Depth of Thermal Penetration: Effect of Relaxation and Thermalization

Tzou, D. Y.; Chiu, Kevin K. S.

doi:10.2514/2.6431

Cited by 6 publications

(3 citation statements)

References 8 publications

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“…The results are straightforward after the growth history of Δ(β) is determined, Tzou and Chiu (1999), with the deviations from the Laplace transform solutions (with Riemann-sum approximation) within 10 percent in small times. The results are straightforward after the growth history of Δ(β) is determined, Tzou and Chiu (1999), with the deviations from the Laplace transform solutions (with Riemann-sum approximation) within 10 percent in small times.…”

Section: Depth Of Thermal Penetrationmentioning

confidence: 85%

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Lagging Behavior

2014

Macro‐ to Microscale Heat Transfer

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All microstructural interactions require finite periods of time to accomplish, which are in the range from femtoseconds (10 -15 s, phonon-electron interactions in metals) to seconds or even longer (slow conducting materials with subscale constituents). Effect of such finite times may not be noticeable as the process time is much greater, exemplified by the nanosecond (10 -9 s) transient as compared to the picosecond (10 -12 s) delay in support of sufficient collisions between electrons and phonons as they approach thermal equilibrium. As the process time enters the same domain of time for the microstructural interactions to take place, however, individual (and yet statistically meaningful) behaviors of the energy carriers would result in new physical responses that are not observed in macroscale. This chapter is dedicated to the physical foundation and mathematical methods describing the lagging response in times comparable to the phase lags characterizing the microstructural interactions.Accommodating the finite times required for completing the physical interactions in microor nanoscale, the lagging response describes the heat flux vector and the temperature gradient that occur at different instants of time in the heat-transfer process. If the heat flux precedes the temperature gradient in the time history, the heat flux is the cause and the temperature gradient is the effect of heat flow. If the temperature gradient precedes the heat flux, on the other hand, the temperature gradient becomes the cause and the heat flux becomes the effect. This concept of precedence does not exist in the classical theory of diffusion because the heat flux vector and the temperature gradient are assumed to occur simultaneously. This chapter establishes the theoretical foundation for the lagging response in times comparable to the characteristic times in micro/nanoscale heat transfer. It results in a new type of energy equation in heat transport, capturing the classical theories of diffusion (macroscopic in both space and time), thermal waves (macroscopic in space but microscopic in time), phonon scattering, and phononelectron interaction (microscopic in both space and time) in the same framework of thermal lagging. The resulting model employing the two phase lags in describing the transient process is called the dual-phase-lag model. Universality of the dual-phase-lag model facilitates a consistent approach describing the intrinsic transition from one type of behavior (diffusion,

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Section: Depth Of Thermal Penetrationmentioning

confidence: 85%

“…Typically, a polynomial in (ξ/Δ) is used for this purpose, which is exemplified here by a cubic polynomial to illustrate the process (Tzou and Chiu, 1999), Typically, a polynomial in (ξ/Δ) is used for this purpose, which is exemplified here by a cubic polynomial to illustrate the process (Tzou and Chiu, 1999), …”

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confidence: 99%

Lagging Behavior

2014

Macro‐ to Microscale Heat Transfer

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“…In this context, we have to take into account that for the Fourier conduction equation the penetration depth d 0 = (a 1 t) 1/2 [2n(n + 1)] 1/2 introduces a singularity because d d 0 /dt is infinite at t = 0. This problem has been studied by Tzou and Chiu [12] for the thermal penetration estimated by the heat-balance integral (a cubic profile with n = 3) when the heat conduction is modelled by a dual-phase lag equation (hyperbolic) with two relaxation times. Now, with the estimate (9b) we have: (1) Qualitatively, we have a model with a relaxation that assumes a finite speed of the heat penetration into the medium.…”

Section: Integral-balance Solutionmentioning

confidence: 99%

A note on the integral approach to non-linear heat conduction with Jeffrey’s fading memory

Hristov

2013

THERM SCI

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Integral approach by using approximate profile is successfully applied to heat conduction equation with fading memory expressed by a Jeffrey's kernel. The solution is straightforward and the final form of the approximate temperature profile clearly delineates the "viscous effects" corresponding to the classical Fourier law and the relaxation (fading memory). The optimal exponent of the approximate solution is discussed in case of Dirichlet boundary condition.

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Determination of the limit of applicability of the parabolic equation of heat conduction

Kostanovskii

Kostanovskaya

2008

Meas Tech

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Depth of Thermal Penetration: Effect of Relaxation and Thermalization

Cited by 6 publications

References 8 publications

Lagging Behavior

Lagging Behavior

A note on the integral approach to non-linear heat conduction with Jeffrey’s fading memory

Determination of the limit of applicability of the parabolic equation of heat conduction

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