Solution Of Heat Conduction Research Articles

Improving convergence of summations in heat conduction

This paper addresses exact, transient heat-conduction solutions in two-dimensional rectangles heated at a boundary. The standard method of separation of variables (SOV) solution has two parts, steady-state (or quasi-steady) and complementary transient. The steady-state component frequently converges slowly at the heated surface, which is usually the one of greatest interest. New procedures are given to construct a steady solution in the form of a single summation, one having eigenvalues in the homogeneous direction (yielding the same result as the standard SOV solution) and the other having eigenvalues in the non-homogeneous direction (called the non-standard solution). The non-standard solutions have much better convergence behavior at and near the heated boundary than the standard forms. Examples are given.

Weighted iterative solutions of linear differential equations and heat conduction of ideal gases in moment theory

An iterative method is proposed to find a particular solution of a system of linear differential equations, in the form of a fixed-point problem, with no boundary conditions. To circumvent the unboundedness of differential operators, iterative approximation with gradually decreasing weight is used. Conditions for convergence that can easily be checked in numerical iterations are established. Furthermore, for the numerical iterative scheme, uniqueness and stability theorems are proved. These results are applied to heat conduction of ideal gases in moment theory.

Finite element solution of transient heat conduction using iterative solvers

PurposeTo provide an analysis of transient heat conduction, which is solved using different iterative solvers for graduate and postgraduate students (researchers) which can help them develop their own research.Design/methodology/approachThree‐dimensional transient heat conduction in homogeneous materials using different time‐stepping methods such as finite difference (Θ explicit, implicit and Crank‐Nicolson) and finite element (weighted residual and least squared) methods. Iterative solvers used in the paper are conjugate gradient (CG), preconditioned gradient, least square CG, conjugate gradient squared (CGS), preconditioned CGS, bi‐conjugate gradient (BCG), preconditioned BCG, bi‐conjugate gradient stabilized (BCGSTAB), reconditioned BCGSTAB and Gaussian elimination with incomplete Cholesky factorization.FindingsProvides information on which time‐stepping method is the most accurate, which solver is the fastest to solve a symmetric and positive system of linear matrix equations of all those considered.Practical implicationsFortran 90 code given as an abstract can be very useful for graduate and postgraduate students to develop their own code.Originality/valueThis paper offers practical help to an individual starting his/her research in the finite element technique and numerical methods.

Unsteady conjugate laminar heat transfer of a rotating non-uniformly heated disk: Application to the transient experimental technique

Presented in the paper are the results of an investigation of 2D heat conduction effects on the transient heat transfer of a rotating disk heated up to a non-uniform initial temperature and suddenly subjected to unsteady cooling by still air. A self-similar solution of the transient laminar convective heat transfer confirmed that the heat transfer coefficient rapidly becomes time-independent and equal to its value at steady-state conditions. An analytical solution of the unsteady two-dimensional heat conduction inside a disk made of Plexiglas ® confirmed that the known infinite-slab approach can still be used as a transient technique for determining heat transfer coefficients. Use of the regular heat transfer regime theory for the same purpose can be recommended only for the cases with the moderate initial temperature non-uniformity.

Green’s function solution for transient heat conduction in concrete-filled CHS subjected to fire

The heat transfer analysis of concrete-filled steel circular hollow sections (CHS) subjected to fire is essentially classified as the process of deriving a solution for transient heat conduction in a composite medium in a cylindrical coordinate system. In this paper, an efficient numerical approach based on an analytical Green’s function (GF) solution is developed, where the spatial variable is analytically tractable without recourse to spatial discretization. The thin steel layer is conveniently treated as a lumped heat capacitance with uniform temperature distribution to avoid tedious analytical solutions for hollow cylinders. A computational model is developed based on Duhamel’s principle, incorporating time-varying fire conditions. The choice of sampling time is not constrained by numerical stability and can be as large as 30 min in one time step. The numerical approach can be used to predict the temperature field inside the entire composite domain and the heat flux at the fire and the steel-concrete interfaces, as well as to verify more sophisticated numerical codes, e.g. finite element codes, of heat transfer.

Sensitivity Analysis of Classical Heat Conduction Solutions Applied to Materials Characterization

This paper presents the analysis of classical heat conduction solutions applied to materials characterization. The formulas for parametric derivatives are obtained and illustrated to demonstrate the evolution of the relative sensitivity functions in time. The potential of using both front-surface and rear-surface solutions for determining material thermal properties, sample thickness, and surface heat exchange parameters is discussed. The roots of the well-known transcendent equation for a non-adiabatic plate are approximated in a polynomial form. Some practical applications of the proposed formulas are reported.

