Exergy of Blackbody Radiation and Monochromatic Photon

  • Zhijun Zhou
  • Shiquan Shan
  • Liping Chen
  • Yanwei ZhangEmail author


The study of radiation exergy has important significance for solar energy and high-temperature engineering. In this paper, several exergy expressions of blackbody radiation were discussed and the differences between Petela’s expression of exergy and two other expressions were analyzed. Considering that radiant energy and thermal energy are different, the radiation machine model was established; furthermore, the validity of Petela’s formula was indicated by this model. Based on the concept of radiation equivalent temperature, the integral form expression of monochromatic photon exergy was put forward by establishing the infinite-staged Carnot heat engine model. At the same time, an approximate relation between equivalent temperature and radiation wavelength was given. The error of this relation is negligible when calculating the exergy of blackbody radiation within the temperature range of the engineering field. Finally, the monochromatic photon entropy was discussed by considering the infinite-staged Carnot heat engine model, and an expression of photon entropy with integral form was given. The monochromatic photon entropy and exergy proposed in our paper satisfy the thermodynamic relation and can reflect the differences between radiant energy and thermal energy.


Blackbody radiation Entropy Exergy Infinite-staged Carnot heat engine Monochromatic photon 

List of symbols


Radiation constant \((7.561 \times 10^{-19}\, \hbox {kJ}\cdot \hbox {m}^{-3}\cdot \hbox {K}^{-4})\)


Blackbody radiation exergy flux \((\hbox {W}\cdot \hbox {m}^{-2})\)


Radiation propagation speed in vacuum \((2.988 \times 10^{8}\,\hbox {m}\cdot \hbox {s}^{-1})\)


First radiation constant \((3.74 \times 10^{-16}\,\hbox {W}\cdot \hbox {m}^{2})\)


Second radiation \(\hbox {constant}(1.4388 \times 10^{-2}\,\hbox {m}\cdot \hbox {K})\)


Exergy (J)


Coefficient in equivalent temperature expression


Planck’s constant \((6.626 \times 10^{-34} \,\hbox {J}\cdot \hbox {s}^{-1})\)


Enthalpy of substance (J)


Enthalpy of substance under ambient conditions (J)

\(i_{b,\lambda }\)

Monochromatic radiation intensity of blackbody \((\hbox {W}\cdot \hbox {m}^{-3})\)


Coefficient in equivalent temperature expression


Coefficient in equivalent temperature expression


Coefficient in equivalent temperature expression


Ambient pressure (Pa)


Heat (J)


Heat flow \((\hbox {W}\cdot \hbox {m}^{-2})\)

\(R_{in}, R_{out}\)

Blackbody radiation energy flow input to radiation engine model or output from radiation engine model \((\hbox {W}\cdot \hbox {m}^{-2})\)


Entropy (\(\hbox {J}\cdot \hbox {K}^{-1}\))


Entropy under ambient conditions (\(\hbox {J}\cdot \hbox {K}^{-1}\))


Entropy flow \((\hbox {W}\cdot \hbox {m}^{-2}\cdot \hbox {K}^{-1})\)

\(S_{in}, S_{out}\)

Blackbody radiation entropy flow input to radiation engine model or output from radiation engine model \((\hbox {W}\cdot \hbox {m}^{-2}\cdot \hbox {K}^{-1})\)


Entropy of monochromatic photon with a frequency of v or a critical frequency of \(v_{0}\;(\hbox {J}\cdot \hbox {s}\cdot \hbox {K}^{-1})\)


Temperature of substance (K)


Ambient temperature (K)


Temperature used to characterize the exergy of the photon (K)

\(T_{(\lambda )}, T_{(v)}\)

Equivalent temperature of the photon with a wavelength of \(\lambda \) or a frequency of v (K)


Blackbody radiation energy (J)


Exergy of monochromatic photon (J)


Volume \((\hbox {m}^{3})\)


Frequency \((\hbox {s}^{-1})\)


Work (J)


Work flow \((\hbox {W}\cdot \hbox {m}^{-2})\)


Useful work (J)


Work spent for ‘compression of the environment’ (J)

Greek symbols

\(\varDelta \)


\(\varepsilon _{1}\)

Error for radiation temperature above ambient

\(\varepsilon _{2}\)

Error for radiation temperature below ambient

\(\eta \)

Exergy-to-energy coefficient

\(\eta _{J}\)

Exergy-to-energy coefficient of blackbody radiation proposed by Jeter

\(\eta _{p}\)

Exergy-to-energy coefficient of blackbody radiation proposed by Petela

\(\eta _{S}\)

Exergy-to-energy coefficient of blackbody radiation proposed by Spanner

\(\eta _{u}\)

Exergy-to-energy coefficient of monochromatic radiation photon

\(\lambda \)

Wavelength \((\upmu \hbox {m})\)

\(\lambda _{0}\)

Critical wavelength \((\upmu \hbox {m})\)

\(\upsigma \)

Stefan–Boltzmann constant \((5.67 \times 10^{8}\, \hbox {W}\cdot \hbox {m}^{-2}\cdot \hbox {K}^{-1})\)



Blackbody radiation


Heat engine



The support of this work by the National Key Technology Support Program of China (No. 2015BAA04B03) is gratefully acknowledged.


