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ARTICLE IN PRESS

Journal of Quantitative Spectroscopy & Radiative Transfer 96 (2005) 423–437 www.elsevier.com/locate/jqsrt

Entropy generation through radiative transfer in participating media: analysis and numerical computation Miguel Caldas, Viriato Semiao Mechanical Engineering Department, Instituto Superior Te´cnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal Received 20 July 2004; accepted 16 November 2004

Abstract Thermodynamics’ second law analysis is the gateway for optimization in thermal equipments and systems. Through entropy minimization techniques it is possible to increase the efﬁciency and overall performance of all kinds of thermal systems. This approach is becoming common practice in the analysis and/or design of thermal equipments. However, evaluation of entropy generation due to radiative transfer in participating media seems to be lacking. Since radiation is the dominant mechanism of heat transfer in high-temperature systems, such omission seems quite unjustiﬁable. Although the subject of entropy production through radiative transfer has been dealt with for quite some time, notably by Max Planck himself, it has not been approached in the perspective of its numerical calculation in a way that is compatible and coherent with the standard heat transfer approach. In the present work, the issue of entropy generation by radiative transfer in participating media is approached from the view-points of its mathematical modeling and numerical calculation using standard radiative heat transfer techniques, namely the discrete ordinates method. Effects from emission, absorption and scattering are isolated and considered independently. r 2005 Elsevier Ltd. All rights reserved. Keywords: Entropy; Entropy generation; Radiative transfer; Participating medium

Corresponding author. Tel.: +351 218417726; fax: +351 218475545.

E-mail address: [email protected] (V. Semiao). 0022-4073/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2004.11.008

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Nomenclature A c G h I Ib k ka ke ks L Q S s T u V F O Z ~ n ~ s r

area speed of light in vacuum irradiation Planck’s constant radiation intensity black body radiation intensity Boltzmann’s constant absorption coefﬁcient extinction coefﬁcient scattering coefﬁcient radiation entropy intensity heat entropy or length traveled radiative entropy density temperature radiative energy density volume phase function solid angle wave number unit normal vector vector deﬁning a direction vector representing a point in space

1. Introduction Entropy generation is an extremely interesting and promising ﬁeld of the thermal sciences, even if frequently neglected. In fact, the study of entropy generation, or thermodynamics’ second law analysis, is the gateway for optimization studies in thermal equipment and systems. Through entropy minimization techniques it is possible to increase the efﬁciency and overall performance of all kinds of thermal systems. This becomes quite transparent when one notices that a thermal apparatus producing less entropy through irreversibilities will destroy less available work and, therefore, will present an increased efﬁciency: see the work of Bejan [1] for a more thorough discussion on the theme of entropy generation minimization. This kind of second law approach is now starting to be used in the analysis and/or design of thermal equipment (see, for instance, Kock and Herwig [2]). However, the authors of the present work have no knowledge of such a work that includes the effect of radiative transfer in the analysis, although entropy generation through radiative transfer acquires a very signiﬁcant relevance when dealing with high-temperature thermal equipment, since radiation is the dominant form of heat transfer in high-temperature systems.

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The subject of entropy production through radiative transfer has been dealt with for quite some time, notably by Max Planck himself [3] and soon after by Oxenius [4] and Kroll [5]. However, such a subject has not—at least to the extent of the present work authors’ knowledge—been approached in the perspective of its numerical calculation in a way that is compatible and coherent with the standard heat transfer approach. For instance, Fort [6] used the maximum entropy formalism applied to radiative transfer, but the procedure is intended to provide a nearequilibrium solution of the radiative ﬁeld. Therefore, in that case, entropy analysis was used as a tool to get the solution of the intensity distribution, not in an analysis and optimization perspective. An exception to such a trend can be found in Wright et al. [7], where the subject of radiative entropy transfer and generation in engineering systems is approached. However, the analysis is restrained to the case of gray radiation in non-participating media. This means that only entropy generation at solid boundaries was considered and all the interaction of the radiative ﬁeld with the participating medium was not accounted for. In this work, the issue of entropy generation by radiative transfer in participating media is dealt with. By different magnitudes, entropy generation is due to all the different ways in which the radiative ﬁeld interacts with the material medium, i.e., emission, absorption and scattering. The approach is consistent with the standard radiative transfer calculation in engineering systems like, e.g., the discrete ordinates method. Since the entropy generation due to the interaction of the radiative ﬁeld with solid boundaries was already treated by Wright et al. [7], it is not dealt with in the present work.

