RADIATION HEAT TRANSFER IN COMBUSTION SYSTEMS heat transfer 99 An adequate treatment of thermal...

Prog. Energy Combust. Sci. 1987. Voh 13, pp. 97-160. 0360-1285/87 $0.00 +.50 Printed in Great Britain. All rights reserved. Copyright O 1987 Pergamon Journals Ltd.

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.

Nomenclature 1. 2.

CONTENTS

Introduction 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

98 98

100 100 104 104 106 107 107 107 109 109 111 113 114 115 116 119 121 122 122 123 123 124 125 126 127 128 129 130 133 133 133 134 135 136 138 139 140 141 142 143 145 145 146 149 151 151 153 154 154

apses 13:z -x 97

98 R. VISKANTA and M. P. MENGf3t~

NOMENCLATURE

A a r e a [m 2) B mass transfer number, (Q Y~o/vowo-h . ) /L C concentration D diameter of particles (am) or burner exit

diameter (m) D, dimensionless heat of combustion.

(2 V,,®/v ,, W,,h,,. Eh blackbody emitted flux defined by aT4(W/m z) E,, exponential integral function,

E,,tx)= So~" - 2exp( -x/la)d/d f size distribution, Eq. (3.11) or phase function

coefficient, Eq. (3.25) J/0) dimensionless stream function at the surhce

./v volume fraction (m3/m 3) g phase function coefficient, Eq. (3.25) h enthalpy or Planck'sconstant I radiation intensity(W/m 2-sr) J radiative flux in radial direction (W/m 2)

K radiative flux in axial direction (W/m 2) k thermal conductivity (W/mK) or imaginary

part of the complex index of refraction Ko Konokov number

L radiative flux in angular direction (W/m z) or effective latent heat of pyrolysis

L,, mean beam length (m) Mp pyrolysis rate Nn radiation-conduction parameter, kh /oT 3 N~ conduction-gaseous radiation parameter,

k ~ x / a T 3 N 2 conduction-ambient radiation parameter

defined as limy~® (k~o/aT3Xu~/xv®) 1/2 h complex index of refraction (= n - i k ) n real part of the complex index of refraction P pressure or probability density function

PN N-th order spherical harmonics approximation

Q Mie efficiency factor or energy released by combustion of v moles of gas phase fuel

Qv heat of reaction per unit mass of oxygen Re burner Reynolds number

q heat flux (W/m 2) r mass consumption number S source function

S~ N-th order discrete ordinates approximation s coordinate along the direction of propa-

gation of radiation or stoichiometric ratio, v s, W / v o W,,

St Stanton number T temperature (K)

T,o surface temperature of load (sink) (K) v velocity (m/sec) V volume (m 3)

W~ molecular weight of species i x size parameter, riD~2 y~ mass fraction of species i

Greek letters

6

absorptivity extinction coefficent (m - ~ ) Dirac delta function emissivity emission coefficient, Eq. (2.4); direction co-

sine, 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

Subscripts

b e

i m

n

o

P r

II'

2 V

Stlperscripts

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

CO-

refers to blackbody refers to effective mean refers to Planck's internal mean refers to mean values refers to narrow-band model oxygen refers to Planck's mean refers to spatial coordinates or radiation refers to wall conditions, fuel surface or wide-

band model refers to wavelength dependent properties refers to frequency dependent properties

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 in- dustrialized country account for a significant fraction of the country's GNP. 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 motivat ion 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 optimizat ion of more cost effective and environ- mentally 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 two- phase 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 prepared?-7 Signif- icant 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 99

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) thermo- chemistry, (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 = - V . p e ~ - V . P - ~ - V . ~ + S (1.1)

Ft

where the heat flux vector, ~, is defined as

~= -kVT+~k+~n ih iV 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 inad- vertently overlooked.

100 R. MISKANTA and 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 absorbing- emitting-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.


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

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 . (2.1)

The loss of energy from this pencil of rays due to absorption and scattering in the cylinder is

( K, + ~r,)l ,dsdA d~dvdt. (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. (2.3)

If Kirchhoff's law is valid, the emission coefficient q~ can be expressed as

rl~ = x~n~lh~ (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 ' f l ' = ' 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).

