Two-dimensional Transient Heat Conduction Problems Research Articles

Mathematical Modeling Study on Combined Side and Top Blowing AOD Refining Process of Stainless Steel

Mathematical modeling of the combined side and top blowing AOD refining process of stainless steel has further been studied. A new model proposed is generally based on the analysis and assumptions made for the process in our previous investigation. Particularly, the heat transfer characteristics of the vessel are analyzed in terms of the two-dimensional transient heat-conduction problems of composite walls. The heat and mass balances of the system are more comprehensively and precisely performed. The model has been applied to 28 heats of 304-grade austenitic stainless steel refining in a 120 t AOD converter. The results present that the changes in the composition and temperature of the liquid steel with the time during the whole process can be accurately predicted using the model. The competitive oxidation among the elements in the steel during the oxidative refining with the relevant oxygen distribution ratios, and the competitive reduction of the oxides in the slag during the Ar agitating and reductive refining with the appropriate oxygen supply ratios can be well characterized and reasonably determined using the Gibbs free energies of the reactions. The critical carbon concentrations (after which the decarburization alters to be limited by the mass transfer of carbon in liquid steel) for the top, side and combined blowing of 304-grade steel in this work are in the ranges of 0.895 to 0.942, 0.078 to 0.224, 0.144 to 0.255 mass%, respectively. The effects of some factors on the refining result and the optimization of blowing technology have been considered from the model predictions. The model can provide a reliable basis and useful information for determining and optimizing the technology of this AOD process of stainless steel, and controlling the process in real time and on-line.

Local RBF-DQ method for two-dimensional transient heat conduction problems

The meshless local radial basis function-based differential quadrature (RBF-DQ) method is applied on two-dimensional heat conduction for different irregular geometries. This method is the combination of differential quadrature approximation of derivatives and function approximation of radial basis function. Four different geometries with regular and irregular boundaries are considered, and numerical results are compared with those gained by finite element (FE) solution achieved by COMSOL commercial code. Outcomes prove that current technique is in very good agreement with FEM and this fact that RBF-DQ method is an accurate and flexible method in solution of heat conduction problems.

Two-Dimensional Inverse Heat Transfer Analysis of Functionally Graded Materials in Estimating Time-Dependent Surface Heat Flux

Two-dimensional transient inverse heat conduction problem (IHCP) of functionally graded materials (FGMs) is studied herein. A combination of the finite element (FE) and differential quadrature (DQ) methods as a simple, accurate, and efficient numerical method for FGMs transient heat transfer analysis is employed for solving the direct problem. In order to estimate the unknown boundary heat flux in solving the inverse problem, conjugate gradient method (CGM) in conjunction with adjoint problem is used. The results obtained show good accuracy for the estimation of boundary heat fluxes. The effects of measurement errors on the inverse solutions are also discussed.

Complex variable reproducing kernel particle method for transient heat conduction problems

On the basis of reproducing kernel particle method(RKPM), the complex variable reproducing kernel particle method (CVRKPM) is discussed. The advantage of the CVRKPM is that the correction function of a 2-D problem is formed with 1-D basis function when the shape function is obtained. Then, we apply the complex variable method to two-dimensional transient heat conduction problems. In combination with the Galerkin weak form of transient heat conduction problems, the penalty method is employed to enforce the essential boundary conditions, the CVRKPM for transient heat conduction problems is investigated and the corresponding formulae are obtained. Compared with the conventional RKPM, the CVRKPM introduced in this paper has a higher precision and a lower computation cost. Some examples given in this paper verify the effectivity of the proposed method.

Modeling Transient Heat Transfer Using SPH and Implicit Time Integration

In this article, a two-dimensional transient heat conduction problem is modeled using smoothed particle hydrodynamics (SPH) with a Crank-Nicolson implicit time integration technique. The main feature of this work is that it applies implicit time stepping, an unconditionally stable Crank-Nicolson approach, in the thermal conduction simulation of liquid-phase diffusion (LPD) semiconductor crystal growth. This SPH simulation is compared with the equivalent finite-volume results. As well, two transient thermal conduction test problems are simulated using both explicit and Crank-Nicolson schemes, and their results compared with the analytical solutions. One of the current drawbacks of SPH is that explicit time-stepping algorithms, such as predictor-corrector methods or leapfrog methods, require extremely small time steps for a stable simulation. Using implicit time integration opens SPH up to a much larger class of practical problems in applied mechanics.

Applying the Hopfield Neural Network to the Solution of the Temperature Distribution Field in Heat Conduction Problems

This article employs the continuous-time analog Hopfield neural network (CHNN) to compute the temperature distribution in one- and two-dimensional transient heat conduction problems. The relationship between the CHNN synaptic connection weights and the governing equations of the problems is established and a corresponding network connectivity circuit design scheme proposed. The CHNN algorithm is initially applied to the solution of conventional problems and is then used to solve more complicated problems involving time-varying heat flux profiles. The results confirm that the CHNN scheme provides an accurate means of solving the transient temperature distributions of heat conduction problems on a real-time basis.