2. Radiation thermodynamics Thermal radiation ﬁlling a given region of space constitutes a thermodynamic system, although in some ways different from a conventional material only system. In fact, spectral and directional effects must be accounted for and these contain counter-intuitive aspects that are sometimes difﬁcult to grasp. As far as the directional effects are concerned, several radiation thermodynamic properties are deﬁned for each pencil of radiation in terms of its spectral intensity, like temperature, for instance. Each radiation pencil has its own temperature (if it is a monochromatic or black radiation pencil, otherwise it has an inﬁnite number of temperatures, one for each wave number). Therefore, in a given point of space there are inﬁnite radiation temperatures, since this property depends simultaneously on direction and wave number. Radiation temperature at a given wave number is deﬁned as the temperature that would be achieved by a body that would totally absorb radiation at the wave number under consideration, while being totally transparent to radiation at all other wave numbers, i.e., the body would behave as a black body only at the considered wave number [8]. The resulting temperature would clearly be the temperature that would produce a black body intensity of magnitude equal to that in which the body is immersed at the considered wavelength. Obviously, this imaginary body could not be under the inﬂuence of any other phenomenon for this deﬁnition to be applicable. In order to express radiation temperature in terms of spectral intensity, one can start from the spectral black body intensity expression, Eq. (1), reverse it and substitute the black body intensity by an arbitrary monochromatic intensity. Then, Eq. (2) for the spectral radiation temperature is obtained (an equivalent expression formulated in terms of frequency can be

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found in [3]), which, like radiation intensity, is a property presenting both directional and spectral characters: I b;Z ðrÞ ¼

sÞ ¼ T Z ðr;~

2hc2 Z3 , expðhcZ=kTðrÞÞ 1

hcZ 1 . 2 k ln½2hc Z3 =I Z ðr;~ sÞ þ 1

(1)

(2)

In the above equations h is Planck’s constant, c is the speed of light in vacuum, k is Boltzmann’s constant, Z is the wave number, r represents a point in space and ~ s deﬁnes a direction. sÞ; is A radiation beam also carries entropy. The spectral radiation entropy intensity, LZ ðr;~ deﬁned in a manner similar to the radiation intensity (more accurately, radiation energy intensity) and, like the former, presents both directional and spectral characters. The expression that allows for the calculation of the entropy intensity can be found in [3] and is reproduced herein through Eq. (3). Its derivative with respect to radiation intensity equals the inverse of radiation temperature—Eq. (4). This can be found in [3] or veriﬁed by differentiation of Eq. (3): I Z ðr;~ sÞ I Z ðr;~ sÞ I Z ðr;~ sÞ I Z ðr;~ sÞ 2 sÞ ¼ 2kcZ þ 1 ln þ1 ln , (3) LZ ðr;~ 2hc2 Z3 2hc2 Z3 2hc2 Z3 2hc2 Z3 2 3 qLZ ðr;~ sÞ k 2hc Z 1 ln ¼ þ1 ¼ . T Z ðr;~ sÞ hcZ I Z ðr;~ sÞ sÞ qI Z ðr;~

(4)

From radiation intensity it is possible to calculate the radiative energy density, u,—see [3] or [9]— like in Eq. (5), were O represents the solid angle and the integral is to be taken over the entire sphere, i.e., over all directions. Likewise, radiative entropy density, s, can be calculated from radiative entropy intensity—Eq. (6). Note that these quantities retain their spectral character but have lost their directional dependence. Z 1 I Z ðr;~ sÞ dO, (5) uZ ðrÞ ¼ c 4p 1 sZ ðrÞ ¼ c

Z LZ ðr;~ sÞ dO.

