Heat Conduction Problem Research Articles

Orthogonal Methods of Weighted Residuals in Thermal Conductivity Problems for Multilayer Structures

An analytical solution of the heat conduction problem for a two-layer plate based on the Galerkin method and usage of additional boundary conditions (ABC) is obtained. The ABCs are found in such a way that their implementation is to be adequate to the implementation of the original equations at the boundary points. Execution of equations at boundary points leads to their execution over the entire range of spatial coordinates. The use of ABC allows us to obtain chain systems of algebraic equations for unknown solution coefficients. The equations of these systems have highly sparse well-conditioned square matrices. In this connection, their solutions are so simplified that with a large number of approximations, in general form, a system of only two algebraic equations should be solved. A high accuracy of finding the eigenvalues is observed that is explained by the use of a special ABC design.

Estimation of thermal input in thermite reaction for innovative wellbore plugging & abandonment techniques

In this work, a methodology is presented to simulate heat transfer in wellbores for plugging & abandonment operations, where the thermal input is provided by a thermite reaction. The operation of burning a thermite column inside the well's production tubing to form a metal plug is studied. The objective is to eliminate the need to simulate the thermite domain and chemical reaction during the process, simplifying the physical model and reducing computational cost. In this model, it is assumed that the thermal input from the thermite reaction is provided to the model through multiple heat flux boundary conditions along the inner wall of the production tubing. The unknown heat flux from the thermite to the inner wall of the tube is estimated by solving an inverse heat conduction problem (IHCP). The Adaptive Function Specification Method is used to estimate multiple heat flux functions at the boundary through information from multiple temperature sensors located on the outer surface along the tube's height. A characteristic behavior of the heat flux curve was verified in all segments of the tube, and the average heat flux curve was used to simulate heat transfer during a 15 m thermite burning process inside a well.

Sequential Regularization Method for the Identification of Mold Heat Flux during Continuous Casting Using Inverse Problem Solutions Techniques

A two-dimensional transient inverse heat-conduction problem (2DIHCP) was established to determine the mold heat flux using observed temperatures. The sequential regularization method (SRM) was used with zeroth-, first-, and second-order spatial regularization to solve the 2DIHCP. The accuracy of the 2DIHCP was investigated under two strict test conditions (Case 1: heat flux with time-spatial periodically varying, and Case 2: that with sharp variations). The effects of the number of future time steps, regularization parameters, order of regularization, discrete grids, and time step size on the accuracy of the 2DIHCP were analyzed. The results showed that the minimum relative error (epred) of the predicted Case 1 heat flux is 5.05%, 5.39%, and 5.88% for zeroth-, first-, and second-order spatial regularization, respectively. The corresponding values for the predicted Case 2 heat flux are 6.31%, 6.30%, and 6.36%. Notably, zeroth- and first-order spatial regularization had higher accuracy than second-order spatial regularization, while zeroth-order spatial regularization was comparable to first-order. Additionally, first-order spatial regularization was more accurate in reconstructing heat flux containing sharp spatial variations. The CPU time of the predicted Case 2 heat flux is 1.71, 1.71, and 1.70 s for zeroth-, first-, and second-order spatial regularization, respectively. The corresponding values for the predicted Case 1 heat flux are 6.18, 6.15, and 6.17 s. It is noteworthy that the choice of spatial regularization order does not significantly impact the required computing time. Lastly, the minimum epred of Case 2 heat flux with zeroth-order spatial regularization is 7.96%, 6.42%, and 7.87% for time step sizes of 1/fs, 1/2fs, and 1/5fs, respectively. The accuracy of the inverse analysis displays an initial improvement followed by degradation as the time step size decreases. A recommended time step size is 1/2fs, where fs denotes the temperature-sampling rate.

