Transient Heat Conduction Problems Research Articles

A precise integration boundary element method for solving transient heat conduction problems

Abstract In this paper, a combined approach of boundary element method and precise integration method is presented for solving transient heat conduction problems with heat sources. The boundary integral equation is derived by means of the Green’s function for the Laplace equation, and as a result, two domain integrals are involved in the derived integral equations. Firstly, the radial integration method is used to convert the domain integrals into equivalent boundary integrals, so the system of ordinary differential equations on the boundary integral equation can be obtained by the boundary element method. Then, the precise integration method is adopted to solve the system of ordinary differential equations. Finally, several numerical examples are given to demonstrate the performance of the present method. The results show that the present approach gives satisfactory performance even for very large time step size, and the results given by the present approach are independent of the time step size if some integrals involves boundary conditions and heat sources can be integrated analytically.

Parametric integral equations systems in 2D transient heat conduction analysis

Currently the most popular numerical methods used for solving transient heat conduction problems, finite element method (FEM) and boundary element method (BEM), have one fundamental defect – the necessity of discretizing the boundary or the domain. This problem escalates even more, when using an iterative process. An alternative to avoid the mentioned problem are parametric integral equations systems (PIES), which do not require classical discretization of the boundary and the domain while being numerically solved. PIES method was previously used with success to solve steady-state problems. The purpose of this paper is to present PIES method for 2D transient heat conduction problems and present results obtained by solving few numerical examples. Accuracy and effectiveness of the method are shown in comparison with analytical solutions and FEM.

Improved numerical prediction and reliability-based optimization of transient heat conduction problem with interval parameters

In this paper, a high-order interval parameter perturbation method (HIPPM) and a reliability-based optimization model are proposed to solve the transient heat conduction problem with uncertainties in both the material properties and initial/boundary conditions. Interval variables are used to quantitatively describe the uncertain parameters with limited information. A modified stability theory is proposed and used to select space step and time step in the interval discrete schemes. Compared with the traditional first-order perturbation method, HIPPM can yield more accurate ranges of the uncertain temperature field by adopting the higher order terms of the Neumann series to approximate the interval matrix inverse. In the following investigated optimization model, a satisfaction degree of interval is employed to deal with the interval constraint functions. Given a reliability index representing the confidence level, uncertain constraints can be transformed into deterministic ones. The proposed HIPPM is used to predict the intervals of the constraints, and whereby eliminate the optimization nesting. A numerical example modeling a thermal protection system is presented to demonstrate the feasibility and efficiency of the proposed method.

Fuzzy finite difference method for heat conduction analysis with uncertain parameters

A new numerical technique named as fuzzy finite difference method is proposed to solve the heat conduction problems with fuzzy uncertainties in both the physical parameters and initial/boundary conditions. In virtue of the level-cut method, the difference discrete equations with fuzzy parameters are equivalently transformed into groups of interval equations. New stability analysis theory suited to fuzzy difference schemes is developed. Based on the parameter perturbationmethod, the interval ranges of the uncertain temperature field can be approximately predicted. Subsequently, fuzzy solutions to the original difference equations are obtained by the fuzzy resolution theorem. Two numerical examples are given to demonstrate the feasibility and efficiency of the presented method for solving both steady-state and transient heat conduction problems.

Finite block method for transient heat conduction analysis in functionally graded media

SUMMARYBased on the one‐dimensional differential matrix derived from the Lagrange series interpolation, the finite block method is proposed first time to solve both stationary and transient heat conduction problems of anisotropic and functionally graded materials. The main idea is to establish the first order one‐dimensional differential matrix constructed by using Lagrange series with uniformly distributed nodes. Then the higher order of derivative matrix for one‐dimensional problem is obtained. By introducing the mapping technique, a block of quadratic type is transformed from Cartesian coordinate (xyz) to normalised coordinate (ξης) with 8 seeds or 20 seeds for two or three dimensions. Then the differential matrices in physical domain are determined from that in normalised transformed coordinate system. In addition, the time dependent partial differential equations are analysed in the Laplace transformed domain, and the Durbin inversion method is used to determine the values in time domain. Illustrative two‐dimensional and three‐dimensional numerical examples are given, and comparisons have been made with analytical solutions. Copyright © 2014 John Wiley & Sons, Ltd.

