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.     

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.     

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.     

Transient free convection from a vertical plate subjected to a change in surface heat flux in porous media

In this paper we examine the general transient natural convection response arising due to a sudden change of the level of uniform flux dissipation rate from a vertical surface which is embedded in a porous medium. From an analytical investigation of the governing boundary-layer equations both a series solution which is valid at small values of the non-dimensional time and a solution which is valid at large times, when the transport of energy is steady, are derived. A numerical, transient formulation of the full unsteady boundary-layer equations is developed using an explicit finite-difference scheme. The numerical temperature profiles are observed to closely follow the small time solution initially and evolve along a curve which approaches the steady-state solution asymptotically. Results are presented to illustrate the occurrence of transients from both an increase and a decrease in the levels of existing energy inputs.     

Temperature and heat flux solutions due to steady and non-steady periodic-type surface temperatures in a semi-infinite solid

An analytical solution for the temperature and heat flux distribution in the case of a semi-infinite solid of constant properties is investigated. The solutions are presented for time-dependent, surface temperatures of the forms: (i)T ₁(t)=T ₀(1+a cos t), and (ii)T ₂(t)=T ₀(1+b t cos t), wherea andb are controlling factors of the periodic oscillations about the constant surface temperatureT ₀. The dimensionless (or reduced) temperature and heat flux solutions are presented in terms of decompositionsC andS of the generalized representation of the incomplete Gamma function. It is demonstrated that the present analysis covers the limiting case for large times which is discussed in several textbooks, for the case of steady periodic-type surface temperatures.

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Lösungen für Temperatur- und Wärmeflußfeld in einem halbunendlichen Körper mit nichtstationär und quasistationär oszillierenden Oberflächentemperaturen

Zusammenfassung Die Untersuchung liefert analytische Lösungen für das Temperatur- und Wärmeflußfeld in einem halbunendlichen Körper mit konstanten Stoffwerten. Die Lösungen werden in dimensionsloser Form für veränderliche Oberflächentemperaturen ermittelt, welche nach folgenden Zeitgesetzen oszillieren: (i)T ₁(t)=T ₀(1+a cos t), (ii)T ₂ (t)=T ₀ (1+b t cos t). Die Darstellung erfolgt unter Verwendung der ZerlegungsfunktionenC undS für die unvollständige Gammafunktion. Für große Zeiten münden die hier mitgeteilten Ergebnisse in die bekannten quasistationären periodischen Lösungen ein.

     

Exact solution of heat transfer from a stretching surface with variable heat flux

An exact expression of the temperature distribution is constructed for the heat transfer from a stretching surface with prescribed power law heat flux. The stretching velocity is inversely proportional to the one third power of the distance measured along the surface from a thin slit. The final result is expressed in terms of hypergeometric functions. Although the exact solution is accomplished, some physically unrealistic phenomena are encounters for specific conditions. The temperature parameter which prescribe the surface heat flux, strongly affects those situations. Two types of temperature distribution are discussed: dimensionless temperatures with and without scaling to the dimensionless surface temperature. The expression of the temperature distribution without scaling is lucid to understand the heat transfer characteristics. Received on 23 July 1997     

Transient boundary-layer heat transfer from a flat plate subjected to a sudden change in heat flux

We examine the transient forced convection heat transfer from a fixed, semi-infinite, flat plate situated in a fluid which, at large distances, is moving with a constant velocity parallel to the plate. Both the fluid and the plate are initially at a constant temperature and the transients are initiated when the zero heat flux at the plate is suddenly changed to a constant value. The thermal boundary-layer equations are solved using numerical techniques to extend a series which is valid for small times and describe fully the development from the initial unsteady state solution (small times) to the ultimate steady state solution (large time).     

变几何域传热的表面热流反演方法

变几何域的表面热流反演是一类特殊的热传导逆问题,在再入飞行器烧蚀型防热材料的表面热流反演中具有工程实用价值.本文首先对变几何域传热的正问题计算方法进行了校核验证,然后建立了求解变几何域表面热流反演问题的顺序函数法和共轭梯度法;给出了这两种反演方法的基本思想和算法推导,并针对典型算例进行了仿真.结果表明:两种反演方法都能计算出较好的反演结果,并且算法受测量噪声的影响较小,具有较好的鲁棒性;反演算法能适应不同的几何域变化函数,但几何域变化量的测量误差在表面热流的反演结果中会有较为直接的反映.     

