Maximum Work Rate Extractable from Energy Fluxes

Viorel Badescu

doi:10.1515/jnet-2021-0039

Article

Maximum Work Rate Extractable from Energy Fluxes

Viorel Badescu

Published/Copyright: December 21, 2021

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Journal of Non-Equilibrium Thermodynamics Volume 47 Issue 1

Abstract

A general formalism is developed to evaluate the amount of work extractable from energy fluxes. It covers nonequilibrium cases when the concept of exergy is not relevant. The rate of work deficiency, which has been previously introduced as the total loss of exergy, is defined here as the total loss of work, which would have resulted if all the work were lost to the environment. New performance indicators are proposed. First, the work content factor gives the proportion of extractable work in a given amount of energy. Second, the work deficiency factor is a measure of the potential of improvement for the operation of energy conversion systems. Previous results reported in literature are particular cases of the general results obtained here. The formalism is used to evaluate the work rate extractable from the solar energy flux. Results are shown in cases where solar radiation interacts with materials without energy bandgap (metals) and with energy bandgaps (semiconductors), respectively.

Keywords: energy flux; work rate; exergy; work deficiency

Acknowledgments

The author thanks the reviewer and the Editor-in-Chief for useful comments and suggestions.

Appendix A

Several different approaches based on the Chapman–Enskog treatment of Boltzmann equations have been used to theorize the macroscopic evolution equations of many body quantum systems [13], [22], [23], [24], [25]. It has been shown that the main fluxes involved in the transport processes in quantum systems consist of species particle fluxes, mass flux, internal energy flux, heat flux, mechanical energy flux, and entropy flux [23]. A similar approach to radiation and matter systems arrived to similar results, differentiated for the photon gas and matter [24]. The same type of approach has been used to quantify the fluxes in degenerate semiconductors and metals [22]. In this particular case (of solids), the (electro-) chemical potential has been defined with respect to the conduction and valence band electrons, and the species particle fluxes are associated with a flux of electrical energy. This has been also found in a more specific study related with the transport theory of semiconductor energy conversion [25]. An explicit expression for the electrical energy density flux has been presented in [20], where the thermodynamics of solar cell efficiency has been considered and in [13] where the statistical thermodynamics foundation of photovoltaic and photothermal conversion has been theorized.

The results of these researches are condensed in the following energy and entropy balances for the primary work extractor, which are given by, respectively:

(A1) U c ′ = ∫ A c J U n e t + J Q + J G n e t + J W e l a s t n e t d A ,

(A2) S c ′ = ∫ A c J S n e t + J Q T c d A + S ˙ c , g e n .

The net flux densities entering and exiting the primary work extractor and other quantities used in eqs. (A1) and (A2) are defined in Table A1. The surface integrals in eqs. (A1) and (A2) cover the whole surface area A c of the primary work extractor. Equation (A1) takes into account that the boundary of the primary work extractor is crossed by fluxes of particles from the source, by heat fluxes, and by fluxes of electrical and chemical energy and elastic energy.

Table A1

Energy and entropy per unit volume and net energy and entropy flux densities for the primary work extractor.

	1. Quantity	2. Symbol	3. Shortcut	4. Units
1	Energy of primary work extractor	U c = ∫ V c u c d V	U c	J
2	Entropy of primary work extractor	S c = ∫ V c s c d V	S c	J/K
3	Energy of primary work extractor per unit volume	u c r , t	u c	J/m³
4	Entropy of primary work extractor per unit volume	s c r , t	s c	J/(Km³)
5	Net energy flux density	J U n e t T s , T c , t	J U	W/m²
6	Net entropy flux density	J S n e t T s , T c , t	J S	W/(Km³)
7	Heat flux density	J Q T c , t	J Q	W/m²
8	Rate of entropy generation	S ˙ c , g e n = ∫ V c s ˙ c , g e n d V	S ˙ c , g e n	W/K
9	Entropy generation per unit volume	s ˙ c , g e n r , t	s ˙ c , g e n	W/(Km³)
10	Net electrical and chemical energy flux density	J G n e t t	J G	W/m²
11	Net elastic energy flux density	W ˙ e l a s t n e t t	J e l a s t	W/m²

