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Predicting radiation emitted from planar and fractal surfaces
International Journal of Thermal Sciences 89 (2015) 357e361
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Predicting radiation emitted from planar and fractal surfaces N. Zekri a, c, *, Y. Baara a, Y. Pizzo b, M. Mense b, A. Kaiss b, B. Porterie b, J-P. Clerc b Universit e des Sciences et de la Technologie d'Oran Mohamed Boudiaf, D epartement de Physique Energ etique, LEPM, BP 1505, El Mnaouer, Oran, Algeria Aix Marseille Universit e, CNRS, IUSTI UMR 7343, 13453 Marseille, France c UNESCO Unitwin/ TVET, Char-Network Department, USTO-MB, LEPM, BP 1505 Oran, Algeria a
b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 June 2014 Received in revised form 14 November 2014 Accepted 19 November 2014 Available online
Analytical solutions for the radiation view factor between either a planar surface or a plane-based fractal surface and an arbitrarily-positioned and arbitrarily-oriented receptive element were obtained. Deterministic fractal surfaces, whose planar cross sections are the Koch curve or a Cantor set, were considered. For surfaces facing each other the view factor exhibits a monotonic behavior as a function of the distance separating them. Otherwise, a maximum appears. Experiments using planar emitting surfaces were conducted which confirm this trend. It was also found that the non-monotonic behavior is strengthened for Cantor and Koch fractal surfaces. Size effects were discussed. © 2014 Elsevier Masson SAS. All rights reserved.
Keywords: View factor Radiation emission Planar surface Fractal surface
1. Introduction In applications where radiation is the dominant mechanism of heat transfer (e.g. wildland and industrial fires, engineering applications, or illumination problems), the determination of view factors between surfaces is crucial. These include a wide variety of fire safety issues (e.g. evaluation of firefighter safety zones, assessment of fuelbreak efficiency or flame radiation exposure at the wildlandeurban interface). For surfaces facing each other analytical and numerical studies showed that radiation decreases monotonically with the distance between them [1e6]. However, these studies considered only canonical surfaces, generally planes or cylinders. In most applications the surface shape may be irregular or even fractal. For example in Ref. [7], Caldarelli et al. revealed the fractal nature of Mediterranean fire scars. Using the box counting method on satellite images [8], they found a fractal dimension of the fire perimeter of about 1.3. In the case of fractal surfaces, some parts of the emitting surface are screened by other parts, and thus shadow effects lead to a reduction in the radiation received by the target. A question arises: does this monotonic behavior still apply when the receptive element is arbitrarily positioned in space? The aim of this paper is to identify the conditions under which non-monotonic behavior occurs.
Analytical view factor solutions are determined for planar and planebased fractal emitting surfaces whatever the position of the receptive element in 3D space. In the present study we focus on deterministic fractal surfaces whose planar cross sections are either a Cantor set [9] or the Koch curve [10]. The former, with a fractal dimension of the cross section of 0.64 (less than 1), is representative of discontinuous surfaces, whereas the latter, with a fractal dimension of 1.26 (close to that obtained by Caldarelli et al. [7]), may mimic continuous rough surfaces with possible shadowing effects. 2. Analytical solutions of the view factor between a single panel and a receptive element The knowledge of the view factor F from a finite surface S2 to an infinitesimal surface element dS1 allows to relate the heat flux (in W/m2) leaving S2 directly toward and intercepted by dS1 (denoted 00 q1 ) to the total heat flux (in W/m2) leaving S2 into all directions 00 (denoted q2 ). 00
00
q1 ¼ F q2 The view factor (VF) is thus defined as.
Z des Sciences et de la * Corresponding author. Tel./fax: þ213 41 627141. Universite partement de Physique Energe tique, Technologie d'Oran Mohamed Boudiaf, De LEPM, BP 1505, El Mnaouer, Oran, Algeria. E-mail address: [email protected] (N. Zekri). http://dx.doi.org/10.1016/j.ijthermalsci.2014.11.024 1290-0729/© 2014 Elsevier Masson SAS. All rights reserved.
(1)
F¼ S2
cos q1 cos q2 dS2 pr 2
(2)
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N. Zekri et al. / International Journal of Thermal Sciences 89 (2015) 357e361
! ! ! ! where cos q1 ¼ n 1 $ r =r and cos q2 ¼ n 2 $ r =r. The notations used are given in Fig. 1a where the orthonormal coordinate system !!! (O x y z ) is attached to the receptive surface element dS1. In the present study we consider situations where the receptor is either ! ! parallel ( n 1 ¼ y ) or perpendicular to the emission surface ! ! ! ! ! ! ( n 1 ¼ z or n 1 ¼ x ), with n 2 ¼ y . After integrating over the emitting surface S2, we obtain
surface [3e6]. For xmin ¼ zmin ¼ 0, Eq. (3) or Eq. (4) reduce to the view factor solution derived by Hamilton [3], and Eq. (1) to that given by Hollands [4]. For xmin ¼ b/2 and zmin ¼ a/2, Eq. (2) is similar to McGuire's expression [5].
