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Entropy generation through combined non-grey gas radiation and forced convection between two parallel plates
ARTICLE IN PRESS
Energy 33 (2008) 1169–1178 www.elsevier.com/locate/energy
Entropy generation through combined non-grey gas radiation and forced convection between two parallel plates F. Ben Nejma, A. Mazgar, N. Abdallah, K. Charrada Unite´ de recherche: Etudes des Milieux Ionise´s et Re´actifs (EMIR), Institut Pre´paratoire aux Etudes d’Inge´nieur de Monastir, Avenue Ibn Eljazar, Monastir 5019, Tunisia Received 27 December 2006
Abstract The current study presents a numerical computation of combined gas radiation and forced convection through two parallel plates. A laminar flow of a temperature-dependent and non-grey gas in the entrance region of the channel was investigated. Over-heated water vapor was chosen as a gas because of its large absorption bands. Some special attention was given to entropy generation and its dependence on geometrical and thermodynamic parameters. The radiative part of the study was solved using the ‘‘Ray Tracing’’ method through S4 directions, associated with the ‘‘statistical narrow band correlated-k’’ (SNBCK) model. The temperature fields were used to calculate the distributions of local and global entropy generation. r 2008 Elsevier Ltd. All rights reserved. Keywords: Entropy generation; Combined radiation and convection; SNBCK model; Non-grey gas radiation
1. Introduction Entropy generation is associated with thermodynamic irreversibility, which is common in all types of heat transfer processes. Bejan [1,2] and San et al. [3] proposed different analytical solutions for the entropy generation equation in several simple flow situations. They introduced the concept of entropy production number, irreversibility distribution ratio, and presented spatial distribution profiles of entropy production. Ever since these studies were carried out, entropy generation has been used as a reference for evaluating the significance of irreversibility related to heat transfer and friction in a thermal system. This has become the main concern in many fields such as heat exchangers, turbo machinery, electronic cooling, porous media and combustion. Based on the concept of efficient energy use and the minimal entropy generation principle, optimal designs of thermodynamic systems have been widely proposed by the thermodynamic second law [4]. It is Corresponding author. Tel.: +216 73 532 007; fax: +216 73 500 512.
E-mail address: [email protected] (F. Ben Nejma). 0360-5442/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2008.02.004
possible to improve the efficiency and overall performance of all kinds of thermal systems through entropy minimization techniques. The analysis of energy utilization and entropy generation has become one of the primary objectives in designing a thermal system. Ko and Ting [5,6], Ko [7] and Mahmud et al. [8–11] performed numerous investigations to calculate entropy production and irreversibility profiles for different geometrical configurations, flow situations and thermal boundary conditions. Several studies have thoroughly dealt with fluid flow irreversibility due to viscous effect and heat transfer by conduction. However, the evaluation of entropy production due to radiative heat transfer in participating media has often been overlooked. Arpaci [12,13] and Arpaci and Selamet [14–16] evaluated the radiative entropy production by analogy to that due to conduction, which is true in the case of an optically thick medium. In their study of entropy generation due to interaction of the radiative field with solid boundaries, Wright et al. [17] extended and generalized this formalism for grey and non-participating media. Recently, Caldas and Semiao [18,19] and Liu and Chu [20] have adapted this approach for non-grey and participating
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Nomenclature
Greek symbols
(A, B) c Cp F g h I Ib k L Lx 2Ly P q s S T Tb u v w x, y
n e t k Dn ~ O dO l m r
SNB model parameters light velocity (m s1) specific heat at constant pressure (J kg1 K1) molar fraction of H2O (F ¼ 1) cumulative function Plank’s constant (h ¼ 6.626 1034 J s) radiation intensity (W m2 sr1) radiation intensity of black body (W m2 sr1) Boltzmann’s constant (k ¼ 1.38 1023 JK1) path length channel length (m)(Lx ¼ 2 m) channel width (m) pressure (Pa) heat flux (W m2) local entropy production (W K1 m3) global entropy production (W K1 m1) temperature (K) bulk temperature (K) axial velocity (m s1) transverse velocity (m s1) weight parameter cartesian coordinates
media. The present paper is aimed at inquiring about entropy generation due to combined gas radiation and forced convection through two parallel plates. The medium’s radiative behavior, which is considered as temperature-dependent and non-grey, is given by solving the ‘‘radiative transfer equation’’ (RTE). Different kinds of numerical methods have actually been developed in order to solve this equation. Modest [21] has cited most of these methods such as the spherical harmonic method, the zonal method, Monte Carlo method and the ‘‘discrete ordinate method (DOM)’’. The latter, which has received great attention during the last few years, was proposed by Chandrasekhar [22] and later developed by Truelove [23] and Carlson and Lathrop [24]. Like the DOM, the ‘‘Ray Tracing’’ method [25], applied in our study through the Sn quadratures, is based on the selection of transfer directions and their associated weights. Using the finite volume technique, we chose the 1-D radiative analysis ‘‘slab problem’’ because of difficulties encountered when planning the discretization of the RTE in a 2-D radiation analysis. This is due to the absence of accurate radiative boundary conditions at the entry and exit of the flow. Anyway, it will later be shown that this has but little effect on global entropy production when the height of the channel is small compared to its length, which is fixed at 2 m in the present study. Today, grey gas approximation is believed to be inaccurate [21] and many approximate radiative property models have been developed to represent molecular spectra of gases. While ‘‘line by line (LBL)’’ models are useful for
wave number (cm1) wall emissivity gas tranmissivity absorption coefficient (m1) spectral resolution (cm1) ray direction ~ elementary solid angle around O thermal conductivity (W m1 K1) dynamic viscosity (N s m2) density of the fluid
Subscripts c r 0 w n
conductive exchange radiative exchange ambiant wall spectral
Superscripts i
partial non-grey medium
providing benchmark solutions, they are not practical in multidimensional applications particularly when the radiative transfer is combined with other transfer modes, since it requires great computational times and storage volumes. Several other models have been introduced to calculate radiative properties of gases like the ‘‘statistical narrow band (SNB)’’ [26,27], ‘‘correlated-k (CK)’’ [28], ‘‘Correlated-k with fictitious gases (CKFG)’’ [29,30], ‘‘exponential wide-band (EWB)’’ [31], ‘‘spectral line-based weightedsum-of-grey-gases (WSGG)’’ model [32,33] and ‘‘absorption distribution function (ADF)’’ [34]. In the absence of LBL results, the SNB model is frequently considered as the most accurate non-grey gas radiation model. However, the implementation of this model into the RTE [35,36] has some shortcomings as it provides transmissivity instead of absorption coefficients [37] and contains a correlation term which requires a long computing time [38]. Initially developed for atmospheric modeling applications, the CK model has the advantage over the SNB model in that scattering effects can be accounted for. The implementation of the SNBCK model for radiation computation previously described by Liu et al. [39] represents an effective approach to avoid the expensive on-line inversion of the cumulative distribution function [28,40,41]. In this case, the distribution function is obtained by inverse Laplace transformation of the SNB gas transmissivity in order to deduce gas absorption coefficients. Heat transfer by simultaneous radiation and convection in absorbing and emitting media is of great interest and appears in many engineering applications such as boilers,
ARTICLE IN PRESS F. Ben Nejma et al. / Energy 33 (2008) 1169–1178
burners, industrial furnaces, combustion chambers, hightemperature heat exchangers, etc. The interaction of thermal radiation and forced convection in the thermal entrance region has been examined by several researchers. Ben Nejma et al. [25] applied the SNB method in rectangular enclosures and Sediki et al. [42,43] used the ADF and the correlated-k band models in cylindrical enclosures. They all could quantify radiative transfers, but did not mention entropy production.
the system: gðkin Þ ¼ gi
~ ¼ hn=kLnð2hn3 =c2 I n ðOÞ ~ þ 1Þ represents the where T n ðOÞ directional and spectral radiative temperature. Using the SNBCK4 method, the local radiative entropy production for non-grey gases is " 4 XZ X ~ sr ðx; yÞ ¼ wi kin ðI bn ðTÞ I in ðOÞÞ
The application of this method by Goutie`re et al. [41] to 2D non-grey gas radiation transfer has demonstrated that the ‘‘statistical narrow band correlated-k (SNBCK)’’ method is an efficient narrow-band method for radiative transfer calculation as well as for low-resolution spectral intensity prediction. Using the n-point Gauss quadrature, where gas radiative properties are represented by n nongrey mediums at each non-overlapping band, the SNBCK method could overcome the difficulties of the SNB model by extracting the gas absorption coefficients from the gas transmissivity given by Malkmus law [26,35] as " !# rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pB 4AL tn ðLÞ ¼ exp 1þ 1 (1) 2 pB
bands
wi fðgi Þ
4p i¼1
1 1 i ~ T T n ðOÞ
!# dO Dn
(6)
We note that the scattering effect is neglected compared to absorption and emission in gas radiation since the medium contains no particles. The gas radiative properties are obtained by integration over the bands with a 100 cm1 spectral resolution, where the covered spectral range is 187.5–4137.5 cm1. Water vapor absorption spectrum is certainly larger than the one we used, but given the temperatures are not too high (maximum 800 K), this produces very few errors.
