Conduction Heat Transfer Problems Research Articles

Solution of direct and inverse conduction heat transfer problems using the method of fundamental solutions and differential evolution

PurposeThe study of heat transfer mechanisms is an area of great interest because of various applications that can be developed. Mathematically, these phenomena are usually represented by partial differential equations associated with initial and boundary conditions. In general, the resolution of these problems requires using numerical techniques through discretization of boundary and internal points of the domain considered, implying a high computational cost. As an alternative to reducing computational costs, various approaches based on meshless (or meshfree) methods have been evaluated in the literature. In this contribution, the purpose of this paper is to formulate and solve direct and inverse problems applied to Laplace’s equation (steady state and bi-dimensional) considering different geometries and regularization techniques. For this purpose, the method of fundamental solutions is associated to Tikhonov regularization or the singular value decomposition method for solving the direct problem and the differential Evolution algorithm is considered as an optimization tool for solving the inverse problem. From the obtained results, it was observed that using a regularization technique is very important for obtaining a reliable solution. Concerning the inverse problem, it was concluded that the results obtained by the proposed methodology were considered satisfactory, as even with different levels of noise, good estimates for design variables in proposed inverse problems were obtained.Design/methodology/approachIn this contribution, the method of fundamental solution is used to solve inverse problems considering the Laplace equation.FindingsIn general, the proposed methodology was able to solve inverse problems considering different geometries.Originality/valueThe association between the differential evolution algorithm and the method of fundamental solutions is the major contribution.

Effect of enrichment functions on GFEM solutions of time-dependent conduction heat transfer problems

This article investigates the effect of the selection of enrichment functions on the formulation of the Generalized Finite Element Method (GFEM) for the solutions of transient heat conduction problems. We present the study of an a-posteriori error estimate with the aim to show it as a reliable tool for the selection of enrichment functions to efficiently capture the sharp thermal gradients of the solutions. Problems in two- and three-dimensional domains are considered to demonstrate the robustness of the proposed error estimate. Numerical experiments consider two different types of enrichment functions that mimic the solution behaviour and capture the time-varying thermal gradients. The presented study shows that the error estimate is independent of the heuristically selected enrichment functions and can be used for any type of enrichment functions. It is concluded that the proposed error estimate efficiently reflects the errors in the GFEM solutions for both types of enrichment functions and can be used as an effective tool for the selection of more suitable enrichment functions that produce lower errors under the considered thermal conditions.

An Exact Solution for Transient Anisotropic Heat Conduction in Composite Cylindrical Shells

Abstract In this study, an analytical solution is proposed for the problem of transient anisotropic conductive heat transfer in composite cylindrical shells. The composite shells are considered to have directional heat transfer properties, which is due to the existence of fibers which can be winded in any direction. The composite shells usually show high conductivity in the direction parallel to fiber direction and low conductivity in other two orthogonal directions. To solve the heat transfer partial differential equation, finite Fourier transform and separation of variables method are used. The present solution is used to find the temperature distribution in a composite cylindrical vessel for which the composite material is graphite/epoxy and the vessel is prone to an external heat flux and also ambient flow. The analytical solution is verified perfectly by the data obtained from a second-order finite difference solution. The solution is used to investigate the effects of values of fiber angle and material conductivity coefficients on temperature distribution of the composite cylindrical vessel. The results show the important role of fiber angle values on the temperature distribution of vessel.

A double distribution lattice Boltzmann scheme for unsteady Conjugate Heat Transfer: The DD-CHT LB method

In this paper, we propose an innovative formulation of the “well-known” thermal lattice Boltzmann method for solving transient conduction heat transfer problems in heterogeneous media: the DD-CHT LB method for Double Distribution – Conjugate Heat Transfer. The main idea of this new formulation is to bring out a second distribution function that represent the effect of thermal inertia and its variation in a heterogeneous media. By adopting this original formulation we achieved several goals, mainly conservation of local properties of the studied media and ensuring the stability of the proposed formulation by avoiding any alteration of the “well-known” TLBM collision step. Validation of the formulation is fulfilled through three different benchmark conduction heat transfer problems in heterogeneous media. All results are compared with those obtained by the Control Volume Method (CVM) and excellent agreements were found.

