Two-dimensional Heat Conduction Research Articles

Generalized thermoelastic dissipation in micro/nano-beams with two-dimensional heat conduction

The precise prediction of quality factor based on thermoelastic dissipation (TED) is of great importance for the optimal design of micro/nano-resonators. In this work, employing the nonlocal-dual-phase-lag (NDPL) model and the modified-couple-stress (MCS) model, analytical TED expressions are developed for the first time for micro/nano-beam resonators with two-dimensional (2D) heat conduction along the thickness and length directions. Initially, the governing equation of NDPL coupled thermoelasticity is established, and then the solutions of 2D fluctuation temperature are obtained by utilizing the Galerkin approach. Subsequently, the natural frequency involving the MCS size-dependence length-scale parameter is achieved based on the governing equation of vibration. Finally, analytical generalized TED models are attained according to the energy-definition approach. Three representative types of constraint boundary conditions for the beam resonator, namely, clamped-clamped, clamped-free, and simply supported, are considered in this study. And the present developed TED models are confirmed to possess perfect generality and excellent convergence performance. Moreover, an expression of equivalent thermal relaxation time in the 2D case is introduced and expressed in terms of the length- and thickness-directional thermal relaxation times. Two typical materials, silicon and gold, are selected in the simulation part. The exploration of fluctuation temperature, TED spectra, and thermal relaxation times considering multiple physical effects is conducted as an emphasis. Results reveal that the effect of heat-conduction dimension is crucial for micro/nano-beams with a small ratio of length to thickness, while the significant NDPL effect is dependent upon the ultra-high vibration frequency. Additionally, the MCS size-dependence effect can enhance the natural frequency and improve the quality factor of micro/nano-beam resonators.

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.

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.

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.

Analytical solution of two-dimensional unsteady heat conduction in multi-layered cylindrical sectors applied to thermal plug and abandonment of oil wells

In this work, a hybrid analytical and numerical solution for transient heat conduction across a composite cylindrical sector is presented. A two-dimensional domain consisting of a multi-layer circular sector of angle φ was investigated (φ<2π). The Separation of Variables Method (SVM) was applied to solve the partial differential equation with non-homogeneous boundary conditions of the first, second and third kinds prescribed in the radial direction. Homogeneous boundary conditions of first and second kinds were arbitrated in the angular direction. A spatial time-independent source term gi(r,θ) was considered. The radial eigenvalues problem for the (r,θ) domain returns only real quantities and depends implicitly on the angular eigenvalues. Results for time dependent temperatures using the Separation of Variables Method were compared with numerical results, showing good accuracy. A second set of results was developed to investigate boundary conditions, material properties and the thermal source power required to rise temperature levels (mainly around the mid-angle φ/2) high enough to promote melting of certain layers of materials. These results might be useful for investigating a novel technology for the decommissioning of oil wells using thermal sources, often referred to in the literature as Thermal Plug and Abandonment (TP&A).

Prediction identification method of time-varying internal heat source based on inverse heat transfer problem

Purpose This paper aims to discuss the inverse problems that arise in various practical heat transfer processes. The purpose of this paper is to provide an identification method for predicting the internal boundary conditions for thermal analysis of mechanical structure. A few examples of heat transfer systems are given to illustrate the applicability of the method and the challenges that must be addressed in solving the inverse problem. Design/methodology/approach In this paper, the thermal network method and the finite difference method are used to model the two-dimensional heat conduction inverse problem of the tube structure, and the heat balance equation is arranged into an explicit form for heat load prediction. To solve the matrix ill-conditioned problem in the process of solving the inverse problem, a Tikhonov regularization parameter selection method based on the inverse computation-contrast-adjustment-approach was proposed. Findings The applicability of the proposed method is illustrated by numerical examples for different dynamically varying heat source functions. It is proved that the method can predict dynamic heat source with different complexity. Practical implications The modeling calculation method described in this paper can be used to predict the boundary conditions for the inner wall of the heat transfer tube, where the temperature sensor cannot be placed. Originality/value This paper presents a general method for the direct prediction of heat sources or boundary conditions in mechanical structure. It can directly obtain the time-varying heat flux load and thtemperature field of the machine structure.

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.

An elliptical inhomogeneity under nonuniform heat flux

We use complex variable techniques to study the decoupled two-dimensional steady-state heat conduction and thermoelastic problems associated with an elliptical elastic inhomogeneity perfectly bonded to an infinite matrix subjected to a nonuniform heat flux at infinity. Specifically, the nonuniform remote heat flux takes the form of a linear distribution. It is found that the internal temperature and thermal stresses inside the elliptical inhomogeneity are quadratic functions of the two in-plane coordinates. Explicit closed-form expressions of the analytic functions characterizing the temperature and thermoelastic field in the matrix are derived.

