SOLUTION OF RADIATIVE HEAT TRANSFER EQUATION II
RECENT BENCHMARKINGS OF RADIATIVE HEAT TRANSFER WITHIN
NONHOMOGENEOUS PARTICIPATING MEDIA AND THE IMPROVED YIX METHOD
Pei-feng HSU1, Zhiqiang TAN2
1 Mechanical and Aerospace Engineering Programs Florida Institute of
Technology Melbourne, Florida, U.S.A.
2 Aerospace Engineering and Engineering Mechanics Department
University of Texas Austin, Texas, U.S.A.
A review of the recent benchmark efforts since the First Symposium on Solution
Methods for Radiative Heat Transfer in Participating Media is presented. The
Symposium was first held at 1992 28th National Heat Transfer Conference and
then at 1994 6th. AIAA/ASME Joint Thermophysics and Heat Transfer Conference.
Also presented is the continuing effort to improve the solution accuracy of the YIX method for benchmarking. The latest work is focused on multi-dimensional,
gray, and nonhomogeneous participating media.
Solution of radiative heat transfer in nonhomogeneous participating media has
been an important research subject for many engineering and scientific
applications. Most of the applications encountered in actual systems contain
participating media with nonuniform radiative property distributions in the
multi-dimensional geometries. Computational limitations are presently a major
factor in dictating the present state-of-the-art since modeling real properties
and geometries is very computationally expensive, usually much more expensive
than the flow simulations in the same system. Due to the integro-differential
nature of the radiation transport, many algorithms, many more than those for
the Navier-Stokes equations, have been developed in the past to solve the
radiation transfer equation (RTE). Although many methods, such as the discrete
ordinates method, the Monte Carlo method, the finite element method, the finite
volume method, and the YIX method, can be applied to arbitrarily complex
geometry and spectral properties in principle, little is known on their
efficiency and accuracy relative to each other.
Because of the complexity of the RTE and the wide variety of methods available,
error estimation of the computations is usually unavailable, and it is not
uncommon that large differences can be found in the results for the same
problem using different methods. This became apparent at the above mentioned
Symposium. In that Symposium, participants were asked to solve a
three-dimensional problem consisting of a nongray mixture of spherical carbon
particles and CO2 gas contained in a rectangular enclosure. The problem was
intended to model a coal-fired furnace. Despite major property and geometry
simplifications made in defining the problem, larger than expected variations
in the predictions were found. Similar situations can be found in journal
publications where large differences among results are not unusual.
Therefore, it is critical to provide some benchmark solutions to the radiation
heat transfer community. The need for benchmarks has been also reflected in a
recent workshop on the use of high-performance computing to solve participating
media radiative heat transfer problems (held in March 1993 at the Sandia
National Laboratories), in which participants were asked to identify 5 classes
of highly challenging, nationally important problems relating to the use of
high-performance computing in the participating media radiative heat transfer.
The first problem identified is the development of benchmark solutions for
Following the first Symposium in 1992, the YIX method has been used in a
separate benchmarking effort with the Monte Carlo and the finite element
methods. They are used to solve the radiative heat transfer within a unit
cubical enclosure with nonhomogeneous participating media. With the first order
accurate distance quadrature and piecewise constant integrand, the YIX
solutions show 1 to 3% difference of surface heat fluxes in cases E1 and E2
as compared with the finite element solutions on the same grid.1
Cases E1 and E2 have uniform temperature distribution inside the cold and black
enclosure. Using the same piecewise constant radiative property distribution as
the YIX solutions, The Monte Carlo solutions have the same order of error. It
is also found that the YIX solutions have bigger difference at the core region
where the optical thickness is larger.
In this paper, three higher order interpolation schemes, i.e., piecewise
linear, trilinear, and parabolic, are presented to improve solution accuracy in
generating benchmarks with YIX method. Detail and systematic error analysis
indicate that superconvergence exists for these interpolations. Especially
notable is the piecewise trilinear interpolation. It has excellent convergent
rate as compared with the order of its interpolation error. The result shows
that one order of magnitude reduction in error can be achieved without
resorting to finer grid. Significant computational time and memory can be saved
with high order interpolation. This has important implications when coupling
the RTE calculation with the flow code.
Other than the interpolation error, the integration errors of the YIX method,
which include distance and angular quadrature, are also examined. The use of
discrete ordinates set in the angular quadrature is studied rigorously and
compared with the use of Simpson rule. The convergence rate of the discrete
ordinates sets is superior due to its spherical symmetry.
