Two-dimensional Steady-state Heat Conduction Research Articles

Bayesian Inference for Estimating Heat Sources Through Temperature Assimilation

Abstract This paper utilizes a Bayesian inference framework to address the two-dimensional (2D) steady-state heat conduction problem, focusing on the estimation of unknown distributed heat sources in a thermally conducting medium with uniform conductivity. The goal is to infer the locations, strength, and shape of heaters by assimilating temperature data in Euclidean space, employing a Fourier series to represent each heater's shape. The Markov Chain Monte Carlo (MCMC) method, incorporating the random-walk Metropolis–Hasting (MH) algorithm and parallel tempering, is utilized for posterior distribution exploration in both unbounded and wall-bounded domains. It is found that multiple solutions arise in cases where the number of temperature sensors is less than the number of unknown states. Moreover, smaller heaters introduce greater uncertainty in estimated strength. To address the challenge of estimating the heater's strength and shape simultaneously due to their strong correlation, our method incorporates sharp priors on one to ensure accurate and feasible solutions of the other. The diffusive nature of heat conduction smooths out any deformations in the temperature contours, especially in the presence of multiple heaters positioned near each other, impacting convergence. In wall-bounded domains with Neumann boundary conditions, the inference of heater parameters tends to be more accurate than in unbounded domains.

Topology optimization and numerical validation for heat transfer improvement in a packed-bed reactor with monolithic catalyst

This study focuses on optimizing heat transfer in packed-bed reactors by simplifying the problem to a two-dimensional steady-state heat conduction scenario. The objective is to efficiently arrange a limited volume of high-conductivity material to transport heat from the source to the low-conductivity heat-absorbing materials, representing the reacting fluid phase. The topology optimization problem is tackled using a density-based method that relies on a gradient-based algorithm. The optimized design is extruded and compared to a honeycomb internal structure using high-fidelity simulations for steam methane reforming. Results show a 6.04 % improvement in CH4 conversion for the optimized structure, highlighting the potential of this method to enhance monolithic catalysts, particularly in cases where heat transfer critically influences the reaction.

Boundary reconstruction in two-dimensional steady-state anisotropic heat conduction

Boundary reconstruction in two-dimensional steady-state anisotropic heat conduction

Modelling 2-D Heat Transfer Problem for Different Material Combination

The analysis of heat transfer problems can be highly complex due to factors such as temperature, position, and time. Most heat transfers are typically two-dimensional as conduction is often negligible in the third dimension. Two-dimensional heat conduction problems can be solved analytically or numerically. In steady-state conditions, the Laplace equation can be applied to solve two-dimensional heat conduction problems analytically, in which the separation of variables method is used to solve the Laplace equation under fixed boundary conditions to determine the temperature at a specific point. The Laplace equation plays a significant role in the solution of heat transfer problems, as it demonstrates the behavior of linear and non-linear equations in the computational fluid dynamics domain. Despite their inability to provide exact results at any point, numerical methods are superior to analytical methods when handling complex geometries with various boundary conditions. This project involves the development of a computational code using MATLAB to solve two-dimensional steady-state heat conduction problems using Gauss-Seidel iterations. Comparing analytical solutions from Excel with numerical solutions from MATLAB and ANSYS, specifically the developed MATLAB code, revealed an accuracy level of 99.902% for the Laplace equation. An analysis of the produced code from MATLAB found that it could solve two-dimensional steady-state heat conduction across different combinations of materials while allowing users to specify initial and boundary conditions to produce a contour plot similar to ANSYS.

