[16] MAZUMDER, S. and MODEST M.F. Application of the Full Spectrum Correlated-k Distribution...

Application of the Full Spectrum Correlated-k DistributionApproach to Modeling Non-Gray Radiation in

Combustion Gases

SANDIP MAZUMDER*CFD Research Corporation, Huntsville, AL 35805, USA

and

MICHAEL F. MODESTDepartment of Mechanical and Nuclear Engineering, The Pennsylvania State University,

University Park, PA 16802, USA

The treatment of radiative transport through combustion gases is rendered extremely difficult by the strongspectral variation of the absorption coefficients of molecular gases. In the full spectrum correlated-kdistribution (FSCK) approach, a transformation is invoked, whereby the radiative transfer equation (RTE) istransformed from wavenumber to non-dimensional Planck-weighted wavenumber space after reordering of thespectrum. The reordering results in a relatively smooth spectrum, allowing accurate spectral integration withvery few quadrature points. The numerical procedures, required to use the FSCK model for full-scalecombustion applications, have been outlined in this article. The FSCK model was first coupled with the DiscreteOrdinates Method (DOM) for solution of the transformed RTE. The accuracy of the model was then examinedfor a variety of cases ranging from homogeneous one-dimensional media to inhomogeneous multi-dimensionalmedia with simultaneous variations in both temperature and concentrations. Comparison with line-by-linecalculations shows that the FSCK model is exact for homogeneous media, and that its accuracy ininhomogeneous media is limited by the accuracy of the scaling approximation. Several approaches for effectivescaling of the absorption coefficient are examined. The model is finally used for radiation calculations in afull-scale combustor, with full coupling to fluid flow, heat transfer and multi-species chemistry. Thecomputational savings resulting from use of the FSCK model is found to be more than four orders of magnitudewhen compared with line-by-line calculations. 2002 by The Combustion Institute

NOMENCLATURE

a base function (Eq. 22)f absorption coefficient distribution

function (Eq. 2)g,g0 nondimensional Planck-weighted

wavenumberi partial Planck function (Eq. 15)I radiation intensity (W/m2/sr)Ib Planck function (W/m

2/sr)k recordered absorption coefficient

(m1 or m1 bar1)Lm mean beam length (m)s path length (m)T temperature (K)u scaling functionV volume (m3)

Greek

absorbtivity wavenumber (m1) absorption coefficient, linear or pressure

based (m1 or m1 bar1)p Planck-mean absorption coefficient,

linear or pressure based (m1 or m1

bar1)s scattering coefficient (m

1) transmissivity

Subscripts

ref global reference valueW wall (or boundary)

INTRODUCTION

Radiation is one of the dominant modes of heattransfer in most combustion and fire applica-* Corresponding author. E-mail: [email protected]

COMBUSTION AND FLAME 129:416438 (2002)0010-2180/02/$see front matter 2002 by The Combustion InstitutePII S0010-2180(02)00359-0 Published by Elsevier Science Inc.

tions. With an increasing trend towards cleanercombustion (lean combustion with little soot),accurate prediction of radiation from moleculargases is beginning to occupy a position of cen-tral importance in the design of gas-fired com-bustion appliances. Radiative heat transfer incombustion gases is rendered extremely com-plex because of the line structure exhibited bythe emission and absorption spectra of mostcombustion gases. Typical absorption spectra ofgases such as CO2, H2O, and CO containroughly 105 to 106 lines [13]. At combustiontemperatures ( 2000 K), the number of linespresent are well over a million [4, 5]. From amodeling perspective, this implies that to accu-rately predict radiative transport in flames, onewould have to solve the radiative transfer equa-tion (RTE) for a million or so spectral intervals.Such computations are referred to as line-by-line (LBL) calculations, and require extraordi-narily large amounts of computer memory andtime. The applicability of LBL computations[611] is currently limited to extremely simplegeometries, specifically designed for validationof reduced models only. For combustion appli-cations, where there are several additional keyissues to deal with (including turbulence andchemical reactions), radiation is only one smallpiece of the gigantic puzzle, and it is impracticalspending bulk of the computational time inperforming LBL calculations. Despite tremen-dous advances in computer technology, judgingby the time required for LBL calculations insimple one-dimensional gas layers, it is fair tosay that such calculations will not be feasible forthree-dimensional combustor geometries in theforeseeable future, and approximate models arenecessary.

Approximate non-gray models, used to date,are based on spectral averaging either overmany lines contained within a vibration-rotationband (wide-band models [3]), or over relativelyfewer lines (narrow-band models [1, 12, 13]).This averaging is necessary to reduce the num-ber of spectral intervals for which the RTE hasto be solved. Spectral averaging results in anexpression for the band emissivity and/or ab-sorptivity, commonly referred to as the bandmodel. These band models make use of simplestatistical relations based on the mean absorp-tion line-strength and half-width distributions

and require analytical integration of the RTEover small spectral intervals. The accuracy ofsuch band models, even for homogeneous me-dia, is largely determined by the narrowness ofthe bands, and the statistical models employedduring analytic integration within the band.Wide-band models are known to be inaccuratefor combustion applications, because of theinherent inaccuracy in treating the so-calledwings of the band, which usually requirespectral resolution much finer than a few tens ofwavenumbers [14]. On the other hand, althoughnarrow-band models can be sometimes quitefruitful, accurate modeling of radiation from aCO2-H2O mixture at high temperature wouldrequire several hundreds of narrow-bands. Fur-thermore, there is no universally applicableproven statistical model available for narrow-band integration. On account of all these short-comings, despite the existence of both narrow-band and wide-band models for more thanthree decades, there has been little success inmodeling radiative transport in combustiongases.

