Skip to main content
SpringerLink
Log in

Discrete ordinates method applied to radiative transfer equation in complex geometries meshed by structured and unstructured grids

  • Technical Paper
  • Published:
Journal of the Brazilian Society of Mechanical Sciences and Engineering Aims and scope Submit manuscript

Abstract

In this paper, discrete ordinates method is used for solving the 2-D radiative transfer equation (RTE). To consider complex 2-D geometries, Cartesian and unstructured grids are used. Geometries with straight edges, inclined and curvilinear boundaries are considered. A participating medium which absorbs and emits radiation is considered. Block off and embedded boundary procedures are used to mesh the complex geometry with the Cartesian grid. For further precision, unstructured grid is used to deal with the complexity of the boundaries. This paper also aims at solving temperature fields for discontinuous heat loads in radiative equilibrium problems. It is shown that the incorporation of the unstructured grid in solving RTE is suitable for all geometries with simple, inclined and curvilinear boundaries.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Abbreviations

A x,y,z :

Area of the control volume in x, y, z direction, respectively

D mj :

Scalar product of mth discrete ordinate \(\vec{s}_{m}\) and outward wall normal vector \(\vec{n}_{j}\)

f :

Interpolation factor for inlet and outlet intensity in the control volume

I :

Radiation intensity

I j :

Radiation intensity at jth face of cell

I P,m :

Radiation intensity in mth direction and the cell center (P)

K :

Absorption coefficient

L w i,j :

Embedded boundary segment

M :

Large number 1020

Q :

Heat released within the medium by chemical reaction

S m :

Block off source function in mth direction

S P :

Block off linear slope source

T :

Temperature

V :

Volume of the cell

α :

Spatial differencing factor for unstructured

β :

Extinction coefficient

Δx or Δy or Δz :

Depends on normal direction of boundary surface

ε w :

Wall emissivity

ζ :

Direction cosine in x, y, z direction

η :

Direction cosine for in y direction

μ :

Direction cosine in x direction

ξ :

Direction cosine in z direction

σ s :

Scattering coefficient

Φ mj :

Scattering phase function

ψ :

Fraction of self-absorbed heat

ω :

Scattering albedo

ω j :

Weight coefficient of direction j

Ω:

Spherical angle

References

  1. Siegel R, Howell JR (2002) Thermal radiation heat transfer, 4th edn. Taylor & Francis, London

    Google Scholar 

  2. Hottel HC, Cohen ES (1958) Radiant heat exchange in a gas-filled enclosure: allowance for nonuniformity of gas temperature. Am Inst Chem Eng J 4:3–14

    Article  Google Scholar 

  3. Howell JR, Perlmutter M (1961) The calculation of nonlinear radiation transport by a Monte Carlo method: statistical physics. Method Comput Phys 1:43–65

    Google Scholar 

  4. Shah NG (1979) New method of computation of radiation heat transfer combustion chambers. PhD Thesis, Imperial College, University of London

  5. Chandrasekhar S (1960) Radiative transfer. Dover Publications, New York

    Google Scholar 

  6. Raithby GD, Chui EHA (1990) Finite-volume method for predicting a radiant heat transfer in enclosures with participating media. ASME J Heat Transf 112:415–423

    Article  Google Scholar 

  7. Thynell ST (1998) Discrete-ordinates method in radiative heat transfer. Int J Eng Sci 36(12–14):1651–1675

    Article  MATH  Google Scholar 

  8. Fiveland WA (1984) Discrete ordinates solutions of the radiative transport equation for rectangular enclosures. J Heat Transf 106(4):699–706

    Article  Google Scholar 

  9. Fiveland WA (1988) Three dimensional radiative heat transfer solution by the discrete ordinates method. J Thermophys Heat Transf 2(4):309–316

    Article  Google Scholar 

  10. Jamaluddin AS, Smith PJ (1988) Predicting radiative heat transfer in axisymmetrical cylindrical enclosures using the discrete ordinates method. Combust Sci Technol 62(4–6):173–186

    Article  Google Scholar 

  11. Jamaluddin AS, Smith PJ (1992) Discrete ordinates solution of radiative transfer equation in nonaxisymmetric cylindrical enclosures. J Thermophys Heat Transf 6(2):242–245

    Article  Google Scholar 

  12. Tsai JR, Ozisik MN (1990) Radiation in cylindrical symmetry with anisotropic scattering and variable properties. Int J Heat Mass Transf 33(12):2651–2658

    Article  Google Scholar 

  13. Chai JC, Lee HS, Patankar SV (1994) Treatment of irregular geometries using a cartesian coordinates finite-volume radiation heat transfer procedure. Numer Heat Transf Part B 26(2):225–235

