Prog. Energy Combust. Sci. 1987. Voh 13, pp. 97-160.
0360-1285/87 $0.00 +.50
Copyright O 1987 Pergamon Journals Ltd.
Printed in Great Britain. All rights reserved.
RADIATION HEAT TRANSFER IN COMBUSTION SYSTEMS
R. VISKANTA* a n d M . P. M E N G O q t
*School oJ Mechanical Engineering, Pttrdue University, West LaJ~tyette, IN 47907, U.S.A.
tDepartment q/Mechanical Engineeriny, University of Kentucky, Lexington, K Y40506, U.S.A.
Abstract An adequate treatment of thermal radiation heat transfer is essential to a mathematical model
of the combustion process or to a design of a combustion system. This paper reviews the fundamentals of
radiation heat transfer and some recent progress in its modeling in combustion systems. Topics covered
include radiative properties of combustion products and their modeling and methods of solving the
radiative transfer equations. Examples of sample combustion systems in which radiation has been
accounted for in the analysis are presented. In several technologically important, practical combustion
systems coupling of radiation to other modes of heat transfer is discussed. Research needs are identified
and potentially promising research topics are also suggested.
CONTENTS
Nomenclature
1. Introduction
2. Radiative Transfer
2.1. Radiative transfer equation
2.2. Conservati_on of radiant energy equation
2.3. Turbulence/radiative interaction
3. Radiative Properties of Combustion Products
3.1. Radiative properties of combustion gases
3.1.1. Narrow-band models
3.1.2. Wide-band models
3.1.3. Total absorptivity emissivity models
3.1.4. Absorption and emission coefficients
3.l.5. Effect of absorption coefficient on the radiative heat flux predictions
3.2. Radiative properties of polydispersions
3.2.1. Types and shapes of polydispersions
3.2.2. Prediction methods of the particle radiative properties
3.2.3. Simplified approaches
3.2.4. Scattering phase function
3.3. Total properties
4. Solution Methods
4.1. Exact models
4.2. Statistical methods
4.3. Zonal method
4.4. Flux methods
4.4.1. Multiflux models
4.4.2. Moment methods
4.4.3. Spherical harmonics approximation
4.4.4. Discrete ordinates approximation
4.4.5. Hybrid and other methods
4.5. Comparison of methods
5. Applications to Simple Combustion Systems
5.1. Single-droplet and solid-particle combustion
5.2. Contribution of radiation to flame wall-quenching of condensed fuels
5.3. Effect of radiation on one-dimensional char flames
5.4. Radiation in a combusting boundary layer along a vertical wall
5.5. Interaction of convection-radiation in a laminar diffusion flame
5.6. Effect of radiation on a planar, two-dimensional turbulent-jet diffusion flame
5.7. Radiation from flames
5.8. Combustion and radiation heat transfer in a porous medium
6. Applications to Combustion Systems
6.1. Industrial furnaces
6.1.1. Stirred vessel model
6.1.2. Plug flow model
6.1.3. Multi-dimensional models
6.2. Coal-fired furnaces
6.3. Gas turbine combustors
6.4. Internal combustion engines
6.5. Fires as combustion systems
7. Concluding Remarks
Acknowledgements
References
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98
R. VISKANTAand M. P. MENGf3t~
NOMENCLATURE
A
B
C
D
D,
Eh
E,,
f
J/0)
./v
g
h
I
J
K
k
Ko
L
L,,
Mp
Nn
N~
N2
h
n
P
PN
Q
Qv
Re
q
r
S
S~
s
St
T
T,o
v
V
W~
x
y~
direction cosine, Eq. 12.8)
p density (kg/m 3)
a Stefan-Boltzmann constant; scattering
efficient (m- 1)
T beam transmittance, Eq. (2.18)
r optical depth, i,y
~P scattering angle, Eq. 2.7)
scattering phase function
05 azimuthal angle
solid angle
t,) single scattering albedo, a/fl
a r e a [m 2)
mass transfer number, (Q Y~o/vowo-h.)/L
concentration
diameter of particles (am) or burner exit
diameter (m)
dimensionless heat of combustion.
(2 V,,®/v ,, W,,h,,.
blackbody emitted flux defined by aT4(W/m z)
exponential integral function,
E,,tx)= So~" - 2exp( -x/la)d/d
size distribution, Eq. (3.11) or phase function
coefficient, Eq. (3.25)
dimensionless stream function at the surhce
volume fraction (m3/m 3)
phase function coefficient, Eq. (3.25)
enthalpy or Planck'sconstant
radiation intensity(W/m 2-sr)
radiative flux in radial direction (W/m 2)
radiative flux in axial direction (W/m 2)
thermal conductivity (W/mK) or imaginary
part of the complex index of refraction
Konokov number
radiative flux in angular direction (W/m z) or
effective latent heat of pyrolysis
mean beam length (m)
pyrolysis rate
radiation-conduction parameter, k h / o T 3
conduction-gaseous radiation parameter,
k~x/aT 3
conduction-ambient radiation parameter
defined as limy~® (k~o/aT3Xu~/xv®) 1/2
complex index of refraction (= n - i k )
real part of the complex index of refraction
pressure or probability density function
N-th order spherical harmonics approximation
Mie efficiency factor or energy released by
combustion of v moles of gas phase fuel
heat of reaction per unit mass of oxygen
burner Reynolds number
heat flux (W/m 2)
mass consumption number
source function
N-th order discrete ordinates approximation
coordinate along the direction of propagation of radiation or stoichiometric ratio,
vs, W / v o W,,
Stanton number
temperature (K)
surface temperature of load (sink) (K)
velocity (m/sec)
volume (m 3)
molecular weight of species i
size parameter, riD~2
mass fraction of species i
Greek letters
absorptivity
extinction coefficent (m - ~)
6 Dirac delta function
emissivity
emission coefficient, Eq. (2.4); direction cosine, Eq. (2.8); dimensionless coordinate
defined as x/g~I~(u®/tt)dy
zenith angle; normalized temperature
absorption coefficient (m - ~)
wavelength of radiation (/tm)
direction cosine, Eq. (2.8)
frequency of radiation; kinematic viscosity:
stoichiometric coefficient
CO-
Subscripts
refers to blackbody
refers to effective mean
i refers to Planck's internal mean
m
refers to mean values
n
refers to narrow-band model
o
oxygen
P refers to Planck's mean
r
refers to spatial coordinates or radiation
II'
refers to wall conditions, fuel surface or wideband model
2 refers to wavelength dependent properties
V
refers to frequency dependent properties
b
e
Stlperscripts
refers to incoming radiation beam
refers to turbulent mean properties
I. I N T R O D U C T I O N
Expenditures on fossil energy by individuals,
commerce, transportation and industry in an industrialized country account for a significant fraction
of the country's G N P . Improved understanding of
combustion systems which use fossil fuels such as
natural gas, oil and coal may result in improved
energy efficiency. The potential improvement in the
thermal performance of such systems could make a
significant impact on the country's economy. This
provides the motivation and economic incentive for
research and development in combustion technology.
An important goal, then, is to develop computational
models which could be used for the design and
optimization of more cost effective and environmentally friendly combustion systems with improved
performance.
Combustion is one of the most difficult processes
to model mathematically since it generally involves
the simultaneous processes of three-dimensional twophase fluid dynamics, turbulent mixing, fuel evaporation, radiative and convective heat transfer, and
chemical kinetics. In order to design combustion
systems based on fundamental principles, comprehensive models incorporating all of these factors are
required. State-of-the-art reviews of modeling some
combustion systems have been p r e p a r e d ? - 7 Significant progress has been made in detailed modeling of
combustion systems, but major problems such as
turbulence in reactive flows, particle formation and
others remain to be solved.
Radiation heat transfer
An adequate treatment of thermal radiation is
essential to develop a mathematical model of the
combustion system. The level of detail required for
radiative transfer depends on whether one is interested in determining the instantaneous spectral local
radiative flux, flame structure, scalar properties of the
flame, formation of flame-generated particles (largely
soot), local radiative flux and its divergence or the
temperature distribution. For example, when the
model is used to predict pollutant concentrations,
accurate temperatures are especially important since
the chemical kinetics involved are extremely temperature dependent.
The fraction of the total heat transfer due to
radiation grows with combustor size, attaining
prominence for gaseous firing at characteristic combustion lengths of about 1 m. Radiation heat transfer, then, plays a dominant role in most industrial
furnaces. Unfortunately, it is governed by a complex
integrodifferential equation which is time consuming
to solve. Economic measures are a necessity, even at
the loss of some accuracy.
In a combustion chamber, radiation heat transfer
from the flame and combustion products to the
surroundings walls can be predicted if the radiative
properties and temperature distributions in the
medium and on the walls are available. Usually,
however, temperature itself is an unknown parameter, and as a result of this, the total energy and
radiant energy conservation equations are coupled,
as in many heat transfer applications. Solution of the
thermal energy equation can be obtained if several
other physical and chemical processes can be
modeled. The major processes which need to be
considered in a combustion system in addition to
radiation include? (i) chemical kinetics, (ii) thermochemistry, (iii) molecular diffusion, (iv) laminar and
turbulent fluid dynamics, (v) nucleation, (vi) phase
transitions, such as evaporation and condensation
and (vii) surface effects. Since the physical and
chemical processes occurring in combustion chambers are very complicated and cannot be modeled on
the microscale, there is a need for physical models to
simulate these processes. Each of these models needs
an extensive and separate treatment, which is outside
the scope of this work. The interested reader is
referred to more specialized publications. ~- v
In nonrelativistic problems of an engineering
nature, radiation does not contribute any terms to
the conservation of mass, momentum and species
conservation. The classical conservation of energy
equation Ls'9 is modified by a contribution which
accounts for radiation heat transfer. This equation
can be written as
~pe
Ft
= -V.pe~-V.P-~-V.~+S
where the heat flux vector, ~, is defined as
(1.1)
99
~= - k V T + ~ k + ~ n i h i V j + ~ a _ ,.
(1.2)
J
In Eq. (1.1) p, pc, ~ and P are the total mass, energy
density, fluid velocity, and pressure, respectively. The
{n;} and {V~} are the number density and diffusion
velocities of the individual chemical species, and ,~"
is the radiation heat flux vector. The first, second,
third and fourth terms in Eq. (1.2) account for
molecular conduction, radiation, interdiffusion and
diffusion-thermo contributions, respectively, to the
heat flux vector. In Eq. (1.1) S is the local volumetric
heat source/sink from other processes, if any. When
radiation heat transfer needs to be accounted for in
the energy equation, it is preferable to use temperature as the dependent variable rather than the
stagnation enthalpy. The divergence of the radiative
flux vector, V..~-~', can be obtained from the radiant
energy equation.
The purpose of this paper is to acquaint the reader
with the basic principles and methods related to
modeling radiation heat transfer in combustion
systems. The importance of radiative transfer in coal
combustion, 3 pulverized coal-fired boilers,'* industrial furnaces, 5 gas turbine combustors 6 and fires 7
has been recognized for some time. Radiative
transfer in some of these systems has received
considerable research attention and a high degree of
organization has been attained.
The paper is organized to give a systematic and
easy-to-follow approach to the major building blocks
of radiative transfer in combustion systems. Section
2 of the paper introduces the fundamentals of
radiation heat transfer, and Section 3 discusses the
radiative properties of gases and particles encountered in combustion systems. These two sections
provide the background necessary for understanding
the specific techniques for solving the radiative
transfer equation discussed in Section 4. Examples of
simple combustion systems in which radiative transfer has been accounted for are discussed in Section 5.
Section 6 reviews modeling of radiation heat transfer
in practical combustion systems and deals with
coupling of radiation to other transport processes in
system models.
There exists a very large body of literature relevant
to radiation heat transfer in combustion systems, and
it is not possible to cover it thoroughly. Emphasis in
the paper is on fundamentals and applications to
simple systems. Reference is made to the original
publications for a more complete discussion. A
review process is a rather arbitrary activity, because
of the decision the authors have to make on what to
include, what to omit, and where to start and end.
This article is no exception, and it reflects the
authors' biases. Because of the broad range of topics
covered, details can not be included, and no claim is
made as to the completeness of the review. In these
days of many journals and other publications, it is
possible that relevant work may have been inadvertently overlooked.
100
R. MISKANTAand M. P. MENG0q
2. RADIATIVE TRANSFER
2.1. Radiative Transfer Equation
Two theories have been developed for the study of
the propagation and interaction of electromagnetic
radiation with matter, namely, the classical electromagnetic wave theory and the radiative transfer
theory. The theories were developed independently
and there is no similarity in their basic formulations.
Conceptually, they are completely distinct; however,
both theories describe the same physical phenomenon. The classical electromagnetic theory has
approached the study of propagation and interaction
of matter with radiation from the microscopic point
of view and the radiative transfer theory from the
macroscopic (or phenomenological) point of view.
The study of the detailed interaction of electromagnetic radiation with matter on the microscopic
level from both the classical and quantum mechanics
point of view yields the interaction cross-sections of
the particles making up the matter. This fundamental
approach predicts the macroscopic properties of the
media, and these properties appear as coefficients in
the radiative transfer equation.
The quantitative study, on the phenomenologicai
level, of the interaction of radiation with matter that
absorbs, emits, and scatters radiant energy is the
concern of the radiative transfer theory. The theory
ignores the wave nature of radiation and visualizes it
in terms of light rays of photons. These are concepts
of geometrical optics. The geometrical optics theory
is the study of electromagnetism in the limiting case
of extremely small wavelengths or of high frequency.
The detailed mechanism of the interaction process
involving atoms or molecules and the radiation field
is not considered. Only the macroscopic problem
consisting of the transformation suffered by the field
of radiation passing through a medium is examined.
Thus, there is a considerable simplification over the
electromagnetic wave theory.
The radiative transfer equation (RTE) forms the
basis for quantitative study of the transfer of radiant
energy in a partici, pating medium. The equation is a
mathematical statement of the conservation principle
applied to a monochromatic pencil (bundle) of
radiation and can be derived from many viewpoints.
Some authors l°A~ have derived the radiative
transfer equation from the Boltzmann equation of
the molecular theory of gases by adopting the
corpuscular ' picture of radiation and recognizing
close analogy between molecules and photons.
Preisendorfer ~2 has presented a development primarily from the standpoint of geometrical optics by
starting from a set of physically motivated axioms
from which the features of radiative transfer were
deduced. Several papers have also considered the
derivation of the equation from quantum mechanics.
Harris and Simon ~3 used the Liouville equation to
consider coherent radiation from a plasma by a
statistical treatment of both plasma particles and the
magnetic field, and Osborn and Klevans ~4 have
refined and generalized their work. The Eulerian
point of view is adopted here and the traditional
intuitive derivation of the RTE found in the
radiative transfer literature 15 - 20 is given.
Rather than presenting the most general derivation of the RTE, certain constraints which help to
avoid complications that obscure the physical significance of the phenomenon are imposed in this
discussion. The treatment presented here constitutes
a reasonable compromise between the generality
needed for engineering applications and clarity of the
development. The idealizing assumptions and constraints imposed are: (1) the discussion is restricted to
a continuous, homogeneous and isotropic absorbingemitting-scattering medium at rest, (2) the state of
polarization is neglected, and (3) the medium is
considered to be in local thermodynamic equilibrium
(LTE).
The RTE is based on application of an energy
balance on an elementary volume taken along the
direction of a pencil of rays and confined within an
elementary solid angle. The detailed mechanism of
the interaction processes involving particles and the
field of radiation is not considered here. On the
phenomenological level only the transformation
suffered by the radiation field passing through a
participating medium is examined. The derivation
accounts mathematically for the rate of change of
radiation intensity along the path in terms of
physical processes of absorption, emission, and
scattering.
Consider a cylindrical volume element, Fig. 1, of
cross-section dA and length ds in an absorbing,
emitting, and scattering medium characterized by the
spectral absorption coefficient xv, scattering coefficient try and true emission coefficient r/v. The axis of
the cylinder is in the direction of the unit vector ~,
i.e. ds is measured along ~. The spectral intensity of
radia.tion (spectral radiance) in the ~-direction
incident normally on one end of the cylinder is Iv
and the intensity of radiation emerging, through the
second end in the same direction is Iv + dlv. Here, v is
the frequency and is related to the wavelength 2 by
v = c/2, where c is the velocity of radiation.
FIG. 1. Coordinates for derivation of the radiative mmsfer
equation.
Radiation heat transfer
It follows from the definition of the spectral
intensity I, that radiant energy incident normally on
the infinitesimally small cross-section dA during time
interval dr, in frequency range dv and within the
elementary solid angle dD about the direction of the
unit vector ~ is
101
Since in this case the sum of probability over all
directions must equal unity, we must have
1
1
4rt
n'=4,
a=4,
1
=-J" ~,(W)dfl= 1.
4n n=4,
l ,dAdfldvdt.
The emerging radiant energy at the other face of the
cylinder in the same direction equals
(1 ~+ dl ,)dAdDdvdt.
The net gain of radiant energy, i.e. the difference
between energy crossing the two faces of the cylinder,
is then given by
(I ~+ dl ~- dl,)dAdf~dvdt = dl,dAdfldvdt.
( K, + ~r,)l ,dsdA d~dvdt.
(2.3)
If Kirchhoff's law is valid, the emission coefficient q~
can be expressed as
rl~ = x~n~lh~
cos W = cos0cos0' + sin0sin0'cos(~ - ~b') (2.6)
or
(2.4)
where lb, is Planck's spectral blackbody intensity of
radiation, and n, is the spectral index of refraction of
the medium. The increase in energy of the pencil of
rays (~,df~) due to in-scattering of radiation by the
matter into the elementary cylindrical volume from
all possible directions ~' is
• ,(s ~ s ; v ~ v )
&v' fl' ='l-x
x l¢(-~')dD'dv']dAd~dvdt.
In this expression the phase function
t~,(g'---,g;v'---~v)df~'dv'/4n represents the probability
that radiation of frequency v' propagating in the
direction g' and confined within the solid angle dfg
is scattered through the angle (g,g) into the solid
angle dD and the frequency interval dr. This probability is determined by the scattering mechanism.
For coherent scattering the phase function is independent of frequency v' and reduces to ~v(~"--*g).
cos W = ~¢'+ qq' + I#~'
(2.7)
~=sin0cos4~, q=sin0sin~b, /~=cos0
(2.8)
where
(2.2)
The emission by the matter inside the cylindrical
volume element dV, in the time interval dt, in the
frequency range dr, confined in the solid angle dD
about the direction g equals
q~d V df~dvdt.
This implies that for coherent scattering the spectral
phase function is normalized to unity. The scattering
angle W, i.e. the angle between g' and g can be
expressed as
(2.1)
The loss of energy from this pencil of rays due to
absorption and scattering in the cylinder is
(2.5)
are the direction cosines in any orthagonal coordinate system. Reference to Fig. 1 shows that
g'(0',~b') represents the incoming direction of the
pencil of rays, and g(0,~b), is the direction of the
pencil after scattering.
Equating the change of energy in the cylindrical
volume element to the net gain or loss of energy
along the traversal path of the cylinder in terms of
the processes of attenuation, emission and inscattering yields
d/,dA dfldvdt = - (x, + a ,)l ,dsdA dDdv dt
+q,dVdlldvdt + a , d s
II ~
~ A,"
S
,, ~,
q~,(s ---~s;v---w)
fl' = 4It
x l¢(-~')dt)'dv'] dAdfldv dt.
(2.9)
Dividing this equation by dAdsdf~dvdt and recalling
that the distance ds traversed by the pencil of rays is
cdt, where c is the velocity of light in the medium,
yields the equation of transfer in a Lagrangian
coordinate system
1 dlv
. . . .
c dt
(x, +tr,)lv + q ,
O" v
+U.I
I
@,(s" -*s;v-*v)l,,(s
)dD' dv.'
~'
"'
(2.10)
Av" 11" = 4 x
Clearly, the left-hand side of this integrodifferential
equation represents the net change in Iv per unit
102
R. V1SKANTAand M. P. MENGOt;
length along the path ds=cdt. Equation (2.10) is a
statement of the conservation of energy principle for
a monochromatic pencil of radiation (in the direction g) and is generally called the "radiative
transfer equation" (RTE). In some literature, the
steady-state form of this equation is called Bouguer's
law probably due to the fact that the constitutive
(Bouguer's) law enters as the first term on the righthand side of Eq. (2.10).
The substantial derivative d/dt refers to the rate of
change of spectral intensity as seen by an observer
propagating along with the velocity of radiation
(Lagrangian coordinates). In terms of a coordinate
system fixed in space (Eulerian coordinates), RTE
may be written as
1 dl,
1 01,
c dt
c 0t
~-(V.-~)I,=fl,(S,-I,)
Z,
(a)
T
/ f /
A s
(b)
(2.11)
where the source function S, represents the sum of
emitted and in-scattered radiation and is defined as
radiant energy leaving an element of volume of
matter in the direction (g,df]) per unit volume, per
unit solid angle, per unit frequency, and per unit
ti me,
(c)
sv = (,lv//L) + (a,//LX1/4~)
O,(s
s;v
v)/¢(s )df~dv.
(2.12)
:.,y
Av' fl' = 4ft
It is evident on inspection of Eq. (2.11) that there is
no net rate of change of Iv at a point, if and only if,
lv=Sv. If I~>S, then dIv/dt<O so that as t
increases, I , is decreasing toward S , and if I , < S ,
then dl,/dt > 0, so that I~ is increasing toward S,.
Equation (2.11) can be written explicitly using the
analytical forms of 07. g)l, which are given in Table
1. The direction cosines ~, q, and/~ are defined by Eq.
(2.8), and they are functions of angular variables 0
and ~b. Usually, the polar and aximuthal angles for
space variables are also designated by 0 and ~b. To
avoid confusion and to be able to use the conventional nomenclature at the same time, we choose
to use subscript r for space variables 0, and ~b, when
appropriate (see Table I).
In Fig. 2, three orthogonal coordinate systems and
the corresponding nomenclature are shown, where
the spherical coordinate system for angular variation
of intensity is superimposed on either a rectangular,
cylindrical or spherical coordinate system for spatial
variables. In general the intensity is a function of
three spatial coordinates, two angles and time; of
course, the seventh independent variable required to
define the radiation intensity is the wavelength or the
frequency of radiation.
For most practical calculations, it is possible to
assume cylindrical or spherical symmetry. The cylindrical symmetry requires that the radiation intensity
FIG. 2. Coordinates for Cartesian (a), cylindrical (b) and
spherical (c) systems.
remains invariant under the rotation about the zdirection. This allows us to combine the two
azimuthal angles, namely ~b and ~b,, to obtain a single
azimuthal angle. If, one writes ~b'=q~-~b,, then
07. g)l, for an axisymmetric cylindrical system can
be given as
7
01, rl 01,
01,
• ~ ) 1 , = ¢ ~ - - ; o ¢, ~u ez"
(2.13)
In a spherically symmetric system, the radiation
intensity depends on only two parameters, i.e. the
radial distance r measured from the origin and the
direction cosine/~ of the angle between the direction
of the radiation beam and the radius vector it . The
analytical expression for 07. g)l, corresponding to a
spherically symmetric system is the last expression of
Table 1, which can be simplified further to read
07" ~)1 __u 0/,+1 - u 2 0t,
(2.14)
Radiation heat transfer
A number of assumptions have been made in the
derivation of the RTE, and for the sake of completeness it is desirable to discuss them briefly. The first
assumption concerning the restriction that the participating media be continuous, homogeneous and
isotropic has been relaxed by Preisendorfer. 12 Although the assumption of a medium at rest is open to
criticism on physical grounds, this approximation
correctly describes all engineering problems where
the fluid velocity is much smaller than the velocity of
light. The absorption and scattering coefficients are
calculated or measured in a laboratory reference
system in which the local macroscopic velocity of
matter is zero, and because of this xv, av and T are
independent of g. It has, however, been shown that
in any frame of reference Iv satisfies the same
equation of radiative transfer. 22 The intensity Iv
changes at points along the path, where the index of
refraction n, changes continuously or discontinuously. Such changes can be systematically accounted
for by simply adopting a new function l,/n~ rather
than Iv. 12 Hence, there is no need to include the
index of refraction explicitly in the transfer equation.
The second assumption concerning the fact that
neglect of polarization is not generally valid is well
recognized, and it is clear that polarization must be
accounted for in any rigorous treatment of radiative
transfer when scattering is present. The radiative
transfer theory has been extended to include the
phenomenon of polarization of radiation. 12'15 It is
also well recognized that the third assumption for the
medium to be in LTE may be invalid under the
conditions where densities and optical thicknesses
become small, scattering becomes an important
mechanism, rapid time variations occur or large
temperature gradients are in evidence? 2 Therefore,
before making the LTE assumption, the conditions
for a given physical system should be carefully
examined.
The radiative transfer equation, Eq. (2.11), is an
integrodifferentiai equation, and because of this it is
very difficult to solve exactly for multidimensional
geometries. Therefore, some simplifications of this
equation are necessary. A close look at the source
term given in Eq. (2.12) reveals that the in-scattering
term (the second term of the right hand side) yields
the integral nature of the RTE. If scattering is
negligible in the medium, then the Eq. (2.11) will be a
linear differential equation, which is much easier to
solve than the linear integrodifferential equation.
A formal solution of the quasi-steady state RTE,
Eq. (2.11), can readily be written. Consider a pencil of
radiation in the direction g (Fig. 3). If the coordinate
s is laid in the direction g, the quasi-steady RTE is
given by
(V- ~)1,= ~g*= f l , ( S v - I O
(2.15)
where the direction of the pencil of rays is understood to be g. The intensity, however, may be a
103
FiG. 3. Coordinates for radiative transfer along a line-ofsight.
function of time indirectly through the source
function if qv is time dependent [see Eq. (2.12)].
Suppose that at some point on the boundary of
matter So, as shown in Fig. 3, the spectral intensity Iv
is known
lv(s)=l,(so)=lo~
at
s = s o.
(2.16)
The integral form of the equation of transfer may be
derived from the integrodifferential equation by
imagining the latter to be an ordinary differential
equation in the unknown Iv and with S, a known
source function. The integrating factor for this
differential equation is exp (Sflvds) and the integral of
Eq. (2.15) with the boundary condition Eq. (2.16) may
be written as
I v(s) = I o ffv(S,So)
i
t
¢
+ f f s (s)T (s s )fl (s)ds
v
v
~
!
v
(2.17)
where s' is a dummy variable of integration and
T(s,s') is the beam transmittance of an arbitrary path
from s' to s along the direction
T~(s,s)=exp
if':/ ,]
-
flv(Od
=
.
(2.18)
•
The concept of the beam transmittance can be made
clearer by the following interpretation. If lov represents the intensity of radiation in some direction g at
some initial point So and Iv(s) is the intensity of the
transmitted radiation at point s in the same direction
over the path from the initial to the terminal points,
then the two intensities are related by
l,(s)= T~(s,so)lov.
(2.19)
Thus, the beam transmittance represents the fraction
of the initial intensity which is transmitted without
104
R. VISKANTAand M. P. MENGO~3
emission or scattering contributions to the intensity
along the path length. Equation (2.17) gives the
spectral intensity of radiation at a point and in a
given direction. Its physical meaning can be more
readily interpreted by referring to Fig. 3. It shows
that Iv(s) is a sum of two contributions: (1) the
transmitted intensity, and (2) the path intensity. The
first term on the right hand side of Eq. (2.17) is the
contribution to Iv due to the initial intensity at point
So in the direction of propagation of the radiation g,
attenuated by the factor Tv(s,s0) to account for
absorption, scattering and induced emission in the
intervening matter. The second term results from
both emission and scattering from elements of the
matter at all interior points, each elementary contribution being attenuated by the factor Tv(s,s') while
the rest is absorbed and scattered along the path.
These elementary contributions are integrated over
all the elements between the boundary of the body s o
and the point s.
We note that the integral form, Eq. (2.17), of the
radiative transfer equation is referred to as "'formal
solution" in the sense that I v is expressed in terms of
integrals that can be evaluated only if the state of the
matter and the radiation field, i.e. Sv is known. This
does not mean that the equation of transfer in a
participating medium has been solved. It is clear that
if the source function depends on the intensity Iv in
some specified way, then one can convert Eq. (2.17)
into an integral equation for Iv. ~5 However, before
we do this it is desirable to derive the conservation of
radiant energy equation.