In recent years, there has been a lot of activityin trying to model non-gray radiation in molec-ular gases using global approaches, as opposedto band models. The correlated-k (CK) methodis based on the fact that inside a spectralinterval, which is sufficiently narrow to assume aconstant Planck function, the precise position ofa spectral line is unimportant for spectral inte-gration. If the medium is homogeneous or theabsorption coefficient obeys the so-called scal-ing approximation, the absorption coefficientcan be reordered into a smooth monotonicallyincreasing function. Because of the presence ofhot lines, the CK method is known to givepoor accuracy in cases with extreme tempera-ture variation. For such scenarios, Riviere andco-workers [10, 15, 16] developed the so-calledcorrelated-k fictitious gas model. Starting with ahigh-resolution database, they grouped linesaccording to the values of their lower energylevel and found the k-distribution for each ofthe fictitious gases, making the CK methodmore accurate when applied to each fictitiousgas separately than when applied to the real gas.They further assumed that the positions of linesbelonging to different classes are statisticallyuncorrelated. Unfortunately, the method only

417FSCK APPROACH TO MODELING NON-GRAY RADIATION

supplies the mean transmissivity for a gas layer,that is, it loses the most important advantages ofusing k-distributions, limiting its application tononscattering media within black enclosures.

Modest and Zhang [9] have demonstrated,based on analysis, that the CK method is, inessence, a close relative of the weighted-sum-of-gray gases (WSGG) method [17]. Denisonand Webb [7, 8, 18, 20] have improved consid-erably on the WSGG method by developing aspectral-line-weighted (SLW) WSGG modelbased on detailed spectral line data. They alsoextended the SLW model to inhomogeneousmedia by introducing a cumulative distributionfunction of the absorption coefficient, calcu-lated over the whole spectrum and weighted bythe Planck function [21]. The method developedby the French researchers [10, 11] is almostidentical to the SLW model, and differs only inthe way the fictitious gas weights are calculated.These weights are chosen in such a manner thatemission by an isothermal gas column is pre-dicted exactly. Pierrot et al. [11] developed aglobal fictitious gas model to improve the treat-ment for strongly inhomogeneous media. Theideas used to develop the full spectrum corre-lated-k distribution (FSCK) model are derived,in part, from the work of Webb and co-workersand the French group.

In this work, the FSCK model has beenvalidated extensively for multidimensional ge-ometry with combined temperature and concen-tration discontinuities. Several different scalingtechniques have been examined to determinewhich of these is the most accurate. Finally, themodel has been used for radiation calculationsin a full-scale combustor with full coupling tothe conservation equations for mass, momen-tum, energy, and species. The article advancesthe state of the art by demonstrating the feasi-bility of using the FSCK model for full-scalecalculation of radiative transport in combustionsystems.

THEORY

Spectral Reordering and k-distribution

Consider a participating molecular gas boundedwithin a black-walled enclosure. In such a situ-

ation, the spectral intensity at any point withinthe gas depends on the temperature (throughthe Planck function) and the spectral absorptioncoefficient of the gas. Over a small spectralinterval (so-called narrow-band), such as afew tens of wavenumbers, the Planck function isessentially constant. Thus, across such a smallspectral interval, the spectral intensity is a func-tion of the absorption co-efficient only. Figure 1shows the absorption coefficient of CO2 across avery small part of its 4.3 m band. It is clearfrom this figure that the absorption coefficientvaries strongly even within such a narrow spec-tral interval. It attains the same value severaltimes within the same spectral interval, eachtime producing an identical spectral intensityfield within the medium. This implies that line-by-line calculations of the radiation intensitywould be wasteful, because the same calcula-tions would be performed several times. If theabsorption coefficient can be somehow reor-dered such that the intensity calculations areperformed only once for each absorption coef-ficient value, the resulting computations wouldbe immensely more efficient without sacrificingthe accuracy. Although this reordering conceptwas first reported in the western literature aboutthree decades ago by Arking and Grossman[22], it has received significant attention withinthe atmospheric science and heat transfer com-munities only in the last decade or so [6, 14, 23,24].

The narrow-band transmissivity (i.e., spectralaverage over a narrow-band) of a homogeneousgas layer of width X, is written as:

Fig. 1. Pressure-based absorption coefficient of a small partof the CO2 4.3 m band at 1500K computed using theHITEMP database [5].

418 S. MAZUMDER AND M. F. MODEST

X 1

ekXd (1)

Let us now define a function f (k), such that

fk 1

k d (2)

where (k ) is the Dirac-delta function.Substituting Eq. 2 into Eq. 1, the narrow-bandtransmissivity may be re-written as:

X 0

ekXfkdk (3)

Noting that, in the vicinity of each point where k, we can replace d by (d/d)d, theintegration in Eq. 2 over gives a weightedsum of the number of points where k [6]:

fk 1

i

dd

i

(4)

From Eq. 4, it is clear that for every maximumor minimum of the absorption coefficient ,the function f(k) will assume a value of infinity,because d/d 0. This implies that using Eq.3 for integration is numerically not amenable.The problem can be circumvented by using theintegral of f(k), which happens to be a smoothfunction. Equation 3 can then be written as:

X 0

ekXfkdk 0

1

ek gXdg (5)

where the so-called cumulative distribution func-tion g(k) is defined as [6]:

gk 0

k

f kdk (6)

It is evident from Eq. 5 that computation of thetransmissivity requires evaluation of k(g). The

function k(g) is commonly referred to as thek-distribution, and is obtained by numericalinversion of Eq. 6. Algorithms for numericalevaluation of the k-distribution will be discussedin sections to follow.