The analytical expression of the cumulative function has been derived by Lacis and Oinas [28] as pffiffiffi 1 a 1 erf pffiffiffi b k gðkÞ ¼ 2 k pffiffiffi 1 a (2) þ 1 erf pffiffiffi þ b k exppB 2 k pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi where a ¼ 12 pAB, b ¼ 12 pB=A and erf is the error function. Using the cumulative function g, the narrow-band average of any radiative variable fn can be calculated as n X
(4)
For the spectral, directional and local radiative entropy production, Caldas and Semiao [18,19] and Liu and Chu [20] used the following expression: ! 1 1 b ~ ~ srn ðOÞ ¼ kn ðI n ðTÞ I n ðOÞÞ: (5) ~ T T n ðOÞ
2. The hybrid SNBCK model
fn ¼
1171
3. Problem formulation The physical model under study is shown in Fig. 1. We consider a stationary flow of over-heated water vapor maintained between two parallel plates. The fluid is supposed to be perfect gas, Newtonian and in laminar flow. We have neglected the effect of buoyancy forces and considered only the axial variation of pressure. The mass, momentum and energy balance equations written in Cartesian coordinates will be
(3)
i¼1
qðruÞ qðrvÞ þ ¼0 qx qy
The calculation of the various monochromatic absorption coefficients is thus reduced to the resolution of
(7)
y Tw H2O u0 T0 P0
x
Lx Fig. 1. Geometry and boundary conditions.
Tw
2Ly
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qu qu dP q qu þ m ru þ rv ¼ qx qy dx qy qy
(8)
sðx; yÞ ¼ sc ðx; yÞ þ sr ðx; yÞ þ sf ðx; yÞ
qT qT þ rC p v qx qy dP q qT þ lðTÞ ¼u divð~ qr Þ dx qy qy
rC p u
(9)
As shown in Refs. [41,45] the radiative source term is given as 4 XZ X ~ dO Dn divð~ qr Þ ¼ wi kin ðI bn I in ðOÞÞ (10) bands
At a given location, the local entropy generation is as follows:
4p i¼1
These equations are associated to the conservation equation of massic flow rate on the various cross-sections of the stream discharge Eq. (11), which is useful for calculating the pressure gradient. Z Ly _ mðxÞ ¼ rðTðx; yÞÞ uðx; yÞ dy ¼ 2rðT 0 ÞU 0 Ly (11) Ly
The boundary conditions for the considered problem are summarized by the system (12): 8 vðx; y ¼ LyÞ ¼ vðx; y ¼ LyÞ ¼ 0 > < vðx ¼ 0; yÞ ¼ 0; uðx ¼ 0; yÞ ¼ U 0 ; uðx; y ¼ LyÞ ¼ uðx; y ¼ LyÞ ¼ 0 > : Tðx ¼ 0; yÞ ¼ T ; Tðx; y ¼ LyÞ ¼ Tðx; y ¼ LyÞ ¼ T
where sf ðx; yÞ is the local entropy generation due to friction [5–11], sc ðx; yÞ is the local entropy production caused by conduction and sr ðx; yÞ is the local entropy generation due to radiation Eq. (6). m m qu 2 sf ðx; yÞ ¼ F T T qy
70 analytical
numerical
60
(12)
50 Nu (x)
Dimensionless numbers have not been utilized because of the radiative flux and the enormous variation of fluid proprieties. The physical properties are regarded as variables and are published in Ref. [25].
(14)
where F is the viscous dissipation. Irreversibility due to viscous effect is neglected compared to entropy production due to heat transfer. The local entropy production due to conduction [1,4–7] can be expressed as " 2 # lðTÞ lðTÞ qT 2 qT 2 sc ðx; yÞ ¼ 2 ðrTÞ ¼ 2 þ (15) qx qy T T
w
0
(13)
40 2Ly=15cm 30
2Ly=10cm
20
4. Entropy generation
2Ly=5cm
10
In this paper, we studied combined radiation and forced convection for a temperature-dependent and non-grey gas through two parallel plates. Like all thermodynamic systems, entropy is generated from the irreversibility due to heat transfer and the friction of fluid flow.
Table 1 Effect of axial grid nature (2Ly ¼ 5 cm; Lx ¼ 2 m; Tw ¼ 800 K; T0 ¼ 500 K; P0 ¼ 1 atm; U0 ¼ 0.1 m s1; e ¼ 1) Sinusoidal
Nu=7.54
0 0
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x (m) Fig. 2. Validation of the model. Lx ¼ 2 m, U0 ¼ 0.1 m/s Tw ¼ 600 K, T0 ¼ 400 K, P ¼ 1 atm, physical properties constant at (Tw+T0)/2.
Table 3 Validation of the radiative model Ref. [45]
Uniform
Grid (x, y) (300,50) (3000,50) (300,50) (3000,50) (6000,50) 3.08 2.32 2.77 2.91 S0x (without radiation) 3.02
0.2
2Ly ¼ 0.1 m 2Ly ¼ 1 m
Present work (SNBCK4)
SNB
SNBCK4
S4
S6
S8
14.2 30.3
14.3 30.6
14.60 31.04
14.46 30.91
14.40 30.85
Table 2 Effect of grid (2Ly ¼ 10 cm; Lx ¼ 2 m; Tw ¼ 600 K; T0 ¼ 400 K; P0 ¼ 1 atm; U0 ¼ 0.5 m s1; e ¼ 1) Grid (x, y) S0x (without radiation) S0x (with radiation)
(100,50) 2.39 6.13
(200,100) 2.78 6.51
(300,200) 3.02 6.75
(400,300) 3.12 6.85
(500,400) 3.18 6.92
(600,500) 3.19 6.92
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Table 4 Effect of friction on global entropy production (2Ly ¼ 10 cm; Lx ¼ 2 m; Tw ¼ 800 K; T0 ¼ 400 K; P0 ¼ 1 atm) Friction is considered
S0x (without radiation)
Friction is not considered
U0 ¼ 0.1 m s1
U0 ¼ 1 m s1
U0 ¼ 10 m s1
U0 ¼ 0.1 m s1
U0 ¼ 1 m s1
U0 ¼ 10 m s1
4.69266
11.43624
27.05439
4.69266
11.43623
27.05159
36 90 1D radiation 2D radiation Tin=Tout=0K 2D radiation Tin=T0 Tout=Tw 2D radiation Tin=Tout=Tw
radiative X=0.5m
70
24 S0-x (W/mK)
conductive
80 s (x,y) (W/m3K)
30
18
60 50 40 30 20 10
12
0 0
10
20
30
40
6
50 60 %y/Ly
70
80
90
100
90
100
300 0 0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x (m) Fig. 3. Effect of the 2-D radiation. Lx ¼ 2 m, 2Ly ¼ 10 cm, U0 ¼ 0.5 m/s, Tw ¼ 800 K, T0 ¼ 400 K, P ¼ 2 atm, e ¼ 1.
