Radiation Heat Transfer in Combustion Systems

RADIATION HEAT TRANSFER IN COMBUSTION SYSTEMS

R. VISKANTA* and M. P. MENGOqt *School oJ Mechanical Engineering, Pttrdue University, West LaJ~tyette, IN 47907, U.S.A. tDepartment q/Mechanical Engineeriny, University of Kentucky, Lexington, K Y40506, U.S.A.

Abstract An adequate treatment of thermal radiation heat transfer is essential to a mathematical model of the combustion process or to a design of a combustion system. This paper reviews the fundamentals of radiation heat transfer and some recent progress in its modeling in combustion systems. Topics covered include radiative properties of combustion products and their modeling and methods of solving the radiative transfer equations. Examples of sample combustion systems in which radiation has been accounted for in the analysis are presented. In several technologically important, practical combustion systems coupling of radiation to other modes of heat transfer is discussed. Research needs are identified and potentially promising research topics are also suggested.

CONTENTS Nomenclature 98 1. Introduction 98 2. Radiative Transfer 100 2.1. Radiative transfer equation 100 2.2. Conservati_on of radiant energy equation 104 2.3. Turbulence/radiative interaction 104 3. Radiative Properties of Combustion Products 106 3.1. Radiative properties of combustion gases 107 3.1.1. Narrow-band models 107 3.1.2. Wide-band models 107 3.1.3. Total absorptivity emissivity models 109 3.1.4. Absorption and emission coefficients 109 3.l.5. Effect of absorption coefficient on the radiative heat flux predictions 111 3.2. Radiative properties of polydispersions 113 3.2.1. Types and shapes of polydispersions 114 3.2.2. Prediction methods of the particle radiative properties 115 3.2.3. Simplified approaches 116 3.2.4. Scattering phase function 119 3.3. Total properties 121 4. Solution Methods 122 4.1. Exact models 122 4.2. Statistical methods 123 4.3. Zonal method 123 4.4. Flux methods 124 4.4.1. Multiflux models 125 4.4.2. Moment methods 126 4.4.3. Spherical harmonics approximation 127 4.4.4. Discrete ordinates approximation 128 4.4.5. Hybrid and other methods 129 4.5. Comparison of methods 130 5. Applications to Simple Combustion Systems 133 5.1. Single-droplet and solid-particle combustion 133 5.2. Contribution of radiation to flame wall-quenching of condensed fuels 133 5.3. Effect of radiation on one-dimensional char flames 134 5.4. Radiation in a combusting boundary layer along a vertical wall 135 5.5. Interaction of convection-radiation in a laminar diffusion flame 136 5.6. Effect of radiation on a planar, two-dimensional turbulent-jet diffusion flame 138 5.7. Radiation from flames 139 5.8. Combustion and radiation heat transfer in a porous medium 140 6. Applications to Combustion Systems 141 6.1. Industrial furnaces 142 6.1.1. Stirred vessel model 143 6.1.2. Plug flow model 145 6.1.3. Multi-dimensional models 145 6.2. Coal-fired furnaces 146 6.3. Gas turbine combustors 149 6.4. Internal combustion engines 151 6.5. Fires as combustion systems 151 7. Concluding Remarks 153 Acknowledgements 154 References 154 apses 13:z-x 97 98 R. VISKANTAand M. P. MENGf3t~

NOMENCLATURE direction cosine, Eq. 12.8) p density (kg/m 3) A area [m 2) a Stefan-Boltzmann constant; scattering CO- B mass transfer number, (Q Y~o/vowo-h.)/L efficient (m- 1) C concentration T beam transmittance, Eq. (2.18) D diameter of particles (am) or burner exit r optical depth, i,y diameter (m) ~P scattering angle, Eq. 2.7) D, dimensionless heat of combustion. scattering phase function (2 V,,®/v ,, W,,h,,. 05 azimuthal angle Eh blackbody emitted flux defined by aT4(W/m z) solid angle E,, exponential integral function, t,) single scattering albedo, a/fl E,,tx)= So~" - 2exp( -x/la)d/d f size distribution, Eq. (3.11) or phase function coefficient, Eq. (3.25) Subscripts J/0) dimensionless stream function at the surhce ./v volume fraction (m3/m 3) b refers to blackbody g phase function coefficient, Eq. (3.25) e refers to effective mean h enthalpy or Planck'sconstant i refers to Planck's internal mean I radiation intensity(W/m 2-sr) m refers to mean values J radiative flux in radial direction (W/m 2) n refers to narrow-band model K radiative flux in axial direction (W/m 2) o oxygen k thermal conductivity (W/mK) or imaginary P refers to Planck's mean part of the complex index of refraction r refers to spatial coordinates or radiation Ko Konokov number II' refers to wall conditions, fuel surface or wide- L radiative flux in angular direction (W/m z) or band model effective latent heat of pyrolysis 2 refers to wavelength dependent properties L,, mean beam length (m) V refers to frequency dependent properties Mp pyrolysis rate Nn radiation-conduction parameter, kh/oT 3 N~ conduction-gaseous radiation parameter, Stlperscripts k~x/aT 3 N 2 conduction-ambient radiation parameter refers to incoming radiation beam defined as limy~® (k~o/aT3Xu~/xv®) 1/2 refers to turbulent mean properties h complex index of refraction (= n -ik) n real part of the complex index of refraction P pressure or probability density function I. INTRODUCTION PN N-th order spherical harmonics approximation Expenditures on fossil energy by individuals, Q Mie efficiency factor or energy released by commerce, transportation and industry in an in- combustion of v moles of gas phase fuel Qv heat of reaction per unit mass of oxygen dustrialized country account for a significant fraction Re burner Reynolds number of the country's GNP. Improved understanding of q heat flux (W/m 2) combustion systems which use fossil fuels such as r mass consumption number natural gas, oil and coal may result in improved S source function energy efficiency. The potential improvement in the S~ N-th order discrete ordinates approximation s coordinate along the direction of propa- thermal performance of such systems could make a gation of radiation or stoichiometric ratio, significant impact on the country's economy. This vs, W/vo W,, provides the motivation and economic incentive for St Stanton number research and development in combustion technology. T temperature (K) T,o surface temperature of load (sink) (K) An important goal, then, is to develop computational v velocity (m/sec) models which could be used for the design and V volume (m 3) optimization of more cost effective and environ- W~ molecular weight of species i mentally friendly combustion systems with improved x size parameter, riD~2 performance. y~ mass fraction of species i Combustion is one of the most difficult processes to model mathematically since it generally involves Greek letters the simultaneous processes of three-dimensional two- absorptivity phase fluid dynamics, turbulent mixing, fuel evapor- extinction coefficent (m - ~) ation, radiative and convective heat transfer, and 6 Dirac delta function chemical kinetics. In order to design combustion emissivity systems based on fundamental principles, compre- emission coefficient, Eq. (2.4); direction cosine, Eq. (2.8); dimensionless coordinate hensive models incorporating all of these factors are defined as x/g~I~(u®/tt)dy required. State-of-the-art reviews of modeling some zenith angle; normalized temperature combustion systems have been prepared?-7 Signif- absorption coefficient (m - ~) icant progress has been made in detailed modeling of wavelength of radiation (/tm) combustion systems, but major problems such as direction cosine, Eq. (2.8) frequency of radiation; kinematic viscosity: turbulence in reactive flows, particle formation and stoichiometric coefficient others remain to be solved. Radiation heat transfer 99

An adequate treatment of thermal radiation is ~= -kVT+~k+~nihiVj+~a_ ,. (1.2) essential to develop a mathematical model of the J combustion system. The level of detail required for In Eq. (1.1) p, pc, ~ and P are the total mass, energy radiative transfer depends on whether one is inter- density, fluid velocity, and pressure, respectively. The ested in determining the instantaneous spectral local {n;} and {V~ } are the number density and diffusion radiative flux, flame structure, scalar properties of the velocities of the individual chemical species, and ,~" flame, formation of flame-generated particles (largely is the radiation heat flux vector. The first, second, soot), local radiative flux and its divergence or the third and fourth terms in Eq. (1.2) account for temperature distribution. For example, when the molecular conduction, radiation, interdiffusion and model is used to predict pollutant concentrations, diffusion-thermo contributions, respectively, to the accurate temperatures are especially important since heat flux vector. In Eq. (1.1) S is the local volumetric the chemical kinetics involved are extremely temper- heat source/sink from other processes, if any. When ature dependent. radiation heat transfer needs to be accounted for in The fraction of the total heat transfer due to the energy equation, it is preferable to use temper- radiation grows with combustor size, attaining ature as the dependent variable rather than the prominence for gaseous firing at characteristic com- stagnation enthalpy. The divergence of the radiative bustion lengths of about 1 m. Radiation heat trans- flux vector, V..~-~', can be obtained from the radiant fer, then, plays a dominant role in most industrial energy equation. furnaces. Unfortunately, it is governed by a complex The purpose of this paper is to acquaint the reader integrodifferential equation which is time consuming with the basic principles and methods related to to solve. Economic measures are a necessity, even at modeling radiation heat transfer in combustion the loss of some accuracy. systems. The importance of radiative transfer in coal In a combustion chamber, radiation heat transfer combustion, 3 pulverized coal-fired boilers,'* indus- from the flame and combustion products to the trial furnaces, 5 gas turbine combustors 6 and fires 7 surroundings walls can be predicted if the radiative has been recognized for some time. Radiative properties and temperature distributions in the transfer in some of these systems has received medium and on the walls are available. Usually, considerable research attention and a high degree of however, temperature itself is an unknown para- organization has been attained. meter, and as a result of this, the total energy and The paper is organized to give a systematic and radiant energy conservation equations are coupled, easy-to-follow approach to the major building blocks as in many heat transfer applications. Solution of the of radiative transfer in combustion systems. Section thermal energy equation can be obtained if several 2 of the paper introduces the fundamentals of other physical and chemical processes can be radiation heat transfer, and Section 3 discusses the modeled. The major processes which need to be radiative properties of gases and particles encoun- considered in a combustion system in addition to tered in combustion systems. These two sections radiation include? (i) chemical kinetics, (ii) thermo- provide the background necessary for understanding chemistry, (iii) molecular diffusion, (iv) laminar and the specific techniques for solving the radiative turbulent fluid dynamics, (v) nucleation, (vi) phase transfer equation discussed in Section 4. Examples of transitions, such as evaporation and condensation simple combustion systems in which radiative trans- and (vii) surface effects. Since the physical and fer has been accounted for are discussed in Section 5. chemical processes occurring in combustion cham- Section 6 reviews modeling of radiation heat transfer bers are very complicated and cannot be modeled on in practical combustion systems and deals with the microscale, there is a need for physical models to coupling of radiation to other transport processes in simulate these processes. Each of these models needs system models. an extensive and separate treatment, which is outside There exists a very large body of literature relevant the scope of this work. The interested reader is to radiation heat transfer in combustion systems, and referred to more specialized publications. ~- v it is not possible to cover it thoroughly. Emphasis in In nonrelativistic problems of an engineering the paper is on fundamentals and applications to nature, radiation does not contribute any terms to simple systems. Reference is made to the original the conservation of mass, momentum and species publications for a more complete discussion. A conservation. The classical conservation of energy review process is a rather arbitrary activity, because equation Ls'9 is modified by a contribution which of the decision the authors have to make on what to accounts for radiation heat transfer. This equation include, what to omit, and where to start and end. can be written as This article is no exception, and it reflects the authors' biases. Because of the broad range of topics ~pe covered, details can not be included, and no claim is = -V.pe~-V.P-~-V.~+S (1.1) Ft 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- where the heat flux vector, ~, is defined as vertently overlooked. 100 R. MISKANTAand M. P. MENG0q

2. RADIATIVE TRANSFER magnetic field, and Osborn and Klevans ~4 have refined and generalized their work. The Eulerian 2.1. Radiative Transfer Equation point of view is adopted here and the traditional Two theories have been developed for the study of intuitive derivation of the RTE found in the the propagation and interaction of electromagnetic radiative transfer literature 15 - 20 is given. radiation with matter, namely, the classical electro- Rather than presenting the most general deriv- magnetic wave theory and the radiative transfer ation of the RTE, certain constraints which help to theory. The theories were developed independently avoid complications that obscure the physical signifi- and there is no similarity in their basic formulations. cance of the phenomenon are imposed in this Conceptually, they are completely distinct; however, discussion. The treatment presented here constitutes both theories describe the same physical phen- a reasonable compromise between the generality omenon. The classical electromagnetic theory has needed for engineering applications and clarity of the approached the study of propagation and interaction development. The idealizing assumptions and con- of matter with radiation from the microscopic point straints imposed are: (1) the discussion is restricted to of view and the radiative transfer theory from the a continuous, homogeneous and isotropic absorbing- macroscopic (or phenomenological) point of view. emitting-scattering medium at rest, (2) the state of The study of the detailed interaction of electro- polarization is neglected, and (3) the medium is magnetic radiation with matter on the microscopic considered to be in local thermodynamic equilibrium level from both the classical and quantum mechanics (LTE). point of view yields the interaction cross-sections of The RTE is based on application of an energy the particles making up the matter. This fundamental balance on an elementary volume taken along the approach predicts the macroscopic properties of the direction of a pencil of rays and confined within an media, and these properties appear as coefficients in elementary solid angle. The detailed mechanism of the radiative transfer equation. the interaction processes involving particles and the The quantitative study, on the phenomenologicai field of radiation is not considered here. On the level, of the interaction of radiation with matter that phenomenological level only the transformation absorbs, emits, and scatters radiant energy is the suffered by the radiation field passing through a concern of the radiative transfer theory. The theory participating medium is examined. The derivation ignores the wave nature of radiation and visualizes it accounts mathematically for the rate of change of in terms of light rays of photons. These are concepts radiation intensity along the path in terms of of geometrical optics. The geometrical optics theory physical processes of absorption, emission, and is the study of electromagnetism in the limiting case scattering. of extremely small wavelengths or of high frequency. Consider a cylindrical volume element, Fig. 1, of The detailed mechanism of the interaction process cross-section dA and length ds in an absorbing, involving atoms or molecules and the radiation field emitting, and scattering medium characterized by the is not considered. Only the macroscopic problem spectral absorption coefficient xv, scattering coef- consisting of the transformation suffered by the field ficient try and true emission coefficient r/v. The axis of of radiation passing through a medium is examined. the cylinder is in the direction of the unit vector ~, Thus, there is a considerable simplification over the i.e. ds is measured along ~. The spectral intensity of electromagnetic wave theory. radia.tion (spectral radiance) in the ~-direction The radiative transfer equation (RTE) forms the incident normally on one end of the cylinder is Iv basis for quantitative study of the transfer of radiant and the intensity of radiation emerging, through the energy in a partici, pating medium. The equation is a second end in the same direction is Iv + dlv. Here, v is mathematical statement of the conservation principle the frequency and is related to the wavelength 2 by applied to a monochromatic pencil (bundle) of v = c/2, where c is the velocity of radiation. 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 FIG. 1. Coordinates for derivation of the radiative mmsfer statistical treatment of both plasma particles and the equation. Radiation heat transfer 101

It follows from the definition of the spectral Since in this case the sum of probability over all intensity I, that radiant energy incident normally on directions must equal unity, we must have the infinitesimally small cross-section dA during time interval dr, in frequency range dv and within the 1 1 elementary solid angle dD about the direction of the 4rt n'=4, a=4, unit vector ~ is 1 l ,dAdfldvdt. =-- J" ~,(W)dfl= 1. (2.5) 4n n=4, The emerging radiant energy at the other face of the This implies that for coherent scattering the spectral cylinder in the same direction equals phase function is normalized to unity. The scattering angle W, i.e. the angle between g' and g can be (1 ~ + dl ,)dAdDdvdt. expressed as The net gain of radiant energy, i.e. the difference cos W = cos0cos0' + sin0sin0'cos(~ - ~b') (2.6) between energy crossing the two faces of the cylinder, is then given by or (I ~ + dl ~ - dl,)dAdf~dvdt = dl,dAdfldvdt. (2.1) cos W = ~¢'+ qq' + I#~' (2.7) The loss of energy from this pencil of rays due to where absorption and scattering in the cylinder is ~=sin0cos4~, q=sin0sin~b, /~=cos0 (2.8) ( K, + ~r,)l ,dsdA d~dvdt. (2.2) are the direction cosines in any orthagonal co- The emission by the matter inside the cylindrical ordinate system. Reference to Fig. 1 shows that volume element dV, in the time interval dt, in the g'(0',~b') represents the incoming direction of the frequency range dr, confined in the solid angle dD pencil of rays, and g(0,~b), is the direction of the about the direction g equals pencil after scattering. Equating the change of energy in the cylindrical q~d V df~dvdt. (2.3) volume element to the net gain or loss of energy along the traversal path of the cylinder in terms of If Kirchhoff's law is valid, the emission coefficient q~ the processes of attenuation, emission and in- can be expressed as scattering yields rl~ = x~n~lh~ (2.4) d/,dA dfldvdt = - (x, + a ,)l ,dsdA dDdv dt where lb, is Planck's spectral blackbody intensity of II ~ ,, ~, radiation, and n, is the spectral index of refraction of +q,dVdlldvdt +a,ds ~ A," S q~,(s ---~s;v---w) the medium. The increase in energy of the pencil of fl' = 4It rays (~,df~) due to in-scattering of radiation by the matter into the elementary cylindrical volume from all possible directions ~' is x l¢(-~')dt)'dv'] dAdfldv dt. (2.9)

• ,(s ~s;v ~v) &v' fl' ='l-x 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, x l¢(-~')dD'dv']dAd~dvdt. yields the equation of transfer in a Lagrangian coordinate system

In this expression the phase function 1 dlv t~,(g'---,g;v'---~v)df~'dv'/4n represents the probability .... (x, +tr,)lv +q, that radiation of frequency v' propagating in the c dt direction g' and confined within the solid angle dfg is scattered through the angle (g,g) into the solid O" v +U.I I @,(s" -*s;v-*v)l,,(s~' "' )dD' dv.' (2.10) angle dD and the frequency interval dr. This proba- Av" 11" = 4x bility is determined by the scattering mechanism. For coherent scattering the phase function is inde- Clearly, the left-hand side of this integrodifferential pendent of frequency v' and reduces to ~v(~"--*g). equation represents the net change in Iv per unit 102 R. V1SKANTAand M. P. MENGOt;

length along the path ds=cdt. Equation (2.10) is a Z, statement of the conservation of energy principle for (a) T 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). / f / 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 (b) A s may be written as

1 dl, 1 01, ~-(V.-~)I,=fl,(S,-I,) (2.11) c dt c 0t 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 (c) ti me, sv = (,lv//L) + (a,//LX1/4~)

O,(s s;v v)/¢(s )df~dv. (2.12) Av' fl' = 4ft :.,y

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, FIG. 2. Coordinates for Cartesian (a), cylindrical (b) and lv=Sv. If I~>S, then dIv/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 remains invariant under the rotation about the z- 1. The direction cosines ~, q, and/~ are defined by Eq. direction. This allows us to combine the two (2.8), and they are functions of angular variables 0 azimuthal angles, namely ~b and ~b,, to obtain a single and ~b. Usually, the polar and aximuthal angles for azimuthal angle. If, one writes ~b'=q~-~b,, then space variables are also designated by 0 and ~b. To 07. g)l, for an axisymmetric cylindrical system can avoid confusion and to be able to use the con- be given as ventional nomenclature at the same time, we choose to use subscript r for space variables 0, and ~b, when 7 01, rl 01, 01, (2.13) appropriate (see Table I). • ~)1,=¢~--; o ¢, ~u ez" In Fig. 2, three orthogonal coordinate systems and the corresponding nomenclature are shown, where the spherical coordinate system for angular variation In a spherically symmetric system, the radiation of intensity is superimposed on either a rectangular, intensity depends on only two parameters, i.e. the cylindrical or spherical coordinate system for spatial radial distance r measured from the origin and the variables. In general the intensity is a function of direction cosine/~ of the angle between the direction three spatial coordinates, two angles and time; of of the radiation beam and the radius vector i t . The course, the seventh independent variable required to analytical expression for 07. g)l, corresponding to a define the radiation intensity is the wavelength or the spherically symmetric system is the last expression of frequency of radiation. Table 1, which can be simplified further to read For most practical calculations, it is possible to assume cylindrical or spherical symmetry. The cylin- 07" ~)1 __u 0/,+1 -u 2 0t, (2.14) drical symmetry requires that the radiation intensity Radiation heat transfer 103

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 FiG. 3. Coordinates for radiative transfer along a line-of- in any frame of reference Iv satisfies the same sight. equation of radiative transfer. 22 The intensity Iv changes at points along the path, where the index of refraction n, changes continuously or discontin- function of time indirectly through the source uously. Such changes can be systematically accounted function if qv is time dependent [see Eq. (2.12)]. for by simply adopting a new function l,/n~ rather Suppose that at some point on the boundary of than Iv. 12 Hence, there is no need to include the matter So, as shown in Fig. 3, the spectral intensity Iv index of refraction explicitly in the transfer equation. is known The second assumption concerning the fact that neglect of polarization is not generally valid is well lv(s)=l,(so)=lo~ at s=s o. (2.16) recognized, and it is clear that polarization must be accounted for in any rigorous treatment of radiative The integral form of the equation of transfer may be transfer when scattering is present. The radiative derived from the integrodifferential equation by transfer theory has been extended to include the imagining the latter to be an ordinary differential phenomenon of polarization of radiation. 12'15 It is equation in the unknown Iv and with S, a known also well recognized that the third assumption for the source function. The integrating factor for this medium to be in LTE may be invalid under the differential equation is exp (Sflvds) and the integral of conditions where densities and optical thicknesses Eq. (2.15) with the boundary condition Eq. (2.16) may become small, scattering becomes an important be written as mechanism, rapid time variations occur or large temperature gradients are in evidence? 2 Therefore, I v(s) = I o ffv(S,So) before making the LTE assumption, the conditions for a given physical system should be carefully i t ¢ ! examined. + ffs v (s)T v (s~ s )flv (s)ds (2.17) The radiative transfer equation, Eq. (2.11), is an integrodifferentiai equation, and because of this it is very difficult to solve exactly for multidimensional where s' is a dummy variable of integration and geometries. Therefore, some simplifications of this T(s,s') is the beam transmittance of an arbitrary path equation are necessary. A close look at the source from s' to s along the direction term given in Eq. (2.12) reveals that the in-scattering term (the second term of the right hand side) yields T~(s,s)=exp - flv(Od . (2.18) the integral nature of the RTE. If scattering is if':/= • ,] negligible in the medium, then the Eq. (2.11) will be a linear differential equation, which is much easier to The concept of the beam transmittance can be made solve than the linear integrodifferential equation. clearer by the following interpretation. If lov repre- A formal solution of the quasi-steady state RTE, sents the intensity of radiation in some direction g at Eq. (2.11), can readily be written. Consider a pencil of some initial point So and Iv(s) is the intensity of the radiation in the direction g (Fig. 3). If the coordinate transmitted radiation at point s in the same direction s is laid in the direction g, the quasi-steady RTE is over the path from the initial to the terminal points, given by then the two intensities are related by