FSCK Model

The above arguments for reordering hold only ifthe Planck function does not vary within thespectral interval in question. While this is a verygood assumption within a narrow band, it isinvalid for a spectral interval larger than a fewtens of wavenumbers. The evaluation of overallradiative heat fluxes, however, requires spectralintegration over the whole spectrum. Althoughsavings can be achieved by employing k-distri-butions within every narrow-band for spectralintegration, extending this concept in its originalform to the full spectrum would still requirethousands of quadrature points and prohibi-tively large computational resources. The chal-lenge is to reorder the absorption coefficient ofthe entire spectrum once and for all by account-ing for the Planck function variation across thespectrum, as well. The FSCK model overcomesthis challenge. The theory underlying this modelis described here briefly to facilitate discussionof numerical algorithms and results. Furtherdetails pertaining to this model may be obtainedfrom the articles by Modest and co-workers [9,25].

Consider a participating gas within a blackenclosure. For simplicity in mathematical deri-vation, we will assume that the medium onlyabsorbs and emits, and does not scatter. It willbe shown later that the model is equally accu-rate for gray scattering media with gray walls.The RTE, under these assumptions, is given by[26]:

dIds

Ib I (7)

where I is the spectral intensity varying along apath s, is the wavenumber, Ib is the Planckfunction, and is the spectral absorption co-efficient. The spectral absorption coefficient, ,is a function (besides ) of temperature, pres-sure, and species concentrations. The formalsolution of Eq. 7 is [26]:


Is Ibw exp 0

s

ds 0

s

IbTs)

exp s

s

dssds (8)where Ibw denotes the Planck function evalu-ated at the wall temperature. Integrated overthe whole spectrum, Eq. 8 becomes

IS 0

Id Ibw1 Tw,03 S

0

s

Ibs

sTs,s3 sds (9)

where

T,s s 1

IbT0

IbT

1 exp s

s

dsd (10)At this point, it is convenient to introduce the

so-called scaling approximation [26]:

,T,pi kuT,pi (11)

which states that the wavenumber dependence ofthe absorption coefficient is unique and does notdepend on temperature and partial pressure.While being a good assumption for soot, it is notvalid for molecular gas mixtures in general. Theaccuracy of this approximation will be evaluatedand discussed thoroughly for a major part of thisarticle, as it constitutes a very important aspect ofmodeling radiation in inhomogeneous paths. Sub-stitution of Eq. 11 into Eqs. 8 and 10 yields:

Is IbwekX0,s

0

s

IbsekXs,skusds (12)

and

T,s s 1

IbT0

IbT1 ekXs,sd

(13)

where

Xs,s s

s

usds (14)

A similar reordering argument can now beapplied to the entire spectrum. Defining a frac-tional Planck function as

iT, 1

IbT0

Ibd (15)

it is obvious that i(T,0) 0 and i(T, ) 1, andthat the fractional Planck function increasesmonotonically from 0 to 1. Furthermore, thereis a single value of k for each value of i, butmany values of i for each value of k. Thus, Eq.1 can be reordered the same way as Eq. 5 toyield:

0

1

ekiXdi0

ekX fT,kdk0

1

ekT, gXdg

(16)

where is the overall transmittance (includingall wavenumbers). It is very important to notehere that g is no longer an equivalent wavenum-ber, but an equivalent Planck-weighted wavenum-ber. Because i is a function of T, so is k, such thatk k(T, g). The goal in using the reorderingapproach is to transform the RTE in such a waythat a reordered absorption coefficient of theform k k(g) can be used instead of ().Thus, the fact that k is now a function of both Tand g, is inconvenient to implement for arbitrarysolution techniques of the RTE. To circumvent


this problem, following Modest [17], an addi-tional reordering step is performed, such that:

0

1

ekT,gXdg 0

1

aT,g0ekT0,g0Xdg0

(17)

where k k(T0, g0) is the k-distribution evalu-ated at T T0. In other words, the temperaturedependence of k has been moved to the basefunction a. Setting k(T, g) k(T0, g0), anddifferentiating leads to

dg kT0,g0/g0kT,g/g

dg0 fT,k g

fT0,k g0dg0

aT,g0dg0 (18)

where a(T0, g0) has been set to unity at thereference condition. For simplicity of notation,henceforth, the subscript 0 will be droppedfrom g0. Also, k(T0, g0) will simply be written ask(g), with the understanding that k(g) is thek-distribution evaluated at the reference state.Equation 17 can be differentiated to yield:

s

s

0

1

aT, gek gXs,sk gXs

dg

0

1

aT, gek gXk gusdg (19)

Substitution into Eq. 12 yields:

Is 0

1

Igsdg 0

1

aTw,gIbwekX0,s

0

s

aTs,gIbsekXs,skusdsdg

(20)

Comparison of Eq. 20 with Eq. 9, we find thatthe integral form of the generalized RTE hasbeen recovered, except that the Planck functionhas been replaced by a weighted Planck func-tion aIb. In differential form, the transformedRTE, with the inclusion of scattering, can bere-written as [9]:

dIgds

k gusaIb Ig sIg

s4

4

Igss,sd

0 g 1 (21)

Equation 21 is our new transformed RTE andneeds to be solved for several different g valuesand finally integrated to obtain the overall radi-ative flux.

The treatment of soot is an important as-pect of modeling radiation in combustionsystems. The FSCK model does not precludeexistence of soot particles. Modest and Zhang[9] have already demonstrated that the modelis applicable to situations where soot is mixedwith gas components. In combustion scenar-ios, prediction of soot volume fraction, parti-cle size, and morphology is in itself a topicof current research. To perform radiationcalculations with soot, such data is necessary,and this is beyond the scope of the currentstudy.

Computation of k-distributions, BaseFunctions, and Scaling Functions

Before solution of Eq. 21 it is necessary tocompute k(g), u(s), and a(T, g). This constitutesa major task in using the FSCK model forradiation calculations. The central idea is togenerate databases for the three functions onceand for all, and re-use this database for allproblems of interest. This would imply that oncethe databases have been generated, the compu-tational requirements would be limited to thatfor solving the RTE for a few quadrature points(g values) only.


k-distributions and Base Function

There are several numerical techniques tocompute the k-distribution from raw line-by-line absorption coefficient data. For the cur-rent study, we employed an algorithm that isconvenient and fast. The steps are outlinedbelow.

