Two-dimensional Steady-state Heat Research Articles

Nodeless variable finite element method for heat transfer analysis by means of flux-based formulation and mesh adaptation

Based on flux-based formulation, a nodeless variable element method is developed to analyze two-dimensional steady-state and transient heat transfer problems. The nodeless variable element employs quadratic interpolation functions to provide higher solution accuracy without necessity to actually generate additional nodes. The flux-based formulation is applied to reduce the complexity in deriving the finite element equations as compared to the conventional finite element method. The solution accuracy is further improved by implementing an adaptive meshing technique to generate finite element mesh that can adapt and move along corresponding to the solution behavior. The technique generates small elements in the regions of steep solution gradients to provide accurate solution, and meanwhile it generates larger elements in the other regions where the solution gradients are slight to reduce the computational time and the computer memory. The effectiveness of the combined procedure is demonstrated by heat transfer problems that have exact solutions. These problems are: (a) a steady-state heat conduction analysis in a square plate subjected to a highly localized surface heating, and (b) a transient heat conduction analysis in a long plate subjected to a moving heat source.

Evaluation of inside surface condensation in double glazing window system with insulation spacer: A case study of residential complex

Aluminum spacer used to keep the glass panes apart acts as a thermal bridge that increases the risk of inside surface condensation due to its high thermal conductivity. In this study, the inside surface condensation in double glazing window systems with conventional aluminum spacer and insulation spacers made of thermally broken aluminum and thick-walled plastic, respectively, is evaluated. Evaluation method and judgment criteria for preventing inside surface condensation are suggested. Thermal characteristics of window system in residential unit are analyzed and two-dimensional steady state heat transfer simulation is carried out in order to obtain the lowest inside surface temperature. The results show that the application of insulation spacer can substantially increase the lowest inside surface temperature, temperature difference ratio and inside air humidity for preventing inside surface condensation and satisfy the required minimum temperature difference ratio.

Design of Micro-Temperature Sensor Array With Thin Film Thermocouples

We are in the process of developing a micro-temperature sensor array with T-type (copper–constantan) thin film thermocouples (TFTCs) to measure the chip temperature distribution of electronic packages. A thin aluminum nitride (AlN) layer of 100 nm thickness was deposited on a silicon substrate. AlN acts not only as an electrical insulator but also as a thermal conductor between the silicon substrate and thin film thermocouples. Copper thin film with a thickness of 50 nm and constantan thin film with the same thickness were deposited on the AlN layer. The sensor array has 10×10 junctions within a 9mm×9mm area, and each junction covers a 100μm×100μm area. Electro-thermal forces measured by TFTCs using one-dimensional steady-state heat conduction were compared with the electro-thermal forces measured by standard thermocouples, and the difference between the Seebeck coefficients of the copper material and the constantan thin film was calculated according to these measurements. In order to verify the sensor array, it was placed under two-dimensional steady-state heat conduction, and electro-thermal forces were measured and converted to temperatures. Finite element analysis simulation results were compared with the temperatures, and with experimental measurements were found to be in agreement with the simulated values.

Numerical analysis of two-dimensional steady-state and transient heat transfer in a parallel duct filled with pressurized He II

Analysis on steady-state and transient heat transfer on a flat plate at the middle of a parallel duct immersed in He II was performed for bath temperatures from 1.8 to 2.1 K at 101.3 kPa. Two-dimensional computer code named SUPER-2D developed by the authors based on the two-fluid model and the theory of mutual friction was used. Steady-state critical heat flux (CHF) and the time lag from the application of a step heat input to λ transition, that is called a lifetime, were obtained numerically for various step heat fluxes and for the channel gaps from 2 to 20 mm. Effect of the gap restriction on the CHF and the lifetime were clarified. The solutions were compared with the experimental data for the ducts with the same structures and the corresponding conditions. They agreed well with the experimental data. The heat transport mechanism in the parallel duct was clarified.

