# Institute of Mathematics

## Mathematical and numerical analysis of radiative heat transfer in semi-transparent media

#### Yao-Chuang Han, Yu-Feng Nie, Zhan-Bin Yuan

###### Received October 22, 2017.   Published online January 8, 2019.

Abstract:  This paper is concerned with mathematical and numerical analysis of the system of radiative integral transfer equations. The existence and uniqueness of solution to the integral system is proved by establishing the boundedness of the radiative integral operators and proving the invertibility of the operator matrix associated with the system. A collocation-boundary element method is developed to discretize the differential-integral system. For the non-convex geometries, an element-subdivision algorithm is developed to handle the computation of the integrals containing the visibility factor. An efficient iterative algorithm is proposed to solve the nonlinear discrete system and its convergence is also established. Numerical experiment results are also presented to verify the effectiveness and accuracy of the proposed method and algorithm.
Keywords:  radiative heat transfer; existence and uniqueness; collocation-boundary element method; shadow detection; iterative nonlinear solver
Classification MSC:  65M38, 45K05, 80A20, 47G10
DOI:  10.21136/AM.2019.0276-17

References:
[1] M. L. Adams, E. W. Larsen: Fast iterative methods for discrete-ordinates particle transport calculations. Progr. Nucl. Energy 40 (2002), 3-159. DOI 10.1016/S0149-1970(01)00023-3
[2] V. Agoshkov: Boundary Value Problems for Transport Equations. Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston (1998). DOI 10.1007/978-1-4612-1994-1 | MR 1638817 | Zbl 0914.35001
[3] Z. Altaç, M. Tekkalmaz: Benchmark solutions of radiative transfer equation for three-dimensional rectangular homogeneous media. J. Quant. Spect. Rad. Transfer 109 (2008), 587-607. DOI 10.1016/j.jqsrt.2007.07.016
[4] Z. Altaç, M. Tekkalmaz: Exact solution of radiative transfer equation for three-dimensional rectangular, linearly scattering medium. J. Thermophys. Heat Transf. 25 (2011), 228-238. DOI 10.2514/1.50910
[5] K. Atkinson, G. Chandler: The collocation method for solving the radiosity equation for unoccluded surfaces. J. Integral Equations Appl. 10 (1998), 253-290. DOI 10.1216/jiea/1181074231 | MR 1656533 | Zbl 0914.65137
[6] K. Atkinson, D. D.-K. Chien, J. Seol: Numerical analysis of the radiosity equation using the collocation method. ETNA, Electron. Trans. Numer. Anal. 11 (2000), 94-120. MR 1799026 | Zbl 0961.65118
[7] R. A. Białecki, Ł. Grela: Application of the boundary element method in radiation. Mech. Teor. Stosow. 36 (1998), 347-364. Zbl 0934.74076
[8] J. Blobner, R. A. Białecki, G. Kuhn: Boundary-element solution of coupled heat conduction-radiation problems in the presence of shadow zones. Numer. Heat Transfer, Part B 39 (2001), 451-478. DOI 10.1080/104077901750188840
[9] S.-S. Chen, B.-W. Li, X.-Y. Tian: Chebyshev collocation spectral domain decomposition method for coupled conductive and radiative heat transfer in a 3D L-shaped enclosure. Numer. Heat Transfer, Part B 70 (2016), 215-232. DOI 10.1080/10407790.2016.1193398
[10] M. F. Cohen, J. R. Wallace: Radiosity and Realistic Image Synthesis. Academic Press Professional, Boston (1993). Zbl 0814.68138
[11] A. L. Crosbie, R. G. Schrenker: Exact expressions for radiative transfer in a three-dimensioanl rectangular geometry. J. Quant. Spect. Rad. Transfer 28 (1982), 507-526. DOI 10.1016/0022-4073(82)90017-6
[12] A. L. Crosbie, R. G. Schrenker: Radiative transfer in a two-dimensional rectangular medium exposed to diffuse radiation. J. Quant. Spect. Rad. Transfer 31 (1984), 339-372. DOI 10.1016/0022-4073(84)90095-5
