Two-dimensional Heat Conduction Research Articles

A new analytical model for the conduction shape factor of annulus sectors

Heat transfer devices, such as heat pipes, vapor chambers, thermosiphons, microchannel heat sinks, and Peltier cooling plates, rely on two-dimensional steady heat conduction to thermally manage telecommunications, aerospace, and microelectronics heat-generating components. The conduction shape factor can evaluate these devices’ two-dimensional steady heat conduction. The geometry of the device’s annulus and its mechanical attachment to the heat-generating component can vary. Given the prominence of single-sided heating and cooling, the two-dimensional heat conduction is commonly through an annulus sector. For the first time, an analytical model is developed to predict the conduction shape factor of annulus sectors. The present model is an extension of the previously developed equivalent concentric circular annulus model and applies the equivalent concentric circular annulus sector. The model is validated with results from finite element modeling for parametric boundary geometries, capturing most of the data across a variety of sectors within a relative difference of 10%. The present model provides a simple, closed-form analytical solution for the shape factor of an annulus sector formed between concentric arbitrarily shaped isothermal boundaries. More importantly, it provides a unified platform for designing and optimizing novel heat transfer devices.

Identifying Varying Thermal Diffusivity of Inhomogeneous Materials Based on a Hybrid Physics-Informed Neural Network

Inhomogeneous materials are characterized by uneven material properties with changing dimensions. Effective and precise identification of nonuniform material properties is essential for studying material inhomogeneity. In this study, we propose a hybrid physics-informed neural network (PINN) to identify the inhomogeneous thermal diffusivity only using temperature data collected in a transient heat conduction problem. A PINN and a fully connected neural network are combined in the proposed model. The PINN model is trained to learn the law of physics governed by the transient heat conduction equation. The identification network is trained to uncover the thermal diffusivity variation with changing dimensions. The proposed model successfully identifies two types of inhomogeneity: temperature-dependent and space-dependent thermal diffusivity, and both two-dimensional and three-dimensional heat conduction problems are investigated. The proposed model is examined to be robust to small datasets and noisy training data. In addition to heat conduction problems, the proposed model can be adopted to identify the material properties of various types of inhomogeneous materials.

Thermomechanical investigation of the continuous casting of ingots using the element-based Finite-Volume Method

A computational thermomechanical simulator using the Element-based Finite-Volume Method (EbFVM) has been elaborated to investigate the solidification of continuous casting ingots. The thermal behavior was evaluated using the two-dimensional transient heat conduction equation with phase change in conjunction with numerical-experimental film coefficients. In addition, the thermal-mechanical approach includes three inelastic representations, which are based at the Ramberg-Osgood plastic and Odqvist's viscoplastic models. The simulator is validated using several benchmarks tests, along with the analytical solution of the Stefan's problem. In the context of continuous casting, in order to enhance the cooling stage in the spray zone, as well as the crack formation, a new set of sprays were proposed for the investigated case. The thermomechanical study provided by EbFVM is able to provide good results for the temperature profile and thermomechanical state that could be used to address problems caused by the cooling process of continuous casting of ingots.

Numerical simulation of the heating process of an NbN film by a current pulse at low temperatures based on the two-dimensional heat-conduction equation

The simulation of the process of heat propagation in the NbN film during the passage of a current pulse through it was carried out on the basis of an inhomogeneous two-dimensional equation of heat conduction. The initial-boundary value problem for longitudinal section of the film is solved numerically, and the analysis of heat transport in the system contacts-film-substrate-thermostat is carried out. The temperature evolution of the film cross section from the onset of the pulse to temperature leveling is visualized. It is shown that the contact material (beryllium bronze) provides efficient heat removal from a superconducting film in a resistive state when a high-density current flows through it. The proposed modeling method and contact material can be used to study other metal or semiconductor films.

