Two-dimensional Heat Conduction Research Articles

Analysis of forming mechanism and influencing factors of thermoacoustic plate end temperature difference

In order to explore the formation mechanism and influencing factors of the temperature difference between two ends of the plate stack, an expression of the temperature change of the stack with time was established based on the two-dimensional heat conduction equation. Based on this, the finite element model of heat transfer between a single thermoacoustic plate stack and the gas above it is established in Ansys, and the temperature of the plate stack is solved. When the sound field is constant, the variation law of the temperature of the stack with the working time and space is obtained, and the formation mechanism of the temperature difference between the two ends of the plate stack is revealed. From the calculation results, it is found that the net heat transfer between the gas and the plate stack is mainly reflected in the two ends of the plate stack, and the contribution of the air mass in the middle part is mainly the relay heat transfer. In the process of working, part of the sound work is converted into the internal energy of the air mass, which makes the gas temperature on the surface of the plate rise as a whole. The working frequency, stack length and stack thermal conductivity are taken as the influencing factors. When no load is added, the variation of the temperature of the high and low end of the stack with the working time under different working conditions is analyzed. And the theory of series between short plates is put forward to explain the formation mechanism of large temperature difference between the two ends of the plates. In order to further reduce the cooling temperature of thermoacoustic refrigerator, a new research method and exploration direction are proposed.

Numerical method research and characteristics analysis on frozen start-up process of high-temperature heat pipe

Future space exploration technology requires a long-life and reliable power source that is not reliant on solar energy. Space micro-reactors are able to meet this need, with Heat Pipe Cooled Reactors (HPR) emerging as a notable type of space micro-reactor that has attracted widespread attention in recent years. The HPR utilizes high-temperature alkali metal heat pipes for heat transfer, which presents certain complexities due to the solid state of the alkali metals working medium at room temperature. This results in a three-phase transition during the high-temperature heat pipes start-up process, which significantly impacts the heat transfer characteristics and dynamic behavior of the HPR start-up process. Consequently, thorough research is necessary in this area. Numerical simulation is a crucial tool that can effectively analyze, predict, and guide experiments. This article utilizes the Finite Volume Method (FVM) to develop a simulation code for high-temperature heat pipe frozen start-up. Various physical models are integrated to describe different components of the heat pipe: the container is represented by a two-dimensional axisymmetric heat conduction equation, the wick region utilizes a Fixed Grid Method (FGM) to depict the melting process of the medium, and the vapor channel is described through a two-dimensional axisymmetric compressible laminar flow. The wick region and vapor channel are coupled through the evaporation and condensation of the medium. For the vapor channel, numerical methods such as SIMPLE and PISO are used for solving. Adaptive time step and OpenMP acceleration are employed in the code to enhance computational efficiency. Finally, by comparing the calculated results with experimental data, the feasibility and accuracy of the code are assessed, highlighting special phenomena during the start-up process. The findings confirm that the developed code accurately predicts parameter changes during start-up, and can serve as a heat pipe analysis module for multi-physics coupling analysis of HPR.

A three-dimensional criticality safety analysis code for spent fuel solution system

The nuclear criticality safety analysis is crucial for the nuclear safety of spent fuel reprocessing plants. Thus, a set of numerical tools with high accuracy and efficiency to predict the criticality and simulate the hypothetical accidents in the reprocessing procedure is of great importance. Among them, this paper focus on the analysis of the storage tank which is used for dissolution and storage of the spent fuel. According to the characteristics of spent fuel solution system, a parallel 3D critical safety analysis tool for fissile solution system, Hydra-TD, is developed. Generally, it contains the cross-section generation model, the three-dimensional space-time neutron kinetics model, and the R-Z two-dimensional heat conduction and radiolysis gas simulation model. The verifications based on the experiments of the SILENE and TRACY facility are conducted. The results show good accuracy of the prediction of the first fission power peak, multiplication time and total fission energy, indicating the reliability of these numerical models.