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. (2.5)

4n n=4,

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

cos W = cos0cos0' + sin0sin0'cos(~ - ~b') (2.6)

o r

cos W = ~¢'+ qq' + I#~' (2.7)

where

~=sin0cos4~, q=sin0sin~b, /~=cos0 (2.8)

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 in- scattering yields

d/,dA dfldvdt = - (x, + a ,)l ,dsdA dDdv dt

I I ~ , , ~ , +q,dVdl ldvd t +a ,ds ~ A," S q~,(s ---~s;v---w)

f l ' = 4 I t

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 . . . . (x , +tr,)lv +q , c dt

O" v @,(s -*s;v-*v)l, ,(s )dD dv. (2.10) + U . I I " ~ ' " ' ' '

Av" 11" = 4 x

Clearly, the left-hand side of this integrodifferential equation represents the net change in Iv per unit

102 R. V1SKANTA and 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 right- hand 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 d l , 1 01, c dt c 0t

~-(V.-~)I,=fl,(S,-I,) (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,

sv = (,lv//L) + (a, / /LX1/4~)

O,(s s;v v)/¢(s )df~dv. (2.12) Av ' f l ' = 4 f t

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 con- ventional 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

(a)

/

Z,

f

T

/

(b) A s

(c)

:.,y

FIG. 2. Coordinates for Cartesian (a), cylindrical (b) and spherical (c) systems.

remains invariant under the rotation about the z- direction. 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, (2.13) • ~ ) 1 , = ¢ ~ - - ; o ¢, ~u ez"

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 i t . 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

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 Al- though 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 discontin- uously. 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

heat transfer 103

FiG. 3. Coordinates for radiative transfer along a line-of- sight.

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)

+ f f s (s)T (s s )fl (s)ds i t ¢ !

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

if':/ ,] T~(s,s)=exp - 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. VISKANTA and 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 / = l f l,df2 (2.21a) C JQ=4~

c~v= S lvdfl=cq/v (2.21b) 1 2 = 4 ~

~ , = S 1 ,~dn (2.21c) D = 4 x

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

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. (2.22) ~t ~o

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 propa- gates 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 How- ever, 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.

2.3. Turbulence~Radiation Interaction

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 mean- square 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


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.

Turbulence can influence radiative transfer through fluctuations in temperature and radiating species concentrat ions 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

TABLE 1. Analytical forms of (V.~)l in common orthogonal geometries 2~ (¢= sin0cos~, t I = sin0sin~b, p = cos0)

Space Direction Geometry wmables cosincs (V. ~)1

FI ?1 ?1 Rectangular .\.y.: ~..q ,u ,~ - - + ll-- E- + p - -

.\" ( y , r -

?1 ; I -\'.Y ~,q ~ - + 'I S - ~.\- ~ y

- I t ; I

P - 5 ( E

Cylindrical F ( r l ) tl FI ; 1 1 ?UII)

r,dp~.: ~..q.fl - - - -~ - - - + p , r ; r r ? # ) , ~ - r ?oh

; t r l ) ; / I ? ( q l ) r . - ;~.q,ll - - - - + I t

r ? r ; z r ; ( ~

? ( r l ) ~1 ? l 1 ?Of f ) r.4~, ~. l l - - - +

r ; r r i ' q ~ , r ;¢b

;~ ? ( r l ) 1 ? [ q l ) r ~,l I

• r ?r r , "~

Sphcrical I t 70"21) g. ? ( s i n O f l )

r.O,.q'), ¢'.q.p - - - - - - + r 2 Fr rsinO~ i'O,

r.O, ~dl . l I

q 21 I ? [ { I - p 2 ) l ]

rsin0, ;q~, r ?/ I

cotO ?(q l )

r ;q~

p ~(r21) ~ ; .(sin(lf l) - - _ _ q - _ _

r 2 ~r rsin0, ; 0 ,

1 ? [ t l - p 2 ) l ] cotO, flql) +

r ; p r ; ~

p ?(r21) I ; [ ( I - / ~ 2 ) I ] - - -b

r 2 2 r r ?I~

106 R. VISKANTA and 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 0

s s s

+ I~(s')exp[ - k~ICds"] {exp[ - kaIC'ds"] 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-of- sight 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.

s

+(qffr /])exp[-kaIC'ds"] }ds'. (2.24) 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 ) ~ = - xa la + rlx. (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

