Heat Conduction Process Research Articles

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.

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.

Fully coupled finite strain consolidation analysis for soft clay treated by prefabricated horizontal drain with vacuum and heat preloading

This paper established a one-dimensional fully coupled finite strain consolidation analysis for soft clay treated by thermal prefabricated horizontal drain under vacuum and heat preloading with the consideration of thermal elastic viscoplastic (TEVP) behaviors of soft clay. Firstly, the governing equations for finite strain consolidation, heat conduction, and heat convention process have been carefully derived based on the well-established Gibson’s large strain consolidation theory and TEVP constitutive model. Secondly, the accuracy and reliability of the numerical method have been verified with benchmark cases and three physical model tests. Then, improvement of consolidation efficiency rate (IER) was defined and used to study the effect of three variables on consolidation efficiency. Last but not least, a graphical user interface (GUI) was provided to support researchers and engineers in utilizing the proposed numerical method for conducting consolidation analysis of soft soil treated in the same or similar fashion.

Construction of mathematical models of thermal conductivity for modern electronic devices with elements of a layered structure

This paper considers a heat conduction process for isotropic layered media with internal thermal heating. As a result of the heterogeneity of environments, significant temperature gradients arise as a result of the thermal load. In order to establish the temperature regimes for the effective operation of electronic devices, linear and non-linear mathematical models for determining the temperature field have been constructed, which would make it possible to further analyze the temperature regimes in these heat-active environments. The coefficient of thermal conductivity for the above structures was represented as a whole using asymmetric unit functions. As a result, the conditions of ideal thermal contact were automatically fulfilled on the surfaces of the conjugation of the layers. This leads to solving one heat conduction equation with discontinuous and singular coefficients and boundary conditions at the boundary surfaces of the medium. For linearization of nonlinear boundary value problems, linearizing functions were introduced. Analytical solutions to both linear and nonlinear boundary value problems were derived in a closed form. For heat-sensitive environments, as an example, the linear dependence of the coefficient of thermal conductivity of structural materials on temperature, which is often observed when solving many practical problems, was chosen. As a result, analytical relations for determining the temperature distribution in these environments were obtained. Based on this, a numerical experiment was performed, and it was geometrically represented depending on the spatial coordinates. This proves that the constructed linear and nonlinear mathematical models testify to their adequacy to the real physical process. They make it possible to analyze heat-active media regarding their thermal resistance. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and their elements but also the entire structure

Local high temperature phenomena near the complex interfaces of thermal barrier coatings based on non-Fourier model

Thermal barrier coatings (TBCs) consist of a metal substrate (Sub), a bond coat (BC), and a ceramic top coat (TC). The ceramic coating is a porous dielectric layer with a significant relaxation time that cannot be ignored. In high-temperature environments, an irregularly shaped thermally grown oxide layer (TGO) forms between the BC and the TC. Therefore, using non-Fourier model to describe the thermal contact process of TBCs with complex interface can get more accurate results. However, it is difficult to obtain the analytical solution of this model. This paper introduces a finite volume method based on unstructured meshes to address the non-Fourier heat transfer process in the complex interfaces of multilayer TBCs. The gradient reconstruction technique is employed to calculate the cell derivatives on the surface. The method's reliability is confirmed through a comparison with a two-dimensional analytical solution. The results indicate that the thermal wave travels at a limited speed within the TC layer. After passing through the interface, the thermal wave travels very fast within the BC and Sub layers, due to the influence of thermal wave reflection and superposition, obvious local high temperature will be formed near the interface. In addition, the effects of thermal conductivity, relaxation time and interface shapes on the non-Fourier heat conduction process of TBCs are also discussed. The results of this paper are useful for the structural design and failure mechanism research in TBCs.

Optimizing micro power generation with blended fuels and porous media for H2-fueled combustion

To enhance the combustion stability and thermal performance of micro-thermophotovoltaic system, a combustor inserted porous media and varying channel heights for premixed H2/CH4/Air burning is proposed. The experimental and numerical methods are conducted to investigate the effects of CH4 addition, equivalence ratio, porous media, and chamber setting on combustion characteristics, heat transfer, and flame stability. The results indicate that a small fraction of CH4 addition and porous media insertion both enhance heat release and transfer of H2-fueled combustion, and the optimal CH4 blending ratio is influenced by chemical energy input. Appropriately increasing the equivalence ratio under high flow rates contributes to stabilize the flame and improve the combustor thermal performance. Furthermore, porous media effectively promotes flame stability and heat transfer efficiency, with longer porous media enhancing heat conduction and convection processes, thereby increasing the radiation temperature. Besides, increasing the channel height of the porous media combustor appropriately enhances the electrical power output of the power system. Besides, the maximum electrical output of 9.7 W is obtained in the porous media combustor with a channel height of 11 mm at mc = 20 %, Ф = 1.0 and mf = 9.448 × 10−5 kg/s, which is 7.2 W higher than the free flame combustor with h = 7 mm.

Regularization in space–time topology optimization for additive manufacturing

In additive manufacturing, the fabrication sequence has a large influence on the quality of manufactured components. While planning of the fabrication sequence is typically performed after the component has been designed, recent developments have demonstrated the possibility and benefits of simultaneous optimization of both the structural layout and the corresponding fabrication sequence. This is particularly relevant in multi-axis additive manufacturing, where rotational motion offers enhanced flexibility compared to planar fabrication. The simultaneous optimization approach, called space–time topology optimization, introduces a pseudo-time field to encode the manufacturing process order, alongside a pseudo-density field representing the structural layout. To comply with manufacturing principles, the pseudo-time field needs to be monotonic, i.e., free of local minima. However, explicitly formulated constraints proposed in prior work are not always effective, particularly for complex structural layouts that commonly result from topology optimization.In this paper, we introduce a novel method to regularize the pseudo-time field in space–time topology optimization. We conceptualize the monotonic additive manufacturing process as a virtual heat conduction process starting from the surface upon which a component is constructed layer by layer. The virtual temperature field, which shall not be confused with the actual temperature field during manufacturing, serves as an analogy for encoding the fabrication sequence. In this new formulation, we use local virtual heat conductivity coefficients as optimization variables to steer the temperature field and, consequently, the fabrication sequence. The virtual temperature field is inherently free of local minima due to the physics it resembles. We numerically validate the effectiveness of this regularization in space–time topology optimization under process-dependent loads, including gravity and thermomechanical loads.

Constructing mathematical models of thermal conductivity in individual elements and units of electronic devices at local heating considering thermosensitivity

This paper considers a heat conduction process for an isotropic medium with local external and internal thermal heating. It was necessary to construct linear and non-linear mathematical models for determining the temperature field, and consequently, for the analysis of temperature regimes in these heat-active environments. To solve the linear boundary value problems and the resulting linearized boundary value problems with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which the analytical solutions to these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, analytical relations for determining the temperature distribution in this environment were established. To determine the numerical values of the temperature and analyze the heat exchange processes in the given structure, caused by the external heat load, a geometric image of the temperature distribution was constructed depending on spatial coordinates. The resulting linear and non-linear mathematical models testify to their adequacy to the real physical process. They make it possible to analyze heat-active media regarding their thermal resistance. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and their elements but the entire structure as well

Spatiotemporal-response-correlation-based model predictive control of heat conduction temperature field

In many engineering fields, controlling the transient temperature field of the heat conduction process is significant practically. For the temperature field control problem, a spatiotemporal-response-correlation-based model predictive control (STRC-MPC) method is developed. In this method, a spatiotemporal mapping eigenvector of control inputs to temperature field is determined by transient heat conduction equations. According to the correlation degree between the temporal mapping eigenvectors of different spatial points, a finite number of representative spatial points (RPs) are extracted, of which can cover the full mapping characteristic of the temperature field. Meanwhile, the predictive model of the temperature field is reduced offline to temperature predictive models of the RPs. Then, the predictive models of the RPs are applied to design a model predictive controller of the temperature field. In addition, a correlation formulation between the temperature responses of the RPs and that of the measurement points (MPs) is derived by making the control inputs as intermediate variable, and a correlation model between the predictive errors of the two kinds of points is established. Combining the correlation model and the predictive errors of the MPs, the predictive errors at the RPs are estimated and the feedback correction of the predictive model of the RPs is achieved. The STRC-MPC method is employed to control the preheating temperature field of a die casting mold by numerical simulations. The model predictive controller and the feedback correction scheme involved in the proposed control method are verified respectively.

Effect of squared, cubic and quartic on-site potentials on heat conduction in a nonlinear silicon lattice

This paper explains the process of heat conduction in a nonlinear silicon lattice governed by squared, cubic and quartic on-site potentials along with squared and quartic interaction potentials. The tanh method is employed to construct solutions to the resulting nonlinear equations. The results show the transfer of heat in the form of solitary waves which are presented graphically.

A semi-analytic method for buoyancy-induced thermal stratification in hot water tanks during standby periods

Buoyancy-induced thermal stratification is a spontaneous phenomenon arising from standby periods of hot water tanks. Accurate estimation of temperature levels is crucial for energy-efficient operations of hot water tanks and associated water heating systems. However, temperature estimation by numerical schemes requires sophisticated algorithms and considerable computation resources. This study proposes a novel semi-analytic method to address buoyancy-induced thermal stratification conveniently without iterative calculations. An explicit water temperature function to tank height and time is analytically derived from the modified energy balance equation where the heat convection term is disregarded. The convection effect is compensated for by artificial enhancement of heat conduction, which is indicated by a synthetic parameter, amplification factor. A comprehensive numerical study is carried out to investigate static and transient characteristics of buoyant flows and temperature distribution in a hot water tank during standby periods. The findings imply a strong analogy between buoyancy-induced thermal stratification and a bottom-initiated heat conduction process, which reinforces the physical rationale for the semi-analytic method. The experimental verification of the semi-analytic method is conducted against published experiment data, the proposed method shows excellent estimation accuracy with the rooted mean square error of less than 0.42 °C. Furthermore, suitable case-specific amplification factors are determined with numerical simulations, and a ready-for-use correlation equation of the amplification factor is formulated for a wide range of tank volumes and aspect ratios. This novel semi-analytic method substantially reduces computation time compared with numerical solvers, and the simple form of the water temperature function can help in formulating more detailed hot water tank models in energy system studies.

Applying Proper Orthogonal Decomposition to Parabolic Equations: A Reduced Order Numerical Approach

n this paper we present a low-order numerical scheme developed using the Proper Orthogonal (POD) method to address non-homogeneous parabolic equations in both one and two dimensions. The proposed schemes leverage the POD technique to reduce the computational complexity associated with solving these equations while maintaining accuracy. By employing POD, the high-dimensional problem is approximated by a reduced set of models, allowing for a more efficient representation of the system dynamics. The application of this method to non-homogenous parabolic equations offers a promising approach for enhancing the computational efficiency of simulations in diverse fields, such as fluid dynamics, heat conduction, and reaction-diffusion processes. The presented numerical scheme demonstrates its efficacy in achieving accurate results with significantly reduced computational costs, making it a valuable tool for applications demanding efficient solutions to non-homogeneous parabolic equations in one and two dimensions.

Analysis of local exergy losses in heat conduction processes for an air-heating heat exchanger of a boiler plant

Analysis of local exergy losses in heat conduction processes for an air-heating heat exchanger of a boiler plant

Linear modeling analysis of the heat balance of the transmission line in high frequency critical ice melting state

Abstract The icing of transmission lines can cause many problems such as increased line load, unbalanced tension, and galloping, posing a serious threat to the reliable operation of the power system. Therefore, linear modeling analysis of the heat balance under a high-frequency critical ice melting state of transmission lines is studied. The thermal conduction states of melting, heating, twisting, rotating melting, and ice layer detachment of ice melting conductors are analyzed. Based on the heat conduction process of thermal melting of ice on transmission lines, a simplified calculation formula for thermal melting ice time is derived to calculate times for heating, ice layer torsion, and air gap increase. At the same time, based on the law of conservation of energy, thermodynamic boundary conditions for iced conductors were established and corresponding physical models were constructed. Based on this model, we conduct a linear modeling of the heat balance in the melting state in order to provide a more accurate prediction of temperature distribution. The experimental results show that there is a negative correlation between environmental temperature and critical melting current, and the melting time decreases with the increase of environmental temperature.

