Unsteady-state Heat Conduction Problems Research Articles

Registration of melting temperature at phase boundaries in melting powder mixtures

An integral constituent of the induction surfacing process for parts hardening pertains to the thermal treatment of rigid alloy particulates and the flux contained within the surfacing amalgam. This scholarly exposition delineates the outcomes of a comprehensive investigation, oriented towards the quantification and simulation of thermal gradients at interphase boundaries within intricate amalgamations of melting and thermosetting powdery substrates. In order to monitor the thermal dynamics during induction surfacing, the application of the CA-microthermocouple methodology and the thermal indication technique utilizing self-propagating SHS-process compositions is posited. The study encompasses both computational simulations and the resolution of the unsteady-state heat conduction problem within the chosen model of composite contacting materials. The devised methodologies for the intricate tracking of thermal profiles during induction surfacing offer the capability to ascertain the temperatures at which the surfacing amalgam and its individual constituents undergo liquefaction, in conjunction with the temperature gradient exhibited by the surface of the targeted component. This facet holds marked significance in the context of optimizing the hardening regimen and ensuring the requisite attributes of the amalgamated materials post-fusion. The outcomes of this research bear practical relevance in industrial applications, wherein enhancements to the caliber and dependability of hardened components are sought, simultaneously facilitating the curtailment of production overheads.

Simulation of Heat Transfer in Laser Transmission Welding

Abstract A numerical simulation of the heat transfer during laser transmission welding is presented. A finite difference approach was used to solve the one-dimensional unsteady-state heat conduction problem and to investigate the effect of welding conditions on the time-dependent temperature profiles for PA 6. For the needs of the simulation, the process was divided into heating and heat redistribution periods. The absorption coefficient of the laser-transparent part was measured experimentally and that of the laser-absorbing part was fitted using experimental data. The predicted temperature profiles were combined with experimental meltdown data to estimate the heat-affected zone thickness in the welded specimens. Good agreement was found between the estimated and measured heat-affected zone thickness values.

An analytical approach of heat transfer modelling with thermal stresses in circular plate by means of Gaussian heat source and stress function

In the proposed research work we have used the Gaussian circular heat source. This heat source is applied with the heat flux boundary condition along the thickness of a circular plate with a finite radius. The research work also deals with the formulation of unsteady-state heat conduction problems along with homogeneous initial and non-homogeneous boundary condition around the temperature distribution in the circular plate. The mathematical model of thermoelasticity with the determination of thermal stresses and displacement has been studied in the present work. The new analytical method, Reduced Differential Transform has been used to obtain the solution. The numerical results are shown graphically with the help of mathematical software SCILAB and results are carried out for the material copper.

Additional boundary conditions in unsteady-state heat conduction problems

Using some additional sought function and boundary conditions, a precise analytical solution of the heat conduction problem for an infinite plate was obtained using the integral heat balance method with symmetric first-order boundary conditions. The additional sought function represents the variation of temperature with time at the center of a plate and, due to an infinite heat propagation velocity described with a parabolic heat conduction equation, changes immediately after application of a first-order boundary condition. Hence, the range of its time and temperature variation completely incorporates the ranges of unsteadystate process times and temperature changes. The additional boundary conditions are such that their fulfilment is equivalent the fulfilment of a differential equation at boundary points. It has been shown that the fulfilment of an equation at boundary points leads to its fulfilment inside the region. The consideration of an additional sought function in the integral heat balance method provide a possibility to confine the solution of an equation in partial derivatives to the integration of an ordinary differential equation, so this method can be applied to the solution of equations, which do not admit the separation of variables (nonlinear, with variable physical properties of a medium, etc.).

Fast thermal simulation of a heated crude oil pipeline with a BFC-Based POD reduced-order model

A proper orthogonal decomposition (POD) Galerkin reduced-order model (ROM) for unsteady-state heat conduction problems is proposed based on body-fitted coordinate (BFC), and the discrete format of spatial derivative of basis function which could enhance the calculation precision is employed in the present paper. Applying the POD reduced-order model into the fast thermal calculation of oil pipeline, such as the batch transportation of cold and hot oil as well as pipeline shutdown, the computing speed of POD reduced-order model is about 259 times and 177 times faster than that of traditional finite volume method (FVM) respectively. And from the calculation results, the reduced-order model is proved to enjoy good precision. Thus the reduced-order model holds great engineering application value and would be a very helpful tool to solve the time-consuming problem involved in the optimal design and risk assessment of oil pipeline.

