chapter 8 thermal radiation modeling in ﬂ ames and ﬁ res · chapter 8 thermal radiation...

CHAPTER 8

Thermal radiation modeling in fl ames and fi res

S. Sen1 & I.K. Puri21Department of Mechanical Engineering, Jadavpur University, India.2Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, USA.

Abstract

An overview of thermal radiation models for different combustion processes is presented. The pertinent constitutive equations and associated radiative property models are discussed. Various solution techniques for the radiative transfer equation are described. Finally, the implications for fl ames and fi res are discussed in the context of the role of different thermal radiation models.

1 Introduction

Radiation is the dominant energy transport mechanism to surrounding surfaces during many combustion processes, particularly when entrained particles are present. It plays an important role in practical systems such as furnaces [1], during fundamental fl ame phenomena, such as the radiation-induced extinction of fl ames at low stretch [2], radiation-turbulence interactions [1], and in fi res [3]. Typically, the radiation intensity on a wall becomes very signifi cant when the combustion length scale approaches a meter [4].

Thermal radiation is strongly coupled with soot kinetics and the fl ame structure through the local temperatures. Characterizing radiative energy transport is therefore a crucial element in modeling combustion systems. However, this is also a very complex problem. For example, in a typical coal fi red furnace, radiation includes contributions from particulates (like coal/char, soot, ash) as well as from gases (such as CO2, H2O). The accuracy of a radiation simulation depends on a combination of the accuracy of the calculation method and the accuracy with which the radiative properties of the media and surfaces are known [5].

The current focus for radiative heat transfer models is on describing radiative interactions with a participating medium. The process is characterized by absorption, emission, and scat-tering of radiant energy, and mathematical models and computer codes are necessary to solve various radiation problems. Products of combustion gases, such as carbon dioxide, water vapor, and their mixtures with soot are considered within these radiation problems. An impor-tant issue of gas radiation is the description of the radiative properties of these gases (or the so-called nongray gases) [6].

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302 Transport Phenomena in Fires

2 Basic equations

2.1 Energy conservation equation

The mass, momentum, energy, and species balance equations must be solved to address com-bustion problems. These equations retain their usual form, while the energy equation addresses thermal radiation in a participating medium through an appropriate term, i.e. [6]

ˆ ˆ( ) ,

hvh q S

tr r

∂+ ∇⋅ = −∇⋅ +

∂

(1)

where h denotes the enthalpy of the fl uid and S. the sources of heat generation, viscous heat dis-

sipation, pressure, and body force work. The heat fl ux vector

1

ˆˆ ˆ .N

i i ii

q K T h v Fr w=

= − ∇ + +∑

(2)

The terms on the right-hand side represent heat conduction, inter-species heat diffusion, and radiative heat fl ux that can be written as:

0 4

ˆ ˆ ˆ ˆ ˆ( , ) ( , , ) d dF r t I r s t slw

l.∞

= π

= Ω∫ ∫

(3)

2.2 Radiative transfer equation

The spectral intensity of radiation, Il (r, s , t), at wavelength l is found from the radiative transfer equation (RTE). The interaction of electromagnetic radiation with matter that absorbs, emits, and scatters thermal radiation is described by the RTE. For the sake of brevity the derivation of RTE has not been repeated here, since it is well explained in the literature [7].

If we consider the local thermodynamic equilibrium of a medium capable of absorbing, emitting, and scattering radiation, the spectral radiation intensity Il (r, s, t) of the radiation fi eld is [7]

4

ˆ ˆ ˆ ˆ( ) ( , )d ,4

sb i i is I k I I I s s sl

l l l l l l ls

bπ

⋅∇ = − + Φ Ωπ ∫

(4)

where k denotes the absorption coeffi cient, b the extinction coeffi cient, s the scattering coeffi -cient, and φ the scattering phase function that can be interpreted as the ratio of scattered radiative intensity in a given direction to the scattered radiative intensity in the same direction by isotropic scattering. The time dependent term for the radiation intensity can be usually neglected (as we have done).

Equation (4) is the radiation balance for an infi nitesimal pencil of rays in which absorption, extinction and scattering terms are considered. Thus, in order to obtain a volume balance, the equation must be integrated over all solid angles, i.e.

4 4 4 4 4

ˆ ˆ ˆ ˆd d d ( ) ( , )d d .4

sb i i is I k I I I s s sl

l l l l l l ls

bπ π π π π

⋅∇ Ω = Ω − Ω + Φ Ω Ωπ∫ ∫ ∫ ∫ ∫

(5)



Thermal Radiation Modeling in Flames and Fires 303

After rearranging, this relation can be written in the form [7]

4 4

ˆ ˆ4 ( )d ( )d .b s i iq k I I s I sl l l l l l lb sπ π

∇⋅ = π − Ω + Ω∫ ∫

(6)

Using the relation kl = bl – ssl, the spectral relationship

( )4

4 d 4 ,b bq k I I k I Gl l l l l l lπ

∇⋅ = π − Ω = π −

∫

(7)

where Gl denotes the incident radiation. Integration over the entire spectrum can be carried out to obtain the divergence of radiation heat fl ux, F.

3 Solution of the RTE

The solution of the RTE can be substituted in the thermal energy equation to determine the temperature distribution in a reacting fl ow. This requires knowledge of the spectral radiative properties of the gases and an effi cient method to solve the RTE. The evaluation of the radiative fl ux vector F or its divergence represents a fundamental problem that is related to the treatment of the spectral radiative properties of gases, the solution of the RTE, and integration over the spectrum.

3.1 Radiative property models

The accuracy of any solution of the RTE depends upon an accurate knowledge of the radiation properties of the combustion product gases and the entrained/generated particles. Models used to defi ne the radiative properties of combustion gases in radiation calculations can be roughly sorted in three groups [8], i.e.

Spectral line-by-line (LBL) models;1. Spectral band models; and2. Global models.3.

Each has its merits and drawbacks that prescribe its area of application. Historically, the old-est and the simplest concept for the prediction of radiation by a gas is the gray gas model [9]. It assumes that the gas absorption coeffi cient is constant and therefore, when compared with other models, provides predictions with poorer accuracy [10].

The LBL model provides the best accuracy. With this method the RTE is integrated over the detailed molecular spectrum for the gases [11]. Because of the enormous computational require-ments, this model is used mainly for benchmark solutions. It has two main disadvantages. First, the required spectral resolution should be smaller than the line width (i.e. 10−3−10−2 cm−1), which corresponds to roughly 106 spectral discretization intervals to cover the entire infrared spectrum. Second, at higher temperatures, accurate descriptions of a large number of lines associated with high energy levels and their quantum states are required.

Recent developments in gas molecular spectroscopy have improved the database of radiative properties. Rothman et al. [12] have successfully implemented the LBL method to predict radiative transfer in the low temperature atmosphere. The 1992 version of the HITRAN database [12] con-tains information about spectroscopic transition line parameters for 32 molecules at the reference




temperature of 296 K. The molecules included in the database are H2O, CO2, O3, N2O, CO, CH4, O2, NO, SO2, NO2, NH3, HNO3, OH, HF, HCl, HBr, ClO, OCS, H2CO, HOCl, N2, HCN, CH3C, HCl, H2O2, C2H2, C2H6, PH3, COFr, SF6, H2S, and HCO2H. It covers a spectral range from 10−6 to 22,656 cm−1 (i.e. 0.4414−1,010 µm). LBL calculations have been reported for H2O−CO2−CO semitransparent gas mixtures up to 2,000 K [13, 14]. In this model the spectral absorption coeffi -cient for a gas mixture

,iji j

k kl l= ∑∑

(8)

where the subscript i refers to the absorbing species and subscript j to all the lines of all the bands in these species. The line mixing and collision effects can be considered using phenomenologi-cal functions [12, 15]. The key problem is to obtain all center locations, lower level energy, and intensities of all lines contributing to absorption. The HITRAN database [12] is available for atmospheric and higher temperatures [16, 17].

Approximate models that have been developed to characterize the spectral structure of radia-tion can be classifi ed into two groups, namely, (1) band models and (2) global or total models. In band models, the entire spectrum is divided into a number of bands. Thereafter, radiative proper-ties, averaged over each band, are calculated from the absorption spectra or statistical properties of a line. These bands are assumed to be suffi ciently narrow such that variations of radiative properties can be neglected.

When a narrow band model is used, the RTE must be averaged over a band to yield the band-averaged radiation intensity. To obtain spectrally-averaged or narrow band values of the absorp-tion coeffi cient, some information must be available on the spacing of individual lines within the group and their strengths. A number of models have been proposed for this purpose and the sta-tistical model is most widely used. It assumes that the spectral lines are not equally spaced within a narrow band. This can be a true representation for complex molecules for which lines from different rotational modes overlap in an irregular fashion [18].

An extensive description of narrow band models can be found in refs [15, 18]. These models require a database containing the measurements of both the reciprocal mean line spacing param-eter and the mean absorption coeffi cient over the entire infrared spectrum at different tempera-tures. The most complete set, of measured narrow band parameters for H2O and CO2 is that of Ludwig and coworkers. Improved narrow band parameters have been published by Taine and coworkers [13, 14, 19] and Phillips [20, 21]. To date, the general dynamic database enables radiative property calculations for H2O, CO2, CO, OH, NO, HF, and CN. For CH4, C2H2, and soot, a narrow band model calculation can be performed using the approaches outlined by Bros-mer and Tien [22, 23] and Ludwig et al. [18].

The statistical narrow band (SNB) model is more accurate for predictions of radiative transfer in high temperature gases. It provides the spectral transmissivity averaged over a narrow band. However, because of this it is diffi cult to couple the model with a solution methodology such as the fi nite volume method (FVM), which requires values of spectral (monochromatic) absorption coeffi cients or their averages over wavelength intervals. Another disadvantage is that the model requires a large number of bands and is therefore computationally very expensive. The wide band model (WBM) is a simplifi cation of the SNB model. Instead of spectral lines, it considers bands and so is more economical but less accurate. It yields wideband absorptance while the solution of RTE by FVM operates either with spectral or averaged absorption coeffi cients. Therefore the WBM cannot easily and simply be incorporated in a FVM. In addition, WBM requires knowl-edge of the path length in the model as well as the associated spectral parameters.




