Solution Of Heat Conduction Equation Research Articles

Data-driven and physics-informed deep learning operators for solution of heat conduction equation with parametric heat source

Deep neural networks as universal approximators of partial differential equations (PDEs) have attracted attention in numerous scientific and technical circles with the introduction of Physics-informed Neural Networks (PINNs). However, in most existing approaches, PINN can only provide solutions for defined input parameters, such as source terms, loads, boundaries, and initial conditions. Any modification in such parameters necessitates retraining or transfer learning. Classical numerical techniques are no exception, as each new input parameter value necessitates a new independent simulation. Unlike PINNs, which approximate solution functions, DeepONet approximates linear and nonlinear PDE solution operators by using parametric functions (infinite-dimensional objects) as inputs and mapping them to different PDE solution function output spaces. We devise, apply, and compare data-driven and physics-informed DeepONet models to solve the heat conduction (Poisson's) equation, one of the most common PDEs in science and engineering, using the variable and spatially multi-dimensional source term as its parameter. We provide novel computational insights into the DeepONet learning process of PDE solution with spatially multi-dimensional parametric input functions. We also show that, after being adequately trained, the proposed frameworks can reliably and almost instantly predict the parametric solution while being orders of magnitude faster than classical numerical solvers and without any additional training.

Analysis of a new numerical approach to solutions of heat conduction equations arising from heat diffusion

In this work, a new numerical finite difference scheme with the aim of obtaining a new numerical scheme that will be used to solve for the solution of Partial Differential Equations (PDE) arising from heat conduction equation is developed. This is significant because in recent times there is a growing interest in literatures to obtain a continuous numerical method for solving PDE. The numerical accuracy of this new approach is also studied. Detailed numerical results have shown that the new method provides better results than the known explicit finite difference method by Schmidt. And in terms of stability, the new scheme has been able to clearly shown that it is more stable than the old Schmidt explicit method. There is no semi-discretization involved and no reduction of PDE to a system of ODEs in the new approach, but rather a system of algebraic equations is directly obtained. MATLAB software wasused to solve for the desired solutions and the results obtained has shown that the method is near exact solutions.

From Two- to Three-Dimensional Model of Heat Flow in Edge-Emitting Laser: Theory, Experiment and Numerical Tools

Mathematical modeling of thermal behavior of edge-emitting lasers requires the usage of sophisticated time-consuming numerical methods like FEM (Finite Element Method) or very complicated 3D analytical approaches. In this work, we present an approach, which is based on a relatively simple 2D analytical solution of heat conduction equation. Our method enables extremely fast calculation of two crucial physical quantities; namely, junction and mirror temperature. As an example subject of research, we chose self-made p-side-down mounted InGaAs/GaAs/AlGaAs laser. Purpose-designed axial heat source function was introduced to take into account various mirror heating mechanisms, namely, surface recombination, reabsorption of radiation, Joule, and bulk heating. Our theoretical investigations were accompanied by experiments. We used micro-Raman spectroscopy for measuring the temperature of the laser front facet. We show excellent convergence of calculated and experimental results. In addition, we present links to freely available self-written Matlab functions, and we give some hints on how to use them for thermal analysis of laser bars or quantum cascade lasers.

Multiphysics Design of Pet-Coke Burner and Hydrogen Production by Applying Methane Steam Reforming System

Pet-coke (petroleum coke) is identified as a carbon-rich and black-colored solid. Despite the environmental risks posed by the exploitation of pet-coke, it is mostly applied as a boiling and combusting fuel in power generation, and cement production plants. It is considered as a promising replacement for coal power plants because of its higher heating value, carbon content, and low ash. A computational fluid dynamics (CFD) computational model of methane steam reforming was developed in this research. The hydrogen production system is composed from a pet-coke burner and a catalyst bed reactor. The heat released, produced by the pet-coke combustion, was utilized for convective and radiative heating of the catalyst bed for maintaining the steam reforming reaction of methane into hydrogen and carbon monoxide. This computational algorithm is composed of three steps—simulation of pet-coke combustion by using fire dynamics simulator (FDS) software coupled with thermal structural analysis of the burner lining and a multiphysics computation of the methane steam reforming (MSR) process taking place inside the catalyst bed. The structural analysis of the burner lining was carried out by coupling the solutions of heat conduction equation, Darcy porous media steam flow equation, and structural mechanics equation. In order to validate the gaseous temperature and carbon monoxide mole fraction obtained by FDS calculation, a comparison was carried out with the literature results. The maximal temperature obtained from the combustion simulation was about 1440 °C. The calculated temperature is similar to the temperature reported, which is also close to 1400 °C. The maximal carbon dioxide mole fraction reading was 15.0%. COMSOL multi-physics software solves simultaneously the catalyst media fluid flow, heat, and mass with chemical reaction kinetics transport equations of the methane steam reforming catalyst bed reactor. The methane conversion is about 27%. The steam and the methane decay along the catalyst bed reactor at the same slope. Similar values have been reported in the literature for MSR temperature of 510 °C. The hydrogen mass fraction was increased by 98.4%.