(V- ~)1,= ~g*= fl,(Sv-IO (2.15) l,(s)= T~(s,so)lov. (2.19)

where the direction of the pencil of rays is under- Thus, the beam transmittance represents the fraction stood to be g. The intensity, however, may be a of the initial intensity which is transmitted without 104 R. VISKANTAand M. P. MENGO~3 emission or scattering contributions to the intensity energy from an element of matter per unit of volume along the path length. Equation (2.17) gives the and per unit of frequency. The term 4nqv(= 4nxJb,) spectral intensity of radiation at a point and in a represents the local rate of emission, and x,ff, given direction. Its physical meaning can be more represents the local rate of absorption of radiation readily interpreted by referring to Fig. 3. It shows per unit of volume. The meaning of the terms can be that Iv(s) is a sum of two contributions: (1) the further clarified when we note that 4nlbv is the transmitted intensity, and (2) the path intensity. The product of the spectral radiant energy density of a first term on the right hand side of Eq. (2.17) is the black body at the local temperature, times the local contribution to Iv due to the initial intensity at point velocity of light c, while c~v is related to the local So in the direction of propagation of the radiation g, radiant energy density of space as defined by Eq. attenuated by the factor Tv(s,s0) to account for (2.21b). In deriving Eq. (2.20) the scattering terms absorption, scattering and induced emission in the have canceled out. This just confirms the physical intervening matter. The second term results from fact that scattered energy is not stored and should both emission and scattering from elements of the not appear in the conservation of radiant energy matter at all interior points, each elementary contri- equation. Integration of Eq. (2.20) over the entire bution being attenuated by the factor Tv(s,s') while spectrum results in the conservation equation of total the rest is absorbed and scattered along the path. radiant energy These elementary contributions are integrated over all the elements between the boundary of the body s o oo 0°d +V.,~= f 1%[4nlbv(T)-f~v]dv. (2.22) and the point s. ~t ~o We note that the integral form, Eq. (2.17), of the radiative transfer equation is referred to as "'formal For reasons that were explained in a previous solution" in the sense that I v is expressed in terms of subsection, the time rate of change of radiant energy integrals that can be evaluated only if the state of the density q/ can be neglected. Note that there is no matter and the radiation field, i.e. Sv is known. This convective term in Eq. (2.22), since radiation propa- does not mean that the equation of transfer in a gates inependently of the local material velocity. The participating medium has been solved. It is clear that equation describing the local change of radiant if the source function depends on the intensity Iv in energy density must be modified in the relativistic some specified way, then one can convert Eq. (2.17) treatment of electromagnetic radiation.l°'23 How- into an integral equation for Iv. ~5 However, before ever, the additional terms which arise in the we do this it is desirable to derive the conservation of conservation of radiant energy equation can gener- radiant energy equation. ally be ignored in engineering applications. It is worth noting that the spectral dependence of radiative properties is denoted either by subscript v 2.2. Conservation of Radiant Energy Equation (frequency) or ~. (wavelength). If the matter through Integration of the RTE, Eq. (2.11), over all which radiation is propagating is not homogeneous directions results in 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 t3ail* + V',~rv=x,[4nlbv(T)--f~v] (2.20) medium, whereas the frequency remains constant dt everywhere. Therefore, the frequency is a more fundamental measure than the wavelength of radi- where the spectral radiant energy density q/v, the ation, and because of this, here, the spectral depend- irradiance aJv and the radiation flux vector "~'v are ence is denoted by v. It is also useful to remember the defined as identity, -lvdv = lad2, between frequency and wavelength based definitions of radiation intensity. q/=lf l,df2 (2.21a) C JQ=4~ 2.3. Turbulence~Radiation Interaction c~v= S lvdfl=cq/v (2.21b) Interaction of convection and radiation has been 12=4~ recognized for some time, but the fact that turbulence can influence radiative transfer and vice versa has ~,= S 1,~dn (2.21c) been recognized more recently. The first attempt at D=4x combined analysis of the equations for the mean- square fluctuations of the velocity and temperature respectively. The physical meaning of Eq. (2.20) is fields with the radiation field is due to Townsend. 24 clear. It is the conservation equation of spectral Applications in which radiation/turbulence inter- radiant energy. The right-hand-side of Eq. (2.20) action may affect flow and heat transfer include represents the net rate of loss or gain of radiant industrial furnaces, gas combustors, flames and Radiation heat transfer 105 fires.25- 3o Most studies concerned with modeling of Turbulence can influence radiative transfer through radiative transfer in combustion chambers and fluctuations in temperature and radiating species furnaces have ignored the turbulence/radiation inter- concentrations which, in turn, influence Planck's action. 3's An up-to-date discussion of the interaction function lba(T) and the special absorption and in flames is available 3~ and need not be repeated scattering coefficients. The fluctuations of the Planck here. Suffice it to mention that the interactions and function and the spectral absorption and scattering coupled effects are more important for luminous than coefficients can be given in terms of the temperature for nonluminous flames. Little is known concerning and species fluctuations by means of Taylor series temporal aspects of radiative transfer in turbulent expansions about the values evaluated at the mean flames as these effects have not been studied properties. Evaluation of the instantaneous intensity extensively. of radiation in terms of the mean and fluctuating

TABLE 1. Analytical forms of (V.~)l in common orthogonal geometries 2~ (¢= sin0cos~, tI = 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 ~,q ~- + 'I S- -\'.Y ~.\- ~ y

;I - It P-5 (E

F(rl) tl FI ;1 1 ?UII) Cylindrical r,dp~.: ~..q.fl - -- -~ - -- + p, r ;r r?#), ~- r ?oh

;trl) ;/ I ?(ql) r.- ;~.q,ll ---- + It r ?r ;z r ;(~

?(rl) ~1 ?l 1 ?Off) r.4~, ~.ll --- + r ;r ri'q~, r ;¢b

;~ ?(rl) 1 ?[ql) r ~,l I • r ?r r ,"~

It 70"21) g. ?(sinOfl) Sphcrical r.O,.q'), ¢'.q.p ------+ r 2 Fr rsinO~ i'O,

q 21 I ?[{I-p2)l] rsin0, ;q~, r ?/I

cotO ?(ql ) r ;q~

p ~(r21) ~ ;.(sin(lfl) r.O, ~dl.l I --__ q-__ r 2 ~r rsin0, ;0,

1 ?[tl-p2)l] cotO, flql) + r ;p r ;~

p ?(r21) I ;[(I-/~2)I] -- -b r 2 2r r ?I~ 106 R. VISKANTAand M. P. MENGf3~:

properties and the time-averaging is straight-forward thin. a° We further assume that the properties of the but tedious. 27 If the absorption coefficient can be fluctuating eddies are statistically independent, and expressed as this implies that there is no correlation between the temperature and concentration within each eddy. xa(s,t) = ~,kai Ci(s,t) (2.23) Under these conditions radiation is transmitted i through an eddy with little change so that the radiance at a local point is affected little by the local the turbulent fluctuations in the absorption coef- fluctuation of xx. Hence, the time-average RTE can ficient can be related to those in the concentrations be approximated as a° Ci of the radiating species. The precise evaluation of the time-average would utilize the joint probability (V' g)la = - xala + qa. (2.26) density function P(T,Ci,s) of the temperature and species concentrations for all points s along the line Following a similar argument, the spectral radiant of sight g in Eq. (2.17). Unfortunately, that infor- energy Eq. (2.20) can be expressed as mation is not available. Those properties of the flow field that are available are the mean temperature T,, V" "#'a = - x-a~a + 4r~Oa. (2.27) species concentrations C, and the second order correlations, T '2, C~T'. To illustrate the nature of the Information necessary to solve Eq. (2.25) for the problem we restrict ourselves to a single radiating time-averaged spectral radiance I~ is not available, species and neglect scattering. Applying Reynolds' and the integration of Eq. (2.24) along the line-of- averaging techniques to Eq. (2.17) but omitting the sight is too time consuming. Some clever way of details, one can obtain 30 ensemble averaging the radiance or developing correlation coefficients for time-averaged quantities s s will be required to enable solution of the integral or ia(s) = loaexp[ - kaIC(s')ds']exp[ -- kaSC'(s')ds'] differential forms of the RTE in turbulently fluctu- 0 0 ating media. The significance of the turbulence/ s s s radiation interaction will be assessed later. + I~(s')exp[ - k~ICds"] {exp[ - kaIC'ds"] 0 s' s'

s 3. RADIATIVE PROPERTIES OF COMBUSTION PRODUCTS +(qffr/])exp[-kaIC'ds"] }ds'. (2.24) s' The accuracy of radiative transfer predictions in combustion systems cannot be better than the This equation can be written in a more useful form in accuracy of the radiative properties of the combus- terms of the two-point correlation coefficients. 28 The tion products used in the analysis. These products representation of the random concentration and usually consist of combustion gases such as water temperature by Gaussian variables is convenient, but vapor, carbon dioxide, carbon monoxide, sulfur it must be noted that they encompass unrealistic dioxide, and nitrous oxide, and particles, like soot, negative values of the variables whose probability fly-ash, pulverized-coal, char or fuel droplets. Before must be kept small in proportion. Comparison of attempting to tackle radiation heat transfer in Eqs (2.17) and (2.24) reveals that consideration of practical combustion systems, it is necessary to know turbulence (i.e. time-averaging) would greatly in- the radiative properties of the combustion products. crease the computational effort of an already difficult Considering the diversity of the products and the problem. probability of having all or some of these in any An alternative to time-averaging the spectral volume element of the system, it can easily be radiance would be to time-average the quasi-steady perceived that the prediction of radiative properties form of RTE, Eq. (2.15), and the radiant energy in combustion systems is not an easy task. The equation, Eq. (2.20), at the start. Time-averaging of wavelength dependence of these properties and Eq. (2.15) results in uncertainties about the volume fractions and size and shape distribution of particles cause additional 07"g)~ = -xala + rlx. (2.25) complications. In order to present a systematic methodology for The difficulty with this equation is the evaluation of the prediction the radiative properties of combustion the coupled correlation xala because instantaneous products, the discussion in this section will be I~ is expressed in terms of an integration along the divided into several subsections in which the re- path as indicated in Eq. (2.17). To simplify the lations for obtaining the properties of the combus- absorption coefficient-radiance correlation x~la we tion gases and different particles are discussed and can assume that the individual eddies are homogen- the simplifications are introduced. Afterwards, some eous and that the radiating gas of a typical size eddy relations will be given to employ these expressions as (i.e. based on macroscale of turbulence) is optically building blocks to determine the radiative properties Radiation heat transfer 107 of the mixture of combustion products. Note that charged particles, then the Stark profile yields a more usually the level of simplification for the properties is accurate representation of the spectral line radi- to be determined by the user, and it should be ation. 33 Note that it is also possible to superpose consistent with the level of sophistication of the these line profiles to incorporate the effects of radiative transfer and total heat transfer models. different physical conditions on the line radi- Also, the relations for the radiative properties of ation.33. 3'* individual constituents should be compatible with There are basically two ~different line arrangements each other as well as with the radiative transfer for narrow band models used extensively in the models. 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 3.1. Radiative Properties of Combustion Gases random exponential line intensity distribution and a Every combustion process produces combustion random line position selected from a uniform gases, such as water vapor, carbon dioxide, carbon probability distribution. For practical engineering monoxide, and others. The partial pressures of these calculations both of these models yield reasonably gases in the combustion products are determined by accurate results. Usually there is less than 8~o the type of the fuel used and the conditions of the discrepancy between the predictions of these two combustion environment, such as fuel-air ratio, total models. 35 A detailed discussion of the narrow band pressure and ambient temperature. These gases do models has been given by Ludwig et al. 34 and in the not scatter radiation significantly, but they are strong review articles by Tien 33 and Edwards. 3S selective absorbers and emitters of radiant energy. Narrow-band model predictions generally require Consequently, the variation of the radiative proper- an extensive and detailed library of input data, and ties with the electromagnetic spectrum must be the calculations cannot be performed with reason- accounted for. Spectral calculations are performed by able computational effort. On the other hand, as long dividing the entire wavelength (or frequency) spec- as the concentration distributions of gaseous species trum into several bands and assuming that the are not accurately known the high accuracy obtained absorption/emission characteristics of each species for the spectral radiative gas properties from narrow remain either uniform or change smoothly in a given band models would not necessarily increase the functional form over these bands. As one might accuracy of radiation heat transfer predictions. Also, expect, the accuracy of the predictions increases as it is not always convenient to use detailed, complex the width of these bands becomes narrower. Exact models for the spectral radiative gas properties. 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 3.1.2. Wide-band models levels of gas molecules. Line-by-line calculations are not practical for most engineering purposes but are Since gaseous radiation is not continuous but is usually required for the study of radiative transfer in concentrated in spectral bands, it is possible to define the atmosphere. Therefore, detailed line-by-line calcu- wide-band absorptivity and/or emissivity models. lations will not be discussed here. The radiation absorption characteristics for each band of any gas can be obtained from experiments and then empirical relations can be fitted to those 3.1.1. Narrow-band models data. The profile of the band absorption may be box Narrow-band models are constructed from spec- or triangular shaped or an exponentially decaying tral absorption-emission lines of molecular gases by function can be used by curve fitting. These types postulating a line shape and an arrangement of lines. of empirical models are known as wide-band models, The shape (profile) of spectral lines is quite important and among them the exponential wide-band model as it yields information for the effect of pressure, of Edwards and Menard 36 is commonly used. For an temperature, optical path length, and intrinsic prop- isothermal medium, several approximate expressions erties of radiating gas on the absorption and for the total band absorptivity and emissivity (see emission characteristics. The Lorentz profile 32 is the Refs 37-40) as well as the reviews of the wide-band most commonly used line shape to describe gases as models are available in the literature. 35.41 -43 moderate temperatures under the conditions of the Recently, Yu et al. 44 have devised a new "'super- local thermodynamic equilibrium, and it is also band" model to correlate total emissivity and Planck known as a collision-broadened line profile. 33 If the mean absorption coefficient data of infrared radi- temperature is high and the pressure is low, the ating gases. In this model, the Edwards exponential Doppler line profile would be more appropriate to band model has been used to approximate the use. 33 If there are ionized gases and plasmas in the emissivities. The spectral lines of the various infrared medium and they are influenced by interactions absorption bands of a radiating gas are rearranged between the radiating particles and surrounding and combined into a single, combined band. 108 R. VISKANTAand M. P. M~GO~

In Figs 4 and 5, the spectral band absorptivity medium. It is clear that the relative importance of distributions from a narrow band model a'* are short wavelength band radiation (i.e. from 1.38 pm compared with those from a wide-band model 35 for (oJ~7000 cm -t) and 1.89/zm (ta~5300 cm -t) H20 two isothermal media. 45 In general, the wide band bands) becomes larger as the temperature of the model is in good agreement with the narrow band medium increases (see Fig. 5 for T=2000 K). The model, especially for-2.7 /~m H20 and CO2 bands error introduced by approximating the short wave- (co=3700 cm-Z), 4.3 pm CO2 band (to~2300 cm-1), length band absorption by wide-band models is and 6.3/~m H20 band (ta~1600cm-')~ In these marginal, since the temperature of a typical combus- figures, the normalized Planck blackbody function tion chamber is usually not as high as 2000 K, and corresponding to the temperature of the medium is the other gas bands absorb radiation more strongly also plotted to show the relative contribution of each than short wavelength bands. gas band to the total radiation absorbed by the It should be mentioned that some of the detailed

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

~.0 [i t S

Q SO.O 'l! / fvl, .....I, (%) : ,.,..,-

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

a m.o I /l /

• ~,~ 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. Radiation heat transfer 109 spectral properties of the combustion gases will be I suppressed when they are combined with those of the ~= ~, as.i [I-e-~,PL]. (3.1) particles. Because of this, use of very accurate i~0 spectral properties of gases may not increase the The weighting factor ae,~ may be interpreted as the accuracy of radiative transfer predictions. In Figs 4 fractional amount of black body energy in the and 5 the soot absorptivities are plotted for two spectral regions where "gray gas absorption coef- different soot-volume fractions. 4s Note that if ficient" xi exists, and they are functions of temper- Iv =Jv.~ = 1.0 x 10-7, then the gas and soot absorptiv- ature. Usually the absorption coefficient for i=0 is ities are of the same order of magnitude, especially assigned a value of zero to account for the transfor longer wavelengths. However, asJ~ increases (see parent windows in the spectrum. The expressions for the curves for J~.2=l.0 x 10-6), the soot absorption the total emissivity and absorptivity of a gas in terms becomes dominant. The soot absorptivity also in- of the weighted sum of gray gases are useful creases with increasing wave number, i.e. decreasing especially for the zonal method of analysis of wavelength, since soot absorption coefficient is radiative transfer. almost inversely proportional to the wavelength of There are several curve-fitted expressions available radiation; we will return to this topic later. 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- 3.1.3. Total absorptivity-emissivity models gray gases. 5~-54 In only two of these expressions soot contribution is accounted for along with the gas A detailed modeling of the radiative properties of contribution. 49's° All of these models are restricted combustion gases may not be warranted for the to the total pressure of one atmosphere, except that accuracy of total heat transfer predictions in combus- of Leckner, 48 and all of them are for the gas tion chambers, but definitely increase the compu- radiation along a homogeneous path, i.e. uniform tational effort. An in-depth review of the world temperature and/or uniform pressure. literature on the thermal radiation properties of If the path is inhomogeneous then the equivalent gaseous combustion products (H20, CO2, CO, SO2, line model 39 or the total transmittance non- NO and N20 ) has recently been prepared,'* and homogeneous method s5 can be used to predict therefore the discussion will not be repeated. For radiation transmitted along the path. However, in engineering calculations it is always desirable to have multidimensional geometries or if scattering par- some reliable yet simple models for predicting the ticles are present in the system, the use of these radiative properties of the gases. Here, we review models for practical calculations becomes prohibi- some of the available models. tive as the equations are much more complicated. One way of obtaining radiative properties easily is to use Hottel's charts which are presented as functions of temperature, pressure and concentration 3.1.4. Absorption and emission coeJflcients of a gas. '.6 Some scaling rules for the total absorptivity and emissivity of combustion gases can be used The total absorptivities and emissivities are useful to extend the range of applicability of Hottel's charts. for zero or one-dimensional radiative transfer ana- For example, the scaling rules given by Edwards and lyses as well as zonal methods for radiative transfer. Matavosian 47 can be employed to predict gas However, for differential models of radiative transfer emissivity at different pressures as well as gas the absorption and emission coefficients are required absorptivity for different wall temperatures and at rather than the total absorptivities and emissivities. gas pressures different than one atmosphere. Of Since scattering is not important for combustion course, in order to use these charts in computer gases (and soot particles), the gray absorption/ models, curve-fitted correlations are desirable. Other emission coefficient can be obtained from the sources for continuous expressions are the narrow Bouguer's or Beer-Lambert's law. For a given mean and wide band models. The spectral or band beam length Lm one can write absorptivities from these models are first integrated ~= (- 1/L,,,) In (1 -e). (3.2) over the entire spectrum for a given temperature and pressure to obtain total absorptivity and emissivity The mean absorption coefficients obtained from curves. Afterwards, appropriate polynomials are spectral calculations as well as curve-fitted contin- curve-fitted to these families of curves at different uous correlations were compared with measurements temperatures and pressures using regression tech- from a smoky ceiling layer formed in a room fire and niques. Sometimes, these curve-fitted expressions can very good agreement was found. 4a It is possible to be so arranged that the resulting expressions would determine the so called "gray" absorption and be presented as the sum of total emissivity or emission coefficients for each temperature, pressure, absorptivity of clear and gray gases. These are known and path-length, which yield approximately the same as the "weighted sum-of-gray-gases" models and are total absorptivity or emissivity of the CO2-H20 given as '.6 mixture. 110 R. VISKANTAand M. P. MENG0~:

It is worth noting that instead of using only the decreases the gas becomes thinner, and eventually in absorption coefficient, absorption as well as emission the limit of optically thin gas the mean absorption coefficients should be employed. Since the total gas coefficient becomes identical to the Planck's mean emissivity differs from total gas absorptivity, it is absorption coefficient. With an increasing size of the quite logical to define and use two separate coeffi- enclosure, the gas becomes optically thicker and the cients. The importance of this fact has been first mean absorption coefficient approaches Rosseland's discussed by Viskanta# 6 He has shown that the mean absorption coefficient. arbitrariness associated with an absorption coeffi- Planck's and Rosseland's mean coefficients are cient can be eliminated by the introduction of a independent of the beam length and are valid only in mean emission coefficient and a mean absorption the thin and thick gas limits, respectively. They are coefficient, which can be related to the spectral defined as absorption coefficient by the following definitions: oo oo

oo oo "Kp= S K;.Ibl, dJ./ S I b~,d~ (3.5) ~s = ~ ~:a~ad)~/~ ffad2, o 0 0 0 ao

ao 1/-~a= ~(lflcz)(dlbz/dT)d,l/~(dlbz/dT)d,t. (3.6) ~,= Sgan2Ebxdg/Sn~Ebad2. (3.3) 0 0 0 o Similar to Rosseland's mean absorption coefficient, Here, ~a is the spectral irradiance. If the index of we can define Pianck's internal mean coefficient 35 as refraction na of the medium is unity, then the mean 0o oo emission coefficient will be equivalent to Planck's -~,= Sr.~,(dIb~,/dT)dg/S(dIh~,/dT)d2 (3.7) mean absorption coefficient, s6 Also, if Ka is indepen- 0 0 dent of wavelength or the medium is in radiative equilibrium, i.e. ffa=n]Ebx for all wavelengths, then which is also appropriate for an optically thick the mean emission and absorption coefficients will be medium. Several other definitions of the mean equal to each other. coefficients were discussed in greater detail by The use of these mean coefficients is justified as Traugott. 5 long as there are no large temperature gradients in Another mean absorption coefficient was defined the medium, s6"57 Therefore, they can be calculated by Patch s a as separately for each zone where the temperature can oo at) be assumed uniform. If the soot volume fraction is ~¢(L) = Slbagzexp(- r2,L)d2/~lb~,exp(- gaL)d2. (3.8) high in the medium, the use of the mean absorption 0 o coefficient would be sufficient, since ~ca (for soot +combustion gases) would be a weak function of Unlike the first three mean absorption coefficients wavelength. defined above, this so-called effective mean coeffi- In order to determine the absorption and emission cient is a function of path length as it contains the coefficients from total absorptivity-emissivity data, beam transmittance [see Eq. (2.18)] in its definition. the corresponding mean beam length must be Therefore, Eq. (3.8) is expected to yield more accurate properly evaluated. The definition of the mean beam predictions for the absorption coefficients of gas length for a volume of a gas radiating to its entire mixtures having intermediate optical thicknesses surface is given as provided that the path length is known. It is also possible to write a mean absorption L,,, = 4 C V/A, (3.4) coefficient based on the narrow band model of Ludwig et al., 34 where C is the correction factor and for an arbitrary k~u geometry its magnitude is 0.9. 46 In general, the +~-aJ (3.9) 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 where k, are tabulated coefficients, u is the product of also be functions of the mean beam length. Therefore, the mean beam length and total pressure, and a is the if the total emissivity of a gas volume is fixed, then fine structure parameter. Note that this expression is the corresponding absorption coefficient decreases also a function of path length. with increasing physical path length or pressure [see The mean absorption coefficients for a water Eq. (3.2)]. The use of absorption/emission coefficients vapor-carbon dioxide-air mixture at two different related to the mean beam length is convenient for the temperatures are presented in Fig. 6 as a function of scaling of radiation heat transfer in practical systems, path length. 45'59 The mean absorption coefficient As the characteristic dimension of the enclosure (~q.w,,) as calculated from Modak's model,'.9 using the Radiation heat transfer 111

o _ - ....-~..-~,~.. ~ ® ~P.~ - .,~:.-,,i ,~ ® ,q _ T=,OOOK ®Ke K 2- ..~.~\ (E) Kl,n -- (~-,) "'..'.% ® ~,,..

I ---- '''%'%_%~ -- T= 2000 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 agree- z0 = 3.0 m, which gives Lm = 1.08 m. The solution of ment with the absorption coefficient calculated from the radiative transfer equation is obtained using the the narrow-band model (Kt.,) or Patch's effective P3-approximation,61 which will be discussed in the mean absorption coefficient (~). In this figure, xe., is next chapter. Radiative transfer calculations are Planck's mean absorption coefficient based on the performed on the spectral basis using the wide-band narrow-band model, 34 xe.w and ri are Planck's mean model of Edwards and Balakrishnan (see Edwards; 3s and internal mean absorption coefficients, respec- Table X). The thirteen spectral bands used for the tively, based on the wide-band model. 35 The mean absorption coefficient are shown in Figs 4 and 5 by absorption coefficient xt.we is a function of path- dotted lines. length and is calculated using Edwards' wide-band In Fig. 7 the radiation heat flux distributions model parameters. 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 3.1.5. Effect of absorption coefficient on the radiative use of the Pianck mean absorption coefficient yields heat flux predictions about six times higher radiative fluxes compared to In preceding sections, we have compared absorp- the detailed spectral calculations. On the other hand, tion coefficients calculated from spectral narrow the mean absorption coefficients calculated from the band models with those obtained from total emissiv- total emissivity model of Modak 49 yield only a small ity models as well as with the Planck mean and overprediction of radiative fluxes in comparison to internal mean absorption coefficients. It is also the spectral results, and the use of Planck's internal desirable to examine the effect of different definitions mean absorption coefficients slightly underpredicts of absorption coefficients on radiative transfer pre- the radiative flux distribution along the wall. In Fig. dictions. For this reason, an axisymmetric cylindrical 8 the same kind of comparisons are given for the enclosure is considered. It is assumed that the second enclosure, which has Lm= 1.08 m. Basically, medium is a homogeneous, uniform gas (H20-CO 2- the trends are the same as those shown in Fig. 7, air) mixture at atmospheric pressure. The partial however, the agreement between spectral and total pressures of water vapor and carbon dioxide are the calculations is better in this case. same and equal to 0.1 atm., and the medium Indeed, the trends in the results predicted using temperature is either 1000 K or 2000 K. The en- different absorption coefficients, as illustrated in closure walls are assumed to be at a temperature of these figures, can be also deduced from the compar- 600 K and diffusely emitting, with emissivity ew=0.8. isons of the absorption coefficients given in Fig. 6. Two different sets of dimensions for the cylindrical For example, for Lm=0.5 m, at T= 1000 K, xt.,., is enclosure are examined. The first one has a mean somewhat larger than the x~ but it is about six times beam length (Lm = 3.6 V/A) of 0.5 m, where ro = 0.4 m, smaller than the re. This is also evident from Figs 7 and Zo=0.9m. For the second one, ro=0.9m, and 8. From Fig. 6 we can conclude that the use of 112 R. VISKANTAand M. P. MENGOq

50 I I I I I I i 0) T= IO00K b ) T = 2000 K 250 A 40

200

x 30 / KP,w / \ L~. / \ \ - 150 o "1" 20 I00 o ...... "0 o KsplctroI n~ lO ~'~ \Kspsctre I 50

0 I I I I t , t i 0 0 I 2 0 I 2 Z/ o FI6. 7. Comparison of radiative flux distributions on the cylidrical walls calculated spectrally and using different mean absorption coefficients, L=0.5 m (see text for the delinitions).

