Skip to main content
SpringerLink
Log in

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

  • Published:
Applications of Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. L. Adams, E. W. Larsen: Fast iterative methods for discrete-ordinates particle transport calculations. Progr. Nucl. Energy 40 (2002), 3–159.

    Article  Google Scholar 

  2. V. Agoshkov: Boundary Value Problems for Transport Equations. Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, 1998.

    Book  MATH  Google Scholar 

  3. Z. Alta¸c, M. Tekkalmaz: Benchmark solutions of radiative transfer equation for three-dimensional rectangular homogeneous media. J. Quant. Spect. Rad. Transfer 109 (2008), 587–607.

    Article  Google Scholar 

  4. Z. Alta¸c, M. Tekkalmaz: Exact solution of radiative transfer equation for threedimensional rectangular, linearly scattering medium. J. Thermophys. Heat Transf. 25 (2011), 228–238.

    Article  Google Scholar 

  5. K. Atkinson, G. Chandler: The collocation method for solving the radiosity equation for unoccluded surfaces. J. Integral Equations Appl. 10 (1998), 253–290.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  7. R. A. Bia lecki, L. Grela: Application of the boundary element method in radiation. Mech. Teor. Stosow. 36 (1998), 347–364.

    Google Scholar 

  8. J. Blobner, R. A. Bia lecki, 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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  10. M. F. Cohen, J. R. Wallace: Radiosity and Realistic Image Synthesis. Academic Press Professional, Boston, 1993.

    MATH  Google Scholar 

  11. A. L. Crosbie, R. G. Schrenker: Exact expressions for radiative transfer in a threedimensioanl rectangular geometry. J. Quant. Spect. Rad. Transfer 28 (1982), 507–526.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  13. U. Eberwien, C. Duenser, W. Moser: Efficient calculation of internal results in 2D elasticity BEM. Eng. Anal. Bound. Elem. 29 (2005), 447–453.

    Article  MATH  Google Scholar 

  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.

    Google Scholar 

  15. O. Hansen: The local behavior of the solution of the radiosity equation at the vertices of polyhedral domains in R3. SIAM J. Math. Anal. 33 (2001), 718–750.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. R. Howell, M. P. Mengüc, R. Siegel: Thermal Radiation Heat Transfer. CRC Press, Boca Raton, 2010.

    Book  Google Scholar 

  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.

    Article  Google Scholar 

  18. R. Kress: Linear Integral Equations. Applied Mathematical Sciences 82, Springer, New York, 2014.

    Book  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  21. W. M. Malalasekera, E. H. James: Radiative heat transfer calculations in threedimensional complex geometries. ASME J. Heat Transfer 118 (1996), 225–228.

    Article  Google Scholar 

  22. M. F. Modest: Radiative Heat Transfer. Academic Press, Oxford, 2013.

    Book  Google Scholar 

  23. N. A. Qatanani, A. Daraghmeh: Asymptotic error analysis for the heat radiation boundary integral equation. Eur. J. Math. Sci. 2 (2013), 51–61.

    Google Scholar 

  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.

    Article  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  27. T. Tiihonen: Stefan-Boltzmann radiation on non-convex surfaces. Math. Methods Appl. Sci. 20 (1997), 47–57.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  29. R. Viskanta: Radiation transfer and interaction of convection with radiation heat transfer. Adv. Heat Transfer 3 (1966), 175–251.

    Article  MATH  Google Scholar 

  30. A. Watt: Fundamentals of Three-Dimensional Computer Graphics. Addison-Wesley Publishing Company, Wokingham, 1989.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xian, Shaanxi, 710129, China

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

Authors
  1. Yao-Chuang Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu-Feng Nie
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhan-Bin Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yao-Chuang Han.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Han, YC., Nie, YF. & Yuan, ZB. Mathematical and numerical analysis of radiative heat transfer in semi-transparent media. Appl Math 64, 75–100 (2019). https://doi.org/10.21136/AM.2019.0276-17

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/AM.2019.0276-17

Keywords

MSC 2010

Search

Navigation