Absorbancia y Emisividad

Pergamon Prog. Energy Combtut. Sri’. Vol. 22, pp. 543-574, 1996

0 1997 Ekvier Science Ltd. All rights reserved. Printed in Great Britain.

OMO-1285/% 529.00

PII: !30360-1285(96)ooo10-x

EVALUATION OF EMISSIVITY CORRELATIONS FOR H20-C02-N2/AIR MIXTURES AND COUPLING WITH SOLUTION METHODS OF THE

RADIATIVE TRANSFER EQUATION

N. Lallemant*, A. Sayret and R. Weber International Flame Research Foundation, P.O. Box 10000, 1970 CA IJmuidpn, The Netherlands

Ah&act-This study is directed towards the limitations of applying total emissivity corre&.ions in computational fluid dynamics (CFD) computer codes for game modeling. The predictions of nine widely applied total emissivity models for HrO-CO2 homogeneous mixtures are compared with the exponential wide band model (EWBM) calculations. The comparison covers a range of total pressures, temperatures and path lengths which are suitable for the use of tine numerical grids in CFD simulations of atmospheric and high pressure combustors.

Attention is paid to coupling of the property models with the radiative transfer equation (RTE) and their performance in non-homogeneous applications. In this respect both the total transmittance non-homogeneous (TINH) model and the spectral group model (SGM) ate used. The latter model is combined with five weighted sum of gray gases models (WSGGM), the single line based sum of gray gases model (SLW) and the k-distribution model. The non-homogeneous validation tests used in situ total radiance measurements in two non-luminous natural gas games repmsem.ing two industrial situations, a water cooled furnace and a refractory lined fumacx. The main wnchtsions am as follows. The spectral group model provides an elegant and accurate method of coupling WSGGM, k-distribution and SLW property models to the equation of radiative transfer. Both homogeneous and non-homogeneous tests indicate the advantage of using the Smith, Shen and Friedman weighted sum of gray gases model over polynomial correlations and the SLW model. It has been shown that in the near burner region of a natural gas diffusion game, the water vapor to carbon dioxide partial pressure ratio departs significantly from the value expected for the complete combustion of methane in air. This tinding emphasizes the limitation of existing WSGGM to Hz0 to COr partial pressure ratios of one and two only. 6 1997 Elsevier Science Ltd. All rights reserved.

CONTENTS

Nomenclature Abbreviations 1. Introduction 2. Benchmark Data to Validate HrO-CO2 Emissivity Predictions

2.1. Standard and Total Emissivity of Gases 2.2. Benchmark Data and Models for H20-COr-Nr Radiative Property Calculations

2.2.1. HITRAN database for line-by-line calculations 2.2.2. General dynamic database for narrow band model calculations 2.2.3. Exponential wide band model database 2.2.4. k-distribution database for calculations of gas radiative properties 2.2.5. Benchmark total emissivity data 2.2.6. Model accuracy for total emissivity calculations

3. Correlations for Predicting the Total Emissivity and Absorptivity of Combustion Gases 3.1. Weighted Sum of Gray Gases Model 3.2. Polynomial Approximations 3.3. Hybrid Models

3.3.1. Steward and Kocaefe ‘Box Model’ 3.3.2. Single line weighted sum of gray gases model

4. Assessment of the Accuracy of Several Total Emissivity Correlations (Homogeneous Calculations) 4.1. Generalities 4.2. Emissivity Tests at Atmospheric Pressure 4.3. Total Emissivity Tests at Elevated Pressure 4.4. Effective Linear Absorption Coefficient at Atmospheric Pressure 4.5. Runtime Comnarison between the EWBM and Emissivitv Correlations

5. Coupling of Total Emissivity Correlations with the RTE * 5.1. Spectral Versus Gray Calculations

5.1.1. The spectral and angular dependence of the RTE 5.1.2. A simple classification of radiation problems 5.1.3. Search for an &cient coupling procedure between existing radiative property

models and the RTE 5.2. Gray and Semi-Gray Gas Approaches

544 544 544 546 546 546 547 547 547 548 549 549 549 551 551 552 552 552 553 553 555 556 556 557 557 557 557 557

558 559

* Corresponding Author. t Presently working at Babcox and Wilcox, U.S.A.

543

544 N. Lallemant et al

a

;i,p

ki

ka

L p, P T

5.3. 5.4. 5.5.

Patch Treatment of the RTE in Homogeneous Non-Gray Media 559 Total Transmittance Non-Homogeneous Model 560 Spectral Group Model ,562 5.5.1. Model formulation 562

5.5. I. 1. Spectrally averaged RTE 562 5.5. I .2. Domain of validity of spectral group model 563 5.5.1.3. Computation of parameters wi and kj 564 5.5.1.4. Advantages and limitations of spectral group model 564

6. Assessment of Radiative Property Models for Non-Homogeneous Calculations 564 6. I. The BERL Experiment73,74

6.1. I. Exoerimental furnace 6. I .2. Natural gas flames 6. I .3. In-flame measurements 6. I .4. Total radiance measurements

6.2. Water Vapor to Carbon Dioxide Partial Pressure R .atio 6.3. Evaluation of Gray Gas and Patch Model 6.4. Evaluation of TTNH Model 6.5. Evaluation of Spectral Group Model

7. Conclusion 8. Some Unsolved Problems Acknowledgments References

56.5 565 565 566 566 567 567 568 570 571 572 57’ 57

NOMENCLATURE

weighting coefficients volumetric concentration pressure absorption coefficient of spectral group ‘i’ = ki,p p: linear absorption coefficient of spectral group ‘i’ linear absorption coefficient path length total pressure partial pressure and sum of partial pressures absolute gas temperature

Greek symbols E emissivity V frequency u Stefan-Boltzmann constant

Subscripts C carbon dioxide car calculated using a given emissivity correlation g gas i absorbing gas ‘i’ j remaining gas ‘j’ r all remaining gases but gas ‘i’ W water vapor

