Hyperbolic Heat Conduction Equation Research Articles

Causal explicit algorithm for heat conduction in a plasma

Hyperbolic heat conduction extends standard Spitzer-Harm heat conduction by including a term proportional to the time derivative of the heat flux. The new term arises from a kinetic derivation of the heat flux that includes higher order corrections. We present a causal explicit numerical algorithm for solving the nonlinear hyperbolic heat conduction equation in an unmagnetized plasma. The maximum stable timestep for the causal explicit algorithm scales linearly with the cell size, owing to the hyperbolic nature of the problem. This is in contrast to the quadratic scaling of the maximum stable timestep with the cell size for the parabolic forward time centered space algorithm. The favorable scaling of the timestep with the cell size enables a practical explicit implementation of heat conduction in high-performance massively parallel plasma codes. In particular, we have implemented the causal explicit algorithm in the laser plasma interaction code pF3D. We verify the CE algorithm and analyze its convergence rate by simulating a harmonic mode, which has an analytic solution within the context of the HHC model. We also compare simulations using the CE algorithm to those using the forward time centered space algorithm on a pair of test problems: evolution in time of a Gaussian temperature perturbation in a uniform plasma and heat transport in the presence of inverse bremsstrahlung heating by a Gaussian laser speckle.

Heat transfer analysis in a longitudinal porous trapezoidal fin by non-Fourier heat conduction model: An application of artificial neural network with Levenberg–Marquardt approach

Thermal critical issues commonly occur in advanced electrical devices as a response to excessive heat generation or a loss in efficient surface area for heat exclusion. This issue can be addressed by utilizing the extended surface to improve the heat transfer performance of the devices. Thus, the present analysis is devoted to scrutinizing the non-Fourier unsteady heat transference of a trapezoidal porous fin. Levenberg–Marquardt technique of backpropagation artificial neural network (LMT-BANN) is employed here to analyze the thermal variation in the fin. The developed governing equation is the hyperbolic heat conduction equation (HHCE), which is transformed into a dimensionless partial differential equation (PDE) using dimensionless variables. LMT-BANN is employed on thermal numerical data and is developed to trace numerical approximation of fin problem using a methodology that includes testing, training, and validation. A graphical visualization of the consequences of thermal variables on the temperature field is presented. The notable evidence of this study reveals that as the magnitude of the convection factor increases, thermal dissipation through the fin gradually decreases. Further, the LMT-BANN technique has been determined to be an effective, reliable, and rapidly convergent stochastic computational solver that can be used effectively for examining the thermal model.

Numerical simulation of the thermal wave induced by a moving interfacial heat source with respect to Christov–Cattaneo's equation

AbstractThe hyperbolic wave equation of heat conduction with respect to Christov's formulation is utilized with the Streamline‐ Upwind‐Petrov‐Galerkin method in space and the Θ, Houbolt, linear acceleration, Wilson‐Θ and Newmark methods in time. The derivation of this equation and its matrix formulation are shown. A 2D transient finite element simulation of a generic asperity with an infinite line heat source in an interface, either as a heat flux density q̇ or temperature distribution T, is performed for Math = [0.5; 1.0]. A sensitivity study is presented for the mentioned numerical schemes. The temperature jump in the solution is interpreted as an indicator for a thermal shock.

Two-Dimensional C-V Heat Conduction Investigation of an FG-Finite Axisymmetric Hollow Cylinder

In the present work, we implement a graded finite element analysis to solve the axisymmetric 2D hyperbolic heat conduction equation in a finite hollow cylinder made of functionally graded materials using quadratic Lagrangian shape functions. The graded FE method is verified, and the simple rule of the mixture with power-law volume fraction is found to enhance the effective thermal properties’ gradation along the radial direction, including the thermal relaxation time. The effects of the Vernotte numbers and material distributions on temperature waves are investigated in depth, and the results are discussed for Fourier and non-Fourier heat conductions, and homogeneous and inhomogeneous material distributions. The homogeneous cylinder wall made of SUS304 shows faster temperature wave velocity in comparison to the ceramic-rich cylinder wall, which demonstrates the slowest one. Furthermore, the temperature profiles along the radial direction when n = 2 and n = 5 are almost the same in all Ve numbers, and by increasing the Ve numbers, the temperature waves move slower in all the material distributions. Finally, by tuning the material distribution which affects the thermal relaxation time, the desirable results for temperature distribution can be achieved.