Analytical solution to the two-dimensional inverse heat conduction problem in an axisymmetric cylindrical coordinate system

The inverse heat conduction problem involves estimating an unknown cooling or heating action based on internal temperature histories in a body. Such a problem is ill-posed because the estimated results are highly sensitive to input data noise. An analytical solution is developed for the two-dimensional inverse heat conduction problem in an axisymmetric cylindrical coordinate system using the Laplace transform technique. On the basis of the known temperatures at multiple measuring points distributed on two cylindrical measuring surfaces along the axial direction inside the solid, the expressions for the surface temperature and heat flux are explicitly obtained in the form of a power series of time and eigenfunctions of the space variable. A comprehensive estimation error assessment is conducted using the internal temperature distribution that is obtained using a direct heat conduction analysis under the prescribed stationary and moving boundary conditions. The constant thermal condition is well reproduced using the present inverse solution, except in the vicinity of the discontinuity point. In addition, the effective heat flux and effective heat transfer coefficient are introduced to obtain the best estimation of the transient heat transfer in the major cooling area of the rotating cylinder.

Geometry-independent spline finite element method to analyze two-dimensional heat conduction and elasticity problems

Traditional isogeometric analysis involves redundant geometric calculations and is more complex than the finite element method when imposing boundary conditions. To address this, a geometry-independent spline finite element method is proposed, which integrates the geometric construction techniques of isogeometric analysis and the element construction techniques of finite element method. Firstly, the geometrical precision element is constructed by using non-uniform rational B-spline and B-splines to describe the geometry and the physical field, where a transformation matrix is introduced to construct shape functions with interpolation properties. Secondly, calculation formats for two-dimensional heat conduction and elasticity problems are derived by parameter mapping. Finally, the feasibility and accuracy of the method are verified through several numerical examples. Unlike in isogeometric analysis, in the proposed method, geometry and physics are separated, and the shape functions have interpolation properties, which reduce redundant geometry calculations and allow the boundary conditions to be applied directly to the nodes. The numerical results indicate that the method has a higher convergence rate compared to the original isogeometric analysis.

Multidimensional freezing time using a pseudo-one-dimensional method

This study presents a method to calculate the freezing time of multidimensional objects using a pseudo-one-dimensional method. For example, the temperature of a rectangle in (x,y,t) can be simulated from the two-dimensional heat conduction equation to obtain a pseudo-one-dimensional temperature T(x,t), using the space grid Δx=Lx/(n−1) (where n=m) and Δy=ΔxLy/Lx as references. This procedure can be used to calculate the freezing time (tcalc) at a selected point, such as the center of an object. A computer program with a runtime similar to that of a one-dimensional problem has been developed for the proposed model. The freezing times (tcalc) of 212 multidimensional objects (parallelepipeds, rectangles, and finite cylinders) were then compared with the experimental freezing times (texper). The calculations yielded the following parameters for all 212 objects: minimum error Emin=−3.9%, mean error Emean=0.2%, maximum error Emax=5.0%, standard deviation σn−1=1.4%, and mean absolute error Eabs=1.1%. The freezing times (tcalc) of 100 multidimensional objects (parallelepipeds and rectangles) were then compared with the freezing times of computational experiments (computational simulation) obtained from the literature using the finite element method (tcomput). The calculations yielded the following parameters for all 100 objects: Emin=−2.8%, Emean=0.1%, Emax=3.7%, σn−1=1.4%, and Eabs=1.1%.

An n-sided polygonal cell-node-based smoothed finite element method for solving two-dimensional heat conduction problems

An n-sided polygonal cell-node-based smoothed finite element method is proposed to analyze two-dimensional heat conduction problems. Through the gradient smoothing technique, the internal integral of the polygonal element is transformed into the boundary integral based on the smoothing domain, thus reducing the continuity requirement of the trial function, and only requiring the shape function value of the smoothing domain boundary calculated using the Wachspress coordinates. Therefore, when constructing the smoothed finite element form of the heat conduction problems, coordinate mapping is avoided, which greatly improves the computational efficiency. Furthermore, a new division of the cell-based smoothing domain is put forward to improve the accuracy of the solution, where the divisions of the node-based and cell-based smoothing domains are combined in background elements to form cell-node-based smoothing domains. Extensive numerical experiments are carried out to show that the current model performs better in the estimation of the accuracy of the solution, the computational cost, the equivalent energy, and the temperature gradient while maintaining the same precision as the polygonal finite element method (PFEM).