Effect of thermal environment on the asymmetric vibration of temperature-dependent two-dimensional functionally graded annular plate by Chebyshev polynomials

In the present paper, an accurate numerical technique is employed to analyze the effect of two-dimensional temperature variation on the free asymmetric vibrations of functionally graded (FG) annular plates using the first-order shear deformation theory. The temperature-dependent mechanical properties of the plate material are graded along the thickness and radial directions. The governing differential equations for asymmetric motion are derived from Hamilton’s principle and the exact solution for the two-dimensional heat conduction equation is obtained using the separation of variables method considering thermal boundary conditions. The Chebyshev collocation technique is adopted to compute the numerical values of fundamental frequency for different boundary conditions. The impact of volume fraction index, heterogeneity parameter, density parameter, nodal lines, radii ratio, and thermal boundary conditions on the frequencies of the plate is investigated. An excellent agreement between the results obtained by the present approach and other methods available in the literature validated the authors’ model and technique. Three-dimensional normalized mode shapes with different nodal lines are illustrated for the specified plates.

Coupled Analysis of Heat Transfer in a Molten Pool With Three-Layer Configuration

Understanding a reliable description of the heat transfer in the molten pool with three-layer configuration is inevitable for the design of severe accident mitigation measures in research reactors, such as IVR-ERVC. However, investigations on a thick lower metallic layer are scarce. Thus, further studies on heat transfer in a molten pool with a thick lower metallic layer are highly required, based on previous severe accident studies in light water reactors. This study conducts a numerical simulation of heat transfer in the molten pool with a thick lower metallic layer during a severe accident, combining model development, code verification, and code application. First, the MPCAP code has been developed, based on the coupled simulation of natural convection inside the molten pool, two-dimensional heat conduction in the RPV lower head, and convection outside the reactor vessel wall. Then, a verification case has been conducted using three-layer configuration in AP1000. The AP1000 verification case indicates that the MPCAP code predicts important parameters in the molten pool well.

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.

Determination of the gun barrel walls temperature distribution and its experimental validation during multiple-shots firing process

The different parameters of the combustion gases mixture were calculated allowing the calculation of the heat convection coefficient between the gases and the bore surface of the gun barrel. The heat conduction in the barrel was reduced to two axisymmetric problems. It was solved numerically using one-dimensional and two-dimensional heat conduction equations. Experiments were conducted to measure the exterior gun wall temperatures using thermal camera. The agreement between the measured and the calculated results enable the validation of the calculation models for both the gases parameters and the temperature distribution in the gun barrel.

Numerical investigations of the lateral heat transfer in coaxial thermocouples

Coaxial thermocouples have been widely used for transient heat transfer measurements in high-enthalpy shock tunnels. The one-dimensional semi-infinite heat conduction theory is typically used for temperature data processing. However, due to the material difference between the two electrodes and the junction, lateral heat transfer occurs, causing a deviation in the heat flux measurement from the prediction results of the one-dimensional semi-infinite heat conduction theory. Thus, the lateral heat transfer effect has to be investigated to improve the accuracy and reliability of heat flux measurements. In this article, the heat transfer in E-type (chromel-constantan) coaxial thermocouples was analyzed by numerically solving the two-dimensional axisymmetric heat conduction equation with the Du Fort-Frankel scheme. During the heating process, the maximum temperature point on the surface of the coaxial thermocouples moved to the positive electrode over time. The numerical simulation indicated that the surface temperature of the coaxial thermocouples and the derived heat flux were larger than the theoretical value. The heat flux measurement error of the coaxial thermocouples can be reduced by increasing the width of the positive electrode. The results of this study can provide a reference for the design and manufacture of coaxial thermocouples.

Evaluation of defect depth in CFRP composites by long pulse thermography

Long pulse thermography (LPT) is an emerging non-destructive technique that has been used for rapid, cost-effective and accurate damage assessment of composite materials. This paper proposes three analytical models for the in-depth evaluation of defects in a carbon fibre-reinforced plastic (CFRP) specimen by long pulse thermography. The first two models rely on the estimate of the peak temperature-contrast and the peak temperature-contrast slope from the thermal data recorded by an infrared (IR) camera. Both the analytical and experimental results showed that the arriving time of the peak temperature-contrast depends on both the depth and the diameter of defects, whilst the contrast time slope is mainly proportional to the square of the damage depth. The third model here proposed consists of a two-dimensional (2D) axisymmetric heat conduction model based on virtual heat sources. Compared to previously reported analytical models, the proposed 2D one showed higher accuracy on the depth estimation.