The concept underlying the exponential wide band model (EWBM) [24, 25] stems from the experimental observation that absorption by gases is mainly due to four or fi ve strong absorption bands located in the near infrared and infrared regions (l−20 µm). The EWBM does not look at a narrow band but rather at the whole absorption band. This model provides fairly simple math-ematical expressions to predict the temperature and pressure dependence of the most important absorption bands of H2O, CO2, CO, CH4, NO, SO2, N2O, NH3, and C2H2. The EWBM can also be utilized to predict the homogeneous total emissivity of gas-soot mixtures provided the soot concentration is known. This model could be utilized to calculate radiation in nonhomogeneous gas mixtures.

From a practical perspective, the requirement of an almost insignifi cant database is the main advantage of EWBMs as compared to SNB models. The model does not require excessive com-putational power and possesses all the generality required for the radiation simulations in com-bustion at atmospheric and elevated pressures. Its major shortcoming is in treating radiation problems in enclosures, since it does not readily account for wall interactions. This problem originates from the formulation, which requires the calculation of the total band absorptance prior to those of the band transmissivity and bandwidth. Since the width of each absorption band varies with temperature, pressure, and pathlength, a different division of the infrared spectrum is required for each path along which the RTE is solved. However, to properly account for gas-wall interactions, the same spectral division for the wall and the gas phase should be used. The divi-sion of the spectral emissivity at the wall can only be setup once the RTE is solved along each path within the gas volume. This procedure requires a larger computational memory than narrow band models and is more computationally expensive. Some of the main impediments to the cou-pling of the EWBM with classical solution methods for the RTE have been discussed by Edwards [26].

In correlated-k (CK) methods, for any radiative quantity that is solely dependent on the gas absorption coeffi cient (which is true for a narrow band over which the blackbody function can be treated as a constant), the integration over wavenumber can be replaced by integration over the absorption coeffi cient. When integration over wavenumber is replaced with one over the gas absorption coeffi cient, the spectral gas absorption coeffi cient kl is denoted by k, since it now plays the role of an independent variable that is no longer a function of wavenumber. This is equivalent to the concept of the reordered absorption coeffi cient introduced by Lee et al. [27]. The distribution function f(k) can be determined in two ways. One is by analyzing the HITRAN database [28] and another by performing the inverse Laplace transformation of the gas transmis-sivity of a narrow band model, as in the SBN-based correlated-k (SNBCK) method described by Lacis and Oinas [29].

Denison and Webb [30−32] have developed polynomial correlations to calculate the cumula-tive distribution functions for water vapor and carbon dioxide. A set of 30 correlations was developed for H2O. The k-distributions were selected to cover the spectral range 400−12,000 cm−1 (0.83−25 µm) with intervals of 400 cm−1. The correlations were obtained by fi tting spectral line predictions of the cumulative distribution in the temperature and volumetric ranges 400−2,500 K and 0−1, respectively. The k-distribution model suffers from the disadvantage that new correlations must be computed whenever the total pressure or the temperature is changed (unlike narrow band models and EWBMs).

The most widely used global model method is the weighted sum of gray gases (WSGG). The concept was fi rst presented by Hottel and Sarofi m [9]. In this method, the nongray gas is replaced by a number of gray gases. For the gray gases, the heat transfer rates can be calculated indepen-dently. The total heat fl ux is then found by summing these heat fl uxes with appropriate weights.




This approach can be used in a directional equation of transfer, as shown by Modest [33], and can therefore be used with any solution technique for the RTE.

3.2 Radiative properties of entrained and generated particles

The evaluation of radiative properties of entrained and generated particles is important in com-bustion processes, since in almost all the cases, the fl ow is laden with particles like soot, coal, or ash. The radiative properties required for an entrained particle phase are the absorption coef-fi cients and the scattering phase function which depend on the particle concentration, size dis-tribution, and effective complex refractive indices. However, the optical properties of particulates, such as soot and coal, are not well characterized [34, 35] and considerable uncertainties exist, e.g. regarding the size and concentration of soot and the refractive index of ash. The absorption and scattering effi ciencies are strongly dependent on the concentration and size distributions. Gener-ally, to simplify the calculation of radiative properties, particles are assumed to be spherical and homogeneous, which is an oversimplifi cation, although the radiative characteristics of a cloud of irregularly shaped particles are not very sensitive to the geometric shapes of the particles (as for pulverized coal) [4].

Given these assumptions, the absorption cross-sections can be calculated using Mie theory [36, 37] based on a specifi ed particle size distribution, wavelength of radiation, and the complex refractive index. Once the absorption and scattering effi ciencies for individual particle sizes are known from Mie theory, the absorption and scattering coeffi cients of the particulates can be evaluated [5]. Unfortunately, Mie theory is strictly applicable only to isolated particles interact-ing with plane waves. Because of this limitation and due to the highly forward-scattering prop-erties of entrained or combustion-generated particulates, scattering intensities are often approximated by phase functions. The scattering phase function can be modeled using the Dirac-delta approximation [38]. The overall absorption coeffi cient for the volume can then be obtained as

p g ,k k k= +

(9)

where kp and kg are the absorption coeffi cients for particle and gas, and the total radiative source for the gas enthalpy equation can be written as

b

4

ˆ d 4 ,F k I Ewπ

= −

∫

(10)

where Eb denotes the blackbody emission of the gas. For a gray analysis, the blackbody emis-sive power for the gas is provided by Stefan−Boltzmann law of radiation [39]. Particle radiative properties and radiative emission can be determined using the source-in-cell technique [40] for Eulerian radiation fi eld calculations.

Almost all the soot models assume that the soot number density decreases as a result of par-ticle agglomeration into spherical aggregates. The only exception is perhaps the study of Eze-koye and Zhang [41] who investigated the effect of particle agglomeration. It has been established that soot aggregates consist of more or less identical primary soot particles but the primary soot particle number density remains almost constant in the growth region. For either a Mie or a Rayleigh−Debye−Gans calculation, the overall properties must be integrated over either particle or aggregate size distribution. However, the fractal aggregate nature of soot has not found its way into most computational studies.




3.3 Solution methodologies

The RTE is an integro-differential equation for which exact solutions are not available for prac-tical engineering applications. Multi-dimensionality, nonhomogeneous media, and the spectral variation of radiative properties make solutions of the RTE quite diffi cult. However, reasonably accurate numerical solutions can be obtained by introducing certain approximations. Because it is not possible to develop a single solution method that is equally applicable to a wide variety of different systems, several methods with varying degrees of approximation have been developed according to the nature of the physical systems, characteristics of the medium, degree of accuracy required, and the availability of computer facilities [4].

The major approaches are: (1) statistical or Monte Carlo methods; (2) zonal methods; (3) fl ux methods including the discrete-ordinates approximation; (4) moment methods; (5) spherical har-monics approximation; and (6) hybrid methods. Description of the statistical methods for radia-tive heat transfer calculations have been provided by Modest [7] and Haji-Sheikh [42]. These can be used for complex geometries and spectral effects can be accounted for without much diffi culty. In its simplest form, a statistical approach simulates the histories of a fi nite but very large number of photons which originate from specifi ed volume/surface elements, propagate in all directions, and are absorbed and scattered based on local values for absorption and scattering coeffi cients. Different researchers have applied these methods to combustion problems [43−45].

The zonal method (usually referred to as Hottel’s zonal method) is a widely employed model for calculating radiative transfer in enclosures such as combustion chambers. Hottel and Sarofi m [9] and Hottel and Cohen [46] fi rst described this approach. In this method, the surface and vol-ume of the combustion chamber are divided into a number of zones, each of which has a uniform temperature and radiative properties. An energy balance is written for each zone, which leads to a set of simultaneous equations for the unknown heat fl uxes or temperatures. The radiative heat fl ux generated by the exchange between the zones is determined using a radiosity method based on appropriate shape factors. The radiosities of the zones are determined by solving simultane-ous equations. When an absorbing−emitting medium is involved, the calculation of direct exchange areas becomes complicated by the attenuation of radiation along a path connecting two zones. Because the approach is practical and powerful, it is attractive for many engineering cal-culations. This method is not however computationally effi cient when coupled with fi nite-vol-ume reactive fl uid fl ow approaches usually used in comprehensive combustion models [47, 48].

Flux methods are based on separating the angular dependence of the radiation intensities, which arise from the spatial dependence of the in-scattering source term. By employing the assumption that intensities are uniform over defi ned intervals of the solid angle, the integro-differential equa-tion can be simplifi ed into a series of coupled, linear, differential equations expressed in terms of average radiative intensities or fl uxes. Different fl ux models arise based on the number of solid angles used to approximate the directional dependence of the radiant intensity, such as two-fl ux approach (i.e. forward and backward scattering) for a one-dimensional approximation, four-fl ux for two dimensions, or six-fl ux models for three dimensions. Within each of the different fl ux models, various approximations and simplifi cations are employed to relate the angular depen-dence of the fl uxes and characterize scattering phase functions to arrive at a closed set of solvable equations.

The fl ux methods have been particularly effective for simultaneous use with reacting fl ow fi eld solutions. The discrete-ordinates approach is a particular case of a fl ux method that was origi-nally developed by Chandrasekhar [49] for astrophysical applications. It has also been used extensively in analyzing neutron transport [50, 51]. In the discrete-ordinates model, a quadrature scheme is used to integrate the in-scattering term. The quadrature set consists of ordinates




(radiant energy directions) and weights which are chosen by applying appropriate constraints, such as symmetry and moments matching, as well as conserving radiant energy within a control angle and the total control volume. One quadrature scheme commonly used is the SN approach, in which the entire solid angle is divided into N(N + 2) angular divisions, where N can be evalu-ated on the basis of the order of the quadrature scheme.

Applications of the SN approach have been shown by Fiveland [48, 52, 53] to produce accu-rate and computationally effi cient results as compared with other approaches. Raithby and Chui [54, 55] and Chai et al. [56] have presented quadrature approaches based on a control volume scheme for fl uid mechanics and convective heat transfer calculations. Control volumes can be based on either structured or unstructured grids. Because of its capabilities to consider common control volumes and grid structures to couple the radiation and reactive fl uid fl ow solutions, the discrete ordinates method is the method of choice in comprehensive combustion models where thermal radiation has an important role.