Structured Multiblock Grid Solution of Two-Dimensional Transient Inverse Heat Conduction Problems in Cartesian Coordinate System

ABSTRACT In this study a structured multiblock grid is used to solve two-dimensional transient inverse heat conduction problems. The multiblock method is implemented for geometric decomposition of the physical domain into regions with blocked interfaces. The finite-element method is employed for direct solution of the transient heat conduction equation in a Cartesian coordinate system. Inverse algorithms used in this research are iterative Levenberg-Marquardt and adjoint conjugate gradient techniques for parameter and function estimations. The measured transient temperature data needed in the inverse solution are given by exact or noisy data. Simultaneous estimation of unknown linear/nonlinear time-varying strengths of two heat sources in two joined surfaces with equal and different heights is obtained for the solution of the inverse problems, and the results of the present study for unknown heat source functions are compared to those of exact functions. This study is an attempt to challenge the goal of combining a multiblock technique with inverse analysis methods. In fact, the structured multiblock grid is capable of providing accurate solutions of inverse heat conduction problems (IHCPs) in industrial configurations, including composite structures. In addition, the multiblock IHCP solver is suitable to estimate unknown parameters and functions in these structures.

Transient heat conduction in anisotropic and functionally graded media by local integral equations

Reliable computational techniques are developed for the solution of two-dimensional (2-d) transient heat conduction problems in anisotropic media with continuously variable material coefficients. Two kinds of the domain-type interpolation, namely the standard domain elements and the meshless point interpolation, are adopted for the approximation of the spatial variation of the temperature field or its Laplace-transform. The coupling among the nodal values of the approximated field is given by integral equations considered on local sub-domains. Three kinds of local integral equations are derived from physical principles instead of using a weak-form formulation. The accuracy and the convergence of the proposed techniques are tested by several examples and compared with exact benchmark solutions, which are derived too.

Modified smoothed particle hydrodynamics method and its application to transient problems

A modification to the smoothed particle hydrodynamics method is proposed that improves the accuracy of the approximation especially at points near the boundary of the domain. The modified method is used to study one-dimensional wave propagation and two-dimensional transient heat conduction problems.

923 二次元温度分布を受ける片持ちはりの熱誘起振動問題

This study is concerned with theoretical development for thermal induced vibration problem of beam. We suppose that thermal and mechanical material properties in beam are given as power function in term of transverse coordinate z. Firstly, we have developed two-dimensional transient heat conduction problem of homogeneous beam due to cyclic heat supply. Then, we have developed thermal induced vibration problem of beam clamped at one edge and free at opposite edge. Next, we have carried out numerical calculations and have discussed about effect of cyclic temperature fluctuation on thermal displacements, thermal stresses and resonance characteristics in beam.

On the Use of Spreadsheets in Heat Conduction Analysis

Detailed methodology and different techniques for simply utilizing the widely available and user friendly spreadsheet programs in heat conduction analysis are presented. Evaluation of analytical and numerical solution of heat conduction problems via spreadsheets is investigated. Detailed techniques of obtaining spreadsheet numerical solutions for one- and two-dimensional steady and transient heat conduction problems are introduced. A new technique of marching the transient numerical solution with time, in a single layer spreadsheet, for one- and two-dimensional heat conduction is explained. Creating macros that automate the spreadsheet processes, particularly calculations, is detailed. Utilization of the powerful graphical facility that is built in the spreadsheets to graphically represent the obtained solutions is outlined.

An overdeterrnined two-dimensional transient inverse heat conduction problem

The method of solving the two-dimensional nonlinear inverse heat conduction problems is presented. The time- and space temperature distribution inside a solid and heat transfer coefficient distribution on the boundary of the solid is determined based on the temperature histories measured in several selected inside locations. The problem is overdetermined because the number of measuring points is higher than the number of calculated parameters (components of heat transfer coefficient).

Model reduction in numerical heat transfer problems

Four recent model reduction schemes, namely, balanced truncation, singular perturbation balanced truncation, Hankel norm approximation, and H ∞ norm approximation are applied to a two-dimensional transient heat conduction problem. The approximations of these four model reduction methods for step response in temperature variation were compared with the finite difference model. It is found that a substantial model order reduction for heat transfer problem is possible. The results are encouraging and attractive especially for the purpose of simulation and control problems.

Time-stepping boundary element method applied to 2-D transient heat conduction problems

This paper is concerned with a new boundary element method based on the time-stepping approximation of the time derivative in the governing differential equation. The reduced differential equation at an arbitrary time step is transferred into an equivalent boundary integral equation, and it is solved in a step-by-step manner by means of the standard boundary element method. The boundary integral formulation is presented in detail, and its numerical implementation is used to develop a computer code for two-dimensional transient heat conduction problems in isotropic, homogeneous media. The computer code developed is applied to several example problems, which demonstrate the effectiveness of the proposed solution procedure. The paper also demonstrates how to determine an appropriate time step under a given boundary element mesh by using the charts of relationship between the diffusion number and the boundary element mesh obtained by example computations.

Application of the Integral Method to Two-Dimensional Transient Heat Conduction Problems

Application of the Integral Method to Two-Dimensional Transient Heat Conduction Problems

Heat conduction model for the calculation of the volume of molten metal in electric discharges

The craters obtained by electric sparks in electro-discharge machining are numerically calculated by use of a thermal model. The computational procedure is based on the exact solution of the related two-dimensional transient heat conduction problem in a finite or semi-infinite body subjected to a stepwise or time-dependent uniform heat flux over a small circular part of its upper surface. Optimal geometrical conditions, relative to a fast convergence and negligible influence of the boundaries, are demonstrated. By superposition of rectangular heat flux pulses over circular parts with different diameters, the influence of the channel widening can be investigated. It is shown that a theoretical model should include the possibility of accounting for the plasma channel widening in order to obtain acceptable agreement with experimental data or to predict the optimal operating conditions.

Formulation of the two-dimensional transient heat-conduction problem for the bladed disk of a gas turbine

A system of differential equations is derived for the transient heat conduction associated with a cooled gas turbine disk equipped with shrouded blades on a shank.