Transient Heat Conduction Research Articles

Constitutive modeling of thermo-chemical decomposition and thermo-mechanical deformation, coupled with transient heat conduction, in ablative matrix composite

Several thermal protection systems employ sacrificial composite layer that undergoes thermo-chemical decomposition in high-temperature environment. This results in the pyrolysis gas formation (endothermic reaction) that gets trapped inside the voids generated in the ablative matrix phase. These trapped gases apply pore pressure on the structure, along with the mechanical loading, thus significantly influencing the structure failure. A novel thermo-chemical (TC) decomposition and thermo-mechanical (TM) deformation-based coupled multi-physics formulation, applicable to ablative composite systems, is thus presented. A novel shrinkage expression, due to ablative matrix decomposition, is derived. The TC + TM coupled formulation is converted to stress update process, and its results are validated against the available experimental data. The proposed formulation is also converted to boundary value problem employing non-linear finite element framework (NL-FEM). The Jacobian matrices, for one- and two-dimensional cases, are systematically derived, and the proposed NL-FEM formulation is successfully verified against several benchmark problems.The transient heat conduction equation is finally coupled with the proposed TC + TM formulation (one-way coupling) thus enabling the analysis of more realistic situations where the constant heating rate assumption is not valid. The coupled formulation is finally implemented for several test cases and it is demonstrated that, it has a significant influence on pore pressure and porosity evolution (through pore volumetric strain) within the ablative matrix phase.

Investigation into the mechanism of ignition of magnesium alloy plates by high-temperature molten droplet

The ignition and spread of magnesium alloy fires are mainly attributed to the ignition of high-temperature molten metal droplets, but the underlying ignition mechanism remains unclear. This study addresses this knowledge gap by employing metallic aluminum and copper particles heated to temperatures over 1200 °C. Systematically, these heated particles were released at a fixed rate onto a magnesium alloy plate located inside a controlled heating furnace that maintained a constant temperature of 500 °C. Experimental results show that whether there are irregular pores and cracks in the resulting molten product can be used as a criterion for judging whether the magnesium alloy sample ignites. Furthermore, it is worth noting that molten copper droplets of the same size exhibit a higher heat-carrying capacity than their aluminum counterparts when both are heated to the same temperature. In addition, this study expanded the research scope through numerical simulation and analysis, using the PATO (Porous-Material Analysis Toolbox) solver to clarify the transient heat conduction process between the high-temperature droplet and the magnesium alloy plate. Simulations show that heat transfer upon droplet impact is primarily longitudinal, directed toward the base of the material. This results in a significantly higher temperature in the center of the affected magnesium alloy plate than at its periphery. In summary, this study helps to improve the understanding of the dangers of high-temperature molten droplets, and it is of great significance to investigate fire accidents caused by high-temperature molten droplets.

Analytical modelling of transient conduction heat transfer in tubes for industrial applications

Analytical modelling of transient conduction heat transfer in tubes for industrial applications

Variationally consistent magnetodynamic computational homogenization of particulate composites using an incremental potential

If composites made of conductive particles embedded in a nonconductive matrix are subject to highly dynamic magnetization processes, then microscopic eddy currents may lead to high magnetodynamic losses and heating of the material. Less obvious, the microscopic eddy currents also induce a dynamic macroscopic magnetization in accordance with Lenz’ law. Conventional homogenization theories based on volume averaging of the magnetic field strength disregard this effect and thus fail to properly predict the dynamic material response. In this work, the variationally consistent homogenization framework by Larsson et al. (2010), originally developed for transient heat conduction, is transferred to the aforementioned particulate composites, in order to properly reproduce this dynamic effect. It turns out that this approach predicts the macroscopic magnetization to be the volume averaged sum of the particles’ (conventional) magnetization and a non-standard dynamic magnetization due to microscopic eddy currents. Some first numerical examples illustrate the improved predictability of the variationally consistent homogenization method with respect to experimental complex permeability data. The resulting theory is shown to exhibit a two-scale incremental potential structure, which is exploited in several ways.

