Two-dimensional Heat Transfer Problem Research Articles

Assessment of Multigrid Schemes with Fixed Patterns for a Two-Dimensional Heat Transfer Problem

Heat transfer problems and their solutions are of critical importance in almost all areas of engineering and technology, while many real-world problems are inherently three-dimensional. Simplifying them to 2D models offers practical advantages with reasonable models. With being the fundamental class of problem of heat transfer, the 2D thermal diffusion problem was selected for the study. Multigrid methods were referred to as standing out in terms of cost reduction while keeping solution accuracy. The effectiveness of multigrid methods employing fixed pattern schemes was subject to investigation. To set up numerical experimentation, an authentic code generation effort was given that implements a basic finite volume method, intergrid operations and iterative solvers. A reference case with an analytical Laplace solution was selected and properly validated by the results. A variety of multigrid schemes with fixed patterns were explored around parameters, iterations per sweep and maximum coarsening level. Results were compiled on the focal points of cost and performance. A comparison of the direct iterative methods with multigrid schemes proved the effectiveness of multigrid schemes. With the assessment of the cost and performance outcomes, it is concluded that any multigrid scheme should visit the maximum coarsest level possible while keeping a minimum number of iterations on each grid resolution.

Research on multi-heat source arrangement optimization based on equivalent heat source method and reconstructed variational autoencoder

The variational autoencoder (VAE) architecture has significant advantages in predictive image generation. This study proposes a novel RFCNN-βVAE model, which combines residual-connected fully connected neural networks with VAE to handle multi-heat source arrangements. By integrating analytical solutions, polynomial fitting, and temperature field superposition, we accurately simulated the temperature rise distribution of a single heat source. We further explored the use of multiple equivalent heat sources to replace adiabatic boundary conditions. This enables the analytical method to effectively solve the two-dimensional conjugate convective heat transfer problem, providing a reliable alternative to traditional numerical solutions. Our results show that the model achieved high predictive accuracy. By adjusting the β parameter, we balanced reconstruction accuracy and latent space generalization. During the stable phase of the multi-heat source optimization iteration, 73.4% of the results outperformed the dense dataset benchmark, indicating that the model successfully optimized heat source coordinates and minimized peak temperature rise. This validates the feasibility and effectiveness of deep learning in thermal management system design. This research lays the groundwork for optimizing complex thermal environments and contributes valuable perspectives on the effective design of systems with multiple thermal sources or for applications like multi-beam lighting equalization.

Multi-Objective Topology Optimization of Conjugate Heat Transfer Using Level Sets and Anisotropic Mesh Adaptation

This study proposes a new computational framework for the multi-objective topology optimization of conjugate heat transfer systems using a continuous adjoint approach. It relies on a monolithic solver for the coupled steady-state Navier–Stokes and heat equations, which combines finite elements stabilized by the variational multi-scale method, level set representations of the fluid–solid interfaces and immersed modeling of heterogeneous materials (fluid–solid) to ensure that the proper amount of heat is exchanged to the ambient fluid by solid objects in arbitrary geometry. At each optimization iteration, anisotropic mesh adaptation is applied in near-wall regions automatically captured by the level set. This considerably cuts the computational effort associated with calling the finite element solver, in comparison to traditional topology optimization algorithms operating on isotropic grids with a comparable refinement level. Given that we operate within the constraint of a specified number of nodes in the mesh, this allows not only to improve the accuracy of interface representation and motion but also to retain the high fidelity of the numerical solutions at the grid points just adjacent to the interface. Finally, the remeshing and resolution steps both run within a highly parallel environment, which makes it possible for the proposed algorithm to tackle large-scale problems in three dimensions with several tens of millions of state degrees of freedom. The developed solver is validated first by minimizing dissipation in a flow splitter device, for which the method delivers relevant optimal designs over a wide range of volume constraints and flow rate distributions over the multiple outlet orifices but yields better accuracy compared to reference data from literature obtained using uniform meshes (in the sense that the layouts are more smooth, and the solutions are better resolved). The scheme is then applied to a two-dimensional heat transfer problem, using bi-objective cost functionals combining flow resistance and thermal recoverable power. A comprehensive parametric study reveals a complex arrangement of optimal solutions on the Pareto front, with multiple branches of symmetric and asymmetric designs, some of them previously unreported. Finally, the algorithmic developments are substantiated with several three-dimensional numerical examples tackled under fixed weights for heat transfer and flow resistance, for which we show that the optimal layouts computed at low Reynolds number, that are intrinsically relevant to a broad range of microfluidic application, can also serve as smooth solutions to high-Reynolds-number engineering problems of practical interest.

