A comparison between weighted sum of gray gases and statistical narrow-band radiation models for combustion applications

240

COMBUSTION AND FLAME 97:240-250 (1994)

A Comparison between Weighted Sum of Gray Gases and Statistical Narrow-Band Radiation Models for Combustion Applications A. SOUFIANI* Laboratoire d'Energ~tique Moldculaire et Macroscopique, Combustion, du CNRS et de I'ECP, Ecole Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry, France.

and

E. DJAVDAN Air Liquide, Centre de Recherche Claude-Delorme, 78350 Jouy-en-Josas, France. The weighted sum of gray gases (WSGG) and the statistical narrow-band (SNB) models are implemented for radiative transfer calculations in realistic combustion gas mixtures and their results are compared. The WSGG model parameters are generated from SNB emissivity calculations in the [300, 2500 K] temperature range for a partial pressure ratio Pw/Pc = 2. In addition, the same methods are used for the resolution of the transfer equation associated with both models. Comparisons are made for the cases of planar geometry and an axisymmetrical methane-oxygen furnace. When the gas mixture is practically isothermal and surrounded by cold walls, small errors are introduced by the use of the WSGG model. On the other hand, in the case of significant temperature gradients, the inaccurate representation of gas absorptivities by the WSGG model leads to important errors.

INTRODUCTION Radiation represents the prevailing mode of heat transfer in furnaces. Its importance increases in oxygen-fuel systems where maximum temperatures reach values over 2000 K. Consequently, an accurate evaluation of radiative fluxes is of great interest if one attempts to improve the efficiencies of such systems, to predict the wall temperature or to simulate pollutant formation. Radiation calculations in flames are complex due to the spectral structure of radiative properties of the combustion products and the huge calculation times required by the resolution of the transfer equation. Significant simplifications can nevertheless be obtained in some situations. For an optically thin (respectively thick) medium, the Planck (respectively, Rosseland) mean absorption coefficients are suitable (see, for example, Siegel and Howell [1]). In the configurations where soot radiation phenomenon is dominant,

* Corresponding author. 0010-2180/94/$7.00

constant absorption coefficients may be used for wide bands since the soot spectrum is continuous and smooth. However, in many engineering applications it is necessary to account for the fine spectral structure of gas radiative properties. In spite of their accuracy, line-by-line calculations are not used because of their computational costs. Some approximate models have been proposed. Statistical narrow-band models have been used to compute radiative intensities along lines of sight in different flame configurations by Faeth and coworkers [2-5], or to study radiative transfer in planar [6-9] or in axisymmetrical geometries [10, 11]. The total transmittance, nonhomogeneous model [12] and the effective angle model [13, 14] simplify considerably the spectral and geometrical integrations in radiative transfer calculations. But the radiation models that are mostly used for industrial configurations are based on the weighted sum of gray gases or even simple gray gas concepts [15-18]. The weighted sum of gray gases model accounts for the presence of windows in the Copyright © 1994 by The Combustion Institute Published by Elsevier Science Inc.

COMPARISON BETWEEN RADIATION MODELS

241

spectrum. It also predicts accurately the emission from a homogeneous medium since the accuracy in this case is controlled by the fitting procedure. However, absorption, which depends on the spectral structure of the incident radiation, cannot be tabulated in a general way. Moreover, the spectral dependence of wall radiative properties, which is of prime importance in glass furnaces for example, cannot be taken into account in such models. Although the weighted sum of gray gases model is commonly used for radiation calculations in flames, little work has been done to validate this approach. Modest [19] has shown the accuracy of this model in one-dimensional configurations but for an hypothetical, spatially independent, spectral absorption coefficient. The objective of the present study is to evaluate the errors introduced by the weighted sum of gray gases model by comparing its results with those of a statistical narrow-band model. The comparison is in the context of realistic configurations such as oxygen-fuel furnaces and concerns the wall radiative flux or the volumetric radiative power. In order to derive significant conclusions, it is necessary to handle similar tools. Thus the same method of resolution of the transfer equation is used in association with both models. In addition, the radiative properties are as coherent as possible. In particular, both models predict the same total emissivity for homogeneous and isothermal columns. First, the main assumptions of the weighted sum of gray gases and of the statistical narrow-band model are briefly presented. Next, the generation of new parameters for the weighted sum of gray gases model is described. The results of the two models are then compared in the case of a one dimensional geometry. Finally, we present the results obtained from both models for an axisymmetrical methane-oxygen furnace.

