The effect of turbulenceâradiation interaction on ...

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ARTICLE IN PRESS

Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 12–23 www.elsevier.com/locate/jqsrt

The effect of turbulence–radiation interaction on radiative entropy generation and heat transfer Miguel Caldas, Viriato Semia˜o Mechanical Engineering Department, Instituto Superior Te´cnico, Technical University of Lisbon, Avenue Rovisco Pais, 1049 001 Lisboa, Portugal Received 8 May 2006; received in revised form 31 July 2006; accepted 9 August 2006

Abstract The analysis under the second law of thermodynamics is the gateway for optimisation in thermal equipments and systems. Through entropy minimisation techniques it is possible to increase the efﬁciency and overall performance of all kinds of thermal systems. Radiation, being the dominant mechanism of heat transfer in high-temperature systems, plays a determinant role in entropy generation within such equipments. Turbulence is also known to be a major player in the phenomenon of entropy generation. Therefore, turbulence–radiation interaction is expected to have a determinant effect on entropy generation. However, this is a subject that has not been dealt with so far, at least to the extent of the authors’ knowledge. The present work attempts to ﬁll that void, by studying the effect of turbulence–radiation interaction on entropy generation. All calculations are approached in such a way as to make them totally compatible with standard engineering methods for radiative heat transfer, namely the discrete ordinates method. It was found that turbulence–radiation interaction does not signiﬁcantly change the spatial pattern of entropy generation, or heat transfer, but does change signiﬁcantly their magnitude, in a way approximately proportional to the square of the intensity of turbulence. r 2006 Elsevier Ltd. All rights reserved. Keywords: Entropy generation; Turbulence–radiation interaction; Radiative transfer; Participating medium

1. Introduction On a recent paper by the present authors [1], the generation of entropy through radiative transfer in participating media was analysed and a numerical method was developed for its calculation. As advocated then, this subject is of utmost interest and importance since radiation, being the dominant form of heat transfer in high-temperature systems, plays a major role in entropy generation that takes place inside such systems. More generally, and independently of the generation process, entropy generation is an extremely interesting and promising ﬁeld of the thermal sciences, although it has frequently been neglected. In fact, the study of entropy generation, that is, the analysis under the second law of thermodynamics, is the gateway for Corresponding author. Tel.: +351 218417726; fax: +351 218475545.

E-mail address: [email protected] (V. Semia˜o). 0022-4073/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2006.08.006

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Nomenclature c G h I Ib k ka ke ks Q S T V F O Z ~ s

speed of light in vacuum irradiation Planck’s constant Radiation intensity blackbody radiation intensity Boltzmann’s constant absorption coefﬁcient extinction coefﬁcient scattering coefﬁcient heat entropy temperature volume phase function solid angle wave number vector deﬁning a direction

optimisation studies in thermal equipments and systems. Through entropy minimisation techniques it is possible to increase the efﬁciency and overall performance of all kinds of thermal systems. This becomes quite evident when one notices that a thermal apparatus producing less entropy through irreversibility will destroy less available work and, therefore, will present an increased efﬁciency (see the work of Bejan [2] for a more thorough discussion on theme of entropy generation minimisation). A mechanism that is renowned for its contribution for entropy generation is turbulence. For instance, the effect of turbulence in the generation of entropy has been addressed by Kock and Herwig [3] and Adeyinka and Naterer [4]. Besides its contribution for entropy generation, turbulence is also known to interact in a decisive manner with radiative transfer. This interaction has been addressed by several authors, either concerning heat transfer alone, like in the work of Hall and Vranos [5], Hartick et al. [6], Liu e Xu [7], Snegirev [8] or Coelho [9], or taking into account the effect of turbulence on the chemical reactions of combustion, like in Baurle e Girimaji [10] or in Coelho et al. [11]. In all previously cited works, their authors concluded that turbulence–radiation interaction plays an important role in the overall heat transfer inside combustion chambers. Notice, however, that the radiation–turbulence interaction has been appreciated as an important physical phenomenon since the 1980s. See, for instance [12] for an elderly reference. However, although the importance of turbulence–radiation interaction on radiative transfer and of turbulence as a mechanism for entropy generation is well acknowledged, the effect of turbulence–radiation interaction in radiative entropy generation has never been reported in the open literature, at least at the extent of the authors’ knowledge. This is not surprising, since it was already seen in the above-mentioned work of the present authors [1] that works in the area of entropy generation through radiative transfer are sparse. Notice, however, that the subject of entropy generation through radiative transfer has been dealt with within the ﬁeld of atmospheric physics [13–15], though not considering the interaction with turbulence. In the present paper, the work presented in [1] about entropy generation by radiative transfer in participating media is developed in order to account for turbulence–radiation interaction effects. The methodology used here is identical to that applied to the former work: the only source of irreversible entropy generation is assumed to be that due to interaction between radiation and matter, entropy ﬂuxes at the walls are not analysed, (for a discussion of these topics the reader is referred to [16]), numerical simulation is not assumed to be accurate from a physical reality point of view (since the modelling is overly simpliﬁed in some aspects not relevant for the scope of the paper) and is used for illustration purposes only. However, opposite