2.2. Conservation of Radiant Energy Equation
Integration of the RTE, Eq. (2.11), over all
directions results in
t3ail* + V',~rv=x,[4nlbv(T)--f~v]
(2.20)
dt
where the spectral radiant energy density q/v, the
irradiance aJv and the radiation flux vector "~'v are
defined as
q/=lf
l,df2
energy from an element of matter per unit of volume
and per unit of frequency. The term 4nqv(= 4nxJb,)
represents the local rate of emission, and x,ff,
represents the local rate of absorption of radiation
per unit of volume. The meaning of the terms can be
further clarified when we note that 4nlbv is the
product of the spectral radiant energy density of a
black body at the local temperature, times the local
velocity of light c, while c~v is related to the local
radiant energy density of space as defined by Eq.
(2.21b). In deriving Eq. (2.20) the scattering terms
have canceled out. This just confirms the physical
fact that scattered energy is not stored and should
not appear in the conservation of radiant energy
equation. Integration of Eq. (2.20) over the entire
spectrum results in the conservation equation of total
radiant energy
oo
0°d + V . , ~ = f 1%[4nlbv(T)-f~v]dv.
~t
~o
(2.22)
For reasons that were explained in a previous
subsection, the time rate of change of radiant energy
density q/ can be neglected. Note that there is no
convective term in Eq. (2.22), since radiation propagates inependently of the local material velocity. The
equation describing the local change of radiant
energy density must be modified in the relativistic
treatment of electromagnetic radiation.l°'23 However, the additional terms which arise in the
conservation of radiant energy equation can generally be ignored in engineering applications.
It is worth noting that the spectral dependence of
radiative properties is denoted either by subscript v
(frequency) or ~. (wavelength). If the matter through
which radiation is propagating is not homogeneous
and uniform, then the index of refraction, and, as a
result of this, the wavelength and speed of light
would be different at different locations in the
medium, whereas the frequency remains constant
everywhere. Therefore, the frequency is a more
fundamental measure than the wavelength of radiation, and because of this, here, the spectral dependence is denoted by v. It is also useful to remember the
identity, -lvdv = lad2, between frequency and wavelength based definitions of radiation intensity.
(2.21a)
C JQ=4~
2.3. Turbulence~Radiation Interaction
c~v= S lvdfl=cq/v
(2.21b)
12=4~
~,=
S 1,~dn
(2.21c)
D=4x
respectively. The physical meaning of Eq. (2.20) is
clear. It is the conservation equation of spectral
radiant energy. The right-hand-side of Eq. (2.20)
represents the net rate of loss or gain of radiant
Interaction of convection and radiation has been
recognized for some time, but the fact that turbulence
can influence radiative transfer and vice versa has
been recognized more recently. The first attempt at
combined analysis of the equations for the meansquare fluctuations of the velocity and temperature
fields with the radiation field is due to Townsend. 24
Applications in which radiation/turbulence interaction may affect flow and heat transfer include
industrial furnaces, gas combustors, flames and
Radiation heat transfer
fires.25- 3o Most studies concerned with modeling of
radiative transfer in combustion chambers and
furnaces have ignored the turbulence/radiation interaction. 3's An up-to-date discussion of the interaction
in flames is available 3~ and need not be repeated
here. Suffice it to mention that the interactions and
coupled effects are more important for luminous than
for nonluminous flames. Little is known concerning
temporal aspects of radiative transfer in turbulent
flames as these effects have not been studied
extensively.
105
Turbulence can influence radiative transfer through
fluctuations in temperature and radiating species
concentrations which, in turn, influence Planck's
function lba(T) and the special absorption and
scattering coefficients. The fluctuations of the Planck
function and the spectral absorption and scattering
coefficients can be given in terms of the temperature
and species fluctuations by means of Taylor series
expansions about the values evaluated at the mean
properties. Evaluation of the instantaneous intensity
of radiation in terms of the mean and fluctuating
TABLE1. Analytical forms of (V.~)l in common orthogonal geometries 2~
(¢= sin0cos~, tI = sin0sin~b,p = cos0)
Geometry
Rectangular
Space Direction
wmables cosincs
.\.y.:
(V. ~)1
FI
?1
?1
,~-- + ll--E- + p - -
~..q ,u
.\"
( y
,r
-\'.Y
~,q
?1
;I
~ - + 'I S ~.\~y
-
It
;I
P-5
-
(E
Cylindrical
r,dp~.:
~..q.fl
;trl)
Sphcrical
r.-
;~.q,ll
r.4~,
~.ll
r
~,l I
r.O,.q'),
¢'.q.p
1 ?UII)
F(rl)
tl FI
;1
- --~ - -+ p,
r ;r
r?#),
~-
----
r
+ It
?r
;/
I ?(ql)
;z
r
?(rl)
~1 ?l
--+
r ;r
ri'q~,
•
r ?oh
;(~
1 ?Off)
r ;¢b
;~ ?(rl)
1 ?[ql)
r ?r
r ,"~
It 70"21)
-----+
r 2 Fr
q
g.
?(sinOfl)
rsinO~ i'O,
21
I ?[{I-p2)l]
rsin0, ;q~,
r
?/I
cotO ?(ql )
r
;q~
p ~(r21)
r.O,
~dl.l I
- - _ _
~
r 2 ~r
+
rsin0,
1 ?[tl-p2)l]
r
r
p ?(r21)
r2
2r
;0,
cotO, flql)
;p
- -
;.(sin(lfl)
q - _ _
I ;[(I-/~2)I]
-b
r
?I~
;~
106
R. VISKANTAand M. P. MENGf3~:
properties and the time-averaging is straight-forward
but tedious. 27 If the absorption coefficient can be
expressed as
xa(s,t) = ~,kai Ci(s,t)
(2.23)
i
the turbulent fluctuations in the absorption coefficient can be related to those in the concentrations
Ci of the radiating species. The precise evaluation of
the time-average would utilize the joint probability
density function P(T,Ci,s) of the temperature and
species concentrations for all points s along the line
of sight g in Eq. (2.17). Unfortunately, that information is not available. Those properties of the flow
field that are available are the mean temperature T,,
species concentrations C , and the second order
correlations, T '2, C~T'. To illustrate the nature of the
problem we restrict ourselves to a single radiating
species and neglect scattering. Applying Reynolds'
averaging techniques to Eq. (2.17) but omitting the
details, one can obtain 30
s
s
ia(s) = loaexp[ - kaIC(s')ds']exp[ -- kaSC'(s')ds']
0
s
0
s
s
thin. a° We further assume that the properties of the
fluctuating eddies are statistically independent, and
this implies that there is no correlation between the
temperature and concentration within each eddy.
Under these conditions radiation is transmitted
through an eddy with little change so that the
radiance at a local point is affected little by the local
fluctuation of xx. Hence, the time-average RTE can
be approximated as a°
(V' g)la = - xala + qa.
(2.26)
Following a similar argument, the spectral radiant
energy Eq. (2.20) can be expressed as
V" "#'a = - x-a~a + 4r~Oa.
(2.27)
Information necessary to solve Eq. (2.25) for the
time-averaged spectral radiance I~ is not available,
and the integration of Eq. (2.24) along the line-ofsight is too time consuming. Some clever way of
ensemble averaging the radiance or developing
correlation coefficients for time-averaged quantities
will be required to enable solution of the integral or
differential forms of the RTE in turbulently fluctuating media. The significance of the turbulence/
radiation interaction will be assessed later.
+ I~(s')exp[ - k~ICds"] {exp[ - kaIC'ds"]
0
s'
s'
s
+(qffr/])exp[-kaIC'ds"] }ds'.
3. R A D I A T I V E
s'
This equation can be written in a more useful form in
terms of the two-point correlation coefficients. 28 The
representation of the random concentration and
temperature by Gaussian variables is convenient, but
it must be noted that they encompass unrealistic
negative values of the variables whose probability
must be kept small in proportion. Comparison of
Eqs (2.17) and (2.24) reveals that consideration of
turbulence (i.e. time-averaging) would greatly increase the computational effort of an already difficult
problem.
An alternative to time-averaging the spectral
radiance would be to time-average the quasi-steady
form of RTE, Eq. (2.15), and the radiant energy
equation, Eq. (2.20), at the start. Time-averaging of
Eq. (2.15) results in
0 7 " g ) ~ = - x a l a + rlx.
PROPERTIES
OF COMBUSTION
PRODUCTS
(2.24)
(2.25)
The difficulty with this equation is the evaluation of
the coupled correlation xala because instantaneous
I~ is expressed in terms of an integration along the
path as indicated in Eq. (2.17). To simplify the
absorption coefficient-radiance correlation x~la we
can assume that the individual eddies are homogeneous and that the radiating gas of a typical size eddy
(i.e. based on macroscale of turbulence) is optically
The accuracy of radiative transfer predictions in
combustion systems cannot be better than the
accuracy of the radiative properties of the combustion products used in the analysis. These products
usually consist of combustion gases such as water
vapor, carbon dioxide, carbon monoxide, sulfur
dioxide, and nitrous oxide, and particles, like soot,
fly-ash, pulverized-coal, char or fuel droplets. Before
attempting to tackle radiation heat transfer in
practical combustion systems, it is necessary to know
the radiative properties of the combustion products.
Considering the diversity of the products and the
probability of having all or some of these in any
volume element of the system, it can easily be
perceived that the prediction of radiative properties
in combustion systems is not an easy task. The
wavelength dependence of these properties and
uncertainties about the volume fractions and size and
shape distribution of particles cause additional
complications.
In order to present a systematic methodology for
the prediction the radiative properties of combustion
products, the discussion in this section will be
divided into several subsections in which the relations for obtaining the properties of the combustion gases and different particles are discussed and
the simplifications are introduced. Afterwards, some
relations will be given to employ these expressions as
building blocks to determine the radiative properties
Radiation heat transfer
of the mixture of combustion products. Note that
usually the level of simplification for the properties is
to be determined by the user, and it should be
consistent with the level of sophistication of the
radiative transfer and total heat transfer models.
Also, the relations for the radiative properties of
individual constituents should be compatible with
each other as well as with the radiative transfer
models.
3.1. Radiative Properties o f Combustion Gases
Every combustion process produces combustion
gases, such as water vapor, carbon dioxide, carbon
monoxide, and others. The partial pressures of these
gases in the combustion products are determined by
the type of the fuel used and the conditions of the
combustion environment, such as fuel-air ratio, total
pressure and ambient temperature. These gases do
not scatter radiation significantly, but they are strong
selective absorbers and emitters of radiant energy.
Consequently, the variation of the radiative properties with the electromagnetic spectrum must be
accounted for. Spectral calculations are performed by
dividing the entire wavelength (or frequency) spectrum into several bands and assuming that the
absorption/emission characteristics of each species
remain either uniform or change smoothly in a given
functional form over these bands. As one might
expect, the accuracy of the predictions increases as
the width of these bands becomes narrower. Exact
results, however, can be obtained only with line-byline calculations which require the analysis of each
discrete absorption--emission line produced as a
result of the transitions between quantized energy
levels of gas molecules. Line-by-line calculations are
not practical for most engineering purposes but are
usually required for the study of radiative transfer in
the atmosphere. Therefore, detailed line-by-line calculations will not be discussed here.
3.1.1. Narrow-band models
Narrow-band models are constructed from spectral absorption-emission lines of molecular gases by
postulating a line shape and an arrangement of lines.
The shape (profile) of spectral lines is quite important
as it yields information for the effect of pressure,
temperature, optical path length, and intrinsic properties of radiating gas on the absorption and
emission characteristics. The Lorentz profile 32 is the
most commonly used line shape to describe gases as
moderate temperatures under the conditions of the
local thermodynamic equilibrium, and it is also
known as a collision-broadened line profile. 33 If the
temperature is high and the pressure is low, the
Doppler line profile would be more appropriate to
use. 33 If there are ionized gases and plasmas in the
medium and they are influenced by interactions
between the radiating particles and surrounding
107
charged particles, then the Stark profile yields a more
accurate representation of the spectral line radiation. 33 Note that it is also possible to superpose
these line profiles to incorporate the effects of
different physical conditions on the line radiation.33. 3'*
There are basically two ~different line arrangements
for narrow band models used extensively in the
literature. The Elsasser or regular model assumes that
the lines are of uniform intensity and are equally
spaced. The Goody or statistical model postulates a
random exponential line intensity distribution and a
random line position selected from a uniform
probability distribution. For practical engineering
calculations both of these models yield reasonably
accurate results. Usually there is less than 8~o
discrepancy between the predictions of these two
models. 35 A detailed discussion of the narrow band
models has been given by Ludwig et al. 34 and in the
review articles by Tien 33 and Edwards. 3S
Narrow-band model predictions generally require
an extensive and detailed library of input data, and
the calculations cannot be performed with reasonable computational effort. On the other hand, as long
as the concentration distributions of gaseous species
are not accurately known the high accuracy obtained
for the spectral radiative gas properties from narrow
band models would not necessarily increase the
accuracy of radiation heat transfer predictions. Also,
it is not always convenient to use detailed, complex
models for the spectral radiative gas properties.
3.1.2. Wide-band models
Since gaseous radiation is not continuous but is
concentrated in spectral bands, it is possible to define
wide-band absorptivity and/or emissivity models.
The radiation absorption characteristics for each
band of any gas can be obtained from experiments
and then empirical relations can be fitted to those
data. The profile of the band absorption may be box
or triangular shaped or an exponentially decaying
function can be used by curve fitting. These types
of empirical models are known as wide-band models,
and among them the exponential wide-band model
of Edwards and Menard 36 is commonly used. For an
isothermal medium, several approximate expressions
for the total band absorptivity and emissivity (see
Refs 37-40) as well as the reviews of the wide-band
models are available in the literature. 35.41 -43
Recently, Yu et al. 44 have devised a new "'superband" model to correlate total emissivity and Planck
mean absorption coefficient data of infrared radiating gases. In this model, the Edwards exponential
band model has been used to approximate the
emissivities. The spectral lines of the various infrared
absorption bands of a radiating gas are rearranged
and combined into a single, combined band.
108
R. VISKANTAand M. P. M~GO~
In Figs 4 and 5, the spectral band absorptivity
distributions from a narrow band model a'* are
compared with those from a wide-band model 35 for
two isothermal media. 45 In general, the wide band
model is in good agreement with the narrow band
model, especially for-2.7 /~m H 2 0 and CO2 bands
(co=3700 cm-Z), 4.3 pm CO2 band (to~2300 cm-1),
and 6.3/~m H 2 0 band ( t a ~ 1 6 0 0 c m - ' ) ~ In these
figures, the normalized Planck blackbody function
corresponding to the temperature of the medium is
also plotted to show the relative contribution of each
gas band to the total radiation absorbed by the
'[I ~ ~ - - - r
,,
too,o
~.0 [i
Q
(%)
medium. It is clear that the relative importance of
short wavelength band radiation (i.e. from 1.38 pm
(oJ~7000 cm - t ) and 1.89/zm (ta~5300 cm - t ) H 2 0
bands) becomes larger as the temperature of the
medium increases (see Fig. 5 for T = 2 0 0 0 K). The
error introduced by approximating the short wavelength band absorption by wide-band models is
marginal, since the temperature of a typical combustion chamber is usually not as high as 2000 K, and
the other gas bands absorb radiation more strongly
than short wavelength bands.
It should be mentioned that some of the detailed
I
tS
'l! /
SO.O
f v ,l
:
,.,..,-
.....I,
WB
2S.O
"~"*"
0.0
o
2000
qooo
6o00
8000
t~>.~
w (cm-')
FIG. 4. Spectral absorptivities of H20-CO2-air mixture calculated from the narrow band (NB) and the
wide band (WB) models, spectral soot absorptivities (],,A = 1.0 x 10- 7 ma/m 3 and j~,.2= 1.0 x 10 - (' m3/m 3)
and normalized Planck's function (Iba/lha.m.): T = 1000 K, P( = 1 atm.,/M20 = Pco2= 0.1 atm., L = I m.
100.0
--
.0
a
I
///
m.o
I
/l
/
•
H.I
%,
o
~ , ~
" Y
,
kq: t
200o
t...:
nN
,,.,
d~' ~
~
~'
rl
,:
qooo
~
~
l(XXX)
o~ ( c m " )
FIG. 5. Spectral absorptivities of H20-CO2-air mixtures as calculated from the narrow band (NB) and
wide band (WB) models, spectral soot absorptivities I]i,.t = 1.0 x 10 - 7 m3/m "*and J,,.., = 1.0 x 10- ~' m "~m "~)
and normalized Phmck's function (lh,t/lha.=**): T=2000 K, P,= 1 atm., ptt2o=Pco2=O.1atm.. L=0.5 m.
Radiation heat transfer
spectral properties of the combustion gases will be
suppressed when they are combined with those of the
particles. Because of this, use of very accurate
spectral properties of gases may not increase the
accuracy of radiative transfer predictions. In Figs 4
and 5 the soot absorptivities are plotted for two
different soot-volume fractions. 4s Note that if
Iv =Jv.~ = 1.0 x 10-7, then the gas and soot absorptivities are of the same order of magnitude, especially
for longer wavelengths. However, asJ~ increases (see
the curves for J~.2=l.0 x 10-6), the soot absorption
becomes dominant. The soot absorptivity also increases with increasing wave number, i.e. decreasing
wavelength, since soot absorption coefficient is
almost inversely proportional to the wavelength of
radiation; we will return to this topic later.
3.1.3. Total absorptivity-emissivity models
A detailed modeling of the radiative properties of
combustion gases may not be warranted for the
accuracy of total heat transfer predictions in combustion chambers, but definitely increase the computational effort. An in-depth review of the world
literature on the thermal radiation properties of
gaseous combustion products (H20, CO2, CO, SO2,
N O and N 2 0 ) has recently been prepared,'* and
therefore the discussion will not be repeated. For
engineering calculations it is always desirable to have
some reliable yet simple models for predicting the
radiative properties of the gases. Here, we review
some of the available models.
One way of obtaining radiative properties easily is
to use Hottel's charts which are presented as
functions of temperature, pressure and concentration
of a gas. '.6 Some scaling rules for the total absorptivity and emissivity of combustion gases can be used
to extend the range of applicability of Hottel's charts.
For example, the scaling rules given by Edwards and
Matavosian 47 can be employed to predict gas
emissivity at different pressures as well as gas
absorptivity for different wall temperatures and at
gas pressures different than one atmosphere. Of
course, in order to use these charts in computer
models, curve-fitted correlations are desirable. Other
sources for continuous expressions are the narrow
and wide band models. The spectral or band
absorptivities from these models are first integrated
over the entire spectrum for a given temperature and
pressure to obtain total absorptivity and emissivity
curves. Afterwards, appropriate polynomials are
curve-fitted to these families of curves at different
temperatures and pressures using regression techniques. Sometimes, these curve-fitted expressions can
be so arranged that the resulting expressions would
be presented as the sum of total emissivity or
absorptivity of clear and gray gases. These are known
as the "weighted sum-of-gray-gases" models and are
given as '.6
109
I
~= ~, as.i [ I - e - ~ , P L ] .
(3.1)
i~0
The weighting factor ae,~ may be interpreted as the
fractional amount of black body energy in the
spectral regions where "gray gas absorption coefficient" xi exists, and they are functions of temperature. Usually the absorption coefficient for i = 0 is
assigned a value of zero to account for the transparent windows in the spectrum. The expressions for
the total emissivity and absorptivity of a gas in terms
of the weighted sum of gray gases are useful
especially for the zonal method of analysis of
radiative transfer.
There are several curve-fitted expressions available
in the literature for use in computer codes. Some of
them are given in terms of polynomials 4s- 50 and the
others are expressed in terms of the weighted sum-ofgray gases. 5~-54 In only two of these expressions
soot contribution is accounted for along with the gas
contribution. 49's° All of these models are restricted
to the total pressure of one atmosphere, except that
of Leckner, 48 and all of them are for the gas
radiation along a homogeneous path, i.e. uniform
temperature and/or uniform pressure.
If the path is inhomogeneous then the equivalent
line model 39 or the total transmittance nonhomogeneous method s5 can be used to predict
radiation transmitted along the path. However, in
multidimensional geometries or if scattering particles are present in the system, the use of these
models for practical calculations becomes prohibitive as the equations are much more complicated.
3.1.4. Absorption and emission coeJflcients
The total absorptivities and emissivities are useful
for zero or one-dimensional radiative transfer analyses as well as zonal methods for radiative transfer.
However, for differential models of radiative transfer
the absorption and emission coefficients are required
rather than the total absorptivities and emissivities.
Since scattering is not important for combustion
gases (and soot particles), the gray absorption/
emission coefficient can be obtained from the
Bouguer's or Beer-Lambert's law. For a given mean
beam length Lm one can write
~= ( - 1/L,,,) In (1 -e).
(3.2)
The mean absorption coefficients obtained from
spectral calculations as well as curve-fitted continuous correlations were compared with measurements
from a smoky ceiling layer formed in a room fire and
very good agreement was found. 4a It is possible to
determine the so called "gray" absorption and
emission coefficients for each temperature, pressure,
and path-length, which yield approximately the same
total absorptivity or emissivity of the C O 2 - H 2 0
mixture.
110
R. VISKANTAand M. P. MENG0~:
It is worth noting that instead of using only the
absorption coefficient, absorption as well as emission
coefficients should be employed. Since the total gas
emissivity differs from total gas absorptivity, it is
quite logical to define and use two separate coefficients. The importance of this fact has been first
discussed by Viskanta# 6 He has shown that the
arbitrariness associated with an absorption coefficient can be eliminated by the introduction of a
mean emission coefficient and a mean absorption
coefficient, which can be related to the spectral
absorption coefficient by the following definitions:
decreases the gas becomes thinner, and eventually in
the limit of optically thin gas the mean absorption
coefficient becomes identical to the Planck's mean
absorption coefficient. With an increasing size of the
enclosure, the gas becomes optically thicker and the
mean absorption coefficient approaches Rosseland's
mean absorption coefficient.
Planck's and Rosseland's mean coefficients are
independent of the beam length and are valid only in
the thin and thick gas limits, respectively. They are
defined as
oo
oo
~s =
~ ~:a~ad)~/~ ffad2,
0
o
ao
1/-~a= ~(lflcz)(dlbz/dT)d,l/~(dlbz/dT)d,t.
~,= Sgan2Ebxdg/Sn~Ebad2.
(3.5)
0
0
ao
0
oo
"Kp= S K;.Ibl,dJ./ S I b~,d~
oo
(3.3)
0
(3.6)
0
o
Here, ~a is the spectral irradiance. If the index of
refraction na of the medium is unity, then the mean
emission coefficient will be equivalent to Planck's
mean absorption coefficient, s6 Also, if Ka is independent of wavelength or the medium is in radiative
equilibrium, i.e. ffa=n]Ebx for all wavelengths, then
the mean emission and absorption coefficients will be
equal to each other.
The use of these mean coefficients is justified as
long as there are no large temperature gradients in
the medium, s6"57 Therefore, they can be calculated
separately for each zone where the temperature can
be assumed uniform. If the soot volume fraction is
high in the medium, the use of the mean absorption
coefficient would be sufficient, since ~ca (for soot
+combustion gases) would be a weak function of
wavelength.
In order to determine the absorption and emission
coefficients from total absorptivity-emissivity data,
the corresponding mean beam length must be
properly evaluated. The definition of the mean beam
length for a volume of a gas radiating to its entire
surface is given as
L,,, = 4 C V/A,
(3.4)
where C is the correction factor and for an arbitrary
geometry its magnitude is 0.9. 46 In general, the
absorption and emission coefficients are functions of
the medium temperature, pressure and gas concentrations. If these coefficients are obtained from the
total emissivity and absorptivity models, they will
also be functions of the mean beam length. Therefore,
if the total emissivity of a gas volume is fixed, then
the corresponding absorption coefficient decreases
with increasing physical path length or pressure [see
Eq. (3.2)]. The use of absorption/emission coefficients
related to the mean beam length is convenient for the
scaling of radiation heat transfer in practical systems,
As the characteristic dimension of the enclosure
Similar to Rosseland's mean absorption coefficient,
we can define Pianck's internal mean coefficient 35 as
0o
oo
-~,= Sr.~,(dIb~,/dT)dg/S(dIh~,/dT)d2
0
(3.7)
0
which is also appropriate for an optically thick
medium. Several other definitions of the mean
coefficients were discussed in greater detail by
Traugott. 5
Another mean absorption coefficient was defined
by Patch s a as
at)
oo
~¢(L) = Slbagzexp(- r2,L)d2/~lb~,exp(- gaL)d2.
0
(3.8)
o
Unlike the first three mean absorption coefficients
defined above, this so-called effective mean coefficient is a function of path length as it contains the
beam transmittance [see Eq. (2.18)] in its definition.
Therefore, Eq. (3.8) is expected to yield more accurate
predictions for the absorption coefficients of gas
mixtures having intermediate optical thicknesses
provided that the path length is known.
It is also possible to write a mean absorption
coefficient based on the narrow band model of
Ludwig et al., 34
k~u
+~-aJ
(3.9)
where k, are tabulated coefficients, u is the product of
the mean beam length and total pressure, and a is the
fine structure parameter. Note that this expression is
also a function of path length.
The mean absorption coefficients for a water
vapor-carbon dioxide-air mixture at two different
temperatures are presented in Fig. 6 as a function of
path length. 45'59 The mean absorption coefficient
(~q.w,,) as calculated from Modak's model, '.9 using the
Radiation heat transfer
111
o
_
-
_
K
(~-,)
.... -~..-~,~..
~
.,~:.-,,i
T=,OOOK
2-
,~
~P.~
®
,q
-
®Ke
..~.~\
(E) Kl,n
"'..'.%
I ----
--
® ~,,..
'''%'%_%~
T= 2 0 0 0 K
,
10-4
®
,
I
I0-s
l
--
" . "..N,~
- .
i
i
i
-
I0"2
i
I
t 't-v-r-~-q--=~-~
IO-I
I
I0I
L (m)
F]~J. 6. Comparison of different gas aborption coefficients as a function of pathlength: P,= 1.0 alto.,
Pn,o=Pco. =0.I atm. "~
Felske-Tien 6° wide-band model, is in good agreement with the absorption coefficient calculated from
the narrow-band model (Kt.,) or Patch's effective
mean absorption coefficient (~). In this figure, xe., is
Planck's mean absorption coefficient based on the
narrow-band model, 34 xe.w and ri are Planck's mean
and internal mean absorption coefficients, respectively, based on the wide-band model. 35 The mean
absorption coefficient xt.we is a function of pathlength and is calculated using Edwards' wide-band
model parameters.
3.1.5. Effect of absorption coefficient on the radiative
heat flux predictions
In preceding sections, we have compared absorption coefficients calculated from spectral narrow
band models with those obtained from total emissivity models as well as with the Planck mean and
internal mean absorption coefficients. It is also
desirable to examine the effect of different definitions
of absorption coefficients on radiative transfer predictions. For this reason, an axisymmetric cylindrical
enclosure is considered. It is assumed that the
medium is a homogeneous, uniform gas ( H 2 0 - C O 2air) mixture at atmospheric pressure. The partial
pressures of water vapor and carbon dioxide are the
same and equal to 0.1 atm., and the medium
temperature is either 1000 K or 2000 K. The enclosure walls are assumed to be at a temperature of
600 K and diffusely emitting, with emissivity ew=0.8.
Two different sets of dimensions for the cylindrical
enclosure are examined. The first one has a mean
beam length (Lm = 3.6 V/A) of 0.5 m, where ro = 0.4 m,
and Zo=0.9m. For the second one, ro=0.9m,
z0 = 3.0 m, which gives Lm= 1.08 m. The solution of
the radiative transfer equation is obtained using the
P3-approximation,61 which will be discussed in the
next chapter. Radiative transfer calculations are
performed on the spectral basis using the wide-band
model of Edwards and Balakrishnan (see Edwards; 3s
Table X). The thirteen spectral bands used for the
absorption coefficient are shown in Figs 4 and 5 by
dotted lines.
