Solution Of Heat Conduction Problem Research Articles

A finite volume scheme with an adaptive heat flux calculation for fast temperature prediction for wire arc additive manufacturing

Thermal simulations are an essential tool for optimizing the wire arc additive manufacturing process. However, their adoption in the process optimization workflow is still challenged by the long computational time required for the numerical solution of the thermal models. In this article, a Finite Volume scheme is proposed for predicting the temperature field evolution during wire arc additive manufacturing. A novel adaptive calculation of the heat flux is used for the acceleration of the numerical solution of heat conduction problems with a moving heat source. Numerical tests and experimental data are utilized to assess the performance of the proposed method in comparison to an explicit Finite Element scheme and a standard explicit Finite Volume scheme. Compared to the reference schemes, the accuracy loss is demonstrated to be negligible for the application to process planning. Depending on the simulated case, more than twice the computational efficiency of the reference schemes is achieved. The results demonstrate the suitability of the method for the simulation of heat conduction problems with a moving heat source and particularly for fast temperature predictions for wire arc additive manufacturing.

Дополнительные граничные условия в задачах теплопроводности с переменным по координате начальным условием

It is exceedingly difficult to obtain mathematically accurate analytical solutions of heat conduction problems with a variable initial condition. Known solutions of these problems are expressed by cumbersome functional series that converge poorly in the range of small values of time and space variables. Thus, to obtain simpler and more effective solutions of these problems is an urgent issue. The authors have used an additional required function and additional boundary conditions to obtain solutions of the problem. Application of the additional required function allows us to reduce the original partial differential equation to the integration of an ordinary differential equation. Additional boundary conditions are in such a form that their fulfillment using the resulting solution is equivalent to the fulfillment of the equation at the boundary points. The authors have developed a technique to obtain an analytical solution of the heat conduction problem under a linear change of the initial condition, based on an additional required function and additional boundary conditions. Solution of an ordinary differential equation with respect to the additional required function determines the eigenvalues. In classical methods these eigenvalues are found in the solution of the Sturm–Liouville boundary value problem. The authors have proposed another, simpler solution to determine eigenvalues. An accurate analytical solution of the heat conduction problem for an unbounded plate with a coordinate-variable initial condition is obtained. The scientific and practical value of the proposed analytical solution is the development of an innovative approach to determine eigenvalues, as well as elimination of complex integrals when we solve the equation and initial conditions of the boundary value problem. It makes possible to simplify the use of the solution obtained in engineering applications.

Experimental estimation of the local heat transfer coefficient for thin liquid film evaporation in a capillary tube

• Local heat transfer coefficient was estimated for thin liquid film evaporation in a capillary tube. • Temperature infrared measurements were employed as input data of the inverse heat conduction problem. • A robust estimation procedure was adopted. • Procedure validation through the application to both synthetic and experimental data. • Original insight of thin liquid film evaporation was given. Two-phase heat transfer devices consisting of mini and microtubes represent a powerful solution to the continuous reduction of electronic components' dimensions and to the increase of the power to dissipate. Despite many studies on the two-phase heat transfer phenomenon in minitubes, most provided outcomes regarding the average heat transfer coefficient or Nusselt number. A technique to assess the local convective wall heat flux and heat transfer coefficient is proposed, validated, and then applied to the experimental case of thin-film evaporation in a heated capillary tube. The evaporation of the thin film of liquid plays a vital part in the two-phase heat transfer phenomenon inside mini/microtubes. Our study investigated this phenomenon by examining a semi-infinite slug flow. The estimation method used is represented by the inverse heat conduction problem solution in the tube wall, applying the Tikhonov regularisation method. The temperature distributions on the outer surface of the channel measured by an infrared camera were employed as input data for the inverse problem. The estimated heat transfer coefficient distributions confirm the available correlations in average values and provide insight into these complex phenomena in local and instantaneous quantities.

Analytical solutions for heat conduction problems with three kinds of periodic boundary conditions and their applications

In the present study, a general analytic expression for solving heat conduction problem in finite domain under periodic boundary conditions was suggested by using the variable separation method. Heat conduction problems under three kinds of periodic upper boundary conditions and constant temperature or zero-flux as bottom boundary conditions were solved respectively. By applying the analytical solutions, an equivalent method for transferring the periodic heat flux and convection combination boundary condition to the Dirichlet boundary condition was proposed. In addition, the proposed solution was generalized to solve the heat conduction problem infinite domain with periodic sine-like law boundary conditions. The present study can provide references for investigating the evolution law of temperature field under complex periodic boundary conditions.