(6)

4p

Knowing both energy and entropy densities, it is possible to make use of the well known and already mentioned (see Eq. (4)) thermodynamic relation expressed by Eq. (7)—see [3] or [8]—to ﬁnd the radiation temperature for a given point in space, which means to ﬁnd a radiation temperature that does not present directional dependence. Intuitively, one would expect this temperature to be deﬁned by some averaging of the directional temperature via some spherical integration, as expressed by Eqs. (5) and (6). In fact, it will be seen that the result is not far from this. qsZ ðrÞ 1 ¼ . T Z ðrÞ quZ ðrÞ

(7)

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Substituting Eqs. (5) and (6) into Eq. (7), we get R q 4p LZ ðr;~ sÞ dO 1 . ¼ R T Z ðrÞ q 4p I Z ðr;~ sÞ dO

427

(8)

Eq. (8) presents the differentiation of a functional with respect to another functional. The function sÞ: Therefore, Eq. (8) can be expanded into subjacent to both functionals is I Z ðr;~ R q LZ ðr;~ sÞ dO=qI Z ðr;~ sÞ 1 . (9) ¼ R4p T Z ðrÞ q 4p I Z ðr;~ sÞ dO=qI Z ðr;~ sÞ Eq. (9) presents two cases of differentiation of functional with respect to its subjacent function. Therefore, it becomes necessary to make use of Fre´chet derivative, or Gaˆteaux derivative [10]. In this case these two derivatives will be identical. As a matter of fact, if the Gaˆteaux derivative exists, it will always be identical to the Fre´chet derivative [10]. After performing the derivatives, the result obtained is that presented in Eq. (10): R ½1=T Z ðr;~ sÞhZ ðr;~ sÞ dO 1 . (10) ¼ 4p R T Z ðrÞ sÞ dO 4p hZ ðr;~ As we can see from Eq. (10), the radiation point temperature is not univocally deﬁned, since the Gaˆteaux derivative depends on the increment function, h, and this could be any function whatsoever [10]. So, apart from the case of isotropic radiation where the radiation temperature at a point is uniquely deﬁned, it is possible to deﬁne several radiation temperatures at a point of space. The deﬁnition of different radiation temperatures at a point will in fact arise naturally, as the subsequent analysis will show.

3. Absorption and emission If a pencil of monochromatic radiation hits an elementary volume of matter at a given temperature and presenting a certain absorption coefﬁcient (see Fig. 1) two different events will occur: there will be an incremental increase (or decrease) in the pencil’s radiation intensity and the volume of matter will lose (gain) a given amount of energy. The energy transferred from the matter to the radiation ﬁeld is sÞ ¼ dI Z ð~ sÞ dAð~ s ~ nÞ dO dZ. dQZ ð~

(11)

dS Iη ( s ) TM

n dA

ka,η

Iη ( s ) + dIη ( s )

Fig. 1. Interaction between an elementary volume of matter and incident monochromatic radiation.

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In the previous equation, dO is the elementary solid angle around de-propagation direction, dA is an elementary area and Q represents energy transferred through heat. It should be noticed that from this point forward the dependence of location (through the r symbol) is dropped for simplicity. In order to consider the area normal to the propagation direction, the area element, dA; is multiplied by the inner product of the unit propagation direction vector, ~ s; with the area element unit normal, ~ n: The intensity variation is sÞ ¼ ka;Z I b;Z dI Z ð~

dS dS sÞ ka;Z I Z ð~ . ð~ s ~ nÞ ð~ s ~ nÞ

(12)