Surface transfer coefficients estimation for heat conduction problem using the Bayesian framework

This work deals with an inverse two-dimensional nonlinear heat conduction problem to determine the top and lateral surface transfer coefficients. For this, the Bayesian framework with the Markov Chain Monte Carlo algorithm is used to determine the posterior distribution of unknown parameters. To handle the computational burden, a lumped one-dimensional model is proposed. The lumped model approximations are considered within the parameter estimation procedure thanks to the Approximation Error Model. The experiments are carried out for several configurations of chamber ventilator speed. Experimental observations are obtained through a complete measurement uncertainty propagation. By solving the inverse problem, accurate probability distributions are determined. Additional investigations are performed to demonstrate the reliability of the lumped model, in terms of accuracy and computational gains.

A new multi-objective optimization approach for process parameters optimization during numerical simulation of quenching steel parts

The paper presents the numerical simulation of quenching cylindrical steel sample immersed in three different quenchants: water, 5 % aquatensid solution, and isorapid oil. The quenching process starts from the initial temperature of the cylinder at 850 °C and moves through the air until it reaches the quenching bath. The quenchant is held at constant temperature of 40 °C. The cylinder is made of carefully selected steel which does not change its structure during quenching and heating. Cylindrical samples were manufactured in three different dimensions (R, H), (mm): (12.5 × 100), (25 × 150) and (37.5 × 225), so that four measuring points were installed in each sample. Each measuring point consists of thermocouple installed beneath the cylinder surface, capable of measuring the temperature every half second. Based on the experiment, the numerical simulation is recognized as transient and nonlinear two-dimensional heat conduction problem consisting of the two main tasks: direct heat transfer problem (DHTP) and inverse heat transfer problem (IHTP). The paper proposes a new multi-objective optimization approach for the estimation of heat transfer coefficients during the numerical simulation of quenching cylindrical steel sample. The proposed approach gained better results and less convergence time compared to the results from the literature. The paper includes methods, algorithms and the source code for the calculation of the temperature fields in time and heat transfer coefficient estimation of the IHTP. The simulation software has been implemented in C# programming language and can be found at http://github.com/bhrnjica/quenching_simulation.

Investigation of the transient heat transfer to a supersonic air jet impinging on a high-temperature plate based on a discrimination-experiment method.

A discrimination-experiment method is developed to investigate the transient heat transfer of air jet impingement by discretizing the solid domain into mutually adiabatic test cylinders. This method can not only reduce the influence of the transverse heat transfer of a solid domain on the heat transfer characteristics of the jet but can also simplify the two-dimensional or three-dimensional heat conduction problem into a one-dimensional problem. Moreover, the discrimination-experiment method eliminates the embedment of thermocouples into the solid domains, further improving the accuracy and reliability of the proposed method. The transient heat transfer characteristics of a supersonic air jet impinging on a high-temperature target (860°C) and the effects of thermo physical parameters, such as the density, specific heat capacity, thermal conductivity and nozzle-to-target distance (H/D = 3, 4, and 5) are analyzed in detail using the discrimination-experiment method. The results provide important guidance for the thermal design of supersonic air jet impingement.

