Prog. Energy Combust. Sci. 1987. Voh 13, pp. 97-160. 0360-1285/87 $0.00 +.50 Printed in Great Britain. All rights reserved. Copyright O 1987 Pergamon Journals Ltd. RADIATION HEAT TRANSFER IN COMBUSTION SYSTEMS R. VISKANTA* 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 approxi- mation 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 co- sine, 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.