Heat-transfer and solidification model of continuous slab casting: CON1D

A simple, but comprehensive model of heat transfer and solidification of the continuous casting of steel slabs is described, including phenomena in the mold and spray regions. The model includes a one-dimensional (1-D) transient finite-difference calculation of heat conduction within the solidifying steel shell coupled with two-dimensional (2-D) steady-state heat conduction within the mold wall. The model features a detailed treatment of the interfacial gap between the shell and mold, including mass and momentum balances on the solid and liquid interfacial slag layers, and the effect of oscillation marks. The model predicts the shell thickness, temperature distributions in the mold and shell, thickness of the resolidified and liquid powder layers, heat-flux profiles down the wide and narrow faces, mold water temperature rise, ideal taper of the mold walls, and other related phenomena. The important effect of the nonuniform distribution of superheat is incorporated using the results from previous three-dimensional (3-D) turbulent fluid-flow calculations within the liquid pool. The FORTRAN program CONID has a user-friendly interface and executes in less than 1 minute on a personal computer. Calibration of the model with several different experimental measurements on operating slab casters is presented along with several example applications. In particular, the model demonstrates that the increase in heat flux throughout the mold at higher casting speeds is caused by two combined effects: a thinner interfacial gap near the top of the mold and a thinner shell toward the bottom. This modeling tool can be applied to a wide range of practical problems in continuous casters.

Numerical solutions of conjugate heat transfer and thermal stresses in a circular pipe externally heated with non-uniform heat flux

Conjugate heat transfer by forced convection through an externally heated pipe has many important engineering applications. In the present work, the radial and axial heat conductions and thermal stresses in a pipe with uniform or non-uniform wall heat flux of fully developed laminar forced convective conjugate heat transfer have been considered for analysis. The analysis is based on the two-dimensional steady-state heat conduction equation and laminar boundary layer equation for the flowing fluid by using a finite difference scheme. Water has been used as a fluid. Numerical calculations have been performed by using the FLUENT 4.5 and HEATING7 computer codes. The temperature and stress ratio distributions inside the pipe wall, heated from the outer surface by applying uniform and non-uniform heat fluxes, have been presented for two different mean flow velocities. The temperature distributions of the flowing fluid inside the pipe have also been presented for all investigated cases.

B223 Excelの表計算機能を利用した数値計算プロセスの可視化

A new visual-oriented numerical method which applies the spreadsheet of Excel has been proposed. In the present study, a programming process on two-dimensional steady-state heat conduction problem in the rectangular coordinate system is explained by way of a specific example, where a volumetrically heat generating rectangular region with a concave is heated uniformly at one wall and other walls are kept at some temperatures. The present steady-state numerical calculation has been realized by introducing icon cells in which difference equations concerning the inner, boundary and singular points etc. are embedded. Through the programming process using this specific model, the proposed method is shown one of the useful numerical methods.

Numerical analysis for steady-state two-dimensional heat transfer from a flat plate at one side of a duct containing pressurized He II

A computer code of two-dimensional heat transfer in superfluid helium named SUPER-2D was developed based on the two fluid model and the theory of the mutual friction. Critical heat fluxes (CHFs) on a flat plate located at one end of six rectangular ducts having different cross-sectional areas were calculated by using the SUPER-2D for temperatures ranging from 1.8 to 2.07 K in pressurized He II. The ratios of the cross-sectional area to heated area, A d/ A h, were varied from 1.0 to 3.5. As the ratio increased, the CHFs rapidly increased and approached a constant value. The solutions agreed with the experimental data by Tatsumoto et al. [Adv. Cryog. Eng. 45 (1999) 1073] for the same structures of the ducts and corresponding experimental conditions. It was found from the analysis that several vortices are generated around the heated surface and play an important role in determining the CHF.

Numerical solution of a Cauchy problem for the Laplaceequation

We consider a two-dimensional steady state heat conductionproblem. The Laplace equation is valid in a domain with a hole.Temperature and heat-flux data are specified on the outerboundary, and we wish to compute the temperature on the innerboundary. This Cauchy problem is ill-posed, i.e. the solutiondoes not depend continuously on the boundary data, and smallerrors in the data can destroy the numerical solution. Weconsider two numerical methods for solving this problem. Astandard approach is to discretize the differential equation byfinite differences, and use Tikhonov regularization on thediscrete problem, which leads to a large sparse least squaresproblem. We propose to use a conformal mapping that maps theregion onto an annulus, where the equivalent problem is solvedusing a technique based on the fast Fourier transform. Theill-posedness is dealt with by filtering away high frequenciesin the solution. Numerical results using both methods are given.

