- Email: [email protected]
Discrete transfer method with the concept of blocked-off region for irregular geometries
ARTICLE IN PRESS
Journal of Quantitative Spectroscopy & Radiative Transfer 98 (2006) 238–248 www.elsevier.com/locate/jqsrt
Discrete transfer method with the concept of blocked-off region for irregular geometries Prabal Talukdara, a
Institute of Fluid Mechanics (LSTM), University of Erlangen-Nuremberg, Cauerstrasse 4, D-91058 Erlangen, Germany Received 22 December 2004; accepted 9 May 2005
Abstract The discrete transfer method (DTM) is applied to irregular geometries with a concept of blocked-off region previously applied in the problems of computational fluid dynamics. This gives a new alternative to the DTM for its implementation to irregular structures. The Cartesian coordinate-based ray-tracing algorithm can be applied to the geometries with inclined or curved boundaries. Some test problems are considered and results are validated with the available results in the literature. Both radiative and nonradiative equilibrium situations are considered. The medium is assumed to be both participating and nonparticipating. Results are found to be accurate for all kinds of situations. r 2005 Elsevier Ltd. All rights reserved. Keywords: Discrete transfer method; Participating media; Radiation; Blocked-off region concept
1. Introduction Radiative heat transfer plays a dominant a role in many of the high-temperature applications as in combustion chambers, nuclear fusion, greenhouses, rocket plume sensing, etc. Due to the complexities in solving the radiative transfer equation, its applicabilty to multi-dimensional problems is always a challenging task. There are limited numbers of literatures available dealing with the radiative transfer problems with complex 2D and 3D geometries. Tel.: +49 9131 8529489; fax: +49 9131 8529503.
E-mail address: [email protected]. 0022-4073/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2005.05.087
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Sanchez and Smith [1] discussed the radiative exchange of a square geometry with a square obstacle at the middle using the discrete ordinates method for a non-participating media. In participating media, Chai et al. [2,3] discussed different possibilities of solving radiative transfer problems in irregular structures using the discrete ordinates method, the finite volume method and the Monte Carlo method. They implemented blocked-off region procedure in the discrete ordinates method [2] and the finite volume method [3] for different types of irregular structures. They applied the Monte Carlo method [4] for irregular geometries and considered a rhombus, a quadrilateral and an enclosure with curved and straight edged boundaries. The medium considered was absorbing, emitting and anisotropically scattering. They also implemented the finite volume method in curvilinear coordinate with multiblocking to handle irregular geometries. Recently, Talukdar et al. [5] implemented the finite volume method in curvilinear coordinate with multiblocking for 3D irregular geometries. Murthy and Mathur [6] worked with unstructured meshes to implement the finite volume method for different complex geometries. Koo et al. [7] studied the effect of three different discrete ordinates methods applied to 2D curved geometries. In another paper, Koo et al. [8] discussed the first-order and second-order interpolation scheme in context with the irregular geometries. Sakami and Charette [9] discussed a modified discrete ordinates method based on triangular grids with a new differencing scheme applicable to different complex geometries. They intended to eliminate the ray effect inherent with the discrete ordinates method breaking the intensity into two parts: the wall-related intensity and the medium-related intensity. The discrete transfer method (DTM) as founded by Shah [10] has been implemented to irregular structures by some of the works in the past. Meng et al. [11] used the DTM for irregular geometries using a finite element formulation. Malalasekera and James [12] applied the DTM for 3D irregular geometries using non-orthogonal body-fitted coordinate system. Malalasekera and Lockwood [13] also used the DTM in conjunction with a cell-blocking procedure based on Cartesian coordinate to model combustion and radiative heat transfer in complex 3D tunnel geometry. In the present work, the blocked-off region procedure is implemented to the DTM. The concept of blocked-off region was previously applied in the computational fluid dynamics (CFD)[14] problems. As mentioned before, in radiative heat transfer, Chai et al. first implemented this concept with the finite volume method [3] and the discrete ordinates method [2]. They found very promising results for different 2D problems. Looking to the prospect of this concept, the current work tries to demonstrate its applicability to the DTM. As mentioned before, the DTM has been widely used for variety of problems. Being a ray-tracing method, the blocked-off region procedure is straight forward to implement in case of this method. The Cartesian based 2D algorithm can be used to calculate radiative fluxes for irregular geometries by dividing the region into active and inactive regions. It is easier and convenient way of handling 2D irregular geometries than to write an algorithm in curvilinear coordinates for a ray tracing method. The present work demonstrates the applicability and usefulness of the DTM with the current approach.
2. Formulation and solution procedure In this section, the basic equations of the DTM used in this work are discussed and the solution strategy is explained.