50 ! I I I I I 500 a) T = I000 K b) T =2000 K

A 4O 400

/ \ x 30 300

-1- 20 200 zo ,~~ KItWm

spsctrel I00 spectre S K i 0 I I I I I I 0 0 I 2 3 0 I 2 3 4 Z/4o FIG. 8. Comparison of radiative flux distributions on the cylindrical walls as calculated spectrally and using three different mean absorption coefficients,L= 1.08 m (see text for the definitions}~

Planck's mean absorption coefficient would be predictions obtained from the spectral and gray acceptable only if the physical dimension or the total analyses. In some earlier parametric studies it has also pressure of the system under consideration was very been shown that the change of the center and width of small. the spectral absorption bands may yield large vari- The spectral radiative fluxes depicted in Figs 7 and ations of the total radiative flux predictions. 57,62 8 do not always yield identical results with those Since the temperature and characteristic length of the calculated from other mean absorption coefficients gas volume have a strong effect on both the center and such as r,,w, or r~, and the difference between them the width of the bands, in practical systems the gas may be as much as 100 %. Clearly it is difficult to have radiative properties are expected to show large a simple correlation between the radiative transfer differences from location to location. Use of a single, Radiation heat transfer 113

40 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 50 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 qr 2- "benchmark" for the purpose of comparisons here. (kW/m2) u 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, I0 1.87 pm and 6.3 #m H20 ) are neglected the error introduced would be on the order of 10-20yo; 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. 0 2 4 6 8 I0 For most practical calculations simple "mean" z(m) absorption coefficients are widely used and preferred F~;. 9. Comparison of radiative fluxes at the wall based on over the detailed spectral radiative properties of spectral and mean absorption coefficient calculations. combustion gases. Therefore, it is desirable to Water-vapor and carbon-dioxide only: "a" from all six compare the accuracy of the results predicted using bands: "b" for 2.7 and 6.3 pm H20 and 2.7 t+m and 4.3/~m the mean coefficients with the benchmark results. In 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 Fig. 9, the radiative heat flux distributions calculated CO2 bands: "'e" 6.3 pm H20 and 4.3 :~m CO2 bands; "f" for using Planck's mean absorption coefficient (curve Planck's mean absorption coefficient; "g" Planck's internal "f"), Planck's internal mean absorption coefficient mean absorption coefficient; +'h" for Edwards" wide-band (curve "g"), and mean absorption coefficients ob- model.59 tained 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 mean absorption coefficient for combustion gas- same figure; therefore, the results obtained using mixtures, in which large temperature gradients exist, Planck's mean absorption coefficient are not in is not expected to predict radiative transfer realistic- agreement with the spectral calculations. Although ally. Consequently, gray calculations employing the curves "g" and "h" yield 20-30 % errors in compar- mean absorption coefficients are not recommended ison to "benchmark" curve "a", they agree better with for predicting radiative transfer in a medium com- the spectral results than those based on Pianck's prised of only combustion gases, if good accuracy is mean. required. It is desirable to discuss the contribution of each major CO2 and H20 band on the radiation heat 3.2. Radiation Properties of Polydispersions fluxes. Figure 9 depicts the radiation heat flux distributions on the cylindrical wall of a combustion Analysis of radiation heat transfer in coal-fired chamber calculated spectrally as well as using mean furnaces, combustion chambers, and other utilization values. 59 The contributions by particles have been systems requires accounting of the effects of particu- neglected in obtaining the results presented in this lates, such as pulverized coal, char, fly-ash and soot, figure in order to determine the relative importance which are present in these systems. For this reason, it of each gas band. Only water vapor and carbon is necessary to have a knowledge of the radiative dioxide are assumed to be present. The mole fraction properties of polydispersions which, in turn, depend distributions of these gases in the furnace were on the particle size distribution, the spectral depend- obtained from the literature 63 for burning of low- ence of the complex index of refraction, and the volatile coal (anthracite); therefore, the water-vapor number density for each type of particle in the fraction in the medium was not high. The absorption combustion products. It is also necessary to know the coefficient of the gas mixture in every zone of the spatial distribution of all the particles in the medium is calculated from Edwards and combustion chamber. Even with all the data on hand, Balakrishnan's wide band model. 35'39 Each spectral it is difficult and time-consuming to predict the band corresponding to a different zone has a radiation characteristics required in radiation heat

JPgCS 13:2-B 114 R. VISKANTAand M. P. MENGOI~ transfer analysis. Most of the time, some simplifying co ao assumptions are made to reduce the difficulties; D,,= If(D)DdD/ If(D)dD however, the simplifications must be reasonable for 0 0 realistic modeling of physical processes. = ZJi(Di)DiADi/~.ji(Di)&d)i (3.13a) With the increasing coal utilization, the need for i i radiative properties of particles formed in coal-fired combustion systems has become more demanding. A which is also expressed as D~ o or rto if a mean radius state-of-the-art review of the type of particles and is needed. Other definitions of the mean diameter their effect on radiative transfer in combustion (radius) are also used in the literature, 69 including the chambers has been given by Sarofim and Hotte164 Rosin-Rammler mean and Sauter mean, which is and Blokh. 4 By assuming that particles are homo- given by geneous and spherical, the radiation characteristics oo co of a cloud of particles can be predicted from the Mie 032 = Jf(D)D3dD/Jf(D)D2dD. (3.13b) (or Lorenz-Mie) theory. 65'66 It should be noted that 0 0 pulverized coal (char) and other particles which exist in combustion chambers are neither homogeneous Sometimes this definition of the Sauter mean is nor spherical. 67 Nevertheless, the extension of the modified to express it as volume to surface area ratio. Mie theory to nonspherical (i.e. cylindrical, ellip- soidal) particles has shown that the radiation characteristics of a cloud of irregular shaped particles 3.2.1. Types and shapes of polydispersions are not very sensitive to the geometrical shape of the particles. 66,6a Therefore, the use of the equivalent In combustion chambers, soot, pulverized coal, spherical particles assumption and the Mie theory char, and fly-ash are the polydispersions to be for coal combustion systems appears to be a considered. Soot is one of the most important reasonable compromise. In this section the method- contributors to radiation heat transfer in practical ology for calculating the radiative properties of systems. Typical diameter of the soot particles is polydispersions is given. Some simple expressions about 30 nm to 65 nm, 64 yet the sizes of soot are also suggested for use in practical calculations. aggiomorates may be much larger.'* Mainly because Following the Mie theory, the spectral absorption, of the small size of the soot particles, scattering of extinction, and scattering coefficients needed for radiation by soot is negligible in comparison to radiative transfer analysis can be evaluated from the absorption, and its radiative properties can be equation, calculated easily provided that the complex index of refraction and volume fraction distribution data are oo available. Numerous experimental studies (see, for qa(ha,N) = I Q~(D,A,haX1tD2/4)f(D)NdD, (3.10) example, Refs 4, 70-73 for citations) have reported a 0 complex index of refraction as well as volume fraction data of soot. Recently, Felske et al. 74 where qa stands either for spectral extinction coeffi- discussed the effect of different soot particle shapes cient fla, the spectral absorption coefficient xa, or for on the scattering characteristics of radiation and the spectral scattering coefficient tra, and Q, is the presented a framework for determining the character- corresponding efficiency factor which is a function of istics of soot agglomerates using those of spherical the size (diffraction) parameter (x=nD/2) and the particles. They demonstrated the sensitivity of the wave-length of radiation 2. Here, h a is the refractive soot radiative properties on the inhomogeneity of index of particles, N represents the particle density, the particles by using the coated sphere model (see and f(D) is the normalized size distribution function, Subsection 3.2.3).

oo The spectral complex index of refraction is the §f(D)dD= 1. (3.11) most fundamental optical property required to 0 calculate radiative characteristics of polydispersions. It is computationally time-consuming to take into For most practical problems, a discrete size distri- account the dependence of the index of refraction on bution of polydispersions is required. Hence, it is wavelength, but conceptually it is straight-forward. better to replace the integral of Eq. (3.10) by a finite In the literature, there are some published data series. Then, using the "step-size distribution", the for the complex index of refraction of various radiative properties can be expressed as coals. 69,75,76 An extensive compilation of the complex index of refraction data, including that for qa(na,N) = ~Q,v,, (D,,2,haX~D~/4)fi(D,)NAD,, (3.12) different Soviet Union coals, is also available.4 The 1 early experiments for determining ha were usually based on the "Fresnel reflection" method. More where i designates the diameter ranges. The mean recently, Brewster and Kunitomo 76 proposed a new diameter of particles in the cloud can be obtained as method, the so-called "particle extinction" technique. Radiation heat transfer 115

Their results show that there is approximately one real and the imaginary parts of the refractive index order of magnitude difference between the impinging on composition were also examined. A semi-empirical part of the complex index of refraction measured mixture rule was developed to allow prediction of the with these two techniques. In brief, there are large real part of the refractive index from 1 #m to 8 #m in differences between the reported spectral data for terms of the weight percents of the major oxide the complex index of refraction of coal particles components SiO2, Al2Oa, CaO, MgO, TiO2, and reported by different investigators, 4'69'7'~-76 and, Fe203. The mixture rule is based on the refractive therefore, more research attention is needed in this indices of the pure oxide components, with two small area. It is also bdieved 64 that the radiative proper- modifications to improve the agreement with the ties of char particles do not show distinctive measured refractive index data. differences from those of other pulverized-coal Shape of a particle is another important indepen- particles. Unfortunately, to the authors' knowledge, dent parameter that should be considered in pre- there is no fundamental study which supports this dicting the radiative properties. For the particles in conclusion for various coals and at different wave- combustion chambers, it is difficult to imagine a lengths of radiation. single, unique shape. Usually shapes of pulverized- The contribution of fly-ash particles to radiation coal particles or soot agglomerates are irregular and heat transfer in pulverized-coal flames exceeds that of random; yet, sometimes, surprisingly uniform and combustion gases or soot substantially; 4 therefore, simple shapes are observed. For example, fly-ash special attention must be given to the radiative particles from coal-fired boilers show fairly smooth, properties of these particles. Although limited, some spherical shapes, a4"8~ The soot, on the other hand, data for radiative properties of fly-ash particles have may agglomerate to form relatively long tails of radii been reported in the literature. 4"77- s3 on the order of the coal particle radius due to the slip The refractive index of fly-ash is sensitive to its velocity between the coal particle and surrounding chemical composition, and this is attributed primarily gases. 86's7 These tails can be considered as infinitely to the varying amounts of oxides of silicon, alumin- long cylinders. The simple shapes are most desirable ium, iron and calcium (i.e. SiO2, A1203, Fe203 and for the simplicity of calculations as the computational CaO) in the ash. The experimental studies have effort is reduced significantly for uniform, symmetric shown that the index of refraction of different fly-ash shapes. However, a large fraction of particles sus- samples from the same flame may be drastically pended in combustion products have totally irregular different, probably indicative of the microscopic shapes. Experimental measurements show that there conditions for their formation. 77 According to Wall are some differences in the scattering properties of et al. 78 the complex refractive index of fly-ash is in these particles in comparison to Mie theory calcu- the range from ha= 1.43 -0.307i to ha= 1.50-0.005i. lations, as where for irregular shape particles: (a) These numerical values of the imaginary part of the oscillations of efficiency factors vs angle and vs size complex index of refraction correspond approxi- parameter are damped; (b) more side scattering mately to the values measured by Blokh, 4 whereas (60°-120 °) is observed; (c) less backscattering is the values of the real part of the complex index of observed and; (d) the agreement with Mie theory :refraction are somewhat lower than those reported. becomes worse for other radiative properties as the The imaginary part of the refractive index of fly-ash size parameter increases past x = 3 or 5. sa For a cloud particles formed during combustion of pulverized- of irregular shape particles, however, the observed coal in a fluidized-bed furnace was of the order of differences in comparison to those for spherical 0.01. 76 This clearly indicates the uncertainty in the particles are less significant. 66'a8 complex index of refraction of fly-ash particles formed in pulverized-coal combustion systems. 3.2.2. Prediction methods of the particle radiative Recently, Goodwin 83 has reported extensive results properties of an experimental study of the bulk optical constants of coal slags. The effects of chemical compos- When radiative properties of particles are needed, ition, wavelength, and temperature were examined. the following quantities, arranged in order of Both synthetic slags, prepared from oxide power increasing complexity are to be considered: 8s (i) mixtures, and "natural" slags, prepared by re-mdting extinction cross-section, (ii) scattering cross-section, fly-ash or gasifier slag, were used. Transmittance and (iii) absorption cross-section, (iv) single-scattering near-normal reflectance measurements were made on albedo, (v) radiation pressure cross-section, (vi) their polished wafers cut from the slags, from which asymmetry factor, (vii) unpolarized phase function, optical constants were determined. The imaginary (viii) Legendre coefficients of unpolarized phase part of the refractive index was shown to depend function, (ix) parallel and perpendicularly polarized primarily on iron, silica and OH content of the slag. scattered intensities, (x) Stokes parameters, (xi) Iron is primarily responsible for absorption in the Mudler matrix, and (xii) Legendre coefficients of short-wavdength infrared region (1 #m<3.<4/Jm), Mueller matrix dements. The last four quantities in and silica is responsible for absorption at longer this list may not be critical for studying radiative (.k > 4/~m) wavelengths. The dependences of both the transfer in combustion systems. However, the other 116 R. VISKANTAand M. P. MENG0~ quantities are definitely needed for radiation heat with the complexity (or asymmetry) of the shape of transfer calculations. the particles. Recently, Wiscombe and Mugnai as.l i l One of the most extensively used models to predict developed a vector algorithm for the EBCM code of the radiative properties of particles is the Mie Barber t°4 and obtained the scattering properties for theory. 65'~6 Although it is widely known by this various axisymmetric particles whose shapes are name after Mie's exact solution of Maxweil's equa- determined from Chebyshev polynomials. Their tions for the scattering of an incident plane wave on results show that there are significant differences a sphere, s9 the solution was also obtained independ- between the radiative properties of spheres and ently by Lorenz and Debye (see Kerker 66 for detailed arbitrary shaped particles depending on the irreg- historical discussion). The exact solution for a right ularity of the surface characteristics. The compu- circular cylinder with radiation incident normal to tational time required for these calculations is too the cylinder axis was given by Rayleigh. Basically, in formidable as to justify the extensive use of the T- the Mie theory the vector Heimholtz equation is matrix method for practical problems. 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 3.2.3. Simplified approaches evaluate; however, for spheres and infinitely-long circular cylinders they can be reduced to single series, One of the simplifications usually made in calcu- and exact solutions can be obtained. The Mie lating the radiative properties of particles is related theory for spheres has been treated extensively in to their shape. If it is possible to assume that the the literature, 65'66'9° and some formulations for particles are spherical, then exact solutions from Mie cylinders,9°-9'* for elliptic cylinders 95 and for theory can be obtained effectively and with much less spheroids 96 have been given. There is no need to computational effort in comparison, for example, to repeat the details of Mie theory here; the interested the T-matrix method. The properties of irregular reader is referred to one of the classical refer- shaped particles can be obtained by assuming them ences 65'66'90 on the subject. as equal-volume spheres if the size parameter The Mie theory has been used extensively, es- (x = riD~A) is small or equal-projected-area spheres if pecially during the last two decades, with the help of the size parameter is large, as The nonsphericity of the computer algorithms which have been devel- particles can be traded off against inhomogeneity by oped 90'97-99 as well as widespread use of digital assuming that the index of refraction varies from the computers. Its restriction to simple, smooth particles core to the periphery. 66 By picking a functional form has led researchers to investigate some other possi- for this variation that allows a reasonably simple bilities to model the scattering of radiation by radial solution with one or two adjustable para- irregular shaped particles. Several new approaches to meters, it may be possible to match nonspherical the solution of the problem have been proposed over scattering properties. Then, the solution for an the years, including exact differential equation inhomogeneous sphere can be obtained rather than approaches ~oo.lot as well as exact integral equation for an irregular-shaped particle, and this is signifi- methods. 1°2-~°* In addition to these, there are cantly simpler. It is also worth noting that the effect several approximate techniques available, including of shape becomes less critical if there is a size the geometrical theory of diffraction ~°s for pre- distribution of particles, as size-averaging in ob- dicting the scattering by sharp-edged particles; the taining the radiative properties "washes out" the fine method of moment for scattering by a perfectly details of nonspherical scattering. 66.as conducting body; 1°6 as well as perturbation1°7 and The Mie calculations for the efficiency factors of point matching methods l°s for nearly spherical spheres are relatively less time-consuming and easier particles. Some empirical models have also been to use than the other exact models. However, the size proposed and shown to be very accurate provided of the particles in combustion chambers are func- that some experimental data are available, t°9 The tions of time and space, and the properties must be details of these methods and others can be found in calculated for each new set of size distributions. In the literature, ss'9°A t o multidimensional and spectral radiative transfer Among these models, the integral equation method analyses use of Mie codes for this purpose is or as more widely known, T-matrix or extended impractical. Because of this, it is desirable to have boundary condition method (EBCM), 1oz- ~o, seems simple approximations for the efficiency factors. One to be the most promising as it is capable of solving such approximation has been given by Mengii~ and the scattering of radiation by any irregular shape Viskanta, lt2 where the efficiency factors for poly- particle. In the EBCM, the incident and scattered dispersions are obtained starting from the anomal- electric fields are expanded in vector spherical ous diffraction theory ~5 and are expressed in conven- harmonics, and then by making use of analytic ient, closed form. In Fig. 10 the Mie theory continuation techniques the integral representation predictions for the normalized extinction and scat- of the fields is reduced to a set of linear algebraic tering coefficients are compared with those of the equations. The complexity of these equations increase simplified model, and in Fig. 11 the predictions of Radiation heat transfer 117

x F(~) I0 ~ tO-t I0 0 tO i I0 e IO : I0 q tO 6 ...... 10-7 10-7 ...... I ...... i ' ' ' ..... I ...... I ' • ' ..... I ...... I

10-.8 lO-a ® CP,RBON a P,NTHRRC I TE [n~q + BITUMINOU$ [m"] 10-e x BITUMINOUB-K i0-9 • LIGNITE • FLY-fiSH

lo-iO i0-I0

lO-ii ...... i ...... i ...... i ...... I ...... , ...... i ...... lO-ii

lO-t lO o 10 I 10 ! lO 3 10 ,I lO 5 10 e xF(~) FIG. 10. Comparison of Mie theory results (points) for the normalized extinction and the scattering coefficients with those calculated from tin approximate analysis (lines).tj 2

1.000

c) CRRBON 0.800 A P,NTHRRC ITE + BITUMINOUS COx X BITUMINOUS-I( 0.600 ,i.~ ,..OO O& • • LIGNITE + FLY-fiSH ...... I I0 "2 I0 -i IO o 10 i I0 ~ I0 : IO N

x F('~) FIG. 1I. Comparison of Mie theory results (points) for the single scattering albedo with those calculated using approximate analysis (lines).~ t 2

the scattering albedo from the Mie theory and the in the volume fraction of polydispersions and the simple model are given.l~ 2 In these figures, flz and aa complex index of refraction data, the agreement are normalized spectral extinction and scattering between the model and exact calculations appears to coefficients, respectively. The normalization factor is be remarkably good, and, because of this, these NxF(ha), with N being the number of particles per simplified models would be useful for radiation heat unit volume, x is the size parameter, and F(h,t) is a transfer calculations in combustion chambers. Note function of the complex index of refraction. Note that the single scattering albedo 09 is related to that absorption, extinction and scattering coefficients by

F 24naka ] <.o = o/,a = 1 -- ~://7. (3.15) Q,=xF(ha)=x 2 ~ 2 2 (3.14) L(na-ka+2) +4naka.J In the literature, there are also some empirical is the absorption efficiency factor for very small size relations available for the radiative properties of spherical particles (x--g)) as obtained from the polydispersions. Buckius and Hwang 113 calculated Rayleigh limit of the Mie theory. The discrete points absorption and extinction coefficients as well as the shown in Figs 10 and 11 are the results obtained asymmetry factor of several coal polydispersions from the rigorous Mie theory for the corresponding using Mie theory and showed that they were almost index of refraction of specific particles. The lines are independent of the size distribution and were from the analytical, closed form expressions given by functions of average radii r32 [see Eq. (3.13b)] and Mengii~ and Viskanta. 112 Considering the uncertainty the complex index of refraction. They obtained some 118 R. V[SKANTAand M. P. MENG0(;

10-2 , , ,

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

i0 "s I0 "2

10"4 ~ i0"3

i0-4 I0x I02 I03 i04 r,zT [/.t.m K] FIG. 12. Phmck and Rossehmd mean coefficients for coal. The shaded area represents results for w~riations in temperature between 750 and 250 K and three coals. ~ 3 empirical correlations for the radiative properties of worth noting that the anomalous diffraction theory coal particles which could be readily used for used for spheres also yielded accurate and simple predicting radiation heat transfer in coal-fired com- relations (see Ref. 112). Most recently, Mackowski et bustion systems. Also, they plotted the normalized a/. 117 derived the same kind of relations for the Planck and Rosseland mean absorption and extinc- spectral radiative properties of cylindrical soot tion coefficients as functions of the mean radius- agglomerates. They showed that small size cylin- temperature product (Fig. 12) and obtained some drical particles extincted radiation two to five times empirical relations for these coefficients. As seen more than spheres. At large radii, on the other hand, from this figure, for small radii particles, extinction the ratio of cylindrical extinction and absorption and absorption coefficients are identical; however, coefficients to those for spherical particles approach with increasing radius the scattering of radiation constant values regardless of the wavelength of also becomes important, and fl and x diverge from radiation} 17 Also, some empirical relations similar each other. Viskanta et al. 1~4 aIso obtained similar to those obtained for spherical particles are pre- results and discussed the effects of several indepen- sented. It is also possible to extend the relations to dent parameters, such as size distribution, coal type mixtures of different types and shapes of particles and wavelength of radiation on the radiative proper- using the T-matrix method. For a specific (coal) ties of polydispersions. It is worth noting that combustion problem, a library of empirical relations although different definitions of mean radius are can be constructed. The use of these relations will used in these studies, i.e. rio [see Eq. (3.13a)] 112 and speed up the calculations significantly, since there r32 [see Eq. (3.13b)], 11a'1~4 still similar results will be no need for lengthy and time consuming Mie independent of size distribution are obtained. This or T-matrix method calculations. indicates that a polydispersion can be often des- When the size parameter (x=nD/2) becomes cribed by a weighted particle radiusJ 15 vanishingly small (x-*0) the size of the particle All of the studies discussed above used the becomes less important. In this limiting case, the spherical particle assumption in obtaining the re- absorption efficiency factor is a function ofx [see Eq. lations for radiative properties of particles. Perfect (3.14)], whereas the scattering efficiency factor varies spheres are not encountered in nature, and, therefore, with x 4, such as it is desirable to obtain similar relations for other than spherical shape particles. Stephens ]~6 has r~-I 4 shown that the anomalous diffraction theory devel- 0s = 3 ~ X . (3.16a) oped by van de Hulst 65 can be extended to infinite- length cylinders. The absorption and extinction efficiency factors calculated from this simplified The extinction efficiency factor is written as theory are in good agreement with those obtained from a rigorous solution of Maxwell's equations. It is Q,. = Q,, + Qs. (3.16b) Radiation heat transfer 119

These expressions are obtained from the Rayleigh 3.2.4. Scattering phase function limit of the Mie theory. 6s Here, ha=na-ika is the complex index of refraction. It is worth noting that In modeling radiation heat transfer in a partici- with decreasing x (or D), the scattering efficiency pating medium, the scattering of radiation by particles must be properly accounted for. This factor becomes negligible in comparison to the requires the use of the scattering phase function absorption efficiency factor. Indeed, these expres- (scattering diagram), which represents the probability sions yield the extensively used soot absorption coefficient, such as that radiation propagating in a given direction is scattered into another direction because of the inhomogeneities and/or particles along the path of xa = 7fJ2 (3.17) radiation. In combustion chambers, the scattering of radiation takes place mainly because of the particles. wherefv is the volume fraction of soot particles and The phase function, along with other radiative the value of "7" was suggested by Hottel and properties, such as absorption, extinction and scat- Sarofim 46 for typical soot particles observed in tering coefficients, can be obtained either exactly combustion chambers. After studying the available from the solution of Maxweli's equations for spher- experimental data for several flames Siegel l~s has ical or infinite-length cylindrical particles 65,66'9° or shown that the coefficient in Eq. (3.17) is between 3.7 from some approximations such as the extended and 7.5 for coal flames; 6.3 for oil flames, and 4.9 and boundary element method (EBCM) for arbitrary 4.0 for propane and acetylene soot, respectively. A shaped particles ~°2- lo4 as functions of wavelength, detailed discussion of the spectral and total absorp- characteristic particle dimension and complex index tion characteristics of uniform-diameter, spherical of refraction. The phase function is written as soot particles covering a very wide range of sizes (0.001