The raw absorption coefficient ( vs. ) is firstread in from a file for a given temperature.This is the reference temperature. Additionalnodes are then introduced in wavenumberspace. For example, if the raw absorptioncoefficient is stored at intervals of 0.01 cm1,additional nodes may be introduced at inter-vals of 0.001cm1.

The values at these intermediate additionalnodes are computed by linear interpolation.These are henceforth referred to as -sam-ples.

The axis is then broken up into samplingbins (henceforth, referred to as -bins). Thenumber of bins to be used is quite arbitrary.The smaller the number of bins, the smootherthe resultant k-distribution will be. For thecurrent study, 100 bins were taken for eachdecade of increase of .

The spectrum is then scanned for all the-samples. Depending on which -bin thesample belongs to, the fractional Planck func-tion is computed and cumulatively added tothe bin value for every temperature. For thecurrent study, temperature nodes were placedevery 100 K starting from 300 K and going upto 2000 K.

After all -samples have been scanned, thebin values for all temperatures are multipliedby /Ti

4. At this point, the bin values repre-sent f (T, kj)kj.

The function g(k) is next evaluated using:g(Ti, kj) g (Ti, kj1) f (Ti, kj) kj.

Of interest to us is k versus g rather than gversus k. The function g(T, k) is next invertednumerically by linear interpolation to getk(T, g) at all temperature nodes, and stored at1000 g-values.

Once f(T, kj) has been computed for all T, thebase function a(T, kj) can easily be computedusing

aT,kj fT,kjfT0,kj

(22)

Once again, the function is inverted numeri-cally to obtain a(T, g) versus g at all T, andstored at 1000 g values. Figure 2 shows k-distributions for CO2 and H2O at differenttemperatures, computed using the above al-gorithm. It is clear from Fig. 2 that thefunction k(g) is much smoother than (),shown in Fig.1. This implies that the integra-

Fig. 2. k-distributions for CO2 and H2O at different refer-ence temperatures computed from the HITEMP database[5].


tion over g-space in Eq. 20 would require veryfew quadrature points (typically less than 20)as opposed to the integration of Eq. 9, whichwould require millions of quadrature points.The computational benefits of using theFSCK model are, thus, unquestionable. Fig-ure 3 shows the base function a(T, g) for CO2and H2O at a reference temperature of 500 K.a(T, g) is not as smooth as the k-distributionsshown in Fig. 2, and high frequency oscilla-tions are observed. Modest and Zhang [9]suggest smoothing of the base function toimprove accuracy. For the current study, thesmoothing was performed using least squarespline fits to the data.

Scaling Function

It is obvious from the above section that thescaling approximation, as described by Eq. 11,is necessary to reorder the absorption co-efficient for the whole spectrum. This is not arestriction that has been imposed by the cur-rent FSCK model, but is, rather, an approxi-mation that cannot be easily bypassed if onewere to treat radiation through inhomoge-neous paths. For example, the exact samescaling technique is applied in the popularCurtis-Godson scaling approximation to cal-culate narrow-band absorptivities for inhomo-geneous paths [26]. While its use has been

prolific, its accuracy is questionable, and re-quires further examination.

Determination of a scaled absorption co-efficient from line-by-line data consists of twomajor steps. First, an appropriate referencetemperature must be chosen, at which the ab-sorption coefficient is set to coincide with thatof the database, that is, (, Tref, pi,ref) k()and u(Tref, pi,ref) 1. Secondly, based on energybalance, a relationship between the scalingfunction, the local absorption coefficient, andthe reference absorption coefficient needs to bedeveloped. Because radiative heat fluxes from agaseous layer are governed by emission ratesattenuated by self-absorption, we evaluate thescaling function in this research from the rela-tionship

0

IbTref)exp T, pi, refLmd

0

IbTrefexp kuT, pi, refLmd

(23)

where Lm is the mean beam length within thegas layer ( 4V/A). For optically thin situations,Eq. 23 ensures that the Planck mean absorptioncoefficient is recovered by scaling. For opticallythick situations, Eq. 23 ensures that the scalinghas the correct heat flux escaping a layer ofthickness Lm. The mean beam length is problemdependent, and therefore, to enable use of atable for the scaling function, it is necessary totabulate values at several different mean beamlengths, as well, in addition to several differentreference temperatures, reference concentra-tions (or partial pressures) and local tempera-tures. For the current study, for each Tref andpji,ref, values of u were tabulated for several Tvalues and Lm values. The T and Tref values werevaried from 300 K to 2000 K, with intervals of100 K. The Lm values were varied from 1 cm to10 m with 20 unequally spaced intermediatevalues. The reference mole fraction values werevaried from 4% to 80% with 20 intermediate

Fig. 3. Base function a(T, g) cmputed using the HITEMPdatabase for CO2 and H2O at 2000K with a referencetemperature of 500K.


values. Such high values of mole fraction weredeemed necessary, because state-of-the-artcombustion appliances are often fired with pureoxygen rather than air. From the above discus-sion, it is clear that Eq. 23 needs to be solved129,600 times (18 Tref 18 T 20 pi,ref 20Lm) for each gas in question. This is a formida-ble task in light of the fact that Eq. 23 is anon-linear equation in u, and iteration is neces-sary. Furthermore, because the absorption co-efficient is stored at more than a million wave-numbers, the integral reduces to a summationover a million terms and is, thus, expensive toevaluate. For the present study, Eq. 23 wassolved using the Newton-Raphson iterative pro-cedure, with particular emphasis on making theinitial guess very close to the actual solution.The solution of Eq. 23 required 8 s of CPU ona Pentium III 733 MHz processor for each case,implying that the calculation for all 129,600cases required about 20 days of CPU for eachgas. It must be emphasized here that once thescaling function has been tabulated for thewhole range of reference temperature, refer-ence partial pressure, local temperature, andmean beam length, it can be used for anyproblem. The solution of Eq. 23 is a one-timeaffair for each gas, and therein lies the advan-tage of the FSCK method. Figure 4 showsscaling functions for CO2 and H2O for variouslocal temperatures and mean beam lengths.

