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Least-squares natural element method for radiative heat transfer in graded index medium with semitransparent surfaces
International Journal of Heat and Mass Transfer 66 (2013) 349–354
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
Least-squares natural element method for radiative heat transfer in graded index medium with semitransparent surfaces Yong Zhang, Hong-Liang Yi ⇑, He-Ping Tan ⇑ School of Energy Science and Engineering, Harbin Institute of Technology, 92 West Dazhi Street, Harbin 150001, PR China
a r t i c l e
i n f o
Article history: Received 21 April 2013 Received in revised form 9 July 2013 Accepted 10 July 2013 Available online 7 August 2013 Keywords: Radiative heat transfer Graded index Semitransparent Least-squares natural element method
a b s t r a c t The least-squares natural element method (LSNEM) is employed for solving radiative heat transfer problem in two-dimensional semitransparent graded index medium with semitransparent and diffusely reflecting surfaces. LSNEM is an extension of natural element method (NEM). Unlike the NEM based on Galerkin discretization, the least-squares weighted residuals approach is employed to spatially discretize the radiative heat transfer equations in LSNEM. Radiative heat transfer problem in rectangular enclosure filled with graded index medium having opaque surfaces is examined to verify the LSNEM. Afterwards, radiative heat transfer in the semicircular enclosure with an inner circle filled with graded index medium is studied. Two kinds of refractive index distributions are considered. The studies show that the refractive index distributions and optical boundary conditions of the surfaces greatly influence the radiative transfer. Effects of the extinction coefficient are investigated on the temperature distributions and radiative heat fluxes. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The media with a spatial variable refractive index distribution (graded index, GRIN) exist broadly in nature. Typical examples include the earth’s atmosphere [1], the stellar atmosphere [2], jellyfish eyes [3] and human eyes [4]. Artificial GRIN materials have a variety of promising applications [4]. Researchers produced various types of GRIN materials to improve the quality of optical systems. Like conventional optical materials, heat-transfer processes must be considered in the manufacture and application of the GRIN materials. Radiative heat transfer in semitransparent GRIN medium is of great importance in thermo-optical systems, and has evoked the wide interest of many researchers. Because a ray goes along a curved path determined by the Fermat principle, the solution of radiative transfer in a GRIN medium is more difficult than that in a uniform refractive index medium. Since 2000, researchers have been working on thermal radiative transfer within GRIN media. Ray tracing and Monte-Carlo techniques are particularly well adapted to solving radiative transfer in the medium with specular boundaries and widely used to solve radiative transfer in the graded index medium. Ben Abdallah and coworkers [5–8] developed a curved ray-tracing technique to analyze radiative heat transfer in an absorbing–emitting ⇑ Corresponding authors. Tel.: +86 451 86412674; fax: +86 451 86221048. E-mail addresses: [email protected] (H.-L. Yi), [email protected] (H.-P. Tan). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.07.034
semitransparent medium with variable spatial refractive index. Based on the works of Ben Abdallah and coworkers, Huang et al. [9,10] and Xia et al. [11] presented a combined curved ray-tracing and pseudo-source adding method for radiative heat transfer in one-dimensional semitransparent medium with graded refractive index. Because of a large number of rays to be launched, the ray tracing method is time consuming and difficult to settle the problem of radiative transfer in multidimensional complex geometries. In order to overcome these disadvantages, other approaches different from the ray-tracing based methods which are based on the discretization of radiative transfer equation were proposed. Lemonnier et al. [12] proposed a discrete ordinates method (DOM) which could be used in the 1D slab. Afterwards, Liu derived another kind of DOM for graded index radiative transfer, expressed in a 3D Cartesian coordinates system [13] and a cylindrical one [14] respectively. Based on the DOM proposed by Liu, different numerical techniques have been developed for solving the radiative transfer equation (RTE) in multi-dimensional graded index media, including the finite volume method [13], proposed a modified FVM to solve the problem of radiative transfer in a graded index media. Most recently, Zhang et al. [15] extended the hybrid finite volume with finite element method (hybrid FVM/FEM) to solve the radiative transfer in a graded index medium. Using a diffuse approximation meshless (DAM) method, Wang [16] studied the radiative transfer in a graded index medium. Recently, some researchers also introduced meshless methods to solve thermal radiative transfer within GRIN media [16]. The
