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Journal of Energy Resources Technology | Volume 137 | Issue 2 | research-article
Research Papers

Electromagnetic Radiation: A Carrier of Energy and Entropy *

[+] Author and Article Information
Gian Paolo Beretta

Università di Brescia,
Brescia, Italy

Elias P. Gyftopoulos

Massachusetts Institute of Technology,
Cambridge, Massachusetts

*Proceedings of the Winter Annual Meeting of the American Society of Mechanical Engineers, Dallas, Texas, November 25–30, 1990, in Fundamentals of Thermodynamics and Exergy Analysis, edited by G. Tsatsaronis, R. A. Gaggioli, Y. M. El-Sayed, and M. K. Drost, ASME book G00566, AES-Vol. 19, pp. 1–6 (1990). Reprinted with permission.

J. Energy Resour. Technol 137(2), 021005 (Mar 01, 2015) (6 pages) Paper No: 05-BerettaGyftopoulos-AS; doi: 10.1115/1.4026381 History: Online December 08, 2014
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Starting from the properties of the electromagnetic radiation field at stable equilibrium, we derive expressions for the flows of energy and entropy between two black bodies at different temperatures, interacting only through electromagnetic radiation. We find that in general the interaction through radiation is nonwork but not heat. It is heat only if the temperature difference between the interacting systems is infinitesimal.
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Grahic Jump Location
Fig. 1

Graph of dimensionless spectral energy densityy=uν8πhν3/c3=uλ8πhc/λ5versus dimensionless spectral entropy densityx=sν8πhν2/c3=sλ8πk/λ4for the stable equilibrium states of radiation modes with frequency ν and wavelength λ = c/ν. The equation of the curve isx=y[(1+1y)ln(1+1y)-1yln1y]

+Graph of dimensionless spectral energy densityy=uν8πhν3/c3=uλ8πhc/λ5versus dimensionless spectral entropy densityx=sν8πhν2/c3=sλ8πk/λ4for the stable equilibrium states of radiation modes with frequency ν and wavelength λ = c/ν. The equation of the curve isx=y[(1+1y)ln(1+1y)-1yln1y]
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Graph of dimensionless spectral energy densityy=uν8πhν3/c3=uλ8πhc/λ5versus dimensionless spectral entropy densityx=sν8πhν2/c3=sλ8πk/λ4for the stable equilibrium states of radiation modes with frequency ν and wavelength λ = c/ν. The equation of the curve isx=y[(1+1y)ln(1+1y)-1yln1y]
Grahic Jump Location
Fig. 2

Graphs as functions of x=hν/kT. Curve (e) is scaled by a factor of 1/10. The area under each if the curves (a), (b), (c), and (d) is unity. The maxima occur respectively at (a) x = 2.82144. (b) x = 2.53823. (c) x = 1.83030 and (d) x = 1.59362. kThuνu=15π4x3ex-1(a)kThsνs=454π4x2(xex-1+ln11-e-x)(b)kThpνp=45π4x2ln11-e-x(c)kThnνn=12ζ(3)x2ex-1(d)pνnνkT=(ex-1)ln11-e-x(e)

+Graphs as functions of x=hν/kT. Curve (e) is scaled by a factor of 1/10. The area under each if the curves (a), (b), (c), and (d) is unity. The maxima occur respectively at (a) x = 2.82144. (b) x = 2.53823. (c) x = 1.83030 and (d) x = 1.59362. kThuνu=15π4x3ex-1(a)kThsνs=454π4x2(xex-1+ln11-e-x)(b)kThpνp=45π4x2ln11-e-x(c)kThnνn=12ζ(3)x2ex-1(d)pνnνkT=(ex-1)ln11-e-x(e)
View Large  |  Save Figure  |  Download Slide (.ppt)
Graphs as functions of x=hν/kT. Curve (e) is scaled by a factor of 1/10. The area under each if the curves (a), (b), (c), and (d) is unity. The maxima occur respectively at (a) x = 2.82144. (b) x = 2.53823. (c) x = 1.83030 and (d) x = 1.59362. kThuνu=15π4x3ex-1(a)kThsνs=454π4x2(xex-1+ln11-e-x)(b)kThpνp=45π4x2ln11-e-x(c)kThnνn=12ζ(3)x2ex-1(d)pνnνkT=(ex-1)ln11-e-x(e)
Grahic Jump Location
Fig. 3

Graphs as functions of y=λkT/hc. The area under each of the curves (a), (b), (c), and (d) is unity. The maxima occur respectively at (a) y = 0.201405, (b) y = 0.208713. (c) y = 0.252417, and (d) y = 0.255057.hckTuλu=15π41/y5e1/y-1(a)hckTsλs=454π41y4(1/ye1/y-1+ln11-e-1/y)(b)hckTpλp=45π41y4ln11-e-1/y(c)hckTnλn=12ζ(3)1/y4e1/y-1(d)pλnλkT=(e1/y-1)ln11-e-1/y(e)

+Graphs as functions of y=λkT/hc. The area under each of the curves (a), (b), (c), and (d) is unity. The maxima occur respectively at (a) y = 0.201405, (b) y = 0.208713. (c) y = 0.252417, and (d) y = 0.255057.hckTuλu=15π41/y5e1/y-1(a)hckTsλs=454π41y4(1/ye1/y-1+ln11-e-1/y)(b)hckTpλp=45π41y4ln11-e-1/y(c)hckTnλn=12ζ(3)1/y4e1/y-1(d)pλnλkT=(e1/y-1)ln11-e-1/y(e)
View Large  |  Save Figure  |  Download Slide (.ppt)
Graphs as functions of y=λkT/hc. The area under each of the curves (a), (b), (c), and (d) is unity. The maxima occur respectively at (a) y = 0.201405, (b) y = 0.208713. (c) y = 0.252417, and (d) y = 0.255057.hckTuλu=15π41/y5e1/y-1(a)hckTsλs=454π41y4(1/ye1/y-1+ln11-e-1/y)(b)hckTpλp=45π41y4ln11-e-1/y(c)hckTnλn=12ζ(3)1/y4e1/y-1(d)pλnλkT=(e1/y-1)ln11-e-1/y(e)

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