Application of the variational iteration method to nonlinear non‐Fourier conduction heat transfer equation with variable coefficient

In this paper, a variational iteration method (VIM) has been applied to nonlinear non‐Fourier conduction heat transfer equation with variable specific heat coefficient. The concept of the variational iteration method is introduced briefly for applying this method for problem solving. The proposed iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. The results of VIM as an analytical solution are then compared with those derived from the established numerical solution obtained by the fourth order Runge–Kutta method in order to verify the accuracy of the proposed method. The results reveal that the VIM is very effective and convenient in predicting the solution of such problems, and it is predicted that VIM can find a wide application in new engineering problems. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20362     

Analytical approach with Laplace transform to the inverse problem of one‐dimensional heat conduction transfer: Application to second and third boundary conditions

An analytical method using Laplace transformation has been developed for one‐dimensional heat conduction. This method succeeded in explicitly deriving the analytical solution by which the surface temperature for the first kind of boundary condition can be well predicted. The analytical solutions for the surface temperature and heat flux are applied to the second and third of the boundary conditions. These solutions are also found to estimate the corresponding surface conditions with a high degree of accuracy when the surface conditions smoothly change. On the other hand, when these conditions erratically change such as the first derivative of temperature with time, the accuracy of the estimation becomes slightly less than that for a smooth condition. This trend in the estimation is similar irrespective of any kind of boundary condition. © 2002 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(1): 29–41, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10069     

A solution to the two‐dimensional inverse heat conduction problem

A simple method is developed in this paper to solve two‐dimensional nonlinear steady inverse heat conduction problems. The unknown boundary conditions can be numerically obtained by using the iteration and modification method. The effect of measurement errors of the wall temperature on the algorithm is numerically tested. The results prove that this method has the advantages of fast convergence, high precision, and good stability. The method is successfully applied to estimate the convective heat transfer coefficient in the case of a fluid flowing in an electrically heated helically coiled tube. © 2000 Scripta Technica, Heat Trans Asian Res, 29(2): 113–119, 2000     

Conjugate heat transfer from a circular cylinder to a power‐law non‐Newtonian fluid

In this paper, conjugate heat transfer from a circular cylinder with a heat source to a non‐Newtonian power‐law fluid is studied. Numerical calculations are carried out in an unconfined computational domain for Reynolds numbers (), power‐law indices (), and Prandtl numbers () with different heat source values. The pressure coefficient, value, and position of maximum temperature inside the cylinder and the local and average Nusselt number are calculated. Also, the effects of Re, Pr, n, and heat source value on the thermal characteristics in the solid cylinder and the fluid around it are studied and discussed.     

Lattice Boltzmann simulation of heat conduction problems in non‐isothermally heated enclosures

A numerical study following the lattice Boltzmann method (LBM) is performed to solve transient heat conduction problems with and without volumetric heat generation/absorption in 2D and 3D Cartesian geometries. Uniform lattices are considered for both geometries. To validate the correctness of LBM, a finite difference method (FDM) is also used to solve the 2D problem without heat generation/absorption and results are compared with that of LBM. For both 2D and 3D geometries one of the walls is heated and cooled with a sinusoidal function and the rest of the walls are cooled isothermally. Effects of amplitude of the sinusoidal function and volumetric heat generation/absorption on temperature profiles are analyzed. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20406     

A finite element analysis for inverse heat conduction problems

This paper presents an efficient inverse analysis technique based on a sensitivity coefficient algorithm to estimate the unknown boundary conditions of multidimensional steady and transient heat conduction problems. Sensitivity coefficients were used to represent the temperature response of a system under unit loading conditions. The proposed method, coupled with the sensitivity analysis in the finite element formulation, is capable of estimating both the unknown temperature and heat flux on the surface provided that temperature data are given at discrete points in the interior of a solid body. Inverse heat conduction problems are referred to as ill-posed because minor inaccuracy or error in temperature measurements cause a drastic effect on the predicted surface temperature and heat flux. To verify the accuracy and validity of the new method, two-dimensional steady and transient problems are considered. Their surface temperature and heat flux are evaluated. From a comparison with the exact solution, the effects of measurement accuracy, number and location of measuring points, a time step, and regularization terms are discussed. © 1998 Scripta Technica. Heat Trans Jpn Res, 26(6): 345–359, 1997     

