Two-dimensional Heat Conduction Research Articles

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.

Influence of time-dependent wall temperature fluctuation on conjugate mixed convection inside a chamber with an oscillating cylinder

The current work is an attempt to examine the mixed convective heat flow in a square chamber subjected to time-dependent sinusoidal wall temperature containing a heat-conducting rotationally oscillating circular cylinder at the center. The right wall is kept at a cold temperature and the left wall is subjected to time-varying wall temperature, whereas the remaining walls are thermally insulated. A sinusoidal oscillating velocity is applied on the surface of the oscillating cylinder. This problem has practical relevance for several applications, notably solar collectors, microelectronic cooling systems, chemical reactors, radiators, and heaters. The main goal of this study is to promote heat transfer by investigating optimum system characteristics. The pressure-velocity formulation of the Navier-Stokes and heat energy equations is used to model the fluid domain inside the chamber, and heat transfer inside the cylinder is modeled by the two-dimensional heat conduction equation. Those equations are discretized using the Galerkin finite element method after converting them into non-dimensional forms. The instantaneous and time-averaged Nusselt numbers of the heated wall are examined for three different frequencies of the cylinder oscillation and sinusoidal wall temperature variation. Parametric simulation is carried out for three different cases within a mixed convection regime through a combination of Reynolds (31.62–316.23), Grashof (103–105), and Richardson (0.1–10) numbers. For each different combination, the optimum values of cylinder frequency are determined. Due to the thermal boundary layer's repeated contraction and expansion, the instantaneous Nusselt number demonstrates an oscillating pattern over time. Moreover, the maximum wall temperature frequency is found to maximize the heat transfer process.

Response spatiotemporal correlation and transient temperature field direct reconstruction for heat conduction system

In this paper, according to a spatiotemporal correlation characteristic of the temperature response of a transient heat conduction system, a response spatiotemporal correlation (RSTC) model of the temperatures between measurement space and estimation space of the system is proposed. Furthermore, a RSTC based direct reconstruction (RSTC-DR) method is developed for a transient temperature field reconstruction problem of a heat conduction system with unknown heat sources. In the offline stage, firstly, mapping eigenvectors of a heat conduction system are determined by the transfer equations without specific heat source information. Then, according to the correlation degree between the mapping eigenvectors, a finite number of representative spatial points (RPs) are extracted. Meanwhile, a fuzzy membership relationship between the mapping characteristics of the RPs and that of the temperature field is determined. Namely, effective mapping characteristic coverage of the RPs to the whole temperature field is realized. Moreover, a RSTC model between the measurement spatial points (MPs) and the RPs is obtained by the step response model of the temperature field. In the online stage, based on the RSTC model, the temperature responses of the RPs are estimated by employing the time series vector of the temperature measurements of the MPs. Next, according to the fuzzy membership relationship, the temperature field can be reconstructed in real time by synthesizing the temperature estimations of the RPs. The RSTC-DR method is employed to achieve the non-destructive reconstruction of the transient temperature field of a two-dimensional heat conduction system. The temperature field is reconstructed by the numerical experiment to verify the effectiveness of the proposed method and to discuss its adaptability. Furthermore, the practical effectiveness of the method is demonstrated by applying the existing experimental data.

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.

Numerical Analysis of Thermal Runaway in Copper-Plated Multifilament Coated Conductor

We have developed a numerical analysis model to simulate the thermal runaway in copper-plated multifilament coated conductors. We combined a circuit model to calculate current distribution in filaments and copper and a two-dimensional heat conduction model to calculate temperature distribution. The thermal runaway simulations of multifilament coated conductor with local defects were performed, and compared with the experimental results to validate the developed model. According to the simulation results, the effect of current sharing on the initiation of thermal runaway was determined.

The polygonal finite element method for solving heat conduction problems

In this paper, a general polygonal finite element method (FEM-polygon) based on the Wachspress shape function is used to solve two-dimensional heat conduction problems. The finite element variational form of the heat conduction problem is first established based on the weighted residual method. Then the formulations of the thermal stiffness matrix, thermal damping matrix, and nodal heat vector are derived using the Wachspress coordinates. The global stiffness matrix is further assembled and the well-behaved algebraic equations of the thermal system are established. Finally, intensive numerical tests are conducted to verify the super convergence in temperature and the high precision in equivalent energy and temperature gradient of the present FEM-polygon for 2D heat conduction problems. Besides, the proposed model also performs excellently for traditional triangular and quadrilateral elements.

An Analytic Solution for 2D Heat Conduction Problems with Space–Time-Dependent Dirichlet Boundary Conditions and Heat Sources

This study proposes a closed-form solution for the two-dimensional (2D) transient heat conduction in a rectangular cross-section of an infinite bar with space–time-dependent Dirichlet boundary conditions and heat sources. The main purpose of this study is to eliminate the limitations of the previous study and add heat sources to the heat conduction system. The restriction of the previous study is that the values of the boundary conditions and initial conditions at the four corners of the rectangular region should be zero. First, the boundary value problem of 2D heat conduction system is transformed into a dimensionless form. Second, the dimensionless temperature function is transformed so that the temperatures at the four endpoints of the boundary of the rectangular region become zero. Dividing the system into two one-dimensional (1D) subsystems and solving them by combining the proposed shifting function method with the eigenfunction expansion theorem, the complete solution in series form is obtained through the superposition of the subsystem solutions. Three examples are studied to illustrate the efficiency and reliability of the method. For convenience, the space–time-dependent functions used in the examples are considered separable in the space–time domain. The linear, parabolic, and sine functions are chosen as the space-dependent functions, and the sine, cosine, and exponential functions are chosen as the time-dependent functions. The solutions in the literature are used to verify the correctness of the solutions derived using the proposed method, and the results are completely consistent. The parameter influence of the time-dependent function of the boundary conditions and heat sources on the temperature variation is also investigated. The time-dependent function includes exponential type and harmonic type. For the exponential time-dependent function, a smaller decay constant of the time-dependent function leads to a greater temperature drop. For the harmonic time-dependent function, a higher frequency of the time-dependent function leads to a more frequent fluctuation of the temperature change.