It is evident from Eq. 23 that the scalingfunction will be dependent upon the referencetemperature and concentrations used. Thechoice of an appropriate reference temperatureand an appropriate reference concentration is atricky matter and needs further investigation. Inlater sections, this issue will be evaluated anddiscussed comprehensively.

Solution of the Transformed RTE

Several different well-known techniques areavailable to solve the transformed RTE, (Eq.21). These include the method of sphericalharmonics (PN approximation [27]), the dis-crete ordinates method [28, 29], the discretetransfer method [30], and the Monte Carlomethod [31]. Over the last decade or so, thediscrete ordinates method (DOM) has be-come the RTE solver of choice for commer-

cial CFD codes. A discussion of its strengthsor weaknesses is beyond the scope of thisarticle, and may be found elsewhere [28].CFD Research Corporations commercialsoftware, CFD-ACE, is already equippedwith a DOM-based RTE solver for multi-dimensional geometry with arbitrary grid to-pology [32]. The model is fully coupled withthe overall energy equation, and allows solu-tion of the RTE for both gray and non-graycases with spatially varying extinction coeffi-cient. This model served as the starting pointfor implementation of the solution techniqueof the transformed RTE. The overall solutionprocedure is depicted schematically in Fig. 5.It is worth noting that the solution strategiesdepicted in Fig. 5 can be applied to anygeneral-purpose flow solver. The solution al-gorithm also depicts that the database ofk-distributions, base functions and scalingfunctions is a stand-alone piece, which isgenerated only once, and is used repeatedlyfor any possible application. The solution ofEq. 23, which happens to be computationallyexpensive, is a one-time affair only.

Fig. 4. Scaling functions for CO2 and H2O computed usingthe HITEMP database [5].


RESULTS AND DISCUSSION

The FSCK model was verified and validated fora number of different scenarios ranging fromhomogeneous paths to inhomogeneous pathswith simultaneous variation of both concentra-tion and temperature fields. This section de-scribes in detail each validation study that wasperformed, and the conclusions drawn from theresults. All results presented in this section wereobtained using the HITEMP database [5],which is a high-temperature rendition of theHITRAN96 spectroscopic database.

Homogeneous Path

The only approximation introduced in develop-ment of the FSCK model is the scaling approx-imation. If the radiation path is homogeneous(i.e., the domain of interest has uniform tem-perature and/or concentration fields), the scal-

ing approximation becomes mute, that is, thescaling function is equal to unity everywhere bydefinition. Under this circumstance, it is ex-pected that the FSCK model will produce re-sults, which match perfectly with exact LBLcalculations.

To verify this matter, a simple problem wasconcocted. In this problem, a hot gas comprisingof 10%CO2 and 90%N2 (assumed to be non-participating) at 1500 K was confined betweentwo infinitely long parallel plates at 0K. Thetemperature and concentration fields were fixedand not allowed to change. The distance be-tween the plate was a parameter, which waschanged to vary the optical thickness of themedium.

The RTE can be solved analytically for thisproblem [26]. As a first step, our goal was toinvestigate the accuracy of the discrete ordi-nates method and other RTE solution tech-niques for this problem. To do so, we used the

Fig. 5. Solution strategies employed to accommodate the FSCK model in the commercial CFD code, CFD-ACE.


FSCK model in conjunction with exact analyti-cal solution of the RTE, several variations ofDOM, and the P-1 approximation. In each casea 10 point Gauss-Legendre quadrature [33] wasemployed for integration in g-space. The resultsare shown in Fig. 6 for two different opticalthicknesses. As expected, the DOM S6 methodproduces the most accurate results, closely fol-lowed by the DOM S4 results. The P-1 methodoverpredicts the divergence significantly, espe-cially for the lower optical thickness case.

Once it was determined that the discreteordinates method is accurate for such computa-tions, we proceeded to verify the accuracy of theFSCK model by comparison with exact line-by-line calculations. In the context of using thek-distribution approach for global heat transfercalculations, one question, which has beenraised over the years by several researchers[9,11], is what quadrature scheme is most suitedfor integration over g-space. This is particularlycritical for full-spectrum calculations, where we

are replacing millions of spectral quadraturepoints by a few tens of points. Obviously, withsuch few points, the quadrature to be used isexpected to affect the results of integrationsignificantly. Literature review revealed thatGauss-Legendre qudrature schemes and Lo-batto quadrature schemes were the two mostwidely used. Careful study of k-distributions(Fig. 2) revealed that k-distributions of all mo-lecular gases tend of have a steep gradient closeto g 1. This implies that the quadrature pointsneed to be clustered around g 1 to produceaccurate integration results. In light of theseobservations, we started experimenting withGaussian Quadrature of Moments (GQM)schemes [33], and soon found that they areindeed very accurate for this type of application.Figure 7 shows comparison of the FSCK modelwith exact line-by-line calculations for the same

Fig. 6. Divergence of radiative heat flux computed usingdifferent solution techniques in conjunction with the FSCKmodel. L is the gap between the two parallel plates.