Optimization of tree-shaped fin structures towards enhanced discharging performance of metal hydride reactor for thermochemical heat storage based on entransy theory

Integrating fin structures into metal hydride (MH) bed can enhance heat discharging performance of the reactor for thermochemical heat storage. In this study, based on the entransy theory, the entransy dissipation rate (Eh,st) or the entransy dissipation (Eh,tr) of a simplified two-dimensional (2D) steady-state or transient heat conduction model for the MH bed was used to be an objective to optimize the tree-shaped fin (TSF) structures. Furthermore, response surface method (RSM) was adopted to explore the relationship between Eh,st or Eh,tr and structural parameters (i.e., branch number N, branch level M, length ratio α, width index β, and angle ratio γ). The results show that RSM can effectively predict influences of structural parameters of TSFs on heat discharging performance. Based on RSM with entransy theory, the obtained optimal combination parameters were N = 8, M = 3, α = 1, β = 0.5, and γ = 1, corresponding to the maximum gravimetric exergy-output rate (GEOR) of 20.98 W/kg. Furthermore, the GEOR with optimized fins was 11.5 % higher than that with unoptimized TSFs. The optimization method based on the entransy theory was effective to optimize the TSF structures by solving 2D model instead of three-dimensional model, and shorten the calculation time for optimization.

An elliptical inhomogeneity under nonuniform heat flux

We use complex variable techniques to study the decoupled two-dimensional steady-state heat conduction and thermoelastic problems associated with an elliptical elastic inhomogeneity perfectly bonded to an infinite matrix subjected to a nonuniform heat flux at infinity. Specifically, the nonuniform remote heat flux takes the form of a linear distribution. It is found that the internal temperature and thermal stresses inside the elliptical inhomogeneity are quadratic functions of the two in-plane coordinates. Explicit closed-form expressions of the analytic functions characterizing the temperature and thermoelastic field in the matrix are derived.

Neutral Inhomogeneities in a Two-dimensional Steady-state Heat Conduction Problem

A steady-state heat conduction problem is considered in a two-dimensional solid body which is filled up composite circular inclusions. The composite circular inclusions consist of a core and a coating both of which are cylindrically orthotropic. In this paper the neutral inhomogeneity is defined as a foreign body (inclusion) which can be introduced into the host body (matrix) without distributing the temperature field in the originally homogeneous body. The perfect thermal contacts are assumed to be between the different components of nonhomogeneous bodies.

Comparative Analysis of Two-Dimensional Steady State Heat Flow Problem of a Rectangular Plate by Analytical and Numerical Approach

Heat transfer problems play a very vital role in many fields of engineering. This present analysis work focuses on analytical and numerical approaches to resolve a two-dimensional steady-state heat conduction problem of a rectangular plate. The temperatures at the inside points of the rectangular plate are calculated analytically and the results are verified numerically. For numerical solution of the problems programming is done in using METLAB software. The material of the rectangular plate used is steel. It will be seen from the literature survey that rigorous analytical solutions are available only for a very few simple boundary conditions and these conditions does not seem to be favourable for complex boundaries. However, majority of engineering problems with simple as well as complex boundary conditions will be solved with the help of numerical approaches. In this analysis work, an attempt is made to obtain solutions using numerical method for a two-dimensional steady state heat conduction problems of a steel rectangular plate. The mathematical formulation of problems is done using Laplace method. The results obtained from the two different methods that is analytical and numerical are then compared with each other. The agreement between the analytical and numerical results is a sign of the accuracy of solution method.

Two-dimensional numerical manifold method for heat conduction problems

The numerical manifold method (NMM) is a calculation method based on Galerkin's variation and contains a dual covering system. Therefore, the NMM can easily deal with the construction of high-order manifold elements and is convenient for adaptive analysis. This study employed the NMM to simulate steady-state and transient heat conduction problems. The system equations were derived and a penalty function method was applied to deal with the boundary conditions. By simulating calculation examples for a one-dimensional bar, cuboid rock specimens, and thick-walled cylinders, the process of solving steady-state and transient heat-conduction problems and the influence law of the temperature pattern are demonstrated. Moreover, the convergence and effectiveness of the NMM in handling two-dimensional (2D) steady-state and 2D transient heat conduction problems was verified.

