Research Papers: Conduction

Entropy Generation in Thin Films Evaluated From Phonon Radiative Transport

[+] Author and Article Information
T. J. Bright

George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332

Z. M. Zhang1

George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332zhuomin.zhang@me.gatech.edu


Corresponding author.

J. Heat Transfer 132(10), 101301 (Jul 27, 2010) (9 pages) doi:10.1115/1.4001913 History: Received June 05, 2009; Revised April 05, 2010; Published July 27, 2010; Online July 27, 2010

One of the approaches for micro/nanoscale heat transfer in semiconductors and dielectric materials is to use the Boltzmann transport equation, which reduces to the equation of phonon radiative transfer under the relaxation time approximation. Transfer and generation of entropy are processes inherently associated with thermal energy transport, yet little has been done to analyze entropy generation in solids at length scales comparable with or smaller than the mean free path of heat carriers. This work extends the concept of radiation entropy in a participating medium to phonon radiation, thus, providing a method to evaluate entropy generation at both large and small length scales. The conventional formula for entropy generation in heat diffusion can be derived under the local equilibrium assumption. Furthermore, the phonon brightness temperature is introduced to describe the nature of nonequilibrium heat conduction. A diamond film is used as a numerical example to illustrate the distribution of entropy generation at the walls and inside the film at low temperatures. A fundamental knowledge of the entropy generation processes provides a thermodynamic understanding of heat transport in solid microstructures; this is particularly important for the performance evaluation of thermal systems and microdevices.

Copyright © 2010 by American Society of Mechanical Engineers
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Figure 1

Schematic showing directional intensity distribution at a single frequency inside a medium between two black walls at temperatures T1 and T2. The intensity at a vertical plane inside of the medium (dashed line) is shown qualitatively for T2>T1 in the acoustically thin limit. Each wall acts as a thermal reservoir, thus, heat is transported as a radiation process inside the medium.

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Figure 2

Phonon intensity at different frequencies corresponding to two black walls at temperatures T1=50 K and T2=100 K in the ballistic limit. The equilibrium temperature of the medium is denoted by T∗ (which is 85.4 K in this case) and Iavg is the average intensity of the two wall distributions.

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Figure 3

Temperature distribution inside of medium between two black walls as given by the EPRT (solid) and the diffusion approximation (dashed) for various Kn.

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Figure 4

Phonon intensity Iω (red dashed) and Iω0(T∗) (black dotted) distributions inside a medium with Kn=1 when T1=10 K and T2=50 K, at three different locations and two different frequencies, i.e., ω1=4×1013 rad/s and ω2=1×1011 rad/s. The unit of intensity is J/(m2 sr rad).

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Figure 5

Phonon intensity Iω (red dashed) and Iω0(T∗) (black dotted) distributions at the center of the medium: (a) Kn=100, (b) Kn=1, and (c) Kn=0.01. The unit of intensity is J/(m2 sr rad), the wall temperatures are T1=10 K and T2=50 K, and the angular frequency is set to ω=1.7×1013 rad/s.

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Figure 6

Polar plots of the brightness temperature Tω(ω,θ) at two frequencies, at the center of the medium, for (a) Kn=100, (b) Kn=1, and (c) Kn=0.01. The wall temperatures are T1=10 K and T2=50 K.

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Figure 7

Brightness temperature for different direction cosine in center of medium with Kn=1 as a function of frequency

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Figure 8

Normalized entropy flux at different locations across the medium for different Kn. Note the jump at the walls corresponding to wall entropy generation.




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