Analytical solution of non-Fourier heat conduction in a 3-D hollow sphere under time-space varying boundary conditions

In the current research, a comprehensive analytical technique is presented to evaluate the non-Fourier thermal behavior of a 3-D hollow sphere subjected to arbitrarily-chosen space and time dependent boundary conditions. The transient hyperbolic thermal diffusion equation is driven based upon the Cattaneo-Vernotte (C-V) model and nondimensionalized using proper dimensionless parameters. The conventional procedure of separation of variables is applied for solving the 3-D hyperbolic heat conduction equation with general boundary conditions. In order to handle the time dependency of the boundary conditions, Duhamel's theorem is employed. Furthermore, for the purpose of demonstrating the applicability of the obtained general solution, two particular cases with different time-space varying boundary conditions are considered. Subsequently, their respective non-Fourier thermal characteristics are elaborately discussed and compared with the Fourier case. The quantitative analysis is carried out, including the profiles of the time-dependent temperature and 3-D distributions of temperature at different time frames. Eventually, the influences of the controlling factors such as Fourier number and Vernotte number on the temperature field distributions within a hollow sphere for both cases are assessed. The findings reveal that the lag time in the hyperbolic thermal propagation diminishes with a decrement of Vernotte number, and it asymptotically vanishes for the Fourier case. Also, the number and severity of the jump points that occurred in the non-Fourier cases decrease by increasing the Fourier number, and these points finally vanish at particular Fourier numbers.

Investigation of the fracture problem of functionally graded materials with an inclined crack under strong transient thermal loading

In this paper, a special method for solving transient thermal stress intensity factors (TSIFs) in functionally graded materials (FGMs) with inclined cracks under strong, transient thermal shock loading is established based on the hyperbolic heat conduction equation. First, the Newmark method is employed to obtain the strong transient temperature field of FGMs containing inclined cracks. Then, the transient interaction energy integral method (IEIM) is used to solve the TSIFs for inclined cracks. In addition, the influences of the distribution patterns of the thermal relaxation parameter, the crack length and the crack angle on the TSIFs, and the crack growth angle are investigated. Moreover, we discuss the difference between using the hyperbolic heat conduction theory and the classical, Fourier heat conduction theory in calculating the TSIFs of FGMs with inclined cracks. The results show that the present, combined IEIM-Newmark method can be used to solve strong transient thermal shock crack problems with high efficiency. This work can be used for the thermal design, evaluation, and engineering applications of FGM structures.

Mathematical modeling of temperature fields of spacecraft antenna reflectors with a new inte-gral transformation

Prediction of the mechanical properties of the reflector, and above all, the deviation of the highly accurate shape of the reflecting surface (RSS) from the given one is the main goal of designing spacecraft antennas. Distortion of the RSS is determined by the stress-deformed state of the elements of the reflector structures under the conditions of orbital operation. At the same time, the main factor determining the distortion of reflectors with RSS in open space is temperature deformation due to the uneven distribution of solar heat fluxes among structural elements. Therefore, the development of methods and models for calculating temperature fields in reflectors during heat flows on the surface is relevant. In the article, for the first time, a new finite integral transformation for the Laplace equation in a cylindrical coordinate system is constructed for a region bounded by several closed piecewise smooth contours. The inverse transformation formula is given. In the article, for the first time, a mathematical model for the calculation of temperature fields in a paraboloid rotating with a constant angular velocity is constructed, taking into account the finite speed of heat propagation in the form of a boundary value problem of mathematical physics for the hyperbolic equation of heat conduction with Dirichlet boundary conditions. With the help of the developed integral transformation, the temperature fields in the paraboloid were found in the form of convergent series according to the Fourier functions. The found solution to the generalized boundary value problem of the heat transfer of the paraboloid of rotation can find application in modulating the temperature fields that arise in the antenna reflectors of space vehicles. The developed integral transformation makes it possible to obtain solutions to complex boundary value problems of mathematical physics.

Space‐time stabilized FEM applied on the hyperbolic heat conduction equation for high speed frictional contact

AbstractFrictional contact results in dissipation of energy, wherein one part is transformed into thermal energy. This leads to heating under the surface, where the resulting transient temperature fields at short times and high Peclet numbers are of interest. Therefore, the numerical method of finite elements with space‐time stabilization technique is applied on the 2D hyperbolic heat conduction equation of Cattaneo‐Vernotte with respect to a moving frame. It is shown, that the combination of the θ‐Method with Streamline‐Upwind‐Petrov‐Galerkin‐FEM gives satisfying solutions for a wide broadband of Peclet numbers.