Analytical Solution for Hyperbolic Heat Conduction in a Hollow Sphere

Received16September2004;revisionreceived29January2005;acceptedfor publication 21 February 2005. Copyright c 2005 by the American In-stitute of Aeronautics and Astronautics, Inc. All rights reserved. Copies ofthis paper may be made for personal or internal use, on condition that thecopier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc.,222 Rosewood Drive, Danvers, MA 01923; include the code 0887-8722/05$10.00 in correspondence with the CCC.

A comparison of chemical reference materials for solution calorimeters

Solution calorimeters are based on semi-adiabatic or isothermal heat-conduction principles and differ in the way they record data. They also have different measuring sensitivities and require different quantities of solute and solvent. As such, the choice of chemical test substance is not straightforward. Usually the dilution of KCl is recommended; it is possible to purchase a reference sample of KCl that has a certified enthalpy of solution and this standard material is usually used to test semi-adiabatic instruments. Here, we review the suitability of a range of chemical test substances (KCl, sucrose and Tris) for an isothermal heat-conduction solution calorimeter. It was found that KCl was not the best test material because its relatively high enthalpy of solution (Δ sol H) necessitated the use of small samples (2 mg), resulting in a relatively large standard deviation ( σ n − 1 ) in the values recorded (Δ sol H = 17.14 ± 0.49 kJ mol −1); furthermore, KCl data must be corrected to account for the effect of dilution, although the correction was found to be small (0.07 kJ mol −1) under the experimental conditions employed here. Sucrose appears to be a much more robust test material for isothermal heat-conduction instruments because its lower enthalpy of solution allows the use of much larger samples (20 mg), which minimises experimental errors. The Δ sol H value returned (6.14 ± 0.08 kJ mol −1) is in excellent agreement with the literature. It is also cheap, readily available and requires minimal preparation although its widespread use would require the preparation of a certified reference sample.

An inverse boundary element method/genetic algorithm based approach for retrieval of multi-dimensional heat transfer coefficients within film cooling holes/slots

An inverse methodology is developed as a means of determining heat transfer coefficient distributions in film cooling holes/slots. Thermal conditions are over-specified at exposed surfaces amenable to measurement, while the temperature and surface heat flux distributions are unknown at the film cooling hole/slot walls. The latter are determined in an iterative manner by solving an inverse problem whose objective is to adjust the film-cooling hole/slot wall temperatures and heat flux distributions until the temperature and heat fluxes at the measurement surfaces are matched in an overall heat conduction solution. The heat conduction problem is solved using boundary element methods, and the inverse problem is solved using a genetic algorithm. The resulting film coefficient distributions are fit to a correlation reflecting dependency on position, the Prandtl and Reynolds numbers.

Performance Of Cables Subjected To Elevated Temperatures

The time to damage, i.e., time to short circuit, and the corresponding temperature within the cable were recorded for two different data cables and one low voltage cable when subjecting the cables to an elevated surrounding temperature. In the experiments it was found that short-circuiting occurred at a certain temperature in the core of the studied cables. The heating of the cables was modeled with the aid of computer programs for thermal analysis and an analytical solution of heat conduction. The experimental results were used to evaluate the different models. Both the analytical solution and the use of the thermal analysis programs turned out to be promising. However, both the analytical solution and the use of computer programs such as TASEF require data of the thermal properties of the cables. This is a complication as such data are not easily accessible. To some extent the thermal properties of the cables could be estimated from the experimental results.

On a posteriori pointwise error estimation using adjoint temperature and Lagrange remainder

The pointwise estimation of heat conduction solution as a function of truncation error of a finite difference scheme is addressed. The truncation error is estimated using a Taylor series with the remainder in the Lagrange form. The contribution of the local error to the total pointwise error is estimated via an adjoint temperature. It is demonstrated that the results of numerical calculation of the temperature at an observation point may thus be refined via adjoint error correction and that an asymptotic error bound may be found.

Transient conduction in spherical fruits: method to estimate the thermal conductivity and volumetric thermal capacity

A methodology to estimate the thermal conductivity ( κ) and volumetric thermal capacity ( ρc p ) of apples is described. The methodology is based in the inverse problem of heat conduction solution. The apples are considered as spherical objects. This approach relies on an analytical solution of the problem and on an optimization technique to estimate the two thermophysical properties. Typically, a test to measures the temperature response of a specimen subjected to transient heating by hot air to determine the properties directly as a function of temperature.

Solutions for multi-dimensional transient heat conduction with solid body motion

The analytical solution for the problem of transient thermal conduction with solid body movement is developed for an orthotropic parallelepiped. Transformations are used to eliminate the flow terms and the orthotropic dependence. The solution uses two types of Green's functions: one coming from the Laplace transform method and the other from the method of separation of variables. The solution method is powerful because it incorporates internal verification of the numerical results by varying the partition time between the short and long components. An example is given for a multi-dimensional case involving both prescribed heat flux and temperature boundary conditions.