In Fig. 7 the radiation heat flux distributions
on the cylindrical walls of the small enclosure
(L,~=0.5 m) are given for two different medium
temperatures. It is clear from these figures that the
use of the Pianck mean absorption coefficient yields
about six times higher radiative fluxes compared to
the detailed spectral calculations. On the other hand,
the mean absorption coefficients calculated from the
total emissivity model of Modak 49 yield only a small
overprediction of radiative fluxes in comparison to
the spectral results, and the use of Planck's internal
mean absorption coefficients slightly underpredicts
the radiative flux distribution along the wall. In Fig.
8 the same kind of comparisons are given for the
second enclosure, which has Lm= 1.08 m. Basically,
the trends are the same as those shown in Fig. 7,
however, the agreement between spectral and total
calculations is better in this case.
Indeed, the trends in the results predicted using
different absorption coefficients, as illustrated in
these figures, can be also deduced from the comparisons of the absorption coefficients given in Fig. 6.
For example, for Lm=0.5 m, at T = 1000 K, xt.,., is
somewhat larger than the x~ but it is about six times
smaller than the re. This is also evident from Figs 7
and 8. From Fig. 6 we can conclude that the use of
112
R. VISKANTAand M. P. MENGOq
50
I
I
I
I
I
0) T= IO00K
i
I
b ) T = 2000 K
250
A
40
200
x
30
/
o
"0
o
n~
KP,w
\
/
L~.
o
"1"
/
\
\
-
150
20
I00
........
KsplctroI
\Kspsctre
~ ' ~
lO
..........
I
50
I
0
I
I
I
I
0
t
2
,
I
0
t
i
2
0
Z/ o
FI6. 7. Comparison of radiative flux distributions on the cylidrical walls calculated spectrally and using
different mean absorption coefficients,L=0.5 m (see text for the delinitions).
50
!
I
I
I
A
500
I
400
4O
/
\
x
I
b) T = 2 0 0 0 K
a) T = I 0 0 0 K
30
300
-1- 20
200
zo
,~~KItWm
spsctrel
I00
spectre
S
Ki
0
0
I
I
I
I
2
3
0
I
I
I
I
2
3
0
4
Z/4o
FIG. 8. Comparison of radiative flux distributions on the cylindrical walls as calculated spectrally and
using three different mean absorption coefficients,L= 1.08 m (see text for the definitions}~
Planck's mean absorption coefficient would be
acceptable only if the physical dimension or the total
pressure of the system under consideration was very
small.
The spectral radiative fluxes depicted in Figs 7 and
8 do not always yield identical results with those
calculated from other mean absorption coefficients
such as r,,w, or r~, and the difference between them
may be as much as 100 %. Clearly it is difficult to have
a simple correlation between the radiative transfer
predictions obtained from the spectral and gray
analyses. In some earlier parametric studies it has also
been shown that the change of the center and width of
the spectral absorption bands may yield large variations of the total radiative flux predictions. 57,62
Since the temperature and characteristic length of the
gas volume have a strong effect on both the center and
the width of the bands, in practical systems the gas
radiative properties are expected to show large
differences from location to location. Use of a single,
Radiation heat transfer
40
50
qr
2-
(kW/m2) u
I0
0
2
4
6
8
I0
z(m)
F~;. 9. Comparison of radiative fluxes at the wall based on
spectral and mean absorption coefficient calculations.
Water-vapor and carbon-dioxide only: "a" from all six
bands: "b" for 2.7 and 6.3 pm H20 and 2.7 t+m and 4.3/~m
CO 2 bands: "c'" for 2.7/tm H20 and 2.7 pm and 4.3/~m
CO2 bands: "d'" for 2.7/~m and 6.3/~m H20 and 2.7 l~m
CO2 bands: "'e" 6.3 pm H20 and 4.3 :~m CO2 bands; "f" for
Planck's mean absorption coefficient;"g" Planck's internal
mean absorption coefficient;+'h" for Edwards" wide-band
model.59
mean absorption coefficient for combustion gasmixtures, in which large temperature gradients exist,
is not expected to predict radiative transfer realistically. Consequently, gray calculations employing the
mean absorption coefficients are not recommended
for predicting radiative transfer in a medium comprised of only combustion gases, if good accuracy is
required.
It is desirable to discuss the contribution of each
major CO2 and H 2 0 band on the radiation heat
fluxes. Figure 9 depicts the radiation heat flux
distributions on the cylindrical wall of a combustion
chamber calculated spectrally as well as using mean
values. 59 The contributions by particles have been
neglected in obtaining the results presented in this
figure in order to determine the relative importance
of each gas band. Only water vapor and carbon
dioxide are assumed to be present. The mole fraction
distributions of these gases in the furnace were
obtained from the literature 63 for burning of lowvolatile coal (anthracite); therefore, the water-vapor
fraction in the medium was not high. The absorption
coefficient of the gas mixture in every zone of the
medium is calculated from
Edwards and
Balakrishnan's wide band model. 35'39 Each spectral
band corresponding to a different zone has a
JPgCS 1 3 : 2 - B
113
different band-width because the temperatures are
different in each zone. In the calculations an average
band-width of each spectral band was employed.
Then, the intensity of each band was adjusted
accordingly. The water vapor rotational band was
not included in these calculations.
In Fig. 9 curve "a" stands for the radiative heat
flux distribution obtained, including all six spectral
gas bands, i.e. 1.38, 1.87, 2.7, 6.38/~m H20 and 2.7,
4.3/~m CO2 bands. This curve is considered as the
"benchmark" for the purpose of comparisons here.
In order to determine whether it is necessary to
include all the bands or not, the number of spectral
bands used is reduced systematically:9 It is worth
noting that if all three minor bands (i.e. 1.38 pm,
1.87 pm and 6.3 #m H20 ) are neglected the error
introduced would be on the order of 1 0 - 2 0 yo;
however, neglect of either of the major bands (i.e.
2.7/~m H20, 2.7/~m and 4.3 #m CO2) in addition to
the minor ones (see curves "c", "d" and "e") would
yield up to 50 % smaller radiation heat fluxes.
For most practical calculations simple "mean"
absorption coefficients are widely used and preferred
over the detailed spectral radiative properties of
combustion gases. Therefore, it is desirable to
compare the accuracy of the results predicted using
the mean coefficients with the benchmark results. In
Fig. 9, the radiative heat flux distributions calculated
using Planck's mean absorption coefficient (curve
"f"), Planck's internal mean absorption coefficient
(curve "g"), and mean absorption coefficients obtained from the wide band model (curve "h") are
shown. The radiative flux denoted by curve "f" has
been multiplied by a factor of 0.4 to include it on the
same figure; therefore, the results obtained using
Planck's mean absorption coefficient are not in
agreement with the spectral calculations. Although
curves "g" and "h" yield 20-30 % errors in comparison to "benchmark" curve "a", they agree better with
the spectral results than those based on Pianck's
mean.
3.2. Radiation Properties of Polydispersions
Analysis of radiation heat transfer in coal-fired
furnaces, combustion chambers, and other utilization
systems requires accounting of the effects of particulates, such as pulverized coal, char, fly-ash and soot,
which are present in these systems. For this reason, it
is necessary to have a knowledge of the radiative
properties of polydispersions which, in turn, depend
on the particle size distribution, the spectral dependence of the complex index of refraction, and the
number density for each type of particle in the
combustion products. It is also necessary to know the
spatial distribution of all the particles in the
combustion chamber. Even with all the data on hand,
it is difficult and time-consuming to predict the
radiation characteristics required in radiation heat
114
R. VISKANTAand M. P. MENGOI~
transfer analysis. Most of the time, some simplifying
assumptions are made to reduce the difficulties;
however, the simplifications must be reasonable for
realistic modeling of physical processes.
With the increasing coal utilization, the need for
radiative properties of particles formed in coal-fired
combustion systems has become more demanding. A
state-of-the-art review of the type of particles and
their effect on radiative transfer in combustion
chambers has been given by Sarofim and Hotte164
and Blokh. 4 By assuming that particles are homogeneous and spherical, the radiation characteristics
of a cloud of particles can be predicted from the Mie
(or Lorenz-Mie) theory. 65'66 It should be noted that
pulverized coal (char) and other particles which exist
in combustion chambers are neither homogeneous
nor spherical. 67 Nevertheless, the extension of the
Mie theory to nonspherical (i.e. cylindrical, ellipsoidal) particles has shown that the radiation
characteristics of a cloud of irregular shaped particles
are not very sensitive to the geometrical shape of the
particles. 66,6a Therefore, the use of the equivalent
spherical particles assumption and the Mie theory
for coal combustion systems appears to be a
reasonable compromise. In this section the methodology for calculating the radiative properties of
polydispersions is given. Some simple expressions
are also suggested for use in practical calculations.
Following the Mie theory, the spectral absorption,
extinction, and scattering coefficients needed for
radiative transfer analysis can be evaluated from the
equation,
oo
qa(ha,N) = I Q~(D,A,haX1tD2/4)f(D)NdD,
(3.10)
0
where qa stands either for spectral extinction coefficient fla, the spectral absorption coefficient xa, or for
the spectral scattering coefficient tra, and Q, is the
corresponding efficiency factor which is a function of
the size (diffraction) parameter (x=nD/2) and the
wave-length of radiation 2. Here, h a is the refractive
index of particles, N represents the particle density,
and f(D) is the normalized size distribution function,
oo
§f(D)dD= 1.
(3.11)
0
For most practical problems, a discrete size distribution of polydispersions is required. Hence, it is
better to replace the integral of Eq. (3.10) by a finite
series. Then, using the "step-size distribution", the
radiative properties can be expressed as
qa(na,N) = ~Q,v,,
(D,,2,haX~D~/4)fi(D,)NAD,,
(3.12)
1
where i designates the diameter ranges. The mean
diameter of particles in the cloud can be obtained as
co
ao
D,,= If(D)DdD/ If(D)dD
0
0
= ZJi(Di)DiADi/~.ji(Di)&d)i
i
(3.13a)
i
which is also expressed as D~ o or rto if a mean radius
is needed. Other definitions of the mean diameter
(radius) are also used in the literature, 69 including the
Rosin-Rammler mean and Sauter mean, which is
given by
oo
co
032 = Jf(D)D3dD/Jf(D)D2dD.
0
(3.13b)
0
Sometimes this definition of the Sauter mean is
modified to express it as volume to surface area ratio.
3.2.1. Types and shapes of polydispersions
In combustion chambers, soot, pulverized coal,
char, and fly-ash are the polydispersions to be
considered. Soot is one of the most important
contributors to radiation heat transfer in practical
systems. Typical diameter of the soot particles is
about 30 nm to 65 nm, 64 yet the sizes of soot
aggiomorates may be much larger.'* Mainly because
of the small size of the soot particles, scattering of
radiation by soot is negligible in comparison to
absorption, and its radiative properties can be
calculated easily provided that the complex index of
refraction and volume fraction distribution data are
available. Numerous experimental studies (see, for
example, Refs 4, 70-73 for citations) have reported a
complex index of refraction as well as volume
fraction data of soot. Recently, Felske et al. 74
discussed the effect of different soot particle shapes
on the scattering characteristics of radiation and
presented a framework for determining the characteristics of soot agglomerates using those of spherical
particles. They demonstrated the sensitivity of the
soot radiative properties on the inhomogeneity of
the particles by using the coated sphere model (see
Subsection 3.2.3).
The spectral complex index of refraction is the
most fundamental optical property required to
calculate radiative characteristics of polydispersions.
It is computationally time-consuming to take into
account the dependence of the index of refraction on
wavelength, but conceptually it is straight-forward.
In the literature, there are some published data
for the complex index of refraction of various
coals. 69,75,76 An extensive compilation of the complex index of refraction data, including that for
different Soviet Union coals, is also available.4 The
early experiments for determining ha were usually
based on the "Fresnel reflection" method. More
recently, Brewster and Kunitomo 76 proposed a new
method, the so-called "particle extinction" technique.
Radiation heat transfer
Their results show that there is approximately one
order of magnitude difference between the impinging
part of the complex index of refraction measured
with these two techniques. In brief, there are large
differences between the reported spectral data for
the complex index of refraction of coal particles
reported by different investigators, 4'69'7'~-76 and,
therefore, more research attention is needed in this
area. It is also bdieved 64 that the radiative properties of char particles do not show distinctive
differences from those of other pulverized-coal
particles. Unfortunately, to the authors' knowledge,
there is no fundamental study which supports this
conclusion for various coals and at different wavelengths of radiation.
The contribution of fly-ash particles to radiation
heat transfer in pulverized-coal flames exceeds that of
combustion gases or soot substantially; 4 therefore,
special attention must be given to the radiative
properties of these particles. Although limited, some
data for radiative properties of fly-ash particles have
been reported in the literature. 4"77- s3
The refractive index of fly-ash is sensitive to its
chemical composition, and this is attributed primarily
to the varying amounts of oxides of silicon, aluminium, iron and calcium (i.e. SiO2, A1203, Fe203 and
CaO) in the ash. The experimental studies have
shown that the index of refraction of different fly-ash
samples from the same flame may be drastically
different, probably indicative of the microscopic
conditions for their formation. 77 According to Wall
et al. 78 the complex refractive index of fly-ash is in
the range from ha= 1.43 -0.307i to ha= 1.50-0.005i.
These numerical values of the imaginary part of the
complex index of refraction correspond approximately to the values measured by Blokh, 4 whereas
the values of the real part of the complex index of
:refraction are somewhat lower than those reported.
The imaginary part of the refractive index of fly-ash
particles formed during combustion of pulverizedcoal in a fluidized-bed furnace was of the order of
0.01. 76 This clearly indicates the uncertainty in the
complex index of refraction of fly-ash particles
formed in pulverized-coal combustion systems.
Recently, Goodwin 83 has reported extensive results
of an experimental study of the bulk optical constants of coal slags. The effects of chemical composition, wavelength, and temperature were examined.
Both synthetic slags, prepared from oxide power
mixtures, and "natural" slags, prepared by re-mdting
fly-ash or gasifier slag, were used. Transmittance and
near-normal reflectance measurements were made on
their polished wafers cut from the slags, from which
optical constants were determined. The imaginary
part of the refractive index was shown to depend
primarily on iron, silica and OH content of the slag.
Iron is primarily responsible for absorption in the
short-wavdength infrared region (1 #m<3.<4/Jm),
and silica is responsible for absorption at longer
(.k> 4/~m) wavelengths. The dependences of both the
115
real and the imaginary parts of the refractive index
on composition were also examined. A semi-empirical
mixture rule was developed to allow prediction of the
real part of the refractive index from 1 #m to 8 #m in
terms of the weight percents of the major oxide
components SiO2, Al2Oa, CaO, MgO, TiO2, and
Fe203. The mixture rule is based on the refractive
indices of the pure oxide components, with two small
modifications to improve the agreement with the
measured refractive index data.
Shape of a particle is another important independent parameter that should be considered in predicting the radiative properties. For the particles in
combustion chambers, it is difficult to imagine a
single, unique shape. Usually shapes of pulverizedcoal particles or soot agglomerates are irregular and
random; yet, sometimes, surprisingly uniform and
simple shapes are observed. For example, fly-ash
particles from coal-fired boilers show fairly smooth,
spherical shapes, a4"8~ The soot, on the other hand,
may agglomerate to form relatively long tails of radii
on the order of the coal particle radius due to the slip
velocity between the coal particle and surrounding
gases. 86's7 These tails can be considered as infinitely
long cylinders. The simple shapes are most desirable
for the simplicity of calculations as the computational
effort is reduced significantly for uniform, symmetric
shapes. However, a large fraction of particles suspended in combustion products have totally irregular
shapes. Experimental measurements show that there
are some differences in the scattering properties of
these particles in comparison to Mie theory calculations, as where for irregular shape particles: (a)
oscillations of efficiency factors vs angle and vs size
parameter are damped; (b) more side scattering
(60°-120 °) is observed; (c) less backscattering is
observed and; (d) the agreement with Mie theory
becomes worse for other radiative properties as the
size parameter increases past x = 3 or 5. sa For a cloud
of irregular shape particles, however, the observed
differences in comparison to those for spherical
particles are less significant. 66'a8
3.2.2. Prediction methods of the particle radiative
properties
When radiative properties of particles are needed,
the following quantities, arranged in order of
increasing complexity are to be considered: 8s (i)
extinction cross-section, (ii) scattering cross-section,
(iii) absorption cross-section, (iv) single-scattering
albedo, (v) radiation pressure cross-section, (vi)
asymmetry factor, (vii) unpolarized phase function,
(viii) Legendre coefficients of unpolarized phase
function, (ix) parallel and perpendicularly polarized
scattered intensities, (x) Stokes parameters, (xi)
Mudler matrix, and (xii) Legendre coefficients of
Mueller matrix dements. The last four quantities in
this list may not be critical for studying radiative
transfer in combustion systems. However, the other
116
R. VISKANTAand M. P. MENG0~
quantities are definitely needed for radiation heat
transfer calculations.
One of the most extensively used models to predict
the radiative properties of particles is the Mie
theory. 65'~6 Although it is widely known by this
name after Mie's exact solution of Maxweil's equations for the scattering of an incident plane wave on
a sphere, s9 the solution was also obtained independently by Lorenz and Debye (see Kerker 66 for detailed
historical discussion). The exact solution for a right
circular cylinder with radiation incident normal to
the cylinder axis was given by Rayleigh. Basically, in
the Mie theory the vector Heimholtz equation is
solved exactly by expanding the electric field in an
infinite series of eigenfunctions. In general, these
series are double series, and they are not easy to
evaluate; however, for spheres and infinitely-long
circular cylinders they can be reduced to single series,
and exact solutions can be obtained. The Mie
theory for spheres has been treated extensively in
the literature, 65'66'9° and some formulations for
cylinders,9°-9'* for elliptic cylinders 95 and for
spheroids 96 have been given. There is no need to
repeat the details of Mie theory here; the interested
reader is referred to one of the classical refere n c e s 6 5 ' 6 6 ' 9 0 o n the subject.
The Mie theory has been used extensively, especially during the last two decades, with the help of
the computer algorithms which have been developed 9 0 ' 9 7 - 9 9 a s well as widespread use of digital
computers. Its restriction to simple, smooth particles
has led researchers to investigate some other possibilities to model the scattering of radiation by
irregular shaped particles. Several new approaches to
the solution of the problem have been proposed over
the years, including exact differential equation
approaches ~oo.lot as well as exact integral equation
methods. 1°2-~°* In addition to these, there are
several approximate techniques available, including
the geometrical theory of diffraction ~°s for predicting the scattering by sharp-edged particles; the
method of moment for scattering by a perfectly
conducting body; 1°6 as well as perturbation1°7 and
point matching methods l°s for nearly spherical
particles. Some empirical models have also been
proposed and shown to be very accurate provided
that some experimental data are available, t°9 The
details of these methods and others can be found in
the literature, ss'9°A t o
Among these models, the integral equation method
or as more widely known, T-matrix or extended
boundary condition method (EBCM), 1oz- ~o, seems
to be the most promising as it is capable of solving
the scattering of radiation by any irregular shape
particle. In the EBCM, the incident and scattered
electric fields are expanded in vector spherical
harmonics, and then by making use of analytic
continuation techniques the integral representation
of the fields is reduced to a set of linear algebraic
equations. The complexity of these equations increase
with the complexity (or asymmetry) of the shape of
the particles. Recently, Wiscombe and Mugnai as.l i l
developed a vector algorithm for the EBCM code of
Barber t°4 and obtained the scattering properties for
various axisymmetric particles whose shapes are
determined from Chebyshev polynomials. Their
results show that there are significant differences
between the radiative properties of spheres and
arbitrary shaped particles depending on the irregularity of the surface characteristics. The computational time required for these calculations is too
formidable as to justify the extensive use of the Tmatrix method for practical problems.
3.2.3. Simplified approaches
One of the simplifications usually made in calculating the radiative properties of particles is related
to their shape. If it is possible to assume that the
particles are spherical, then exact solutions from Mie
theory can be obtained effectively and with much less
computational effort in comparison, for example, to
the T-matrix method. The properties of irregular
shaped particles can be obtained by assuming them
as equal-volume spheres if the size parameter
(x = riD~A) is small or equal-projected-area spheres if
the size parameter is large, as The nonsphericity of
particles can be traded off against inhomogeneity by
assuming that the index of refraction varies from the
core to the periphery. 66 By picking a functional form
for this variation that allows a reasonably simple
radial solution with one or two adjustable parameters, it may be possible to match nonspherical
scattering properties. Then, the solution for an
inhomogeneous sphere can be obtained rather than
for an irregular-shaped particle, and this is significantly simpler. It is also worth noting that the effect
of shape becomes less critical if there is a size
distribution of particles, as size-averaging in obtaining the radiative properties "washes out" the fine
details of nonspherical scattering. 66.as
The Mie calculations for the efficiency factors of
spheres are relatively less time-consuming and easier
to use than the other exact models. However, the size
of the particles in combustion chambers are functions of time and space, and the properties must be
calculated for each new set of size distributions. In
multidimensional and spectral radiative transfer
analyses use of Mie codes for this purpose is
impractical. Because of this, it is desirable to have
simple approximations for the efficiency factors. One
such approximation has been given by Mengii~ and
Viskanta, lt2 where the efficiency factors for polydispersions are obtained starting from the anomalous diffraction theory ~5 and are expressed in convenient, closed form. In Fig. 10 the Mie theory
predictions for the normalized extinction and scattering coefficients are compared with those of the
simplified model, and in Fig. 11 the predictions of
Radiation heat transfer
117
x F(~)
I0 0
tO-t
I0 ~
10-7
........
I
........
i
tO i
'
' ' ..... I
I0 e
........
I0 q
IO :
'
I
• ' .....
I
10-.8
........
tO 6
10-7
......
I
lO-a
[n~q
10-e
®
CP,RBON
a
P,NTHRRCI TE
+
BITUMINOU$
x
BITUMINOUB-K
•
LIGNITE
•
FLY-fiSH
[m"]
i0-9
i0-I0
lo-iO
lO-ii
........
lO-t
i
lO o
........
i
10 I
........
i
10
........
I
!
lO
........
3
,
10
........
,I
i
lO
......
lO-ii
5
10
e
xF(~)
FIG. 10. Comparison of Mie theory results (points) for the normalized extinction and the scattering
coefficients with those calculated from tin approximate analysis (lines).tj 2
1.000
c) CRRBON
0.800
COx
,i.~
0.600
,..OOO&•
A
P,NTHRRCITE
+
BITUMINOUS
X
BITUMINOUS-I(
•
LIGNITE
+
FLY-fiSH
I0 ~
I0 :
........ I
I0 "2
I 0 -i
IO o
10
i
IO N
x F('~)
FIG. 1I. Comparison of Mie theory results (points) for the single scattering albedo with those calculated
using approximate analysis (lines).~t 2
the scattering albedo from the Mie theory and the
simple model are given.l~ 2 In these figures, flz and aa
are normalized spectral extinction and scattering
coefficients, respectively. The normalization factor is
NxF(ha), with N being the number of particles per
unit volume, x is the size parameter, and F(h,t) is a
function of the complex index of refraction. Note
that
Q,=xF(ha)=x
F
24naka
]
2 ~
2 2 (3.14)
L ( n a - k a + 2 ) +4naka.J
is the absorption efficiency factor for very small size
spherical particles (x--g)) as obtained from the
Rayleigh limit of the Mie theory. The discrete points
shown in Figs 10 and 11 are the results obtained
from the rigorous Mie theory for the corresponding
index of refraction of specific particles. The lines are
from the analytical, closed form expressions given by
Mengii~ and Viskanta. 112 Considering the uncertainty
in the volume fraction of polydispersions and the
complex index of refraction data, the agreement
between the model and exact calculations appears to
be remarkably good, and, because of this, these
simplified models would be useful for radiation heat
transfer calculations in combustion chambers. Note
that the single scattering albedo 09 is related to
absorption, extinction and scattering coefficients by
<.o = o/,a = 1
--
~://7.
(3.15)
In the literature, there are also some empirical
relations available for the radiative properties of
polydispersions. Buckius and Hwang 113 calculated
absorption and extinction coefficients as well as the
asymmetry factor of several coal polydispersions
using Mie theory and showed that they were almost
independent of the size distribution and were
functions of average radii r32 [see Eq. (3.13b)] and
the complex index of refraction. They obtained some
118
R. V[SKANTAand M. P. MENG0(;
10-2
,
,
,
. . . . '~ -'~. . . . J. . . . . . . . . . . . . . K.'lX,•)
"IN,, ~ .
i0"s
I0"2
10"4 ~
i0"3
I0x
I02
I03
i0-4
i04
r,zT [/.t.m K]
FIG. 12. Phmck and Rossehmd mean coefficients for coal. The shaded area represents results for w~riations
in temperature between 750 and 250 K and three coals. ~ 3
empirical correlations for the radiative properties of
coal particles which could be readily used for
predicting radiation heat transfer in coal-fired combustion systems. Also, they plotted the normalized
Planck and Rosseland mean absorption and extinction coefficients as functions of the mean radiustemperature product (Fig. 12) and obtained some
empirical relations for these coefficients. As seen
from this figure, for small radii particles, extinction
and absorption coefficients are identical; however,
with increasing radius the scattering of radiation
also becomes important, and fl and x diverge from
each other. Viskanta et al. 1~4 aIso obtained similar
results and discussed the effects of several independent parameters, such as size distribution, coal type
and wavelength of radiation on the radiative properties of polydispersions. It is worth noting that
although different definitions of mean radius are
used in these studies, i.e. rio [see Eq. (3.13a)] 112 and
r32 [see Eq. (3.13b)], 11a'1~4 still similar results
independent of size distribution are obtained. This
indicates that a polydispersion can be often described by a weighted particle radiusJ 15
All of the studies discussed above used the
spherical particle assumption in obtaining the relations for radiative properties of particles. Perfect
spheres are not encountered in nature, and, therefore,
it is desirable to obtain similar relations for other
than spherical shape particles. Stephens ]~6 has
shown that the anomalous diffraction theory developed by van de Hulst 65 can be extended to infinitelength cylinders. The absorption and extinction
efficiency factors calculated from this simplified
theory are in good agreement with those obtained
from a rigorous solution of Maxwell's equations. It is
worth noting that the anomalous diffraction theory
used for spheres also yielded accurate and simple
relations (see Ref. 112). Most recently, Mackowski et
a/. 117 derived the same kind of relations for the
spectral radiative properties of cylindrical soot
agglomerates. They showed that small size cylindrical particles extincted radiation two to five times
more than spheres. At large radii, on the other hand,
the ratio of cylindrical extinction and absorption
coefficients to those for spherical particles approach
constant values regardless of the wavelength of
radiation} 17 Also, some empirical relations similar
to those obtained for spherical particles are presented. It is also possible to extend the relations to
mixtures of different types and shapes of particles
using the T-matrix method. For a specific (coal)
combustion problem, a library of empirical relations
can be constructed. The use of these relations will
speed up the calculations significantly, since there
will be no need for lengthy and time consuming Mie
or T-matrix method calculations.
When the size parameter ( x = n D / 2 ) becomes
vanishingly small (x-*0) the size of the particle
becomes less important. In this limiting case, the
absorption efficiency factor is a function ofx [see Eq.
(3.14)], whereas the scattering efficiency factor varies
with x 4, such as
r~-I
0s = 3 ~
4
X .
(3.16a)
The extinction efficiency factor is written as
Q,. = Q,, + Qs.
