Investigation of transient conduction–radiation heat transfer in a square cavity using combination of LBM and FVM

In this paper, the effect of surface radiation in a square cavity containing an absorbing, emitting and scattering medium with four heated boundaries is investigated, numerically. Lattice Boltzmann method (LBM) is used to solve the energy equation of a transient conduction–radiation heat transfer problem and the radiative heat transfer equation is solved using finite-volume method (FVM). In this work, two different heat flux boundary conditions are considered for the east wall: a uniform and a sinusoidally varying heat flux profile. The results show that as the value of conduction–radiation decreases, the dimensionless temperature in the medium increases. Also, it is clarified that, for an arbitrary value of the conduction–radiation parameter, the temperature decreases with decreasing scattering albedo. It is observed that when the boundaries reflect more, a higher temperature is achieved in the medium and on boundaries.     

Transient non‐linear heat conduction–radiation problems—a boundary element formulation

A novel boundary‐only formulation for transient temperature fields in bodies of non‐linear material properties and arbitrary non‐linear boundary conditions has been developed. The option for self‐irradiating boundaries has been included in the formulation. Heat conduction equation has been partially linearized by Kirchhoff's transformation. The result has been discretized by the dual reciprocity boundary element method. The integral equation of heat radiation has been discretized by the standard boundary element method. The coupling of the resulting two sets of equations has been accomplished by static condensation of the radiative heat fluxes arising in both sets. The final set of ordinary differential equations has been solved using the Runge–Kutta solver with automatic time step adjustment. The algorithm proved to be robust and stable. Numerical examples are included. Copyright © 1999 John Wiley & Sons, Ltd.     

Transient heat conduction analysis in a piecewise homogeneous domain by a coupled boundary and finite element method

A coupled finite element–boundary element analysis method for the solution of transient two‐dimensional heat conduction equations involving dissimilar materials and geometric discontinuities is developed. Along the interfaces between different material regions of the domain, temperature continuity and energy balance are enforced directly. Also, a special algorithm is implemented in the boundary element method (BEM) to treat the existence of corners of arbitrary angles along the boundary of the domain. Unknown interface fluxes are expressed in terms of unknown interface temperatures by using the boundary element method for each material region of the domain. Energy balance and temperature continuity are used for the solution of unknown interface temperatures leading to a complete set of boundary conditions in each region, thus allowing the solution of the remaining unknown boundary quantities. The concepts developed for the BEM formulation of a domain with dissimilar regions is employed in the finite element–boundary element coupling procedure. Along the common boundaries of FEM–BEM regions, fluxes from specific BEM regions are expressed in terms of common boundary (interface) temperatures, then integrated and lumped at the nodal points of the common FEM–BEM boundary so that they are treated as boundary conditions in the analysis of finite element method (FEM) regions along the common FEM–BEM boundary. Copyright © 2002 John Wiley & Sons, Ltd.     

Parameter estimation in heat conduction using a two-dimensional inverse analysis

This article is concerned with a two-dimensional inverse steady-state heat conduction problem. The aim of this study is to estimate the thermal conductivity, the heat transfer coefficient, and the heat flux in irregular bodies (both separately and simultaneously) using a two-dimensional inverse analysis. The numerical procedure consists of an elliptic grid generation technique to generate a mesh over the irregular body and solve for the heat conduction equation. This article describes a novel sensitivity analysis scheme to compute the sensitivity of the temperatures to variation of the thermal conductivity, the heat transfer coefficient, and the heat flux. This sensitivity analysis scheme allows for the solution of inverse problem without requiring solution of adjoint equation even for a large number of unknown variables. The conjugate gradient method (CGM) is used to minimize the difference between the computed temperature on part of the boundary and the simulated measured temperature distribution. The obtained results reveal that the proposed algorithm is very accurate and efficient.     