TABL[ 2. Comparisons of spectral F(ha) functions for N different shape small soot particles • a(W) = ~ a,.~P,(~P) (3.21a) /(/am) na ka F,~,,, F~,n~®r Fomt,,ola rl=O 0.50 1.92 0.55 0.754 1.871 1.700 where 1.50 1.88 0.73 1.007 2.450 2.134 2.50 2.10 1.09 1.140 3.683 2.888 1 5.00 2.69 1.57 0.863 6.073 3.888 an,a-2n+l J" ~a(W)P,(~)dD (3.21b) O=4f 120 R. VISKANTAand M. P. MENO0q is the expansion coefficient and P, is the Legendre 1 polynomial of degree n. By changing the upper limit A=U~ I o~(,I,)dn t~=2~ N of the series, any phase function can be written in the form of Eq. (3.21). The coefficients a,.a can be =½+½ ~ (-1)"a2"+'(2m)! (3.26a) obtained by employing the orthogonality relations of r.=0 22"+im!(m+ 1)t Legendre polynomials. In order to accurately repre- sent the phase function of highly forward scattering ba = 1 -fa. (3.26b) particles, however, as many as 100 terms may be required in the series. For the multidimensional The factors ba and fa are especialy useful when radiative transfer calculations, the use of such a obtaining solutions of the radiative transfer equation complicated scattering phase function is not prac- using flux methods. tical either. Consequently, some further simplifi- Brewster and Tien 125 have given a different cations are required. If N=0, the phase function is definition of the backward scattering coefficient for written as an azimuthally symmetric layer such as

(IDa(W)= 1 (3.22) 1 o Ba=½~ ~ ~a(/~,/~')d/~'du (3.27) which is for isotropic scattering. If N= 1, then the 0 -1 linearly anisotropic scattering phase function is obtained, This expression is valid for a plane-parallel layer of particles, whereas Eqs (3.26) are appropriate for ~a(~) = 1 + a l,acos W. (3.23) scattering from a single particle. It has been shown that for a cloud of non-absorbing spherical particles For N = 2 the phase function corresponds to second- (with h=1.33, x=6.0), b~(=0.036) is drastically degree anisotropic scattering, and if the expansion smaller than Ba(= 0.137). coefficients are set arbitrarily such that at.,t=0 and In atmospheric studies, the Henyey~3reenstein a2,a= 1/2, this yields the Rayleigh scattering phase phase function approximation is often used 121 and is function.~ 9 expressed as Most of the particles encountered in combustion chambers (pulverized coal, char or fly-ash) scatter radiation predominantly in the forward direction. @n-o,a( W)= [1 +O]-2gacos tF]3/2" (3.28) Such a scattering behavior can be modeled using a Dirac-delta function. The transport, delta-M and Here, g~ is the asymmetry factor and is defined as the delta-Eddington approximations are of that form. TM The delta-Eddington approximation is written as ~22 ga= (cosW)= S la(g)cosWd~/ S la(g)df~ (3.29) fl=4x O=4f

~a(W) = 2f~6(1 - cos W) + (1 -faX1 + 39acos W) (3.24) which can be directly obtained from the Mie theory. Although it approximates the Mie phase function where fz and 0a are related to the expansion quite accurately, the application of the Henyey- coefficients defined by Eq. (3.21b) as 59 Greenstein phase function approximation to multidimensional geometries may be quite tedious. fa = {i:al i f_ ~ 2)12 ii:: ,~ 1)/2 (3.25a) Several different approximations for the scattering phase function, such as linearly anisotropic scattering, delta-M, delta-Eddington, transport or Henyey- and Greenstein approximations, have been reviewed in detail by McKellar and Box)21 They have concluded al ,,a -fa that for highly forward scattering particles the 9a = (3.25b) 1-fa delta-Eddington approximation ~22 is the most accurate and the simplest of all the approximations provided that al,a>a2,a. A detailed account of mentioned. In modeling radiative transfer in coal- Dirac-deita phase approximations has recently been fired furnaces the delta-Eddington approximation given by Crosbie and Davidson? 23 for the scattering phase function is desirable for two In the heat transfer literature, another phase reasons: (1) it represents the highly forward-directed function approximation has found wide application. scattering of radiation by the pulverized coal and fly- The phase function is expressed in terms of the ash particles; and (2) it is compatible with differential forward (fa) and backward (ba) scattering coefficients, approximations such as the spherical harmonics and they are written in terms of a.'s of Eq. (3.21b) for approximation used to model the radiative transfer an azimuthally symmetric medium such as 124 equation. Radiation heat transfer 121

The scattering phase function of particles is the concept may be of limited utility for predicting directly related to the size (or diameter) of the radiation heat transfer in multidimensional combus- particles. Therefore, for polydispersions there should tion systems which contain particles, If the emissivity be as many scattering phase functions as the number (or absorptivity) of a particle laden flame is known of size intervals considered. For the sake of simplicity, then the extinction coefficient of the medium can be it is desirable to have a single, mean scattering phase written as function over the entire particle size range. Then, the mean scattering phase function can be written as fla.tot= - L~ In (3.36) 1 N ~a = :- ~ a~,i ¢Pa.~ , (3.30) 0"2 i provided that the mean-beam-length L,, is known. However, fla,tot and toa are interrelated properties where N is the number of the intervals. If the [see Eq. (3.34)]. If the mean coefficients are to be delta-Eddington phase function approximation is used, the equations given above should be rewritten used for the scattering phase function, the corres- by dropping the subscript 2, and appending the ponding mean parameters are defined similarly, appropriate mean coefficient subscript. These definitions require the mean beam length of 1 N 1 N radiation Lm, which is a vague concept. 59 It is .'/-~- ~- ~O'2.,if]i, "g~-~- ~- EU2..iOa.i. (3.31) defined 2° as a radius of a gas hemisphere which G~ i G'l i 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 3.3. Total Properties accurate results for simple systems, for complicated geometries it needs additional research attention. Once the absorption, scattering and extinction Recently, Scholand and Schenkel ~27 have calculated coefficients of polydispersions, such as pulverized coal, char, fly-ash, and those of soot and combustion the mean beam length of radiation between a volume element and the surfaces of rectangular paral- gases are known, the total radiative properties are lelepiped enclosures. Cartigny ~28 has extended the written as definition of the mean beam length to an optically thin scattering medium, which can be used for ra.tot = ~xa.~ly-i + xa.,,**,+ ~xa.,,,,,_.i, (3.32) calculation of radiative transfer in sooty flames. i j The empirical relations for the total mean extinction and absorption coefficients for fly-ash, pulverized coal and char particle polydispersions have also fla,to,= xa.tot+ ~aa.vo,y-i, (3.33) i been reported. 4 It has been found, for example, that the mean extinction coefficient fl of fly-ash can be and expressed by an empirical equation of the form,

fl= g,, F(CL)ApC (3.37) o93 = ~tra,~ly_ i/fl~.,tol (3.34) i where Q,, is the extinction efficiency factor; F(CL) is where "poly-i" refers to i-th type polydispersion, and the function which accounts for the dependence of "gas-j" refers to j-th gas species. Note that if there the extinction coefficient on the product of the are no scattering particles in the medium, then concentration C and the layer thickness L, and At, is fla.tot= K~.,tot and t~a = 0. the surface area of a particle. The total extinction An alternative formulation of total absorptivity efficiency factor Q~ has been found to depend, as and emissivity of a scattering medium has been might be expected, on the type of coal burned, fly-ash recently given as ~26 particle size and the spectral distribution of incident radiation determined by the black body temperature F tg'l''t°t [1 --exp(-fl~.,totLm) ] . (3.35) T used as the radiation source. Based on experi- ~. = C£~ = Lfl,l.,tot -I mental data it is possible to express Q,. by an empirical equation, In writing this expression it was assumed that spectral irradiance was equal to the spectral emitted Q,.=0.07 A(xT) 1/3. (3.38) flux of the surroundings. The spectral absorptivities of polydispersions of coal and fly-ash particles have The empirical constant A depends on the type of coal been predicted using a two-flux approximationJ ~4 burned and the shape of the fly-ash particle, and x is The expression for the absorptivity 1~4 is not as the size parameter based on the mean particle simple as that given by Eq. (3.35). This suggests that diameter. The function F(CL) has been determined 122 R. VISKANTAand M. P. MENG0~'

empirically and was found to depend on the type of 4.1. Exact Models coal burned. 4 The total effective absorptivity of a fly-ash layer of The most desirable solution of any equation is its thickness L calculated from the expression exact closed form solution. The exact solution of the integrodifferential radiative transfer equation can cZ®fe = 1 - exp( - ~L) (3.39) only be obtained after some simplifying assumptions, such as uniform radiative properties of the medium has been found to agree well with the experimental and homogeneous boundary conditions. For one- data.'* It was determined from the data that the dimensional, plane-parallel media, exact solution of optical thickness zL(= ~L) of the layer varies linearly the RTE has received much attention in the atmos- with CL only for moderate values of CL(<20 g/m2). pheric sciences, 12a 5,t 6 neutron transport 13 ~- ~33 and At higher values of CL the mean extinction coef- heat transfer ~9'2°'~34 literature. A detailed review of ficient ~ starts to depend on CL, because the one-dimensional exact solution methods is avail- radiative properties of fly-ash particles depend on able? 35 However, there have only been a few wavelength. This leads to the departure of the attempts to formulate and solve the RTE for function zL(CL) from linearity. multidimensional geometries. The Hottel charts for the emissivity and absorptiv- One of the earliest accounts to formulate the ity of combustion gases are very convenient for radiative transfer equation in a three-dimensional practical calculations. Skocypec and Buckius ~29 space with anisotropic scattering was that of and Skocypec et a/. 13° extended these charts to Hunt? 36 He considered a phase function comprised include isotropically scattering particles. In their of three terms in Legendre polynomials and reduced calculations, they obtained the gas properties from the integrodifferential radiative transfer equation to the Edwards wide-band model 35 and presented an integral equation. Cheng ~37 used a rigorous hemispherical emissivities in graphical form and approach to solve the RTE for an absorbing- discussed the effects of optical thickness, pressure, emitting medium in rectangular enclosures, and Dua temperature and single scattering albedo. These and Cheng ~38 extended this method to cylindrical charts yield accurate radiative properties without geometries. For an absorbing, emitting, and scat- any additional calculations; however, they cannot be tering medium Crosbie and his co-workers presented used directly for predicting the local radiation heat exact formulations of the RTE for three-dimensional flux in a combustion system. rectangular ~39 as well as three-dimensional cylin- dricaia4° enclosures. The solution of these equations for cylindrical geometry was obtained by the method 4. SOLUTION METHODS of subtracting the singularity? 4a The exact solutions of RTE for an absorbing and emitting medium were The radiative transfer equation is an integro- also solved by Selcuk ~42 in a three-dimensional differential equation, and its solution even for a one- rectangular enclosure employing a numerical scheme. dimensional, planar, gray medium is quite difficult. In a cylindrical geometry, the radiative transfer Most engineering systems, on the other hand, are equation is obtained from Eq. (2.11). Then, the multidimensional. In addition, spectral variation of integral form of the source function, for an absorbing, the radiative properties must be accounted for in the emitting and isotropically scattering medium with solution of the RTE for accurate prediction of incident diffuse radiation source on one of the end radiation heat transfer. These considerations make surfaces of the cylinder can be written as 14° the problem even more complicated. Therefore, it is almost necessary to introduce some simplifying S2(r,z,q~) = (1 -- 0)a)/ba[ T(r,z,q~)] assumptions for each application before attempting to solve the RTE in its general form. It is not possible 0) 2 r 2x to develop a single general solution method for the +-- J S l d,~(r',d?')e-P~"z'~)X; (x~ ) - 3zr'dq ~'dr' equation which would be equally applicable to 4no o different systems. Consequently, several different solution methods have been developed over the +m2 z~ r~ ~Sa(r,,z,,(a,)flae_Pa%x~ 2r,dda,dr,dz ' (4.1) years. According to the nature of the physical system, 4nooo characteristics of the medium, the degree of accuracy required, and the available computer facilities, one of where several different methods can be adopted for the solution of the problem considered. Before choosing x; = [r 2 + (r') 2 - 2rr'cos(q~ -- qS')+ z2] 1/2 (4.2a) one solution method over another one, it is important to know the advantages and disadvantages of Xp = [r 2 + (r')2 - 2rr'cos(cb - ~b')+ (z- z')2] I/2. (4.2b) each method. In this section, several radiative transfer models of interest to combustion systems are Here, the primes are used to denote the dummy discussed, and their features are highlighted. variables, and l~,a is the spectral diffuse radiation Radiation heat transfer 123 source on the end of the cylinder at z=O. Note that Monte Carlo method can be used for any complex ld. a can also be interpreted as the diffuse emission geometry, and spectral effects can be accounted for and reflection from the walls. Some additional without much difficulty. Mainly for this reason, the integral terms are to be added to these expressions to method has been used extensively in atmos- account for the other surface effects` After some pheric 143'144 and neutron transport 133 studies. It lengthy and tedious algebra, the implicit expressions has also been successfully employed to solve some for the radiative fluxes in the r, z, and ~b directions general radiation heat transfer problems 14~,z46 as can be derived: 14° well as radiative transfer problems in multidimensional enclosures ~,17 and furnaces. ~4s,1,,9 ,[ 2[ -P'~;(x;) -~ There is no single Monte Carlo method. Rather, F,.a(r,z,ck) = ld,,t(~,~b')¢ there are many different statistical approaches. In its 00 simplest form, the method consists of simulating a z r 2z finite number of photon (energy packet) histories x Jr- r'cos(~b - ~b')]zr'dc~'dr'+ [ [ I Sa(r',z',c~') through the use of a random number generator. 133 000 For each photon, random numbers are generated and used to sample appropriate probability distributions x flae-P~%(xp)- air - r'cos(~b - t~')] r'dq~'dr'dz' (4.3) for scattering angles and pathlengths between col-

r 21~ lisions. If it is assumed that the problem is time- ~,~(,,z,~)= [ I l d,a(r' ,c~' )e -p ~x+ ~ (xp+ ) -, dependent, each photon history is started by 00 assigning a set of values to the photon, its initial energy, position and direction. Following this, the g o r o 21 x z2r'ddp'dr'+ ~ ~ I Sa(r',z',ck') number of mean free paths that the photon propa- 000 gates is determined stochastically. Then, the cross- section (or absorption and scattering coefficients) x flae-Pa~r~xp) - a(z-z')r'dc~'dr'dz' (4.4) data are sampled, and it is determined whether the collided photon is absorbed or scattered by the gas r 2x molecules or particles in the medium. If it is Fo,,(r,z,¢)= [ ~ ld,a(r',¢')e-~,~;(x;) -" absorbed, the history is terminated. If it is scattered, 00 the distribution of scattering angles is sampled and a

z o r o 21t new direction is assigned to the photon. In the case x r'sin(~b- ~')zr'ddp'dr'+ ~ J ~ Sa(r',z',q~') of elastic scattering, a new energy is determined by 000 conservation of energy and momentum. With the new set of assigned energy, position, and direction x flze-P*%(x~)- ar'sin(q~-dp')]r'dO'dr'dz '. (4.5) the procedure is repeated for successive collisions until the photon is absorbed or escapes from the When ld,a is interpreted as the wall function which system. includes the diffuse emission and reflection from the Monte Carlo calculations yield answers that walls, the additional integral terms will appear on the fluctuate around the "'real" answer. As the number of right-hand-side of these equations. It should be noted photons initiated from each surface and/or volume that in deriving these expressions, the medium is element increases this method is expected to con- assumed to be homogeneous. The evaluation of the verge to the exact solution of the problem. Since the integrals in these equations yields exact results for directions of the photons are obtained from a the radiative flux distributions in the medium. These random number generator, the method is always equations can be integrated numerically, as closed subject to statistical errors and the lack of guaranteed form solutions are not possible unless further simpli- convergence. ~46 However, as next generation com- fications are introduced. Considering that in most puters become more readily available, Monte Carlo engineering systems the medium is inhomogeneous methods are expected to become more attractive for and radiative properties are spectral in nature, it can engineering applications. It has already been shown be concluded that the exact solutions for RTE are not that vectorization of the Monte Carlo computer code practical for engineering applications. Nevertheless, yields significant improvements in efficiency using exact solutions for simple geometries and systems are supercomputers such as CYBER-205 and more needed, as they can serve as benchmarks against precise results are obtained. 1s o which the accuracy of other approximate solutions are checked.

4.3. Zonal Method 4.2. Statistical Methods The zonal method, which is usually known as The purely statistical methods, such as the Monte Hottel's zonal method, TM is one of the most widely Carlo method, usually yield radiation heat transfer used methods for calculating radiation heat transfer predictions as accurate as the exact methods. The in practical engineering systems. In this method, the 124 R. VISKANTA and M. P. MENG0~ surface and the volume of the enclosure is divided As originally formulated TM the zonal method has into a number of zones, each assumed to have a some inherent limitations, such as the treatment of uniform distribution of temperature and radiative non-gray, temperature dependent radiative proper- properties. Then, the direct exchange areas (factors) ties of combustion gases. The effects of temperature, between the surface and volume dements are pressure and different species on gas properties can evaluated and the total exchange areas are deter- be accounted for by weighted sum-of-gray-gases mined using matrix inversion techniques. For an moddg 5~'5z In addition, it is usually difficult to absorbing and emitting medium, the calculation of couple the zonal method with the flow field and direct exchange areas becomes complicated as the energy equations which are usually solved using attenuation of radiation along the path connecting finite difference or finite element techniques. This is two area (area-volume and volume--volume) elements mainly because of the different size of the control must be taken into account. volumes required; the zonal method can be comp- The zonal method reduces the radiative transfer utationally prohibitive if the same grid scheme used problem to the solution of a set of nonlinear by the finite difference equations is adopted• Steward algebraic equations. The set of energy balances for and Tennankore ~57 have coupled the zonal method the zones in a closed radiaton system is written as with finite difference equations in modeling a combustor by adapting two different grid schemes; SE = Q (4.6) one for the radiation part and the other for the flow and temperature fields. Recently, Smith et al. ~54 have where combined the zone method with momentum and energy equations to predict heat transfer in an -Xs, S2St ... S,,Sl absorbing, emitting, and scattering medium flowing J in a cylindrical duct. The zonal method can not be readily adopted for Sj S2 S2S2 -~S2Sj ..• S,,S2 problems having complicated geometries, since J numerous exchange factors between the zones must S = S~ S3 S2S 3 ... SnS 3 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 SLS,, s,s ... s,s,-Es,,s Subsection 4.5.5• Note that the direct-exchange areas J for rectangular enclosures have been recently calculated by Siddal115a who employed a new approach 12¢hl I Qt [ for the evaluation of the multiple integrals• With this I.~h2 ] Q~ I technique, it is possible to obtain these factors with E = Eh3 and Q= Q3I 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 .Elm. .Q.J alternatives, and therefore the method is attractive for practical engineering caiculations.l 56

Here, SiS~ is the total exchange area which is the 4.4. Flux Methods ratio of the radiant energy emitted by a zone Si which is absorbed by zone S~ (directly or after The radiation intensity is a function of the multiple reflections from other zones) to the total location, the direction of propagation of radiation energy emitted by zone S, Eu is the total blackbody and of wavelength• Usually the angular dependence emitted flux and Q~ is the imposed heat flux at zone of the intensity complicates the problem since all Si.152 possible directions must be taken into account. It is, Although the formulation of the zonal method for therefore, desirable to separate the angular (direct- an absorbing, emitting, and scattering medium has io.nal) dependence of the intensity from its spatial been available 153 for a long time, it has been only dependence to simplify the governing equationg If it recently applied to the solution of radiation heat is assumed that the intensity is uniform on given transfer problems in a system containing scattering intervals of the solid angle, then the radiative transfer particles. 154 Larsen and Howell 155 presented an equation can be significantly simplified as the alternative formulation of the zonal method and integrodifferential RTE equation would be reduced accounted for only the isotropic out-scattering from to a series of coupled linear differential equations in each volume element. This new approach, however, terms of average radiation intensities or fluxes. This does not show any computational advantage over the procedure yields the flux methods. By changing the classical zonal method.~ 56 number of solid angles over which radiative intensity Radiation heat transfer 125 is assumed constant, one can obtain different flux an absorbing, emitting, and scattering medium are methods, such as two-flux, four-flux or six-flux comprised of six coupled partial differential equ- methods. Intuitively, one can deduce that as the ations. 165 The equations are quite complex and number of fluxes increases the accuracy of the lengthy; therefore, they are not given here. method would increase. Indeed, if the number of In general, the accuracy of the flux approximation solid angles and corresponding directions are deter- depends on the choice of solid-angle subdivisions. If mined from basic mathematical principles (see, e.g. there is no intersection between two adjacent sub- Whitney 159) more accurate and efficient flux divisions, more accurate results are expected. 165 This methods can be warranted. It is also possible to use has also been observed by Selcuk and Siddall a66 for non-uniform solid angle divisions in the spherical rectangular enclosures. If the distribution of radi- space. For example, if the direction and size of the ation intensity is assumed for each subdivision, the solid angles are determined from the Gaussian or general equations given by Abramzon and Lisin t65 Lobatto quadratures, a non-uniform flux approxi- can be simplified and solved simultaneously. If the mation is developed and the resulting expressions are fluxes in each subdivision are assumed constant, a called the discrete ordinates approximation to the simpler six-flux model can be obtained from the RTE. 15 general flux equations. For an absorbing, emitting Another way of avoiding complicated expressions and scattering medium, Spalding ~67 suggested a of the RTE due to the angular dependence of the similar six flux model for cylindrical geometry, which intensity is to integrate the radiative transfer is written as equation over the space after first multiplying it by certain directional cosines. The resulting expressions ld are called moment approximations. The spherical + r drr [rJ~a] = -(xa + crx)~a + xaEoa(T) harmonics approximation is developed similarly, but a more elegant and mathematically sound method of (4.7) integration of RTE is employed. If the integrations r are performed over hemispheres or quarter-spheres, then double or quadruple spherical harmonics - d~ (K~) = - (xa + o'a)K:~ + xj.Eba(T) approximations are obtained, respectively. The first order moment, spherical harmonics, and first-order (4.8) discrete ordinate methods are identical for the one- +6(J~" +J~- +K~" +K~- +L~" +Lj-) dimensional, planar geometry; 16° however, they differ from each other slightly for multidimensional 1 d geometries. -+ r cl~ (L~) = - (xa + (ra)L~ + xaEba(T) Due to the simplicity of the governing equations, several flux methods have been developed for one- +~(J~ +J~- +K~ +K j +L~ +Lj) (4.9) dimensional plane-parallel media. They are reviewed 0 elsewhere, TM and those which can be extended to multidimensional geometries are compared against where J~, Jj are spectral fluxes in positive and experiments ~62 as well as against exact solu- negative radial (r) directions; K~, K j- in positive and tions. TM In this discussion, the focus is on negative axial (z) directions; L~, L~- in positive and multidimensional models. negative angular (4)) directions. These equations can be manipulated to obtain three second order differential equations: 4.4.1. Multiflux models Ever since the publication of the pioneering works 1 dfl- r -Id + } 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 = tCa[J~" + Jf - 2 Eba(T)] transfer calculations. With the advances in computers, the extensions of flux models for the application to multidimensional systems have become + ~ra [2(J~ +J;)-K~ -K~ - L~ -L~] (4.10) r possible, and consequently several different versions have been proposed over the years. Recently, 1 d 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 = xaEK~ +K~ - 2Eba(T)] obtained from this general formulation. The governing equations for the general flux approximation in +3[-J~ -J~- +2(K~" +K~)-L] -L;] (4.11) a three-dimensional cylindrical enclosure containing 126 R. VISKANTAand M. P. MLmG09

ldfl- 1 -Id + } where, J,~, J~', K~ and K~- have the same meanings as defined before. The four unknown fluxes in Eqs (4.13)-(4.16) are = x~[ LI + L~- - 2 E~( r)] determined from the four equations, and then the radiative fluxes and the divergence of the radiative

