.     

Propagation of blast waves with exponential heat release and internal heat conduction and thermal radiation

The problem of reactive blast waves in a combustible gas mixture, where the heat release at the detonation front decays exponentially with the distance from the center, is analyzed. The central theme of the paper is on the propagation of reactive blast into a uniform, quiescent, counterpressure atmosphere of a perfect gas with constant specific heats. The limiting cases of Chapman-Jouguet detonation waves are considered in the phenomenon of point explosion. In order to deal with this problem, the governing equations including thermal radiation and heat conduction were solved by the method of characteristics using a problem-specific grid and a series expansion as start solution. Numerical results for the distribution of the gas-dynamic parameters inside the flow field are shown and discussed.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Analytical solution of the overdetermined inverse heat conduction problem with an application to monitoring thermal stresses

The paper presents a technique for determining the transient temperature distribution, when input data as thermocouple responses are known at several interior locations. If the temperature field is known, then the thermal stresses can be calculated. The problem is overdetermined and is solved using a least squares method that minimizes the error between the computed and measured thermocouple temperatures. The present method incorporates the advantages of simplicity and accuracy of analytical solutions. Several numerical examples and measurements are presented as an indication of the accuracy of the presented method. Received on 10 January 1997     

Transient heat conduction with uniform heat generation in a slab subjected to convection and radiation cooling

A finite difference scheme with fourth order Runge-Kutta method is employed to determine the unsteady state temperature distribution in a plane slab with uniform heat generation. The plane slab is insulated on one face and subjected to convective and radiative cooling at the other face. The plane slab has a uniform initial temperature and the ambient environment as well as the fluid temperatures are assumed to be constant. Heat conduction is considered to be one dimensional. Results are presented in dimensionless charts over a wide range of parameters.

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Instationäre Wärmeleitung mit gleichförmiger Wärmeerzeugung in einer Platte bei Kühlung durch Konvektion und Abstrahlung

Zusammenfassung Zur Ermittlung der instationären Temperaturverteilung in einer Platte mit gleichförmiger Wärmeerzeugung wird ein Differenzverfahren vierter Ordnung nach Runge-Kutta angewendet. Die ebene Platte ist einseitig isoliert und wird auf der anderen Seite durch Konvektion und Abstrahlung gekühlt. Zu Beginn befindet sich die Platte gleichförmig auf einer bestimmten Anfangstemperatur, die Temperaturen der umgebenden Objekte sowie des Fluids sind ebenfalls konstant. Der Wärmeleitungsvorgang sei eindimensional. Die Ergebnisse sind in dimensionsloser Form für einen weiten Parameterbereich in Diagrammform dargestellt.

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Nomenclature Bi Biot number - c specific heat - h heat transfer coefficient - k thermal conductivity - L half thickness of plate - Nr Radiation number - T absolute temperature - t time - F _o dimensionless time - T _e environment temperature - T _f fluid temperature - T _i initial plate temperature - u internal energy generation number - u internal energy generation - x space coordinate - x* dimensionless space coordinate Greek symbols thermal diffusivity - emissivity - dimensionless temperature - density - Stephan-Boltzman constant     

Perturbation solution to heat conduction in melting or solidification with heat generation

The Stefan problem involving a source term is considered in this technical note. As an example, planar solidification with time-dependent heat generation in a semi-infinite plane is solved by use of a perturbation technique. The perturbation solution is validated by reducing the problem to the case without heat generation whose exact solution is available. An application to the case with constant heat generation is presented, for which a closed-form solution is obtained. The effects of heat generation and Stefan number on the evolution of solidification are examined using the perturbation solution.     

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.     

Contribution to the heat conduction in composite infinite solids

Summary Comparatively little attention has been paid to problems on heat conduction in a homogeneous and isotropic infinite solid bounded internally by a circular cylinder. And it is almost impossible to find some results on composite media of this kind. The present paper brings new facts on the subject.     

Analysis of heat conduction in a disk brake system

In this paper, the governing heat equations for the disk and the pad are extracted in the form of transient heat equations with heat generation that is dependant to time and space. In the derivation of the heat equations, parameters such as the duration of braking, vehicle velocity, geometries and the dimensions of the brake components, materials of the disk brake rotor and the pad and contact pressure distribution have been taken into account. The problem is solved analytically using Green’s function approach. It is concluded that the heat generated due to friction between the disk and the pad should be ideally dissipated to the environment to avoid decreasing the friction coefficient between the disk and the pad and to avoid the temperature rise of various brake components and brake fluid vaporization due to excessive heating.