A comparative numerical study of a semi-infinite heat conductor subject to double strip heating under non-Fourier models

The present work considers a two-dimensional (2D) heat conduction problem in the semi-infinite domain based on the classical Fourier model and other non-Fourier models, e.g., the Maxwell–Cattaneo–Vernotte (MCV) equation, parabolic, hyperbolic, and modified hyperbolic dual-phase-lag (DPL) equations. Using the integral transform technique, Laplace, and Fourier transforms, we provide a solution of the problem (Green’s function) in Laplace domain. The thermal double-strip problem, allowing the wave interference within the heat conductor, is considered. A numerical technique, based on the Durbin series for inverting Laplace transform and the trapezoidal rule for calculating an integral form of the solution in the double-strip case, is adopted to recover the solution in the physical domain. Finally, discussions for different non-Fourier heat transfer situations are presented. We compare among the speeds of hyperbolic heat transfer models and shed light on the concepts of flux-precedence and temperature-gradient-precedence, hallmarks of the lagging response idea. Otherwise, we emphasize the existence of a relationship between the waves speed and the time instant of interference onset, underlying the five employed heat transfer models.

Comparison of uniform and piecewise-uniform heatings when estimating thermal properties of high-conductivity materials

A d-optimum criterion is applied to the three-layer apparatus used for simultaneously estimating thermal properties (for example, thermal conductivity and effusivity) through the plane source method. The objective is to perform a comparison between uniform heating and piecewise-uniform heating of high-conductivity solid samples. In particular, the latter case is modeled through a two-dimensional heat conduction problem in which a rectangular plate (i.e. the sample) is partially heated at the front boundary through a surface heat flux, while all the other boundaries are kept insulated. The optimal experiment is designed for different set-ups of the experimental apparatus (width of the heated region, number of sensors and their locations). The convergence and the computational efficiency of the estimation iterative procedure are the terms of the comparison, as well as the expected standard deviations of thermal conductivity and effusivity. The results indicate that the use of a piecewise-uniform heating is not completely beneficial for isotropic materials. In fact, if on one hand it may offer standard deviations reduced up to about 40%, on the other hand it would require an experiment about six times longer than that required by a uniform heating to ensure a good convergence of the estimation iterative procedure. Therefore, a major computational effort is required.

A self-scaling sequential quasi-Newton method for estimating the heat transfer coefficient distribution in the air jet impingement

Accurate knowledge of the heat transfer coefficient (HTC) distribution is of great significance to the control and optimization of the heat transfer system in the air jet impingement. For this purpose, the estimation of the temporally-spatially varying HTC in the air jet impingement is studied based on the inverse heat conduction problem. The transient two-dimensional heat conduction model of the target disc under a single cooling air jet is established and utilized as the direct problem. A self-scaling sequential quasi-Newton method (SS-SQNM) is developed for accurately and efficiently solving the inverse problem. As a modification of the sequential quasi-Newton method (SQNM), the SS-SQNM adopts the self-scaling updating equation to accelerate the single-step convergence rate during iteration, and thus attains higher inversion efficiency than the SQNM. A series of numerical tests are carried out to investigate the performance of the SS-SQNM and the effects of the regularization parameters on inverse results. The results reveal that the SS-SQNM can give stable and accurate estimates of the HTC when regularization parameters are appropriately selected through the discrepancy principle. The mean absolute percentage error of the estimated HTC distribution is generally less than 10.2% in the presence of measurement noises. Additionally, the iteration number required by the SS-SQNM for inversion is only about 7.81%∼27.78% of that of the SQNM under the same conditions, which saves lots of computational time for the SS-SQNM. Therefore, the SS-SQNM developed in this study is a promising inverse algorithm, which can be employed to estimate the HTC distribution in the air jet impingement with reasonable accuracy and efficiency.

A polygonal element differential method for solving two-dimensional transient nonlinear heat conduction problems

A novel polygonal element differential method (PEDM) is presented for solving two-dimensional nonlinear transient heat conduction problems for the first time. New shape functions as well as their derivatives with respect to isoparametric coordinates are derived to treat the polygonal elements with an internal node. System equations of the PBEM are formulated in terms of the governing equation and heat flux equilibrium condition, in which, a finite difference scheme is executed for calculating the transient term. Then, the nonlinearity system equations are dealt with by the Newton iterative method. Finally, examples with different structural complexities are designed to examine the property of the proposed method. The results show that the PEDM can effectively solve general two-dimensional transient nonlinear heat conduction problems with excellent accuracy and efficiency.