Superscripts n spectral 0 standard reference point (Pt = 1 atm, pi = 0) also

designates the blackbody radiation

ABBREVIATIONS

BA BEA BERL CFD DOM EWBM

block approximation band energy approximation Burner Engineering Research Laboratory computational fluid dynamics discrete ordinate method exponential wide band model

ERZ IFRF IRZ LBLM NBM NBZ NG RTE SCM SLW

TTNH WBM WSGGM

external recirculation zone International Flame Research Foundation internal recirculation zone line-by-line models narrow band models near burner zone natural gas radiative transfer equation spectral group model single line weighted sum of gray gases model total transmittance non-homogeneous wide band models weighted sum of gray gases models

1. INTRODUCTION

Over the last 30 years, several approximate models for total emissivity and absorptivity calculations of water vapor and carbon dioxide mixtures have been developed. The development of these models is attributable to the advent of computer codes for heat transfer calculations and to the necessity of speeding up the computation of radiative properties without recourse to spectral integration. To date, most computational fluid dynamic (CFD) codes for flame modeling make use of such simplified models. The simplicity of the models’ mathematical formulation and their ability to simulate the non-grayness of gases in a wide range of temperature and partial pressures has greatly contributed to their adoption in CFD codes. Most total emissivity and absorptivity correlations for H20-CO2 mixtures exist either in the form of polynomial approximations’-3 or weighted sum of gray gases models (WSGGM).4-‘5 Steward and Kocaefe’6*‘7 also proposed a total emissivity correlation that possesses some characteristics of a spectral model, while Denison and Webb’8-20 developed the single line weighted sum of gray gases based

Evaluation of emissivity correlations for HZO-COz-NJair 545

on the k-distribution model. The difficulty of selecting an efficient total emissivity correlation for use with CFD codes for flame calculations arises for three reasons.

First, although Hottel’s charts are still the most referenced benchmark source for total emissivity data of Hz0 and COz, it is no longer the most accurate one. Since their publication in 1954,2’ both spectral databases and calculation techniques for line-by-line models (LBLM), narrow band models (NBM) and wide band models (WBM) have considerably improved. To date, these models provide reliable ways of estimating the total emissivity of gases in the regions where Hottel charts are based on extrapolation rather than on measured data. However, a sound knowledge of the spectral database used in conjunc- tion with these models is of crucial importance to a full understanding of the origin of discrepancies between various emissivity correlations and the benchmark data used in the comparison.

Second, the accuracy attributed to the total emissivity correlations is often neither a sufficient nor a relevant criterion for selection. The models’ accuracy specified in the original papers cannot be used as a selection criterion since most of these correlations are based on different benchmark emissivity data. Moreover, it is often not clearly specified whether the model accuracy refers to an average or maximum discrepancy and how this has been calculated. Eventually, although the notion of total gas emissivity is generally well understood, that of standard gas emissivity is often more evasive. The clear distinction between these two emissivities is crucial since, theoretically, only the total emissivity is a relevant quantity in radiative heat transfer calculations. While most H20-COz emissivity correlations are developed for total emissivity predictions, 2*3~6-8J3-15 some correlations are to compute the standard as well as the total emissivity,2’3”3 while others apply only to the standard emissivity.““*

Third, the coupling of these emissivity correlations with existing solution methods of the radiative transfer equation (RTE) remains one of the major stumbling blocks in radiative heat transfer for non-homogeneous media. This issue is of particular importance with models based on the differential form of the RTE (discrete ordinate, spherical harmonics, moment methods), since these models require a linear emission/absorption coefficient which is not explicitly related to the total emissivity of homogeneous non- gray mixtures. Therefore, in the mathematical modeling of flames, it has become customary to evade this problem by approximating the absorption coefficient by an effective linear absorption coefficient defined as k, = -1/L x ln(1 - es(L)), where es is the total emissivity of a column of length L for a HzO-COz- Nz gas mixture. In a furnace enclosure, L is generally approximated by the mean beam length calculated from the furnace dimensions. Although this procedure appears to provide fairly accurate total flux predictions

in a number of configurations,22’23 it has no sound theoretical foundations. Working with differential solution methods of the RTE implies that all the quantities appearing in the flux equations are defined locally. Therefore, the use of a single emission/ absorption coefficient that is valid over the whole computational domain is theoretically unsound. Aver- aging the RTE spectrally appears to be a more appropriate approach. However, as shown by Vis- kanta, the resulting equation is intractable since it requires two radiative coefficients, the Planck emission coefficient and an absorption coefficient which can be estimated only if the solution of the spectral RTE is known.

The prime objective of this study is to present a critical analysis of the existing relevant information about the development and use of total emissivity correlations in CFD modeling of radiative heat transfer in combustion systems. Also, this study is aimed at providing combustion specialists with a practical and comprehensive view of the limitations and advantages of total gas radiative property models in radiative heat transfer calculations. The paper is divided into four sections. In Section 1, the various sources of benchmark emissivity data used to develop approximate emissivity models are presented and critically analyzed. The distinction between the notions of standard and total gas emissivity is also provided. In Section 2, a table comparing the benchmark data, mathematical formulation, applicability domain and accuracy attributed to several well-known emissivity correlations for H20-CO2 mixtures is presented. The total emissivity predictions of nine widely applied emissivity correlations are compared with the exponential wide band model (EWBM) calculations.25’26 These correlations are tested over a range of temperatures, total pressures and partial pressure path lengths that are wider than their prescribed applicability domain. Indeed, since high pressure processes are becoming more wide- spread, there is a need to validate existing models at elevated pressures. Besides, recent advances in flame modeling, including NO, predictions, indicate that accurate predictions of the flame front can only be accomplished using very fine computational grids. For instance, to resolve the near burner zone (NBZ) of a 2MW swirling flame of natural gas (NG), it is necessary to resort to computational cells as small as 0.5 mm.27 Since total emissivity correlations of H&- C02-N2 mixtures are generally applicable to gas volumes whose characteristic sizes are greater than about one centimeter, particular emphasis is drawn on the importance of quantifying the error introduced when calculating the effective absorption coefficient at very small path lengths. Computing times for HzO- CO2 total emissivity calculations using the EWBM are presented, as is a detailed comparison with the times required by the models of Leckner,* Modak,3 Steward and Kocaefe” and Smith et a1.13 In Section 3, the coupling of existing total emissivity correlations