In other approximations of the RTE, the radiative intensity is expressed as a series of products of angular and spatial functions. With it, the integral part of the equation can be eliminated and a series of equations in terms of different orders of moments can be generated. A moment is defi ned as the integral of intensity multiplied by a power of a directional cosine over a predeter-mined solid angle division. If the angular dependence is expressed using a Taylor power series expansion, then the method is called the moment method, and if spherical harmonics are used to express the intensity, the spherical harmonics (PN) approximation results. The fi rst moment, the fi rst order spherical harmonics (P1), and the S2 discrete ordinates approximation are essentially identical. Applications of the moment method have been discussed by several researchers [57−60].

The general equations for the solution of the P1 and P3 approximations for absorbing, emit-ting, and anisotropically scattering in cylindrical and three-dimensional rectangular enclosures were developed by Menguc and Viskanta [61, 62]. The P1 approximation is particularly simple, since it can be cast as a single, second-order differential equation, but this simplicity is at the expense of accuracy. The prediction error can be as large as 50% for media with small optical thicknesses. Finally, combinations of various methods for solving the RTE (described above) have been used to formulate hybrid methods, which attempt to compensate for the fl aws of one approach with the strengths of another. Several of these hybrid approaches have been summarized by Viskanta and Menguc [4]. As an example of a hybrid approach, the ‘discrete transfer’ model proposed by Lockwood and Shah [58] combines the advantages of the zonal, Monte Carlo, and discrete ordi-nates methods. Although designed for computing radiation in absorbing, emitting, and scattering media, no results have been reported for scattering in media in multi-dimensional enclosures. This method has been applied in furnace simulations [63, 64] and has been adapted for coupling with complex fl uid mechanics and heat transfer grid topologies [65].

4 Radiation from fl ames

Although the importance of radiation heat transfer as a fl ame heat-loss mechanism is recognized, the radiation absorption and nongray radiation models have not been adequately implemented in combustion simulations. It has become almost common practice, mainly due to computational constraints, to assume (often without justifi cation) that the optically thin approximation (OTA) is valid to simulate various laboratory (sooting or nonsooting) fl ames due to their relatively small optical dimensions [66]. In fact, several numerical calculations of such fl ames using detailed chemistry and the OTA for radiation heat transfer have been performed [67−72].




Kennedy et al. [67] developed a model for a laminar soot-laden ethylene diffusion cofl ow fl ame and compared it with measurements in sooting and nonsooting fl ames, studied by Santoro et al. [73]. They incorporated the divergence of the radiative heat fl ux term in the energy equation and calculated it using an optically thin model for emission from soot alone [74, 75]. They ignored the gas radiation in view of the strong luminosity of the soot particles and compared calculated-temperature values for the nonsooting fl ame with experimentally obtained data [76]. This comparison shows good agreement at a lower height above the burner. At a large distance above the burner, their model predicts a higher temperature. Since the model considers radiation from only soot, it predicts a smaller radiation loss than if gas radiation had also been included.

Smooke et al. [68] included an optically thin radiation model in their calculations of laminar cofl ow fl ames. They assumed that for methane−air mixtures the signifi cant radiating species are H2O, CO, CO2, and soot. The OTA method was also employed by Hall [77, 78] whose results predict the general shape and structure of fl ames (e.g. locations of peak fl ame temperatures) reasonably well. Temperature is a critical fl ame parameter since it has a direct infl uence on virtu-ally all the other fl ame properties. The Hall model underpredicts experimental measurements along the centerline by about 100 K at all heights within the fl ame. While the location and value of the peak temperature are reasonably well-described by the model up to a certain height for a fl ame, temperatures are overpredicted at the outside fl ame edge (i.e. predicted temperatures do not fall off as rapidly as the data with increasing radius). In addition, the laboratory fl ame closes to the centerline more rapidly than their model predicts. Figure 1 shows the temperature profi les at different heights in the fl ame from ref. [68].

CalculatedMeasured

0.00

500

1000

Tem

pera

ture

(K

)

1500

2000z = 1.0 cm

0.1 0.2 0.3 0.4r (cm)

0.5 0.6 0.7 0.8

CalculatedMeasured

0.00

500

1000

Tem

pera

ture

(K

)

1500

2000z = 1.5 cm

0.1 0.2 0.3 0.4r (cm)

0.5 0.6 0.7 0.8

CalculatedMeasured

0.00

500

1000

Tem

pera

ture

(K

)

1500

2000z = 2.0 cm

0.1 0.2 0.3 0.4r (cm)

0.5 0.6 0.7 0.8

CalculatedMeasured

0.00

500

1000

Tem

pera

ture

(K

)

1500

2000z = 2.5 cm

0.1 0.2 0.3 0.4r (cm)

0.5 0.6 0.7 0.8

Figure 1: Computed and measured temperature profi le at various heights in the fl ame. (Reprint-ed with permission from Elsevier Science [68]. Copyright 1999 by The Combustion Institute.)




One possible explanation for the low centerline temperatures is the lack of inclusion in the model of absorption by methane of the energy radiated from the fl ame front. This lack of par-ticipation by different components induces a smaller radiation loss from the outer surface of the fl ame. Without this additional radiation component, the predicted peak temperatures are higher by an increment that is large enough to affect predicted soot levels and species concentrations. The signifi cance of thermal radiation in a lightly sooting cofl ow fl ame arises fromresidence times that are relatively long compared to those for counterfl ow diffusion fl ames, in which radiation effects are signifi cant only for extremely low strain rates [79].

In nonbuoyant fl ames, radiation plays an important role since it is the major heat transfer mechanism (in absence of convection). Walsh et al. [71] performed a numerical and experimen-tal investigation of an axisymmetric laminar diffusion fl ame to assess the role of buoyancy and dilution on various fl ame properties, such as temperature, fuel, and oxygen concentration, and soot volume fraction. Their predicted temperature profi les are in excellent agreement with mea-surement in both normal gravity and microgravity fl ames for low dilution levels. Kaplan and Kailashnath [70] investigated fl owfi eld effects on soot formation in normal and inverse methane−air diffusion fl ames. Because methane is a relatively low sooting fuel, the radiation transport submodel is based on the simplifying assumption that the medium is optically thin. The absorp-tion coeffi cient for soot was assumed based on Kent and Honnery [80], while that for CO2 and H2O was derived from Magnussen and Hjertager [81]. These were combined to provide an over-all Planck mean absorption coeffi cient.

Microgravity experiments on fl ame spread over thermally thick fuels have been important for assessments of fi re hazards in orbiting spacecraft [82]. Such experiments on fl ame spread have been conducted over thermally thick foam fuels (that have relatively low densities and thermal conductivities, and thus higher spread rates as compared to dense fuels such as PMMA [83]). These experiments suggest that steady spread can occur over thick fuels in quiescent micrograv-ity environments, especially when a radiatively active diluent such as CO2 is employed. This is due to the dominance of radiation transfer from the fl ame to the fuel surface over conduction heat transfer from the fl ame to the fuel bed. Radiative effects become more signifi cant at microgravity due to a larger fl ame thickness volume of radiating combustion products.

Liu et al. [84] investigated an atmospheric cofl ow laminar moderately sooting ethylene-air diffusion fl ame. They calculated radiation heat transfer using both the optically thin model and the discrete-ordinates method coupled with a SNB model. The radiation source term in the energy equation was obtained using the discrete-ordinates method in an axisymmetric cylin-drical geometry as described by Truelove [85]. The T3 quadrature [86] was used for the angu-lar discretization. Spatial discretization of the transfer equation was achieved using the FVM along with the central difference scheme. The SNBCK-based WBM developed by Liu et al. [87, 88] was employed to obtain the abso rption coeffi cients of the combustion products con-taining CO, CO2, and H2O at each wide band. They assumed the spectral absorption coeffi -cient of soot to be 5.5fv /l (fv being the soot volume fraction and l the wavelength). The wide bands considered in the calculations were formed by lumping 10 successive uniform narrow bands of 25 cm−1 to obtain a bandwidth of 250 cm−1 for each wide band. The blackbody inten-sity at each wide band was evaluated at the band center. The SNB parameters for CO, CO2, and H2O were those compiled by Soufi ani and Taine [89] based on LBL calculations. At overlap-ping bands, the approximate treatment based on the optically thin limit developed by Liu et al. [90] was employed. To further speed up the calculations without losing accuracy, the 4-point Gaussian−Legendre quadrature was used to invert the cumulative distribution function to obtain the absorption coeffi cients based on Liu et al. [91]. The radiation source term was cal-culated by summing up contributions of all the 36 wide bands (from 150 to 9,150 cm−1) con-sidered in the calculations.




To evaluate the results of the optically thin model using those of the WBM discussed above, numerical calculations were also conducted [84] using the optically thin radiation model. Under OTA, the radiation source term

5 4

v pˆ 4 ,F Cf T k Ts= − −

(11)

where C is a constant that is calculated based on the spectral absorption coeffi cient of soot, s the Stefan−Boltzmann constant, and kp the Planck mean absorption coeffi cient of the gas mixture including contributions from CO, CO2, and H2O. The Planck mean absorption coeffi cients of these three species were calculated based on the SNB model given by Ju et al. [92].

The predicted temperature fi eld with both gas and soot radiation accounted for (using SNBCK) was in qualitative agreement with measurements, as shown in Fig. 2. However, the predicted peak fl ame and centerline temperatures were more than 100 K lower than the corresponding measured values. In addition, the predicted maximum temperature annulus was found to be thinner than the measurement. The causes of these discrepancies were attributed to the use of a simplifi ed soot model. The peak temperature predicted using the optically thin model was only about 5 K lower than that of the band model. The centerline temperatures were also underpredicted due to neglect of radiation absorption by CO2 (and to a lesser degree by CO), especially in CO burnout regions just above the fl ame tip where the concentration of CO2 is high. The optically thin model under-predicted the temperatures in these regions by more than 50 K compared to the band model. When radiation by gases was neglected, the predicted temperature levels were in reasonably good agreement with experimental values. However, the peak fl ame temperature was now higher, indicating that radiation by gases is also important and should be accounted for. The results [84] suggest that soot radiation is more important than gas radiation in such a fl ame, since its inclu-sion has a greater impact on the predicted temperature distributions. The comparison of mea-sured and predicted soot volume fraction is shown in Fig. 3.