Thermo-vibrational analyses of skin tissue subjected to laser heating source in thermal therapy

Laser-induced thermal therapy, due to its applications in various clinical treatments, has become an efficient alternative, especially for skin ablation. In this work, the two-dimensional thermomechanical response of skin tissue subjected to different types of thermal loading is investigated. Considering the thermoelastic coupling term, the two-dimensional differential equation of heat conduction in the skin tissue based on the Cattaneo–Vernotte heat conduction law is presented. The two-dimensional differential equation of the tissue displacement coupled with the two-dimensional hyperbolic heat conduction equation in the tissue is solved simultaneously to analyze the thermal and mechanical response of the skin tissue. The existence of mixed complicated boundary conditions makes the problem so complex and intricate. The Galerkin-based reduced-order model has been utilized to solve the two-sided coupled differential equations of vibration and heat transfer in the tissue with accompanying complicated boundary conditions. The effect of various types of heating sources such as thermal shock, single and repetitive pulses, repeating sequence stairs, ramp-type, and harmonic-type heating, on the thermomechanical response of the tissue is investigated. The temperature distribution in the tissue along depth and radial direction is also presented. The transient temperature and displacement response of tissue considering different relaxation times are studied, and the results are discussed in detail.

On the Determination of Thermal Conductivity From Frequency Domain Thermoreflectance Experiments

Abstract The Fourier and the hyperbolic heat conduction equations were solved numerically to simulate a frequency-domain thermoreflectance (FDTR) experiment. Numerical solutions enable isolation of pump and probe laser spot size effects and use of realistic boundary conditions. The equations were solved in time domain and the phase lag between the temperature of the transducer (averaged over the probe laser spot) and the modulated pump laser signal was computed for a modulation frequency range of 200 kHz–200 MHz. Numerical calculations showed that extracted values of the thermal conductivity are sensitive to both the pump and probe laser spot sizes, while analytical solutions (based on Hankel transform) cannot isolate the two effects. However, for the same effective (combined) spot size, the two solutions are found to be in excellent agreement. If the substrate (computational domain) is sufficiently large, the far-field boundary conditions were found to have no effect on the computed phase lag. The interface conductance between the transducer and the substrate was found to have some effect on the extracted thermal conductivity. The hyperbolic heat conduction equation yielded almost the same results as the Fourier heat conduction equation for the particular case studied. The numerically extracted thermal conductivity value (best fit) for the silicon substrate considered in this study was found to be about 82–108 W/m/K, depending on the pump and probe laser spot sizes used.

Fractional Modeling of Fin on non-Fourier Heat Conduction via Modern Fractional Differential Operators

The enhancement of heat transfer for electronic kits and automobiles has become highly dependent on the finned heat exchangers; this is because fin provides high heat transfer rate and superior performance with a significant temperature reduction. In this manuscript, a fractional modeling of non-Fourier heat conduction problem of a fin is proposed within the periodic temperature boundary condition. The mathematical modeling is performed via classical theory of heat conduction that is directly proportional to temperature gradient through which hyperbolic heat conduction equation for a fin is generated. The hyperbolic heat conduction equation for a fin is fractionalized via modern approaches of fractional differentiations, namely Atangana–Baleanu and Caputo–Fabrizio differential operators. In order to have analyticity of hyperbolic heat conduction equation for a fin, we invoked the mathematical techniques of Laplace transform. The exact solutions of temperature distribution have been obtained in terms of Fox-H and Mittag–Leffler functions with the product of convolution. The solutions of temperature distribution have been classified into integer verses non-integer theories by making fractional parameters $$ \alpha = \beta = 1 $$ and $$ \alpha \ne \beta \ne 1 $$ , respectively. Our results suggest that due to variability of different rheological parameters on temperature distribution, the cooling process is faster via fractional models in comparison to non-fractional model. Additionally, it is also observed that thermal wave propagates at a specific time results the reciprocal trend in temperature distribution.