Therefore the total entropy generation is obtained by integrating Eq. (13). Z x Z Ly S 0x ¼ sðx; yÞ dy dx (16) 0
Ly
Because of the strong variability of the physical properties and the non-explicit form of radiative entropy production term according to the channel’s dimensions as well as the variables conditioning the flow, the analysis of the entropy minimization seems impossible (absence of similarity). This is why we will confine ourselves to presenting the evolution of entropy creation according to the various variables. 5. Grid, validation and assumption justifications Because of symmetry, the study was conducted on half of the field. The finite volume method was applied to solve the conservation equations. As shown in Table 1 the utilization of a sinusoidal grid through axial direction is essential to take into account the great variation in entropy production in the vicinity of the entrance. Table 2 shows that a mixed grid system defined by 200 uniform nodes through transverse direction, and 300 sinusoidal nodes through axial direction gives rather precise results and the error rate does not exceed 3%.
conductive
250 s (x,y) (W/m3K)
0
radiative
200 X=0.5m
150 100 50 0 0
10
20
30
40
50 60 %y/Ly
70
80
Fig. 4. Comparison between conductive and radiative transfer effects in local entropy production. (a). Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 800 K, T0 ¼ 500 K, P ¼ 1 atm, e ¼ 1, X ¼ 0.5, 1, 1.5, 2 m. (b) 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 500 K, T0 ¼ 800 K, P ¼ 1 atm, e ¼ 1, X ¼ 0.5, 1, 1.5, 2 m.
In order to validate our numerical code we compared the local Nusselt number, found by considering the physical properties as constant, with the analytical model published in Ref. [44] and given by Eq. (17). Fig. 2 shows that the difference is very weak: NuðxÞ ¼
P1 32=3Fn2 xþ 4Ly qT 8 n¼0 Gne P ¼ 2 32=3Fn2 xþ T w T b ðxÞ qy y¼0 3 1 n¼0 Gn=Fn e (17)
To validate the radiative model chosen in our computation, we utilized a configuration previously investigated by Liu et al. [45] where water vapor kept at 1 atm and 1000 K is introduced in the enclosure of two black body plates
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maintained at 0 K. Table 3 presents a comparison between heat flux density kept at the wall, given by Liu et al. in SNB and SNBCK models and our results using different Sn quadratures. It can be noted that the results are very close (less than 3%) and the S4 quadrature seems sufficient. In order to justify the fact of having neglected the entropy production due to viscous frictions we draw up in Table 4 a comparison between the global entropy production when the viscosity effect is taken in consideration and that when frictions are neglected. It is clear that the difference is negligible. This is expected since the Eckert number ðEc ¼ u20 =C P jT w T 0 jÞ is quite small. In order to justify the choice of a one-way radiative model, we draw up in Fig. 3 the evolution of the total entropy production when radiation is regarded as 1-D and that when the radiative exchanges are supposed to be bi-directional. When radiation is considered as 2-D, the duct’s entry and exit are supposed to be like fictitious black walls at Tin and Tout temperatures. We applied the DOM with the S8 quadrature, which seems to be sufficient. In the three studied combinations of Tin and Tout the difference with 1-D model is weak. This is due to the
120
6. Results and discussion This study is focused on the change in total entropy generation according to the main parameters that define the flow. For different thermal and geometrical boundary conditions, a selected set of graphical results is presented in Figs. 4–11 with and without radiation. In order to observe fast variations close to entry, we chose flow conditions, which do not always correspond to forced convection. Fig. 4 shows the evolution of radiative and conductive fields of local entropy generation as function of axial and transverse location in heating and cooling conditions. It can be noted that radiative contribution permits direct heating transfers with the central zone of the flow. This effect is rapid because of interaction between fluid particles on the one hand and between fluid and boundaries on the other. It should also be pointed out that entropy generation rate by radiation in the heating case is high in the central part of the channel, but for the cooling case, it remains more or less the same across the height of the channel. This is probably due to the fact that the heating process is faster than the cooling one (see Fig. 6).
with radiation
without radiation 100
X=0.5m
200000
80
150000
60 -div(q) (W/m3)
s (x,y) (W/m3K)
fact that the channel’s length is great compared to its thickness.
40 20 0 0
10
20
30
40
50
60
70
80
90
100
X=0.5m
100000 50000 0 0
10
20
30
-100000 -150000
without radiation
80
90 100
radiative
%y/Ly
150000
X=0.5m
conductive
radiative
100000
200 -div(q) (W/m3)
s (x,y) (W/m3K)
70
with radiation
300 250
60
conductive
%y/Ly 350
50
40
-50000
150 100 50 0 0
10
20
30
40
50
60
70
80
90
100
%y/Ly
50000 0 0
10
20
30
40
50
60
70
80
90
100
-50000 -100000 X=0.5m -150000 %y/Ly
Fig. 5. Evolution of local entropy generation with and without radiation. (a) Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 800 K, T0 ¼ 500 K, P ¼ 1 atm, e ¼ 1, X ¼ 0.5, 1, 1.5, 2 m. (b) Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 500 K, T0 ¼ 800 K, P ¼ 1 atm, e ¼ 1, X ¼ 0.5, 1, 1.5, 2 m.
Fig. 6. Comparison between conductive and radiative source term. (a) Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 800 K, T0 ¼ 500 K, P ¼ 1 atm, e ¼ 1, X ¼ 0.5, 1, 1.5, 2 m. (b) Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 500 K, T0 ¼ 800 K, P ¼ 1 atm, e ¼ 1, X ¼ 0.5, 1, 1.5, 2 m.
ARTICLE IN PRESS F. Ben Nejma et al. / Energy 33 (2008) 1169–1178
12
8 Tw=600K
7
Tw=700K
Tw=800K
2Ly=5cm
10
6
with radiation
5
without radiation
S0-x (W/mK)
S0-x (W/mK)
1175
4 3
2Ly=10cm
2Ly=15cm
with radiation without radiation
8 6 4
2 2
1
0
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0
2
0.2
0.4
0.6
0.8
x (m)
1.4
1.6
1.8
2
9
7 Tw=500K
6
Tw=600K
8
Tw=700K
with radiation without radiation
2Ly=5cm
7 S0-x (W/mK)
5 S0-x (W/mK)
1 1.2 x (m)
4 3 2
2Ly=10cm
2Ly=15cm
with radiation without radiation
6 5 4 3 2
1
1
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x (m)
0 0
0.2
0.4
0.6
0.8
1 1.2 x (m)
1.4
1.6
1.8
2
Fig. 7. Effect of boundary temperature on global entropy production. (a) U0 ¼ 0.1 m/s, P ¼ 1 atm, Lx ¼ 2 m, 2Ly ¼ 5 cm, T0 ¼ 500 K¸ e ¼ 1. (b) U0 ¼ 0.1 m/s, P ¼ 1 atm, Lx ¼ 2 m¸2Ly ¼ 5 cm, T0 ¼ 800 K, e ¼ 1.
Fig. 8. Effect of medium thickness on global entropy production. (a) U0 ¼ 0.1 m/s, P0 ¼ 1 atm, Lx ¼ 2 m, Tw ¼ 800 K, T0 ¼ 500 K, e ¼ 1. (b) U0 ¼ 0.1 m/s, P0 ¼ 1 atm, Lx ¼ 2 m, Tw ¼ 500 K, T0 ¼ 800 K, e ¼ 1.
In the heating case, the effect of conductive contribution on local entropy generation reaches a peak in the vicinity of the walls because of the important temperature gradient value in this zone. This peak is absent in the cooling case. Fig. 5 illustrates local entropy generation with and without radiation as a function of transverse distance for different values of axial location. We note the important effect of radiation in local entropy production, which is distinctly developed for sections close to the entrance region of the channel where transfers are very important as shown in Fig. 6. This is explained in profiles of total entropy production by important gradient values. We also notice, in case of heating, the existence of a relative maximum whose value decreases and whose position migrates towards the central zone of the flow as we go further along the channel. This phenomenon is not visible in case of cooling. A similitude can be observed in the same configuration between fields of local entropy production presented in Fig. 4 and 5 and profiles of source term plotted in Fig. 6.