O% flux vector are obtained readily. This method was +3 [ -JI -J~- -K~ -K~- +2(L + +L~-)]. (4.12) used to predict non-gray radiation heat transfer in an axisymmetric furnace and good accuracy was ob- These are the simplest forms of the flux equations tained) 72 Note that, although scattering in the and can be easily written for axisymmetric enclosures medium was neglected in deriving these expressions, as a four-flux approximation. The derivation of these it can be accounted for in the formulation. Also, expressions is based on the Schuster-Hamaker these equations can be modified to relax the method ~63'~64 which is the crudest and the least axisymmetry assumption to obtain a more general accurate flux approximation for one-dimensional formulation. systems. Whitacre and McCann~6S showed that the One of the oldest multi-flux methods is the six-flux four-flux version of this model t69 predicted the method of Chu and Churchill. 173 Although it was temperature field accurately, whereas the radiation developed for a one-dimensional, plane-paralld fluxes were usually underestimated in comparison to medium, it is possible to modify this method for Hottel's zonal method. A close examination of Eqs multidimensional enclosures. Varma ~4 obtained a (4.7) to (4.9) reveals that the fluxes for one direction four-flux model for axisymmetric cylindrical enclos- are not coupled with those of the other directions if ures starting from this six-flux method. However, the the medium is nonscattering. A similar type, un- comparisons with more accurate models show that coupled four-flux model was also developed by this version of the four flux method is not very Richter and Quack IT° and applied to a pulverized reliable. 17s Note that both the four- and six-flux coal-fired furnace. methods account for the scattering of radiation. In one-dimensional systems, the Schuster- Another six-flux model was proposed for three- Schwarzchild two-flux approximation or its modified dimensional enclosures containing absorbing and form ~@*'17~ yields more accurate results. Lowes et emitting gases. ~e6 A comparison of the predictions al) 72 extended this method to axisymmetric en- based on this model with the Monte Carlo results closures and derived an alternative four-flux model. showed that the maximum error in the radiation heat The corresponding equations can also be obtained flux was not more than 23 % and could be reduced to from the general relations by assuming axial sym- about 1% if the subdivisions of the solid angles were metry and defining the boundaries for the sub- adjusted according to the geometry of the furnace. divisions, t65 Then the governing equations* be- There are mainly three objections to the multi-flux come 17 2 approximations of the radiative transfer equation developed and used by some investigators for d + (JI - practical problems (see Smoot and Smith 3 and 2 drr [J~ - J~ ] ÷ 4 r 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 = -Ttxa(J~ +J~)+2xaEba(T ) (4.13) physically unrealistic. Second, the approximation of the intensity distribution from which the flux ~/~-~ d + ~/~ (J~ -J~ -r~ -K~-) equations are obtained is arbitrary. Third, the model 2 dr [Jz+J~-]4 4 r equations cannot approximate highly anisotropic scattering correctly, although it is theoretically possible. = -nxz(J~" + J~') (4.14)

x/~n d + x/~S-~n (J; - J~) 4.4.2. Moment methods 2 dzEK~ -K;]-~ 4 r In the moment methods, the radiation intensity is expressed as a series in products of angular and = -nxx(Kf +Kf)+2xaEba(T) (4.15) spatial functions:

d I(x0,,z,0,+) 2 dz[K~+Kf]=--nxa(K~-K;) (4.16) N =Ao+ ~ [~'A,.~+q"A,.,+II'A..j (4.17) n=l

*Note that these equations are modified slightly to follow where A's are functions of location only; ~,~/, and g a consistent nomenclature. are direction cosines in x, y, and z-directions, Radiation heat transfer 127 respectively [see Eq. (2.8)1. Although this equation is mation, is one of the most tedious and cumbersome written in Cartesian coordinates, it can be given for of the radiative transfer approximations; however, it any orthogonal system. As the upper limit of the may be the most elegant one because of its sound series N approaches infinity, this expression con- mathematical foundation. The method was originally verges to the exact solution for the radiation developed, as most other approximations, for study of intensity. Note that Eq. (4.17) can be considered as radiative transfer in the atmosphere, 17s later modified the Taylor series expansion of the intensity in terms for the solution of neutron transport problems, TM of direction cosines. and extensively used for one-dimensional radiative The simplest moment expression for the intensity transfer problems. 15-17Ag"2°'za2'179 Although the can be obtained by taking N = 1. This is called the formulation of the spherical harmonics approxi- first-order moment method. The AI.~, A~.y and A~,.. mation for multidimensional geometries was dis- coefficients can be obtained by integrating the cussed some time ago, m3t only during the last decade intensity over the entire space. DeMarco and has the method been extended to two- and three- Lockwood 176 have suggested some modifications of dimensional systems. For non-scattering Cartesian, the moment method using the flux definitions of the cylindrical and spherical media the first-order (P~) Schuster-Schwarzchild model, and defined the coef- and third-order (P3) spherical harmonics approxi- ficients as mations,~ ao.~ 81 for an isotropically scattering cylindrical medium the PI -approximation, 182.184 and for Ao=0 an isotropicaily scattering two-dimensional rectangular medium Pt- and P3-approximations ~ss A~ ..,.=(J; -J;)/2, have been formulated and solved. Meanwhile, the A~.,=(I<;-K;)/2, A~.:=(L~-L;)/2 (4.18) first-order spherical harmonics approximation has A2.x=(J~ + J~-)/2, also been formulated to study the effect of cuboidal A2.r=(K~ +K~-)/2, A2.:=(L~" +L~-)/2 clouds on radiative transfer in the atmosphere. 144't s6 Most recently, Menguc and Viskanta 61'187 reported where A's are implicit functions of the wavelengths of the general formulations of the PI- and P3- radiation 2, and J**, K,a ±, La + are integrated approximations for absorbing, emitting, and aniso- spectral radiation intensities over appropriate solid tropically scattering medium in two-dimensional, angles in the _+x, +y, +z-directions, respectively. finite cylindrical as well as three-dimensional rect- These equations were solved by dividing the total angular enclosures. solid angle 4n into six equal angles of 4n/6, each one In the spherical harmonics approximation, the having the coordinate directions as its symmetry radiation intensity is expressed by a series of axis. Another solution scheme was also adopted by spherical harmonics instead of a Taylor series and is choosing a magnitude of 2n for each solid angle. .76 written as t a2 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 t~(x,y,z,O,¢) = F~ A~Ax,y,z)r~(O,¢)(4.20) three-dimensional rectangular enclosure. ~76 A further n~O m = --n improvement of this method was recommended by with allowing some flexibility in the magnitude of solid angle corresponding to each direction. 177 For a r.~(O,O)= ( - 1 ~" ÷ I,.j~/2 medium with a minimum optical thickness (absorption coefficient-characteristic length product) of r2.+ 1 (.-Iml)!-I '/=.j.j, .. ,.,, x _ ,~..,./ r, ~cosvle (4.21) 2 this modified method yielded more accurate results L 4= ~,,+lml)!j in comparison to the earlier versions. In neither of these models ~76.177 was scattering of radiation in the where Y~ are the spherical harmonics, and P~ are the medium accounted for. It is interesting to note that if associated Legendre polynomials which are related the A-coefficients of this formulation are approxi- to the Legendre polynomials. mated as In Eq. (4.20} the upper limit N for the index n is known as the order of the approximation. Exact A2.x = A2..v= As,". (4.19) solution of the RTE is obtained if N is taken as infinity; however, for practical calculations a finite N then the first-order moment method will be obtained value is assigned. N=I results in P1- and N=3 (as ~2+r/2+/t2= 1), which is equivalent to the first- results in P3-approximations. Usually, the odd order spherical harmonics P~-approximation. 19 orders of spherical harmonics approximation are employed, although there are occasionally some others which use even order approximations, lsa The 4.4.3. Spherical harmonics approximation reason for using the odd-order approximation is The spherical harmonics (Ps) approximation, simply to avoid the mathematical singularity of the which is also known as the differential approxi- intensity at directions parallel to the boundaries. The 128 R. VISKANTAand M. P. MENGOt; radiation intensity is usually discontinuous at the radiation fields, but at the expense of additional interfaces; therefore, it is not possible to have a single computational effort. It is shown 61 that the accuracy value of intensity at the boundary. Consequently, it is of P3- as well as Pl-approximations can be sub- not desirable to have an angular grid point just on stantially improved by using "exact" boundary the interface. The roots of Legendre polynomials conditions, rather than somewhat arbitrarily defined used in spherical-harmonics approximation yield Mark's or Marshak's boundary conditions (see Refs Gaussian quadrature points, where the N-th order 19, 20, 131 and 132 for definitions and 61, 185 and polynomial gives the N-th order Gaussian quad- 187 for implementations of the Marshak's boundary rature scheme. If N is even, one of the quadrature conditions). points will have a value of zero, which corresponds It is also possible to improve the accuracy of the to an angular grid point on the boundary, whereas, if spherical harmonics approximation by obtaining the N is odd there will be no quadrature point on the moments of radiation intensity in half or quarter boundary. Therefore, an odd-order spherical har- spheres, xsg-19a Since the angular variation of monics approximation yields a more stable solution. moments is allowed for in this method, the aniso- The above discussion can be easily followed for a tropy of the radiation field can be modeled more plane-parallel geometry. accurately than by the Pt-approximation. On the The Pt-approximation is comprised of a single other hand, the governing equations are simpler than elliptic partial differential equation ~a7 those for the Pa-approximation.

V210,a = Aa[10,a-4nlh4(T)] (4.22) 4.4.4. Discrete-ordinate approximation where 10.4 is the spectral zeroth-order moment of A discrete-ordinate approximation to the radiative intensity I-irradiance cga, see Eq. (2.21b)1, lb4 is transfer equation is obtained, as the name suggests, Planck's blackbody function, and A4 is the coefficient by discretizing the entire solid angle (f2=4n) using a which is a function of single scattering albedo 094, finite number of ordinate directions and corres- extinction coefficient il4, and phase function para- ponding weight factors. The RTE is written for each metersJa and ga: ordinate and the integral terms are replaced by a quadrature summed over each ordinate. Originally 10.4 = S ladle, suggested by Chandrasekhar 15 for astrophysical n=4x problems, the discrete-ordinates method has been A 4 = 3fl](1 - 094)[ 1 - o~4(ja + 04 -J]94)'] • (4.23) extensively applied to the problems of neutron transport. 21`13a'lq'*A95 A simpler version of this In writing the above approximation, the delta- method, which is called SN-approximation, was Eddington phase function is employed [see Eq. obtained by dividing the spherical space into N (3.24)-]. In the P3-approximation, higher order equal solid angles, a96 However, more accurate form- moments of intensity, i.e. the integrals of radiation ulations were obtained later using Gaussian or intensity-direction cosine products over all direc- Lobatto quadratures. These are also called SN- tions within solid angle 4n are employed. Naturally, approximations to symbolize the discrete-ordinates the resulting equations are more complicated than approximation in which there are N discrete values those of the Pt-approximation. For axisymmetric of direction cosines ~., q., it., which always satisfy the cylindrical geometry, there are four second order identity ~.2 + q.2 +/~.2 = 1. elliptic partial differential equations for the P3- In one-dimensional plane-parallel media, the discrete approximation; 6~ whereas, for three-dimensional ordinates approximation has found many appli- rectangular enclosures six equations are needed, ls7 cations (see, e.g. Viskanta, a 34 Houf and Incropera, 197 These equations are solved simultaneously for the Khalil et al.t9s). Recently, the SN-approximation has second-order moments, and afterwards the other been applied to two-dimensional cylindrical and moments, radiation intensity, radiation heat fluxes rectangular radiative transfer problems with combus- and the divergence of radiation heat flux are tion chamber applications in mind, and reasonably calculated. accurate results were obtained in comparison to Although the P~-approximation is very accurate if exact solutions? 75,199 the optical dimension (i.e. the product of extinction The radiative transfer equation for an axisym- coefficient and characteristic length) of the medium is metric cylindrical enclosure is written for each large (i.e. greater than 2), it yields inaccurate results quadrature point n as for thinner media, especially near the boundaries. Also, if the radiation field is anisotropic, i.e. there are large temperature and/or particle concentration r Or r ~dp q-p,~-;+fl4la., gradients in the medium, the P~-approximation becomes less reliable. The P3-approximation, how- 0" 2 ever, can yield accurate results for an optical =xalb4+7~,w., ~.,.14,.. (4.24) dimension as small as 0.5 61.~a5 and for anisotropic °t/l: n" Radiation heat transfer 129 where w, is the weight of the Gaussian quadrature Here, we discuss only those which are applicable to points. Integrating Eq. (4.24) over an arbitrary combustion problems. control volume and rearranging yields, The basic flaw of the zonal method is the computational effort required to calculate the exchange factors between various volume and surface { ~.(ANI a,.,N -- Asla,.,s) + 14,(ArJ a..,v. - Awl~,~,w) elements in complex geometries. This difficulty can be overcome using the Monte Carlo method to 1 calculate the direct exchange areas. Is2 If the radi- - (As - As)-- (~. + 1/2I~.~ + 1/2,c Wn ative properties of the medium are known and do not depend on temperature, it is possible to calculate -~,-1/2I~., - 1/2,c)}/Vc 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 compu- 0",1 = - flalx.n.c + xxlba.c + 7- ~, Wnn' ~,m'lx,,',C (4.25) tational time required by the zonal method to predict t'l'Tt n' radiation heat transfer in complex geometries is decreased substantially. However, the computer stor- where A is the corresponding area of control-volume age requirements can become prohibitive if the side for N, S, E or W, i.e. for north, south, east, or number of zones is large. west side, respectively; V is the volume of the control The Monte Carlo method suffers from statistical volume, C is for the central node, and or-terms are to error as well as the extensive computational time preserve the conservation of intensity in the curved required for the calculations. If the direction of each coordinate, which are determined from the radiative ray is given deterministically rather than statistically equilibrium condition.~ 99 These governing equations and if all the directions constitute an orthogonal set, are solved numerically, for example, using a finite- then the solution would be less time-consuming and difference scheme. 175'199 A finite element solution the accuracy would increase with the increase in the scheme was also developed to solve the discrete number of directions. With this in mind, Lockwood ordinates approximation equations in two-dimensional and Shah 2°3'2°4 proposed a "'discrete transfer" Cartesian geometry for radiative transfer in the model which combines the virtues of the zonal, atmosphere. 2°° Monte Carlo, and discrete ordinates methods. They If the resulting equations of the discrete ordinates showed that very accurate results could be obtained approximation are carefully coded, they can result in with this method in one- and two-dimensional computer algorithms that combine minimum com- geometries by increasing the number of directions. puter memory requirements with few arithmetic Although this method is claimed to be capable of operations per space-angle grid point. 133 However, accounting for scattering in the medium, no results this approximation is not flawless, but suffers from have been reported or compared against other the so called "ray effects" which yield anomalies in benchmark methods for scattering media in multi- the scalar flux distributiori.TM .202 The ray effects are dimensional enclosures. The results for a one- especially pronounced if there are localized radiation dimensional scattering medium did not show the sources in the medium and scattering is less im- same level of agreement with the benchmark results portant in comparison to absorption. As the single as did the non-scattering medium predictions. 2°3 scattering albedo increases, the radiation field be- This method is also likely to yield erroneous results comes more isotropic and the ray effects become less due to the "ray effects" discussed in Subsection 4.4.4. noticeable. However, with increasing single scat- A similar approach to the solution of the RTE for tering albedo and/or optical thickness of the multidimensional enclosures has also been presented medium, the convergence rate may become very by Taniguchi et al. 2°s for absorbing-emitting media. slow. 133 Considering the flame in combustion cham- This so called "'radiant heat ray method" is based on bers as a localized radiation source, it is natural to the Beer-Lambert's or Bouguer's law and yields the anticipate the ray effects in the solution of the RTE radiant energy absorption distribution in noniso- in combustion chambers, if the discrete-ordinates thermal enclosures containing combustion gases. approximation is used. If the combustion chamber is Comparisons of the predictions based on this a pulverized-coal fired furnace in which there are method with other results show that the method is scattering particles present, the results are expected more accurate and less time consuming than both to be more reliable. 199 zonal and Monte-Carlo techniques if the radiative properties such as the absorption coefficient and wall 4.4.5. Hybrid and other methods emissivities are constant. 2os Another hybrid model based on the Monte Carlo Almost all methods discussed have some flaws. In method and generalized radiosity-irradiation ap- order to take advantage of the desirable features of proach has been suggested by Edwards. 2°6 This the different models, various hybrid radiative trans- method accounts for the volumetric scattering, yields fer models have been developed in the literature. accurate results for optical dimensions as small as JPBCS 13:2-e 130 R. VISKANTAand M. P. M~,~GOt;

0.5, and is computationally faster than the Monte accuracy and computational costs. In order to decide Carlo method. whether a model is appropriate for a given problem, The main reason why the discrete ordinate one has to compare its predictions against the approximation suffers from ray effects is because of benchmark results obtained from either experiments the inability of the low-order Sn-quadrature to or exact solutions. Zonal and Monte Carlo methods integrate accurately over the angular flux. 133 If are extensively used as the benchmark for compar- piecewise continuous approximations of the angular isons as they generally yield accurate predictions of flux are given in terms of directional variables, and radiation heat transfer. approximate spatial equations are obtained by In one-dimensional systems, comparisons of differ- integrating over appropriate solid angles, these ray ent radiation models have been effects can be avoided. The resulting expressions can given. 125']62-164"197']98 However, the accuracy of a be considered as hybrid models which combine method in predicting radiative transfer in a simple discrete ordinates or multiflux approximations with system may not always warrant its use in more the spherical-harmonics approximation. Indeed, the complicated systems. Therefore, it is important to double or quadruple spherical-harmonics approxi- evaluate radiative transfer models for multidimen- mations described in Subsection 4.4.3 can be con- sional geometries, preferably for practical situations. sidered 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 1.0 accuracy and reliability of the predictions as well as Finite Element to decrease the computational effort, t 32.133 0.8 ~L/H ,m o Zonal Flux models can also be coupled with the moment or spherical harmonics approximation to improve the accuracy of the radiation heat transfer pre- 0.6~ dictions. A model which combines the Pt-approximation with a two-flux method was proposed by O.4 Selcuk and Siddall 2°7 and applied to a two- ~o dimensional axisymmetric cylindrical furnace. The comparisons of the temperature and heat flux distri- O.2 i5 butions in the medium with those obtained with the zonal method showed very good agreement. Since O | I I ~ I I this model was developed for a gas-fired furnace, 0 0.2 0.4 0.6 0.8 1.0 scattering of radiation was not taken into account. DimensionlessPosition, x/L Another similar hybrid method was derived by FiG. 13. Dimensionless centerline temperature profiles in Harshvardhan et a/. 2°8 who combined the modified rectangular enclosures of different aspect ratio with black two-flux method]7~ with the P~-approximation. In walls; bottom wall at dimensionless temperature 0= 1.0, this method, the linearly anisotropic scattering other walls at 0=0. 2~ 2 medium assumption was made, and the method was used to predict radiative transfer through three- ~" ILlH .O.t dimensional cuboidal clouds. Comparisons of the predictions with the Monte Carlo results showed ..'-- reasonably good agreement. ,'7" 0.8 ~- .... ~"*"* 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 .o 0.6 de Hulst 2It). The predictions for the radiative flux "0 distribution in cuboidal clouds obtained by this ZlU 0.4 Finite Element method compare very well with those of the Monte ---P3 Carlo method. It should be noted, however, that the _~ o Zonal assumption of homogeneous and symmetric bound- .~_ ary conditions simolify these problems considerably. 0.2

4.5. Comparison of Methods Oo o'.2 a,I ols o8I ,.o! Although there are several radiative transfer models available, it is difficult to choose a "best" DimensionlessPosition, x,/L model for different applications. For a given physical FIG. 14. Dimensionless net radiation heat flux at the lower wall in rectangular enclosures of different aspect ratio with situation, one of the several models can be used black walls, bottom wall at dimensionless temperature according to the applicability of the model, desired 0= 1.0, other walls at 0=0. 2~2 Radiation heat transfer 131

2.8 accuracy of the surface net radiation heat flux • • • Zonal decreases with decreasing optical thickness. Similar "•"2.4 .~ -'-Ps conclusions have also been reported by different .o --- ~ researchers.6~'l s5.1 s7 2.0 In Figs 15 and 16, comparisons between zonal, ,5 spherical harmonics (P3), and discrete ordinates (SN) (J - 1.6 approximations are presented for a purely scattering o medium with different wall emissivities. 199 The P3- and S6-results for the centerline irradiance distri- .~_ 1.2 :'"i" ~ ...... bution are in very good agreement with the zonal method (see Fig. 15). The Ps-approximation, how- ,,, 0.8 e, ever, overestimates the radiation heat flux at the .=_o walls for large wall emissivities, although both S4- a4 and S6-approximations yield accurate results (Fig. E 16). 0 The lower-order spherical harmonics approxi- 0 o', & & ,o mations generally yield more accurate predictions if Dimensionless Position, y/H the radiation field in the medium is almost isotropic, FIG. 15. Comparison of irradiances in a two-dimensional which is the case if the optical thickness is large, square cross-section enclosure with a gray scattering medium (ic=0), Ehl = 1 and El,_,= EI,.~= El,4=0.199 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, • • • Zonal the Pi- and P3-approximations are to be used for 1.0 ~'-,,, --'-- Ps media having optical thicknesses of 1.0 and 0.5 •,~ S,, Ss or larger, respectively, ls°'lsSas7 The main reason for this inaccuracy for anisotropic radiation fields ~ 08 h is use of arbitrarily defined boundary conditions, like Marshak's condition) 9 In Fig. 17, the P3- :I: O.6 approximation results are compared against those of

o an exact model 139 for a cylindrical enclosure, 61 where both Marshak's (m) and "exact" analytical (a)

~ O.4 .,,.::.~---;--~---~ ...... boundary conditions are used. Here, it is assumed that there is a uniform, diffuse radiation source incident on one of the end surfaces of a cylindrical

(,0.1 0.8 0 I I I I 0 (11 0.2 03 0.4 05 Dimensionless Position, x//L O.6 FIG. 16. Comparison of radiation heat fluxes at a wall of a qr (r -- to)0.4 two-dimensional square cross-section gray enclosure with a .•-m Exact scattering medium (h=0), Eht = 1. Eh2 = E~,3= Et,4 =0.199 : a ...... PI 0.2 ~... ------PS N~. Unfortunately, this is not always possible because of 0 0 2 4 z/ro6_ 8 K) the analytical or numerical difficulties. The radiative equilibrium (heat transfer by radiation alone) assumption yields the most simple case la --/ m.-~ -,'--t ...... :," for solving the radiative transfer equation in two- or o.,I-...... N ...... :: / oo,o three-dimensional enclosures. In Figs 13 ad 14 the qz r- ...... qz (z=O) _ La_~-A_m"'~ ~ (Z=Zo) centerline dimensionless temperature and net radi- o u F----'~'_.-_-~-__-_..... ~__.=.__ OOOS ation heat flux distributions at one of the walls of a I rectangular enclosure containing a gray, absorbing- ozl l , ; ...... i ..... "'-'1o.oo o emitting medium are compared for three different o oz o.4 o.6 as ,.o ,/to methods. 212 The zonal and finite element methods are in good agreement with each other. The third FIG. 17. Comparison of P,- and P.,-approximation results with exact benchmark solution: 1",)=0.1 m. #=l.0m-', order spherical harmonics (Ps) approximation yields co=0.5, T~,.=O, 8,= 1.0 (m refers to Marshak's boundary good results for the distribution; however, the conditions, a refers to analytical boundary conditions). 6~ 132 R. VISKANTAand M. P. MENG0q

i I , i i 0 Meosured Values P,C Approximation

I I I I i I a I i IO 2.0 3,0 40 50 z (ml FIG. 18. Comparison of local radiation fluxes at the wall based on P.~-approximation results for a combustion chamber with experimental data and discrete ordinates method: r. =0.45 m, :, = 5.1 m. "~

enclosure containing a homogenous, absorbing and N 150 scattering medium. Radiative flux disribution on the cylinrical walls (upper panel) and on the end walls °°'°\ (lower panel) are plotted from the results obtained x I00 .o using different boundary condition models. 61 It is Lt. O.Im "l clear from the figure that the use of Marshak's ~ boundary condition yields substantially higher local ,,.~,li~d~ ~- '1 cleat • 2 qtay errors in radiation heat flux than the use of analytical conditions. This suggests that with a careful and I I I I I I more rigorous treatment of the boundary conditions, 0 I 2 3 4 5 6 even the very simple P~-approximation can be Furnoce Lenqth (m) employed to predict the radiation heat transfer in FIG. 19. Comparison of predicted radiation heat fluxes combustion chambers accurately. using the four flux model (lines) with different absorption There are also some accounts in the literature coefficient formuhltions,1 v2 Points are zonal method results. which compare different radiation models for practical systems, such as large scale furnaces where there is a gradual temperature variation along the cham- obtained very good agreement between the experi- ber.61.168,172.175.213 In these comparisons, radiation mental data and one-dimensional radiation models is decoupled from the energy equation, since the using the measured radiative property data in the temperature distribution as well as the radiative models. From these findings, one may conclude that properties of the medium are assumed to be given. In for complicated systems such as combustion cham- Fig. 18, the predicted radiation heat flux distribution bers, the accuracy of the radiative properties is as along the cylindrical walls of a furnace is shown. 61 important, or even more so, than the accuracy of the Both the $4- and P3-approximations are seen to be models. Additional sensitivity studies must be per- accurate for a given "uniform" absorption coefficient formed for combustion chambers to determine which value of 0.3 m -1 However, it is difficult (almost properties are the most important and under what impossible) to assign a single "gray" absorption conditions. For example, Menguc and Viskanta 2~4 coefficient for the entire furnace. When x is changed have shown that the index of refraction of coal from 0.3 m -~ to 0.35 m -~ (see Fig. 18) the P3- particles does not play a significant role in predicting approximation yields improved agreement between the radiation heat flux distributions along the the data and predictions. If the value of x were furnace walls. In their model the pulverized coal changed to 0.25, the agreement would have been particles were assumed to be only in the flame zone, poorer. The sensitivity of the results to the radiative and the predictions were obtained using the P3- properties was also shown by Lowes et aL 172 They approximation. On the other hand, Piccirelli et compared the predictions obtained from zonal and al. 215 presented a similar analysis using the PI- four-flux models for different absorption coefficients approximation for a one-dimensional cylindrical against the experimental data obtained for a gas- system and showed that the complex index of fired furnace. As seen from Fig. 19, the results are refraction was a very important parameter in more sensitive to the radiative properties than to the predicting the emissivity and absorptivity accurately. models. Indeed, in predicting the radiation flux These contradictory conclusions are not due to distribution in a large furnace, Selcuk et a/. 213 different solution techniques, but basically result Radiation heat transfer 133 from assumptions related to the radiation property External distributions in the medium. In the former 2~4 the Radiotion coal particles were assumed to be only in the flame zone, whereas in the latter 2t5 the coal particles filled Thin the entire combustion chamber. ~. VOlOtlle r~loua,~ Flame