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meshless methods are alternative approaches to the mesh based numerical methods to solve partial differential equations (PDE). A common feature of all the meshless methods is that the approximation of field variables is constructed entirely based on a group of discrete nodes and no predefined nodal connectivity is required. The natural element method (NEM) proposed by Braun and Sambridge [17] and Sukumar et al. [18] is a relatively new meshless Galerkin procedure based on the natural neighbor interpolation scheme, which in turn relies on the concepts of Voronoi diagrams and Delaunay triangulation to build Galerkin trial and test functions. Compared to the MLS approximation, some of the most important advantages of natural neighbor interpolants are the properties of interpolation of nodal data, easiness of imposing essential boundary conditions, and a well-defined and robust approximation with no user-defined parameter on non-uniform grids. Most recently, we has been introduced the NEM to solve radiative heat transfer problems [19,20]. In our work [19], several benchmark problems of two dimensions were solved using NEM, and the results shows that the NEM for radiative heat transfer is efficient, accurate and stable. Afterwards, we [20] extended the natural element solutions to deal with the coupled heat transfer problem in an irregular enclosure under mixed boundary conditions. The research works mentioned above are all dealing with radiative heat transfer in semitransparent medium with opaque boundaries. To the knowledge of the authors, there are no research works have been carried out to analyze the radiative heat transfer in multidimensional graded index media having semitransparent surfaces. For semitransparent surfaces, radiative energy will be reflected and transmitted when it reaches the semitransparent surfaces, and total reflection will occur therein when the incident angle is greater than the critical angle. Therefore, the problem caused by semitransparent surfaces becomes more complex. A few papers discussed the problems in uniform index medium with semitransparent boundary conditions [21,22]. In the treatment of boundary conditions of semitransparent surfaces, complex phenomena such as refraction through a semitransparent limit and (or) convective heat exchange with the surroundings were taken into account. In this paper, we aims at further extending the application of NEM to solve radiative transfer problems in two-dimensional GRIN media having semitransparent and diffusely reflecting surfaces. The least-squares weighted residuals approach is employed to spatially discretize the radiative heat transfer equations. To distinguish with the traditional NEM that is called as a meshless Galerkin procedure, we name the method least-squares natural element method (LSNEM). We validate the correctness of the solution of the LSNEM to radiative transfer in semitransparent medium with graded refractive index by comparing with the results obtained from the literature. The radiative heat transfer problems involving semitransparent surfaces are investigated. For the semicircular medium with an inner circle, we consider two kinds of refractive index distributions in the medium. Effects of various parameters such as the extinction coefficient and the scattering albedo are examined on the temperature distributions. 2. Mathematical formulation We consider radiative heat transfer problems in enclosures filled with absorbing, emitting and scattering medium. The surfaces bounding the physical domain are assumed gray and diffusely reflecting semitransparent or opaque boundaries. The discrete ordinate equation of radiative transfer in a multidimensional graded index medium can be written as [13]:
sm rIm ðr; sÞ þ
1 @ sin hm @h
Im ðr; sÞðnX kÞ
rn
n 1 @ rn Im ðr; sÞ s1 þ n sin hm @ u
þ ðja þ js ÞIm ðr; sÞ ¼ n2 ja Ib þ
M 0 0 js X 0 Im Um ;m wm 4p m0 ¼1
ð1Þ
where s = il + jg + kn = isinhcosu + jsinhsinu + kcosh. The radiative boundary condition for a reflecting semitransparent surface can be expressed as:
Im w ¼ ð1 qO ÞI0 þ
qI X 0 m0 n sm Im nw s m < 0 ww ; p n sm0 >0 w
ð2Þ
w
where qO is the external diffuse reflectivity, qI is the internal diffuse reflectivity, Te is the temperature of the environment. The external diffuse reflectivity qO can be expressed as [23]:
1 2
qo ðn0 Þ ¼ þ
ð3n0 þ 1Þðn0 1Þ 6ðn0 þ 1Þ 03
02
2
þ
n02 ðn02 1Þ 3
ðn02 þ 1Þ
2
ln
ðn0 1Þ ðn0 þ 1Þ
0
2n ðn þ 2n 1Þ 8n04 ðn04 þ 1Þ lnðn0 Þ þ 2 02 ðn02 þ 1Þðn04 1Þ ðn þ 1Þðn04 1Þ
ð3Þ
where n0 = n/1 = n. Considering the effect of total reflection, the internal diffuse reflectivity of is [23]
qI ðn0 Þ ¼ 1
1 ½1 qo ðn0 Þ n02
ð4Þ
Using the piecewise constant angular (PCA) quadrature and the step scheme for the treatment of the angular redistribution terms in Eq. (1)
~m;n Im;n ¼ Sm;n sm;n rIm;n þ b
ð5Þ
~ where the effective extinction coefficient bðrÞ and effective source term Sm,n(r) are defined as
~m;n ðrÞ ¼ 1 max vmþ1=2;n ; 0 þ 1 max vm1=2;n ; 0 b h h m m wh wh 1 1 m;nþ1=2 ; 0 þ n max vm;n1=2 ;0 þ n max vu u wu wu þ ðja þ js Þ Sm;n ðrÞ ¼ n2 ja Ib þ
ð6aÞ
Nu Nh X 0 0 0 0 js X 0 n0 Im ;n Um ;n ;m;n wm h wu 4p m0 ¼1n0 ¼1
1 mþ1=2;n max vh ; 0 Imþ1;n m wh 1 þ m max vm1=2;n ; 0 Im1;n h wh 1 þ n max vm;nþ1=2 ; 0 Im;nþ1 u wu þ
þ
1 max vm;n1=2 ; 0 Im;n1 u n wu
The recursion formulas for following:
vmþ1=2;n vm1=2;n ¼ h h N þ1=2;n v1=2;n ¼ vh h ¼0 h
ð6bÞ
vm1=2;n and vm1=2 are giving as u h
wm @ðnXÞ rn h m @h n X¼Xm;n sin h
ð7Þ
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Y. Zhang et al. / International Journal of Heat and Mass Transfer 66 (2013) 349–354
wnu @s1 rn m;n1=2 vm;nþ1=2 v ¼ u u sin hm @ u n X¼Xm;n
1 rn m;Nu þ1=2 vm;1=2 ¼ v ¼ j u u n sin hm
ð8Þ
By using the natural neighbor shape function, an approximate solution of Im is written as the following expression:
Im ¼
N X Im j /j
ð9Þ
j¼1
