General Heat Conduction Equation Research Articles

Femtosecond laser ablation of aluminium

We investigate femtosecond laser ablation of aluminium using a hybrid simulation scheme. Two equations are solved simultaneously: one for the electronic system, which accounts for laser energy absorption and heat conduction, the other for the dynamics of the lattice where the ablation process takes place. For the electron-temperature a generalized heat-conduction equation is solved by applying a finite difference scheme. For the lattice properties, e.g. pressure, density or temperature, we use common molecular dynamics. Energy transfer between the subsystems is allowed by introducing an electron–phonon coupling term. This combined treatment of the electronic and atomic systems is an extension of the well known two-temperature model [Anisimov, Kapeliovich, Perel’man, Electron emission from metal surfaces exposed to ultra short laser pulses, JETP Lett. 39 (2)].

Uniqueness in exterior domains for the generalized heat conduction

In this note, we prove uniqueness of those solutions of the generalized heat conduction equation that increase as an exponential of the square of the distance from the origin. Continuous dependence results with respect to initial data and supply terms are also proved. The results are obtained with the help of the weighted energy method. We also prove uniqueness of the solutions of the backward in time problem. This second result is obtained by means of the weighted Lagrange identities method.

Green's Function Approach to Solution of Transient Temperature for Thermal Stresses of Functionally Graded Material.

The transient temperature solution for a functionally graded material (FGM) is formulated by Green’s function based on the Galerkin method. An approximate solution that satisfies the homogeneous boundary condition is substituted into the governing equation to yield an eigenvalue problem. To solve the eigenvalue problem, the eigenfunctions are approximated by a series of polynomials satisfying the homogeneous boundary condition. The Galerkin method is used to determine the coefficients of eigenfunctions. The transient temperature solution for a general heat conduction equation with a source and nonhomogeneous boundary conditions is obtained by using Green’s function, which is expressed by eigenvalues and corresponding eigenfunctions. Transient thermal stresses in a FGM plate and a FGM hollow circular cylinder are discussed.

Study of through-thickness residual stress by numerical and experimental techniques

The quenching process of aluminium alloys is modelled using the finite element method. The study of residual stress field induced by quenching is divided into two: the thermal and mechanical aspects. In the thermal problem, the general heat conduction equation is solved and the temperature field during quenching is calculated. In the mechanical problem, the calculated temperature field and mechanical properties are used to predict the residual stress field. In this paper, the two different boundary conditions used in the thermal problem are examined. The first is surface convection using the appropriate heat transfer coefficient. The second is the temperature variation measured at the surface of the part. These boundary conditions are compared, and the advantages and the drawbacks of each are shown. The influence of different quenching parameters on the level of residual stress is studied. To validate the quenching modelling, the incremental hole drilling and neutron diffraction methods are used to measure the residual stress field in the studied parts. The hole drilling technique has been adapted to measure the residual stress through a larger thickness of the part. The aim of this paper is the combination of numerical and experimental techniques for the investigation of the through-thickness residual stress field.

Control strategy using functions of sensitivity for thermal processing of sausages

The proposed strategy permits the on-line control of processes with arbitrary fluctuations of the heating-medium temperature from its nominal value without using an additional sausage-centre temperature probe. This approach using a finite-difference algorithm for integrating the generalized heat conduction equation can be also applied to thermal processing of solid foodstuffs with an irregular shape. Computed functions of sensitivity related to three kinds of cooked sausages are given. An illustrative example indicates the effect of the proposed strategy.

A method of numerical solution of one-dimensional stefan problems of two types

A one-dimensional conjugate problem of heat transfer with phase transitions of two types (with interfaces of two phases and with a two-phase zone) is solved by a finite difference method based on the general initial heat conduction equation written with the Dirac delta function. A calculating scheme is developed using a nonuniform spatial net with floating nodes and the method of oppositely directed pivots.

Criteria for finite element algorithm of generalized heat conduction equation

To eliminate oscillation and overbounding of finite element solutions of classical heat conduction equation, the author and Xiao have put forward two new concepts of monotonies and have derived and proved several criteria. This idea is borrowed here to deal with generalized heat conduction equation and finite element criteria for eliminating oscillation and overbounding are also presented. Some new and useful conclusions are drawn.