(3.16b)
Radiation heat transfer
These expressions are obtained from the Rayleigh
limit of the Mie theory. 6s Here, ha=na-ika is the
complex index of refraction. It is worth noting that
with decreasing x (or D), the scattering efficiency
factor becomes negligible in comparison to the
absorption efficiency factor. Indeed, these expressions yield the extensively used soot absorption
coefficient, such as
xa = 7f J2
(3.17)
wherefv is the volume fraction of soot particles and
the value of "7" was suggested by Hottel and
Sarofim 46 for typical soot particles observed in
combustion chambers. After studying the available
experimental data for several flames Siegel l~s has
shown that the coefficient in Eq. (3.17) is between 3.7
and 7.5 for coal flames; 6.3 for oil flames, and 4.9 and
4.0 for propane and acetylene soot, respectively. A
detailed discussion of the spectral and total absorption characteristics of uniform-diameter, spherical
soot particles covering a very wide range of sizes
(0.001 <D<10/~m) is given by BlokE'*
Equations (3.14) and (3.16) were obtained for
spherical particles; therefore, the approximation
given by Eq. (3.17) may not yield accurate results
for arbitrary shaped small particles. 1~° For nonspherical particles, an expression for the average
absorption efficiency factor was derived by integrating over a distribution of shape parameters in
the Rayleigh-ellipsoid approximation, 119 such as
119
3.2.4. Scattering phase function
In modeling radiation heat transfer in a participating medium, the scattering of radiation by
particles must be properly accounted for. This
requires the use of the scattering phase function
(scattering diagram), which represents the probability
that radiation propagating in a given direction is
scattered into another direction because of the
inhomogeneities and/or particles along the path of
radiation. In combustion chambers, the scattering of
radiation takes place mainly because of the particles.
The phase function, along with other radiative
properties, such as absorption, extinction and scattering coefficients, can be obtained either exactly
from the solution of Maxweli's equations for spherical or infinite-length cylindrical particles 65,66'9° or
from some approximations such as the extended
boundary element method (EBCM) for arbitrary
shaped particles ~°2- lo4 as functions of wavelength,
characteristic particle dimension and complex index
of refraction. The phase function is written as
Oa(g'---,g) = 4la(-g)/x2Qs
(3.19)
where la(g) is the incident radiation intensity and x
is the size parameter with the effective diameter D
and radiation of wavelength 2. Note that the
radiative properties are not only functions of the size
of the particle or the wavelength of radiation, but
they are functions of the size parameter x, which can
be considered as a scaling factor. In Eq. (3.19) Qs is
the scattering efficiency factor, which is defined as 9°
2ha
Q,=xF(ha)=x Im [hA-- 1 (log na-ika)]
(3.18)
where x depends on the effective diameter D(--- V/A).
This relation yielded very good agreement with the
experimental data for quartz particles.119
In Table 2 a comparison is given of spectral F(ha)
functions for spheres [see Eq. (3.14)1, infinite-length
cylinders 117 and ellipsoids [see (Eq. 3.18),1 at four
different wavelengths. It is important to note that the
results for ellipsoids are between those for spheres
and cylinders. The spectral complex indices of
refraction used in this comparison are from the
dispersion relations developed by Lee and Tien 71 for
acetylene and propane soot at 1700 K.
Qs = CJA = (Ws/I,)/A
where A is the particle cross-sectional area projected
onto a plane perpendicular to the incident beam li
(e.g. A = nD2/4 for a sphere of diameter D); Ws is the
energy scattering rate by the particle, and C s is the
scattering cross-section. Similar expressions can be
written for extinction and absorption efficiency
factors by replacing the subscripts "s" in Eq. (3.20) by
"e" for extinction and "a" for absorption. These
quantities are obtained from the Mie theory or
approximate models. 9°
Use of the phase function in the form of Eq. (3.19)
would be a very time consuming procedure, A more
convenient form of the phase function is obtained by
expanding it in a series of Legendre polynomials) 20
TABL[2. Comparisons of spectral F(ha)functions for
different shape small soot particles
/(/am)
na
ka
F,~,,,
0.50
1.50
2.50
5.00
1.92
1.88
2.10
2.69
0.55
0.73
1.09
1.57
0.754
1.007
1.140
0.863
N
• a(W) = ~ a,.~P,(~P)
1.700
2.134
2.888
3.888
(3.21a)
rl=O
F~,n~®r Fomt,,ola
1.871
2.450
3.683
6.073
(3.20)
where
1
an,a-2n+l
J" ~a(W)P,(~)dD
O=4f
(3.21b)
120
R. VISKANTAand M. P. MENO0q
is the expansion coefficient and P, is the Legendre
polynomial of degree n. By changing the upper limit
N of the series, any phase function can be written in
the form of Eq. (3.21). The coefficients a,.a can be
obtained by employing the orthogonality relations of
Legendre polynomials. In order to accurately represent the phase function of highly forward scattering
particles, however, as many as 100 terms may be
required in the series. For the multidimensional
radiative transfer calculations, the use of such a
complicated scattering phase function is not practical either. Consequently, some further simplifications are required. If N = 0 , the phase function is
written as
(IDa(W)= 1
1
A=U~ I o~(,I,)dn
t~=2~
=½+½ ~ (-1)"a2"+'(2m)!
r.=0 22"+im!(m+ 1)t
(3.26a)
ba = 1 - f a .
(3.26b)
The factors ba and fa are especialy useful when
obtaining solutions of the radiative transfer equation
using flux methods.
Brewster and Tien 125 have given a different
definition of the backward scattering coefficient for
an azimuthally symmetric layer such as
(3.22)
1
o
Ba=½~ ~ ~a(/~,/~')d/~'du
which is for isotropic scattering. If N = 1, then the
linearly anisotropic scattering phase function is
obtained,
~ a ( ~ ) = 1 + a l,acos W.
(3.23)
For N = 2 the phase function corresponds to seconddegree anisotropic scattering, and if the expansion
coefficients are set arbitrarily such that at.,t=0 and
a2,a= 1/2, this yields the Rayleigh scattering phase
function.~ 9
Most of the particles encountered in combustion
chambers (pulverized coal, char or fly-ash) scatter
radiation predominantly in the forward direction.
Such a scattering behavior can be modeled using a
Dirac-delta function. The transport, delta-M and
the delta-Eddington approximations are of that
form. T M The delta-Eddington approximation is
written as ~22
This expression is valid for a plane-parallel layer of
particles, whereas Eqs (3.26) are appropriate for
scattering from a single particle. It has been shown
that for a cloud of non-absorbing spherical particles
(with h=1.33, x=6.0), b~(=0.036) is drastically
smaller than Ba(= 0.137).
In atmospheric studies, the Henyey~3reenstein
phase function approximation is often used 121 and is
expressed as
@n-o,a( W)= [1 +O]-2gacos tF]3/2"
where fz and 0a are related to the expansion
coefficients defined by Eq. (3.21b) as 59
fa = {i:al i f_ ~ 2)12 i i : : ,~ 1)/2
(3.25a)
and
9a =
a l ,,a - f a
1-fa
(3.25b)
provided that al,a>a2,a. A detailed account of
Dirac-deita phase approximations has recently been
given by Crosbie and Davidson? 23
In the heat transfer literature, another phase
function approximation has found wide application.
The phase function is expressed in terms of the
forward (fa) and backward (ba) scattering coefficients,
and they are written in terms of a.'s of Eq. (3.21b) for
an azimuthally symmetric medium such as 124
(3.28)
Here, g~ is the asymmetry factor and is defined as
ga= ( c o s W ) =
S la(g)cosWd~/ S la(g)df~ (3.29)
fl=4x
~a(W) = 2f~6(1 - cos W) + (1 -faX1 + 39acos W) (3.24)
(3.27)
0 -1
O=4f
which can be directly obtained from the Mie theory.
Although it approximates the Mie phase function
quite accurately, the application of the HenyeyGreenstein phase function approximation to multidimensional geometries may be quite tedious.
Several different approximations for the scattering
phase function, such as linearly anisotropic scattering,
delta-M, delta-Eddington, transport or HenyeyGreenstein approximations, have been reviewed in
detail by McKellar and Box)21 They have concluded
that for highly forward scattering particles the
delta-Eddington approximation ~22 is the most
accurate and the simplest of all the approximations
mentioned. In modeling radiative transfer in coalfired furnaces the delta-Eddington approximation
for the scattering phase function is desirable for two
reasons: (1) it represents the highly forward-directed
scattering of radiation by the pulverized coal and flyash particles; and (2) it is compatible with differential
approximations such as the spherical harmonics
approximation used to model the radiative transfer
equation.
Radiation heat transfer
The scattering phase function of particles is
directly related to the size (or diameter) of the
particles. Therefore, for polydispersions there should
be as many scattering phase functions as the number
of size intervals considered. For the sake of simplicity,
it is desirable to have a single, mean scattering phase
function over the entire particle size range. Then, the
mean scattering phase function can be written as
121
the concept may be of limited utility for predicting
radiation heat transfer in multidimensional combustion systems which contain particles, If the emissivity
(or absorptivity) of a particle laden flame is known
then the extinction coefficient of the medium can be
written as
fla.tot= - L~ In
(3.36)
1 N
~a = : - ~ a~,i ¢Pa.~,
0"2 i
(3.30)
where N is the number of the intervals. If the
delta-Eddington phase function approximation is
used for the scattering phase function, the corresponding mean parameters are defined similarly,
1
N
1
.'/-~- ~- ~O'2.,if]i,
G~ i
N
"g~-~- ~- EU2..iOa.i.
G'l i
(3.31)
3.3. Total Properties
Once the absorption, scattering and extinction
coefficients of polydispersions, such as pulverized
coal, char, fly-ash, and those of soot and combustion
gases are known, the total radiative properties are
written as
ra.tot = ~xa.~ly-i + xa.,,**,+ ~xa.,,,,,_.i,
i
j
(3.32)
fla,to,= xa.tot+ ~aa.vo,y-i,
i
(3.33)
o93= ~tra,~ly_ i/fl~.,tol
i
(3.34)
and
where "poly-i" refers to i-th type polydispersion, and
"gas-j" refers to j-th gas species. Note that if there
are no scattering particles in the medium, then
fla.tot= K~.,tot and t~a = 0.
An alternative formulation of total absorptivity
and emissivity of a scattering medium has been
recently given as ~26
F tg'l''t°t [1 --exp(-fl~.,totLm) ] .
(3.35)
~. = C£~= Lfl,l.,tot -I
In writing this expression it was assumed that
spectral irradiance was equal to the spectral emitted
flux of the surroundings. The spectral absorptivities
of polydispersions of coal and fly-ash particles have
been predicted using a two-flux approximationJ ~4
The expression for the absorptivity 1~4 is not as
simple as that given by Eq. (3.35). This suggests that
provided that the mean-beam-length L,, is known.
However, fla,tot and toa are interrelated properties
[see Eq. (3.34)]. If the mean coefficients are to be
used, the equations given above should be rewritten
by dropping the subscript 2, and appending the
appropriate mean coefficient subscript.
These definitions require the mean beam length of
radiation Lm, which is a vague concept. 59 It is
defined 2° as a radius of a gas hemisphere which
radiates a flux to the center of its base equal to the
average flux radiated to the area of interest by the
actual volume of gas. Although the concept yields
accurate results for simple systems, for complicated
geometries it needs additional research attention.
Recently, Scholand and Schenkel ~27 have calculated
the mean beam length of radiation between a volume
element and the surfaces of rectangular parallelepiped enclosures. Cartigny ~28 has extended the
definition of the mean beam length to an optically
thin scattering medium, which can be used for
calculation of radiative transfer in sooty flames.
The empirical relations for the total mean extinction and absorption coefficients for fly-ash, pulverized coal and char particle polydispersions have also
been reported. 4 It has been found, for example, that
the mean extinction coefficient fl of fly-ash can be
expressed by an empirical equation of the form,
fl= g,, F(CL)ApC
(3.37)
where Q,, is the extinction efficiency factor; F(CL) is
the function which accounts for the dependence of
the extinction coefficient on the product of the
concentration C and the layer thickness L, and At, is
the surface area of a particle. The total extinction
efficiency factor Q~ has been found to depend, as
might be expected, on the type of coal burned, fly-ash
particle size and the spectral distribution of incident
radiation determined by the black body temperature
T used as the radiation source. Based on experimental data it is possible to express Q,. by an
empirical equation,
Q,.=0.07 A(xT) 1/3.
(3.38)
The empirical constant A depends on the type of coal
burned and the shape of the fly-ash particle, and x is
the size parameter based on the mean particle
diameter. The function F(CL) has been determined
122
R. VISKANTAand M. P. MENG0~'
empirically and was found to depend on the type of
coal burned. 4
The total effective absorptivity of a fly-ash layer of
thickness L calculated from the expression
cZ®fe= 1 - exp( - ~L)
(3.39)
has been found to agree well with the experimental
data.'* It was determined from the data that the
optical thickness zL(= ~L) of the layer varies linearly
with CL only for moderate values of CL(<20 g/m2).
At higher values of CL the mean extinction coefficient ~ starts to depend on CL, because the
radiative properties of fly-ash particles depend on
wavelength. This leads to the departure of the
function zL(CL) from linearity.
The Hottel charts for the emissivity and absorptivity of combustion gases are very convenient for
practical calculations. Skocypec and Buckius ~29
and Skocypec et a/. 13° extended these charts to
include isotropically scattering particles. In their
calculations, they obtained the gas properties from
the Edwards wide-band model 35 and presented
hemispherical emissivities in graphical form and
discussed the effects of optical thickness, pressure,
temperature and single scattering albedo. These
charts yield accurate radiative properties without
any additional calculations; however, they cannot be
used directly for predicting the local radiation heat
flux in a combustion system.
4. SOLUTION M E T H O D S
The radiative transfer equation is an integrodifferential equation, and its solution even for a onedimensional, planar, gray medium is quite difficult.
Most engineering systems, on the other hand, are
multidimensional. In addition, spectral variation of
the radiative properties must be accounted for in the
solution of the RTE for accurate prediction of
radiation heat transfer. These considerations make
the problem even more complicated. Therefore, it is
almost necessary to introduce some simplifying
assumptions for each application before attempting
to solve the RTE in its general form. It is not possible
to develop a single general solution method for the
equation which would be equally applicable to
different systems. Consequently, several different
solution methods have been developed over the
years. According to the nature of the physical system,
characteristics of the medium, the degree of accuracy
required, and the available computer facilities, one of
several different methods can be adopted for the
solution of the problem considered. Before choosing
one solution method over another one, it is important to know the advantages and disadvantages of
each method. In this section, several radiative
transfer models of interest to combustion systems are
discussed, and their features are highlighted.
4.1. Exact Models
The most desirable solution of any equation is its
exact closed form solution. The exact solution of the
integrodifferential radiative transfer equation can
only be obtained after some simplifying assumptions,
such as uniform radiative properties of the medium
and homogeneous boundary conditions. For onedimensional, plane-parallel media, exact solution of
the RTE has received much attention in the atmospheric sciences, 12a 5,t 6 neutron transport 13~- ~33 and
heat transfer ~9'2°'~34 literature. A detailed review of
one-dimensional exact solution methods is available? 35 However, there have only been a few
attempts to formulate and solve the RTE for
multidimensional geometries.
One of the earliest accounts to formulate the
radiative transfer equation in a three-dimensional
space with anisotropic scattering was that of
Hunt? 36 He considered a phase function comprised
of three terms in Legendre polynomials and reduced
the integrodifferential radiative transfer equation to
an integral equation. Cheng ~37 used a rigorous
approach to solve the RTE for an absorbingemitting medium in rectangular enclosures, and Dua
and Cheng ~38 extended this method to cylindrical
geometries. For an absorbing, emitting, and scattering medium Crosbie and his co-workers presented
exact formulations of the RTE for three-dimensional
rectangular ~39 as well as three-dimensional cylindricaia4° enclosures. The solution of these equations
for cylindrical geometry was obtained by the method
of subtracting the singularity? 4a The exact solutions
of RTE for an absorbing and emitting medium were
also solved by Selcuk ~42 in a three-dimensional
rectangular enclosure employing a numerical scheme.
In a cylindrical geometry, the radiative transfer
equation is obtained from Eq. (2.11). Then, the
integral form of the source function, for an absorbing,
emitting and isotropically scattering medium with
incident diffuse radiation source on one of the end
surfaces of the cylinder can be written as 14°
S2(r,z,q~)= (1 -- 0)a)/ba[ T(r,z,q~)]
0) 2 r 2x
+ - - J S l d,~(r',d?')e-P~"z'~)X; (x~ ) - 3zr'dq~'dr'
4no
o
+m2 z~ r~ ~Sa(r,,z,,(a,)flae_Pa%x~ 2r,dda,dr,dz ' (4.1)
4nooo
where
x ; = [r 2 + (r') 2 - 2rr'cos(q~ -- qS')+ z2] 1/2 (4.2a)
Xp =
[r 2 + (r')2 - 2rr'cos(cb - ~b')+ (z- z')2] I/2. (4.2b)
Here, the primes are used to denote the dummy
variables, and l~,a is the spectral diffuse radiation
Radiation heat transfer
source on the end of the cylinder at z=O. Note that
ld. a can also be interpreted as the diffuse emission
and reflection from the walls. Some additional
integral terms are to be added to these expressions to
account for the other surface effects` After some
lengthy and tedious algebra, the implicit expressions
for the radiative fluxes in the r, z, and ~b directions
can be derived: 14°
F,.a(r,z,ck) = ,[ 2[ld,,t(~,~b')¢-P'~;(x;) -~
00
z
r
2z
x J r - r'cos(~b - ~b')]zr'dc~'dr'+ [ [ I Sa(r',z',c~')
0 0 0
x flae-P~%(xp)- air - r'cos(~b - t~')] r'dq~'dr'dz' (4.3)
r
21~
~,~(,,z,~)= [ I l d,a(r' ,c~' )e - p ~x +~(xp+ ) - ,
00
g o r o 21
x z2r'ddp'dr'+ ~ ~ I Sa(r',z',ck')
0 0 0
x flae-Pa~r~xp) - a(z-z')r'dc~'dr'dz'
r
(4.4)
2x
Fo,,(r,z,¢)= [ ~ ld,a(r',¢')e-~,~;(x;) -"
00
z o r o 21t
x r'sin(~b- ~')zr'ddp'dr'+ ~ J ~ Sa(r',z',q~')
0 0 0
x flze-P*%(x~)- ar'sin(q~-dp')]r'dO'dr'dz '. (4.5)
When ld,a is interpreted as the wall function which
includes the diffuse emission and reflection from the
walls, the additional integral terms will appear on the
right-hand-side of these equations. It should be noted
that in deriving these expressions, the medium is
assumed to be homogeneous. The evaluation of the
integrals in these equations yields exact results for
the radiative flux distributions in the medium. These
equations can be integrated numerically, as closed
form solutions are not possible unless further simplifications are introduced. Considering that in most
engineering systems the medium is inhomogeneous
and radiative properties are spectral in nature, it can
be concluded that the exact solutions for RTE are not
practical for engineering applications. Nevertheless,
exact solutions for simple geometries and systems are
needed, as they can serve as benchmarks against
which the accuracy of other approximate solutions
are checked.
4.2. Statistical Methods
The purely statistical methods, such as the Monte
Carlo method, usually yield radiation heat transfer
predictions as accurate as the exact methods. The
123
Monte Carlo method can be used for any complex
geometry, and spectral effects can be accounted for
without much difficulty. Mainly for this reason, the
method has been used extensively in atmospheric 143'144 and neutron transport 133 studies. It
has also been successfully employed to solve some
general radiation heat transfer problems 14~,z46 as
well as radiative transfer problems in multidimensional enclosures ~,17 and furnaces. ~4s,1,,9
There is no single Monte Carlo method. Rather,
there are many different statistical approaches. In its
simplest form, the method consists of simulating a
finite number of photon (energy packet) histories
through the use of a random number generator. 133
For each photon, random numbers are generated and
used to sample appropriate probability distributions
for scattering angles and pathlengths between collisions. If it is assumed that the problem is timedependent, each photon history is started by
assigning a set of values to the photon, its initial
energy, position and direction. Following this, the
number of mean free paths that the photon propagates is determined stochastically. Then, the crosssection (or absorption and scattering coefficients)
data are sampled, and it is determined whether the
collided photon is absorbed or scattered by the gas
molecules or particles in the medium. If it is
absorbed, the history is terminated. If it is scattered,
the distribution of scattering angles is sampled and a
new direction is assigned to the photon. In the case
of elastic scattering, a new energy is determined by
conservation of energy and momentum. With the
new set of assigned energy, position, and direction
the procedure is repeated for successive collisions
until the photon is absorbed or escapes from the
system.
Monte Carlo calculations yield answers that
fluctuate around the "'real" answer. 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 the problem. Since the
directions of the photons are obtained from a
random number generator, the method is always
subject to statistical errors and the lack of guaranteed
convergence. ~46 However, as next generation computers become more readily available, Monte Carlo
methods are expected to become more attractive for
engineering applications. It has already been shown
that vectorization of the Monte Carlo computer code
yields significant improvements in efficiency using
supercomputers such as CYBER-205 and more
precise results are obtained. 1s o
4.3. Zonal Method
The zonal method, which is usually known as
Hottel's zonal method, T M is one of the most widely
used methods for calculating radiation heat transfer
in practical engineering systems. In this method, the
124
R. VISKANTA and M. P. MENG0~
surface and the volume of the enclosure is divided
into a number of zones, each assumed to have a
uniform distribution of temperature and radiative
properties. Then, the direct exchange areas (factors)
between the surface and volume dements are
evaluated and the total exchange areas are determined using matrix inversion techniques. For an
absorbing and emitting medium, the calculation of
direct exchange areas becomes complicated as the
attenuation of radiation along the path connecting
two area (area-volume and volume--volume) elements
must be taken into account.
The zonal method reduces the radiative transfer
problem to the solution of a set of nonlinear
algebraic equations. The set of energy balances for
the zones in a closed radiaton system is written as
(4.6)
SE = Q
where
-Xs,
S2St
...
S,,Sl
S2S2 - ~ S 2 S j ..•
J
S,,S2
J
Sj S2
S=
S~ S3
S2S 3
...
SLS,,
s,s
... s,s,-Es,,s
SnS 3
J
Qt [
12¢hl I
Q~ I
I.~h2 ]
E=
Eh3
.Elm.
and
Q=
Q3I
.Q.J
Here, SiS~ is the total exchange area which is the
ratio of the radiant energy emitted by a zone Si
which is absorbed by zone S~ (directly or after
multiple reflections from other zones) to the total
energy emitted by zone S, Eu is the total blackbody
emitted flux and Q~ is the imposed heat flux at zone
Si.152
Although the formulation of the zonal method for
an absorbing, emitting, and scattering medium has
been available 153 for a long time, it has been only
recently applied to the solution of radiation heat
transfer problems in a system containing scattering
particles. 154 Larsen and Howell 155 presented an
alternative formulation of the zonal method and
accounted for only the isotropic out-scattering from
each volume element. This new approach, however,
does not show any computational advantage over the
classical zonal method.~ 56
As originally formulated TM the zonal method has
some inherent limitations, such as the treatment of
non-gray, temperature dependent radiative properties of combustion gases. The effects of temperature,
pressure and different species on gas properties can
be accounted for by weighted sum-of-gray-gases
m o d d g 5~'5z In addition, it is usually difficult to
couple the zonal method with the flow field and
energy equations which are usually solved using
finite difference or finite element techniques. This is
mainly because of the different size of the control
volumes required; the zonal method can be computationally prohibitive if the same grid scheme used
by the finite difference equations is adopted• Steward
and Tennankore ~57 have coupled the zonal method
with finite difference equations in modeling a
combustor by adapting two different grid schemes;
one for the radiation part and the other for the flow
and temperature fields. Recently, Smith et al. ~54 have
combined the zone method with momentum and
energy equations to predict heat transfer in an
absorbing, emitting, and scattering medium flowing
in a cylindrical duct.
The zonal method can not be readily adopted for
problems having complicated geometries, since
numerous exchange factors between the zones must
be evaluated and stored in the computer memory.
However, this difficulty can be overcome by adapting
a hybrid solution scheme which employs both zonal
and Monte Carlo methods. This will be discussed in
Subsection 4.5.5• Note that the direct-exchange areas
for rectangular enclosures have been recently calculated by Siddal115a who employed a new approach
for the evaluation of the multiple integrals• With this
technique, it is possible to obtain these factors with
any degree of accuracy desired• It is worth noting
that the computer time required by the zonal method
in predicting radiative transfer in enclosures is
usually smaller than the time required by its
alternatives, and therefore the method is attractive
for practical engineering caiculations.l 56
4.4. Flux Methods
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 (directio.nal) dependence of the intensity from its spatial
dependence to simplify the governing equationg If it
is assumed that the intensity is uniform on given
intervals of the solid angle, then the radiative transfer
equation can be significantly simplified as the
integrodifferential RTE equation 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
Radiation heat transfer
is assumed constant, one can obtain different flux
methods, such as two-flux, four-flux or six-flux
methods. Intuitively, one can deduce that as the
number of fluxes increases the accuracy of the
method would increase. Indeed, if the number of
solid angles and corresponding directions are determined from basic mathematical principles (see, e.g.
Whitney 159) more accurate and efficient flux
methods can be warranted. It is also possible to use
non-uniform solid angle divisions in the spherical
space. For example, if the direction and size of the
solid angles are determined from the Gaussian or
Lobatto quadratures, a non-uniform flux approximation is developed and the resulting expressions are
called the discrete ordinates approximation to the
RTE. 15
Another way of avoiding complicated expressions
of the RTE due to the angular dependence of the
intensity is to integrate the radiative transfer
equation over the space after first multiplying it by
certain directional cosines. The resulting expressions
are called moment approximations. The spherical
harmonics approximation is developed similarly, but
a more elegant and mathematically sound method of
integration of RTE is employed. If the integrations
are performed over hemispheres or quarter-spheres,
then double or quadruple spherical harmonics
approximations are obtained, respectively. The first
order moment, spherical harmonics, and first-order
discrete ordinate methods are identical for the onedimensional, planar geometry; 16° however, they
differ from each other slightly for multidimensional
geometries.
Due to the simplicity of the governing equations,
several flux methods have been developed for onedimensional plane-parallel media. They are reviewed
elsewhere, T M and those which can be extended to
multidimensional geometries are compared against
experiments ~62 as well as against exact solutions. T M
In this discussion, the focus is on
multidimensional models.
125
an absorbing, emitting, and scattering medium are
comprised of six coupled partial differential equations. 165 The equations are quite complex and
lengthy; therefore, they are not given here.
In general, the accuracy of the flux approximation
depends on the choice of solid-angle subdivisions. If
there is no intersection between two adjacent subdivisions, more accurate results are expected. 165 This
has also been observed by Selcuk and Siddall a66 for
rectangular enclosures. If the distribution of radiation intensity is assumed for each subdivision, the
general equations given by Abramzon and Lisin t65
can be simplified and solved simultaneously. If the
fluxes in each subdivision are assumed constant, a
simpler six-flux model can be obtained from the
general flux equations. For an absorbing, emitting
and scattering medium, Spalding ~67 suggested a
similar six flux model for cylindrical geometry, which
is written as
ld
+ r drr [rJ~a] = -(xa + crx)~a + xaEoa(T)
(4.7)
r
- d~ (K~) = - (xa + o'a)K:~ + xj.Eba(T)
+ 6 ( J ~ " +J~- +K~" +K~- +L~" +Lj-)
(4.8)
1 d
-+ r cl~ (L~) = - (xa + (ra)L~ + xaEba(T)
+ ~ ( J ~ +J~- + K ~ + K j + L ~ + L j )
0
(4.9)
where J~, J j are spectral fluxes in positive and
negative radial (r) directions; K~, K j- in positive and
negative axial (z) directions; L~, L~- in positive and
negative angular (4)) directions. These equations can
be manipulated to obtain three second order differential equations:
4.4.1. Multiflux models
Ever since the publication of the pioneering works
of Schuster (in 1905) and Sehwarzchild (in 1906) on
the two-flux approximation, as flux models have been
one of the most used methods for radiative heat
transfer calculations. With the advances in computers, the extensions of flux models for the application to multidimensional systems have become
possible, and consequently several different versions
have been proposed over the years. Recently,
Abramzon and Lisin 165 have presented a general
analysis for flux models in a three-dimensional
rectangular enclosure and have shown that most
other models reported in the literature can be
obtained from this general formulation. The governing equations for the general flux approximation in
a three-dimensional cylindrical enclosure containing
1 dfl-
r
-Id
= tCa[J~" + J f
+
}
- 2 Eba(T)]
+ ~ra [2(J~ + J ; ) - K ~ - K ~ - L~ - L ~ ]
(4.10)
r
1
d
= xaEK~ + K ~ - 2Eba(T)]
+3[-J~
-J~- +2(K~" + K ~ ) - L ] - L ; ]
(4.11)
126
R. VISKANTAand M. P. MLmG09
ldfl-
1
-Id
+
}
= x~[ L I + L~- - 2 E ~ ( r ) ]
O%
- J ~ - - K ~ - K ~ - +2(L + +L~-)]. (4.12)
+3 [ -JI
These are the simplest forms of the flux equations
and can be easily written for axisymmetric enclosures
as a four-flux approximation. The derivation of these
expressions is based on the Schuster-Hamaker
method ~63'~64 which is the crudest and the least
accurate flux approximation for one-dimensional
systems. Whitacre and McCann~6S showed that the
four-flux version of this model t69 predicted the
temperature field accurately, whereas the radiation
fluxes were usually underestimated in comparison to
Hottel's zonal method. A close examination of Eqs
(4.7) to (4.9) reveals that the fluxes for one direction
are not coupled with those of the other directions if
the medium is nonscattering. A similar type, uncoupled four-flux model was also developed by
Richter and Quack IT° and applied to a pulverized
coal-fired furnace.
