Heat Conduction Equation Research Articles

Application of pulsed laser ablation thermal model in nanosecond pulsed laser removal of the epoxy resin paint film

The effect of laser surface paint removal depends on the mechanism of paint removal. The mechanism of nanosecond laser removal of epoxy resin paint film on aluminum alloy surfaces was studied theoretically and experimentally. Firstly, a laser thermal paint removal model was established based on the heat conduction equation, Hertz-Knudsen equation, and Hooke's law. The adsorption relationship between the paint film and the aluminum alloy substrate molecules and the adsorption force between the paint film and the substrate were analyzed and calculated. A laser thermal paint removal model was solved using COMSOL finite element software. By analyzing the changes in temperature and thermal stress with laser fluence, we obtained the theoretical threshold for laser paint removal. Subsequently, experimental research was conducted on laser paint removal. Through theoretical and experimental studies, it was found that the paint removal threshold of nanosecond pulse laser is 0.47 J/cm2. The primary mechanism of nanosecond laser paint removal is thermal stress stripping. When the laser fluence is low, the nanosecond pulse laser can only ablate the paint on the surface, and the thermal stress between the paint film and the substrate makes it difficult to remove the paint film. As the laser fluence increases, the nanosecond laser will ablate the surface paint film, and the thermal stress between the paint film and the substrate will cause the stripping of the underlying paint film and the substrate. The primary mechanism of laser paint removal is stripping.

Topography optimisation using a reduced-dimensional model for transient conjugate heat transfer between fluid channels and solid plates with volumetric heat source

Consideration of transient effects is important for industrial applications of heat transfer structure optimisation studies; however, the huge computational cost associated with transient problems is a pressing concern. This paper proposes an extension of a previous reduced-dimensional model to transient conjugate heat transfer between a fluid flow and solid-heated plates in a plate heat exchanger. The extended reduced-dimensional model introduces the temperature field of the plate governed by the heat conduction equation, which is coupled to the temperature field of the fluid, governed by the convection-diffusion equation, through the heat flux balance equation at the contact surface. The model is based on assumptions of fully developed flow and constant temperature profile, reducing the three-dimensional problem to a planar problem and significantly reducing computational costs. The accuracy of the model for the simulation of transient heat transfer is verified by comparison with a three-dimensional model. In this paper, the topography of the heat exchanger plate is optimised for both steady-state and transient conditions by applying the reduced-dimensional model. The effectiveness of the optimised design was demonstrated by the cross-check of both the reduced-dimensional and full three-dimensional models. Furthermore, this work considers the effect of time-independent boundary conditions and time-dependent boundary conditions on transient optimisation. The transient and steady-state optimised designs are analysed and compared for both conditions, and the necessity of transient optimisation is discussed.

Fast explicit time integration schemes for parabolic problems in mechanics

AbstractWe present a family of fast explicit time integration schemes of first, second and third order accuracy for parabolic problems in mechanics solved via standard numerical methods that have considerable higher computational efficiency versus existing explicit methods of the same order. The derivation of the new explicit schemes is inspired on the finite increment calculus (FIC) procedure used for obtaining stabilized numerical schemes in fluid and solid mechanics. The new (so‐called) explicit FIC‐Time (EFT) schemes allow considerable larger time steps than the standard first order forward Euler (FE) scheme and the second and third order Adams–Bashforth schemes. The comparison with Runge–Kutta schemes also favors the FIC‐Time schemes in terms of the limit time step size (for second order schemes) and the total number of matrix‐vector multiplications per time step (for second and third order schemes). The new first order explicit schemes have a faster convergence to steady‐state than the FE scheme. The accuracy and efficiency of the new EFT schemes are verified in examples of application to the transient heat conduction equation using the finite element method.

Nonlinear dynamic response of a temperature-dependent FGM spherical shell under various boundary conditions and thermal shocks: Examination of dynamic snap-through

In this study, a comprehensive exploration of the nonlinear dynamic snap-through phenomenon observed in shallow spherical shells composed of functionally graded materials (FGMs) is examined. The analysis encompasses the interplay of thermomechanical characteristics within metal and ceramic phases, taking into account their temperature-dependent behaviors. To address this problem, the transient heat conduction equation in a nonlinear form, accounting for the temperature dependence factor is derived. To solve this complexity, the Generalized Differential Quadrature (GDQ) method is employed, coupled with the Crank-Nicolson time integrating scheme, leveraging an iterative approach for precision. The formulation of nonlinear dynamic motion equations is based on Hamilton's principle and incorporates the utilization of nonlinear strain-displacement equations and uncoupled thermoelasticity. In the spatial domain, GDQ proves invaluable in solving these nonlinear, coupled equations. In the temporal domain, the Newmark procedure in conjunction with the Newton-Raphson iterative method are utilized to navigate through dynamic equations, yielding insights into shell's behavior over time. In the initial stages of the investigation, we validate our spherical shell formulation and methodologies by comparing shell's response with existing, simpler works. This step ensures the reliability and accuracy of our model. The primary objective is to assess the presence and characteristics of the thermal snap-through phenomenon within the shell under these diverse conditions.

Assessment of Axisymmetric Dynamic Snap-Through and Thermally Induced Vibrations in FGM Cylindrical Shells Under Instantaneous Heating

This paper addresses the investigation of axisymmetric thermally induced vibrations in Functionally Graded Material (FGM) cylindrical shells. It considers temperature-dependent (TD) properties and geometric non-linearity (the von Karman effect). The study systematically solves a transient heat conduction equation using finite differences method and the Crank–Nicolson method. During the heating stages, the evaluation of thermal forces and moments takes place. Equations of motion are derived through the application of Hamilton’s principle. Spatial dependencies are discretized using the generalized Ritz method, while temporal dependencies are approximated using the [Formula: see text]-Newmark method with Newton–Raphson linearization. A comparative analysis validates the procedure’s efficiency and precision. Parametric studies explore the influence of parameters, including the temperature-dependency material properties, geometric nonlinearity, and shell’s power-law index, providing valuable insights into FGM shell behavior under thermal shock.

Magnetic-thermal-mechanical coupling behavior simulation for a porous high-temperature superconductor based on fractal derivatives

In this paper, the magnetic-thermal-mechanical coupling behavior for a porous high-temperature superconductor placed in a pulsed field is studied numerically within the framework of finite element analysis and based on fractal derivatives. Firstly, as a kind of oxide ceramic material, the porous properties of high-temperature superconductors (SCs) are characterized through fractal methods. Then, we obtained the Maxwell equation and heat conduction equation in the form of fractal derivatives, and thus the difficulties brought by the multi connectivity of materials to finite element (FEM) modeling can be overcome. The FEM simulation results indicate that the porous properties have a significant impact on the maximum trapped magnetic field, surface temperature of superconductors, and maximum stress inside superconductors. The presented method can also provide a more efficient solution for the multi-field coupling simulation of other porous electromagnetic media.

Assessment of continuous laser ablation model for lightweight quartz fiber reinforced phenolic composite

AbstractA comprehensive investigation into the interaction between a continuous wave (CW) laser and lightweight quartz fiber reinforced phenolic (LQFRP) composite would provide valuable insights into the characteristics and performance of ablation. To simulate the CW laser ablation of LQFRP composite, a 2D symmetrical element model was developed, incorporating the heat conduction equation, resin pyrolysis mechanism, and deformed geometry module. This model was the first to examine the internal pyrolysis degree and gas flow field of LQFRP composite under CW laser irradiation, and its predictions were in reasonable agreement with experimental measurements. Additionally, the model was used to simulate CW laser ablation with varying spot sizes and power densities. The results revealed that during the laser ablation process, an internal high‐pressure zone formed on the side of the ablation hole. The volatilization of SiO2 was identified as the dominant dissipation mechanism for material removal. The width of the ablation hole was positively correlated with the laser spot size, while the depth of the ablation hole was positively correlated with the laser power density.Highlights The model first studied the internal pyrolysis degree and flow field of LQFRP composite under laser irradiation. The volatilization of SiO2 was identified as the dominant dissipation mechanism for material removal. An internal high‐pressure zone formed on the side of the ablation hole during the laser ablation process. The width of the ablation hole was positively correlated with the laser spot size, while the depth of the ablation hole was positively correlated with the laser power density.

Analysis of the Cooling Modes of the Lining of a Ferroalloy-Casting Ladle

Analysis of the Cooling Modes of the Lining of a Ferroalloy-Casting Ladle

Nonlinear motion characteristics of the high-temperature superconducting levitation system with boundary crack

High-temperature superconducting levitation systems have nonlinear behaviors, such as the period-doubling bifurcation and chaotic vibration under external excitation, in connection with the nonlinear hysteresis interaction between the high-temperature superconductors (HTSC) and permanent magnet. The HTSC is a non-ideal type II superconductor in which the fracture is generated internally during manufacturing, and the material properties are brittle. The HTSC tends to crack under a strong magnetic field and electromagnetic force, and its internal defects are likely to cause structural damage with the variable Lorentz force. The fracture performance of the superconducting magnetic levitation system will affect the temperature variation of HTSC, in which the temperature is coupled with the internal electromagnetic force. In this paper, we analyze the fracture characteristics of the HTSC in nonlinear vibration with thermal effect. A superconducting magnetic levitation system model with boundary cracks is applied to study the coupling of multiple physical fields in dynamic processes. The Maxwell equation and superconducting electromagnetic constitutive equation are used to obtain the superconductor’s current density, and the temperature is calculated through the heat conduction equation. The superconducting magnetic flux flow and creep model is applied to analyze the magnetic flux motion inside the superconductors. We compare the superconducting levitation system’s electromagnetic force, temperature, and current density with four critical current densities. The numerical results show that the critical current density of superconductors significantly impacts bifurcation motion, and the temperature of the crack tip of the superconductors varies greatly during vibration, causing the superconductor to lose its superconductivity.

Regularization with two differential operators and its application to inverse problems

We introduce a regularization method with two differential operators for solving a linear ill-posed operator equation system. The existence and uniqueness of regularized solutions to the problem are derived. With an a priori as well as an a posteriori parameter choice strategy, convergence and convergence rates of the regularized solution are also obtained. As an application, we apply the regularization to a simultaneous inversion of the source term and the initial value problem for a heat conduction equation, numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Analytical solution of a pollutant transport model for unsaturated soil considering the effects of consolidation and temperature

The migration behavior of pollutants is affected by consolidation and temperature when using thermal desorption technology to clean contaminated sites. Based on a one-dimensional consolidation model for unsaturated soil and the traditional heat conduction equation, a pollutant transport model accounting for the combined effects of consolidation and temperature was established in this paper. An analytical solution was obtained by using the separation of variables method and the integral transformation method. In addition, the correctness of the proposed model was verified via a comparison between the existing analytical solution and the theoretical model. Finally, adopting benzene as the research object, the influence of different factors on pollutant migration was studied. It was found that the growth rate of the pollutant concentration increased with increasing consolidation pressure, and the final pollutant concentration decreased with increasing consolidation pressure. The pollutant concentration increment due to temperature first increased and then decreased with increasing migration distance. The higher the Soret coefficient and volumetric moisture content are, the higher the pollutant concentration.

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.

Optimizing Cloud Service Load Balancing Through Heat Conduction Equation Applications

Optimizing Cloud Service Load Balancing Through Heat Conduction Equation Applications

IMPROVEMENT OF THE METHODOLOGY FOR OPTIMIZING THE PARAMETERS OF COMPLEX SYSTEMS UNDER THE INFLUENCE OF THERMAL LOAD SOURCES

The article investigates some aspects of solving the problem of constructing mathematical models for optimizing the parameters of local laser action on multilayer microbiological systems. The constraints on the desired laser parameters and on the temperature field in the material subject to laser fission are set. Verification of compliance with the restrictions on not exceeding the maximum value of the temperature field requires multiple calculations of the corresponding temperature field in the microbiological material. Control over the temperature field limits ensures the viability of its parts during the biotechnological process of laser fission. To construct adequate optimization mathematical models, the author substantiates the adequacy of the computational mathematical models describing the state of a microbiological system under the action of laser radiation sources. A multi-point boundary value problem with a system of inhomogeneous differential heat conduction equations for a multilayer microbiological medium is investigated and the correctness of this problem is substantiated for minor perturbations of the right-hand side of the differential equation. The results obtained in the article guarantee the adequacy of applied optimization mathematical models for finding rational values of technical parameters of laser emitters. In order to improve the accuracy of optimization of the technical parameters of laser emitters, the article presents a mathematical modeling of the preparatory stage of embryo defrosting. The presented mathematical models and methods of their implementation are necessary to improve the quality of embryo defrosting. Improvement of the methodology for solving applied biotechnological problems will certainly lead to the complication of mathematical models, but it will increase the accuracy of calculation and optimization of technical parameters of the biotechnological process of laser embryo division.

Fractional Heat Conduction with Heat Absorption in a Solid with a Spherical Cavity under Time-Harmonic Heat Flux

The central-symmetric time-fractional heat conduction equation with heat absorption is investigated in a solid with a spherical hole under time-harmonic heat flux at the boundary. The problem is solved using the auxiliary function, for which the Robin-type boundary condition with a prescribed value of a linear combination of a function and its normal derivative is fulfilled. The Laplace and Fourier sine–cosine integral transformations are employed. Graphical representations of numerical simulation results are given for typical values of the parameters.

Thermoelastic transient memory response analysis of non-localized nano-piezoelectric plates based on Moore-Gibson-Thompson thermoelasticity theory

With their abilities to induce electric charge, stress or deformation in response to external forces as well as the ease of fabrication, design flexibility and excellent electromechanical properties, piezoelectric materials have been widely used in actuators, sensors and even in nano/micro-electro-mechanical systems. With the rapid advancement of nano/micro-technology, from the perspective of theoretical analysis, the priority is to develop applicable models to describe the piezoelectric-thermoelastic responses of piezoelectric nano/micro-structures suffering transient heat conduction by taking the size-dependent effect and the memory-dependent effect into consideration. In this work, a new model based on the existing piezoelectric-thermoelastic model is established by introducing the nonlocality into the constitutive equations and the memory-dependent derivative into Moore-Gibson-Thompson (MGT) heat conduction equation respectively. Then, this new model is applied to investigating the dynamic responses of a piezoelectric nanoplate subjected to thermal shock. The corresponding governing equations are formulated and then solved by Laplace transform and its numerical inversion. In calculation, the influences of the time delay factor and the kernel function of MDD and the non-local parameter on the thermal, electric and elastic fields in the piezoelectric nanoplate are examined. Meanwhile, the predictions of transient response among different thermoelastic models are compared. The numerical results are illustrated graphically and discussed in detail. The obtained results show that MDD has a significant effect on the transient response, where the effect of the kernel function is more pronounced. This work may provide a theoretical reference for strength design, thermal protection and thermal processing strategies for piezoelectric nanodevices.

Active control of thermally induced vibrations of temperature-dependent FGM circular plate with piezoelectric sensor/actuator layers

This study investigates the novel application of active control methods in mitigating thermally induced vibrations of functionally graded material (FGM) circular plates subjected to thermal shock. The top surface of the FGM circular plate is exposed to rapid surface heating, resulting in thermally induced vibrations. To counteract these deflections, two piezoelectric layers, one functioning as a sensor and the other as an actuator, are strategically integrated into the plate opposite sides. This layered configuration effectively distances the direct rapid surface heating from piezoelectric layers. All thermos-electro-mechanical properties of the used materials are considered temperature-dependent. The research employs the Generalized differential quadrature element (GDQE) method and the Crank-Nicolson time marching procedure to solve the nonlinear Fourier-type heat conduction equations for the layered structure. After obtaining the transient temperature profile, the study calculates thermal moments, forces, and pyroelectric terms. These thermally induced effects are then integrated into the motion and electrostatic equations, which inherently exhibit nonlinearity due to the von Kármán large deformations theory. The plate displacements are approximated using the first-order shear deformation theory (FSDT) along the plate thickness, while a quadrature distribution is considered for approximating the electric potential within the piezoelectric layers. The highly coupled electro-mechanical equations are solved using the GDQ method and the Newmark approach, with the Newton-Raphson iterative scheme utilized to address nonlinearity. A significant contribution of this research lies in the exploration of three distinct active control strategies: proportional control, derivative control, and proportional-derivative control. The study thoroughly analyzes the impact of each control type on the plate response, considering various boundary conditions.

Математичні моделі локального нагрівання елементів електронних пристроїв

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in isotropic spatial heat-active media sub-jected to internal and external local heat load, have been developed. In the case of nonlinear boundary-value problems, the Kirchhoff transformation was applied, using which the original nonlinear heat conduction equations and nonlinear boundary conditions were linearized, and as a result, linearized second-order differential equations with partial derivatives and a discontinu-ous right-hand side and partially linearized boundary conditions were obtained. For the final linearization of the partially linearized differential equation and boundary conditions, the ap-proximation of the temperature according to one of the spatial coordinates on the boundary sur-faces of the inclusion was performed by piecewise constant functions. To solve linear bounda-ry-value problems, as well as obtained linearized boundary-value problems with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which analytical solutions of 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 envi-ronment were obtained. Numerical analysis of temperature behavior as a function of spatial co-ordinates for given values of geometric and thermophysical parameters was performed. The in-fluence of the power of internal heat sources and environmental materials on the temperature distribution was studied. To determine the numerical values of the temperature in the given structure, as well as to analyze the heat exchange processes in the middle of these structures, caused by the internal and external heat load, software tools were developed, using which a ge-ometric image of the temperature distribution depending on the spatial coordinates was made.

Numerical Model of Simultaneous Multi-Regime Boiling Quenching of Metals

This work presents a heat transfer and boiling model that computes the evolution of the temperature field in a representative steel workpiece quenched from 850 or 930 °C by immersion in water flowing at average velocities of 0.2 or 0.6 m/s, respectively. Under these conditions, all three boiling regimes were present during cooling: stable vapor film, nucleate boiling, and single-phase convection. The model was based on the numerical solution of the heat conduction equation coupled to the solution of the energy and momentum equations for water. The mixture phase approach was adopted using the Lee model to compute the rates of water evaporation–condensation. Heat flux at the wall was calculated for all regimes using a single semi-mechanistic model. Therefore, the evolution of boiling regimes at every position on the wall surface was automatically determined. Predictions were validated using laboratory results, namely: (a) videorecording the upward motion of the wetting front along the workpiece wall surface; and (b) cooling curves obtained with embedded thermocouples in the steel probe. Wall heat flux calculations were used to determine the importance of the simultaneous presence of all three boiling regimes on the heat flux distribution. It was found that this simultaneous presence leads to high heat flux variations that should be avoided in production lines. In addition, it was determined that the corresponding inverse heat conduction problem to estimate the active heat transfer boundary condition must be set-up for 2D heat flow.

Simplified expression for transverse mode instability threshold in high power fiber lasers.

In this work, we propose an analytical expression for calculating the transverse mode instability (TMI) threshold power, which clearly shows the role of various fiber parameters and system parameters. The TMI threshold expression is obtained by solving the heat conduction equation and the nonlinear coupling equation using the fundamental mode fitted by Gaussian functions. The calculation results of the proposed TMI threshold expression are consistent with the experimental phenomena and simulation results from the well-recognized theoretical model. The influence of some special parameters on the TMI threshold and the power scaling is also investigated. This work will be helpful for fiber design and TMI mitigation of high-power fiber lasers.