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The DTM is a ray-tracing method and it uses the integral form of the radiative transfer equation. To find out the radiative flux q or incident radiation G at a particular point, the intensities coming from all directions have to be considered. They are calculated as Z 2p Z p iðf; yÞ sin y dy df, (1) G¼ f¼0
Z
2p
Z
y¼0 p
q¼
iðf; yÞ sin y cos y dy df. f¼0
(2)
y¼0
Here, in the above equations, y and f are the polar angle and the azimuthal angle, respectively. The intensity i depends on both angular directions (y and f) and spatial locations. In the DTM, the intensities are always traced from the boundaries. If the boundary temperature T b and emissivity b are known, then the boundary intensity for a diffuse-gray surface can be calculated as Z Z b sT 4b ð1 b Þ 2p p=2 þ iðf; yÞ cos y sin y dy df. (3) i0 ¼ p p f¼0 y¼0 When the intensity travels through a participating medium, it always gets changed due to absorption, emission and scattering by the media. The intensity at a point n þ 1 can be calculated if the intensity at the upstream point n is known, inþ1 ¼ in expðtÞ þ S½1 expðtÞ.
(4)
Here, in the above equation, t ð¼ b L, b is the extinction coefficient of the medium and L is the physical length that the intensity propagates) is the optical thickness of the medium and S is the source function. In the above equation, the source function S is assumed to be constant in a particular control volume. The source function itself is a function of the intensity and has to be calculated in an iterative way if either the medium temperature T is unknown or if the medium scatters. For an isotropically scattering medium, it can be expressed as sT 4 o (5) þ G. 4p p Here, o is the scattering albedo whose value is 0 for a purely absorbing medium and 1 for a purely scattering medium. In the present work, the blocked-off region procedure is implemented with the DTM to handle irregular geometries. The same 2D rectangular ray-tracing algorithm is improvised to handle curved or inclined boundaries. The whole 2D region is divided into two parts: active and inactive or blocked-off regions. The region where solutions are sought is known as the active region and the remaining portion is known as the inactive or the blocked-off region. When an intensity enters an inactive region its magnitude becomes zero and it takes another boundary condition if it enters again to an active region. In Fig. 1(a) and (b), two sample geometries are presented to show how they are treated to simulate from a rectangular geometry. The shaded portion is called as inactive or blocked-off region where solutions are not of importance. The remaining portion is the active region which is the real domain of interest. The rectangular domain can be termed as the simulated domain. In Fig. 2, how a domain is described in this problem is shown. This case is in S ¼ ð1 oÞ
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Inactive or blocked-off region (shaded portion)
(a)
Real domain
Simulated domain
(b)
Real domain
Simulated domain
Fig. 1. Sample irregular geometries. 1st boundary condition of the intensity a 1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
0
0
0
1
1
0
0
0
0
c 1
1
0
0
0
0
1
1
0
0
0
0
1
1
0
0
1 0
b
d 2nd boundary condition of the intensity
Fig. 2. Ray tracing in a domain with blocked-off/inactive region.
reference to Fig. 1(a) of the T-shaped enclosure. The whole simulated domain is discretised into several control volumes and the control volumes which are inside the active region are designated as one (1) and otherwise they are zero (0). A typical intensity path ad is shown in this figure. The intensity originates from point a with the known boundary condition. As soon as the intensity passes through the point b, it enters to a domain with a zero value. As a result, its history terminates and it becomes zero till it reaches the point c. At point c, which is in the interface of the inactive and active region, it gets a second boundary condition. The calculations for the path ac is meaningless and only the cd path contributes to the flux calculations at the point d.
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For every type of geometry, a domain file has to be created as shown in Fig. 2. By changing the value of a control volume from 1 to 0, it can be made inactive. This domain file has to be created as per as the grid file for a particular problem. The other important additional task for this approach is to define the additional boundary condition referred as the second boundary condition in Fig. 2. Depending on the shape of the geometry, a boundary condition file has to be specified.
3. Results and discussion To show the validity and the accuracy of the current method, different test problems are considered. Some problems are self-validated and some are compared with the available results of the literature. Different types of irregular geometries like boundaries with inclined and curved surfaces and geometries with obstacles are considered. Both radiative and non-radiative equilibrium conditions are considered with the medium assumed to be both participating and non-participating. Test Problem 1: To check the validity of the proposed approach, a test problem is considered with a square domain. The real geometry is extended in one side by half of the original length to simulate it as a blocked-off or inactive region. In Fig. 3(a), the blocked-off region is shown by the shaded lines. The whole geometry is simulated with shaded region considered as blocked-off and then the results are compared with the results of the real square geometry which is shown in the Inactive/blocked-off region T=0 T = 0.5
T = 0.5
T=0 d
x
x
T=1 d Real Geometry
(a)
T=1
1.5d Simulated Geometry
1 0.9
β = 0.1
Heat Flux
0.8 β=1
0.7 0.6
Blocked-off Real
0.5 β = 10
0.4 0.3 0.2 (b)
0
0.2
0.4 0.6 Distance x
0.8
1
Fig. 3. (a) Sample geometry and (b) comparison of heat flux results at the south (hot) boundary for different extinction coefficients b; radiative equilibrium.