THERMAL ANALYSIS OF BI-DIRECTIONAL FUNCTIONALLY GRADED PLATES SUBJECTED TO HEAT GENERATION

The two-dimensional steady-state heat transfer problem is analyzed under the most general boundary conditions in a heterogeneous environment. It is assumed that the internal heat generation varies arbitrarily in two directions, while the thermal conductivity coefficient of the material is bi-directly (thickness-length) graded with the Mori-Tanaka model. The linear inhomogeneous partial differential equations produced under these conditions may not be solved by conventional methods. The governing equation of the system is numerically solved with high accuracy using the 2D pseudospectral Chebyshev method. Benchmark problems are used to check the accuracy of the method. It has been observed that the effect of volumetric heat generation on the temperature distribution is more than the heat transmission coefficient. Besides, this method is simple and effective and can be easily applied to such problems.

Analysis of plane elasticity problems using the dual mesh control domain method

The dual mesh control domain method (DMCDM) proposed by Reddy combines the advantages of the finite element method (interpolation of variables) and the finite volume method (satisfaction of the global form of the governing equations). Currently, the method has been used to solve two-dimensional heat transfer, fluid and solid mechanics problems on structured meshes of linear right triangular and bilinear rectangular elements. However, such meshes are not useful from the standpoint of most practical problems which are posed on irregular domains. The present study extends the DMCDM for the analysis of plane elasticity problems involving non-rectangular domains meshed with arbitrarily shaped elements. The approach makes use of the isoparametric formulation, in which the primal mesh used for the approximation of the geometry is also used for approximating the dependent unknowns. Several boundary value problems of elasticity are solved using the DMCDM, and the results are compared with those obtained using the finite element method (FEM). The results show that the DMCDM exhibits excellent convergence and accuracy while requiring less computational cost when compared with the FEM.

Multifidelity Physics-Constrained Neural Networks With Minimax Architecture

Abstract Data sparsity is still the main challenge to apply machine learning models to solve complex scientific and engineering problems. The root cause is the “curse of dimensionality” in training these models. Training algorithms need to explore and exploit in a very high-dimensional parameter space to search the optimal parameters for complex models. In this study, a new scheme of multifidelity physics-constrained neural networks with minimax architecture is proposed to improve the data efficiency of training neural networks by incorporating physical knowledge as constraints and sampling data with various fidelities. In this new framework, fully connected neural networks with two levels of fidelities are combined to improve the prediction accuracy. The low-fidelity neural network is used to approximate the low-fidelity data, whereas the high-fidelity neural network is adopted to approximate the correlation function between the low-fidelity and high-fidelity data. To systematically search the optimal weights of various losses for reducing the training time, the Dual-Dimer algorithm is adopted to search high-order saddle points of the minimax optimization problem. The proposed framework is demonstrated with two-dimensional heat transfer, phase transition, and dendritic growth problems, which are fundamental in materials modeling. With the same set of training data, the prediction error of the multifidelity physics-constrained neural network with minimax architecture can be two orders of magnitude lower than that of the multifidelity neural network with minimax architecture.

Analysis of Two-Dimensional Heat Transfer Problem Using the Boundary Integral Equation

In this paper, we examine the problem of two-dimensional heat equations with certain initial and boundary conditions being considered. In a two-dimensional heat transport problem, the boundary integral equation technique was applied. The problem is expressed by an integral equation using the fundamental solution in Green’s identity. In this study, we transform the boundary value problem for the steady-state heat transfer problem into a boundary integral equation and drive the solution of the two-dimensional heat transfer problem using the boundary integral equation for the mixed boundary value problem by using Green’s identity and fundamental solution.