Inaccuracies in the inverse heat conduction problem solution and their effect on the estimation of heat fluxes during quenching

Solving the Inverse Heat Conduction Problem (IHCP) is a common approach to estimate the surface heat fluxes involved in transient, quenching experiments. The inverse problem presents several challenges, such as accounting for the non-uniqueness of its solution, for the effects of noise, or for other practical issues affecting the experimental input data, such as the thermal contact of the internal temperature sensors. In this paper, possible sources of inaccuracy in the IHCP solution and their effect on the estimated surface heat flux in experiments on quenching by water jet impingement are systematically investigated. A “virtual experiment” approach is followed to analyze the effect of a noise cancelling technique, the ambiguity in initial conditions and the quality of the thermocouple contact on the accuracy of the heat flux estimations. The results show that the invalid assumption of perfect thermal contact between thermocouple and test plate leads to overestimation of surface temperature in the initial stages of quenching. Based on these results, two measures are proposed to avoid misinterpretation of quenching heat flux estimations when solving the IHCP.

Analytical Solutions of Heat Conduction Problems for Anisotropic Solid Body

Exact analytical solutions of heat conduction problems in anisotropic twodimensional solid bodies are presented in this study. Timedependent and steadystate problems are considered. A linear coordinate transformation is introduced which reduces the anisotropic heat conduction problem to an equivalent isotropic one. The solution of the anisotropic heat conduction problem is expressed in terms of solutions of the corresponding isotropic heat conduction problem. The connection of data of the applied linear coordinate transformation and the thermal material properties of anisotropic solid body is analysed. All result of the paper is based on the Fourier’s theory of heat conduction in solid bodies. Examples illustrate the applications of the developed method

Novel numerical method for heat conduction using superposition of exact solutions

A novel formulation for numerical solution of heat conduction problems using superposition of exact solutions (SES) to represent temperature on sub-elements of a region is described and demonstrated. A simple 1-D linear problem is used to describe the method and highlight potential benefits; however, extensions to higher geometric dimensions and linearization for temperature-dependent properties are possible. Significant new features of this method which are not part of conventional approaches are: local functions for temperature are both time- and space-dependent; both heat flux and temperature at the boundaries of each element have continuous representation; temperature and heat flux can be evaluated at every location in the domain; and heat fluxes on both internal and external boundaries are represented as linear functions of time. The SES method is validated against several exact solutions for the linear case (constant thermophysical properties) and is shown to have accuracy orders of magnitude greater than conventional Crank-Nicolson method. Extension to higher geometries and temperature-dependent properties are discussed.

Calculation of temperature oscillations in thermal layer of regular solids at unsteady coefficient of heat transfer

The temperature oscillations in thermal layer of regular solids, caused by periodic oscillations of the ambient temperature and heat transfer coefficient, are analysed. A halfspace, a plate, a space with cylindrical channel, a cylinder, a space with spherical cavity are considered as solids. Exact solutions of the corresponding cyclic problems of heat conduction in the complex and real forms of trigonometric Fourier series are obtained. Methods of simplified solution of heat conduction problem based on boundary condition modification are considered. Comparison of exact solutions with approximate ones is carried out and the admissibility of the proposed approximate solutions is shown.

Homogenization of time dependent boundary conditions for multi-layer heat conduction problem in cylindrical polar coordinates

In eigenfunction based solutions for heat conduction problems, the inhomogeneous boundary conditions (BCs) and the eigenfunctions (corresponding to homogeneous BCs) are incompatible with each other at the boundaries. Therefore, homogenization of BCs is essential to avoid the abovementioned incompatibility. However, homogenization is usually quite tedious especially for time dependent BCs. The process becomes even more difficult for multi-dimensional multi-layer problems. These difficulties come from the fact that, in general, an auxiliary function (quasi-static part of temperature) needs to be subtracted from the temperature of each layer of the composite. This treatment yields a set of conditions on auxiliary functions. In this work it is demonstrated that these auxiliary functions are not unique and subsequently, two formulations of auxiliary function are proposed here for cylindrical polar (r, θ) coordinates. The second formulation leads to relatively simple computational procedure. The computational ease of the proposed methodology arises from the fact that auxiliary functions are defined only in the innermost and the outermost layers of the composite in this approach. The solutions of a few illustrative examples are obtained without homogenization as well as with homogenization (using two distinct approaches). The results without homogenization of BCs exhibits large mismatch near the boundaries, while such mismatch goes way in case of homogenized BCs. Moreover, the novel (second) approach proposed here for homogenization is easy to implement and efficient compared to the first formulation approach.