The net energy and entropy flux density, J U n e t T s , T c , t and J S n e t T s , T c , t , respectively, in Table A1 and the net energy and entropy flux, Φ U n e t T s , T c , t and Φ S n e t T s , T c , t , respectively, in Table 1 take into account incoming particles from the source at temperature T s and the outgoing particles of the same kind as those of the source that are emitted by stimulation by the primary work extractor, being characterized by the lower temperature T c . Particles from the source that do not exchange energy and entropy with the primary work extractor and keep their temperature T s are not counted. They include particles elastically reflected by the primary work extractor or particles transmitted through the primary work extractor. The net electrical and chemical energy flux density and elastic energy flux density, J G n e t t and W ˙ e l a s t n e t t , respectively, in Table A1 and the net flux of electrical and chemical energy and elastic energy, G ˙ n e t t and W ˙ e l a s t n e t t , respectively, in Table 1, take into account that the accumulators of electrical, chemical, and potential mechanical energy may transfer work to the primary work extractor, depending on the difference of electrical, chemical, or mechanical potential created across the primary work extractor [13].

Notice that the entropy transported by the flux densities J G n e t t and W ˙ e l a s t n e t t is zero. Therefore, there is no contribution from these flux densities to the entropy balance eq. (A2).

Usage of the Gauss–Ostrogradsky theorem allows to transform the surface integrals in eqs. (A1) and (A2) into integrals over the volume V c of the primary work extractor. The definitions in Table A1 are used and the local form of the energy and entropy balance equations is, respectively:

(A3) u c ′ = ∇ · J U + ∇ · J Q + ∇ · J G + ∇ · J e l a s t ,

(A4) s c ′ = ∇ · J S + ∇ · J Q T c + s ˙ c , g e n .

A case of academic interest involves an environment in thermal equilibrium at (constant) temperature T 0 . This allows stating the usual definition of the exergy per unit volume ( x c ), which is

(A5) x c = u c − T 0 s c .

Multiplying eq. (A4) by T 0 and subtracting eq. (A4) from eq. (A3) yields the following local exergy balance equation:

(A6) x c ′ = ∇ · J X t o t − T 0 s ˙ c , g e n ,

where the total exergy flux density J X t o t is defined as follows:

(A7) J X t o t ≡ J U − T 0 J S + 1 − T 0 T c J Q + J G + J e l a s t .

The first parenthesis in the r. h. s. of eq. (A7) is the exergy flux density of the incoming particles inside the primary work extractor, whereas the second term in the r. h. s. of eq. (A7) is the exergy transported by the heat flux density. The last two terms in the r. h. s. of eq. (A7) show that the flux densities of electrical and chemical energy and elastic energy, respectively, are indeed exergy flux densities.

References

[1] B. Andresen, M. H. Rubin and R. Stephen Berry, Availability for finite-time processes. General theory and a model, J. Phys. Chem. 87 (1983), 2704–2713.10.1021/j100238a006Search in Google Scholar

[2] K. H. Hoffmann, B. Andresen and P. Salamon, Measures of dissipation, Phys. Rev. A 39 (1989), 3618–3621.10.1103/PhysRevA.39.3618Search in Google Scholar PubMed

[3] V. Badescu, Optimal paths for minimizing lost available work during usual heat transfer processes, J. Non-Equilib. Thermodyn. 29 (2004), 53–73.10.1515/JNETDY.2004.005Search in Google Scholar

[4] V. Badescu, Lost available work and entropy generation: heat versus radiation reservoirs, J. Non-Equilib. Thermodyn. 38 (2013), 313–333.10.1515/jnetdy-2013-0017Search in Google Scholar

[5] V. Badescu, Unified upper bound for photothermal and photovoltaic conversion efficiency, J. Appl. Phys. 103 (2008), 054903.10.1063/1.2890145Search in Google Scholar

[6] V. Badescu, Exergy transported by particle fluxes, Physica A 387 (2008), 1818–1826.10.1016/j.physa.2007.11.043Search in Google Scholar

[7] V. Badescu and D. Isvoranu, Classical statistical thermodynamics approach for the exergy of nuclear radiation, Europhys. Lett. 80 (2007), 30003.10.1209/0295-5075/80/30003Search in Google Scholar

[8] V. Badescu, Exact and approximate statistical approaches for the exergy of blackbody radiation, Cent. Eur. J. Phys. 6 (2008), 344–350.10.2478/s11534-008-0033-1Search in Google Scholar