Heat‐flux‐specified boundary treatment for gas flow and heat transfer in microchannel using direct simulation Monte Carlo method

Direct simulation Monte Carlo (DSMC) method has been widely used to study gaseous flow and heat transfer in micro‐fluidic devices. For flows associated with microelectromechanical systems (MEMS), the heat‐flux‐specified (HFS) boundary condition broadly exists. However, problems with HFS boundary have not been realized in the simulation of microchannel flows with DSMC method. To overcome this problem, a new technique named as inverse temperature sampling (ITS) is developed. This technique provides an approach to calculate the molecular reflective characteristic temperature from the specified heat flux at the wall boundary. Coupling with DSMC method, the ITS technique can treat the HFS boundary condition in DSMC method for both simple gas and gas mixtures. For validation, heat flux obtained from two‐dimensional Poiseuille flows with wall‐temperature‐specified (WTS) boundary condition is employed as the initial thermal boundary condition of our new method. Sampled wall temperature by the ITS method agrees well with the expected value. Pressure, velocity and temperature distributions under these two thermal boundary conditions (WTS and HFS) are compared. Effects of molecule collision model and gas–surface interaction model are also investigated. Results show that the proposed ITS method could accurately simulate gaseous flow and heat transfer in MEMS. Copyright © 2007 John Wiley & Sons, Ltd.     

The method of fundamental solutions for inverse source problems associated with the steady‐state heat conduction

This paper presents the use of the method of fundamental solutions (MFS) for recovering the heat source in steady‐state heat conduction problems from boundary temperature and heat flux measurements. It is well known that boundary data alone do not determine uniquely a general heat source and hence some a priori knowledge is assumed in order to guarantee the uniqueness of the solution. In the present study, the heat source is assumed to satisfy a second‐order partial differential equation on a physical basis, thereby transforming the problem into a fourth‐order partial differential equation, which can be conveniently solved using the MFS. Since the matrix arising from the MFS discretization is severely ill‐conditioned, a regularized solution is obtained by employing the truncated singular value decomposition, whilst the optimal regularization parameter is determined by the L‐curve criterion. Numerical results are presented for several two‐dimensional problems with both exact and noisy data. The sensitivity analysis with respect to two solution parameters, i.e. the number of source points and the distance between the fictitious and physical boundaries, and one problem parameter, i.e. the measure of the accessible part of the boundary, is also performed. The stability of the scheme with respect to the amount of noise added into the data is analysed. The numerical results obtained show that the proposed numerical algorithm is accurate, convergent, stable and computationally efficient for solving inverse source problems in steady‐state heat conduction. Copyright © 2006 John Wiley & Sons, Ltd.     

Analytic study of radiative heat transfer in ultra‐fine powder insulations with dependent scattering and absorption

Abstract

In this work, the radiative heat transfer of the ultra‐fine powder insulation Aerosil 380, with dependent scattering and absorption, is investigated theoretically. The radiative transport process is modeled by the two‐flux model and the diffusion approximate method to solve the government equations of transfer. The radiative properties of Aerosil 380 have been determined by the Rayleigh scattering theory because of the small values of particle size parameter. The results show that the dependent effect of scattering will reduce the scattering efficiency; however, the absorption efficiency will be increased due to the dependent absorption. The overall thermal radiation resistance is increased by the dependent effect. A comparison of radiative thermal conductivity has been calculated by the two models. The comparison reveals that the difference is small at a mean temperature of 300°K, but that the difference goes up to about 30 percent at a mean temperature of 400°K.     

