Determining boundary heat flux profiles in an enclosure containing solid conducting block

A numerical implementation of estimating boundary heat fluxes in enclosures is proposed in the present work. Particularly, the flow field is dynamically coupled with the heat convection in the fluid and the heat conduction in the solid domain. An iterative conjugate gradient method is applied such that the gradient of the cost function is introduced when the appropriate sensitivity and adjoint problems are defined. In this approach, no a priori information is needed about the unknown function to be determined. Numerical solutions are obtained for the case of a square enclosure centrally-inserted with a solid block and subjected to an unknown heat flux on one side and to known conditions on the remaining sides. Fluid and heat transports are visualized by the streamlines and heatlines respectively, which are evidently affected by the thermal Rayleigh number, solid body size and thermal conductivity of solid phase, and the functional form of the imposed heat flux. The accuracy of the heat flux profile estimations is shown to depend strongly on the thermal Rayleigh number, body size and relative thermal conductivity of the solid material. Effects of functional form of the unknowns, sensors number and position, and measurement errors on the accuracy of estimation are also investigated. The present work is significant for the flow control simultaneously involving the heat conduction and convection.     

Non-iteration estimation of thermal conductivity using finite volume method

The finite volume approach is developed for the inverse estimation of thermal conductivity in one-dimensional domain. The differential governing equation of heat conduction is converted to a system of linear equations in matrix form using the temperature data and heat generation at the discrete grid points as well as surface heat flux. The unknown thermal conductivities are obtained by solving the system equations directly. The features of the present method are that no prior information about the functional form of the thermal conductivity is required and no iterations in the calculation process are needed. The accuracy and robust of the present method are verified by comparing examples of inverse estimation of spatially and temperature-dependent thermal conductivities with the exact solutions.     

The determination temperature-dependent thermal conductivity as inverse steady heat conduction problem

The paper deals with the non-iterative inverse determination of the temperature-dependent thermal conductivity in 2-D steady-state heat conduction problem. The thermal conductivity is modeled as a polynomial function of temperature with the unknown coefficients. The identification of the thermal conductivity is obtained by using the boundary data and additionally from the knowledge of temperature inside the domain. The method of fundamental solutions is used to solve the 2-D heat conduction problem. The golden section search is used to find the optimal place for pseudo-boundary on which are placed the singularities in the frame of method of fundamental solutions.     

A simple direct estimation of temperature-dependent thermal conductivity with kirchhoff transformation

A direct method is proposed to estimate the temperature-dependent thermal conductivity without internal measurements. In the proposed method, the steady-state nonlinear heat conduction equation is transformed into the Laplace equation via the Kirchhoff transformation. The thermal conductivity is modeled as a linear combination of known functions with unknown coefficients, which are directly determined from the imposed heat flux and measured temperatures at the boundary. Several inverse heat conduction problems are successfully introduced to confirm the validity of the proposed method.     

INVERSE HEAT CONDUCTION PROBLEM OF SIMULTANEOUSLY ESTIMATING SPATIALLY VARYING THERMAL CONDUCTIVITY AND HEAT CAPACITY PER UNIT VOLUME

An inverse heat conduction method for simultaneously estimating spatially varying thermal conductivity and heat capacity per unit volume under the conditions of a flash method type of experiment is developed. The unknown thermal properties are assumed to vary only in the space dimension normal to the slab sample and are modeled with piecewise linear representations. Lacking in the literature are specific requirements that must be satisfied by the number of measurements in the spatial domain in order to ensure uniqueness of the inverse solution. We prepared a series of numerical experiments to provide a better understanding of this issue. Multiple temperature sensors are shown to be necessary to determine spatially varying properties. The effectiveness of the method is illustrated through simulated experimental applications of the method.     

APPLICATION OF THE NETWORK METHOD TO HEAT CONDUCTION PROCESSES WITH POLYNOMIAL AND POTENTIAL-EXPONENTIALLY VARYING THERMAL PROPERTIES

Nonlinear transient heat conduction in a finite slab with potential-exponential temperature-dependent specific heat and thermal conductivity is investigated numerically by using the network method. A general network model for this process is proposed, whatever the exponent of the temperature-dependent functions may be, including initial and boundary conditions. With this network model and using the electrical circuit simulation program PSPICE, time-dependent temperature and heat flux profiles at any location can be obtained. This approach allows us to solve this conduction problem by a general, efficient, and relatively simple method. To show the accuracy of the network method, a comparison is made of the present results and those obtained by other methods for a particular case.     

One-step GPS for the estimation of temperature-dependent thermal conductivity

In this paper we are concerned with the estimation of temperature-dependent thermal conductivity of a one-dimensional inverse heat conduction problem. First, we construct a one-step group-preserving scheme (GPS) for the semi-discretization of quasilinear heat conduction equation, and then derive a quasilinear algebraic equation to determine the unknown thermal conductivity under a given initial temperature and a measured temperature perturbed by noise at time T. The new method does not require any prior information on the functional form of thermal conductivity. Several examples are examined to show that the new approach has high accuracy and efficiency, and the number of iterations spent in solving the quasilinear algebraic equation is smaller than five even in a large temperature range.     