3. R A D I A T I V E P R O P E R T I E S O F C O M B U S T I O N P R O D U C T S

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


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-by- line 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

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 "'super- band" 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. VISKANTA and 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~1600cm- ' )~ 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

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 approximat ing 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

too,o ,, '[I ~ ~ - - - r I

~.0 [ i t S

' l f v l Q SO.O ! / , . . . . . 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 / / / I

/ l / a m.o I

• ~ , ~ d~' ~ ~ ~ '

H.I " Y %, , t...: nN rl kq: t ,,., ,:

o 200o qooo ~ ~ l(XXX) o~ ( cm" )

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.1 atm.. L=0.5 m.


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, NO 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

I

~= ~, as.i [ I - e - ~ , P L ] . i ~ 0

(3.1)

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-of- gray 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 CO2-H20 mixture.

110 R. VISKANTA and 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:

oo oo

~s = ~ ~:a~ad)~/~ ffad2, 0 0

ao

~,= Sgan2Ebxdg/Sn~Ebad2. (3.3) 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

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

"Kp= S K;.I bl, dJ./ S I b~,d~ ( 3 . 5 )

o 0

ao

1/-~a= ~(lflcz)(dlbz/dT)d,l/~(dlbz/dT)d,t. (3.6) 0 0

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 (3.7) 0 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

o o at)

~¢(L) = Slbagzexp(- r2,L)d2/~lb~,exp(- gaL)d2. (3.8) 0 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


o _ - .... -~..-~,~.. ~ ® ~P.~ -

. ,~: . - , , i ,~ ® ,q _ T = , O O O K ® K e

K 2 - . .~ .~\ (E) Kl,n - - (~-,) "'..'.% ® ~,,..

I ---- ' ' ' % ' % _ % ~ - - T= 2 0 0 0 K " " . .N,~ . - .

, , I l i i i - i I t ' t - v - r - ~ - q - - = ~ - ~ 10 -4 I0 -s I0 "2 IO -I I I0 I

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 (H20-CO 2- air) 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. VISKANTA and M. P. MENGOq

A

x

L~.

o "1"

o " 0 o

n~

50

40

30

20

lO

I I

0) T= IO00K

I I

/ KP,w / \

/ \ \

K s p l c t r o I

I I I I I 2

I I i

b ) T = 2 0 0 0 K

........ ..........

~ ' ~ \ K s p s c t r e I

0 t , t i 0 0 0 I 2

Z/ o

250

200

- 1 5 0

I00

5 0

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).

50 500

A 4 O

x 30

-1- 20

zo

! I I

a) T = I 0 0 0 K

\

, ~ ~ K I tWm

s p e c t r e

I I I

b) T =2000 K

/

s p s c t r e l

S K i

I I I I 2 3

0 I I I 0 0 I 2 3 0 4

Z/4o

400

300

200

I00

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,


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 gas- mixtures, in which large temperature gradients exist, is not expected to predict radiative transfer realistic- ally. 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 H20 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 low- volatile 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

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 10-20y o; 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

JPgCS 1 3 : 2 - B

114 R. VISKANTA and 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, ellip- soidal) 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 i

(3.13a)

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. (3.13b) 0 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.


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, alumin- ium, 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 pulverized- coal 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

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 pulverized- coal 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 sus- pended 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. VISKANTA and 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 refer- e n c e s 65 '66 '90 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 possi- bilities 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), 1 oz- ~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 irreg- ularity of the surface characteristics. The computational time required for these calculations is too formidable as to justify the extensive use of the T- matrix 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


I0 ~ 10-7

x F ( ~ )

tO-t I0 0 tO i I0 e

........ I ........ i ' ' ' ..... I ........ I

IO : ' • ' . . . . . I

I0 q ........ I

tO 6 . . . . . . 10 -7

10-.8

[n~q 10-e

lo-iO

® CP, RBON

a P, NTHRRC I TE

+ BITUMI NOU$

x BITUMINOUB-K

• LIGNITE

• FLY-fiSH

lO-a

[m"] i0-9

i 0 - I 0

l O - i i . . . . . . . . i . . . . . . . . i . . . . . . . . i . . . . . . . . I . . . . . . . . , . . . . . . . . i . . . . . . l O - i i

lO - t lO o 10 I 10 ! l O 3 1 0 ,I l O 5 1 0 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

