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## DISCRETE ORDINATES AND FINITE VOLUME METHODS

Following from: Radiation transfer theory and the computational methods

The discrete ordinates method (DOM) was originally developed in the astrophysics community to study radiation transfer in stellar and planetary atmospheres (Chandrasekhar, 1950). Later on, it was employed in nuclear engineering to study neutron transport (Lee, 1962; Lathrop, 1966; Carlson and Lathrop, 1968). Despite a few early works on 1D heat transfer problems published between 1965 and 1975 (see Modest, 2003), the DOM only began to be used by the heat transfer community after the works of Hyde and Truelove (1977), Truelove (1987, 1988), Fiveland (1982, 1984, 1987, 1988), and Jamaluddin and Smith (1988a,b).

The finite volume method (FVM), a popular approach for the solution of partial differential equations, widely employed in computational fluid dynamics (see, e.g., Patankar, 1980), was first proposed by Raithby and Chui (1990) for the solution of the radiative transfer equation, and further developed in Chui and Raithby (1992, 1993) and Chui et al. (1992, 1993). The original formulation of the FVM might seem quite different from the DOM, which had only been developed for Cartesian or cylindrical enclosures, due to the use of irregular hexahedral control volumes and a new spatial discretization scheme. However, the FVM formulation reported in Chai et al. (1994) for Cartesian coordinates and based on a simpler discretization scheme revealed that the DOM and FVM share many features, and the differences between them are small (see article “Mathematical formulation”). Chai et al. (1995) extended the FVM to irregular geometries, and the DOM was also extended to irregular enclosures (Liu et al., 1997; Sakami et al., 1998), further illuminating the similarities between the two methods. The mathematical formulation of the DOM and FVM is described in the article “Mathematical formulation.” Both methods require the evaluation of the radiation intensity at the boundaries of the control volumes, based on the radiation intensities at the grid nodes. Methods available to perform this evaluation are described in the article “Spatial discretization schemes.” The article “Angular discretization methods” describes methods to select the ordinates and weights employed in the DOM. The boundary conditions required to close the system of discretized governing equations are discussed in the article “Boundary conditions,” while numerical methods available to solve that system of equations are addressed in “Solution algorithm.” The designation “discrete ordinates” in the DOM refers to the angular discretization. Different methods can be used to perform the spatial discretization. The FVM is commonly used for this purpose, but other methods may be employed, and these are addressed in the article “Alternative formulations.” In contrast, the FVM generally relies on the FVM for both spatial and angular discretization.

Nowadays, the DOM and the FVM are among the most widely used methods for the solution of the radiative transfer problems, since they provide a good compromise of accuracy, flexibility, and computational economy. Nevertheless, they suffer from some drawbacks, namely, ray effects, which are due to the angular discretization, and false scattering, also referred to as false diffusion, numerical scattering, numerical diffusion, or numerical smearing, which is due to spatial discretization. These problems are described in the article “Ray effects and false scattering.” A successful remedy to mitigate ray effects is the modified DOM, which is described in the article “Modified discrete ordinates and finite volume methods.” A similar modification may be applied to the FVM, as mentioned in the same article.

The solution of the radiative transfer equation by means of the DOM or FVM may be a time-consuming task, particularly if the number of grid nodes and angular directions is large, and the medium is nongray. In addition, in the case of combined heat transfer modes or combined fluid flow/heat transfer problems, the radiative transfer equation often needs to be solved many times during a global iterative procedure. These computationally intensive calculations are often performed using parallel computers or clusters of workstations or PCs. To take full advantage of the computer power available, the radiation solver needs to be parallelized. This is the subject of the article “Parallel implementation.”

#### REFERENCES

Carlson, B. G. and Lathrop, K. D., Transport Theory–The Method of Discrete Ordinates, Computing Methods in Reactor Physics, H. Greenspan, C. N. Kelber, and D. Okrent, Eds., New York: Gordon & Breach, 1968.

Chai, J. C., Lee, H. S., and Patankar, S. V., Finite Volume Method for Radiation Heat Transfer, J. Thermophys. Heat Transfer, vol. 8, no. 3, pp. 419−425, 1994.

Chai, J. C., Parthasarathy, G., Lee, H. S., and Patankar, S. V., Finite Volume Radiative Heat Transfer Procedure for Irregular Geometries, J. Thermophys. Heat Transfer, vol. 9, no. 3, pp. 410−415, 1995.

Chandrasekhar, S., Radiative Transfer, London: Oxford University Press, 1950.