Fig. 7. Comparison of FSCK model with exact solution.Exact LBL represents analytical solution of the RTE inconjunction with line-by-line integration. The other fourcurves were computed using the FSCK model in conjunc-tion with S4 DOM.


problem described earlier. The correspondingwall heat fluxes are tabulated in Table 1.

From Fig. 7, it is clear that the 8-pointGaussian quadrature of moments proved to bethe most accurate integration scheme at leastfor this particular problem for both opticalthickness. The 10-point Gauss-Legendrescheme was also found to be quite accurate,although it requires two extra points (20% moreCPU). The Lobatto quadrature scheme wasdeemed unsatisfactory for the optically thinnercase.

From the results presented above, it is clearthat the FSCK model indeed produces almostexact answers. The small errors are attributed todifferences between exact solutions and DOM,and errors because of interpolation and numer-ical integration using less than 10 points. Thenext task undertaken was to validate the modelfor inhomogeneous paths.

Inhomogeneous Path

Temperature Inhomogeneity

The first problem we studied is that of a gas ofuniform concentration confined between infi-nitely large parallel plates, with a jump intemperature in the middle of the gas layer. Thisis shown schematically in Fig. 8. The cold andhot layer thicknesses were retained as parame-ters. Scattering was neglected and the wallswere black. LBL (with S4-DOM) and FSCKcalculations were performed for various coldlayer thicknesses. Comparison of the radiativeheat flux divergence is shown in Fig. 9. Thecorresponding heat fluxes are depicted in Fig.10.

As discussed earlier, the scaling approxima-

tion plays a critical role in the accuracy of FSCKresults. In particular, the choice of an appropri-ate reference temperature is important for pro-ducing accurate answers for inhomogeneouspaths. Although the scenario described aboveappears to be rather benign on account of thesimple geometry and boundary conditions, atemperature discontinuity of 1000 K is anythingbut benign in terms of scaling.

There are several approaches for fixing thereference temperature. Modest and Zhang [9]has suggested use of volume average of thefourth-power of temperature (denote in Figs. 9and 10 as VAT) defined as:

Tref4

1V

V

T4dV (24)

They also indicated [9] that the Planck-meantemperature (denoted by PMT in Figs. 9 and10), defined by the implicit relationship

TABLE 1

Wall Heat Fluxes: Comparison Against Exact Solution. The Percentage Errors areShown in Parentheses

Model

Non-dimensional Wall Heat Flux

L 1 cm L 1 m

Exact LBL 0.0153 0.1333810 point Gauss-Legendre 0.01596 (4.3%) 0.13547 (1.6%)10 point Lobatto 0.01692 (10.6%) 0.14313 (7.3%)8 point Gaussian Moment with 3rd order fit 0.01677 (9.6%) 0.13457 (0.9%)8 point Gaussian Moment with 4th order fit 0.0162 (5.8%) 0.13498 (1.2%)

Fig. 8. Schematic of the 2-layer case used to validate theFSCK model.


Fig. 9. Comparison of FSCK model with line-by-line (LBL) calculations for various cold layer thickness. The quantity, L, isthe sum of the thicknesses of the hot and cold layers.


pT4ref

1V

v

pT4dV (25)

is more appropriate. Both of these options wereexamined. Their studies also showed that if thetemperature at which the Planck function isevaluated in Eq. 23 is the average emissiontemperature of the domain, and the absorptioncoefficient is evaluated at the Planck-mean tem-perature, the scaling is even more accurate. Thisis consistent with the physical argument that ifTref Temission in Eq. 23, the emission from thehot layer is treated more accurately, while usingthe Planck mean temperature for the absorp-tion coefficient implies that absorption withinthe cold layer is treated accurately. In principle,it is possible to use separate temperatures forevaluating the absorption coefficient and thePlanck function in scaling Eq. 23. This, however,poses additional problems in terms of creatingdatabase for the scaling function. It was stated

in an earlier section, that currently 129,600cases need to be evaluated to construct thedatabase for each gas. If the emission tempera-ture is considered a separate independent vari-able, it would result in 129,600 18 cases,leading to 20 18 days of CPU, which isprohibitive. In light of these observations, thisparticular scaling technique was deemed im-practical for general use. For the problem underconsideration, the emission-averaged tempera-ture happens to be very close to 1500 K, that is,the hot layer temperature. In other words, using1500 K to evaluate the Planck function in Eq. 23is equivalent to using Temission. Figure 9 clearlyshows that for Tref Temission Thot the resultsare quite accurate (denoted by triangles) for thehot layer. However, Fig. 10 shows that the heatfluxes on the wall next to the cold layer (x/L 1) are not so accurate with this approach. Theheat fluxes on the same wall are more accuratewith the hybrid model (denoted by circles),which uses two independent temperatures forscaling. It is also evident that except for the casewhere Tref Tcold, all other choices of referencetemperature produce fairly accurate results. Inlight of the fact that a 1000 K jump in temper-ature is quite challenging for scaling, the resultslend tremendous credibility to the FSCK modelfor inhomogeneous temperature fields.

The FSCK model was next validated for atwo-dimensional (2D) axisymmetric geometryinvolving several different boundary tempera-tures and with temperature jumps as high as1500 K. The geometry, boundary conditions,and operating parameters for this 2D case aredepicted in Fig. 11. In this simulation, theemissivity at all walls was set to 0.5, and grayisotropic scattering was turned on, with a scat-tering coefficient of 1 m1. The gas in thedomain consists of 10% CO2 (by mole) and 90%N2 (assumed non-participating). The divergenceof heat flux at the three plotting locations forthe three simulations are shown in Fig. 12. Inthis case too, it is seen that Tref Temission Thotaccurately predicts the divergence in the hotregions but underpredicts the divergence in thecolder regions (x 0.825). In colder regions,more accurate results are produced either bysetting the reference temperature as the Planck-mean temperature or by setting it equal to thevolume-averaged temperature. For this particu-

Fig. 10. Wall heat fluxes computed using the FSCK modelwith different scaling techniques.


lar case, the volume of the cold region is muchlarger than the volume of the hot region, andthe Planck mean temperature is only about 714K. Thus, the scaling works well for the coldregion if the reference temperature is close to500 K. The results also prove the validity of themodel for gray walls and gray scattering. Theconclusion to be drawn from these results is thatthe predictions are very accurate if the Planck-weighted temperature of the entire computa-tional domain is chosen as the reference tem-perature, and the emission-weightedtemperature is used for computation of thePlanck function in Eq. 23. In practice, it isprohibitively expensive to generate the databasewith two different temperatures, and the bestchoice is to use the emission-weighted temper-ature as the reference temperature as well as tocompute the Planck function in Eq. 23.