! ! For the parallel case: n 1 ¼ y
1 F ¼ 2p k
( " ! !# xmin þ b 1 zmin þ a 1 zmin tan tan Cxb Cxb Cxb x z þa z tan1 min min tan1 min Cx Cx Cx zmin þ a 1 xmin þ b 1 xmin tan tan þ Cza Cza Cza ) Z x þb x tan1 min min tan1 min Cz Cz Cz
(3)
! ! For the perpendicular case with n 1 ¼ z
Fz⊥ ¼
c 1 x þb x arctanþ min arctanþ min 2p Cz Cz Cz 1 x þb x a arctanþ min a atanþ min Cz Cz Cza
(4)
! ! For the perpendicular case with n 1 ¼ x
Fx⊥ ¼
c 2p
(
1 z þa z arctanþ min arctanþ min Cx Cx Cx " ! !#) 1 z þa z b arctanþ min b arctanþ min Cxb Cx Cx (5)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Cz ¼ c2 þ z2min , Cza ¼ c2 þ ðzmin þ aÞ2 , Cx ¼ c2 þ x2min , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and Cxb ¼ c2 þ ðxmin þ bÞ2 . The function arctanþ corresponds to the positive part of the inverse tangent function (i.e, arctanþ ðxÞ ¼ tan1 ðxÞ if x > 0, and 0 otherwise). It is easy to show that the VFs given by Eqs. (3)e(5) only depend on the dimensionless ratios a/c, b/c, xmin/c, and zmin/c, which is particularly useful for scaling purposes. The solution for an arbitrarily-oriented and arbitrarilypositioned receptive element is a combination of perpendicular and parallel VFs.
! ! ! ! F ¼ max½ð n 1 $ x Þ; 0Fx⊥ þ max½ð n 1 $ y Þ; 0F k ! ! þ max½ð n 1 $ z Þ; 0Fz⊥
(6)
Eq. (6) generalizes the analytical expressions found in the literature for receptive elements located in front of the emitting
Fig. 1. (a) Geometry and notations used for view factor calculations; (b) and (c) cross sections of Koch-like and Cantor-like surfaces at successive iterations. The length of the radiant system is b ¼ 3 m.
N. Zekri et al. / International Journal of Thermal Sciences 89 (2015) 357e361
3. Results The emitting surfaces considered here are a planar surface and two fractal surfaces. They are assumed to emit radiation uniformly. Fractal surfaces are composed of a collection of deterministic sets of panels whose projection in the xy plane is either the Koch curve or a Cantor set, both constructed from an iterative process (Fig. 1b and c). Although they are deterministic, these fractal surfaces are representative of major geometrical surface features (e.g., roughness, discontinuity or shadowing). For the sake of simplicity and without any loss of generality we consider here the perpendicular ! ! ! ! case ( n 1 ⊥ n 2 ), with n 1 ¼ z and zmin ¼ 0, unless otherwise specified. The dimensions of the emitting panel are a ¼ 1 m and b ¼ 3 m.
359
emitting plane), while it decreases asymptotically as a power law with an exponent of 3 (2 for the parallel case, not shown) for larger values of c. This power-law behavior can be easily deduced from a Taylor expansion of the analytical equations as c approaches infinity. When the receptive element is staggered from the emitting panel in the x-direction (xmin > 0 or xmin < b), the view factor dependence on c exhibits a maximum. It increases linearly with c close to the panel (with c2 in the parallel case). Far from the panel all data collapse. The y-position of the VF maximum increases linearly with xmin, with a slope of about 0.64 (Fig. 2a) (1.13 for the parallel case). Although not shown, it is found i) that the same trends are observed in the z-direction for zmin > 0 or zmin < a, and 1:2 ii) that the maximum magnitude of the VF decreases as x1:9 min (xmin for the parallel case). 3.2. Radiation emission from fractal surfaces
3.1. Radiation emission from single panels The view factor is plotted as a function of the distance c in Fig. 2 for different values of xmin. For planar emitting surfaces (Fig. 2a), when the receptive element is in front of the emitting surface (b xmin 0), the view factor decreases monotonically. A characteristic distance x from the emitting panel appears, comparable to its smallest side length (here x~a). For values of c of less than x, the VF decreases slowly (the receptor does not “view” the edges of the
Let us now consider a fractal emitting surface composed of Cantor panels. As shown in Fig. 2b, the non-monotonic behavior is also observed when the receptive element is placed between two successive panels (see Fig. 1c for the positions of the receptors), independently of the number of iterations. The view factor increases with the number of iterations since the emitting panels are closer and closer to the receptor and the contribution of remote panels is negligible. The asymptotic behavior of the VF is similar to
Fig. 2. Diagrams (a)e (c): view factor vs. c (open symbols) and y-position of the VF maximum vs. xmin (lines with filled symbols) for planar (a) and fractal surfaces based on (b) Cantor-like panels and (c) Koch-like panels at different iterations. For Cantor-like surfaces, the receptive element is placed either in front of the first panel (xmin ¼ 0) or between the two first panels (xmin ¼ eb/2, eb/6 and eb/18, as shown in Fig. 1c for iterations 1e3). Diagram (d): comparison between analytical solutions (lines) and measurements (symbols) using a radiant panel measuring 44 cm 44 cm.