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

Finite Difference Method Applied in Two-Dimensional Heat Conduction Problem in the Permanent Regime in Rectangular Coordinates

The solution of many conduction heat transfer problems is found by two-dimensional simplification using the analytical method since different points has different initial temperatures. The temperature at each point of a given element can be analyzed through the Heat Equation that, in some cases, converges to analytical solutions without precision and is far from the real. However, with the application of the Finite Difference Method (FDM), it is possible to solve it numerically in a relatively fast way, providing satisfactory results for the most varied boundary conditions and diverse geometries, characteristics of heat transfer problems by conduction. This study solved two problems inside a plate with and without heat generation involved in temperature distribution. Algorithms were built with the aid of the Matlab programming language, and applied to obtain a numerical solution using the FDM numerical method. The computational and analytical solutions were then compared. Under certain conditions of the parameters involved in the phenomenon of each problem, the numerical method was very efficient for presenting errors less than or equal to 0.003 and 0.03, respectively, for cases without and with heat generation.

Study of two-dimensional transient heat transfer through a multi-layers wall: Application to the buildings thermal insulation

The two-dimensional transient heat conduction through a multilayers wall made of different materials and thicknesses was numerically resolved. The equations system resolution was carried out by Alternating direction implicit method (ADI). The outdoor and indoor temperatures and the convective coefficients used as boundary conditions in the developed Fortran program were from the Algerian regulatory technical document. After validation of the Fortran program with literature, it was used to studying the influence of different boundary conditions (bottom and top sides), on the thermal insulation in the building, for many configurations of external walls, usually used in building construction at Batna city (Algeria). Results showed that for the configurations that give bad thermal insulation, the conditions imposed on the top and bottom of the wall have practically no influence on the internal temperature of the multilayers wall, however, the opposite is observed for the configurations that ensure good thermal insulation.

Semi-analytical model for the heat conduction resistance of a single spherical condensate droplet

The evolution and heat transfer of a single spherical condensate droplet on a superhydrophobic surface (SHS) can be modeled by the droplet thermal resistance network. The calculation of the heat conduction resistance of a single spherical condensate droplet (Rd) is a fundamental problem in analyzing the droplet thermal resistance network. Till now, the majority of researchers have used Kim's formula to calculate Rd. However, the results from Kim's formula have errors above 85% compared to those from the CFD software (ANSYS Fluent). In this paper, we analyze the reason for inaccuracy of Kim's formula. Taking two-dimensional droplet heat conduction into consideration, we gain the theoretical temperature field function inside the condensate droplet by utilizing the conformal mapping. Basing on calculation of heat transfer rate through the droplet and ANSYS Fluent analysis, we put forward a new semi-analytical model to calculate Rd. We verify and find that the results from our new model have errors less than 13% compared to those from ANSYS Fluent. It means that our new semi-analytical model is more accurate than Kim's formula. We calculate the Rd proportion of the overall heat transfer resistance and find that the application of our model can obviously decrease heat transfer calculation errors for big droplets (r>16μm) with large contact angles (θ>120∘). The new semi-analytical model for the heat conduction resistance of a single spherical condensate droplet is promising in the sense that it might replace Kim's model to become a novel design criterion of enhanced heat transfer SHSs.

Thermo-vibrational analyses of skin tissue subjected to laser heating source in thermal therapy

Laser-induced thermal therapy, due to its applications in various clinical treatments, has become an efficient alternative, especially for skin ablation. In this work, the two-dimensional thermomechanical response of skin tissue subjected to different types of thermal loading is investigated. Considering the thermoelastic coupling term, the two-dimensional differential equation of heat conduction in the skin tissue based on the Cattaneo–Vernotte heat conduction law is presented. The two-dimensional differential equation of the tissue displacement coupled with the two-dimensional hyperbolic heat conduction equation in the tissue is solved simultaneously to analyze the thermal and mechanical response of the skin tissue. The existence of mixed complicated boundary conditions makes the problem so complex and intricate. The Galerkin-based reduced-order model has been utilized to solve the two-sided coupled differential equations of vibration and heat transfer in the tissue with accompanying complicated boundary conditions. The effect of various types of heating sources such as thermal shock, single and repetitive pulses, repeating sequence stairs, ramp-type, and harmonic-type heating, on the thermomechanical response of the tissue is investigated. The temperature distribution in the tissue along depth and radial direction is also presented. The transient temperature and displacement response of tissue considering different relaxation times are studied, and the results are discussed in detail.