The CPU requirements for performing thesecalculations need to be mentioned at this point.The LBL calculations, involving about a millionbands, requires about 2 days of CPU on an IntelPentium III 733 MHz processor. This is for the2D case described above with just 144 cells. Incomparison, the FSCK calculations with 8quadrature points, requires only about 35 sec-

onds of CPU. Of course, the use of the FSCKmodel requires generation of databases, but thisis a one-time affair, and once a database isgenerated, it can be used for all cases. In fact,plans are underway to publish the databasesthat we have generated in the open literature, sothat future researchers do not have to regener-ate these data, but can use the data directly.

Combined Concentration and TemperatureInhomogeneities

For combustion applications, for which the cur-rent model is intended, it is of utmost impor-tance to treat concentration variations along theradiation path. The absorption coefficient of amixture of gases is approximately linearly pro-portional to the partial pressures of the constit-uents:

i

ipi (27)

where i is the pressure-based absorption co-efficient of species i, and pi their partial pres-sures. This, however, does not apply to reor-dered k-distributions. For such a case, thek-distribution changes for each mole-fraction

Fig. 11. Geometry, boundary conditions, and operating parameters for 2D axisymmetric test case.


Fig. 12. Comparison of FSCK results with LBL results for 2D axisymmetric case with gray walls (emissivity 0.5) andisotropic scattering (scattering coefficient 1 m1)


combination, and the relationship between k-distributions at various concentrations cannotbe described adequately by a linear scaling law.

To study the effect of scaling on non-uniformcomposition fields, we concocted two separateproblems. The geometry for both cases is thesame as that shown schematically in Fig. 11. Thetwo cases are different by virtue of havingconstant versus non-constant CO2/H2O ratio.The exact composition of the mixtures con-tained within the three zones for the two casesare tabulated in Table 2. It is worth noting thatboth cases have strong discontinuities in bothtemperature and concentrations.

Calculations were performed for both casesusing the FSCK model with various strategiesfor scaling. The results for the uniform CO2/H2O ratio case are shown in Fig. 13. Twodifferent reference concentration fields werechosen. The first of these was that correspond-ing to that of the large outer cold region (Zone3). The second of these was that correspondingto the middle hot layer (Zone 2). Interestinglyenough, the results produced by both of thesereference conditions is identical (triangles andcircles). If the ratio of CO2/H2O is constant, theresulting k-distributions at the two referencestates have identical profiles, except that theyget scaled by a constant factor. For such ascenario, the resultant scaling function is simplya mole-fraction-weighted sum of the individualscaling functions.

When the concentrations of CO2 and H2Oare varied independently, resulting in non-con-stant CO2/H2O ratio, the k-distributions for thetwo reference states may be vastly different, andlead to different results (Fig. 14). For suchcases, the resultant scaling function is not amole fraction weighted sum of the individualscaling functions. For a full-scale combustion

application, the reference concentration is ex-pected to change with iteration. As long as theratio of the mole fractions of the constituentspecies remain constant, the scaling as appliedto the FSCK model is linear, and the results aregoing to be independent of the reference molefractions. This is a scenario, which is likely tohappen in cases where single-step kinetics isused. For example, for a single step propane-airreaction to CO2H2O, the CO2/H2O ratio willalways be 3:4, although the actual magnitude ofthe mole fractions may vary from problem toproblem and iteration to iteration. If multi-stepkinetics is used, the ratio will no longer remainconstant because of the production of interme-diate species, and the linear scaling argumentwill be violated. Also, if the fuel is treated as aparticipating gas, its concentration variationwith space is completely uncorrelated with theconcentration variation of either CO2 or H2O,that is, the concentration of fuel is high wher-ever the concentration of CO2 and H2O is lowand vice versa. In such a case, the fuel to CO2and/or H2O ratio is guaranteed to be non-constant. Despite this shortcoming of the FSCKmodel, the results produced even for the case ofnon-uniform CO2/H2O ratio are quite accuratein most cases.

Demonstration for Full-Scale Combustor

In the validation studies presented in the pre-ceding section, the temperature and/or concen-tration fields were fixed, and not allowed tochange. This was mainly done because LBLcalculations are prohibitive for changing condi-tions because solution of the RTE itself takes afew days of CPU. For a full-scale combustionproblem, the temperature and concentrationfields will change with every iteration, and LBL

TABLE 2

Composition of Various Spatial Zones for the Two Test Cases with CombinedTemperature and Concentration Inhomogeneities

Case 1 (Uniform CO2/H2O ratio) Case 2 (Non-uniform CO2/H2O ratio)

T(K) CO2 (% mole) H2O (% mole) T(K) CO2 (% mole) H2O (% mole)

Zone 1 500 5 10 500 30 10Zone 2 2000 10 20 2000 20 20Zone 3 500 15 30 500 10 30


Fig. 13. Comparison of FSCK model with LBL calculations for uniform CO2/H2O ratio


Fig. 14. Comparison of FSCK model with LBL calculations for non-uniform CO2/H2O ratio.


calculations cannot be performed for such acase. Furthermore, each cell will have a differ-ent temperature and concentration value, and itis not possible to store that many absorptioncoefficient arrays in run-time memory. Thus,LBL calculations are prohibitive from a mem-ory perspective, as well. The FSCK code, how-ever, was designed in a manner such that itaccounts for changing reference conditions.This is evident in the solution flowchart de-picted in Fig. 5, where it is clearly shown thatthe reference states are recomputed at eachiteration of the global solution loop, and thecorresponding k-distributions, scaling functionsand base functions are simply extracted fromthe database by linear interpolation.