A Bayesian model to solve a two-dimensional inverse heat transfer problem of gas turbine discs

Blade tip clearances in gas turbine engines are strongly affected by heat transfer on rotating disc surfaces as it determines the temperature and in turn the radial growth of rotating discs. In experiments, disc heat transfer is often calculated from disc temperature measurements. This creates a typical ill-posed inverse problem where small uncertainties of temperature measurements can cause large uncertainties of the calculated heat fluxes. In this paper, a Bayesian method is built to calculate heat transfer on both the upstream and downstream surfaces of discs from simulated temperature measurements, reducing the ill-posedness of the problem. A Matérn covariance matrix is employed in the Gaussian prior distribution, and parametric studies are conducted to investigate the accuracy, stability and robustness of the Bayesian method. It is shown that, compared to traditional curve-fitting methods, the Bayesian method offers better accuracy, robustness, and stability. With a temperature uncertainty of 5e-3, the relative errors of the calculated Biot number from the Bayesian method are less than 10%, whereas those from the curve-fitting methods are in the range of 40–90%. Using different initial guesses of the Biot number, the variations of the relative errors are less than 0.8%. The accuracy of the Bayesian method depends on the selection of the standard deviation in the prior distribution, and the best accuracy is obtained when it is twice the maximum Biot number. This study provides guidelines for subsequent experimental research on gas turbine disc heat transfer and data processing.

Curvature-based R-Adaptive Planar NURBS Parameterization Method for Isogeometric Analysis Using Bi-Level Approach

Localized and anisotropic features extensively exist in various physical phenomena. The present work focuses on the r -adaptive parameterization technique for isogeometric analysis (IGA), which aims to acquire higher numerical accuracy while keeping the degrees of freedom constant. The principal feature is utilizing the so-called absolute principal curvature of the IGA solution surfaces to characterize numerical errors instead of posteriori error estimations, which establishes the relation between analysis results and geometric quantity. The bijectivity is a fundamental requirement for analysis-suitable parameterization. With the cooperation of a minor regularization and common line search criteria, the proposed method guarantees the bijectivity of the resulting parameterizations. The bi-level approach with two refinement levels of the same geometry is employed: a coarse level (design model) to update the parameterization and a fine level (analysis model) to perform the isogeometric simulation. Moreover, we develop several detailed algorithms for explaining the sensitivity propagation from the design model to the analysis model and analytically computing the sensitivity, which allows accurate calculation of sensitivity and enhances the robustness during a gradient-based optimization. Several examples and comparisons are presented to demonstrate the effectiveness and efficiency of the proposed method. As an application, we apply the proposed method to a two-dimensional linear heat transfer problem with a moving Gaussian heat source, which is a simplified model for the additive manufacturing application. The proposed r -adaptive technique effectively captures the thermal history of the problem. • An r-adaptivity based on absolute principal curvature is proposed for isogeometric analysis. • Bi-level approach using coarse and fine refinements is employed to balance efficiency and accuracy. • Bijectivity is guaranteed by regularization technique and common line search criteria. • Two detailed algorithms are given to explain the sensitivity propagation and analytical sensitivity analysis. • A simple additive manufacturing application demonstrates the effectiveness of our method.