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

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

Determination of transient heat transfer by cooling channel in high-pressure die casting using inverse method

Complex shapes of aluminum castings are typically manufactured during the short cycle process known as the high-pressure die casting (HPDC). High productivity is ensured by introducing die cooling through a system of channels, die inserts or jet coolers. Die cooling can also effectively help in reducing internal porosity in cast components. Accurate simulations based on sophisticated numerical models require accurate input data such as material properties, initial and boundary conditions. Although the heat is dominantly dissipated through die cooling, indicating the importance of knowing precise thermal boundary conditions, open literature lacks a detailed information about the spatial distribution of heat transfer coefficient. This study presents an inverse method to determine accurate heat transfer coefficients of a die insert based on temperature measurements in multiple points by 0.5 mm K-type thermocouples and a subsequent solution of the two-dimensional inverse heat conduction problem. The solver was built in the open-source CFD code OpenFOAM and the free library for nonlinear optimization NLopt. The results are presented for the commonly used 10 mm die insert with a hemispherical tip and coolant flow rates ranging from 100 l/h to 200 l/h. Heat transfer coefficients reach values well above 50 kW/m2K in the hemispherical tip, which is followed by a secondary peak and then a gradual drop to values around 1 kW/m2K further downstream.

Application of the adaptive method to determine the process noise in the extended Kalman filter to estimate the parameters of the two dimensional inverse heat transfer problem

To deal with inverse nonlinear heat transfer problems in a two-dimensional heat conduction problem, this paper presents a hybrid approach combining fuzzy logic and the extended Kalman filter (EKF). The proposed algorithm has been applied to the real-time reconstruction of the temperature field, time-varying convective heat transfer coefficient, and time-varying boundary heat flux using temperature data from a set of sensors. The effects of the covariance of the initial estimation error, the time step of data acquisition by the sensors, and the number of installed sensors on the precision and stability of the estimation results has been investigated by numerical experiments. To compare and validate the results, all these parameters were investigated using the EKF method. The results show that the adaptive fuzzy extended Kalman filter (FEKF) method for estimating temperature, convection coefficient, and heat flux at the boundaries is precise and stable. The comparison shows that the performance of the FEKF method in parameter estimation is better than the EKF method, and the highest increase in precision and stability of the results is related to the convective heat transfer coefficient.

The Two-Dimensional Conduction Heat Transfer Equation on a Square Plate: Explicit vs. Crank-Nicolson Method in MS Excel Spreadsheet

In this paper, the two-dimensional conduction heat transfer equation on a square plate is analyzed using a finite difference method. We have developed both the forward time-centered space (FTCS) and Crank-Nicolson (CN) finite difference schemes for the two-dimensional heat equation, employing Taylor series. Subsequently, these schemes were employed to solve the governing equations. The primary objective of this study is to compare the efficiency of the two methods in solving the conduction heat transfer equation. This was accomplished by implementing Spreadsheet Excel instructions. The results are presented, highlighting a comparison between the exact and approximate solutions. Furthermore, to demonstrate the convergence of the numerical schemes, we estimated the error between the actual and approximate solutions for a specific numerical problem and presented the results graphically. The data utilized in this research included the thermal conductivity of the medium of the square plate concerning width, grid, and compliance with initial and boundary conditions. The findings indicate that the Crank-Nicolson method is more accurate than the forward time-centered space method, as it approaches the exact solution more effectively. Furthermore, this study confirms that the solution and simulation of the heat transfer equation on a square plate can be accurately performed using an Excel spreadsheet as well as other numerical software.