where 1 is the column length, /(k is the absorption coefficient of the kth gray gas, Pa is the partial pressure of the absorbing species and ak(T), whose variations account for the temperature dependence of the emissivity, is the weighting factor for the kth gas. ak(T) may be interpreted as the fraction of blackbody energy in a spectral region in which the absorption coefficient is close to /(k. In the case of spatially independent radiative properties, nonscattering medium and black walls, Modest [19] has recently shown that the radiation transfer equation can be replaced by a system of n equations of transfer for the n gray gases:

MODEL ASSUMPTIONS

I k = eak(Tw)Ib(Tw)

In the weighted sum of gray gases model, the total emissivity Er of a column of gas mixture at temperature T is represented by [15]

er(T,l, pa) = ~ ak[1 -- e--KkP°~], k=l

(1)

I = ~ Ik

(2)

k=l d1 k d$

= KkPa(akib

-- Ik)"

(3)

Equations 2 and 3 are commonly used for practical systems in which radiative properties vary with the spatial position. The weighting factor a k in Eq. 3 is then taken at the local temperature. This assumption implies that absorption by cold regions of the medium is accounted for by using the weighting factors at the temperature of the emitting region. To avoid this assumption, the model may be formulated in terms of the exact total absorptivity, which depends, for each couple of interacting points, on the whole scalar fields between these points. This total absorptivity depends on the spectral structure of the incident radiation intensity and cannot be tabulated in a general way. It is generally assumed equal to the total emissivity ~r, which is a function of only a local temperature. In order to use the model in practical situations, the weighted sum of gray gases model is extended here to nonblack but gray walls by assuming diffuse wall emission and reflection:

+(1-- e)l f

Ik'inCcosOd~q,

(4)

"IT " 2 ~rsr

where E designates the wall (gray) emissivity and I k' inc is the incident radiative intensity for the gas k.

242

A. SOUFIANI AND E. DJAVDAN

Statistical narrow-band models generally lead to analytical expressions of the transmissivity Yv averaged over a spectral range A v containing many absorption lines, but small enough to assume that Ibv remains constant inside it. Statistical assumptions are made for line locations, shapes and intensities. The result for homogeneous columns is a simple expression of Yv as a function of few parameters characterizing the high resolution spectrum. For polyatomic molecules such as H 2 0 and CO2, it has been shown [6] that the best agreement with line by line calculations is achieved when the following assumptions are used: (i) the absorption lines are randomly located inside A v, and (ii) line intensities obey the exponential-tailed inverse probability distribution law introduced by Malkmus [20]. ~'v is then given, for an isothermal and homogeneous column of length I and for Lorentz line shapes, by Yv=exp - , ,

1

~

1

,

(5)

where k is the mean line intensity to spacing ratio and ~ is the line-overlapping parameter. For nonisothermal columns, we use the Curtis-Godson approximation which leads to accurate results for moderate line-width variations along the path [6, 21]. For mixtures containing more than one absorbing species in the same spectral region, the mixture transmissivity is given by the product of individual transmissivities, since spectra of different gases are not correlated. Radiative transfer equation, when averaged over A v may be written in the integral form ,s

0~

iv(s) = Iv(0)%(0, s) + )nlbv(s')-~s, (S', s) ds'. (6) Rigorously speaking, the reflected part of the leaving wall intensity Iv(0) is spectrally correlated with the transmissivity between 0 and s; however, the decorrelation hypothesis Iv(0)%(0, s) = Iv(0)%(0, s),

(7)

which is used in this study, introduces small errors in many practical situations (between

0% and 4% on wall fluxes for wall reflectivities up to 0.9 in the configurations considered in Ref. 9). WEIGHTED SUM PARAMETERS

OF

GRAY

GASES

WSGG parameters have been calculated by several authors. Using a statistical narrow-band model and experimental spectral data, Taylor and Foster [22, 23] have fitted C O 2 - H 2 0 and C O z - H 2 0 - s o o t mixture parameters by using, respectively, a one-clear 3-gray gases and a 3-gray gases model for temperatures between 1200 K and 2400 K, Pw/Pc = 1 or 2 (Pw and Pc are respectively, H 2 0 and CO 2 partial pressures). Nakra and Smith [24] have used the H 2 0 - C O 2 emissivities given by Hottel and Sarofim [15] for a one-clear 3-gray gases model. Third-order polynomials have been used for ak(T) in the case Pw/Pc = 1 and temperatures between 560 and 1960 K. Smith et al. [25] proposed a one-clear 3-gray gases model and third-order polynomial fit of ak(T) for H 2 0 - C O 2 mixture emissivities and absorptivities for temperatures between 600 and 2400 K, Pw/Pc -- 1 or 2. Absorptivity fittings are also given for Pc --' 0 and p,~ --, 0. The initial emissivities are obtained in this study from the exponential wide-band model [26, 27]. For temperatures over 2000 K, Coppalle and Vervisch [28] have adjusted total C O 2 - H 2 0 mixture emissivities calculated from the wide-band model of Edwards [26], for P,,/Pc = 1 or 2. The dependence of the weighting factors on temperature is linear as in [22]. As the objective of this study is to evaluate the accuracy of WSGG model when applied to radiative transfer calculations, the parameters of this model and those of the reference model (SNB) must lead to the same total emissivities for homogeneous and isothermal columns in wide ranges of temperature, partial pressures and optical thickness. Thus, although WSGG parameters, for the conditions considered in this study, are available in the literature, we generate new WSGG parameters by fitting total emissivities calculated from the reference SNB model described in the previous section. This is done for H20-CO2-transparent gas mixtures at a total pressure p = 1 arm and a

COMPARISON B E T W E E N R A D I A T I O N MODELS ratio of partial pressures Pw/Pc = 2, corresponding, for example, to methane-air or methane-oxygen combustion. SNB model parameters have been derived from line-by-line calculations using a spectroscopic data basis extended to high-temperature applications [6, 29, 30]. The low resolution used both for WSGG parameter generation and for the SNB calculations presented in the following sections is A v = 25 cm -~. The covered spectral range is [150-7000 cm-1]. It includes the most influential absorption bands of H 2 0 and CO 2. We use a weighted sum of three gases and one clear gas, as is most often done: 3

er = Y'. ak[1

e-rkPwl],

-

(8)

k=l

the fourth gas is the clear gas with a weighting coefficient: 3

a 4 = 1 -- ~ a k k=1

(9)

Note that the CO 2 partial pressure does not appear in Eq. 8. However, this expression can be used since the ratio Pw/Pc is fixed in this study. The fitting procedure adopted is similar to the method described in Ref. 22. The ranges of variation of water vapor partial pressure and of the length are respectively [0.01, 0.66 atm] and [1 cm, 200 cm]. A regular mesh (20 × 20) in these ranges is used for the adjustments. The coefficients Kk and a k (k = 1, 2, 3) are first fitted for T = 1700 K by using a nonlinear least-square procedure. For other temperatures in the range [300, 2500 K] (with a temperature step AT = 200 K), the Kk values are fixed to those obtained at 1700 K and the coefficients a k are fitted for each temperature value. The dependence of these coefficients versus temperature is finally adjusted by using

243

a fifth-order polynomial: 5

ak(T) = Y'~ akjTJ.

(10)

j=0

The parameters Kk and ak; are given in Table 1 and the original and adjusted functions ak(T) are shown in Fig. 1. The use of the adjusted parameters leads to relative errors on the emissivities calculated for the same values of (Pw, l) used for the adjustment, characterized by a quadratic mean value of 6.3% in the whole temperature range and of 4.4% in the temperature range [1100-2500 K]. The highest errors (about 20%-30%) occur for very small emissivity values (Pw = 0.01, l = 1 cm). It seems difficult to achieve a better WSGG parameter adjustment over the whole (Pw, 1) range since the expression for Er in Eq. 1 takes into account only the variations of the product pwl and not the variations of Pw and l separately. More precisely, the important difference between self and foreign broadening of H 2 0 absorption lines does not appear in this expression. The expression of the average value of H 2 0 line widths in H 2 0 - C O 2 - N 2 - O 2 mixtures [6], ~H2O = 0.46p~--~-- + × [0.079p~ + O.lpc + 0.079pN 2 + 0.036po 2],

(11)

shows that the use of the product Pw l alone in the expression of e r introduces necessarily significant errors, even if the number of gray gases is increased. Nevertheless, these errors decrease as the temperature increases since the resonant term 0.46p,(296/T) decreases quickly with temperature and line widths become less sensitive to Pw. Finally, it is worth noticing that parameter adjustments over reduced temperature ranges

TABLE 1 Weighted Sum of Gray Gases Parameters x k ( m - 1 a t m - 1) and akj (to be used in Eq. 10 with T in Kelvin).

k

l
OtkO

Otkl

Otk2

Olk3

Olk4

Otk5

1 2 3

1.2531 8.4258 87.064

1.6879 E-1 4.9577 E-2 2.7890 E-1

2.5682 E-4 9.3954 E-4 -5.1265 E-4

9.5161 E-8 - 1.6416 E-6 6.7320 E-7

- 3.1660 E-10 1.1478 E-9 -5.1488 E-10

1.4834 E-13 - 3.7600 E-13 1.8887 E-13

- 2.1560 E-17 4.7503 E-17 -2.5856 E-17

244

A. SOUFIANI AND E. DJAVDAN

0.4

I

0.3 f

6

i

I

the volumetric radiative power is written in a discretized form:

i

Aqr i Pri . . . . .

Ay~

500

1000

1500

2000

2500

(for example partial adjustments in [300, 900 K] and [1100, 2500 K] ranges) do not improve appreciably the quality of the fit.

APPLICATION TO CONFIGURATION

ONE DIMENSIONAL

Resolution of the Transfer Equation We consider a planar H 2 0 - C O 2 gas mixture at atmospheric pressure between two parallel, infinite, diffuse plates at temperatures T 1 and T N. The ratio P~/Pc is equal to 2 and wall emissivities el and eN are gray, as required by the use of the WSGG model. We calculate the radiative field given a fixed temperature profile in the gas. The solution of the monochromatic radiative transfer equation (RTE) is well known in this case (see, for example, Ref. 1) and its extension to the WSGG model may be achieved in a straightforward manner [19]. In order to avoid differences between the results due to the use of different resolution methods, the same numerical method is used for the resolution of the RTE associated to both WSGG and SNB models. The numerical procedure used here is described in detail in Ref. 6 where it was applied in conjunction with the SNB model. We show in the following how this procedure is extended to the WSGG model. The medium is subdivided into (N - 2) infinite parallel layers. For the gaseous layer i of width Ay~ between the abscissa Yi-1 and y;,

(12)

where k designates the kth gray gas. The radiative flux difference Aq,~ may be written as a sum of interaction terms with the walls (j = 1, N) and the other layers (j = 2 , . . . , N - 1):

T (K)

Fig. 1. WSGG model weighting factors ak(T). Points obtained from individual fits at each temperature. Solid curves correspond to polynomial adjustment (Eq. 10, Table 1).

Aq k 2~ ~ k Ay~

N

Aq~ = E a j ki B )k•

(13)

]=1

Ajk represents the geometrically averaged emission, transmission and absorption processes. It is a combination of terms such as f0~ri~/~ dt~ where ~ is the cosine of the angle 0 between the normal to the plates (Oy) and a given propagation direction and rikj designates the transmissivity of the column of length (Yj - Yi)/lz. As the WSGG absorption coefficients Kk have fixed values, this spatially averaged transmissivity is given by 1 k

£ riil.~dtx=Ea[Kkp,,(yj-yi)],

(14)

where E 3 is the third integroexponential function. For j = 2 , . . . , N - 1, n~ is proportional to the fraction of the blackbody emitted flux for the kth gray gas:

B k = ak(Tj)trTj 4,

j -- 2 , . . . , N - 1,

(15)

where Ty is the temperature of layer j. For j = 1, N, B~ designates wall radiosities: N

B k = elak(T1)trT 4 + (1 - el ) ~ A j l B k, j=2

(16) N-1

B~ = eNak(TN)trT ~ + (I -- e~V) ~ AjNB k. j=l

(17)

COMPARISON BETWEEN RADIATION MODELS The knowledge of all the variables Bf and A1k leads to the volumetric radiative power (Eqs. 12, 13) and to wall fluxes:

245 '\

~,

-100

I

",,,

k k A j l B j - ak(T1)o"

k

\j=2

=

EEN k

E

~

SNB

""~"'................

-200 -250x1~

N-1

qrN

(]8)

,

I

..... WSGG

"..

-150

q~l = ~ 1

1

50

)

k k A::,:Bj

ak(TN)~r~

100 L (cm)

.

~50

20o

(a)

j=l

(19)

"-..

