Critical parameters for thermal conduction equations

Thermal stability of two models, whose thermal conductivity is a function of temperature, is investigated. Studies are made for the cylindrical geometry for both exponetial and power forms for the temperature dependence of conductivity. The maximum value of a certain parameter, δ_cr, defines criticality and this is the condition for explosion. The purpose of this paper is to ascertain the way in which explosion is affected by boundary conditions and how changes in the values of a certain constant, β, affect the critical parameter δ_cr.     

Heat conduction and thermal stress induced by an electric current in an infinite thin plate containing an elliptical hole with an edge crack

A uniform electric current at infinity was applied to a thin infinite conductor containing an elliptical hole with an edge crack. The electric current gives rise to two states, i.e., uniform and uneven Joule heat. These two states must be considered to analyze the heat conduction problem. The uneven Joule heat gives rise to uneven temperature and thus to heat flux, and to thermal stress.Using a rational mapping function, problems of the electric current, the Joule heat, the temperature, the heat flux, the thermal stress are analyzed, and each of their solutions is obtained as a closed form. The distributions of the electric current, the Joule heat, the temperature, the heat flux and the stress are shown in figures.The heat conduction problem is solved as a temperature boundary value problem. Solving the thermal stress problem, dislocation and rotation terms appear, which complicates this problem. The solutions of the Joule heat, the temperature, the heat flux and the thermal stress are nonlinear in the direction of the electric current. The crack problems are also analyzed, and the singular intensities at the crack tip of each problem are obtained. Mode II (sliding mode) stress intensity factor (SIF) is produced as well as Mode I (opening mode) SIF, for any direction of the electric current. The relations between the electric current density and the melting temperature and between the electric current density and SIF are investigated for some crack lengths in an aluminum plate.     

The interaction of thermal conduction and infrared gaseous radiation

Summary The present investigation is concerned with energy transfer in gases which transmit heat by the combined mechanisms of molecular conduction and infrared radiation. For illustrative purposes an analysis is presented for a gas bounded by two parallel black plates and within which there is a uniform heat source. Radiation-conduction interaction parameters, appropriate to both the optically thin and large path length limits, are presented for a number of gases. Numerical solutions for the gas centerline temperature have been obtained for CO, CO₂, H₂O, and CH₄, and these are compared with limiting solutions.Nomenclature A _i total band absorptance of the ith band, cm^–1 - A _oi correlation quantity, cm^–1 - _i dimensionless band absorptance, A _i /A _oi - c speed of light - C _oi ^/2 correlation quantity, atm^–1 cm^–1 - C _oi ² Planck's function evaluated at center of ith band - h Planck's constant - k Boltzmann's constant - L distance between plates, cm - P gas pressure, atm - q _R radiation heat flux, W/cm² - Q heat source or sink, W/cm³ - T temperature, °K - T ₁ wall temperature - T _e centerline temperature - u _i dimensionless coordinate, C _oi ^/2 Py - u _oi dimensionless path length, C _oi ^/2 PL - y physical coordinate, cm - dimensionless coordinate, y/L - thermal conductivity, W (cm °K)^–1     

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.     

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.     

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.     

Heat conduction in a cylinder crossed by an electric current with skin effect

The steady periodic temperature distribution in an infinitely long solid cylinder crossed by an alternating current is evaluated. First, the time dependent and non-uniform power generated per unit volume by Joule effect within the cylinder is determined. Then, the dimensionless temperature distribution is obtained by analytical methods in steady periodic regime. Dimensionless tables which yield the amplitude and the phase of temperature oscillations both on the axis and on the surface of copper or nichrome cylindrical electric resistors are presented.

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Wärmeleitung in einem stromdurchflossenen Zylinder unter Berücksichtigung des Skin-Effektes

Zusammenfassung Es wird die periodische Temperaturverteilung für den eingeschwungenen Zustand in einem unendlich langen, von Wechselstrom durchflossenen Vollzylinder ermittelt. Zuerst erfolgt die Bestimmung der zeitabhängigen, nichgleichförmigen Energiefreisetzung pro Volumeneinheit des Zylinders infolge Joulescher Wärmeentwicklung und anschließend die Ermittlung der quasistationären Temperaturverteilung auf analytischem Wege. Amplitude und Phasenverzögerung der Temperaturschwingungen werden für die Achse und die Oberfläche eines Kupfer- oder Nickelchromzylinders tabellarisch in dimensionsloser Form mitgeteilt.