The Method of Calculating the Heat Transfer Coefficient in the Heliosystems with Laminar and Transient Modes of Heat Carrier Flow Movement Structured Into Parts

In this study, a new method of choosing classical empirical equations for calculating heat transfer coefficients in the tubes of a shell-and-tube heat exchanger in the transient mode is proposed. This method is based on the fact that the flow is structured into a laminar boundary layer (LBL) zone and a turbulized part, and the heat transfer coefficient is calculated through the transient and turbulent heat conductivity, as well as the average thickness of the LBL and, accordingly, the average thickness of the rest of the coolant flow. At the same time, the key point of this method is the condition that the transient thermal conductivity of the LBL should be lower than the thermal conductivity of the turbulized part. If this condition is not fulfilled, it is concluded that the corresponding classical empirical equation is not suitable for calculating the heat transfer coefficient. A 45% aqueous solution of propylene glycol was taken as a model liquid, which can be widely used in solar collectors, in particular with nanofillers. This coolant is interesting because at a constant speed of V = 0.93 m/s, and the linear size (diameter) of the "live section" of the flow D = 0.021 m in the temperature range of 243–273 K it moves in the laminar mode, in the temperature range of 283–323 K — in transient mode and 333–353 K — in turbulent mode. A new formula is proposed for calculating the coefficient of turbulence of the coolant flow a, the numerical values of which are experimentally found in literary sources only for the air coolant.

T-phPINN: Physics-informed neural networks for solving 2D non-Fourier heat conduction equations

Non-Fourier heat conduction plays a dominant role in many extreme transient heat conduction processes, such as laser pulses and heat transfer in biological systems, but the heat wave effect makes it difficult to solve the temperature field accurately and quickly. In order to solve this problem, the first order time derivative enhanced parallel hard constraints physics-informed neural networks (T-phPINN) is proposed. T-phPINN comprises two subnetworks and incorporates a first order time derivative to capture sharp temperature changes. Two numerical cases show that the minimum relative error of T-phPINN is 0.001 % and 0.015 %, which is 1.04 % and 12.30 % of the error of conventional PINN respectively, proving the accuracy of our architecture. A transfer learning framework is established for scenarios of different parameters, the training only requires 1/6 iterations of the basic model, and close accuracy is obtained. The computational cost of T-phPINN is evaluated using the finite element method as the baseline. For the two cases, the single calculation time is 33.43 % and 51.50 % of the baseline, while the multiple calculation time under the acceleration of transfer learning is 11.59 % and 17.75 % of the baseline. This study will be helpful for solving large-scale non-Fourier heat conduction equations precisely and expeditiously.

Analysis of transient heat conduction in tubes under convective boundary conditions

AbstractIn this work, six analytical solutions are given to estimate the energy exchange by transient conduction in pipes with convection conditions. The developed models were adjusted for an interval from 0.2 to 0.8, dimensionless Fourier (Fo) and Biot (Bi) numbers, from 0.05 to 50 and 0.005 to 50, respectively. In each case, 616 tests of temperature distributions were computed using the Heisler approximate method (HAM) and the exact models proposed, for different combinations of . For the comparison made between the analytical solutions and the HAM, 3696 tests were used, detecting that the HAM correlates with the analytical method with an average deviation of and for 71.2%, and 90.4% of the different values of combinations evaluated. The best fit was found for Case 5, with a mean deviation of and for 81.1% and 92.3% of the data used, respectively, while the weaker fit was detected for Case 2, with a mean deviation of and for 67.9% and 88.2% of the available tests.

Accelerating Transient Aerothermal Simulations of the BOLT II Flight Experiment

Aircraft with short flight durations may not reach thermal steady state due to limited time for aeroheating to conduct throughout their internal structure. Consequently, predictions of the solid temperature field require time-accurate simulation of the transient heat conduction. In this work, the use of an explicit super-time-stepping scheme to accelerate the time-accurate solution of the heat equation has been explored. The experience with this technique is documented here because, to the authors’ knowledge, it is not widely used in the hypersonics simulation community. The performance of the super-time-stepping scheme was investigated by comparison to a standard explicit scheme and an implicit time-differencing scheme. Both the super-time-stepping scheme and the implicit time-differencing scheme demonstrate significant speedup over the standard explicit scheme, with comparable wall-clock performance. The solid solver has been coupled with an existing open-source fluid solver to perform conjugate heat transfer simulations. The efficacy of the developed solver in simulating hypersonic aerothermal problems is demonstrated through verification and validation studies. By applying the coupled solver to the aerothermal analysis of the BOLT II flight experiment, the effectiveness of the super-time-stepping scheme in mitigating the computational expense associated with transient conjugate heat transfer simulations is demonstrated.