Effect of convection boundary condition on hyperbolic heat conduction in thermoelastic medium

This paper aims to study the effect of heat transfer by convection on thermoelastic rectangular solid medium in the context of hyperbolic heat conduction. The studied geometry is a two dimensional finite thin rectangular plate without internal heat generation, which is initially at a uniform temperature and subjected to convection heat transfer from the extreme edge(y=a) while the opposite side is kept insulated and remaining two sides are at a constant temperature. The material properties are assumed to be constant. The differential transform method is applied to solve the hyperbolic heat conduction equation to obtain temperature distribution in the spatial and temporal domain. Then by applying the obtained temperature in the thermoelastic equation, the displacement component, and the stress field are calculated. Also, the effect of convection boundary conditions on temperature distribution and thermoelastic field are illustrated numerically and graphically for a copper plate. It is observed that the compressive and tensile stress occurs along y-direction due to heat transfer through convection at one end.

Propagation of heat wave in composite cylinder using Cattaneo- Vernotte theory

Using Cattaneo-Vernotte theory, a general analytical solution for the distribution of non-axisymmetric temperature in a long multi-layer composite cylinder is presented. The temperature distribution due to an initial asymmetric excitation in the cylinder is extracted by solving the hyperbolic heat conduction equation using the separation of variables method. The analytical solution is verified with the results of the finite difference method. In addition, the effect of imperfect interface on the temperature distribution is studied. According to results, the temperature in a multi-layer composite cylinder is dominantly affected by heat waves, and propagate through layers due to the discontinuity between initial and boundary conditions, and their reflections from the interfaces due to the difference in thermal properties of layers. In addition, the heatwaves generated on the imperfect interfaces, due to their thermal resistance, disturb the temperature. Also, the effect of fiber angle on the temperature distribution in radial and circumferential directions is studied. Finally, the temperature distribution is examined in terms of time and it is shown that the temperature distribution in terms of time at a specific point generally has an oscillatory behavior.

Effect of convection boundary condition on hyperbolic heat conduction in thermoelastic medium

This paper aims to study the effect of heat transfer by convection on thermoelastic rectangular solid medium in the context of hyperbolic heat conduction. The studied geometry is a two dimensional finite thin rectangular plate without internal heat generation, which is initially at a uniform temperature and subjected to convection heat transfer from the extreme edge(y=a) while the opposite side is kept insulated and remaining two sides are at a constant temperature. The material properties are assumed to be constant. The differential transform method is applied to solve the hyperbolic heat conduction equation to obtain temperature distribution in the spatial and temporal domain. Then by applying the obtained temperature in the thermoelastic equation, the displacement component, and the stress field are calculated. Also, the effect of convection boundary conditions on temperature distribution and thermoelastic field are illustrated numerically and graphically for a copper plate. It is observed that the compressive and tensile stress occurs along y-direction due to heat transfer through convection at one end.

MATHEMATIC MODEL OF AND METHOD FOR SOLVING THE DIRICHLET HEAT-EXCHANGE PROBLEM FOR ONE-SHEET ROTARY HYPERBOLOID

It is the first generalized 3D mathematical model developed for calculating temperature fields in the thin-wall one-sheet rotary hyperboloid, which rotates with constant angular velocity around the axis OZ, ; the model was created with the help of known equations of generating lines in cylindrical coordinate system with taking into account finite velocity of heat conductivity and in the form of the Dirichlet boundary problem for the hyperbolic equation of heat conduction under condition that heat-conduction properties of the body were constant, and no internal sources of the heat were available. At initial moment of time, the body temperature was constant; values of temperature on outside surfaces of the body were known and presented continuous function of coordinate.The hyperbolic heat-conductivity equation was derived from the generalized energy transfer equation for the moving element of continuous medium with taking into account finiteness of the heat conductivity velocity.In order to solve the boundary problem, the desired temperature field was represented as a complex Fourier series. The obtained boundary problems for the Fourier coefficients were found with the help of Laplace integral transformations and the new integral transformation for two-dimensional finite space. Intrinsic values and intrinsic functions for the integral transformation kernel were found by method of finite element and the Galerkin methods. Besides, the domain was divided into simplex element.As a result, the temperature field in the thin-wall one-sheet rotary hyperboloid was found in the form of convergent series in Fourier functions.

