Combustion and Flame 158 (2011) 893–901

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Combustion and Flame

journal homepage: www.elsevier .com/locate /combustflame

Account for variations in the H2O to CO2 molar ratio when modelling gaseousradiative heat transfer with the weighted-sum-of-grey-gases model

Robert Johansson ⇑, Bo Leckner, Klas Andersson, Filip JohnssonDepartment of Energy and Environment, Chalmers University of Technology, SE-412 96 Göteborg, Sweden

a r t i c l e i n f o a b s t r a c t

Article history:Received 31 May 2010Received in revised form 21 December 2010Accepted 1 February 2011Available online 19 February 2011

Keywords:Gas radiationCombustionModellingSpectral properties

0010-2180/$ - see front matter � 2011 The Combustdoi:10.1016/j.combustflame.2011.02.001

⇑ Corresponding author.E-mail address: robert.johansson@chalmers.se (R.

This work focuses on models suitable for taking into account the spectral properties of combustion gasesin computationally demanding applications, such as computational fluid dynamics. One such model,which is often applied in combustion modelling, is the weighted-sum-of-grey-gases (WSGG) model.The standard formulation of this model uses parameters fitted to a wide range of temperatures, but onlyfor specific ratios of H2O to CO2. Then, the model is limited to gases from fuels with a given compositionof hydrogen and carbon, unless several sets of fitted parameters are used. Here, the WSGG model is mod-ified to account for various ratios of H2O to CO2 concentrations. The range of molar ratios covers both oxy-fuel combustion of coal, with dry- or wet flue gas recycling, as well as combustion of natural gas. The non-grey formulation of the modified WSGG model is tested by comparing predictions of the radiative sourceterm and wall fluxes in a gaseous domain between two infinite plates with predictions by a statisticalnarrow-band model. Two grey approximations are also included in the comparison, since such modelsare frequently used for calculation of gas radiation in comprehensive combustion computations. It isshown that the modified WSGG model significantly improves the estimation of the radiative source termcompared to the grey models, while the accuracy of wall fluxes is similar to that of the grey models orbetter.

� 2011 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction

Radiative heat transfer is a key factor in the design of a combus-tion chamber. The hot combustion products, both gases and parti-cles, emit radiation which is absorbed by the walls or by thesurrounding media. The directional nature of radiation and thescattering caused by particles necessitate the solution of an inte-gro-differential equation known as the radiative transfer equation(RTE). The problem is further complicated by the spectrally depen-dent properties of the combustion gases, mainly H2O and CO2. Afull integration of the RTE involving all spectral lines in the gasspectrum, carried out through line-by-line (LBL) calculations, is ex-tremely demanding and is only realistic in simplified cases. In com-prehensive combustion models, for example computational fluiddynamics (CFD) models, the spectral variation of the gases is com-monly neglected and the spectrum is treated by a single average,i.e. by a grey approximation [1–5].

Apart from the two extremes, LBL calculations and grey approx-imations, there are several approximate models accounting for thespectral properties of gases. Some of the models provide estimatesof transmissivity or emissivity of spectral bands (narrow- and

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Johansson).

wide-band models). As these properties are averaged over bandsof the physical spectrum, a correlated solution of the RTE is re-quired to avoid additional errors [6]. The drawback of the corre-lated formulation is the need for ray-tracing methods, whichbecome computationally expensive in multidimensional geome-tries. An alternative to the transmissivity models are models thatpredict a distribution of the absorption coefficient, either for spec-tral bands or for the full spectrum (global models). These absorp-tion-coefficient models have the advantage that a non-correlatedformulation of the RTE can be applied without any additional lossof accuracy, although in non-uniform media the re-ordering of thespectrum involved in the calculation of the distribution of theabsorption coefficient requires a scaling approximation [7,8].Several works have pointed out the numerical efficiency of theabsorption-coefficient models, their high accuracy, and theirusefulness for computationally demanding applications, such ascombustion modelling [9–11].

In the group of global absorption-coefficient models a numberof examples can be found: the spectral line-based weighted-sum-of-grey-gases model (SLW) [12], the absorption distribution-function (ADF) model [13] and the full spectrum correlated-kdistribution (FSCK) model [14]. The most simplified implementa-tion of this type of model is the banded formulation of theweighted-sum-of-grey-gases (WSGG) model [15]. Existing

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Nomenclature

A area (m2)aj weighting factor of gas j in WSGG model (–)d mean line spacing (cm�1)Im spectral intensity (W m�2 sr�1 cm)I total intensity (W m�2 sr�1)Ib blackbody intensity (W m�2 sr�1)Ibm blackbody spectral intensity (W m�2 sr�1cm)k mean line intensity to typical line-spacing ratio within a

narrow band (cm�1)P total pressure (bar)qþwall incident wall flux (W m�2)S total pathlength used with spectral data in cm�1 (cm)Sm total pathlength (m)Schar characteristic domain length (m)s coordinate along a radiative path (–)s unit vector in a given directionT temperature (K)V volume (m3)Y mole fraction (–)rq radiative source term (W m�3)

Greek symbolse emissivity (–)c mean line half-width (cm�1)j absorption coefficient (m�1 or bar�1 m�1)s transmissivity (–)

m wavenumber (cm�1)Dm bandwidth (cm�1)Ds length of a computational cell (m)

Subscripts0 wall, starting point of a radiative pathwayav averageb blackbodyi cell numberj gas j in WSGG modelk band k in narrow-band modelloc local cell propertyn cell numberref referencem spectral property

AbbreviationsCFD computational fluid dynamicsLBL line-by-lineMR molar ratioRTE radiative transfer equationSNB statistical narrow bandWSGG weighted-sum-of-grey-gases

894 R. Johansson et al. / Combustion and Flame 158 (2011) 893–901

parameters fitted for this model usually account for temperaturevariations but are limited to specific molar ratios (MR) of H2Oand CO2 [16–18], here referred to as the standard WSGG model.Fuels have various contents of hydrogen, and the MR of the com-bustion products varies. Complete combustion of natural gas givesa ratio equal to two, oil a ratio around unity, while the ratio for coaldepends on coal rank and moisture content. Therefore, the generaluse of this model requires several versions of parameters. Applyingthe model to other ratios than the ones specified or to non-homo-geneous media involves uncertainties that are hard to estimate.

The purpose of the present work is to modify the WSGG modelto account for MRs between 0.125 and 2, a range which coversmost relevant fuels and cases. Furthermore, the range of the modelparameters is extended to longer pressure pathlengths to make itapplicable to conditions with high mole fractions, such as couldbe present in large oxy-fired furnaces [19], where the combustionair is replaced by pure oxygen diluted with recycled flue gas, andwhere the MR can vary depending on whether dry or wet flue gasesare recycled. There are no WSGG model parameters available in theliterature for such conditions. The WSGG model is easy to imple-ment in a CFD framework and does not involve the complex calcu-lation of parameters and the optimization procedures required ifonly a few gases are to be considered in the above mentioned glo-bal absorption-coefficient models [20,21]. The intention of thepresent work is to provide a model with parameters ready to beapplied in CFD simulations under various combustion conditions.For combustion problems, this model is efficient when a more re-fined option than a grey approximation is needed to calculate theradiative source term for a wide range of temperatures and con-centrations. However, the model also calculates total emissivitiesif a grey approximation is sufficient. The straightforward imple-mentation of the WSGG model comes at a cost, though, and theaccuracy is likely to be lower compared to the more detailedSLW, ADF and FSCK models.

2. Theory

The modified WSGG model is evaluated by comparing with astatistical narrow-band (SNB) model and two grey gas approxima-tions. Using the models, the radiative source term and the incidentwall fluxes are calculated for a gaseous domain between two infi-nite black plates. Both isothermal and non-isothermal cases areinvestigated, all at atmospheric pressure. The SNB model servesas a reference since it has shown good agreement with LBL-calcu-lations even in strongly non-isothermal media [11], and the database from which the narrow-band parameters are derived repre-sents available experimental data well [22,23]. The parameters ofthe WSGG model are fitted to total emissivities calculated withthe SNB model. Errors introduced by the SNB model for calculationof total emissivities are small compared to the errors of the WSGGmodel, and the SNB data can therefore be regarded as an adequatesource of data. The grey approximations are based on the SNB andthe WSGG models, and the intention of including two grey approx-imations is to investigate the influence of the model used to calcu-late the grey properties. One is a detailed model and the other is arelatively simple model.

An SNB model divides the spectrum into bands, narrow enoughto regard the blackbody radiation as constant but wide enough tocontain a large number of lines to make a statistical treatment pos-sible. In the Malkmus model [24], Eq. (1), which is used in thiswork, two parameters are tabulated for each band: the meanline-intensity to the typical line-spacing ratio kk and the mean linespacing dk, by means of which the band transmittance can be cal-culated. The parameters of the model were presented by Soufianiand Taine with co-workers at the EM2C lab [22,23,25]. They usedthe extensive HITRAN92 database combined with lines from Flaudet al. [26] for H2O near 2700 cm�1 and calculated data for addi-tional hot lines. In a second step, the SNB model parameters werefitted to LBL calculations for spectral intervals with a width of

Table 1

R. Johansson et al. / Combustion and Flame 158 (2011) 893–901 895

25 cm�1. The spectrally averaged transmissivity of a narrow band kis

�smk¼ exp �2c

dk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ YPSkkdk=c

q� 1

� �� �ð1Þ

The half-widths, c, were calculated from (Pref = 1 bar andTref = 296 K)

cH2 O¼P

Pref0:462YH2 O

Tref

T

� �þ Tref

T

� �0:5

0:079 1�YCO2 �YO2

� �þ0:106YCO2 þ0:036YO2

� �( )

ð2Þ

cCO2¼ P

Pref

Tref

T

� �0:7

0:07YCO2 þ 0:058 1� YCO2 � YH2O� �

þ 0:1YH2O

An RTE is solved for each narrow band using a correlated formula-tion, Eq. (3), of the RTE. The intensity of band k at the end of a pathis

�Imk ;n ¼ �Imk ;0�smk ;0!n þ

Xn�1

i¼0

ð�smk ;iþ1!n � �smk ;i!nÞ �Ibtk ;iþ1=2 ð3Þ

Index i refers to the spatial discretization of the path, Fig. 1,where 0 indicates the starting point.

The total transmittance of a band for a mixture of H2O and CO2

is given by the product of the transmissivities of the species. In thecase of non-homogeneous or non-isothermal pathways the Curtis-Godson approximation [27] has been applied, and the total inten-sity is finally calculated as the sum of the intensities of the k bands

In ¼X

k

Dmk�Imk ;n ð4Þ

The WSGG model, originally proposed by Hottel and Sarofim[28] for calculation of the total emissivity as a weighted sum of Jgrey gases and one clear gas, is written

e ¼XJ

j¼0

aj 1� expð�jjSmPðYCO2 þ YH2OÞÞ� �

ð5Þ

Each gas represents spectral regions which have an absorptioncoefficient within a specific range which is assumed to be de-scribed by a constant value, jj. The weight aj is the fraction ofthe blackbody radiation that belongs to the spectral regions ofthe gas. The parts of the spectrum, in which the combustion gasesdo not emit, are called a clear gas having an absorption coefficientj0 = 0. The sum of the weights, including the clear gas, is equal tounity. Usually, the weights depend on temperature, while theabsorption coefficients, jj, are constants chosen for a specific gascomposition [16–18]. A common way of applying the WSGG modelin CFD modelling is to make a grey approximation and calculatetotal emissivities in order to solve the spectrally integrated RTE.An alternative approach is the non-grey or banded formulation[15], where an RTE of each of the grey gases and the clear gas issolved. Emitted blackbody radiation, corresponding to each

Fig. 1. Discretization of a line of sight.

equation, is given by the total blackbody radiation times theweight of the gas. In this formulation the non-correlated recur-rence relation can be applied and for gas j the discretized RTEbecomes

Ij;n ¼ Ij;n�1 expð�jjDsPðYCO2 þ YH2OÞÞþ aj Ib;j;n�1=2ð1� expð�jjDsPðYCO2 þ YH2OÞÞÞ

ð6Þ

where Ds is the length of the computational cell ranging from n � 1to n. The intensity of each gas along a path Sm is calculated succes-sively for the n cells with Eq. (6), and then the total intensity is ob-tained as the sum of the individual intensities of the gases.

In ¼XJ

j¼0

Ij;n ð7Þ

In this work the coefficients of a WSGG model with four greygases, J = 4, and one clear gas have been derived by fitting calcu-lated total emissivities to those calculated by the SNB model. Boththe absorption coefficients and the weights depend on MR, whileonly the weights depend on temperature, according to

aj ¼X

i

cj;iðT=Tref Þi�1 ð8Þ

With Tref = 1200 K. The dependence of the absorption coeffi-cients on the molar ratio is assumed to be linear, but the depen-dence of the coefficients cj,i is expressed as a second orderpolynomial

jj ¼ K1j þ K2jYH2O

YCO2; cj;i ¼ C1j;i þ C2j;i

YH2O

YCO2þ C3j;i

YH2O

YCO2

� �2

ð9Þ

The model coefficients were calculated by first fitting values ofjj and cj,i for a number of MR = YH2O/YCO2 between 0.125 and 2,through minimization of the deviation between the emissivity gi-ven by Eq. (5) and the total emissivity, calculated with the EM2CSNB model

e ¼ 1� 1Ib

Xk

�Ibmk�smk

ð10Þ

For each MR, this procedure was carried out for homogeneouspaths in the temperature range of 500–2500 K and for pressurepathlengths between 0.01 and 60 bar m. The total pressure waskept at 1 bar. The minimization was solved by the fmincon functionin the Matlab 7.5 software, whose algorithm is based on a sequen-tial quadratic programming method. The coefficients K and C,Table 1, were then fitted to the set of jj and cj,i values obtainedby the polyfit function in the same software. The minimization ofthe error in the emissivity of each MR contains several local min-ima, and the solutions chosen for fitting of the K1 and K2 coeffi-cients were selected to represent a linear relation between theabsorption coefficient, jj, and the range of molar ratios involved,

Coefficients of the four grey gases for the modified WSGG model.

j 1 2 3 4

K1j 0.055 0.88 10 135K2j 0.012 �0.021 �1.6 �35C1j,1 0.358 0.392 0.142 0.0798C1j,2 0.0731 �0.212 �0.0831 �0.0370C1j,3 �0.0466 0.0191 0.0148 0.0023C2j,1 �0.165 �0.291 0.348 0.0866C2j,2 �0.0554 0.644 �0.294 �0.106C2j,3 0.0930 �0.209 0.0662 0.0305C3j,1 0.0598 0.0784 �0.122 �0.0127C3j,2 0.0028 �0.197 0.118 0.0169C3j,3 �0.0256 0.0662 �0.0295 �0.0051

896 R. Johansson et al. / Combustion and Flame 158 (2011) 893–901

according to Eq. (9). The choice of a normalised temperature in Eq.(8) significantly simplifies the minimization as the parameters cj,i

all become of the same magnitude. Data for the clear gas are givenby their definition; the absorption coefficient is zero, j0 = 0, andthe weight, a0, makes the sum of the weights equal to unity.

a0 ¼ 1�X4

j¼1

aj ð11Þ

Apart from calculation of total emissivities for the fitting ofWSGG coefficients, Eqs. (5) and (10) have been applied in greymodel formulations for calculation of total intensities. A grey mod-el implies that a single RTE is solved for the spectrally averagedintensity, based on total properties. Here, the spectrally averagedproperties were calculated from Eqs. (5), (10). In CFD applications,a grey model is often used to calculate a local absorption coeffi-cient related to temperature and gas concentrations in each com-putational cell Ds, i.e. a non-correlated approach. This approachhas been followed here with the intensity at position n given by

In ¼ In�1 expð�jDsÞ þ Ib;n�1=2ð1� expð�jDsÞÞj ¼ � logð1� eðTloc;Yloc; ScharÞÞ=Schar

ð12Þ

A length, Schar, is needed to estimate the absorption coefficient forthe grey models. The two most obvious choices are the length ofthe computational cell Ds or a characteristic length of the computa-tional domain, a mean-beam length. The cell-based alternative doesnot yield grid-independent solutions [29] and the errors increase asthe grid is refined. With a domain-based alternative this is avoided,although it will result in erroneous transmissivities at distancesshorter than the domain length. However, the errors are kept withincertain limits and do not increase as the grid is refined. The domainbased-option is, therefore, normally recommended for combustionapplications and has been chosen here. The mean-beam length isusually calculated as 3.6V/A [28], where V is the volume of the do-main and A the corresponding surface area.

In CFD calculations the radiative heat-transfer enters the energyequation as a source term. This source term is the difference betweenthe locally absorbed and emitted radiative energy. A radiative en-ergy balance shows that this energy is equal to the divergence ofthe radiative heat flux, rq. For a grey and non-scattering mediumthe divergence can be written as

r � q ¼ r �Z

4pIðsÞsdX ð13Þ

Fig. 2. Comparison of total emissivity calculated with the SNB model and themodified WSGG model, (a) MR = 0.125 and (b) MR = 1. A version of the WSGGmodel with first order polynomials for the cj,i parameters is also included.

3. Cases investigated

To evaluate the models, the radiative source term and the inci-dent wall flux are calculated for a gaseous domain between twoinfinite black parallel plates, whose distance ranges from 0.4 mto 40 m. Both isothermal and non-isothermal cases are investi-gated, all at atmospheric pressure. The test cases cover a widerange of conditions to illustrate the ability of the model to handlea variety of computational problems in combustion applicationswith flames. The results will therefore indicate the error limitsand give an upper bound of the errors relevant for various applica-tions. In the isothermal cases the gas temperature is 1200 K andthe plate temperature 700 K, while the temperature profile in thenon-isothermal cases is described by a cosine profile

T ¼ 1250� 750 cosð2ps=SmÞ ð14Þ

where Sm is the distance between the plates. This profile rangesfrom 500 K at the plates to 2000 K in the centre of the domain. Caseswith an opposite sign of the cosine term in Eq. (14), yielding hot re-gions near the plates and a colder central region, are also included.

In these cases, the temperature of the plates is equal to that of thegas next to it, either 500 K or 2000 K. The profiles represent ratherextreme temperature differences that may not appear in most com-bustion applications. Therefore, smaller temperature differenceshave also been investigated. These have a maximum temperaturedifference of 800 K instead of 1500 K and a wall temperature of900 K.

T ¼ 1300� 400 cosð2ps=SmÞ ð15Þ

For each temperature profile different molar ratios of H2O toCO2 are investigated: MR = 0.125, MR = 0.5, MR = 1 and MR = 2.The total mole fraction (H2O + CO2) is 0.9 in all these homogeneouscases, which means that not only air-fired conditions, but also con-ditions relevant for oxy-fired furnaces are covered. In addition tothe cases with temperature variations and homogeneous concen-tration, some examples with the mole fractions given by a cosineprofile are examined,

Y ¼ 0:15þ 0:075 cosð2ps=SmÞ ð16Þ

In the non-homogeneous cases, either YH2O or YCO2 is given byEq. (16) while the other is constant at a value of 0.15. This meansthat the MR spans between 0.67 and 2 at constant YH2O and be-tween 0.5 and 1.5 at constant YCO2. For the spatial integration ofthe RTE, the discrete transfer method [30] is applied with anglesand quadrature weights of the rays taken from the S12 scheme

R. Johansson et al. / Combustion and Flame 158 (2011) 893–901 897

[31]. The six paths for which the RTE is solved are discretized into50 computational cells. For this geometry the characteristic do-main-length applied in the grey approximations is Schar = 1.8Sm.

Fig. 4. Radiative source term in cases with MR = 1, a wall distance of 10 m and a

4. Results

The total emissivities in Fig. 2 show how well the fitted param-eters of the modified WSGG model reproduce the original data. Thedeviation from the SNB model is less than 5% and of similar mag-nitude for both low and high temperatures and for the two molarratios. The accuracy can be regarded as good for such a model.Figure 2 also shows that emissivities determined using first orderpolynomials for the cj,i parameters deviated more than 10% fromthe SNB model. The second order polynomials, Eq. (9), are a goodtrade off between accuracy and computational cost and are there-fore chosen for the modified WSGG model. The polynomials of theabsorption coefficients do not influence the results to the same ex-tent, and accordingly a linear dependency has been consideredsufficient.

In Figs. 3 and 4 predictions of the radiative source term are com-pared for the three temperature profiles. The two grey models donot reproduce the spatial variations of the source term in the iso-thermal case, Fig. 3, and in the non-isothermal case with hot areasnear the walls and a colder central region, Fig. 4b. With a hot cen-tral region the grey models agree better with the reference model,although the shape of the profile deviates significantly near thewalls, Fig. 4a. The banded formulation of the WSGG model im-proves substantially the prediction of the profile compared to thegrey models. The results of the two grey models are not far fromeach other, indicating that the errors originate from the greyapproximation itself rather than from the model used to calculatethe total properties. Owing to this observation, only the results ofthe grey approximation based on the SNB model, Eq. (10), will beshown in the following. The trends shown in Figs. 3 and 4 are rep-resentative also for other distances between the plates and otherMRs.

Figure 5 shows an example of the error caused by a WSGG mod-el with parameters fitted to a specific MR but applied to conditionswith different MRs. The WSGG model with parameters fitted toMR = 1, the standard WSGG model, agrees well with the SNB model

Fig. 3. Radiative source term in an isothermal case with MR = 1 and a wall distanceof 10 m. Solid line: SNB model, dashed line: modified WSGG model, dotted line:grey model based on Eq. (10), dashed-dotted line: grey model based on Eq. (5).

temperature profile from (a) Eq. (14) and (b) from Eq. (14) with opposite sign of thecosine term. Solid lines: SNB model, dashed lines: modified WSGG model, dottedlines: grey model based on Eq. (10), dashed-dotted lines: grey model basedon Eq. (5).

when the conditions of the test case correspond to this ratio. If theratio is changed to 0.125 with the same sum of H2O and CO2, thelower content of H2O compared to CO2 decreases the radiative heattransfer as well as the maximum and minimum values of thesource term, as seen from the profiles of the SNB model. The WSGGmodel fitted to MR = 1 cannot account for these changes and givesthe same source terms in both cases. The modified WSGG model,presented in this work, including a dependence on MR in itsparameters, is well capable of predicting the difference in thesource term when MR is changed from 0.125 to 1.

Figure 6 shows the error in the radiative source term and in theincident wall flux compared to the SNB model. The error of thesource term is expressed for the entire domain: the difference fromthe reference model is integrated across the domain and comparedwith the integrated absolute value of the source term

r � qav ¼1R Sm

0 jr � qref jds

Z Sm

0ðr � qref �r � qÞ�� ��ds ð17Þ

Fig. 5. Radiative source term in cases with a wall distance of 10 m and atemperature profile according to Eq. (14). Thick lines are for MR = 1 and thin linesfor MR = 0.125. Solid lines: SNB model, dashed lines: modified WSGG model, dottedline: standard WSGG model with coefficients fitted to MR = 1.

898 R. Johansson et al. / Combustion and Flame 158 (2011) 893–901

This comparison is more representative than an integration ofthe local relative error, which would give too much influence fromregions with a small magnitude of the source term where the localerror is likely to be large but of minor interest. Apart from the devi-ation of the source term resulting from the grey approximation,which is considerable, the errors are relatively small in the isother-mal cases, Fig. 6a. The error in the source term compared to calcu-lations with the reference model is less than 10% for the modifiedWSGG model, and the wall flux is mostly predicted within 5% bothfor the grey approximation and for the WSGG model. The non-iso-thermal conditions, Fig. 6b and c, introduce more complexity to theproblem and result in larger deviations. The error in the sourceterm, applying the modified WSGG model, is less than 15% in mostcases, while the grey approximation results in a significantly largererror. With a hot central region and colder regions next to theplate, Fig. 6b, it is around 30% and more, while the error from thereversed temperature profile, Fig. 6c, is very large, see also Fig. 4.Deviations in the wall flux, calculated with the modified WSGGmodel, are around 20–40% in most cases with the hot region inthe centre, but less than 10% in the cases with a lower temperaturein the centre. The corresponding error of the grey model is in therange of 30–50% for both non-isothermal profiles.

In Figure 7 the modified WSGG model, whose parameters areexpressed by Eq. (9), is compared with a standard WSGG model,whose parameters are fitted to specific molar ratios of water va-pour to carbon dioxide, in the present case MR = 0.125, 1.0, and2.0. The investigated conditions are non-isothermal, but with thesmaller temperature variations given by Eq. (15). The set of param-eters of the standard WSGG model are the ones used to obtain theMR polynomials of the modified WSGG model. As seen in the fig-ure, the modified model gives more or less the same error as thestandard model, which indicates that the polynomials effectivelyaccount for the molar ratio. It is also worthwhile noting that thelower temperature difference, here compared to the ones ofEq. (14), results in smaller errors. Especially the error in the wallfluxes is lower and less than 15% in most cases, Figs. 6b and 7b.

The results presented so far concern cases with a constant molarratio along the path. In practical combustion applications the molarratio may differ between the flame and zones where the gas con-centration corresponds to complete combustion. As a measure ofthis effect, Fig. 8 presents the non-homogeneous cases where MR

varies between 0.5 and 2. The error in the average source term ofthe modified WSGG model is slightly larger than in the homoge-neous cases having the same temperature profile, but it is still lessthan 20%, Figs. 6b and 8a. The deviation of the wall flux has thesame magnitude as in the homogeneous cases, but it is significantlylarger than the error in the source term. Compared to the grey mod-el, the results show the same trends as in the homogeneous cases:the deviation of the source term is significantly lower, but bothmodels give large errors in the wall fluxes. These results indicatethat the modified WSGG model can handle variations of the gascomposition without introducing additional major error.

5. Discussion

The relatively large error in wall flux in the case with a centraltemperature of 2000 K and a temperature near the walls of 500 K,Fig. 6b, can be seen as a drawback of the modified WSGG model.The wall flux is a measure of the global radiative heat balance,and a large error in this quantity could indicate that the radiationmodel results in an erroneous average temperature. However, thetemperature difference, 1500 K, investigated is larger than in mostcombustion applications. In most cases of Fig. 7 with lower tem-perature difference, 800 K, between centre and wall, the error inthe wall flux is less than 15%, and here the modified WSGG modelgives a reasonable accuracy of both the source term and wall flux.The errors in Fig. 7 are likely to be more in line with many practicalcombustion situations where the temperature differences aresmaller or only present in certain regions of the domain.

If the application in mind is combustion modelling and CFDsimulations, the primary objective is to model the radiative sourceterm, which is the input the radiation model provides to the energyequation. For the prediction of the local temperature distribution,the radiative source term should thus be the most relevant mea-sure of the accuracy of the radiation model. Predictions of the radi-ative source term in a CFD simulation depends not only on theaccuracy of the radiation model, but also on many other factorsthat affect the temperature field. Such factors are, for example,convective heat transport and heat release due to combustion reac-tions. If these processes are more important than the radiativesource term in the energy equation, radiation can be treated withless detail, and a grey approximation may be sufficient. In suchapplications the modified WSGG model provides a simple methodto calculate the total emissivities needed in the grey approxima-tion for a broad range of conditions. This grey application of theWSGG model predicts the radiative source term with similar accu-racy as grey models based on detailed spectral data, Figs. 3 and 4.

Once accurate temperature and concentration fields have beendetermined, wall-flux predictions for design of heat transfer sur-faces can be achieved by post-processing with a more detailedradiation model. With such a procedure, the error of the wall fluxwould mainly depend on the ability to predict the temperature andconcentration of radiatively active species.

The calculation time of the radiation model is proportional tothe number of gases, as one radiative intensity equation of eachgas must be solved. This means that the banded formulation ofthe modified WSGG model is five times as costly as a grey approx-imation. Reduction in computational cost could be achieved by fit-ting parameters to a model with a smaller number of gases, at thecost of decreased accuracy. In the end, the choice of model must beevaluated for the specific CFD-application. The additional compu-tational cost of the radiation model depends on many other factors,such as the cost of modelling other processes and how frequentlythe radiative source term has to be updated. It could be a goodalternative to start with a grey formulation and switch to a non-grey model once a solution has been obtained.

Fig. 6. Relative errors for (a) isothermal cases, (b) non-isothermal cases with a temperature according to Eq. (14) and (c) non-isothermal cases with a temperature accordingto Eq. (14) with opposite sign of the cosine term. Solid lines are cases with MR = 0.125, dashed lines MR = 0.5, dotted lines MR = 1 and dashed-dotted lines MR = 2.

R. Johansson et al. / Combustion and Flame 158 (2011) 893–901 899

Compared to the deviation of 5% in the prediction of total emis-sivity, the error in wall flux is relatively large in some of the non-uniform cases. The main reason for the large errors is found in thescaling approximation [7,8], which is applied in this type of modelin non-uniform media, and in addition, in the simplified treatmentof the absorption coefficient. The scaling approximation assumesthat the same spectral regions belong to the same gas, irrespectiveof conditions. This is, however, not always the case, especially not

in non-isothermal media like the cases examined here. Otherabsorption-coefficient models have implemented approximationsto account for non-uniform conditions [13,20,32,33]. However, ifthese methods are to be used in a computationally efficient way,i.e. employing only a few gases, they need optimization of param-eters for the specific conditions of interest. Even if this optimiza-tion can be done by pre-processing with little influence on thecomputational time, it makes the models more difficult to

Fig. 7. Relative errors of (a) the source term and (b) wall flux for cases with atemperature from Eq. (15). Thick lines are the modified WSGG model and thin linesare the standard WSGG model with parameters fitted to MR = 0.125, MR = 1 and toMR = 2.

Fig. 8. Relative errors of (a) the source term and (b) wall flux for non-isothermaland non-homogeneous cases. The temperature is given by Eq. (14). Solid lines: YCO2

from Eq. (16) and YH2O = 0.15, dashed lines: YCO2 = 0.15 and YH2O from Eq. (16).

900 R. Johansson et al. / Combustion and Flame 158 (2011) 893–901

implement for users with no previous experience of the models.The modified WSGG model is based on a more general approachwith parameters valid for all conditions and calculated by simpleexpressions. It is, thus, easy to implement, although some accuracymay be lost compared to optimization of parameters for specificconditions. The generality of the modified WSGG model is also amajor benefit compared to the standard WSGG model with param-eters fitted for specific ratios of H2O to CO2. Conditions with vary-ing concentrations require a more flexible approach than what thestandard WSGG model can offer.

6. Conclusions

In this work the standard WSGG model, which accounts for vari-ations in temperature, was modified to also account for various ra-tios of the concentrations of H2O to CO2. Parameters of the modelare fitted to conditions in the temperature range of 500–2500 K,pressure pathlengths of 0.01–60 bar m and molar ratios between0.125 and 2. The range of molar ratios covers both oxy-fuelcombustion of coal, with dry- or wet flue gas recycling, as well ascombustion of natural gas. The modified banded formulation ofthe WSGG model and two grey models are compared with an SNBmodel for prediction of the radiative source term and the wall flux.The SNB model serves as a reference in the calculation of the param-eters of the approximate models. The conditions investigated cover

isothermal as well as non-isothermal and non-homogeneous cases.The modified WSGG model gives total emissivities for isothermalcases within 5% of the reference model, which indicates that the fit-ting well reproduces the reference data. In the isothermal test casesthe model predicts the radiative source term and wall fluxes within10%, while the error of the source term in the non-isothermal testcases is up to 15%. In test cases with both non-isothermal andnon-homogeneous conditions the error is somewhat higher, butthe source term is still within 20% of the reference model. Com-pared to a grey approximation, where the deviation of the sourceterm is 30% or more, this is a significant improvement. The modifiedWSGG model is therefore a computationally efficient option for CFDsimulations, where there is a need for a simple gas radiation modelthat is more accurate than a grey approximation. In non-isothermalcases with hot central parts and colder wall regions the error in thewall flux of the modified WSGG model is large, more than 40% insome cases with large temperature differences. This error is, how-ever, reduced to less than 15% when the maximum temperature dif-ference in the domain is decreased from 1500 K to 800 K. In theother investigated cases the deviation of the wall flux is less than10%.

Acknowledgments

Vattenfall AB and ALSTOM Power Systems GmbH are acknowl-edged for financial support.

R. Johansson et al. / Combustion and Flame 158 (2011) 893–901 901

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