Chui, E. H., Hughes, P. M. J., and Raithby, G. D., Implementation of the Finite Volume Method for Calculating Radiative Transfer in a Pulverized Fuel Flame, Combust. Sci. Technol., vol. 92, pp. 225−242, 1993.

Chui, E. H. and Raithby, G. D., Implicit Solution Scheme to Improve Convergence Rate in Radiative Transfer Problems, Numer. Heat Transfer, Part B, vol. 22, pp. 251−272, 1992.

Chui, E. H. and Raithby, G. D., Computation of Radiant Heat Transfer on a Nonorthogomal Mesh Using the Finite-Volume Method, Numer. Heat Transfer, Part B, vol. 23, pp. 269−288, 1993.

Chui, E. H., Raithby, G. D., and Hughes, P. M. J., Prediction of Radiative Transfer in Cylindrical Enclosures with the Finite Volume Method, J. Thermophys. Heat Transfer, vol. 6, no. 4, pp. 605−611, 1992.

Fiveland, W. A., A Discrete Ordinates Method for Predicting Radiative Heat Transfer in Axisymmetric Enclosures, ASME Paper No. 82−HT−20, 1982.

Fiveland, W. A., Discrete-Ordinates Solutions of the Radiative Transport Equation for Rectangular Enclosures, J. Heat Transfer, vol. 106, pp. 699−706, 1984.

Fiveland, W. A., Discrete Ordinate Methods for Radiative Heat Transfer in Isotropically and Anisotropically Scattering Media, J. Heat Transfer, vol. 109, pp. 809−812, 1987.

Fiveland, W. A., Three-Dimensional Radiative Heat-Transfer Solutions by the Discrete-Ordinates Method, J. Thermophys. Heat Transfer, vol. 2, no. 4, pp. 309−316, 1988.

Hyde, D. J. and Truelove, J. S., The Discrete Ordinates Approximation for Multidimensional Radiant Heat Transfer in Furnaces, UKAEA Report No. AERE−R8502, AERE Harwell, Oxfordshire, 1977.

Jamaluddin, A. S. and Smith, P. J., Predicting Radiative Transfer in Rectangular Enclosures Using the Discrete Ordinates Method, Combust. Sci. Technol., vol. 59, pp. 321−340, 1988a.

Jamaluddin, A. S. and Smith, P. J., Predicting Radiative Transfer in Axisymmetric Cylindrical Enclosures Using the Discrete Ordinates Method, Combust. Sci. Technol., vol. 62, pp. 173−186, 1988b.

Lathrop, K. D., Use of Discrete-Ordinate Methods for Solution of Photon Transport Problems, Nucl. Sci. Eng., vol. 24, pp. 381−388, 1966.

Lee, C. E., The Discrete Sn Approximation to Transport Theory, Technical Information Series Report No. LA 2595, Lawrence Livermore Laboratory, 1962.

Liu, J., Shang, H. M., Chen, Y. S., and Wang, T. S., Prediction of Radiative Transfer in General Body-Fitted Coordinates, Numer. Heat Transfer, Part B, vol. 31, pp. 423−439, 1997.

Patankar, S. V., Numerical Heat Transfer and Fluid Flow, New York: McGraw-Hill, 1980.

Raithby, G. D. and Chui, E. H., A Finite Volume Method for Predicting a Radiant Heat Transfer in Enclosures with Participating Media, J. Heat Transfer, vol. 112, pp. 415−423, 1990.

Sakami, M., Charrete, A., and Le Dez, V., Radiative Heat Transfer in Three-Dimensional Enclosures of Complex Geometry by Using the Discrete-Ordinates Method, J. Quant. Spectrosc. Radiat. Transfer, vol. 59, pp. 117−136, 1998.

Truelove, J. S., Discrete-Ordinates Solutions of the Radiation Transport Equation, J. Heat Transfer, vol. 109, pp. 1048−1051, 1987.

Truelove, J. S., Three-Dimensional Radiation in Absorbing-Emitting-Scattering Media Using the Discrete-Ordinates Method, J. Quant. Spectrosc. Radiat. Transfer, vol. 39, no. 1, pp. 27−31, 1988.