546 N. Lallemant et al.

Table 1. Existing databases for the calculation of infrared radiative properties of gases

Model Database/research groups Number Temperature range Pressure. range of species (R) (atm)

Refs

Line-by-line HITRAN 92 32 N 300-1500 _ O-20 3536 Narrow band model General dynamics 9 _ 300-3000 N 0.01-20 37,39-41,44,45 Exponential wide band model Edwards and co-workers 9 N 300-3000 N 0.5-20 25,26

Tien and co-workers 30,31 Total emissivity Hottel and co-workers 2 300-2500 1 32-34,29

with simple solution methods of the RTE is discussed. Eventually, an extensive comparison of these methods against total radiance measurements is presented for two 300 kW flames in Section 4. Results of the computations are also compared with the predictions based on the exponential wide band model coupled with the Chan and Tien’* scaling method.

2. BENCHMARK DATA TO VALIDATE H20-CO* EMISSIVITV PREDICTIONS

2.1. Standard and Total Emissivity of Gases

The standard emissivity of a gas, E:, may* be defined as the total emissivity of the gas at atmospheric pressure assuming that the collision effects between the own molecules of that gas on the total emissivity values is neglected. Mathematically this becomes:

CT. T4 (1)

where E,” is the spectral blackbody emissive power at the temperature T of the gas and u is the Stefan- Boltzmann constant. The superscript (“) appearing in the notation of the standard emissivity indicates that it is calculated at the standard reference point Pt = 1 atm, pi = 0 (see Hottel and Egbert).**

In engineering calculations, a pressure correction factor accounts for the difference between the total and standard gas emissivities. This factor is defined as the ratio:

Detailed narrow band CO2 and Hz0 pressure correction factors have been calculated by Farag.29 At temperatures above 5OOK and in the total pressure range l-20 atm, the CO2 pressure correction factor is found to be in the range 0.6-1.4, while it is in the 0.47-2.05 range for H20. For both gases, the

* Although this definition is the one most generally used in the literature, it is by no means unique. Brosmer and Tier?’ defmed the standard emissivity as the total emissivity of a pure gas, pi = 1 and pe = 0, at atmospheric pressure. For further discussion about the different defini-

2.2. Benchmark Data and Models for H20-C02-N2 Radiative Property Calculations

The development and validation of total emissivity tions, see e.g. Ref. 61. correlations are based on four sources of HzO-CO2

o,22 (a) Line-by-line models (LBLM) -_

I Wavelength [pm]

(b) Narrow band models (NBM) 1.2

I I

0.8

0.6 t: in

‘.Ot I. c,

,.,t i”l - I JUY

0 2 4 6 8 10

0.2J- w

Wavelength [km]

(c) Exponential wide band models (EWBM)

1.21 ; 21 1.0 .Z .? 0.8 D ‘5 0.6 no ll ~ I

4 8 8 10 Wavelength [pm]

Fig. 1. Examples of emission/absorption spectra; (a) line-by- line model (LBL); (b) and (c) show predictions of the emissivity of an HrO-CO2 gas mixture using the narrow band model (NBM) and exponential wide band model

(EWBM), respectively.

correction factor tends rapidly towards unity with decreasing total gas pressure and increasing temperature. For example, at atmospheric pressure, the pressure correction factor is nearly 1 for CO2 and in the range 0.6-l .4 for H20.

Evaluation of emissivity correlations for HzO-C02-N2/air 547

emissivity data (see Table 1). The first source of data originates mainly from the total emissivity measurements of Hottel and Mangelsdorf,32 Hottel and Smith,33 and Hottel and Egbert2sV34 (for a more complete list of references, see Ref. 29). The other benchmark data are generated using either line-by-line calculations based on the HITRAN database,35,M the regular and/or statistical narrow band models30,31,37,38 or the exponential wide band mode1.25,26

2.2.1. HITRAN database for line-by-line calculations

The 1992 version of the HITRAN database35’36 contains information about spectroscopic transition line parameters for 32 molecules at the reference temperature of 296 K. The molecules included in the database are H20, C02, 03, N20, CO, CH4, 02, NO, S02, N02, NH3, HN03, OH, HF, HCl, HBr, ClO, OCS, H2C0, HOCL, N2, HCN, CH3C, HCl, Hz02, C2H2, C2H6, PH3, COFr, SF6, H2S, HC02H. It covers a spectral range from 10V6 to 22656cm-’ (i.e. 0.4414 to 10” pm).

The HITRAN database has been used to validate narrow band model calculations as well as compute new narrow band model parameters for Hz0 and C02.3g-4’ Recently, Denison and Webb’8’20 used line-by-line models to compute cumulative k-distribution correlations for Hz0 and CO2 in the temperature range 400-25OOK. Similar models have also been utilized for the modeling of radiation exchanges in the atmosphere and for some detailed flame and plume diagnostics.”

Line-by-line models are computationally expensive due to the large amount of information that needs to be processed to compute radiative exchanges over the whole infrared spectrum [see Fig. l(a)]. Moreover, the generation of spectral data at high temperatures for the narrow band model or k-distribution model is hampered by the necessity to extrapolate the information tabulated in the HITRAN database in order to account for the contribution of hot lines. The accuracy of the estimation procedure to calculate both the location and intensity of hot lines not tabulated in HITRAN decreases with increasing temperature. As pointed out by Denison,‘* at 25OOK, about 50% of the total integrated intensity is associated with hot lines. Denison has evaluated the average error in estimating the total intensity of hot lines for Hz0 as 4% at lOOOK and about 14% at 25OOK, if compared to direct calculations based on the few hot line transitions that are tabulated in the HITRAN database. For CO2, at 296K, the sum of all band intensities obtained using the estimation procedure was found to be 13% higher if compared to the results obtained using tabulated hot lines in HITRAN. Although the error in the sum of intensities is small for both H20 and C02, large errors exist for individual line intensities.‘s Therefore, it is recommended that radiative property calculations at temperatures above approximately 1500K

using LBL models together with the HITRAN 92 database should be examined with circumspection.