where the Im j is the value at the node j and /j is the natural neighbor shape function. In the NEM framework, two different interpolants, natural neighbor (Sibson) interpolation and Laplace interpolation, have been proposed. The Sibson and Laplace shape functions hold some remarkable properties in common such as partition of unity, positivity, interpolation, and regularity of the shape functions [24]. In 2D, the Laplace shape function involves ratio of length measures whereas the Sibson shape function is based on the ratio of areas. Hence, the computational cost favors the use of Laplace interpolant in 2D. In order to reduce the computational costs, the Laplace interpolation is adopted in this paper (see [17,19]). The least-squares weighted residuals approach is used to spatially discretize Eq. (5). The following expression is selected as the weight function:
~m;n / W i ¼ sm;n r/i þ b i
ð10Þ
Substituting Eq. (9) into Eq. (5), then integrating over the spatial solution domain yields
Z N Z X ~m;n / Im;n W i dV ¼ Sm;n W i dV; sm;n r/j þ b j j V
j¼1
i ¼ 1; 2; . . . ; N
V
ð11Þ Substituting Eq. (10) into Eq. (11), we obtain the discretization equation of radiative transfer in a graded index medium by the least-squares natural element method (LSNEM), expressed in a matrix form as:
Km Im ¼ Hm
ð12Þ
where
K m;n ij
Z ~m;n / sm;n r/ ¼ sm;n r/j þ b j i V ~m;n / b ~m;n / dV þ sm;n r/ þ b j
Hm i ¼
Z
j
i
~m;n / dV Sm;n sm;n r/i þ Sm;n b i
ð13aÞ ð13bÞ
V
After solving the RTE, the incident radiation G is obtained from
GðrÞ ¼
Z
Iðr; XÞdX
rectangular enclosure filled with a graded index medium 0:5 nðx; yÞ ¼ 5½1 0:4356ðx2 þ y2 Þ=L2 having opaque and diffuse surfaces from literature. For the following numerical study, the maximum relative error 106 of the incident radiation is taken as the stopping criterion of iteration. The medium is assumed to be anisotropic scattering with a forward scattering phase function as U(X, X0 ) = 1 + XX0 . The numbers of uniform grids 21 21 and discrete directions Mh M/ = 10 20 are used for grid-independent situations. The south boundary of the rectangular is at some finite temperature TS and is thus a radiation source. The other three boundaries are cold (at zero temperature). The medium withunknown temperatures is at the radiative equilibrium. The non-dimensional temperature distributions along the line x/L = 0.325 is presented in Fig. 1 for two values of surface emissivity, namely, 0.5 and 1.0, and the scattering albedo is kept at x = 0.0. The extinction coefficient of b is taken as 1 m1. There are temperature slips at boundary surfaces, which is similar to that in uniform index media. By the comparison, we can see that the LSNEM results agree with those obtained by the MCM [25] very well. 4. Results and discussion The correctness and the reliability of LSNEM solution for radiative transfer in enclosures filled with GRIN medium were proven in the Section 3. Following that, in this section, we apply the LSNEM to solve the radiative heat transfer in graded index media having semitransparent and diffusely reflecting surfaces. Different from the opaque surface, the diffuse reflectivity of the semitransparent surface depends on the refractive index distribution at the boundaries as seen in Eqs. (3) and (4). In this section, we examine radiative heat transfer in a semicircular enclosure with an inner circle filled with absorbing, emitting and isotropically scattering graded index medium as shown in Fig. 2a. The inner circle is centered at x = 0 m and y = 0.4 m. The radiative heat transfer problems for the medium having a uniform refractive index in this enclosure have been investigated [26]. Here we consider two kinds of refractive index distributions as shown in Fig. 2(b and c) for which
GRIN 1 :
h i0:5 nðx; yÞ ¼ 5 1 0:4356ðx2 þ y2 Þ=R2
ð17Þ
GRIN 2 :
nðx; yÞ ¼ 2 þ 0:5ðj cosðpx=2=RÞj þ j sinðpyÞjÞ
ð18Þ
where R is the radius of the outer semicircle. In the following cases, we investigate effects of transmission characteristic of surfaces on the radiative heat transfer in the media with the two kinds of GRIN mentioned above. Only the surface of the outer semicircle is assumed semitransparent. The results are
ð14Þ
0.90
The divergence of radiative heat flux rqr(r) that accounts for the volumetric radiation is given by:
0.85
r qr ðrÞ ¼ ja ð4n2 rT 4 ðrÞ GðrÞÞ
ð15Þ
For radiative equilibrium problem, rqr(r) = 0, and the temperature distribution of the medium can be written as:
TðrÞ ¼
GðrÞ 4n2 r
14
Temperature T/TS
4p
MCM Ref. [25] NEM
0.80 0.75 0.70
ð16Þ
3. Validation of the least-squares natural element method To check the performance and accuracy of the LSNEM presented in this paper, we examine radiative heat transfer problems in a
0.65 0.60 0.0
0.2
0.4
0.6
0.8
1.0
Distance, y/Ly Fig. 1. Non-dimensional temperature distributions along line x/L = 0.325.
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Y. Zhang et al. / International Journal of Heat and Mass Transfer 66 (2013) 349–354
(a) 3.76 3.95 4.14 4.33 4.52 4.71 4.9
2.01 2.16 2.31 2.46 2.61 2.76 2.91
(b)
(c)
Fig. 2. Schematic of semicircular enclosure with inner circle as shown in (a) and graded index distributions given by (b) Eq. (17) and (c) Eq. (18).
0.40 0.50 0.60 0.70 0.80 0.90
(a)
0.40 0.50 0.60 0.70 0.80 0.90
0.65 0.70 0.75 0.80 0.85 0.90 0.95
(b)
0.50 0.58 0.66 0.74 0.82 0.90
Fig. 3. Non-dimensional incident radiation G ðG=rT 4g Þ distribution with GRIN 1: (a) opaque case and (b) semitransparent case and those with GRIN 2:(c) opaque case and (d) semitransparent case.
compared with those for the enclosure whose surfaces are all opaque. For all the cases, the 1144 nodes and discrete directions Mh M/ = 15 30 are considered, and all the opaque surfaces are black. The bottom surface at some finite temperature TS, the other boundaries and environment are cold (at zero temperature). The medium with unknown temperatures is at radiative equilibrium.
