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Comparative Study of Simulation of Temperature Rise in Ring Main Unit
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Nonstationary heat conduction in one-dimensional models with substrate potential.

O V Gendelman1, R Shvartsman, B Madar

  • 1Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel. ovgend@tx.technion.ac.il

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 10, 2012
PubMed
Summary

This study reveals non-Fourier heat conduction in models with substrate potential. Hyperbolic corrections are needed, as temperature decay shifts from oscillatory to diffusive with wavelength.

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Area of Science:

  • Condensed Matter Physics
  • Statistical Mechanics
  • Nonlinear Dynamics

Background:

  • Classical heat conduction is described by the Fourier equation, a parabolic partial differential equation.
  • Non-Fourier heat conduction phenomena, particularly in low-dimensional systems, require modifications to classical models.
  • Substrate potentials introduce complexities in energy transport, necessitating advanced theoretical frameworks.

Purpose of the Study:

  • To investigate nonstationary heat conduction in one-dimensional models featuring substrate potentials.
  • To establish universal characteristic properties of heat conduction across different models.
  • To identify the limitations of existing heat conduction models and propose necessary corrections.

Main Methods:

  • Numerical simulations of three distinct models: Frenkel-Kontorova (FK), phi4+, and phi4-.
  • Analysis of temperature field perturbations and their decay patterns.
  • Comparison of simulation results with predictions from Fourier's law and hyperbolic generalizations.

Main Results:

  • Observed a crossover from oscillatory decay (short waves) to diffusive decay (long waves) in temperature perturbations.
  • Demonstrated that this behavior contradicts the parabolic Fourier equation, highlighting the need for hyperbolic corrections.
  • Found that crossover wavelength decreases with increasing average temperature.
  • Noted exponential relaxation of thermal perturbations, differing from power-law relaxation in linear chains.
  • Determined that the Cattaneo-Vernotte (telegraph) equation is insufficient due to wavelength-dependent relaxation times.

Conclusions:

  • Existing heat conduction models, including the Cattaneo-Vernotte equation, are inadequate for describing nonstationary heat transport in these 1D systems with substrate potentials.
  • The observed crossover behavior necessitates the development of more sophisticated hyperbolic models for heat conduction.
  • The findings suggest that the characteristic relaxation time in oscillatory regimes exhibits a scaling law for some models, warranting further investigation.