Fast calculation of 3D radiofrequency ablation zone based on a closed-form solution of heat conduction equation fitted by ex vivo measurements

Fast calculation or simulation of the ablation zone induced by radiofrequency ablation (RFA) has a critical role in hepatic RFA planning and therapy. However, it remains challenging to approximate the ablation zone in real time, especially when more than one probe is involved in one ablation session. This paper presents a novel computational technique to calculate the 3D ablation zone of one probe RFA and two-probe switching RFA. The main idea is to get an approximate solution of the temperature distribution from a simplified Pennes bioheat equation, and further fit the solution to the coagulation measurements on ex vivo porcine liver. With a closed-form solution of temperature distribution, the calculation of the ablation zone is as simple as the commonly used ellipsoidal model, but it allows a more realistic prediction of combined ablation zones with different inter-probe spacing. The new approximation technique could potentially replace the original ellipsoidal model in the intervention planning step.

A Good Approximate Solution of Heat Conduction Equations in Cylindrical Coordinates Composite Media.(Dept.P)

A Good Approximate Solution of Heat Conduction Equations in Cylindrical Coordinates Composite Media.(Dept.P)

Enhancement of methodology for protection of structures in contradiction of thermal effects

The research paper is devoted to protection of structures against heat and temperature effects. The necessity of improving the calculation of multilayered fence structures is shown. The solution of a one-dimensional unsteady heat conduction equation with constant and variable coefficients allowing to use of inhomogeneous and anisotropic materials as the fence material is given. An example of the solution of a fence made of inhomogeneous and anisotropic material is given. Solution of heat conduction equation is obtained by the recurrence-operator method. The solution of one-dimensional unsteady heat conduction equation with variable coefficients is obtained using the recurrence-operator method. The possibility of using the solution of the equation for multilayered inhomogeneous anisotropic fence materials is indicated.

The development of a three-dimensional finite element method code for the heat flux analysis of tungsten monoblock divertor on EAST

The calculation of heat flux on the divertor is critical to study the characteristics of energy transport in controlled fusion plasma. Since 2014, the upper divertor of Experimental Advanced Superconducting Tokamak (EAST) has been upgraded into water-cooled W/Cu monoblock, and therefore the original two-dimensional(2D) heat calculation code does not apply to the heat flux calculation on the upper divertor due to the asymmetry toroidal geometry. Thus, it is urgent to develop a Three-dimension(3D) heat flux calculation code based on Finite Element Method(FEM) to determine the 2D heat flux distribution at the divertor surface in EAST. This paper presents a detailed design and development of this 3D FEM heat flux calculation method with the full consideration the monoblock geometry of the upper divertor. Based on the surface temperature captured from the IR camera, and the effect of a thin surface layer on the divertor, the heat transmission coefficient was determined in the solution of heat conduction equation. To benchmark this code, the calculation results are compared with the results using ANSYS software with the same conditions(constant temperature boundary and constant heat flux boundary). The comparison results show good agreements that the maximum fractional error between the two methods is less than 4 %. Finally, the characteristics of heat flux distribution on the divertor due to edge localized modes(ELMs) losses on EAST are also demonstrated.

Microexplosive Atomization of Heterogeneous Fuel Drops for Intensifying Their Ignition

Models of microexplosive atomization of two-liquid unmixed drops under high-temperature heating are developed. Two most common mechanisms, namely, superheating of interphase boundary and critical bubble size, are chosen as atomization criteria. Two mathematical models, one based on solution of heat conduction equation and the other on VOF, are presented. The studied drops contain rapeseed oil and aqueous suspension of graphite. Experiments were conducted to determine the time lag of atomization of moving two-liquid unmixed drops under high-temperature heating. The theoretical and experimental atomization time lag values are found to be in satisfactory agreement. The results can be used to develop existing and create new gas-vapor-droplet technologies in chemical and petroleum engineering fields because they help predict the conditions of intense secondary atomization of heterogeneous drops by microexplosive dispersion.