Inverse heat conduction problems by meshless local Petrov–Galerkin method

The meshless local Petrov–Galerkin (MLPG) method is used to solve stationary and transient heat conduction inverse problems in 2-D and 3-D axisymmetric bodies. A 3-D axisymmetric body is generated by rotating a cross section around an axis of symmetry. Axial symmetry of geometry and boundary conditions reduce the original 3-D boundary value problem to a 2-D problem. The analyzed domain is covered by small circular subdomains surrounding nodes randomly spread over the analyzed domain. A unit step function is chosen as test function in deriving the local integral equations (LIEs) on the boundaries of the chosen subdomains. The time integration schemes are formulated based on the Laplace transform technique and the time difference approach, respectively. The local integral equations are non-singular and take a very simple form. Spatial variation of the temperature and heat flux (or of their Laplace transforms) at discrete time instants are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares (MLS) method. Singular value decomposition (SVD) is applied to solve the ill-conditioned linear system of algebraic equations obtained from the LIE after MLS approximation. The Stehfest algorithm is applied for the numerical Laplace inversion, in order to retrieve the time-dependent solutions.     

A Raviart–Thomas mixed finite element scheme for the two‐dimensional three‐temperature heat conduction problems

For the two‐dimensional three‐temperature radiative heat conduction problem appearing in the inertial confinement numerical stimulations, we choose the Freezing coefficient method to linearize the nonlinear equations, and initially apply the well‐known mixed finite element scheme with the lowest order Raviart–Thomas element associated with the triangulation to the linearized equations, and obtain the convergence with one order with respect to the space direction for the temperature and flux function approximations, and design a simple but efficient algorithm for the discrete system. Three numerical examples are displayed. The former two verify theoretical results and show the super‐convergence for temperature and flux functions at the barycenter of the element, which is helpful for solving the radiative heat conduction problems. The third validates the robustness of this scheme with small energy conservative error and one order convergence for the time discretization. Copyright © 2016 John Wiley & Sons, Ltd.     

Investigation of 2D Transient Heat Transfer under the Effect of Dual-Phase-Lag Model in a Nanoscale Geometry

Analytical and numerical solutions of the 2D transient dual-phase-lag (DPL) heat conduction equation are presented in this article. The geometry is that of a simplified metal oxide semiconductor field effect transistor with a heater placed on it. A temperature jump boundary condition is used on all boundaries in order to consider boundary phonon scattering at the micro- and nanoscale. A combination of a Laplace transformation technique and separation of variables is used to solve governing equations analytically, and a three-level finite difference scheme is employed to generate numerical results. The results are illustrated for three Knudsen numbers of 0.1, 1, and 10 at different instants of time. It is seen that the wave characteristic of the DPL model is strengthened by increasing the Knudsen number. It is found that the combination of the DPL model with the proposed mixed-type temperature boundary condition has the potential to accurately predict a 2D temperature distribution not only within the transistor itself but also in the near-boundary region.     

A wall boundary treatment using analytical volume integrations in a particle method

Numerical treatment of complicated wall geometry has been one of the most important challenges in particle methods for computational fluid dynamics. In this study, a novel wall boundary treatment using analytical volume integrations has been developed for two-dimensional (2D) incompressible flow simulations with the moving particle semi-implicit method. In our approach, wall geometry is represented by a set of line segments in 2D space. Thus, arbitrary-shaped boundaries can easily be handled without auxiliary boundary particles. The wall's contributions to the spatial derivatives as well as the particle number density are formulated based on volume integrations over the solid domain. These volume integrations are analytically solved. Therefore, it does not entail an expensive calculation cost nor compromise accuracy. Numerical simulations have been carried out for several test cases including the plane Poiseuille flow, a hydrostatic pressure problem with complicated shape, a high viscous flow driven by a rotating screw, a free-surface flow driven by a rotating cylinder and a dam break in a tank with a wedge. The results obtained using the proposed method agreed well with analytical solutions, experimental observations or calculation results obtained using finite volume method (FVM), which confirms that the proposed wall boundary treatment is accurate and robust.     