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N. Zekri et al. / International Journal of Thermal Sciences 89 (2015) 357e361
that observed for the single panel, with the same power-law exponents. For a given Cantor set (e.g. at the first iteration) and for reasons of symmetry, the y-position of the VF maximum is maximum when the receptor is in the median plane between the two panels. For staggered receptors, the y-position of the VF maximum varies linearly with xmin with a slope of 0.65 (Fig. 2b), which is very close to that obtained for the single panel. This result is independent of the number of iterations, which means that only the nearest panels contribute to the view factor. It is found that the 1:3 maximum magnitude of the VF decreases as x2:1 min and xmin for the perpendicular and parallel cases respectively. Fig. 2c reveals that, as in the case of planar and Cantor-like surfaces, radiation emission from Koch-like surfaces exhibits a non-monotonic behavior for staggered receptive elements, regardless of the number of iterations. However there are two main differences. First, the view factor does not vanish for small values of c. This results from the contributions of the nearest oblique panels (Fig. 1c) due to surface roughness. Second, the y-position of the VF maximum versus xmin shows two different behaviors whatever the iteration order: one is slightly nonlinear for values of xmin corresponding to y-positions of VF maximum less than x, where a roughness effect appears, and the other is linear for larger values. As for planar surfaces, the power-law decrease of the maximum magnitude of the VF is also observed for fractal surfaces. Analytical results were compared with data from current experiments using a radiant panel as an emitting surface, while moving the receptor (a CAPTEC flux meter) away from the panel. The radiative heat flux received by the receptor is related to the total heat flux leaving the emitting panel into all directions by Eq. (1). Fig. 2d shows that an excellent agreement is obtained by varying the emissive power of the radiation source, the receptor being oriented perpendicularly or parallel to the panel surface. The effects of size (height a, width b) of the emitting surface on the view factor for planar surfaces were investigated for a given distance c from the receptor. In the present analysis we consider the ! ! perpendicular case with n 1 ¼ z (Fig. 1) and emitting surfaces located at xmin ¼ b/2, and zmin ¼ a/2. Thus, Eq. (4) reduces to
8 0 1 > > > > B C Bb C 1 < 1 1 C ffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffitan1 B q Fz⊥ ðcÞ ¼ B C 2 2 2p > 2c a a A > @ 1 þ 4c > 1 þ 4c 2 2 > : 19 > > > B C> = B C 1 1 1 B b qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitan B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC C 2 2 2c > @ A> 1 þ 9a 1 þ 9a > 4c2 4c2 > ; 0
(7)
Close to the emitting surface (i.e., c ≪ min(a,b)), a first-order expansion of the right-hand side of Eq. (7) yields a scaling of the VF as 1/aeO(1/a2b), whereas asymptotically far from the emitting surface (i.e., c[max(a,b)), the VF scales as b3 þ O(b3a2). For fixed values of b and c, a characteristic height xa ¼ 2c/3 occurs, so that the VF scales as (a/xa)2 in the limit a≪xa and saturates for very large heights. For fixed values of a and c, a characteristic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi width xb ¼ 4c2 þ 9a2 occurs, so that the VF scales linearly with b in the limit b≪xb, and saturates for very large widths. The same trends are observed for plane-based fractal surfaces. Size effects on the maximum VF position and magnitude were also examined for both planar and fractal surfaces. Since the structure of the radiant system is fractal only in the x-direction, these effects were studied by only varying the side length b. Fig. 3 shows the y-position and magnitude of the VF maximum vs. b. The
Fig. 3. Size effect on the (a) y-position and (b) magnitude of the VF maximum for planar and fractal surfaces.