Generalized frequency shift and attenuation in simply-supported micro/nano-beam resonators with thermoelastic dissipation

The low frequency shift, weak frequency attenuation, and large quality factor (Q) are strongly required for micro/nano-resonators with excellent operation performance. Therefore, the accurate predictions of frequency shift, frequency attenuation, and Q based on thermoelastic dissipation (QTED ) are significant when optimally designing micro/nano-resonators. In this work, employing the modified-couple-stress (MCS) and nonlocal-single-phase-lag (NSPL) models, the generalized analytical expressions of the frequency-shift ratio, frequency-attenuation ratio, and QTED are first proposed for the simply-supported micro/nano-beam resonator with two-dimensional (2D) heat conduction. In the modeling procedure, utilizing the Galerkin approach, the function of 2D fluctuation temperature is solved from the governing equation of NSPL thermoelasticity. For simplicity, the function of one-dimensional (1D) fluctuation temperature is presented directly. Various expressions of complex natural frequency are obtained from the governing equation of MCS vibration with thermoelastic coupling, and then the models of the frequency-shift ratio, frequency-attenuation ratio, and QTED are developed based on the relevant expressions of complex natural frequency. It is confirmed that the present developed models with generality can effectively degenerate into the classical models. In the simulation cases, silicon (Si) that is a representative semiconductor material is selected. Results reveal that the MCS size-dependence effect can affect the frequency shift more markedly than the NSPL and heat-conduction-dimension (HCD) effects. Moreover, for the beam with a small thickness and operated in a high-order mode, the NSPL and HCD effects will significantly determine behaviors of the frequency shift, frequency attenuation, and QTED .

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.

DETERMINATION AND ANALYSIS OF THE TEMPERATURE FIELD IN A THREE-LAYER ISOTROPIC PLATE WITH A GIVENINITIAL TEMPERATURE DISTRIBUTION

An approximate system of two-dimensional heat conduction equations for a multilayer isotropic plate is written down. Boundary conditions for a rectangular plate of finite dimensions are formulated. A general solution of the non-stationary heat conduction problem for this plate was found using integral Fourier transforms in spatial variables and Laplace transform in time.On the basis of the obtained general solutions, the solution of the thermal conductivity problem for a three-layer plate, which at the initial moment of time is heated by a temperature field linear in its thickness, which is uniformly distributed over the surface of the plate in a rectangular region, was analyzed.The numerical analysis of the temperature field was performed for a three-layer plate, the middle layer of which is made of metal, and the outer layers are made of ceramics.

Comprehensive analysis method of acquiring wall heat fluxes in rotating detonation combustors

Accurate perception of the combustor thermal environment is crucial for thermal protection design of a rotating detonation combustor (RDC). In this study, a comprehensive analysis method is established to calculate the non-uniform heat flux distribution of the RDC by utilizing the measured temperature distribution of the combustor outer wall obtained by the high-speed infrared thermal imager. Firstly, a physical model based on the geometric characteristics of the RDC is constructed and its thermal conductive process is simulated, given by different heat flux boundary conditions. The analysis results indicate that the method used in current research for computing the heat flux of RDC leads to significant discrepancies, particularly in regions of obvious axial heat flux gradients near the top of the combustor. Then the Levenberg-Marquardt (L-M) method for inversely solving non-uniform heat flux is developed based on the two-dimensional heat conduction equation, and the wall heat fluxes are inversely calculated by the L-M method based on the above numerical data. Results show that the L-M method can obtain more accurate heat flux distribution even in the zones with large heat flux gradients, considering the axial heat conduction within the combustor outer wall caused by the non-uniform heat flux. Finally, the wall heat flux distribution is analyzed coupling the L-M method together with the experimental measurements in kerosene two-phase RDC. The analyses show that the wall heat flux exhibits non-uniform distributions both circumferentially and axially, with axial non-uniformity being particularly pronounced. The highest temperature of combustor outer surface and the highest wall heat flux occurs within the region of 20 mm from the combustor head, which corresponds to nearly 14 % of the combustor length. The heat flux peak and thermal heat rate are positively correlated with the combustor equivalence ratio in the range between 0.44 and 0.64. In this study, the distribution of wall temperature field in the two-phase RDC is obtained for the first time and the wall heat flux is calculated inversely by L-M method. This work provides a more accurate method for calculating heat flux in RDC and offers references for thermal protection designs in RDCs.