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left side of Fig. 3(a). The south boundary is considered to be at temperature T ¼ 1 and the west boundary is at temperature T ¼ 0:5. The other boundaries are at zero temperature and all boundaries are considered to be black. A radiative equilibrium situation is considered. Radiative heat flux results at the hot south boundary are shown in Fig. 3(b) for a different extinction coefficient b. Results of blocked-off procedure matches exactly with the real geometry results. Test Problem 2: The second test problem considered is also a square. To check the blocked-off region procedure, the square block is extended both on the west and south side as shown in Fig. 4(a). The shaded portion is the blocked-off region. The dimensions are shown in the figure. A non-radiative equilibrium condition is considered with isothermal medium and cold black boundaries. Results are validated for different extinction coefficients b. In Fig. 4(b), heat flux results at boundary 1 are shown for both blocked-off and real geometry simulation. They are found to be exactly matched. Test Problem 3: The third problem considered is an L-shaped enclosure. The L-shaped geometry is extracted from the square geometry by making the north–east quadrant as a blockedoff region as shown in Fig. 5(a). Both radiative and non-radiative equilibrium conditions are considered. In Fig. 5(b), a non-radiative equilibrium situation is considered with an isothermal d
boundary 4
boundary 2
0.5d boundary 3
x boundary 1
0.5d
d
Heat Flux
(a)
(b)
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
β = 10 Blocked-off Real β=1
β = 0.1 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Distance x
Fig. 4. (a) Sample geometry and (b) comparison of heat flux distributions at the boundary 1 for different extinction coefficients b; non-radiative equilibrium.
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medium surrounded by cold black boundaries. Heat flux results at the south boundary is shown in the figure for different scattering albedo o. Results are compared with the finite volume method (FVM) results generated by the author. The FVM code is already validated with the literature. It is to be noted that the FVM code uses a multi-block procedure for simulating this geometry. Results are found to be matching well for both the methods. The FVM results are generated for the same number of control volumes and rays as with the DTM results. d
0.5d
0.5d d East Blocked-off region x
(a)
South
0.6
Heat Flux
0.5 ω=0
0.4 0.3
ω = 0.5
Blocked-off FVM
0.2
ω = 0.9
0.1 0
0
0.2
(b)
0.4 0.6 Distance x
0.8
1
0.92 FVM(step, coarse grid) FVM(step, fine grid) FVM(expo, coarse grid) FVM(expo, fine grid) DTM(coarse grid)
0.9
Heat Flux
0.88 0.86 0.84 0.82 0.8 0.78 0.76 (c)
0
0.2
0.4 0.6 Distance x
0.8
1
Fig. 5. (a) Sample geometry (b) heat flux distributions at the south boundary of the L-shaped geometry with the blocked-off DTM and block-structured FVM; non-radiative equilibrium, and (c) grid independency study of the blocked-off DTM with the block-structured FVM for heat flux at the south (hot) boundary; radiative equilibrium.
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In Fig. 5(c), a radiative equilibrium situation is considered with the south boundary as the hot black boundary and all the other boundaries to be cold black. The heat flux results at the south boundary are shown in the figure. A grid and ray independency test is carried out in this work. It is seen that for both the methods, a 12 16 rays in polar and azimuthal directions, respectively, are found to be optimum. For grid sizes, the DTM requires a less number of control volume ð20 20Þ to get grid-independent results whereas in the FVM, both the step and exponential scheme requires a finer grid with ð40 40Þ control volume to match closely with the DTM results. Test Problem 4: The fourth problem is a quadrilateral geometry previously considered by other researchers for validation of irregular geometries. The geometry is described in Fig. 6(a). Nonradiative equilibrium situation is considered with isothermal medium and cold black boundaries. Three different extinction coefficients are considered with an absorbing-emitting situation. The whole rectangle is simulated with the inclined planes approximated by step-size grids. A grid of 30 40 control volume size in the X –Y directions and 12 24 (y f) intensity directions are found to be sufficient to have accurate results. Results are compared with the exact results available in the literature [8] and also with the finite volume results with a step scheme generated by the author. An excellent agreement has been found for this problem. Test Problem 5: In the fifth problem, a geometry with curved boundary is simulated. This geometry is a common problem previously taken by other researchers. The schematic of the problem is shown in Fig. 7 (a). The curved boundary is simulated with a step-size grid as shown in
Actual domain (shaded portion)
Simulated domain (whole rectangle) (0.5, 1)
(1.5, 1.2)
T=0
T=0
T=0
(0, 0)
(2.2, 0)
T=0
1 β =10
Heat Flux
0.8 0.6
β =1
0.4
β = 0.1
0.2
Blocked-off FVM Exact
0 0
0.5
1 Distance
1.5
2
Fig. 6. (a) A quadrilateral geometry and (b) Comparison of heat flux distributions at the south boundary; non-radiative equilibrium.
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T=0
T=1 1.5 d d Real domain (shaded portion)
Blocked off Region Simulated domain (whole rectangle)
(a)
1 β = 10
Heat Flux
0.8 0.6 0.4
β=1
0.2
β = 0.1
Blocked-off FVM
0 0
0.2
0.4
(b)
0.6
0.8
1
Distance 0.7 0.6 ω=0
Heat Flux
0.5 0.4 0.3
ω = 0.5
Blocked-off FVM
0.2
ω = 0.9
0.1 0 (c)
0
0.2
0.4
0.6
0.8
1
Distance
Fig. 7. (a) A curved geometry with step size grids, (b) heat flux distributions at the north boundary with the effect of b; non-radiative equilibrium and (c) heat flux distributions at the north boundary with the effect of o; non-radiative equilibrium.
