Classical Heat Conduction Equation Research Articles

Fractional non-Fourier modeling of laser drilling process

In this paper, a novel fractional non-Fourier model is employed to simulate the laser drilling process, addressing limitations inherent in classical heat conduction equations, including the well-known heat equation paradox associated with infinite heat propagation velocity. This model approach combines spatial approximation via the Meshless Local Petrov-Galerkin method with temporal approximation using the Grünwald-Letnikov finite difference scheme. The study assesses the impact of employing fractional orders, both constant and variable over time, on numerical results, and validates the model using experimental data.

Grain refinement mechanism in adiabatic shear band of solution treated Ti–10V–2Fe–3Al alloy under high strain rate

The grain refinement mechanism in the adiabatic shear band (ASB) of solution treated Ti–10V–2Fe–3Al alloy was investigated under high strain rates using a split Hopkinson pressure bar and hat-shaped specimens. The TEM results show that elongated and parallel grains are the major characteristics of the transition region between the matrix and the centre of the ASB. The central region of the ASB consists of ultrafine and equiaxed recrystallized grains with a typical size of 100 nm. According to the classical one-dimensional Fourier heat conduction equation and recrystallization theory calculations, it is proved that ultrafine and equiaxed recrystallized grains did not undergo significant growth during the cooling stage after deformation. The thermodynamics and kinetics calculated results indicate that instant grain refinement within the ASB is due to the rotational dynamic recrystallization (RDR) that occurs during the deformation process.

Analytical solution of heat equation taking into account convection with isothermal boundary conditions

The problem of determining the distribution of heat through the reservoir constantly arises in the production of hydrocarbons. Changes of temperature affect the viscosity of oil and consequently the rate of its production. Taking into account the filtration process, additional terms appear in the classical heat conduction equation, including nonlinear ones. Various numerical schemes are used to solve the modified equations. The question of the convergence of such methods often arises. The task of this work is to obtain an analytical solution of the heat equation in cases where it is possible, in order to further compare numerical solutions with them.

Laser-induced thermal emission of carbon microparticles on transparent heat-sink substrates

Thermal emission is an informative tool to study materials’ properties at high temperatures under laser irradiation. The kinetics decay of laser-induced thermal emission from carbon microparticles deposited on heat-sink surfaces of transparent dielectrics (glass and sapphire) was studied. A Q-switched YAG:Nd3+ laser (pulse duration τi = 20 ns, energy/power density 0.5 J·cm–2, 25 MW·cm–2) was employed to excite thermal emission. In calculations, the classical heat conduction equation was used. With increasing the thermal conductivity of substrate (from glass to sapphire), reduction in the emission pulse duration has been observed.

A modified model for thermoelastic stress analysis under low frequency fatigue loading

Thermoelastic stress analysis (TSA) can be used to determine the stress field of structures or materials under cyclic loadings through temperature under adiabatic conditions. So high-frequency cyclic loadings are considered as a fundamental condition. However, loading frequencies in many practical applications do not meet the adiabatic conditions. In order to improve the accuracy of TSA under low-frequency fatigue loadings, the non-adiabatic response in TSA was derived by combining the classical thermoelastic equation and heat conduction equation. This response was used to create a modified model that accounted for heat transfer and temperature distribution under low-frequency loading conditions. Otherwise, fatigue test and finite element method were used to evaluate the correctness of the modified model using epoxy resin material. The results showed that the temperature change obtained from the modified model were in good agreement with the experiment results. The modified model can effectively evaluate the structural stress under low-frequency fatigue loadings.

О возможности создания крупногабаритных конструкций в условиях открытого космоса

The paper gives a brief overview of the history of the development of inflatable space struc-tures. Further development should be the use of inflatable shells that harden in space. The possibility of using this technology in the framework of the use of simple tasks is considered in the article. On the ex-ample of thick-walled spherical and cylindrical shells, using Lame's analytical solutions, it is shown that the thickness of the shell can be thin. The thin shell design is much easier to inflate. No high pressure is required for this. Using the classical heat conduction equation, the temperature distribution inside a cy-lindrical shell is determined. When setting the boundary conditions, the flux of rays incident both from the side of the Sun and from the side of the Earth was taken into account. The radiation on the surface of the cylinder was calculated according to the Stefan-Boltzmann law. It has been established that a tem-perature occurs in the shell, the maximum value of which is in the range from 150 to 200 degrees Celsi-us. Prepreg casings require the use of a high temperature resin suitable for hot curing.

Group Invariant Solutions and Local Conservation Laws of Heat Conduction Equation Arising Laser Heating Carbon Nanotubes Using Lie Group Analysis

In this study, based on the continuous transformations of Lie groups, the exact analytic solutions of the laser heating carbon nanotubes formulated by using the classical heat conduction equation with various physical properties were constructed. These solutions are the type of group invariant solutions. The constructed solutions have expanded and enriched the solution forms of this new model existing in the literature. With the help of the Maple package program, 3D, density, and contour graphs were drawn for the special values of the parameters in the solutions, and the physical structures of the solutions obtained in this way were also observed. The solutions obtained can be used in the explanation of physical phenomena occurring in cancer investigations.

Numerical analysis of dynamic thermoelastic two-dimensional crack problem combined with non-Fourier heat transfer equation by finite-difference time-domain method

In recent years, a variety of microscopes using heat sources, such as photoacoustic microscopes, have been developed. In this observation principle, reflection and diffraction phenomena of two waves with different properties, namely thermal wave and elastic wave, are utilized. However, the thermal wave cannot be reproduced using the classical Fourier-type heat conduction equation. In order to overcome this mathematical difficulty, we had derived a coupled nonlinear partial differential equation consisting of the heat conduction with a delay time and the dynamic thermoelastic equation, and had simulated numerically the propagation process of thermal and elastic waves for one-dimensional problem. In this study, thermal and elastic waves were simulated for a two-dimensional plane problem with and without defect using the finite-difference time-domain method. The results showed that the reflection and interference phenomena of two waves can employ for detecting small internal defects existing near the surface of target sample.

Quantum master equation approach to heat transport in dielectrics and semiconductors

We report on the derivation of the heat transport equation for nonmetals using a quantum Markovian master equation in Lindblad form. We first establish the equations of motion describing the time variation of the on-site energy of atoms in a one dimensional periodic chain that is coupled to a heat reservoir. In the continuum limit, the Fourier law of heat conduction naturally emerges, and the heat conductivity is explicitly obtained. It is found that the effect of the heat reservoir on the lattice is described by a heat source density that depends on the diffusion coefficients of the atoms. We show that the Markovian dynamics is equivalent to the long wavelength approximation for phonons, which is typical for the case of elastic solids. The high temperature limit is shown to reproduce the classical heat conduction equation.

Non-linear free vibrations of the column loaded with a mass element and a local heat source

In this paper, the problem of non-linear vibrations of an articulated column subjected to two loads was considered. The first load is axial compression associated with the action of the mass element. The second is the heat load in the form of a local heat source. In such a system, the problem analysis was divided into two parts. The first illustrates the influence of temperature on the physical and mechanical properties of the structure. The second - concerning non-linear vibrations of the system. The problem of heat flow in the column was formulated on the basis of the classical Fourier heat conduction equation and solved with the use of the Finite Element Method. Distribution of material properties (Young’s modulus) was obtained in subsequent moments of exposure of the column to the heat source. The obtained results were used in the problem of system vibrations, which was formulated on the basis of Hamilton’s principle. Due to the non-linearity, the perturbative small parameter method was used to finally derive the differential equations describing the behaviour of the system during vibrations. Numerical calculations were carried out on the basis of which the characteristic curves of the system (on the load - natural frequency plane) for various stages of heating the column were determined. Moreover, the influence of the heating time of the system on its non-linear natural frequency was presented. The significant influence of the longitudinal inertia of the mass element causing the load on the vibrations of the structure was shown. The mathematical model and the correctness of the performed numerical simulations were verified by experimental research.

Calculation of non-stationary heat exchange in multi-layer media by means of the theory of Markov chains

Calculation of non-stationary heat exchange in multi-layer dissimilar media is the problem, which is often found in many industries including high temperature metallurgic technologies of production and treatment of metals, alloys and their products. The similar problems arise during the design of buildings and structures. The models used for calculation are mainly based on approximate solutions of classical heat conduction equations. The drawback of such models is a complex computational procedure, associated with the need to solve a large number of equations and to define a large number of identification parameters. At present, the calculation of one mode of heat transfer between the sections of a composite structure requires 6–8 hours of operation of a computer of average power. In this regard, the development of discrete models that require less computer time is of current importance. The proposed model of the process is based on the theory of Markov chains. A multi-layer medium is presented as a chain of small but finite cells. Each of them contains a certain amount of heat that can be transferred to the neighboring cells. The part of the transferred heat is directly proportional to the heat conduction coefficient and in inverse proportion to the material heat capacity, material density and the cell length. The matrix model to describe heat transfer between cell in a multi-layer media based on the theory of Markov chains is developed. Construction of the matrix of transition probabilities is described, evolution of the state vectors i.e. distribution of heat and temperature is carried out, and non-uniformity of the heater temperature is taken into account. Comparison of calculated and experimental data has showed the adequate description of the real process using the model. Analysis of identification parameters has given a satisfactory result.