(a) experiment (b) with radiation SNBCK/DOM 2010.5 K

(d) without gas radiation 2040.3 K

(e) without soot radiation 2035.3 K

(f) without radiation 2133.5 K

(c) with radiation: optically thin 2005.2 K2156.1 K

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Figure 2: Comparison of measured and predicted temperature distributions with the peak values indicated. (Reprinted with permission from Elsevier [84]. Copyright 2002 Elsevier.)




Sivathanu and Gore [93] investigated the effect of gas band radiation on soot kinetics in a laminar cofl ow methane-air diffusion fl ame. The local radiative heat loss fraction was obtained using the solution of the energy equation including soot and gas-band radiation [94]. They con-cluded that the contribution of the participating gases (CO2 and H2O) dominates the soot radia-tion by an order of magnitude in their methane/air fl ames and the local radiative heat loss/gain strongly infl uences the soot nucleation, formation, and oxidation rates. The measurements and predictions of area-integrated soot volume fractions as a function of axial distance using three different methods of estimating the local temperatures are shown in Fig. 4 from ref. [93]. The calculations are offset by 20 mm so that soot growth starts at approximately the same location for both the measurements as well as the predictions. These three sets of calculations correspond to: (1) a fully coupled calculation where the local temperatures are obtained by considering the local energy loss or gain by radiation from gas molecules and soot particles, (2) an uncoupled calcula-tion where adiabatic fl ame temperatures are prescribed at all locations as a function of the local mixture fraction, and (3) an uncoupled calculation where the radiative heat loss fraction was assumed to be constant at 19% (close to the experimentally observed value) and temperatures are prescribed at all locations as a function of the local mixture fraction.

Qin et al. [95] characterized gravity and radiation effects on the structure of laminar methane-air partially premixed fl ames through detailed simulations. They modeled radiation using an optically thin assumption that provides a limiting value for the radiation heat transfer. The over-all effect of radiation on the structure of 1-g fl ame is less signifi cant than for the corresponding 0-g fl ame. Due to radiation effects, the fl ame heights of 0-g fl ames increase, and the heat release rate intensity near the premixed reaction zone tip decreases. When radiation effects are not included in the simulations, the peak temperatures are nearly the same for the 1-g and 0-g fl ames. With radiation the difference in these temperatures is signifi cant. The decrease in the peak tem-perature due to radiation for their 0-g fl ame is nearly fi ve times larger than for the 1-g fl ame. The value of the radiation loss fraction for 0-g fl ames without cofl ow can be as large as 50%, although

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(a) experiment (b) with radiation SNBCK/DOM 7.966 ppm

(d) without gas radiation 8.603 ppm

(e) without soot radiation 7.930 ppm

(f) without radiation 8.327 ppm

(c) with radiation: optically thin 7.805 ppm8.021 ppm

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Figure 3: Comparison of measured and predicted soot volume fraction distributions with the peak values indicated. (Reprinted with permission from Elsevier [84]. Copyright 2002 Elsevier.)




it drops signifi cantly in the presence of a cofl ow. While the fl owfi elds upstream of the inner pre-mixed reaction zone are nearly identical for 1-g and 0-g double fl ames, they are markedly differ-ent in the regions between the two reaction zones as well as downstream of the outer nonpremixed reaction zone. Lock et al. [96] also discussed important aspects of thermal radiation effects in microgravity fl ames.

Gas radiative properties vary strongly with wavelength so that use of the gray-gas model or the OTA often introduces large errors. Although LBL models provide very accurate results of spec-tral radiation heat transfer, they are not practical for coupled multi-dimensional calculations of radiation, fl uid fl ow, and chemical reactions due to large computational time required. In the absence of LBL results, results of the SNB model are often suffi ciently accurate, even for bench-mark solutions for the evaluation of other approximate nongray-gas radiation models. Direct implementation of the SNB model in multiple dimensions is computationally very demanding since it is formulated in terms of the transmissivity instead of the absorption coeffi cient [97].

Goutière et al. [98] have shown that the SNBCK method is an effi cient alternative to the SNB model with essentially the same accuracy. The computational time for the standard SNBCK method (using a bandwidth of 25 cm−1 and seven-point Gauss−Lobatto quadrature), though substantially smaller than for the SNB model (6 times), is still quite large, especially in multi-dimensional prob-lems involving two or more radiating gases. In several subsequent studies, Liu et al. [88, 90, 91] considerably improved the computational effi ciency of the SNBCK method with only minimal loss of accuracy by developing approximate treatments for overlapping bands, introducing the band-lumping strategy, and using an optimized quadrature set. Band models of different band-widths can be formulated by easily lumping different numbers of narrow bands. In addition, the order of Gauss quadrature used in SNBCK calculations can also be readily changed. The improved SNBCK

0 for calculations

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Figure 4: Axial variation of the measured and predicted soot volume fraction in a methane/air diffusion fl ame using different radiation models. (Reprinted with permission from El-sevier Science [93]. Copyright 1997 by The Combustion Institute.)




method also offers greater fl exibility to accommodate different levels of compromise between effi -ciency and accuracy without reformulation of the model. Recently, Goutière et al. [99] proposed the so-called optimized band-lumping (regrouping) strategy based on the four large absorbing bands of CO2. This band-lumping method offers better computational effi ciency than the uniform lumping strategy originally proposed by Liu et al. [87, 100]. The spectral-line-based weighted-sum-of-gray-gases model developed by Denison and Webb [31, 32], though computationally more effi cient, can be less accurate than the SNBCK methods in two-dimensional gas radiation calcula-tions [98, 99].

The effects of radiative transfer on the structure and extinction limits of counterfl ow H2/O2/N2 diffusion fl ames have been investigated [101]. The radiative properties of the main emitting spe-cies, H2O and OH in these fl ames were computed using an SNB model. As expected radiative losses decrease the fl ame temperature and width, and signifi cantly reduce the minor species (such as NO) that are sensitive to temperature. The amplitude of the radiative effects increases with the ratio of the global radiative loss to chemical heat release. Quick estimation of this parameter is useful for determining whether radiative transfer can be neglected in the prediction of small-scale fl ames. No effect of radiation on the extinction of high-strain rate extinction limit was found in the study, but suggested a value for the radiation-limited extinction for low strain rate fl ames.

An early review of radiation from turbulent fl ames [102] suggests that the calculation of ther-mal radiation in turbulent combustion involves three key factors: the solution of the RTE, the spectral variation of radiative properties, and the evaluation of turbulence−radiation interactions. Because of the inherent diffi culties in radiation calculations, the common practice for turbulent fl ames has been to neglect turbulence−radiation interactions and to use the OTA or a gray-me-dium assumption, even for luminous sooting fl ames [103−105]. Although the OTA is useful, its application produces errors, since it neglects self-absorption effects [106, 107]. Radiation, along with soot, is such a complex and nonlinear phenomena that gray or OTA models generally are inadequate to describe it, particularly in oxygen-enriched fl ames. Wang et al. [108] implemented two radiation models with self-absorption effects (one that accounts for nongray-gas properties and the other that does not) for an oxygen-enriched, propane-fueled, turbulent, nonpremixed jet fl ame.

One important aspect for combustion studies is the prediction of radiative transport due to particles and gases within fl ames. The RTE is generally used to solve for the radiative heat trans-fer in a semi-transparent media. In order to make proper calculations, information regarding the radiative properties of the medium must be calculated. Soot, or the polycyclic aromatic hydro-carbon particulates formed during coal combustion, can be a major contributor to radiative trans-port. Some calculations indicate that soot can contribute as much as 15% to the total radiative fl ux [109]. Other research indicates that temperature predictions in certain combustor locations can vary by as much as several hundred degrees kelvin based on the inclusion of soot predictions [110, 111]. While this may not be important for the overall energy balance in a combustor, it could be signifi cant for pollutant calculations. Minor changes in temperatures can make a sub-stantial impact on pollutant formation predictions, which tend to be highly temperature-depen-dent. Since the near burner region is of great interest for pollutant prediction calculations (and also tends to be the region of maximum soot concentration), predicting the radiative contribution of soot becomes even more important.

Adams and Smith [110] developed a simple empirical model for turbulent soot formation that related the soot volume fraction to the local equivalence ratio. Soot was assumed to exist where the local equivalence ratio was unity and above, increasing linearly to a maximum value at an equivalence ratio of 2.0 and above. The maximum soot concentration value was calculated as a




direct function of the amount of volatile carbon calculated to exist at that point. The soot volume fraction was then related to the radiative properties. They concluded that the inclusion of a soot radiation model increases predicted radiative transfer; however, the maximum local temperature difference between predictions with and without the soot model was lower than expected (about 50 K). Ahluwalia and Im [109] developed a similar model, and reported that soot signifi cantly enhanced the radiative heat transfer. Both models use an empirical survivability constant of 0.1 which serves to reduce the predicted soot concentrations. A detailed discussion on coal combus-tion and gasifi cation can be found in ref. [112]. Another important aspect is study of the fl ames immersed in porous media. There are many interesting radiation problems in such system. Inter-ested readers can consult the excellent review paper by Howell et al. [113].

5 Radiation from fi res

Two types of fi re analyses are mainly found in the fi re safety literature: for compartment fi res (also called enclosure fi res) and for outdoor fi res (such as forest or wildland fi res). Although ther-mal radiation plays a signifi cant role in these phenomena, radiative mechanisms are often only grossly accounted for in the associated models [3]. Fire simulations use fl uid dynamic models for chemically reacting and radiating fl ows. Due to the complexity of the phenomena, fi re model-ing has been carried out with different degrees of simplifi cation ranging from strictly empirical models to correlations that are variously based on dimensional analysis, semi-empirical models, and theoretical models.

The trend in fi re simulations is to use software packages based on CFD codes for chemically reacting fl ows. A number of fi re simulation codes were developed in the 1980s of which some are based on commercially-available codes [114, 115]. In fi re modeling, the spread of heat and smoke in a complex geometry is required. The complex structure of the combustion products, composed of molecular gases and soot particles, makes the treatment of radiative transport very diffi cult. Hence, researchers initially used grossly simplifi ed models for combustion and/or radiation [116].