Analytical solution of steady state heat conduction equations in irregular domains with various BCs by use of Schwarz-Christoffel conformal mapping

Undoubtedly, analytical solutions of conduction heat transfer problems play an important role in formation of both qualitative and quantitative understanding of the heat transfer phenomenon and other phenomena related to it. Analytical methods for solving this class of problems, especially in irregular domains, contain much complexity, making the solution process extremely difficult. This paper demonstrates, in a general manner, the use of conformal mapping, specifically the Schwarz-Christoffel (SC) mapping, to solve the steady-state conduction heat transfer equations (namely Laplace and Poisson) and to obtain the analytical exact solution. The exact temperature distribution can then be obtained for irregular domains (i.e. domains with irregular boundaries) for homogeneous and non-homogeneous conditions. To get to this goal and to clarify the method more, for a specific case, namely a triangular plate with a right angle on the origin and constant-temperature boundary conditions, the steady-state heat conduction equation has been analytically solved using the conformal mapping method. In the analytical solution process, the importance and capabilities of complex variables and their applications, especially the SC mapping, is highlighted. Finally, in order to validate the exact solution, the problem has been solved using the finite-element software COMSOL Multi-physics 5.0 and the results have been compared to the analytical solution. Good agreement between two solutions has confirmed the accuracy of the analytical solution.

A New Approach to the Analytical Treatment of Steam-Assisted Gravity Drainage: A Prescribed Interface Model

Summary The majority of the models in the literature for the steam-assisted-gravity-drainage (SAGD) process solve the problem of conductive heat transfer ahead of a moving hot interface using a quasisteady-state assumption and extend the solution to the base of the steam chamber where the interface is not moving. This approach, as discussed by Butler (1985) and Reis (1992), results in inaccurate or sometimes infeasible estimations of the oil-production rate, steam/oil ratio (SOR), and steam-chamber shape. In this work, a new approach for the analytical treatment of SAGD is proposed in which the problem of heat transfer is directly solved for a stationary source of heat at the base of the steam chamber, where the oil production occurs. The distribution of heat along the interface is then estimated depending on the geometry of the steam chamber. This methodology is more representative of the heat-transfer characteristics of SAGD and resolves the challenges of those earlier models. In addition, it allows for the extension of the formulations to the early stages of the process when the side interfaces of the chamber are almost stationary, without loss of the solution continuity. The model requires the overall shape of the steam chamber as an input. It then estimates the movement of chamber interfaces using the movement of the uppermost interface point and by satisfying the global material-balance requirements. Oil-production rate and steam demand are estimated by Darcy's law and energy-balance calculations, respectively. The result is a model that is applicable to the entire lifetime of a typical SAGD project and provides more-representative estimations of in-situ heat distribution, bitumen-production rate, and SOR. With the improved knowledge obtained on the fundamentals of heat transfer in SAGD, the reason for the discrepancies between the various earlier models will be clarified. Results of the analytical models developed in this work show reasonable agreement with fine-scale numerical simulation, which indicates that the primary physics are properly captured. In the final section of the paper, the application of the developed models to two field case studies will be demonstrated.

Real-time retrieval of transient heat flux on the surface of participating medium by using the EKF-RLSE technique

A hybrid algorithm of extended Kalman filtering and recursive least-square estimator (EKF-RLSE) is developed to solve the nonlinear inverse heat conduction-radiation problems in participating medium. By measuring the time-resolved temperature signals from a thermocouple installed on the other boundary of the participating medium, the proposed EKF-RLSE algorithm can realize real-time retrieval of the time-varying heat flux on the medium surface. The influence factors of the sampling interval, measurement and process covariance, forgetting factor, and medium thickness on the reconstruction accuracy and stability of the EKF-RLSE are investigated by numerical experiments. All the estimated results show that the present algorithm is robust and accurate in real-time estimating transient heat flux. If the nonlinear coupled conduction and radiation heat transfer problems is regard as the linear problems, the unknown time-dependent heat flux predicted by the KF-RLSE are also illustrated in this work.

Topology optimization of conductive heat transfer problems using parametric L-systems

Generative encodings have the potential of improving the performance of evolutionary algorithms. In this work we apply parametric L-systems, which can be described as developmental recipes, to evolutionary topology optimization of widely studied two-dimensional steady-state heat conduction problems. We translate L-systems into geometries using the turtle interpretation, and evaluate their objective functions, i.e. average and maximum temperatures, using the Finite Volume Method (FVM). The method requires two orders of magnitude fewer function evaluations, and yields better results in 10 out of 12 tested optimization problems (the result is statistically significant), than a reference method using direct encoding. Further, our results indicate that the method yields designs with lower average temperatures than the widely used and well established SIMP (Solid Isotropic Material with Penalization) method in optimization problems where the product of volume fraction and the ratio of high and low conductive material is less or equal to 1. Finally, we demonstrate the ability of the methodology to tackle multi-objective optimization problems with relevant temperature and manufacturing related objectives.