Development of complex methods for exergetic losses calculating in heat transfer through wall in air-heater heat utilizers

Ensuring high economic indicators of power plants of various purposes, which include waste gas heat utilization systems, is possible with the use of effective heat utilization equipment. One of the criteria for evaluating the thermodynamic perfection of power plants is the amount of exergy losses in the structural elements of the plant. A decrease in exergy losses corresponds to an increase in exergy efficiency. Determination of the structural elements in the heat utilization system, in which the maximum exergy losses are localized, will allow to influence them in a targeted manner in order to increase the exergy efficiency of the installation. The paper presents the results of the development of complex methods for calculating local exergy losses in the processes of heat transfer through flat and cylindrical walls of air-heated heat exchangers, which are included in the heat utilization systems of power plants of various purposes. Complex methods are based on integral equations for calculating exergy losses, as well as differential equations of heat transfer theory when applied to heat utilization systems. The classification of exergy losses for heat utilization systems of power plants has been carried out, and the location and type of local exergy losses have been determined. Three types of local exergy losses are considered: losses during heat transfer from the heat carrier to the wall, heat transfer from the wall to the heat carrier, and exergy losses in heat conduction processes. A system of integrodifferential equations was developed and analytical expressions were obtained for the calculation of local exergy losses in heat transfer processes through the flat and cylindrical walls of air-heating heat exchangers. The introduction of new complex research methods expands the possibilities of using exergy analysis methods to increase the exergy efficiency of heat utilization systems. Research using the specified methods will allow to develop heat utilization systems with high indicators of energy and exergetic efficiency.

Thermophysical model of a thermonic cathode with induction heating

A thermophysical model was built and the temperature field of a cylindrical thermionic cathode with induction heating was calculated, taking into account the initial and boundary conditions, based on the adoption of assumptions to simplify the mathematical model. During the induction heating of the cathode, a non-stationary heat conduction process is established, which is described by the differential equation of heat conduction with internal sources of Joule heat. The distribution of internal heat sources in the volume of the cathode is determined by the distribution of the ring induced current. The cylindrical design of the inductive thermionic cathode, due to spatial symmetry, allows to reduce the number of spatial variables, significantly simplify functional dependencies, and limit the algorithm for solving the problem. The problem was solved in the cylindrical coordinate system. The obtained approximate solutions were assessed for the correctness of the accepted simplifications when finding the distribution of the temperature field with a sufficient degree of accuracy. Despite the high thermal conductivity of the cathode material, when the cathode is inductively heated, there can be a significant temperature difference between its outer and inner surfaces. The article shows the permissible temperature difference on the surface of the cathode, which limits the choice of geometric dimensions of the cathode. The temperature difference on the surfaces of the induction thermionic cathode is most affected on the end (annular) surfaces of the cathode, so it is better to apply emitting coatings to the side surfaces of the cylindrical cathode, thus complicating the design of the cathode. The application of induction heating of the thermionic cathode allows to simplify the heating unit, increase the reliability and service life of powerful electronic devices. The obtained results are planned to be used in further research as test data for the analysis of more complex problems of numerical calculation of the thermal regimes of the cathode unit with heat shields and focusing elements of thermoelectron flows.

Analysis of GAAFET’s transient heat transport process based on phonon hydrodynamic equations

