Growth and Decay of a Thermal Pulse Predicted by the Hyperbolic Heat Conduction Equation

1983 ◽  
Vol 105 (4) ◽  
pp. 902-907 ◽  
Author(s):  
B. Vick ◽  
M. N. O¨zisik
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Wave Front ◽  
Heat Conduction Equation ◽  
Heat Propagation ◽  
Heat Diffusion ◽  
Conduction Equation ◽  
Hyperbolic Heat Conduction ◽  
Thermal Front ◽  
Wave Nature

The wave nature of heat propagation in a semi-infinite medium containing volumetric energy sources is investigated by solving the hyperbolic heat conduction equation. Analytic expressions are developed for the temperature and heat flux distributions. The solutions reveal that the spontaneous release of a finite pulse of energy gives rise to a thermal wave front which travels through the medium at a finite velocity, decaying exponentially while dissipating its energy along its path. When a concentrated pulse of energy is released, the temperature and heat flux in the wave front become severe. For situations involving very short times or very low temperatures, the classical heat diffusion theory significantly underestimates the magnitude of the temperature and heat flux in this thermal front, since the classical theory leads to instantaneous heat diffusion at an infinite propagation velocity.

Mesoscopic Heat Conduction and Onset of Periodicity

2003 ◽  
Author(s):  
Kal Renganathan Sharma
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Time Domain ◽  
Heat Conduction Equation ◽  
Higher Order ◽  
Heat Propagation ◽  
Conduction Equation ◽  
Transient Temperature ◽  
Higher Order Terms ◽  
Fourier Heat Conduction

Mesoscopic approach deals with study that considers temporal fluctuations which is often averaged out in a macroscopic approach without going into the molecular or microscopic approach. Transient heat conduction cannot be fully described by Fourier representation. The non-Fourier effects or finite speed of heat propagation effect is accounted for by some investigators using the Cattaneo and Vernotte non-Fourier heat conduction equation: q=−k∂T/∂x−τr∂q/∂t(1) A generalized expression to account for the non-Fourier or thermal inertia effects suggested by Sharma (5) as: q=−k∂T/∂x−τr∂q/∂t−τr2/2!∂2q/∂t2−τr3/3!∂3q/∂t3−…(2) This was obtained by a Taylor series expansion in time domain. Manifestation of higher order terms in the modified Fourier’w law as periodicity in the time domain is considered in this study. When a CWT is maintained at one end of a medium of length L where L is the distance from the isothermal wall beyond which there is no appreciable temperature change from the initial condition during the duration of the study the transient temperature profile is obtained by the method of Laplace transforms. The space averaged heat flux is obtained and upon inversion from Laplace domain found to be a constant for the the case obeying Fourier’s law; 1 − exp(−τ) using the Cattaneo and Vernotte non-Fourier heat conduction equation, and upon introduction of the second derivative in time of the heat flux the expression becomes, 1 − exp(−τ)(Sin(τ) + Cos(τ)). Thus the periodicity in time domain is lost when the higher order terms in the generalized Fourier expression is neglected.

Numerical simulation of non-Fourier effects in combined heat transfer

2010 ◽  
Vol 225 (2) ◽  
pp. 429-436
Author(s):  
E Izadpanah ◽  
S Talebi ◽  
M H Hekmat
Keyword(s):  
Heat Transfer ◽  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Splitting Method ◽  
Thermal Relaxation ◽  
Conduction Equation ◽  
Hyperbolic Heat Conduction ◽  
Wave Nature ◽  
Fourier Heat Conduction ◽  
Combined Heat Transfer

The non-Fourier effects on transient and steady temperature distribution in combined heat transfer are studied. The processes of coupled conduction and radiation heat transfer in grey, absorbing, emitting, scattering, one-dimensional medium with black boundary surfaces are analysed numerically. The hyperbolic heat conduction equation is solved by flux splitting method, and the radiative transfer equation is solved by P1 approximate method. The transient thermal responses obtained from non-Fourier heat conduction equation are compared with those obtained from the Fourier heat conduction equation. The results show that the non-Fourier effect can be important when the conduction to radiation parameter and the thermal relaxation time are larger. Further, the radiation effect is more pronounced at small values of single scattering albedo and conduction to radiation parameters. Analysis results indicate that the internal radiation in the medium significantly influences the wave nature.

scholarly journals Time-Fractional Heat Conduction in a Plane with Two External Half-Infinite Line Slits under Heat Flux Loading

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 689 ◽  
Author(s):  
Yuriy Povstenko ◽  
Tamara Kyrylych
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Graphical Representation ◽  
Integral Transform ◽  
Conduction Equation ◽  
Infinite Plane ◽  
Conservation Of Energy ◽  
Infinite Line ◽  
Fractional Heat Conduction

The time-fractional heat conduction equation follows from the law of conservation of energy and the corresponding time-nonlocal extension of the Fourier law with the “long-tail” power kernel. The time-fractional heat conduction equation with the Caputo derivative is solved for an infinite plane with two external half-infinite slits with the prescribed heat flux across their surfaces. The integral transform technique is used. The solution is obtained in the form of integrals with integrand being the Mittag–Leffler function. A graphical representation of numerical results is given.

Assessment of Models for Extracting Thermal Conductivity From Frequency Domain Thermoreflectance Experiments

2020 ◽  
Author(s):  
Siddharth Saurav ◽  
Sandip Mazumder
Keyword(s):  
Thermal Conductivity ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Frequency Domain ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Hyperbolic Heat Conduction ◽  
Phase Lag ◽  
Computational Domain ◽  
Fourier Heat Conduction

Abstract The Fourier heat conduction and the hyperbolic heat conduction equations were solved numerically to simulate a frequency-domain thermoreflectance (FDTR) experimental setup. Numerical solutions enable use of realistic boundary conditions, such as convective cooling from the various surfaces of the substrate and transducer. The equations were solved in time domain and the phase lag between the temperature at the center of the transducer and the modulated pump laser signal were computed for a modulation frequency range of 200 kHz to 200 MHz. It was found that the numerical predictions fit the experimentally measured phase lag better than analytical frequency-domain solutions of the Fourier heat equation based on Hankel transforms. The effects of boundary conditions were investigated and it was found that if the substrate (computational domain) is sufficiently large, the far-field boundary conditions have no effect on the computed phase lag. The interface conductance between the transducer and the substrate was also treated as a parameter, and was found to have some effect on the predicted thermal conductivity, but only in certain regimes. The hyperbolic heat conduction equation yielded identical results as the Fourier heat conduction equation for the particular case studied. The thermal conductivity value (best fit) for the silicon substrate considered in this study was found to be 108 W/m/K, which is slightly different from previously reported values for the same experimental data.

Analytical Solution for Three-Dimensional Hyperbolic Heat Conduction Equation with Time-Dependent and Distributed Heat Source

2016 ◽  
Vol 33 (1) ◽  
pp. 65-75 ◽  
Author(s):  
M. R. Talaee ◽  
V. Sarafrazi
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Source ◽  
Heat Conduction Equation ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Time Dependent ◽  
Conduction Equation ◽  
Hyperbolic Heat Conduction

AbstractThis paper is devoted to the analytical solution of three-dimensional hyperbolic heat conduction equation in a finite solid medium with rectangular cross-section under time dependent and non-uniform internal heat source. The closed form solution of both Fourier and non-Fourier profiles are introduced with Eigen function expansion method. The solution is applied for simple simulation of absorption of a continues laser in biological tissue. The results show the depth of laser absorption in tissue and considerable difference between the Fourier and Non-Fourier temperature profiles. In addition the solution can be applied as a verification branch for other numerical solutions.

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