    Article  Google Scholar 

  14. Talukdar P (2006) Radiative heat transfer for irregular geometries with the collapsed dimension method. Int J Therm Sci 45(2):103–109

    Article  Google Scholar 

  15. Talukdar P (2006) Discrete transfer method with the concept of blocked-off region for irregular geometries. J Quant Spectrosc Radiat Transf 98(2):238–248

    Article  MathSciNet  Google Scholar 

  16. Kamel G, Naceur BM, Rachid M, Rachid S (2006) Formulation and testing of the FTn finite volume method for radiation in 3-D complex inhomogeneous participating media. J Quant Spectrosc Radiat Transf 98(3):425–445

    Article  Google Scholar 

  17. Howell LH, Beckner VE (1997) Discrete ordinates algorithm for domains with embedded boundaries. AIAA J Thermophys Heat Transf 11(4):549–555

    Article  Google Scholar 

  18. Byun DY, Baek SW, Kim MY (2003) Investigation of radiative heat transfer in complex geometries using block-off, multi block, and embedded boundary treatments. Numer Heat Transf Part A 43(8):807–825

    Article  Google Scholar 

  19. Sun YS, Li BW (2012) Prediction of radiative heat transfer in 2D irregular geometries using the collocation spectral method based on body-fitted coordinates. J Quant Spectrosc Radiat Transf 113(17):2205–2212

    Article  Google Scholar 

  20. Thakur S, Shyy W, Udaykumar HS, Hill L (1998) Multi-block interface treatments in a pressure-based flow solver. Numer Heat Transf B 33(4):367–396

    Article  Google Scholar 

  21. Sakami M, Charette A, Le Dez V (1996) Application of the discrete ordinates method to combined conductive and radiative heat transfer in a two-dimensional complex geometry. J Quant Spectrosc Radiat Transf 56(4):517–533

    Article  Google Scholar 

  22. Sakami M, Charette A (1998) A new differencing scheme for the discrete ordinates method in complex geometries. Revue Générale de Thermique 37(6):440–449

    Article  Google Scholar 

  23. Sakami M, Charette A (2000) Application of a modified discrete ordinates method to two-dimensional enclosures of irregular geometries. J Quant Spectrosc Radiat Transf 64(3):275–298

    Article  Google Scholar 

  24. Liu J, Shang HM, Chen YS, Wang TS (2000) Development of an unstructured radiation model applicable for two dimensional planar, axisymmetric and 3-dimensional geometries. J Quant Spectrosc Radiat Transf 66(1):17–33

    Article  Google Scholar 

  25. Chai JC, Lee HS, Patankar SV (1993) Treatment of irregular geometries using a Cartesian coordinates control angle control volume based discrete ordinate method. In: Proceedings of the international heat transfer conference. American Society of Mechanical Engineers, Atlanta, pp 35–43

  26. Sanchez A, Smith TF (1992) Surface radiation exchange for two-dimensional rectangular enclosures using the discrete-ordinates method. J Heat Transf 114(2):465–472

    Article  Google Scholar 

  27. Ströhle J, Schnell U, Hein KRG (2001) A mean flux discrete ordinates interpolation scheme for general coordinates. In: 3rd International conference on heat transfer, Antalya

  28. Thurgood CP, Pollard A, Becker HA (1995) The T N quadrature set for the discrete ordinates method. J Heat Transf 117:1068–1070

    Article  Google Scholar 

  29. Zhang Y, Yi HL, Tan HP (2013) Natural element method for radiative heat transfer in two-dimensional semitransparent medium. Int J Heat Mass Transf 56(1–2):411–423

    Article  Google Scholar 

  30. Zhang Y, Yi HL, Tan HP (2013) Natural element method for radiative heat transfer in a semitransparent medium with irregular geometries. J Comput Phys 241:18–34

    Article  Google Scholar 

  31. Zhou HC, Ai YH (2006) Effect of radiative transfer of heat released from combustion reaction on temperature distribution: a numerical study for 2-D system. J Quant Spectrosc Radiat Transf 101(1):109–118

    Article  Google Scholar 

  32. Aghanajafi C, Abjadpour A (2013) Solution to radiative transfer equation in a 3-D rectangular enclosure due to discontinuous heat flow divergence. Meccanica 48(5):1127–1138

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran

    C. Aghanajafi & A. Abjadpour

Authors
  1. C. Aghanajafi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Abjadpour
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Abjadpour.

Additional information

Technical Editor: Francis HR Franca.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aghanajafi, C., Abjadpour, A. Discrete ordinates method applied to radiative transfer equation in complex geometries meshed by structured and unstructured grids. J Braz. Soc. Mech. Sci. Eng. 38, 1007–1019 (2016). https://doi.org/10.1007/s40430-015-0397-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40430-015-0397-2

Keywords

Search

Navigation