Detection of flaws in a two-dimensional body through measurement of surface temperatures and use of conjugate gradient method

This paper aims to obtain parameters (i.e. location and dimensions) relevant to flaws in a two-dimensional body by measuring the temperature on its boundaries. In this endeavour, a steady-state heat conduction problem is formulated, and the geometry under study is subjected to a known heat load, resulting in a specific heat distribution in the body. By using a number of heat sensors, the temperature at selected points on the boundary of the body is obtained. Inverse heat conduction methods implement these temperature data, working toward estimating the flaw parameters. The objective function is optimized using conjugate gradients method, and in solving the direct problem, an FEM code is employed. To check the effectiveness of this method, sample cases with one or more circular, elliptical cavities or cracks in the body, and a case with unknown cavity shape is solved. Finally the ensuing results analyzed.     

Global space–time multiquadric method for inverse heat conduction problem

In this paper, a radial basis collocation method (RBCM) based on the global space–time multiquadric (MQ) is proposed to solve the inverse heat conduction problem (IHCP). The global MQ is simply constructed by incorporating time dimension into the MQ function as a new variable in radial coordinate. The method approximates the IHCP as an over‐determined linear system with the use of two sets of collocation points: one is satisfied with the governing equation and another is for the given conditions. The least‐square technique is introduced to find the solution of the over‐determined linear system. The present work investigates two types of the ill‐posed heat conduction problems: the IHCP to recover the surface temperature and heat flux history on a source point from the measurement data at interior locations, and the backward heat conduction problem (BHCP) to retrieve the initial temperature distribution from the known temperature distribution at a given time. Numerical results of four benchmark examples show that the proposed method can provide accurate and stable numerical solutions for one‐dimensional and two‐dimensional IHCP problems. The sensitivity of the method with respect to the measured data, location of measurement, time step, shape parameter and scaling factor is also investigated. Copyright © 2010 John Wiley & Sons, Ltd.     

Uncertainties Quantification of Effective Thermal Conductivity for Ceramic Fiber Blanket

In the present paper, a probabilistic propagation model for assessing the uncertainty of the effective thermal conductivity was developed based on a combined conduction and radiation heat transfer model of a ceramic fiber blanket composite. The Monte Carlo technique was used to cope with the uncertainties in the material density, radiative properties, and boundary temperatures observed in experimental tests. The calculated effective thermal-conductivity distribution for the sample was compared with the experimental measurements performed on multiple samples, and the predicted mean values were in good agreement with the measured data. The result validates the thermal predictive model and demonstrates the suitability of the stochastic model containing statistical distributions in the input variables. Statistical information also indicates that the uncertainty effect can be enlarged at high temperatures. Response sensitivity analysis between the random inputs and the effective thermal conductivity demonstrates that the randomness in the hot-side temperature, the cold-side temperature, and extinction coefficient of the sample has a significant influence on the variability of thermal-conductivity properties. The extinction coefficient becomes more and more important with an increase of temperature due to the dominant radiative heat transfer contribution at high temperature. The analysis provides good insight into the scattering control in the experimental measurement and theoretical prediction of the effective thermal conductivity of a ceramic fiber composite.     

The dimension splitting and improved complex variable element‐free Galerkin method for 3‐dimensional transient heat conduction problems

In this paper, by combining the dimension splitting method and the improved complex variable element‐free Galerkin method, the dimension splitting and improved complex variable element‐free Galerkin (DS‐ICVEFG) method is presented for 3‐dimensional (3D) transient heat conduction problems. Using the dimension splitting method, a 3D transient heat conduction problem is translated into a series of 2‐dimensional ones, which can be solved with the improved complex variable element‐free Galerkin (ICVEFG) method. In the ICVEFG method for each 2‐dimensional problem, the improved complex variable moving least‐square approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions. Finite difference method is used in the 1‐dimensional direction, and the Galerkin weak form of 3D transient heat conduction problem is used to obtain the final discretized equations. Then, the DS‐ICVEFG method for 3D transient heat conduction problems is presented. Four numerical examples are given to show that the new method has higher computational precision and efficiency.     