The ﬁrst term on the right-hand side in Eq. (12) refers to emission, while the second one concerns absorption. I b;Z is the black body intensity at the temperature of the matter, ka;Z is the medium’s absorption coefﬁcient and dS is an elementary length—see Fig. 1. Notice that the distance traveled within the matter is the quotient of dS by the inner product mentioned before. The energy transferred is sÞ ¼ ka;Z ðI b;Z I Z ð~ sÞÞ dV dO dZ, dQZ ð~

(13)

where dV ¼ dS dA: This energy exchange will cause a variation of entropy in the volume of matter under consideration that is given by its deﬁnition s; ZÞ ¼ dS M ð~

dQZ ð~ sÞ ka;Z ðI Z ð~ sÞ I b;Z Þ ¼ dV dO dZ. TM TM

(14)

On the other hand, the increment in radiation entropy is given by dS R;Z ð~ sÞ ¼ dLZ ð~ sÞ dAð~ s ~ nÞ dO dZ and the increment in radiation entropy intensity can be expressed by qLZ dLZ ð~ sÞ ¼ dI Z ð~ sÞ. qI Z I Z ð~sÞ As already seen qLZ 1 . ¼ qI Z I Z ð~sÞ T Z ð~ sÞ

(15)

(16)

(17)

Therefore, dLZ ð~ sÞ ¼

dI Z ð~ sÞ sÞ T Z ð~

(18)

or, by using Eq. (12), it is possible to write the increment of radiative entropy intensity as presented in the following expression where ds is the elementary distance traveled by the radiation beam: sÞ ¼ dLZ ð~

dI Z ð~ sÞ ðI b;Z I Z ð~ sÞÞ ¼ ka;Z ds. T Z ð~ T Z ð~ sÞ sÞ

(19)

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The variation of radiative entropy is equal to dS R;Z ð~ sÞ ¼

dQZ ð~ sÞ ðI b;Z I Z ð~ sÞÞ ¼ ka;Z dV dO dZ. sÞ sÞ T Z ð~ T Z ð~

Therefore, the entropy generated in the process is 1 1 sÞ ¼ dS M;Z ð~ sÞ þ dS R;Z ð~ sÞ ¼ sÞ. dQZ ð~ dS G;Z ð~ T Z ð~ sÞ T M

(20)

(21)

In the previous equation, dQ is positive whenever energy is transferred from the matter to the radiative ﬁeld, i.e., whenever the black body radiative intensity at the matter’s temperature, T M ; is superior to the intensity of incident radiation. This is equivalent to saying that dQ will be positive if T M is higher than the radiation temperature and negative otherwise. So, if T M is in fact higher than the radiation temperature, then dQ will be positive and the term within brackets on the righthand side of Eq. (21) will also be positive. This means that entropy generation will be positive. On the other side, if T M is lower than the radiation temperature, then dQ will be negative but the term within brackets on the right-hand side of Eq. (21) will also be negative. This means that entropy generation will once more be positive. If matter and radiation are in equilibrium, no energy will be exchanged and entropy generation will be nil. As expected, entropy generation is always nonnegative. Energy transferred from matter to the radiative ﬁeld in a point in space is calculated by integrating Eq. (13) along the 4p steradians solid angle, so as to eliminate directional effects. The result is presented below; GZ is the irradiation deﬁned by Eq. (23): dQZ ¼ ka;Z ð4pI b;Z G Z Þ dV dZ,

(22)

Z GZ ¼

I Z ð~ sÞ dO ¼ cuZ .

(23)

4p

In order to calculate the entropy generation in a point in space, a similar procedure is adopted. To do so, it is convenient to start by writing Eq. (21) in a slightly more expanded way: 1 1 sÞ ¼ ka;Z ðI b;Z I Z ð~ sÞÞ dV dO dZ. (24) dS G;Z ð~ T Z ð~ sÞ T M Operating now a directional integration of Eq. (24), one gets R Z sÞI Z ð~ sÞ dO 1 1 Z ð~ 4p ½1=T R dV dZ dO G Z dS G;Z ¼ ka;Z 4pI b;Z 4p 4p T Z ð~ sÞ sÞ dO 4p I Z ð~ 1 ka;Z ð4pI b;Z G Z Þ dV dZ. TM