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to the previous paper, and based on the conclusions drawn from it, scattering is not considered in the present work. Following the philosophy of simplifying to the extreme all the aspects that lie outside the main scope of the paper, which is the development of a methodology for the calculation of radiative entropy generation under the inﬂuence of radiation–turbulence interaction, a simpliﬁed approach is followed for the numerical modelling of turbulence effects: Gaussian probability density functions (pdfs) are assumed and only turbulent temperature ﬂuctuations are accounted for. These assumptions will be further developed in the following sections. 2. Radiative entropy production and heat transfer It was seen in [1] that the heat transferred between the radiative ﬁeld and the medium is given by Eq. (1) and that entropy generation due to radiative transfer is given by Eq. (2). dQZ ¼ ka;Z 4pI b;Z G Z dV dZ, (1)

ka;Z 4pI b;Z

dSG;Z ¼

1 1 1 1 1 ka;Z GZ þks;Z GZ dV dZ. T EM;Z T T EX;Z T T SC;Z T EX;Z 1

(2)

In the above equations, ka,Z is the medium spectral absorption coefﬁcient, ks,Z is the spectral scattering coefﬁcient, Ib,Z is the blackbody spectral intensity, given by Eq. (3), GZ is the irradiance, given by Eq. (4), Z is the wave number, V stands for volume, T is the medium’s temperature and TEM,Z, TEX,Z and TSC,Z are all different radiation point temperatures, which are so designated because they are related to the emission, extinction and scattering phenomena, respectively. Equations for such temperatures are expressed by Eqs. (5)–(7). It should be noted that in the case of isotropic radiation all these temperatures will be identical. 2hc2 Z3 , exp hcZ=kT 1 Z GZ ¼ I Z ð~ sÞ dO,

I b;Z ¼

(3)

(4)

4p

1 T EM;Z 1 T EX;Z 1 T SC;Z

1 ¼ 4p R ¼

Z

1 dO, sÞ 4p T Z ð~

1 ~ 4p T Z ð~ sÞ I Z ðsÞ dO

R

~ 4p I Z ðsÞ dO

R R ¼

1 4p 4p T Z ð~ sÞ FZ

4p

R

(5)

,

0 0 0 ~ s IZ ~ s ;~ s dO dO

~ 4p I Z ðsÞ dO

(6)

.

(7)

In Eq. (3), h is Planck’s constant, c is the speed of light in vacuum and k is Boltzmann’s constant. In Eq. (4) ~ s deﬁnes a direction, I Z ð~ sÞ is thespectral radiation intensity in that direction and dX is the elementary solid angle around it. In Eqs. (5)–(7), FZ ~ s;~ s0 is the medium phase function and T Z ð~ sÞ, related to the radiation intensity through Eq. (8), is the spectral temperature of a pencil of radiation. T Z ð~ sÞ ¼

hcZ 1 . 2 3 k ln 2hc Z =I Z ð~ sÞ þ 1

(8)

Obviously, in order to know the radiation intensity ﬁeld, the radiative heat transfer equation (RHTE)— expressed through Eq. (9)—must be solved by some of the standard methods available in the literature. Z 0 0 0 dI Z ð~ ks;Z sÞ ¼ ke;Z I Z ð~ FZ ~ s IZ ~ (9) sÞ þ ka;Z I b;Z þ s ;~ s dO . ds 4p 4p It should be remembered that it was concluded by the present authors [1] that the scattering phenomenon contributes for the generation of entropy to a much lesser extent than those of emission and absorption. In

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fact, it can be observed from the case study in [1] that the contribution from scattering is about 200 times less than that from absorption and emission. Therefore, the scattering term in Eq. (2) can be neglected. Doing so results in 1 1 1 1 dS G;Z ¼ ka;Z 4pI b;Z GZ dV dZ. (10) T EM;Z T T EX;Z T This equation can be rearranged as 4pI b;Z GZ G Z 4pI b;Z dS G;Z ¼ ka;Z þ dV dZ. T EM;Z T EX;Z T

(11)

In Eq. (11), the ﬁrst two terms of the right-hand side concern entropy changes within the radiative ﬁeld alone while the third term concerns the entropy change occurring in the matter. Substituting the deﬁnitions of GZ, TEM,Z and TEX,Z into Eq. (11) results in Z Z I Z ð~ GZ 4pI b;Z sÞ 1 dS G;Z ¼ ka;Z I b;Z dO dO þ dV dZ. (12) T sÞ sÞ 4p T Z ð~ 4p T Z ð~