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the figure. The actual grid considered is 30 50 although in the figure a coarse grid is shown. This is a case of non-radiative equilibrium with isothermal medium and cold black boundaries. The heat flux results at the north boundary is shown in Fig. 7(b) and (c) for different extinction coefficients b and scattering albedos o, respectively. The results are compared with the selfgenerated FVM results using the step scheme. A good agreement is found as seen from the figures. For FVM, a coarse grid ð20 20Þ is sufficient whereas in the DTM, a large number of control volumes have to be considered to take care of the curved boundary with the step-size grids. The number of intensities considered are equivalent in both the methods. Test Problem 6: The last problem considered is the problem investigated previously by Sanchez and Smith [1] and then Chai et al. [3]. The schematic of the problem is shown in Fig. 8(a). It consists of a square enclosure with a central blockage. The medium is non-participating and all boundaries are assumed black. The left boundary is set at 320 K and all the other boundaries including the boundaries of the central blockage are set at 300 K. The heat flux results at the boundary of the enclosure are presented in Fig. 8(b). A total of 40 40 control volumes and 12 24 (y f) intensity directions are considered. The distance z is measured from the lower left
T = 300K
ζ=2.0
ζ=1.0
ζ=2.5
ζ=0.5 T = 320K
T = 300K
T = 300K ζ T = 300K
(a)
Heat Flux (W/m2)
150
50 0 −50 −100 0.5
(b)
Present RIM
100
1
1.5 Distance ζ
2
2.5
Fig. 8. (a) Schematic of the problem and (b) comparison of heat flux distributions at the enclosure boundary; nonparticipating medium.
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corner of the enclosure. The results compare well with the solution of the RIM (radiosity/ irradiation method) of Sanchez and Smith [1]. 4. Conclusions A blocked-off region concept previously used in computational fluid dynamics is applied to the DTM to handle irregular geometries. Different types of irregularities are considered in 2D geometries. Both radiative and non-radiative equilibrium situations are considered. Results are validated with the finite volume method results and also some of the results already published in the literature. Results are found to be promising for a wide range of extinction coefficients and scattering albedos. The method can be recommended as a good alternative to solve problems with irregular geometries. Although, the method suffers from the unneccessary computation in the blocked-off region, it has the major advantage of using the same 2D rectangular ray-tracing algorithm for all types of 2D geometries.
References [1] Sanchez A, Smith TF. Surface radiation exchange for two-dimensional rectangular enclosures using the discreteordinates method. J Heat Transfer 1992;114:465–72. [2] Chai JC, Lee HS, Patankar SV. Treatment of irregular geometries using a Cartesian coordinates control-angle control-volume-based discrete-ordinate method. In: Proceedings of the national heat transfer conference, Atlanta. American Society of Mechanical Engineers; August 8–11, 1993. p. 35–43. [3] Chai JC, Lee HS, Patankar SV. Treatment of irregular geometries using a Cartesian coordinates finite volume radiation heat transfer procedure. Numer Heat Transfer—Part B 1994;26:225–35. [4] Parthasarathy G, Lee HS, Chai JC, Patankar SV. Monte Carlo solutions for radiative heat transfer in irregular two-dimensional geometries. J Heat Transfer 1995;117:792–4. [5] Talukdar P, Steven M, Issendorff FV, Trimis D. Finite volume method in 3-D curvilinear coordinates with multiblocking procedure for radiative transport problems. Int J Heat Mass Transfer, accepted. [6] Murthy JY, Mathur SR. Radiative heat transfer in axisymmetric geometries using an unstructured finite-volume method. Numer Heat Transfer—Part B 1998;33:397–416. [7] Koo H-M, Vaillon R, Goutiere V, Dez VL, Cha H, Song T-H. Comparison of three discrete ordinates methods applied to two-dimensional curved geometries. Int J Therm Sci 2003;42:343–59. [8] Koo H-M, Cheong K-B, Song T-H. Schemes and applications of first and second-order discrete ordinates interpolation methods to irregular two-dimensional geometries. J Heat Transfer 1997;119:730–7. [9] Sakami M, Charette A. Application of a modified discrete ordinates method to two-dimensional enclosures of irregular geometry. JQSRT 2000;64:275–98. [10] Shah NG. New method of computation of radiation heat transfer combustion chambers. PhD thesis, Imperial College, University of London, England, 1979. [11] Meng FL, McKenty F, Camarero R. Radiative heat transfer by the discrete transfer method using an unstructured mesh. ASME HTD 1993;2044:55–6. [12] Malalasekera WMG, James EH. Radiative heat transfer calculations in three-dimensional complex geometries. J Heat Transfer 118:225–8. [13] Malalasekera WMG, Lockwood FC. Computer simulation of the king’s cross fire: effect of radiative heat transfer on fire spread. Proc Inst Mech Eng 1991;205:201–8. [14] Patankar SV. Numerical heat transfer and fluid flow. Washingtom, DC: Hemisphere Publishing; 1980.