Analytical method of two‐dimensional inverse heat conduction problem using Laplace transformation: Effect of number of measurement points

An analytical method has been developed for the inverse problem of two‐dimensional heat conduction using the Laplace transform technique. The inverse problem is solved for only two unknown surface conditions and the other surfaces are insulated in a finite rectangular body. In actual temperature measurement, the number of points in a solid is usually limited so that the number of temperature measurements required to approximate the temperature change in the solid becomes too small to obtain an approximate function using a half polynomial power series of time and the Fourier series of the eigenfunction. In order to compensate for this lack of measurement points, the third‐order Spline method is recommended for interpolating unknown temperatures at locations between measurement points. Eight points are recommended as the minimum number of temperature measurement points. The calculated results for a number of representative cases indicate that the surface temperature and the surface heat flux can be predicted well, as revealed by comparison with the given surface condition. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 618–629, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10116     

Solution to shape identification problem of unsteady heat‐conduction fields

This paper presents a numerical analysis method for shape determination problems of unsteady heat‐conduction fields in which time histories of temperature distributions on prescribed subboundaries or time histories of gradient distributions of temperature in prescribed subdomains have prescribed distributions. The square error integrals between the actual distributions and the prescribed distributions on the prescribed subboundaries or in the prescribed subdomains during the specified period of time are used as objective functionals. Reshaping is accomplished by the traction method that was proposed as a solution to shape optimization problems of domains in which boundary value problems are defined. The shape gradient functions of these shape determination problems are derived theoretically using the Lagrange multiplier method and the formulation of material derivative. The time histories of temperature distributions are evaluated using the finite‐element method for a space integral and the Crank–Nicolson method for a time integral. Numerical analyses of nozzle and coolant flow passage in a wing are demonstrated to confirm the validity of this method. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(3): 212–226, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10086     

Exact variational principle for 3-D unsteady heat conduction with second sound

The exact variational formulation of the extended unsteady heat conduction equation with finite propagationspeed(the 2nd sound speed)of hyperbolic type is derived herein via a systematic and natural way.Moreover,theboundary- and the physically acceptable initial-value conditions are accommodated in the variational principle bya novel method suggested just recently.In this way a perfect justification of the variational theory of transient heatconduction and a rigorous theoretical basis for the finite element analysis of heat conduction are provided.     

Boundary shape determination of steady‐state heat‐conduction fields

In this paper, a practical method of numerical analysis for the boundary shape design of steady‐state heat‐conduction fields to control the temperature distribution to a prescribed distribution is presented. Reshaping was accomplished by the traction method proposed by one of the authors as a solution to domain optimization problems in which elliptic boundary value problems were defined. In this study, we formulated a temperature square error norm minimization problem between the prescribed distribution and the actual distribution on prescribed boundaries and theoretically derived a shape gradient function for this problem. We developed a convenient numerical method using a general‐purpose FEM program for the temperature prescribed problem. The successful results for the two‐dimensional problems of an erosion surface on a blast furnace hearth and a coolant flow passage in a wing demonstrate the validity of the presented method. © 2001 Scripta Technica, Heat Trans Asian Res, 30(3): 245–258, 2001     

He's homotopy perturbation method for two‐dimensional heat conduction equation: Comparison with finite element method

Heat conduction appears in almost all natural and industrial processes. In the current study, a two‐dimensional heat conduction equation with different complex Dirichlet boundary conditions has been studied. An analytical solution for the temperature distribution and gradient is derived using the homotopy perturbation method (HPM). Unlike most of previous studies in the field of analytical solution with homotopy‐based methods which investigate the ODEs, we focus on the partial differential equation (PDE). Employing the Taylor series, the gained series has been converted to an exact expression describing the temperature distribution in the computational domain. Problems were also solved numerically employing the finite element method (FEM). Analytical and numerical results were compared with each other and excellent agreement was obtained. The present investigation shows the effectiveness of the HPM for the solution of PDEs and represents an exact solution for a practical problem. The mathematical procedure proves that the present mathematical method is much simpler than other analytical techniques due to using a combination of homotopy analysis and classic perturbation method. The current mathematical solution can be used in further analytical and numerical surveys as well as related natural and industrial applications even with complex boundary conditions as a simple accurate technique. © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20292     

Finite element method formulation in polar coordinates for transient heat conduction problems