Isogeometric Boundary Element Method for Two-Dimensional Steady-State Non-Homogeneous Heat Conduction Problem

Isogeometric Boundary Element Method for Two-Dimensional Steady-State Non-Homogeneous Heat Conduction Problem

Heat Transfer Analysis in a Rotating Cavity With Axial Through-Flow

Abstract This paper presents local Nusselt numbers computed from experimental measurements of surface temperature of compressor disks in a multiple rotating cavity test rig with axial throughflow. A validated two-dimensional (2D) steady-state heat conduction analysis methodology is presented, using the actual test geometry, and 95% confidence intervals calculated using Monte Carlo simulation. Sensitivity of the solution to curve fitting types, geometric simplification, and surface instrumentation are explored. The results indicate that polynomial curves fits, while computational simple, are unsuitable especially at higher orders. It is shown that geometric simplifications, that typically simplify the algorithmic implementation, may also omit significant variation in heat flux at critical stress relieving locations. The effect of reducing measurement points in the analysis is to both overpredict heat transfer and increase the uncertainty of the results. Finally, the methodology is applied to previously published thermal data from the University of Sussex, facilitating qualitative discussion on the influence of the governing parameters. While this study does not overcome the inherent uncertainty associated with inverse solutions, it is intended to present a methodology that is readily available to the wider community for the analysis of thermal test data and suggests some guidelines at the planning and postprocessing stages. The range of experiment reported here covers: 1.13 × 105 < Rez < 5.14 × 105, 1.65 × 106 < Reθ < 3.16 × 106, 0.10 < Ro < 0.60, and 3.40 × 1011 < Gr < 1.25 × 1012.

Measurement of effective thermal conductivity of LaNi[formula omitted] powder packed bed

Effective thermal conductivity of LaNi5 powder packed bed was analyzed with a miniaturized guarded hot-plate (GHP) apparatus. Here, GHP was designed for precise measurement of effective thermal conductivity of metal-hydride powders even with small sample amounts (2.12×104 mm3). Dimensions of sample container and apparatus were determined through two-dimensional (2-D) steady-state heat conduction analysis. Calibration experiment and uncertainty analysis were conducted to validate the accuracy of the GHP. Based on the measurements of the residual thermal conductivity of the LaNi5 packed bed, effect of particle size on contact factor of LaNi5 packed bed was estimated. By applying the Yagi and Kunii (YK) model to the effective thermal conductivity of LaNi5 packed bed, effect of contact factor and gas thermal conductivity on characteristic length of gas film were newly analyzed. Factors of YK model were modified in present work and validated through comparison with experimental data from previous literature.

Meshless Analysis of Nonlocal Boundary Value Problems in Anisotropic and Inhomogeneous Media

In this work, meshless methods based on a radial basis function (RBF) are applied for the solution of two-dimensional steady-state heat conduction problems with nonlocal multi-point boundary conditions (NMBC). These meshless procedures are based on the multiquadric (MQ) RBF and its modified version (i.e., integrated MQ RBF). The meshless method is extended to the NMBC and compared with the standard collocation method (i.e., Kansa’s method). In extended methods, the interior and the boundary solutions are approximated with a sum of RBF series, while in Kansa’s method, a single series of RBF is considered. Three different sorts of solution domains are considered in which Dirichlet or Neumann boundary conditions are specified on some part of the boundary while the unknown function values of the remaining portion of the boundary are related to a discrete set of interior points. The influences of NMBC on the accuracy and condition number of the system matrix associated with the proposed methods are investigated. The sensitivity of the shape parameter is also analyzed in the proposed methods. The performance of the proposed approaches in terms of accuracy and efficiency is confirmed on the benchmark problems.

Error estimation of bilinear Galerkin finite element method for 2D thermal problems

This study demonstrates a two-dimensional steady state heat conduction Laplace partial differential equation solution using the bilinear Galerkin finite element method. Heat transfer analysis is of vital importance in many engineering applications and devising computationally inexpensive numerical methods while maintaining accuracy is one of the primary concerns. The method uses structured mesh grid over a two-dimensional rectangular domain and solved using a stiffness matrix for the bilinear elements, calculated using the proposed modified numerical scheme. Several numerical experiments are conducted by controlling the number of nodes and changing element sizes of the presented scheme, and comparison made between analytical solution and software generated solution.