A related issue with angular quadrature is the ray effect. It is shown that for
the problem that could cause ray effect in the solution, the effect can be
eliminated by using large number of angular quadrature points. It is further
determined that an adaptive angular quadrature scheme has the potential of
removing ray effect without significant increase of the computational time. On
the other hand, the use of high order distance quadrature is found that,
without the corresponding higher order interpolation, little benefit can be
obtained from it. The main intent of these results is to provide a verified
set of solutions which can be useful as benchmarks when developing other
1 Hsu, P. and Farmer, J.T., Benchmark Solutions of
Radiative Heat Transfer within Nonhomogeneous Participating Media Using the
Monte Carlo and YIX Methods, presented at the 30th National Heat Transfer
Conference, Portland, Oregon, August 5-8, 1995.
EVALUATIONS OF DISCRETE TRANSFER MODEL FOR RADIATIVE
TRANSFER IN RECTANGULAR FURNACES
Nevin Selçuk and Nuray Kayakol
Department of Chemical Engineering
Middle East Technical University
Ankara 06531, Türkiye
The Discrete Transfer Model was applied to the predictions of the radiative
heat flux density and source term of a box-shaped enclosure problem based on
data reported previously on a large-scale experimental furnace with steep
temperature gradients typically encountered in industrial furnaces. The
rectangular enclosure under consideration has interior black walls and an
absorbing emitting medium of constant properties. The predictive accuracy of
the model was evaluated by comparing its predictions with exact numerical
solutions produced previously for the same enclosure problem. The comparisons
show that the model provides radiative heat flux and energy source term
distributions in close agreement with the benchmark solutions. Evaluations of
the accuracy of the Discrete Transfer Model against exact solutions on a
rectengular enclosure problem with the steep temperature gradients is not
available to date.
Presented in the "“First International Symposium on Radiative Heat
Transfer”" 14-19 August 1995, Kuþadasý, Türkiye
THREE-DIMENSIONAL SPECTRAL RADIATIVE TRANSFER CALCULATION
IN A CYLINDRICAL MODEL COMBUSTOR USING THE DISCRETE ORDINATES METHOD
W. Krebs, R. Koch, H.-J. Bauer and S.
Lehrstuhl und Institut für Thermische Strömungsmaschinen,
Universität Karlsruhe, Kaiserstraße 12, D-76128 Karlsruhe,
One of the basic requirements of modern combustor design is a detailed
knowledge of the heat load of the combustor walls. Since high flame
temperatures and high optical densities due to high pressures or large
geometrical scales are encountered, in most combustion systems a considerable
part of the wall heat load stems from radiation.
For the calculation of the multidimensional spectral radiative heat transfer
quite a lot of models have been suggested to describe the radiative properties
of combustion gases and soot as well as the multidimensional radiative heat
exchange have been evaluated seperately at the Institut für Thermische
Strömungsmaschinen (ITS) in special test cases. It has been found that the
Discrete Ordinates method should be best suited to combustion systems with
respect to accuracy and computation time. However the Discrete Ordinates method
has not been evaluated in real combustion systems.
Therefore, the main objective of this paper is the application of the Discrete
Ordinates method to calculate the multidimensional radiative heat transfer in a
"“real"” propane fired three-dimensional cylindrical model combustor featuring
high gradients of temperature and concentrations of radiative active species.
The information about the reacting flow-field inside this model combustor
has been provided by both experiments and numerical analysis.
The radiation calculations are performed on a spectral basis using narrow-band
models for an accurate representation of the radiative properties of combustion
gases. To avoid "“sweeping"” in angular space the Discrete Ordinates Equations
describing radiative exchange in a three-dimensional cylindrical coordinate
system have been discretized by a “purely” spatially applied Finite Volume
technique for the first time. This has been achieved by formulating the
discrete direction vector in a cylindrical coordinates, too.
The different applied Sn-approximations (S4,
S6, S8 ) are evaluated by comparing the calculated
spectra to measured spectra and calculated spectra obtained from
1D-calculations which serve as reference data. It has been proven, that
multidimensional spectral radiative heat transfer in a real cylindrical
combustion system featuring steep gradients of temperature and concentration of
radiative active species can be predicted accurately using the models
presented. As expected, the calculations of radiative heat transfer through the
reaction zone is most critical, due to the steep gradients of scalar variables.
However, using a fine grid size of 18998 grid points, highly accurate results
are already obtained by application of the S4-approximation.