Trefftz-Type Functions Applied for Subcooled Flow Boiling Heat Transfer Calculations in a Minichannel of Different Spatial Orientation

ABSTRACT The paper describes the results of subcooled flow boiling heat transfer in an asymmetrically heated group of five minichannels 1 mm deep. The heated element for the cooling fluid (FC-72) flowing in the minichannels was a foil made of Haynes-230 alloy. The time-dependent temperature distribution of the outer heated foil surface was recorded as a result of infrared thermography. The test section was oriented in several spatial orientations, that is, 30°, 45°, 60° and 75° inclinations to the horizontal plane. The objective of the numerical calculations, referring to the experimental results, was to determine the time–dependent heat transfer coefficient on the wall-fluid contact surface from the Robin boundary condition. Both the foil and the fluid temperatures were the result of solving the inverse and direct two-dimensional heat transfer problem in two neighboring domains: the heated foil and flowing fluid. The problem was solved using the Trefftz method and the FEM with Trefftz type basis functions. The heat transfer coefficient obtained by two numerical methods appears to be similar. During subcooled flow boiling, at higher minichannel inclination angles, higher values of the heated foil temperature were obtained, as well as higher values of the heat transfer coefficient.

Optimization and Solution of the Temperature Curve of Reflow

In the production process of electronic products such as process integrated circuit boards, some firmware needs to be heated and soldered to the circuit board through a reflow furnace. In this process, the temperature setting of each temperature zone and the control of the furnace passing speed play a vital role in the quality of the product. By consulting the data, it is known that the reflow heats the welding area by impinging jets, and it is considered that the heat transfer in the reflow is carried out by heat convection. Using Newton’s law of cooling, the three-dimensional heat transfer process is simplified to a two-dimensional unsteady heat transfer problem. Taking into account the lag of the center temperature of the circuit board relative to the surface temperature, it is considered that the heat transfer in the circuit board is carried out by heat conduction, and the physical parameters such as the density and heat transfer coefficient of the circuit board and the soldering area are integrated into a constant τ. The known data is obtained, and the temperature change law model is finally established.

Generalized source term multiflux method coupled with Runge-Kutta ray tracing technique for arbitrary radiative intensity of graded-index media.

The generalized source term multiflux method (GSMFM) combined with Runge-Kutta ray tracing technique is developed to calculate arbitrary directional radiative intensity of graded-index media. In this method, the finite volume method is employed to solve source terms along the curved ray path determined by the Fermat principle. Runge-Kutta ray tracing technique is adopted to obtain the ray trajectory numerical solution in graded-index media. And the GSMFM is used to solve radiative intensity to be expected. One-dimensional and two-dimensional radiative heat transfer problems are investigated to verify the performance of this method. The numerical results show that the accuracy of the GSMFM is close to that of backward Monte Carlo (BMC) method, while the efficiency of GSMFM is much higher than that of the BMC. Therefore, the GSMFM developed can be considered as a promising method to solve arbitrary radiative intensity in graded-index media.

A Comprehensive Experiment-Based Approach to Generate Stress Field and Slip Lines in Cutting Process

Abstract This study proposes a comprehensive experiment-based method to determine stress field and slip lines in metal cutting process. The chip geometry and workpiece's strain and strain rate fields are determined using an in situ imaging technique. The two-dimensional (2D) heat transfer problem for the steady-state cutting process is solved to derive the cutting temperature, and the flow stresses of work material in the main deformation zone are calculated based on the plasticity theory. Furthermore, the stress field is comprehensively determined to satisfy the stress equilibrium, friction law along the tool-chip interface, and traction-free boundary condition along the uncut chip surface. In addition, slip lines in the main deformation zone are derived according to the direction of maximum shear stress without the assumption of perfect rigid-plastic material. The proposed method is validated by comparing the cutting forces calculated based on the obtained stress field with the experimental results.

A general procedure for infrared thermography heat transfer measurements in hypersonic wind tunnels

Heat transfer measurements in hypersonic wind tunnels are particularly challenging when dealing with fully three-dimensional geometries and/or flows with high spatial temperature gradients. In the present work, a full and consistent description of a general procedure to perform infrared thermography heat transfer measurements is proposed. Radiometric and optical camera calibration, spectral directional emissivity characterization, model oscillations correction and 3D surface temperature reconstruction problems are theoretically and practically analyzed along with new solutions. A novel hybrid heat transfer problem is proposed. This technique is based on the solution of the direct problem during the wind tunnel unsteady start-up to accurately estimate the initial condition for the inverse problem solution during the quasi-steady test phase. The data reduction procedure was validated through an experimental campaign conducted in the Boeing/AFOSR Mach-6 Quiet Tunnel on a sharp cone at 0 deg angle of attack. The experimental Stanton number obtained by solving a one-dimensional heat transfer problem is in agreement with the theoretical laminar solution for a range of free-stream conditions with an average error between −1.1÷0.64% and a standard deviation of 1.55%. Finally, a two-dimensional heat transfer problem has been solved when the cone is tested at 6 deg angle of attack demonstrating the IR procedure capability to analyze multi-dimensional flows and to take into account tangential conduction effects caused by surface temperature gradients due to the presence of crossflow vortices.