For this case, only the semicircle surface is semitransparent whereas all other surfaces are kept opaque and black. For GRIN 1 and 2, with b = 1.0 m1 and x = 0.5, the non-dimensional incident radiation G distributions are shown in Fig. 3. Owing to higher values of the refractive index, the incident radiation distributions of GRIN 1 are higher than those of GRIN 2. It can been
Y. Zhang et al. / International Journal of Heat and Mass Transfer 66 (2013) 349–354
(a)
353
(b)
(c)
(d)
Fig. 4. Non-dimensional temperature distributions along the circle arc AA0 for different values of the extinction coefficient for semitransparent cases (a) GRIN 1 and (b) GRIN 2, non-dimensional radiative heat flux distribution along the inner circle wall for various the extinction coefficient (c) opaque case and (d) semitransparent case.
seen that, for different graded index medium, the incident radiation distributions have little differences. In Fig. 4(a and b), for semitransparent cases, the curves for nondimensional temperature T/TS distributions along the circle arc AA0 at RL = 0.8 are plotted for different values of the extinction coefficient b, and the scattering albedo is kept at x = 0.5. It can be observed that, for both the two refractive index distribution, a temperature slip is observed at the bottom surface, which is the characteristic of radiative equilibrium situation. As b increases, the medium with increased absorption and scattering coefficients does not allow the radiation from the boundaries to penetrate deep into the medium. As a result, large variations in temperature distribution along AA0 are observed. For GRIN 1, Fig. 4(c and d) depicts the non-dimensional radiative heat flux distribution along the inner circle wall for various the extinction coefficient b. It can be seen that higher the extinction coefficient is, bigger the radiative heat flux is. It can be further observed that, for opaque case, the radiative heat flux at the place near the bottom wall changes little as b increases. While for semitransparent case, the changes of the radiative heat flux at the places near or away from the bottom wall trend to be consistent, which is attributed to the semitransparent semicircle wall.
medium with diffusely reflecting semitransparent or opaque walls. In the present work, we firstly had a successful implementation of the LSNEM to the problems for radiative heat transfer in the rectangular enclosure and a semicircular enclosure with an inner circle involving semitransparent surfaces. The influence of the refractive index distribution and transmission characteristic of surfaces on the results are analyzed. Effects of the extinction coefficient are investigated on the radiative heat fluxes and temperature distributions.
Acknowledgments This work was supported by program for New Century Excellent Talents in University (NCET-09-0067), the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 51121004) and the National Natural Science Foundation of China (Grant No. 51176040). A very special acknowledgement is made to the editors and referees who make important comments to improve this paper.
References 5. Conclusions Based on the discrete-ordinates equation of radiative transfer, the least-squares natural element method is developed for the simulation of radiative heat transfer problems in graded index
[1] C. Reino Gomez, M.V. Rerez, C. Bao, Gradient-Index Optics: Fundamentals and Applications, Springer, Berlin, 2002. [2] P.M. Kowalski, D. Saumon, Radiative transfer in the refractive atmospheres of very cool white dwarfs, Astrophys. J. (607) (2004) 970–981. [3] D.E. Nilsson, L. Gislen, M.M. Coates, C. Skogh, A. Garm, Advanced optics in a jellyfish eye, Nature 435 (2005) 201–205. [4] Y.T. Qiao, Graded Index Optics, Science Press, 1991.
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[5] P. Ben Abdallah, V. Le Dez, Temperature field inside an absorbing–emitting semitransparent slab at radiative equilibrium with variable spatial refractive index, J. Quant. Spectrosc. Radiat. Transfer 65 (2000) 595–608. [6] P. Ben Abdallah, V. Le Dez, Thermal emission of a two-dimensional rectangular cavity with spatial affine refractive index, J. Quant. Spectrosc. Radiat. Transfer 66 (2000) 555–569. [7] P. Ben Abdallah, V. Le Dez, Thermal emission of a semi-transparent slab with variable spatial refractive index, J. Quant. Spectrosc. Radiat. Transfer 67 (2000) 185–198. [8] P. Ben Abdallah, A. Charette, V. Le Dez, Influence of a spatial variation of the thermo-optical constants on the radiative transfer inside an absorbing– emitting semi-transparent sphere, J. Quant. Spectrosc. Radiat. Transfer 70 (2001) 341–365. [9] Y. Huang, X.L. Xia, H.P. Tan, Radiative intensity solution and thermal emission analysis of a semitransparent medium layer with a sinusoidal refractive index, J. Quant. Spectrosc. Radiat. Transfer 74 (2002) 217–233. [10] Y. Huang, X.L. Xia, H.P. Tan, Temperature field of radiative equilibrium in a semitransparent slab with a linear refractive index and gray walls, J. Quant. Spectrosc. Radiat. Transfer 74 (2002) 249–261. [11] X.L. Xia, Y. Huang, H.P. Tan, Thermal emission and volumetric absorption of a graded index semitransparent medium layer, J. Quant. Spectrosc. Radiat. Transfer 74 (2002) 235–248. [12] D. Lemonnier, V. Le Dez, Discrete ordinates solution of radiative transfer across a slab with variable refractive index, J. Quant. Spectrosc. Radiat. Transfer 73 (2002) 195–204. [13] L.H. Liu, Finite volume method for radiation heat transfer in graded index medium, J. Thermophys. Heat Transfer 20 (2006) 59–66. [14] L.H. Liu, L. Zhang, H.P. Tan, Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems, J. Quant. Spectrosc. Radiat. Transfer 97 (2006) 446–456. [15] L. Zhang, J.M. Zhao, L.H. Liu, S.Y. Wang, Hybrid finite volume/finite element method for radiative heat transfer in graded index media, J. Quant. Spectrosc. Radiat. Transfer 113 (2012) 1826–1835.