[13] U. Eberwien, C. Duenser, W. Moser: Efficient calculation of internal results in 2D elasticity BEM. Eng. Anal. Bound. Elem. 29 (2005), 447-453. DOI 10.1016/j.enganabound.2005.01.008 | Zbl 1182.74214
[14] A. F. Emery, O. Johansson, M. Lobo, A. Abrous: A comparative study of methods for computing the diffuse radiation viewfactors for complex structures. J. Heat Transfer 113 (1991), 413-422. DOI 10.1115/1.2910577
[15] O. Hansen: The local behavior of the solution of the radiosity equation at the vertices of polyhedral domains in \$\Bbb R^3\$. SIAM J. Math. Anal. 33 (2001), 718-750. DOI 10.1137/S0036141000378103 | MR 1871418 | Zbl 1001.45002
[16] J. R. Howell, M. P. Mengüç, R. Siegel: Thermal Radiation Heat Transfer. CRC Press, Boca Raton (2010). DOI 10.1201/9781439894552
[17] P.-F. Hsu, Z. Tan: Radiative and combined-mode heat transfer within L-shaped nonhomogeneous and nongray participating media. Numer. Heat Transfer, Part A 31 (1997), 819-835. DOI 10.1080/10407789708914066
[18] R. Kress: Linear Integral Equations. Applied Mathematical Sciences 82, Springer, New York (2014). DOI 10.1007/978-1-4614-9593-2 | MR 3184286 | Zbl 1328.45001
[19] M. T. Laitinen, T. Tiihonen: Integro-differential equation modelling heat transfer in conducting, radiating and semitransparent materials. Math. Methods Appl. Sci. 21 (1998), 375-392. DOI 10.1002/(SICI)1099-1476(19980325)21:5<375::AID-MMA953>3.0.CO;2-U | MR 1608072 | Zbl 0958.80003
[20] B. Q. Li, X. Cui, S. P. Song: The Galerkin boundary element solution for thermal radiation problems. Eng. Anal. Bound. Elem. 28 (2004), 881-892. DOI 10.1016/j.enganabound.2004.01.009 | Zbl 1066.80008
[21] W. M. Malalasekera, E. H. James: Radiative heat transfer calculations in three-dimensional complex geometries. ASME J. Heat Transfer 118 (1996), 225-228. DOI 10.1115/1.2824045
[22] M. F. Modest: Radiative Heat Transfer. Academic Press, Oxford (2013). DOI 10.1016/C2010-0-65874-3
[23] N. A. Qatanani, A. Daraghmeh: Asymptotic error analysis for the heat radiation boundary integral equation. Eur. J. Math. Sci. 2 (2013), 51-61.
[24] B. Sun, D. Zheng, B. Klimpke, B. Yildir: Modified boundary element method for radiative heat transfer analyses in emitting, absorbing and scattering media. Eng. Anal. Bound. Elem. 21 (1998), 93-104. DOI 10.1016/S0955-7997(97)00068-4 | Zbl 0936.80006
[25] Z. Tan: Radiative heat transfer in multidimensional emitting, absorbing, and anisotropic scattering media: mathematical formulation and numerical method. J. Heat Transfer 111 (1989), 141-147. DOI 10.1115/1.3250636
[26] S. T. Thynell: The integral form of the equation of transfer in finite, two-dimensional, cylindrical media. J. Quant. Spect. Rad. Transfer 42 (1989), 117-136. DOI 10.1016/0022-4073(89)90094-0
[27] T. Tiihonen: Stefan-Boltzmann radiation on non-convex surfaces. Math. Methods Appl. Sci. 20 (1997), 47-57. DOI 10.1002/(SICI)1099-1476(19970110)20:1<47::AID-MMA847>3.0.CO;2-B | MR 1429330 | Zbl 0872.35044
[28] D. N. Trivic, C. H. Amon: Modeling the 3-D radiation of anisotropically scattering media by two different numerical methods. Int. J. Heat Mass Transfer 51 (2008), 2711-2732. DOI 10.1016/j.ijheatmasstransfer.2007.10.015 | Zbl 1143.80329
[29] R. Viskanta: Radiation transfer and interaction of convection with radiation heat transfer. Adv. Heat Transfer 3 (1966), 175-251. DOI 10.1016/s0065-2717(08)70052-2 | Zbl 0139.23801
[30] A. Watt: Fundamentals of Three-Dimensional Computer Graphics. Addison-Wesley Publishing Company, Wokingham (1989). Zbl 0702.68099

Affiliations:   Yao-Chuang Han, Yu-Feng Nie, Zhan-Bin Yuan, Department of Applied Mathematics, Northwestern Polytechnical University, Xian, Shaanxi, China, 710129, e-mail: hychuang2013@mail.nwpu.edu.cn, yfnie@mail.nwpu.edu.cn, yzzzb@nwpu.edu.cn

PDF available at: