Heat Conduction Equation Research Articles

Thermal analysis for in-situ consolidation in the AFP process

An analytical approach was developed for assessing the thermal history of thermoplastic composites during tape placement for in-situ consolidation of the automated fiber placement (AFP) technique. In this study, the heat-conduction equation is developed and solved for internal energy instead of temperature because the diffusivity does not significantly change over a wide range of temperatures in the AFP process, despite the significant change in the specific heat and conductivity. The three-dimensional internal energy history is derived in an integrated form using the Green function technique. The temperature field obtained from the internal energy field agreed with the finite element solution, where the temperature-dependent thermal properties were considered. The effects of manufacturing parameters, such as the placement speed and thickness of the placed laminates, on the thermal history of laminated composites during AFP are discussed using the present approach and finite element analysiss.

Numerical simulation of thermal effects in laser diode end-pumped solid-state lasers utilizing finite element method

Based on the finite element numerical analysis method, the temperature field distribution in the crystal with time is simulated by using the PDE-tool in MATLAB. The critical factors affecting the crystal transient temperature distribution are simulated and analyzed. According to the simulation parameters and results, a simplified method for solving the transient heat conduction equation is proposed. The transient thermal focal length is solved to obtain its distribution expression and an approximate analytical solution. The results provide a theoretical basis for the thermal analysis process of suppressing or utilizing the transient thermal effect in high-energy solid-state lasers.

Nonlinear large amplitude vibrations of higher-order functionally graded beams under cooling shock

In this paper, thermally induced vibrations of beams made of functionally graded materials (FGMs) subjected to cooling shocks are investigated. It is considered that the beam has been made of a mixture of stainless steel (SUS 304) and low-carbon steel (AISI 1020). To model the displacement field, the third-order beam theory, known as the Reddy beam theory (RBT), is used. Material properties depend on temperature and distribution of materials, and this dependence is modeled through the temperature and the location of materials along the thickness direction. Considering the uncoupled thermoelasticity theory, the temperature distribution is obtained using a one-dimensional Fourier-type transient heat conduction equation, and the equations of motion governing the higher-order beam are derived utilizing Hamilton's principle. Solving the equations is done numerically; the generalized differential quadrature method (GDQM) is employed to approximate the spatial derivatives, and the Newton-Raphson scheme is applied to linearize the equations. In addition, for approximation of the time derivatives, the Newmark method is utilized. Subsequently, the effects of various parameters on the non-dimensional lateral deflection of the higher-order beam considering two different types of thermal loading are investigated. A comprehensive parametric study is conducted to study the effects of important parameters including beam thickness, thermal load rapidity time, the amount of applied load, and the FG parameter.

Continuous dependence and convergence for Moore–Gibson–Thompson heat equation

In this paper, we investigate how the solutions vary when the relaxation parameter, the conductivity rate parameter, or the thermal conductivity parameter change in the case of the Moore-Gibson-Thompson heat equation. In fact, we prove that they can be controlled by a term depending upon the square of the variation of the parameter. These results concern the structural stability of the problem. We also compare the solutions of the MGT equation with the Maxwell-Cattaneo heat conduction equation and the type III heat equation (limit cases for the first two previous parameters) and we show how the difference between the solutions can be controlled by a term depending on the square of the limit parameter. This result gives a measure of the convergence between the solutions for the different theories.

Study of the model of the phase transition envelope taking into account the process of thermal storage under natural draft and by air injection

The study proposed mathematical modeling of the enclosing structure of the building, which includes an energy-active panel accumulating solar radiation due to the phase transition of the heat-accumulating material. The mathematical model is based on a two-dimensional non-stationary nonlinear heat conduction equation describing the process of heat transfer in the bearing layer of the enclosing structure and the energy-active panel, taking into account the process of heat accumulation under natural and forced traction by air injection. In comparison of the calculation results, it was found. that the efficiency of the use of forced convection relative to natural was 33–44% in the question of the selection of useful heat, depending on the time (second or first day), and the value of heat accumulation showed almost the same results. Taking into account the research carried out by the authors, it is shown by calculation that enclosing structures in which a heat-accumulating material with a phase transition is used to accumulate solar energy are more energy efficient than simple heat accumulation in the carrier layer. The introduction of highly conductive inclusions into the energy-active panel and the use of forced convection instead of natural traction during heat extraction also significantly increase the energy efficiency of the structure, which can subsequently be used as an option for external fencing under appropriate climatic conditions.