Solution of a two-dimensional parabolic model problem in a degenerate angular domain

In this paper, the boundary value problem of heat conduction in a domain was considered, boundary of which changes with time, as well as there is no the problem solution domain at the initial time, that is, it degenerates into a point. To solve the problem, the method of heat potentials was used, which makes it possible to reduce it to a singular Volterra type integral equations of the second kind. The peculiarity of the obtained integral equation is that it fundamentally differs from the classical Volterra integral equations, since the Picard method is not applicable to it and the corresponding homogeneous integral equation has a nonzero solution.

Pyrit: A finite element based field simulation software written in Python

PurposeThe purpose of this paper is to present the freely available finite element simulation software Pyrit.Design/methodology/approachIn a first step, the design principles and the objective of the software project are defined. Then, the software’s structure is established: The software is organized in packages for which an overview is given. The structure is based on the typical steps of a simulation workflow, i.e., problem definition, problem-solving and post-processing. State-of-the-art software engineering principles are applied to ensure a high code quality at all times. Finally, the modeling and simulation workflow of Pyrit is demonstrated by three examples.FindingsPyrit is a field simulation software based on the finite element method written in Python to solve coupled systems of partial differential equations. It is designed as a modular software that is easily modifiable and extendable. The framework can, therefore, be adapted to various activities, i.e., research, education and industry collaboration.Research limitations/implicationsThe focus of Pyrit are static and quasistatic electromagnetic problems as well as (coupled) heat conduction problems. It allows for both time domain and frequency domain simulations.Originality/valueIn research, problem-specific modifications and direct access to the source code of simulation tools are essential. With Pyrit, the authors present a computationally efficient and platform-independent simulation software for various electromagnetic and thermal field problems.

Geometric inverse estimation for the inner wall of a furnace under transient heat conduction based on dual reciprocity boundary element method

A boundary shape estimation problem for the furnace’s inner surface is solved using the dual reciprocity boundary element method (DRBEM) and the conjugate gradient method (CGM) under transient heat conduction. The DRBEM is utilized to eliminate the drawbacks of traditional numerical methods and classical boundary element method of discretizing the entire computational domain, with only boundary discretization. The inversion results are obtained by applying the CGM to minimize the objective function, in which the sensitivity coefficients are calculated with the complex variable derivation method (CVDM), making the calculation precise and independent of step size. To verify the accuracy of the DRBEM in solving the transient heat conduction problem, the influencing factors including radial-basis function, the number of internal collocation points, and time step size are investigated. The influences of measurement time interval, future time step, initial guess, measurement error, and the number and position of measurement points on the inversion results are analyzed. Meanwhile, the effectiveness of the proposed approach is tested by numerical examples, and the inversion results show that it is stable, accurate, and efficient for identifying different and complicated unknown boundary shapes of the furnace.

A Boundary-Type Numerical Procedure to Solve Nonlinear Nonhomogeneous Backward-in-Time Heat Conduction Equations

In this paper, an explicit boundary-type numerical procedure, including a constraint-type fictitious time integration method (FTIM) and a two-point boundary solution of the Lie-group shooting method (LGSM), is constructed to tackle nonlinear nonhomogeneous backward heat conduction problems (BHCPs). Conventional methods cannot effectively overcome numerical instability to solve inverse problems that lack initial conditions and take a long time to calculate, even using different variable transformations and regularization techniques. Therefore, an explicit-type numerical procedure is developed from the FTIM and the LGSM to avoid numerical instability and numerical iterations. First, a two-point boundary solution of the LGSM is introduced into the numerical algorithm. Then, the maximum and minimum values of the initial guess value can be determined linearly from the boundary conditions at the initial and final times. Finally, an explicit-type boundary-type numerical procedure, including a boundary value solution and constraint-type FTIM, can directly avoid the numerical iterative problems of BHCPs. Several nonlinear examples are tested. Based on the numerical results shown, this boundary-type numerical procedure using a two-point solution can directly obtain an approximated solution and can achieve stable convergence to boundary conditions, even if numerical iterations occur. Furthermore, the numerical efficiency and accuracy are better than in the previous literature, even with an increased computational time span without the regularization technique.