Study on a BFC-Based POD-Galerkin Reduced-Order Model for the Unsteady-State Variable-Property Heat Transfer Problem

A proper orthogonal decomposition–reduced-order model (POD-ROM) method for unsteady-state heat conduction problems with variable physical properties is developed based on the body-fitted coordinate. Two mathematically equivalent forms of the ROM are derived, but their performance is quite different for finite volume. It is found that to obtain accurate prediction by the POD-ROM, the operators of the Jacobi factor and other variables in the ROM should be consistent with those in the finite-volume method (FVM). Validated by a complicated unsteady-state heat conduction problem, the established ROM can obtain accurate prediction of the unsteady-state heat conduction problem with variable properties in which the maximum ratio of thermal conductivity reaches 333.

Numerical solution of transient heat conduction problems using improved meshless local Petrov–Galerkin method

An improved meshless local Petrov–Galerkin (MLPG) method is presented and applied to calculate the two-dimensional unsteady state heat conduction problems. In this method, the moving Kriging interpolation is employed instead of the traditional MLS approximation to construct the MLPG shape functions which possess Kronecker delta function property and thus make it easy to implement essential boundary conditions and then, the Heaviside step function is used as the test function over a local sub-domain. Since no mesh is needed either for integration of the local weak form, or for construction of the shape functions, the presently developed MLPG method is a truly meshless method. Several examples are performed to illustrate the accuracy and efficiency of the present method. A good agreement can be found among the proposed, analytical and finite element methods.

Application of the boundary face method for thermal analysis in dam construction

This paper applies the boundary face method (BFM) to solve both steady-state and unsteady-state heat conduction problems in the dam construction. Instead of employing the isoparametric elements as in the boundary and finite element methods, the BFM uses directly geometric models from CAD packages to avoid geometric errors for any complex structures. Moreover, the variable approximations and the boundary integration are all performed in the parametric space of the boundary surfaces, which means that the elements and integral patches are defined in the parametric space. A numerical example of steady-state analysis presented in this paper is a concrete dam structure which consists of several blocks. The example of unsteady-state analysis, which concerns a single block with several holes in it, has also been presented to illustrate the validity and the efficiency of the proposed method.

Analysis of Unsteady State Heat Conduction Performance in Anisotropic Materials

The magnitudes of heat flux for both isotropic materials and anisotropic materials can be determined from Fourier’s law. But the mechanism of heat transfer in anisotropic materials is quite different from that in isotropic materials. The thermal conductivities at one point are not equal in all directions for anisotropic materials. The magnitude of heat transfer rate of anisotropic material in one direction is not only relevant to the temperature gradient in this direction, but also relevant to the temperature gradient which is normal to the direction. Basing on one-dimensional unsteady state heat conduction problem in isotropic materials which can be solved by separation-of-variables method, the performance of unsteady state heat conduction in anisotropic materials was analyzed. And the unsteady state temperature field of the anisotropic plane wall was obtained. The results will provide help for the energy conservation of anisotropic materials.

Analytical Solution of Unsteady State Conduction Problem with Heat Source

Separation-of-variables method is one of the analytical solution methods to solve unsteady state heat conduction problems. But unsteady state conduction with heat source is an inhomogeneous problem. It can not be solved by separation-of-variables method directly. The combination of variables division method and separation-of-variables method was applied in this paper to deal with heat conduction with heat source in an infinite plate wall. The problem was divided to a steady state and inhomogeneous heat conduction problem, and an unsteady state and homogeneous heat conduction problem by variables division method. The steady state and inhomogeneous problem can be integrated directly. The unsteady state and homogeneous problem can be transformed to the problem that can be solved by separation-of-variables method directly through variable substitution. The unsteady state temperature field in the infinite plate wall was then obtained.

Matrix free meshless method for transient heat conduction problems

In the paper, the element free Galerkin method (EFGM) is applied to calculate two-dimensional unsteady state heat conduction problems. As is well known, most of the meshless methods have higher computational cost than that of finite element method (FEM). In order to overcome this shortcoming especially for transient heat conduction problems, mass lumping procedure is adopted in EFGM, which can decrease the computational cost evidently. Moreover, this technique which can simplify the solution procedure makes the essential boundary conditions enforced directly. The results obtained by EFGM combining mass lumping technique are compared with those obtained by finite element method as well as analytical solutions, which shows that the solutions of the present method are in good agreement with FEM’s and analytical solutions.

