Discrete ordinate approximation

The / -approximation is more difficult to solve in multidimensional geometries. It yields improved predictions over those obtained from the Pi-approximation, and its accuracy is comparable to the similar-order discrete ordinates approximation (S4), which is computationally more efficient. For one-dimensional systems, it is possible to develop higher-order PN approximations [51,52] however, for multidimensional geometries, even the P5-approximation is extremely complicated. 

The discrete ordinates (DO) approximation is also a multiflux model. The discrete ordinates approximation was originally suggested by Chandrasekhar [19] for astrophysical applications, and a detailed derivation of the related equations was discussed by several researchers for application to neutron transport problems [33, 57-61], During the last two decades the method has been applied to various heat transfer problems [62-81]. 

J. S. Truelove, Three-Dimensional Radiation in Absorbing-Emitting-Scattering Media Using the Discrete-Ordinates Approximation, Journal of Quantitative Spectroscopy and Radiative Transfer, 39, pp. 27-31,1988. 

Another two-constant approach, based on a discrete ordinate approximation of the radiation transfer equation [17,40] was recently applied to describe the diffuse reflectance in the NIR [41,42]. In this... 

Unlike the Mie theory, the discrete ordinate approximation assumes that the sample is continuous matrix of points that absorb and scatter. The values for the coefficients obtained from the discrete ordinate approximation will also change and deviate substantially from the Mie coefficients as the particle size becomes large. This has been called the hidden mass effect [42]. 

Because of the limitations inherent in representing a real sample as a continuum, the more sophisticated radiative transfer models, such as the discrete ordinate approximation or the diffusion approximation, hold little hope for obtaining for a better understanding of the effects occurring in diffuse reflection spectrometry for the general case. 

The radiative source term is a discretized formulation of the net radiant absorption for each volume zone which may be incorporated as a source term into numerical approximations for the generalized energy equation. As such, it permits formulation of energy balances on each zone that may include conductive and convective heat transfer. For K—> 0, GS —> 0, and GG —> 0 leading to S —> On. When K 0 and S = 0N, the gas is said to be in a state of radiative equilibrium. In the notation usually associated with the discrete ordinate (DO) and finite volume (FV) methods, see Modest (op. cit., Chap. 16), one would write S /V, = K[G - 4- g] = Here H. = G/4 is the average flux... 

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. 

In order to take advantage of moment methods, the radiation intensity is expressed as a series of products of angular and spatial functions. If the angular dependence is expressed using a simple power series, the moment method (MM) is obtained if spherical harmonics are employed to express the intensity, then the method is called the spherical harmonics (SH) approximation. In principle, the first-order moment and the first-order spherical harmonics approximations are identical to each other, as well as to the lowest order discrete ordinates (DO) approximation [34]. 

S2 and S4 Models for Cylindrical Geometries. For the solution of the RTE in cylindrical media, the formulations for the S2 and S4 discrete ordinates (DO) approximations (based on Ref. 69) will be presented here. Note that, in a more recent study, Jendoubi et al. [75] used a similar DO approximation in cylindrical geometry and evaluated the effect of anisotropic phase function on the accuracy of the model. 

The approximate methods used frequently for criticality calculations are classified into several broad categories Monte Carlo, discrete ordinates, integral transport, and diffusion theory. Within each category there exist several computer programs, each with a different treatment of spatial detail and energy group structure. Several of these methods are topics for later papers in this session. In the remainder of this paper we will describe some general aspects of each method and indicate the types of situations where each is used. 

The discrete ordinates method is obtained from the transport equation by approximating the neutron angular distribution in terms of a set of discrete angular ordinates and an appropriate interpolation formula between ordinates. The resulting set of coupled linear equations can be solved rapidly and accurately. Combined with the mulUgroup approximation, the discrete ordinates method is applicable to a wide class of problems which can be described by simple one- or two-dimensional geometry. 

The development of the transport-diffusion al rithm consists of mathematical operations on the transport equation to obtain a relationship between the net current and flux gradient, formially designated as the. transport diffusion coefficient. The procedure is similar to that of Pomraning with the noteworthy differences that (a) no analytical approximations are required, and, (b) the analytical formulation is constrained to yield a computational algorithm that is consistent with those einpipyed in 2-D discrete ordinates and diffusion theory codes e.g., TWOTRAN (Ref. 2) and 2-DB (Ref. 3). 

There are three broad categories of deterministic whole reactor calculation methods currently in use (a) diffusion theory (b) spherical harmonics approximations, or Pn methods, and (c) discrete ordinates or Sn methods. Diffusion theory is widely used for reactor core calculations although the faster Pn and Sn methods developed in recent years are now replacing them. 

Finite-difference discrete ordinate codes provide the most rigorous solution of the transport equation. However, truly 3D finite-difference Sn methods are still too inefficient to be used as routine tools for fast reactor analysis. Therefore, nodal transport methods have been developed intensively in recent years, because these methods, although approximate, provide the... 

An analytic solution to Eq. (9.1.24) has been obtained by the method of discrete ordinates (Samuelson, 1983), which is a generalization of the two-stream approximation considered in this chapter. The solution is then used to obtain expressions for the flux from Eq. (9.1.18) and the Planck intensity from Eq. (9.1.17). The upper boundary condition... 

Figure 4.26. Electrophoretic mobility as a function of the potential at the slip plane, after O Brien and White (1978). Smooth, dielectric spherical particles (1-1) electrolyte. Conduction behind the slip plane neglected has its classical meaning as the potential of a discrete slip plane - - approximation (4.6.44] for high Ka. (Redrawn from R.W. O Brien, R.J. Hunter, loc. cit. (In the original the factor 3/2 on the ordinate is missing.).)...

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