2.2.2. General dynamic database for narrow band model calculations

An extensive description of narrow band models (NBM) [see Fig. l(b)] can be found in Goody43 and Ludwig et aL3’ These models require a database containing the measurements of both the reciprocal mean line spacing parameter and the mean absorption coefficient &, over the entire infrared spectrum at different temperatures. The most complete set, of measured narrow band parameters for H 0 and CO2 is that of Ludwig and co-workers. 3i? Improved narrow band parameters have been published by Taine and co-workers3g-4’ and more recently by Phillips.“‘45 To date, the general dynamic database enables radiative property calculations for H20, CO2, CO, OH, NO, HF and CN. For CH4, C2H2 and soot, the narrow band model calculation can be performed using the approaches outlined in Refs 30, 31 and 37, respectively.

Although narrow band models together with line- by-line models are the most general and accurate models to predict radiative properties of mixtures of gases, they still are computationally too expensive to be utilized in CFD codes for flame calculations. Generally, narrow band models are used to produce benchmark data for other radiant properties models. Recently, Wieringa46 applied such a model to analyze radiative heat transfer in gas-fired furnaces, while Faeth et a1.47 employed the statistical narrow band model for the analysis of radiation/ turbulence interactions in turbulent diffusion flames.

2.2.3. Exponential wide band model database

The key idea underlying the development of the EWBM25,48 stems from the experimental observation that absorption by gases is mainly due to four or five strong absorption bands located in the near infrared and infrared regions( l-20 pm). The EWBM does not look at a narrow band but rather at the whole absorption band [see Fig. l(c)]. This model provides fairly simple mathematical expressions to predict the temperature and pressure dependence of the most important absorption bands of H20, C02, CO, CH4, NO, SO2, N20, NH3 and &Hz. The EWBM can also be utilized to predict the homogeneous total emissivity of gas-soot mixtures, provided the soot concentration is known. Perhaps more importantly, this model can be utilized to calculate radiation in non- homogeneous gas mixtures. From a practical viewpoint, one of the main advantages of the exponential wide band model stems from the fact that the model requires an almost insignificant database when compared to that needed by statistical narrow band models. The EWBM does not require excessive computational power and possesses all the generality


required for the computation of radiation in combustion at atmospheric and elevated pressures.

The main shortcoming of the exponential wide band model in treating radiation problems in enclosures is that it does not easily account for the wall interactions. This problem originates from the formulation of the model, which requires calculation of the total band absorptance prior to that of the band transmissivity and band width. Since the width of each absorption band varies with temperature, pressure and path length, a different division of the infrared spectrum is required for each path along which the equation of radiative transfer is solved. Yet, to account properly for gas-wall interactions, it is necessary to use the same spectral division for the wall and for the gas. The division of the spectral emissivity at the wall can only be set up once the equation of radiative transfer has been solved along each path within the gas volume. It may be inferred that this procedure will require a larger memory than the use of the NBM, and will computationally be more expensive. Some of the main impediments of coupling the EWBM to classical solution methods of the RTE have been discussed by Edwards.49

2.2.4. k-distribution database,for calculations of gas radiative properties

The aim of the k-distribution model is to transform the spectral integration appearing in the definition of the mean radiative properties with an integration over absorption coefficients. This transformation is defined for a non-gray homogeneous medium only. The key physical property in the k-distribution model is the so called kernel functionh(k) for the band ‘i’ of the gas studied. Physically, the quantity f;(k) dk may be interpreted as the fraction of absorption coefficient which has a value between k and k + dk in the spectral interval [Vr,ij Vu,i], where VL,~ and Vu,i are the lower and upper wavenumber limits of band ‘i’ and v is the wavenumber (cm-‘). This function may be obtained for a particular gas, either by reordering the results of a line-by-line calculation18 or, if the narrow band transmissivity is known, by taking the inverse Laplace transform of the band transmissivity.50 As shown by the calculations of Lacis and 0inas,5’ the profiles of the cumulative k-distribution, gi(k) = $kf;(k) dk, are free of the discontinuities present in the plots of the frequency distribution A(k) and are thus more amenable to numerical handling. Therefore, the k- distribution model is based on the cumulative function of the absorption coefficient rather than on the kernel functionA( Physically, the quantity gi(k) is the fraction of absorption coefficient which has a value less than k within the interval AVi. It is an increasing monotonic function of k with values in the interval 0 5 gi(k) < 1.

Denison and Webb18-20 have developed polynomial correlations to calculate the cumulative distribution functions for water vapor and carbon dioxide.

1 E-3

___ SLW - 20 Groups CKF 20 Groups

0 500 1WM 1500 2000 2500

TEMPERATURE [K]

Fig. 2. Comparison between the Hz0 standard emissivity predictions using EWBM (solid lines) and Farag’s bench-

mark data (symbols).

1

0 500 1000 1500 zoo0 2500

TEMPERATURE [1<1

Fig. 3. Comparison between the CO2 standard emissivity predictions using EWBM (solid lines) and Farag’s bench-

mark data (symbols).