The difference that exists is visible near the boundaries and is due to the influence caused by directional and spectral radiative temperatures. Fig. 7 shows the distribution of total entropy generation in heating and cooling configurations for different wall temperatures. As seen in this figure, increasing the temperature difference between gas and boundaries increases the entropy generation. The radiative contribution enhances this phenomenon. It permits to considerably increase heat transfers in the central zone of the flow and consequently amplifies entropy production, which is more pronounced in heating conditions. The effect of the medium thickness on entropy production is given in Fig. 8. It can be noted that total entropy generation is considerably affected by this parameter essentially in heating mode, where boundary emission is very important compared to gas emission. In this case, the radiative effect increases. Furthermore, the evolutions of total entropy creation according to thickness, with and without radiation, are opposite. In the absence of radiation
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30
14
25
U0=0.1m/s
P=1atm
12
U0=10m/s
with radiation without radiation
15 10
P=2atm
8 6 4
5
2
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0
2
0
0.2
0.4
0.6
0.8
x (m) 30
1 1.2 x (m)
1.4
1.6
1.8
2
1.6
1.8
2
12 P=1atm U0=0.1m/s
25
U0=1m/s
P=2atm
P=3atm
10
U0=10m/s S0-x (W/mK)
with radiation without radiation
20 S0-x (W/mK)
P=3atm
with radiation without radiation
10 S0-x (W/mK)
20 S0-x (W/mK)
U0=1m/s
15
with radiation without radiation
8 6 4
10 2 5
0 0
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x (m) Fig. 9. Effect of flow rate on global entropy production. (a). Lx ¼ 2 m, 2Ly ¼ 5 cm, P ¼ 1 atm, Tw ¼ 800 K, T0 ¼ 500 K, e ¼ 1. (b) Lx ¼ 2 m, 2Ly ¼ 5 cm, P ¼ 1 atm, Tw ¼ 500 K, T0 ¼ 800 K, e ¼ 1.
and after a determined length, the velocity profile is almost parabolic. The velocity of particles near the boundaries is all the weaker as the width of channel is important, which makes renewing the particles near the boundaries a slow process. This leads to a decrease in global entropy creation. With radiation, the heat transfer increase is high enough to compensate this phenomenon, increasing the total entropy creation with the medium thickness. Thus, according to these figures, the use of smaller spacing between plates, results in a significantly lower radiative entropy creation. When radiation is taken into account, the profiles of total entropy production are also seen to be greater in case of heating. In Fig. 9, the variations of total entropy generation are plotted for different values of the flow rate. It should be noted that entropy generation goes up when velocity flow increases. In fact, increasing velocity flow increases the difference between boundary temperature and the bulk temperature of the fluid. This creates an important temperature gradient, which accentuates heat transfers by
0.2
0.4
0.6
0.8
1 1.2 x (m)
1.4
Fig. 10. Effect of pressure on global entropy production. (a). Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 800 K, T0 ¼ 500 K, e ¼ 1. (b) Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s Tw ¼ 500 K, T0 ¼ 800 K, e ¼ 1.
conduction and radiation, raising entropy production in the medium. Fig. 10 illustrates the effect of pressure on entropy production. The pressure increase leads to a rise in fluid density generating the growth of entropy production due to conduction effect. Moreover, and according to Malkmus law, the rise in pressure decreases gas transmissivity and amplifies its absorption coefficient improving radiative transfers, and thus causing a greater entropy generation. The effect of pressure on entropy production is much less significant when radiation is not considered, and the entropy production shows a higher sensitivity to pressure variation at lower pressures when radiation is considered. This is expected since the optically thick limit is gradually approached as the pressure increases. Computation results using different values of emissivity are presented in Fig. 11. Decreasing the wall emissivity decreases wall radiation heat loss. As shown, decreasing the emissivity coefficient results in a significantly lower entropy generation. It is noted that the variation in the entropy production seems to display a constant sensitivity
ARTICLE IN PRESS F. Ben Nejma et al. / Energy 33 (2008) 1169–1178
8
the calculated results, the following conclusions can be drawn:
ε=0.4 ε=0.8 without Radiation
ε=0.2 ε=0.6 ε=1
7
S0-x (W/mK)
6 5 4 3 2 1 0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x (m) 7
ε=0.2 ε=0.6 ε=1
6
ε=0.4 ε=0.8 without radiation
5 S0-x (W/mK)
1177
1. The radiative contribution enhances heat transfer in the central region of the flow and permits to accelerate the establishment of the flow. 2. Increasing the temperature difference between gas and boundaries results in a rise in entropy generation. 3. The creation of entropy is definitely higher in the heating mode. 4. The greater the spacing between plates, the higher entropy production. 5. Radiative entropy generation is reduced when decreasing pressure. 6. The use of a lower flow rate results in a significantly lower entropy generation. 7. Diminishing the walls’ emissivity leads to a decrease in entropy generation.
References
4 3 2 1 0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x (m) Fig. 11. Effect of the walls’ emissivities on global entropy production. (a). Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 800 K, T0 ¼ 500 K, P ¼ 1 atm. (b) Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 500 K, T0 ¼ 800 K, P ¼ 1 atm.
to the wall emissivity regardless of its value in the heating case. In the cooling case, however, the entropy production shows a higher sensitivity to the wall emissivity as the latter increases. 7. Conclusion In this paper, the numerical calculation of entropy generation due to combined gas radiation and forced convection in participating media is investigated. This study is focused on a laminar flow of temperaturedependent and non-grey gas in the entrance region of the duct. An association between the ‘‘Ray Tracing’’ method through S4 directions and the SNBCK method is used to solve the studied problem. We examined the contribution of radiation in coupled heat transfer generated by interaction between fluid particles on the one hand and between fluid and boundaries on the other. Entropy generation fields with a variety of boundary conditions are presented in heating and cooling modes. Based on
[1] Bejan A. Entropy generation through heat and fluid flow. New York: Wiley Interscience; 1982. [2] Bejan A. A study of entropy generation in fundamental convective heat transfer. J Heat Transfer 1979;101:718–25. [3] San JY, Worek WM, Lavan Z. Entropy generation in convective heat transfer and isothermal convective mass transfer. J Heat Transfer 1987;109:647–52. [4] Bejan A. Entropy generation minimization. Boca Raton, FL: CRC Press; 1996. [5] Ko TH, Ting K. Entropy generation and thermodynamic optimization of fully developed laminar convection in a helical coil. Int Commun Heat Mass Transfer 2005;32:214–23. [6] Ko TH, Ting K. Entropy generation and optimal analysis for laminar forced convection in curved rectangular ducts: a numerical study. Int J Therm Sci 2006;45:138–50. [7] Ko TH. Analysis of optimal Reynolds number for developing laminar forced convection in double sine ducts based on entropy generation minimization principle. Energy Conver Manage 2006; 47:655–70. [8] Mahmud S, Fraser RA. Analysis of mixed convection–radiation interaction in a vertical channel: entropy generation. Exergy 2002;2:330–9. [9] Mahmud S, Islam AKMS. Laminar free convection and entropy generation inside an inclined wavy enclosure. Int J Therm Sci 2003; 42:1003–12. [10] Mahmud S, Fraser RA. Magnetohydrodynamic free convection and entropy generation in a square porous cavity. Int J Heat Mass Transfer 2004;47:3245–56. [11] Mahmud S, Fraser RA. The second law analysis in fundamental convective heat transfer problems. Int J Therm Sci 2003;42:177–86. [12] Arpaci VS. Radiative entropy production. AIAA J 1986;24:1859–60. [13] Arpaci VS. Radiative entropy production-heat lost to entropy. Adv Heat Transfer 1991;21:239–76. [14] Arpaci VS, Selamet A. Radiative entropy production. In: 8th international heat transfer conference, vol. 2; 1986. p. 729–34. [15] Arpaci VS, Selamet A. Entropy production in flames. Combust Flame 1988;73:254–9. [16] Arpaci VS, Selamet A. Entropy production in boundary layers. J Thermophys Heat Transfer 1990;4:404–7. [17] Wright SE, Scott DS, Haddow JB, Rosen MA. On the entropy of radiative heat transfer in engineering thermodynamics. Int J Eng Sci 2001;39:1691–706.