In one-dimensional systems, the SchusterSchwarzchild two-flux approximation or its modified
form ~@*'17~ yields more accurate results. Lowes et
al) 72 extended this method to axisymmetric enclosures and derived an alternative four-flux model.
The corresponding equations can also be obtained
from the general relations by assuming axial symmetry and defining the boundaries for the subdivisions, t65 Then the governing equations* become 1 7 2
d
2
+
(JI
drr [J~ - J~ ] ÷ 4
-
r
= -Ttxa(J~ +J~)+2xaEba(T ) (4.13)
~/~-~ d
+
~ / ~ (J~ -J~ - r ~ -K~-)
2 dr [ J z + J ~ - ] 4
4
r
= -nxz(J~" + J~')
(4.14)
4.4.2. Moment methods
x/~n d
+
x/~S-~n (J; - J~)
2 dzEK~ - K ; ] - ~ 4
r
= - n x x ( K f +Kf)+2xaEba(T)
(4.15)
dz[K~+Kf]=--nxa(K~-K;)
In the moment methods, the radiation intensity is
expressed as a series in products of angular and
spatial functions:
I(x0,,z,0,+)
d
2
where, J,~, J~', K~ and K~- have the same meanings
as defined before.
The four unknown fluxes in Eqs (4.13)-(4.16) are
determined from the four equations, and then the
radiative fluxes and the divergence of the radiative
flux vector are obtained readily. This method was
used to predict non-gray radiation heat transfer in an
axisymmetric furnace and good accuracy was obtained) 72 Note that, although scattering in the
medium was neglected in deriving these expressions,
it can be accounted for in the formulation. Also,
these equations can be modified to relax the
axisymmetry assumption to obtain a more general
formulation.
One of the oldest multi-flux methods is the six-flux
method of Chu and Churchill. 173 Although it was
developed for a one-dimensional, plane-paralld
medium, it is possible to modify this method for
multidimensional enclosures. Varma ~4 obtained a
four-flux model for axisymmetric cylindrical enclosures starting from this six-flux method. However, the
comparisons with more accurate models show that
this version of the four flux method is not very
reliable. 17s Note that both the four- and six-flux
methods account for the scattering of radiation.
Another six-flux model was proposed for threedimensional enclosures containing absorbing and
emitting gases. ~e6 A comparison of the predictions
based on this model with the Monte Carlo results
showed that the maximum error in the radiation heat
flux was not more than 23 % and could be reduced to
about 1% if the subdivisions of the solid angles were
adjusted according to the geometry of the furnace.
There are mainly three objections to the multi-flux
approximations of the radiative transfer equation
developed and used by some investigators for
practical problems (see Smoot and Smith 3 and
Khalil 5 for extensive lists of references and applications). First, there may be no coupling between the
axial and radial fluxes, which makes the equations
physically unrealistic. Second, the approximation of
the intensity distribution from which the flux
equations are obtained is arbitrary. Third, the model
equations cannot approximate highly anisotropic
scattering correctly, although it is theoretically
possible.
(4.16)
N
= A o + ~ [~'A,.~+q"A,.,+II'A..j
(4.17)
n=l
*Note that these equations are modified slightly to follow
a consistent nomenclature.
where A's are functions of location only; ~,~/, and g
are direction cosines in x, y, and z-directions,
Radiation heat transfer
respectively [see Eq. (2.8)1. Although this equation is
written in Cartesian coordinates, it can be given for
any orthogonal system. As the upper limit of the
series N approaches infinity, this expression converges to the exact solution for the radiation
intensity. Note that Eq. (4.17) can be considered as
the Taylor series expansion of the intensity in terms
of direction cosines.
The simplest moment expression for the intensity
can be obtained by taking N = 1. This is called the
first-order moment method. The AI.~, A~.y and A~,..
coefficients can be obtained by integrating the
intensity over the entire space. DeMarco and
Lockwood 176 have suggested some modifications of
the moment method using the flux definitions of the
Schuster-Schwarzchild model, and defined the coefficients as
Ao=0
A~ ..,.=(J; -J;)/2,
A~.,=(I<;-K;)/2, A~.:=(L~-L;)/2
(4.18)
A2.x=(J~ + J~-)/2,
A2.r=(K~ +K~-)/2, A2.:=(L~" +L~-)/2
where A's are implicit functions of the wavelengths of
radiation 2, and J**, K,a ±, La + are integrated
spectral radiation intensities over appropriate solid
angles in the _+x, +y, +z-directions, respectively.
These equations were solved by dividing the total
solid angle 4n into six equal angles of 4n/6, each one
having the coordinate directions as its symmetry
axis. Another solution scheme was also adopted by
choosing a magnitude of 2n for each solid angle. .76
Although the latter assumption produces overlapping
of the solid angles, the predictions based on it yielded
better agreement with the Monte Carlo results for a
three-dimensional rectangular enclosure. ~76 A further
improvement of this method was recommended by
allowing some flexibility in the magnitude of solid
angle corresponding to each direction. 177 For a
medium with a minimum optical thickness (absorption coefficient-characteristic length product) of
2 this modified method yielded more accurate results
in comparison to the earlier versions. In neither of
these models ~76.177 was scattering of radiation in the
medium accounted for. It is interesting to note that if
the A-coefficients of this formulation are approximated as
A2.x = A2..v= As,".
(4.19)
then the first-order moment method will be obtained
(as ~2+r/2+/t2= 1), which is equivalent to the firstorder spherical harmonics P~-approximation. 19
4.4.3. Spherical harmonics approximation
The spherical harmonics (Ps) approximation,
which is also known as the differential approxi-
127
mation, is one of the most tedious and cumbersome
of the radiative transfer approximations; however, it
may be the most elegant one because of its sound
mathematical foundation. The method was originally
developed, as most other approximations, for study of
radiative transfer in the atmosphere, 17s later modified
for the solution of neutron transport problems, T M
and extensively used for one-dimensional radiative
transfer problems. 15-17Ag"2°'za2'179 Although the
formulation of the spherical harmonics approximation for multidimensional geometries was discussed some time ago, m3t only during the last decade
has the method been extended to two- and threedimensional systems. For non-scattering Cartesian,
cylindrical and spherical media the first-order (P~)
and third-order (P3) spherical harmonics approximations,~ ao.~81 for an isotropically scattering cylindrical medium the PI -approximation, 182.184 and for
an isotropicaily scattering two-dimensional rectangular medium Pt- and P3-approximations ~ss
have been formulated and solved. Meanwhile, the
first-order spherical harmonics approximation has
also been formulated to study the effect of cuboidal
clouds on radiative transfer in the atmosphere. 144't s6
Most recently, Menguc and Viskanta 61'187 reported
the general formulations of the PI- and P3approximations for absorbing, emitting, and anisotropically scattering medium in two-dimensional,
finite cylindrical as well as three-dimensional rectangular enclosures.
In the spherical harmonics approximation, the
radiation intensity is expressed by a series of
spherical harmonics instead of a Taylor series and is
written as t a2
A~Ax,y,z)r~(O,¢)(4.20)
t~(x,y,z,O,¢) = F~
n~O
m = --n
with
r.~(O,O)= ( - 1 ~" ÷ I,.j~/2
r2.+ 1 (.-Iml)!-I
'/=.j.j,
..
,.,,
,~..,./
r , ~cosvle
L 4= ~,,+lml)!j
x _
(4.21)
where Y~ are the spherical harmonics, and P~ are the
associated Legendre polynomials which are related
to the Legendre polynomials.
In Eq. (4.20} the upper limit N for the index n is
known as the order of the approximation. Exact
solution of the RTE is obtained if N is taken as
infinity; however, for practical calculations a finite N
value is assigned. N = I results in P1- and N=3
results in P3-approximations. Usually, the odd
orders of spherical harmonics approximation are
employed, although there are occasionally some
others which use even order approximations, lsa The
reason for using the odd-order approximation is
simply to avoid the mathematical singularity of the
intensity at directions parallel to the boundaries. The
128
R. VISKANTAand M. P. MENGOt;
radiation intensity is usually discontinuous at the
interfaces; therefore, it is not possible to have a single
value of intensity at the boundary. Consequently, it is
not desirable to have an angular grid point just on
the interface. The roots of Legendre polynomials
used in spherical-harmonics approximation yield
Gaussian quadrature points, where the N-th order
polynomial gives the N-th order Gaussian quadrature scheme. If N is even, one of the quadrature
points will have a value of zero, which corresponds
to an angular grid point on the boundary, whereas, if
N is odd there will be no quadrature point on the
boundary. Therefore, an odd-order spherical harmonics approximation yields a more stable solution.
The above discussion can be easily followed for a
plane-parallel geometry.
The Pt-approximation is comprised of a single
elliptic partial differential equation ~a7
V210,a = Aa[10,a-4nlh4(T)]
(4.22)
where 10.4 is the spectral zeroth-order moment of
intensity I-irradiance cga, see Eq. (2.21b)1, lb4 is
Planck's blackbody function, and A4 is the coefficient
which is a function of single scattering albedo 094,
extinction coefficient il4, and phase function parametersJa and ga:
10.4 =
S ladle,
n=4x
A 4 = 3fl](1 - 094)[1 -
o~4(ja + 04 -J]94)'] • (4.23)
In writing the above approximation, the deltaEddington phase function is employed [see Eq.
(3.24)-]. In the P3-approximation, higher order
moments of intensity, i.e. the integrals of radiation
intensity-direction cosine products over all directions within solid angle 4n are employed. Naturally,
the resulting equations are more complicated than
those of the Pt-approximation. For axisymmetric
cylindrical geometry, there are four second order
elliptic partial differential equations for the P3approximation; 6~ whereas, for three-dimensional
rectangular enclosures six equations are needed, ls7
These equations are solved simultaneously for the
second-order moments, and afterwards the other
moments, radiation intensity, radiation heat fluxes
and the divergence of radiation heat flux are
calculated.
Although the P~-approximation is very accurate if
the optical dimension (i.e. the product of extinction
coefficient and characteristic length) of the medium is
large (i.e. greater than 2), it yields inaccurate results
for thinner media, especially near the boundaries.
Also, if the radiation field is anisotropic, i.e. there are
large temperature and/or particle concentration
gradients in the medium, the P~-approximation
becomes less reliable. The P3-approximation, however, can yield accurate results for an optical
dimension as small as 0.5 61.~a5 and for anisotropic
radiation fields, but at the expense of additional
computational effort. It is shown 61 that the accuracy
of P3- as well as Pl-approximations can be substantially improved by using "exact" boundary
conditions, rather than somewhat arbitrarily defined
Mark's or Marshak's boundary conditions (see Refs
19, 20, 131 and 132 for definitions and 61, 185 and
187 for implementations of the Marshak's boundary
conditions).
It is also possible to improve the accuracy of the
spherical harmonics approximation by obtaining the
moments of radiation intensity in half or quarter
spheres, xsg-19a Since the angular variation of
moments is allowed for in this method, the anisotropy of the radiation field can be modeled more
accurately than by the Pt-approximation. On the
other hand, the governing equations are simpler than
those for the Pa-approximation.
4.4.4. Discrete-ordinate approximation
A discrete-ordinate approximation to the radiative
transfer equation is obtained, as the name suggests,
by discretizing the entire solid angle (f2=4n) using a
finite number of ordinate directions and corresponding weight factors. The RTE is written for each
ordinate and the integral terms are replaced by a
quadrature summed over each ordinate. Originally
suggested by Chandrasekhar 15 for astrophysical
problems, the discrete-ordinates method has been
extensively applied to the problems of neutron
transport. 21`13a'lq'*A95 A simpler version of this
method, which is called SN-approximation, was
obtained by dividing the spherical space into N
equal solid angles, a96 However, more accurate formulations were obtained later using Gaussian or
Lobatto quadratures. These are also called SNapproximations to symbolize the discrete-ordinates
approximation in which there are N discrete values
of direction cosines ~., q., it., which always satisfy the
identity ~.2 + q.2 +/~.2 = 1.
In one-dimensional plane-parallel media, the discrete
ordinates approximation has found many applications (see, e.g. Viskanta, a34 Houf and Incropera, 197
Khalil et al.t9s). Recently, the SN-approximation has
been applied to two-dimensional cylindrical and
rectangular radiative transfer problems with combustion chamber applications in mind, and reasonably
accurate results were obtained in comparison to
exact solutions? 75,199
The radiative transfer equation for an axisymmetric cylindrical enclosure is written for each
quadrature point n as
r
Or
r ~dp q-p,~-;+fl4la.,
0" 2
=xalb4+7~,w., ~.,.14,.. (4.24)
°t/l: n"
Radiation heat transfer
where w, is the weight of the Gaussian quadrature
points. Integrating Eq. (4.24) over an arbitrary
control volume and rearranging yields,
{ ~.(ANI a,.,N -- Asla,.,s) + 14,(ArJ a..,v. - Awl~,~,w)
1
- (As - As)-- (~. +
Wn
1/2I~.~ + 1/2,c
-~,-1/2I~., - 1/2,c)}/Vc
0",1
= - flalx.n.c + xxlba.c + 7 - ~, Wnn' ~,m'lx,,',C
(4.25)
t'l'Tt n'
where A is the corresponding area of control-volume
side for N, S, E or W, i.e. for north, south, east, or
west side, respectively; V is the volume of the control
volume, C is for the central node, and or-terms are to
preserve the conservation of intensity in the curved
coordinate, which are determined from the radiative
equilibrium condition.~ 99 These governing equations
are solved numerically, for example, using a finitedifference scheme. 175'199 A finite element solution
scheme was also developed to solve the discrete
ordinates approximation equations in two-dimensional
Cartesian geometry for radiative transfer in the
atmosphere. 2°°
If the resulting equations of the discrete ordinates
approximation are carefully coded, they can result in
computer algorithms that combine minimum computer memory requirements with few arithmetic
operations per space-angle grid point. 133 However,
this approximation is not flawless, but suffers from
the so called "ray effects" which yield anomalies in
the scalar flux distributiori. T M .202 The ray effects are
especially pronounced if there are localized radiation
sources in the medium and scattering is less important in comparison to absorption. As the single
scattering albedo increases, the radiation field becomes more isotropic and the ray effects become less
noticeable. However, with increasing single scattering albedo and/or optical thickness of the
medium, the convergence rate may become very
slow. 133 Considering the flame in combustion chambers as a localized radiation source, it is natural to
anticipate the ray effects in the solution of the RTE
in combustion chambers, if the discrete-ordinates
approximation is used. If the combustion chamber is
a pulverized-coal fired furnace in which there are
scattering particles present, the results are expected
to be more reliable. 199
4.4.5. Hybrid and other methods
Almost all methods discussed have some flaws. In
order to take advantage of the desirable features of
the different models, various hybrid radiative transfer models have been developed in the literature.
JPBCS 1 3 : 2 - e
129
Here, we discuss only those which are applicable to
combustion problems.
The basic flaw of the zonal method is the
computational effort required to calculate the exchange factors between various volume and surface
elements in complex geometries. This difficulty can
be overcome using the Monte Carlo method to
calculate the direct exchange areas. Is2 If the radiative properties of the medium are known and do not
depend on temperature, it is possible to calculate
these exchange factors only once and store them in
the memory of a host computer for later use in the
zonal method predictions. By doing this, the computational time required by the zonal method to predict
radiation heat transfer in complex geometries is
decreased substantially. However, the computer storage requirements can become prohibitive if the
number of zones is large.
The Monte Carlo method suffers from statistical
error as well as the extensive computational time
required for the calculations. If the direction of each
ray is given deterministically rather than statistically
and if all the directions constitute an orthogonal set,
then the solution would be less time-consuming and
the accuracy would increase with the increase in the
number of directions. With this in mind, Lockwood
a n d S h a h 2°3'2°4 proposed a "'discrete transfer"
model which combines the virtues of the zonal,
Monte Carlo, and discrete ordinates methods. They
showed that very accurate results could be obtained
with this method in one- and two-dimensional
geometries by increasing the number of directions.
Although this method is claimed to be capable of
accounting for scattering in the medium, no results
have been reported or compared against other
benchmark methods for scattering media in multidimensional enclosures. The results for a onedimensional scattering medium did not show the
same level of agreement with the benchmark results
as did the non-scattering medium predictions. 2°3
This method is also likely to yield erroneous results
due to the "ray effects" discussed in Subsection 4.4.4.
A similar approach to the solution of the RTE for
multidimensional enclosures has also been presented
by Taniguchi et al. 2°s for absorbing-emitting media.
This so called "'radiant heat ray method" is based on
the Beer-Lambert's or Bouguer's law and yields the
radiant energy absorption distribution in nonisothermal enclosures containing combustion gases.
Comparisons of the predictions based on this
method with other results show that the method is
more accurate and less time consuming than both
zonal and Monte-Carlo techniques if the radiative
properties such as the absorption coefficient and wall
emissivities are constant. 2os
Another hybrid model based on the Monte Carlo
method and generalized radiosity-irradiation approach has been suggested by Edwards. 2°6 This
method accounts for the volumetric scattering, yields
accurate results for optical dimensions as small as
130
R. VISKANTAand M. P. M~,~GOt;
0.5, and is computationally faster than the Monte
Carlo method.
The main reason why the discrete ordinate
approximation suffers from ray effects is because of
the inability of the low-order Sn-quadrature to
integrate accurately over the angular flux. 133 If
piecewise continuous approximations of the angular
flux are given in terms of directional variables, and
approximate spatial equations are obtained by
integrating over appropriate solid angles, these ray
effects can be avoided. The resulting expressions can
be considered as hybrid models which combine
discrete ordinates or multiflux approximations with
the spherical-harmonics approximation. Indeed, the
double or quadruple spherical-harmonics approximations described in Subsection 4.4.3 can be considered as this kind of hybrid model. In neutron
transport literature there were several accounts
which discussed the possibility of combining the Snmethod with the Pn_~-method to improve the
accuracy and reliability of the predictions as well as
to decrease the computational effort, t 32.133
Flux models can also be coupled with the moment
or spherical harmonics approximation to improve
the accuracy of the radiation heat transfer predictions. A model which combines the Pt-approximation with a two-flux method was proposed by
Selcuk and Siddall 2°7 and applied to a twodimensional axisymmetric cylindrical furnace. The
comparisons of the temperature and heat flux distributions in the medium with those obtained with the
zonal method showed very good agreement. Since
this model was developed for a gas-fired furnace,
scattering of radiation was not taken into account.
Another similar hybrid method was derived by
Harshvardhan et a/. 2°8 who combined the modified
two-flux method]7~ with the P~-approximation. In
this method, the linearly anisotropic scattering
medium assumption was made, and the method was
used to predict radiative transfer through threedimensional cuboidal clouds. Comparisons of the
predictions with the Monte Carlo results showed
reasonably good agreement.
Recently, a new three-dimensional radiative transfer model was proposed 2o9.2~o by extending the onedimensional adding-doubling technique (see, e.g. van
de Hulst 2It). The predictions for the radiative flux
distribution in cuboidal clouds obtained by this
method compare very well with those of the Monte
Carlo method. It should be noted, however, that the
assumption of homogeneous and symmetric boundary conditions simolify these problems considerably.
accuracy and computational costs. In order to decide
whether a model is appropriate for a given problem,
one has to compare its predictions against the
benchmark results obtained from either experiments
or exact solutions. Zonal and Monte Carlo methods
are extensively used as the benchmark for comparisons as they generally yield accurate predictions of
radiation heat transfer.
In one-dimensional systems, comparisons of different
radiation
models
have
been
given. 125']62-164"197']98 However, the accuracy of a
method in predicting radiative transfer in a simple
system may not always warrant its use in more
complicated systems. Therefore, it is important to
evaluate radiative transfer models for multidimensional geometries, preferably for practical situations.
1.0
FiniteElement
0.8~ L / H
0
~o
i5
,m
.
6
Zonal
~
O.4
O.2
O
|
0
I
I
~
I
0.2 0.4
0.6 0.8
DimensionlessPosition,x/L
I
1.0
FiG. 13. Dimensionless centerline temperature profiles in
rectangular enclosures of different aspect ratio with black
walls; bottom wall at dimensionless temperature 0= 1.0,
other walls at 0=0. 2~2
~"
ILlH .O.t
..'-,'7"
0.8
~ - ....
~"*"*
.o 0.6
"0
ZlU 0.4
_~
.~_
0.2
FiniteElement
---P3
o Zonal
4.5. Comparison of Methods
Although there are several radiative transfer
models available, it is difficult to choose a "best"
model for different applications. For a given physical
situation, one of the several models can be used
according to the applicability of the model, desired
o
Oo
o'.2
I
a,
ols
I
o8
!
,.o
DimensionlessPosition,x,/L
FIG. 14. Dimensionless net radiation heat flux at the lower
wall in rectangular enclosures of different aspect ratio with
black walls, bottom wall at dimensionless temperature
0= 1.0, other walls at 0=0. 2~2
Radiation heat transfer
2.8
"•"
2.4 . ~
.o
• • •
-'-Ps
Zonal
---
~
2.0
,5
(J
-
o
1.6
.~_ 1.2 :'"i"
~. . . . . . . .
,,, 0.8
e,
.=_o
a4
E
0
o',
0
&
&
,o
Dimensionless Position, y/H
FIG. 15. Comparison of irradiances in a two-dimensional
square cross-section enclosure with a gray scattering
medium (ic=0), Ehl = 1 and El,_,= EI,.~= El,4=0.199
1.0
• • • Zonal
- - ' - - Ps
~'-,,,
•,~
~
S,, Ss
08
h
:I:
O.6
o
~
O.4
.,,.::.~---;--~---~
........
131
accuracy of the surface net radiation heat flux
decreases with decreasing optical thickness. Similar
conclusions have also been reported by different
researchers.6~'l s5.1 s7
In Figs 15 and 16, comparisons between zonal,
spherical harmonics (P3), and discrete ordinates (SN)
approximations are presented for a purely scattering
medium with different wall emissivities. 199 The P3and S6-results for the centerline irradiance distribution are in very good agreement with the zonal
method (see Fig. 15). The Ps-approximation, however, overestimates the radiation heat flux at the
walls for large wall emissivities, although both S4and S6-approximations yield accurate results (Fig.
16).
The lower-order spherical harmonics approximations generally yield more accurate predictions if
the radiation field in the medium is almost isotropic,
which is the case if the optical thickness is large,
the medium is predominantly scattering or the
surfaces are diffusely reflecting. If the radiation field
is highly anisotropic, the P3- and especially P t "
approximations become less reliable. Because of this,
the Pi- and P3-approximations are to be used for
media having optical thicknesses of 1.0 and 0.5
or larger, respectively, ls°'lsSas7 The main reason
for this inaccuracy for anisotropic radiation fields
is use of arbitrarily defined boundary conditions,
like Marshak's condition) 9 In Fig. 17, the P3approximation results are compared against those of
an exact model 139 for a cylindrical enclosure, 61
where both Marshak's (m) and "exact" analytical (a)
boundary conditions are used. Here, it is assumed
that there is a uniform, diffuse radiation source
incident on one of the end surfaces of a cylindrical
(,0.1
0
0
I
I
I
I
(11
0.2
03
0.4
0.8
05
Dimensionless Position, x//L
FIG. 16. Comparison of radiation heat fluxes at a wall of a
two-dimensional square cross-section gray enclosure with a
scattering medium (h=0), Eht = 1. Eh2 = E~,3= Et,4=0.199
O.6
qr
(r --to)0.4
.•-m
:
0.2
Unfortunately, this is not always possible because of
the analytical or numerical difficulties.
The radiative equilibrium (heat transfer by radiation alone) assumption yields the most simple case
for solving the radiative transfer equation in two- or
three-dimensional enclosures. In Figs 13 ad 14 the
centerline dimensionless temperature and net radiation heat flux distributions at one of the walls of a
rectangular enclosure containing a gray, absorbingemitting medium are compared for three different
methods. 212 The zonal and finite element methods
are in good agreement with each other. The third
order spherical harmonics (Ps) approximation yields
good results for the distribution; however, the
0
Exact
a
..........
~...
N~.
0
2
PI
------ PS
4 z/ro6_
la - - / m.-~
8
-,'--t
....... : , "
o.,I-. .............N ................ :: /
qz
r- . . . . . . . . . . . .
(z=O) _ La_~-A_m"'~
o u F----'~'_.-_-~-__-_.. . . .
K)
~
~__.=.__
oo,o
qz
(Z=Zo)
OOOS
I
ozl
o
l
oz
,
; ......... i ..... "'-'1o.oo o
o.4
o.6
as
,.o
,/to
FIG. 17. Comparison of P,- and P.,-approximation results
with exact benchmark solution: 1",)=0.1 m. # = l . 0 m - ' ,
co=0.5, T~,.=O, 8 , = 1.0 (m refers to Marshak's boundary
conditions, a refers to analytical boundary conditions). 6~
132
R. VISKANTAand M. P.
i
I
,
MENG0q
i
i
0
Meosured Values
P,C A p p r o x i m a t i o n
%
v
I
I
IO
I
I
i
2.0
I
3,0
a
I
i
40
50
z (ml
FIG. 18. Comparison of local radiation fluxes at the wall based on P.~-approximation results for a
combustion chamber with experimental data and discrete ordinates method: r. =0.45 m, :, = 5.1 m."~
enclosure containing a homogenous, absorbing and
scattering medium. Radiative flux disribution on the
cylinrical walls (upper panel) and on the end walls
(lower panel) are plotted from the results obtained
using different boundary condition models. 61 It is
clear from the figure that the use of Marshak's
boundary condition yields substantially higher local
errors in radiation heat flux than the use of analytical
conditions. This suggests that with a careful and
more rigorous treatment of the boundary conditions,
even the very simple P~-approximation can be
employed to predict the radiation heat transfer in
combustion chambers accurately.
There are also some accounts in the literature
which compare different radiation models for practical systems, such as large scale furnaces where there
is a gradual temperature variation along the chamber.61.168,172.175.213 In these comparisons, radiation
is decoupled from the energy equation, since the
temperature distribution as well as the radiative
properties of the medium are assumed to be given. In
Fig. 18, the predicted radiation heat flux distribution
along the cylindrical walls of a furnace is shown. 61
Both the $4- and P3-approximations are seen to be
accurate for a given "uniform" absorption coefficient
value of 0.3 m -1 However, it is difficult (almost
impossible) to assign a single "gray" absorption
coefficient for the entire furnace. When x is changed
from 0.3 m -~ to 0.35 m -~ (see Fig. 18) the P3approximation yields improved agreement between
the data and predictions. If the value of x were
changed to 0.25, the agreement would have been
poorer. The sensitivity of the results to the radiative
properties was also shown by Lowes e t aL 172 They
compared the predictions obtained from zonal and
four-flux models for different absorption coefficients
against the experimental data obtained for a gasfired furnace. As seen from Fig. 19, the results are
more sensitive to the radiative properties than to the
models. Indeed, in predicting the radiation flux
distribution in a large furnace, Selcuk et a/. 213
150
N
°°'°\
x
I00
Lt.