An Efficient Method for Solving Two-Dimensional Partial Differential Equations with the Deep Operator Network

Partial differential equations (PDEs) usually apply for modeling complex physical phenomena in the real world, and the corresponding solution is the key to interpreting these problems. Generally, traditional solving methods suffer from inefficiency and time consumption. At the same time, the current rise in machine learning algorithms, represented by the Deep Operator Network (DeepONet), could compensate for these shortcomings and effectively predict the solutions of PDEs by learning the operators from the data. The current deep learning-based methods focus on solving one-dimensional PDEs, but the research on higher-dimensional problems is still in development. Therefore, this paper proposes an efficient scheme to predict the solution of two-dimensional PDEs with improved DeepONet. In order to construct the data needed for training, the functions are sampled from a classical function space and produce the corresponding two-dimensional data. The difference method is used to obtain the numerical solutions of the PDEs and form a point-value data set. For training the network, the matrix representing two-dimensional functions is processed to form vectors and adapt the DeepONet model perfectly. In addition, we theoretically prove that the discrete point division of the data ensures that the model loss is guaranteed to be in a small range. This method is verified for predicting the two-dimensional Poisson equation and heat conduction equation solutions through experiments. Compared with other methods, the proposed scheme is simple and effective.

Nonlinear two-dimensional transient thermoelastic analysis of functionally graded composite plates subjected to localised cooling loads

In this study, for the first time, localized thermal cooling is considered for the transient thermoelastic response of an axisymmetric plate, addressing the underexplored area of localized low-temperature thermoelasticity. Functionally graded composite plates are analyzed which consist of a mixture of stainless steel (SUS304) and low-carbon steel (AISI1020). A two-dimensional transient heat conduction equation is employed to capture localized cooling effects, featuring various adaptive time-dependent thermal boundary conditions which mirror both loading and unloading scenarios, effectively simulating real-world phenomena. Novel parameters are introduced to facilitate a comprehensive understanding of various aspects of thermal load handling and their impact on thermoelastic responses. The heat conduction equations are solved using the Generalized Differential Quadrature Method (GDQM) and the Crank-Nicolson scheme. The nonlinear governing equations, incorporating geometrical nonlinearity through the von Kármán assumption within the First Shear Deformation Theory (FSDT) framework, are solved using the GDQM and the Picard's technique. The model is validated using Finite Element Method (FEM) through Abaqus. A comprehensive analysis is provided that considers influence of the ratio of thermally affected and unaffected plate surface, thermal load magnitude, rapidity of thermal loading and unloading, duration of the cooling load, geometrical nonlinearity, and temperature dependence of material. The study shows that the maximum von Mises stress within the structure remains consistent regardless of the duration of the cooling load, as long as the rate at which the cooling load is applied stays the same. Additionally, the findings reveal that even minor localized thermal loads can produce substantial stress, especially at the intersection of thermally affected and non-affected zones on the exposed surface in some situations.

Simultaneous determination of the space-dependent source and initial value for a two-dimensional heat conduction equation

Simultaneous determination of the space-dependent source and initial value for a two-dimensional heat conduction equation

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.

Finite Difference Method for Solving the Two-Dimensional Laplace Equation in Curvilinear Coordinates

Abstract This paper presents a numerical method for solving the two-dimensional Laplace equation in curvilinear coordinates using a finite difference scheme. Laplace equation is a differential equation used to describe various physical phenomena studied within different branches of engineering. The two-dimensional Laplace equation has analytic solutions only for simple shaped domains and simple boundary conditions (first and second kind). Even so, the results are usually complicated to interpret and use. The present paper proposes a method to solve the equation for domains bounded by complex inner and outer curves. The boundary curves are defined by a set of points from which the equations of curves are approximated by using Fourier series. The domain is generated by blending the two curves using a blending function. The results are a set of continuous parametric equations which are defining the domain in angle and radius variables. Following a method presented in the literature, the Laplace equation is transformed from Cartesian coordinates to curvilinear coordinates, as the Fourier series is easily differentiable. Next, the new Laplace equation is approximated using the finite difference method and a MATLAB code is developed to solve the system of linear equations using successive over-relaxation method (SOR). The presented method is used to simulate steady two-dimensional heat conduction in a polygonal shape domain, with uniform and non-uniform Dirichlet boundary conditions. Results are analyzed in order to determine the model’s precision, speed and convergence criteria.