Numerical Investigation of Melting in Pressurized Water Reactor Fuel Rod Considering Operational Parameters of the Core

The meltdown of a fuel rod is a severe accident resulting from the overheating of the reactor core. In the present study, a numerical investigation of this process, focusing on the loss of coolant has been conducted. The objective of this study is to conduct a numerical simulation of transient heat conduction and melting at various points within a typical pressurized water reactor fuel rod. In this analysis, heat conduction in the radial direction of a fuel rod, including the UO2 fuel pellet, the gap, and the zircaloy cladding, is investigated. The FRAPCON steady-state code is employed to calculate the operational parameters of the fuel rod. The calculated parameters, such as coolant and fuel temperatures, fission gas fraction, gap heat transfer coefficient, and burnup, are utilized to evaluate and compare the melting phenomena at different time intervals. In the investigation of the phase change in various parts of the pellet and fuel rod, the explicit finite difference (FD) method is utilized with enthalpy instead of temperature-dependent equations. Finally, the temperature history, phase change, and melting map at different points along the radial and axial directions of the fuel rod during coolant loss and heat transfer coefficient reduction are evaluated based on various operating parameters of the core. To enhance the quality of the results, an uncertainty analysis of effective parameters is conducted. According to this analysis, the heat transfer coefficient of the coolant under accident conditions (0.2 ± 5% kWm−2K−1) and the thermal conductivity of the fuel have the most significant impact on the temperature history and melting process. Highlights include the following: 1. The meltdown of a nuclear fuel rod is analyzed under a loss-of-coolant accident. 2. The enthalpy formula is discretized by the explicit FD numerical method. 3. Effective parameters in melting, such as coolant temperature, burnup, and gap heat transfer coefficient, are obtained by FRAPCON. 4. The temperature history, phase changes, and melting map of various radial points within the fuel pellet and cladding along the axial direction of the fuel rod are determined.

Quantum coupling and self-heating impacts on threshold voltages of GaN metal–insulator-semiconductor high-electron-mobility transistors

A transient heat conduction equation has been proposed to describe the cooling process of a hot two-dimensional electron gas layer in a GaN-based transistor. This equation serves as the basis for a physical analytical model that explains the impacts of hot electrons in the hot two-dimensional electron gas layer via quantum coupling on the threshold voltage of GaN-based transistors. The temperature of the two-dimensional electron gas layer decreases to ambient temperature as time increases, and the threshold voltage shifts caused by such a temperature will recover with time. The proposed model accurately reflects the observed recovery of threshold voltage over time in experiments. At the same time, it offers insight into how the gate voltage, the source-drain voltage, and the ambient temperature can shift the threshold voltage in these transistors. The simplicity and analytic nature of this proposed model not only provide a physical origin of the threshold voltage instability in GaN-based devices but also offer the possibility for improving device reliability by selecting appropriate device physical parameters.

Timestep-dependent element interpolation functions in the method of matched sections on the example of heat conduction problem

The paper is devoted to further elaboration of the method of matched sections as a new technique within the finite element method. Like FEM it supposes that: a) the complex domain is represented as a mesh of nonintersecting simple elements; b) algebraic relations between the main parameters of an element are established from the governing differential equations; c) all relationships from all elements are assembled into one global matrix. On the other hand, it has two distinct features. The first one is that relations between kinematic and inner force parameters (called Connection equations) are derived from the approximate analytical solution of the governing equations rather than by the application of minimization techniques. The second one consists in that the conjugation between elements is provided between the adjacent sides (sections) rather than in the nodes of the elements. In application to the transient 2D heat conduction, it is assumed that for each small rectangular element, the 2D problem can be considered as the combination of two 1D problems – one is x-dependent, and another is y-dependent. Each problem is characterized by two functions – the temperature, T, and heat flux Q. In practical realization for rectangular finite elements, the method is reduced to the determination of eight unknowns for each element – two unknowns on each side, which are related by the connection equations, and the requirement of the temperature continuity at the center of the element. Another salient feature of the paper is an implementation of the original implicit time integration scheme, where the time step becomes the parameter of the element interpolation function within the element, i.e. it determines the behavior of the connection equations. This method was initially proposed by the first author for several 1D problems, and here for the first time, it is applied to 2D problems. The number of tests for rectangular plates exhibits the remarkable properties of the proposed time integration scheme concerning stability, accuracy, and absence of any restrictions as to increasing the time step.