0
I
I
~
.o
,,.~,li~d~
~-
I
2
Furnoce
O.Im "l
I
3
'1 cleat • 2 q t a y
I
4
I
5
I
6
Lenqth (m)
FIG. 19. Comparison of predicted radiation heat fluxes
using the four flux model (lines) with different absorption
coefficient formuhltions,1 v2 Points are zonal method results.
obtained very good agreement between the experimental data and one-dimensional radiation models
using the measured radiative property data in the
models. From these findings, one may conclude that
for complicated systems such as combustion chambers, the accuracy of the radiative properties is as
important, or even more so, than the accuracy of the
models. Additional sensitivity studies must be performed for combustion chambers to determine which
properties are the most important and under what
conditions. For example, Menguc and Viskanta 2~4
have shown that the index of refraction of coal
particles does not play a significant role in predicting
the radiation heat flux distributions along the
furnace walls. In their model the pulverized coal
particles were assumed to be only in the flame zone,
and the predictions were obtained using the P3approximation. On the other hand, Piccirelli et
al. 215 presented a similar analysis using the PIapproximation for a one-dimensional cylindrical
system and showed that the complex index of
refraction was a very important parameter in
predicting the emissivity and absorptivity accurately.
These contradictory conclusions are not due to
different solution techniques, but basically result
Radiation heat transfer
133
from assumptions related to the radiation property
distributions in the medium. In the former 2~4 the
coal particles were assumed to be only in the flame
zone, whereas in the latter 2t5 the coal particles filled
the entire combustion chamber.
External
Radiotion
Thin
~. VOlOtlle r~loua,~ Flame
5. APPLICATIONSTO SIMPLE COMBUSTION SYSTEMS
To illustrate the coupling between radiation heat
transfer, combustion and other transport processes
we consider in this section several simple combustion
situations in which radiative transfer has been
accounted for. The emphasis in the discussion is on
the effects of radiation heat transfer. Since the
physical situations considered are quite simple and
the systems are not large, the effects of radiation on
the results are expected to be small as the optical
dimensions characterizing the systems are also small.
5.1. Single-Droplet and Solid-Particle Combustion
Burning of a single-droplet of liquid fuel or of a
solid particle is a very simple system. Transport
process and not chemical kinetics dominate the
combustion of fuel. This phenomenon has been
studied extensively for many years and experimental
and theoretical accounts are available, s'2~6'2~7 The
theory of single-droplet combustion is complicated
by many factors, such as circulation in the droplet,
finite-rate chemistry in the diffusion flame that
surrounds the droplet, nonsteady accumulation of
fuel between the surface and the flame, etc. In recent
reviews of the theory these complications have been
discussed.S.217 Radiative transfer in single-droplet
combustion has been ignored in most studies reported
in the literature. 8 Recently, a model has been
developed to study coal particle behavior under
simultaneous devolatilization and combustion in
which transport of radiation in the volatile cloud
(radiatively participating medium) surrounding a
coal particle has been accounted for. 218
The spherical volatile cloud, enclosed by a thin
flame sheet whose location is determined by diffusionlimited combustion, is modeled as a radiatively
participating medium. The modeling concept is
similar to that of liquid droplet combustion except
that volatiles emitted by the coal particle form a
concentric luminous mantle (see Fig. 20) and radiative transfer is important in addition to convective
and conductive transport. The model describes the
heat transfer mechanisms between the particle, the
volatile cloud, the flame, and the external environment. The analysis of combined conductionradiation heat transfer in a concentric sphere filled
with a radiatively participating medium first developed by Viskanta and Merriam 2~9 was used. The
absorbing, emitting and scattering medium was
assumed to be confined between two gray, diffuse
isothermal spheres kept at different but uniform
\.
'/r2
.
' "
."x/,Caol.
Parttele
FIG. 20. Schematic diagram of a spherical translucent and
radiating cloud model. 218
temperatures. The spherical transluscent and radiating cloud model is identical to the problem studied
earlier. 219 This complex energy transfer situation for
solid particle combustion is treated rigorously, and
influencing parameters are identified.
The model permits calculation of the steady-state
heat transfer rate when the particle surface temperature, flame-sheet radius and temperature and other
environmental conditions are given. The optical
thickness has been found to be an important model
parameter in calculating radiative transfer rates. A
fair amount of numerical computation was required
to obtain solutions. The model has been shown to be
useful for the interpretation of muiticolor 22° and
two-color T M pyrometry for more accurate experimental data reduction. Ultimately, a simplified
version of the model could be incorporated into a
coal combustion model which explicitly includes
particle heat-up and devolatization rates. However,
for an application to a combustion system the model
must be extended to account for the interaction
between the burning particle and the surroundings
which contain radiating gases, clouds of particles
and the system walls.
A systematic investigation of the effects of thermal
radiation on the combustion behavior of char
particles exposed to an oxygen environment has been
performed 222 using the general mathematical models
developed by Sotirchos and Amundson. 223'224
Pseudo-steady computations have shown that porous
char particles reacting under radiative equilibrium
conditions are found to present considerably lower
burning times (by more than one order of magnitude
in some cases) than their heat-radiating counterparts. 222
5.2. Contribution oJ Radiation to Flame
Wall-Quenchin 9 of Condensed Fuels
The understanding of extinction phenomena has
been greatly improved over the years, and recently
several important mechanisms of extinction or
134
R. VISKANTAand M. P. MENGOt~
quenching phenomena have been proposed. 225.226
Among those proposed are the stretching effect of the
combustion zone, preferential diffusion, buoyancy
and heat losses. The effect of heat loss on quenching
can be more pronounced in the presence of relatively
cold boundaries, due to steep temperature gradients.
The pyrolizing surface of condensed fuels is a
representative example of cold boundaries in a
combustion situation of practical interest. In this as
well as in many other studies on heat transfer in fires,
the significance of thermal radiation has become
increasingly recognized as radiation accounts for a
significant portion of heat losses. Its effect has been
shown to be considerable not only in large-scale and
small-scale fires 227'229 but also in small diesel
engines. 2a°
Radiation blockage by soot layers between the
flame and the fuel is considered to be an important
characteristic of fires. The radiation blockage effect
has been investigated and found to depend on the
type of fuel and size of fires. 227 Using experimentally
obtained data, it is shown that for polymer fuels
of polymethylmethacrylate (PMMA), polypropylene
(PP), and polyoxymethylene (POM) no significant
radiation blockage is present in moderate-scale fires;
however, for sootier fuels such as polystyrene (PS),
radiation blockage has a considerable effect even in
small-scale fires. In Fig. 21, the effects of gas layer
thickness and soot volume fraction on the blockage
of radiation are shown graphically.
The effect of thermal radiation and conduction on
the cold-wall flame-quenching distance in the combustion of condensed fuels has been studied using a
simple physical model. 227 In the analysis a steadystate, no-flow condition, one-dimensional energy
equation for the optically thin quenching layer is
solved employing the singular perturbation technique. The quenching distance is obtained as a
1.0
/,
0.8
0.6
°
E~,- 0.9
.... E.. o . g s ~
~ 0.4
0.,
fr ~
/
'7/' / /
_
10-2
10"1
GAS LAYER THICKNESS ( m )
FIG. 21. Radiation blockage as a function of gas layer
thickness for a plane flame layer model, Lj./Lo=0.4. 227
function of various thermophysical and radiative
parameters such as the conduction-radiation ratio,
optical thickness and heat generation intensity by
chemical reactions. A new dimensionless group, the
modified Damk6hler number (ratio of dimensionless
heat source intensity to the conduction-radiation
parameter), which characterizes the relative strength
of heat generation to radiation transport, emerges
from the analysis. The quenching-layer thickness is
determined primarily by the conduction effect near
the relatively cold surface. However, thermal radiation is still the dominant mode of heat transfer
there. Numerical calculations have shown that the
fraction of radiative heat flux at the fuel surface is
over 85 ~o of the total heat flux. Optical thicknesses
less than 0.5 show little influence on the quenching
distance, and more opaque systems yield shorter
quenching distances.
Radiation blockage may also be desired in other
physical situations to avoid excessive temperatures at
the system boundaries. Siegel ~ls has systematically
studied a one-dimensional system at high temperature, with and without flow, to determine the
governing parameters needed to keep the walls at a
prescribed temperature range. Using an analytical
approach, he concluded that a dimensionless parameter ME= TJ~CLxL/C2) and the ratio of suspension
temperature to source temperature, TraIT,, were the
two important parameters. Here, fv is the soot
volume fraction, CL is the ratio of mean beam length
to layer thickness, x is the constant absorption
coefficient, and C2 is Planck's second radiation
constant. When M ~ 2 the soot (or suspension) layer
absorbs practically all the radiation incident on it,
and when M ~ 0 . 2 half of the radiation is absorbed.
Although at the beginning the soot layer blocks the
radiation, the energy trapped in the layer raises its
temperature. After a while, the layer begins to radiate
energy. This can be avoided using perforated walls
and introducing the cool seeded gas from many holes
along the surface at frequent intervals)~S A similar
approach can also be applied to combustion chambers, like the liners of gas-turbine engines, to predict
the amount of film cooling necessary. It is worth
noting the similarities between these results and
those obtained by Lee et al. 227
5.3. E.JJect of Radiation oll One-Dimensional Char
Flames
In one-dimensional pulverized-char or coal flames,
two different flame types are recognized as "small" or
"long". T M The "small" type flames can be modeled
qualitatively using a conduction-diffusion approximation, whereas for the "'long" type flames, radiation
heat transfer is also an important mechanism. Earlier
attempts to model these types of flames without
including radiation have not been very successful;
however, a model based primarily on radiation
Radiation heat transfer
predicted the flame temperatures and burnout profiles very accurately. TM In this model it is assumed
that reaction is controlled by combined diffusion and
surface chemical reaction for either shrinking, constant density particles or constant diameter, decreasing density particles. Also, the size distribution
of the spherical particles in the flame is accounted
for; however, the particle temperature is assumed to
be equal to that of the surrounding gases. This
approximation can not be justified in physical
systems where particles burn in dilute suspensions
with a large excess of oxygen, yet it is a reasonable
approximation for concentrated suspensions in practical flames (see comments of I. W. Smith to Xieu et
al.231). The radiation heat flux was obtained by
modeling the RTE between two vertical infinite
parallel-plates, and the flux divergence is given by 23~
c3q,
....
dz
4xtr T4( z ) + 2x[ a T41E2( z )
tL
+trT4.2E2('rl,--z) +tr S Ta(t)El(lz-tl) dt]
o
(5.1)
where x is the absorption coefficient for the mixture,
T is the optical distance, tr is the Stefan-Boitzmann
constant, El and E 2 are the first- and second-order
exponential integral functions, respectively. The key
parameter in this model is the absorption coefficient,
which, in general, is a function of location. The
assumption of a constant value for 1( did not yield
accurate predictions for the temperature and burnout
profiles. 232 Recognizing that the absorption coefficient is varying with the projected cross-section of
the particles (see Section 3.2), Xieu et al. TM used a
new expression
(5.2)
~, = 3QD,/4pr
where D, is the dust cloud concentration for the size
interval corresponding to mean particle radius r, p is
'°°1 I
i
J[
/
ca 60 "1
40-1
~0
I
Rosin -Rommter monosize
~ C
~ ~
%
\ I ' ~
I monosize (D~l~"
I
I
I
20
Polysize
I
~-.....__--_.~
I
I
40
60
80
ioo
Distance from tube bonk (cm)
I
J20
FIG. 22. Comparison of prediction and experiment for coal
burn-out, using the given polysize and two alternative,
mort osi ze models. 231
135
the density of coal, and Q is the correction factor
which is near 1.5 for practical flames. Equation (5.2)
is applicable if a particle size distribution is given. It
can be used also if a single mean value for "r" can be
defined. Two different mean values, one from the
Sauter-mean diameter definition (i.e. volume-tosurface ratio, D32 ) and the other from the RosinRammler index were also used in the analysis. The
predictions for coal burn-out are compared against
experimental data in Fig. 22 for polysize as well as
two mean diameter models. The polysize model is in
exceptionally good agreement with the data, suggesting that the accuracy of properties used in a
radiation model are as critical or, maybe, even more
so than the accuracy of the model itself (see also
Section 4.5).
Another model, based on the one-dimensional
Eddington (P1) approximation for predicting the
contribution of radiation in "long" flames was given
by Krezinski et al. 233 They obtained the radiative
properties of coal particles using the complex
refractive index data of Foster and Howarth; v5
however, they did not compare model predictions
with experimental data.
5.4. Radiation in a Combusting Boundary Layer Along
a Vertical Wall
Classical studies of boundary layer diffusion
flames have neglected radiation, a.234.235 To isolate
the effects of radiation in flames from the complexities of fluid motion and to gain better understanding
of radiation heat transfer in fires, analyses have been
made of a laminar, combusting boundary layer along
a vertical wall. 236-23s The rate of upward flame
spread over a vertical combustible surface is an
important parameter in the ranking of the fire hazard
offered by different materials. Typically, when the
flame height reaches about 2 m, the flow becomes
turbulent and radiation heat transfer starts to play an
important role in the overall energy balance. Free,
mixed and forced convection boundary layers along
a vertical, burning wall have been studied analytically. Previous work on thermal radiation from flames
has been reviewed 239-24~ and related experimental
work has also been reported. 242 Here, we discuss the
results of numerical solutions obtained for a steady,
laminar, radiating, combusting, boundary layer over
a vertical pyrolizing fuel slab.
An analysis has been developed for steady free and
forced laminar combusting boundary layers in which
a pyrolysis zone separates the flame from the fuel
surface 23s as shown in Fig. 23. The soot layer is on
the fuel side of the flame zone. Through the
transparent gas the combusting layer exchanges
radiation with a distant black wall, which is
maintained at a specified temperature. The chemical
energy lost to the system in the formation of soot is
neglected. The effects of radiation on the local fields,
and excess pyrolyzate escaping downstream at the
136
R. V1SKANTAand M. P. MENO0~
/
I ecu~o~v
/
/
LAYER EDGE
/
/
/
FUEL /
/
AME ZONE
/
/
/
/
/
AMBIENT AIR
FIG. 23. Schematic of a steady, two-dimensional, laminar.
radiating combusting boundary layer flame with soot on a
pyrolizing fuel slab.238
top of the fuel slab are examined by assuming that
the dominant effect of the soot particles is on the
radiation heat transfer.
The conservation equations for mass, momentum
and species for a radiating-combusting boundary
layer are identical to those for a nonradiating
boundary layer s and therefore are not repeated here.
The energy equation is given as 23s
{ Oh
t3h\
~ {k --I
p / u -- + v--1 = - - / - -
t3h\-dq'+S
(5.3)
where the enthalpy is defined as
7"
(5.4)
I1= f cpdT.
T
Assuming a spectraily gray, homogeneous medium
with a constant absorption coefficient, the local
radiative flux in the y-direction* can be expressed
as 23s
t
q,=2[Eb~.E3(z)-E
~ E 3 ( 17~ - - T)--J'- SEb(t)Ez(z
-- t)dt
0
ta
-- I Eb(t)E2(t - z)dt]. (5.5)
*There is an error in the expression for the radiation flux
given in Ref. 238, but this does not affect the validity of the
results obtained because of the approximations.
Here ~=xy and T6=x6 , where 3 is the boundary
thickness. In writing Eq. (5.5) a one-dimensional
radiative transfer model is used which is consistent
with the boundary-layer approximations.
Numerical solutions were reported for both forced
and free flow along a vertical pyrolizing fuel slab. 23s
The optically-thin approximation was assumed to be
valid for radiation. In the analysis of a combusting
boundary layer with radiation, the pyrolysis rate was
found to depend on nine dimensionless parameters
and the intensity of the external radiation flux. The
dimensionless heat of combustion (Dc) plays a
dominant role in determining the flame temperature
and makes it a significant parameter in radiating
systems. In addition, the optical thickness of the
boundary layer and the radiation parameter (Na)
affect the emission from the combusting boundary
layer. The surface temperature and the emissivity
characterize the surface emission, which can dominate the flame radiation in the boundary layer for
solid fuels of small dimension. A comparison between numerical and experimental pyrolysis rates
shows good agreement for a case where surface
emission dominates flame radiation, i.e. burning of
polymethylmethacrylate (PMMA) in air. Values of a
mean absorption coefficient and soot generation
rates were also obtained using the analysis. This type
of data could be used to quantify soot formation
models. 23s
5.5. Interaction of Convection-Radiation in a Laminar
Diffusion Flame
Most of the earlier and even some recent studies
dealing with high-temperature situations such as
those found in laminar diffusion flames have excluded
the effects of radiation (e.g. Refs 234, 235, 243-245).
In many situations radiation from the hot gases can
significantly alter temperature in both the flame itself
and in the surrounding regions as well as within the
flame structure. The relatively simple character of
diffusion flames in laminar stagnation-point flows
has led to several theoretical and experimental
studies of that system in which thermal radiation has
been incorporated in the analysis. 246- 249
Interaction of convection and radiation on the
temperature and species concentration distributions
in a diffusion flame located in the lower stagnation
region of a porous horizontal cylinder 246 and a
vertical flat plate 249 have been studied experimentally and theoretically. The exponential wide-band gas
radiation model was employed in this inhomogeneous (nonuniform temperature and composition)
problem through the use of scaling techniques. Using
a numerical scheme, the compressible energy, flow,
and species-diffusion equations were solved simultaneously with and without the radiative component.
In the experiment, methane was blown uniformly
from the surface of the porous cylinder, setting up
Radiation heat transfer
Z.O
!
I
|
0 F • 2600 c c / m l n
2.4
I[xponentlol Wide-Bond Ido~l
- - m - - Groy GaS Model K'p-0.15
~-~
2.0
No Rodiotlve Interoction
Q I[xporimento! Run I
O l[aoerimental Run 2
I.G
f,.
I
~ t 1.2
3
~. 0.8
0.4
0 I
0
~-
I.O
2.0
3.0
4.0
5.0
II
FIG. 24. Comparison of theoretically predicted and experimentally measured temperature protiles during methane
combustion around a horizontal porous cylindrical burner.2'.6
(upon ignition) a diffusion flame within the freeconvection boundary layer. Using a Mach-Zehnder
interferometer and a gas chromatograph, temperature and composition measurements were obtained
along the stagnation line. Excellent agreement has
been found between the temperature distributions
based on the nongray wide-band model and experimental data (Fig. 24). Examination of Fig. 24 reveals
that the wide-band model yielded results that were
superior to those that excluded radiation-convection
interaction. It is evident from the figure that
radiation-convection interaction lowers the predicted temperatures in the high-temperature region
of the boundary layer near the flame front and raises
the temperature in the cooler region near the edge of
the boundary layer. Furthermore, this interaction
effect increases for larger fuel flow rates. 246 The effect
of radiation interaction can be interpreted to result
from a transfer of energy by gaseous radiation heat
transfer from the hotter region to the cooler portion
of the boundary layer, thus reducing the higher
temperatures and raising the lower ones.
An attempt was made to determine an arbitrary
value of the Planck mean absorption coefficient ~
which when used in a gray-gas model would yield
temperature profiles matching the experimental
results. 246 Although the results of Fig. 24 demonstrate that matching values of ~e may indeed exist, a
different value of a mean absorption coefficient was
required for each fuel-flow condition. It has been
shown in Section 4.5 that by changing the mean
137
absorption coefficient it is possible to match experimental data with predictions; however, this approach
is not based on first principles and requires data for
each set of conditions.
The experimentally measured convective heat fluxes
at the wall of the cylinder were found to be in better
agreement with the results calculated using the wideband than the gray-gas model. 246 This further
supports the performance of the nongray model and
shows that the model is superior to those based on
the gray-gas model as well as the results that ignore
the effects of radiation. Measurements of convective
and radiative heat fluxes in a diffusion flame
surrounding a porous cylinder burning drops of nheptane have shown that radiation heat transfer to
the cylinder is by no means negligible. 24s Radiation
accounts for about 40 % of the total heat transferred
to the cylinder, but the radiation from gases (CO2
and H 2 0 ) is only 20 % of the total radiation, with the
rest being soot radiation. For those types of flames
where soot radiation is more important than gaseous
radiation, the use of a gray model would yield
reasonable results.
A similarity solution for an opposed laminar
diffusion flame with radiation has also been obtained. 2'.8 In this combustion system a stream of
oxidizer approaches the stagnation point on a
condensed surface and reacts with pyrolyzed fuel in a
thin diffusion flame with a constant-thickness boundary layer. The fuel surface is assumed gray and
diffuse, and the gas is considered gray. Only the
pyrolysis region is considered. Numerical results
were obtained using the exponential kernel and
optically-thin approximation for radiation heat
transfer.
Analysis reveals eight dimensionless parameters
which control the system under investigation. Five
parameters, i.e. the mass consumption number, r; the
mass transfer number, B; the Prandtl number, Pr; a
dimensionless heat of combustion, D o the fuel
surface temperature, 0,., are the combustion groups,
and the three radiation groups, the conduction/
gaseous radiation parameter, N~, the conduction/
ambient radiation parameter, N 2 , and the fuel
surface emissivity, e,., are required to describe the
combusting-radiating system. The parameters D c
and 0w, which were of secondary importance in
nonradiating systems, emerge from the analysis with
new significance, dominating the parameters N~, N 2
and ew.
The effect of radiation on the pyrolysis rate and
unburned fraction of total pyrolyzate is shown in
Fig. 25 for axisymmetric combustion. 248 The pyrolysis rate is seen to increase strongly with increasing
dimensionless heat of combustion, D c. This is
primarily because of an increase in flame temperature (01 ~ O,.Dc). Like the mass fraction of fuel at the
surface, the dimensionless flame temperature 0y,
which for non-radiating flows can be determined a
priori from measurable quantities, depends on all
138
R. VISKANTAand M. P. MENGOq
I.o
....
i.o
t
Rodiotin9
------ Non.r(idiolin
9
o.s N
-~
o.s
0
I.O
DIMENSIONLESS HEATOFCOMBUSTION,Oc
IO
0
FIG. 25. Pyrolysis rate and unburned pyrolyzate vs dimensionless heat of combustion for axisymmetric
flow with B= 1.0, r=0.22, 0,,.=2.0, N~ = 0.05, N 2 = 5 0 . 0 , and e = 1.0. 248
eight parameters and is not predetermined. The
pyrolysis rate with radiation is lower due to the net
efflux of radiation at the surface. In general, the
influx of gaseous radiation is insufficient to cancel
the efflux of surface emission, hence a lower pyrolysis
rate results in comparison to non radiative combustion. Lower pyrolysis rates may result even when a
net influx of radiation prevails because of the
decrease in conduction caused by the lower flame
temperature due to radiant loss from the combustion
zone. At low Dc, the reaction releases little energy to
counter surface emission losses, giving low pyrolysis
rates, whereas at large Dc much energy is released
which easily overcomes surface losses and yields
large pyrolysis rates. 2as The net effect of radiation on
pyrolysis appears to be low for several reasons. The
properties of real opposed diffusion flames are not
yet sufficiently well known to give accurate parameter values. Those chosen for the calculations (see
caption of Fig. 25 and others in Ref. 248) may not be
sufficiently realistic as they all tend to underestimate
the differences between non-radiating and radiating
systems.
As data on radiative properties of stagnationpoint flames become available, the approximation of
a constant absorption coefficient should be replaced
with a nonuniform one based on measured distri-
butions of soot volume fractions and CO 2 and H 2 0
concentrations. The utility of the analysis will then
come from both the proper quantification of radiative effects in opposed-flow diffusion flame experiments and from the use of such systems to refine
techniques for incorporating radiation in combustion modeling.
5.6. EJJect oJ Radiation on a Planar, Two-Dimensional
Turbulent-Jet DiJJusion Flame
A simple combustion situation has been modeled
to assess the importance of thermal radiation in
establishing temperature distribution in a turbulent
diffusion flame. 25° Although, turbulent diffusion
flames have been extensively studied by Bilger and
his co-workers) 51- 253 they have not considered the
effects of radiation. However, radiation heat transfer
modifies the temperature distribution which, in turn,
affects the combustion process. Small changes in
peak temperatures have a large influence upon nitric
oxide production for a given residence time. It is of
interest to determine how the various control
strategies such as lowering combustion air preheat or
recirculating exhaust products into the combustion
air affect the unwanted nitric oxide emissions and
the desired radiation heat transfer.
BLACKPLANEWALL
!11111111111111111111111111111/6
AIR
FUEL
~f
Am
t
i
JET
MIDPLANE
///7"//////////////////////// ///
BLACK PLANEWALL
FlG. 26. Schematicaldiagram ofaphme, radiatingjetconlinedbetweentwoparallelplates.
Radiation heat transfer
The physical model of the problem analyzed by
James and Edwards 25° is shown schematically in
Fig. 26. A planar jet of methane is injected with
velocity ut,~,l into a stream of air flowing with
velocity u,~r parallel to the fuel. Diffusion-controlled
combustion occurs in the mixing region of the jet.
Plane-parallel, isothermal and black walls symmetrically located above and below the jet form the
combustion chamber. A soot-free flame is assumed to
exist so that molecular gas bands determine the
thermal radiative transfer to the walls. Boundarylayer approximations were used to simplify the
conservation equations, and nongray radiation
described by the exponential wide band model for
molecular gas band radiation was added to the
energy equation. The model equations for turbulent
combustion of methan in a planar, enclosed jetdiffusion flame were solved numerically.
The analysis demonstrates that realistic nongray
radiative transfer calculations can be coupled to an
implicit numerical method for solution of the highly
nonlinear partial differential conservation equations
without undue expenditure of computation time. The
results of computations have shown that the larger
channels (A= 1 m and 10 m) have markedly lower
peak temperatures because of greater gaseous radiative transfer. It was also found that a given
reduction in minimum combustion temperature to
reduce nitric oxide formation could be accomplished
with a much less detrimental reduction of heat
transfer by recirculating exhaust product into the
combustion air than by reducing preheat.
5.7. Radiation Ji'om Flames
Gas- and liquid-fueled flames have numerous
applications and flame radiation is an important
aspect of heat transfer in furnaces, internal combustion engines, aircraft propulsion systems, flares,
unwanted fires, etc. This has motivated many studies
of flame radiation and comprehensive, up-to-date
reviews are available.'*a'239.25'*'255 The issues of
concern here are nonluminous and luminous radiation from flames, prediction of radiation characteristics given the instantaneous scalar structure, and
turbulence/radiation interactions in simple laboratory
flames.
Significant progress has been made concerning
structure and prediction of radiation intensity of
nonluminous flames. Narrow band-model predictions T M for nonisothermal mixtures of CO2, H 2 0
and CO are in good agreement with the measurements. The total transmittance nonhomogeneous
model (TTNH) of Grosshandler 256.257 has been
found to be about 500 times faster than narrow-band
models. The model has been applied to several
realistic combustion examples containing variable
concentrations of CO2, H20, CH4, CO and soot. It
was found to be usually within 10% of the more
accurate computation. 2s 7
139
G4
I
°,,,
.t
....
. s A . peso.
H ,1T/'~
-.-
s~e,. ,~o. I ] :W ',.~
,..,ooo
o.2
!
"s
~
iYl
"L
_sJ ~.J.J
0.0
,z, ~4i
.,o ..o
"\
,1 v
",~
I%~J
I
I
I
_
-
"'-'~- . . . .
x/O
~1~\
- 50
-
,q
-
-
0.0
1.0
2.0
WAVELENGTH
3.0
4~
(pm)
FIG. 27. Spectral radiation intensities ~ r radial paths
through aturbulenthydrogen:airdiffusion flame. 25s
Estimates of spectral intensities emerging from
flames, based on predicted mean scalar properties,
are typically within 20-30 % of the measurements of
well-defined laboratory flames. 31,25'*.255 This is comparable to the uncertainties in the narrow-band and
flame-structure models. Measured and predicted
spectral radiation intensities for a turbulent hydrogen/
air diffusion flame are given in Fig. 27. Results are
shown for horizontal radial paths through the flame
at x/D=50 and 90, the latter position being just
below the flame tip. Predictions use both timeaveraged scalar properties along the path and
stochastic methods which take into account turbulence/radiation interactions. The stochastic method
models the interactions by assuming that the flow
field consists of many eddies which are uniform and
statistically independent of each other. Eddy length
varies along the path length, and time-averaged
probability density (PDF) of mixture fraction f for
each eddy is randomly sampled and scalar properties
are found from the state relationships at the
corresponding value of J~ Once the scalar properties
are known, the RTE is solved. The details of solution
can be found elsewhere. 31 Spectral radiation intensities (Fig. 27) are dominated by the 1.38, 1.87 and
2.7/~m water vapor bands in the range of 1-4/~m
shown. The stochastic method yields spectral intensities which sometimes are about a factor of two
higher than the mean property method with the
measurement generally falling between the two
predictions. These results suggest significant effects
of turbulence/radiation interactions. Findings for
carbon monoxide/air and methane/air flames, however, show smaller effects for turbulence/radiation
interactions. 254.255
140
R. VISKANTAand M. P. M~,~GO~
Work on luminous flames has been limited.