Heat conduction in heterogeneous materials and irregular boundaries III: A computer graphics based algorithm for calculating effective thermal parameters

In earlier papers [1–3] a simple and computationally fast method for investigating heat conduction in materials of irregular shapes with heterogeneous thermal conductivity was presented. In that method, dubbed EPIC, the space is tesselated into cubical cell, an interfacial cells with regions of different thermal conductivities inside it is replaced by a cell with an effective uniform conductivity, and the heat conduction equation is solved numerically by using a finite difference method. In this paper, we present a novel computer graphics based approach to calculate effective conductivity which allows EPIC to be used for complex engineering systems generated by using standard CAD/CAM packages, bringing EPIC one step closer to becoming a practical tool.     

Inverse Estimation of the Thermal Conductivity in a One-Dimensional Domain by Taylor Series Approach

The Taylor series approximation is developed for the inverse estimation of thermal conductivity in a one-dimensional domain. The differential governing equation of heat conduction is converted to a discrete system of linear equations in matrix form using the temperature measurement and heat generation at the grid points as well as surface heat flux. The unknown thermal conductivity is estimated by solving the linear algebraic equations directly without iterations. The features of the present method are that no prior information about the functional form of the thermal conductivity is required, nor are any initial guesses or iterations in the calculation process needed. The accuracy and robustness of the present method are verified by comparing the results with the analytical solutions for constant, spatial- and temperature-dependent thermal conductivities. The results show that the inverse solutions are in good agreement with the exact solutions.     

Shape optimization of steady‐state heat‐conduction fields considering temperature dependency of thermal conductivity coefficient

This paper presents a numerical analysis method for shape optimization of domains with steady‐state heat‐conduction fields considering the temperature dependence of the thermal conductivity coefficient. In this paper, we formulate two shape optimization problems, namely, maximization of thermal dissipation on heat transfer boundaries and minimization of heat‐conduction fields. The shape gradient functions for these shape optimization problems are derived theoretically using the Lagrange multiplier method and formulae of the material derivative. Reshaping is accomplished using the traction method proposed as a solution to the shape optimization problems. The proposed method is validated from the results of two‐dimensional numerical analysis. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20374     

多孔介质导热的分形模型

多孔介质中热量传递与多孔介质内部的几何结构有密切的关系，讨论了多孔介质的分形结构和相关的分形维数，利用能量方程，导出了分形维数为D的有限尺度多孔介质中的广义热传导方程，在此基础上，假定热量在多孔介质中的传导路线也是一种分形结构，提出了一个筒化的多孔介质并联通道分形导热模型，求出了基于分形理论的多孔介质有效导热系数表达式。     

Temperature in thermally nonlinear pad–disk brake system

The influence of the thermal sensitivity of pad and disk materials on temperature at braking is under investigation. A mathematical model of process of frictional heating in a pad–disk brake system, which takes into account the temperature-sensitive materials, is proposed. The basic element of this model is the thermal problem of friction—a one-dimensional boundary-value heat conduction problem with temperature-dependent thermal conductivity and specific heat. Contrary to the prior studies of authors, where a simple nonlinearity was considered, in this article the arbitrary nonlinearity of the thermophysical properties of materials is studied. The solution of a nonlinear boundary-value heat conduction problem is obtained by the method of successive approximations. The numerical analysis of temperature is executed for some materials of a pad and a disk with and without taking into account their thermal sensitivity.     

A Precise Time-Domain Expanding Boundary-Element Method for Solving Three-Dimensional Transient Heat Conduction Problems with Variable Thermal Conductivity

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

D.C. scanning thermal microscopy: Characterisation and interpretation of the measurement

The used Scanning Thermal Microscopy (SThM) probe is a thin Pt resistance wire acting as a heat source and as a detector simultaneously. Its energetic balance is investigated by the study of the temperature profile along the probe. A theoretical approach of the measurement, based on this investigation, is then proposed. Simulations with this modelling are shown to predict how the heat, electrically produced in the probe, is dissipated in the probe-sample system. In particular, it is shown that the steady-state of conduction losses to the thermal element support varies versus the thermal conductivity of the sample and can lead to bad interpretations of the measurement.     