Peridynamics for Heat Conduction

Abstract Peridynamics for transient heat conduction problems in general anisotropic materials is developed. In order to develop a new peridynamic governing equation for heat conduction problems, the microconductivity (or microdiffusivity), which contains equivalent information as the constitutive equation for classical heat conduction, is determined by directly requiring the resulting peridynamic equation to converge to a classical heat conduction equation for anisotropic materials as the generalized material horizon approaches zero. Therefore, the convergence proof is built into the theory from the perspective of the governing equation. For the application of the newly obtained peridynamic governing equation, a time-dependent three-dimensional (3D) peridynamic heat equation is analytically solved with two types of heat sources, and the results are discussed. These are believed to be the first exact analytical solutions for peridynamic heat conduction.

Refined bond-based peridynamics for thermal diffusion

PurposeThis paper aims to review the existing bond-based peridynamic (PD) and state-based PD heat conduction models, and further propose a refined bond-based PD thermal conduction model by using the PD differential operator.Design/methodology/approachThe general refined bond-based PD is established by replacing the local spatial derivatives in the classical heat conduction equations with their corresponding nonlocal integral forms obtained by the PD differential operator. This modeling approach is representative of the state-based PD models, whereas the resulting governing equations appear as the bond-based PD models.FindingsThe refined model can be reduced to the existing bond-based PD heat conduction models by specifying particular influence functions. Also, the refined model does not require any calibration procedure unlike the bond-based PD. A systematic explicit dynamic solver is introduced to validate 1 D, 2 D and 3 D heat conduction in domains with and without a crack subjected to a combination of Dirichlet, Neumann and convection boundary conditions. All of the PD predictions are in excellent agreement with the classical solutions and demonstrate the nonlocal feature and advantage of PD in dealing with heat conduction in discontinuous domains.Originality/valueThe existing PD heat conduction models are reviewed. A refined bond-based PD thermal conduction model by using PD differential operator is proposed and 3 D thermal conduction in intact or cracked structures is simulated.

Change of entropy for the one-dimensional ballistic heat equation: Sinusoidal initial perturbation.

This work presents a thermodynamic analysis of the ballistic heat equation from two viewpoints: classical irreversible thermodynamics (CIT) and extended irreversible thermodynamics (EIT). A formula for calculating the entropy within the framework of EIT for the ballistic heat equation is derived. The entropy is calculated for a sinusoidal initial temperature perturbation by using both approaches. The results obtained from CIT show that the entropy is a non-monotonic function and that the entropy production can be negative. The results obtained for EIT show that the entropy is a monotonic function and that the entropy production is nonnegative. A comparison between the entropy behaviors predicted for the ballistic, for the ordinary Fourier-based, and for the hyperbolic heat equation is made. A crucial difference of the asymptotic behavior of the entropy for the ballistic heat equation is shown. It is argued that mathematical time reversibility of the partial differential ballistic heat equation is not consistent with its physical irreversibility. The processes described by the ballistic heat equation are irreversible because of the entropy increase.

Hyperbolic heat conduction at a microscopic sliding contact with account of adhesion-deformational heat generation and wear

Different non-Fourier models were proposed to simulate temperatures in materials subjected to extremely fast thermal disturbances, when the speed of heat propagation should be concerned. The present study investigated temperature and heat balance at a microscopic sliding contact during a single frictional interaction based on the Cattaneo-Vernotte hyperbolic heat conduction equation. Two fundamental features of friction, namely, adhesion-deformational heat generation and wear, were taken into account. By applying the Laplace transform approach, non-stationary temperature expressions were derived for the hyperbolic and classical parabolic heat conduction equations. Parametric analysis was then done for parameter ranges typical of brake materials. It was found that the hyperbolic heat conduction generally results in a higher temperature at the sliding surface compared to the parabolic heat conduction. The influence of the heat propagation speed can be significant for thermal relaxation time of the order above microsecond. It becomes stronger with an increase in the contribution of the adhesive heat generation. Another finding obtained is that a considerable fraction of heat is removed from the contact zone along with wear debris, resulting in a lower temperature. This fraction is larger for the hyperbolic heat conduction.