FINITE ELEMENT METHOD FOR HEAT TRANSFER PHENOMENON ON A CLOSED RECTANGULAR PLATE

In the diagnosis and control of various thermal systems, the philosophy of heat fluxes, and temperatures are very crucial. Temperature as an integral property of any thermal system is understood and also, has well-developed measurement approaches. Though finite difference (FD) had been used to ascertain the distribution of temperature, however, this current article investigates the impact of finite element method (FEM) on temperature distribution in a square plate geometry to compare with finite difference approach. Most times, in industries, cold and hot fluids run through rectangular channels, even in many technical types of equipment. Hence, the distribution of temperature of the plate with different boundary conditions is studied. In this work, let’s develop a finite element method (code) for the solution of a closed squared aluminum plate in a two-dimensional (2D) mixed boundary heat transfer problem at different boundary conditions. To analyze the heat conduction problems, let’s solve the two smooth mixed boundary heat conduction problems using the finite element method and compare the temperature distribution of the plate obtained using the finite difference to that of the plate obtained using the finite element method. The temperature distribution of heat conduction in the 2D heated plate using a finite element method was used to justify the effectiveness of the heat conduction compared with the analytical and finite difference methods

Extending meshless method of approximate particular solutions (MAPS) to two-dimensional convection heat transfer problems.

Natural, forced and mixed convection heat transfer problems are solved by the meshless Method of Approximate Particular Solutions (MAPS). Particular solutions of Poisson and Stokes equations are employed to approximate temperature and velocity, respectively. The latter is used to obtained a closed expression for pressure particular solution. In both cases, the source terms are multiquadric radial basis functions which allow to obtain analytical expressions for these auxiliary problems. In order to couple momentum and energy equations, a relaxation strategy is implemented to avoid convergence problems due to the difference between successive temperature and velocity changes when solving the steady problem from an initial guess. The developed and validated numerical scheme is used to study flow and heat transfer in two two-dimensional problems: natural convection in concentric annulus between a square and a circular cylinder and non-isothermal flow past a staggered tube bundle. Numerical solutions obtained by MAPS are comparable in accuracy to solutions reported by authors who uses denser nodal distributions, showing the capability of the present method to accurately solve heat convection problems with temperature and heat flux boundary conditions as well as curve geometries and internal flow situations.

Numerical investigation on thermoelectric generators in an exhaust automotive line with aluminium foam

In the present paper a two-dimensional convective heat transfer problem in a partially filled channel with metal foam is numerically solved in steady state regime. An external thermoelectric generators (TEG) component is placed on the top surface of the channel. The numerical analyses are accomplished assuming the local thermal equilibrium (LTE) model to simulate the presence of the aluminum foam. The working fluid is exhaust gas with properties equal to the air for fixed temperature of the upper surface of the thermo-electric generator (TEG). The thermophysical properties are assumed temperature independent and the TEG component is considered as a solid with an internal energy generation. The Ansys-Fluent code is applied in order to resolve the governing equations for gas, porous media and TEG. Several mass flow rates of exhaust gas on the inlet section of the channel are considered. Different thicknesses of aluminum foam are assumed into the duct. The foam is characterized by different porosity equal to 0.90, 0.95, 0.97. Moreover, the number of pores per inch also changes and assumes the following values of 5, 20, 40. Results are showed in terms of temperature distributions, pressure drop, thermoelectric efficiency for different exhaust gas flow rates and metal foam characteristics and thicknesses. The results highlight that the use of metal foams significantly increases the heat transfer between the surface of exhaust gas tube and hot gas. Consequently, the effectiveness improves, and it increases between three-ten times with respect to the one for tube without metal foams. It is shown that the pore density does not affect the effectiveness.