Two-dimensional steady-state heat conduction problem for heat networks

The method of determining the level of heat losses through structures of heat networks and the two-dimensional steady-state heat conduction problem for multilayer structures of heat networks have been considered, given that each of the layers has its own physical properties and heat transfer coefficients depend on r and z coordinates. The definition of heat losses is an important practical task, since they are one of the main indicators of energy efficiency in the operating heat networks and are included in heat tariffs.

Simulation of two-dimensional steady-state heat conduction problems by a fast singular boundary method

The study presents a fast direct algorithm for solutions of systems arising from singular boundary method (SBM) in two-dimensional (2D) steady-state heat conduction problems. As a strong-form boundary discretization collocation technique, the SBM is mathematically simple, easy-to-implement, and free of mesh. Similar to boundary element method (BEM), the SBM generates a dense coefficient matrix, which requires O(N3) operations and O(N2) memory to solve the system with direct solvers. In this study, a fast direct solver based on recursive skeletonization factorization (RSF) is applied to solve the linear systems in the SBM. Due to the fact that the coefficient matrix is hierarchically block separable, we can construct a multilevel generalized LU decomposition that allows fast application of the inverse of the coefficient matrix. Combing the RSF and the SBM can dramatically reduce the operations and the memory to O(N). The accuracy and effectiveness of the RSF-SBM are tested through some 2D heat conduction problems including isotropic and anisotropic cases.

Boundary function method for boundary identification in two-dimensional steady-state nonlinear heat conduction problems

In this article, a novel meshless boundary function method (BFM) is proposed for solving the boundary identification problem of steady-state nonlinear heat conduction in arbitrary plane domain. Firstly, the original governing equation is transformed to a new one with homogeneous Cauchy boundary conditions by using a homogenization technique. Secondly, the domain type meshless collocation method is employed to solve the new partial different equation in a reduced domain, in which the numerical solution is expanded by a sequence of boundary functions, automatically satisfying the homogeneous boundary conditions on the known boundary. After that, a nonlinear equation corresponding to each angle is formed and then is solved by the Newton iterative method in order to determine the missing boundary shape. Finally, the accuracy and robustness of the proposed BFM are examined through three numerical examples.

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.

Mechanical and thermal stresses in radially functionally graded hollow cylinders with variable thickness due to symmetric loads

ABSTRACT In this paper, based on the higher-order shear deformation theory, a thermo-elastic analysis for axisymmetric clamped–clamped rotating pressurised thick cylindrical shells with variable thickness made of functionally graded materials subjected to thermal loading is presented. The material properties, except the Poisson’s ratio, are assumed to vary with the power law function in the radial direction of the cylinder. The governing equations are a system of non-homogeneous ordinary differential equations with variable coefficients. Based on two-dimensional steady-state heat conduction equation, temperature distribution in cylinder with variable thickness is given. Using multilayer method, these equations could be converted into systems of differential equations with constant coefficients. The effects of higher-order approximations on the radial and axial displacements, von Mises stress and shear stress have been studied. Finally, the displacements and stresses along the radius and length have been plotted, and distribution of these is compared with the solution using finite-element method.

An optimization algorithm for determining a point heat source position in a 2D domain using a hybrid metaheuristic

Abstract In this paper, we study an inverse problem of determining the unknown heat source position of a point-wise heat source in a two-dimensional steady-state heat conduction problem governed by a linear elliptic equation with the Dirichlet boundary conditions. The problem is solved by the hybridization of particle swarm optimization and the gravitational search algorithm with a newly defined mutation operator. Some empirical studies are also performed to gauge the accuracy of the proposed approach.