Journal of Quantitative Spectroscopy & Radiative Transfer 98 (2006) 238–248 www.elsevier.com/locate/jqsrt
Discrete transfer method with the concept of blocked-off region for irregular geometries Prabal Talukdara, a
Institute of Fluid Mechanics (LSTM), University of Erlangen-Nuremberg, Cauerstrasse 4, D-91058 Erlangen, Germany Received 22 December 2004; accepted 9 May 2005
Abstract The discrete transfer method (DTM) is applied to irregular geometries with a concept of blocked-off region previously applied in the problems of computational fluid dynamics. This gives a new alternative to the DTM for its implementation to irregular structures. The Cartesian coordinate-based ray-tracing algorithm can be applied to the geometries with inclined or curved boundaries. Some test problems are considered and results are validated with the available results in the literature. Both radiative and nonradiative equilibrium situations are considered. The medium is assumed to be both participating and nonparticipating. Results are found to be accurate for all kinds of situations. r 2005 Elsevier Ltd. All rights reserved. Keywords: Discrete transfer method; Participating media; Radiation; Blocked-off region concept
1. Introduction Radiative heat transfer plays a dominant a role in many of the high-temperature applications as in combustion chambers, nuclear fusion, greenhouses, rocket plume sensing, etc. Due to the complexities in solving the radiative transfer equation, its applicabilty to multi-dimensional problems is always a challenging task. There are limited numbers of literatures available dealing with the radiative transfer problems with complex 2D and 3D geometries. Tel.: +49 9131 8529489; fax: +49 9131 8529503.
E-mail address: [email protected]. 0022-4073/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2005.05.087
ARTICLE IN PRESS P. Talukdar / Journal of Quantitative Spectroscopy & Radiative Transfer 98 (2006) 238–248
239
Sanchez and Smith [1] discussed the radiative exchange of a square geometry with a square obstacle at the middle using the discrete ordinates method for a non-participating media. In participating media, Chai et al. [2,3] discussed different possibilities of solving radiative transfer problems in irregular structures using the discrete ordinates method, the finite volume method and the Monte Carlo method. They implemented blocked-off region procedure in the discrete ordinates method [2] and the finite volume method [3] for different types of irregular structures. They applied the Monte Carlo method [4] for irregular geometries and considered a rhombus, a quadrilateral and an enclosure with curved and straight edged boundaries. The medium considered was absorbing, emitting and anisotropically scattering. They also implemented the finite volume method in curvilinear coordinate with multiblocking to handle irregular geometries. Recently, Talukdar et al. [5] implemented the finite volume method in curvilinear coordinate with multiblocking for 3D irregular geometries. Murthy and Mathur [6] worked with unstructured meshes to implement the finite volume method for different complex geometries. Koo et al. [7] studied the effect of three different discrete ordinates methods applied to 2D curved geometries. In another paper, Koo et al. [8] discussed the first-order and second-order interpolation scheme in context with the irregular geometries. Sakami and Charette [9] discussed a modified discrete ordinates method based on triangular grids with a new differencing scheme applicable to different complex geometries. They intended to eliminate the ray effect inherent with the discrete ordinates method breaking the intensity into two parts: the wall-related intensity and the medium-related intensity. The discrete transfer method (DTM) as founded by Shah [10] has been implemented to irregular structures by some of the works in the past. Meng et al. [11] used the DTM for irregular geometries using a finite element formulation. Malalasekera and James [12] applied the DTM for 3D irregular geometries using non-orthogonal body-fitted coordinate system. Malalasekera and Lockwood [13] also used the DTM in conjunction with a cell-blocking procedure based on Cartesian coordinate to model combustion and radiative heat transfer in complex 3D tunnel geometry. In the present work, the blocked-off region procedure is implemented to the DTM. The concept of blocked-off region was previously applied in the computational fluid dynamics (CFD)[14] problems. As mentioned before, in radiative heat transfer, Chai et al. first implemented this concept with the finite volume method [3] and the discrete ordinates method [2]. They found very promising results for different 2D problems. Looking to the prospect of this concept, the current work tries to demonstrate its applicability to the DTM. As mentioned before, the DTM has been widely used for variety of problems. Being a ray-tracing method, the blocked-off region procedure is straight forward to implement in case of this method. The Cartesian based 2D algorithm can be used to calculate radiative fluxes for irregular geometries by dividing the region into active and inactive regions. It is easier and convenient way of handling 2D irregular geometries than to write an algorithm in curvilinear coordinates for a ray tracing method. The present work demonstrates the applicability and usefulness of the DTM with the current approach.
2. Formulation and solution procedure In this section, the basic equations of the DTM used in this work are discussed and the solution strategy is explained.
