Tiago L. Farias*, M.Graça Carvalho*, Ümit Ö. Köylü**, Gerard M. Faeth***
*Mech. Eng. Dep., Instituto Superior Técnico, 1096 Lisbon, Portugal
**Istanbul Tech. Univ., Aeron. and Astron. Faculty, Maslak, Istanbul, Turkey
***Dep. of Aeros. Eng., The University of Michigan, Ann Arbor, Michigan, 48109-2118, U.S.A.

The absorption and scattering properties of soot are needed both to predict the continuum radiation properties of soot and to interpret nonintrusive optical measurements to find soot concentrations and structure. This is a challenging problem because soot consists of small primary particles that combine into branched aggregates that exhibit neither simple Rayleigh, nor Mie scattering behavior.

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 measurements1,2 . 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, xp. 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.

  1. Köylü, Ü.Ö. and Faeth, G.M., Optical properties of Overfire Soot in Buoyant Turbulent Diffusion Flames at Long Residence Times, J.Heat Transfer, Vol. 116, pp. 152-159, 1994.
  2. Köylü, Ü.Ö. and Faeth, G.M., Optical Properties of Soot in Buoyant Laminar Diffusion Flames J. Heat Transfer, Vol. 116, pp. 971-979, 1994.
  3. Iskander, M.F., Chen, H.Y. and Penner, J.E., Optical Scattering and Absorption by Branched Chains of Aerosols, Appl. Optics, Vol. 28, pp. 3083-3091, 1989.
  4. Köylü, Ü.Ö. and Faeth, G.M., Structure of Overfire Soot in Buoyant Turbulent Diffusion Flames at Long Residence Times, Combust. and Flame, Vol. 89, pp. 140-156, 1992.


Jun Yamada, Yasuo Kurosaki
Department of Mechanical and Intelligent Systems Engineering
Tokyo Institute of Technology, Tokyo, JAPAN

This paper numerically investigates radiation transfer in fibrous media with a large size parameter using a pseudo-continuous model. Fibrous media have recently been applied for use in a radiation burner (1) and solar energy collector (2) due to their ability to convert energy between thermal radiation and gas enthalpy. In such energy conversion systems, since radiation is emitted from high-temperature sources, its wavelength is usually less than the medium'’s fiber diameter d. Thus, evaluation of the radiation transfer in fibrous media having a large size parameter (=d/) is essential to improve system performance.

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

Figure 1 schematically shows how an infinitely long fiber scatters radiation when it is irradiated by collimated radiation, where most is scattered along the surface of a cone having a cone angle of 20 (7,8). Such scattering can be characterized by the optical characteristics of a single fiber, namely, its scattering and extinction efficiencies. It is difficult to use conventional methods to determine the optical characteristics of a single fiber having a large size parameter.

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. Echigo, R., Yoshizawa, Y., Hanamura, K. and Tomimura, T., Proc. 8th Int. Heat Transf. Conf., Vol. 2, pp. 827-833, 1986.
  2. Kanayama, K., Baba, H., Koseki, K. and Nakajima, H., Jpn., Jour. Thermophysical Prop., Vol. 6, No. 2, pp. 78-82,1992.
  3. Tien, C.L., Trans. ASME, J. Heat Transf., Vol. 110, Nov., pp. 1230-1240, 1988.
  4. Houston, R.L. and Korpela, S.A., Proc. 7th Int. Heat. Transf., Conf., Vol. 2, pp. 499-504, 1982.
  5. Lee, S.C., J. Quant. Spectrosc. Radiat. Transf., Vol. 36, No.3, pp. 253-263, 1986.
  6. Yamada,J. and Kurosaki,Y., Proc. 28th ASME/AIChE, Proc. Nat. Heat Transf. Conf. at San Diego, HTD-Vol. 203, pp. 63-70, 1992.
  7. Van de Hulst, Light Scattering by Small Particles, Dover, Dover, New York, p. 297, 1957.
  8. Kerker, M., The Scattering of Light, Academic Press, New York, p. 255, 1969.


Leonid A. Dombrovsky
The Scientific-Research Institute of Thermal Processes,
125438, Moscow, Russia

In this paper, high-density disperse systems of randomly placed particles are considered. Applicability of the Mie theory and the radiation transfer theory for these systems is not obvious due to small distances between particles in comparison with the radiation wavelength. Solutions of the following problems are presented:
  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.


Y. Yener and J.W. Cipolla, Jr.
Department of Mechanical Engineering
Northeastern University, Boston, MA 02115, U.S.A.

The interaction of radiation and thermophoresis of radiating aerosols is discussed both in external laminar boundary-layer flow over a cold surface and in the thermally developing laminar flow in a parallel-plate channel. In both cases, the fluid is a radiatively nonparticipating constant-property gas containing emitting, absorbing and isotropically scattering gray aerosol particles, with their absorption and scattering coefficients being proportional to the local concentration of particles in the mixture. Various results are presented to illustrate the effects of the parameters of the problems on the temperature and aerosol concentration distributions, as well as on the heat and particle flux to the plate in the external flow case.

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.


Andrei V. Galaktionov
Institute for High Temperatures Russian Academy of Sciences,
13/19 Izhorskaya, Moscow, 127412, Russia

The interaction between radiative and conductive heat transfer is analyzed in semi-transparent gray non-scattering media under periodical external heat action. Two dimensionless parameters are proposed for classification of radiative-conductive temperature waves. Singular eigenfunction expansion method is used to obtain an exact quasi-steady solution to the system of radiative transport and energy equations. Various boundary conditions are considered. The problem is reduced to a singular integral equation of the Cauchy type. Exact analytical solution is written out in a closed analytical form in case of simplest boundary condition for radiative transport equation by Muskhelishvili’s theory for singular integral equations. Regularization of the singular integral equation is proposed in case of scattered and reflected boundary. Numerical technique is developed for solving the regular equation.

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.

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