COx

0.800

0 . 6 0 0

c) CRRBON

A P, NTHRRC ITE

+ BITUMINOUS

X BITUMINOUS-I(

• LIGNITE

+ FLY-fiSH , i . ~ ,..O O O& •

........ I I0 - i I0 "2 IO o 10 i I0 ~ I0 : IO N

x F('~)

FIG. 1 I. 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

F 24naka ] Q,=xF(ha)=x 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 . 1 5 )

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[SKANTA and M. P. MENG0(;

10-2 , , ,

. . . . '~ -'~ . . . . J . . . . . . . . . . . . . . K.'lX,•) "IN,, ~ .

i 0 "s I 0 "2

10"4 ~ i0"3

i0-4 I0 x I0 2 I0 3 i0 4

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 radius- temperature 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 infinite- length 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 4 0 s = 3 ~ X . (3.16a)

The extinction efficiency factor is written as

Q,. = Q,, + Qs. (3.16b)


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

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.

TABL[ 2. Comparisons of spectral F(ha) functions for different shape small soot particles

/(/am) na ka F,~,,, F~,n~®r Fomt,,ola

0.50 1.92 0.55 0.754 1.871 1.700 1.50 1.88 0.73 1.007 2.450 2.134 2.50 2.10 1.09 1.140 3.683 2.888 5.00 2.69 1.57 0.863 6.073 3.888

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°

Qs = CJA = (Ws/I,)/A (3.20)

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

N

• a(W) = ~ a,.~P,(~P) (3.21a) r l=O

where

1 an ,a -2n+ l J" ~a(W)P,(~)dD (3.21b)

O = 4 f

120 R. VISKANTA and 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 repre- sent 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 (3.22)

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 second- degree 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. TM The delta-Eddington approximation is written as ~ 22

~a(W) = 2f~6(1 - cos W) + (1 -faX1 + 39acos W) (3.24)

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

a l ,,a - f a 9a = (3.25b)

1 - f a

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

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

1 o

Ba=½~ ~ ~a(/~,/~')d/~'du (3.27) 0 - 1

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" (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) f l = 4 x O = 4 f

which can be directly obtained from the Mie theory. Although it approximates the Mie phase function quite accurately, the application of the Henyey- Greenstein 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 Henyey- Greenstein 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 coal- fired 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 fly- ash particles; and (2) it is compatible with differential approximations such as the spherical harmonics approximation used to model the radiative transfer equation.


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

1 N ~a = : - ~ a~,i ¢Pa.~ , (3.30)

0"2 i

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 N . '/-~- ~ - ~O'2. , i f ] i , "g~-~- ~ - EU2.. iOa.i . (3.31)

G~ i G'l i

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

and

ra.tot = ~xa.~ly-i + xa.,,**, + ~xa.,,,,,_.i, (3.32) i j

fla,to, = xa.tot + ~aa.vo,y-i, (3.33) i

o93 = ~tra,~ly_ i/fl~.,tol (3.34) i

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

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)

provided that the mean-beam-length L,, is known. H o w e v e r , 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 paral- lelepiped 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. VISKANTA and 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 METHODS

The radiative transfer equation is an integrodifferential equation, and its solution even for a one- dimensional, 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 one- dimensional, 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 absorbing- emitting 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 cylin- dricaia4° 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'

4 n o o

+m2 z~ r~ ~Sa(r,,z,,(a,)flae_Pa%x~ 2r,dda,dr,dz ' (4.1) 4 n o o o

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. (4.2 b)

Here, the primes are used to denote the dummy variables, and l~,a is the spectral diffuse radiation


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°

,[ 2[ -P'~;(x;) -~ F,.a(r,z,ck) = ld,,t(~,~b')¢ 0 0

z r 2 z

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 ' ' - p x+ + - , l d,a(r ,c~ )e ~ ~ (xp ) 0 0

g o r o 2 1

x z2r'ddp'dr'+ ~ ~ I Sa(r',z',ck') 0 0 0

x flae-Pa~r~xp) - a(z-z')r'dc~'dr'dz' (4.4)

r 2 x

Fo,,(r,z,¢)= [ ~ ld,a(r',¢')e-~,~;(x;) -" 0 0

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

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 time- dependent, 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 propa- gates is determined stochastically. Then, the cross- section (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 con- verge 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. 1 s 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


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

SE = Q (4.6)

where

- X s , J

Sj S2

S = S~ S3

SLS,,

12¢hl I

I.~h2 ]

E = Eh3

. E l m .