#### Referencias

1. Carlson, B. G. and Lathrop, K. D., Transport Theory–The Method of Discrete Ordinates, Computing Methods in Reactor Physics, H. Greenspan, C. N. Kelber, and D. Okrent, Eds., New York: Gordon & Breach, 1968.
2. Chai, J. C., Lee, H. S., and Patankar, S. V., Finite Volume Method for Radiation Heat Transfer, J. Thermophys. Heat Transfer, vol. 8, no. 3, pp. 419−425, 1994.
3. Chai, J. C., Parthasarathy, G., Lee, H. S., and Patankar, S. V., Finite Volume Radiative Heat Transfer Procedure for Irregular Geometries, J. Thermophys. Heat Transfer, vol. 9, no. 3, pp. 410−415, 1995.
4. Chandrasekhar, S., Radiative Transfer, London: Oxford University Press, 1950.
5. Chui, E. H., Hughes, P. M. J., and Raithby, G. D., Implementation of the Finite Volume Method for Calculating Radiative Transfer in a Pulverized Fuel Flame, Combust. Sci. Technol., vol. 92, pp. 225−242, 1993.
6. Chui, E. H. and Raithby, G. D., Implicit Solution Scheme to Improve Convergence Rate in Radiative Transfer Problems, Numer. Heat Transfer, Part B, vol. 22, pp. 251−272, 1992.
7. Chui, E. H. and Raithby, G. D., Computation of Radiant Heat Transfer on a Nonorthogomal Mesh Using the Finite-Volume Method, Numer. Heat Transfer, Part B, vol. 23, pp. 269−288, 1993.
8. Chui, E. H., Raithby, G. D., and Hughes, P. M. J., Prediction of Radiative Transfer in Cylindrical Enclosures with the Finite Volume Method, J. Thermophys. Heat Transfer, vol. 6, no. 4, pp. 605−611, 1992.
9. Fiveland, W. A., A Discrete Ordinates Method for Predicting Radiative Heat Transfer in Axisymmetric Enclosures, ASME Paper No. 82−HT−20, 1982.
10. Fiveland, W. A., Discrete-Ordinates Solutions of the Radiative Transport Equation for Rectangular Enclosures, J. Heat Transfer, vol. 106, pp. 699−706, 1984.
11. Fiveland, W. A., Discrete Ordinate Methods for Radiative Heat Transfer in Isotropically and Anisotropically Scattering Media, J. Heat Transfer, vol. 109, pp. 809−812, 1987.
12. Fiveland, W. A., Three-Dimensional Radiative Heat-Transfer Solutions by the Discrete-Ordinates Method, J. Thermophys. Heat Transfer, vol. 2, no. 4, pp. 309−316, 1988.
13. Hyde, D. J. and Truelove, J. S., The Discrete Ordinates Approximation for Multidimensional Radiant Heat Transfer in Furnaces, UKAEA Report No. AERE−R8502, AERE Harwell, Oxfordshire, 1977.
14. Jamaluddin, A. S. and Smith, P. J., Predicting Radiative Transfer in Rectangular Enclosures Using the Discrete Ordinates Method, Combust. Sci. Technol., vol. 59, pp. 321−340, 1988a.
15. Jamaluddin, A. S. and Smith, P. J., Predicting Radiative Transfer in Axisymmetric Cylindrical Enclosures Using the Discrete Ordinates Method, Combust. Sci. Technol., vol. 62, pp. 173−186, 1988b.
16. Lathrop, K. D., Use of Discrete-Ordinate Methods for Solution of Photon Transport Problems, Nucl. Sci. Eng., vol. 24, pp. 381−388, 1966.
17. Lee, C. E., The Discrete Sn Approximation to Transport Theory, Technical Information Series Report No. LA 2595, Lawrence Livermore Laboratory, 1962.
18. Liu, J., Shang, H. M., Chen, Y. S., and Wang, T. S., Prediction of Radiative Transfer in General Body-Fitted Coordinates, Numer. Heat Transfer, Part B, vol. 31, pp. 423−439, 1997.
19. Modest, M. F., Radiative Heat Transfer, New York: Academic Press, 2003.
20. Patankar, S. V., Numerical Heat Transfer and Fluid Flow, New York: McGraw-Hill, 1980.
21. Raithby, G. D. and Chui, E. H., A Finite Volume Method for Predicting a Radiant Heat Transfer in Enclosures with Participating Media, J. Heat Transfer, vol. 112, pp. 415−423, 1990.
22. Sakami, M., Charrete, A., and Le Dez, V., Radiative Heat Transfer in Three-Dimensional Enclosures of Complex Geometry by Using the Discrete-Ordinates Method, J. Quant. Spectrosc. Radiat. Transfer, vol. 59, pp. 117−136, 1998.
23. Truelove, J. S., Discrete-Ordinates Solutions of the Radiation Transport Equation, J. Heat Transfer, vol. 109, pp. 1048−1051, 1987.
24. Truelove, J. S., Three-Dimensional Radiation in Absorbing-Emitting-Scattering Media Using the Discrete-Ordinates Method, J. Quant. Spectrosc. Radiat. Transfer, vol. 39, no. 1, pp. 27−31, 1988.