Two-dimensional Heat Conduction Equation Research Articles

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

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.

Thermoelastic resonance analysis of BNNT-reinforced nanocomposite beams heated by internal thermal source

The nanocomposite structures due to excellent chemical and thermal resistant have considerable potential for use in an aggressive environment with high corrosion and temperature gradient. This study analyzes subharmonic (Ω/ωL=2 and 3) and superharmonic (Ω/ωL=1/2 and 1/3) resonance behavior of boron nitride nanotube-reinforced composite (BNNT-RC) beams under harmonic excitation and internal thermal source. The thermal analysis is carried out for determining the temperature change using the two-dimensional heat conduction equation with the assumption of an arbitrary variation in the thickness direction. In the framework of adjacent equilibrium configurations and Galerkin method, the modified couple stress Timoshenko beam equations are reduced to one nonlinear decoupled equation. Next, closed-form relations of vibration amplitude (a) and detuning parameter (σ∼) are developed for subharmonic and superharmonic excitations using the multiple-scale perturbation technique. The nonlinear frequency and resonance responses are verified by comparison with the existing ones in the literature. Finally, parametric studies are presented to discuss the effects of length scale parameter, temperature change, energy rate of thermal source, detuning parameter, and geometric ratios. The snap-through buckling behavior is observed in the curves of ωNL/ω∗-ΔT and different unstable paths exist after the occurrence of instability. The thermoelastic effect can increase the stability range of harmonic load amplitude for various values of detuning parameter and it is more pronounced for the temperature variation induced by an internal thermal source.

Analytical model for temperature prediction of hot-rolled strip based on symplectic space Hamiltonian system

The cooling process of hot-rolled strip has an important influence on the mechanical and metallurgical properties of the final product. In particular, the uneven temperature of the hot-rolled strip in the run-out table cooling after rolling will lead to the formation of residual stress and eventually cause the flatness defects. Therefore, this study focused on the temperature distribution prediction of steel strips during run-out table cooling. For this purpose, an analytical model for calculating temperature distribution of strip width and thickness is presented. It is different from the traditional analytical models, this study introduces the two-dimensional heat conduction equation with an internal heat source under the traditional Lagrange system into the symplectic space Hamiltonian system to obtain the symplectic dual equations. Meanwhile, symplectic superposition method is used to deal with complex cooling boundary conditions. The symplectic analytical solution of the two-dimensional temperature field is calculated by strict derivation without any assumptions or predetermination of the solution forms. In order to verify the accuracy of the symplectic analytical solution, a temperature−phase transformation coupled finite element model for run-out table cooling of hot strip was established. Then the accuracy of the finite element model is verified by the measured data. Finally, it has been determined that the symplectic analytical solution exhibits a high degree of consistency with the computed results obtained from the finite element model.

Arbitrary temperature gradient in thermal-postbuckling behavior of polymer nanocomposites reinforced by boron nitride nanotubes

Polymer nanocomposites reinforced by boron nitride nanotubes (BNNTs) have great potential for use in a wide variety of thermal environments. A closed-form solution for nonlinear bending and thermal-postbuckling behavior of BNNT-reinforced nanocomposite beams under arbitrary temperature gradient in the thickness and length directions is presented for the first time. The uniform and functionally graded (FG) distributions of BNNTs through the thickness of macro-or micro-sized beams are considered. For constant temperature or constant heat flux conditions at two ends of the beam, thermal analysis is carried out to obtain temperature field ΔTx,z using the two-dimensional heat conduction equation and the assumption of arbitrary variation in the z direction. The governing equations of modified couple stress Timoshenko beam theory with von Kármán geometric nonlinearity are derived for BNNT-RC beams subjected to thermomechanical loads. Applying change of variable technique, the governing equations are exactly solved to develop closed-form expression for nonlinear transverse deflection. Finally, parametric study is performed to investigate the effects of length scale parameter, aspect ratio (L/h), distribution of BNNTs, and temperature gradient on the nonlinear bending and postbuckling equilibrium paths. It is shown that the thermal-buckling behavior of clamped and cantilever beams is completely different. Under thermomechanical loading, an initial snap-through buckling behavior is observed in the cantilever beam, while there exist both stable and unstable configurations in the static equilibrium path of clamped beam. Moreover, it is found that the assumption of parabolic variation of temperature through thickness is not compatible with real situations, since the temperature field is obtained as harmonic functions with very short wavelengths in the length direction.