An a priori upper bound estimate for conduction heat transfer problems with temperature-dependent thermal conductivity

This article presents an a priori upper bound estimate for the steady-state temperature distribution in a body with any temperature-dependent thermal conductivity, generalizing a previous result (Gama et al., 2013) [1]. The discussion is carried out assuming a large class of nonlinear boundary conditions (for instance representing thermal radiant interchange). These estimates consist of a powerful tool that may avoid an expensive numerical simulation of a nonlinear heat transfer problem, whenever it suffices to know the highest temperature. In these cases the methodology proposed in this work is more effective than the usual approximations that assume thermal conductivities and heat sources as constants.

Existence and uniqueness of a transient state for the coupled radiative–conductive heat transfer problem

This paper deals with existence and uniqueness results for a transient nonlinear radiative–conductive system in three-dimensional case. This system describes the heat transfer for a grey, semi-transparent and non-scattering medium with general boundary conditions. We reformulate the full transient state system as a fixed-point problem. The existence and uniqueness proof is based on Banach fixed point theorem.

A hybrid strategy of lattice Boltzmann method and finite volume method for combined conduction and radiation in irregular geometry

Lattice Boltzmann method (LBM) has been a trendy tool for fluid flow and heat transfer problems. Recently it has been applied to combined heat transfer problems with radiation, while most of the work consider a square enclosure. To extend the LBM to combined heat transfer problems with radiation in irregular geometries, it is coupled with the finite volume method (FVM) to solve the energy equation and radiative transfer equation (RTE), respectively. Blocked-off method is used in the FVM to approximate the curved boundary for radiation modeling. By using the half-way bounce back boundary method in the LBM, it can be perfectly coupled with the blocked-off FVM. Treatments of complex boundaries of these two methods are validated, and then the hybrid method is used to solve combined conduction and radiation heat transfer in enclosures with inclined and curved walls. Effects of the conduction-radiation number, optical thickness, scattering albedo and wall emissivity on the temperature distribution and wall heat flux are analyzed. Results show that this hybrid method can solve combined conduction and radiation heat transfer problems in irregular geometries.

Multiscale analysis and computation for coupled conduction, convection and radiation heat transfer problem in porous materials

This paper discusses the multiscale analysis and numerical algorithms for coupled conduction, convection and radiation heat transfer problem in periodic porous materials. First, the multiscale asymptotic expansion of the solution for the coupled problem is presented, and high-order correctors are constructed. Then, error estimates and their proofs will be given on some regularity hypothesis. Finally, the corresponding finite element algorithms based on multiscale method are introduced and some numerical results are given in detail. The numerical tests demonstrate that the developed method is feasible and valid for predicting the heat transfer performance of periodic porous materials, and support the approximate convergence results proposed in this paper.

Improved social spider optimization algorithms for solving inverse radiation and coupled radiation–conduction heat transfer problems

A novel bio-inspired swarm algorithm, social spider optimization (SSO), is introduced to solve the inverse transient radiation and coupled radiation–conduction problems for the first time. Based on the original model, five improved SSO (ISSO) algorithms are developed to enhance search ability and convergence velocity. The sensitivity analysis of measured signals with respect to the physical parameters of the medium are described. After which, the SSO and ISSO algorithms are applied to solve the inverse estimation problems in a one-dimensional participating medium. Two cases concerns radiative transfer problems are investigated, in which the radiative source term, extinction coefficient, scattering albedo, and scattering symmetry factor are reconstructed. Furthermore, the coupled radiation–conduction heat transfer model is considered and the main parameters such as the conduction–radiation parameter, boundary emissivity, and scattering albedo are retrieved. All retrieval results show that SSO-based algorithms are robust and effective in solving inverse estimation problems even with measurement errors. Findings also show that the proposed ISSO algorithms are superior to the original SSO model in terms of computational accuracy and convergence velocity.

Evaluation of three different radiative transfer equation solvers for combined conduction and radiation heat transfer

This work investigates the performance of P1 method, FVM and SP3 method for 2D combined conduction and radiation heat transfer problem. Results based on the Monte Carlo method coupled with the energy equation are used as the benchmark solutions. Effects of the conduction-radiation parameter and optical thickness are considered. Performance analyses in term of the accuracy of heat flux and temperature predictions and of computing time are presented and analyzed.

Measurement of Gas Thermal Properties Using the Parametric Reduced-Order Modeling Approach

This paper deals with a thermal gas property micro-sensor. The proposed modeling approach of the sensor was based on reduced order modeling, in contrast to the traditional analytical modelling approach, which is the standard for this kind of sensors. This sensor was deployed for the measurement of the thermal conductivity ( $k$ ) and the volumetric heat capacity ( $\rho c_{p}$ ) of gases and works according to the temperature oscillation technique. A proper model is crucial for the measurement accuracy. The scope of this paper was to investigate the applicability of a sensor model based on a reduced-order modeling approach, intending to improve the performance of this sensor, as the behavior of the sensor can be modeled much more accurately than using an analytical model. For this reason, a parametric model-order reduction technique using proper orthogonal decomposition was applied. The main advantage of the reduced-order model is the high accuracy in the modeling of the conductive heat transfer problem, while it requires low computation effort. The approach was tested experimentally, where the model was calibrated in two pure gases and evaluated in 21 gases and gas mixtures. The sensor achieved an accuracy in the thermal conductivity of 6.5% and in the volumetric heat capacity of 3.2%.