The combustion case undertaken to demon-strate the feasibility of using the FSCK modelfor real-life combustion applications is dis-cussed next. For simplicity, we considered anaxisymmetric version (Fig. 15) of the actualthree-dimensional combustor. The combustorhas a single fuel nozzle surrounded by fourindependent air nozzles. There is a secondaryair inlet along the annulus, halfway down thelength of the combustor to supply excess air. Forthe current study, we considered combustion ofpropane in air. The flow rates were adjusted sothat the propane fuel undergoes complete com-bustion, and no excess propane exits the com-bustor (lean conditions). In order to keep theCO2/H2O ratio constant, a single-step globalfinite-rate reaction was used:C3H85O2f 3CO24H2O. The reaction rateconstants were taken from Westbrook andDryer [34]. The inlet streams were injected at923 K. The external walls of the combustor werecooled using an external heat transfer boundary

condition including both convective as well asradiative losses to a 300 K ambient. JANNAFcoefficients were used to compute species ther-modynamic properties. The RNG k- model,which is a standard option in CFD-ACE, wasused to treat turbulence. Turbulence-chemistryand turbulence-radiation interactions were ne-glected. Because the HITEMP database is cur-rently limited to CO2 and H2O only, we alsoassumed that propane is non-participating. Fur-thermore, for temperatures above 2000 K, thedata for 2000 K was used. This was done mainlybecause HITEMP data is questionable beyond2000 K [35,36], and therefore, our database hasan upper cutoff of 2000 K. We did not see anybenefit of trying to compute such data for thisdemonstration problem only.

First, a simulation was performed for thisgeometry without turning on radiation. Next,the Planck-mean absorption coefficient, definedby

p 0

Ibd0

Ibd (28)

was computed for CO2 and H2O at 100 Kinterval from 300 K to 2000 K. This wastabulated, and Eq. 27 was employed to calcu-late the gray Planck-mean absorption coeffi-cient at any given temperature and partialpressure after linear interpolation within thistable. The next simulation used this grayspatially varying Planck-mean absorption co-efficient for radiation calculations. This ap-proach is employed routinely within the com-bustion community for simplistic radiation

Fig. 15. Geometry and grid for axisymmetric combustor used for demonstrating the FSCK model.


calculations, and more or less represents thestate-of-the-art when it comes to calculationof radiation in combustion systems [37]. Fi-nally, the FSCK model was used to computeradiation fluxes. The reference condition cho-sen was the Planck-mean temperature, whichwas automatically recomputed at each itera-tion of the global solution loop. The referenceconcentration was taken to be the volumetricaverage, which was also recomputed. Thetemperature profiles obtained for the threecases are shown in Fig. 16. Obviously, without

radiation, the flame temperature is severelyoverpredicted. With the gray Planck-meanabsorption coefficient, the absorption bycolder regions is overpredicted and the emis-sion by hotter regions is also overpredictedresulting in a much more uniform tempera-ture distribution. With the FSCK model, thesharp flame structure is retained, and theflame is colder by about 200 K compared tothe case without radiation.

The CPU required to simulate the case withthe FSCK model was only 25% more than the

Fig. 16. Steady state temperature profiles for propane-air combustion without radiation and using two different radiationmodels.


CPU required for the case without radiation.The fractional increase in CPU is expected to besignificantly lower for cases with multi-stepchemistry involving many species. The totalCPU required for three order of magnitudeconvergence of this problem involving 11,410cells was about 210 min on a 733 MHz IntelPentium III processor.

SUMMARY AND CONCLUSIONS

The full spectrum correlated k-distribution ap-proach is a powerful approach to modelingnongray radiation in combustion gases. Its ac-curacy is comparable to exact solutions of theRTE for homogeneous gas layers. For inhomo-geneous gas layers with strong discontinuities inboth temperature and concentrations, its accu-racy is limited by the scaling approximation.Several different full-spectrum scaling tech-niques were investigated, and it was found thata reference temperature equal to the emission-weighted temperature of the domain, and areference concentration equal to the volume-averaged concentration yields best results. Forall cases considered under this study, resultswere always within 20% of LBL answers. Globalspectrum approaches, such as the FSCK ap-proach and the SLW approach have the disad-vantage that they are sensitive to the choice ofreference conditions when it comes to treat-ment of radiation in inhomogeneous media.One alternative is to use local spectrum ap-proaches, which are far more computationallyexpensive, but produce accurate results inde-pendent of the choice of reference conditions,as demonstrated recently by Solovjov and Webb[38].

When used in conjunction with the DiscreteOrdinates Method, the FSCK model can proveto be a powerful tool for computing radiativefluxes within full-scale combustors involvingcoupled fluid flow, heat transfer, and chemistryin complex geometries. It was found that theadditional CPU requirement (to compute radi-ative fluxes) is less than 25% for a full-scalecombustion problem involving 5 species, and isexpected to be significantly lower with increasein the number of species.

The support of the National Science Founda-tion through an SBIR Phase I grant (Contract #DMI 0060286, Program Officer: Cheryl Albus) isgratefully acknowledged.

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Received 21 August 2001; revised 25 January 2002; accepted10 February 2002.


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