S2St . . . S,,Sl

S2S2 - ~ S 2 S j ..• S,,S2 J

S 2 S 3 . . . S n S 3

s,s ... s,s,-Es,,s J

and

Qt [

Q~ I Q= Q3I

.Q.J

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 moddg 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

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

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 (direct- io.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


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 one- dimensional, 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 one- dimensional 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.

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

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

l d + r drr [rJ~a] = -(xa + crx)~a + xaEoa(T)

r (4.7)

- 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:

1 d f l - 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. VISKANTA and M. P. MLmG09

l d f l - 1 - I d + }

= x~[ L I + L~- - 2 E~ ( r ) ]

O%

+ 3 [ - J I - J~- - K ~ -K~- +2(L + +L~-)]. (4.12)

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, un- coupled four-flux model was also developed by Richter and Quack IT° and applied to a pulverized coal-fired furnace.

In one-dimensional systems, the Schuster- Schwarzchild 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 17 2

d + ( J I -

2 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)

x/~n d + x/~S-~n (J; - J~) 2 dzEK~ - K ; ] - ~ 4 r

= - n x x ( K f +Kf)+2xaEba(T) (4.15)

d 2 d z [ K ~ + K f ] = - - n x a ( K ~ - K ; ) (4.16)

*Note that these equations are modified slightly to follow a consistent nomenclature.

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 three- dimensional 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.

4.4.2. Moment methods

In the moment methods, the radiation intensity is expressed as a series in products of angular and spatial functions:

I(x0,,z,0,+) N

=Ao+ ~ [~'A,.~+q"A,. ,+II 'A. . j n = l

(4.17)

where A's are functions of location only; ~,~/, and g are direction cosines in x, y, and z-directions,


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 con- verges 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 first- order spherical harmonics P~-approximation. 19

4.4.3. Spherical harmonics approximation

The spherical harmonics (Ps) approximation, which is also known as the differential approxi-

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, m 3t only during the last decade has the method been extended to two- and three- dimensional 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 P3- approximations for absorbing, emitting, and aniso- tropically 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

with

t~(x,y,z,O,¢) = F~ A~Ax,y,z)r~(O,¢) (4.20) n ~ O m = - - n

r . ~ ( O , O ) = ( - 1 ~" ÷ I,.j~/2

r2.+ 1 (.-Iml)!-I '/=.j.j, . . , . , , x _ , ~ . . , . / r , ~cosvle (4.21)

L 4= ~,,+lml)!j

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 l imit 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. VISKANTA and 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 para- metersJa and ga:

10.4 = S ladle, n = 4 x

A 4 = 3fl](1 - 094)[ 1 - o~4(ja + 04 -J]94)'] • (4.23)

In writing the above approximation, the delta- Eddington 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 P3- approximation; 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 aniso- tropy 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 SN- approximations to symbolize the discrete-ordinates approximation in which there are N discrete values of direction cosines ~., q., it., which always satisfy the

~. + q. +/~. = 1. identity 2 2 2 In one-dimensional plane-parallel media, the discrete

ordinates approximation has found many applications (see, e.g. Viskanta, a 34 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"


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 - ( A s - A s ) - - ( ~ . + 1/2I~.~ + 1/2,c

Wn

-~ , -1 /2I~ . , - 1/2,c)}/Vc

0",1 = - flalx.n.c + xxlba.c + 7 - ~, Wnn' ~,m'lx,,',C

t'l'Tt n' (4.25)

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 finite- difference 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

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 stor- age 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 and Shah 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 one- dimensional 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. VISKANTA and 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 f lux. 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 Sn- method 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 two- dimensional 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 three- dimensional 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 one- dimensional 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.

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

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.

~o

i5

1.0 Finite Element

0.8 ~ L / H ,m o Zonal

0 . 6 ~

O.4

O.2

O | I I ~ I I

0 0.2 0.4 0.6 0.8 1.0 Dimensionless Position, x/L

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

~" ILl H . O.t

..'-- ,'7" 0.8 ~ - .... ~"*"*

.o 0.6 "0

lU Z 0.4 Finite Element - - -P3

_~ o Zonal .~_

0.2

I I !