Three approaches to evaluate of the heat dissipated during fatigue crack propagation experiments

This work is devoted to the comparative analysis of three techniques for measurement of energy dissipation in metals under fatigue crack propagation: original contact heat flux sensor, post-processing of the infrared thermography data and lock-in thermography. The contact heat flux sensor allows real-time recording of the heat source value. Non-contact temperature measurements by infrared thermography techniques allows one to calculate the heat source field on the specimen surface using the solution of heat conductivity equation. Lock-in thermography is a well-established technique for measuring of the dissipated energy under cyclic loading based on the analysis of the second harmonic amplitude of the thermal signal. This paper deals with the V-notched flat specimens made of stainless steel AISE 304 subjected to cyclic loading. It was shown that the dissipated energy values estimated by different techniques have a good qualitative agreement. Contact and non-contact measurements can be used for investigation on energy dissipation either in combination or separately. The measured values allows one to propose a relation between the fatigue crack growth rate and dissipated heat near the crack tip.

Passive Solar Heating with Adjustable Heating Modes

This paper discusses different schemes of the performance of passive solar heating of buildings. Based on the solution for heat conduction equation for active building constructions taking into account the influence of convective heat transfer and heat flux, incoming solar radiation, the ways of energy-saving facilities’ operation is analyzed. It is shown that to increase the efficiency of such systems, an individual design should be performed taking into account the climatic features of the construction area, and the choice of rational combinatorics for the outer shell, such as movable shielding heat-protective device, and creating an air gap between the accumulating and heat-insulating layers. Based on the calculations, the energyactive area of the bearing course of buildings, protected from the outer side with the translucent barrier is revealed. The suitable thickness of the accumulation layer for recycling process of solar radiation is determined. The presented innovative technical solutions could help to enhance the efficiency of solar energy utilization and contribute to more active implementation of passive solar heating, expanding the scope of their territorial application

An Analysis of Thermal-Bending Stresses in a Simply Supported Thin Elliptical Plate

In this paper, a transient thermal stress investigation on a simply supported thin elliptical plate during sectional heating with time-dependent temperature supply is considered. The solution of heat conduction equation with corresponding initial and boundary conditions is obtained by employing an integral transform approach. The governing equation solution for the small deflection theory is obtained and utilized to preserve the intensities of thermal bending moments, involving the Mathieu and modified functions and its derivatives. The deflection results show an approximately good agreement with the previously given results. It is also demonstrated that the temperature field in a circular solution could be resulted in a particular case of the present mathematical solution. The obtained numerical results utilizing computational implements are precise enough for practical purposes.

Application of Homotopy Perturbation Method Using Laplace Transform Intended for Determining the Temperature in the Heterogeneous Casting-Mould System

In this article, we established an application of homotopy perturbation method using Laplace transform (LT-HPM) to elaborate the analytical solution of heat conduction equation in the heterogeneous casting-mould system. The solution of the problem is provided with our supposition of an ideal contact between the cast and the mould. In the proposed method, we have chosen initial approximations of unknown constants which can be implemented by imposing the boundary and initial conditions. Examples have been discussed and confirmed the usefulness of this method.

Solving Non-Fourier Heat Conduction-Radiation Problem, a Semi-Analytical Approach

Objectives: In this study, the thermal analysis of a thin film exposed to laser heating, was investigated using Cattaneo Vernotte non-Fourier heat conduction model. Methods/Statistical Analysis: Nonfourier equations were derived based on radiative heat loss and also variable thermal conductivity. Then Variational Iteration Method (VIM) and Reduced Differential Transformation Method (RDTM) were used to solve these equations. Findings: VIM is an efficient choice to find an approximate solution of nonlinear non-Fourier heat conduction and its accuracy is higher than RDTM. Considering the thermal conductivity vary with temperature creates a significant difference in the temperature profiles, which shows the importance of nonlinear analysis of the problem. In certain values of relaxation time, temperature variations take place and out of that certain domain, variations of relaxation time doesn’t effect on temperature at all. Application/Improvements: There is a remarkable difference between the Fourier and non-Fourier solutions of heat conduction equations that demonstrate the importance of non-Fourier analysis of this type of equations. Keywords: Conduction-Radiation, Laser Heating, Non-Fourier, Semi-Analytical, RDTM, VIM

Analytical Solution Based on a New Series along with the Numeric Solution of the Non-Homogeneous Transient Heat Conduction Equation, in Hollow Cylinder under General Mixed Boundary Conditions

In this study, the problem of transient conduction heat transfer for an infinite hollow cylinder under non-homogeneous mixed boundary conditions at the both surfaces simultaneously in radial direction with general heat source depend on time and radius, also general initial condition by the method of superposition and separation variables, is solved and temperature distribution is obtained analytically. A new series based on the Bessel functions is obtained for the problem of transient heat conduction without heat source by using separation of variables. Any function that has expanded conditions by the Fourier series can be expanded by this new series. Then, by expanding the heat source function according to this new series, the problem of transient heat transfer involving the thermal source has been solved and the radial temperature distribution is obtained. Due to the limited case studies, numerical solution of heat conduction equation with implicit finite difference method also is presented. Finally, a numerical example is given to compare between analytical and numerical solutions.