Impact of radiative cooling on the energy performance of courtyards in Mediterranean climate

Radiative cooling has proven to be a useful tool to address the problems of lack of comfort and excessive energy consumption in situations of high temperatures, overheating and heat waves. Likewise, incorporating courtyards in warm climate zones has been found to be highly beneficial in addressing similar challenges. Hence, there is interest in analyzing the combined effects of both: radiative cooling and courtyards. This paper presents an analysis of the impact of the application of radiative cooling on a courtyard using a comprehensive simulation approach that includes a CFD model for the thermodynamic airflow in the adjacent roofs and inside the courtyard, equations for the transient heat conduction through roofs, walls and courtyard slabs, and a hybrid raytracing-radiosity model for the evaluation of the solar radiation reaching the building surfaces and its reflections, both of specular and diffuse origin, and for the calculation of the thermal radiation exchange, especially with the sky. The results show that in the hot season, the courtyard with radiative cooling always provides lower temperatures than the initial courtyard does, with a temperature range of 18.33 °C to 33.78 °C, compared to a range of 19.32 °C to 38.00 °C in the initial courtyard, and producing a greater difference with outdoor temperatures that can reach 12 °C versus 8 °C for the reference case. In addition, it was found that the courtyard with radiative cooling is able to significantly reduce the observed nighttime overheating by providing lower temperatures than the outdoor temperatures in the 50% of the nights studied. It was also found that the thermal loads to achieve indoor thermal comfort in the spaces adjacent to the courtyard were reduced by 63.46% to 69.85%.

Hyper-reduced-order model for estimating convection heat transfer coefficients of turbine rotors

Forced convection heat transfer occurs inside the steam turbine, and it is crucial for thermal and strength analysis to accurately estimate the convection heat transfer coefficients of the turbine rotor. This paper presents a hyper-reduced-order model for efficiently estimating the forced convection heat transfer coefficients of a steam turbine rotor. Unlike the full-order finite element method, the hyper-reduced-order model uses the discrete empirical interpolation method to approximate the domain integration of transient heat conduction simulations. This approach greatly reduces the degrees of freedom of numerical integration, thus substantially improving the computational efficiency of forward heat conduction problems. The hyper-reduced-order model and Levenberg–Marquardt algorithm are combined to minimize temperature errors between the numerical simulation results and remote sensor measurements and iteratively estimate the heat transfer coefficients of forced convection in the rotor. The accuracy and efficiency of the proposed model are verified through a transient heat conduction simulation. Moreover, the effects of the initial guess values and measurement errors are investigated to demonstrate the universality and robustness of the proposed approach in determining the convection heat transfer coefficient. Compared with the full-order finite element model, the proposed hyper-reduced-order model can increase the efficiency of forward heat conduction simulations of the turbine rotor by more than 10 times, and it only takes 0.87 s computer runtime in single transient step simulation, and then rapidly estimate the forced convection heat transfer coefficients. Furthermore, the performance of the proposed approach is insensitive to the initial guess values, and it exhibits great robustness under measurement error. The mean errors are 4.78 %, 2.67 % and 10.51 % when the initial values are 0.015 mW·mm−3·°C−1, 0.05 mW·mm−3·°C−1 and 0.15 mW·mm−3·°C−1 respectively. When there is no measurement error, the estimated error of the convection heat transfer coefficients is 2.67 %, and even if the measurement error reaches 5 %, and the accuracy loss is only 5.57 %. Consequently, this approach is highly suitable for estimating convection heat transfer coefficients during the dynamic operation of steam turbines, and this research provides reliable guidance for the optimization design and safe operation of steam turbines.