Arbitrary temperature gradient in thermal-postbuckling behavior of polymer nanocomposites reinforced by boron nitride nanotubes

Polymer nanocomposites reinforced by boron nitride nanotubes (BNNTs) have great potential for use in a wide variety of thermal environments. A closed-form solution for nonlinear bending and thermal-postbuckling behavior of BNNT-reinforced nanocomposite beams under arbitrary temperature gradient in the thickness and length directions is presented for the first time. The uniform and functionally graded (FG) distributions of BNNTs through the thickness of macro-or micro-sized beams are considered. For constant temperature or constant heat flux conditions at two ends of the beam, thermal analysis is carried out to obtain temperature field ΔTx,z using the two-dimensional heat conduction equation and the assumption of arbitrary variation in the z direction. The governing equations of modified couple stress Timoshenko beam theory with von Kármán geometric nonlinearity are derived for BNNT-RC beams subjected to thermomechanical loads. Applying change of variable technique, the governing equations are exactly solved to develop closed-form expression for nonlinear transverse deflection. Finally, parametric study is performed to investigate the effects of length scale parameter, aspect ratio (L/h), distribution of BNNTs, and temperature gradient on the nonlinear bending and postbuckling equilibrium paths. It is shown that the thermal-buckling behavior of clamped and cantilever beams is completely different. Under thermomechanical loading, an initial snap-through buckling behavior is observed in the cantilever beam, while there exist both stable and unstable configurations in the static equilibrium path of clamped beam. Moreover, it is found that the assumption of parabolic variation of temperature through thickness is not compatible with real situations, since the temperature field is obtained as harmonic functions with very short wavelengths in the length direction.

Analytical Solution of Transient Heat Conduction through a Hollow Spherical Thermal Insulation Material of a Temperature Dependant Thermal Conductivity

The one-dimensional, spherical coordinate, non-linear partial differential equation of transient heat conduction through a hollow spherical thermal insulation material of a thermal conductivity temperature dependent property proposed by an available empirical function is solved analytically using Kirchhoff’s transformation. It is assumed that this insulating material is initially at a uniform temperature. Then, it is suddenly subjected at its inner radius with a step change in temperature. Four thermal insulation materials were selected. An identical analytical solution was achieved when comparing the results of temperature distribution with available analytical solution for the same four case studies that assume a constant thermal conductivity. It is found that the characteristics of the thermal insulation material and the pressure value between its particles have a major effect on the rate of heat transfer and temperature profile.

Nonlinear consecutive dynamic instabilities of thermally shocked composite circular plates on the softening elastic foundation

In this research, the nonlinear dynamics of a clamped circular composite plate placed on a softening elastic foundation under rapid thermal loading is investigated. In this situation, based on the amount of temperature supplied to the structure and the coefficients of softening elastic foundation, two instabilities may happen one after the other. The structure will thermally buckle and deform dynamically if the applied temperature exceeds a critical level. If the softening coefficient of the elastic foundation is critical, the structure will completely lose its stability after a certain deformation range. A polymer containing graphene platelets (GPL) makes up the system. Based on various functions, the volume fraction of fillers varies along the thickness. The system’s nonlinear dynamic equations are obtained by applying Hamilton’s principle and the Von-Kármán theory. The transient heat conduction equation is solved by the cubic B-spline collocation (CBSC) and Crank–Nicolson procedures. The CBSC and the Newmark methods are used to solve spatially and temporally dependent governing nonlinear differential equations. Also, the Newton–Raphson method is used as a powerful tool to solve nonlinear algebraic equations. The temporal evolution, phase-plane, and post-buckling-to-maximum deflection paths are demonstrated to analyze the instabilities of the plate.