-50

RESULTS AND DISCUSSION The configuration studied in this section is an atmospheric homogeneous H 2 0 - C O 2 mixture (pw _-_ 2 atm, Pc = 1 atm). The calculation domain is subdivided into 20 equal layers ( N - 22) and typically 25 directions in the [0, zr/2] 0 range are considered. The first set of calculations is for a parabolic temperature profile between two hot plates. The wall temperatures are T 1 = Tu = 2500 K while the gas centerline is at 500 K. In Fig. 2a is plotted the variation of the wall radiative flux obtained from SNB and W S G G models as a function of the distance between the plates (the emissivities of the walls are = el = eN = 0.8). This figure shows that the wall flux absolute value calculated by using the W S G G model is always underpredicted by at least 30%. The relative difference between the model results increases as the medium becomes optically thin. Due to the formulation of the W S G G model, the gas absorption phenomenon is described by absorption weighting factors taken at the emitting body (here the wall) temperature. Consequently, since the wall temperature is higher than the gas temperature, and since the total emissivity decreases quickly as temperature increases, gas absorption is underpredieted by the W S G G model, resulting in an underprediction of the absolute value of the wall radiative flux. Figure 2b shows, for the same temperature profile, the variation of the wall flux as a function of the wall emissivities. H e r e the distance between the two plates is 1 m. The r e l a t i v e d i s c r e p a n c y 100 x ( q W S G G _

I

"".... ............

I

I

I

..... WSGG

-150 -200 -250x103 I

I

0.2

0.4

I

I

0.8

0.5

1,0

£

Co)

Fig. 2. Wall radiative flux (one dimensional geometry, T1 = TN = 2500 K, centerline temperature = 500 K) (a) versus distance between plates with ~ = eI = ~lv = 0.8, (b) versus wall emissivitywith L = 1 m. qwSNBx-) / q wSNB varies from 30% to 43% and is

higher for small wall emissivity values. The second set of calculations is for a parabolic temperature variation between two cold plates at T 1 = TN = 500 K, the centerline being at 2500 K. This temperature profile is close to industrial furnace configurations. Figure 3a shows the variation of the wall radiative flux computed from both models as a function of the distance between the plates, the emissivities of the walls being again E = e I = •N = 0.8. H e r e the wall radiative flux is overpredicted when the W S G G model is used. This phenomenon is coherent with the fact, mentioned previously, that the W S G G model associates a total absorptivity taken at the emitting body temperature. But here, the principal emitting body is the gas itself in the high temperature region. Absorption occurring in the cold parts of the gas is underpredicted, which leads to an overprediction of wall fluxes. Moreover, since the gas is surrounded by cold walls, gas emission, which is well described by the W S G G model, is the dominant phenomenon. Consequently, the differences between model results

246

A. SOUFIANI AND E. DJAVDAN

120X103~._

I

Calculation ~lanes

.....~0,.,.~.~

1O0~

40

I

D

.°.oO......°...-....................................

!

,Ol-

I

..//"~

..... w s c o

i| U l II~mi

( , 50 100,-7150

200

L (cm)

Fig. 4. Spatial and directional discretizations in axisymmetrical configuration.

(a) I

|

I

I

.-"1

~ 120xlOS1006080~,.......,...-'"°"'""'"'"'"'"'"'°"°'"'" i 40

SNB

20 ~ I

0.2

I

I

0.4

£

0.6

0.8

1.0

Co) Fig. 3. Wall radiative flux (one dimensional geometry, T 1 = T N = 500 K, centerline temperature = 2500 K), (a) versus distance between plates with • = •l = •N = 0.8,

(b) versus wall emissivitywith L = 1 m.

are smaller than the differences obtained in hot plates configuration. Figure 3b shows, for the same temperature profile, the variation of the radiative flux as a function of the wall emissivity (the distance between the two plates is 1 m). For very small wall emissivities, the simple interpretation given in the case of absorbing walls is no more sufficient. Gas to gas radiative transfer becomes predominant as E decreases and the WSGG model may lead to an underestimation of the wall flux.

APPLICATION TO CONFIGURATIONS

x

lab

AXISYMMETRICAL

Resolution of the Transfer Equation

We use here also the same numerical method to resolve the radiative transfer equation associated with the SNB or with the WSGG model. The method uses spectral, spatial, and directional discretizations, as described in detail in Refs. 10 and 11, where the SNB model is used.

Only the modifications introduced by the use of the WSGG model are given here. Calculations are in planes parallel to the system axis and tangential to the coaxial cylinders, defined by the radial discretization points (Fig. 4). For each plane and each transfer direction u characterized by the angle 0, the radiative intensity I k in the kth spectral domain (or the kth gray gas) is step-by-step calculated by using an interpolation method. With the notations of Fig. 4:

Ik(M)

