Thermal Radiation and the Second Law of Thermodynamics

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1 IOS Jounal of Applied Physics (IOS-JAP) e-issn: olue 9, Issue 4 e. III (Jul. Aug. 2017), PP heal adiation and the Second Law of heodynaics * D. akhanlall echanical Engineeing, Anton de Ko Univesity of Suinae, Paaaibo, Suinae Coesponding Autho: D. akhanlall Abstact heal adiation is an ipotant phenoenon in any engineeing systes. Howeve, up to now, adiation theodynaic effects ae geneally neglected o evaluated incoectly by the engineeing counity. heefoe, we dedicate this note to adiation theodynaics: isconceptions suounding adiation theodynaics ae discussed; the second law of theodynaics is defined to explicitly account fo adiation theodynaic effects. and the evesible and ievesible souces of adiation entopy ae discussed. Keywods: Second law of theodynaics; heal adiation; Entopy geneation, Date of Subission: Date of acceptance: Noenclatue Aea A c h I Ib j kb L n A S gen ss, s t Speed of light Planck s constant, o enthalpy Spectal adiation intensity Spectal blackbody intensity Heat cuent density Boltzann s constant Spectal adiation entopy intensity ass flow ate Unit noal vecto of bounday Unit position vecto ate of entopy geneation density Unit diection vectos ass-specific entopy epeatue Spectal adiation tepeatue ie volue a s Geek Sybols Spectal absoption coefficient Spectal scatteing coefficient Wavelength Scatteing phase function Solid angle I. Intoduction In the engineeing counity, seveal isconceptions suound the ole and calculus of adiation entopy geneation [1]. In fact, these isconceptions ae built up step-by-step in undegaduate classes duing the intoduction of the second law of theodynaics. In standad textbooks [2, 3], the second law is foulated fist in tes of the Clausius inequality: DOI: / Page

2 heal adiation and the second law of theodynaics 0 which, when applied to a closed syste that inteacts with the envionent (Fig. 1a), shows the degee of ievesibility of diffeent paths going fo an initial state to a final state, i.e.: Sgen S2 S1 0 DOI: / Page 2 1 (1) (2) Eq. (2) is easily extended to open systes by noting that, in addition to heat, entopy is also tanspoted with ass flow (Fig. 1b). he ate of entopy geneation in an open syste is thus given by the second law in the fo S Sgen s s 0 t (3) Eq. (3) is supposed to goven entopy geneation in all type of systes [2, 3]. By expessing the heat tansfe te in vecto fo, seveal authos show that adiation theodynaic effects ae accounted fo as follows [1]: 1 S out in C 2 2 heat j j j j (4) his equation is in geneal not coect, and it actually points to a fundaental weakness in the setup of the enegy equation in which theal adiation is teated as a souce te. Such an appoach ay not coectly descibe the behavio of theal adiation. Coect extension of the second law of theodynaics is discussed heein. II. Fundaental adiation heodynaics 2.1 Planck s equation of non-equilibiu adiation theodynaics Planck [5] deived the spectal adiative entopy intensity of an incoheent unpolaized adiation bea: 2kbc I I I I L 1 ln 1 ln (5) 2hc 2hc 2hc 2hc Eq. (5) is consideed the ost fundaental equation in non-equilibiu adiation theodynaics [6]. In the special case of an equilibiu adiation field, the spectal intensity is given by Planck s law [5]: 2 2hc 1 Ib (6) 5 exp hc k heal adiation and the second law With Planck s definition fo the spectal adiation entopy intensity, the second law of theodynaics can eadily be extended to account fo theal adiation effects. Conside the open theodynaic syste shown in Fig. 2. In its ost geneal fo, the second law states that the total change of entopy in the syste is caused by the evesible entopy change (tansfeed entopy) and the ievesible entopy change (entopy geneation) [7]. he ate at which the total entopy in the syste changes is thus [7]: b desev disi (7) dt dt dt Now, conside these tes sepaately: (a) he total entopy change If we neglect theal adiation, then the total entopy change in the syste occus in atte, and is given by [7]: s d dt dt (8) t Howeve, when a adiation field is pesent, then the total entopy change in the syste consists of the entopy change in atte plus the entopy change in the adiation field. he entopy change in the adiation field is [8]: 1L,, st d dd dt c t 4 which is negligible fo ost systes because of the facto 1/c. Hence, in the pesence of theal adiation, the total entopy change is (9)

3 heal adiation and the second law of theodynaics s d dt dt dt (10) t (b) he evesible entopy change Again, neglecting theal adiation, and taken into account entopy tansfe into the syste with heat conduction at the bounday and with the ass flow, then desev c s s dt dt dt (11) bounday in out When a adiation field is also pesent, we need to conside that thee is a adiative entopy flow in the adiative field, and a adiative entopy flow associated with the absoption-eission of adiation heat by atte [9, 10]. Hence, in the, the total evesible adiative entopy change is whee, and dt dt dt dt 4 da 4 n s L s, ddda s L, s ddd d d a I Ib d dd dt 4 In the at solid walls, the total evesible adiative entopy change is given by whee, fo an opaque wall suface, and A (12) (13),s (14) dt dt dt wall w w L w, w d dda dt A 4 dt w (15) s n s (16) w A 4 w dda A w w 1 I w, w ddda s n s Hence, in the pesence of theal adiation, the evesible entopy change elation takes on the fo (c) he ievesible entopy change When adiation is neglected, the total entopy geneation is (17) desev c (18) dt dt dt dt DOI: / Page