One-dimensional temperature profile prediction in multi-layered rigid pavement systems using a separation of variables method

This paper presents an analytical solution for prediction of the one-dimensional (1D) time-dependent temperature profile in a multi-layered rigid pavement system. Temperature at any depth in a rigid pavement system can be estimated by using the proposed solution with limited input data, such as pavement layer thicknesses, material thermal properties, measured air temperatures and solar radiation intensities. This temperature prediction problem is modelled as a boundary value problem governed by the classic heat conduction equations, and the air temperatures and solar radiation intensities are considered in the surface boundary condition. Interpolatory trigonometric polynomials, based on the discrete least squares approximation method, are used to fit the measured air temperatures and solar radiation intensities during the time period of interest. The solution technique employs the complex variable approach along with the separation of variables method. A FORTRAN program was coded to implement the proposed 1D analytical solution. Field model validation demonstrates that the proposed solution generates reasonable temperature profile in the concrete slab for a four-layered rigid pavement system during two different time periods of the year.     

Computation of hypersonic flow and radiation of viscous chemically reacting gas in a channel modeling a section of a scramjet

This work is devoted to a consideration of flow and combustion of a hydrogen-air mixture in a channel modeling a section of a supersonic combustion ramjet (scramjet). Fields of concentrations, pressure, and temperature are obtained. Based on them, the thermal radiation of gas within a scramjet combustor is computed. The density of the radiative heat flux to the chamber wall is computed by two methods, i.e., in a P1 approximation of the spherical harmonics method and in an approximation of the plane layer. It has been shown that the radiative heat flux contribute significantly to the total heating of the jet wall.     

A homogenization technique for heat transfer in periodic granular materials

A homogenization technique is proposed to simulate the thermal conduction of periodic granular materials in vacuum. The effective thermal conductivity (ETC) and effective volumetric heat capacity (EVHC) can be obtained from the granular represent volume element (RVE) via average techniques: average heat flux and average temperature gradient can be formulated by the positions and heat flows of particles on the boundaries of the RVE as well as of the contact pairs within the RVE. With the thermal boundary condition imposed on the border region around the granular RVE, the ETC of the granular RVE can be computed from the average heat flux and average temperature gradient obtained from thermal discrete element method (DEM) simulations. The simulation results indicate that the ETC of the granular assembly consisting of simple-cubic arranged spheres coincides with the theoretical prediction. The homogenization technique is performed to obtain the ETC of the RVE consisting of random packed particles and the results exhibit the anisotropy of the thermal conduction properties of the RVE. Both the ETC and EVHC obtained are then employed to simulate the thermal conduction procedure in periodic granular materials with finite element analyses, which give the similar results of temperature profile and conduction properties as the DEM simulations.     

Two‐dimensional unsteady heat conduction analysis with heat generation by triple‐reciprocity BEM

If the initial temperature is assumed to be constant, a domain integral is not needed to solve unsteady heat conduction problems without heat generation using the boundary element method (BEM).However, with heat generation or a non‐uniform initial temperature distribution, the domain integral is necessary. This paper demonstrates that two‐dimensional problems of unsteady heat conduction with heat generation and a non‐uniform initial temperature distribution can be solved approximately without the domain integral by the triple‐reciprocity boundary element method. In this method, heat generation and the initial temperature distribution are interpolated using the boundary integral equation. Copyright © 2001 John Wiley & Sons, Ltd.     

Non-linear heat conduction in composite bodies: A boundary element formulation

The present paper discusses the numerical solution of steady-state non-linear heat conduction problems in composite bodies by using the boundary element method. Two kinds of non-linearities are considered: the temperature dependence of the thermal conductivity and boundary conditions of the radiative type. By introducing the integral of conductivity as a new variable the governing equation of the problem becomes linear in the transform space. Transformed boundary conditions of the Dirichlet and Neumann types are also linear but convective boundary conditions become non-linear. Also, discontinuities arise in the value of the integral of conductivity across the interface between materials with different properties since continuity of temperature is imposed. The problem is numerically solved by discretizing the external and interface boundaries of the region under consideration with constant boundary elements and applying an iterative scheme of the Newton–Raphson type.