Recently a potentially useful approximate theory for soot optical properties (denoted RDG-FA theory in the following) has been developed based on the Rayleigh-Debye-Gans (RDG) scattering approximation while assuming that soot aggregates are mass-fractal objects. Measurements of both soot structure and scattering properties for soot aggregates found in flame environments have exhibit good agreement between RDG-FA predictions based on the structure measurements, and the scattering measurements^1,2 . Unfortunately, these soot populations involved relatively large soot aggregates so that it was not possible to adequately test RDG-FA theory in the small angle (Guinier) regime where the use of the RDG scattering approximation is least reliable.

In the present study the range of validity of the RDG-FA theory for computing the optical properties of soot aggregates is investigated. RDG-FA was evaluated by more exact predictions from the solution of the volume integral equation formulation of the governing equations, based on the ICP algorithm of Iskander et al³. Numerical simulations were used to construct statistically-significant populations of soot aggregates having appropriate fractal properties, prescribed numbers of primary particles per aggregate, N, and primary particle optical size parameters, x_p. Error contour maps were produced for the total scattering and absorption cross sections as well as differential scattering cross sections for different scattering angles along the Guinier regime. Parameters covered were as follows: number of primary particles within an aggregate up to 256; primary particle size parameter between 0.1 and 1.0 and soot complex refractive index modulus, [m-1], between 0.1 and 1.

In addition, the effects of polydisperse aggregate population study the numbers of primary particles per aggregate were chosen to match a log-normal size distribution function with mean values and standard deviations according to experimental data presented by Köylü and Faeth⁴. Primary particle diameter within an aggregate were randomly chosen while satisfying a normal size distribution function with mean values and standard deviations typical of soot aggregates found in the fuel rich and lean regions of laminar and turbulent diffusion flames⁴.

Specific ranges of aggregate properties for the present study were as follows: primary particle mean optical size parameter between 0.1 and 0.3 (standard deviation up to 25 %), mean number of primary particles per aggregate up to 256, log-normal standard deviation of aggregate size distribution between 1 and 2, mean fractal dimension of 1.75 and refractive indices typical of soot.

The following main conclusion was obtained: while in the large angle (power-law) regime results seem to be independent of polydisperse soot aggregate populations and polydisperse primary particle size parameter, in the Guinier regime a strong influence of these effects was noticed.

Such radiation transfer is dependent on the fiber’s optical characteristics, which are characterized by the material’s complex refractive index andits size parameter, as well as the orientation of the fiber, and therefore, to adequately evaluate it, a model is required to establish the relationship between the optical characteristics and fiber orientation.

Towards this end, several pseudo-continuous models (3)-(6) have been developed to clarify the effects of optical characteristics and orientation on the radiation transfer, yet they have not been applied to media with a size parameter greater than 10, i.e., these media have only been studied to evaluate their thermal insulation characteristics at comparatively low-temperature radiation transfer conditions. In addition, the following also causes the lack of the study for radiation transfer in fibrous media with a large size parameter.

Figure 1 Scattering by a single fiber

Figure 2 Employed rectangular aperture model for representing diffraction in a fiber

In this study, we derive an approximation method for estimating this optical characteristics of a single fiber with such a size parameter by considering (i) the Fraunhofer diffraction of a rectangular aperture to represent diffraction in a fiber (Fig. 2) and (ii) the specular reflection off the surface of a fiber, with the effects of available size parameters for fibers being subsequently clarified by comparing these approximated optical characteristics with exact ones.

In addition, we also use the resultant optical characteristics to estimate the radiative properties of and radiation transfer in two types of planar fibrous media having a typical orientations. Based on our results which cover a wide range of size parameters, we subsequently discuss the effects of the complex refractive index and fiber orientation on the radiative properties and radiation transfer associated with a single fiber.

1. Microwave emissivity of a foam on the water surface both for the emulsive monolayer and the honeycomb structure including the spectral range, in which the wavelength is greater than the foam layer thickness.
2. Absorption and scattering of microwave radiation by metal powder in dielectric at the wavelength much greater than distances between particles.
3. Infrared radiative properties of the quartz fibrous material with numerous contacts between fibers. A comparison of the calculations with the experimental data provides a way of estimating the range in which one may use an assumption that effects of large particle concentration are small. The following results are obtained:

• A microwave thermal radiation of a foam on the water surface may be treated as that of rarefied disperse system of water bubbles. Calculations showed that individual particles are near to the Rayleigh range due to very small thickness of the watershell. A good agreement with the experimental data on spectral emissivity in the wavelength range from 2.6 mm to 80 mm argues for an applicability of the theoretical model without taking into account any collective effects. This conclusion does not hold for observations at small angles to an exposed surface.
• It has been found that the microwave properties of spherical aluminum particles of radius from 1 to 100 m in the spectral range from 1 to 100 mm are specified by the magnetic dipole scattering. Simple analytical expressions for disperse system characteristics by taking into account two first partial waves amplitudes in the Mie solution are derived. It was shown that there is a sharp decrease of the specific absorption coefficient in the centimeter range with the decrease of the particle radius when it is less than 2 m. A comparison with the experimental data for a disperse system with an average particle radius about 7m at the wavelength 45mm confirms an applicability of the theoretical model up to particle concentration 100kg/m³.
• Infrared radiative properties of the quartz fibrous material with randomly oriented fibers are calculated by use of the scattering theory for infinite cylinders at oblique incidence of the radiation. A comparison of the calculated values of the radiation diffusion coefficient with the experimental data in semitransparency region shows that a nonsignificant discrepancy in the visible range may be explained by a more intensive scattering by bends and seals of fibers. One can obtain an evaluation of this effect by addition of small amount of spherical particles of radius equal to that of the most thin fibers. Collective effects, which are expected first of all in the longwavelength region, do not observed even for the fibrous material of comparably large density 144kg/m³. An additional verification of the theoretical model is obtained by comparison with the spectral measurements of the optical thickness for samples of fiberglass insulation of density 86kg/m³.

In the channel flow case various results are also presented for the particle deposition efficiency along the channel when the walls are cold and on the development of the particle-free zone along the walls when they are hot.

New effect of temperature wave bifurcation is briefly discussed. The bifurcation is the appearance or disappearance of second temperature wave while the frequency of external heat action or the mean temperature of the medium are changing slowly. the bifurcation does not result from instabilities and nonlinearities, but arises from the competition between radiative and conductive heat transfer. Effect of different type of reflection from the boundary on temperature wave propagation is numerically examined.

Applications of the results to nondestructive absorption coefficient measurements by thermal wave techniques are considered. It is shown that there exists optimal value of absorption coefficient, which maximizes contribution of radiation to the combined heat transfer while other properties of the medium and parameters of the external heat action are fixed. dimensionless estimation of contribution of radiative heat transfer relative to conductive heat transfer is proposed. The estimation is useful for arbitrary nongray medium with unknown optical properties.