Local RBF meshless scheme for coupled radiative and conductive heat transfer

ABSTRACTA local radial basis function meshless (LRBFM) method is developed to solve coupled radiative and conductive heat transfer problems in multidimensional participating media, in which compact support radial basis functions (RBFs) augmented on a polynomial basis are employed to construct the trial function, and the radiative transfer equation (RTE) and energy conservation equation are discretized directly at nodes by the collocation method. LRBFM belongs to a class of truly meshless methods which require no mesh or grid, and can be readily implemented in a set of uniform or irregular node distributions with no node connectivity. Performances of the LRBFM is compared to numerical results reported in the literature via a variety of coupled radiative and conductive heat transfer problems in 1D and 2D geometries. It is demonstrated that the local radial basis function meshless method provides high accuracy and great efficiency to solve coupled radiative and conductive heat transfer problems in multidimensional participating media with uniform and irregular node distribution, especially for coupled heat transfer problems in irregular geometry with Cartesian coordinates. In addition, it is extremely simple to implement.

Industrial application of topology optimization for combined conductive and convective heat transfer problems

This paper presents an industrial application of topology optimization for combined conductive and convective heat transfer problems. The solution is based on a synergy of computer aided design and engineering software tools from Dassault Systemes. The considered physical problem of steady-state heat transfer under convection is simulated using SIMULIA-Abaqus. A corresponding topology optimization feature is provided by SIMULIA-Tosca. By following a standard workflow of design optimization, the proposed solution is able to accommodate practical design scenarios and results in efficient conceptual design proposals. Several design examples with verification results are presented to demonstrate the applicability.

Stability analysis of generalized mass formulation in dynamic heat transfer

ABSTRACTIn this article, the generalized mass formulation is developed in an explicit analysis of transient transport problems. It has been well known that the time step is typically smaller in explicit analysis than in implicit analysis when the same size mesh is used. Further, the over-stiffness of conventional finite-element model may result in poor accuracy with linear triangular or tetrahedral elements. In order to improve the computational efficiency and numerical accuracy, this article proposes a generalized mass formulation by matching the mass matrix to the smoothed stiffness matrix using linear triangular elements in 2-D problems. The proposed mass matrix can be obtained by simply shifting the integration points from the conventional locations. Without loss of generality, several 2-D examples, including conduction, convection, and radiation heat transfer problems, are presented to demonstrate that the generalized mass formulation allows a larger time step in explicit analysis compared with the lumped and consistent mass matrices. In addition, it is found that the maximum allowable time step is proportional to the softened effect of the discretized model in an explicit analysis.

Analysis of combined mode heat transfer in a porous medium using the lattice Boltzmann method

ABSTRACTApplication of the lattice Boltzmann method (LBM) in solving a combined mode conduction, convection, and radiation heat transfer problem in a porous medium is extended. Consideration is given to a 1-D planar porous medium with a localized volumetric heat generation zone. Three particle distribution functions, one each for the solid temperature, the gas temperature, and the intensity of radiation, are simultaneously used to solve the gas- and the solid-phase energy equations. The volumetric radiation source term appears in the solid-phase energy equation, and it is also computed using the LBM. To check the accuracy of the LBM results, the same problem is also solved using the finite volume method (FVM). Effects of convective coupling, flow enthalpy, solid-phase conductivity, scattering albedo porosity, and emissivity on axial temperature distribution are studied and compared with the FVM results. Effects of flow enthalpy, solid-phase conductivity, and emissivity are also studied on radiative output. LBM results are in excellent agreement with those of the FVM.

Analysis of transient conduction and radiation problems using lattice Boltzmann and finite volume methods

In this paper, the lattice Boltzmann method (LBM) and the finite volume method (FVM) are coupled to solve transient conduction and radiation heat transfer problems. They are used to solve the energy equation and get the radiative heat source, respectively. To avoid additional interpolations between these two solvers when adding the radiative heat source to the energy equation, we propose to use the same grid systems for the LBM and FVM. This is achieved by adopting a half way bounce back boundary scheme for LBM. Good agreement with available literature is obtained for both 1D and 2D conditions. Effects of conduction–radiation parameter and scattering albedo on the transient temperature distribution are presented and discussed. The method proposed by this study is efficiently able to solve combined conduction and radiation problems.