The aim of this paper is the formulation of the finite element method in polar coordinates to solve transient heat conduction problems. It is hard to find in the literature a formulation of the finite element method (FEM) in polar or cylindrical coordinates for the solution of heat transfer problems. This document shows how to apply the most often used boundary conditions. The global equation system is solved by the Crank-Nicolson method. The proposed algorithm is verified in three numerical tests. In the first example, the obtained transient temperature distribution is compared with the temperature obtained from the presented analytical solution. In the second numerical example, the variable boundary condition is assumed. In the last numerical example the component with the shape different than cylindrical is used. All examples show that the introduction of the polar coordinate system gives better results than in the Cartesian coordinate system. The finite element method formulation in polar coordinates is valuable since it provides a higher accuracy of the calculations without compacting the mesh in cylindrical or similar to tubular components. The proposed method can be applied for circular elements such as boiler drums, outlet headers, flux tubes. This algorithm can be useful during the solution of inverse problems, which do not allow for high density grid. This method can calculate the temperature distribution in the bodies of different properties in the circumferential and the radial direction. The presented algorithm can be developed for other coordinate systems. The examples demonstrate a good accuracy and stability of the proposed method.     

Application of inverse solution in two‐dimensional heat conduction problem using Laplace transformation

An inverse solution has been explicitly derived for two‐dimensional heat conduction in cylindrical coordinates using the Laplace transformation. The applicability of the inverse solution is checked using the numerical temperatures with a normal random error calculated from the corresponding direct solution. For a gradual temperature change in a solid, the surface heat flux and temperature can be satisfactorily predicted, while for a rapid change in the temperature this method needs the help of a time partition method, in which the entire measurement time is split into several partitions. The solution with the time partitions is found to make an improvement in the prediction of the surface heat flux and temperature. It is found that the solution can be applied to experimental data, leading to good prediction. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 602–617, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10115     

Forced convective heat transfer between assemblages of spherical particles and power‐law non‐Newtonian liquids with velocity slip at the interface

Heat transfer from spheres can be influenced by a varying degree of slip at the fluid‐particle interface along with the rheology of the surrounding continuous liquid and adjacent spheres. Thus in this study, the effects of dimensionless velocity slip parameter (λ) along with power‐law fluid rheology and other pertinent kinematic flow and heat transfer parameters on isotherm contours, local and average Nusselt numbers of assemblages of spherical slip particles are presented. This is done by adopting a segregated approach where dimensionless momentum and energy equations are solved by SMAC algorithm formulated in spherical coordinates within the finite difference formulation. Before obtaining new results, grid independence studies for either extreme values of power‐law consistency index of non‐Newtonian fluids are carried out. Finally, the major contribution of this study is the development of a correlative equation for the average Nusselt number of assemblages of spherical slip particles in power‐law fluids based on the present results (5880 data points) as a function of pertinent dimensionless parameters.     

Numerical analysis of unsteady heat conduction in regular solid bodies comprising natural convection to nearby fluids

The present paper addresses unsteady, unidirectional heat conduction in regular solid bodies (vertical plate, horizontal cylinder, and sphere) that exchange heat by natural convection with a neighboring fluid. From thermal physics, natural convection constitutes a worst-case scenario for forced convection cooling. Under the premises of natural convection heat transfer, the unsteady, 1-dimensional heat conduction equation consists in a linear parabolic partial differential equation with a dominant natural convection boundary condition represented by the mean convective coefficient that depends upon temperature. As expected, the nonlinear unsteady, unidirectional heat conduction problem is complex and does not admit an exact, analytical solution. Instead, the nonlinear unsteady, unidirectional heat conduction problem forcibly necessitates approximate numerical treatment with the finite difference method. The computed dimensionless center, surface, and mean temperatures varying with dimensionless time are obtained numerically and are graphed for 3 solids: iron, aluminum, copper exposed to 3 fluids: air, water, oil; the 6 media are used in numerous engineering applications.     