Thermal management of a battery pack using fin-embedded phase change material

Abstract The present study focuses on the numerical investigation of heat transfer from a battery cell surrounded by phase change material (PCM) with longitudinal fins embedded in it. Three-dimensional transient heat conduction equation is solved numerically in the battery domain, whereas the solidification-melting model adopting Enthalpy-Porosity approach is solved in PCM to obtain liquid fraction and temperature distribution. PCM with 16 fins was found to be quite effective in reducing the battery surface temperature compared to 12, 8, and 4 fins. However, the effectiveness of 16 fins is observed up to 1500 sec till the PCM completely melts into liquid. A maximum reduction of 25 K has been achieved at 1500 sec by adding 16 fins to PCM. Beyond 1500 sec, the temperature rises sharply, exceeding all other cases at 2200 sec. Fins play an essential role in augmenting heat transfer, which benefits in achieving the phase change process in less time to restrict the temperature rise; however, at the same time, when the phase change process completes, fins become detrimental to the system since the temperature of the battery starts rising sharply.

An accelerated sequential function specification method based on LM gradient for transient inverse heat conduction problem

The sequential function specification method (SFSM) and the Levenberg-Marquardt method (LM) are two typical methods for inverse heat conduction problems (IHCP). However, these two methods have poor convergence and low convergence speed. Therefore, the combination of LM with SFSM is depended on to propose a modified Sequential Levenberg-Marquardt gradient method (SLM). The computation results of three shapes of heat fluxes with noise show the SLM is averagely improved over 16.3% more than SFSM in computation accuracy, but equally reduced over 39.9% less than in computation time. On other hand, when the emphasis is on the effect of the Aitken acceleration method, Tikhonov regularization and polynomial order on SLM, a neural network prediction model of the root mean square error is trained by regularization factor, polynomial order and the number of future time steps. Subsequently, the genetic algorithm optimization method is proposed for the minimum root mean square error so that it can improve the computation accuracy and efficiency of IHCP to the maximum extent. Finally, Second-order polynomial order and small order-of-magnitude regularization factor are highly recommended for parameter selection with multiple choices. The ASLM only needs to determine the optimal number of future time steps.

Understanding tool cutting-edge microstructure and deformation mechanism induced by adhesive wear in the turning of nickel-based superalloys

Nickel-based superalloy has great potential in the aerospace industry as key components. However, difficult-to-machine characteristics have further limited its application. Thermal-barrier coatings of the titanium-based family on carbide tools offer exceptional performance in cutting superalloys, combining high-temperature stability and remarkable toughness. This work primarily concentrates on understanding cutting-edge microstructure and deformation induced by tool adhesive wear in the turning of nickel-based superalloys with experiment and modelling. The dominant tool wear mechanisms are revealed to be adhesive wear and abrasive wear by SEM/EDS. The microstructure of the cutting-edge interface is qualitatively and quantitatively investigated by SEM/EDS and EBSD. The adhesive layer thickness at cutting edge is about 10–30 μm. The GND density of WC grains at cutting edge is 15-20 × 1014/m. The nanomechanics properties of the tool wear interface were quantitatively evaluated by nanoindentation. The average hardness of WC and Ni-superalloy at cutting-edge interface is evaluated to be 15–19 GPa and 6.2–6.5 GPa, respectively. Further, the underlying deformation mechanisms induced by tool wear behaviours are revealed through the transient heat conduction model and cutting-edge stress distribution model. This research offers a framework and mechanism for the cutting tool wear surface/interface characteristics targeted to the difficult-to-cut superalloy materials.

A hybrid shape functions based temporal finite element method to solve transient heat conduction problems

A novel temporal finite element method is presented to solve Fourier and non-Fourier heat conduction problems. Two kinds of temporal finite element models are developed using Gurtin variational principle and weighted residual technique, respectively. A kind of hybrid shape function with polynomial and trigonometric basis is stressed to give more flexible and appropriate description of temporal behavior of temperature. A recursive algorithm is developed by which the temperature solution at a specific time can be obtained only via a matrix power product with the initial condition, and a criterion of stability analysis is derived, which is can numerically be conducted when the constitution of shape function and time step size are prescribed. The proposed approach is available for Fourier and non-Fourier heat transfer problems, and can conveniently be combined with well-developed numerical algorithms for the boundary value problem, such as FE, SBFEM, etc. Various numerical examples, including those with the nonlinear thermal conductivity, radiative boundary condition, etc. are provided to illustrate the efficiency of proposed approaches, and impacts of the temporal FE model, constitution of shape functions, and step size, etc. are taken into account. Satisfactory results are achieved in comparison with the analytical and the ABAQUS based numerical solutions.

