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Current Density01:21

Current Density

The total amount of current flowing through one unit value of a cross-sectional area is referred to as current density. If the current flow is uniform, the amount of current flowing through a conductor is the same at all points along the conductor, even if the conductor area varies. The current density consists of the local magnitude and direction of the charge flow, which varies from point to point. Current density is measured in amperes per meter square, and direction is defined as the net...
Conduction, Convection and Radiation: Problem Solving01:20

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There are three methods by which heat transfer can take place: conduction, convection, and radiation. Each method has unique and interesting characteristics, but all three have two things in common: they transfer heat solely because of a temperature difference; and the greater the temperature difference, the faster the heat transfer.
In order to solve a problem related to heat transfer, first of all, the situation needs to be examined to determine the type of heat transfer involved. This could...
Mechanisms of Heat Transfer I01:14

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Just as interesting as the effects of heat transfer on a system are the methods by which the heat transfer occur. Whenever there is a temperature difference, heat transfer occurs. It may occur rapidly, such as through a cooking pan, or slowly, such as through the walls of a picnic ice box. So many processes involve heat transfer that it is hard to imagine a situation where no heat transfer occurs. Yet, every heat transfer takes place by only three methods: conduction, convection, and radiation.
Mechanisms of Heat Transfer II01:20

Mechanisms of Heat Transfer II

In convection, thermal energy is carried by the large-scale flow of matter. Ocean currents and large-scale atmospheric circulation, which result from the buoyancy of warm air and water, transfer hot air from the tropics toward the poles and cold air from the poles toward the tropics. The Earth’s rotation interacts with those flows, causing the observed eastward flow of air in the temperate zones. Convection dominates heat transfer by air, and the amount of available space for the airflow...
Boundary Conditions for Current Density01:25

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Current density becomes discontinuous across an interface of materials with different electrical conductivities. The normal component of the current density is continuous across the boundary.
Thermodynamic Potentials01:26

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Thermodynamic potentials are state functions that are extremely useful in analyzing a thermodynamic system. They have dimensions of energy. The four important thermodynamic potentials are internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy. These thermodynamic potentials can be expressed using two of the following variables: pressure, volume, temperature, and entropy. These two variables are expressed as the rate of change of the thermodynamic potential with respect to other...

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Related Experiment Video

Updated: May 24, 2026

Comparative Study of Simulation of Temperature Rise in Ring Main Unit
04:35

Comparative Study of Simulation of Temperature Rise in Ring Main Unit

Published on: July 5, 2024

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.