The fi re dynamics simulator (FDS) is most popular among these codes. It uses the large eddy simulation (LES) FVM with improved radiation and combustion models [117−119]. An FVM submodel solves the RTE by using a narrow band gas radiation model (RADCAL). There have been many enhancements to RADCAL after its earlier development, e.g. Fuss et al. [120, 121] incorporated high temperature fl ammable gases into RADCAL.

A more sophisticated radiative submodel is available in the JASMINE code [114], which is based on the discrete transfer method (DTM) combined with a simplifi ed model of gas radiative properties. The radiation model works in conjunction with a SIMPLE pressure correction algo-rithm and the standard k−ε method to compute turbulence. The simulation of fi res in enclosures (SOFIE) code was developed to predict fi re propagation in enclosures [122]. It uses either a gray six-fl ux algorithm or a DTM combined with a WSGG to solve the RTE [123].

A Monte Carlo solution method for the gray-formulated RTE has been implemented in a 3D CFD model to investigate compartment fi res. Consalvi et al. [124] developed a CFD FVM code to simulate fi res in enclosures with internal obstacles or partitions. They used a modifi ed Chai et al. blocked-off discrete ordinates model [56] to solve a gray formulation of the RTE. The FDS code has also been used recently with some modifi cations in the combustion and soot models [125] and by incorporating the computation of the radiative fl ux emitted towards the fuel surface and upstream of it [126].

The structure of soot and its radiative properties are important parameters for analyses of fi res. These radiative properties have been extensively investigated by Faeth and coworkers in the




context of fi re modeling [127−131]. The work by Sivathanu et al. [132] on the in situ determina-tion of the absorption coeffi cient of soot particles represents another contribution. The estimation of optical and radiative properties of soot can be carried out using the Rayleigh−Debye−Gans theory for fractal aggregates with acceptable accuracy [133].

Gas radiative properties are also important parameters for fi re modeling. The use of LBL cal-culations is formidable for fi re applications since the absorption spectra of most combustion gases contain a large number of spectral lines for evaluating gas radiative properties. Modak [134, 135] proposed a method for computing gray gas radiative properties for isothermal, homo-geneous mixtures of carbon dioxide, water vapor, and soot. This method is popular due to its simplicity and reasonable accuracy and is still being used in a number of fi re simulation codes. Since the isothermal assumption is not valid in large-scale fi res, Modak’s model might not pro-vide highly accurate results for such cases [136]. Grosshandler [137] proposed remedial correc-tions to Modak’s simplifi ed method and suggested an SNB model based upon the Goody approach [15] and the Curtis−Godson approximation to deal with nonhomogeneous effects. His code uses the data of Ludwig et al. [18] and the parameters for the primary infrared bands of water vapor and carbon dioxide are computed according to Malkmus [138, 139].

Grosshandler’s SNB approach used in conjunction with the DTM were recently evaluated together with three other gas radiative property models, Modak’s approach [135], Truelove’s mixed gray gas model [85], and Edward’s WBM [24]. The test cases were an idealized 1D case with parabolic profi les of temperature, CO2 and H2O concentrations, and two experimental cases, i.e. a laboratory-scale fl ame and a fi eld-scale natural gas jet fi re [140]. The comparison in terms of radiative fl uxes shows a satisfactory level of agreement. However, the performances of all models are diminished when compared to the narrow band model while dealing with complex non-Cartesian meshes. Encouraged by this, researchers have developed ‘fast narrow band mod-els (FASTNB)’ [141, 142]. Used in conjunction with the DTM, FASTNB [143] is 20 times faster than RADCAL for comparable results (and deviations smaller than 1%).

Other useful alternatives to SNB approaches that are based on the absorption coeffi cient con-cept are the WSGG model, CK model, and their recently improved versions (SLW, ADF, ADFFG, FSCK). Detailed presentations of these models can be found in Modest [7]. Most of these predict small/medium scale fi re behavior (like that of compartment fi res) satisfactorily. The large num-ber of grid points required for large/outdoor fi res requires a large amount of computational time and this, in turn, limits the complexity of the radiation model that is employed [144]. Discrete Transfer Models are therefore less attractive to solve the RTE for large fi res and thus simplifi ca-tions are required in the associated radiation models. Usually, the radiation models employed in case of large fi res are gray and scattering is generally neglected [3].

An overview of the recent research on wildland and forest fi res has been provided by Albini [145] and regarding which Grishin and coworkers have made noteworthy contributions [146−148]. Radiation has been unsurprisingly identifi ed as the controlling heat transfer mechanism that fi xes the rate of spread of wildland fi res [149−153]. Fire growth models are now being used in campaign fi re strategic planning. Research is also being performed to develop a gray nonscat-tering DOM for solving the RTE and couple it to a CFD model for wildland fi re propagation [154, 155].

Research is also focused on developing radiation models to simulate fi re spread in urban areas. Mainly due to the complex geometries, most such models are zonal. Radiative transfer is approxi-mated by means of gray gas, gray surface emissivities, and radiative exchange areas or view factors between the cells [156]. The urban−wildland interface fi re is another subject of concern. Again, radiative transfer is mostly treated through rough simplifi ed models. Predictions of the impact of fi re on structures in the urban−wildland interface have been obtained using sophisticated simulation




tools, such as DOM with SLW or gray nonscattering gas models, that have been coupled with CFD codes [157, 158]. The gray gas model (as compared to SLW) has been found to be reasonably accurate. On the other hand, savings in computational time (six times faster than SLW) allow the use of this model as a fi re operational tool [157, 158]. Simeoni et al. [159] reached a compromise between rapidity and accuracy, by developing a semi-physical model of fi re spread across a fuel bed, which is unsteady and two-dimensional along the ground shape. In this model, they assumed that radiation is the prevailing heat transfer mechanism involved in fi re spread [160]. Nevertheless, it was unable to correctly predict the high wind effects on the rate of spread.

6 Summary

Thermal radiation is an important and often dominant heat transfer mode during many combus-tion processes. However, the complexities of the RTE and the associated radiative properties have made the modeling effort quite challenging. Among the different gas property models, the LBL method makes computations time intensive and it is therefore mainly used to obtain benchmark results. Global models are very popular for their simplicity and acceptable accuracy. However, a compromise between the two can be found in spectral band models and their differ-ent variations which have found extensive use by the researchers.

The radiative and optical properties of entrained particles (like soot or ash) play an important role during combustion. These properties are usually found using the Mie theory which, how-ever, is not valid for particle aggregates. Hence other theories such as the Rayleigh−Debye−Gans theory are employed to describe the radiative properties of particulate aggregates.

Various types of solution methods are used to solve the integro-differential RTE. It is not pos-sible to develop a single solution method that is equally applicable for different systems. There-fore, several solution methods (with varying degrees of approximation) have been developed and can be applied according to the nature of the physical system, characteristic of the medium, the degree of accuracy required, and the availability of computer facilities.

References

[1] Viskanta, R., Overview of convection and radiation in high temperature gas fl ows. Intl. J. Engg. Sci., 36, pp. 1677−1699, 1998.

[2] Maruta, K., Yoshida, M., Guo, H., Ju, Y. & Niioka, T., Extinction of Low-stretched Dif-fusion Flame in Microgravity. Combust. Flame, 112, pp. 181−187, 1998.

[3] Sacadura, J.F., Radiative heat transfer in fi re safety science. J. Quant. Spec. Radiat. Transf., 93, pp. 5−24, 2005.

[4] Viskanta, R. & Menguc, M.P., Radiation heat-transfer in combustion systems. Prog. En-ergy Combust. Sci., 13, pp. 97−160, 1987.

[5] Eaton, A.M., Smoot, L.D., Hill, S.C. & Eatough, C.N., Components, formulations, solu-tions, evaluation, and application of comprehensive combustion models. Prog. Energy Combust. Sci., 25, pp. 387−436, 1999.

[6] Kuo, K.K., Principles of Combustion, Wiley: New York, 1986. [7] Modest, M.F., Radiative Heat Transfer, Elsevier Science: USA, 2003. [8] Modest, M.F. & Zhang, H., The full-spectrum correlated-k distribution and itsrelation-

ship to the weighted-sum-of-gray gases-method. Proceedings of the 2000 IMECE, Vol. HTD-366−1, ASME: Orlando, FL, pp. 75−84, 2000.




[9] Hottel, H.C. & Sarofi m, A.F., Radiative Transfer, McGraw-Hill: New York, 1967. [10] Lallemant, N., Sayre, A. & Weber, R., Evaluation of emissivity correlations for H2O-CO2−

N2/air mixtures and coupling with solution methods of the radiative transfer equation. Prog. Energy Combust. Sci., 22, pp. 543−574, 1996.

[11] Ozisik, M.N., Radiative Transfer and Interactions with Conduction and Convection, Wi-ley: New York, 1973.

[12] Rothman, L.S., Gamache, R.R., Tipping, R.H., Rinsland, C.P., Smith, M.A.H., Chris Benner, D., Malathy Devi, V., Flaud, J.-M., Camy-Peyret, C., Perrin, A., Goldman, A., Massie, S.T., Brown, L.R. & Toth, R.A., The HITRAN molecular database: editions of 1991 and 1992. J. Quant. Spec. Radiat. Transf., 48, pp. 469−507, 1992.

[13] Taine, J., A line-by-line calculation of low-resolution radiative properties of CO2-CO-transparent nonisothermal gases mixtures up to 3000 K. J. Quant. Spec. Radiat. Transf., 30, pp. 371−379, 1983.

[14] Hartmann, J.-M., Levi di Leon, R. & Taine, J., Line-by-line and narrow-band statistical model calculations for H2O. J. Quant. Spec. Radiat. Transf., 32, pp. 119−127, 1984.

[15] Goody, R.M. & Yung, Y.L., Atmospheric Radiation, Oxford University Press: Oxford, 1989. [16] Scutaru, D., Rosenmann, L. & Taine, J., Approximate intensities of CO2 hot bands at 2.7

m, 4.3 m and 12 m for high-temperature and medium resolution applications. J. Quant. Spec. Radiat. Transf., 52, pp. 765−781, 1994.