Thermal shock resistance and thermal fracture of a thermopiezoelectric cylinder based on hyperbolic heat conduction

Accurate evaluation of the thermal shock resistance is great importance when estimating both the life and the performance of piezoelectric structures under thermal shock load. Non-Fourier heat conduction law assumed that the speed of heat propagation in a body is finite, which is more exact to evaluate the thermal shock resistance when the thermal shock time reduces to ps/fs scale or the length size of materials reduces to micro/nanoscale. A hyperbolic heat conduction equation was used to study the thermal shock resistance of a cylindrical piezoelectric structure, in which a permeable penny-shaped crack is considered. The analytical solution of non-Fourier temperature filed is determined by the method of separated variable. The formulas of the associating thermal stress and electrical displacement in the piezoelectric cylinder without crack were obtained. The crack problem is solved by the Dual integral equation and Abel integral equation. The thermal shock resistance is evaluated by associating the stress-based and fracture mechanics-based failure criteria.

Extended Nonequilibrium Variables for 1D Hyperbolic Heat Conduction

Abstract We use the Shannon (information) entropy to define an “entropic” temperature for 1D nonequilibrium system with heat flux. In contrast to the kinetic temperature, which is related to the average kinetic energy, the nonequilibrium entropic temperature is related to the changes in entropy and serves as a criterion for thermalization. However, the direction and value of the heat flux is controlled by the gradient of the kinetic temperature, whereas space-time evolution and the space-time evolution of the heat flux are governed by the hyperbolic heat conduction equation. The extended nonequilibrium variables, namely, entropy, entropic temperature, thermal conductivity, and heat capacity demonstrate a third-law-like behavior at high deviation from equilibrium when the heat flux tends to its maximum value, even at nonzero value of the kinetic temperature. The ratio of the heat flux to its maximum possible value plays a role of an order parameter – it varies from zero in the equilibrium (disordered) state to unity in the nonequilibrium (ordered) state.

MATHEMATIC SIMULATION OF GENERALIZED PROBLEM OF HEAT-EXCHANGE IN THE BODIES WITH HEMISPHERICAL SHAPE

The article presents the first mathematical model for calculating temperature fields of hemispherical bodies, which approximately simulates operation of the diamond-drilling bit and takes into account angular velocity of drilling operations and finite velocity of heat conduction, and which was created as a physicomathematical boundary problem for hyperbolic equation of heat conduction with the Dirichlet boundary conditions. Besides, a new integral transformation was formulated for the two-dimensional finite space, with the help of which and with the help of finite element method and Galerkin method a temperature field was found in the form of convergence series.

Phase transition induced by CW-laser in a finite silver selenide (Ag2 Se) according to (HHCE) model

Thermal effects induced in a finite silver selenide slab by CW laser source are studied. The hyperbolic heat conduction equation (HHCE) in dimensionless form is solved using Laplace integral transform technique. Different laser power densities are considered. Expression for the temperature field within the target is obtained. As an illustrative example, the critical time required to initiate phase transition and that required to initiate melting are computed for different pulses.

Effect of hyperbolic heat conduction on the linear and nonlinear vibration of CNT reinforced size-dependent functionally graded microbeams

As a first attempt, the combined application of the differential quadrature method (DQM) and the Newton–Raphson method is used to solve the hyperbolic (non-Fourier) heat conduction equations to obtain temperature, displacements and nonlinear frequency in the functionally graded (FG) nanocomposite Timoshenko microbeam. To do so, we need to follow two steps: (1): solving the hyperbolic heat conduction to obtain the temperature in the spatial and temporal domains by using DQM and Newton–Raphson method; (2): implementation of the obtained temperature in thermoelastic equations of microbeam to obtain displacements and frequency at each time step by direct iterative method. The material length scale parameter is introduced in the non-classical Timoshenko beam model, to interpret the size effect in microstructures. The material properties of the FG nanocomposite beam are estimated using the Eshelby-Mori-Tanaka approach and carbon nanotubes (CNTs) are randomly distributed within the composite. The nonlinear governing equations and boundary conditions are derived using the Hamilton principle and von Kármán geometric nonlinearity. A direct iterative method is employed to determine the nonlinear frequencies and mode shapes of the beams. All material properties such as Young modulus (E), heat capacity (Cp), relaxation time (τ), density (ρ) and thermal conductivity (K) are considered as a function of temperature and CNT volume fraction. The effects of temperature change, thermal conductivity, CNTs volume fraction, length to span ratio, heat wave speed, heat flux, and end support conditions on the nonlinear vibration of the beam are discussed in detail. Unlike all previous publications, the present results show that increasing thickness-to-length scale ratio (h/l) will increase the frequency.