Multi-Fidelity Physics-Constrained Neural Network and Its Application in Materials Modeling

Abstract Training machine learning tools such as neural networks require the availability of sizable data, which can be difficult for engineering and scientific applications where experiments or simulations are expensive. In this work, a novel multi-fidelity physics-constrained neural network is proposed to reduce the required amount of training data, where physical knowledge is applied to constrain neural networks, and multi-fidelity networks are constructed to improve training efficiency. A low-cost low-fidelity physics-constrained neural network is used as the baseline model, whereas a limited amount of data from a high-fidelity physics-constrained neural network is used to train a second neural network to predict the difference between the two models. The proposed framework is demonstrated with two-dimensional heat transfer, phase transition, and dendritic growth problems, which are fundamental in materials modeling. Physics is described by partial differential equations. With the same set of training data, the prediction error of physics-constrained neural network can be one order of magnitude lower than that of the classical artificial neural network without physical constraints. The accuracy of the prediction is comparable to those from direct numerical solutions of equations.

Combination of Isogeometric analysis and level-set method in topology optimization of heat-conduction systems

In this paper, an isogeometrical approach for topology optimization in two-dimensional heat-transfer problems, including a concentrated heat flow and uniformly distributed heat generation has been developed using the level-set method and reaction–diffusion equation for the evolution of the design variables. The NURBS surface illustrates the exact geometry representation and the level sets lead to the identification of topological variation in the NURBS surface. The level-set surface is extended based on the value of the design variables at the control points of the NURBS surface as an implicit dynamic hyper-surface in higher dimension. The same basis functions are employed to approximate the unknown temperatures and geometry modeling. Since the coordinates of the level-set grids are similar to the coordinates of the control point grids, the sensitivity analysis of the control point grids is obtained and used as a velocity field, to evolution the reaction–diffusion equation. Then, any change in the coordinates of the level-set grids is mapped over the physical domain which is parameterized using NURBS. In the optimization formulation, the heat transfer potential capacity is considered as the objective function. Also, the constraint is the maximum volume fraction of the material. Finally, several numerical examples are presented to demonstrate the optimal topology described in CAD systems and its sensitivity to various loading.

Conjugate natural heat transfer scrutiny in differentially heated cavity partitioned with a conducting solid using the lattice Boltzmann method

In the present paper, we numerically investigated the two-dimensional conjugate heat transfer problems in a unitary computational domain containing both the solid and fluid regions. The physical problem configuration consists of two adiabatic horizontal walls of finite thickness and two vertical walls; the left one is maintained at hot temperature Th and the right one is maintained at cold temperature Tc. The lattice Boltzmann method (LBM) based on the BGK model has been used to simulate laminar natural convection in the partitioned air-filled cavity with a heat-conducting solid. In the interface boundaries of the heat-conducting solid, the continuity of temperature and heat transfer is considered. A series of numerical simulation is carried out over a wide range of the Rayleigh number (Ra = 103–106), the thermal conductivity ratio kr and the solid partition thickness (δ = 1–95℅) and its horizontal position. The results show that the partition reduces the heat transfer rate in the cavity. For a centered partition (Xs = 0.5), the average Nusselt number decreases almost linearly with partition thickness for δ ≤ 0.45; however, it increases for δ ≥ 0.45 due to the confinement in the thin fluid regions. For Ra = 105, the heat transfer rate decreases with the partition position until a critical value close to 0.325 and rises slightly until Xs = 0.5. The critical position value decreases with the Ra number increase and it is close to 0.2 for Ra = 106 where Nu = 3.766. The heat transfer rate is enhanced with the increase in thermal conductivity. Correlations of the average Nusselt numbers are obtained as a function of Rayleigh number.