Similar results to those presented in Fig. 27 have
been reported by Gore and Faeth (cited in Refs 254
and 255) for a turbulent ethylene/air diffusion flame.
The spectra are dominated by continuum radiation
from soot, however, the effects of 1.38, 1.87 and
2.7/~m gas bands of the H 2 0 and the 2.7 and 4.3 #m
gas bands of CO2 can still be seen. In this case the
mean-property method has provided the best quantitative agreement with the data, but the agreement is
considered to be fortuitous in view of poorer
extinction predictions obtained using the approach. 255 The predictions of continuum radiation
are very sensitive to local temperature estimates, and
the assumption optically-thin radiative heat losses
are quite crude. Differences between mean property
and stochastic predictions suggest significant effects
of turbulence/radiation interactions in luminous
flames. More exact coupled structure and radiation
analysis could modify the relative performance of the
mean-property and stochastic methods and suggest
that presently available models must be improved.
Measurements and predictions of total radiative
heat fluxes to points surrounding the turbulent
hydrogen/air, 258 carbon monoxide/air, 259 methane/
air, 26° and ethylene/air T M diffusion flames have been
made. The discrepancies between the measured and
predicted total radiation heat fluxes along the axis of
a turbulent methane/air diffusion flame (Fig. 28) are
within the order of 10-30 %. Such levels of error are
similar to the differences between prediction and
measurement for the spectral intensities. Comparable
1.0 I
I
I
I
0.8
E
0.6
X
:3
.-I
t,I.
0.4
m.m
W
V<r,
O.2
O.0t0
800
1600
2400
AXIAL DISTANCE (ram)
FIG. 28. Total radiative heat flux distribution along the axis
of turbulent methane air diffusion flames at NTP. -''°
results have been obtained for other laboratory
diffusion flames. Excellent agreement has been obtained between measured and predicted radiative
heat flux distributiotrs parallel to the axi's of
turbulent carbon monoxide/air diffusion flames. 259
The mean property predictions agree very well with
the measurements because the effects of turbulence
radiation interaction are small. The analysis correctly
predicts maximum heat fluxes near the flame tip as
well as the effects of burner flow rate.
Discussion of the effects of turbulence/radiation
interactions has been given by Faeth et aL 31'255 The
available results show that the interactions are very
significant for hydrogen/air diffusion flames, with
stochastic predictions being as much as twice the
mean property predictions. 2sa In contrast, turbulence/
radiation interactions caused less than a 30~o
increase in spectral radiation intensities for carbon
monoxide/air and methane/air diffusion flames. This
difference is attributed to the relatively rapid variation of radiation parameters (water vapor concentration and temperature) near stoichiometric conditions for hydrogen/air diffusion flames. The
stochastic methods at.254~255 have many ad hoc
features and additional fundamental research effort
is needed to develop more reliable methods not only
for small laboratory flames but also for scaling large
flames containing soot.
5.8. Combustion and Radiation Heat TransJer in a
Porous Medium
Although flame radiation plays an important role
in combustion systems, a furnace requires a sufficient
volume for the heating chamber to increase the
opacity of the flame and furnace for effective
radiation heat transfer to the load. Moreover, the
load must be placed away from the reaction zone to
prevent the emission of unburnt species when its
surface temperature is low. These factors make it
difficult to reduce the size of a combustion chamber
appreciably. Echigo 262,26a has shown that a porous
medium of an appropriate optical thickness placed in
a duct is very effective in converting enthaipy of a
flowing gas stream to thermal radiation directed
toward the higher temperature side. Successful applications to an industrial furnace, 262'263 to a combustor of low calorific gas, 264 to a water tube by
combustion gases in porous media, 265 and to other
systems 26a have been reported. The thermal structure
in the porous medium with internal heat generation
due to chemical reactions has been studied analytically and experimentally,265'2~6 and a review of the
work is available. 26a'26~
A one-dimensional model in which radiative
transfer in a gas-solid two-phase system is treated
rigorously has been constructed, and extensive
numerical calculations have been performed for a
radiation controlled flame. 267,26a The combustion
Radiation heat transfer
I
f
I
I
I
I
I
40
I
50
Significant energy recovery has been achieved from
the burned gas to preheat the combustible mixture
prior to entering the reaction zone by propagation of
thermal radiation against the flow direction. This is
deafly shown in Fig. 29 which compares the
predicted and measured particulate-phase (Te) temperatures in the system. 26s In the figure, both the
distance x and also the optical depth z along the
combustion system are used as the abscissa. The
results show that as the combustion load increases,
both the measured and calculated temperatures
increase uniformly. This is a consequence not only of
the relative reduction of heat loss in comparison to
heat generation during combustion but also due to
the essential nature of radiation heat transfer.
1200
800
• . . . . 1538 W
o .... 1025 W
"~ ~ ' - - 513W
4O0
O
-IO
I
I
0
I
1(3
I
I
20
30
x {mm}
~I P.-t
-6o
aM-=
r
I~
141
PM-n
, 3 5 ,,.35
FIG. 29. Comparison of measured and predicted temperature structures in porous media for different combustion
loads. 2~8 The lower abscissa scale r is optical depth based
on the ~bsorption coefficient of the porous medium, and
PM-I, PM-I1 and PM-III are the abbreviations for porous
media I, I! ~md II!, respectively.
mixture flows through a porous medium and the
combustion reactions take place in the medium. The
results of comprehensive calculations show that the
thermal structure (profiles of temperature, local
radiation flux, etc.) in the high porosity medium
depends strongly on the absorption coefficient and
total optical thickness of the medium as well as the
position of the reaction zone. G o o d agreement
between predicted and measured temperature distributions has been obtained and a drastic temperature
decrease in the porous medium has been revealed. 266"26s The results have also revealed remarkable heat transfer and combustion augmentation.
6. APPLICATIONS
TO COMBUSTION
SYSTEMS
The advent of more powerful digital computers
has provided the means whereby mathematical
modeling can be applied to combustion system
problems to facilitate the arduous task of their
design. This is now of great interest in view of the
current demands which system designers are required
to meet--in particular, efficiency of combustion at a
wide range of operating conditions and strict control
of pollutant emissions. The latter has become
increasingly stringent in recent years for economic
and political reasons. The present trend is away from
the traditional cut-and-try methods, which are expensive and do not necessarily produce the optimum
design, toward fundamental modeling of the physical
and chemical processes occurring within the combustion systems. Multidimensional modeling of twophase combustion is being approached with the aim
of producing algorithms based on fundamental
I Two ° Phase Fluid
Mechanics
(Turbulent)
Chemical
Kinetics
Phase
Transitions
(Evaporation)
(Condensation)
Nucleation
Gas
Particle
Interaction
Par title Phase
Reaction
Devolatilizofion)
eter. Oxidation)
Gas Phase
Reaction
Heat Transfer
(Convective)
(Radiative)
FIG. 30. Schematic representation of submodels for combustion of coal.
142
R. VISKANTAand M. P. MENGi3t~
principles which can correlate all of the details of
combustion systems. 3,269-272 The predictive procedures for a combustion system model require
theoretical and empirical inputs to describe turbulent
flow, chemical kinetics, thermodynamic and thermophysical properties and other transport processes,
including radiation heat transfer (see Fig. 30).
This section of the article discusses application of
the methodology described in the previous sections
to practical combustion systems. The emphasis is on
the methodology and radiation heat transfer results
rather than the application of mathematical techniques for design and performance calculations of
practical systems. Even with the advances in mainframe computers the difficulty of treating infrared
radiation transfer rigorously in nonhomogeneous
gases containing particles lies primarily in the
enormous complications introduced by selective
gaseous emission and absorption of radiation as well
as scattering by irregular-shaped particles. Because of
this complexity practical simplifications are necessary to keep the calculations at a reasonable level. As
a compromise between desired accuracy and computational effort, practical methods which are also
compatible with the numerical algorithms for solving
the transport equations are stressed, and radiation
heat transfer in several different combustion systems
is discussed. The body of literature concerned with
modeling and evaluation of combstion systems is
very large, and it is not practical in an article of
limited scope to discuss even the more recent works.
Most of the work reported has stressed modeling and
evaluation of chemically reacting turbulent flows and
combustion and much less radiation heat transfer.
The emphasis in this review is on the latter.
6.1. Industrial Furnaces
One of the important parameters in assessing the
performance of an industrial furnace is the heat flux
distribution to its thermal load (sink). Methods
based on fundamental principles are now available
using numerical techniques and digital computers,
that permit determinations to be made for both gasand oil-fired industrial furnaces. In such furnaces
heat transfer to the load is predominantly by thermal
radiation. The problems associated with prediction
of radiation heat transfer within the combustion
chamber can be divided in two main types:
(a)
Evaluation of radiation heat transfer at all
locations in the enclosure if the temperature
distribution and radiative properties of the
combustion products are known; and
(b) Evaluation of radiation heat transfer as well as
temperature and radiating species concentration
distributions.
Problems of type (a) are more straight-forward and
require development of radiation heat transfer models~
Problems of type (b) require the coupling of the
radiation model, through the radiative properties of
combustion products, to the mathematical transport
model to predict the temperature and radiating
species concentration distributions. With the presently available algorithms, 3"5'269-271 the latter type
problems require an iterative solution procedure
which is rather time-consuming.
A validated computer model has been used to
construct a detailed energy flow (Sankey) diagram for
an industrial furnace. 273 The diagram (Fig. 31) shows
that more than half of the heat to the load comes
from the refractory wall. Of the balance, part is
convection (4~), part is direct radiation from the
flame (6 ~), and part is flame/wall radiation absorbed
by the gas which has been re-radiated by the wall
(6 9/o). The furnace shows a thermal efficiency of 35 ~o
with the typical high flue loss and indicates the
importance of the wall-to-wall re-radiation effect.
With the exception of different magnitudes, Fig. 31
shows a typical pattern for all industrial natural gas
and oil fired furnaces. As the flame becomes more
opaque and/or the wall temperatures drop there will
be obviously more radiation from the flame and less
from the wall. Also, as the wall temperatures drop
there will be smaller radiation exchange between the
walls and the load.
The close examination of Fig. 31 clearly indicates
why there has been so little attention given to the
calculation of convective heat transfer inside furnaces. In industrial furnaces convective heat transfer
usually accounts for a very small fraction of the total
heat transfer to the load. Local convective heat
transfer coefficients have been measured at a surface
heated by gases 274 and empirical correlations for the
average Nusselt number have been reported for
differently-directed gas streams incident on the
load. 274-276 An interesting finding of the experimental study 274 was that in the absence of combustion the average heat transfer coefficient at the load
surface was about 35 W/m2K, while in the presence
of combustion the values were from 80 to 120 W/m2K,
suggesting almost a threefold enhancement of convective heat transfer by combustion.
Radiation in furnaces predominates over convection; therefore, more emphasis has been given to
radiation over the years and the radiative transfer
theory has been much more fully developed,4-5 •2 7 7 - 2 8 1
and presently capability exists to predict simultaneous
three-dimensional flow, heat transfer and reaction
rates inside furnaces. 269'27! However, the theory has
outstripped experimental validation, which is in a
much more primitive state, but even in this area a
number of papers describing direct comparisons
between predictions and experimental data have
appeared, t 6 9 , 2 8 2 - 2 8 8
The results obtained for a model furnace using
the phenomenological furnace-performance equations
have been used to determine the relative importance
of the model parameters. 2s° Analysis of the results
led to the conclusion that the flame emissivity was of
Radiation heat transfer
~/Au°T//~~~~
CONVECTION
143
TOI.OAD4°/mL
FItJ. 31. Energy flow (Sankey) diagram for one operating point of un industrial furnace, illustrating the
four different contributions to output .'rod the effect of wall-to-wall radi~,tion exchange, z-3
Type 1: Stirred Vessel
'°'
,--t-
~Heot
Flux Oistri~
Type 2: Plug Flow
Type 3: Two - Dimensional
rather specific conditions; and some aspects of the
analysis could be considered arguable.
The models for analyzing heat transfer in
industrial furnaces are of three types (see Fig. 32):
(1)
the
"stirred
vessel"
(zero-dimensional)
mode145.273.277.278,281,285-289 which yields only the
total heat transfer rate without providing information on the local heat flux distribution, (2) the
"plug-flow" (one-dimensional) model z76,2s°,zs2-2ss
which is capable of predicting the local heat flux in
the furnace along the flow direction, and (3) the
multi-dimensional model 269.271 which can predict
two-dimensional heat flux distribution at the load
surface. The first two models are being used routinely
in engineering design calculations, and these models
are discussed here in greater detail.
6.1.1. Stirred vessel model
',x~- .~:- F-~T-j---:t:_F-_a..i
Heo! Flux Oistrib.
HG. 32. Schematic representation of it furnace illustrating
different heat transfer models.2as
second-order importance. The other factors (in the
order of decreased importance): heat transfer to the
load (sink), excess air, process temperature, flame/load
temperature difference, load absorptivity and wall
losses, were of greater significance. These conclusions
were reached, however, by generalizing from some
Let us consider a schematic diagram of a furnace
(Fig. 33) and apply the "stirred vessel" model to
calculate the heat transfer rate to the load. According
to the model '.5.277,278.2al the combustion products
are assumed to be gray and at a uniform temperature.
The temperature and the radiative properties of the
load and of the refractory walls are assumed to be
uniform but different. A steady-state, overall energy
balance on the load can be written as
/~/1 -/~/2 = Q, + Qe
(6.1)
Here/~/~ and/:/2 are the ¢nthalpy inflow and outflow
rates. Within the framework of the zonal approxi-
144
R. VISKANTAand M. P. ME~Gi~t~
,0,,
Y,,'/,//
F s --
,/////'/////////
Qs
A~_,,a
T~
Waste Gases
~aml~stloa
==~.
Products
o.,
(6.5)
T,
Fuel fit AIr
/~>
~o.
//I
Lo,,
/
= [ 0 ~ - O] + ( S t ' K o X O m - 0,)].
~,.r=.~
For the special case when convective heat transfer to
the load is negligible in comparison to radiation
(St=0), Eq. (6.4) simplifies to
e/r,,A '
7111111111111111~
Ko(1 - 0m)= 0~-- 0].
(6.6)
FIG. 33. Schematic diagram of a stirred furnace model.
mation for radiation heat exchange, the heat transfer
rate to the load can be expressed as 277'2al
O~---A, [h(T~ - T~)+ ,fr _ ,,a(T~ - T])]
(6.2)
where h is the average convective heat transfer
coefficient at the load, and " ~ s - m is Hottel's radiation
exchange factor or A ~ - ~ _ m is the total radiation
exchange area. This factor is a rather complicated
function of the gas emissivity, wall emissivity and the
sink-to-refractory area ratio, and expressions are
available in the literature. 45'287 The heat losses
through the walls of the furnace can be expressed as
Qt = UoAo(Tm - T,)
(6.3)
where U0 and Ao are the overall heat transfer
coefficient and area of the refractory walls, respectively; and T,, and T, are the mean combustion
products and ambient air temperatures, respectively.
Substitution of Eqs (6.2) and (6.3) into Eq. (6.1), and
assumption of negligible wall heat losses allows the
resultant equation to be written as
+l + St
0~-0,,, = ( 1 / K o ) ( 0 ~ - 0 ~ )
1~
By eliminating the mean combustion-product temperature, the dimensionless heat transfer rate can be
expressed as
r ~ = Ko[1 - ( F, + 0 ~)z/'*].
Extensive calculations have been reported for the
dimensionless mean gas temperature and heat transfer rate and the results can be found in the
literature. 2aS-za7 Experiments have also been performed and compared with model predictions. 2s6'2a7
Figure 34 shows a comparison between the measured
and the calculated average heat fluxes in an experimental combustion chamber having a 1.25 m long
firing space and two different cross-sections (0.4 m x
0.4 m and 0.4 m x 0.8 m). The results show that the
stirred-vessel heat transfer model can be successfully
applied to those furnaces in which there is no
appreciable axial drop of the mean gas temperature.
This condition is roughly met in combustion chambers fired with high-velocity burners and in furnaces
where the flame length is approximately equal to the
furnace length. Under these conditions, a maximum
error of _+20% can be expected in calculating the
absorbed heat flow to the load being heated. In
predicting the energy consumption of the furnace,
this would mean a maximum error of _+10 ~oo.286
(6.4)
l + St
100
F~naee Crog-Seellon In mmZ=LOO,t.O0 ~Or,9~
I¢o, 06
•
V
where the dimensionless variables and parameters
are defined as
0 =T"
"
0
T~
;nCpm
A ~ . _ ma T 3
7,;
(6.7)
hA~
60
©
-..
\
"1-
"~ d" k
St= :~.
rnC pra
In this equation, T~ is a fictitious gas inlet temperature in which the heat losses through the walls of the
furnace have been accounted for; m and q,,. are the
gas mass flow rate and the mean specific heat of the
gas, respectively, and K o and St are the Konakov and
Stanton numbers, respectively. The heat transfer rate
to the sink, Eq. (6.2), can be expressed in dimensionless form as
0
0
200
too
600
oCo iOoo
1200 'v,OO
Surface Temperature (~)
FIG. 34. Comparison of measured and predicted average
heat fluxes in a furnace as a function of the load
temperature, z8~
Radiation heat transfer
145
where
6.1.2. Plug flow model
A schematic diagram of the "plug-flow" model is
shown in Fig. 35. The temperature of the gas
(combustion products) is assumed to depend on the
coordinate x in the flow direction. This means that
the plug flow model can be considered to consist of
an infinite number of stirred vessels. The temperature
and the radiative properties of the load and walls are
assumed constant but different. Based on a gray-gas
and zonal approximation for radiation heat exchange,
the steady-state energy balance on a control volume
of gas of length dx gives
x
Ko=
+h[T~j(x)- T~] } - P o U o [ T ~ ( x ) - T j
(6.8)
where W and P0 are the furnace width and perimeter,
respectively, and lg is the effective gas emissivity
which accounts for the refractory walls and other
surfaces in the furnace. In dimensionless form, the
energy equation for the gas temperature can be
written as
dO._ (1/KoXO4_O~)_St(O_O~)
d¢
-- ~0(0.--0~)
'
DiCpm
h WL
St . . . . .
WLi,.fl T 3
~ncp~
dPo -
dT~(x)
.
,
,
thepm dx - - W{e,a[T,(x)-- T,]
T~ 0 _ 7 " ;
°"=
UoPoL
thcmn
Analytical solutions of Eq. (6.9) and its special forms
have been obtained and graphical results reported.29°'291
Extensive numerical calculations of the gas temperatures along the furnace using the stirred-vessel,
stirred-vessel-cascade, plug flow and the modified
zonal models have been reported for furnaces having
constant and varying sink temperatures. 292'293 A
comparison of temperature distributions using five
zones (sections) along the furnace is given in Fig. 36.
The published results show that as the number of
sections in the furnace increases, the temperature
distribution predicted employing the stirred-vesselcascade and the modified zonal models approaches
the temperature calculated using the plug-flow model,
As expected for a single section along the furnace, the
stirred-vessel and the modified zonal models predict
practically identical gas temperatures in the furnace.
(6.9)
6.1.3. Multi-dimensional models
~
H
Products
dQ,,
IldO~
....
H
........
The computational methods which have been
developed are able to complement, but not replace,
empirically based design procedures. This is because
chemically reacting turbulent flows are not fully
understood, and it is proving particularly difficult
to eliminate the deficiencies of existing turbulence
models. In the absence of reliable turbulence models
it is hardly possible to subject any of the everincreasing number of combustion and radiative
U I'T!
-.:'H
FIG. 35. Schematic diagram of a plug flow model.
1800
\\
' {o)'
i
'
i
i
\
,~. Plug Flow
1600
ca
'
%--Plug Flow
\
\
1400 , S f i t r e ¢ 1 % %
Vessel ~
E
2x_~~ .,.
~.
Stirred Vessel
;!.,... / :aecade
\
Zonol
1200
Stirred Vessel
~
Zonal
O
it\x)
o
i
02
014
i
0.6
i
08
0
1
0.2
i
0.4
0.6
i
0.8
LO
x~
FIG. 36. Comparison of gas temperature distributions along a one-zone la) and five-zone (b) furnace
predicted by different models: Ko= 1, K=0.1 m- ] i:~j=0.104,~==0.8, h/h=2/l, l/h=20/l, T~= 773 K. 2<j3
J?gCS 13:2-D
146
R. VISKANTAand M. P. MENG(~(;
transfer model proposals to a stringent assessment.
Nevertheless, two-dimensionaP ,294.295 (among others)
and three-dimensional 269.271.296 combustion validation studies reveal, for gaseous combustion at least,
that predictions which are obtainable are sufficiently
reliable to be of interest to combustion engineers.
General computer-based procedures for the prediction of gaseous-fired rectangular and cylindrical
combustion chambers have been developed and a
review is available: The zonal, flux, discreteordinates and first-order spherical harmonics ( P :
approximation) methods have been assessed. For
natural gas and oil fired furnaces only three species
(CO2, HzO and soot) contribute significantly to the
transport of radiation in the infrared. The computations have been carried out on the gray or at most
on a weighed sum-of-the-gray gas bases. Reasonable
agreements are reported between measured and
predicted fluxes (see Ref. 5 for comparisons). Unfortunately, the original references includes little detail
on how the mean absorption coefficients needed in
the radiative transfer models have been determined.
It is suspected that the authors had to do considerable "fine-tuning" of these model parameters to bring
about good agreement between model predictions
and data. The sensitivity of the results to radiative
properties have already been discussed in Section 4.5.
It should be pointed out that in the studies
discussed by Khalil 5 and others 269 the emphasis has
been on modeling chemically reacting turbulent flow
and combustion and much less on realistic modeling
of radiation heat transfer. The general prediction
procedures which describe the computation of flow,
reaction, and heat transfer in the combustion region
of a typical, natural gas-fired industrial glass producing furnace are sufficiently developed to constitute a useful design tool. 269 Economic handling of
three-dimensional geometric features is considerably
enhanced by the use of special grids and the separate
calculation of the burner and bulk combustion
chamber regions in a manner which takes into
account the differing features of their flows. The
predictions demonstrate the value of computations
to furnace designers for the range of operating
parameters. Recently, the radiative transfer has been
treated in sufficient detail using the discrete transfer
method which contains some features of the zone,
discrete ordinates and Monte Carlo procedures. 2°4
The combustion products are treated as gray and
scattering by particles, such as soot agglomerates has
been neglected.
A detailed discussion of analytical modeling of
practical combustion chambers and furnaces, including a very extensive review of the literature has
recently been given by Robinson. 271 A threedimensional mathematical model is constructed of a
large tangentially-fired furnace of the type used in
power-station boilers. The model is based on a set of
13 differential equations governing the transport of
mass,momentum and energy, together with additional
250
i
,
------
i
,
,
i
,
/ T I i " I000 K
2O0
150
.••'•Wilh
5O
Tuth/RQa. Inlw.
Willloul Turb./Ro4. I n t l r .
0
-50
I
0
I
0.2
I
I
I
04.
I
I
0.6
I
0.8
I
1.0
x/L
FIG. 37. Effect of preheated air fuel mixture temperature
and turbulence, radiation interaction on heat flux distribution along a two-dimensional furnace burning methane:
H= 1 m. L=5 m, ~ = 1500 K. e~=0.8, e, =0.6. 3°
equations constituting subsidiary models of turbulence, chemical reaction and radiation heat transfer
phenomena. A six-flux, gray gas model is used to
predict radiative transfer. Computer-memory limitations restrict the amount of geometrical detail that
can be included and prevent the use of a finitedifference grid having the desired fineness. The model
is validated against experimental data acquired on
two large, natural gas-fired furnaces.
Recently, the effect of turbulence/radiation interaction in a two-dimensional, natural gas-fired, industrial furnace has been examined. 3° Based on an
approximate analysis of radiative transfer, the results
of calculations show that the effect of turbulence/
radiation interaction on combustion and scalar
properties is small for a preheated fuel-air mixture
when the flame occupies a small volume of the
furnace. However, when the flame occupies a large
volume fraction of the combustion chamber the
interaction is quite significant. Another reason why
the effect of the interaction is larger for T;= 300 K
than for T~=1000 K is because the temperature
fluctuations are larger for the former case. The effect
of the interaction on the total heat flux along the
furnace shown in Fig. 37 clearly indicates the need to
account for turbulence when predicting radiation
heat transfer in large, high-temperature combustion
systems. The net local heat flux to the sink (load) can
become negative for the case when the turbulence/
radiation interaction is neglected, because the assumed sink temperature (Ts= 1500 K) is higher than
the local effective temperature of the combustion
products.
6.2. Coal-Fired Furnaces
Radiation heat transfer in coal-fired furnaces has
received considerable attention for more than 60 yr
because of the realization that it is the dominant
mode of heat transfer in such systems. The earlier
Radiation heat transfer
work on the subject has been discussed by Doleza1297
and more recent studies have been reviewed by
Blokh. 4 The latter volume in particular contains a
large body of fundamental radiation property data,
measured spectral and total incident radiation fluxes
along the height of different capacity furnaces as well
as empirical correlations for analyzing the thermal
performance of coal-fired boilers. An up-to-date
discussion of coal combustion models in which
radiation heat transfer has also been considered is
available. 3 Despite the considerable progress in the
development of analytical methods of engineering
science and despite an increasing understanding of
fundamental combustion processes, the design or
performance predictions of coal-fired furnaces may
still be considered as an art based primarily on
empirical knowledge and the ingenuity of the combustion engineer. This is particularly true for large
boiler furnaces because of their extremely complicated geometry and boundary conditions 4'272'29s as
well as the lack of confidence in the existing
analytical methods. Scale-up and advanced performance analyses of boiler combustion chambers have
been developed272 using laboratory and/or small
model furnace data. In spite of major improvements
in the analytical methods for predicting the performance of coal-fired furnaces 3'272 there is still distrust
by practical furnace designers of the analytical
methods because of geometrical restrictions, problems
of stability, complexity of the new methods, limited
applicability of the models, etc.
In this section we discuss the use of more recent
models to predict radiation heat transfer in relatively
simple furnaces, for the purpose of gaining improved
understanding of radiative transfer and of the
relative importance of the model parameters. It is
hoped that this would provide the bridge between the
scientific community which is developing comprehensive combustion system models and furnace
designers who are attempting to solve practical
problems based on empirical knowledge. Reference is
made to literature which discusses methods for
evaluation of thermal performance of large boiler
furnaces.
Detailed reviews of radiation heat transfer in
pulverized coal-fired furnaces are available.4"272"299
Radiation heat transfer in furnaces is due to gaseous
and particulate contributions. Emissivity data for the
major emitting gaseous species CO2 and H 2 0 are
generally adequate. 4.64 Other gaseous species (e.g.
CO, SO2, NO, N 2 0 ) are usually of secondary
importance because of low concentration. Local
variations in gas temperature and species composition
are subject to more uncertainty than the emissivity
data. Contributions to particle radiation in pulverized coal-fired systems usually results from coal
(char), soot andfly-ash. Information required for
predicting radiative transfer includes different particle
concentrations, size distributions, complex indices of
refraction and temperature. 2~'* Finally, the mineral
147
matter deposited onto surfaces of coal-fired furnaces
can greatly affect radiation heat transfer due to the
alteration of its emissivity. 7s Mineral matter and ash
deposited on walls of the tubes can also increase
greatly the thermal resistance to heat conduction
across the deposit, and some simple conductance
models have been developed.'*
Data for soot, carbon and coal refractive indices
are generally (but not necessarily very accurately)
available,'*'64 but significant uncertainty exists in the
particle concentration and size distributions. In
gasifiers and staged combustion systems, which
operate fuel-rich for nitrogen oxide pollutant control,
soot radiation may be particularly important. Unless
the soot-volume-fraction distribution in the medium
is known accurately, radiation heat transfer to the
chamber walls can not be predicted with confidence.