Study of the model of the phase transition envelope taking into account the process of thermal storage under natural draft and by air injection

The study proposed mathematical modeling of the enclosing structure of the building, which includes an energy-active panel accumulating solar radiation due to the phase transition of the heat-accumulating material. The mathematical model is based on a two-dimensional non-stationary nonlinear heat conduction equation describing the process of heat transfer in the bearing layer of the enclosing structure and the energy-active panel, taking into account the process of heat accumulation under natural and forced traction by air injection. In comparison of the calculation results, it was found. that the efficiency of the use of forced convection relative to natural was 33–44% in the question of the selection of useful heat, depending on the time (second or first day), and the value of heat accumulation showed almost the same results. Taking into account the research carried out by the authors, it is shown by calculation that enclosing structures in which a heat-accumulating material with a phase transition is used to accumulate solar energy are more energy efficient than simple heat accumulation in the carrier layer. The introduction of highly conductive inclusions into the energy-active panel and the use of forced convection instead of natural traction during heat extraction also significantly increase the energy efficiency of the structure, which can subsequently be used as an option for external fencing under appropriate climatic conditions.

A neural network-based PDE solving algorithm with high precision

The consumption of solving large-scale linear equations is one of the most critical issues in numerical computation. An innovative method is introduced in this study to solve linear equations based on deep neural networks. To achieve a high accuracy, we employ the residual network architecture and the correction iteration inspired by the classic iteration methods. By solving the one-dimensional Burgers equation and the two-dimensional heat-conduction equation, the precision and effectiveness of the proposed method have been proven. Numerical results indicate that this DNN-based technique is capable of obtaining an error of less than 10–7. Moreover, its computation time is less sensitive to the problem size than that of classic iterative methods. Consequently, the proposed method possesses a significant efficiency advantage for large-scale problems.

Optimal design of E-type coaxial thermocouples for transient heat measurements in shock tunnels

Coaxial thermocouples have been widely used for transient heat transfer measurements in high-enthalpy shock tunnels. The one-dimensional semi-infinite heat conduction theory is typically used for temperature data processing. However, lateral heat transfer occurs due to material differences between the two electrodes and the junction, causing a deviation in the heat flux measurement from the prediction results of the one-dimensional semi-infinite heat conduction theory. Thus, the lateral heat conduction effect should be investigated to improve the accuracy and reliability of heat flux measurements. In this paper, the heat transfer in E-type (Chromel-Constantan) coaxial thermocouples was analyzed by numerically solving the two-dimensional axisymmetric heat conduction equation with the Du Fort-Frankel scheme. The maximum temperature point on the surface of the coaxial thermocouples shifted to the positive electrode during the heating process. The numerical simulation indicated that the surface temperature of the coaxial thermocouples and the derived heat flux were larger than the theoretical value. The heat flux measurement error of the coaxial thermocouples can be reduced by increasing the width of the positive electrode. Hence, 13 combinations of the diameters of positive electrode and negative electrode were designed for analysis. With the increase of positive electrode diameter or decrease of negative electrode, the heat flux measurement error kept on decreasing, which can be lower than 0.5% in some cases. The results of this study can provide a reference for the design and optimization of coaxial thermocouples.