3. Averaging procedure As mentioned in the Introduction, although it deals generically with the effect of turbulence–radiation interaction in entropy generation, this paper focuses particularly on the effect of turbulent temperature ﬂuctuations. In fact, the variables that are relevant to this study depend either on the temperature alone, like the blackbody intensity, or, in the case of the absorption coefﬁcient, on the temperature, species concentrations and pressure. Therefore, since ﬂuctuations of other variables besides temperature inﬂuence only the absorption coefﬁcient, two assumptions have to be made in order to isolate the inﬂuence of the temperature ﬂuctuations themselves: (i) temperature ﬂuctuations are statistically independent from every other ﬂuctuation and (ii) the absorption coefﬁcient depends weakly on the temperature. However, the above assumption of statistical independence between temperature and species concentrations is not necessarily realistic. As a matter of fact, it is common practise, when simulating turbulent non-premixed combustion, to make both temperature and species concentrations depend on the mixture fraction, and, therefore, make them correlated among each other. Then, a pdf with a single independent variable, usually the mixture fraction, is used to carry out the averaging procedure. This approach was used by [5,6,8,9] and [11]. Nevertheless, in the present paper the statistical independence between temperature and species concentrations will be assumed in order to enhance the physical insight over the effect of the intensity of turbulent ﬂuctuations of the temperature in radiative entropy generation. Under those assumptions, it is now possible to ﬁnd a simpliﬁed way to express the average value of the product of the absorption coefﬁcient by any given function of the temperature. In order to do so, it is best to start with the deﬁnition of the average value of that product, expressed by Eq. (13), where x represents all variables other than temperature (like pressure and species concentrations), and f represents the averaging pdf. Z Z ka ðT; xÞF ðTÞ ¼ ka ðT; xÞF ðTÞf ðT; xÞ dT dx. (13) DT

Dx

The ﬁrst step is to change the variables of integration from T and x to their ﬂuctuations, which yields Z 1Z 1 ka ðT; xÞF ðTÞ ¼ ka ðT; xÞF ðTÞf ðT; xÞ dT 0 dx0 . (14) 1

1

Now, under the assumption of statistical independence, the joint pdf can be split into the product of two independent pdfs Z 1Z 1 ka ðT; xÞF ðTÞ ka ðT; xÞF ðTÞf T ðTÞf x ðxÞ dT 0 dx0 . (15) 1

1

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Moreover, under the assumption of the absorption coefﬁcient being weakly dependant on the temperature and, therefore, being independent of temperature ﬂuctuations, one gets [16] Z 1 Z 1 ¯ xÞf x ðxÞ dx0 ka ðT; xÞF ðTÞ ka ðT; F ðTÞf T ðTÞ dT 0 . (16) 1

1

This means that it is possible to write, in a fairly approximate way ka ðT; xÞF ðTÞ k¯ a F ðTÞ.

(17)

The result obtained and expressed by Eq. (17) will be used throughout this paper. The averaging of functions depending only on the temperature, F(T), will be carried out according to the following procedure: Z 1 F ðTÞ ¼ F ðTÞf T ðTÞ dT 0 . (18) 1

To represent the temperature pdf, fT, the Gaussean distribution indicated in Eq. (19) was chosen. ¯ 2 =2s2T exp ðT TÞ pﬃﬃﬃﬃﬃﬃ f ðTÞ ¼ . sT 2p

(19)

This is not an uncommon choice. In fact, many other authors, as for example, Baurle e Girimaji [10], have made similar options. The popularity of the Gaussean distribution can be justiﬁed not only due to the fact of its mathematical simplicity but also because this distribution is the one that emerges from the quadratic expansion of a general pdf centred in its modal value. In fact, that quadratic expansion is a Gaussean distribution with the same mean value and standard deviation as those of the original pdf (see Sivia [17] for details on this derivation). Therefore, the Gaussean distribution, Eq. (19), is the most comprehensive and ¯ and of the standard deviation, sT. general one for given values of the mean temperature value, T, The standard deviation is related to the average of the ﬂuctuations square, as indicated in Eq. (20). Notice that in some works, as for example those of Caldeira-Pires et al. [18] and Liu e Xu [7], the standard deviation of the temperature is also called root mean-square temperature (rms temperature) because of the way it depends on the temperatures ﬂuctuations. qﬃﬃﬃﬃﬃﬃﬃ sT ¼

T 02.