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[18] Caldas M, Semiao V. Entropy generation through radiative transfer in participating media: analysis and numerical computation. JQSRT 2005;96:423–37. [19] Caldas M, Semiao V. The effect of turbulence–radiation interaction on radiative entropy generation and heat transfer. JQSRT 2007;104: 12–23. [20] Liu LH, Chu SX. Verification of numerical simulation method for entropy generation of radiative heat transfer in semitransparent medium. JQSRT 2007;103:43–56. [21] Modest MF. Radiative heat transfer. International editions. New York: McGraw-Hill; 1993. [22] Chandrasekhar S. Radiative transfer. Oxford: Clarendon Press; 1950. [23] Truelove JS. Discrete-ordinate solutions of the radiation transport equation. ASME J Heat Transfer 1987;109:1048–51. [24] Carlson BG, Lathrop KD. Transport theory—the method of discrete ordinates. In: Greenspan H, Kelber CN, Okrent D, editors. Computing methods in reactor physics. London: Gorden and Breach; 1993. p. 165–266. [25] Ben Nejma F, Radhouani MS. Study of transfers coupled by convection and radiation in a boundary layer flow of thermodependent non-gray fluid: application of a narrow band model. Rev Gen Therm 1998;37(6):488–99. [26] Malkmus W. Random Lorentz band model with exponential-tailed S1 line-intensity distribution. J Opt Soc Am 1967;57:323–9. [27] Goody RM, Yung YL. Atmospheric radiation: theoretical basis. Oxford: Oxford University Press; 1989. [28] Lacis A, Oinas V. A description of the correlated-k distribution method for modeling non-gray gaseous absorption, thermal emission, and multiple scattering in vertically inhomogeneous atmospheres. J Geophys Res 1991;96:9027–63. [29] Rivie`re P, Soufiani A, Taine J. Correlated-k and fictitious gas methods for H2O near 2.7 mm. J Quant Spectrosc Radiat Transfer 1992;48:187–203. [30] Levi di Leon R, Taine J. Fictive-gas-method for accurate computation of low-resolution IR gas transmissivities: application to the 4.3 mm CO2 band. Rev Phys Appl 1986;21:825–31. [31] Edwards DK. Molecular gas band radiation. In: Hartnett JP, Irvine TF, editors. Advances in heat transfer, vol. 12. New York: Academic Press; 1976. p. 115–93.
[32] Denison MK, Webb B. The spectral-line-based-weighted-sum-of-graygases model for H2O/CO2 mixtures. J Heat Transfer 1995;117:788–92. [33] Denison MK, Webb B. The spectral-line-based-weighted-sum-ofgray-gases model in non-isothermal non-homogeneous media. J Heat Transfer 1995;117:359–65. [34] Taine J, Soufiani A. Gas IR radiative properties: from spectroscopic data to approximate models. In: Hartnett JP, Irvine TF, editors. Advances in heat transfer, vol. 33. New York: Academic Press; 1999. p. 295–414. [35] Ludwig DB, Malkmus W, Reardon JE, Thomson JAL. Handbook of infrared radiation from combustion gases. NASA SP3080; 1973. [36] Soufiani A, Taine J. High temperature gas radiative property parameters of statistical narrow-band model for H2O, CO2 and CO, and correlated-k model for H2O and CO2. Int J Heat Mass Transfer 1997;40:987–91. [37] Liu F, Gu¨lder O¨L, Smallwood GJ. Three-dimensional non-grey gas radiative transfer analysis using the statistical narrow-band model. Rev Gen Therm 1998;37:759–68. [38] Kim TK, Menart JA, Lee HS. Non-gray radiative gas analyses using the S–N discrete-ordinates method. J Heat Transfer 1991;113:946–52. [39] Liu F, Smallwood GJ. An efficient approach for the implementation of the SNB based correlated-k method and its evaluation. JQSRT 2004;84:465–75. [40] Liu F, Smallwood GJ, Gu¨lder O¨L. Radiation heat transfer calculations using the SNBCK method. In: 33rd thermophysics conference, AIAA 99-3679, Norfolk, VA, June 28–July 1, 1999. [41] Goutie`re V, Liu F, Charette A. An assessment of real-gas modeling in 2D enclosures. JQSRT 2000;64:299–326. [42] Sediki E, Soufiani A, Sifaoui MS. Spectrally correlated radiation and laminar forced convection in the entrance region of a circular duct. Int J Heat Mass Transfer 2002;45:5069–81. [43] Sediki E, Soufiani A, Sifaoui MS. Combined gas radiation and laminar mixed convection in vertical circular tubes. Int J Heat Fluid Flow 2003;24:736–46. [44] Taine J, Petit JP. Transferts thermiques: me´canique des fluides anisothermes. Paris: Edition Bordas; 1989. [45] Liu F, Smallwood GJ, Gu¨lder O¨L. Application of the statistical narrow-band correlated-k method to low-resolution spectral intensity and radiative heat transfer calculations—effects of the quadrature scheme. Int J Heat and Mass Transfer 2000;43:3119–35.
Energy 33 (2008) 1169–1178 www.elsevier.com/locate/energy
Entropy generation through combined non-grey gas radiation and forced convection between two parallel plates F. Ben Nejma, A. Mazgar, N. Abdallah, K. Charrada Unite´ de recherche: Etudes des Milieux Ionise´s et Re´actifs (EMIR), Institut Pre´paratoire aux Etudes d’Inge´nieur de Monastir, Avenue Ibn Eljazar, Monastir 5019, Tunisia Received 27 December 2006
Abstract The current study presents a numerical computation of combined gas radiation and forced convection through two parallel plates. A laminar flow of a temperature-dependent and non-grey gas in the entrance region of the channel was investigated. Over-heated water vapor was chosen as a gas because of its large absorption bands. Some special attention was given to entropy generation and its dependence on geometrical and thermodynamic parameters. The radiative part of the study was solved using the ‘‘Ray Tracing’’ method through S4 directions, associated with the ‘‘statistical narrow band correlated-k’’ (SNBCK) model. The temperature fields were used to calculate the distributions of local and global entropy generation. r 2008 Elsevier Ltd. All rights reserved. Keywords: Entropy generation; Combined radiation and convection; SNBCK model; Non-grey gas radiation
1. Introduction Entropy generation is associated with thermodynamic irreversibility, which is common in all types of heat transfer processes. Bejan [1,2] and San et al. [3] proposed different analytical solutions for the entropy generation equation in several simple flow situations. They introduced the concept of entropy production number, irreversibility distribution ratio, and presented spatial distribution profiles of entropy production. Ever since these studies were carried out, entropy generation has been used as a reference for evaluating the significance of irreversibility related to heat transfer and friction in a thermal system. This has become the main concern in many fields such as heat exchangers, turbo machinery, electronic cooling, porous media and combustion. Based on the concept of efficient energy use and the minimal entropy generation principle, optimal designs of thermodynamic systems have been widely proposed by the thermodynamic second law [4]. It is Corresponding author. Tel.: +216 73 532 007; fax: +216 73 500 512.