Irreversibility and Action of the Heat Conduction Process.

Irreversibility (that is, the “one-sidedness” of time) of a physical process can be characterized by using Lyapunov functions in the modern theory of stability. In this theoretical framework, entropy and its production rate have been generally regarded as Lyapunov functions in order to measure the irreversibility of various physical processes. In fact, the Lyapunov function is not always unique. In the represent work, a rigorous proof is given that the entransy and its dissipation rate can also serve as Lyapunov functions associated with the irreversibility of the heat conduction process without the conversion between heat and work. In addition, the variation of the entransy dissipation rate can lead to Fourier’s heat conduction law, while the entropy production rate cannot. This shows that the entransy dissipation rate, rather than the entropy production rate, is the unique action for the heat conduction process, and can be used to establish the finite element method for the approximate solution of heat conduction problems and the optimization of heat transfer processes.

Notes on using temperature-dependent thermal diffusivity—forgotten rules

New materials and structures are applied in order to maintain the engineering products more and more efficient. Different types of thermal analysis/modelling of such materials and structures are also needed for the proper planning or design. The thermal analysis/modelling processes are supported by many commercial codes but in some cases, the experts might have applied a self-developed method also. The success of the analysis/modelling depends on many factors, but the basis is the correct application of the heat conduction theory which is more important in case of transient thermal processes. As the used tools are more and more powerful, the users pay less attention to the theory background, according to the author experience. It seems that the long tradition of the inconsistent application of the temperature-dependent thermal diffusivity is continuing, even strengthened by the increased availability of data of the temperature-dependent thermal properties. Regarding the origination of the thermal diffusivity, this might lead to the misinterpretation or misunderstanding of the nonlinear heat conduction problem solution procedures. This paper calls attention to avoid this, shows what kind of mistakes might occur related to this widely observable practice, summarizes the proper application and interprets where and how one could find room for the temperature-dependent thermal diffusivity.

Estimation of the local heat transfer coefficient in coiled tubes

PurposeMost of the passive techniques for enhancing heat transfer inside pipes (e.g. rough surfaces, swirl-flow devices and coiled tubes) give origin to an irregular distribution of the heat transfer coefficient at the fluid–wall interface along the wall perimeter. This irregular distribution could be critical in some industrial applications, but most of the available research papers, mainly due to the practical difficulty of local measuring heat flux on the internal wall surface of a pipe, present the results only in terms of Nusselt number averaged along the wall circumference. This paper aims to study the application of inverse problem solution techniques, which could overcome this limitation.Design/methodology/approachWith regard to the estimation of the local convective heat transfer coefficient in coiled tubes, two different inverse heat conduction problem solution techniques were considered and compared both by synthetic and experimental data.FindingsThe paper shows the success of two inverse problem solution techniques in the estimation of the local convective heat transfer coefficient in coiled tubes.Originality/valueThis paper fulfills an identified need because most of the available research papers present the results only in terms of average thermal performance, neglecting local behavior.

Determination of transient temperature fields in thick-walled elements using the inverse method

The purpose of this paper is to propose a method of determining the transient temperature of the inner surface of thick-walled elements. The method can be used to determine thermal stresses in pressure elements. An inverse marching method is proposed to determine the transient temperature of the thickwalled element inner surface with high accuracy. The inverse method was validated computationally. The comparison between the temperatures obtained from the direct heat conduction problem solution and the results obtained by means of the proposed inverse method is very satisfactory. The advantage of the method is the possibility of determining the heat transfer coefficient at a point on the exposed surface based on the local temperature distribution measured on the insulated outer surface. The heat transfer coefficient determined experimentally can be used to calculate thermal stresses in elements with a complex shape. The proposed method can be used in online computer systems to monitor temperature and thermal stresses in thick-walled pressure components because the computing time is very short.

