Discrete ordinate method

Stamnes, K S.-C. Tsay, W. Wiscombe, and K. Jayaweera, Numerically Stable Algorithm for Discrete-Ordinate-Method Radiative Transfer in Multiple Scattering and Emitting Layered Media, Appl. Opt., 27, 2502-2509 (1988). 

Stammes K., Tsay S., W. Wiscombe and K. Jayaweera, Numerical stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media, Applied Optics 27, 2502-... 

Fiveland, W. A. and Jamaluddin, A. S. (1991). Three-Dimensional Spectral Radiative Heat Transfer Solutions by the Discrete-Ordinate Method. J. Thermophysics, 5, 335. 

Collin, A., Boulet, P., Lacroix, D., and Jeandel, G. On radiative transfer in water spray curtains using the discrete ordinates method. Journal of Quantitative Spectroscopy Radiative Transfer, 2005. 92, 85-110. 

In particular, Cabrera et al. (1996), Brandi et al. (1999), and Satuf et al. (2005) have determined optical parameters for Ti02 particles of several commercial brands. The determinations were carried out by means of spectrophotometry experiments involving the measurement of specular reflectance and beam transmittance, as well as hemispherical transmittance and reflectance, of catalyst suspensions (Cabrera et al., 1996). By radiative transfer calculations with the discrete ordinates method (DOM), the values of the extinction and absorption coefficient and of the asymmetry parameter that better fitted the results of measurements were found. Actually, the extinction coefficients of Satuf et al. (2005) are the same as those of Brandi... 

To solve Equations (67)-(69), the Discrete Ordinate Method was applied (Duderstadt and Martin, 1979). From the solution of the RTE, the monochromatic radiation intensity at each point and each direction inside the reactor can be obtained. Considering constant optical properties of the catalyst and steady radiation supply by the emitting system, the radiation field can be considered independent of time. 

The discrete ordinate method (Duderstadt and Martin, 1979) was employed to solve the RTE (Equations 82 and 83). Afterward, the LVRPA was obtained according to... 

The discrete ordinates method in a S4-approximation is used to solve the radiation transport equation. Since the intensity of radiation depends on absorption, emission and scattering characteristics of the medium passed through, a detailed representation of the radiative properties of a gas mixture would be very complex and currently beyond the scope of a 3D-code for the simulation of industrial combustion systems. Thus, contributing to the numerical efficiency, some simplifications are introduced, even at the loss of some accuracy. The absorption coefficient of the gas phase is assumed to have a constant value of 0.2/m. The wall emissivity was set to 0.65 for the ceramic walls and to a value of 0.15 for the glass pane inserted in one side wall for optical access. 

Several analytic methods have been proposed to solve the equation of radiative transfer in an absorbing and scattering atmosphere, but they can only be applied for the most simple cases. To obtain quantitative solutions, numerical methods are generally used, such as the Monte-Carlo method, DART method, iterative Gauss, discrete ordinate method, etc. A complete summary of these techniques is provided by Lenoble (1977), and a detailed discussion of multiple scattering processes in plane parallel atmospheres is given in the book by Liou (2002). 

Hendricks and Howell [253] measured the spectral normal transmittance and normal hemispherical reflectance of three sample thicknesses each of reticulated partially stabilized zirconia and silicon carbide at pore sizes of 10, 20, and 65 ppi. The measurements covered a spectral range of 400-500 nm. They used an inverse discrete ordinates method to find the spectrally dependent absorption and scattering coefficients as well as the constants appropri-... 

W. A. Fiveland, A Discrete-Ordinates Method for Predicting Radiative Heat Transfer in Axisymmetric Enclosures, ASME paper No. 82-HT-20, ASME, New York, 1982. 

W. A. Fiveland, Three-Dimensional Radiative Heat-Transfer Solutions by the Discrete-Ordinates Method, AIAA Journal of Thermophysics and Heat Transfer, 2, pp. 309-316,1988. 

A. S. Jamaluddin and P. J. Smith, Predicting Radiative Transfer in Rectangular Enclosures Using the Discrete Ordinates Method, Combustion Science and Technology, 62, p. 173,1988. 

N. E. Wakil and J. F. Sacadura, Some Improvements of the Discrete Ordinates Method for the Solution of the Radiative Transport Equation in Multidimensional Anisotropically Scattering Media, in HTD vol. 203, pp. 119-127, ASME, New York, 1992. 

W. A. Fiveland and J. P. Jessee, A Finite Element Formulation of the Discrete-Ordinate Method For Multidimensional Geometries, in Radiative Heat Transfer Current Research, ASME HTD no. 244, New York, 1993. 

J. C. Chai and S. V. Patankar, Evaluation of Spatial Differencing Practices for the Discrete Ordinates Method, AlAA Journal of Thermophysics and Heat Transfer, vol. 8, pp. 140-144,1994. 

W. Krebs, S. Wittig, and R. Viskanta, A Parabolic Formulation of the Discrete Ordinates Method for the Treatment of Complex Geometries in M. P. Mengii (ed.). Radiative Transfer—I Proceedings of the First International Symposium on Radiative Transfer, pp. 355-371, Begell House, New York, 1996. 

N. Selfuk and N. Kayakol, Evaluation of Angular Quadrature and Spatial Differencing Schemes for Discrete Ordinates Method in Rectangular Furnaces, in R. D. Skocypec, S. T. Thynell, D. A. Kaminski, A. M. Smith, and T. Tong (eds.), Solution Methods for Radiative Transfer in Participating Media, ASME HTD vol. 325, ASME, New York, 1996. 

N. Selkirk and N. Kayakol, Evaluation of Discrete Ordinates Method for Radiative Transfer in Rectangular Furnaces, International Journal of Heat and Mass Transfer, 40(2), pp. 213-222,1997. 

To study scattering effects by solid particles in a fluid and adapt previous existing methods in generalized transport theory (the discrete ordinate method or DOM) (Duderstadt and Martin, 1979) to solve the RTE (Alfano etai, 1995). 

Expressions (173) and (174) provide efficient formulations for the solution of deep-penetration problems whose geometrical irregularity is in the vicinity of the detector. The unperturbed flux distribution is calculated with a low-order, low-dimensional calculational method (such as a one-dimensional discrete-ordinates method). The irregularity in the geometry is treated as an alteration to the unperturbed system. The 5 or distribution is then obtained from the solution of Eq. (44a) or Eq. (163), respectively, for the local region of the detector and the alteration. High-order and multidimensional calculational methods (such as Monte Carlo) can be used for this local solution. Equation (173) provides an efficient formulation for the calculation of the response of many different detectors (such as different reaction rates and the spatial distribution of a given reaction rate) in a... 

Abe Takashi. 1997. Derivation of the lattice Boltzmann method by means of the discrete ordinate method for the Boltzmann equation. Journal of Computational Physics. 131 (1). 

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