Heat and mass transfer of drops of concentrated solutions during spray dehydration under conditions of convective-radiation energy supply

The results of modeling the dehydration of drops of a concentrated liquid, on the example of ceramics, with convective-radiation energy supply under conditions of direct-flow and counter-current phase motion, as well as pulsed counter-flows of gas, are presented. A model for the dehydration of a single drop is formulated based on the equations of heat conduction with a source term and diffusion of moisture, taking into account the change in its size due to evaporation. This takes into account the influence of the convective vapor flow from the evaporating droplet surface (Stefan flow), as well as the blowing of evaporating vapor into the hot gas flow on the heat transfer coefficient (Spalding correction). The impact of infrared radiation is described by the Bouguer equation. The equation of motion of a drop in a gas flow takes into account the forces due to gravity, the difference in velocities and phase densities. As a result of numerical simulation, it was found that with countercurrent phase movement, the intensity of dehydration is higher than with cocurrent flow. This is due to both an increase in the relative velocity of the phases and an increase in the residence time of the drop in the intense region of infrared radiation. It is shown that further intensification of evaporation is possible due to the creation of pulsed counter gas flows. The calculated results are compared with the experimental data, which confirms the adequacy of the model. The results of the study can be useful in the development of new heat technologies and devices for dehydration of concentrated solutions and suspensions.

Temperature Effects on Electrical Response of FinFET Transistors in the Static Regime

The aim of this work is to develop an electrothermal model capable of predicting the FinFET operating temperature in different drain and gate biases with different gate lengths. A new effective electron mobility that depends on mobility degradation due to phonon scattering, surface roughness, and to Colombian interaction is used to enhance the drift-diffusion (D-D) model. To better investigate the electrical characteristics, heat conduction equation is coupled with the D-D model. To achieve high evidence to the proposed model, the simulation of ID-VG and ID-VD is compared with experimental data and Monte Carlo (MC) simulation, and a good concordance is observed. The output characteristics degradation due to device temperature is analyzed. We demonstrated that a drain current of 1 mA/_m is obtained when the operating temperature is around 360 K for a 20-nm FinFET. The effect of FinFET biases on power dissipation and device temperature along the channel region and in the drain side channel region interface is investigated for 10-and 20-nm FinFET gate lengths. We found that: 1) the power dissipation is four times greater when the gate length decreases from 20 to 10 nm and 2) the critical temperature, which is responsible to vanish the drain current, is from 380 to 460 K for Lg <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$=$</tex-math> </inline-formula> 20 and 10 nm, respectively. This work addresses a key reliability constraint for CMOS-based FinFET circuit designers while designing tuned gate and drain biases circuit.

Heat and mass transfer in the processes of drying thin natural leathers pasted on smooth surfaces

The results of the study of drying thin natural leather pasted on a smooth surface are presented. The results of solving the differential equation of non-stationary heat conduction with constant thermophysical coefficients are used to determine the average temperature during the period of decreasing drying rate. The calculated values of temperatures for moisture-proof and moisture-free skin surfaces are given. Methods for simplifying solutions of nonlinear equations with the aim of linearizing these equations for approximating solutions with variable transport coefficients are considered. The use of the method of piecewise stepwise approximation of the transfer coefficients with constant values of these coefficients over short time intervals showed that for low-intensity drying processes, the solutions of the linear heat equation are implemented quite accurately, confirming the regularities obtained empirically. The calculation of the duration of drying according to the method of B. S. Sazhin is given. The results of the processing of experimental curves for drying leathers pasted are presented. The reliability of the obtained equations is verified and the experiment is compared with the calculated values using the formulas. The obtained approximate analytical solutions with reliable values of the transfer coefficients, confirmed by experimentally established regularities, are of practical importance. Together with experimental methods, analytical methods make it possible to establish optimal drying regimes and more accurately generalize experimental data.