(20)

pﬃﬃﬃ 2sT , Substituting Eq. (19) into Eq. (18) and performing a change in the integration variable, i.e., z ¼ T one gets Z 1 1 F ðTÞ ¼ pﬃﬃﬃ F ðTÞ expðz2 Þ dz. (21) p 1 ¯ and standard deviation, One can also express the instantaneous temperature in terms of its mean value, T, sT, as follows: Z 1 1 sT pﬃﬃﬃ F ðTÞ ¼ pﬃﬃﬃ F T¯ 1 þ (22) 2z expðz2 Þ dz. p 1 T¯ 0

Through simple inspection of Eq. (22), it is possible to conclude that the parameter that determines the ¯ intensity of turbulence is the quotient between the temperature standard deviation and its mean value, sT =T. Focussing now on Eq. (12), it is easy to notice that some quantities, namely T and Ib,Z, the latter being a function of the former, present an exclusively local character, while other quantities, like I Z ð~ sÞ and T Z ð~ sÞ, the latter being again a function of the former, present an obviously global character, depending both on local conditions and on the conditions of the entire domain. So, it is reckonable that it is senseless to average such quantities that depend globally on the entire domain using an exclusively local pdf. Therefore, for those parameters presenting global character, i.e., for functions of the radiation intensity, average values should be calculated using a different pdf, as indicated in Eq. (23). Z 1 sI;Z ð~ sÞ pﬃﬃﬃ 1 F Z I Z ð~ F Z I¯ Z ð~ (23) sÞ ¼ pﬃﬃﬃ sÞ 1 þ 2z expðz2 Þ dz. p 1 sÞ I¯ Z ð~

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It should be noted that the standard deviation of the intensity presents both spectral and directional character, and that it will also present a global character, as it is obvious. Therefore, it will not be possible to calculate the standard deviation of the intensity based on local properties only and a transport equation will have to be solved for it. The exact procedure will be described in the next section. 4. Mean values of entropy production and heat transfer The average value of heat transfer due to radiation can be found by averaging Eq. (1). Using the procedure described before, one gets ¯ Z ¼ k¯ a;Z 4pI¯ b;Z G ¯ Z dV dZ. dQ (24) The average black body intensity can be found simply by substituting Eq. (3) into Eq. (22), resulting in 2 3Z 1 Z expðz2 Þ ¯I b;Z ¼ 2hc pﬃﬃﬃ pﬃﬃﬃ dz. (25) ¯ 2z 1 p 1 exp hcZ kT¯ 1 þ ðsT =TÞ The average irradiance, being a linear function of the intensity, is simply given by Z ¯ I¯ Z ð~ sÞ dO. GZ ¼

(26)

4p

In order to ﬁnd the average intensity of radiation, I¯ Z ð~ sÞ, it is necessary to take the average of the RHTE, which results in Z Z 0 0 0 k¯ s;Z 0 0 0 0 0 dI¯ Z ð~ sÞ 1 0 0 ¯ ¯ ¯ ¼ ke;Z I Z ð~ ks;Z FZ ~ s IZ ~ ks;Z FZ ~ s IZ ~ sÞ ke;Z I Z ð~ sÞ þ ka;Z I b;Z þ s ;~ s dO þ s ;~ s dO . 4p 4p 4p 4p ds (27) It is possible to abridge Eq. (27) by introducing a simpliﬁcation performed by [8,9,11], among others, consisting of neglecting all the terms that involve a correlation between intensity ﬂuctuations and local properties ﬂuctuations. This procedure assumes that turbulent eddies are homogeneous, statistical independent from each other and optically thin. Another way to state the same thing is to say that the ﬂuctuation distance is much shorter than the inverse of the absorption coefﬁcient. Coelho et al. [9], estimated that the error introduced through this assumption is not greater than 4%. Therefore, Eq. (27) will assume the following form: Z 0 0 0 dI¯ Z ð~ sÞ 1 ¼ k¯ e;Z I¯ Z ð~ ks;Z FZ ~ s I¯ Z ~ (28) sÞ þ ka;Z I b;Z þ s ;~ s dO . 4p 4p ds Finally, using the assumptions presented in the previous section, the following form will result: Z 0 0 0 dI¯ Z ð~ sÞ 1 ¼ k¯ e;Z I¯ Z ð~ ks;Z FZ ~ s I¯ Z ~ sÞ þ k¯ a;Z I¯ b;Z þ s ;~ s dO . 4p 4p ds

(29)

At this point, one possesses all that is required to compute the radiative heat transfer. Focusing now on entropy generation, taking the average of Eq. (12) results in ( ) Z Z ~ I G I b;Z ð s Þ 1 Z Z dS¯ G;Z ¼ k¯ a;Z I b;Z dO dO þ 4p dV dZ. (30) ~ ~ T T T T ð s Þ ð s Þ 4p Z 4p Z As discussed in the last section, and in order to calculate the averages indicated on the ﬁrst terms inside sÞ I¯ Z ð~ sÞ must be known. The denominator of that fraction offers no brackets in Eq. (30), the value of sI;Z ð~ difﬁculty, since it is already known. The numerator, the standard deviation of the intensity, is given by the qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ square root of the intensity ﬂuctuations autocorrelation, i.e., sI;Z ð~ sÞ ¼ I 0Z 2 ð~ sÞ. Therefore, it is necessary to calculate the value of I 0Z 2 ð~ sÞ in order to evaluate the average entropy generation. To do so, let us take the ﬂuctuating part of the RHTE, Eq. (31), multiply it by the ﬂuctuation of the intensity, I 0Z ð~ sÞ, and take the