[17] Riviere, Ph., Soufi ani, A. & Taine, J., Correlated-k fi ctitious gas model for H2O infrared radiation in the Voigt regime. J. Quant. Spec. Radiat. Transf., 53, pp. 335−346, 1995.

[18] Ludwig, D.B., Malkmus, W., Reardon, J.E. & Thomson, J.A.L., Handbook of Infrared Radiation from Combustion Gases, NASA SP3080, 1973.

[19] Soufi ani, A., Hartmann, J.M. & Taine, J., Validity of band-model calculations for CO2 and H2O applied to radiative properties and conductive-radiative transfer. J. Quant. Spec. Radiat. Transf., 33, pp. 243−257, 1985.

[20] Phillips, W.J., Band-model parameters of the 2.7 µm band of H2O. J. Quant. Spec. Ra-diat. Transf., 43, pp. 13−31, 1990.

[21] Phillips, W.J., Band-model parameters for 4.3 µm CO2 band in the 300−1000 K tempera-ture region. J. Quant. Spec. Radiat. Transf., 48, pp. 91−104, 1992.

[22] Brosmer, M.A. & Tien, C.L., Thermal radiation properties. of acetylene. J. Heat Transf., 107, pp. 943−948, 1985.

[23] Brosmer, M.A. & Tien, C.L., Infrared radiation properties of methane at elevated tem-peratures. J. Quant. Spec. Radiat. Transf., 33, pp. 521−532, 1985.

[24] Edwards, D.K., Molecular gas band radiation. Advances in Heat Transfer, Vol. 12, eds T.F. Irvine, Jr. & J.P. Harnett, Academic Press: New York, 1976.

[25] Edwards, D.K. & Menard W.A., Comparison of models for correlation of total band ab-sorption (Spectral band absorption vs mass path length and pressure, using models based on goody model). Appl. Optics, 3, pp. 621−625, 1964.

[26] Edwards, D.K., Numerical Methods in Radiation Heat Transfer. Proceedings of the 2nd Symposium on Numerical Properties and Methodologies in Heat Transfer, ed. T.M. Shih, Hemisphere Publishing, pp. 479−496, 1983.

[27] Lee, P.Y.C., Hollands, K.G.T. & Raithby, G.D., Reordering the absorptioncoeffi cient within the wide band for predicting gaseous radiant exchange. J. Heat Transf., 118, pp. 394−400, 1996.

[28] Tang, K.C. & Brewster, M.Q., Analysis of molecular gas radiation: real gas property ef-fects. ASME Proceedings of the 7th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, ed. B.F. Armaly, Vol. 1, pp. 32−32, 1998.




[29] Lacis, A.A. & Oinas, V., J. A description of the correlated-k distribution method for mod-eling nongray gaseous absorption, thermal emission, and multiple scattering in vertically inhomogeneous atmospheres. Geophys. Res., 96, pp. 9027−9063, 1991.

[30] Denison, M.K., A Spectral Line-Based Weighted Sum of Gray Gases Model for Arbitrary RTE Solvers, PhD Dissertation, Department of Mechanical Engineering, Brigham Young University, Provo, UT, 1994.

[31] Denison, M.K. & Webb, B.W., An absorption-line blackbody distribution function for effi cient calculation of total gas radiative transfer. J. Quant. Spec. Radiat. Transf., 50, pp. 499−510, 1993.

[32] Denison, M.K. & Webb, B.W., k-distribution and weighted-sum-of-gray-gases − a hy-brid model. J. Heat Transf., 2, pp. 19−24, 1994.

[33] Modest, M.E., The weighted-sum-of-gray-gases model for arbitrary solution methods in radiative transfer. J. Heat Transf., 113, pp. 650−656, 1991.

[34] Brewster, M.Q. & Kunimoto, T., The optical constants of coal, char, and limestone. J. Heat Transf., 106, pp. 678−683, 1984.

[35] Baxter, L.L., Fletcher, T.H. & Otteson, D.K., Spectral emittance measurements of coal particles. Energy and Fuels, 2, pp. 423−430, 1988.

[36] Van De Hulst, H.C., Light Scattering by Small Particles, Dover: New York, 1981. [37] Kerker, M., The Scattering of Light, Academic Press: New York, 1969. [38] Crosbie, A.L. & Davidson, B.W., Dirac-delta function approximations to the scattering

phase function. J. Quant. Spec. Radiat. Transf., 33, pp. 391−409, 1985. [39] Siegel, R. & Howell, J.R., Thermal Radiation Heat Transfer, Hemisphere: Washington,

DC, 1981. [40] Crowe, C.T., Gas-particle fl ow (Chapter 6). Pulverized Coal Combustion and Gasifi ca-

tion, eds L.D. Smoot & D.T. Pratt, Plenum: New York, 1979. [41] Ezekoye, O.A. & Zhang, Z., Soot oxidation and agglomeration processes in a micrograv-

ity diffusion fl ame. Combust. Flame, 110, pp. 127−139, 1997. [42] Haji-Sheikh, A., Monte carlo methods. Handbook of Heat Transfer, eds W.J. Minkow-

ycz, E.M. Sparrow, R.H. Pletcher & G.E. Schneider, Wiley: New York, 1999. [43] Taniguchi, H. & Funazu, M., The numerical analysis of temperature distributions in a

three dimensional furnace. Bull JSME, 13, pp. 1458−1468, 1970. [44] Xu, X.C., Mathematical Modeling of Three-Dimensional Heat Transfer from the Flame

in Combustion Chamber. Proc. Combust. Inst., 18, pp. 1919−1926, 1981. [45] Gupta, R.P., Wall, T.F. & Truelove, J.S., Radiative scatter by fl y ash in pulverized-coal-

fi red furnaces: application of the Monte Carlo method to anisotropic scatter. Intl. J. Heat Mass Transf., 26, pp. 1649−1660, 1983.

[46] Hottel, H.C. & Cohen, E.S., Radiant Heat Exchange in a Gas-Filled Enclosure: Allow-ance for Nonuniformity of Gas Temperature. AIChE J., 4, pp. 3−14, 1958.

[47] Smoot, L.D. & Smith, P.J., Coal Combustion and Gasifi cation, Plenum: New York, 1985.

[48] Fiveland, W.A., Discrete-ordinates solutions of the radiative transport equation for rect-angular enclosures. J. Heat Transf., 106, pp. 699−706, 1984.

[49] Chandrasekhar, S., Radiative Transfer, Dover: New York, 1960. [50] Carlson, B.G. & Lathrop, K.D., Transport Theory: The Method of Discrete-Ordinates.

Computing Methods in Reactor Physics, eds H. Greenspan, C. Kelber & D. Okrent, pp. 171−266, Gordon and Breach: New York, 1968.

[51] Lewis, E.E. & Miller, Jr., W.F., Computational Methods of Neutron Transport, Wiley: New York, 1984.




[52] Fiveland, W.A., Discrete ordinates methods for predicting radiative heat transfer in axy-metric enclosure. ASME Paper No. HT-20, 1982.

[53] Fiveland, W.A., Three-dimensional radiative heat-transfer solutions by the discrete ordi-nates method. J. Thermophys. Heat Transf., 2, pp. 309−316, 1988.

[54] Raithby, B.D. & Chui, E.H., A fi nite-volume method for predicting a radiant heat transfer in enclosures with participating media. J. Heat Transf., 112, pp. 415−423, 1990.

[55] Chui, E.H. & Raithby, G.D., Computation of radiant heat transfer on a nonorthogonal mesh using the fi nite-volume method. Num. Heat Transf., Part B, 23, pp. 269−288, 1993.

[56] Chai, J.C., Lee, H.S. & Patankar, S.V., Finite volume method for radiation heat transfer. J. Thermophys. Heat Transf., 8, pp. 419−425, 1994.

[57] DeMarco, A.G. & Lockwood, F.C., A new fl ux model for the calculation of radiation in furnaces. La Rivista dei Combustible, 29, pp. 184−204, 1975.

[58] Lockwood, F.C. & Shah, N.G., Heat Transfer, Hemisphere: Washington, DC, 1978. [59] Higenyi, J. & Bayazitoglu, Y., Differential approximation of radiactive heat transfer in a gray

medium, J Heat Transfer. J. Heat Transf., 102, pp. 719−723, 1980. [60] Ratzel, III, A.C. & Howell, J.R., Two-dimensional radiation in absorbing-emitting media

using the P-N approximation. J. Heat Transf., 105, pp. 333−340, 1983. [61] Menguc, M.P. & Viskanta, R., Radiative transfer in three-dimensional rectangular en-

closures containing inhomogeneous, anisotropically scattering media. J. Quant. Spec. Radiat. Transf., 33, pp. 533−549, 1985.

[62] Menguc, M.P. & Viskanta, R., Radiative Heat Transfer in Axisymmetric, FiniteCylindri-cal Enclosure. J. Heat Transf., 108, pp. 271−276, 1986.

[63] Fiveland, W.A. & Wessel, R.A., A numerical model for predicting performance of three-dimensional pulverized-fuel fi red furnaces. ASME Paper No. 86-HT-35, 1986.

[64] Gosman, A.D., Lockwood, F.C., Megahed, I.E.A. & Shah, N.G., Prediction of the fl ow, reaction, and heat transfer in a glass furnace. J. Energy, 6, pp. 353−360, 1982.

[65] Murthy, J.Y. & Choudhury, D., Developments in radiative heat transfer. ASME HTD Vol. 203, pp. 153−160, 1992.

[66] Liu, F., Guo, H. & Smallwood, G.J., Effects of radiation model on the modeling of a laminar cofl ow methane/air diffusion fl ame. Combust. Flame, 138, pp. 136−154, 2004.

[67] Kennedy, I.M., Yam, C., Rapp, D.C. & Santoro, R.J., Modeling and measurements of soot and species in a laminar diffusion fl ame. Combust. Flame, 107, pp. 368−382, 1996.

[68] Smooke, M.D., McEnally, C.S., Pfefferle, L.D., Hall, R.J. & Colket, M.B., Computa-tional and experimental study of soot formation in a cofl ow, laminar diffusion fl ame. Combust. Flame, 117, pp. 117−139, 1999.