Analytical and numerical radially symmetric solutions to a heat equation with arbitrary nonlinearity

The paper deals with the construction of radially symmetric heat waves, which are solutions to the heat conduction equation with an arbitrary form of nonlinearity under nonzero boundary condition specified on a moving manifold. The boundary value problem under study is a generalization of those solved by us earlier. Firstly, the class of the considered parabolic equations is extended; secondly, the boundary condition generating a heat wave in a space of arbitrary dimensionality has a more general form. A new theorem of the existence and uniqueness of the heat-wave-type analytical solution is proved for this problem. An approximate method of constructing solutions of the required form is proposed, which is based on expansion in radial basis functions combined with the collocation method. At each time step, the solution is constructed in two stages. The first stage is solving a problem in the region bounded by a specified moving manifold and a heat wave front, which is a priori unknown and evaluated during solving. Herewith, a special substitution is used, i.e. the required function and the spatial variable change their roles. In the second stage, the solution is completed in the region bounded by the positions of the specified moving manifold on a current step and at the initial time. The boundary conditions are defined from the first-step solution. In the test example, the solutions constructed by the developed algorithm are compared with the known exact solution. Calculations show a good accuracy of the numerical solutions at various values of the numerical parameters, including space dimensionality. The observed numerical convergence with respect to the time step shows the correctness of the proposed computational procedure.

Transient Nonlinear Heat Conduction in Concrete Structures: A Semi-Analytical Approach.

Thermal loading, especially in fire scenarios, challenges the safety and long-term durability of concrete structures. The resulting heat propagation within the structure is governed by the heat conduction equation, which can be difficult to solve analytically because of the nonlinearity related to the thermophysical properties of concrete. A semi-analytical approach for the transient nonlinear heat conduction problem in concrete structures was established in the present work. The nonlinearity related to the temperature-dependent thermal conductivity, mass density, and specific heat capacity of heated concrete was taken into consideration. A Taylor series approximate solution was first established within a small neighborhood, employing the Boltzmann transformation in combination with the mean value theorem. Thereafter, it was extended to the whole domain by utilizing the Bernstein polynomial. The semi-analytical approach was validated by comparing it with the numerical results of two independent Finite Element simulations of nonlinear heat conduction along concrete plates, subjected to either moderate or fierce thermal loading. Absolute values of the relative errors are smaller than 5%. The validated semi-analytical approach was further applied to prediction of the temporal evolution of the temperature field of a scaled model of a subway station, subjected to fire disaster. The nonlinearities, related to the time-dependent surface temperature and the temperature-dependent thermophysical properties of concrete, were taken into consideration. The predictions agree well with the experimental measurements. The established semi-analytical approach exhibits good accuracy and stability, providing insight into the interaction between the thermophysical properties of concrete in the heat conduction process.

One-dimensional inverse problems of determining the kernel of the integro-differential heat equation in a bounded domain

Abstract The integro-differential equation of heat conduction with the time-convolution integral on the right side is considered. The direct problem is the initial-boundary problem for this integro-differential equation. Two inverse problems are studied for this direct problem consisting in determining a kernel of the integral member on two given additional conditions with respect to the solution of the direct problems, respectively. The problems are replaced with the equivalent system of the integral equations with respect to unknown functions and on the basis of contractive mapping the unique solvability inverse problem.

Generalized MGT Heat Transfer Model for an Electro-Thermal Microbeam Lying on a Viscous-Pasternak Foundation with a Laser Excitation Heat Source

In this study, the effects of laser light on the heat transfer of a thin beam heated by an applied current and voltage are investigated. Laser heating pulses are simulated as endogenous heat sources with discrete temporal properties. The heat conduction equation is developed using the energy conservation equation and the modified Moore–Gibson–Thompson (MGT) heat flow vector. Thermal and structural analysis of Euler–Bernoulli microbeams is provided with the support of visco-Pasternak’s base with three parameters. Using the Laplace transform method, an approximation of an analytical solution is found for the field variables being examined. A comparison was made of the impacts of laser pulse length, the three foundation coefficients, and the thermal parameters on the responses to changes in measured thermophysical fields, such as deflection and temperature.

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.

An investigation on responses of thermoelastic interactions of transversely isotropic thick circular plate due to ring load with memory-dependent derivatives

The present investigation has focus on the variations in a transversely isotropic thick circular plate subjected to ring loading. The modified Green Nagdhi (GN) heat conduction equation with and without energy dissipation by introducing memory-dependent derivatives (MDD) with two temperatures has been used to model the problem. General solutions to the field equations have been found using the Hankel and Laplace transform. The analytical expressions of stress, conductive temperature, and components of displacement are obtained in the transformed domain. Physical solutions have been obtained using numerical inversion techniques. The effects of Kernel functions of memory-dependent derivatives have been depicted graphically. The present investigation also reveals some specific cases.