=

Ik(p).gk(pM) +(1 - ~J'(PM))I~(E),

(20)

where Ik(p) is obtained by an interpolation between Ik(B) and Ik(C) and E is the middle point of the segment PM at temperature TE. I~(E) s t a n d s for A vlbvk(Te) (resp. ak(Te)o'T~/~r) when the SNB (respectively WSGG) model is used. In Eq. 20, the spectral correlation between transmissivity and radiative intensity is neglected. This is coherent with the WSGG model but not with the SNB model. In the latter case, the spectral effects due to the correlations between absorption line positions for different column elements are computed by using the Curtis-Godson approximation for some principal directions and approximated for other directions by using a continuous correlation function [10, 11]. The use of Eq. 20 for each direction inside each plane, and of the boundary condition for diffuse walls lead to a linear system where the unknowns are the wall leaving intensities. This system is resolved for each kth spectral range (or each kth gray gas). The volumetric radiative power may be computed practically in the

COMPARISON BETWEEN RADIATION MODELS W S G G model either from the divergence of the radiative flux "

OIk

Pr(M) = -- k J47r ~ d~'~

247

02

Temperature (K), Tw = 1850 K

US o~

1900

2000

1900

B.lS

(21)

o,os o ,

_

~ .

(a)

or directly from the intensity field

Temperature (K), Tw = 500 K

Pr(M)

=

--~kf4~rsrKkPw .~

×(lk-ak(M) trT~t) dl~'qr

(22)

o.ls

~ o.,o.o5L o

~

1600 "~.'~_._.~4..00~ -.

-

Oo)

The first method is used, since there is no physically meaningful absorption coefficient compatible with the SNB model. RESULTS AND DISCUSSION

H 2 0 Molar Fraction, Tw = 1850 K

IIJ o.0

=

o

In this section we consider an axisymmetrical furnace (0.6 m diameter, 2 m long). Combustion o c c u r s in a t u r b u l e n t d i f f u s i o n methane-oxygen flame. The fuel and the oxident are introduced in the furnace through a coaxial burner. The inlet velocities are 60 m s -1 for methane and 40 m s - t for oxygen corresponding to mass flow rates of respectively 7.62 × 10 -4 kg s - t and 30.5 1 0 - 4 kg s-1. The diameter of the outlet is 0.2 m. Calculations of the flow are performed using a finite volume-SIMPLE computational code. Note that due to the high temperatures prevailing in oxygen-fuel furnaces, endothermic dissociations of CO 2 and H 2 0 are not negligible. However, the only reaction and species considered here are CH 4 + 2 0 2 ~ C O 2 "k- 2 H 2 0 ,

(R1)

but the specific heats of the combustion products take into account the presence, at equilibrium, of species such as CO, H 2, O H . . . Moreover, since the study is focused on radiation models, only a simple turbulent combustion model is used to yield representative temperature and concentration fields. In particular, the turbulent rate of reaction (R1) is determined by a mixing rate controlling model [31]. Two sets of calculations are performed corresponding to fixed wall temperatures of 1850 and 500 K. The emissivity of the walls is 0.6 and the

51 .6

. . . . o.5

1 X (m)

__,__L__X_. 1.s

2

(c)

Fig. 5. Axisymmetrical temperature and molar fraction fields used for comparisons, (a) temperature field (Tw = 1850 K), (b) temperature field (T~.= 500 K), (c) H20 molar fraction field = 1850 K).

(Tw

radiation model used to obtain the flow field is a gray gas-discrete transfer model as described in Ref. 18. The numbers of cells for the flow field calculations are 146 and 74, respectively, for the axial and the radial directions. The SNB and W S G G calculations use the results of the flow field calculations. For computational time considerations, and in order to focus the attention on radiation models, only the radiation field is calculated and the effect of the new results on the flow field is not considered (uncoupled calculations). Moreover, for computational reasons the flow field is represented, for radiation calculations, by a 30 × 15 grid obtained after linear interpolations from the initial grid. In each calculation plane, 25 regularly spaced 0 values are considered in the range [0, 7r/2]. In Figs. 5a and 5b are displayed the temperature fields of the flow calculations corresponding respectively to wall temperatures of 1850 and 500 K. The gas is almost isothermal in the major part of the furnace. Figure 5c shows the H 2 0 molar fractions field (Tw = 1850 K). The gas is homogeneous, apart from

