Hyperbolic heat conduction in the semi-infinite body with the heat source which capacity linearly depends on temperature

It is commonly taken for granted that long-time solutions of Cattaneo's hyperbolic heat conduction equation tend to overlap the related parabolic solutions. Usually, for small times considerable qualitative as well as quantitative differences between the hyperbolic and parabolic solutions are observed which vanish completely for large times. However, in the case of a heat conducting body with the temperature dependent inner heat generation, the quantitative differences may grow with time. This arises from the feedback between the temperature and the source capacity. To illustrate this effect, Cattaneo's hyperbolic equation for the semi-infinite body, with the heat source which capacity is linearly dependent on temperature, is solved analytically by the Laplace transforms method. Received on 1 July 1997     

Heat conduction in a semi-infinite solid subject to time-dependent surface heat fluxes: an analytical study

A closed-form model for the computation of the transient temperature and heat flux distribution in the case of a semi-infinite solid of constant properties is investigated. The temperature and heat flux solutions are presented for time-dependent, surface-heat flux of the forms: (i) , (ii) , and (iii) , where is a real number and is a positive real number. The dimensionless (or reduced) temperature and heat flux solutions are presented in terms of the Whittaker function, the generalized representation of an incomplete Gamma functionI (b, x) which can also be expressed by the complementary error functions. It is also demonstrated that the present analysis covers some well known (classical) solutions as well as a family of new solutions for the heat transfer through a semi-infinite solid.

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Wärmeleitung in einem halbunendlichen Festkörper bei zeitveränderlichem Randwärmefluß: Eine analytische Untersuchung

Zusammenfassung Es wird ein geschlossenes Modell zur Berechnung der nichtstationären Temperatur- and Wärmestromverteilung für einen halbunendlichen Festkörper mit konstanten Stoffwerten untersucht. Die Lösungen für das Temperatur- und Wärmeflußfeld basieren auf folgenden Zeitgesetzen für den Randwärmefluß: (i) , (ii) , und (iii) wobei eine reelle Zahl und eine positive reelle Zahl ist. Die dimensionslosen Lösungen für das Temperatur- und Wärmflußfeld lassen sich in Form der Whittaker- Funktionen, der verallgemeinerten Darstellung einer unvollständigen Gamma-FunktionI (b, x) angeben, welche auch durch das komplementäre Fehlerintegral ausgedrückt werden kann. Es wird ferner gezeigt, daß die hier durchgeführte Untersuchung sowohl einige bekannte (klassische) Lösungen für die Wärmeleitung im halbunendlichen Festkörper liefert, wie auch eine Familie von neuen Lösungen.

     

Heat conduction in a semi-infinite solid when subjected to spatially decaying instantaneous laser source

A closed-form model for the computation of temperature and heat flux distribution in a semi-infinite solid when subjected to spatially decaying, instantaneous laser source is investigated. The appropriate dimensionless parameters are identified and the reduced temperature and heat flux as a function of these parameters are presented in the graphic form. Some special cases of practical interest are also discussed. It is demonstrated that the present analysis covers the classical case of no heat generation in the solid as well as some new solutions.

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Wärmeleitung in einem halbunendlichen Körper, der plötzlich durch eine mit zunehmender Eindringtiefe abklingende Laser-Wärmequelle beaufschlagt wird

Zusammenfassung Es wird ein geschlossenes Rechenmodell zur Ermittlung der Temperatur- und Wärmeflußverteilung in einem halbunendlichen Körper untersucht, der plötzlich durch eine mit der Eindringtiefe abklingende Laserquelle beaufschlagt wird. Die normierten Temperaturen und Wärmeflüsse sind als Funktionen relevanter Kenngrößen graphisch dargestellt. Spezialfälle von praktischem Interesse werden diskutiert. Die vorliegende Untersuchung liefert neben einigen neuen Lösungen auch das Ergebnis für den klassischen Fall, daß keine Wärmequellen wirksam sind.

     

The approximate solutions to the non-linear heat conduction problems in a semi-infinite medium

The approximate solutions to the non-linear heat conduction problems in a semi-infinite medium are investigated. The entire temperature range is divided into a number of small sub-regions where the thermal properties can be approximated to be constant. The resulting problems can be considered as the Stefan’s problem of a multi-phase with no latent heat and the exact solutions called Neumann’s solution are available. In order to obtain the solutions, however, a set of highly non-linear equations in determining the phase boundaries should be solved simultaneously. This work presents a semi-analytic algorithm to determine the phase boundaries without solving the highly non-linear equations. Results show that the solutions for a set of highly non-linear equations depend strongly on the initial guess, bad initial guess leads to the wrong solutions. However, the present algorithm does not require the initial guess and always converges to the correct solutions.     

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.     

半无限介质瞬态热传导的同伦分析法

In the current work, transient heat conduction in a semi-infinite medium is considered for its many applications in various heat fields. Here, the homotopy analysis method （HAM） is applied to solve this problem and analytical results are compared with those of the exact and integral methods results. The results show that the HAM can give much better approximations than the other approximate methods： Changes in heat fluxes and profiles of temperature are obtained at different times and positions for copper, iron and aluminum.     