ð25Þ

The ﬁrst parcel on the right-hand side of Eq. (25) represents the entropy increase from the radiation side, while the second parcel represents the decrease in entropy veriﬁed in the matter. It should be noticed that the decrease in entropy occurring in the matter is simply dS M ¼ dQZ =T M ; although in the radiation side, the expression is far more complicated (except in the particular case of an isotropic radiation ﬁeld; in this case it is simply dS R;Z ¼ dQZ =TZ).

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In the special case of isotropic radiation, both terms inside straight parenthesis in Eq. (25) are identical to the point radiative temperature. In this special case Eq. (25) assumes the following, particularly simple, form: 1 1 (26) dQZ . dS G;ZISO ¼ TZ TM

4. Scattering When a beam of monochromatic radiation irradiates an elemental volume of matter, presenting a given scattering coefﬁcient, there will be a decrement in the radiation intensity of the original beam due to the effect of out-scattering—see Fig. 2. This means that a portion of the original radiative energy will be scattered along all directions in space. The exact proportion of the radiative energy that will be scattered in each direction is determined by the medium phase function. When a portion of matter is irradiated from all directions, besides the decrease in intensity due to out-scattering, a given beam of radiation will experience an increase in intensity on account of the radiative energy originally traveling in other directions that was scattered into the direction under analysis. This phenomenon is called in-scattering. It should be stressed that the scattering phenomenon does not involve any energy transfer between the radiative ﬁeld and matter, or vice-versa. It merely reorganizes the directional distribution of energy within the radiative ﬁeld. Therefore, the radiative ﬁeld energy level, the matter energy level and the matter entropy level will remain unaltered. The only change presenting a non-directional character that will take place is the change in the entropy level of the radiative ﬁeld, i.e., scattering is an entropy generator phenomenon. The decrease in the original beam’s intensity due to out-scattering is given by the following equation, where ks;Z is the scattering coefﬁcient dI Z ð~ sÞ ¼ ks;Z I Z ð~ sÞ dS.

(27)

d Is,η (s' )

Iη (s) – d Iη (s)

Iη (s) dA

ks,η dS

Fig. 2. Out-scattering phenomenon.

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Energy conservation requires energy out-scattered from its original direction to be somewhere in all other directions. This conservation principle is expressed by Eq. (28): Z sÞ ¼ dI s;Z ð~ s0 Þ dO0 . (28) dI Z ð~ 4p

Therefore, the increment in radiative intensity observed in all other directions is given by Eq. (29), s;~ s0 Þ is the medium’s phase function. It is easy to verify that, because of the phase where FZ ð~ function normalization, Eqs. (28) and (29) are perfectly compatible. FZ ð~ s;~ s0 Þ sÞ. dI Z ð~ 4p The combined decrement of intensity due to both out-scattering and in-scattering is Z sÞ ¼ ks;Z I Z ð~ sÞ dS dI s;Z ð~ s0 ;~ sÞ dO0 . dI Z ð~ s0 Þ ¼ dI s;Z ð~

(29)

(30)

4p

Substituting Eq. (29) into this, one gets Z 1 0 0 0 sÞ ¼ ks;Z I Z ð~ sÞ FZ ð~ s ;~ sÞI Z ð~ s Þ dO dS. dI Z ð~ 4p 4p This energy intensity decrement has associated an entropy intensity decrement equal to Z I Z ð~ sÞ 1 1 0 0 0 dLZ ð~ sÞ ¼ ks;Z FZ ð~ s ;~ sÞI Z ð~ s Þ dO dS. sÞ 4p T Z ð~ sÞ 4p T Z ð~

(31)

(32)

In order to calculate the entropy generated in a point in space due to scattering, one must take into account that Eq. (32) represents a decrement, multiply it by dA and dZ and integrate it along all directions. The result is Z Z Z FZ ð~ s;~ s0 Þ I Z ð~ sÞ 1 0 dO dO I Z ð~ sÞ dS G;Z ¼ ks;Z dO dV dZ. (33) 4p 4p sÞ s0 Þ 4p T Z ð~ 4p T Z ð~ Please notice that if the radiation ﬁeld is isotropic, Eq. (33) will be identically nil. This means that in an isotropic radiative ﬁeld the scattering phenomenon does not generate entropy.