For H20, a set of 30 correlations was developed. The k-distributions were selected to cover a spectral interval of 400cm-’ over the spectral range 400- 12 OOOcm-’ (0.83-25 pm). The correlations were obtained by fitting spectral line predictions of the

Evaluation of emissivity correlations for HzO-COz-NJair 549

cumulative distribution in the temperature and volumetric range 400-2500 K and 0- 1, respectively. For C02, the spectral width of the k-distributions was chosen to be 3OOcm-‘. A set of 23 polynomial correlations covering the spectral range 300- 6900cm-’ was developed. As for HzO, the correlations are applicable at atmospheric pressure and in the temperature range 400-2500K. The accuracy of those correlations should be expected to decrease with increasing temperature since they are based on line-by-line calculations. Unlike the NBM and EWBM, the k-distribution model suffers from the fact that new correlations must be computed whenever the total pressure or the temperature domain are changed.

2.2.5. Benchmark total emissivity data

Hottel’s data have been reviewed in a number of studies.2*29.38,52-s4 All agree on the inaccuracy of Hottel’s charts in regions where the latter are based on extrapolation rather than on measured data. For carbon dioxide, at temperatures above around 120OK, differences as large as 30% were shown to exist between Hottel’s charts and emissivities computed using either the NBM or the EWBM.29,52 Many studies also suggest the unreliability of Hottel’s data for water vapor at temperatures above 1200K.2X29,38 For both gases, the discrepancies were found to be largest at small partial pressure-path lengths. This is not surprising, since accurate measurements of gas radiative properties are difficult to perform under such conditions. Therefore, the best agreement between model predictions and experimental data should be expected at medium and large pressure- path lengths.

2.2.6. Model accuracy for total emissivity calculations

The accuracy of the k-distribution model is closely related to that of the line-by-line models. Both are expected to be fairly accurate at temperatures up to about 1500 K. At higher temperatures, however, these models are expected to underestimate the total emissivity when compared to the predictions of the EWBM and Farag’s benchmark data. This is well illustrated in Figs 2 and 3. For H20, at temperatures above 1500 K, the average discrepancy between Farag’s standard emissivity data and predictions based on Denison’s k-distribution is 40% for HzO, while it is only 20% in the 400-1500 K range. For CO*, the average discrepancy in the temperature range 600-1500 K is found to be 14%, while it is 12% in the temperature range 1500- 1800 K. However, as shown in Fig. 3, larger discrepancies exist at temperatures above approximately 1800 K.

Both the EWBM and the NBM are reliable models for predicting the total emissivity of gases. However, it is difficult to estimate the real accuracy of these

models since the accuracy of the spectrally measured data upon which they are based is often unknown or difficult to quantify. The average accuracy of band emissivity predictions using these models is often estimated to be within f20%.25,29,37 In the regions where measured total emissivity data exist, these models typically reproduce the measured values to within *lo%. Outside this region, there does not seem to exist enough reliable total emissivity measurements to estimate correctly the accuracy of any of the aforementioned models and one has to rely on comparisons between the model predictions. Farag’s thorough comparisons of the EWBM and NBM predictions for CO2 shows excellent agreement between the two models at temperatures above 1200K. As for H20, a comparison between Farag’s NBM predictions for the H20 standard emissivities and the authors’ calculations using the EWBM show that for temperatures below 2400K and pw L less than about 1 atm.m, the departure between the two models does not exceed f 15%. In view of the lack of total emissivity data to validate the predictions of the EWBM and the NBM at high temperatures and considering the good agreement between the two model predictions, it seems justified to place the same degree of confidence in both models.* In the present study, the EWBM is retained to generate benchmark emissivity data to compare predictions from several well-known emissivity correlations.

3. CORRELATIONS FOR PREDICHNG THE TOTAL EMISSIVITY AND ABSORPTIVITY OF

COMBUSTION GASES

Emissivity correlations are usually limited to calculations of the CO, and H20 total emissivity. Mathematically, these models appear either in the form of the weighted sum of gray gases model (WSGGM)4-‘5 or in the form of polynomials.‘-3 Existing WSGGM are somewhat less general than the polynomial correlations since coefficients for the WSGGM have to be recalculated for each H20/ CO1 partial pressure ratio. Polynomial correlations such as those of Leckner2 and Modak3 do not feature such shortcomings; they involve many more fitted coefficients (e.g. 48 for each species in Modak’s model) but retain all the generality required to

*Although it is a common perception that the narrow band model is more accurate than the exponential wide band model, the authors believe that, at the present state of knowledge, it is not possible to draw any impartial conclusion whether it also applies to total gas emissivity calculations. Indeed, (1) the paucity of total emissivity data at high temperatures and small path lengths, (2) the semi- empirical nature of both models, (3) the good agreement between the models predictions and experimental data in regions where total emissivity measurements exist, and (4) the small departure between the predictions of the two models at high temperatures,2g represent clear evidence that any conclusion in favor of either model would be premature and speculative.

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80

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700

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Evaluation of emissivity correlations for HzO-CO*--N2/air 551

model total emissivity of gas mixtures. Both types of correlations are accurate enough and simple to use in engineering calculations. However, they are often limited to total emissivity calculations in volumes of gas with a mean beam length greater than 1 cm. This section surveys the total emissivity correlations presented in Table 2. Only the models which have been widely applied in CFD modeling of flames and engineering combustion problems are described.

3.1. The Weigh ted Sum of Gray Gases Model

The weighted sum of gray gases model (WSGGM) was originally developed to provide engineers with easy-to-use mathematical expressions for estimating the total radiative properties of gases.4-5 The total emissivity of a real gas mixture is represented by a weighted sum of a sufficient number of gray-gases’ expressions, i.e. by a relation of the form:

Q(T,~,L) = cwi(T)- (1 -ePk,,p@) (3) i=l

where w,(T) is a weighting coefficient that is a function of temperature only and ki,p is a fitting constant. For a single radiating gas, p denotes the partial pressure of the gas. For a mixture of two or more gases, p represents the sum of the partial pressure of the radiating gases and L is the path length. Although, mathematically, Eqn (3) is nothing more than a non-linear quadrature curve fitting it is possible to relate the coefficients ki = ki,p p and v to physical quantities. Using the /c-distribution model, it can be inferred that ki is the optimal mean absorption coefficient over a group of bands with a nearly identical spectral absorption coefficient. Similarly, the weighting factor w,(T) must be viewed as the sum of the fraction of blackbody radiation over a group of bands having identical ki values and whose locations in the spectrum are not known. It may further be shown that:

M ,!$#.A% wj(T) g Cz ’ [Ei(kj+l) -Ei(kj)] (4)

i=l

where the index ‘i’ relates to the infrared band centred at Vc,i and of width A4. T is the absolute temperature K, Q is the Stefan-Boltzmann constant and E& is the blackbody emissive power calculated at temperature T and spectral wavenumber V,-i. The function gi(kj) is the fraction of absorption coefficient in the band ‘i’ which has a value less than kj.

3.2. Polynomial Approximations

The two most well-known and general total emissivity correlations using polynomials are those developed by L.eckne2 and Modak.3 Prior to these publications, Hadvig’ derived polynomial expressions to calculate the total emissivity of HzO-CO2

552

0.20

0.15

P 5 ; 0.10

; z

0.05

0.00

N. Lallemant et al.

H O-CO total emissivity vs. temperature

0 250 500 750 low 1250 1500 1750 2cKm 2250 2500 27%

TemperehrmM

Temperature [KJ

Fig. 4. Comparison of the computed total emissivity of HzO-CO2 mixtures using various total emissivity correlations with the predictions of the exponential wide band model (EWBM).

gas mixtures for pW/pC = 1 and pW/pC = 2. How- ever, in view of the limited range of applicability of this model, it is excluded from the assessment in Section 4.

3.3. Hybrid Models

3.3.1. Steward and Kocaefe ‘Box Model’

Steward and Kocaefe proposed a total emissivity model for water vapor and carbon dioxide mixtures16V’7 based on the Penner box mode1.5Vs6 The model combines some of the characteristics of the EWBM,

polynomial correlations and WSGGM. Only the latest and most complete version of that model is examined in this study.”

3.3.2. Single line weighted sum of gray gases model

Denison and Webb’8-20 developed the single line weighted sum of gray gases model (SLW model) as a simplification of the k-distribution model. Since the model is relatively new, a more comprehensive description of the model is given below.

The key idea behind the SLW model is to reduce the computational time involved in the calculation

Evaluation of emissivity correlations for H20-CO*-Nz/air 553

Table 3. Maximum absolute discrepancy and average absolute discrepancy (arithmetic mean) in the calculation of total emissivity of H,0/C02 mixtures using various total emissivity correlations. The benchmark emissivity data are the

predictions using the EWBM. X, = x, = 0.1, P,/P, = 1, path length L = 0.1 m

Temperature range (J‘?

discrepancy (“/I

Average discrepancy

(%I

Modak [I9791 300 < T 5 2500 13.0 7.2 Leckner [1972] 300 5 7- 5 2500 27.5 10.4 Smith et al. [1982] 300 5 T 5 2500 29.4 5.9 Steward et al. [ 19861 300 5 T 5 2500 22.6 4.9 Truelove, 2 gray terms [1976] 300 5 T 5 2500 42.8 18.3 Truelove, 3 gray terms [1976] 300 5 T 5 2500 42.5 19.8 Taylor et al. [I9741 300 2 T 5 2500 38.1 17.1 Johnson [1971] 300 5 T < 2500 108.6 27.2 Coppale et al. [ 19831 300 5 T < 2500 41.6 21.9

Temperature ran e (K?

300 5 T 5 2000 400 5 T < 2500 600 5 T 5 2400 600 5 T 5 2000 600 5 T 5 2400 6005 T12400

12005 T12400 lOOO~T<20OO 2000 5 T 5 2500

discrepancy (%J

Average discrepancy

(%)

13.0 6.7 27.5 10.6 16.4 2.9 10.3 4.0 42.8 17.5 42.5 17.9 38.1 25.2 24.1 15.7

7.5 2.3

’ As specified in the original papers.

of the cumulative distribution function gi(kj) by reducing the number of such distributions to one. Rather than developing a cumulative distribution that applies to the whole infrared spectrum, the authors introduced a new quantity which they termed the blackbody absorption line distribution function. Physically, this function represents the fraction of blackbody radiation emitted in the portion of the spectrum where the absorption coefficient k,” is less than the value k,. For a given species ‘s,’ the blackbody distribution function is defined as:

(5)

where the subscript ‘i’ refers to the ith spectral interval where k,” 5 k and Tb is the background temperature. The blackbody absorption line distribution function is integrated over the whole spectrum so that it has values within the range 0 5 Fs 5 1. Denison’* developed mathematical correlations of the function F, for Hz0 and COz. Detailed information about these correlations may be found in Refs 18 and 19. It may be shown that the total emissivity of a gas is related to the blackbody distribution function by the relation:

Es E eWj(T)’ (1 -lXp(-ij.L)) j=l

with Wj(T) E F,(kj+l) - F,(kj). (6)

The analogy between the mathematical formulation of Eqn (6) and that of the WSSGM is striking. As reported by Denison and Webb,lg the advantage of the SLW model over the WSGGM model in homogeneous media is that it provides an efficient means of calculating total radiative heat transfer rates in gases with an accuracy comparable to line-by-line calculations.

Although the use of an LBL model to estimate band model properties is sound, it may substantially

underestimate total emissivity values at high temperatures for the reasons already discussed in Section 2.2.1. Hence, there are questions about the accuracy of the SLW model at high temperatures. Denison found that, in the temperature range 40&2500K, total emissivity predictions based on the SLW model agree to within 5% with the data extracted from Hottel’s charts. However, Hottel’s data are known to underpredict the true gas emissivity by about 30% at high temperatures.2g Thus, Denison’s finding appears fortuitous. The good agreement between the SLW model and Hottel’s data should be attributed to the inappropriate choice of the benchmark data rather than to the good performance of the model predictions. Plots of standard emissivity predictions for Hz0 and CO2 using the SLW model are shown in Figs 2 and 3. The predicted standard emissivity based on the SLW model are consistently lower than those of the k-distribution model. The latter are themselves 20-40% lower than Farag’s benchmark data. The accuracy attributed to the SLW model for total emissivity calculations should be somewhat less than that of the k-distribution and LBL models.