[69] McEnally, C.S., Schaffer, A.M., Long, M.B., Pfefferle, L.D., Smooke, M.D., Colket, M.B. & Hall, R.J., Computational and Experimental Study of Soot Formation in a Cofl ow, Laminar Ethylene Diffusion Flame Proc. Combust. Inst., 27, pp. 1497−1505, 1998.

[70] Kaplan, C.R. & Kailasanath, K., Flow-fi eld effects on soot formation in normal and in-verse methane-air diffusion fl ames. Combust. Flame, 124, pp. 275−294, 2001.

[71] Walsh, K.T., Fielding, J., Smooke, M.D. & Long, M.B., Experimental and computational study of temperature, species, and soot in buoyant and non-buoyant cofl ow laminar dif-fusion fl ames. Proc. Combust. Inst., 28, pp. 1973−1979, 2000.

[72] McEnally, C.S., Pfefferle, L.D., Schaffer, A.M., Long, M.B., Mohammed, R.K., Smooke, M.D. & Colket, M.B., Characterization of a cofl owing methane/air nonpremixed fl ame with computer modeling, Rayleigh-Raman imaging, and online mass spectrometry. Proc. Combust. Inst., 28, pp. 2063−2070, 2000.




[73] Santoro, R.J., Semerjian, H.G. & Dobbins, R.A., Soot Particle Measurements in Diffu-sion Flames. Combust. Flame, 51, pp. 203−218, 1983.

[74] Villasenor, R. & Kennedy, I.M., Soot formation and oxidation in laminar diffusion fl ame. Proc. Combust. Inst., 24, pp. 1023−1030, 1992.

[75] Kennedy, I.M., Kollmann, W. & Chen, J.Y., A model for soot formation in a laminar dif-fusion fl ame. Combust. Flame, 81, pp. 73−85, 1997.

[76] Santoro, R.J., Yeh, T.T., Horvath, J.J. & Semerjian, H.G., The transport and growth of soot particles in laminar diffusion fl ames. Combust. Sci. Technol., 53, pp. 89−115, 1987.

[77] Hall, R.J., The radiative source term for plane-parallel layers of reacting combustion gases. J. Quant. Spec. Radiat. Transf., 49, pp. 517−523, 1993.

[78] Hall, R.J., Radiative dissipation in planar gas-soot mixtures. J. Quant. Spec. Radiat. Transf., 51, pp. 635−644, 1994.

[79] Garcia, C., Tanoff, M.A., Smooke, M.D. & Hall, R.J., Application of a Non-Optically Thin Radiation Model to the Study of Low-Strain Laminar Counterfl ow Flames. Eastern States Meeting of the Combustion Institute, Hilton Head, December 1996.

[80] Kent, J.H. & Honnery, D.R., A soot formation rate map for a laminar ethylene diffusion fl ame. Combust. Flame, 79, pp. 287−298, 1990.

[81] Magnussen, B.F. & Hjertager, B.H., On mathematical modeling of turbulent combus-tion with special emphasis on soot formation and combustion. Proc. Combust. Inst., 16, pp. 727−729, 1977.

[82] Son, Y., Honda, L., Ronney, P.D. & Gokoglu, S., Radiation-driven fl ame spread experi-ments in the combustion integrated rack on ISS. Conference and Exhibit on ISS Utiliza-tion, AIAA, Reston, VA, AIAA-2001−5082, pp. 1−12, 2001.

[83] Son, Y. & Ronney, P.D., Radiation-driven fl ame spread over thermally-thick fuels in qui-escent microgravity environments. Proc. Combust. Inst., 29, pp. 2587−2594, 2002.

[84] Liu, F., Guo, H., Smallwood, G.J. & Gülder, Ö.L., Effects of gas and soot radiation on soot formation in a cofl ow laminar ethylene diffusion fl ame. J. Quant. Spec. Radiat. Transf., 73, pp. 409−421, 2002.

[85] Truelove, J.S., Evaluation of a multi-fl ux model for radiative heat transfer in cylindrical furnaces. AERE-R-9100, AERE, Harwell, UK, 1978.

[86] Thurgood, C.P., Pollard, A. & Becker, H.A. &, The T[N] quadrature set for the discrete ordinates method. J. Heat Transf., 117, pp. 1068−1070, 1995.

[87] Liu, F., Smallwood, G.J. & Gülder, O.L., Radiation heat transfer calculations using the SNBCK method. 33rd Thermophysics Conference, Norfolk, VA, AIAA 99−3679, 1999.

[88] Liu, F., Smallwood, G.J. & Gülder, Ö.L., Band lumping strategy for radiation heat trans-fer calculations using a narrowband model. J. Thermophys. Heat Transf., 14, pp. 278−281, 2000.

[89] Soufi ani, A. & Taine, J., High temperature gas radiative property parameters of statistical narrow-band model for H2O, CO2 and CO, and orrelated-k model for H2O and CO2. Intl. J. Heat Mass Transf., 40, pp. 987−991, 1997.

[90] Liu, F., Smallwood, G.J. & Gülder, Ö.L., Application of the statistical narrow-band cor-related-K method to non-grey gas radiation in CO2−H2O mixtures: approximate treat-ments of overlapping bands. J. Quant. Spec. Radiat. Transf., 68, pp. 401−417, 2001.

[91] Liu, F., Smallwood, G.J. & Gülder, Ö.L., Application of the statistical narrow-band cor-related-k method to low-resolution spectral intensity and radiative heat transfer calcula-tions Ð e.ects of the quadrature scheme. Intl. J. Heat Mass Transf., 43, pp. 3119−3135, 2000.




[92] Ju, Y., Guo, H., Liu, F. & Maruta, K., Effects of the Lewis number and radiative heat loss on the bifurcation and extinction of CH4/O2−N2−He fl ames. J. Fluid Mech., 379, pp. 165−190, 1999.

[93] Sivathanu, Y.R. & Gore, J.Y., Effects of gas-band radiation on soot kinetics in laminar methane/air diffusion fl ames. Combust. Flame, 110, pp. 256−263, 1997.

[94] Sivathanu, Y.R. & Gore, J.P., Coupled radiation and soot kinetics calculations in laminar acetylene/air diffusion fl ames. Combust. Flame, 97, pp. 161−172, 1994.

[95] Qin, X., Puri, I.K., Aggarwal, S.K. & Katta, V.R., Gravity, radiation and cofl ow effects on partially premixed fl ames. Phys. Fluids, 16, pp. 2963−2974, 2004.

[96] Lock, A.J., Ganguly, R., Puri, I.K., Aggarwal, S.K. & Hegde, U., Gravity Effects on Par-tially Premixed Flames: An Experimental-Numerical Investigation. Proc. Combust. Inst., 30, pp. 511−518, 2005.

[97] Liu, F., Gülder, Ö.L., Smallwood, G.J. & Ju, Y., Non-grey gas radiative transfer analyses using the statistical narrow-band model. Intl. J. Heat Mass Transf., 41, pp. 2227−2236, 1998.

[98] Goutière, V., Liu, F. & Charette, A., An Assessment of Real Gas Modeling in 2D Enclo-sures. J. Quant. Spec. Radiat. Transf., 64, pp. 299−332, 2000.

[99] Goutière, V., Charette, A. & Kiss, L., Comparative performance of non-gray gas model-ing techniques. Num. Heat Transf., Part B, 41, pp. 361−381, 2002.

[100] Liu, F., Gülder, Ö.L. & Smallwood, G.J., Three-dimensional non-grey gas radiative trans-fer analysis using the statistical narrow-band model. Rev. Gen. Therm., 37, pp. 759−768, 1998.

[101] Jeewon, L., Parulekar, S.J., Daguse, T., Croonenbroek, T., Rolon, J.C., Darabiha, N. & Soufi ani, A., Study of radiative effects on laminar counterfl ow H2/O2/N2 diffusion fl ames. Combust. Flame, 106, pp. 271−287, 1996.

[102] Faeth, G.M., Gore, J.P., Chuech, S.G. & Jeng, S.M., Radiation from turbulent diffusion fl ames. Annual Review of Numerical Fluid Mechanics and Heat Transfer, Vol. 2, eds. C.L. Tien & T.C. Chawla, Hemisphere: Washington, DC, 1989.

[103] Bai, X.S., Balthasar, M., Mauss, F. & Fuchs, L., Detailed soot modeling in turbulent jet diffusion fl ames. Proc. Combust. Inst., 27, pp. 1623−1630, 1998.

[104] Gore, J.P. & Jang, J.H., Transient Radiation Properties of a Subgrid Scale Eddy. J. Heat Transf., 114, pp. 234−242, 1992.

[105] Pitsch, H., Riesmeier, E. & Peters, N., Unsteady fl amelet modeling of soot formation in turbulent diffusion fl ames. Combust. Sci. Technol., 158, pp. 389−406, 2000.

[106] Frank, J.H., Barlow, R.S. & Lundquist, C., Radiation and nitric oxide Formation in non-premixed jet fl ames. Proc. Combust. Inst., 28, pp. 447−454, 2000.

[107] Zhu, X.L., Gore, J.P., Karpetis, A.N. & Barlow, R.S., The effects of selfabsorption of ra-diation on an opposed fl ow partially premixed fl ame. Combust. Flame, 129, pp. 342−345, 2002.

[108] Wang, L., Haworth, D.C., Turns, S.R. & Modest, M.F., Interactions among soot, ther-mal radiation, and NOx emissions in oxygen-enriched turbulent nonpremixed fl ames: a computational fl uid dynamics modeling study. Combust. Flame, 141, pp. 170−179, 2005.

[109] Ahluwalia, R.K. & Im, K.H., Spectral radiative heat-transfer in coal furnaces using a hybrid technique. J. Inst. Energy, 67, p. 23−29, 1994.

[110] Adams, B.R. & Smith, P.J., Modeling effects of soot and radiative transfer in turbulent gaseous combustion. Combust. Sci. Tech., 109, pp. 121−140, 1995.




[111] Kollmann, W., Kennedy, I.M., Metternich, M. & Chen, J.-Y., Application of a Soot Mod-el to a Turbulent Ethylene Diffusion Flame. Springer Series in Chemical Physics, 59, pp. 503−526, 1994.

[112] Smoot, L.D. & Smith, P.J., Coal Combustion and Gasifi cation, Plenum Press: New York, 1985.