5. Radiative heat and entropy transfer equations Adding up the intensity increments and decrements resulting from the above-discussed phenomena (absorption, emission, out-scattering and in-scattering) occurring along a path ds in which a radiation beam interacts with a lump of matter, one gets the well-known radiative heat and transfer equation (RHTE) (see, for instance, [11]), here presented in its stationary form—Eq. (34). Notice that the extinction coefﬁcient (equal to the sum of the absorption and scattering coefﬁcients: ke;Z ¼ ka;Z þ ks;Z ) has been introduced. Z dI Z ð~ sÞ ks;Z sÞ þ ka;Z I b;Z þ FZ ð~ s0 ;~ sÞI Z ð~ s0 Þ dO0 . (34) ¼ ke;Z I Z ð~ 4p 4p ds

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Recalling the relationship that exists between dI Z ð~ sÞ and dLZ ð~ sÞ—Eq. (4)—an equation for the radiative entropy intensity follows immediately: Z dLZ ð~ sÞ I Z ð~ sÞ ka;Z I b;Z ks;Z I Z ð~ s0 Þ FZ ð~ s0 ;~ sÞ (35) ¼ ke;Z þ þ dO0 . ds T Z ð~ T Z ð~ sÞ T Z ð~ sÞ 4p 4p sÞ Upon integration of Eq. (35) over the entire 4p steradians, Eq. (36) is obtained, which, after multiplication by dA and dZ; provides an equation for the local increment of entropy generation due to the radiative ﬁeld: Z dLZ ð~ sÞ dO ds 4p Z Z Z Z I Z ð~ sÞ ks;Z I Z ð~ s0 Þ 1 ¼ ke;Z FZ ð~ s0 ;~ sÞ ð36Þ dO þ ka;Z I b;Z dO þ dO0 dO. T Z ð~ sÞ sÞ 4p 4p 4p sÞ 4p T Z ð~ 4p T Z ð~ Notice that this equation could also have been easily derived from the addition of the entropy intensity increments and decrements resulting from absorption, emission and scattering—Eqs. (19) and (32)—just as the RHTE. It is now appropriate to introduce, for simplicity sake, three different radiative temperatures, that arise quite naturally from inspection of Eq. (36), that will be referred to as T EM;Z ; T EX;Z and T SC;Z because they are related to the emission, extinction and scattering phenomena, respectively. Expressions for these temperatures can be found in Eqs. (37)–(39). In the case of isotropic radiation all these temperatures will be identical. Z 1 1 1 dO, (37) ¼ T EM;Z 4p 4p T Z ð~ sÞ 1 T EX;Z 1 T SC;Z

R ¼

ð~ sÞI Z ð~ sÞ dO 4p ½1=T R Z sÞ dO 4p I Z ð~

,

(38)

R R ¼

sÞFZ ð~ s0 ;~ sÞI Z ð~ s0 Þ dO0 dO 4p 4p ½1=T Z ð~ R 4p

sÞ dO 4p I Z ð~

.