Free axisymmetric vibrations of heated non-uniform Bi-directional FGM Mindlin rings employing quadrature approaches

The present article investigates the axisymmetric vibrations of non-uniform bi-directional functionally graded rings under two-dimensional temperature distribution on the basis of first-order shear deformation theory. The linear variation of thickness along the radial direction is controlled by a taper parameter. The temperature-dependent mechanical properties are assumed to be graded in thickness as well as radial direction. The separation of variables method is applied to obtain the classical solution of two-dimensional heat conduction equation by imposing the thermal boundary conditions. The governing differential equations of thermoelastic equilibrium and axisymmetric motion together with boundary conditions are extracted from Hamilton’s principle. The quadrature technique is used to obtain the frequency equations and then the lowest three roots of these equations are calculated by MATLAB. After validating the results for the present approach, numerical results are given to analyze the effects of thermal environment, taper parameter, volume fraction index, heterogeneity and density parameters on the frequency parameter. Mode shapes for the specified rings are illustrated.

A global model for fast calculation of the thermal response factor of large-scale boreholes heat exchangers combining the FLS model, the 2D heat equation and a three-points method

In order to progress towards more energy efficient buildings, vertical ground coupled heat pumps are a promising solution. Optimisation of both the design and operation of boreholes heat exchangers is a key factor to reduce energy consumption of such systems. This requires a fast evaluation of the thermal response factor of the ground heat exchanger, particularly if it contains numerous boreholes and operates for multiple years. To overcome this challenge, this article reports a new global model combining the finite line source (FLS) model, the two-dimensional (2D) heat conduction equation, and a newly developed three-points method. The borehole field is sorted in increasing distance categories, each being simulated with varying timesteps. The 2D heat conduction equation is used to determine: 1) when the detailed calculation needs to be performed; and 2) the growth of the timestep. A three-points method avoiding double integration of the temperature profiles is proposed to evaluate the borehole wall temperature. The global model calculation time and accuracy were evaluated. The thermal response factor calculation for a square field of 26×26 boreholes for one simulated year took 4s, showing a calculation time reduction factor of around 1,000,000, and relative errors smaller than 2 % compared to the original FLS model with superposition principle. For 20 simulated years, the proposed model took only 1min. It is appropriate for various boreholes configurations. Its features such as accuracy, speed and load-independency are essential for its integration into building energy simulation tools.

Numerical simulation of flow and heat transfer performance during supercritical water injection in vertical wellbore: A parameter sensitivity analysis

Supercritical water injection is a promising technology for heavy oil thermal recovery. Predicting and regulating the thermophysical parameters of supercritical water at bottomhole are the prerequisite for achieving high recovery efficiency. In this paper, a novel numerical model was proposed to simulate wellbore flow and heat transfer of supercritical water injection. A modified correlation of frictional coefficient was developed to calculate water flow resistance near its critical point, where its properties change abruptly. The unsteady heat loss to the formation was calculated directly by solving two-dimensional unsteady heat conduction equations. They were respectively coupled in momentum and energy balance equations using an iterative scheme. This model was proved to be accurate by two oilfield cases in which the relative errors of wellbore fluid pressure and temperature are less than 1%. Then parameters sensitivity analysis of the injection pressure, temperature, mass flux and the apparent heat conductivity of insulating tube was conducted. The results indicated that the temperature variation of wellbore fluid depended on both enthalpy drop (or heat loss) and Joule-Thomson effect. An abnormal phenomenon that the fluid temperature increased with wellbore depth near the critical and pseudo-critical points was found because of the sudden increase in high heat capacity and Joule-Thomson coefficient of water. Raising the bottomhole fluid temperature was the key to enhanced oil recovery by supercritical water injection. Low apparent heat conductivity of insulating tube contributed richly to raise bottomhole fluid temperature by enlarging thermal resistance and reducing wellbore heat loss. There existed an optimal mass flux for maximizing bottomhole temperature, because when the mass flux increased, the shortened resident time within wellbore and the decreased fluid pressure favored temperature increase and decrease respectively. Selecting an injection pressure near the critical or pseudo-critical point and raising the injection temperature would increase the bottomhole temperature and reduce relative fluid heat loss.