above discussed scaling behavior for planar surfaces is confirmed for the magnitude of the VF maximum that increases linearly and saturates for b[xb (Fig. 3b), the contribution of the edge zones of the panel being negligible. For fractal surfaces, the same linear increase of the VF maximum appears for small widths, but the saturation disappears. For Cantor-like surfaces, the y-position of the VF maximum increases linearly with b, while its magnitude decreases asymptotically as a power law (Fig. 3b). In this case, increasing b leads to an increase in the distance between the two nearest panels along the x-axis, i.e. b/3, and thus xmin ¼ b/6. The mean slope is equal to 0.12, which can be related to that of the linear increase of the y-position with xmin, as previously found, 0.65 xmin ¼ 0.11 b (Fig. 2b). Regarding the magnitude of the VF maximum, we found that it behaves as b1.45±0.07 (b0.71±0.04 for the parallel case). Within the statistical errors, the exponent value of 0.71 observed in the parallel case is compatible with the fractal dimension of the Cantor set, 0.63. For Koch-like surfaces the yposition of the VF maximum increases linearly with b (Fig. 3a), while its magnitude saturates for large values of b (Fig. 3b). Increasing b causes the foremost position of the emitting surface to shift by a distance of ðb=3Þ cos 30 ¼ 0:29 b (Fig. 1c). The coefficient 0.29 corresponds to the slope of the line yposition(b), independently of the iteration order (Fig. 3a). As the receptor moves away from the emitting surface, the VF magnitude decreases. This results from the competition between the increase of the VF due to the contribution of new emitting panels (Fig. 1b) and its decrease with distance from the individual panels (Fig. 3b). The maximum magnitude saturates above xb for the first iteration, in agreement with single panel results (Fig. 3b) since only the nearest panels contribute to the VF (the other panels being far from the receptor). As the order of iterations increases, the decreasing and saturation phases of the VF magnitude occur for wider systems, since the nearest panels are smaller and smaller (at the i-th iteration, the size of each elementary panel is b/3i). A comparative analysis of the results obtained for planar and plane-based fractal surfaces shows that, for the same value of b, the y-position of the VF maximum for Koch-like surfaces is greater than those calculated for Cantor-like and planar surfaces. The magnitude of VF maximum is larger for planar surfaces.
4. Conclusions The VF between a planar or a fractal surface and an arbitrarilypositioned and arbitrarily-oriented receptive element was
N. Zekri et al. / International Journal of Thermal Sciences 89 (2015) 357e361
analytically determined and validated, extending the classical expressions found in the literature. The following conclusions can be drawn from the present study. Unlike in the case of facing surfaces, the VF exhibits a non-monotonic behavior with distance when the receptor is staggered with respect to the emitting panels. The maximum VF decreases as a power law with the lateral shift, whereas its y-position varies linearly. The roughness induced by Koch-like surfaces leads to a nonlinear behavior of both the magnitude and the y-position of the maximum VF for small values of the lateral shift. Size effects of the maximum VF reveal saturation for single panels and Koch-like surfaces, while for Cantor-like surfaces the VF maximum decreases as a power law with size. The yposition of the VF maximum varies linearly with distance for Cantor-like or Koch-like emitting surfaces, whereas it saturates for single panels. Regarding applications, the present expressions allow the estimation of safety distances for wildland fires [11]. Studies including semi-transparent model of the flames in radiation heat transfer are in course to account for this realistic effect. Furthermore, the non-monotonic behavior can lead to unexpected fire consequences. This may explain, in particular, the small-world effects of the fractal patterns of large wildland fires, such as lacunarity or fingering [12].
361
Acknowledgments NZ and YB would like to acknowledge the Algerian Ministry of Education and Research for partial financial support through the CNEPRU project D01920120035. References [1] Y. Billaud, A. Kaiss, J.L. Consalvi, B. Porterie, Int. J. Therm. Sci. 50 (2011) 2. [2] K. Mudan, Prog. Energy Combust. Sci. 10 (1984) 59. [3] D.C. Hamilton, W.R. Morgan, Radiant-interchange configuration factors, 1952. Reportnaca-tn-2836. [4] K.G.T. Hollands, J. Heat Transfer. 117 (1995) 241. [5] J.H. McGuire, Heat Transfer by Radiation, Fire Research Special Rep. No. 2, Her Majesty’s Stationery Office, London, 1953. [6] J.R. Howell, A Catalog of Radiation Heat Transfer. Configuration Factors, Department of Mechanical Engineering, University of Texas, Web site: http:// www.me.utexas.edu/~howell/index.html. [7] G. Caldarelli, R. Frondoni, A. Gabrielli, M. Montuori, R. Retzlaff, C. Ricotta, Europhys. Lett. 56 (2001) 510. [8] B. Mandelbrot, The Fractal Geometry of Nature, Freeman and Company, New York, 1977. [9] G. Cantor, Acta Math. 4 (1884) 381. [10] H. von Koch, Acta Math. 30 (1906) 145. [11] L. Zarate, J. Arnaldos, J. Casal, Fire Saf. J. 43 (2008) 565. [12] R. Albert, A.L. Barabasi, Rev.Mod. Phys. 74 (2002) 47.