[16] C.A. Wang, H. Sadat, V. Le Dez, Meshless method for solving multidimensional radiative transfer in graded index medium, Appl. Math. Model. 36 (2012) 5309–5319. [17] J. Braun, M. Sambridge, A numerical method for solving partial differential equations on highly irregular evolving grids, Nature 376 (1995) 655–660. [18] N. Sukumar, B. Moran, T. Belytschko, The natural element method in solid mechanics, Int. J. Numer. Methods Eng. 43 (1998) 839–887. [19] Y. Zhang, H.L. Yi, H.P. Tan, Natural element method for radiative heat transfer in two-dimensional semitransparent medium, Int. J. Heat Mass Transfer 56 (2013) 411–423. [20] Y. Zhang, H.L. Yi, H.P. Tan, Natural element method for radiative heat transfer in a semitransparent medium with irregular geometries, J. Comput. Phys. 241 (2013) 18–34. [21] J.F. Luo, H.L. Yi, S.L. Chang, H.P. Tan, Radiation and conduction in an isotropic scattering rectangular medium with one semitransparent and diffusely reflection boundary, ASME J. Heat Transfer 132 (2010) 112701. [22] L. David, B. Nacer, B. Pascal, J. Gérard, Transient radiative and conductive heat transfer in non-gray semitransparent two-dimensional media with mixed boundary conditions, Heat Mass Transfer 42 (2006) 322–337. [23] R. Siegel, Variable refractive index effects on radiation in semitransparent scattering multilayered regions, J. Thermophys. Heat Transfer 7 (1993) 624– 630. [24] N. Sukumar, B. Moran, A. Yu Semenov, V.V. Belikov, Natural neighbor Galerkin methods, Int. J. Numer. Methods Eng. 50 (2001) 1–27. [25] L.H. Liu, Benchmark numerical solutions for radiative heat transfer in twodimensional medium with graded index distribution, J. Quant. Spectrosc. Radiat. Transfer 102 (2006) 293–303. [26] H. Amiri, S.H. Mansouri, A. Safavinejad, Combined conductive and radiative heat transfer in an anisotropic scattering participating medium with irregular geometries, Int. J. Therm. Sci. 49 (2010) 492–503.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
Least-squares natural element method for radiative heat transfer in graded index medium with semitransparent surfaces Yong Zhang, Hong-Liang Yi ⇑, He-Ping Tan ⇑ School of Energy Science and Engineering, Harbin Institute of Technology, 92 West Dazhi Street, Harbin 150001, PR China
a r t i c l e
i n f o
Article history: Received 21 April 2013 Received in revised form 9 July 2013 Accepted 10 July 2013 Available online 7 August 2013 Keywords: Radiative heat transfer Graded index Semitransparent Least-squares natural element method
a b s t r a c t The least-squares natural element method (LSNEM) is employed for solving radiative heat transfer problem in two-dimensional semitransparent graded index medium with semitransparent and diffusely reflecting surfaces. LSNEM is an extension of natural element method (NEM). Unlike the NEM based on Galerkin discretization, the least-squares weighted residuals approach is employed to spatially discretize the radiative heat transfer equations in LSNEM. Radiative heat transfer problem in rectangular enclosure filled with graded index medium having opaque surfaces is examined to verify the LSNEM. Afterwards, radiative heat transfer in the semicircular enclosure with an inner circle filled with graded index medium is studied. Two kinds of refractive index distributions are considered. The studies show that the refractive index distributions and optical boundary conditions of the surfaces greatly influence the radiative transfer. Effects of the extinction coefficient are investigated on the temperature distributions and radiative heat fluxes. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The media with a spatial variable refractive index distribution (graded index, GRIN) exist broadly in nature. Typical examples include the earth’s atmosphere [1], the stellar atmosphere [2], jellyfish eyes [3] and human eyes [4]. Artificial GRIN materials have a variety of promising applications [4]. Researchers produced various types of GRIN materials to improve the quality of optical systems. Like conventional optical materials, heat-transfer processes must be considered in the manufacture and application of the GRIN materials. Radiative heat transfer in semitransparent GRIN medium is of great importance in thermo-optical systems, and has evoked the wide interest of many researchers. Because a ray goes along a curved path determined by the Fermat principle, the solution of radiative transfer in a GRIN medium is more difficult than that in a uniform refractive index medium. Since 2000, researchers have been working on thermal radiative transfer within GRIN media. Ray tracing and Monte-Carlo techniques are particularly well adapted to solving radiative transfer in the medium with specular boundaries and widely used to solve radiative transfer in the graded index medium. Ben Abdallah and coworkers [5–8] developed a curved ray-tracing technique to analyze radiative heat transfer in an absorbing–emitting ⇑ Corresponding authors. Tel.: +86 451 86412674; fax: +86 451 86221048. E-mail addresses: [email protected] (H.-L. Yi), [email protected] (H.-P. Tan). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.07.034