4 heal adiation and the second law of theodynaics i dt In the pesence o adiation, Eq.(17) ust also be consideed. In this case, i dt i i S d (19) gen wall S gend S da 20) A Hence, the second law of theodynaics, estated to account fo theal adiation, assues the following fo: C Sgen L s s 0 dt out in (21) he fist te in the backet is fo the adiative entopy flow in the adiative field, while second one is fo the adiative entopy flow in atte, as shown in Fig he evesible and ievesible souces of adiation entopy he extended second law of theodynaics deived in the pevious section can in pinciple be sepaated into thee pats, i.e. one pat fo each of the caies of entopy: S S S c s s in out C 0 dt L he geneation tes on the LHS of above expession ae the ievesible souces of entopy, while the HS backet contains the evesible (tansfeed) entopy souces. he adiation ievesible souce te accounts fo adiation entopy geneation in the and at the solid walls. he following elations holds: with and S S S S 0 S wall L 0 wall Lwall 0 wall he HS expessions in these equations ae the evesible (tansfeed) adiation entopy souces. hey can be deteined fo the elations deived in the pevious sections: (a) evesible adiative entopy souces in : L L, dd d dt 4 a I Ib d dd dt 4 (22) (23) (24) (25) s s (26),s (27) DOI: / Page

5 heal adiation and the second law of theodynaics (b) evesible adiative entopy souces at solid walls: wall w w dt w A 4 s n s (28) L L, ddda 1, ddda I dt w wall w A 4 w s nw s With eq. (5), and using the adiative enegy tansfe equation (EE) [11]: to deive the adiative entopy tansfe equation (EnE):, ( ), s I s I s I a s a b s I,, d 4 s s s 4 I s, Ib s L, s L, s I s I, s ( a s ) a, s, s I, s s, d 4 ss s, 4 we can estate the evesible adiative entopy souces in thei local fos, which can be solved nueically by the pocedues given in [10], and eploy the to evaluate the local ievesible adiative entopy souces: (c) Local ievesible and evesible adiative entopy souces in : S L I, Ib s ( a s ) a, 4, s s I s ss 4, 4 s 4 s, (d) Local ievesible and evesible adiative entopy souces at walls:, d d a I Ib dd,s wall I w, s S L, d d wall L w w s n s wall 4 w he spectal adiative entopy intensity at solid boundaies can be witten in tes of wall eissivity using Planck s law: whee, L b (29) (30) (31) (32) (33) 2kc w,s 4 1 ln 1 ln (34) w exphc k w 1 (35) DOI: / Page

6 heal adiation and the second law of theodynaics III. Conclusion adiation theodynaics has significant iplications fo engineeing systes. It is howeve not pat of standad univesity cuiculu, neithe is it coectly teated in any textbooks. As long as adiation theodynaics is neglected, engineeing students will be insufficiently educated in the liitations of Clausius inequality and the second law of theodynaics. In addition, by neglecting adiation theodynaics we fail to addess the flaw in the fist law of theodynaics. In the setup of the fist law, all phenoena that ae difficult to teat at fist instance, including adiation, ae consideed to be souces. his appoach does not coectly descibe the behavio of theal adiation. Fo this eason, the heat cuent density in the enegy equation does not give an accuate account fo adiation heat tansfe. his is also why the conventional extension of the second law is incoect. he ipleentation of adiation theodynaic effects in the second law is an still initial step in teating the theodynaics of electoagnetic phenoena. It is ipotant to ipleent all the physics descibed by axwell equation in the second law. he intedisciplinay field of theodynaics, fluid dynaics, and electoagnetis has ipotant engineeing application such as agneto hydodynaic casting, and plasa flow. efeences [1] L. H. Liu, S. X. Chu. On the entopy geneation foula of adiation heat tansfe pocesses, Jounal of Heat ansfe, 2006, 128: [2] Y. A. Cengel,. Boles. heodynaics: An Engineeing Appoach New Yok: cgaw-hill Highe Education, 2012 [3]. J. oan, H. N. Shapio, D. D. Boettne,. B. Bailey. Fundaentals of Engineeing heodynaics, New Yok: Wiley and Sons, Inc., 2010 [4] D. akhanlall. Ievesible theodynaics of diffusion cobustion pocesses and the ole of theal adiation, singhua Univesity, 2015, Unpublished epot [5]. Planck. he heoy of Heat adiation, New Yok: Dove Publications, 1959 [6] D. akhanlall. heodynaic second-law analysis of adiative heat tansfe in two-phase (paticulate) edia, 2013, Jounal of heophysics and Heat ansfe [7] I. Pigogine. Intoduction to heodynaics of Ievesible Pocesses, New Yok: John Wiley & Sons, Inc., 1967 [8] D. akhanlall, L.H. Liu, H.C. Zhang. SLA (second-law analysis) of tansient adiative tansfe pocesses, Enegy, 2010 (35): [9] F. Caldas,. Seiao. Entopy geneation though adiative tansfe in paticipating edia: analysis and nueical coputation, Jounal of uantitative Spectoscopy & adiative ansfe, 2005, 96: [10] L. H. Liu, S. X. Chu. eification of nueical siulation ethod fo entopy geneation of adiation heat tansfe in seitanspaent, Jounal of uantitative Spectoscopy & adiative ansfe, 2007, 103: [11]. F. odest. adiative Heat ansfe, San Diego, CA: Acadeic Pess, 2003 DOI: / Page

7 heal adiation and the second law of theodynaics Fig. 1 Entopy tansfe in closed and open theodynaic systes DOI: / Page

8 heal adiation and the second law of theodynaics Fig. 2 Open theodynaic syste with adiation entopy tansfe *D. akhanlall. " heal adiation and the Second Law of heodynaics." IOS Jounal of Applied Physics (IOS-JAP) 9.4 (2017): DOI: / Page