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Predicting radiation emitted from planar and fractal surfaces N. Zekri a, c, *, Y. Baara a, Y. Pizzo b, M. Mense b, A. Kaiss b, B. Porterie b, J-P. Clerc b Universit e des Sciences et de la Technologie d'Oran Mohamed Boudiaf, D epartement de Physique Energ etique, LEPM, BP 1505, El Mnaouer, Oran, Algeria Aix Marseille Universit e, CNRS, IUSTI UMR 7343, 13453 Marseille, France c UNESCO Unitwin/ TVET, Char-Network Department, USTO-MB, LEPM, BP 1505 Oran, Algeria a
b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 June 2014 Received in revised form 14 November 2014 Accepted 19 November 2014 Available online
Analytical solutions for the radiation view factor between either a planar surface or a plane-based fractal surface and an arbitrarily-positioned and arbitrarily-oriented receptive element were obtained. Deterministic fractal surfaces, whose planar cross sections are the Koch curve or a Cantor set, were considered. For surfaces facing each other the view factor exhibits a monotonic behavior as a function of the distance separating them. Otherwise, a maximum appears. Experiments using planar emitting surfaces were conducted which confirm this trend. It was also found that the non-monotonic behavior is strengthened for Cantor and Koch fractal surfaces. Size effects were discussed. © 2014 Elsevier Masson SAS. All rights reserved.
Keywords: View factor Radiation emission Planar surface Fractal surface
1. Introduction In applications where radiation is the dominant mechanism of heat transfer (e.g. wildland and industrial fires, engineering applications, or illumination problems), the determination of view factors between surfaces is crucial. These include a wide variety of fire safety issues (e.g. evaluation of firefighter safety zones, assessment of fuelbreak efficiency or flame radiation exposure at the wildlandeurban interface). For surfaces facing each other analytical and numerical studies showed that radiation decreases monotonically with the distance between them [1e6]. However, these studies considered only canonical surfaces, generally planes or cylinders. In most applications the surface shape may be irregular or even fractal. For example in Ref. [7], Caldarelli et al. revealed the fractal nature of Mediterranean fire scars. Using the box counting method on satellite images [8], they found a fractal dimension of the fire perimeter of about 1.3. In the case of fractal surfaces, some parts of the emitting surface are screened by other parts, and thus shadow effects lead to a reduction in the radiation received by the target. A question arises: does this monotonic behavior still apply when the receptive element is arbitrarily positioned in space? The aim of this paper is to identify the conditions under which non-monotonic behavior occurs.
Analytical view factor solutions are determined for planar and planebased fractal emitting surfaces whatever the position of the receptive element in 3D space. In the present study we focus on deterministic fractal surfaces whose planar cross sections are either a Cantor set [9] or the Koch curve [10]. The former, with a fractal dimension of the cross section of 0.64 (less than 1), is representative of discontinuous surfaces, whereas the latter, with a fractal dimension of 1.26 (close to that obtained by Caldarelli et al. [7]), may mimic continuous rough surfaces with possible shadowing effects. 2. Analytical solutions of the view factor between a single panel and a receptive element The knowledge of the view factor F from a finite surface S2 to an infinitesimal surface element dS1 allows to relate the heat flux (in W/m2) leaving S2 directly toward and intercepted by dS1 (denoted 00 q1 ) to the total heat flux (in W/m2) leaving S2 into all directions 00 (denoted q2 ). 00
00
q1 ¼ F q2 The view factor (VF) is thus defined as.
Z des Sciences et de la * Corresponding author. Tel./fax: þ213 41 627141. Universite partement de Physique Energe tique, Technologie d'Oran Mohamed Boudiaf, De LEPM, BP 1505, El Mnaouer, Oran, Algeria. E-mail address: [email protected] (N. Zekri). http://dx.doi.org/10.1016/j.ijthermalsci.2014.11.024 1290-0729/© 2014 Elsevier Masson SAS. All rights reserved.
(1)
F¼ S2
cos q1 cos q2 dS2 pr 2
(2)
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N. Zekri et al. / International Journal of Thermal Sciences 89 (2015) 357e361
! ! ! ! where cos q1 ¼ n 1 $ r =r and cos q2 ¼ n 2 $ r =r. The notations used are given in Fig. 1a where the orthonormal coordinate system !!! (O x y z ) is attached to the receptive surface element dS1. In the present study we consider situations where the receptor is either ! ! parallel ( n 1 ¼ y ) or perpendicular to the emission surface ! ! ! ! ! ! ( n 1 ¼ z or n 1 ¼ x ), with n 2 ¼ y . After integrating over the emitting surface S2, we obtain
surface [3e6]. For xmin ¼ zmin ¼ 0, Eq. (3) or Eq. (4) reduce to the view factor solution derived by Hamilton [3], and Eq. (1) to that given by Hollands [4]. For xmin ¼ b/2 and zmin ¼ a/2, Eq. (2) is similar to McGuire's expression [5].