ARTICLE IN PRESS P. Talukdar / Journal of Quantitative Spectroscopy & Radiative Transfer 98 (2006) 238–248
240
The DTM is a ray-tracing method and it uses the integral form of the radiative transfer equation. To find out the radiative flux q or incident radiation G at a particular point, the intensities coming from all directions have to be considered. They are calculated as Z 2p Z p iðf; yÞ sin y dy df, (1) G¼ f¼0
Z
2p
Z
y¼0 p
q¼
iðf; yÞ sin y cos y dy df. f¼0
(2)
y¼0
Here, in the above equations, y and f are the polar angle and the azimuthal angle, respectively. The intensity i depends on both angular directions (y and f) and spatial locations. In the DTM, the intensities are always traced from the boundaries. If the boundary temperature T b and emissivity b are known, then the boundary intensity for a diffuse-gray surface can be calculated as Z Z b sT 4b ð1 b Þ 2p p=2 þ iðf; yÞ cos y sin y dy df. (3) i0 ¼ p p f¼0 y¼0 When the intensity travels through a participating medium, it always gets changed due to absorption, emission and scattering by the media. The intensity at a point n þ 1 can be calculated if the intensity at the upstream point n is known, inþ1 ¼ in expðtÞ þ S½1 expðtÞ.
(4)
Here, in the above equation, t ð¼ b L, b is the extinction coefficient of the medium and L is the physical length that the intensity propagates) is the optical thickness of the medium and S is the source function. In the above equation, the source function S is assumed to be constant in a particular control volume. The source function itself is a function of the intensity and has to be calculated in an iterative way if either the medium temperature T is unknown or if the medium scatters. For an isotropically scattering medium, it can be expressed as sT 4 o (5) þ G. 4p p Here, o is the scattering albedo whose value is 0 for a purely absorbing medium and 1 for a purely scattering medium. In the present work, the blocked-off region procedure is implemented with the DTM to handle irregular geometries. The same 2D rectangular ray-tracing algorithm is improvised to handle curved or inclined boundaries. The whole 2D region is divided into two parts: active and inactive or blocked-off regions. The region where solutions are sought is known as the active region and the remaining portion is known as the inactive or the blocked-off region. When an intensity enters an inactive region its magnitude becomes zero and it takes another boundary condition if it enters again to an active region. In Fig. 1(a) and (b), two sample geometries are presented to show how they are treated to simulate from a rectangular geometry. The shaded portion is called as inactive or blocked-off region where solutions are not of importance. The remaining portion is the active region which is the real domain of interest. The rectangular domain can be termed as the simulated domain. In Fig. 2, how a domain is described in this problem is shown. This case is in S ¼ ð1 oÞ
ARTICLE IN PRESS P. Talukdar / Journal of Quantitative Spectroscopy & Radiative Transfer 98 (2006) 238–248
241
Inactive or blocked-off region (shaded portion)
(a)
Real domain
Simulated domain
(b)
Real domain
Simulated domain
Fig. 1. Sample irregular geometries. 1st boundary condition of the intensity a 1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
0
0
0
1
1
0
0
0
0
c 1
1
0
0
0
0
1
1
0
0
0
0
1
1
0
0
1 0
b
d 2nd boundary condition of the intensity
Fig. 2. Ray tracing in a domain with blocked-off/inactive region.
reference to Fig. 1(a) of the T-shaped enclosure. The whole simulated domain is discretised into several control volumes and the control volumes which are inside the active region are designated as one (1) and otherwise they are zero (0). A typical intensity path ad is shown in this figure. The intensity originates from point a with the known boundary condition. As soon as the intensity passes through the point b, it enters to a domain with a zero value. As a result, its history terminates and it becomes zero till it reaches the point c. At point c, which is in the interface of the inactive and active region, it gets a second boundary condition. The calculations for the path ac is meaningless and only the cd path contributes to the flux calculations at the point d.
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For every type of geometry, a domain file has to be created as shown in Fig. 2. By changing the value of a control volume from 1 to 0, it can be made inactive. This domain file has to be created as per as the grid file for a particular problem. The other important additional task for this approach is to define the additional boundary condition referred as the second boundary condition in Fig. 2. Depending on the shape of the geometry, a boundary condition file has to be specified.
3. Results and discussion To show the validity and the accuracy of the current method, different test problems are considered. Some problems are self-validated and some are compared with the available results of the literature. Different types of irregular geometries like boundaries with inclined and curved surfaces and geometries with obstacles are considered. Both radiative and non-radiative equilibrium conditions are considered with the medium assumed to be both participating and non-participating. Test Problem 1: To check the validity of the proposed approach, a test problem is considered with a square domain. The real geometry is extended in one side by half of the original length to simulate it as a blocked-off or inactive region. In Fig. 3(a), the blocked-off region is shown by the shaded lines. The whole geometry is simulated with shaded region considered as blocked-off and then the results are compared with the results of the real square geometry which is shown in the Inactive/blocked-off region T=0 T = 0.5
T = 0.5
T=0 d
x
x
T=1 d Real Geometry
(a)
T=1
1.5d Simulated Geometry
1 0.9
β = 0.1
Heat Flux
0.8 β=1
0.7 0.6
Blocked-off Real
0.5 β = 10
0.4 0.3 0.2 (b)
0
0.2
0.4 0.6 Distance x
0.8
1
Fig. 3. (a) Sample geometry and (b) comparison of heat flux results at the south (hot) boundary for different extinction coefficients b; radiative equilibrium.