(39)

Using the nomenclature just introduced, it is possible to write for the entropy generated by radiative phenomena the following expression: GZ 4pI b;Z GZ ð4pI b;Z G Z Þ dV dZ. (40) þ ka;Z þ ks;Z ka;Z dS G;Z ¼ ke;Z TM T EX;Z T EM;Z T SC;Z In Eq. (40), the ﬁrst three terms on the right-hand side concern entropy changes within the radiative ﬁeld alone and the fourth term concerns the entropy change happening in the matter. Eq. (40) can be rewritten in order to isolate the entropy generation due to each of the emission, extinction and scattering phenomenon, like in Eq. (41). Notice that there is no contribution of T M

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for the entropy generation due to scattering, as expected. 1 1 1 1 ka;Z GZ dS G;Z ¼ ka;Z 4pI b;Z T EM;Z T M T EX;Z T M 1 1 dV dZ. þ ks;Z G Z T SC;Z T EX;Z

433

ð41Þ

6. Case study In order to test the proposed methodology for the numerical calculation of entropy generation due to radiative transfer in participating medium, a simple case was tested: a rectangular combustion chamber, presenting a hotter central zone, to simulate the presence of a ﬂame, and with constant wall properties. The geometry and properties of this case study can be seen in Fig. 3. Taking advantage of the symmetry of the problem only half of the domain was simulated. Anisotropic scattering was considered and gray behavior was assumed. For this case study the irradiation, G, is shown in Fig. 4. As it can be seen, this parameter presents approximately a two-fold variation within the domain and is maximum roughly at the center of the ﬂame. In Fig. 5 a plot of the total entropy generated in both the radiative ﬁeld and material medium can be seen. Notice that entropy generated at the walls of the chamber is not considered and, therefore, does not show in the plot. What can be seen in this plot is that the entropy generated in the cooler, less participative part of the domain is rather small, and then increases steeply at the zone where strong gradients of temperature and absorption/scattering coefﬁcients exist. This is not surprising at all, since it is well established that interfaces and strong gradient zones are the preferential sites for entropy generation. It then decreases slightly toward the interior of the simulated ﬂame, but remains at levels much higher than those occurring in the cooler region of the combustion chamber. This behavior is certainly due to the much larger interaction between radiation and matter—that can be inferred by the values of the absorption/scattering coefﬁcients—happening within the ﬂame. Entropy appears to be generated mainly through this interaction. TW=1000 K, ε=0.8 T=1200 K ka=0.25 m-1 ks=0.01 m-1, Φ=1+0.6 cos(θ)

T=2000 K ka=0.4 m-1 ks=0.2 m-1, Φ=1+0.6 cos(θ)

Fig. 3. Geometry and properties of the case study.

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6.2E+05

G [W m-2]

5.6E+05

5.6E+05-6.2E+05

5.0E+05

5.0E+05-5.6E+05

4.4E+05

4.4E+05-5.0E+05 3.8E+05

3.8E+05-4.4E+05

3.2E+05

3.2E+05-3.8E+05 2.6E+05-3.2E+05 0.025 0.075 0.125 0.175 0.225 0.275 0.325 0.375 0.425 0.475 0.525 0.575 0.625 0.675 0.725 0.775 0.825 0.875 0.925 0.975

2.6E+05 0.325 0.225 0.125 ] 0.025 [%

Y

X [%]

Fig. 4. Irradiation ﬁeld.

500

SG [W K-1 m-3]

400 300

400-500 300-400 200-300 100-200 0-100

200 100

0.025 0.075 0.125 0.175 0.225 0.275 0.325 0.375 0.425 0.475 0.525 0.575 0.625 0.675 0.725 0.775 0.825 0.875 0.925 0.975

0 0.325 0.225 0.125 ] 0.025 [%

Y

X [%]

Fig. 5. Total entropy generation ﬁeld.

Next, the entropy generation due to emission and absorption is shown in Fig. 6. As it can be seen, it is very similar to total entropy generation, thus hinting that these mechanisms are the main contributors to entropy generation and that entropy generation due to scattering is much smaller. This will be conﬁrmed by inspection of Fig. 7, which shows a plot of entropy generated due to scattering. Note that in this case the temperature and properties gradient has an even stronger effect than in the previous case. The mechanisms of emission and absorption were not analyzed separately. This is due to the fact that these two mechanisms have no separated existence, i.e., it is impossible to ﬁnd an actual physical situation in which one of these phenomena happens independently of the other. This

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500

SG [W K-1 m-3]

400 300

400-500 300-400 200-300 100-200 0-100

200 100

0.025 0.075 0.125 0.175 0.225 0.275 0.325 0.375 0.425 0.475 0.525 0.575 0.625 0.675 0.725 0.775 0.825 0.875 0.925 0.975

0 0.325 0.225 0.125 ] [% 0.025

Y

X [%]

Fig. 6. Entropy generation ﬁeld due to emission and absorption.