! ! For the parallel case: n 1 ¼ y
1 F ¼ 2p k
( " ! !# xmin þ b 1 zmin þ a 1 zmin tan tan Cxb Cxb Cxb x z þa z tan1 min min tan1 min Cx Cx Cx zmin þ a 1 xmin þ b 1 xmin tan tan þ Cza Cza Cza ) Z x þb x tan1 min min tan1 min Cz Cz Cz
(3)
! ! For the perpendicular case with n 1 ¼ z
Fz⊥ ¼
c 1 x þb x arctanþ min arctanþ min 2p Cz Cz Cz 1 x þb x a arctanþ min a atanþ min Cz Cz Cza
(4)
! ! For the perpendicular case with n 1 ¼ x
Fx⊥ ¼
c 2p
(
1 z þa z arctanþ min arctanþ min Cx Cx Cx " ! !#) 1 z þa z b arctanþ min b arctanþ min Cxb Cx Cx (5)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Cz ¼ c2 þ z2min , Cza ¼ c2 þ ðzmin þ aÞ2 , Cx ¼ c2 þ x2min , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and Cxb ¼ c2 þ ðxmin þ bÞ2 . The function arctanþ corresponds to the positive part of the inverse tangent function (i.e, arctanþ ðxÞ ¼ tan1 ðxÞ if x > 0, and 0 otherwise). It is easy to show that the VFs given by Eqs. (3)e(5) only depend on the dimensionless ratios a/c, b/c, xmin/c, and zmin/c, which is particularly useful for scaling purposes. The solution for an arbitrarily-oriented and arbitrarilypositioned receptive element is a combination of perpendicular and parallel VFs.
! ! ! ! F ¼ max½ð n 1 $ x Þ; 0Fx⊥ þ max½ð n 1 $ y Þ; 0F k ! ! þ max½ð n 1 $ z Þ; 0Fz⊥
(6)
Eq. (6) generalizes the analytical expressions found in the literature for receptive elements located in front of the emitting
Fig. 1. (a) Geometry and notations used for view factor calculations; (b) and (c) cross sections of Koch-like and Cantor-like surfaces at successive iterations. The length of the radiant system is b ¼ 3 m.
N. Zekri et al. / International Journal of Thermal Sciences 89 (2015) 357e361
3. Results The emitting surfaces considered here are a planar surface and two fractal surfaces. They are assumed to emit radiation uniformly. Fractal surfaces are composed of a collection of deterministic sets of panels whose projection in the xy plane is either the Koch curve or a Cantor set, both constructed from an iterative process (Fig. 1b and c). Although they are deterministic, these fractal surfaces are representative of major geometrical surface features (e.g., roughness, discontinuity or shadowing). For the sake of simplicity and without any loss of generality we consider here the perpendicular ! ! ! ! case ( n 1 ⊥ n 2 ), with n 1 ¼ z and zmin ¼ 0, unless otherwise specified. The dimensions of the emitting panel are a ¼ 1 m and b ¼ 3 m.
359
emitting plane), while it decreases asymptotically as a power law with an exponent of 3 (2 for the parallel case, not shown) for larger values of c. This power-law behavior can be easily deduced from a Taylor expansion of the analytical equations as c approaches infinity. When the receptive element is staggered from the emitting panel in the x-direction (xmin > 0 or xmin < b), the view factor dependence on c exhibits a maximum. It increases linearly with c close to the panel (with c2 in the parallel case). Far from the panel all data collapse. The y-position of the VF maximum increases linearly with xmin, with a slope of about 0.64 (Fig. 2a) (1.13 for the parallel case). Although not shown, it is found i) that the same trends are observed in the z-direction for zmin > 0 or zmin < a, and 1:2 ii) that the maximum magnitude of the VF decreases as x1:9 min (xmin for the parallel case). 3.2. Radiation emission from fractal surfaces
3.1. Radiation emission from single panels The view factor is plotted as a function of the distance c in Fig. 2 for different values of xmin. For planar emitting surfaces (Fig. 2a), when the receptive element is in front of the emitting surface (b xmin 0), the view factor decreases monotonically. A characteristic distance x from the emitting panel appears, comparable to its smallest side length (here x~a). For values of c of less than x, the VF decreases slowly (the receptor does not “view” the edges of the
Let us now consider a fractal emitting surface composed of Cantor panels. As shown in Fig. 2b, the non-monotonic behavior is also observed when the receptive element is placed between two successive panels (see Fig. 1c for the positions of the receptors), independently of the number of iterations. The view factor increases with the number of iterations since the emitting panels are closer and closer to the receptor and the contribution of remote panels is negligible. The asymptotic behavior of the VF is similar to
Fig. 2. Diagrams (a)e (c): view factor vs. c (open symbols) and y-position of the VF maximum vs. xmin (lines with filled symbols) for planar (a) and fractal surfaces based on (b) Cantor-like panels and (c) Koch-like panels at different iterations. For Cantor-like surfaces, the receptive element is placed either in front of the first panel (xmin ¼ 0) or between the two first panels (xmin ¼ eb/2, eb/6 and eb/18, as shown in Fig. 1c for iterations 1e3). Diagram (d): comparison between analytical solutions (lines) and measurements (symbols) using a radiant panel measuring 44 cm 44 cm.