Oo o'.2 a , ols o8 ,.o Dimensionless Position, 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


2.8

"•" 2.4

2.0

,5 ( J

- 1 . 6 o

.~_ 1.2

,,, 0.8 e,

.=_o a4

E

0 0

• • • Zonal . ~ - ' - P s

.o - - - ~

:'"i" ~ . . . . . . . .

o', & & ,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

~ 08 h

:I: O.6

o

~ O.4

0 0

• • • Zonal ~'-,,, - - ' - - Ps

• ,~ S,, S s

.,,.::.~---;--~---~ ........

( ,0.1

I I I I

(11 0.2 03 0.4 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

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, absorbing- emitting 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

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 P3- and 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 S4- and 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 Pt" 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 condit ion) 9 In Fig. 17, the P3- approximation 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

0.8

O.6

qr (r -- to)0.4

0.2

0 0

.•-m Exact

: a . . . . . . . . . . P I

~... ------ PS N~.

2 4 z/ro6_ 8 K)

la - - / m.-~ -,'--t ....... : , " o.,I-. ............. N ................ :: / oo,o

qz r- . . . . . . . . . . . . qz (z=O) _ La_~-A_m"'~ ~ (Z=Zo)

o u F----'~'_.-_-~-__-_. . . . . ~__.=.__ OOOS I

o z l l , ; ......... i . . . . . " '- '1o.oo o o oz o.4 o.6 as ,.o

, / t o

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. VISKANTA and M. P. MENG0q

%

v

i I , i i

0 Meosured Values

P,C Approx imat ion

I I I I i I a I i

IO 2.0 3,0 40 50

z ( m l

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 P3- approximation 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 et aL 172 They compared the predictions obtained from zonal and four-flux models for different absorption coefficients against the experimental data obtained for a gas- fired 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.

°°'°\ .o ~ O.Im "l

, , . ~ , l i ~ d ~ ~- '1 c lea t • 2 q tay

I I I I I I 0 I 2 3 4 5 6

F u r n o c e Lenq th (m)

FIG. 19. Comparison of predicted radiation heat fluxes using the four flux model (lines) with different absorption

1 2 coefficient formuhltions, v 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 P3- approximation. On the other hand, Piccirelli et

al. 215 presented a similar analysis using the PI- approximation 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


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.

5. APPLICATIONS TO 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 diffusion- limited 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 conduction- radiation 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

External Radiotion

Thin ~. VOlOtlle r~loua,~ Flame

\ . ' / r 2 . ' " ."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 counter- parts. 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. VISKANTA and 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 steady- state, 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 E~,- 0.9

0.6 .... E.. o . g s ~

° '7/' / / ~ 0.4

fr ~

0., / _

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


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, . . . . 4xtr T4( z ) + 2x[ a T41E2( z ) dz

t L

+trT4.2E2( ' r l , - - z ) +tr S Ta(t)El( lz- t l ) dt] (5.1) o

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

~, = 3QD,/4pr (5.2)

where D, is the dust cloud concentration for the size interval corresponding to mean particle radius r, p is

'°°1 I i

J [ / Rosin -Rommter monosize

ca 60 "1 ~ C

40-1 ~ ~ % I \ I ' ~ Polysize

~- I monosize ( D ~ l ~ " ~ - . . . . . _ _ - - _ . ~ I I I I I I I

0 20 40 60 80 ioo J20 Distance from tube bonk (cm)

FIG. 22. Comparison of prediction and experiment for coal burn-out, using the given polysize and two alternative,

mort osi ze models. 231

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-to- surface ratio, D32 ) and the other from the Rosin- Rammler 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 complex- ities 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. V1SKANTA and M. P. MENO0~

/

/ /

/ / /

FUEL / / / / / / /

I ecu~o~v LAYER EDGE

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 t3h\-dq'+S p / u - - + v - - 1 = - - / - - (5.3)

where the enthalpy is defined as

7"

I1= f cpdT. T

(5.4)