E-mail address: [email protected] (F. Ben Nejma). 0360-5442/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2008.02.004
possible to improve the efficiency and overall performance of all kinds of thermal systems through entropy minimization techniques. The analysis of energy utilization and entropy generation has become one of the primary objectives in designing a thermal system. Ko and Ting [5,6], Ko [7] and Mahmud et al. [8–11] performed numerous investigations to calculate entropy production and irreversibility profiles for different geometrical configurations, flow situations and thermal boundary conditions. Several studies have thoroughly dealt with fluid flow irreversibility due to viscous effect and heat transfer by conduction. However, the evaluation of entropy production due to radiative heat transfer in participating media has often been overlooked. Arpaci [12,13] and Arpaci and Selamet [14–16] evaluated the radiative entropy production by analogy to that due to conduction, which is true in the case of an optically thick medium. In their study of entropy generation due to interaction of the radiative field with solid boundaries, Wright et al. [17] extended and generalized this formalism for grey and non-participating media. Recently, Caldas and Semiao [18,19] and Liu and Chu [20] have adapted this approach for non-grey and participating
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Nomenclature
Greek symbols
(A, B) c Cp F g h I Ib k L Lx 2Ly P q s S T Tb u v w x, y
n e t k Dn ~ O dO l m r
SNB model parameters light velocity (m s1) specific heat at constant pressure (J kg1 K1) molar fraction of H2O (F ¼ 1) cumulative function Plank’s constant (h ¼ 6.626 1034 J s) radiation intensity (W m2 sr1) radiation intensity of black body (W m2 sr1) Boltzmann’s constant (k ¼ 1.38 1023 JK1) path length channel length (m)(Lx ¼ 2 m) channel width (m) pressure (Pa) heat flux (W m2) local entropy production (W K1 m3) global entropy production (W K1 m1) temperature (K) bulk temperature (K) axial velocity (m s1) transverse velocity (m s1) weight parameter cartesian coordinates
media. The present paper is aimed at inquiring about entropy generation due to combined gas radiation and forced convection through two parallel plates. The medium’s radiative behavior, which is considered as temperature-dependent and non-grey, is given by solving the ‘‘radiative transfer equation’’ (RTE). Different kinds of numerical methods have actually been developed in order to solve this equation. Modest [21] has cited most of these methods such as the spherical harmonic method, the zonal method, Monte Carlo method and the ‘‘discrete ordinate method (DOM)’’. The latter, which has received great attention during the last few years, was proposed by Chandrasekhar [22] and later developed by Truelove [23] and Carlson and Lathrop [24]. Like the DOM, the ‘‘Ray Tracing’’ method [25], applied in our study through the Sn quadratures, is based on the selection of transfer directions and their associated weights. Using the finite volume technique, we chose the 1-D radiative analysis ‘‘slab problem’’ because of difficulties encountered when planning the discretization of the RTE in a 2-D radiation analysis. This is due to the absence of accurate radiative boundary conditions at the entry and exit of the flow. Anyway, it will later be shown that this has but little effect on global entropy production when the height of the channel is small compared to its length, which is fixed at 2 m in the present study. Today, grey gas approximation is believed to be inaccurate [21] and many approximate radiative property models have been developed to represent molecular spectra of gases. While ‘‘line by line (LBL)’’ models are useful for
wave number (cm1) wall emissivity gas tranmissivity absorption coefficient (m1) spectral resolution (cm1) ray direction ~ elementary solid angle around O thermal conductivity (W m1 K1) dynamic viscosity (N s m2) density of the fluid
Subscripts c r 0 w n
conductive exchange radiative exchange ambiant wall spectral
Superscripts i
partial non-grey medium
providing benchmark solutions, they are not practical in multidimensional applications particularly when the radiative transfer is combined with other transfer modes, since it requires great computational times and storage volumes. Several other models have been introduced to calculate radiative properties of gases like the ‘‘statistical narrow band (SNB)’’ [26,27], ‘‘correlated-k (CK)’’ [28], ‘‘Correlated-k with fictitious gases (CKFG)’’ [29,30], ‘‘exponential wide-band (EWB)’’ [31], ‘‘spectral line-based weightedsum-of-grey-gases (WSGG)’’ model [32,33] and ‘‘absorption distribution function (ADF)’’ [34]. In the absence of LBL results, the SNB model is frequently considered as the most accurate non-grey gas radiation model. However, the implementation of this model into the RTE [35,36] has some shortcomings as it provides transmissivity instead of absorption coefficients [37] and contains a correlation term which requires a long computing time [38]. Initially developed for atmospheric modeling applications, the CK model has the advantage over the SNB model in that scattering effects can be accounted for. The implementation of the SNBCK model for radiation computation previously described by Liu et al. [39] represents an effective approach to avoid the expensive on-line inversion of the cumulative distribution function [28,40,41]. In this case, the distribution function is obtained by inverse Laplace transformation of the SNB gas transmissivity in order to deduce gas absorption coefficients. Heat transfer by simultaneous radiation and convection in absorbing and emitting media is of great interest and appears in many engineering applications such as boilers,
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burners, industrial furnaces, combustion chambers, hightemperature heat exchangers, etc. The interaction of thermal radiation and forced convection in the thermal entrance region has been examined by several researchers. Ben Nejma et al. [25] applied the SNB method in rectangular enclosures and Sediki et al. [42,43] used the ADF and the correlated-k band models in cylindrical enclosures. They all could quantify radiative transfers, but did not mention entropy production.
the system: gðkin Þ ¼ gi
~ ¼ hn=kLnð2hn3 =c2 I n ðOÞ ~ þ 1Þ represents the where T n ðOÞ directional and spectral radiative temperature. Using the SNBCK4 method, the local radiative entropy production for non-grey gases is " 4 XZ X ~ sr ðx; yÞ ¼ wi kin ðI bn ðTÞ I in ðOÞÞ
The application of this method by Goutie`re et al. [41] to 2D non-grey gas radiation transfer has demonstrated that the ‘‘statistical narrow band correlated-k (SNBCK)’’ method is an efficient narrow-band method for radiative transfer calculation as well as for low-resolution spectral intensity prediction. Using the n-point Gauss quadrature, where gas radiative properties are represented by n nongrey mediums at each non-overlapping band, the SNBCK method could overcome the difficulties of the SNB model by extracting the gas absorption coefficients from the gas transmissivity given by Malkmus law [26,35] as " !# rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pB 4AL tn ðLÞ ¼ exp 1þ 1 (1) 2 pB
bands
wi fðgi Þ
4p i¼1
1 1 i ~ T T n ðOÞ
!# dO Dn
(6)
We note that the scattering effect is neglected compared to absorption and emission in gas radiation since the medium contains no particles. The gas radiative properties are obtained by integration over the bands with a 100 cm1 spectral resolution, where the covered spectral range is 187.5–4137.5 cm1. Water vapor absorption spectrum is certainly larger than the one we used, but given the temperatures are not too high (maximum 800 K), this produces very few errors.
The analytical expression of the cumulative function has been derived by Lacis and Oinas [28] as pffiffiffi 1 a 1 erf pffiffiffi b k gðkÞ ¼ 2 k pffiffiffi 1 a (2) þ 1 erf pffiffiffi þ b k exppB 2 k pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi where a ¼ 12 pAB, b ¼ 12 pB=A and erf is the error function. Using the cumulative function g, the narrow-band average of any radiative variable fn can be calculated as n X
(4)
For the spectral, directional and local radiative entropy production, Caldas and Semiao [18,19] and Liu and Chu [20] used the following expression: ! 1 1 b ~ ~ srn ðOÞ ¼ kn ðI n ðTÞ I n ðOÞÞ: (5) ~ T T n ðOÞ
2. The hybrid SNBCK model
fn ¼
1171
3. Problem formulation The physical model under study is shown in Fig. 1. We consider a stationary flow of over-heated water vapor maintained between two parallel plates. The fluid is supposed to be perfect gas, Newtonian and in laminar flow. We have neglected the effect of buoyancy forces and considered only the axial variation of pressure. The mass, momentum and energy balance equations written in Cartesian coordinates will be
(3)
i¼1
qðruÞ qðrvÞ þ ¼0 qx qy
The calculation of the various monochromatic absorption coefficients is thus reduced to the resolution of
(7)
y Tw H2O u0 T0 P0
x
Lx Fig. 1. Geometry and boundary conditions.
Tw
2Ly
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qu qu dP q qu þ m ru þ rv ¼ qx qy dx qy qy
(8)
sðx; yÞ ¼ sc ðx; yÞ þ sr ðx; yÞ þ sf ðx; yÞ
qT qT þ rC p v qx qy dP q qT þ lðTÞ ¼u divð~ qr Þ dx qy qy
rC p u
(9)
As shown in Refs. [41,45] the radiative source term is given as 4 XZ X ~ dO Dn divð~ qr Þ ¼ wi kin ðI bn I in ðOÞÞ (10) bands
At a given location, the local entropy generation is as follows:
4p i¼1
These equations are associated to the conservation equation of massic flow rate on the various cross-sections of the stream discharge Eq. (11), which is useful for calculating the pressure gradient. Z Ly _ mðxÞ ¼ rðTðx; yÞÞ uðx; yÞ dy ¼ 2rðT 0 ÞU 0 Ly (11) Ly
The boundary conditions for the considered problem are summarized by the system (12): 8 vðx; y ¼ LyÞ ¼ vðx; y ¼ LyÞ ¼ 0 > < vðx ¼ 0; yÞ ¼ 0; uðx ¼ 0; yÞ ¼ U 0 ; uðx; y ¼ LyÞ ¼ uðx; y ¼ LyÞ ¼ 0 > : Tðx ¼ 0; yÞ ¼ T ; Tðx; y ¼ LyÞ ¼ Tðx; y ¼ LyÞ ¼ T
where sf ðx; yÞ is the local entropy generation due to friction [5–11], sc ðx; yÞ is the local entropy production caused by conduction and sr ðx; yÞ is the local entropy generation due to radiation Eq. (6). m m qu 2 sf ðx; yÞ ¼ F T T qy
70 analytical
numerical
60
(12)
50 Nu (x)
Dimensionless numbers have not been utilized because of the radiative flux and the enormous variation of fluid proprieties. The physical properties are regarded as variables and are published in Ref. [25].