Deep Learning Method Based on Physics-Informed Neural Network for 3D Anisotropic Steady-State Heat Conduction Problems

This paper uses the physical information neural network (PINN) model to solve a 3D anisotropic steady-state heat conduction problem based on deep learning techniques. The model embeds the problem’s governing equations and boundary conditions into the neural network and treats the neural network’s output as the numerical solution of the partial differential equation. Then, the network is trained using the Adam optimizer on the training set. The output progressively converges toward the accurate solution of the equation. In the first numerical example, we demonstrate the convergence of the PINN by discussing the effect of the neural network’s number of layers, each hidden layer’s number of neurons, the initial learning rate and decay rate, the size of the training set, the mini-batch size, the amount of training points on the boundary, and the training steps on the relative error of the numerical solution, respectively. The numerical solutions are presented for three different examples. Thus, the effectiveness of the method is verified.

A hybrid numerical method for non-linear transient heat conduction problems with temperature-dependent thermal conductivity

This paper introduces a hybrid numerical technique aimed at effectively addressing two-dimensional (2D) non-linear transient heat conduction problems, featuring temperature-dependent thermal conductivity. This scheme involves integrating the Krylov deferred correction (KDC) method in the temporal discretization and the generalized finite difference method (GFDM) in the spatial discretization. The core of the KDC method introduces a new unknown variable in the form of the first-order time derivative of temperature, leading to a non-linear equation after temporal discretization. Subsequently, the corresponding non-linear equation is solved in the spatial domain using the GFDM, supported by the Jacobian-free Newton–Krylov (JFNK) solver and fourth-order Taylor series expansion. The accuracy and stability of the hybrid approach are verified through two numerical experiments, demonstrating the strong performance of the present algorithm known as the KDC-GFDM for the simulations of the interested problems in long-time intervals.

MLS-based numerical manifold method based on IPIM for 3D transient heat conduction of FGMs

In this study, the moving least squares based numerical manifold method (MLS-NMM) is presented to solve three-dimensional (3D) transient heat conduction problems of functionally graded materials (FGMs), where the increment dimensional precise integration method (IPIM) is applied to integrate the ordinary differential equations derived from the semi-discrete form. In the 3D MLS-NMM, the influence domains of the MLS-nodes are taken as the mathematical patches which are used to construct the mathematical cover (MC), and the shape functions of the MLS-nodes are employed as the weight functions subordinate to the MC. MLS-NMM is utilized for spatial discretization of the weak form of the problem. The IPIM overcomes the shortcomings of traditional backward difference scheme dependent on the time step and greatly improves the computing efficiency. Meanwhile, a mass lumping technique is proposed for the 3D MLS-NMM. Furthermore, a number of numerical experiments on transient thermal conduction are conducted, suggesting that the 3D MLS-NMM based on IPIM and the proposed mass lumping has excellent accuracy, absolutely stable and high computing efficiency.

A Molecular Dynamics Model for Studying the Influence of High Temperatures under Laser Irradiation on Changes in a BCC Crystal Structure

The paper discusses the changes in the structures of BCC crystals subjected to high-temperature exposure. The interest in the study is explained by the processes occurring in the liquid surface layer and their subsequent impact on layer crystallization. They will further affect various physical and geometric characteristics of the material surface as a whole. The presented model helps observe the imperfections of the structure caused by the appearance of pores on the surface layers of the metal. The computational cell temperature in the designed model is distributed according to the solution of the linear problem of heat conduction. The model allows for revealing a surface layer continuity violation when the excess free volume localizes in the form of a group of spherical pores. The dimensions of such imperfections, as well as the duration of their existence, differ when modeling different laser radiation energy densities. Further research reveals the conditions for the pores to remain stable throughout the entire simulation time, as well as the relationship between the crystallographic orientation of the “solid-liquid” interphase boundary and the sizes of the formed pores” interphase boundary and the sizes of the pores formed. Keywords: bcc crystal, molecular dynamics model, interface boundary, porosity.