248

A. SOUFIANI AND E. DJAVDAN

the region close to the injection. A similar field of H20 is obtained in the case T~ = 500 K. Figure 6 shows the wall radiative flux computed from the SNB and WSGG models corresponding to wall temperatures of 1850 K. It is seen that the flux at the upper wall is underpredicted by more than 30% by the WSGG model, while the same model overpredicts drastically the flux at the injection side wall. It can be seen from Fig. 5a that the temperature radial gradient is small, indicating that for the radial direction the gas can be considered as almost isothermal at a temperature close to 2100 K. The underprediction of the wall flux with the WSGG model is due again to an inaccurate representation of gaseous absorption. The flux emitted by the walls and absorbed by the gas is overestimated, since the walls are at a lower temperature. The tendencies are inverted for the injection side wall. In fact, it appears on Fig. 5a that the axial variation of gas temperature is not negligible, especially close to the burner. Moreover, the gas is optically thicker in the axial direction. Considering for simplicity two zones in the gas, one (Z1) in the center of the furnace at about 2100 K and another (Z2) close to the injection side wall at about 1900 K, and assuming, in a first approach, that the walls are black, the wall radiation flux (injection side) can be decomposed into (i) the flux emitted by Z1 and transmitted by Z2, (ii) the flux emitted by Z2, and (iii) the opposite of the flux emitted by the wall and absorbed by the gas. As the

WSGG model attributes to the gas a total absorptivity at the emitting body temperature, the first term (i) is overpredicted while the third term (iii) is underpredicted by using the WSGG model. In this condition, the global radiative flux can increase or decrease depending on the relative weight of the different terms. The sign of this flux may be reversed as shown on Fig. 6 for points close to the flow inlet. Figure 7a shows the axial variation of the radiative power Pr computed from the SNB model for different radii, and Fig. 7b represents the axial variation of the relative difference: p(SNB)

_

p(WSGG)

(23)

A P, =

for the same radii. On Fig. 7a it appears that the gas mixture near the inlet absorbs more radiation than it emits (negative values of Pr). As absorption of radiation emitted by the hot region is underestimated by the WSGG model, the absolute value of the radiative power is also underestimated. This result is in agreei

I

i

soo

~ 0 .~ ~ -so0

'

---

-I000 0.0

t--0.!

I

I

I

0.5

1.0

1.5

2.0

x (m) 15

i

i

i

(a)

i

1.o

....

i

i

i

10 0.8

~

. . . . . r--0.225 m

o~

:.~,

-- ~.o~

5 "'*'**.,.. "~

of_. -5

a

0.0

-SNB ...... WSGG

0.4

0.0 c

b l

I

I

I

0.5

1.0

1 .S

2.0

s (m)

Fig. 6. Wall radiative flux in the case Tw = 1850 K. (ab) corresponds to radius of injection side wall and (bc) is outer wall parallel to the axis of combustion chamber.

-0.2

0.0

\j I 0.5

I 1.0

I 1.5

2.0

X(m)

Co) Fig. 7. Radiative power versus axial coordinate for different radii (T~, = 1850 K), (a) SNB model, (b) relative difference p}SNm _ p(wsGG)/p}SNn)

COMPARISON BETWEEN RADIATION MODELS 35

10 0.0

I

I

I

I

I 0.5

I 1.0

I 1.5

I 2.0

s (m)

Fig. 8. Wall radiative flux for TW= 500 K. Same convention for s as in Fig. 7.

ment with the overprediction of the radiative flux on the injection side wall. Figure 8 shows the wall radiative flux calculated by the SNB and the WSGG models for wall temperatures equal to 500 K. The previous tendencies are found again in this case but the differences between the WSGG and SNB results are less important. Gaseous emission is here the dominant phenomenon and the differences between model results, due to the inaccurate representation of absorption when using the WSGG model decrease. CONCLUDING REMARKS Weighted sum of gray gases (WSGG) parameters have been generated to investigate the ability of this model to predict radiative transfer in systems involving typical combustion products. Radiative heat fluxes and volumetric power computed from this model have been compared with those obtained from a statistical narrow-band (SNB) model. The same numerical methods have been implemented to resolve the radiative transfer equation, associated with the WSGG or with the SNB model, in planar and axisymmetrical systems. These systems have necessarily gray walls, as required by the WSGG model. Important errors in radiative wall fluxes and power per unit volume are found, especially when significant temperature variations are encountered. The errors are smaller when the medium is hot, almost isothermal, and surrounded by cold walls. The usual implementa-