semitransparent medium with variable spatial refractive index. Based on the works of Ben Abdallah and coworkers, Huang et al. [9,10] and Xia et al. [11] presented a combined curved ray-tracing and pseudo-source adding method for radiative heat transfer in one-dimensional semitransparent medium with graded refractive index. Because of a large number of rays to be launched, the ray tracing method is time consuming and difficult to settle the problem of radiative transfer in multidimensional complex geometries. In order to overcome these disadvantages, other approaches different from the ray-tracing based methods which are based on the discretization of radiative transfer equation were proposed. Lemonnier et al. [12] proposed a discrete ordinates method (DOM) which could be used in the 1D slab. Afterwards, Liu derived another kind of DOM for graded index radiative transfer, expressed in a 3D Cartesian coordinates system [13] and a cylindrical one [14] respectively. Based on the DOM proposed by Liu, different numerical techniques have been developed for solving the radiative transfer equation (RTE) in multi-dimensional graded index media, including the finite volume method [13], proposed a modified FVM to solve the problem of radiative transfer in a graded index media. Most recently, Zhang et al. [15] extended the hybrid finite volume with finite element method (hybrid FVM/FEM) to solve the radiative transfer in a graded index medium. Using a diffuse approximation meshless (DAM) method, Wang [16] studied the radiative transfer in a graded index medium. Recently, some researchers also introduced meshless methods to solve thermal radiative transfer within GRIN media [16]. The
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meshless methods are alternative approaches to the mesh based numerical methods to solve partial differential equations (PDE). A common feature of all the meshless methods is that the approximation of field variables is constructed entirely based on a group of discrete nodes and no predefined nodal connectivity is required. The natural element method (NEM) proposed by Braun and Sambridge [17] and Sukumar et al. [18] is a relatively new meshless Galerkin procedure based on the natural neighbor interpolation scheme, which in turn relies on the concepts of Voronoi diagrams and Delaunay triangulation to build Galerkin trial and test functions. Compared to the MLS approximation, some of the most important advantages of natural neighbor interpolants are the properties of interpolation of nodal data, easiness of imposing essential boundary conditions, and a well-defined and robust approximation with no user-defined parameter on non-uniform grids. Most recently, we has been introduced the NEM to solve radiative heat transfer problems [19,20]. In our work [19], several benchmark problems of two dimensions were solved using NEM, and the results shows that the NEM for radiative heat transfer is efficient, accurate and stable. Afterwards, we [20] extended the natural element solutions to deal with the coupled heat transfer problem in an irregular enclosure under mixed boundary conditions. The research works mentioned above are all dealing with radiative heat transfer in semitransparent medium with opaque boundaries. To the knowledge of the authors, there are no research works have been carried out to analyze the radiative heat transfer in multidimensional graded index media having semitransparent surfaces. For semitransparent surfaces, radiative energy will be reflected and transmitted when it reaches the semitransparent surfaces, and total reflection will occur therein when the incident angle is greater than the critical angle. Therefore, the problem caused by semitransparent surfaces becomes more complex. A few papers discussed the problems in uniform index medium with semitransparent boundary conditions [21,22]. In the treatment of boundary conditions of semitransparent surfaces, complex phenomena such as refraction through a semitransparent limit and (or) convective heat exchange with the surroundings were taken into account. In this paper, we aims at further extending the application of NEM to solve radiative transfer problems in two-dimensional GRIN media having semitransparent and diffusely reflecting surfaces. The least-squares weighted residuals approach is employed to spatially discretize the radiative heat transfer equations. To distinguish with the traditional NEM that is called as a meshless Galerkin procedure, we name the method least-squares natural element method (LSNEM). We validate the correctness of the solution of the LSNEM to radiative transfer in semitransparent medium with graded refractive index by comparing with the results obtained from the literature. The radiative heat transfer problems involving semitransparent surfaces are investigated. For the semicircular medium with an inner circle, we consider two kinds of refractive index distributions in the medium. Effects of various parameters such as the extinction coefficient and the scattering albedo are examined on the temperature distributions. 2. Mathematical formulation We consider radiative heat transfer problems in enclosures filled with absorbing, emitting and scattering medium. The surfaces bounding the physical domain are assumed gray and diffusely reflecting semitransparent or opaque boundaries. The discrete ordinate equation of radiative transfer in a multidimensional graded index medium can be written as [13]:
sm rIm ðr; sÞ þ
1 @ sin hm @h
Im ðr; sÞðnX kÞ
rn
n 1 @ rn Im ðr; sÞ s1 þ n sin hm @ u
þ ðja þ js ÞIm ðr; sÞ ¼ n2 ja Ib þ
M 0 0 js X 0 Im Um ;m wm 4p m0 ¼1
ð1Þ
where s = il + jg + kn = isinhcosu + jsinhsinu + kcosh. The radiative boundary condition for a reflecting semitransparent surface can be expressed as:
Im w ¼ ð1 qO ÞI0 þ
qI X 0 m0 n sm Im nw s m < 0 ww ; p n sm0 >0 w
ð2Þ
w
where qO is the external diffuse reflectivity, qI is the internal diffuse reflectivity, Te is the temperature of the environment. The external diffuse reflectivity qO can be expressed as [23]:
1 2
qo ðn0 Þ ¼ þ
ð3n0 þ 1Þðn0 1Þ 6ðn0 þ 1Þ 03
02
2
þ
n02 ðn02 1Þ 3
ðn02 þ 1Þ
2
ln
ðn0 1Þ ðn0 þ 1Þ
0
2n ðn þ 2n 1Þ 8n04 ðn04 þ 1Þ lnðn0 Þ þ 2 02 ðn02 þ 1Þðn04 1Þ ðn þ 1Þðn04 1Þ
ð3Þ
where n0 = n/1 = n. Considering the effect of total reflection, the internal diffuse reflectivity of is [23]
qI ðn0 Þ ¼ 1
1 ½1 qo ðn0 Þ n02
ð4Þ
Using the piecewise constant angular (PCA) quadrature and the step scheme for the treatment of the angular redistribution terms in Eq. (1)
~m;n Im;n ¼ Sm;n sm;n rIm;n þ b
ð5Þ
~ where the effective extinction coefficient bðrÞ and effective source term Sm,n(r) are defined as
~m;n ðrÞ ¼ 1 max vmþ1=2;n ; 0 þ 1 max vm1=2;n ; 0 b h h m m wh wh 1 1 m;nþ1=2 ; 0 þ n max vm;n1=2 ;0 þ n max vu u wu wu þ ðja þ js Þ Sm;n ðrÞ ¼ n2 ja Ib þ
ð6aÞ
Nu Nh X 0 0 0 0 js X 0 n0 Im ;n Um ;n ;m;n wm h wu 4p m0 ¼1n0 ¼1
1 mþ1=2;n max vh ; 0 Imþ1;n m wh 1 þ m max vm1=2;n ; 0 Im1;n h wh 1 þ n max vm;nþ1=2 ; 0 Im;nþ1 u wu þ
þ
1 max vm;n1=2 ; 0 Im;n1 u n wu
The recursion formulas for following:
vmþ1=2;n vm1=2;n ¼ h h N þ1=2;n v1=2;n ¼ vh h ¼0 h
ð6bÞ
vm1=2;n and vm1=2 are giving as u h
wm @ðnXÞ rn h m @h n X¼Xm;n sin h
ð7Þ
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wnu @s1 rn m;n1=2 vm;nþ1=2 v ¼ u u sin hm @ u n X¼Xm;n
1 rn m;Nu þ1=2 vm;1=2 ¼ v ¼ j u u n sin hm
ð8Þ
By using the natural neighbor shape function, an approximate solution of Im is written as the following expression:
Im ¼
N X Im j /j
ð9Þ
j¼1
where the Im j is the value at the node j and /j is the natural neighbor shape function. In the NEM framework, two different interpolants, natural neighbor (Sibson) interpolation and Laplace interpolation, have been proposed. The Sibson and Laplace shape functions hold some remarkable properties in common such as partition of unity, positivity, interpolation, and regularity of the shape functions [24]. In 2D, the Laplace shape function involves ratio of length measures whereas the Sibson shape function is based on the ratio of areas. Hence, the computational cost favors the use of Laplace interpolant in 2D. In order to reduce the computational costs, the Laplace interpolation is adopted in this paper (see [17,19]). The least-squares weighted residuals approach is used to spatially discretize Eq. (5). The following expression is selected as the weight function:
~m;n / W i ¼ sm;n r/i þ b i
ð10Þ
Substituting Eq. (9) into Eq. (5), then integrating over the spatial solution domain yields
Z N Z X ~m;n / Im;n W i dV ¼ Sm;n W i dV; sm;n r/j þ b j j V
j¼1
i ¼ 1; 2; . . . ; N
V
ð11Þ Substituting Eq. (10) into Eq. (11), we obtain the discretization equation of radiative transfer in a graded index medium by the least-squares natural element method (LSNEM), expressed in a matrix form as:
Km Im ¼ Hm
ð12Þ
where
K m;n ij
Z ~m;n / sm;n r/ ¼ sm;n r/j þ b j i V ~m;n / b ~m;n / dV þ sm;n r/ þ b j
Hm i ¼
Z
j
i
~m;n / dV Sm;n sm;n r/i þ Sm;n b i
ð13aÞ ð13bÞ
V
After solving the RTE, the incident radiation G is obtained from
GðrÞ ¼
Z
Iðr; XÞdX
rectangular enclosure filled with a graded index medium 0:5 nðx; yÞ ¼ 5½1 0:4356ðx2 þ y2 Þ=L2 having opaque and diffuse surfaces from literature. For the following numerical study, the maximum relative error 106 of the incident radiation is taken as the stopping criterion of iteration. The medium is assumed to be anisotropic scattering with a forward scattering phase function as U(X, X0 ) = 1 + XX0 . The numbers of uniform grids 21 21 and discrete directions Mh M/ = 10 20 are used for grid-independent situations. The south boundary of the rectangular is at some finite temperature TS and is thus a radiation source. The other three boundaries are cold (at zero temperature). The medium withunknown temperatures is at the radiative equilibrium. The non-dimensional temperature distributions along the line x/L = 0.325 is presented in Fig. 1 for two values of surface emissivity, namely, 0.5 and 1.0, and the scattering albedo is kept at x = 0.0. The extinction coefficient of b is taken as 1 m1. There are temperature slips at boundary surfaces, which is similar to that in uniform index media. By the comparison, we can see that the LSNEM results agree with those obtained by the MCM [25] very well. 4. Results and discussion The correctness and the reliability of LSNEM solution for radiative transfer in enclosures filled with GRIN medium were proven in the Section 3. Following that, in this section, we apply the LSNEM to solve the radiative heat transfer in graded index media having semitransparent and diffusely reflecting surfaces. Different from the opaque surface, the diffuse reflectivity of the semitransparent surface depends on the refractive index distribution at the boundaries as seen in Eqs. (3) and (4). In this section, we examine radiative heat transfer in a semicircular enclosure with an inner circle filled with absorbing, emitting and isotropically scattering graded index medium as shown in Fig. 2a. The inner circle is centered at x = 0 m and y = 0.4 m. The radiative heat transfer problems for the medium having a uniform refractive index in this enclosure have been investigated [26]. Here we consider two kinds of refractive index distributions as shown in Fig. 2(b and c) for which