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left side of Fig. 3(a). The south boundary is considered to be at temperature T ¼ 1 and the west boundary is at temperature T ¼ 0:5. The other boundaries are at zero temperature and all boundaries are considered to be black. A radiative equilibrium situation is considered. Radiative heat flux results at the hot south boundary are shown in Fig. 3(b) for a different extinction coefficient b. Results of blocked-off procedure matches exactly with the real geometry results. Test Problem 2: The second test problem considered is also a square. To check the blocked-off region procedure, the square block is extended both on the west and south side as shown in Fig. 4(a). The shaded portion is the blocked-off region. The dimensions are shown in the figure. A non-radiative equilibrium condition is considered with isothermal medium and cold black boundaries. Results are validated for different extinction coefficients b. In Fig. 4(b), heat flux results at boundary 1 are shown for both blocked-off and real geometry simulation. They are found to be exactly matched. Test Problem 3: The third problem considered is an L-shaped enclosure. The L-shaped geometry is extracted from the square geometry by making the north–east quadrant as a blockedoff region as shown in Fig. 5(a). Both radiative and non-radiative equilibrium conditions are considered. In Fig. 5(b), a non-radiative equilibrium situation is considered with an isothermal d
boundary 4
boundary 2
0.5d boundary 3
x boundary 1
0.5d
d
Heat Flux
(a)
(b)
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
β = 10 Blocked-off Real β=1
β = 0.1 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Distance x
Fig. 4. (a) Sample geometry and (b) comparison of heat flux distributions at the boundary 1 for different extinction coefficients b; non-radiative equilibrium.
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medium surrounded by cold black boundaries. Heat flux results at the south boundary is shown in the figure for different scattering albedo o. Results are compared with the finite volume method (FVM) results generated by the author. The FVM code is already validated with the literature. It is to be noted that the FVM code uses a multi-block procedure for simulating this geometry. Results are found to be matching well for both the methods. The FVM results are generated for the same number of control volumes and rays as with the DTM results. d
0.5d
0.5d d East Blocked-off region x
(a)
South
0.6
Heat Flux
0.5 ω=0
0.4 0.3
ω = 0.5
Blocked-off FVM
0.2
ω = 0.9
0.1 0
0
0.2
(b)
0.4 0.6 Distance x
0.8
1
0.92 FVM(step, coarse grid) FVM(step, fine grid) FVM(expo, coarse grid) FVM(expo, fine grid) DTM(coarse grid)
0.9
Heat Flux
0.88 0.86 0.84 0.82 0.8 0.78 0.76 (c)
0
0.2
0.4 0.6 Distance x
0.8
1
Fig. 5. (a) Sample geometry (b) heat flux distributions at the south boundary of the L-shaped geometry with the blocked-off DTM and block-structured FVM; non-radiative equilibrium, and (c) grid independency study of the blocked-off DTM with the block-structured FVM for heat flux at the south (hot) boundary; radiative equilibrium.
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In Fig. 5(c), a radiative equilibrium situation is considered with the south boundary as the hot black boundary and all the other boundaries to be cold black. The heat flux results at the south boundary are shown in the figure. A grid and ray independency test is carried out in this work. It is seen that for both the methods, a 12 16 rays in polar and azimuthal directions, respectively, are found to be optimum. For grid sizes, the DTM requires a less number of control volume ð20 20Þ to get grid-independent results whereas in the FVM, both the step and exponential scheme requires a finer grid with ð40 40Þ control volume to match closely with the DTM results. Test Problem 4: The fourth problem is a quadrilateral geometry previously considered by other researchers for validation of irregular geometries. The geometry is described in Fig. 6(a). Nonradiative equilibrium situation is considered with isothermal medium and cold black boundaries. Three different extinction coefficients are considered with an absorbing-emitting situation. The whole rectangle is simulated with the inclined planes approximated by step-size grids. A grid of 30 40 control volume size in the X –Y directions and 12 24 (y f) intensity directions are found to be sufficient to have accurate results. Results are compared with the exact results available in the literature [8] and also with the finite volume results with a step scheme generated by the author. An excellent agreement has been found for this problem. Test Problem 5: In the fifth problem, a geometry with curved boundary is simulated. This geometry is a common problem previously taken by other researchers. The schematic of the problem is shown in Fig. 7 (a). The curved boundary is simulated with a step-size grid as shown in
Actual domain (shaded portion)
Simulated domain (whole rectangle) (0.5, 1)
(1.5, 1.2)
T=0
T=0
T=0
(0, 0)
(2.2, 0)
T=0
1 β =10
Heat Flux
0.8 0.6
β =1
0.4
β = 0.1
0.2
Blocked-off FVM Exact
0 0
0.5
1 Distance
1.5
2
Fig. 6. (a) A quadrilateral geometry and (b) Comparison of heat flux distributions at the south boundary; non-radiative equilibrium.
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T=0
T=1 1.5 d d Real domain (shaded portion)
Blocked off Region Simulated domain (whole rectangle)
(a)
1 β = 10
Heat Flux
0.8 0.6 0.4
β=1
0.2
β = 0.1
Blocked-off FVM
0 0
0.2
0.4
(b)
0.6
0.8
1
Distance 0.7 0.6 ω=0
Heat Flux
0.5 0.4 0.3
ω = 0.5
Blocked-off FVM
0.2
ω = 0.9
0.1 0 (c)
0
0.2
0.4
0.6
0.8
1
Distance
Fig. 7. (a) A curved geometry with step size grids, (b) heat flux distributions at the north boundary with the effect of b; non-radiative equilibrium and (c) heat flux distributions at the north boundary with the effect of o; non-radiative equilibrium.
