PARTICLES, FIBERS, THERMOPHORESIS, AND WAVES
THE RANGE OF VALIDITY OF THE
RAYLEIGH-DEBYE-GANS/FRACTAL-AGGREGATE THEORY FOR COMPUTING THE OPTICAL
PROPERTIES OF SOOT
Tiago L. Farias*,
Ümit Ö. Köylü**,
Gerard M. Faeth***
*Mech. Eng. Dep., Instituto Superior Técnico, 1096 Lisbon,
**Istanbul Tech. Univ., Aeron. and Astron. Faculty, Maslak,
***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
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.
- 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.
- Köylü, Ü.Ö. and Faeth, G.M., Optical Properties of Soot in Buoyant Laminar Diffusion Flames J. Heat Transfer, Vol. 116, pp. 971-979,
- Iskander, M.F., Chen, H.Y. and Penner, J.E., Optical Scattering and
Absorption by Branched Chains of Aerosols, Appl. Optics, Vol. 28, pp.
- 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.
RADIATION TRANSFER IN FIBROUS MEDIA WITH LARGE SIZE
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
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
Figure 1 Scattering by a single fiber
Figure 2 Employed rectangular aperture model for representing diffraction in a
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.
- Echigo, R., Yoshizawa, Y., Hanamura, K. and Tomimura, T., Proc. 8th Int.
Heat Transf. Conf., Vol. 2, pp. 827-833, 1986.
- Kanayama, K., Baba, H., Koseki, K. and Nakajima, H., Jpn., Jour.
Thermophysical Prop., Vol. 6, No. 2, pp. 78-82,1992.
- Tien, C.L., Trans. ASME, J. Heat Transf., Vol. 110, Nov., pp. 1230-1240,
- Houston, R.L. and Korpela, S.A., Proc. 7th Int. Heat. Transf., Conf., Vol.
2, pp. 499-504, 1982.
- Lee, S.C., J. Quant. Spectrosc. Radiat. Transf., Vol. 36, No.3,
pp. 253-263, 1986.
- Yamada,J. and Kurosaki,Y., Proc. 28th ASME/AIChE, Proc. Nat. Heat Transf.
Conf. at San Diego, HTD-Vol. 203, pp. 63-70, 1992.
- Van de Hulst, Light Scattering by Small Particles, Dover, Dover, New York,
p. 297, 1957.
- Kerker, M., The Scattering of Light, Academic Press, New York, p. 255, 1969.
THE MIE THEORY ANALYSIS OF COMPARABLY DENSE DISPERSE
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
- 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.
- Absorption and scattering of microwave radiation by metal powder in
dielectric at the wavelength much greater than distances between particles.
- 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
- 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
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.
RADIATIVE-CONDUCTIVE TEMPERATURE WAVES IN HALF-INFINITE
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
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