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that observed for the single panel, with the same power-law exponents. For a given Cantor set (e.g. at the first iteration) and for reasons of symmetry, the y-position of the VF maximum is maximum when the receptor is in the median plane between the two panels. For staggered receptors, the y-position of the VF maximum varies linearly with xmin with a slope of 0.65 (Fig. 2b), which is very close to that obtained for the single panel. This result is independent of the number of iterations, which means that only the nearest panels contribute to the view factor. It is found that the 1:3 maximum magnitude of the VF decreases as x2:1 min and xmin for the perpendicular and parallel cases respectively. Fig. 2c reveals that, as in the case of planar and Cantor-like surfaces, radiation emission from Koch-like surfaces exhibits a non-monotonic behavior for staggered receptive elements, regardless of the number of iterations. However there are two main differences. First, the view factor does not vanish for small values of c. This results from the contributions of the nearest oblique panels (Fig. 1c) due to surface roughness. Second, the y-position of the VF maximum versus xmin shows two different behaviors whatever the iteration order: one is slightly nonlinear for values of xmin corresponding to y-positions of VF maximum less than x, where a roughness effect appears, and the other is linear for larger values. As for planar surfaces, the power-law decrease of the maximum magnitude of the VF is also observed for fractal surfaces. Analytical results were compared with data from current experiments using a radiant panel as an emitting surface, while moving the receptor (a CAPTEC flux meter) away from the panel. The radiative heat flux received by the receptor is related to the total heat flux leaving the emitting panel into all directions by Eq. (1). Fig. 2d shows that an excellent agreement is obtained by varying the emissive power of the radiation source, the receptor being oriented perpendicularly or parallel to the panel surface. The effects of size (height a, width b) of the emitting surface on the view factor for planar surfaces were investigated for a given distance c from the receptor. In the present analysis we consider the ! ! perpendicular case with n 1 ¼ z (Fig. 1) and emitting surfaces located at xmin ¼ b/2, and zmin ¼ a/2. Thus, Eq. (4) reduces to
8 0 1 > > > > B C Bb C 1 < 1 1 C ffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffitan1 B q Fz⊥ ðcÞ ¼ B C 2 2 2p > 2c a a A > @ 1 þ 4c > 1 þ 4c 2 2 > : 19 > > > B C> = B C 1 1 1 B b qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitan B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC C 2 2 2c > @ A> 1 þ 9a 1 þ 9a > 4c2 4c2 > ; 0
(7)
Close to the emitting surface (i.e., c ≪ min(a,b)), a first-order expansion of the right-hand side of Eq. (7) yields a scaling of the VF as 1/aeO(1/a2b), whereas asymptotically far from the emitting surface (i.e., c[max(a,b)), the VF scales as b3 þ O(b3a2). For fixed values of b and c, a characteristic height xa ¼ 2c/3 occurs, so that the VF scales as (a/xa)2 in the limit a≪xa and saturates for very large heights. For fixed values of a and c, a characteristic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi width xb ¼ 4c2 þ 9a2 occurs, so that the VF scales linearly with b in the limit b≪xb, and saturates for very large widths. The same trends are observed for plane-based fractal surfaces. Size effects on the maximum VF position and magnitude were also examined for both planar and fractal surfaces. Since the structure of the radiant system is fractal only in the x-direction, these effects were studied by only varying the side length b. Fig. 3 shows the y-position and magnitude of the VF maximum vs. b. The
Fig. 3. Size effect on the (a) y-position and (b) magnitude of the VF maximum for planar and fractal surfaces.