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average of the product. The result is expressed in Eq. (32). dI 0Z ð~ sÞ ds

I 0Z ð~ sÞ

1 ¼ k¯ e;Z I 0Z ð~ sÞ k0e;Z I¯ Z ð~ sÞ þ k¯ a;Z I 0b;Z þ k0a;Z I¯ b;Z þ 4p Z 0 0 0 0 1 þ ks;Z FZ ~ s I¯ Z ~ s ;~ s dO . 4p 4p

dI 0Z ð~ sÞ ds

Z 4p

0 0 0 0 ks;Z FZ ~ s ; s0 I Z ~ s dOZ

¼ k¯ e;Z I 0Z 2 ð~ sÞ I¯ b;Z k0e;Z I 0Z ð~ sÞ þ k¯ a;Z I 0b;Z I 0Z ð~ sÞ þ I¯ b;Z k0a;Z I 0Z ð~ sÞ Z Z 0 0 0 0 0 0 0 0 0 1 1 ks;Z FZ ~ s IZ ~ ks;Z FZ ~ s I Z ð~ þ s ;~ s I Z ð~ sÞ dO0 þ s ;~ sÞI¯ Z ~ s dO , 4p 4p 4p 4p

Using the simplifying assumptions referred earlier, Eq. (32) simpliﬁes into Z 0 0 0 0 dI 0Z ð~ sÞ 1 ¼ k¯ e;Z I 0Z 2 ð~ I 0Z ð~ ks;Z FZ ~ s IZ ~ sÞ sÞ þ k¯ a;Z I 0b;Z I 0Z ð~ sÞ þ s ;~ s I Z ð~ sÞ dO0 . 4p 4p ds

ð31Þ

ð32Þ

(33)

Since intensity ﬂuctuations are history dependant, i.e., they depend on the properties of the entire path travelled, it is not expected to exist any kind of correlation between the ﬂuctuations of the intensity of two different pencils of radiation. Therefore, the last term of Eq. (33) can be neglected. Also, considering that dI 0Z ð~ sÞ d 02 , I Z ð~ sÞ ¼ 2I 0Z ð~ sÞ ds ds the ﬁnal transport equation for I 0Z 2 ð~ sÞ becomes

d 02 I ð~ sÞ ¼ 2k¯ e;Z I 0Z 2 ð~ sÞ þ 2k¯ a;Z I 0b;Z I 0Z ð~ sÞ. (34) ds Z Notice that the structure of the present equation is identical to that of the RHTE. Therefore, Eq. (34) can be solved with any of the standard known methods for solving the RHTE. Also notice that this equation should be solved with nil Drichlet-type boundary conditions, i.e., I 0Z 2 ð~ sÞ ¼ 0, at the wall. This conclusion results from the fact that emission from the wall is not subjected to turbulent ﬂuctuations and, since diffuse reﬂection is assumed, turbulent ﬂuctuations of reﬂected radiation will annihilate each other because of their statistical independence. However, the transport equation for I 0Z 2 ð~ sÞ contains an unknown quantity, I 0b;Z I 0Z ð~ sÞ. In order to solve Eq. (34) this variable must be known, which means that an extra transport equation has to be deduced and solved. In order to ﬁnd this new transport equation, one can again start with Eq. (31), multiply it by the ﬂuctuation of the blackbody intensity, I 0b;Z , and take the average of the product. The result of such procedure is expressed in Eq. (35). I 0b;Z

dI 0Z ð~ sÞ ds

¼ k¯ e;Z I 0b;Z I 0Z ð~ sÞ þ k¯ a;Z I 0 2b;Z I¯ Z ð~ sÞk0e;Z I 0b;Z þ I¯ b;Z k0a;Z I 0b;Z Z Z 0 0 0 0 0 0 0 0 0 1 1 0 þ ks;Z FZ ~ s IZ ~ ks;Z FZ ~ s I b;Z I¯ Z ~ s ;~ s I b;Z dO þ s ;~ s dO . 4p 4p 4p 4p