Compared to the classical Fourier’s law, the phonon hydrodynamic model has demonstrated significant advantages in describing ultrafast phonon heat transport at the nanoscale. The gate-all-around field-effect transistor (GAAFET) greatly optimizes its electrical performance through its three-dimensional channel design, but its nanoscale characteristics also lead to challenges such as self-heating and localized overheating. Therefore, it is of great significance to study the internal heat transport mechanism of GAAFET devices to obtain the thermal process and heat distribution characteristics. Based on this, this paper conducts theoretical and numerical simulation analyses on the phonon heat transfer characteristics within nanoscale GAAFET devices. Firstly, based on the phonon Boltzmann equation, the phonon hydrodynamic model and boundary conditions are rigorously derived, establishing a numerical solution method based on finite elements. For the novel GAAFET devices, the effects of factors such as surface roughness, channel length, channel radius, gate dielectric, and interface thermal resistance on their heat transfer characteristics are analyzed. The research results indicate that the larger the surface roughness, the smaller the channel length and the channel radius, the larger the interface thermal resistance leads to the higher hot spot peak temperature. The non-Fourier heat analysis method based on the phonon hydrodynamic model and temperature jump condition within the continuous medium framework constructed in this paper can accurately predict the non-Fourier phonon heat conduction process inside GAAFET and reveal the mechanisms of resistive scattering and phonon/interface scattering. This work provides important theoretical support for further optimizing the thermal reliability design of GAAFET, improving its thermal stability, and operational performance.

Coupled thermo-mechanical field-enriched finite element method for simulating the thermal failure process of quasi-brittle solids

Thermal loads and temperature changes significantly impact the deformation and fracture of quasi-brittle solids, and the computational simulation of such a process is challenging. In this paper, a coupled thermo-mechanical field-enriched finite element method (TM-FE-FEM) is proposed to simulate the thermal fracture process of quasi-brittle solids. The accuracy and correctness of the proposed coupled TM-FE-FEM are validated by four benchmark examples, and the ability to simulate thermal induced crack is validated by the plate containing single edge flaw under coupled thermo-mechanical loading. Finally, the thermal fracture process of the Lac du Bonnet (LdB) granite specimen is reproduced by the proposed method. It can be concluded from the numerical results that the proposed coupled TM-FE-FEM has the ability to simulate the processes of heat conduction, thermal deformation and thermal fracture in quasi-brittle solids.

Approximate Solution of Optimal Pulse Control Problem Associated with the Heat Conduction Process

Abstract We consider the approximate solution of the control problem with minimum energy for an object described by the heat equation, with the process described by the linear equation of parabolic type and the system controlled by impulsive external influences. Our optimal control problem deals with finding a control parameter belonging to the class of admissible controls that provides the desired temperature distribution in a finite time with minimal energy consumption (energy consumption is described by the quadratic functional). Previous works dedicated to optimal impulse control problems have mostly used the Pontryagin’s maximum principle. However, from a practical point of view, this approach does not lead to satisfactory results. This is due to the fact that the corresponding boundary value problems in this case have no solution in a traditional class of absolutely continuous trajectories. In this work, we propose a method based on the moment relations. We seek for the approximate solution of the corresponding boundary value problem in the form of finite Fourier sum and state our optimal control problem in a finite-dimensional phase space. As a result, we obtain an optimal impulse control problem in a finite-dimensional function space. Taking into account the given condition for a finite time, we reduce the obtained problem to the L-problem of moments. Thus, the problem of finding a control parameter is reduced to the solution of the system of Fredholm integral equations of the first kind, with the norm of the sought solution not exceeding a given number. By Levi’s theorem, every element of Hilbert space can be represented by the sum of the elements of two orthogonal subspaces. This assertion makes it possible to find control parameters in analytical form. We also establish the convergence of the chosen approximation.

Using transducerless time-domain thermoreflectance technique to measure in- and cross-plane thermal conductivity of nanofilms

Time-domain thermoreflectance (TDTR) and frequency-domain thermoreflectance techniques have been widely used to measure thermal properties. However, the existence of the metal sensor brings some limitations to the experimental measurement, such as temperature limits, disability to measure low in-plane thermal conductivity, in situ measurement cannot be achieved, etc. This paper proposes a transducerless time-domain thermoreflectance method to measure in- and cross-plane thermal conductivity of nanofilms, in which the optical absorption depth and thermal conductivity tensor are considered to establish a new differential equation that can describe the heat conduction process in multilayer structures. This thermal model can also calculate the effects of spot ellipticity and spot offset distance. Then, the analytical solution and relative deviation of this new model and the surface heat flow boundary model used in conventional TDTR are compared by calculating the phase signals. In terms of experimental measurement, this model is successfully used to derive cross- and in-plane thermal conductivity of PdSi and IrNiTa amorphous alloy nanofilms without a metal sensor.