Shape optimization of steady‐state heat‐conduction fields considering temperature dependency of thermal conductivity coefficient

This paper presents a numerical analysis method for shape optimization of domains with steady‐state heat‐conduction fields considering the temperature dependence of the thermal conductivity coefficient. In this paper, we formulate two shape optimization problems, namely, maximization of thermal dissipation on heat transfer boundaries and minimization of heat‐conduction fields. The shape gradient functions for these shape optimization problems are derived theoretically using the Lagrange multiplier method and formulae of the material derivative. Reshaping is accomplished using the traction method proposed as a solution to the shape optimization problems. The proposed method is validated from the results of two‐dimensional numerical analysis. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20374     

Numerical Methods for Inverse Problem of Heat Conduction with Unknown Boundary Based on Variational Principles with Variable Domain

In this article a variable-domain variational approach to the entitled problem is presented.A pair of comple-mentary variational principles with a variable domain in terms of temperature and heat-streamfunction are firstestablished.Based on them,two methods of solution—generalized Ritz method and variable-domain FEM—both capable of handling problems with unknown boundaries,are suggested.Then,three sample numericalexamples have been tested.The computational process is quite stable,and the results are encouraging.Thisvariational approach can be extended straightforwardly to 3-D inverse problems as well as to other problems inmathematical physics.     

Critical heat flux in non‐uniformly heated tube under low‐pressure and low‐mass‐flux condition

In an actual boiling channel, e.g., a boiler water‐tube, the circumferential heat flux is not uniform. Thus, the critical heat flux (CHF) of a non‐uniformly heated tube becomes an important design factor for conventional boilers, especially for a compact water‐tube boiler with a tube‐nested combustor. A small compact boiler is operated under low‐pressure and low‐mass‐flux conditions compared with a large‐scale boiler, thus the redistribution of liquid film strongly affects the characteristics of CHF. In this investigation, non‐uniform heat flux distribution along the circumferential direction was generated by using the Joule heating of SUS304 tubes with the wall thickness distribution. The heated length of test‐section was 900 mm with an inner diameter of 20 mm and an outer diameter of 24 mm. The center of the inner tube surface was shifted by ε=0, 0.5, 1.0, 1.5 mm from the center of the outer tube surface. The heat flux ratio between maximum and minimum heat flux of these tubes corresponded to 1.0, 1.7, 3.0, and 7.0, respectively. The experimental conditions were as follows: system pressure at 0.3 and 0.4 MPa, mass flux of 10–100kg/(m²s), inlet temperatures at 30° and 80°. The experimental results showed an increase in the critical heat flux substantiated by the existence of the redistribution of the flow. These characteristics are explained by using a concept similar to that of Butterworth's spreading model. © 2005 Wiley Periodicals, Inc. Heat Trans Asian Res, 35(1): 47–60, 2006; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20095     

Effect of variable viscosity and thermal conductivity on heat transfer in a circular porous passage with axial conduction

In this study, heat transfer in a circular porous passage with axial conduction and variable viscosity and thermal conductivity numerically has been investigated. Heat transfer for the different values of Da and Pe is investigated. The results found show that Nusselt distribution with variable viscosity does not significantly differ from that with constant viscosity in both thermally developing and fully developed regions. However, in the developing region there is a significant difference between Nusselt distribution with variable thermal conductivity and Nusselt distribution with constant thermal conductivity. In the fully developed region Nusselt distribution with variable and constant thermal conductivities does not drastically differ. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21028     

Study on heat conduction of functionally graded plate with the variable gradient parameters under the H(t) heat source

Abstract

A hybrid numerical method for heat conduction of functionally graded plate with the variable gradient parameters under the H(t) heat source was studied. A weighted residual equation of heat conduction was considered under thermal boundary conditions. In order to calculate temperature distribution of functionally graded plate with variable gradient parameters, the Fourier transform and inverse Fourier transform were applied and the temperature field was obtained under the H(t) heat source. Results show that the influences of the gradient parameters on temperature distribution are dramatic. But with the increase of gradient parameters, the influences of parameters on the temperature distribution are gradually reduced. When the gradient parameters reach a certain critical value, the temperature does not change anymore. By comparing the temperature distribution of the upper and lower surfaces, it is seen that the temperature presents a gentle downward trend with the increase of the heat source distance, while the temperature does not change with the time in farther distance from heat source. Also, the results show that the influence of the heat source has only partial and limited influence on the temperature, which is in accordance with St. Venant’s Principle. The law of the temperature distribution of the lower surface varies with the gradient parameters, which is also discussed, an optimal gradient parameter with the thermal insulation effect of the functionally graded plate is obtained.