ð35Þ

The left-hand side of Eq. (35) can be altered if one takes into account the following expansion: dI 0b;Z dI 0Z ð~ sÞ d 0 0 þ I 0b;Z . (36) I b;Z I Z ð~ sÞ ¼ I 0Z ð~ sÞ ds ds ds Combining Eq. (35) and (36) results in d 0 0 I I ð~ sÞ ¼ k¯ e;Z I 0b;Z I 0Z ð~ sÞ þ k¯ a;Z I 0 2b;Z I¯ Z ð~ sÞk0e;Z I 0b;Z þ I¯ b;Z k0a;Z I 0b;Z ds b;Z Z Z Z 0 0 dI 0b;Z 0 0 0 0 0 0 0 0 1 1 þ þ ks;Z FZ ~ s I b;Z I¯ Z ~ ks;Z FZ ~ s I b;Z I Z ~ s ;~ s dO þ I 0Z ð~ sÞ s ;~ s dO . 4p 4p 4p 4p ds ð37Þ

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0 Notice that I 0Z ð~ sÞdI 0b;Z =ds; can also be written as I 0Z ð~ sÞ dI b;Z =ds : This is a correlation between ﬂuctuation of the intensity of radiation and ﬂuctuation of a local property (the derivative of the blackbody intensity with respect to the distance is clearly local) and, consistently with the assumption used throughout this paper that the ﬂuctuation distance is much shorter than the inverse of the absorption coefﬁcient, can be neglected. An alternative derivation of why this term can be neglected can be found in [16]. Therefore, using the assumptions consistently used in this paper, one gets the following result for the equation of transport of I 0b;Z I 0Z ð~ sÞ: Z

0 0 0 0 0 d 0 0 1 I b;Z I Z ð~ ks;Z FZ ~ s I b;Z I Z ~ (38) sÞ ¼ k¯ e;Z I 0b;Z I 0Z ð~ sÞ þ k¯ a;Z I 0 2b;Z þ s ;~ s dO . ds 4p 4p This equation, similarly to Eq. (34) and due to the same reasons, should be solved by one of the known methods for solving the RHTE subjected to zero Drichelet-type boundary conditions.

At this point, a method for the computation of sI;Z ð~ sÞ I¯ Z ð~ sÞ is available, and one can turn his attention back to Eq. (30). Doing so, and according to the reasoning previously presented, one concludes that the ﬂuctuations of local properties, like temperature, are independent from the ﬂuctuations of global properties, like radiation intensity. Therefore, when averaging a term that contains a product of a local quantity by a global quantity, it is legitimate to take the product of the respective averages. Therefore, the ﬁnal expression for entropy generation will be ( ) Z Z ~ I I b;Z ð s Þ 1 1 Z ¯Z dS¯ G;Z ¼ k¯ a;Z I¯ b;Z 4p dV dZ. (39) dO dO þ G ~ ~ T T T T ð s Þ ð s Þ Z Z 4p 4p In Eq. (39), the terms I¯ b;Z , 1=T and I b;Z =T present a local character and should be calculated using the general Eq. (22). The terms 1=T Z ð~ sÞ and I Z ð~ sÞ=T Z ð~ sÞ clearly exhibit a global character and should be ¯ Z results directly from the average intensity, given by the averaged RHTE. calculated though Eq. (23). G 5. Case study In order to test the proposed methodology for the numerical calculation of radiative entropy generation and heat transfer under the inﬂuence of turbulence, a simple case was chosen: a rectangular combustion chamber, presenting a hotter central zone, to simulate the presence of a ﬂame and possessing constant wall properties. The geometry and properties of this case study are shown in Fig. 1. Taking advantage of the symmetry of the problem only half of the domain was simulated. Due to its small inﬂuence on the issues under study (two orders of magnitude inferior to other phenomena, as it can be seen in detail in [1]), scattering was neglected. Also, grey behaviour was assumed for the sake of simplicity. As far as turbulent ﬂuctuations are concerned, several distinct cases were studied. First, a case with no turbulence was simulated in order to provide a reference case against which other cases could be compared. Next, turbulent ﬂuctuations were introduced, while keeping constant the average values. According to the experimental work of Caldeira-Pires et al. [18], the value of sT =T¯ in the ﬂame zone is approximately 0.25. This value should be expected to decrease as one moves to zones of the combustion chamber that are more distant from the ﬂame. Therefore, in the present case study, the value of sT =T¯ in the remaining zone of the combustion chamber (other than the ﬂame zone) was set equal to 0.1. In order to investigate the rate of change of the quantities under study with the intensity of turbulence, a number of intermediate values of sT =T¯ were also simulated. These are depicted in Fig. 1. Notice that in all the cases studied where turbulence is present, the reason between the value of sT =T¯ inside the ﬂame (which assumes values between 0.05 and 0.30) and outside of it (between 0.02 and 0.12), was kept constant and equal to 2.5. Fig. 2 depicts the heat exchange distribution between the medium and the radiative ﬁeld for the nonturbulent case. The radiative entropy generation for the same case is presented in Fig. 3. It can be very easily veriﬁed that both the energy exchanged and the entropy generation are much greater in the zone of the ﬂame, where the interaction between the medium and the radiative ﬁeld is higher. In fact, both parameters assume an almost nil value elsewhere in the domain. It can also be observed that the entropy generation is also inﬂuenced by the presence of strong gradients of temperature and absorption coefﬁcients, as it was already concluded in [1].