Therefore considering CPU-time consumption and accuracy, the
S4-approximation is best suited to radiation calculations in
THREE-DIMENSIONAL ANALYSIS OF RADIATION HEAT TRANSFER IN A
RADIANT COOLED SPACE
Toshiyuki MIYANAGA, Yukio NAKANO, Toshiharu
Laser & Optical Radiation Group, Electrophysics Department
Central Research Institute of Electric Power Industry, Tokyo,
The objective of this paper is to develop an analysis method of local radiation heat transfer in a radiant cooled space containing arbitrarily shaped objects.
In this paper, the following methods are proposed. The surfaces of the objects
are divided into small quadrilateral subsurfaces. Radiation view factors
between subsurfaces are calculated by combining the Mitalas-Stephenson contour
integration method and a simple method for judging obstructions. Then the
radiation heat exchange on each subsurface is obtained using Gebhart’s
enclosure analysis method. Heat transfer by convection, conduction and
ventilation is considered in the calculation in addition to radiation. We can
obtain the steady-state temperature of each subsurface by solving the nonlinear
heat balance equations using Newton-Raphson and Gauss-Seidel methods.
By using above methods, the steady-state cooling environment of a meeting room
with a cooled ceiling panel is analyzed. The local thermal influence of a
heated window on the cooling environment is examined. In the analysis, a
three-dimensional model of a human body is also used. It is divided into three
parts: head, body and legs. The skin temperature and clothing combination
can be assigned to each part of the body independently. The thermal sensation i
of the human body in the room is evaluated quantitatively by calculating the
heat loss of the human body and the predicted mean vote (PMV).
Our analysis method contributes to the systematic design of the radiant cooled
space. It can determine the arrangement of furniture and the temperature of the cooling panel to obtain satisfactory thermal sensation in the space. It also
enables us to improve the environment where thermal radiation sources such as
heated windows cause thermal discomfort.
DIRECT EXCHANGE AREAS FOR AN INFINITE RECTANGULAR DUCT BY
K.H. Byun*, Theodore F.
* Department of Mechanical Engineering, Dongguk University, SEOUL
100-715, Republic of Korea
** Department of Mechanical Engineering, The University of Iowa,
Iowa 52242, U.S.A.
The purpose of this study is to compute direct exchange areas (DEA) for an
infinite rectangular duct by using the discrete-ordinates method. A gray
absorbing and emitting medium is enclosed by opaque black walls. The system may
have a protrusion on the surfaces and that causes shadings between surfaces.
DEA’s of the system are calculated. The discrete-ordinates weights are
expressed in terms of the product of cross-sectional weights and axial
directional weights. For the cross-sectional weights and directions, Sanchezand
Smith’s method or Chevyshef method is used. The abscissas and wights of
Gaussian quadrature are utilized for axial direction integration to incorporate
absorbing and emitting effects by the infinite layer.
The effects of optical thickness as well as the number of spatial and angular
divisions on the accuracy of the DEA results are studied. The results are
presented at the optical thickness values of 0, 0.1, 1, and 10. When optical
thickness is greater than 5, if the relative errors of DEA values should be
less than about 2%, then it is needed more spatial and angular divisions than
current study. As optical thickness increases at a given spatial and angular
divisions, the errors increase. For a given optical thickness, the errors are
reduced as the number of spatial and angular divisions increase. If there is
shading due to protrusion in the system, the accuracy of the results depends on
both the number of spatial and angular divisions.
The results are compared with the DEA prediction by the S-N discrete-ordinates
methods of order upto S-10. The results indicate that the higher order S-N
method than S-10 is necessary to increase the accuracy of computation. The
results are also compared with the numerical integration values of the DEA’s
expression. In conclusion, whether there is shading in the system or not,
direct exchange areas can be accurately obtained by using the
RADIANT FLASH MELTING OF MICRON POLYMER-PARTICLES
K. Hanamura, M. Kumada
Department of Mechanical Engineering
Gifu University, 1-1 Yanagido, Gifu, Japan
The transitions of radiation properties of a toner-particle bed during the
fusing process in electronic printing were investigated. The bidirectional
transmittance and reflectance for the bed before fusing, a slightly fused bed
and the completely fixed bed were measured. The radiation properties were
estimated, as an inverse problem, from a comparison between the experimental
and numerical results. The results show that the optical thickness and the
scattering albedo vary during the fusing process, depending on the change of
the shape of the toner particle.