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the figure. The actual grid considered is 30 50 although in the figure a coarse grid is shown. This is a case of non-radiative equilibrium with isothermal medium and cold black boundaries. The heat flux results at the north boundary is shown in Fig. 7(b) and (c) for different extinction coefficients b and scattering albedos o, respectively. The results are compared with the selfgenerated FVM results using the step scheme. A good agreement is found as seen from the figures. For FVM, a coarse grid ð20 20Þ is sufficient whereas in the DTM, a large number of control volumes have to be considered to take care of the curved boundary with the step-size grids. The number of intensities considered are equivalent in both the methods. Test Problem 6: The last problem considered is the problem investigated previously by Sanchez and Smith [1] and then Chai et al. [3]. The schematic of the problem is shown in Fig. 8(a). It consists of a square enclosure with a central blockage. The medium is non-participating and all boundaries are assumed black. The left boundary is set at 320 K and all the other boundaries including the boundaries of the central blockage are set at 300 K. The heat flux results at the boundary of the enclosure are presented in Fig. 8(b). A total of 40 40 control volumes and 12 24 (y f) intensity directions are considered. The distance z is measured from the lower left
T = 300K
ζ=2.0
ζ=1.0
ζ=2.5
ζ=0.5 T = 320K
T = 300K
T = 300K ζ T = 300K
(a)
Heat Flux (W/m2)
150
50 0 −50 −100 0.5
(b)
Present RIM
100
1
1.5 Distance ζ
2
2.5
Fig. 8. (a) Schematic of the problem and (b) comparison of heat flux distributions at the enclosure boundary; nonparticipating medium.
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corner of the enclosure. The results compare well with the solution of the RIM (radiosity/ irradiation method) of Sanchez and Smith [1]. 4. Conclusions A blocked-off region concept previously used in computational fluid dynamics is applied to the DTM to handle irregular geometries. Different types of irregularities are considered in 2D geometries. Both radiative and non-radiative equilibrium situations are considered. Results are validated with the finite volume method results and also some of the results already published in the literature. Results are found to be promising for a wide range of extinction coefficients and scattering albedos. The method can be recommended as a good alternative to solve problems with irregular geometries. Although, the method suffers from the unneccessary computation in the blocked-off region, it has the major advantage of using the same 2D rectangular ray-tracing algorithm for all types of 2D geometries.
References [1] Sanchez A, Smith TF. Surface radiation exchange for two-dimensional rectangular enclosures using the discreteordinates method. J Heat Transfer 1992;114:465–72. [2] Chai JC, Lee HS, Patankar SV. Treatment of irregular geometries using a Cartesian coordinates control-angle control-volume-based discrete-ordinate method. In: Proceedings of the national heat transfer conference, Atlanta. American Society of Mechanical Engineers; August 8–11, 1993. p. 35–43. [3] Chai JC, Lee HS, Patankar SV. Treatment of irregular geometries using a Cartesian coordinates finite volume radiation heat transfer procedure. Numer Heat Transfer—Part B 1994;26:225–35. [4] Parthasarathy G, Lee HS, Chai JC, Patankar SV. Monte Carlo solutions for radiative heat transfer in irregular two-dimensional geometries. J Heat Transfer 1995;117:792–4. [5] Talukdar P, Steven M, Issendorff FV, Trimis D. Finite volume method in 3-D curvilinear coordinates with multiblocking procedure for radiative transport problems. Int J Heat Mass Transfer, accepted. [6] Murthy JY, Mathur SR. Radiative heat transfer in axisymmetric geometries using an unstructured finite-volume method. Numer Heat Transfer—Part B 1998;33:397–416. [7] Koo H-M, Vaillon R, Goutiere V, Dez VL, Cha H, Song T-H. Comparison of three discrete ordinates methods applied to two-dimensional curved geometries. Int J Therm Sci 2003;42:343–59. [8] Koo H-M, Cheong K-B, Song T-H. Schemes and applications of first and second-order discrete ordinates interpolation methods to irregular two-dimensional geometries. J Heat Transfer 1997;119:730–7. [9] Sakami M, Charette A. Application of a modified discrete ordinates method to two-dimensional enclosures of irregular geometry. JQSRT 2000;64:275–98. [10] Shah NG. New method of computation of radiation heat transfer combustion chambers. PhD thesis, Imperial College, University of London, England, 1979. [11] Meng FL, McKenty F, Camarero R. Radiative heat transfer by the discrete transfer method using an unstructured mesh. ASME HTD 1993;2044:55–6. [12] Malalasekera WMG, James EH. Radiative heat transfer calculations in three-dimensional complex geometries. J Heat Transfer 118:225–8. [13] Malalasekera WMG, Lockwood FC. Computer simulation of the king’s cross fire: effect of radiative heat transfer on fire spread. Proc Inst Mech Eng 1991;205:201–8. [14] Patankar SV. Numerical heat transfer and fluid flow. Washingtom, DC: Hemisphere Publishing; 1980.