4. ASSESSMENT OF THE ACCURACY OF SEVERAL TOTAL EMISSIVITY CORRELATIONS (HOMOGENOUS

CALCULATIONS)

4.1. Generalities

In this section, the exponential wide band model (EWBM)25,26 is used to provide benchmark data to validate the total emissivity models developed by Johnson6 Leckner,2 Taylor and Foster,’ Modak,3 Smith et a1.,13 Coppale and Vervish14 and Steward and Kocaefe” (see Table 2). The models of Hadvig’ and Bartelds and co-workersgjtO are excluded from the comparison tests since the former model cannot be easily adapted for CFD calculations, while the latter is limited to total emissivity predictions of H20-CO* mixtures at a single partial pressure ratio. Farag and Allam’s” and Farag’s models’2 are also excluded


1 0.8

E Q) 0.7 3 I! 0.8

lE4 lE-3 0.01 0.1 1 10 100

mkngth Ml

unear Abgorption Coeffiit VS. pathlength

lE4 lE-3 0.01 10 100

Fig. 5. Comparison between H20-CO2 total emissivity predictions using various emissivity correlations and the exponential wide band model (EWBM). Calculations for the conditions L = 0.1 m, P, = 1 atm,

XHIO = k0, = 0.l~ PH20fPC02 = I.

since these models are limited to the standard emissivity of gases.

There are three series of tests. In the first, the model predictions are validated against the EWBM calculations in the temperature range 300-25OOK, water vapor to carbon dioxide partial pressure ratios PWIPC = lb/ = xc = 0.1) and p,/p, = 2(x,,, = 2. xc = 0.2), path lengths L = 0.0001, 0.001, 0.01, 0.1, 1 and 10m. The total pressure is taken as 1 atm. Results of the calculations were presented in the form of graphs.” An example of such a graph is given in Fig. 4. In order to facilitate comparison with the EWBM results, the relative discrepancy

between the predictions of the EWBM and the emissivity correlations has been computed. It is calculated as:

Discrepancy (%) = 100 x kE~EM-M%~r) (q

where +waM represents the EWBM total emissivity, while ecor is the total emissivity obtained using the emissivity correlation. For each test, the average (arithmetic average of the absolute value of Eqn. (7)) and maximum discrepancies (maximum absolute value of Eqn (7)) are also computed. The results are

Evaluation of emissivity correlations for HrO-COr-Nr/air 555

Table 4. Maximum absolute discrepancy and arithmetic average discrepancy between the linear absorption coefficient k, = - 1 /L x 1 n( 1 - c) calculated using various total emissivity correlations and the EWBM predictions. This table refers to

the data plotted in Fig. 5. T = 2000 K, pw/pC = 1, pr = 1 atm

Path length Maximum Average Path length’ Maximum Average (m) discrepancy discrepancy (m) discrepancy discrepancy

(“/] (“/) (%) (%)

Modak [1979] 10-4 < L < 100 48 20 0.006 5 L 5 20 39 Leckner (1972) 10-4 7 L ? 100 117 28 0.006 5 L 5 50 36 Smith et al. [1982] 10-4 7 L 7 100 19 5

low4 < L 2 100 0.005 5 L 5 50 9

Truelove, 2 gray terms [I9761 64 40 0.05 5 L 5 50 50 Truelove, 3 gray terms [1976] 10m4 5 L 5 100 111 52 43 Taylor and Foster [ 19741 10-4 < L 5 100

0.033 5 L 5 33.3 60 35 0.005 5 L 5 5 59

Steward and Kocaefe [1986] 10m4 5 L 5 100 5038 209 0.03 5 L 5 10 15 Johnson [ 19701 10-4 5 L 5 100 49 36 36 Coppale and Vervish [I9821 10-4 5 L 5 100

0.11L17 17 6 0.05 5 L 5 10 17

12 15 4

22 21 33

7 28

7

’ As specified in the original papers.

Table 5. Absolute discrepancy (%) between the emissivity predictions using Leckner’s total emissivity correlation and the EWBM computed emissivities. L = 0.0001 m,

x,=x,=0.1

Table 7. Absolute discrepancy (X) between the emissivity predictions using Smith and co-workers’ total emissivity correlation and the EWBM computed emissivities.

L=O.OOOlm, x,=x,=0.1

T(K)

1

Total pressure (atm)

5 10 15 20

T(K) Total pressure (atm) -

1 5 10 15 20 -

1000 63.7 17.1 7.8 10.0 15.0 1000 8.1 19.3 34.5 44.6 51.3 1200 74.6 28.2 14.0 13.0 16.0 1200 2.3 14.6 31.0 42.2 49.7 1400 81.4 38.8 22.0 19.2 20.6 1400 0.1 12.3 29.3 41.2 49.4 1600 85.1 46.5 30.0 25.3 25.7 1600 0.1 12.3 29.6 41.8 50.3 1800 86.7 50.7 34.2 29.4 29.3 1800 2.2 14.2 31.1 43.4 52.0 2000 86.9 51.8 35.6 30.8 30.6 2000 4.6 16.2 32.8 45.0 53.6

Table 6. Absolute discrepancy (%) between the emissivity predictions using Modak’s total emissivity correlation and the EWBM computed emissivities. L = 0.0001 m,

x,=x, =O.l

T(K)

1000 1200 1400 1600 1800 2000

1

9.6 21.3 33.8 41.7 41.2 24.6

Total pressure (atm)

5 10 15

4.4 5.0 15.3 1.8 3.9 12.7 4.0 5.4 12.3 7.7 6.0 11.4 4.7 1.6 6.7

11.7 13.4 6.7

20

24.0 20.9 19.6 18.0 13.3 1.6

pressure ratios pw/p, = 1 (xw =x, = 0.1) and pw/p, = 2 (x, = 2. xc = 0.2) and temperatures T = 1000 K and T = 2000 K. An example of calculation results for T=20OOK and pw = pC = 0.1 atm is presented in Fig. 5, together with the relative errors given in Table 4.

presented in Table 3 for the test corresponding to Fig. 4. Columns three and four show, respectively, the maximum and average discrepancies calculated over the temperature range 300-25OOK. Column six and seven show the discrepancies calculated over the temperature ranges as specified in Table 3.