  1. 1.
    S. Wright, Int. J. Thermodyn. 10, 27 (2007)Google Scholar
  2. 2.
    M.J. Moran, E. Sciubba, J. Eng. Gas. Turbines Power 116, 285 (1994)CrossRefGoogle Scholar
  3. 3.
    R. Petela, J. Heat Transf. 86, 187 (1964)CrossRefGoogle Scholar
  4. 4.
    V. Badescu, Int. J. Sol. Energy 20, 149 (2000)CrossRefGoogle Scholar
  5. 5.
    R. Petela, Sol. Energy 74, 469 (2003)ADSCrossRefGoogle Scholar
  6. 6.
    S.M. Jeter, Sol. Energy 26, 231 (1981)ADSCrossRefGoogle Scholar
  7. 7.
    D.C. Spanner, Introduction to Thermodynamics (Academic Press, London, 1964)Google Scholar
  8. 8.
    R.H. Edgerton, Energy 5, 693 (1980)ADSCrossRefGoogle Scholar
  9. 9.
    L.N.M. Duysens, Brookhaven Symp. Biol. 11, 18 (1958)Google Scholar
  10. 10.
    R.S. Knox, in Primary Processes of Photosynthesis, ed. by J. Barber (Elsevier, Amsterdam, 1977)Google Scholar
  11. 11.
    J.D. Lewins, Int. J. Mech. Eng. Educ. 31, 283 (2003)CrossRefGoogle Scholar
  12. 12.
    A. De Vos, H. Pauwels, J. Phys. C Solid State 16, 6897 (1983)ADSCrossRefGoogle Scholar
  13. 13.
    P.T. Landsberg, G. Tonge, J. Phys. A Math. Gen. 12, 551 (1979)ADSCrossRefGoogle Scholar
  14. 14.
    P.T. Landsberg, P. Baruch, J. Phys. A Math. Gen. 22, 1911 (1989)ADSCrossRefGoogle Scholar
  15. 15.
    V. Badescu, J. Phys. D Appl. Phys. 23, 289 (1990)ADSCrossRefGoogle Scholar
  16. 16.
    S. Sieniutycz, P. Kuran, Int. J. Heat Mass Transf. 49, 3264 (2006)CrossRefGoogle Scholar
  17. 17.
    V.I. Laptev, J. Appl. Phys. 98, 124905 (2005)ADSCrossRefGoogle Scholar
  18. 18.
    S.E. Wright, M.A. Rosen, J. Sol. Energy Eng. 126, 673 (2004)CrossRefGoogle Scholar
  19. 19.
    A. Bejan, J. Sol. Energy Trans. ASME 109, 46 (1987)CrossRefGoogle Scholar
  20. 20.
    S.E. Wright, M.A. Rosen, D.S. Scott, J.B. Haddow, Int. J. Exergy 2, 24 (2002)CrossRefGoogle Scholar
  21. 21.
    A.H. Carter, Classical and Statistical Thermodynamics (Pearson Education, London, 2001)Google Scholar
  22. 22.
    R. Petela, Engineering Thermodynamics of Thermal Radiation for Solar Power Utilization (McGraw Hill, New York, 2010)zbMATHGoogle Scholar
  23. 23.
    V. Badescu, Cent. Eur. J. Phys. 6, 344 (2008)ADSGoogle Scholar
  24. 24.
    T. Markvart, G.H. Bauer, Appl. Phys. Lett. 101, 193901 (2012)ADSCrossRefGoogle Scholar
  25. 25.
    S. Lems, H.J. Van Der Kooi, J. De Swaan Arons, Int. J. Exergy 7, 333 (2010)CrossRefGoogle Scholar
  26. 26.
    D. Juretić, P. Županović, Comput. Boil. Chem. 27, 541 (2003)CrossRefGoogle Scholar
  27. 27.
    J. Lavergne, P. Joliot, in Energy Transduction in Membranes, ed. by W.A. Cramer (Biophysical Society, Rockville, 2000)Google Scholar
  28. 28.
    G. Meszéna, H.V. Westerhoff, O. Somsen, J. Phys. A. Math Gen. 33, 1301 (2000)ADSCrossRefGoogle Scholar
  29. 29.
    Z. Chen, S. Mo, Prog. Nat. Sci. 17, 1250 (2007)Google Scholar
  30. 30.
    R. Petela, Int. J. Exergy 7, 89 (2010)CrossRefGoogle Scholar
  31. 31.
    M.F. Modest, Radiative Heat Transfer, 3rd edn. (Elsevier, New York, 2013)Google Scholar
  32. 32.
    M. Planck, The Theory of Heat Radiation (Blakistons, Philadelphia, 1914)zbMATHGoogle Scholar
  33. 33.
    E. Ito, T. Komatsu, H. Suzuki, Biophys. Chem. 74, 59 (1998)CrossRefGoogle Scholar
  34. 34.
    A.D. Kirwan Jr., Int. J. Eng. Sci. 42, 725 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Zhijun Zhou
    • 1
  • Shiquan Shan
    • 1
  • Liping Chen
    • 1
  • Yanwei Zhang
    • 1
    Email author
  1. 1.State Key Laboratory of Clean Energy UtilizationZhejiang UniversityHangzhouPeople’s Republic of China

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