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TW=1000 K, ε=0.8

T=1200 K, T T=0.00; 0.02; …; 0.10; 0.12 ka=0.25 m-1 ks=0.00 m-1 y

x

T=2000 K, T T=0.00; 0.05; …; 0.25; 0.30 ka=0.4 m-1 ks=0.0 m-1

Fig. 1. Geometry, wall and medium properties and conditions of the case study.

1.5E+03

Q [kW m-3]

1.2E+03

1.2E+03-1.5E+03 9.0E+02-1.2E+03 6.0E+02-9.0E+02 3.0E+02-6.0E+02 0.0E+00-3.0E+02

9.0E+02 6.0E+02 3.0E+02

2.5E-02 7.5E-02 1.3E-01 1.8E-01 2.3E-01 2.8E-01 3.3E-01 3.8E-01 4.3E-01 4.8E-01 5.3E-01 5.8E-01 6.3E-01 6.8E-01 7.3E-01 7.8E-01 8.3E-01 8.8E-01 9.3E-01 9.8E-01

0.0E+00 3.3E-01 2.3E-01 1.3E-01 ] [% 2.5E-02 Y

X [% ]

Fig. 2. Energy transferred between the material medium and the radiative ﬁeld for the non-turbulent case—sT =T¯ fl ¼ 0.

In Figs. 4 and 5, the distributions of radiative energy transfer and radiative entropy generation, respectively, can be seen for the case sT =T¯ fl ¼ 0:25, which is the closest to the measurements of [18]. As it can be seen, the pattern of those distributions is very similar to that that of the case without turbulence, varying only the magnitude of the parameters. The same feature is present in every other values of the parameter sT =T¯ fl , being therefore useless to present ﬁgures referring to other cases. Therefore, the following analysis will be centred in the total energy transfer and entropy generation throughout the problem’s domain. In order to allow the referred analysis of both the total energy transfer and entropy generation values, such values are displayed in Table 1 for different values of turbulence intensity, keeping the ratio between the turbulence intensity in the ﬂame and that outside the ﬂame equal to 2.5. These data becomes more easily interpretable when directly compared with the reference case of no turbulence. This is done in Fig. 6, where the values of the relative change in total radiative heat transfer and entropy generation are plotted against the intensity of the turbulence within the ﬂame. In the same ﬁgure curves presenting the functional form y ¼ axb,

ARTICLE IN PRESS M. Caldas, V. Semia˜o / Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 12–23

21

500

SG [W K-1 m-3]

400

400-500 300-400 200-300

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

X [%]

0.325 0.225 0.125 ] [% 0.025 Y

Fig. 3. Entropy generated through radiative transfer for the non-turbulent case—sT =T¯ fl ¼ 0.

1.8E+03

Q [kW m-3]

1.5E+03

1.5E+03-1.8E+03 1.2E+03-1.5E+03 9.0E+02-1.2E+03 6.0E+02-9.0E+02 3.0E+02-6.0E+02 0.0E+00-3.0E+02

1.2E+03 9.0E+02 6.0E+02 3.0E+02

2.5E-02 7.5E-02 1.3E-01 1.8E-01 2.3E-01 2.8E-01 3.3E-01 3.8E-01 4.3E-01 4.8E-01 5.3E-01 5.8E-01 6.3E-01 6.8E-01 7.3E-01 7.8E-01 8.3E-01 8.8E-01 9.3E-01 9.8E-01

0.0E+00 3.3E-01 2.3E-01 ] 1.3E-01 [% 2.5E-02 Y

X [%

]

Fig. 4. Energy transferred between the material medium and the radiative ﬁeld for a representative turbulent case—sT =T¯ fl ¼ 0:25.

ﬁtted to the previous data, are displayed. These ﬁtted curves show that, curiously, both the total heat transfer and the total entropy generation depend very closely in a quadratic way with the intensity of turbulence. It is also observable that the rate of growth of total entropy generation is about twice of that of total heat transfer. This marked inﬂuence of turbulence in radiative heat transfer is already known and has been researched (see, for instance, Coelho [11]). However, the even stronger effect of turbulence on radiative entropy generation remains an unexplored area of research and might come to have a signiﬁcant impact on the design, conception and operation of high-temperature devices.