A New Approach to Thermal Analysis of a Multilayered Cylindrical Structure With Imperfect Bonds and Internal Heat Source

In the present research, a new and straightforward mathematical model, named augmented state-space method, is introduced to solve the heat conduction equation for a multilayered orthotropic hollow cylinder with bonding imperfection in the presence of heat source. Since such problems including heat source are inherently inhomogeneous and complex, augmented state-space method converts these inhomogeneous equations into homogeneous ones. The transient solution will be achieved by present method based on laminate approximation theory in the Laplace domain, and then the solutions obtained are retrieved into the time domain by applying the numerical Laplace transform inversion. All material properties can be considered to vary continuously within the cylinder along the radial direction with arbitrary grading pattern. Based on the proposed method, the solution of heat conduction problem can be also obtained for general boundary conditions which may be included various combinations of arbitrary temperature, flux, or convection. Due to lack of any data on the transient thermal analysis corresponding to problems with imperfect bonds in the cylindrical coordinate system (r,θ), comparison is carried out with the available results for the three-layer semi-circular annular region with perfect bonds in the literature. Finally, the influence of orthotropy and interface imperfection on the distribution of the temperature field for three-layer hollow cylinder, in which the second layer is made of orthotropic functionally graded material (FGM), will be visualized.

The empirical evaluation of thermal conduction coefficient of some liquid composite heat insulating materials

We experimentally determined the coefficients of thermal conductivity of some ultra thin liquid composite heat insulating coatings, for sample #1 λ = 0.086 W/(m·°C), for sample #2 λ = 0.091 W/(m·°C). We performed the measurement error calculation. The actual thermal conduction coefficient of the studied samples was higher than the declared one. The manufactures of liquid coatings might have used some "ideal" conditions when defining heat conductivity in the laboratory or the coefficient was obtained by means of theoretical solution of heat conduction problem in liquid composite insulating media. However, liquid insulating coatings are of great interest to builders, because they allow to warm objects of complex geometric shapes (valve chambers, complex assemblies, etc.), which makes them virtually irreplaceable. The proper accounting of heating qualities of paints will allow to avoid heat loss increase above the specified limits in insulated pipes with heat transfer materials or building structures, as well as protect them from possible thawing in the period of subzero weather.

A novel solution for heat conduction problems by extending scaled boundary finite element method

A technique to evaluate the temperature field is developed by extending the scaled boundary finite element method (SBFEM). Firstly, the method for the steady state problem with prescribed side-face uniform temperature is developed, so only the partial boundary of the problem domain is discretized. Secondly, the effect of the heat capacity on the element interior temperature field is taken into account for the transient heat conduction problem. Thirdly, an improved time domain precise integration method is developed for computing the boundary temperature of the SBFE, meanwhile, a finite-difference scheme is adopted to obtain the element interior temperature at different time. Finally, three examples are presented to illustrate feasibility and good accuracy of the proposed methods.

Numerical solution of heat conduction problems using orthogonal collocation on finite elements

Technique of orthogonal collocation along with finite elements has been presented to solve the linear and non linear heat conduction problems numerically. Choice of Lagrangian interpolation polynomials as base function has been opted to discretize the trial function. Error analysis has been discussed in terms of element size for both the linear and non linear problems. Proposed technique has been applied on different types of linear and non linear heat conduction problems and the numerical values are plotted using 2D and 3D graphs.

3D Unsteady Heat Transfer in Multi-Connected Domains via LSFEM: A Case Study

This paper aims in particular to do a case study of the numerical efficiency of the application of LSFEM (Least Squares Finite Element Method) in the solution of heat conduction problems in multi-connected domains. To demonstrate this study two cases (the first with exact solution for comparison of results) are presented in the same multi-connected geometry, of easy construction, to facilitate the comparison of the results of this paper with future studies of other researchers.

Transient thermal analysis in 2D orthotropic FG hollow cylinder with heat source

A straightforward implementation is applied to solve heat conduction equation for a 2D hollow cylinder made of orthotropic functionally graded material (FGM) in the presence of time-dependent heat source. All material properties are considered to vary continuously within the cylinder along the radial direction with arbitrary law. The transient solution can be obtained by augmented state space method, which leads to carry out the results easily, based on laminate approximation theory in the Laplace domain, and then the results obtained are converted into the time domain by applying the numerical Laplace transform inversion. By this method, the solution of heat conduction problem is obtained for general boundary conditions which can be included various combinations of arbitrary temperature, flux, or convection. Comparison of obtained results with special cases in the literatures shows the capability of the new presented method. Finally, in the presence of time-dependent heat source, the effects of circumferential to the radial thermal conductivity coefficient ratio and heat source size on temperature field are graphically shown.

Analytical Solutions of Heat-Conduction Problems with Time-Varying Heat-Transfer Coefficients

Analytical Solutions of Heat-Conduction Problems with Time-Varying Heat-Transfer Coefficients