Numerical solution of stochastic linear heat conduction problem by using new algorithms

We consider a stochastic linear heat conduction problem so, it is reduced to a special integral equation of the second kind. Our aim, in this paper is to give stable numerical algorithms for approximating solution of integral equation and heat conduction.

Transient laminar conjugate heat transfer of a rotating disk: theory and numerical simulations

The study considers transient laminar heat transfer of a rotating disk heated to a constant initial temperature and sudden subjection to unsteady cooling by still air. Both numerical simulations of conjugate heat transfer inside the disk and convection between the disk and air, as well as a self-similar solution show that the heat transfer coefficient becomes time independent very quickly and equal to its value at steady-state conditions. It appeared that the widely employed solution of unsteady one-dimensional heat conduction in a semi-infinite plate overestimates significantly the disk temperature and consequently the heat transfer coefficient calculated from this solution using known experimental instant disk surface temperature.

Algebraically explicit analytical solutions of unsteady conduction with variable thermal properties in cylindrical coordinate*

Abstract The analytical solutions of unsteady heat conduction with variable thermal properties (thermal conductivity, density and specific heat are functions of temperature or coordinates) are meaningful in theory. In addition, they are very useful to the computational heat conduction to check the numerical solutions and to develop numerical schemes, grid generation methods and so forth. Such solutions in rectangular coordinates have been derived by the authors. Some other solutions for 1-D and 2-D axisymmetrical heat conduction in cylindrical coordinates are given in this paperto promote the heat conduction theory and to develop the relative computational heat conduction.

Study of transient heat conduction in 2.5D domains using the boundary element method

This paper presents the solution for transient heat conduction around a cylindrical irregular inclusion of infinite length, inserted in a homogeneous elastic medium and subjected to heat point sources placed at some point in the host medium. The solution is computed in the frequency domain for a wide range of frequencies and axial wavenumbers, and time series are then obtained by means of (fast) inverse Fourier transforms into space–time. The method and the expressions presented are implemented and validated by applying them to a cylindrical circular inclusion placed in an infinite homogeneous medium and subjected to a point heat source, for which the solution is calculated in closed form. The boundary elements method is then used to evaluate the temperature field generated by a point source in the presence of a cylindrical inclusion, with a non-circular cross-section, inserted in an unbounded homogeneous medium. Simulation analyses using this model are then performed to study the transient heat conduction in the vicinity of these inclusions.

Analytical solution of transient heat conduction in a two-layer anisotropic cylindrical slab excited superficially by a short laser pulse

This paper presents an exact analytical solution of 2D transient temperature, caused by a non-periodical pulse heating of a cylindrical two-layer slab. The corresponding partial differential equations are solved step-by-step, using the separation of variables technique. Analytical/computational procedure for eigenvalue derivation is considered separately. General initial assumptions, such as existence of thermal contact resistance, thermal conduction and diffusion anisotropy, radiative heat losses, non-uniform heating, and finite absorption depth are accounted in the solution. Numerical simulations of temperature distribution throughout the sample are given at the end of the paper.

A boundary element method for anisotropic coupled thermoelasticity

A boundary element formulation is presented for the solution of the equations of fully coupled thermoelasticity for materials of arbitrary degree of anisotropy. By employing the fundamental solutions of anisotropic elastostatics and stationary heat conduction, a system of equations with time-independent matrices is obtained. Since the fundamental solutions are uncoupled and time-independent, a domain integral remains in the representation formula which contains the time-dependence as well as the thermoelastic coupling. This domain integral is transformed to the boundary by means of the dual reciprocity method. By taking this approach, the use of dynamic fundamental solutions is avoided, which enables an efficient calculation of system matrices. In addition, the solution of transient processes as well as, free and forced vibration analysis becomes straightforward and can be carried out with standard time-stepping schemes and eigensystem solvers. Another important advantage of the present formulation is its versatility, since it includes a number of simplified thermoelastic theories, viz. the theory of thermal stresses, coupled and uncoupled quasi-static thermoelasticity, and stationary thermoelasticity. The accuracy of the new thermoelastic boundary element method is demonstrated by a number of example problems.

Improvement of inverse heat conduction solution using Laplace transformation: Method of partial division of time

AbstractAny solution for an inverse heat conduction problem makes the estimation of surface temperature and surface heat flux worsen in the case where these values behave like a triangular shape change with time. In order to compensate for this defect, Monde and colleagues, who succeeded in obtaining analytical inverse solutions using the Laplace transform technique, introduce a new idea where these changes over the entire measurement time can be split into several parts depending on the behavior. Therefore, an approximate equation to trace the measured temperature change can be derived, resulting in good estimation of surface temperature and surface heat flux even in the case of the triangular shape change and sharp change. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 630–638, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.10117