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ABSTRACT
Approximate analysis is a major application of variational principles for heat conduction. Recently, O’Toole’s variational principle for Fourier’s law has been extended to non-Fourier heat conduction models, which are applied to approximate analyses based on the Rayleigh–Ritz method. Suitable trial functions satisfying boundary conditions are sought, and then substituted into the variational principles to obtain the undetermined coefficients. From the inverse Laplace transforms, the approximate solutions are obtained. Examples are provided for 1D problems for different heat conduction models. The largest calculation errors are one or two orders of magnitude smaller than the equilibrium temperature, which will tend to be zero.
Nomenclature
| a | = | thermal diffusivity in Fourier’s law, m2/s |
| c | = | isothermal first-sound velocity in the GK model, m/s |
| CE | = | heat wave velocity, m/s |
| cV | = | specific heat, J/K · kg |
| D | = | spatial domain |
| F | = | Laplace transform of temperature, |
| Fo | = | Fourier number |
| k | = | total thermal conductivity in the Jeffrey model, W/K · m |
| kF | = | thermal conductivity for Fourier heat conduction in the Jeffrey model, W/K · m |
| l | = | boundary coordinate, m |
| q | = | heat flux, W/m2 |
| t | = | time coordinate, s |
| T | = | temperature, K |
| Te | = | electron temperature in the TT model, K |
| x | = | space coordinate, m |
| αe | = | thermal diffusivity of the electrons in the TT model, m2/s |
| αE | = | equivalent thermal diffusivity in the TT model, m2/s |
| λ | = | thermal conductivity in Fourier’s law, W/K · m |
| ρ | = | mass density, kg/m3 |
| τ | = | thermal relaxation time in the CV and Jeffrey models, s |
| τN | = | single-phonon relaxation time for normal processes in the GK model, s |
| τR | = | momentum loss relaxation time in the GK model, s |
Nomenclature
| a | = | thermal diffusivity in Fourier’s law, m2/s |
| c | = | isothermal first-sound velocity in the GK model, m/s |
| CE | = | heat wave velocity, m/s |
| cV | = | specific heat, J/K · kg |
| D | = | spatial domain |
| F | = | Laplace transform of temperature, |
| Fo | = | Fourier number |
| k | = | total thermal conductivity in the Jeffrey model, W/K · m |
| kF | = | thermal conductivity for Fourier heat conduction in the Jeffrey model, W/K · m |
| l | = | boundary coordinate, m |
| q | = | heat flux, W/m2 |
| t | = | time coordinate, s |
| T | = | temperature, K |
| Te | = | electron temperature in the TT model, K |
| x | = | space coordinate, m |
| αe | = | thermal diffusivity of the electrons in the TT model, m2/s |
| αE | = | equivalent thermal diffusivity in the TT model, m2/s |
| λ | = | thermal conductivity in Fourier’s law, W/K · m |
| ρ | = | mass density, kg/m3 |
| τ | = | thermal relaxation time in the CV and Jeffrey models, s |
| τN | = | single-phonon relaxation time for normal processes in the GK model, s |
| τR | = | momentum loss relaxation time in the GK model, s |