Evaluation of the interfacial conduction heat transfer coefficient in two-temperature macroscopic models of homogenous porous media using a fully developed unsteady microscopic model in periodic unit cells

A method is proposed for the evaluation of the interfacial conduction heat transfer coefficient in two-temperature macroscopic models of homogeneous fluid-saturated porous media. It is based on the numerical solutions of a microscopic model of unsteady conduction heat transfer in periodic unit cells, with different uniform initial temperatures of the fluid and solid. A novel formulation of the microscopic model in the fully developed regime is also proposed. Results for the variation of interfacial conduction Nusselt number with porosity, fluid–solid thermal conductivity ratio, and fluid–solid thermal diffusivity ratio are presented and discussed for four two-dimensional and two three-dimensional cases.     

Combined conduction and radiation heat transfer with variable thermal conductivity and variable refractive index

This article deals with the solution of conduction–radiation heat transfer problem involving variable thermal conductivity and variable refractive index. The discrete transfer method has been used for the determination of radiative information for the energy equation that has been solved using the lattice Boltzmann method. Radiatively, medium is absorbing, emitting and scattering. To validate the formulation, transient conduction and radiation heat transfer in a planar participating medium has been considered. For constant thermal conductivity and constant and variable refractive indices, results have been compared with those available in the literature. Effects of conduction–radiation parameter and scattering albedo on temperature have been studied for variable thermal conductivity and constant and/or variable refractive index. Lattice Boltzmann method and the discrete transfer method have been found to successfully deal with the complexities introduced due to variable thermal conductivity and variable refractive index.     

3-Omega Method for Thin Films with the Non-Fourier Boundary Condition

The effect of non-Fourier boundary condition on the 3-omega method for measuring the thermal conductivity of microscale thin films using the hyperbolic heat conduction equation and the Fourier equation is examined. Non-Fourier boundary condition with the Fourier equation leads to 80% error in the temperature oscillations and increases the error to 85% in the case of non-Fourier boundary condition with the hyperbolic heat conduction equation. The solution of the non-Fourier boundary condition with the hyperbolic heat conduction equation gives the most accurate thermal conductivity expression. The analysis also provides a method for determining the relaxation time of thin films.     

基于最小二乘有限元的导热系数直接反算方法

导热系数是温度场计算分析中的重要参数,针对基于优化方法反演导热系数计算量大、收敛慢、易陷入局部最优解等问题,提出了一种基于最小二乘有限元的直接反算方法,可快速得到材料导热系数。该方法是将有限单元法和最小二乘法引入热传导方程,直接求解材料导热系数时空分布,不需要进行大量的正算和迭代过程,大大提高了计算效率,且由于不需要事先假定导热系数分布函数形式,保证了计算精度。最后,以水泥砂浆导热系数测试试验为例,验证了最小二乘有限单元法在导热系数反算中的适用性和可靠性。     

Review on effective thermal conductivity of bubble type porous media

多孔介质内部结构中发生的质量,动量,能量的传递是众多自然现象和生产,生活领域中发生的基本过程.有关多孔介质中的传热问题涉及许多科学领域,早就引起人们的广泛关注,研究人员也对其进行了长期的研究.等效导热系数方法即将多孔介质视为一种连续介质,将实际多孔介质中固体骨架与各种流体的传热模式(导热,对流,辐射)折合成一个综合的传热问题.此方法已成为研究多孔介质内部传热问题最常用的方法.最近一二十年,泡沫型多孔介质(如泡沫金属,泡沫碳等)的出现引起人们广泛的关注.本文针对此种新型多孔介质等效导热系数的研究做了综述,介绍了3种常用的研究方法,分别是实验测试法,理论推导法及数值模拟法,探讨了各种研究方法存在的问题.实验测试法准确性高,但每种传热模式对等效导热系数的影响很难确定,且成本高;理论推导法虽然物理意义明确,适用性广,但与实测结果有较大的偏差,且有些方法还需与实验相结合;数值模拟法建模较复杂,但模拟结果与实验数据较接近.     

Thermal fatigue on pistons induced by shaped high power laser. Part II: Design of spatial intensity distribution via numerical simulation

In the laser induced thermal fatigue simulation test on pistons, the high power laser was transformed from the incident Gaussian beam into a concentric multi-circular pattern with specific intensity ratio. The spatial intensity distribution of the shaped beam, which determines the temperature field in the piston, must be designed before a diffractive optical element (DOE) can be manufactured. In this paper, a reverse method based on finite element model (FEM) was proposed to design the intensity distribution in order to simulate the thermal loadings on pistons. Temperature fields were obtained by solving a transient three-dimensional heat conduction equation with convective boundary conditions at the surfaces of the piston workpiece. The numerical model then was validated by approaching the computational results to the experimental data. During the process, some important parameters including laser absorptivity, convective heat transfer coefficient, thermal conductivity and Biot number were also validated. Then, optimization procedure was processed to find favorable spatial intensity distribution for the shaped beam, with the aid of the validated FEM. The analysis shows that the reverse method incorporated with numerical simulation can reduce design cycle and design expense efficiently. This method can serve as a kind of virtual experimental vehicle as well, which makes the thermal fatigue simulation test more controllable and predictable.