(14)
where F is the viscous dissipation. Irreversibility due to viscous effect is neglected compared to entropy production due to heat transfer. The local entropy production due to conduction [1,4–7] can be expressed as " 2 # lðTÞ lðTÞ qT 2 qT 2 sc ðx; yÞ ¼ 2 ðrTÞ ¼ 2 þ (15) qx qy T T
w
0
(13)
40 2Ly=15cm 30
2Ly=10cm
20
4. Entropy generation
2Ly=5cm
10
In this paper, we studied combined radiation and forced convection for a temperature-dependent and non-grey gas through two parallel plates. Like all thermodynamic systems, entropy is generated from the irreversibility due to heat transfer and the friction of fluid flow.
Table 1 Effect of axial grid nature (2Ly ¼ 5 cm; Lx ¼ 2 m; Tw ¼ 800 K; T0 ¼ 500 K; P0 ¼ 1 atm; U0 ¼ 0.1 m s1; e ¼ 1) Sinusoidal
Nu=7.54
0 0
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x (m) Fig. 2. Validation of the model. Lx ¼ 2 m, U0 ¼ 0.1 m/s Tw ¼ 600 K, T0 ¼ 400 K, P ¼ 1 atm, physical properties constant at (Tw+T0)/2.
Table 3 Validation of the radiative model Ref. [45]
Uniform
Grid (x, y) (300,50) (3000,50) (300,50) (3000,50) (6000,50) 3.08 2.32 2.77 2.91 S0x (without radiation) 3.02
0.2
2Ly ¼ 0.1 m 2Ly ¼ 1 m
Present work (SNBCK4)
SNB
SNBCK4
S4
S6
S8
14.2 30.3
14.3 30.6
14.60 31.04
14.46 30.91
14.40 30.85
Table 2 Effect of grid (2Ly ¼ 10 cm; Lx ¼ 2 m; Tw ¼ 600 K; T0 ¼ 400 K; P0 ¼ 1 atm; U0 ¼ 0.5 m s1; e ¼ 1) Grid (x, y) S0x (without radiation) S0x (with radiation)
(100,50) 2.39 6.13
(200,100) 2.78 6.51
(300,200) 3.02 6.75
(400,300) 3.12 6.85
(500,400) 3.18 6.92
(600,500) 3.19 6.92
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Table 4 Effect of friction on global entropy production (2Ly ¼ 10 cm; Lx ¼ 2 m; Tw ¼ 800 K; T0 ¼ 400 K; P0 ¼ 1 atm) Friction is considered
S0x (without radiation)
Friction is not considered
U0 ¼ 0.1 m s1
U0 ¼ 1 m s1
U0 ¼ 10 m s1
U0 ¼ 0.1 m s1
U0 ¼ 1 m s1
U0 ¼ 10 m s1
4.69266
11.43624
27.05439
4.69266
11.43623
27.05159
36 90 1D radiation 2D radiation Tin=Tout=0K 2D radiation Tin=T0 Tout=Tw 2D radiation Tin=Tout=Tw
radiative X=0.5m
70
24 S0-x (W/mK)
conductive
80 s (x,y) (W/m3K)
30
18
60 50 40 30 20 10
12
0 0
10
20
30
40
6
50 60 %y/Ly
70
80
90
100
90
100
300 0 0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x (m) Fig. 3. Effect of the 2-D radiation. Lx ¼ 2 m, 2Ly ¼ 10 cm, U0 ¼ 0.5 m/s, Tw ¼ 800 K, T0 ¼ 400 K, P ¼ 2 atm, e ¼ 1.
Therefore the total entropy generation is obtained by integrating Eq. (13). Z x Z Ly S 0x ¼ sðx; yÞ dy dx (16) 0
Ly
Because of the strong variability of the physical properties and the non-explicit form of radiative entropy production term according to the channel’s dimensions as well as the variables conditioning the flow, the analysis of the entropy minimization seems impossible (absence of similarity). This is why we will confine ourselves to presenting the evolution of entropy creation according to the various variables. 5. Grid, validation and assumption justifications Because of symmetry, the study was conducted on half of the field. The finite volume method was applied to solve the conservation equations. As shown in Table 1 the utilization of a sinusoidal grid through axial direction is essential to take into account the great variation in entropy production in the vicinity of the entrance. Table 2 shows that a mixed grid system defined by 200 uniform nodes through transverse direction, and 300 sinusoidal nodes through axial direction gives rather precise results and the error rate does not exceed 3%.
conductive
250 s (x,y) (W/m3K)
0
radiative
200 X=0.5m
150 100 50 0 0
10
20
30
40
50 60 %y/Ly
70
80
Fig. 4. Comparison between conductive and radiative transfer effects in local entropy production. (a). Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 800 K, T0 ¼ 500 K, P ¼ 1 atm, e ¼ 1, X ¼ 0.5, 1, 1.5, 2 m. (b) 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 500 K, T0 ¼ 800 K, P ¼ 1 atm, e ¼ 1, X ¼ 0.5, 1, 1.5, 2 m.
In order to validate our numerical code we compared the local Nusselt number, found by considering the physical properties as constant, with the analytical model published in Ref. [44] and given by Eq. (17). Fig. 2 shows that the difference is very weak: NuðxÞ ¼
P1 32=3Fn2 xþ 4Ly qT 8 n¼0 Gne P ¼ 2 32=3Fn2 xþ T w T b ðxÞ qy y¼0 3 1 n¼0 Gn=Fn e (17)
To validate the radiative model chosen in our computation, we utilized a configuration previously investigated by Liu et al. [45] where water vapor kept at 1 atm and 1000 K is introduced in the enclosure of two black body plates
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maintained at 0 K. Table 3 presents a comparison between heat flux density kept at the wall, given by Liu et al. in SNB and SNBCK models and our results using different Sn quadratures. It can be noted that the results are very close (less than 3%) and the S4 quadrature seems sufficient. In order to justify the fact of having neglected the entropy production due to viscous frictions we draw up in Table 4 a comparison between the global entropy production when the viscosity effect is taken in consideration and that when frictions are neglected. It is clear that the difference is negligible. This is expected since the Eckert number ðEc ¼ u20 =C P jT w T 0 jÞ is quite small. In order to justify the choice of a one-way radiative model, we draw up in Fig. 3 the evolution of the total entropy production when radiation is regarded as 1-D and that when the radiative exchanges are supposed to be bi-directional. When radiation is considered as 2-D, the duct’s entry and exit are supposed to be like fictitious black walls at Tin and Tout temperatures. We applied the DOM with the S8 quadrature, which seems to be sufficient. In the three studied combinations of Tin and Tout the difference with 1-D model is weak. This is due to the
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6. Results and discussion This study is focused on the change in total entropy generation according to the main parameters that define the flow. For different thermal and geometrical boundary conditions, a selected set of graphical results is presented in Figs. 4–11 with and without radiation. In order to observe fast variations close to entry, we chose flow conditions, which do not always correspond to forced convection. Fig. 4 shows the evolution of radiative and conductive fields of local entropy generation as function of axial and transverse location in heating and cooling conditions. It can be noted that radiative contribution permits direct heating transfers with the central zone of the flow. This effect is rapid because of interaction between fluid particles on the one hand and between fluid and boundaries on the other. It should also be pointed out that entropy generation rate by radiation in the heating case is high in the central part of the channel, but for the cooling case, it remains more or less the same across the height of the channel. This is probably due to the fact that the heating process is faster than the cooling one (see Fig. 6).
with radiation
without radiation 100
X=0.5m
200000
80
150000
60 -div(q) (W/m3)
s (x,y) (W/m3K)
fact that the channel’s length is great compared to its thickness.