Application of GDQ method to large amplitude response of FGM joined spherical-conical shells under rapid surface heating

Geometrically non-linear thermally induced vibrations of functionally graded material (FGM) joined spherical-conical shells are analyzed in the current research. Thermo-mechanical properties of the shells are assumed to be temperature and position dependent. The system of joined spherical-conical shells is subjected to thermal shock on the ceramic-rich surface, whereas the metal-rich one is kept at reference temperature. The one-dimensional transient heat conduction equation is established and solved via the generalized differential quadrature (GDQ) and Crank-Nicolson methods. This equation is non-linear since the thermo-mechanical properties of the shells are temperature dependent. The total functional of the shells is obtained under the assumptions of uncoupled thermoelasticity laws, first order shear deformation shell theory, and the von Kármán type of geometrical non-linearity. Non-linear coupled equations of motion are solved via the iterative Picard method accompanied with the β-Newmark time approximation technique. Numerical results are well-validated with the available data for the case of single FGM spherical shell. Parametric studies are conducted to examine the influences of conical and spherical shell geometries, material composition, temperature dependence, in-plane and out-of-plane mechanical boundary conditions, various configuration of conical/spherical shell system, and thermal boundary conditions. It is highlighted that thermally induced vibrations indeed exists.

Heat flow variations in 2 km deep borehole Litoměřice, Czechia

Temperature in 2 km deep borehole Litoměřice, drilled in 2007, was repeatedly logged down to 1700 m in the period 2007 – 2020. We were able to monitor a return of the temperature to the equilibrium temperature-depth profile undisturbed by drilling. The uppermost part of the profile contains signal of the recent warming manifested by a negative temperature gradient close to the surface and a temperature minimum at a depth of about 40 m. The minimum has been migrating downward at a rate of 1.5 – 2 m per year in the period 2015 – 2020.A detailed knowledge of temperature gradient together with thermal conductivity, diffusivity and heat production measurements on the drill-core samples of mica-schist that occurs below 900 m depth enabled us to analyze the heat flow vertical variations in the lithologically homogeneous depth section 900 – 1700 m. We came to the conclusion that temperature-depth profile in this section contains a robust climate signal of the last glacial cycle. The reconstructed ground surface temperature history indicates the magnitude of the last glacial – Holocene warming 13 -15 K and existence of a minimum 15 – 20 ka. The long-term mean ground surface temperature +1 - +2 °C suggests that the borehole site was permafrost free for most of the glacial cycle. Existence of about 100 m deep permafrost is possible in the coldest part of the last glacial.The steady-state surface heat flow has been estimated at 88 mW/m2. The reconstructed ground surface temperature history used as a surface forcing function in a numerical solution of the transient heat conduction equation provided an estimate of the present-day heat flow in the well. The estimate is practically independent from the poorly constrained conductivity of the 900 m thick sedimentary cover. According to it the present-day heat flow is lower than the steady-state one by 20 - 30 mW/m2 in the first hundreds of meters below the surface and still by about 10 mW/m2 at a depth of 1 km.

Research on MEMS Solid-State Fuse Logic Control Chip Based on Electrical Explosion Effect

A microelectromechanical systems (MEMS) solid-state logic control chip with three layers-diversion layer, control layer, and substrate layer-is designed to satisfy fuse miniaturization and integration requirements. A mathematical model is established according to the heat conduction equation, and the limit conditions of different structures are presented. The finite element multi-physical field simulation method is used to simulate the size and the action voltage of the diversion layer of the control chip. Based on the surface silicon process, fuse processing, and testing with the MEMS solid-state fuse-logic control chip, a diversion layer constant current, maximum current resistance test, and a control layer of different bridge area sizes, the bridge area size is 200 × 30 μm, and the minimum electrical explosion voltage is 23.6 V. The theoretical calculation results at 20 V and 100 μF demonstrate that the capacitor energy is insufficient to support the complete vaporization of the bridge area, but can be partially vaporized, consistent with the experimental results.