Physics-informed neural networks for studying heat transfer in porous media

Numerous efforts have been devoted to studying heat transfer problems in porous media. Physics-based models, numerical methods and experiments are commonly adopted to obtain the temperature and heat flux fields, along with effective thermophysical properties like effective thermal conductivity for heat conduction, which exert significant impact on analyzing the heat transfer efficiency in porous systems. Recently, using data-driven machine learning methods to predict temperature/heat flux fields and effective thermal conductivity of porous media has gained attention, demonstrating the potential to achieve higher accuracy than physics-based models while requiring less computational time than numerical methods. However, machine learning approaches are commonly restricted by the requirement for sufficient labeled training data, which can be difficult and time-consuming to acquire. In this work, we apply physics-informed neural networks to investigate heat conduction in porous media. We show that, without any labeled training data, accurate predictions for temperature/heat flux fields in porous media can be achieved. The obtained effective thermal conductivity values for an ensemble of porous media samples have an average relative error of only 2.49%. Compared with numerical calculations, a computation acceleration of 5 orders of magnitude has been achieved. Compared with data-driven machine learning methods, this method offers enhanced flexibility since no labeled data is required. Furthermore, we also illustrate that physics-informed neural networks can be easily extended to predict nonlinear heat conduction in porous media. Our work demonstrates that physics-informed neural networks are promising tools for studying heat conduction problems and can also be possibly extended to study other complex heat transfer problems in porous media.

A Difference Method for Calculating the Heat Flux on an Inaccessible Boundary in a Heat Conduction Problem

A Difference Method for Calculating the Heat Flux on an Inaccessible Boundary in a Heat Conduction Problem

Temperature Distribution in a Finite-Length Cylindrical Channel Filled with Biomass Transported by Electrically Heated Auger

The heat conduction problem for a cylindrical ring reactor of finite length, filled with biomass, which is transported at a constant speed by means of a rotating screw, is considered. The screw is assumed to be mounted on a circular shaft and is inductively heated by the Joule–Lenz effect. The surfaces of the channel and the shaft are thermally insulated. At the entrance and exit of the channel, boundary conditions of the third kind are formulated. The surface of the screw is replaced by uniformly distributed point heat sources. The problem is solved using the decomposition of the investigated temperature into Fourier–Bessel series over space variables and the integral Laplace transform over time. It is shown that the temperature has a quasi-stationary character with a short-term transient process. A numerical analysis of the spatio-temporal structure of temperature and its relationship with the thermophysical, kinematic and geometric parameters of the screw and biomass was carried out. In particular, it was found that the temperature along the reactor increases almost linearly starting from 400 K. It is shown that as in the case of an infinitely long channel, the condition of space–time resonance of the temperature field is fulfilled here.

A numerical investigation of 2D transient heat conduction problems in anisotropic FGMs with time-dependent conductivity

A class of two-dimensional (2D) transient heat conduction problems for anisotropic Functionally Graded Materials (FGMs) of space–time dependent conductivity and coefficient of variable separable forms is considered. The problems are reduced to a boundary-only integral equation which can be solved numerically using a standard Boundary Element Method (BEM). Some examples are considered to show the validity of the analysis and to examine the accuracy of the numerical method.

Inverse heat conduction problem with a nonlinear source term by a local strong form of meshless technique based on radial point interpolation method

Inverse heat conduction problem with a nonlinear source term by a local strong form of meshless technique based on radial point interpolation method

Experimental investigation of the influence of nanoparticles on droplet spreading dynamics and heat transfer during early stage cooling

