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Radiation Transfer Calculation Methods

The theory of thermal radiation is very complex and an exact solution, even for reasonably simple situations, is generally impossible. The combustion products and soot, acting as participating media, add further complexity to the situation. Therefore, a wide range of calculation methods and mathematical models with varying levels of complexity and accuracy have been developed.

A number of radiation solution methods exist for solving the equation governing the transfer of thermal radiation (SiegelR:1993). The methods differ in complexity and accuracy of the calculation of view factors and economy of the solution algorithm. The most commonly used radiation solution methods and their key features are summarised below:

Factor Zonal or View Calculation Method

This popular method has been widely used by engineers to estimate the radiative transfer in the absence of detailed knowledge of participating media. The walls and interior of the enclosure are divided into zones of finite size. View factors are defined which are measures of the radiation exchange occurring between pairs of zones. This procedure results in n simultaneous equations for each of the n zones and leads to a system with a n2 x n2 matrix. This method is very accurate but its main drawback is that the view factor, for each geometry, must be worked in advance and for complex geometries the view factors are not available (Hottel and Cohen 1958).

Statistical or Monte-Carlo Method

The purely statistical methods, such as the Monte-Carlo method, usually yield radiation heat transfer predictions as accurate as the exact method. There is no single Monte-Carlo method because there are many different statistical approaches.

The simplest Monte-Carlo method is based on simulating a finite number of photons (discretized energy bundles) histories through the use of a random number generator. For each photon, random numbers are generated and used to sample appropriate probability distributions for scattering angles and path lengths between collisions. As the number of photons initiated from each surface and/or volume element increases, this method is expected to converge to the exact solution of a problem. Thus, this statistical method is numerically precise provided the number of photons is large and the random number generator of the computer good enough.

Furthermore, in contrast to the zonal method, the Monte-Carlo method does also not suffer from the calculation of view factors in advance because the view factors are automatically calculated as the randomly chosen energy release bundles are tracked through the enclosure containing the fire. However, since the directions of photons are obtained from a random number generator of the computer used, the method is always subjected to statistical errors and lack of guaranteed convergence but this would improve as the next generation of computers (with more powerful random number generators) become more readily available (HowellJR:1964).

The Flux (or Multi-Flux) Method

The radiation intensity is a function of the location, the direction of propagation of radiation and of wavelength. Usually the angular dependence of the intensity complicates the problem since all possible directions must be taken into account. It is, therefore desirable to separate the angular (directional) dependence of the radiation intensity from its spatial dependence to simplify the governing radiation transfer equation (RTE). If it is assumed that the intensity is uniform on given intervals of the solid angle, then the RTE can be significantly simplified as the integro-differential RTE would be reduced to a series of coupled linear differential equations in terms of average radiation intensities or fluxes. This procedure yields the flux methods. By changing the number of solid angles over which radiative intensity is assumed constant, one can obtain different flux methods, such as two-flux for one-dimensional geometry, four-flux for two-dimensional geometry or six-flux methods for three-dimensional geometry. The accuracy of flux-methods will increase by increasing the number of fluxes. The six-flux methods have been reasonably successful for fire spread and smoke movement inside compartments (KumarS:1989a); (KumarS:1991). They are not suitable for predicting flame spread over surfaces or flames projecting outside openings, where finer discretization of the solid angle than offered by the six-flux method would be necessary (GosmanAD:1972).

The Discrete Ordinate Method

The discrete-ordinate method (e.g., (ChandrasekharS:1950); (LockwoodFC:1978); (FivelandWA:1982)) was originally suggested in (ChandrasekharS:1950) for astrophysical problems. It is derived by applying discrete-ordinate approximation to the RTE through discretising the entire solid angle (Ω = 4π) using a finite number of ordinate directions and corresponding factors. A simpler version of the method is also called SN-approximation because it is obtained by dividing the spherical space into N equal solid angles. More accurate SN-approximations of the N discrete ordinates are obtained by using Gaussian or Lobatto quadratures and choosing N discrete values of the direction cosines ζn, ηn, μn such that they satisfy the identity ζn2 + ηn2 + μn2 =1.The SN-approximation has been used successfully for two-dimensional cylindrical and rectangular radiative transfer problems with combustion chamber applications, where reasonably accurate results were obtained in comparison to exact solutions (FivelandWA:1982)(FivelandWA:1984). However, the method suffers from the so-called “ray effects”, causing anomalies in the scalar flux distribution (LathropKD:1968)(LathropKD:1971). The ray effects are particularly more pronounced when there are localised radiation sources in the medium (e.g., flame in an enclosure) and radiation is less important in comparison to absorption. Clearly, as scattering increases and radiation field becomes more isotropic, they become less noticeable. However, with increasing scattering and/or optical thickness, the convergence rate may become slow (LewisEE:1984).

The Discrete Transfer Method

The discrete transfer method is a mixture of the Monte-Carlo, zone and flux methods (LockwoodFC:1981). Similar to the zone method, the enclosure is divided into cells and equation is analytically integrated along rays in each cell, but the method is much faster and the calculation of the view factors is an inherent feature of the procedure. The only drawback of the method is that to obtain ray-insensitive solution the method may require more rays than affordable on economy grounds for practical problems (CumberPS:2000).

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Page last modified on February 21, 2009, at 01:36 AM