Discrete element thermomechanical modelling of rock cutting with valuation of tool wear

The paper presents a thermomechanical discrete element model of rock cutting process. The thermomechanical formulation of the discrete element method considers mechanical and thermal phenomena and their reciprocal influence. The thermal model developed for transient heat conduction problems takes into account conductive heat transfer at the contact between particles and convection on the free surface. The thermal and mechanical problems are coupled by consideration of: (1) heat generated due to friction which is calculated in the mechanical problem and passed to the thermal solution, (2) influence of thermal expansion on mechanical interaction between particles. Estimation of temperature dependent wear has been included into the contact model. The coupled problem is solved using the staggered scheme.The thermomechanical algorithm has been implemented in a discrete element program and applied to simulation of rock cutting with single pick of a dredge cutter head. Numerical results confirm good performance of the developed algorithm.

Transient Heat Conduction Between Rough Sliding Surfaces

When two rough bodies slide against each other, asperities on the opposing surfaces interact with each other, defining a transient contact and heat conduction problem. We represent each body by a Greenwood and Williamson asperity model with a Gaussian height distribution of identical spherical asperities. The heat transfer during a typical asperity interaction is analyzed, and the results are combined with the height distributions to determine the mean heat flux and the mean normal contact pressure as functions of the separation between reference planes in the two surfaces. We find that the effective thermal conductance is an approximately linear function of nominal contact pressure, but it also increases with the square root of the sliding speed and decreases with the 3/4 power of the combined RMS roughness. The results can be used to define an effective thermal contact resistance and division of frictional heat in macroscale (e.g., finite element) models of engineering components, requiring as input only the measured roughness and material properties.

Bézier versus Lagrange polynomials-based finite element analysis of 2-D potential problems

In this paper two types of tensor product finite macro-elements are contrasted, the former being the well known Lagrange type and the latter the Bézier (Bernstein) type elements. Although they have a different mathematical origin and seemingly are irrelevant, they both are based on complete polynomials thus sharing the same functional space, i.e. the classes {xn} and {yn}. Therefore, from the theoretical point of view it is anticipated that they should lead to numerically identical results in both static and dynamic analysis. For both types of elements details are provided concerning the main computer programming steps, while selective parts of a typical MATLAB® code are presented. Numerical application includes static (Laplace, Poisson), eigenvalue (acoustics) and transient (heat conduction) problems of rectangular, circular and elliptic shapes, which were treated as a single macroelement. In agreement to the theory, in all six examples the results obtained using Bézier and Lagrange polynomials were found to be identical and of exceptional accuracy.

Installation of roof-top solar PV modules and their impact on building cooling load

It has been shown by many researchers that over a long term there has been a slow but steady rise of ambient temperature within the Indian sub-continent. Due to an increased economic prosperity there has been an accompanied increase in the urban heat island effect. Furthermore, urbanisation of large cities in India has also led to higher population densities. The above factors had the combined effect of a significant increase of cooling load of buildings. The high density of dwellings and other building construction has resulted in shading of walls. However, the flat roof spaces are exposed to an uninterrupted solar radiation regime and this in turn leads to generation of high sol-air temperatures which cause higher cooling loads. Presently, it has been argued that roof spaces are one of the major contributors to building cooling load. In this article, the reasons behind the phenomenal rise in the installation of air-conditioners in India are reviewed. The dual role of roof-top PV systems in electricity generation and reduction of building cooling load due to the shading they provide is then investigated. For this purpose, the CIBSE method to obtain sol-air temperature with solar radiation and outdoor ambient temperature has been used. Sol-air temperature for five key Indian locations (Delhi, Bhopal, Ahmedabad, Bhubaneswar and Chennai), based on the recently presented data by the NREL-India Meteorological Department consortium, were then obtained. A computer simulation routine was presently developed for solving the classical transient heat conduction problem with hourly sol-air temperature data and roof construction details provided to the routine. This program was executed to obtain the cooling load profile for each of the five Indian locations for the respective design day. Practical application: The present work reviews the reasons behind the phenomenal rise in the installation of air-conditioners in India. The dual role of roof-top PV systems in electricity generation and reduction of building cooling load due to the shading they provide has been investigated. The computer simulation demonstrated that the energy required for roof-induced cooling load decreased between 73% and 90% after installation of the PV system. The method used in this work has the advantage that it enables the user to obtain cooling load estimates using a general transient heat conduction approach. Moreover, all software was developed within MS-Excel environment, this is also an additional advantage as the cost associated with purchase and training of proprietary building energy software can be prohibitive for many consultants who are based in developing countries.