Evaluation of the thermal processing of meat products in cylindrical cans using the General and Computer methods

Cylindrical cans (73x109 mm) were filled with luncheon meat and corned beef were sterilized with steam at 121.1°C for 96 and 91 min, respectively. The evaluation of the thermal processing was made using the General method and a Computer method. The General method used the experimental temperatures recorded by thermocouples while the Computer method predicted the internal temperatures of foods in cylindrical cans, using the finite difference numerical solution of heat conduction equation. By the end of heating at 121.1 °C and following cooling, comparison of the (Fo) value and the decimal reduction (log D121.1) of the initial population of the Clostridium botulinum at the geometric centre of the can for both examined meat products, estimated by the General method with those estimated by the Computer method, showed that there was no any significant difference between them (P>0.05). Comparison of recorded temperatures at the geometric centre of cans of both meat products with those estimated by the Computer method showed that, there was no any significant difference between them (P>0.05) during heating at 121.1 ° C, but a significant difference between them (P<0.05) was observed by the end of cooling.

A simplified approach for the thermoelastic large deflection in the thin clamped annular sector plate

ABSTRACTThe present article is concerned with analysis of large deflection of a heated thin annular sector plate with clamped edges under transient temperature distribution using Berger’s approximate methods. The prescribed surface temperature is at the top face of the plate whereas the bottom face is kept at zero temperature. In this study, the Laplace transform as well as the classical method have been used for the solution of heat conduction equation. The thermal moment is derived on the basis of temperature distribution, and its stresses are obtained using resultant bending moment and resultant forces per unit length. The calculations are obtained for the aluminium plate in the form of an infinite series involving Bessel functions, and the numerical results for temperature, deflection, resultant bending moments, and thermal stresses have been illustrated by graphs.

Information entropy and its application in heat conduction

In this paper, heat conduction problems are analyzed from the view of information entropy. The variation law of temperature with time in the lumped-heat-capacity system can be expressed in the form of exponential distribution. It is proved that exponential distribution is the solution of a conditional extreme value problem with information entropy as an objective function. The general analysis procedure of heat conduction problems from the view of information entropy is proposed. Fundamental solution of heat conduction equation can be written in the form of normal distribution, and normal distribution can be derived from the maximum entropy principle.

An exact analysis for transient anisotropic heat conduction in truncated composite conical shells

This paper presents an exact analytical solution for transient anisotropic heat conduction in a truncated composite conical shell for the first time. The fibers of composite conical shell are winded around the body in any arbitrary direction. Heat convection between the ambient fluid flow and composite conical shell is modeled and the exact solution of heat conduction equation is obtained by combining two popular analytical methods. First, by means of an integral transform in angular direction, the heat conduction equation is transformed and after that the transferred heat equation is solved using separation of variables method. Eventually, an inverse transformation is used in order to achieve the final exact solution of heat conduction equation. Validation of the present analytical solution is performed by comparing the analytical results with the solution of second order finite different method. Besides, the capability of present solution in solving the industrial cases is tested. The present solution is applicable in some industrial cases such as cooling pin fins and aeronautical instruments.

Study on random walk and its application to solution of heat conduction equation by Monte Carlo method

In this paper optimized and fast numerical method based on Monte Carlo simulation are introduced to achieve several algorithms for solving the steady state and transient heat conduction problems with any prescribed convection boundary conditions and interior thermal source distribution. The methods are mesh-less and thus can solve problems with complex geometries, non-uniform heat source or porous bodies with no inherent limitation. In the proposed approach the temperature of any point in the studying body can be found independently from other points without solution of related system of linear algebraic equations. As part of this study, an effort was undertaken to summarize various random walk methods in Monte Carlo. The Monte Carlo methods that are introduced in this paper use Semi Floating Random Walk (SFRW) and Full Floating Random Walk (FFRW) to estimate temperature in the domain of the definition. Also, Fixed Random Walk (FRW) method is applied to solve transient heat conduction equation. Each method has its own unique features that will be discussed during this work. It was observed that FFRW method is the fastest method to steady state conditions and also FRW method has capability to run both steady state and transient heat conduction equation. It was showed that the results are in good agreement with those of the finite difference method.