Thermomechanical investigation of the continuous casting of ingots using the element-based Finite-Volume Method

A computational thermomechanical simulator using the Element-based Finite-Volume Method (EbFVM) has been elaborated to investigate the solidification of continuous casting ingots. The thermal behavior was evaluated using the two-dimensional transient heat conduction equation with phase change in conjunction with numerical-experimental film coefficients. In addition, the thermal-mechanical approach includes three inelastic representations, which are based at the Ramberg-Osgood plastic and Odqvist's viscoplastic models. The simulator is validated using several benchmarks tests, along with the analytical solution of the Stefan's problem. In the context of continuous casting, in order to enhance the cooling stage in the spray zone, as well as the crack formation, a new set of sprays were proposed for the investigated case. The thermomechanical study provided by EbFVM is able to provide good results for the temperature profile and thermomechanical state that could be used to address problems caused by the cooling process of continuous casting of ingots.

Numerical simulation of the heating process of an NbN film by a current pulse at low temperatures based on the two-dimensional heat-conduction equation

The simulation of the process of heat propagation in the NbN film during the passage of a current pulse through it was carried out on the basis of an inhomogeneous two-dimensional equation of heat conduction. The initial-boundary value problem for longitudinal section of the film is solved numerically, and the analysis of heat transport in the system contacts-film-substrate-thermostat is carried out. The temperature evolution of the film cross section from the onset of the pulse to temperature leveling is visualized. It is shown that the contact material (beryllium bronze) provides efficient heat removal from a superconducting film in a resistive state when a high-density current flows through it. The proposed modeling method and contact material can be used to study other metal or semiconductor films.

Effect of thermal environment on the asymmetric vibration of temperature-dependent two-dimensional functionally graded annular plate by Chebyshev polynomials

In the present paper, an accurate numerical technique is employed to analyze the effect of two-dimensional temperature variation on the free asymmetric vibrations of functionally graded (FG) annular plates using the first-order shear deformation theory. The temperature-dependent mechanical properties of the plate material are graded along the thickness and radial directions. The governing differential equations for asymmetric motion are derived from Hamilton’s principle and the exact solution for the two-dimensional heat conduction equation is obtained using the separation of variables method considering thermal boundary conditions. The Chebyshev collocation technique is adopted to compute the numerical values of fundamental frequency for different boundary conditions. The impact of volume fraction index, heterogeneity parameter, density parameter, nodal lines, radii ratio, and thermal boundary conditions on the frequencies of the plate is investigated. An excellent agreement between the results obtained by the present approach and other methods available in the literature validated the authors’ model and technique. Three-dimensional normalized mode shapes with different nodal lines are illustrated for the specified plates.

Determination of the gun barrel walls temperature distribution and its experimental validation during multiple-shots firing process

The different parameters of the combustion gases mixture were calculated allowing the calculation of the heat convection coefficient between the gases and the bore surface of the gun barrel. The heat conduction in the barrel was reduced to two axisymmetric problems. It was solved numerically using one-dimensional and two-dimensional heat conduction equations. Experiments were conducted to measure the exterior gun wall temperatures using thermal camera. The agreement between the measured and the calculated results enable the validation of the calculation models for both the gases parameters and the temperature distribution in the gun barrel.

Numerical investigations of the lateral heat transfer in coaxial thermocouples

Coaxial thermocouples have been widely used for transient heat transfer measurements in high-enthalpy shock tunnels. The one-dimensional semi-infinite heat conduction theory is typically used for temperature data processing. However, due to the material difference between the two electrodes and the junction, lateral heat transfer occurs, causing a deviation in the heat flux measurement from the prediction results of the one-dimensional semi-infinite heat conduction theory. Thus, the lateral heat transfer effect has to be investigated to improve the accuracy and reliability of heat flux measurements. In this article, the heat transfer in E-type (chromel-constantan) coaxial thermocouples was analyzed by numerically solving the two-dimensional axisymmetric heat conduction equation with the Du Fort-Frankel scheme. During the heating process, the maximum temperature point on the surface of the coaxial thermocouples moved to the positive electrode over time. The numerical simulation indicated that the surface temperature of the coaxial thermocouples and the derived heat flux were larger than the theoretical value. The heat flux measurement error of the coaxial thermocouples can be reduced by increasing the width of the positive electrode. The results of this study can provide a reference for the design and manufacture of coaxial thermocouples.