Fly-ash particles greatly influence the radiative
properties of the flame and of the combustion
products in a pulverized-coal fired furnace. Data for
fly-ash are much less certain. 4'79-83 There is significant variation in the refractive indices of pulverizedcoal and fly-ash with the type of coal, mineral matter
in the coal, as well as the combustion process itself.
Experiments have revealed that the refractive index
of fly-ash particles formed during the combustion of
even one coal shows quite large differencesfl 7 Lowe
et al. 3°° have shown that in large boilers fly-ash
exerts a much greater effect on heat transfer to the
heat-absorbing surfaces in a furnace than the aerodynamics and kinetic characteristics of a pulverized
coal burn-out. Radiation from fly-ash particles
exceeds substantially the contribution of both triatomic combustion gases, as well as char and soot
particles'* Contribution to radiative transfer by char
particles is essentially over the length of the flame. At
the end of the furnace the concentration of the char
particles is small, and there they exert very little effect
on the radiation heat flux at the wall.
400
g
300
qr, z,,,
(kW/m
c+s+g
200
I00
0
0
2
4
6
z On)
8
I0
FIG, 38. Effect of combustion products composition on the
radiation heat flux distribution along the wall of a
pulverized coal-fired furnace; (c=coal, f=fly-ash, s=soot,
g= combustion gases), for soot J,, = 2 m - 1.2,4
148
R. VISKANTAand M. P. MENGOt;
Radiation heat transfer in a cylindrical, pulverized
coal-fired combustion furnace has been predicted
both on a gray 214 and nongray s9 basis. The
calculations were carried out by assuming the
temperature and radiating species concentration
distributions in the furnace. The radiative characteristics of the coal particles were predicted from the
Mie theory, after first assuming a coal particle size
distribution. Details of radiation heat transfer and
sensitivity calculations can be found elsewhere. 214
The contributions of the different constituents (coal,
fly-ash, soot and combustion gases) on the local
radiative flux along the furnace are shown in Fig. 38.
It is clear from the figure that neglect of the fly-ash
contribution and inclusion of soot absorption yields
a dramatic change in the radiative transfer in the
medium and at the cylindrical walls (see curves
denoted as c + f + g and c + s + g ) . The main reason
for this discrepancy is the replacement of strongly
scattering fly-ash particles by strongly absorbing
soot particles. The addition of soot to coal +fly-ash
+ gas mixtures (c + f + g) simply decreases the radiative flux on the cylindrical wall since a greater
fraction of the radiant energy is being absorbed by
the medium itself. It should be mentioned, however,
that the effects predicted 6°'2~4 in this way may be
exaggerated since in these calculations the energy
equation is not solved. When radiative transfer is
taken into account in the energy equation, the
temperature would change in a manner that would
partially compensate for the effects of changes in
radiative properties.
The results of sensitivity studies 214 have shown
that accurate knowledge of number density, temperature and particle concentration distributions are
more critical than the detailed information about the
index of refraction of particles and gas concentration
distributions. The type of coal used affects radiative
transfer relatively little; however, the neglect of flyash outside of the flame zone has been shown to have
a potential for large errors. Apparently, the accuracy
of radiative transfer predictions is not only limited
by the solution techniques of the radiative transfer
equation or the prediction of radiative properties,
but mostly by the accuracy of particle concentration
and combustion product temperature distributions
which are more time-consuming to evaluate in the
needed detail.
The importance of the spatial distribution of
radiative properties of pulverized-coal and fly-ash in
predicting radiation heat transfer accurately was also
shown by Lowe e t a / . 3°° In their analysis they
employed Hottel's zonal method to solve for radiative transfer in a utility type pulverized, coal-fired
furnace. They showed that furnace heat transfer was
insensitive to the type of coal and coal fineness and
concluded that combustion data were adequate for
calculation of radiative heat transfer. Lowe et al. 3°°
recommended research on ignition, combustion
stability and radiative properties of fly-ash.
Current reviews of coal-fired combustion models
are available 2'3 there is no need to repeat these
comprehensive discussions. Recent radiative transfer
modeling for inclusion in comprehensive multidimensional combustion codes has focused on more
efficient differential and flux methods, 3°1-303 but
there are exceptions. For example, Truelove 3°4 used
a discrete-ordinates method, which is more timeconsuming to evaluate; however, to simplify the
procedure the gas was considered to be gray and the
particles were assumed to be black and nonscattering. The classical Hottel zonal method is computationaUy inefficient for use in multidimensional
codes. In addition, there are conceptual and numerical difficulties in adopting the method when anisotropically scattering particles are present in the
combustion products.
Available computer models for scale-up and performance predictions of boiler combustion chambers
have been reviewed. 272 The state-of-the-art model for
predicting radiation heat transfer in a complicated
boiler combustion furnace is based on advanced
Monte Carlo type techniques. The model is described in more detail elsewhere together with examples of its practical application. 272 It is shown how
pilot plant-scale results can be scaled up with the
help of the model to predict full-scale performance of
particular boiler furnaces. The uncertainties in predicting temperatures and heat fluxes are also discussed. It is pointed out that for pulverized coal-fired
boilers major uncertainties are caused by the unknown slagging and fouling patterns in the furnace,
and an ash deposition model could help to reduce
these uncertainties.
Recently, Fiveland and Wesse1298 have developed
a very detailed and extensive computer model to
predict the performance of three-dimensional pulverized coal-fired furnaces. They have accounted for
almost all of the important physical phenomena that
can be expected in such systems, including turbulence, chemical reactions, devolatilization, char oxidation as well as radiation heat transfer. Although
they have considered different size particles (e.g.
polydispersions) and evaluated the radiative properties of particles from Mie theory, scattering in the
medium has been considered isotropic. The combustion gas properties have been obtained using the
Edwards wide-band model, a5 and the average
properties of the gas-particle mixture have been
calculated using the averaging technique proposed
by Wessel. '26 The radiative transfer equation has
been solved using the discrete transfer method of
Lockwood and Shah; 2°3"2°4 however, the method
has been revised first to avoid arbitrary radiative
source/sink terms encountered in certain volume
elements due to numerical diffusion. Wall emissivity
and thermal conductance of ash deposits can provide
a major resistance to heat transfer from the flamecombustion products to the walls of the furnace, and
these factors were accounted for in the analysis. Flow
Radiation heat transfer
149
FIG. 39. Heat flux isopleths on furnace walls (in W/m2). 29s
patterns, gas temperature, concentration and heat
flux distributions have been predicted. In Fig. 39 the
heat flux distribution on the walls of the furnace is
depicted. Note that this figure shows the furnace as
unfolded. These types of results can be helpful in
identifying potential slagging/fouling problems on
membrane walls or convection-pass elements.
Models of this type are essential to understand the
complex, large-scale, pulverized coal-fired furnaces
and are valuable engineering design tools. The
radiation heat transfer model needs to be improved
to make it more realistic. Anisotropic scattering by
particles has been neglected and soot has not been
taken into account; therefore, the enhancement or
blockage of radiation by the soot layer is not
considered. However, as the authors claim, the model
is still in the initial stages of validation, and further
modifications in the radiation model would definitely
improve its reliability.
6.3. Gas Turbine Combustors
It is well established that in gas turbine combustors a large fraction of the heat transferred from the
gases to the liner walls is by radiation. The radiation
is due to two contributions: (1) the nonluminous
radiation emitted by gases such as CO2, H20, CO
and others, and (2) the luminous radiation emitted by
soot particles in the flame. The luminous contribution from the soot depends on the number and size
of the soot particles. In the primary combustion zone
most of the radiation emanates from the soot
particles produced in the fuel-rich regions of the
flame. At high pressures encountered in modern
150
R. VISKANTAand M. P. MENGO~:
turbines, the concentrations of soot particles is
sufficiently large to produce high enough opacities
and consequently soot radiates as a blackbody. It is
under these conditions that radiant heating of the
liner walls is most severe and poses serious problems
to liner'durability.3°5
An excellent up-to-date review of radiation heat
transfer from the flame in gas turbine combustors has
been prepared. 6 Methods for estimating nonluminous
radiation together with various analytical (global)
models for flame radiation in enclosures are discussed,
but attention is focused mainly on the factors that
govern total radiation heat transfer to the liner wall.
The impact of radiation heat transfer on combustor
design features, combustor operating conditions, fuel
composition and fuel spray characteristics are discussed. A need for better understanding of radiative
transfer to establish realistic models for predicting
local heat flux distribution is emphasized. The
understanding can be useful in developing analytical
tools which may lead to improved liner durability in
future designs by prescribing optimum arrangements
for the quantity and distribution of film-cooling air.
In turn, this approach can also lead to reductions in
the time and cost of liner development. 3°5
The simple, global methods based on the meanbcam-length concept for predicting flame emissivity
reviewed 6 are not capable of predicting local radiation heat flux distribution along the liner wall.
Furthermore, simple methods cannot account properly for radial and axial nonuniformities of temperature, species concentration and radiative properties
of the soot-gas mixtures. This is a serious shortcoming because the combustor designer allocates film
cooling air based on the total heat flux at the liner
wall. In the absence of reliable heat flux predictions,
the designer must overprotect the liner. Too much
cool air near the walls, however, can reduce combustion efficiency, increase pollutant emissions, and
distort the temperature pattern at the combustor
outlet, which stresses the turbine blades.
The local radiative flux distributions at the liner
wall of a typical gas turbine combustor have been
predicted using the Ps-approximation for radiation
transfer. 3°6 The mean temperature and soot concentration distributions along the combustor were based
on experimental data. 3°7 The effects of axial and
radial temperature and soot concentration distributions, type of fuel, and scattering by fuel droplets
were investigated. It was found that the axial and
radial temperature and soot concentration distributions impacted the local radiative flux along the
liner wall in several ways. In Fig. 40, the radiative
fluxes to the cylindrical wall calculated for radially
uniform (solid lines) and radially nonuniform (dashed
lines) soot concentration distributions are compared.
The medium with a uniform radial soot concentration yielded larger radiative flux at the liner walls,
at peak, than the nonuniform profile. The temperature distribution was assumed uniform for both
1600
I
J
t
12oo
R~
~
, ,\//
I
0
0
2
4
~
6
8
z/r,
FIG. 40. Effect of fuel type (K-kerosine and R50-fuel blend)
and of radial soot concentration distribution on radiation
heat flux at the cylindrical gas turbine combustor wall IIuniform and 2,3-nonuniform radial soot distributions). 3""
soot profiles. In practice, such a nearly uniform soot
concentration profile, though unlikely, might come
about if film cooling air of the combustor penetrated
into the combustion zone sufficiently to quench the
soot oxidation process.
The results suggest that accurate calculation of the
radiation heat flux at the combustor wall would
require both the radial temperature and soot concentration distributions in the products. Indeed, the
radial temperature distribution had greater impact
on the total radiative heat flux than the type of fuel
for the conditions examined in the study. 3°6 However,
scattering of radiation by fuel droplets in a gas
turbine combustor was found to be negligible in
comparison to absorption by soot. The average
radiative heat flux calculated by the P3-approximarion compared reasonably well with results based
on the mean-beam-length calculations used in the gas
turbine combustor industry. 6"a°s However, the P3model results were able to pinpoint locations of
maximum radiative flux at the liner wall.
The problem of three-dimensional two-phase combustion has been approached with the aim of
producing an algorithm based on fundamental
principles which correlate all of the details of
combustion occurring within a gas turbine combustion can. "~°'3°s'3°9 A mathematical model of the
three-dimensional, two-phase reacting flows in gas
turbine combustors has been developed which takes
into account the mass, momentum, and energy
couplings between the phases, The model incorporates an accurate representation of the droplet
distributions encountered in gas turbine combustors,
Radiation heat transfer
and solves the relevant equations for the trajectory
and evaporation of droplets numerically in a
Lagrangian frame of reference, using a finitedifference solution of the governing equations of the
gas. Radiative transfer is modeled using the six-flux
approximation, but information on the radiative
properties of the combustion products used in the
calculations is not provided. The emphasis in the
results reported is on flow and combustion parameters as no results on radiative transfer are given.
6.4. Internal Combustion Engines
Radiation heat transfer in diesel engines is dominated by the continuum radiation emission by soot
particles, which are present during the combustion
process. Radiation also occurs from the carbon
dioxide and water vapor molecules, but because that
energy is concentrated in spectral bands rather than
over the entire spectrum its effect is subordinate with
respect to the energy emitted by the soot. Radiation
is also emitted in bands by many of the intermediate
species formed during combustion, but their effect is
assumed to be even less important.
In spark-ignition engines, where the combustion is
usually soot free, the radiation heat transfer is always
small compared to the convection heat transfer. The
same seems to be the case in diesel engines during
those times in the cycle when soot is not prevalent.
During combustion the radiation heat transfer is of
the same order of magnitude as the convection heat
transfer; whether 25, 50 or 150y,, of the convection
heat transfer is a point argued about even in the
current literature. The arguments stem from the facts
that (1) unequivocal heat transfer measurements are
not possible, and (2) the relative importance of
convection compared to radiation is highly dependent upon the engine design and operating characteristics. In ceramic-lined engines the convection heat
transfer is expected to be reduced more than the
radiation heat transfer, and thus radiation will be
relatively more important than convection.
Parametric studies of radiation heat transfer in
diesel engines have been recently reported, an°-an3
The method developed by Chang et aL 3j°'3~n calculates spectral and total intensity at the chamber walls.
It is based on the integral form of the RTE along the
line-of-sight and uses in-cylinder species and temperature distributions as well as a coordinate transformation to aid in the integrations. The method is
incompatible with the finite-difference combustion
models, but can yield accurate results for radiative
transfer along the line-of-sight. Spherical harmonics
(P~- and P3-) approximations have also been applied
to predict radiation heat transfer for the conditions
encountered in a diesel engine. It has been shown
that the P~-approximation is computationally very
cost effective in comparison to the P3-approximation,
although it overpredicts the total radiation heat loss
to the engine walls by 20 ~ and the local radiation
151
12
• Pu
Lo - e p,,
~O.G
o.z.
/ _ ~ _
//
~f
"~O"
~', ¢onstan!
....#."
O"
I0"
20"
CA
FIG. 41. Comparison of total radiation heat losses to a
diesel engine cyJinder wall as a function of crank angle
(CA)..~13
heat flux to the walls adjacent to thin gas zones by as
much as 1 0 0 ~ . 313
The most important advantage of differential
models (like the spherical harmonics approximation)
is their flexibility to allow for variation of radiative
properties within the medium. In Fig. 41 the total
radiative flux to diesel engine walls is compared at
different crank angles for constant and spatially
varying extinction coefficient distributions 3n 3 which
were obtained from published experimental data. 3~4
It is clear from this figure that using a mean
extinction coefficient to simplify the radiative transfer calculations can not always be justified, as the
fluxes may be underpredicted by about a factor of
three. The radiation from soot has been found to be
much stronger than that from the gases. 3~J In
addition, the spectral results also reveal distinct
spectral selectivity due to the strong gas radiation
bands of CO2 and H 2 0 at elevated pressures.
As in gas turbines, scattering of radiation by fuel
droplets in diesel engines was also found to be
negligible compared to absorption by soot. 3z 3 Use of
an average homogeneous (position independent)
absorption coefficient in the engine to simplify
radiation calculations was found to be unjustifiable. 3~a It was also shown that the distribution of
radiative flux at the head and piston was incorrectly
predicted and that the total heat loss could be
underpredicted by as much as 60 ~o.
6.5. Fires as Combustion Systems
Flame radiation plays an important role in the
flame structure, spread and heat transfer from
unwanted fires. A recent review 7 has focused on basic
aspects of fire and has presented an elementary but
unified treatment of the phenomenon by considering
both urban and wildland fires. Several other reviews 31"43'239'240'315'316 have treated aspects of
flame radiation and have contributed greatly to the
phenomenoiogy. The interested reader is referred to
these reviews for books and original research papers
152
R. VISKANTAand M. P. MENGOg:
in the field, and the special issues of Combustion
Science and Technology (Vol. 39, Nos. 1--6 and Vol.
40, Nos. 1-4, 1984) on Fire Science for Fire Safety,
honoring Professor Howard W. Emmons in which
numerous papers concerned with fires are included.
It has now been accepted that radiation is the
dominant mode of heat transfer in fires of large scale,
whereas convection (or conductionI is the dominant
mode of heat transfer of very small scale fires.
Detailed heat transfer measurements have demonstrated that radiation heat transfer from fuel surfaces
typically exceeds free convection heat transfer for
characteristic fuel lengths greater than 0.2 m. 239
Nonluminous and luminous radiation from turbulent
diffusion flames has been recently discussed and
the importance of turbulence/radiation interactions
has been recently pointed out by Faeth et
al. 31'254'255 During the last decade there have been
numerous contributions to the literature concerned
with radiation heat transfer in fires, and it is not
possible to do justice to them in this very short
account.
Buoyant enclosure flows have applications to
furnaces and in such phenomena as fire spread in
rooms and buildings. Numerical and experimental
studies of two-dimensional and three-dimensional
turbulent buoyant, simple and complex enclosures
have been summarized by Yang and Lloyd. 317 The
results obtained have demonstrated that firstprinciple numerical finite-difference calculations,
together with a simple, yet rational algebraic turbulence model, can provide reasonable predictions to a
variety of buoyancy-driven vented enclosure-flow
phenomena when compared to corresponding experimental data. The geometries considered unvented
and vented enclosures, aircraft cabin compartments
and others, but the effects of radiation were neglected.
At higher temperatures thermal radiation generally plays a significant role in affecting the heat
transfer in enclosures such as rooms and buildings,
and interactions between thermal radiation and
natural or mixed convection must be accounted for
in the description of the pertinent momentum and
energy transfer processes. Recent discussions on
numerical modeling of natural convection-radiation
interactions in multidimensional enclosures are
available. 3~a'319 The interactions depend on the
radiative properties of the absorbing, emitting and
scattering media filling the enclosure, a method of
calculating multidimensional radiative transfer and
the numerical solution of the governing equations
for buoyant flows. Current knowledge in these subareas has been discussed. On the basis of these
reviews,3~ s.319 it is apparent that natural convectionradiation interactions in buoyant enclosure flows are
still in the developing stage. An efficient overall
computational scheme is still lacking, and methodologies which have been developed for naturalconvection interaction studies do not appear to have
been applied to gain improved understanding or
modeling of fire phenomena. Several studies are
mentioned here.
Cooper studied fires in enclosures and described
the ceiling jet resulting from the fire,32° the effect of
buoyant source in stratified layers, 32~ and the effect
of side walls in growing fires. 322 However, only in the
last paper did he consider the effect of radiation
using simple expressions for radiative transfer to
estimate the wall temperature. Bagnaro et al. 323
developed a model to predict experimental room fires
under steady and transient conditions. They used a
moment method ~77 to solve the radiative transfer
equation in three-dimensional enclosures. To represent the combustion gas contribution they employed
a sum-of-gray-gases model. Their results showed
good agreement with experimental data. Also,
Markatos and Pericleous 324 studied the effect of
radiation on fires in three-dimensional enclosures.
They employed the six-flux model of Spalding (see
Subsection 4.4.1) for the solution of RTE. However,
in neither of these studies is the dependence of the
radiative properties on the position (i.e. concentration and temperature) in the medium considered
in detail.
Tien and Lee 43 have provided a comprehensive
summary of the radiative properties of nonhomogeneous and particulate containing media typical of
the flame environment. These data can then be used
in radiation-energy transfer models, which, in turn,
determine the characteristics of ignition and fire
spread for the condensed fuel. T M 6.325 331 During
the combustion of condensed fuels, pyrolysis at the
fuel surface produces numerous and varied hydrocarbon gases and soot. The fuel vapors diffuse to the
flame zone where they react exothermically with
oxygen diffusing from the other side of the flame
zone. Energy released from the flame zone heats the
fuel surface, thus maintaining the existing pyrolysis,
creating new areas of pyrolysis, and spreading the
fire. The pyrolyzed gases absorb energy in the
infrared and attenuate the feedback radiation to the
fuel surface. This feedback mechanism becomes
important when the gases are strongly absorbing and
are sooty or when the pathlength becomes large, as in
large-scale fires. For solid and liquid fires, the
combustion rate is controlled by the heat transfer
from the combustion zone to the fuel surface. In
large-scale fires (L>0.7 m) fire energy is dominated
by radiation,Qnd the combustion rate is controlled
by radiant feedback from the flame to the fuel
surface. Blockage effects by the pyrolized gases and
particulates near the fuel surface (discussed in
Section 5.2) can attenuate significantly the incoming
radiation flux. Current analytical models for predicting the radiation heat flux to the fuel surface
consistently overpredict the pyrolysis rate because
the blockage effect is not accounted for. The
assumption of an isothermal and homogeneous flame
for large scale fires may also lead to significant errors.
The lack of radiative property data for radiation
-
Radiation heat transfer
heat transfer calculations is a major limitation in
improving current fire models. Radiative properties
of common combustion gases and optical constants
for soot and simple calculation schemes for determining the emission coefficients of luminous flames
have been reviewed. 43 The properties for some of the
hydrocarbon gas species which are evolved by the
pyrolysis of condensed fuels, such as plastics, have
been published recently. 332- 33,, Radiative properties
of such gas species as ethylene (C2H4), ethane (C2H¢,),
propane (C3Hs), methylmethacrylate (C3HsO2), and
others which are major species in pyrolized gases are
needed. The wide-band 35 and super-band 4'~ model
parameters need to be generated from experimental
data for the radiatively important gases. Total
emissivity charts can be developed for each gas once
the band parameters have been determined. These
charts graphically express the dependence of total
emissivity on the temperature, pressure, and optical
pathlength of the emitting gas and greatly simplify
the calculation of flame radiation problems. However, band information becomes necessary when
different gases are combined which have overlapping
bands in order to determine the correction. Predictions of radiative transfer in large-scale fires based
on data from small-scale flames in laboratory
experiments, however, have been very limited in
accuracy and require much more research attention.
The turbulence/radiation interactions and coupled
effects of radiation and flame structure for small
laboratory flames were discussed in Section 5.7. They
were found to be more important for luminous than
for nonluminous flames. Since smoke (soot) is
generated in open, compartment and building fires
which are much larger in scale than small laboratory
flames, the turbulence/radiation interactions are
expected to be even more significant because of the
large and highly variable local opacities that may be
encountered in these types of systems. The buoyant
smoke plume generated by a large fire also involves
radiation exchange within itself and with its environment. The heat and particulates released by a fire
create complex flow patterns which are determined
by a variety of factors. The interactions of radiation,
turbulence and flow structure as well as the feedback
between them in large fires are topics which have
received practically no research attention and are not
understood.
7. C O N C L U D I N G REMARKS
By highlighting recent developments in modeling
radiative transfer, the present review aims to increase
recognition that very often radiation plays an
important, if not the dominant, role in heat transfer
not only in large and intermediate but also in small
combustion systems. Neglect of radiation cannot
be justified in modeling combustion phenomena.
Modeling of radiative transfer in combustion systems can be rather "forgiving" because radiation is a
JPEC8 13 : 2 - g
153
"'long range" or "action at a distance" transport
process. In many physical situations radiation can be
modeled without detailed input of complex chemistry,
chemically reacting turbulent flow and knowledge of
the flame and the reaction region.
This review has concentrated on radiation heat
transfer in combustion systems. It is clear from the
review that radiation from flames and combustion
products requires detailed information on the radiative properties of the combustion gases and particulates. Despite the many efforts which have been
devoted to the problem, the methods developed for
radiation heat transfer in multidimensional geometries are far from satisfactory, particularly when
temperatures and gas partial pressures and particulate concentrations are varying along the path
length. The calculation of radiation in combustion
systems is quite involved, and most of the techniques,
except those which are called flux or differential
approximations, are incompatible with the numerical algorithms for solving the fluid dynamicstransport equations.
During the course of the review, a number of
problem areas have been identified and are discussed
in the article. Some specific recommendations for
work in modeling radiative transfer in combustion
systems are the following:
(1)
Radiative property data of less common gases
such as ethylene (C2H4), ethane (C2H~,), as well
as propane (C3Hs) and other more important
radicals are needed. Radiative properties of
particulates encountered in pulverized coal combustion such as fly-ash, char and others need to
be predicted and verified experimentally. There
is a very large uncertainty in the radiative
properties of these types of particulates that
have been reported in the literature. Most of the
properties of particles have been obtained at
conditions much different than those encountered in flames; therefore, it is still not clear
whether these data can be used with confidence
for combustion studies.
(2) There has been progress in modeling the thermal
radiation properties of gases and particulates.
However, more research effort is needed, especially on physically and analytically wellfounded representations that are simple and
convenient for use in computer codes of combustion systems. Considering that a characteristic length is always required for use in the
models and that such a length can not be
rigorously defined for most practical multidimensional systems, it is clear that the concept
needs additional research attention.
(3) The nongray effects have been recognized as
being very important and it is known that the
gray approximation overpredicts the emission
of radiation from flames with low soot content.
The calculations of radiative transfer for non-
154
(4)
(5)
(6)
(7)
R. VISKANTAand M. P. MENG0~
homogeneous, nonisothermai flames on a nongray basis would enable accurate predictions of
flame emission for a wide range of pathlengths~
The results could then be used to establish
scaling relations and to assess the range of
validity of the gray analysis.
In combustion systems involving the burning of
solid fuels such as pulverized coal, the particles
and gases surrounding them are at different
temperatures. Analytical models based on
experiment need to be developed to predict
radiative transfer and temperatures in such
systems. The slip between particles and gases
must be considered. This is not only important
for predicting accurately the flow and temperature fields, but also necessary for the understanding of soot formation and soot volume
fraction distribution in the medium.
In most practical, large-scale combustion systems the chemically reacting flow is turbulent.
The question needing an answer is to what
extent the interaction of turbulence and radiation will modify the flow properties, radiative
transfer and temperature in the combustion
system. In turn, this will affect the chemical
reactions, radiating species concentrations and
their distributions as well as the flame structure.
This may be particularly important in largescale, highly turbulent flames and fires containing soot.
Rigorous and relatively simple models for
handling radiative transfer in one-dimensional
and some two-dimensional geometries are available; however, there is still a need for effective,
accurate, and simple-to-use multidimensional
models. Radiative transfer should be accounted
for in the thermal energy equation when modeling combustion phenomena. The accuracy of the
radiation model should be compatible with that
of the combustion model. The calculations
should be interactive in nature, that is, radiative
properties should be predicted from the knowledge of the gas and particle concentrations, and
these properties should then be used in calculating local radiation heat transfer, temperature
distributions and local radiating species concentrations. Because of the nonlinearities of the
processes, such calculations will, most likely,
have to be carried out iteratively.
Effort should be devoted to develop approximate, but physically sound, relations Claws") for
scaling radiative transfer in combustion systems.
Such relations are needed for scaling small
laboratory flames (combustion systems) to large
scale ones typical of real or practical combustion systems. Most likely some of these laws will
be empirical in nature; therefore, experimental
data will be needed for small laboratory, prototypes as well as full-scale systems to validate the
relations.
(8)
Research effort should be devoted to experimentally validating the radiative transfer model(s) in
order to demonstrate the potential usefulness of
the methods to the analysis and design of
practical systems.
Acknowledgements Much of the author's recent work
reported in this review was supported by CONOCO. Inc.
through a grant to the Coal Research Center of Purdue
University. It is a pleasure to acknowledge CONOCO's
interest in fundamental radiation heat transfer research
rehlted to combustion systems. The authors wish to express
their appreciation to Miss Nancy Rowe for her dedicated
help in transforming their notes into a polished manuscript.
The authors are also indebted to the anonymous reviewers
for pointing out typographical errors and for suggesting
improvements in the presentation.
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