Conduction of heat in a semi-infinite solid with an exponential-type initial temperature profile: temperature and heat flux solutions due to an instantaneous laser source

A closed-form model for the computation of temperature and heat-flux distribution in a semi-infinite solid with an exponentially-decaying, initial-temperature profile is investigated. The solutions are presented for the case of an instantaneous laser source which is absorbed partially in the surface layers followed by an exponential decay with position in the material itself. The appropriate dimensionless parameters are identified and the reduced temperature and heat flux solutions are presented in the graphic form as a function of these parameters. Some limiting cases of practical interest are also discussed. It is demonstrated that the present analysis covers the classical case of no heat generation in the solid as well as some new solutions.     

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.     

A general solution of 3-D quasi-steady-state problem of a moving heat source on a semi-infinite solid

The well-known Jaeger–Rosenthal asymptotic particular solution for the quasi-steady-state problem of moving heat source is proven to be inconsistent with the source constant intensity, especially at dimensionless trailing edge coordinates vx/a < −2. The problem is reduced to an equivalent Poisson’s equation by exponential transformation of moving coordinate scale. Using the method of images, the fundamental solution is found; the temperature rise function exponentially approximates to 0 along negative semi-axis. The temperature field in a semi-infinite solid for the general case of surface power intensity distribution is expressed, using the found Green’s function. The cases of point, line, and circular heat sources are considered. The found fundamental solution and particular solution for moving circular heat source explain the phenomena of martensite transformation in low-carbon steel substrate at relatively low source velocity 1.7 cm/s.     

Study of the analytical solution to the heat transfer problem and surface temperature in a semi-infinite body with a constant heat flux at the surface and an initial temperature distribution

In many practical cases, one heats a semi-infinite solid with a constant heat flux source. For such an unsteady heat transfer problem, if the body has a uniform initial temperature, the analytical solution has been given by Carslaw and Jaeger. The surface temperature of the semi-infinite body follows the $\sqrt t $ -rule, that is, the surface temperature changes in proportion to square root of heating time. But if, instead of the uniform initial temperature, the body has a temperature distribution at the beginning of heating, the analytical solution has not yet been developed. Analytical solutions to the same problem with an exponential or a linear initial temperature distribution are obtained in this paper. It is shown, that in the case of a linear initial temperature distribution the surface temperature also changes according to $\sqrt t $ -rule Approximating the initial temperature distribution near the surface by its tangent at the surface, it is found that the surface temperature within a short time after the start of heating should also satisfy the $\sqrt t $ -rule, in spite of an arbitrary initial temperature distribution. The experimental data support this argument. Furthermore, the constant heat flux can be calculated after relationship between the surface temperature and heating time according to the equation derived in this paper, if the initial temperature distribution or its first-order derivative at the surface is known.     

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.     

Green's functions for a piezoelectric semi-infinite body with a fixed conductor surface

By using Stroh's formalism, simple explicit compact expressions of Green's functions for a piezoelectric semi-infinite body, with a fixed conductor surface electrode, subject to a singularity (i.e., a generalized line dislocation and a generalized line force at a point z°) are presented. Coulomb forces acted on the free line charge at z° due to the boundary polarization charges of the medium and the induction charges of the conductor together with the electromechanical coupling effects inside the region are analyzed in detail. The obtained results are valid not only for plane and antiplane problems but also for the coupled problems between inplane and outplane deformations.     

Thermographic validation of a novel,laminate body,analytical heat conduction model

The two-region fin model captures the heat spreading behaviour in multilayered composite bodies (i.e., laminates), heated only over a small part of their domains (finite heat source), where there is an inner layer that has a substantial capacity for heat conduction parallel to the heat exchange surface (convection cooling). This resulting heat conduction behaviour improves the overall heat transfer process when compared to heat conduction in homogeneous bodies. Long-term heat storage using supercooling salt hydrate phase change materials, stovetop cookware, and electronics cooling applications could all benefit from this kind of heat-spreading in laminates. Experiments using laminate films reclaimed from post-consumer Tetra Brik cartons were conducted with thin rectangular and circular heaters to confirm the laminate body, steady-state, heat conduction behaviour predicted by the two-region fin model. Medium to high accuracy experimental validation of the two-region fin model was achieved in Cartesian and cylindrical coordinates for forced external convection and natural convection, the latter for Cartesian only. These were conducted using constant heat flux finite heat source temperature profiles that were measured by infrared thermography. This validation is also deemed valid for constant temperature heat sources.     

Temperature field in a microperiodic two-layered composite caused by a circular laser heat source

The paper is connected with the determination of␣temperature field in a periodically layered half-space caused by a laser heating. Two types of thermal loadings distributed on a boundary of the body is considered: uniform and normal (Gaussian). The influence of geometrical and thermal parameters of the layered composite on the temperature field is analyzed. Received on 27 November 1997     

Non-Fourier heat conduction in a finite hollow cylinder with periodic surface heat flux

Analytical solution of the non-Fourier axisymmetric temperature field within a finite hollow cylinder exposed to a periodic boundary heat flux is investigated. The problem studied considering the Cattaneo–Vernotte (CV) constitutive heat flux relation. The material is assumed to be homogeneous and isotropic with temperature-independent thermal properties. The standard method of separation of variables is used for solving the problem with time-independent boundary conditions, and the Duhamel integral is used for applying the time dependency. The solution is applied for the special cases of harmonic uniform heat flux and an exponentially pulsed heat flux with Gaussian distribution in outer surface for modeling a laser pulse, and their respective non-Fourier thermal behavior is studied.