With the increasing interest in improving high-performance cooling systems, research on droplet cooling requires more complex methods for enhancing heat transfer. Hence, we investigate the effect of nanoparticle presence and concentration on droplet spreading dynamics and heat transfer during the early stages of spreading on a heated surface. For this purpose, water and three water-based TiO2 nanofluid droplets (0.2, 0.5, and 1 wt%) are tested at four release heights impinging on a sapphire substrate below boiling point (80 °C). With an emissive TiAlN coating in the infrared (IR) domain, the surface temperature of the substrate is measured via a high-speed IR thermal camera. Droplet spreading is simultaneously monitored with two high-speed black/white cameras. The thermal and rheological properties are experimentally characterized for accurate results and to investigate only the nanoparticle presence in the fluid. The captured temperature fields are analyzed by solving the inverse heat conduction problem. We observe that all nanofluid droplets spread on a heated surface marginally less than water droplets. To the best of the authors’ knowledge, this study is the first to reveal that during the early stages of droplet spreading, the presence of nanoparticles and the resulting change in viscosity primarily influence heat transfer through the modification of the droplet spreading diameter, rather than changes in thermal properties.

Transient thermomechanical axisymmetric analysis of multilayer thermoplastic composite pipes

Thermoplastic composite pipe (TCP) is a promising design concept for offshore sour oil fields. The bonded structure typically consists of inner and outer homogeneous isotropic polymeric layers and intermediate transversely isotropic laminate ply layers. A theoretical transient thermomechanical analysis under axisymmetric loadings and thermal gradient subjected to convective heat transfers at the inner and outer surfaces is presented. For the heat conduction problem in a multilayer composite pipe, the transient temperature is expressed as a combination of steady state and filtered homogeneous transient solutions employing Bessel's functions. The general solution for the TCP thermomechanical axisymmetric model is also derived. An instantaneous temperature interpolation of the exact interlayer values is employed for the thermal stress analysis to overcome the difficulty of the double integration of Bessel's functions. The von Mises stress and Maximum Stress are respectively applied as failure criteria for the isotropic and laminate layers. During the warming up and cooling down, the through-thickness temperatures and stresses gradually progress from the initial towards the final steady state regime. The thermomechanical transient response is dominated by mechanically induced stresses. The transient failure indexes are below the initial and steady state values for the axisymmetric mechanical and thermal loadings during the selected warm up or cool down conditions.

Generalized entransy dissipation and its application in heat conduction optimizations with arbitrary boundary conditions

Heat conduction optimization with arbitrary boundary conditions is a challenging problem that lacks a universal optimization criterion. In the present work, the concept of generalized entransy dissipation (GED) is proposed through transforming heat conduction optimization problems with arbitrary boundaries into their homogeneous counterparts. It is demonstrated that minimizing GED leads to optimal thermal performance of heat conduction problems with arbitrary boundary conditions. In addition, GED-based continuous optimization problems are convex, guaranteeing the uniqueness and global optimality of the solution and benefitting numerical calculations. Two typical problems with complex boundary conditions are studied by applying the minimum principle of GED, and the results are compared with other optimization objectives. The numerical results show that GED achieves better thermal performance than entropy generation- (EG) and entransy dissipation- (ED) based optimizations. For the optimization of boundary average temperature under the given input heat flux of ▪, GED achieves the best result, where the optimized average temperature is ▪ and ▪ lower compared with EG and ED optimizations, respectively. In general, GED offers a reasonable and easy to implement optimization principle for heat conduction processes with arbitrary boundaries and may provide new insights for heat conduction optimization.

MLS-based numerical manifold method for steady-state three-dimensional heat conduction problems of functionally graded materials

In this study, the moving least squares based numerical manifold method (MLS-NMM) is firstly applied to discretize three-dimensional (3D) steady heat conduction problems of functionally graded materials (FGMs). In the 3D MLS-NMM, the influence domains of nodes in the moving least squares (MLS) are used as the mathematical patches to construct the mathematical cover (MC); while the shape functions of MLS-nodes as the weight functions subordinate to the MC. Compared with the traditional NMM using finite elements to form the MC, the cutting operations in generating the physical cover are unnecessary and the computation complexity is decreased significantly. Based on the Galerkin method, the discrete form of 3D heat conduction is derived. Finally, a series of numerical experiments concerning steady heat conduction problems are performed, suggesting that the proposed MLS-NMM enjoys advantages of both MLS and NMM in solving steady heat conduction.