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700

SG [W K-1 m-3]

600 500

600-700

400

400-500

500-600 300-400

300

200-300 100-200

200

0-100

100

[% ]

0.325 0.225 0.125 0.025 Y

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

X [%]

Fig. 5. Entropy generated through radiative transfer for a representative turbulent case—sT =T¯ fl ¼ 0:25.

Table 1 Total energy transfer and entropy generation for several levels of turbulence intensity Case

sT =T¯ in the ﬂame

sT =T¯ outside the ﬂame

QTOT (kW)

STOT (W/K)

1 2 3 4 5 6 7

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.00 0.02 0.04 0.06 0.08 0.1 0.12

59.11 59.88 62.20 66.11 71.65 78.90 87.96

17.04 17.53 19.03 21.33 24.86 28.24 33.68

100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% 0.05

ChgS≈10.2 x1.949

Chg Q≈5.5 x2.021

0.1

0.15

0.2

0.25

0.3

T T fl Fig. 6. Change of total energy transferred and entropy generated in relation to the reference case of sT =T¯ fl ¼ 0.

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6. Conclusions This paper deals with the modelling and numerical calculation method of entropy generation and heat transfer by radiation under the inﬂuence of turbulence. 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 equipments functioning in the turbulent regime or as a tool for improving it through entropy minimisation techniques. It was concluded that turbulence–radiation interaction does not signiﬁcantly change the pattern of entropy generation, or heat transfer, but does change signiﬁcantly their magnitude. The results here presented hint that both heat transfer and entropy generation increases are approximately proportional to the square of the turbulence intensity, although the rate of increase of entropy is about twice the rate of increase of heat transfer. 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 PhD scholarship SFRH/BD/4833/2001. References [1] Caldas M, Semia˜o V. Entropy generation through radiative transfer in participating media: analysis and numerical computation. JQSRT 2005;96:423–7. [2] Bejan A. Fundamentals of exergy analysis, entropy generation minimization, and the generation of ﬂow architecture. Int J Energy Res 2002;26:545–65. [3] Kock F, Herwig H. Local entropy production in turbulent shear ﬂows: a high-Reynolds number model with wall functions. Int J Heat Mass Transfer 2004;47:2205–15. [4] Adeyinka OB, Naterer GF. Modeling of entropy production in turbulent ﬂows. J Fluids Eng 2004;126:893–9. [5] Hall RJ, Vranos A. Efﬁcient calculations of gas radiation from turbulent ﬂames. Int J Heat Mass Transfer 1994;37:2745–50. [6] Hartick JW, Tacke M, Fruchtel G, Hassel EP, Janicka J. Interaction of turbulence and radiation in conﬁned diffusion ﬂames. Proc Combust Inst 1996;26:75–82. [7] Liu LH, Xu X. Monte Carlo ray-tracing simulation for radiative heat transfer in turbulent ﬂuctuating media under the optically thin ﬂuctuation approximation. JQSRT 2004;84:349–55. [8] Snegirev AY. Statistical modelling of thermal radiation transfer in buoyant turbulent diffusion ﬂames. Combust Flame 2004;136:51–71. [9] Coelho PJ. Detailed numerical simulation of radiative transfer in a nonluminous turbulent jet diffusion ﬂame. Combust Flame 2004;136:481–92. [10] Baurle RA, Girimaji SS. Assumed PDF turbulence-chemistry closere with temperature–composition correlations. Combust Flame 2003;134:131–48. [11] Coelho PJ, Teerling OJ, Roekaerts D. Spectral radiative effects and turbulence/radiation interaction in a non-luminous turbulent jet diffusion ﬂame. Combust Flame 2003;133:75–91. [12] Song TH, Viskanta R. Interaction of radiation with turbulence: application to a combustion system. J Thermophys Heat Transfer 1987;1:56–62. [13] Goody RM, Abdou W. Reversible and irreversible sources of atmospheric entropy. QJR Meteorolog Soc 1996;122:493–4. [14] Goody RM. Sources and sinks of climate entropy. QJR Meteorolog Soc 2000;126:1953–70. [15] Pauluis O, Held IM. Entropy budget of an atmosphere in radiative–convevtive equilibrium. Part I: maximum work and frictional dissipation. J Atmos Sci 2002;59:125–39. [16] Caldas M, Modelac- a˜o da Interacc- a˜o da radiac- a˜o com o meio em sistemas de combusta˜o, Phd. thesis, Instituto Superior Te´cnico, Universidade Te´cnica de Lisboa, 2005. [17] Sivia DS. Data analysis, a Bayesian tutorial. Oxford: Oxford University Press; 1996. [18] Caldeira-Pires A, Heitor MV, Moreira ALN. On the analysis of temperature dissipation in a turbulent jet propane ﬂame. Exp Therm Fluid Sci 1998;18:116–21.

Examining the effect of binary interaction parameters on ...