249

tion of the WSGG model leads to an inaccurate prediction of gas total absorptivities since the weighting factors are always taken at the temperature of the emitting body. Absorption by cold gases of radiation emitted by hot walls or gases is generally underestimated. In realistic configurations, this may lead to important errors and even to the inversion of the radiative flux sign. This difficulty cannot be circumvented when the radiative transfer equation is used in a differential form. When the latter is written in an integral form, some authors (e.g., Modest [19]) recommend the use of a total absorptivity, depending on the whole temperature and concentration fields along the line of sight. This is an attractive idea but its implementation is not easy since the weighting factors for absorption must be functions of the scalar fields or of the spectral distribution of the incident radiation. This type of implementation requires further research. SNB models are of course more CPU time consuming than the WSGG. model. In this study, about 200 narrow-bands have been used while only four gray gases are sufficient for a satisfactory prediction of homogeneous and isothermal columns. The CPU time ratio is then about 50. However, the width of the narrow spectral ranges A v = 25 cm -t may be increased with small loss of accuracy. REFERENCES 1. Siegel, R., and Howell, J. R., Thermal Radiation Heat Transfer, McGraw-Hill, New York, 1981. 2. Sivathanu, Y. R., Kounalakis, M. E., and Faeth, G. M., Twenty Third Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1990, p. 1543. 3. Jeng, S. M., Lai, M. C., and Faeth, G. M., Combust. Sci. Technol. 40:41-43 (1984). 4. Faeth, G. M., Jeng, S. M., and Gore, J. P., in Heat Transfer in Fire and Combustion Systems (C. K. Law et al., Ed.), ASME, New York, 1985, Vol. 45, p. 137. 5. Kounalakis, M. E., Sivathanu, Y. R., and Faeth, G. M., J. Heat Transl. 113:437-445 (1991). 6. Soufiani, A., Hartmann, J. M., and Taine, J., J. Quant. Spectrosc. Radiat. Transl. 33:243-257 (1985). 7. Soufiani, A., and Taine, J., Int. J. Heat Mass Transl. 30:437-447 (1987). 8. Kim, T. K., Menart, J. A., and Lee, H. S., J. Heat Transl. 113:946 (1991). 9. Menart, J. A., Lee, H. S., and Kim, T. K., J. Heat Transf. 115:184 (1993).

250 10. Zhang, L., Souliani, A., and Taine, J., Int. J. Heat Mass Transl. 31:2261-2272 (1988). 11. Soutiani, A., and Taine, J., The Sixth International Symposium on Transport Phenomena in Thermal Engineering, Seoul, Korea, 1993, p. 181. 12. Grosshandler, W. L., Int. J. Heat Mass Transl. 23:1447-1459 (1980). 13. Bhattacharjee, S., and Grosshandler, W. L., Int. J. Heat Mass Transl. 33:507-516 (1990). 14. Bhattacharjee, S., and Grosshandler, W. L., Combust. Flame 77:347-357 (1989). 15. Hottel, H. C., and Sarofim, A. F., Radiative Transfer, McGraw-Hill, New York, 1967. 16. Johnson, T. R., and Beer, J. M., Fourteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1973, p. 639. 17. Carvalho, M. G., and Nogueira, M., in Heat Transfer in Radiating and Combusting Systems (Carvalho et ai., Eds.), Springer-Vedag, 1991, p. 374. 18. Lockwood, F. C., and Shah, N. G., Eighteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1980, p. 1405. 19. Modest, M. F., J. Heat Transl. 113:650-656 (1991). 20. Malkmus, W., J. Opt. Soc. Am. 57:323-329 (1967). 21. Young, S. J., J. Quant. Spectrosc. Radiat. Transl. 18:1-28 (1977).

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22. Taylor, P. B., and Foster, P. J., Int. J. Heat Mass Transl. 17:1591-1605 (1974). 23. Taylor, P. B., and Foster, P. J., Int. J. Heat Mass Transl. 18:1331-1332 (1975). 24. Nakra, N. K., and Smith, T. F., J. Heat Transl. 99:60-65 (1977). 25. Smith, T. F., Shen, Z. F., and Friedman, J. N., J. Heat Transf. 104:602-608 (1982). 26. Edwards, D. K., in Advances in Heat Transfer (T. F. Irvine and J. P. Hartnett, Eds.), Academic Press, New York, 1976, Vol. 12. 27. Modak, A. T., J. Quant. Spectrosc. Radiat. Transl. 21:131-142 (1979). 28. Coppalle, A., and Vervisch, P., Combust. Flame 49:101-108 (1983). 29. Taine, J., J. Quant. Spectrosc. Radiat. Transl. 30:371-379 (1983). 30. Hartmann, J. M., Levi Di Leon, R., and Taine, J., J. Quant. Spectrosc. Radiat. Transl. 32:119-127 (1984). 31. Magnussen, B. F., and Hjertagger, B. H., The SIXteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1976, p. 719. Received 2 August 1993; revised 12 December 1993