Electronic printing is performed through the processes of corona charging,
exposure with a light and lenses, development into a visible image with a toner
powder, transfer to a paper, fusing and fixing of the toner of the paper,
cleaning and erasing. The toner is a mixture of resinous binders, coloring
agents, magnetites, lubricants, and other additives; where the thermoplastic
resins are the main components of the toner. The physical properties of the
toner particle are not easily available. The best combination is empirically
determined, taking the types of the development and fusing into account. the
printing speed is principally controlled by fusing process of the powder on a
paper. Four kinds of methods, i.e., hot rolling, fusing in an oven, pressurized
rolling and radiant flash fusing, are investigated, associated with several
kinds of the toner powder, for the past few decades. Of these, it is considered
that the radiant flash fusing is the most useful method for high speed
printing. However, no phenomenological work has been reported on the fusing
process through the radiant heating. The toner powder on the paper is regarded
as a packed bed. A regime map for for independent and dependent scattering is
depicted with respect to the size parameter and the volume fraction. However,
the results are for sphere particles. On the other hand, the toner particle
before fusing is not a sphere, and the shape is expected to change at a
temperature above the softening point. In addition, some additives are also
able to absorb and scatter the radiation. In the present study, the transitions
of radiation properties of the toner bed during the fusing process have been
investigated through the measurements of the bidirectional transmittance
and reflectance and through the numerical calculation.
The spectral bidirectional transmittance and reflectance, and the spectral
normal directional transmittance are measured for a toner particle bed before
fusing, for a slightly fused bed and for a completely fixed bed. The
wavelengths of the incident beam are 0.6, 0.8 and 1
m. The mean size (equivalent diameter)
of a particle and the mean thickness of the bed, as measured by a scanning
laser microscope, are 9.2 and 31.7 m,
respectively. The volume fraction is 0.28. The scattered light is measured
using a photo sensor mounted on an optical rail. Thereby, the azimuthally
symmetric radiation field can be measured by traversing a single plane, where
the azimuthal symmetry is confirmed through the comparison between scattered
light intensities at two different azimuths.
Thermal radiation is absorbed and scattered by toner particles, i.e., the
thermoplastic resin, magnetites, coloring agents and other additives.
Furthermore, the shape of the toner particle changes from a non-spherical to
almost a spherical particle, and then to almost a slab. Therefore, it is
impossible to specify what is the main scattering and/or absorbing materials,
and what is the dominant phenomena, i.e., the surface reflection, refraction or
Mie scattering by small additives, in each step during the fusing process. For
analysis, in the present study, the medium is treated as a pseudo-continuum to
clarify the global absorbing and scattering processes. The emission of
radiation is assumed negligibly small. The Henyey-Greenstein approximation is
used for the scattering phase function. As a result, the unknown parameters in
the radiative transfer equation are the optical thickness, the scattering
albedo and the asymmetry parameter of the phase function. Of these, the optical
thickness is estimated from the normal directional transmittance on the basis
of the Beer’s law using a fine incident beam. Other parameters are estimated
from a comparison between the profiles of the bidirectional reflectance and
transmittance for measurement and calculation.
4. RESULTS AND DISCUSSION
Through fusing process, the shape of the particle changes from a non-spherical
to a spherical one, resulting in the decrease in the projected area of the
particle. Simultaneously, the transparency of the particle becomes increased.
On the other hand, the completely fixed bed dose not have so many pores
transmitting the radiation through the bed. As a result, the optical thickness
is strongly dependent on the total projected area of the particles.
Furthermore, both the scattering albedo and the asymmetry parameter of the
slightly fused bed are higher than those of the bed before fusing since surface
of the particle becomes smooth through fusing. For the completely fixed bed,
the scattering albedo is considerably small compared with those of other beds.
For the bed before fusing, the surface of the particle has a roughness with
several different characteristic heights. The surface, which is optically rough
for short wavelengths, can be smooth at long wavelengths. As a result, the
bidirectional transmittance and reflectance increase with wavelength ranging
from 0.6 to 1 m. On the other hand,
the values of the normal directional transmittances are almost the same over
the range of the wavelength since the scattering cross section dose not change.
Although the effect of the additives, such as magnetites and coloring agents,
is not clarified in the present study, it is expected that the materials
strongly contribute to the scattering and absorption of the radiation when the
transparency of the toner particle becomes high; that is, in the cases of the
slightly fused bed and the completely fixed bed. Consequently, the amount of
the radiant energy absorbed by toner bed varies with time in the flash melting
The authors acknowledge the financial support from the Grant-in-aid for
Scientific Research of the Ministry of Education of Japan (No. 06230209).