5. APPLICATIONSTO SIMPLE COMBUSTION SYSTEMS To illustrate the coupling between radiation heat transfer, combustion and other transport processes \. '/r2 . ' " ."x/,Caol. we consider in this section several simple combustion Parttele situations in which radiative transfer has been accounted for. The emphasis in the discussion is on the effects of radiation heat transfer. Since the FIG. 20. Schematic diagram of a spherical translucent and physical situations considered are quite simple and radiating cloud model. 218 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. temperatures. The spherical transluscent and radiating cloud model is identical to the problem studied earlier. 219 This complex energy transfer situation for 5.1. Single-Droplet and Solid-Particle Combustion solid particle combustion is treated rigorously, and Burning of a single-droplet of liquid fuel or of a influencing parameters are identified. solid particle is a very simple system. Transport The model permits calculation of the steady-state process and not chemical kinetics dominate the heat transfer rate when the particle surface temper- combustion of fuel. This phenomenon has been ature, flame-sheet radius and temperature and other studied extensively for many years and experimental environmental conditions are given. The optical and theoretical accounts are available, s'2~6'2~7 The thickness has been found to be an important model theory of single-droplet combustion is complicated parameter in calculating radiative transfer rates. A by many factors, such as circulation in the droplet, fair amount of numerical computation was required finite-rate chemistry in the diffusion flame that to obtain solutions. The model has been shown to be surrounds the droplet, nonsteady accumulation of useful for the interpretation of muiticolor 22° and fuel between the surface and the flame, etc. In recent two-color TM pyrometry for more accurate experi- reviews of the theory these complications have been mental data reduction. Ultimately, a simplified discussed.S.217 Radiative transfer in single-droplet version of the model could be incorporated into a combustion has been ignored in most studies reported coal combustion model which explicitly includes in the literature. 8 Recently, a model has been particle heat-up and devolatization rates. However, developed to study coal particle behavior under for an application to a combustion system the model simultaneous devolatilization and combustion in must be extended to account for the interaction which transport of radiation in the volatile cloud between the burning particle and the surroundings (radiatively participating medium) surrounding a which contain radiating gases, clouds of particles coal particle has been accounted for. 218 and the system walls. The spherical volatile cloud, enclosed by a thin A systematic investigation of the effects of thermal flame sheet whose location is determined by diffusion- radiation on the combustion behavior of char limited combustion, is modeled as a radiatively particles exposed to an oxygen environment has been participating medium. The modeling concept is performed 222 using the general mathematical models similar to that of liquid droplet combustion except developed by Sotirchos and Amundson. 223'224 that volatiles emitted by the coal particle form a Pseudo-steady computations have shown that porous concentric luminous mantle (see Fig. 20) and radi- char particles reacting under radiative equilibrium ative transfer is important in addition to convective conditions are found to present considerably lower and conductive transport. The model describes the burning times (by more than one order of magnitude heat transfer mechanisms between the particle, the in some cases) than their heat-radiating counter- volatile cloud, the flame, and the external en- parts. 222 vironment. The analysis of combined conduction- radiation heat transfer in a concentric sphere filled 5.2. Contribution oJ Radiation to Flame with a radiatively participating medium first devel- Wall-Quenchin 9 of Condensed Fuels oped by Viskanta and Merriam 2~9 was used. The absorbing, emitting and scattering medium was The understanding of extinction phenomena has assumed to be confined between two gray, diffuse been greatly improved over the years, and recently isothermal spheres kept at different but uniform several important mechanisms of extinction or 134 R. VISKANTAand M. P. MENGOt~

quenching phenomena have been proposed. 225.226 function of various thermophysical and radiative Among those proposed are the stretching effect of the parameters such as the conduction-radiation ratio, combustion zone, preferential diffusion, buoyancy optical thickness and heat generation intensity by and heat losses. The effect of heat loss on quenching chemical reactions. A new dimensionless group, the can be more pronounced in the presence of relatively modified Damk6hler number (ratio of dimensionless cold boundaries, due to steep temperature gradients. heat source intensity to the conduction-radiation The pyrolizing surface of condensed fuels is a parameter), which characterizes the relative strength representative example of cold boundaries in a of heat generation to radiation transport, emerges combustion situation of practical interest. In this as from the analysis. The quenching-layer thickness is well as in many other studies on heat transfer in fires, determined primarily by the conduction effect near the significance of thermal radiation has become the relatively cold surface. However, thermal radi- increasingly recognized as radiation accounts for a ation is still the dominant mode of heat transfer significant portion of heat losses. Its effect has been there. Numerical calculations have shown that the shown to be considerable not only in large-scale and fraction of radiative heat flux at the fuel surface is small-scale fires 227'229 but also in small diesel over 85 ~o of the total heat flux. Optical thicknesses engines. 2a° less than 0.5 show little influence on the quenching Radiation blockage by soot layers between the distance, and more opaque systems yield shorter flame and the fuel is considered to be an important quenching distances. characteristic of fires. The radiation blockage effect Radiation blockage may also be desired in other has been investigated and found to depend on the physical situations to avoid excessive temperatures at type of fuel and size of fires. 227 Using experimentally the system boundaries. Siegel ~ls has systematically obtained data, it is shown that for polymer fuels studied a one-dimensional system at high temper- of polymethylmethacrylate (PMMA), polypropylene ature, with and without flow, to determine the (PP), and polyoxymethylene (POM) no significant governing parameters needed to keep the walls at a radiation blockage is present in moderate-scale fires; prescribed temperature range. Using an analytical however, for sootier fuels such as polystyrene (PS), approach, he concluded that a dimensionless para- radiation blockage has a considerable effect even in meter ME= TJ~CLxL/C2) and the ratio of suspension small-scale fires. In Fig. 21, the effects of gas layer temperature to source temperature, TraIT,, were the thickness and soot volume fraction on the blockage two important parameters. Here, fv is the soot of radiation are shown graphically. volume fraction, CL is the ratio of mean beam length The effect of thermal radiation and conduction on to layer thickness, x is the constant absorption the cold-wall flame-quenching distance in the com- coefficient, and C2 is Planck's second radiation bustion of condensed fuels has been studied using a constant. When M~2 the soot (or suspension) layer simple physical model. 227 In the analysis a steady- absorbs practically all the radiation incident on it, state, no-flow condition, one-dimensional energy and when M~0.2 half of the radiation is absorbed. equation for the optically thin quenching layer is Although at the beginning the soot layer blocks the solved employing the singular perturbation tech- radiation, the energy trapped in the layer raises its nique. The quenching distance is obtained as a 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 1.0 along the surface at frequent intervals)~S A similar approach can also be applied to combustion cham- /, bers, like the liners of gas-turbine engines, to predict 0.8 E~,- 0.9 the amount of film cooling necessary. It is worth noting the similarities between these results and 0.6 .... E.. o.gs~ those obtained by Lee et al. 227

5.3. E.JJect of Radiation oll One-Dimensional Char ~ 0.4 Flames fr ~ In one-dimensional pulverized-char or coal flames, two different flame types are recognized as "small" or °0., / '7/' / /_ "long". TM The "small" type flames can be modeled qualitatively using a conduction-diffusion approximation, whereas for the "'long" type flames, radiation 10-2 10"1 heat transfer is also an important mechanism. Earlier GAS LAYER THICKNESS (m) attempts to model these types of flames without FIG. 21. Radiation blockage as a function of gas layer including radiation have not been very successful; thickness for a plane flame layer model, Lj./Lo=0.4. 227 however, a model based primarily on radiation Radiation heat transfer 135 predicted the flame temperatures and burnout pro- the density of coal, and Q is the correction factor files very accurately. TM In this model it is assumed which is near 1.5 for practical flames. Equation (5.2) that reaction is controlled by combined diffusion and is applicable if a particle size distribution is given. It surface chemical reaction for either shrinking, con- can be used also if a single mean value for "r" can be stant density particles or constant diameter, de- defined. Two different mean values, one from the creasing density particles. Also, the size distribution Sauter-mean diameter definition (i.e. volume-to- of the spherical particles in the flame is accounted surface ratio, D32 ) and the other from the Rosin- for; however, the particle temperature is assumed to Rammler index were also used in the analysis. The be equal to that of the surrounding gases. This predictions for coal burn-out are compared against approximation can not be justified in physical experimental data in Fig. 22 for polysize as well as systems where particles burn in dilute suspensions two mean diameter models. The polysize model is in with a large excess of oxygen, yet it is a reasonable exceptionally good agreement with the data, sug- approximation for concentrated suspensions in prac- gesting that the accuracy of properties used in a tical flames (see comments of I. W. Smith to Xieu et radiation model are as critical or, maybe, even more al.231). The radiation heat flux was obtained by so than the accuracy of the model itself (see also modeling the RTE between two vertical infinite Section 4.5). parallel-plates, and the flux divergence is given by 23~ Another model, based on the one-dimensional Eddington (P1) approximation for predicting the c3q, contribution of radiation in "long" flames was given .... 4xtr T4( z ) + 2x[ a T41E2( z ) dz by Krezinski et al. 233 They obtained the radiative properties of coal particles using the complex t L refractive index data of Foster and Howarth; v5 +trT4.2E2('rl,--z) +tr S Ta(t)El(lz-tl) dt] (5.1) o however, they did not compare model predictions with experimental data. where x is the absorption coefficient for the mixture, T is the optical distance, tr is the Stefan-Boitzmann 5.4. Radiation in a Combusting Boundary Layer Along constant, El and E 2 are the first- and second-order a Vertical Wall exponential integral functions, respectively. The key parameter in this model is the absorption coefficient, Classical studies of boundary layer diffusion which, in general, is a function of location. The flames have neglected radiation, a.234.235 To isolate assumption of a constant value for 1( did not yield the effects of radiation in flames from the complex- accurate predictions for the temperature and burnout ities of fluid motion and to gain better understanding profiles. 232 Recognizing that the absorption coef- of radiation heat transfer in fires, analyses have been ficient is varying with the projected cross-section of made of a laminar, combusting boundary layer along the particles (see Section 3.2), Xieu et al. TM used a a vertical wall. 236-23s The rate of upward flame new expression spread over a vertical combustible surface is an important parameter in the ranking of the fire hazard ~, = 3QD,/4pr (5.2) offered by different materials. Typically, when the flame height reaches about 2 m, the flow becomes where D, is the dust cloud concentration for the size turbulent and radiation heat transfer starts to play an interval corresponding to mean particle radius r, p is 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 '°°1 has been reviewed 239-24~ and related experimental I i work has also been reported. 242 Here, we discuss the J[ / Rosin -Rommter monosize results of numerical solutions obtained for a steady, laminar, radiating, combusting, boundary layer over ca 60 "1 ~C a vertical pyrolizing fuel slab. An analysis has been developed for steady free and 40-1 ~ ~ % forced laminar combusting boundary layers in which I \ I'~ Polysize a pyrolysis zone separates the flame from the fuel surface 23s as shown in Fig. 23. The soot layer is on ~- I monosize (D~l~" ~-.....__--_.~ the fuel side of the flame zone. Through the transparent gas the combusting layer exchanges I I I I I I I 0 20 40 60 80 ioo J20 radiation with a distant black wall, which is Distance from tube bonk (cm) maintained at a specified temperature. The chemical FIG. 22. Comparison of prediction and experiment for coal energy lost to the system in the formation of soot is burn-out, using the given polysize and two alternative, neglected. The effects of radiation on the local fields, mort osi ze models. 231 and excess pyrolyzate escaping downstream at the 136 R. V1SKANTAand M. P. MENO0~

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. I ecu~o~v / Numerical solutions were reported for both forced / LAYER EDGE 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 FUEL / dominant role in determining the flame temperature / AME ZONE 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 / AMBIENT AIR 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 FIG. 23. Schematic of a steady, two-dimensional, laminar. polymethylmethacrylate (PMMA) in air. Values of a radiating combusting boundary layer flame with soot on a mean absorption coefficient and soot generation pyrolizing fuel slab.238 rates were also obtained using the analysis. This type of data could be used to quantify soot formation top of the fuel slab are examined by assuming that models. 23s the dominant effect of the soot particles is on the radiation heat transfer. The conservation equations for mass, momentum 5.5. Interaction of Convection-Radiation in a Laminar and species for a radiating-combusting boundary Diffusion Flame layer are identical to those for a nonradiating boundary layer s and therefore are not repeated here. Most of the earlier and even some recent studies The energy equation is given as 23s 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). { Oh t3h\ ~ {k --I t3h\-dq'+S p/u -- + v--1 = --/-- (5.3) 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 where the enthalpy is defined as flame structure. The relatively simple character of

7" diffusion flames in laminar stagnation-point flows I1= f cpdT. (5.4) has led to several theoretical and experimental T studies of that system in which thermal radiation has been incorporated in the analysis. 246- 249 Assuming a spectraily gray, homogeneous medium Interaction of convection and radiation on the with a constant absorption coefficient, the local temperature and species concentration distributions radiative flux in the y-direction* can be expressed in a diffusion flame located in the lower stagnation as 23s region of a porous horizontal cylinder 246 and a vertical flat plate 249 have been studied experiment- t ally and theoretically. The exponential wide-band gas q,=2[Eb~.E3(z)-E ~E3( 17~ -- T)--J'- SEb(t)Ez(z -- t)dt radiation model was employed in this inhomo- 0 geneous (nonuniform temperature and composition) t a -- I Eb(t)E2(t - z)dt]. (5.5) problem through the use of scaling techniques. Using a numerical scheme, the compressible energy, flow, and species-diffusion equations were solved simul- *There is an error in the expression for the radiation flux taneously with and without the radiative component. given in Ref. 238, but this does not affect the validity of the In the experiment, methane was blown uniformly results obtained because of the approximations. from the surface of the porous cylinder, setting up Radiation heat transfer 137

Z.O ! I | absorption coefficient it is possible to match experimental data with predictions; however, this approach

0 F • 2600 cc/mln is not based on first principles and requires data for 2.4 each set of conditions. I[xponentlol Wide-Bond Ido~l The experimentally measured convective heat fluxes --m-- Groy GaS Model K'p-0.15 at the wall of the cylinder were found to be in better 2.0 ~-~ No Rodiotlve Interoction agreement with the results calculated using the wide- Q I[xporimento! Run I band than the gray-gas model. 246 This further O l[aoerimental Run 2 supports the performance of the nongray model and I.G shows that the model is superior to those based on the gray-gas model as well as the results that ignore f,. I the effects of radiation. Measurements of convective ~t 1.2 and radiative heat fluxes in a diffusion flame surrounding a porous cylinder burning drops of n- 3 heptane have shown that radiation heat transfer to ~. 0.8 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

0.4 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 0 I ~- radiation, the use of a gray model would yield 0 I.O 2.0 3.0 4.0 5.0 reasonable results. II A similarity solution for an opposed laminar diffusion flame with radiation has also been ob- FIG. 24. Comparison of theoretically predicted and experimentally measured temperature protiles during methane tained. 2'.8 In this combustion system a stream of combustion around a horizontal porous cylindrical burner.2'.6 oxidizer approaches the stagnation point on a condensed surface and reacts with pyrolyzed fuel in a thin diffusion flame with a constant-thickness boun- (upon ignition) a diffusion flame within the free- dary layer. The fuel surface is assumed gray and convection boundary layer. Using a Mach-Zehnder diffuse, and the gas is considered gray. Only the interferometer and a gas chromatograph, temper- pyrolysis region is considered. Numerical results ature and composition measurements were obtained were obtained using the exponential kernel and along the stagnation line. Excellent agreement has optically-thin approximation for radiation heat been found between the temperature distributions transfer. based on the nongray wide-band model and experi- Analysis reveals eight dimensionless parameters mental data (Fig. 24). Examination of Fig. 24 reveals which control the system under investigation. Five that the wide-band model yielded results that were parameters, i.e. the mass consumption number, r; the superior to those that excluded radiation-convection mass transfer number, B; the Prandtl number, Pr; a interaction. It is evident from the figure that dimensionless heat of combustion, D o the fuel radiation-convection interaction lowers the pre- surface temperature, 0,., are the combustion groups, dicted temperatures in the high-temperature region and the three radiation groups, the conduction/ of the boundary layer near the flame front and raises gaseous radiation parameter, N~, the conduction/ the temperature in the cooler region near the edge of ambient radiation parameter, N2, and the fuel the boundary layer. Furthermore, this interaction surface emissivity, e,., are required to describe the effect increases for larger fuel flow rates. 246 The effect combusting-radiating system. The parameters D c of radiation interaction can be interpreted to result and 0w, which were of secondary importance in from a transfer of energy by gaseous radiation heat nonradiating systems, emerge from the analysis with transfer from the hotter region to the cooler portion new significance, dominating the parameters N~, N 2 of the boundary layer, thus reducing the higher and ew. temperatures and raising the lower ones. The effect of radiation on the pyrolysis rate and An attempt was made to determine an arbitrary unburned fraction of total pyrolyzate is shown in value of the Planck mean absorption coefficient ~ Fig. 25 for axisymmetric combustion. 248 The pyroly- which when used in a gray-gas model would yield sis rate is seen to increase strongly with increasing temperature profiles matching the experimental dimensionless heat of combustion, D c. This is results. 246 Although the results of Fig. 24 demon- primarily because of an increase in flame temper- strate that matching values of ~e may indeed exist, a ature (01 ~ O,.Dc). Like the mass fraction of fuel at the different value of a mean absorption coefficient was surface, the dimensionless flame temperature 0y, required for each fuel-flow condition. It has been which for non-radiating flows can be determined a shown in Section 4.5 that by changing the mean priori from measurable quantities, depends on all 138 R. VISKANTAand M. P. MENGOq

I.o .... i.o

------Non.r(idiolinRodiotin9 9 t

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, N2 = 50.0, and e= 1.0. 248 eight parameters and is not predetermined. The butions of soot volume fractions and CO 2 and H20 pyrolysis rate with radiation is lower due to the net concentrations. The utility of the analysis will then efflux of radiation at the surface. In general, the come from both the proper quantification of radi- influx of gaseous radiation is insufficient to cancel ative effects in opposed-flow diffusion flame experi- the efflux of surface emission, hence a lower pyrolysis ments and from the use of such systems to refine rate results in comparison to non radiative combus- techniques for incorporating radiation in combustion. Lower pyrolysis rates may result even when a tion modeling. net influx of radiation prevails because of the decrease in conduction caused by the lower flame 5.6. EJJect oJ Radiation on a Planar, Two-Dimensional temperature due to radiant loss from the combustion Turbulent-Jet DiJJusion Flame zone. At low D c, the reaction releases little energy to counter surface emission losses, giving low pyrolysis A simple combustion situation has been modeled rates, whereas at large Dc much energy is released to assess the importance of thermal radiation in which easily overcomes surface losses and yields establishing temperature distribution in a turbulent large pyrolysis rates. 2as The net effect of radiation on diffusion flame. 25° Although, turbulent diffusion pyrolysis appears to be low for several reasons. The flames have been extensively studied by Bilger and properties of real opposed diffusion flames are not his co-workers) 51- 25 3 they have not considered the yet sufficiently well known to give accurate para- effects of radiation. However, radiation heat transfer meter values. Those chosen for the calculations (see modifies the temperature distribution which, in turn, caption of Fig. 25 and others in Ref. 248) may not be affects the combustion process. Small changes in sufficiently realistic as they all tend to underestimate peak temperatures have a large influence upon nitric the differences between non-radiating and radiating oxide production for a given residence time. It is of systems. interest to determine how the various control As data on radiative properties of stagnation- strategies such as lowering combustion air preheat or point flames become available, the approximation of recirculating exhaust products into the combustion a constant absorption coefficient should be replaced air affect the unwanted nitric oxide emissions and with a nonuniform one based on measured distri- 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. Radiation heat transfer 139

The physical model of the problem analyzed by G4 James and Edwards 25° is shown schematically in °,,, .t .,o ..o Fig. 26. A planar jet of methane is injected with .... .sA. peso. H ,1T/'~ velocity ut,~,l into a stream of air flowing with -.- s~e,. ,~o. I] :W ',.~ velocity u,~r parallel to the fuel. Diffusion-controlled I o.2 ! ,..,ooo iYl "\ combustion occurs in the mixing region of the jet. "s "L ,1 v ",~ Plane-parallel, isothermal and black walls sym- metrically located above and below the jet form the ~ 0.0 _sJ ~.J.J I%~J I "'-'~- .... combustion chamber. A soot-free flame is assumed to I I exist so that molecular gas bands determine the thermal radiative transfer to the walls. Boundary- ,z, ~4i _ x/O - 50 layer approximations were used to simplify the - ~1~\ - conservation equations, and nongray radiation described by the exponential wide band model for ,q 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 0.0 1.0 2.0 3.0 4~ radiative transfer calculations can be coupled to an implicit numerical method for solution of the highly WAVELENGTH (pm) nonlinear partial differential conservation equations FIG. 27. Spectral radiation intensities ~r radial paths without undue expenditure of computation time. The through aturbulenthydrogen:airdiffusion flame. 25s results of computations have shown that the larger channels (A= 1 m and 10 m) have markedly lower peak temperatures because of greater gaseous radi- Estimates of spectral intensities emerging from ative transfer. It was also found that a given flames, based on predicted mean scalar properties, reduction in minimum combustion temperature to are typically within 20-30 % of the measurements of reduce nitric oxide formation could be accomplished well-defined laboratory flames. 31,25'*.255 This is com- with a much less detrimental reduction of heat parable to the uncertainties in the narrow-band and transfer by recirculating exhaust product into the flame-structure models. Measured and predicted combustion air than by reducing preheat. spectral radiation intensities for a turbulent hydrogen/ air diffusion flame are given in Fig. 27. Results are 5.7. Radiation Ji'om Flames shown for horizontal radial paths through the flame at x/D=50 and 90, the latter position being just Gas- and liquid-fueled flames have numerous below the flame tip. Predictions use both time- applications and flame radiation is an important averaged scalar properties along the path and aspect of heat transfer in furnaces, internal combus- stochastic methods which take into account turb- tion engines, aircraft propulsion systems, flares, ulence/radiation interactions. The stochastic method unwanted fires, etc. This has motivated many studies models the interactions by assuming that the flow of flame radiation and comprehensive, up-to-date field consists of many eddies which are uniform and reviews are available.'*a'239.25'*'255 The issues of statistically independent of each other. Eddy length concern here are nonluminous and luminous radi- varies along the path length, and time-averaged ation from flames, prediction of radiation character- probability density (PDF) of mixture fraction f for istics given the instantaneous scalar structure, and each eddy is randomly sampled and scalar properties turbulence/radiation interactions in simple laboratory are found from the state relationships at the flames. corresponding value of J~ Once the scalar properties Significant progress has been made concerning are known, the RTE is solved. The details of solution structure and prediction of radiation intensity of can be found elsewhere. 31 Spectral radiation inten- nonluminous flames. Narrow band-model predic- sities (Fig. 27) are dominated by the 1.38, 1.87 and tions TM for nonisothermal mixtures of CO2, H20 2.7/~m water vapor bands in the range of 1-4/~m and CO are in good agreement with the measure- shown. The stochastic method yields spectral inten- ments. The total transmittance nonhomogeneous sities which sometimes are about a factor of two model (TTNH) of Grosshandler 256.257 has been higher than the mean property method with the found to be about 500 times faster than narrow-band measurement generally falling between the two models. The model has been applied to several predictions. These results suggest significant effects realistic combustion examples containing variable of turbulence/radiation interactions. Findings for concentrations of CO2, H20, CH4, CO and soot. It carbon monoxide/air and methane/air flames, how- was found to be usually within 10% of the more ever, show smaller effects for turbulence/radiation accurate computation. 2s 7 interactions. 254.255 140 R. VISKANTAand M. P. M~,~GO~