[9] V. Badescu, Exergy of boson and fermion fluxes, Int. J. Exergy 6 (2009), 749–759.10.1504/IJEX.2009.027500Search in Google Scholar

[10] V. Badescu, Exergy of nuclear radiation – a quantum statistical thermodynamics approach, Cent. Eur. J. Phys. 7 (2009), 141–146.10.2478/s11534-008-0136-8Search in Google Scholar

[11] C. Essex, D. C. Kennedy and R. S. Berry, How hot is radiation?, Am. J. Phys. 71 (2003), 969.10.1119/1.1603268Search in Google Scholar

[12] P. T. Landsberg and G. Tonge, Thermodynamic energy conversion efficiencies, J. Appl. Phys. 51 (1980), R1, DOI: 10.1063/1.328187.Search in Google Scholar

[13] V. Badescu and P. T. Landsberg, Statistical thermodynamic foundation for photovoltaic and photothermal conversion. I. Theory, J. Appl. Phys. 78 (1995), 2782, DOI: 10.1063/1.360755.Search in Google Scholar

[14] C. Essex, D. C. Kennedy and S. A. Bludman, The Nonequilibrium Thermodynamics of Radiation Interaction, in: S. Sieniutycz and H. Farkas (eds.), Variational and Extremum Principles in Macroscopic Systems, Elsevier, Amsterdam (2005), 603–626. ch. 12.10.1016/B978-008044488-8/50032-1Search in Google Scholar

[15] C. Essex, R. McKitrick and B. Andresen, Does a global temperature exist?, J. Non-Equilib. Thermodyn.. 32 (2007), 1–27.10.1515/JNETDY.2007.001Search in Google Scholar

[16] C. Essex and B. Andresen, The principal equations of state for classical particles. photons, and neutrinos, J. Non-Equilib. Thermodyn. 38 (2013), 293–312.10.1515/jnetdy-2013-0005Search in Google Scholar

[17] S. Karlsson, The exergy of incoherent electromagnetic radiation, Phys. Scr.. 26 (1982), 329–332.10.1088/0031-8949/26/4/009Search in Google Scholar

[18] V. Badescu, Thermodynamics of Photovoltaics, in: Reference Module in Earth Systems and Environmental Sciences, Elsevier (2013), 1–39. DOI: 10.1016/B978-0-12-409548-9.04806-5.Search in Google Scholar

[19] P. T. Landsberg and V. Badescu, Carnot factor in solar cell efficiencies, J. Phys. D, Appl. Phys. 33 (2000), 3004–3008.10.1088/0022-3727/33/22/320Search in Google Scholar

[20] J. E. Parrott, Thermodynamics of solar cell efficiency, Sol. Energy Mater. Sol. Cells 25 (1992), 73–85.10.1016/0927-0248(92)90017-JSearch in Google Scholar

[21] V. Badescu and P. T. Landsberg, Statistical thermodynamic foundation for photovoltaic and photothermal conversion. II. Application to photovoltaic conversion, J. Appl. Phys. 78 (1995), 2793.10.1063/1.360077Search in Google Scholar

[22] K. M. Van Vliet, Entropy production and choice of heat flux for degenerate semiconductors and metals, Phys. Status Solidi B 78 (1976), 667–676.10.1002/pssb.2220780227Search in Google Scholar

[23] B. C. Eu, Kinetic theory and irreversible thermodynamics of nonlinear transport processes in quantum systems, J. Chem. Phys. 80 (1984), 2123, DOI: 10.1063/1.446978.Search in Google Scholar

[24] B. C. Eu and K. Mao, Kinetic theory approach to irreversible thermodynamics of radiation and matter, Physica A 180 (1992), 65–114.10.1016/0378-4371(92)90109-4Search in Google Scholar

[25] J. E. Parrott, Transport theory of semiconductor energy conversion, J. Appl. Phys. 53 (1982), 9105, DOI: 10.1063/1.330422.Search in Google Scholar

Received: 2021-05-25

Revised: 2021-11-02

Accepted: 2021-11-10

Published Online: 2021-12-21

Published in Print: 2022-01-31

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/jnet-2021-0039

Keywords for this article

energy flux; work rate; exergy; work deficiency