2

SG [W K-1 m-3]

1.6 1.2

1.6-2 1.2-1.6 0.8-1.2 0.4-0.8 0-0.4

0.8 0.4

0.025 0.075 0.125 0.175 0.225 0.275 0.325 0.375 0.425 0.475 0.525 0.575 0.625 0.675 0.725 0.775 0.825 0.875 0.925 0.975

0 0.325 0.225 0.125 ] [% 0.025 Y

X [%]

Fig. 7. Entropy generation ﬁeld due to scattering.

intimate relation, which can be very intuitively inferred by the fact that both phenomena depend on the same coefﬁcient, means that it is senseless to try to analyze emission and absorption separately. If one tries to do so, one will arrive at unrealistic results, i.e., the entropy generated by one of the two phenomena will be negative. Finally, the partition between entropy change from the radiative ﬁeld side and from the matter side is examined. Fig. 8 depicts the entropy change from the radiation side, while Fig. 9 shows the entropy change in the material participating medium.

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1200

SG [W K-1 m-3]

1000 800 1000-1200 800-1000 600-800 400-600 200-400 0-200

600 400 200

0.025 0.075 0.125 0.175 0.225 0.275 0.325 0.375 0.425 0.475 0.525 0.575 0.625 0.675 0.725 0.775 0.825 0.875 0.925 0.975

0 0.325 0.225 0.125 %] [ 0.025 Y

X [%]

Fig. 8. Entropy change in the radiative ﬁeld.

SG [W K-1 m-3]

720 600 480

600-720 480-600 360-480 240-360 120-240 0-120

360 240 120

0.025 0.075 0.125 0.175 0.225 0.275 0.325 0.375 0.425 0.475 0.525 0.575 0.625 0.675 0.725 0.775 0.825 0.875 0.925 0.975

0 0.325 0.225 0.125 ] [% 0.025

Y

X [%]

Fig. 9. Entropy change in the material medium.

It can be seen that the entropy change in the radiation side is much higher than that resulting from the participating medium. It can also be perceived that the radiation related entropy change is more susceptible to gradient effects than matter related entropy changes. In fact, radiation related entropy changes react not only to level of interaction between radiation and matter, but also to the presence of strong gradients in the material properties. Matter related entropy changes seem more inﬂuenced by the level of interaction between radiation and matter than by the presence of strong gradients in the material properties.

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7. Conclusions In this paper the numerical calculation of entropy generation by radiative transfer in participating media was approached. The procedure is completely compatible with standard radiative heat transfer calculation in engineering systems like the discrete ordinates method. The methodology presented here can be used either to assess the efﬁciency of high-temperature thermal equipment or as a tool for improving it through entropy minimization techniques. The separation performed herein from the start between the several phenomena involved in radiative heat transfer in participating media, i.e., emission and absorption and scattering, will allow for each of these aspects to be looked at individually, thus simplifying the analysis procedure. It was concluded that entropy generated through emission and absorption is much larger than that generated through scattering. It was also concluded, not surprisingly, that entropy generation will be higher at zones presenting strong properties gradients and higher interaction between radiation and matter. Finally, it was veriﬁed that the entropy change in the radiation side is much larger than that on the matter side. Acknowledgements This work has been performed with the ﬁnancial support of Fundac- a˜o para a Cieˆncia e a Tecnologia, Programa PRAXIS XXI, under the Ph.D. scholarship SFRH/BD/4833/2001.

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