The MLS-based numerical manifold method for nonlinear transient heat conduction problems in functionally graded materials

In this study, we aim to numerically solve two-dimensional nonlinear transient heat conduction problems of functionally gradient materials (FGMs), where the governing equation is a quasilinear partial differential equation of second order. For this purpose, the weak form of the initial boundary value problem is first established. Then influence domains of nodes in the moving least squares (MLS) instead of finite elements are used as the mathematical patches to construct the mathematical cover of the numerical manifold method (NMM); while the shape functions of MLS-nodes as the weight functions subordinate to the mathematical cover, leading to the moving least squares based numerical manifold method (MLS-NMM). With the weak form, the spatial discretization is carried out using MLS-NMM, whereas temporal discretization follows the Euler backward difference. Finally, a series of numerical experiments are conducted concerning transient heat conduction problems of FGMs with heat source and heat convection, suggesting that the proposed MLS-NMM enjoys advantages of both MLS and NMM in solving nonlinear transient heat conduction of FGMs.

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.

A posteriori regularization method for the two-dimensional inverse heat conduction problem

Abstract In this article, we consider a two-dimensional inverse heat conduction problem that determines the surface temperature distribution from measured data at the fixed location. This problem is severely ill-posed, i.e., the solution does not depend continuously on the data. A quasi-boundary value regularization method in conjunction with the a posteriori parameter choice strategy is proposed to solve the problem. A Hölder-type error estimate between the approximate solution and its exact solution is also given. The error estimate shows that the regularized solution is dependent continuously on the data.

Size determination of interior defects by reconstruction of subsurface virtual heat flux for step heating thermography

Step heating thermography is a non-destructive testing (NDT) technique that inspects interior defects by observing surface temperature rises during a long pulse heat stimulation. Due to the heat diffusion induced blurring of defect shapes, it is a challenging task to accurately determine the sizes of deep and small defects. This work focuses on estimating the size of a subsurface defect by an inversion of temperature images for the step heating thermography, to overcome the information loss during the propagation of thermal waves. We assume a space- and time-dependent virtual heat flux on the defect surface, which is reconstructed from surface temperature measurements by solving a two-dimensional inverse heat conduction problem. The local future time concept is combined with the Tikhonov's regularization to stabilize the inverse solution. A prudent choice of the regularization parameter is carried out through the L-curve. The feasibility of the proposed approach is assessed by numerical simulations and experiments. It is shown that the inversion brings an improvement in defect sizing accuracy for defects with low width-to-depth ratios and an efficient suppression of measurement noises.

Stress intensity factor for a circumferential crack in a coke drum under two-dimensional transient temperature distribution

Coke drums are subjected to extreme temperature cycling in service, leading to circumferential cracking in the drum wall. For a deeper understanding in the cracking mechanism, stress intensity factor (SIF) for a circumferential crack in a coke drum was investigated in this paper. Based on two-dimensional heat conduction theory and classical thin shell theory, analytical solutions for the transient temperature distribution, displacement, and thermal stress in the drum wall during hot feed filling and water quench steps were determined. The weight function method (WFM) was applied to investigate the mode I SIFs based on the analytical thermal stress solutions. Numerical simulations using finite element method (FEM) were also performed to validate the analytical solutions for both the thermos-elastic field and the SIFs. The effects of shell geometry, operating parameters and material properties on the SIF during the water quench step were discussed. It is shown that the SIF increases significantly as crack depth increases. A greater shell thickness or lower diameter to thickness ratio results in lower SIFs. Lower rising rate and higher temperature of quench water can reduce the SIF. The SIF increases as the thermal conductivity and thermal expansion coefficient of the drum wall materials increase. Finally, the SIFs for Cr–Mo steel and carbon steel coke drums are also discussed.

Incorporating two-dimensional conduction modeling techniques into an energy simulation program: The EnergyPlus radiant system example

Due to the inherent complexity of multi-dimensional solutions for transient conduction in a multilayered building element, most heat balance-based algorithms assume that conduction occurs only in one dimension—through the building element between the inside and outside environments. Usually, since the length and width of the elements are much greater than the thickness of the element, this is a valid assumption. However, there are some cases such as hydronic radiant slabs where this assumption may not lead to an accurate estimate of the energy consumption of the system. In a hydronic radiant slab, water tubes are embedded in a slab, and hot or chilled water is circulated to meet any heating or cooling loads. The energy consumption of such a system will depend highly on the temperature of the water being sent to the slab (i.e., produced by the boiler or chiller). The temperature of the water that must be sent to the slab will depend on the spacing of the tubes, the multilayer construction of the building element, and the boundary conditions to which the construction is exposed. Thus, to accurately predict the energy consumption of such a system, a transient two-dimensional heat conduction model must be integrated with a heat balance-based simulation. This paper demonstrates the potential loss of accuracy associated with the one-dimensional radiant system models by providing a comparison of one- and two-dimensional solutions using EnergyPlus.

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.