Development and Investigation of Nitrate Phase Change Filler Material for Solar Process Heating Applications

Thermocline thermal energy storage system is an imperial system for solar process heating applications. The thermal energy storage system comprises filler material and heat transfer fluid. In the present study, the commercial phase change material (PCM) (filler material) solar salt (60%NaNO3+40%KNO3) is synthesized, and it is encapsulated with a 1mm thickness of stainless steel 316L encapsulation with the macro-size of 60mm; the encapsulated PCMs is considered as a single particle in the thermocline thermal energy storage system. The performance of the single encapsulated PCM is studied with a constant heat transfer fluid (Therminol-VP1) flow rate of 0.00015 kg/s during the charging process. The specific heat of the solar salt is measured in Differential Scanning Calorimeter (DSC) in the solid phase is found as 1.381 J/gK, and liquid phase is identified as 1.551 J/g-K, the enthalpy during the phase change process is found as 116.72 J/g. As well, the thermal conductivity of the solar salt is measured in Transient Heat Conduction method, and the thermal conductivity follows the linear relationship of -0.0025(T)+0.4023 (45ºC<T<100ºC) in the sensible region. The temperature variations and the phase change phenomena for the solar salt is identified and the contours are provided. The charging time for the encapsulated solar salt is identified as ~138 minutes. The energy storage cost for the single filler material (Solar Salt + SS316L encapsulation) is identified as Rs.5.93/- (USD 0.071) for the period of charging duration.

Analytic Modelling of 2-D Transient Heat Conduction with Heat Source Under Mixed Boundary Constraints by Symplectic Superposition

Abstract Many studies have been conducted on 2-D transient heat conduction, but analytic modelling is still uncommon for the cases with complex boundary constraints due to the mathematical challenge. With an unusual symplectic superposition method, this paper reports new analytic solutions to 2-D isotropic transient heat conduction problems with heat source over a rectangular region under mixed boundary constraints at an edge. With the Laplace transform, the Hamiltonian governing equation is derived. The applicable mathematical treatments, e.g., the variable separation and the symplectic eigenvector expansion in the symplectic space, are implemented for the fundamental solutions whose superposition yields the ultimate solutions. Benchmark results obtained by the present method are tabulated, with verification by the finite element solutions. Instead of the conventional Euclidean space, the present symplectic-space solution framework has the superiority on rigorous derivations without pre-determining solution forms, which may be extended to more issues with the complexity caused by mixed boundary constraints.

Numerical Computation of 2D Domain Integrals in Boundary Element Method by (α, β) Distance Transformation for Transient Heat Conduction Problems

When the time-dependent boundary element method, also termed the pseudo-initial condition method, is employed for solving transient heat conduction problems, the numerical evaluation of domain integrals is necessitated. Consequently, the accurate calculation of the domain integrals is of crucial importance for analyzing transient heat conduction. However, as the time step decreases progressively and approaches zero, the integrand of the domain integrals is close to singular, resulting in large errors when employing standard Gaussian quadrature directly. To solve the problem and further improve the calculation accuracy of the domain integrals, an (α, β) distance transformation is presented. Distance transformation is a simple and efficient method for eliminating near-singularity, typically applied to nearly singular integrals. Firstly, the (α, β) coordinate transformation is introduced. Then, a new distance transformation for the domain integrals is constructed by replacing the shortest distance with the time step. With the new method, the integrand of the domain integrals is substantially smoothed, and the singularity arising from small time steps in the domain integrals is effectively eliminated. Thus, more accurate results can be obtained by the (α, β) distance transformation. Different sizes of time steps, positions of source point, and shapes of integration elements are considered in numerical examples. Comparative studies of the numerical results for the domain integrals using various methods demonstrate that higher accuracy and efficiency are achieved by the proposed method.