40 20 0 0
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X=0.5m
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%y/Ly
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300 250
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conductive
%y/Ly 350
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-50000 -100000 X=0.5m -150000 %y/Ly
Fig. 5. Evolution of local entropy generation with and without radiation. (a) Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 800 K, T0 ¼ 500 K, P ¼ 1 atm, e ¼ 1, X ¼ 0.5, 1, 1.5, 2 m. (b) Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 500 K, T0 ¼ 800 K, P ¼ 1 atm, e ¼ 1, X ¼ 0.5, 1, 1.5, 2 m.
Fig. 6. Comparison between conductive and radiative source term. (a) Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 800 K, T0 ¼ 500 K, P ¼ 1 atm, e ¼ 1, X ¼ 0.5, 1, 1.5, 2 m. (b) Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 500 K, T0 ¼ 800 K, P ¼ 1 atm, e ¼ 1, X ¼ 0.5, 1, 1.5, 2 m.
ARTICLE IN PRESS F. Ben Nejma et al. / Energy 33 (2008) 1169–1178
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8 Tw=600K
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Tw=800K
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4 3
2Ly=10cm
2Ly=15cm
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with radiation without radiation
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Fig. 7. Effect of boundary temperature on global entropy production. (a) U0 ¼ 0.1 m/s, P ¼ 1 atm, Lx ¼ 2 m, 2Ly ¼ 5 cm, T0 ¼ 500 K¸ e ¼ 1. (b) U0 ¼ 0.1 m/s, P ¼ 1 atm, Lx ¼ 2 m¸2Ly ¼ 5 cm, T0 ¼ 800 K, e ¼ 1.
Fig. 8. Effect of medium thickness on global entropy production. (a) U0 ¼ 0.1 m/s, P0 ¼ 1 atm, Lx ¼ 2 m, Tw ¼ 800 K, T0 ¼ 500 K, e ¼ 1. (b) U0 ¼ 0.1 m/s, P0 ¼ 1 atm, Lx ¼ 2 m, Tw ¼ 500 K, T0 ¼ 800 K, e ¼ 1.
In the heating case, the effect of conductive contribution on local entropy generation reaches a peak in the vicinity of the walls because of the important temperature gradient value in this zone. This peak is absent in the cooling case. Fig. 5 illustrates local entropy generation with and without radiation as a function of transverse distance for different values of axial location. We note the important effect of radiation in local entropy production, which is distinctly developed for sections close to the entrance region of the channel where transfers are very important as shown in Fig. 6. This is explained in profiles of total entropy production by important gradient values. We also notice, in case of heating, the existence of a relative maximum whose value decreases and whose position migrates towards the central zone of the flow as we go further along the channel. This phenomenon is not visible in case of cooling. A similitude can be observed in the same configuration between fields of local entropy production presented in Fig. 4 and 5 and profiles of source term plotted in Fig. 6.
The difference that exists is visible near the boundaries and is due to the influence caused by directional and spectral radiative temperatures. Fig. 7 shows the distribution of total entropy generation in heating and cooling configurations for different wall temperatures. As seen in this figure, increasing the temperature difference between gas and boundaries increases the entropy generation. The radiative contribution enhances this phenomenon. It permits to considerably increase heat transfers in the central zone of the flow and consequently amplifies entropy production, which is more pronounced in heating conditions. The effect of the medium thickness on entropy production is given in Fig. 8. It can be noted that total entropy generation is considerably affected by this parameter essentially in heating mode, where boundary emission is very important compared to gas emission. In this case, the radiative effect increases. Furthermore, the evolutions of total entropy creation according to thickness, with and without radiation, are opposite. In the absence of radiation
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30
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P=1atm
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U0=10m/s
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with radiation without radiation
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P=3atm
with radiation without radiation
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x (m) Fig. 9. Effect of flow rate on global entropy production. (a). Lx ¼ 2 m, 2Ly ¼ 5 cm, P ¼ 1 atm, Tw ¼ 800 K, T0 ¼ 500 K, e ¼ 1. (b) Lx ¼ 2 m, 2Ly ¼ 5 cm, P ¼ 1 atm, Tw ¼ 500 K, T0 ¼ 800 K, e ¼ 1.
and after a determined length, the velocity profile is almost parabolic. The velocity of particles near the boundaries is all the weaker as the width of channel is important, which makes renewing the particles near the boundaries a slow process. This leads to a decrease in global entropy creation. With radiation, the heat transfer increase is high enough to compensate this phenomenon, increasing the total entropy creation with the medium thickness. Thus, according to these figures, the use of smaller spacing between plates, results in a significantly lower radiative entropy creation. When radiation is taken into account, the profiles of total entropy production are also seen to be greater in case of heating. In Fig. 9, the variations of total entropy generation are plotted for different values of the flow rate. It should be noted that entropy generation goes up when velocity flow increases. In fact, increasing velocity flow increases the difference between boundary temperature and the bulk temperature of the fluid. This creates an important temperature gradient, which accentuates heat transfers by
0.2
0.4
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1 1.2 x (m)
1.4
Fig. 10. Effect of pressure on global entropy production. (a). Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 800 K, T0 ¼ 500 K, e ¼ 1. (b) Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s Tw ¼ 500 K, T0 ¼ 800 K, e ¼ 1.
conduction and radiation, raising entropy production in the medium. Fig. 10 illustrates the effect of pressure on entropy production. The pressure increase leads to a rise in fluid density generating the growth of entropy production due to conduction effect. Moreover, and according to Malkmus law, the rise in pressure decreases gas transmissivity and amplifies its absorption coefficient improving radiative transfers, and thus causing a greater entropy generation. The effect of pressure on entropy production is much less significant when radiation is not considered, and the entropy production shows a higher sensitivity to pressure variation at lower pressures when radiation is considered. This is expected since the optically thick limit is gradually approached as the pressure increases. Computation results using different values of emissivity are presented in Fig. 11. Decreasing the wall emissivity decreases wall radiation heat loss. As shown, decreasing the emissivity coefficient results in a significantly lower entropy generation. It is noted that the variation in the entropy production seems to display a constant sensitivity
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the calculated results, the following conclusions can be drawn:
ε=0.4 ε=0.8 without Radiation
ε=0.2 ε=0.6 ε=1
7
S0-x (W/mK)
6 5 4 3 2 1 0 0
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ε=0.2 ε=0.6 ε=1
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1. The radiative contribution enhances heat transfer in the central region of the flow and permits to accelerate the establishment of the flow. 2. Increasing the temperature difference between gas and boundaries results in a rise in entropy generation. 3. The creation of entropy is definitely higher in the heating mode. 4. The greater the spacing between plates, the higher entropy production. 5. Radiative entropy generation is reduced when decreasing pressure. 6. The use of a lower flow rate results in a significantly lower entropy generation. 7. Diminishing the walls’ emissivity leads to a decrease in entropy generation.
References
4 3 2 1 0 0
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x (m) Fig. 11. Effect of the walls’ emissivities on global entropy production. (a). Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 800 K, T0 ¼ 500 K, P ¼ 1 atm. (b) Lx ¼ 2 m, 2Ly ¼ 5 cm, U0 ¼ 0.1 m/s, Tw ¼ 500 K, T0 ¼ 800 K, P ¼ 1 atm.
to the wall emissivity regardless of its value in the heating case. In the cooling case, however, the entropy production shows a higher sensitivity to the wall emissivity as the latter increases. 7. Conclusion In this paper, the numerical calculation of entropy generation due to combined gas radiation and forced convection in participating media is investigated. This study is focused on a laminar flow of temperaturedependent and non-grey gas in the entrance region of the duct. An association between the ‘‘Ray Tracing’’ method through S4 directions and the SNBCK method is used to solve the studied problem. We examined the contribution of radiation in coupled heat transfer generated by interaction between fluid particles on the one hand and between fluid and boundaries on the other. Entropy generation fields with a variety of boundary conditions are presented in heating and cooling modes. Based on
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