Construction of a solution of a mixed problem for the generalized heat conduction equation by the fourier method

By the Fourier method a solution of the equation $$\frac{{\partial u(x,t)}}{{\partial t}} = a(t)\sum\limits_{j,k = 1}^n {\frac{\partial }{{\partial x_j }}\left( {A_{jk} \frac{{\partial u(x,t)}}{{\partial x_k }}} \right)} + b(t)\sum\limits_{j,k = 1}^n {\frac{\partial }{{\partial x_j }}\left( {B_{jk} \frac{{\partial u(x,t - \tau (t))}}{{\partial x_k }}} \right)} ,$$ is obtained, which satisfies a homogeneous boundary condition and the initial condition u(x,t)=ϕ (x,t), t∈ E0, x ∈ G. A justification for the Fourier method is given. The following problem was considered in [1]: u(x,t)=ϕ(x,t), tε[−τ,0], x∈[0,l].

Rigorous thermodynamic treatment of heat generation and conduction in semiconductor device modeling

A treatment of the self-heating problem is presented. It is based on the laws of phenomenological irreversible thermodynamics (e.g. Onsager's relations and conservation of total energy) and is also consistent with the physical models usually considered in the isothermal drift diffusion approximation. The classical isothermal device equations are extended and completed by a generalized heat-conduction equation involving heat sources and sinks which, besides Joule and Thomson heat, reflect the energy exchanged through recombination (radiative and nonradiative) and optical generation. Thus the extended model also applies to direct semiconductors (e.g., optoelectronic devices) and accounts for effects caused by the ambient light intensity. It fully allows for low temperature since the case of incomplete ionization of donors and acceptors (impurity freeze-out) is properly incorporated in the theory. A critical comparison with previous work is made, showing that, in the steady state, some of the heuristic models of heat generation, thermal conductivity, and heat capacity could indeed approximate the correct results within an error bound of 1-10%. In the transient regime, however, none of the models used previously seems to be reliable, particularly, if short switching times ( >

Stress intensity factor for transient thermal stresses in a transversely isotropic infinite body with an external circular crack

The present work deals with the transient thermal stress in a transversely isotropic infinite body with an external circular crack. The surface cooling of the crack depends on position and time. Since it it usually very difficult to obtain an analytical solution for the temperature field, a finite difference formulation with respect to a tive variable is introduced. In the first step, applying this method to the general heat conduction equation in an orthotropic body, a very compact difference equation with respect to the spatial variables is obtained. In the second step, this method is applied to the transient thermoelastic problem in a transversely isotropic infinite body with an external circular crack subjected to heat exchange on the crack surface. Thermal stresses are analyzed by means of the transversely isotropic potential functions method.

A model of sputtering from spikes

A simple model of sputtering from spikes is described based upon the solution of a general heat conduction equation for spherical symmetry. The model accounts for many anomalies observed in energy spectra of sputtered atoms.

Determination of the parameters in the generalized heat-conduction equation from transient experimental data

The synthesis of an iterative algorithm for determining the parameters in the generalized heat-conduction equation from the results of temperature measurements is discussed.

Some remarks on a generalized heat conduction equation

The author makes some remars on the theory of heat conducted in solids.. (AIP)

Generalized heat conduction equation

The linear constitutive relation between heat flux and temperature gradient known as Fourier’s law is modified to be nonlocal in time; i.e., the heat flux depends on the past history of the temperature gradient. It is shown that this leads to a generalized heat conduction equation which can be hyperbolic and thus possess a wavelike solution with finite velocity of propagation of a heat pulse. In two extreme limits of very short and very long memory the equation reduces, respectively, to the usual parabolic heat conduction equation and to a hyperbolic wave equation. For in between memory the equation has the properties of a telegraph equation characterized by a relaxation time. Such equations have arisen in treatments of second sound, exciton transport, and quasiparticle transport in 3He.

Solution of the nonlinear inverse problem for a generalized heat-conduction equation in a region with moving boundaries

In an analysis of the numerical solution of the nonlinear inverse heat-conduction problem in a region with moving boundaries, a regularization method is used to construct an algorithm for smoothing the experimental data in a compilation of the input data for the inverse problem.

On the stability of steady temperature distribution in a transparent solid with internal energy source

The stability of the steady solution of general heat conduction equation with temperature-dependent as well as position-dependent internal energy source, radiation, and physical properties is studied. It is shown by way of a simple example that a temperature-dependent energy source may cause the temperature to increase without limit. In case of locally one-dimensional steady-state temperature and property distributions, the condition for the temperature distribution in an infinite solid to be stable is derived. The result is discussed as to its validity in different cases.