Excelの表計算機能を利用した数値計算プロセスの可視化 : 第5報 2次元円筒座標系で記述された非定常熱伝導問題へのアイコン化セルの拡張(一般セッション 伝熱解析及び理論 I)

Excelの表計算機能を利用した数値計算プロセスの可視化 : 第5報 2次元円筒座標系で記述された非定常熱伝導問題へのアイコン化セルの拡張(一般セッション 伝熱解析及び理論 I)

B211 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 newly developed visual-oriented numerical method for solving the two-dimensional unsteady-state heat conduction problem which applies the spreadsheet of Excel is explained through a simple physical model. The present unsteady-state numerical calculation has been realized by introducing a dummy sheet with empty cells and a sheet with icon cells, in which difference equations concerning the inner, boundary and singular points and so on are embedded. By organizing and expanding such sorts of icon cells, the proposed method is expected to get applied against much wider range of heat transfer problems in the future.

On the analysis of the electrochemical spark machining process

The electrochemical spark machining (ECSM) process has been proved as a potential process for machining of low machinability high-strength electrically non-conducting materials, but the mechanism of material removal during the process, by and large, is not yet understood. In the present work, the electrochemical discharge is modelled as a phenomenon similar to that which occurs in arc discharge valves. This phenomenon is used to explain various experimental results, on the basis of circuit and arc discharge valve characteristics. The spark energy and the approximate order of hydrogen gas bubble diameter are computed by the proposed valve theory. Material removal rate is evaluated by modelling the problem as a 3-D unsteady state heat conduction problem. The problem is solved by the finite element method to compute the temperature distribution which is post-processed for estimating material removal per spark, overcut obtained in the machined cavity, and attainable maximum penetration depth. The conclusion drawn is that the application of valve theory to the ECSM process seems to be realistic. Estimated material removal rate, overcut and maximum penetration depth show a good agreement with experimental findings.

Error minimization in nonlinear variational unsteady-state heat conduction problems

The article considers a variational method of solving unsteady-state heat conduction problems.

Determination of unsteady-state temperature fields in multilayered orthotopic plates

A new hypothesis on temperature distribution with respect to the thickness of a multilayered plate is proposed. An analytical solution of the unsteady-state heat conduction problem is obtained for rectangular multilayered orthotropic plates.

Solution of the unsteady-state heat conduction problem for a two-dimensional region with a moving boundary

With the use of the convolution-type functional a variational description is given for the process of unsteady-state heat conduction with the first-kind boundary conditions for a two-dimensional region whose boundary moves in time according to the familiar arbitrary law. Based on the Galerkin-Kantorovich method, a corresponding system of Euler equations is written the solution of which (numerical or analytical) is required to determine the temperature field in each specific case. As an example, the first and second analytic approximations to the solution of the above problem are obtained for the case of the deformation of a prism having initially a circular cross-section.

Some basic problems of heat conduction in an anisotropic finite medium

It is shown that Galerkin's variational formulation can be advantageously used when solving certain unsteady state heat conduction problems in two-dimensional anisotropic media. The case of heat sources is also considered. The methodology developed in this study can be used also for three-dimensional problems.

Numerical models for casting solidification: Part I. The coupling of the boundary element and finite difference methods for solidification problems

An investigation on the coupling technique for boundary element and finite difference methods was carried out to simulate solidification processes. In the coupling model, solidification problems in casting and unsteady-state heat conduction problems in mold regions were analyzed by explicit finite difference and boundary element methods, respectively. A comparison was made between the coupling model and the finite difference method on the solidification of castings in metal and sand molds. The proper range of time steps for boundary elements in transient problems was presented for a simple geometry. And two types of time marching schemes were proposed for application of the boundary element method to solidification processes.

Radial temperature distribution in isotachophoresis columns of circular cross-section

An analysis of the unsteady-state heat-conduction problem in isotachophoretic columns of circular cross-section is given, based on a linear variation of electrica conductivity with temperature and taking into account the effects of a finite wall thickness. It is seen that previous analyses, based on constant electrical conductivity, tend to underpredict actual temperature distributions considerably. The use of thick-walled columns is seen to be advantageous both from the viewpoint of higher resolution and of increased production.