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Nomenclature A integration constant introduced in Eq. (2) - ber, bei Thomson functions of order zero - Bi Biot numberhr ₀/ - c speed of light in empty space - c ₁,c ₂ integration constants introduced in Eq. (46) - c _p specific heat at constant pressure - E electric field - E _z component ofE alongz - E time independent part ofE, defined in Eq. (1) - f function ofs and defined in Eq. (11) - g function ofs and defined in Eq. (37) - h convection heat transfer coefficient - H magnetic field - i imaginary uniti=(–1)^1/2 - I electric current - I _eff effective electric currentI _eff=I/2^1/2 - Im imaginary part of a complex number - J _n Bessel function of first kind and ordern - J electric current density - q _g power generated per unit volume - time average of the power generated per unit volume - time averaged power per unit length - r radial coordinate - R electric resistance per unit length - r ₀ radius of the cylinder - Re real part of a complex number - s dimensionless radial coordinates=r/r ₀ - s, s integration variables - t time - T temperature - time averaged temperature - T _f fluid temperature outside the boundary layer - time average of the surface temperature of the cylinder - u, functions ofs, and defined in Eqs. (47) and (48) - W Wronskian - x position vector - x real variable - Y _n Bessel function of second kind and ordern - z unit vector parallel to the axis of the cylinder - z axial coordinate - · modulus of a complex number - equal by definition Greek symbols amplitude of the dimensionless temperature oscillations - electric permittivity - dimensionless temperature defined in Eq. (16) - ₀, ₁, ₂ functions ofs defined in Eq. (22) - thermal conductivity - dimensionless parameter=(2)^1/2 - magnetic permeability - ₀ magnetic permeability of free space - function of defined in Eq. (59) - dimensionless parameter=c _p/() - mass density - electric conductivity - dimensionless time=t - phase of the dimensionless temperature oscillations - function ofs:= ₁+i ₂ - angular frequency - dimensionless parameter=()^1/2 r ₀     

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     

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.     

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 test fixture for mapping thermal conduction in composites under transient and steady-state conditions

A test fixture has been designed that allows the measurement of steady-state and transient temperature profiles in composites or other types of anisotropic or orthotropic materials under pure conduction in a vacuum. The uniqueness of this approach is the ability to study heat source to composite interface techniques by comparison of surface temperature contours and temperature versus time profiles. Localized changes in these conditions can be evaluated and the most successful technique can be adapted to larger parts, i.e., printed wiring boards (PWBs), electronic enclosure thermal walls, etc.     

On the viscosity and thermal conduction of fluids with multivalued internal energy

This work is concerned with an extension of the classical compressible Euler model of fluid dynamics in which the fluid internal energy is a measure-valued quantity. This model can be derived from the hydrodynamic limit of a kinetic model involving a specific class of collision operators. In the present paper, we investigate diffusive corrections of this fluid dynamical model derived from a Chapman–Enskog expansion of the kinetic model, in the case where the collision time depends on the particle energy in the fluid frame. We show that the closure relations for the stress tensor and heat flux vector differ from their expression in the usual Navier–Stokes model. We argue why such a feature could be used as a tool towards an understanding of fluid turbulence from kinetic theory.     

A dynamic theory for magnetizable elastic solids with thermal and electrical conduction

A general dynamical theory of magnetizable, electrically and thermally conducting media is developed for soft ferromagnetic or paramagnetic materials in external electromagnetic fields. The general equations are linearized by assuming infinitesimal strains, linear constitutive equations and that all field variables may be divided into two parts: a "rigid body state" and a "perturbation state". The former is the same as the one in rigid body electrodynamics, and the latter which accounts for electromagnetic interaction with the deformable continuum is coupled with stress and strain through linearized field equations. The theory is developed for general anisotropy but specialized for materials with uniaxial, or higher, symmetry.     

Heat transfer by Hagen-Poiseuille flow in the thermal development region with axial conduction

The heat transfer in the region of circular pipes close to the beginning of the heating section is investigated for low-Péclet-number flows with fully developed laminar velocity profile. Axial heat conduction is included and its effect on the temperature distribution is studied not only for the region downstream of the start of heating but also for that upstream. The energy equation is solved numerically by a finite difference method. Results are presented graphically for various Péclet numbers between 1 and 50. The boundary conditions are uniform wall temperature and uniform wall heat flux with step change at a certain cross-section. For the latter case, also some results for the region near the end of the heating section are reported. The solutions are applicable for the corresponding mass transfer situations where axial diffusion is important if the temperature is replaced by the concentration andPe byReSc.     

On the non-classical problem of thermal conduction on anisotropic layers in R3

We propose a solution methodology for boundary problems of parabolic and hyperbolic thermal conduction on anisotropic layers in R³. We study the wave nature of heat transfer with pulse thermal effects in bounded bodies with cavity. We compare the solutions of the parabolic and hyperbolic equations of thermal conduction and we show that the assumption on the wave nature of energy transfer is justified under the conditions for high-speed processes.     

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.     

Investigation of the mechanisms of the onset of instability in weakly conduction media in electric fields

The initial stable state (of rest or laminar flow) of a weakly-conducting viscous multicomponent fluid in a plane channel, whose walls are electrodes with electrochemical properties varied in the process of investigation, are considered. The conventional electrohydrodynamic models of a medium with one or two charged particle components, whose concentrations are low compared with that of the neutral particles of the carrying fluid, are used. The time-dependent two-dimensional problem of the formation of the medium structure in the inter-electrode space after the voltage from the electrodes has been turned on is numerically solved. The obtained distributions of the Coulomb forces acting on the medium make it possible to develop several possible scenarios of the disturbance development, similar with those known in fluid dynamics. The realization of any of these scenarios considerably depends on the conditions at the electrode surfaces.