Wave Heat Transfer in the Orthotropic Half-Space Under the Action of a Nonstationary Point Source of Thermal Energy

A new analytical solution to the problem of wave heat transfer in the orthotropic half-space under the action of a time-dependent point heat flux is obtained and studied. The heat transfer is described by a hyperbolic heat conduction wave equation, in which the directions of the thermal conductivity coincide with the Cartesian coordinate system axes (the orthotropic solid). The obtained analytical solution has allowed us to trace the behavior of the point temperature profile in the vicinity of the initial time moment during a number of relaxation times, which is impossible to do when the classical parabolic heat conduction equation is used.

A Theoretical Model of Laser Heating Carbon Nanotubes.

We present a theoretical model of laser heating carbon nanotubes to determine the temperature profile during laser irradiation. Laser heating carbon nanotubes is an essential physics phenomenon in many aspects such as materials science, pharmacy, and medicine. In the present article, we explain the applications of carbon nanotubes for photoacoustic imaging contrast agents and photothermal therapy heating agents by evaluating the heat propagation in the carbon nanotube and its surrounding. Our model is constructed by applying the classical heat conduction equation. To simplify the problem, we assume the carbon nanotube is a solid cylinder with the length of the tube much larger than its diameter. The laser spot is also much larger than the dimension of carbon nanotubes. Consequently, we can neglect the length of tube dependence. Theoretically, we show that the temperature during laser heating is proportional to the diameter of carbon nanotube. Based on the solution of our model, we suggest using the larger diameter of carbon nanotubes to maximize the laser heating process. These results extend our understanding of the laser heating carbon nanotubes and provide the foundation for future technologically applying laser heating carbon nanotubes.

Heat transfer of micro-droplet during free fall in drop tube

To explore the heat transfer of micro-droplet during free fall in the drop tube, the falling velocity and microgravity level are calculated. The Newtonian heat transfer formulation is coupled with the classical heat conduction equation to predict the heat transfer process within micro-droplet. Based on the numerical solution by finite difference method with implicit Euler scheme, the temporal evolution of thermal information inside micro-droplet is obtained including the temperature distribution, cooling rate, temperature difference and gradient. To quantitatively reveal the mechanism how the various factors affect the heat transfer of micro-droplet, the effects of physical properties of liquid metal and cooling gas as well as the micro-droplet size are studied. As the important indicators of heat transfer process, the cooling rate and temperature difference are acquired to systematically investigate the relationship between thermophysical properties and heat transfer process of different metallic micro-droplets.

Integrating two-temperature and classical heat accumulation models to predict femtosecond laser processing of silicon

A robust, computationally efficient modeling method describing multi-pulse femtosecond laser-material interaction is required to determine the optimal laser parameters to control and differentiate non-thermal ablation and heat accumulation processes for surface structuring and laser welding applications. We establish a three-dimensional, two-temperature model (TTM) and a heat-accumulation model based on classical heat generation and conduction equations to evaluate their efficacy and efficiency in simulating non-thermal ablation and heat accumulation during multi-pulse femtosecond laser processing of silicon. Only the TTM is capable of accurately predicting the laser fluences required to achieve non-thermal ablation, which is experimentally validated. Both the TTM and the classical heat accumulation model can predict heat accumulation. The TTM can accurately predict heat accumulation, but requires lengthy simulation times on the order of several hours. The classical heat accumulation model consistently predicts heat accumulation with the TTM and is time efficient, but is case specific to interaction parameters, requiring input of an experimentally-determined absorption coefficient. For the first time to our knowledge, an integrated modeling method is devised to accurately and efficiently simulate laser-processing-induced heat accumulation by using the TTM to determine an absorption coefficient to feed back to the heat accumulation model to extend it to the general case. This integrated modeling method enables the accurate prediction of heat accumulation with simulation times on the order of a minute per pulse, defining a path to determine laser parameters to control heat accumulation for specific processing applications.

Reconstruct the Heat Conduction Model with Memory Dependent Derivative

The classical heat conduction equation is derived from the assumption that the temperature increases immediately after heat transfer, but the increase of temperature is a slow process, so the memory-dependent heat conduction model has been reconstructed. Numerical results show that the solution of the initial boundary value problem of the new model is similar to that of the classical heat conduction equation, but its propagation speed is slower than that of the latter. In addition, the propagation speed of the former is also affected by time delay and kernel function.