Optimal experimental designs for linear inverse problems

Optimal experimental designs for inverse and ill-posed problems are investigated. Given discrete points s t, for i=1,…, m, a Fredholm integral equation of the first kind can be discretized into a semi-discrete form by using a collocation method, and into a fully discrete form by restricting the unknown function within an n-dimensional subspace. Our optimal design problem is to determine a distribution of the discrete points s i, i = l,...,m maximizing the nth singular value of the semi-discrete or the fully discrete operators, where the number n plays the role of the regularization parameter (it could be fixed a priori or be selected a posteriori). The optimal experimental designs for various inverse problems -numerical differentiation, inversion of Laplace transformation, small particles sizing by light extinction and two-dimensional steady-state inverse heat conduction problem, are studied numerically in detail. Several new and interesting features for the considered problems are found.

Thermal Analysis of Hot Roller in a Dry Film Laminator

The thermal analysis of the hot roller in a dry film laminator is studied numerically by steady-state two-dimensional heat transfer. In the laminating process for PDP glass or PCB, the temperature distributions in a hot roller are presented considering the effects of the roller rotation speed and the inner and outer radii of the roller. The results show that the temperature distributions are strongly dependent on Peclet number. If Pe number becomes larger, the iso-thermal lines are more concentric about the rotating axis and the temperature difference on the hot roller surface decreases exponentially. It also shows that if the contact angle between the roller and the film becomes smaller the temperature difference becomes smaller. However, the changes of the rollers inner or outer radius have little effect on the temperature difference.

A new finite element for treating plane thermomechanical heterogeneous solids

This study is concerned with the development and implementation of a new finite element which is capable of treating the problem of interacting circular inhomogeneities in heterogeneous solids under mechanical and thermal loadings. The general form of the element, which is constructed from a cell containing a single circular inhomogeneity in a surrounding matrix, is derived explicitly using the complex potentials of Muskhelishvili and the Laurent series expansion method. The newly proposed eight-noded plane element can be used to treat quite readily the two-dimensional steady-state heat conduction and thermoelastic problems of an elastic circular inclusion embedded in an elastic matrix with different thermomechanical properties. Moreover, the devised element may be applied to deal with arbitrarily and periodically located multiple inhomogeneities under general mechanical and thermal loading conditions using a very limited number of elements. The current element also enables the determination of the local and effective thermoelastic properties of composite materials with relative ease. Three numerical examples are given to demonstrate its versatility, accuracy and efficiency. Copyright © 1999 John Wiley & Sons. Ltd.

Multinode finite element based on boundary integral equations

A finite element constructed on the basis of boundary integral equations is proposed. This element has a flexible shape and arbitrary number of nodes. It also has good approximation properties. A procedure of constructing an element stiffness matrix is demonstrated first for one-dimensional case and then for two-dimensional steady-state heat conduction problem. Numerical examples demonstrate applicability and advantages of the method. © 1998 John Wiley & Sons, Ltd.

A HARMONIC-SINC SOLUTION OF THE LAPLACE EQUATION FOR PROBLEMS WITH SINGULARITIES AND SEMI-INFINITE DOMAINS

In this article, a recently derived harmonic sine approximation method is used to obtain approximate solutions to two-dimensional steady-state heat conduction problems with singularities and semi-infinite domains and Dirichlet boundary conditions. The first problem is conduction in a square geometry, and the second one involves a semi-infinite medium with a rectangular cavity. In the case of square geometry, results show that the harmonic sine approximation method performs better than the finite-difference and multigrid methods everywhere within the computational domain, especially at points close to the singularity at the upper left and right corners of the square. The results from the harmonic sine approximation method for the semi-infinite domain problem with a very shallow rectangular cavity agree well with the analytical solution for a semi-infinite domain without the cavity. The results obtained from the harmonic sine approximation also agree well with the results from the finite-element package ANSYS for the semi-infinite medium conduction problem with a rectangular cavity of aspect ratio 1.