[113] Howell, J.R., Hall, M.J. & Ellzey, J.L., Combustion of hydrocarbon fuels with porous inert media. Prog. Energy Comb. Sci., 22, pp. 121−145, 1996.

[114] Markatos, N.C., Malin, M.R. & Cox, G., Mathematical modeling of buoyancy induced smoke fl ow in enclosures. Intl. J. Heat Mass Transf., 25, pp. 63−75, 1982.

[115] Galea, E.R. & Markatos, N.C., The mathematical modeling and computer simulation of fi re development in aircraft. Intl. J. Heat Mass Transf., 34, pp. 181−197, 1991.

[116] Jia, F., Galea, E.R. & Patel, M.K., The prediction of re propagation in enclosure fi res. Proceedings of the 5th International Symposium on Fire Safety Science, Melbourne, Australia, pp. 439−450, 1997.

[117] McGrattan, K.B., Floyd, J.E., Forney, G.P., Baum, H.R. & Hostikka, S., Improved radiation and combustion routines for a LES fi re model. Proceedings of the 7th In-ternational Symposium on Fire Safety Science, Worcester, MA, USA, pp. 827−438, 2002.

[118] McGrattan, K.B., Baum, H.R., Rehm, R.G., Hamins, A., Forney, G.P., Floyd, J.E. & Hostikka, S., Fire dynamics simulator (Version 2). Technical reference guide. National Institute of Standards and Technology. Report NISTIR 6783, 2001.

[119] Hostikka, S., McGrattan, K.B. & Hamins, A., Numerical modeling of pool fi res using LES and fi nite volume method for radiation. Proceedings of the 7th International Sym-posium on Fire Safety Science, Worcester, MA, USA, pp. 383−394, 2002.

[120] Fuss, S.P., Ezekoye, O.A. & Hall, M.J., The Absorptance of Infrared Radiation by Meth-ane at Elevated Temperatures. J. Heat Transfer, 118, pp. 918−923, 1996.

[121] Fuss, S.P., Hall, M.J. & Ezekoye, O.A., Band-Integrated Infrared Absorptance of Low-Molecular-Weight Paraffi n Hydrocarbons at High Temperatures. Applied Optics, 38, pp. 2895−2904, 1999.

[122] Lewis, M.J., Moss, M.B. & Rubini, P., CFD modeling of combustion and heat transfer in compartment fi res. Proceedings of the 5th International Symposium on Fire Safety Sci-ence, Melbourne, Australia, pp. 463−474, 1997.

[123] Bressloff, N.W., Moss, J.B. & Rubini, P.A., CFD prediction of coupled radiation heat transfer and soot production in turbulent fl ames. Proc. Combust. Inst., 2, p. 2371−2386, 1996.

[124] Consalvi, J.L., Porterie, B. & Loraud, J.C., Method for computing the interaction of fi re and solid internal regions. Num. Heat Transf., Part A, 43, pp. 777−805, 2003.

[125] Wang, H.Y. & Joulain, P., Numerical simulation of wind-aided turbulent fi res in a ventilat-ed model tunnel. Proceedings of the 7th International Symposium on Fire Safety Science, Worcester, MA, USA, pp. 161−172, 2002.

[126] Legros, G., Etude du comportement radiatif d’une fl amme de diffusion en micropesan-teur (Study of the radiative behavior of a diffusion fl ame in microgravity). PhD thesis, Université de Poitiers, 2003.

[127] Koylu, U.O. & Faeth, G.M., Structure of overfi re soot in buoyant turbulent diffusion soot fl ames at long residence times. Combust. Flame, 89, pp. 140−156, 1992.

[128] Koylu, U.O. & Faeth, G.M., Radiative properties of fl ame-generated soot. J. Heat Transf., 115, pp. 409−417, 1993.




[129] Koylu, U.O. & Faeth, G.M., Optical properties of overfi re soot in buoyantturbulent diffu-sion soot fl ames at long residence times. J. Heat Transf., 116, pp. 152−159, 1994.

[130] Koylu, U.O. & Faeth, G.M., Spectral extinction coeffi cient of soot aggregates from turbulent diffusion fl ames. J. Heat Transf., 118, pp. 415−421, 1996.

[131] Wu, J.S., Krishnan, S.S. & Faeth, G.M., Refractives indices at visible wavelengths of soot emitted from buoyant turbulent diffusion fl ames. J. Heat Transf., 119, pp. 230−237, 1997.

[132] Sivathanu, Y.R., Gore, J.P., Jansen, J.M. & Senser, D.W., A study of in situ specifi c ab-sorption coeffi cients of soot particles in laminar fl ames. J. Heat Transf., 115, pp. 653−669, 1993.

[133] Baillis, D. & Sacadura, J.F., Thermal radiation properties of dispersed media: theoreti-cal prediction and experimental characterization. J. Quant. Spec. Radiat. Transf., 67, pp. 327−363, 2000.

[134] Modak, A.T., Thermal radiation from pool fi res. Combust. Flame, 29, pp. 177−192, 1977.

[135] Modak, A.T., Radiation from products of combustion. Fire Res., 1, pp. 339−361, 1978/79.

[136] De Ris J., Fire radiation−a review. Proc. Combust. Inst., 17, pp. 1003−1016, 1979.[137] Grosshandler, W.L., Transfer in non-homogeneous gases: a simplifi ed approach. Intl. J.

Heat Mass Transf., 23, pp. 1447−1459, 1980.[138] Malkmus, W., Infrared emissivity of carbon dioxide (2.7 µm band). General Dynamics-

Astronautics, AE63−0047, 1963.[139] Malkmus, W., Infrared emissivity of carbon dioxide (4.3 µm band). J. Opt. Soc. Am., 53,

pp. 951−961, 1963.[140] Cumber, P.S., Fairweather, M. & Ledin, H.S., Application of wide band radiation models

to non-homogeneous combustion systems. Intl. J. Heat Mass Transf., 4, pp. 1573−1584, 1998.

[141] Yan, Z. & Holmstedt, G., CFD simulation of upward fl ame spread over fuel surface. Pro-ceedings of the 5th International Symposium on Fire Safety Science, Melbourne, Australia, pp. 345−356, 1997.

[142] Yan, Z. & Holmstedt, G., Three-dimensional computation of heat transfer from fl ames between vertical and parallel walls. Combust. Flame, 117, pp. 574−578, 1999.

[143] Yan, Z. & Holmstedt, G., Fast narrow-band computer model for radiation calculations. Num. Heat Transf., Part B, 31, pp. 61−71, 1997.

[144] Baum, H.R. & McGrattan, K.B., Simulation of large industrial outdoor fi res. Proceed-ings of the 6th International Symposium on Fire Safety Science, Poitiers, France, pp. 611−622, 1999.

[145] Albini, F.A., An overview of research on wildland fi re. Proceedings of the 5th Interna-tional Symposium on Fire Safety Science, Melbourne, Australia, pp. 59−74, 1997.

[146] Grishin, A.M., Mathematical Modeling of Forest Fires, Publishing House of the Univer-sity of Tomsk, Russia, 1981.

[147] Grishin, A.M., Zverev, V.G. & Shevelev, S.V., Steady state propagation of top crown for-est fi re. Fizika Goreniya i Vzryva, 22, pp.101−108, 1986.

[148] Grishin, A.M. Mathematical modeling of forest fi res and new methods of fi ghting them. Ed. Publishing House of the University of Tomsk: Russia, 1997.

[149] Emmons, E.W., Fire in the forest. Fire Res. Abstr. Rev., 5, pp.163−178, 1964.[150] Hottel, H.C., Williams, G.C. & Steward, F.R., Modeling of fi re spread through a fuel bed.

Proc. Combust. Inst., 10, pp. 997−1009, 1964.




[151] Telitsin, H.P., Flame radiation as a mechanism of fi re spread in forests. Heat Transfer in Flames, eds. N.H. Afgan & J.M. Beer, Wiley: New York, pp. 441−449, 1974.

[152] Albini, F.A., Wildland fi re spread by radiation − a model including fuel cooling by natu-ral convection. Combust. Sci. Technol., 45, pp. 101−113, 1986.

[153] Carrier, G.F., Fendell, F.E. & Wolff, M.F., Wind-aided fi re spread across arrays of dis-crete fuel elements. Combust. Sci. Technol., 75, pp. 31−51, 1991.

[154] Porterie, B., Morvan, D., Loraud, J.C. & Larini, M., Firespread through fuel beds: mod-eling of wind-aided fi res and induced hydrodynamics. Phys. Fluids, 12, pp. 1762−1782, 2000.

[155] Morvan, D. & Dupuy, J.L., Numerical simulation of the propagation of a surface fi re through a Mediterranean schrub. Proceedings of the 7th International Symposium on Fire Safety Science, Worcester, MA, USA, pp. 557−567, 2002.

[156] Himoto, K. & Tanaka, T., A physically-based model for urban fi re spread. Proceed-ings of the 7th International Symposium on Fire Safety Science, Worcester, MA, USA, pp. 129−140, 2002.

[157] Porterie, B., Nicolas, S., Consalvi, J.L., Loraud, J.C., Giroud, F. & Picard, C., Modeling thermal impact of wildland fi res on structures in the urban interface—Part 1: Radiative and convective impact of fl ames representative of vegetation fi res. Num. Heat Transf., Part A, 47, pp. 471−489, 2005.

[158] Consalvi, J.L., Porterie, B., Nicolas, S., Loraud, J.C. & Kaiss, A., Modeling thermal im-pact of wildland fi res on structures in the urban interface—Part 2: Radiative impact of a flre front. Num. Heat Transf., Part A, 47, pp. 491−503, 2005.

[159] Simeoni, A., Santoni, P.A., Larini, M. & Balbi, J.H., Reduction of a multiphase formula-tion to include a simplifi ed fl ow in a semi-physical model of fi re spread across a fuel bed Intl. J. Thermal Sciences, 42, pp. 95−105, 2003.

[160] Morandini, F., Santoni, P.A. & Balbi, J.H., Analogy between wind and slope effects on fi re spread across a fuel bed - modelling and validation at laboratory scale. Proceedings of the 3rd International Seminar on Fire and Explosion Hazards, Edinburgh, Scotland, UK, 2000.



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