Work on luminous flames has been limited. results have been obtained for other laboratory Similar results to those presented in Fig. 27 have diffusion flames. Excellent agreement has been ob- been reported by Gore and Faeth (cited in Refs 254 tained between measured and predicted radiative and 255) for a turbulent ethylene/air diffusion flame. heat flux distributiotrs parallel to the axi's of The spectra are dominated by continuum radiation turbulent carbon monoxide/air diffusion flames. 259 from soot, however, the effects of 1.38, 1.87 and The mean property predictions agree very well with 2.7/~m gas bands of the H20 and the 2.7 and 4.3 #m the measurements because the effects of turbulence gas bands of CO2 can still be seen. In this case the radiation interaction are small. The analysis correctly mean-property method has provided the best quan- predicts maximum heat fluxes near the flame tip as titative agreement with the data, but the agreement is well as the effects of burner flow rate. considered to be fortuitous in view of poorer Discussion of the effects of turbulence/radiation extinction predictions obtained using the ap- interactions has been given by Faeth et aL 31'255 The proach. 255 The predictions of continuum radiation available results show that the interactions are very are very sensitive to local temperature estimates, and significant for hydrogen/air diffusion flames, with the assumption optically-thin radiative heat losses stochastic predictions being as much as twice the are quite crude. Differences between mean property mean property predictions. 2sa In contrast, turbulence/ and stochastic predictions suggest significant effects radiation interactions caused less than a 30~o of turbulence/radiation interactions in luminous increase in spectral radiation intensities for carbon flames. More exact coupled structure and radiation monoxide/air and methane/air diffusion flames. This analysis could modify the relative performance of the difference is attributed to the relatively rapid vari- mean-property and stochastic methods and suggest ation of radiation parameters (water vapor concen- that presently available models must be improved. tration and temperature) near stoichiometric con- Measurements and predictions of total radiative ditions for hydrogen/air diffusion flames. The heat fluxes to points surrounding the turbulent stochastic methods at.254~255 have many ad hoc hydrogen/air, 258 carbon monoxide/air, 259 methane/ features and additional fundamental research effort air, 26° and ethylene/air TM diffusion flames have been is needed to develop more reliable methods not only made. The discrepancies between the measured and for small laboratory flames but also for scaling large predicted total radiation heat fluxes along the axis of flames containing soot. 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 5.8. Combustion and Radiation Heat TransJer in a measurement for the spectral intensities. Comparable Porous Medium Although flame radiation plays an important role in combustion systems, a furnace requires a sufficient 1.0 I I I I 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 0.8 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 E appreciably. Echigo 262,26a has shown that a porous 0.6 medium of an appropriate optical thickness placed in a duct is very effective in converting enthaipy of a X :3 flowing gas stream to thermal radiation directed .-I t,I. toward the higher temperature side. Successful appli- 0.4 cations to an industrial furnace, 262'263 to a combus- m.m tor of low calorific gas, 264 to a water tube by

W combustion gases in porous media, 265 and to other V- systems 26a have been reported. The thermal structure in the porous medium with internal heat generation r,< O.2 due to chemical reactions has been studied analytically and experimentally,265'2~6 and a review of the work is available. 26a'26~ O.0t- A one-dimensional model in which radiative 0 800 1600 2400 transfer in a gas-solid two-phase system is treated AXIAL DISTANCE (ram) rigorously has been constructed, and extensive FIG. 28. Total radiative heat flux distribution along the axis numerical calculations have been performed for a of turbulent methane air diffusion flames at NTP. -''° radiation controlled flame. 267,26a The combustion Radiation heat transfer 141

I f I I I I Significant energy recovery has been achieved from 1200 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 800 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 • .... 1538 W combustion system are used as the abscissa. The 4O0 o .... 1025 W "~ ~'-- 513W results show that as the combustion load increases, both the measured and calculated temperatures increase uniformly. This is a consequence not only of O I I I I I I -IO 0 1(3 20 30 40 50 the relative reduction of heat loss in comparison to I x {mm} heat generation during combustion but also due to ~I P.-t aM-= I~ PM-n the essential nature of radiation heat transfer. -6o ,35 ,,.35 r

FIG. 29. Comparison of measured and predicted temper- 6. APPLICATIONS TO COMBUSTION SYSTEMS ature structures in porous media for different combustion loads. 2~8 The lower abscissa scale r is optical depth based The advent of more powerful digital computers on the ~bsorption coefficient of the porous medium, and has provided the means whereby mathematical PM-I, PM-I1 and PM-III are the abbreviations for porous media I, I! ~md II!, respectively. 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 mixture flows through a porous medium and the current demands which system designers are required combustion reactions take place in the medium. The to meet--in particular, efficiency of combustion at a results of comprehensive calculations show that the wide range of operating conditions and strict control thermal structure (profiles of temperature, local of pollutant emissions. The latter has become radiation flux, etc.) in the high porosity medium increasingly stringent in recent years for economic depends strongly on the absorption coefficient and and political reasons. The present trend is away from total optical thickness of the medium as well as the the traditional cut-and-try methods, which are ex- position of the reaction zone. Good agreement pensive and do not necessarily produce the optimum between predicted and measured temperature distri- design, toward fundamental modeling of the physical butions has been obtained and a drastic temperature and chemical processes occurring within the combus- decrease in the porous medium has been re- tion systems. Multidimensional modeling of two- vealed. 266"26s The results have also revealed remark- phase combustion is being approached with the aim able heat transfer and combustion augmentation. of producing algorithms based on fundamental

I Two ° Phase Fluid Mechanics (Turbulent) Phase Chemical Transitions Kinetics (Evaporation) (Condensation)

Gas Nucleation Particle Interaction

Par title Phase Reaction Gas Phase Devolatilizofion) Reaction eter. Oxidation)

Heat Transfer (Convective) (Radiative)

FIG. 30. Schematic representation of submodels for combustion of coal. 142 R. VISKANTAand M. P. MENGi3t~ principles which can correlate all of the details of radiation model, through the radiative properties of combustion systems. 3,269-272 The predictive pro- combustion products, to the mathematical transport cedures for a combustion system model require model to predict the temperature and radiating theoretical and empirical inputs to describe turbulent species concentration distributions. With the pres- flow, chemical kinetics, thermodynamic and thermo- ently available algorithms, 3"5'269 -271 the latter type physical properties and other transport processes, problems require an iterative solution procedure including radiation heat transfer (see Fig. 30). which is rather time-consuming. This section of the article discusses application of A validated computer model has been used to the methodology described in the previous sections construct a detailed energy flow (Sankey) diagram for to practical combustion systems. The emphasis is on an industrial furnace. 273 The diagram (Fig. 31) shows the methodology and radiation heat transfer results that more than half of the heat to the load comes rather than the application of mathematical tech- from the refractory wall. Of the balance, part is niques for design and performance calculations of convection (4~), part is direct radiation from the practical systems. Even with the advances in main- flame (6 ~), and part is flame/wall radiation absorbed frame computers the difficulty of treating infrared by the gas which has been re-radiated by the wall radiation transfer rigorously in nonhomogeneous (6 9/o). The furnace shows a thermal efficiency of 35 ~o gases containing particles lies primarily in the with the typical high flue loss and indicates the enormous complications introduced by selective importance of the wall-to-wall re-radiation effect. gaseous emission and absorption of radiation as well With the exception of different magnitudes, Fig. 31 as scattering by irregular-shaped particles. Because of shows a typical pattern for all industrial natural gas this complexity practical simplifications are necess- and oil fired furnaces. As the flame becomes more ary to keep the calculations at a reasonable level. As opaque and/or the wall temperatures drop there will a compromise between desired accuracy and compu- be obviously more radiation from the flame and less tational effort, practical methods which are also from the wall. Also, as the wall temperatures drop compatible with the numerical algorithms for solving there will be smaller radiation exchange between the the transport equations are stressed, and radiation walls and the load. heat transfer in several different combustion systems The close examination of Fig. 31 clearly indicates is discussed. The body of literature concerned with why there has been so little attention given to the modeling and evaluation of combstion systems is calculation of convective heat transfer inside fur- very large, and it is not practical in an article of naces. In industrial furnaces convective heat transfer limited scope to discuss even the more recent works. usually accounts for a very small fraction of the total Most of the work reported has stressed modeling and heat transfer to the load. Local convective heat evaluation of chemically reacting turbulent flows and transfer coefficients have been measured at a surface combustion and much less radiation heat transfer. heated by gases 274 and empirical correlations for the The emphasis in this review is on the latter. average Nusselt number have been reported for differently-directed gas streams incident on the load. 274-276 An interesting finding of the experi- 6.1. Industrial Furnaces mental study 274 was that in the absence of combus- One of the important parameters in assessing the tion the average heat transfer coefficient at the load performance of an industrial furnace is the heat flux surface was about 35 W/m2K, while in the presence distribution to its thermal load (sink). Methods of combustion the values were from 80 to 120 W/m2K, based on fundamental principles are now available suggesting almost a threefold enhancement of con- using numerical techniques and digital computers, vective heat transfer by combustion. that permit determinations to be made for both gas- Radiation in furnaces predominates over con- and oil-fired industrial furnaces. In such furnaces vection; therefore, more emphasis has been given to radiation over the years and the radiative transfer heat transfer to the load is predominantly by thermal 4-5 277 281 radiation. The problems associated with prediction theory has been much more fully developed, • - of radiation heat transfer within the combustion and presently capability exists to predict simultaneous chamber can be divided in two main types: three-dimensional flow, heat transfer and reaction rates inside furnaces. 269'27! However, the theory has (a) Evaluation of radiation heat transfer at all outstripped experimental validation, which is in a locations in the enclosure if the temperature much more primitive state, but even in this area a distribution and radiative properties of the number of papers describing direct comparisons combustion products are known; and between predictions and experimental data have (b) Evaluation of radiation heat transfer as well as appeared, t 69,282 - 288 temperature and radiating species concentration The results obtained for a model furnace using distributions. the phenomenological furnace-performance equations Problems of type (a) are more straight-forward and have been used to determine the relative importance require development of radiation heat transfer models~ of the model parameters. 2s° Analysis of the results Problems of type (b) require the coupling of the led to the conclusion that the flame emissivity was of Radiation heat transfer 143

~/Au°T//~~~~ CONVECTION TOI..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 rather specific conditions; and some aspects of the analysis could be considered arguable. '°' ,--t- 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 ~Heot Flux Oistri~ total heat transfer rate without providing infor- Type 2: Plug Flow mation 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 Type 3: Two - Dimensional are discussed here in greater detail.

6.1.1. Stirred vessel model

',x~- .~:- F-~T-j---:t:_F-_ a..i 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

Heo! Flux Oistrib. are assumed to be gray and at a uniform temperature. HG. 32. Schematic representation of it furnace illustrating The temperature and the radiative properties of the different heat transfer models. 2as load and of the refractory walls are assumed to be uniform but different. A steady-state, overall energy second-order importance. The other factors (in the balance on the load can be written as order of decreased importance): heat transfer to the load (sink), excess air, process temperature, flame/load /~/1 -/~/2 = Q, + Qe (6.1) temperature difference, load absorptivity and wall losses, were of greater significance. These conclusions Here/~/~ and/:/2 are the ¢nthalpy inflow and outflow were reached, however, by generalizing from some rates. Within the framework of the zonal approxi- 144 R. VISKANTAand M. P. ME~Gi~t~

,0,, Qs F s -- Y,,'/,// ,/////'///////// A~_,,a T~ Waste Gases ==~. ~aml~stloa = [0~- O] + (St'KoXOm- 0,)]. (6.5) Products ~,.r=.~ o., T, Fuel fit AIr For the special case when convective heat transfer to /~> ~o. e/r,,A ' the load is negligible in comparison to radiation //I Lo,, (St=0), Eq. (6.4) simplifies to / 7111111111111111~ Ko(1 - 0m) = 0~-- 0]. (6.6) FIG. 33. Schematic diagram of a stirred furnace model. By eliminating the mean combustion-product temperature, the dimensionless heat transfer rate can be expressed as mation for radiation heat exchange, the heat transfer rate to the load can be expressed as 277'2al r~= Ko[1 -( F, + 0 ~)z/'*]. (6.7)

O~--- A, [h(T~ - T~)+ ,fr _ ,,a(T~ - T])] (6.2) Extensive calculations have been reported for the dimensionless mean gas temperature and heat trans- where h is the average convective heat transfer fer rate and the results can be found in the coefficient at the load, and "~s-m is Hottel's radiation literature. 2aS-za7 Experiments have also been per- exchange factor or A~-~_m is the total radiation formed and compared with model predictions. 2s6'2a7 exchange area. This factor is a rather complicated Figure 34 shows a comparison between the measured function of the gas emissivity, wall emissivity and the and the calculated average heat fluxes in an experi- sink-to-refractory area ratio, and expressions are mental combustion chamber having a 1.25 m long available in the literature. 45'287 The heat losses firing space and two different cross-sections (0.4 m x through the walls of the furnace can be expressed as 0.4 m and 0.4 m x 0.8 m). The results show that the stirred-vessel heat transfer model can be successfully Qt = UoAo(Tm - T,) (6.3) applied to those furnaces in which there is no appreciable axial drop of the mean gas temperature. where U0 and Ao are the overall heat transfer This condition is roughly met in combustion cham- coefficient and area of the refractory walls, respec- bers fired with high-velocity burners and in furnaces tively; and T,, and T, are the mean combustion where the flame length is approximately equal to the products and ambient air temperatures, respectively. furnace length. Under these conditions, a maximum Substitution of Eqs (6.2) and (6.3) into Eq. (6.1), and error of _+20% can be expected in calculating the assumption of negligible wall heat losses allows the absorbed heat flow to the load being heated. In resultant equation to be written as predicting the energy consumption of the furnace, this would mean a maximum error of _+ 10 ~oo.286 +- 0~-0,,, =(1/Ko)(0~-0~) (6.4) l + St 1~ l + St 100 F~naee Crog-Seellon In mmZ= LOO,t.O0 ~Or,9~ where the dimensionless variables and parameters I¢o, 06 • V are defined as

hA~ "1- "~ d" k St= :~. rnC pra

In this equation, T~ is a fictitious gas inlet temperature in which the heat losses through the walls of the furnace have been accounted for; m and q,,. are the 0 gas mass flow rate and the mean specific heat of the 0 200 too 600 oCo iOoo 1200 'v,OO gas, respectively, and Ko and St are the Konakov and Surface Temperature (~) Stanton numbers, respectively. The heat transfer rate FIG. 34. Comparison of measured and predicted average to the sink, Eq. (6.2), can be expressed in dimension- heat fluxes in a furnace as a function of the load less form as temperature, z8~ Radiation heat transfer 145

6.1.2. Plug flow model where A schematic diagram of the "plug-flow" model is x T~ 0_7"; 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 Ko= DiCpm h WL St ..... an infinite number of stirred vessels. The temperature WLi,.fl T 3 ~ncp~ and the radiative properties of the load and walls are assumed constant but different. Based on a gray-gas UoPoL dPo - and zonal approximation for radiation heat exchange, thcmn the steady-state energy balance on a control volume of gas of length dx gives Analytical solutions of Eq. (6.9) and its special forms have been obtained and graphical results reported.29°'291 dT~(x) . , , thepm dx - - W{e,a[T,(x)-- T,] Extensive numerical calculations of the gas temperatures along the furnace using the stirred-vessel, +h[T~j(x)- T~] }-PoUo[T~(x)- Tj (6.8) stirred-vessel-cascade, plug flow and the modified zonal models have been reported for furnaces having constant and varying sink temperatures. 292'293 A where W and P0 are the furnace width and perimeter, comparison of temperature distributions using five respectively, and lg is the effective gas emissivity zones (sections) along the furnace is given in Fig. 36. which accounts for the refractory walls and other The published results show that as the number of surfaces in the furnace. In dimensionless form, the sections in the furnace increases, the temperature energy equation for the gas temperature can be distribution predicted employing the stirred-vessel- written as 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 dO._ (1/KoXO4_O~)_St(O_O~) stirred-vessel and the modified zonal models predict d¢ practically identical gas temperatures in the furnace.

-- ~0(0.--0~) (6.9) 6.1.3. Multi-dimensional models ~ dQ,, The computational methods which have been developed are able to complement, but not replace, empirically based design procedures. This is because H Products IldO~ H U I'T! chemically reacting turbulent flows are not fully understood, and it is proving particularly difficult to eliminate the deficiencies of existing turbulence ...... -.:'H models. In the absence of reliable turbulence models it is hardly possible to subject any of the ever- FIG. 35. Schematic diagram of a plug flow model. increasing number of combustion and radiative

1800 i i i \\ ' ' {o)' ' \ ,~. Plug Flow %--Plug Flow 1600 \ \ ca 2x_~~ .,. Stirred Vessel 1400 , Sfitre¢1%% ;!.,... / :aecade Vessel ~ ~. E \ 1200 Zonol ~ Stirred Vessel O Zonal it\x) 014 i i 1 i i o 02i 0.6 08 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

transfer model proposals to a stringent assessment. 250 i , i , , i , Nevertheless, two-dimensionaP ,294.295 (among others) ------/TIi" I000 K and three-dimensional 269.271.296 combustion valid- 2O0 ation studies reveal, for gaseous combustion at least, that predictions which are obtainable are sufficiently 150 reliable to be of interest to combustion engineers. General computer-based procedures for the prediction of gaseous-fired rectangular and cylindrical 5O combustion chambers have been developed and a review is available: The zonal, flux, discrete- 0 Tuth/RQa. Inlw. ordinates and first-order spherical harmonics (P: .••'•WilhWillloul Turb./Ro4. Intlr. approximation) methods have been assessed. For -50 I I I I I I I I I natural gas and oil fired furnaces only three species 0 0.2 04. 0.6 0.8 1.0 (CO2, HzO and soot) contribute significantly to the x/L transport of radiation in the infrared. The compu- FIG. 37. Effect of preheated air fuel mixture temperature tations have been carried out on the gray or at most and turbulence, radiation interaction on heat flux distri- on a weighed sum-of-the-gray gas bases. Reasonable bution along a two-dimensional furnace burning methane: H= 1 m. L=5 m, ~= 1500 K. e~=0.8, e, =0.6. 3° 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 equations constituting subsidiary models of turbu- the radiative transfer models have been determined. lence, chemical reaction and radiation heat transfer It is suspected that the authors had to do consider- phenomena. A six-flux, gray gas model is used to able "fine-tuning" of these model parameters to bring predict radiative transfer. Computer-memory limit- about good agreement between model predictions ations restrict the amount of geometrical detail that and data. The sensitivity of the results to radiative can be included and prevent the use of a finite- properties have already been discussed in Section 4.5. difference grid having the desired fineness. The model It should be pointed out that in the studies is validated against experimental data acquired on discussed by Khalil 5 and others 269 the emphasis has two large, natural gas-fired furnaces. been on modeling chemically reacting turbulent flow Recently, the effect of turbulence/radiation inter- and combustion and much less on realistic modeling action in a two-dimensional, natural gas-fired, indus- of radiation heat transfer. The general prediction trial furnace has been examined. 3° Based on an procedures which describe the computation of flow, approximate analysis of radiative transfer, the results reaction, and heat transfer in the combustion region of calculations show that the effect of turbulence/ of a typical, natural gas-fired industrial glass pro- radiation interaction on combustion and scalar ducing furnace are sufficiently developed to consti- properties is small for a preheated fuel-air mixture tute a useful design tool. 269 Economic handling of when the flame occupies a small volume of the three-dimensional geometric features is considerably furnace. However, when the flame occupies a large enhanced by the use of special grids and the separate volume fraction of the combustion chamber the calculation of the burner and bulk combustion interaction is quite significant. Another reason why chamber regions in a manner which takes into the effect of the interaction is larger for T;= 300 K account the differing features of their flows. The than for T~=1000 K is because the temperature predictions demonstrate the value of computations fluctuations are larger for the former case. The effect to furnace designers for the range of operating of the interaction on the total heat flux along the parameters. Recently, the radiative transfer has been furnace shown in Fig. 37 clearly indicates the need to treated in sufficient detail using the discrete transfer account for turbulence when predicting radiation method which contains some features of the zone, heat transfer in large, high-temperature combustion discrete ordinates and Monte Carlo procedures. 2°4 systems. The net local heat flux to the sink (load) can The combustion products are treated as gray and become negative for the case when the turbulence/ scattering by particles, such as soot agglomerates has radiation interaction is neglected, because the as- been neglected. sumed sink temperature (Ts= 1500 K) is higher than A detailed discussion of analytical modeling of the local effective temperature of the combustion practical combustion chambers and furnaces, in- products. cluding a very extensive review of the literature has recently been given by Robinson. 271 A three- 6.2. Coal-Fired Furnaces dimensional mathematical model is constructed of a large tangentially-fired furnace of the type used in Radiation heat transfer in coal-fired furnaces has power-station boilers. The model is based on a set of received considerable attention for more than 60 yr 13 differential equations governing the transport of because of the realization that it is the dominant mass,momentum and energy, together with additional mode of heat transfer in such systems. The earlier Radiation heat transfer 147 work on the subject has been discussed by Doleza1297 matter deposited onto surfaces of coal-fired furnaces and more recent studies have been reviewed by can greatly affect radiation heat transfer due to the Blokh. 4 The latter volume in particular contains a alteration of its emissivity. 7s Mineral matter and ash large body of fundamental radiation property data, deposited on walls of the tubes can also increase measured spectral and total incident radiation fluxes greatly the thermal resistance to heat conduction along the height of different capacity furnaces as well across the deposit, and some simple conductance as empirical correlations for analyzing the thermal models have been developed.'* performance of coal-fired boilers. An up-to-date Data for soot, carbon and coal refractive indices discussion of coal combustion models in which are generally (but not necessarily very accurately) radiation heat transfer has also been considered is available,'*'64 but significant uncertainty exists in the available. 3 Despite the considerable progress in the particle concentration and size distributions. In development of analytical methods of engineering gasifiers and staged combustion systems, which science and despite an increasing understanding of operate fuel-rich for nitrogen oxide pollutant control, fundamental combustion processes, the design or soot radiation may be particularly important. Unless performance predictions of coal-fired furnaces may the soot-volume-fraction distribution in the medium still be considered as an art based primarily on is known accurately, radiation heat transfer to the empirical knowledge and the ingenuity of the com- chamber walls can not be predicted with confidence. bustion engineer. This is particularly true for large Fly-ash particles greatly influence the radiative boiler furnaces because of their extremely compli- properties of the flame and of the combustion cated geometry and boundary conditions 4'272'29s as products in a pulverized-coal fired furnace. Data for well as the lack of confidence in the existing fly-ash are much less certain. 4'79-83 There is signifi- analytical methods. Scale-up and advanced perform- cant variation in the refractive indices of pulverized- ance analyses of boiler combustion chambers have coal and fly-ash with the type of coal, mineral matter been developed272 using laboratory and/or small in the coal, as well as the combustion process itself. model furnace data. In spite of major improvements Experiments have revealed that the refractive index in the analytical methods for predicting the perform- of fly-ash particles formed during the combustion of ance of coal-fired furnaces 3'272 there is still distrust even one coal shows quite large differencesfl 7 Lowe by practical furnace designers of the analytical et al. 3°° have shown that in large boilers fly-ash methods because of geometrical restrictions, problems exerts a much greater effect on heat transfer to the of stability, complexity of the new methods, limited heat-absorbing surfaces in a furnace than the aero- applicability of the models, etc. dynamics and kinetic characteristics of a pulverized In this section we discuss the use of more recent coal burn-out. Radiation from fly-ash particles models to predict radiation heat transfer in relatively exceeds substantially the contribution of both tri- simple furnaces, for the purpose of gaining improved atomic combustion gases, as well as char and soot understanding of radiative transfer and of the particles'* Contribution to radiative transfer by char relative importance of the model parameters. It is particles is essentially over the length of the flame. At hoped that this would provide the bridge between the the end of the furnace the concentration of the char scientific community which is developing compre- particles is small, and there they exert very little effect hensive combustion system models and furnace on the radiation heat flux at the wall. designers who are attempting to solve practical problems based on empirical knowledge. Reference is made to literature which discusses methods for 400 evaluation of thermal performance of large boiler g furnaces. Detailed reviews of radiation heat transfer in 300 pulverized coal-fired furnaces are available.4"272"299 c+s+g Radiation heat transfer in furnaces is due to gaseous and particulate contributions. Emissivity data for the qr, z,, major emitting gaseous species CO2 and H20 are (kW/m , 200 generally adequate. 4.64 Other gaseous species (e.g. CO, SO2, NO, N20 ) are usually of secondary importance because of low concentration. Local I00 variations in gas temperature and species composition are subject to more uncertainty than the emissivity data. Contributions to particle radiation in pulver- 0 0 2 4 6 8 I0 ized coal-fired systems usually results from coal z On) (char), soot andfly-ash. Information required for FIG, 38. Effect of combustion products composition on the predicting radiative transfer includes different particle radiation heat flux distribution along the wall of a concentrations, size distributions, complex indices of pulverized coal-fired furnace; (c=coal, f=fly-ash, s=soot, refraction and temperature. 2~'* Finally, the mineral g= combustion gases), for soot J,, = 2 m - 1.2,4 148 R. VISKANTAand M. P. MENGOt;

