HEAT CONDUCTION IN ONE-DIMENSIONAL GEOMETRIES WITH NONUNIFORM INTERNAL HEAT GENERATION

1970 ◽  
Author(s):  
Gilbert B. Melese-d'Hospital ◽  
J. Ernest Wilkins, Jr.
Keyword(s):  
Heat Conduction ◽  
Heat Generation ◽  
Internal Heat ◽  
Internal Heat Generation ◽  
One Dimensional

Simulation of Nanoscale Multidimensional Transient Heat Conduction Problems Using Ballistic-Diffusive Equations and Phonon Boltzmann Equation

2005 ◽  
Vol 127 (3) ◽  
pp. 298-306 ◽  
Author(s):  
Ronggui Yang ◽  
Gang Chen ◽  
Marine Laroche ◽  
Yuan Taur
Keyword(s):  
Heat Conduction ◽  
Boltzmann Equation ◽  
Boundary Conditions ◽  
Heat Generation ◽  
Internal Heat ◽  
Internal Heat Generation ◽  
Computational Time ◽  
Discrete Ordinates ◽  
Fourier Law ◽  
Phonon Boltzmann Equation

Heat conduction in micro- and nanoscale and in ultrafast processes may deviate from the predictions of the Fourier law, due to boundary and interface scattering, the ballistic nature of the transport, and the finite relaxation time of heat carriers. The transient ballistic-diffusive heat conduction equations (BDE) were developed as an approximation to the phonon Boltzmann equation (BTE) for nanoscale heat conduction problems. In this paper, we further develop BDE for multidimensional heat conduction, including nanoscale heat source term and different boundary conditions, and compare the simulation results with those obtained from the phonon BTE and the Fourier law. The numerical solution strategies for multidimensional nanoscale heat conduction using BDE are presented. Several two-dimensional cases are simulated and compared to the results of the transient phonon BTE and the Fourier heat conduction theory. The transient BTE is solved using the discrete ordinates method with a two Gauss-Legendre quadratures. Special attention has been paid to the boundary conditions. Compared to the cases without internal heat generation, the difference between the BTE and BDE is larger for the case studied with internal heat generation due to the nature of the ballistic-diffusive approximation, but the results from BDE are still significantly better than those from the Fourier law. Thus we conclude that BDE captures the characteristics of the phonon BTE with much shorter computational time.

Heat Conduction in an Eccentrically Hollow, Infinitely Long Cylinder With Internal Heat Generation

1961 ◽  
Vol 83 (4) ◽  
pp. 510-512 ◽  
Author(s):  
M. R. El-Saden
Keyword(s):  
Heat Conduction ◽  
Temperature Distribution ◽  
Hollow Cylinder ◽  
Heat Generation ◽  
Internal Heat ◽  
Internal Heat Generation ◽  
Theoretical Solution ◽  
Steady Temperature ◽  
Special Case ◽  
Long Cylinder

This paper discusses the steady temperature distribution in an infinitely long, eccentrically hollow cylinder with uniform rate of internal heat generation. An exact theoretical solution is presented. The result is applied to the special case of no internal heat generation, and the rate of heat conduction is obtained.

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