above discussed scaling behavior for planar surfaces is confirmed for the magnitude of the VF maximum that increases linearly and saturates for b[xb (Fig. 3b), the contribution of the edge zones of the panel being negligible. For fractal surfaces, the same linear increase of the VF maximum appears for small widths, but the saturation disappears. For Cantor-like surfaces, the y-position of the VF maximum increases linearly with b, while its magnitude decreases asymptotically as a power law (Fig. 3b). In this case, increasing b leads to an increase in the distance between the two nearest panels along the x-axis, i.e. b/3, and thus xmin ¼ b/6. The mean slope is equal to 0.12, which can be related to that of the linear increase of the y-position with xmin, as previously found, 0.65 xmin ¼ 0.11 b (Fig. 2b). Regarding the magnitude of the VF maximum, we found that it behaves as b1.45±0.07 (b0.71±0.04 for the parallel case). Within the statistical errors, the exponent value of 0.71 observed in the parallel case is compatible with the fractal dimension of the Cantor set, 0.63. For Koch-like surfaces the yposition of the VF maximum increases linearly with b (Fig. 3a), while its magnitude saturates for large values of b (Fig. 3b). Increasing b causes the foremost position of the emitting surface to shift by a distance of ðb=3Þ cos 30 ¼ 0:29 b (Fig. 1c). The coefficient 0.29 corresponds to the slope of the line yposition(b), independently of the iteration order (Fig. 3a). As the receptor moves away from the emitting surface, the VF magnitude decreases. This results from the competition between the increase of the VF due to the contribution of new emitting panels (Fig. 1b) and its decrease with distance from the individual panels (Fig. 3b). The maximum magnitude saturates above xb for the first iteration, in agreement with single panel results (Fig. 3b) since only the nearest panels contribute to the VF (the other panels being far from the receptor). As the order of iterations increases, the decreasing and saturation phases of the VF magnitude occur for wider systems, since the nearest panels are smaller and smaller (at the i-th iteration, the size of each elementary panel is b/3i). A comparative analysis of the results obtained for planar and plane-based fractal surfaces shows that, for the same value of b, the y-position of the VF maximum for Koch-like surfaces is greater than those calculated for Cantor-like and planar surfaces. The magnitude of VF maximum is larger for planar surfaces.
4. Conclusions The VF between a planar or a fractal surface and an arbitrarilypositioned and arbitrarily-oriented receptive element was
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analytically determined and validated, extending the classical expressions found in the literature. The following conclusions can be drawn from the present study. Unlike in the case of facing surfaces, the VF exhibits a non-monotonic behavior with distance when the receptor is staggered with respect to the emitting panels. The maximum VF decreases as a power law with the lateral shift, whereas its y-position varies linearly. The roughness induced by Koch-like surfaces leads to a nonlinear behavior of both the magnitude and the y-position of the maximum VF for small values of the lateral shift. Size effects of the maximum VF reveal saturation for single panels and Koch-like surfaces, while for Cantor-like surfaces the VF maximum decreases as a power law with size. The yposition of the VF maximum varies linearly with distance for Cantor-like or Koch-like emitting surfaces, whereas it saturates for single panels. Regarding applications, the present expressions allow the estimation of safety distances for wildland fires [11]. Studies including semi-transparent model of the flames in radiation heat transfer are in course to account for this realistic effect. Furthermore, the non-monotonic behavior can lead to unexpected fire consequences. This may explain, in particular, the small-world effects of the fractal patterns of large wildland fires, such as lacunarity or fingering [12].
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Acknowledgments NZ and YB would like to acknowledge the Algerian Ministry of Education and Research for partial financial support through the CNEPRU project D01920120035. References [1] Y. Billaud, A. Kaiss, J.L. Consalvi, B. Porterie, Int. J. Therm. Sci. 50 (2011) 2. [2] K. Mudan, Prog. Energy Combust. Sci. 10 (1984) 59. [3] D.C. Hamilton, W.R. Morgan, Radiant-interchange configuration factors, 1952. Reportnaca-tn-2836. [4] K.G.T. Hollands, J. Heat Transfer. 117 (1995) 241. [5] J.H. McGuire, Heat Transfer by Radiation, Fire Research Special Rep. No. 2, Her Majesty’s Stationery Office, London, 1953. [6] J.R. Howell, A Catalog of Radiation Heat Transfer. Configuration Factors, Department of Mechanical Engineering, University of Texas, Web site: http:// www.me.utexas.edu/~howell/index.html. [7] G. Caldarelli, R. Frondoni, A. Gabrielli, M. Montuori, R. Retzlaff, C. Ricotta, Europhys. Lett. 56 (2001) 510. [8] B. Mandelbrot, The Fractal Geometry of Nature, Freeman and Company, New York, 1977. [9] G. Cantor, Acta Math. 4 (1884) 381. [10] H. von Koch, Acta Math. 30 (1906) 145. [11] L. Zarate, J. Arnaldos, J. Casal, Fire Saf. J. 43 (2008) 565. [12] R. Albert, A.L. Barabasi, Rev.Mod. Phys. 74 (2002) 47.