Assuming a spectraily gray, homogeneous medium with a constant absorption coefficient, the local radiative flux in the y-direction* can be expressed a s 2 3 s

q,=2[Eb~.E3(z)-E t

~ E 3 ( 17~ - - T)--J'- SEb(t)Ez(z -- t)dt 0

t a

-- 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


Z.O

2 .4

2.0

I.G

f,. I

~ t 1.2

3 ~. 0.8

! I |

0 F • 2600 c c / m l n

I[xponentlol Wide-Bond Ido~l

- - m - - Groy GaS Model K'p-0.15 ~ - ~ No Rodiotlve Interoction

Q I[xporimento! Run I

O l[aoerimental Run 2

0.4

0 I ~ - 0 I.O 2.0 3.0 4.0 5.0

I I

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 free- convection 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

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 wide- band 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 n- heptane 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 H20 ) 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, N2 , 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. VISKANTA and M. P. MENGOq

I . o .... i . o

Rodiotin9 t ------ Non.r(idiolin 9

o.s o.s -~ N

0 0 I.O IO

DIMENSIONLESS HEAT OF COMBUSTION, O c

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 = 50 .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 D c, 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 stagnation- point 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- 25 3 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.

BLACK PLANE WALL !11111111111111111111111111111/6

AIR t i JET

FUEL ~ MIDPLANE f

Am ///7"//////////////////////// ///

BLACK PLANE WALL FlG. 26. Schematicaldiagram ofaphme, radiatingjetconlinedbetweentwoparallelplates.


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 sym- metrically 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. Boundary- layer 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 jet- diffusion 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, H20 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

G4

I o.2 !

"s

~ 0 . 0

,z, ~4i

,q

0.0

° , , , .t . , o . . o . . . . . sA . peso. H ,1T/'~ - . - s~e,. ,~o. I ] :W ',.~ ,..,ooo iYl "\

"L ,1 v ",~

_sJ ~.J.J I % ~ J I " ' - '~- . . . .

I I

_ x / O - 5 0

- ~ 1 ~ \ -

- -

1.0 2 .0 3.0 4 ~

WAVELENGTH (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 time- averaged 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. VISKANTA and 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 H20 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- < O.2 r,

O.0t- 0 800 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

1200

Chemical Kinetics

Nucleation

800

4O0

I f I I I I

• . . . . 1538 W o .... 1025 W "~ ~ ' - - 513W

O I I I I I I -IO 0 1(3 20 30 40 50

I x {mm} ~I P. - t aM-= I ~ PM-n

- 6 o , 35 ,,.35 r

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. Good 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 remark- able heat transfer and combustion augmentation.

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.

6. A P P L I C A T I O N S T O C O M B U S T I O N S Y S T E M S

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 ex- pensive 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 two- phase combustion is being approached with the aim of producing algorithms based on fundamental

I Two ° Phase Fluid Mechanics (Turbulent)

Par t i t le Phase Reaction

Devolatilizofion) eter. Oxidation)


Heat Transfer (Convective) (Radiative)

Phase Transitions

(Evaporation) (Condensation)

Gas Part ic le

Interaction

Gas Phase Reaction

FIG. 30. Schematic representation of submodels for combustion of coal.

142 R. VISKANTA and 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 main- frame 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 gas- and 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

4-5 2 7 7 2 8 1 theory has been much more fully developed, • - 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


~ / A u ° T / / ~ ~ ~ ~ CONVECTION TO I..OAD 4°/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 - '°'

~ H e o t Flux O i s t r i ~ Type 2: Plug Flow

Type 3: Two - Dimensional

' ,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

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

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. VISKANTA and M. P. ME~Gi~t~

Fuel fit AIr

,0,,

Y , , ' / , / / , / / / / / ' / / / / / / / / /

~aml~stloa Products

o., T,

/~> ~o. e / r , , A '

/ / I Lo,, /

7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ~

Waste Gases = = ~ .

~,.r=.~

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

+- 0~-0,,, = (1 /Ko) (0~-0~) (6.4) l + St 1 ~ l + St

where the dimensionless variables and parameters are defined as

0 = T " 0 T~ ;nCpm

" 7 , ; A ~ . _ ma T 3

hA~ S t = : ~ .