Slow flow past periodic arrays of cylinders with application to heat transfer

Solutions for the slow flow past a square and a hexagonal array of cylinders are determined using a somewhat non-conventional numerical method. The calculated values of the drag on a cylinder as a function of $\text{c}$, the volume fraction of the cylinders, are shown to be in excellent agreement with the corresponding asymptotic expressions for $\text{c ? 1}$ and for $\text{c → c}{}_{\mathrm{<mtext>max</mtext>}}$, the maximum volume fraction. These solutions are then used to calculate the average temperature difference between the bulk and the cylinders which are heated uniformly under conditions of small Reynolds and Péclet numbers.     

Transient heat conduction in functionally graded thick hollow cylinders by analytical method

In this article, transient heat conduction in a cylindrical shell of functionally graded material is studied by using analytical method. The shell is assumed to be in axisymmetry conditions. The material properties are considered to be nonlinear with a power law distribution through the thickness. The temperature distribution is derived analytically by using the Bessel functions. To verify the proposed method the obtained numerical results are compared with the published results. The comparisons of temperature distribution between various time and material properties are presented.     

Convection heat transfer of closely-spaced spheres with surface blowing

A validated computer simulation model has been developed for the analysis of colinear spheres in a heated gas stream. Using the Galerkin finite element method, the steady-state Navier-Stokes and heat transfer equations have been solved describing laminar axisymmetric thermal flow past closely-spaced monodisperse spheres with fluid injection. Of interest are the coupled nonlinear interaction effects on the temperature fields and ultimately on the Nusselt number of each sphere for different free stream Reynolds numbers (20 ≤Re ≤ 200) and intersphere distances (1.5 ≤d _ij ≤ 6.0) in the presence of surface blowing (0 ≤v _b < 0.1). Fluid injection (i.e. blowing) and associated wake effects generate lower average heat transfer coefficients for each interacting sphere when the Reynolds number increases (Re>100). Heat transfer is also reduced at small spacings especially for the second and third sphere. A Nusselt number correlation for each interacting (porous) sphere has been developed based on computer experiments.     

Free convection from a vertical permeable circular cone with non-uniform surface heat flux

 In this paper, the problem of laminar free convection from a vertical permeable circular cone maintained with non-uniform surface heat flux is considered. The governing boundary layer equations are reduced non-similar boundary layer equations with surface heat flux proportional to x ⁿ (where x is the distance measured from the leading edge). The solutions of the reduced equations are obtained by using three distinct solution methodologies; namely, (i) perturbation solution for small transpiration parameter, ξ, (ii) asymptotic solution for large ξ, and (iii) the finite difference solutions for all ξ. The solutions are presented in terms of local skin-friction and local Nusselt number for smaller values of Prandtl number and heat flux gradient and are displayed in tabular form as well as graphically. Effects of pertinent parameters on velocity and temperature profiles are also shown graphically. Solutions obtained by finite difference method are also compared with the perturbation solutions for small and large ξ and found to be in excellent agreement. Received on 1 October 1999     

Some remarks on neutron transport in homogeneous slabs and spheres

Summary In this work we are concerned with the stationary neutron transport Boltzmann equation (in its integral form) in plane and spherical symmetry; both the study of the properties of the exact and approximate solution are based on functional analysis. We have taken advantage of the fact that the kernel of the integral equation is even, to semplify the general discussion; among the results, we quote that relating to the continuity of the solution with respect to the optical thickness. Moreover we emphasized that the Ritz method seems the most convenient one for the approximate evaluation of the solution; in this connexion, some observations have been made about the convergence of the approximate solutions to the exact one.

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Sommario In questo lavoro consideriamo l'equazione stazionaria di Boltzmann (nella forma integrale) in simmetria piana e sferica; l'uso di alcuni metodi dell'analisi funzionale ci consente di studiare sia le proprietà della soluzione esatta che di quella approssimata. La semplice osservazione che il nucleo dell'equazione integrale è pari, ci ha permesso di semplificare la discussione generale; tra i risultati segnaliamo quello riguardante la continuità della soluzione rispetto allo spessore ottico. Inoltre abbiamo sottolineato che il metodo di Ritz ci sembra il più opportuno punto di partenza per il calcolo approssimato della soluzione; in relazione a ciò, vengono fatte alcune osservazioni a proposito della convergenza delle soluzioni approssimate verso quella esatta.

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Work performed under contract C.N.R.