Radiation heat transfer in a cylindrical, pulverized Current reviews of coal-fired combustion models coal-fired combustion furnace has been predicted are available 2'3 there is no need to repeat these both on a gray 214 and nongray s9 basis. The comprehensive discussions. Recent radiative transfer calculations were carried out by assuming the modeling for inclusion in comprehensive multi- temperature and radiating species concentration dimensional combustion codes has focused on more distributions in the furnace. The radiative character- efficient differential and flux methods, 3°1-303 but istics of the coal particles were predicted from the there are exceptions. For example, Truelove 3°4 used Mie theory, after first assuming a coal particle size a discrete-ordinates method, which is more time- distribution. Details of radiation heat transfer and consuming to evaluate; however, to simplify the sensitivity calculations can be found elsewhere. 214 procedure the gas was considered to be gray and the The contributions of the different constituents (coal, particles were assumed to be black and nonscat- fly-ash, soot and combustion gases) on the local tering. The classical Hottel zonal method is compu- radiative flux along the furnace are shown in Fig. 38. tationaUy inefficient for use in multidimensional It is clear from the figure that neglect of the fly-ash codes. In addition, there are conceptual and numer- contribution and inclusion of soot absorption yields ical difficulties in adopting the method when aniso- a dramatic change in the radiative transfer in the tropically scattering particles are present in the medium and at the cylindrical walls (see curves combustion products. denoted as c+f+g and c+s+g). The main reason Available computer models for scale-up and per- for this discrepancy is the replacement of strongly formance predictions of boiler combustion chambers scattering fly-ash particles by strongly absorbing have been reviewed. 272 The state-of-the-art model for soot particles. The addition of soot to coal +fly-ash predicting radiation heat transfer in a complicated + gas mixtures (c + f+ g) simply decreases the radiat- boiler combustion furnace is based on advanced ive flux on the cylindrical wall since a greater Monte Carlo type techniques. The model is des- fraction of the radiant energy is being absorbed by cribed in more detail elsewhere together with exam- the medium itself. It should be mentioned, however, ples of its practical application. 272 It is shown how that the effects predicted 6°'2~4 in this way may be pilot plant-scale results can be scaled up with the exaggerated since in these calculations the energy help of the model to predict full-scale performance of equation is not solved. When radiative transfer is particular boiler furnaces. The uncertainties in pre- taken into account in the energy equation, the dicting temperatures and heat fluxes are also dis- temperature would change in a manner that would cussed. It is pointed out that for pulverized coal-fired partially compensate for the effects of changes in boilers major uncertainties are caused by the un- radiative properties. known slagging and fouling patterns in the furnace, The results of sensitivity studies 214 have shown and an ash deposition model could help to reduce that accurate knowledge of number density, temper- these uncertainties. ature and particle concentration distributions are Recently, Fiveland and Wesse1298 have developed more critical than the detailed information about the a very detailed and extensive computer model to index of refraction of particles and gas concentration predict the performance of three-dimensional pulver- distributions. The type of coal used affects radiative ized coal-fired furnaces. They have accounted for transfer relatively little; however, the neglect of fly- almost all of the important physical phenomena that ash outside of the flame zone has been shown to have can be expected in such systems, including turbu- a potential for large errors. Apparently, the accuracy lence, chemical reactions, devolatilization, char oxi- of radiative transfer predictions is not only limited dation as well as radiation heat transfer. Although by the solution techniques of the radiative transfer they have considered different size particles (e.g. equation or the prediction of radiative properties, polydispersions) and evaluated the radiative proper- but mostly by the accuracy of particle concentration ties of particles from Mie theory, scattering in the and combustion product temperature distributions medium has been considered isotropic. The combus- which are more time-consuming to evaluate in the tion gas properties have been obtained using the needed detail. Edwards wide-band model, a5 and the average The importance of the spatial distribution of properties of the gas-particle mixture have been radiative properties of pulverized-coal and fly-ash in calculated using the averaging technique proposed predicting radiation heat transfer accurately was also by Wessel. '26 The radiative transfer equation has shown by Lowe eta/. 3°° In their analysis they been solved using the discrete transfer method of employed Hottel's zonal method to solve for radi- Lockwood and Shah; 2°3"2°4 however, the method ative transfer in a utility type pulverized, coal-fired has been revised first to avoid arbitrary radiative furnace. They showed that furnace heat transfer was source/sink terms encountered in certain volume insensitive to the type of coal and coal fineness and elements due to numerical diffusion. Wall emissivity concluded that combustion data were adequate for and thermal conductance of ash deposits can provide calculation of radiative heat transfer. Lowe et al. 3°° a major resistance to heat transfer from the flame- recommended research on ignition, combustion combustion products to the walls of the furnace, and stability and radiative properties of fly-ash. these factors were accounted for in the analysis. Flow Radiation heat transfer 149

FIG. 39. Heat flux isopleths on furnace walls (in W/m2). 29s patterns, gas temperature, concentration and heat modifications in the radiation model would definitely flux distributions have been predicted. In Fig. 39 the improve its reliability. heat flux distribution on the walls of the furnace is depicted. Note that this figure shows the furnace as 6.3. Gas Turbine Combustors unfolded. These types of results can be helpful in identifying potential slagging/fouling problems on It is well established that in gas turbine combus- membrane walls or convection-pass elements. tors a large fraction of the heat transferred from the Models of this type are essential to understand the gases to the liner walls is by radiation. The radiation complex, large-scale, pulverized coal-fired furnaces is due to two contributions: (1) the nonluminous and are valuable engineering design tools. The radiation emitted by gases such as CO2, H20, CO radiation heat transfer model needs to be improved and others, and (2) the luminous radiation emitted by to make it more realistic. Anisotropic scattering by soot particles in the flame. The luminous contri- particles has been neglected and soot has not been bution from the soot depends on the number and size taken into account; therefore, the enhancement or of the soot particles. In the primary combustion zone blockage of radiation by the soot layer is not most of the radiation emanates from the soot considered. However, as the authors claim, the model particles produced in the fuel-rich regions of the is still in the initial stages of validation, and further flame. At high pressures encountered in modern 150 R. VISKANTAand M. P. MENGO~:

turbines, the concentrations of soot particles is 1600 I J t 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 12oo to liner'durability.3°5 An excellent up-to-date review of radiation heat transfer from the flame in gas turbine combustors has R~ 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 dis- I cussed. A need for better understanding of radiative transfer to establish realistic models for predicting local heat flux distribution is emphasized. The 0 ~ understanding can be useful in developing analytical 0 2 4 6 8 tools which may lead to improved liner durability in z/r, future designs by prescribing optimum arrangements FIG. 40. Effect of fuel type (K-kerosine and R50-fuel blend) for the quantity and distribution of film-cooling air. and of radial soot concentration distribution on radiation In turn, this approach can also lead to reductions in heat flux at the cylindrical gas turbine combustor wall II- the time and cost of liner development. 3°5 uniform and 2,3-nonuniform radial soot distributions). 3"" The simple, global methods based on the mean- bcam-length concept for predicting flame emissivity reviewed 6 are not capable of predicting local radi- soot profiles. In practice, such a nearly uniform soot ation heat flux distribution along the liner wall. concentration profile, though unlikely, might come Furthermore, simple methods cannot account prop- about if film cooling air of the combustor penetrated erly for radial and axial nonuniformities of temper- into the combustion zone sufficiently to quench the ature, species concentration and radiative properties soot oxidation process. of the soot-gas mixtures. This is a serious short- The results suggest that accurate calculation of the coming because the combustor designer allocates film radiation heat flux at the combustor wall would cooling air based on the total heat flux at the liner require both the radial temperature and soot concen- wall. In the absence of reliable heat flux predictions, tration distributions in the products. Indeed, the the designer must overprotect the liner. Too much radial temperature distribution had greater impact cool air near the walls, however, can reduce combus- on the total radiative heat flux than the type of fuel tion efficiency, increase pollutant emissions, and for the conditions examined in the study. 3°6 However, distort the temperature pattern at the combustor scattering of radiation by fuel droplets in a gas outlet, which stresses the turbine blades. turbine combustor was found to be negligible in The local radiative flux distributions at the liner comparison to absorption by soot. The average wall of a typical gas turbine combustor have been radiative heat flux calculated by the P3-approxi- predicted using the Ps-approximation for radiation marion compared reasonably well with results based transfer. 3°6 The mean temperature and soot concen- on the mean-beam-length calculations used in the gas tration distributions along the combustor were based turbine combustor industry. 6"a°s However, the P3- on experimental data. 3°7 The effects of axial and model results were able to pinpoint locations of radial temperature and soot concentration distri- maximum radiative flux at the liner wall. butions, type of fuel, and scattering by fuel droplets The problem of three-dimensional two-phase com- were investigated. It was found that the axial and bustion has been approached with the aim of radial temperature and soot concentration distri- producing an algorithm based on fundamental butions impacted the local radiative flux along the principles which correlate all of the details of liner wall in several ways. In Fig. 40, the radiative combustion occurring within a gas turbine combus- fluxes to the cylindrical wall calculated for radially tion can. "~°'3°s'3°9 A mathematical model of the uniform (solid lines) and radially nonuniform (dashed three-dimensional, two-phase reacting flows in gas lines) soot concentration distributions are compared. turbine combustors has been developed which takes The medium with a uniform radial soot concen- into account the mass, momentum, and energy tration yielded larger radiative flux at the liner walls, couplings between the phases, The model incorpor- at peak, than the nonuniform profile. The temper- ates an accurate representation of the droplet ature distribution was assumed uniform for both distributions encountered in gas turbine combustors, Radiation heat transfer 151 and solves the relevant equations for the trajectory 12 • Pu and evaporation of droplets numerically in a Lo - e p,, /_~_ 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 ~O.G // ~', ¢onstan! calculations is not provided. The emphasis in the results reported is on flow and combustion parameters as no results on radiative transfer are given. o.z. ~f ....#."

6.4. Internal Combustion Engines "~O" O" I0" 20" CA Radiation heat transfer in diesel engines is dominated by the continuum radiation emission by soot FIG. 41. Comparison of total radiation heat losses to a particles, which are present during the combustion diesel engine cyJinder wall as a function of crank angle (CA). .~13 process. Radiation also occurs from the carbon dioxide and water vapor molecules, but because that energy is concentrated in spectral bands rather than heat flux to the walls adjacent to thin gas zones by as over the entire spectrum its effect is subordinate with much as 100~. 313 respect to the energy emitted by the soot. Radiation The most important advantage of differential is also emitted in bands by many of the intermediate models (like the spherical harmonics approximation) species formed during combustion, but their effect is is their flexibility to allow for variation of radiative assumed to be even less important. properties within the medium. In Fig. 41 the total In spark-ignition engines, where the combustion is radiative flux to diesel engine walls is compared at usually soot free, the radiation heat transfer is always different crank angles for constant and spatially small compared to the convection heat transfer. The varying extinction coefficient distributions 3n 3 which same seems to be the case in diesel engines during were obtained from published experimental data. 3~4 those times in the cycle when soot is not prevalent. It is clear from this figure that using a mean During combustion the radiation heat transfer is of extinction coefficient to simplify the radiative trans- the same order of magnitude as the convection heat fer calculations can not always be justified, as the transfer; whether 25, 50 or 150y,, of the convection fluxes may be underpredicted by about a factor of heat transfer is a point argued about even in the three. The radiation from soot has been found to be current literature. The arguments stem from the facts much stronger than that from the gases. 3~J In that (1) unequivocal heat transfer measurements are addition, the spectral results also reveal distinct not possible, and (2) the relative importance of spectral selectivity due to the strong gas radiation convection compared to radiation is highly depen- bands of CO2 and H20 at elevated pressures. dent upon the engine design and operating character- As in gas turbines, scattering of radiation by fuel istics. In ceramic-lined engines the convection heat droplets in diesel engines was also found to be transfer is expected to be reduced more than the negligible compared to absorption by soot. 3z 3 Use of radiation heat transfer, and thus radiation will be an average homogeneous (position independent) relatively more important than convection. absorption coefficient in the engine to simplify Parametric studies of radiation heat transfer in radiation calculations was found to be unjustifi- diesel engines have been recently reported, an°-an3 able. 3~a It was also shown that the distribution of The method developed by Chang et aL 3j°'3~n calcu- radiative flux at the head and piston was incorrectly lates spectral and total intensity at the chamber walls. predicted and that the total heat loss could be It is based on the integral form of the RTE along the underpredicted by as much as 60 ~o. line-of-sight and uses in-cylinder species and temperature distributions as well as a coordinate transfor- 6.5. Fires as Combustion Systems mation to aid in the integrations. The method is incompatible with the finite-difference combustion Flame radiation plays an important role in the models, but can yield accurate results for radiative flame structure, spread and heat transfer from transfer along the line-of-sight. Spherical harmonics unwanted fires. A recent review 7 has focused on basic (P~- and P3-) approximations have also been applied aspects of fire and has presented an elementary but to predict radiation heat transfer for the conditions unified treatment of the phenomenon by considering encountered in a diesel engine. It has been shown both urban and wildland fires. Several other re- that the P~-approximation is computationally very views 31"43'239'240'315'316 have treated aspects of cost effective in comparison to the P3-approximation, flame radiation and have contributed greatly to the although it overpredicts the total radiation heat loss phenomenoiogy. The interested reader is referred to to the engine walls by 20 ~ and the local radiation these reviews for books and original research papers 152 R. VISKANTAand M. P. MENGOg:

in the field, and the special issues of Combustion modeling of fire phenomena. Several studies are Science and Technology (Vol. 39, Nos. 1--6 and Vol. mentioned here. 40, Nos. 1-4, 1984) on Fire Science for Fire Safety, Cooper studied fires in enclosures and described honoring Professor Howard W. Emmons in which the ceiling jet resulting from the fire,32° the effect of numerous papers concerned with fires are included. buoyant source in stratified layers, 32~ and the effect It has now been accepted that radiation is the of side walls in growing fires. 322 However, only in the dominant mode of heat transfer in fires of large scale, last paper did he consider the effect of radiation whereas convection (or conductionI is the dominant using simple expressions for radiative transfer to mode of heat transfer of very small scale fires. estimate the wall temperature. Bagnaro et al. 323 Detailed heat transfer measurements have demon- developed a model to predict experimental room fires strated that radiation heat transfer from fuel surfaces under steady and transient conditions. They used a typically exceeds free convection heat transfer for moment method ~77 to solve the radiative transfer characteristic fuel lengths greater than 0.2 m. 239 equation in three-dimensional enclosures. To repre- Nonluminous and luminous radiation from turbulent sent the combustion gas contribution they employed diffusion flames has been recently discussed and a sum-of-gray-gases model. Their results showed the importance of turbulence/radiation interactions good agreement with experimental data. Also, has been recently pointed out by Faeth et Markatos and Pericleous 324 studied the effect of al. 31'254'255 During the last decade there have been radiation on fires in three-dimensional enclosures. numerous contributions to the literature concerned They employed the six-flux model of Spalding (see with radiation heat transfer in fires, and it is not Subsection 4.4.1) for the solution of RTE. However, possible to do justice to them in this very short in neither of these studies is the dependence of the account. radiative properties on the position (i.e. concen- Buoyant enclosure flows have applications to tration and temperature) in the medium considered furnaces and in such phenomena as fire spread in in detail. rooms and buildings. Numerical and experimental Tien and Lee 43 have provided a comprehensive studies of two-dimensional and three-dimensional summary of the radiative properties of nonhomo- turbulent buoyant, simple and complex enclosures geneous and particulate containing media typical of have been summarized by Yang and Lloyd. 317 The the flame environment. These data can then be used results obtained have demonstrated that first- in radiation-energy transfer models, which, in turn, principle numerical finite-difference calculations, determine the characteristics of ignition and fire together with a simple, yet rational algebraic turbu- spread for the condensed fuel. TM 6.325 - 331 During lence model, can provide reasonable predictions to a the combustion of condensed fuels, pyrolysis at the variety of buoyancy-driven vented enclosure-flow fuel surface produces numerous and varied hydro- phenomena when compared to corresponding experi- carbon gases and soot. The fuel vapors diffuse to the mental data. The geometries considered unvented flame zone where they react exothermically with and vented enclosures, aircraft cabin compartments oxygen diffusing from the other side of the flame and others, but the effects of radiation were neglected. zone. Energy released from the flame zone heats the At higher temperatures thermal radiation gener- fuel surface, thus maintaining the existing pyrolysis, ally plays a significant role in affecting the heat creating new areas of pyrolysis, and spreading the transfer in enclosures such as rooms and buildings, fire. The pyrolyzed gases absorb energy in the and interactions between thermal radiation and infrared and attenuate the feedback radiation to the natural or mixed convection must be accounted for fuel surface. This feedback mechanism becomes in the description of the pertinent momentum and important when the gases are strongly absorbing and energy transfer processes. Recent discussions on are sooty or when the pathlength becomes large, as in numerical modeling of natural convection-radiation large-scale fires. For solid and liquid fires, the interactions in multidimensional enclosures are combustion rate is controlled by the heat transfer available. 3~a'319 The interactions depend on the from the combustion zone to the fuel surface. In radiative properties of the absorbing, emitting and large-scale fires (L>0.7 m) fire energy is dominated scattering media filling the enclosure, a method of by radiation,Qnd the combustion rate is controlled calculating multidimensional radiative transfer and by radiant feedback from the flame to the fuel the numerical solution of the governing equations surface. Blockage effects by the pyrolized gases and for buoyant flows. Current knowledge in these sub- particulates near the fuel surface (discussed in areas has been discussed. On the basis of these Section 5.2) can attenuate significantly the incoming reviews,3~ s.319 it is apparent that natural convection- radiation flux. Current analytical models for pre- radiation interactions in buoyant enclosure flows are dicting the radiation heat flux to the fuel surface still in the developing stage. An efficient overall consistently overpredict the pyrolysis rate because computational scheme is still lacking, and metho- the blockage effect is not accounted for. The dologies which have been developed for natural- assumption of an isothermal and homogeneous flame convection interaction studies do not appear to have for large scale fires may also lead to significant errors. been applied to gain improved understanding or The lack of radiative property data for radiation Radiation heat transfer 153 heat transfer calculations is a major limitation in "'long range" or "action at a distance" transport improving current fire models. Radiative properties process. In many physical situations radiation can be of common combustion gases and optical constants modeled without detailed input of complex chemistry, for soot and simple calculation schemes for deter- chemically reacting turbulent flow and knowledge of mining the emission coefficients of luminous flames the flame and the reaction region. have been reviewed. 43 The properties for some of the This review has concentrated on radiation heat hydrocarbon gas species which are evolved by the transfer in combustion systems. It is clear from the pyrolysis of condensed fuels, such as plastics, have review that radiation from flames and combustion been published recently. 332- 33,, Radiative properties products requires detailed information on the radi- of such gas species as ethylene (C2H4), ethane (C2H¢,), ative properties of the combustion gases and partic- propane (C3Hs), methylmethacrylate (C3HsO2), and ulates. Despite the many efforts which have been others which are major species in pyrolized gases are devoted to the problem, the methods developed for needed. The wide-band 35 and super-band 4'~ model radiation heat transfer in multidimensional geo- parameters need to be generated from experimental metries are far from satisfactory, particularly when data for the radiatively important gases. Total temperatures and gas partial pressures and partic- emissivity charts can be developed for each gas once ulate concentrations are varying along the path the band parameters have been determined. These length. The calculation of radiation in combustion charts graphically express the dependence of total systems is quite involved, and most of the techniques, emissivity on the temperature, pressure, and optical except those which are called flux or differential pathlength of the emitting gas and greatly simplify approximations, are incompatible with the numer- the calculation of flame radiation problems. How- ical algorithms for solving the fluid dynamics- ever, band information becomes necessary when transport equations. different gases are combined which have overlapping During the course of the review, a number of bands in order to determine the correction. Pre- problem areas have been identified and are discussed dictions of radiative transfer in large-scale fires based in the article. Some specific recommendations for on data from small-scale flames in laboratory work in modeling radiative transfer in combustion experiments, however, have been very limited in systems are the following: accuracy and require much more research attention. The turbulence/radiation interactions and coupled (1) Radiative property data of less common gases effects of radiation and flame structure for small such as ethylene (C2H4), ethane (C2H~,), as well laboratory flames were discussed in Section 5.7. They as propane (C3Hs) and other more important were found to be more important for luminous than radicals are needed. Radiative properties of for nonluminous flames. Since smoke (soot) is particulates encountered in pulverized coal com- generated in open, compartment and building fires bustion such as fly-ash, char and others need to which are much larger in scale than small laboratory be predicted and verified experimentally. There flames, the turbulence/radiation interactions are is a very large uncertainty in the radiative expected to be even more significant because of the properties of these types of particulates that large and highly variable local opacities that may be have been reported in the literature. Most of the encountered in these types of systems. The buoyant properties of particles have been obtained at smoke plume generated by a large fire also involves conditions much different than those encoun- radiation exchange within itself and with its environ- tered in flames; therefore, it is still not clear ment. The heat and particulates released by a fire whether these data can be used with confidence create complex flow patterns which are determined for combustion studies. by a variety of factors. The interactions of radiation, (2) There has been progress in modeling the thermal turbulence and flow structure as well as the feedback radiation properties of gases and particulates. between them in large fires are topics which have However, more research effort is needed, es- received practically no research attention and are not pecially on physically and analytically well- understood. founded representations that are simple and convenient for use in computer codes of com-

7. CONCLUDING REMARKS bustion systems. Considering that a characteristic length is always required for use in the By highlighting recent developments in modeling models and that such a length can not be radiative transfer, the present review aims to increase rigorously defined for most practical multi- recognition that very often radiation plays an dimensional systems, it is clear that the concept important, if not the dominant, role in heat transfer needs additional research attention. not only in large and intermediate but also in small (3) The nongray effects have been recognized as combustion systems. Neglect of radiation cannot being very important and it is known that the be justified in modeling combustion phenomena. gray approximation overpredicts the emission Modeling of radiative transfer in combustion sys- of radiation from flames with low soot content. tems can be rather "forgiving" because radiation is a The calculations of radiative transfer for non-

JPEC8 13 : 2-g 154 R. VISKANTAand M. P. MENG0~

homogeneous, nonisothermai flames on a non- (8) Research effort should be devoted to experiment- gray basis would enable accurate predictions of ally validating the radiative transfer model(s) in flame emission for a wide range of pathlengths~ order to demonstrate the potential usefulness of The results could then be used to establish the methods to the analysis and design of scaling relations and to assess the range of practical systems. validity of the gray analysis. (4) In combustion systems involving the burning of Acknowledgements Much of the author's recent work solid fuels such as pulverized coal, the particles reported in this review was supported by CONOCO. Inc. and gases surrounding them are at different through a grant to the Coal Research Center of Purdue University. It is a pleasure to acknowledge CONOCO's temperatures. Analytical models based on interest in fundamental radiation heat transfer research experiment need to be developed to predict rehlted to combustion systems. The authors wish to express radiative transfer and temperatures in such their appreciation to Miss Nancy Rowe for her dedicated systems. The slip between particles and gases help in transforming their notes into a polished manuscript. must be considered. This is not only important The authors are also indebted to the anonymous reviewers for pointing out typographical errors and for suggesting for predicting accurately the flow and temper- improvements in the presentation. ature 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 sys- REFERENCES tems the chemically reacting flow is turbulent. The question needing an answer is to what 1. ORAN, E. S. and Bo)us, J. P., Prog. 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