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

Qs F s --

A ~ _ , , a T~

= [ 0 ~ - O] + ( S t ' K o X O m - 0,)]. (6.5)

For the special case when convective heat transfer to the load is negligible in comparison to radiation (St=0), Eq. (6.4) simplifies to

Ko(1 - 0m) = 0~-- 0] . (6.6)

By eliminating the mean combustion-product temperature, the dimensionless heat transfer rate can be expressed as

r~= Ko[1 - ( F, + 0 ~)z/'*]. (6.7)

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

100 F~naee Crog-Seellon In mmZ= LOO,t.O0 ~Or,9 ~

I¢o, 06 • V

60 © - . . \

"1- "~ d" k

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~


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

dT~(x) . , , - - W{e,a[T,(x)-- T,] thepm dx

+h[T~j(x)- T~] } -PoUo[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¢

where

x T~ 0 _ 7 " ; ° " = '

Ko= DiCpm h WL St . . . . . WLi,.fl T 3 ~ncp~

UoPoL dP o -

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-vessel- cascade 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.

-- ~0(0.--0~) (6.9)

~ dQ, ,

H Products IldO~ H U I ' T !

. . . . . . . . . . . . - . : ' H

FIG. 35. Schematic diagram of a plug flow model.

6.1.3. Multi-dimensional models

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 ever- increasing number of combustion and radiative

1800

1600

ca 1400

E

1200

O

i t \ x ) o

\ \ ' ' { o ) ' '

,~. Plug Flow

\ \

, S f i t r e ¢ 1 % % Vessel ~ ~ .

Zonol ~

i i i

\ %--Plug Flow

2x_~ ~ .,. Stirred Vessel ;!.,... / :aecade

\ Stirred Vessel

Zonal

i 014 i i 1 i i 0 2 0.6 0 8 0 0.2 0.4 0.6 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. VISKANTA and 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 avai lable: The zonal, flux, discrete- ordinates 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). Unfor- tunately, 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 three- dimensional 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

2O0

150

5O

0

-50

i , i , , i ,

- - - - - - / T I i " I000 K

.••'•Wilh Tuth/RQa. Inlw. Willloul Turb./Ro4. I n t l r .

I I I I I I I I I

0 0.2 04. 0.6 0.8 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 finite- difference 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


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 develope d272 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 H20 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 andf ly-ash. Information required for predicting radiative transfer includes different particle concentrations, size distributions, complex indices of refraction and temperature. 2~'* Finally, the mineral

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 pulverized- coal 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 aero- dynamics and kinetic characteristics of a pulverized coal burn-out. Radiation from fly-ash particles exceeds substantially the contribution of both tri- atomic 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

300

qr, z,, (kW/m , 200

I00

0 0 I0

g

c+s+g

2 4 6 8 z On)

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. VISKANTA and 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 fly- ash 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 time- consuming 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 compu- tationaUy inefficient for use in multidimensional codes. In addition, there are conceptual and numerical difficulties in adopting the method when aniso- tropically 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 flame- combustion products to the walls of the furnace, and these factors were accounted for in the analysis. Flow


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. VISKANTA and 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 mean- bcam-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 short- coming 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 II- uniform 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-approxi- marion compared reasonably well with results based on the mean-beam-length calculations used in the gas turbine combustor industry. 6"a°s However, the P3- model 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 incorpor- ates an accurate representation of the droplet distributions encountered in gas turbine combustors,


and solves the relevant equations for the trajectory and evaporation of droplets numerically in a Lagrangian frame of reference, using a finite- difference 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 calcu- lates 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

12 • Pu

Lo - e p,, / _ ~ _

~O.G / / ~', ¢onstan!

o . z . ~f ....#."

"~O" 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 100~ . 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 H20 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 unjustifi- able. 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. VISKANTA and 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 first- principle 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 sub- areas has been discussed. On the basis of these reviews,3~ s.319 it is apparent that natural convection- radiation interactions in buoyant enclosure flows are still in the developing stage. An efficient overall computational scheme is still lacking, and metho- dologies which have been developed for natural- convection 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 repre- sent 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


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. How- ever, band information becomes necessary when different gases are combined which have overlapping bands in order to determine the correction. Pre- dictions 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

"'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 dynamics- transport 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 well- founded 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-


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.

(4) 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.

(5) 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 large- scale, highly turbulent flames and fires containing soot.

(6) 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.

(7) 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, proto- types 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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