A Global Heat Compliance Measure Based Topology Optimization for the Transient Heat Conduction Problem

For topology optimization with transient loads, heat compliance varies with transient heat analysis. The peak value of the transient heat compliance should be minimized. Thus, this article proposes a global heat compliance measure to handle this kind of topology optimization for the transient heat conduction problem. The optimization model is then constructed by the global heat compliance measure. The finite-element, equivalent static loads, and continuum shape based sensitivity analyses are derived using the adjoint variable method. Through case studies, the effectiveness of the proposed global heat compliance measure for the transient heat conduction topology optimization is validated.

A Precise Integration Boundary-Element Method for Solving Transient Heat Conduction Problems with Variable Thermal Conductivity

In this article, a combined approach of the radial integration boundary element method (RIBEM) and the precise integration method is presented for solving transient heat conduction problems with variable thermal conductivity. First, the system of ordinary differential equations on the boundary integral equation can be obtained by the RIBEM. Then, the precise integration method is adopted to solve the system of ordinary differential equations. Finally, three numerical examples are presented to demonstrate the performance of the present method. The results show that the present approach can obtain satisfactory performance even for very large time-step size.

Analysis of the possibility of determining the internal structure of composite material by estimating its thermal diffusivity

Abstract The aim of this paper is analysis of the possibility of determining the internal structure of the fibrous composite material by estimating its thermal diffusivity. A thermal diffusivity of the composite material was determined by applying inverse heat conduction method and measurement data. The idea of the proposed method depends on measuring the timedependent temperature distribution at selected points of the sample and identification of the thermal diffusivity by solving a transient inverse heat conduction problem. The investigated system which was used for the identification of thermal parameters consists of two cylindrical samples, in which transient temperature field is forced by the electric heater located between them. The temperature response of the system is measured in the chosen point of sample. One dimensional discrete mathematical model of the transient heat conduction within the investigated sample has been formulated based on the control volume method. The optimal dynamic filtration method as solution of the inverse problem has been applied to identify unknown diffusivity of multi-layered fibrous composite material. Next using this thermal diffusivity of the composite material its internal structure was determined. The chosen results have been presented in the paper.

Transient 3D heat conduction in functionally graded materials by the method of fundamental solutions

In this paper, the three-dimensional transient heat conduction problems in functionally graded materials (FGMs) have been solved using the method of fundamental solutions (MFS). To be more specific, we consider the FGMs with thermal conductivity and specific heat vary exponentially in z-direction. In the numerical simulation, we coupled the fundamental solution of diffusion equation with the method of time–space unification which provides a simple and direct approach for solving time-dependent problems. The parameter transformation technique is also utilized to obtain the fundamental solutions which contain the thermal conductivity and the specific heat conditions. The MFS is very attractive in handling problems with irregular domain due to the simplicity of the method. The numerical results are in good agreement comparing with analytical solution and results obtained from the finite element method.

Cavity detection in a heat conductor using linear sampling method

In this paper, solution of inverse problems in heat conduction transient fields is investigated. For this purpose, a new time-domain version of linear sampling method (TDLSM) is developed for cavity detection in a heat conductor. The linear sampling method (LSM) is an effective approach to image the geometrical features of unknown targets. Although this method has been used in the context of inverse scattering problems such as solid, acoustics, and electromagnetism, there is no specific attempt to apply this method to identification of cavities in heat conductors. This study emphasizes the implementation of the LSM in the time-domain fields using finite element method. A set of numerical simulations on two-dimensional transient heat conduction problems is presented to highlight many effective features of the proposed TDLSM fast qualitative identification method.

Numerical analysis of uncertain temperature field by stochastic finite difference method

Based on the combination of stochastic mathematics and conventional finite difference method, a new numerical computing technique named stochastic finite difference for solving heat conduction problems with random physical parameters, initial and boundary conditions is discussed. Begin with the analysis of steady-state heat conduction problems, difference discrete equations with random parameters are established, and then the computing formulas for the mean value and variance of temperature field are derived by the second-order stochastic parameter perturbation method. Subsequently, the proposed random model and method are extended to the field of transient heat conduction and the new analysis theory of stability applicable to stochastic difference schemes is developed. The layer-by-layer recursive equations for the first two probabilistic moments of the transient temperature field at different time points are quickly obtained and easily solved by programming. Finally, by comparing the results with traditional Monte Carlo simulation, two numerical examples are given to demonstrate the feasibility and effectiveness of the presented method for solving both steady-state and transient heat conduction problems.

Application of the finite pointset method to non-stationary heat conduction problems

This paper proposes the use of the finite pointset method for the numerical solution of two-dimensional transient heat conduction problems. The strong formulation of the parabolic partial differential equation is directly used instead of the corresponding weak form. Moreover, a numerical comparison between the finite pointset method and the corresponding analytical solutions is reported.

A Combined Approach of RIBEM and Precise Time Integration Algorithm for Solving Transient Heat Conduction Problems

A combined approach of the radial integration boundary-element method (RIBEM) and the precise algorithm in the time domain is presented for solving transient heat conduction problems with heat sources. First, by expanding physical quantities at discrete time intervals, the recursive formulations of the governing equation are derived. Then, the recursive equation is solved by the RIBEM. Finally, a self-adaptive check technique is carried out to estimate how many expansion terms are needed in a time-step size. Numerical results show that the present approach can obtain satisfactory performance.

The Interpolating Element-Free Galerkin Method for 2D Transient Heat Conduction Problems

An interpolating element-free Galerkin (IEFG) method is presented for transient heat conduction problems. The shape function in the moving least-squares (MLS) approximation does not satisfy the property of Kronecker delta function, so an interpolating moving least-squares (IMLS) method is discussed; then combining the shape function constructed by the IMLS method and Galerkin weak form of the 2D transient heat conduction problems, the interpolating element-free Galerkin (IEFG) method for transient heat conduction problems is presented, and the corresponding formulae are obtained. The main advantage of this approach over the conventional meshless method is that essential boundary conditions can be applied directly. Numerical results show that the IEFG method has high computational accuracy.

Approximate Analytic Solutions of Transient Nonlinear Heat Conduction with Temperature-Dependent Thermal Diffusivity

A new approach for generating approximate analytic solutions of transient nonlinear heat conduction problems is presented. It is based on an effective combination of Lie symmetry method, homotopy perturbation method, finite element method, and simulation based error reduction techniques. Implementation of the proposed approach is demonstrated by applying it to determine approximate analytic solutions of real life problems consisting of transient nonlinear heat conduction in semi-infinite bars made of stainless steel AISI 304 and mild steel. The results from the approximate analytical solutions and the numerical solution are compared indicating good agreement.

A stochastic analysis of steady and transient heat conduction in random media using a homogenization approach

We present a new stochastic analysis for steady and transient one-dimensional heat conduction problem based on the homogenization approach. Thermal conductivity is assumed to be a random field K consisting of random variables of a total number N. Both steady and transient solutions T are expressed in terms of the homogenized solution T∼ and its spatial derivatives T(x,t)=T∼+∑n=1∞Ln(x)∂nT∼/∂xn, where homogenized solution T∼ is obtained by solving the homogenized equation with effective thermal conductivity. Both mean and variance of stochastic solutions can be obtained analytically for K field consisting of independent identically distributed (i.i.d) random variables. The mean and variance of T are shown to be dependent only on the mean and variance of these i.i.d variables, not the particular form of probability distribution function of i.i.d variables. Variance of temperature field T can be separated into two contributions: the ensemble contribution (through the homogenized temperature T∼); and the configurational contribution (through the random variable Ln(x)). The configurational contribution is shown to be proportional to the local gradient of T∼. Large uncertainty of T field was found at locations with large gradient of T∼ due to the significant configurational contributions at these locations. Numerical simulations were implemented based on a direct Monte Carlo method and good agreement is obtained between numerical Monte Carlo results and the proposed stochastic analysis.