GRIN 1 :
h i0:5 nðx; yÞ ¼ 5 1 0:4356ðx2 þ y2 Þ=R2
ð17Þ
GRIN 2 :
nðx; yÞ ¼ 2 þ 0:5ðj cosðpx=2=RÞj þ j sinðpyÞjÞ
ð18Þ
where R is the radius of the outer semicircle. In the following cases, we investigate effects of transmission characteristic of surfaces on the radiative heat transfer in the media with the two kinds of GRIN mentioned above. Only the surface of the outer semicircle is assumed semitransparent. The results are
ð14Þ
0.90
The divergence of radiative heat flux rqr(r) that accounts for the volumetric radiation is given by:
0.85
r qr ðrÞ ¼ ja ð4n2 rT 4 ðrÞ GðrÞÞ
ð15Þ
For radiative equilibrium problem, rqr(r) = 0, and the temperature distribution of the medium can be written as:
TðrÞ ¼
GðrÞ 4n2 r
14
Temperature T/TS
4p
MCM Ref. [25] NEM
0.80 0.75 0.70
ð16Þ
3. Validation of the least-squares natural element method To check the performance and accuracy of the LSNEM presented in this paper, we examine radiative heat transfer problems in a
0.65 0.60 0.0
0.2
0.4
0.6
0.8
1.0
Distance, y/Ly Fig. 1. Non-dimensional temperature distributions along line x/L = 0.325.
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(a) 3.76 3.95 4.14 4.33 4.52 4.71 4.9
2.01 2.16 2.31 2.46 2.61 2.76 2.91
(b)
(c)
Fig. 2. Schematic of semicircular enclosure with inner circle as shown in (a) and graded index distributions given by (b) Eq. (17) and (c) Eq. (18).
0.40 0.50 0.60 0.70 0.80 0.90
(a)
0.40 0.50 0.60 0.70 0.80 0.90
0.65 0.70 0.75 0.80 0.85 0.90 0.95
(b)
0.50 0.58 0.66 0.74 0.82 0.90
Fig. 3. Non-dimensional incident radiation G ðG=rT 4g Þ distribution with GRIN 1: (a) opaque case and (b) semitransparent case and those with GRIN 2:(c) opaque case and (d) semitransparent case.
compared with those for the enclosure whose surfaces are all opaque. For all the cases, the 1144 nodes and discrete directions Mh M/ = 15 30 are considered, and all the opaque surfaces are black. The bottom surface at some finite temperature TS, the other boundaries and environment are cold (at zero temperature). The medium with unknown temperatures is at radiative equilibrium.
For this case, only the semicircle surface is semitransparent whereas all other surfaces are kept opaque and black. For GRIN 1 and 2, with b = 1.0 m1 and x = 0.5, the non-dimensional incident radiation G distributions are shown in Fig. 3. Owing to higher values of the refractive index, the incident radiation distributions of GRIN 1 are higher than those of GRIN 2. It can been
Y. Zhang et al. / International Journal of Heat and Mass Transfer 66 (2013) 349–354
(a)
353
(b)
(c)
(d)
Fig. 4. Non-dimensional temperature distributions along the circle arc AA0 for different values of the extinction coefficient for semitransparent cases (a) GRIN 1 and (b) GRIN 2, non-dimensional radiative heat flux distribution along the inner circle wall for various the extinction coefficient (c) opaque case and (d) semitransparent case.
seen that, for different graded index medium, the incident radiation distributions have little differences. In Fig. 4(a and b), for semitransparent cases, the curves for nondimensional temperature T/TS distributions along the circle arc AA0 at RL = 0.8 are plotted for different values of the extinction coefficient b, and the scattering albedo is kept at x = 0.5. It can be observed that, for both the two refractive index distribution, a temperature slip is observed at the bottom surface, which is the characteristic of radiative equilibrium situation. As b increases, the medium with increased absorption and scattering coefficients does not allow the radiation from the boundaries to penetrate deep into the medium. As a result, large variations in temperature distribution along AA0 are observed. For GRIN 1, Fig. 4(c and d) depicts the non-dimensional radiative heat flux distribution along the inner circle wall for various the extinction coefficient b. It can be seen that higher the extinction coefficient is, bigger the radiative heat flux is. It can be further observed that, for opaque case, the radiative heat flux at the place near the bottom wall changes little as b increases. While for semitransparent case, the changes of the radiative heat flux at the places near or away from the bottom wall trend to be consistent, which is attributed to the semitransparent semicircle wall.
medium with diffusely reflecting semitransparent or opaque walls. In the present work, we firstly had a successful implementation of the LSNEM to the problems for radiative heat transfer in the rectangular enclosure and a semicircular enclosure with an inner circle involving semitransparent surfaces. The influence of the refractive index distribution and transmission characteristic of surfaces on the results are analyzed. Effects of the extinction coefficient are investigated on the radiative heat fluxes and temperature distributions.
Acknowledgments This work was supported by program for New Century Excellent Talents in University (NCET-09-0067), the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 51121004) and the National Natural Science Foundation of China (Grant No. 51176040). A very special acknowledgement is made to the editors and referees who make important comments to improve this paper.
References 5. Conclusions Based on the discrete-ordinates equation of radiative transfer, the least-squares natural element method is developed for the simulation of radiative heat transfer problems in graded index
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