USE OF TWO DIMENSIONAL FINITE ELEMENTS FOR APPROXIMATE ANALYSIS OF TEMPERATURE DISTRIBUTION IN AN INDUCTION MOTOR

ABSTRACT In this paper the problem of two-dimensional steady state heat flow in an induction motor is solved using finite-element formulation and employing simple triangular elements in the xy plane of the rectangular co-ordinate system. The shape functions and exact solution matrices are derived algebraically for utmost economy in computation. The global system of equations is determined by a suitable algorithm to avoid repeated calculations of the elemental stiffness matrix. The temperature distribution has been computed considering convection form the outer boundary surfaces of the machine.

Spread Sheet Model of Continuous Casting

Spreadsheet programs, such as Microsoft Excel, Informix WINGZ, and Lotus 123, provide a framework for very fast and easy development of simple engineering models. The present paper describes a model of the continuous casting process that has been developed using a spreadsheet program, Microsoft Excel, running on a Macintosh II personal computer. The model consists of two-dimensional (2-D) steady-state finite-difference heat conduction calculations within a continuous casting mold coupled to a one-dimensional (1-D) transient solidification heat transfer model of the solidifying shell. The model structure and equations are described and the model predictions are compared with previous solutions. Practical examples using the model are discussed and sample results are provided. Spreadsheet programs running on personal computers are capable of relatively complex calculations that would require extensive effort using conventional programming languages.

Two-Phase Potentials for the Treatment of an Elastic Inclusion in Plane Thermoelasticity

A solution to the uncoupled two-dimensional steady-state heat conduction and thermoelastic problems of an elastic curvilinear inclusion embedded in an elastic matrix, with different thermomechanical properties, is provided. The proposed analysis describes the heat conduction problem in terms of one holomorphic complex potential and the thermoelastic problem in terms of two holomorphic potentials; known hereafter as two-phase potentials. The general results of the developed analysis are applied to specific examples and explicit forms of the solution are obtained. It is shown that a uniform heat flow at infinity induces a linear stress distribution within the elliptic inclusion.

A local iteration scheme for nonlinear two-dimensional steady-state heat conduction: a BEM approach

An efficient algorithm is proposed to solve the steady-state nonlinear heat conduction equation using the boundary element method (BEM). Nonlinearity of the heat conduction equation arises from nonlinear boundary conditions and temperature dependence of thermal conductivity. Using Kirchhoff's transformation, the case of temperature dependence of thermal conductivity can be transformed to the nonlinear boundary conditions case. Applying the BEM technique, the resulting matrix equation becomes nonlinear. The nonlinearity, however, only involves the boundary nodes that have nonlinearboundary conditions. The proposed local iterative scheme reduces the entire BEM matrix equation to a smaller matrix equation whose rank is the same as the number of boundary nodes with nonlinear boundary conditions. The Newton-Raphson iteration scheme is used to solve the reduced nonlinear matrix equation. The local iterative scheme is first applied to two one-dimensional problems (analytical solutions are possible) with different nonlinear boundary conditions. It is then applied to a two-region problem. Finally, the local iterative scheme is applied to two cavity problems in which radiation plays a role in the heat transfer.

Approximate analysis of steady-state heat conduction in an induction motor

The problem of two-dimensional steady-state heat flow in the stator of an induction motor is solved using finite element formulation and using arch shaped elements in the theta plane of the cylindrical coordinate system. The shape functions and exact solution matrices are derived algebraically for utmost economy in computation. The temperature distribution has been determined considering convection from the stator frame and the cylindrical air-gap surface. The temperatures obtained by this two-dimensional approximation are compared with the actual temperatures obtained by three-dimensional analysis.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Efficient BEM Design Sensitivity Analysis of Two-Dimensional Steady-State Heat Conduction Problems for Fields with Small Circular Holes.

In conventional design sensitivity analysis using the boundary element method, all boundaries must be discretized. Therefore, the size of the coefficient matrix of the resulting system of linear algebraic equations becomes quite large for a field with small circular holes. In this paper, we present an efficient sensitivity analysis approach for two-dimensional steady-state heat conduction problems. In the boundary element formulation, the temperature and the heat flux over the boundary of a circular hole are approximated by simple trigonometrical functions. Then, the boundary integral equation is differentiated directly with respect to a design variable. The resulting boundary integral equation relates the sensitivity coefficients of temperature and heat flux over the boundary. The effectiveness of the proposed approach is illustrated through several numerical examples.