The second series of tests shows a comparison

The last series of tests is concerned with the validations of the total emissivity correlations of Leckner,’ Modak3 and Smith et a1.13 at total pressures up to 20atm. These tests cover the following range of operating conditions: total pressures P, = 1, 5, 10, 15, 20atm, temperatures T = 1000, 1200, 1400, 1600, 1800 and 20OOK, path lengths L = 0.0001, 0.01 and 1 m and partial pressure ratios p,/p, = l(xw =x, = 0.1) and p,,,/p, = 2(x,., = 2. x, = 0.2). Tables 5-7 present some test results at 0.0001 m path length.

4.2. Emissivity Tests at Atmospheric Pressure

between the effective absorption coefficient k, = -l/L x ln(1 - e,(T,p,,p,,P,,L)) calculated using the EWBM and the emissivity correlations. The tests are for a total pressure Pt = 1 atm, path length range O.OOOl-lOm, Hz0 to CO2 partial

An analysis of the results from the first series of tests indicates that all the emissivity correlations possess regions of good agreement with the EWBM. However, the position of those regions can vary considerably with the model, temperature, path length and H20-CO2 volumetric fractions. For all the models, the maximum and average discrepancies


are found to be minimal at intermediary path lengths (L = 0.1 and 1 m). The discrepancy strongly increases with decreasing path length. With the exception of Leckner’s’ and Smith and co-workers’ models,13 all the other total emissivity correlations are found to be unreliable when used outside their path length applicability domain. Smith et al.? WSGGM exhibits a maximum discrepancy of about 10% at 0.001 m path length and a partial pressure ratio pw/p, = 2. For the same path length and pw/p, = 1, Leckner’s model shows a 14% maximum discrepancy. Modak’s model may also be used to predict the total emissivity of carbon dioxide and water vapor mixtures up to temperatures of 2500K. However, it is not recommended outside its specified range of partial pressure-path length (see Table 1). A comparison of the average discrepancies such as those presented in column 5 of Table 2 (see Ref. 55 for the complete set of tables) with the model accuracy listed in Table 1 reveals that, with the exception of Leckner’s and Smith and co-workers’ models, all the prescribed accuracies are lower than the average discrepancy calculated in the present work. This is because the average accuracy given in Table 1 refers to different benchmark emissivity data than that generated using the EWBM. For instance, at temperatures above 12OOK, the NBM predictions of water vapor and carbon dioxide emissivity are known to be higher than those calculated using the EWBM.25 Consequently, the total emissivity correlations based on the NBM data should predict higher emissivities than the EWBM calculations. This is clearly the case for the models of Taylor and Foster,7 Truelove’ and Leckner.’ It is also seen that the model of Steward and Kocaefe does not provide better emissivity predictions than the WSGGM or the polynomial approximations.

4.3. Total Emissivity Tests at Elevated Pressure

The incentive behind the second series of tests is to quantify the error introduced when calculating H20- CO2 total emissivities at elevated pressures using the total emissivity correlations of Leckner,2 Modak3 and Smith et aLI Although the latter models are developed for atmospheric pressure calculations, it may be possible to use them at elevated pressures since the effect of total pressure is partially accounted for in the partial pressure-path length pi L = (Xi Pt) . L. Table 4 shows that Leckner’s model is not reliable at very low path lengths, where discrepancies as large as 86.7% exists. At 0.01 and 1 m path lengths, the model predictions are within 20-30% of the EWBM computed emissivities. The model gives acceptable predictions at total pressures below 5 atm and path lengths higher than 0.01 m. Although Modak’s model yields reasonable predictions at low path lengths, the good agreement with the EWBM should not be misinterpreted. At elevated pressures and at 10e4m path length (see Table 5)

the model departs from the EWBM by not more than 40-50%. However, the latter departures are probably artifacts, since at atmospheric pressure, deviations as large as 40% from the EWBM calculations already exist. The model of Smith and co-workers departs by as much as 53.6% at low path lengths, high temperatures and a partial pressure ratio of two (see Table 6). In all other cases, the model appears to be within 30% of the EWBM predictions.

4.4. Effective Linear Absorption Coefficient at Atmospheric Pressure

The effective absorption coefficient k, is the coefficient that is generally used in flux radiation models (see for instance Adams and Smithz2). Therefore, it is valuable to analyze how k, varies with the temperature, the gas partial pressure and the path length. The path length dependence of k, may be seen, such as in the case of the Patch57 mean emission coefficient, as the expression of gas self-absorption. At small path lengths (L less than about 0.01 m), k, tends towards the Planck emission coefficient, while at larger path lengths, k, starts to decrease rapidly and finally asymptotically tends towards zero. Figure 5 clearly shows that neither polynomial approximations nor Steward and Kocaefe’s hybrid model are reliable at small path lengths. The region between 10m4 m and 0.01 m is important in mathematical modeling of flames, especially in the near burner zone, where the size of the smallest computational cells required to properly model the cornbusting flow can be as small as 0.5mm.27 The effective linear absorption coefli- cients calculated using both Leckner’s2 and Modak’s3 polynomials reach a maximum at path lengths around 0.005m. Then, the curves suddenly reverse and fall off rapidly as L decreases. The predictions using Steward and Kocaefe’s model diverge toward infinity as L approaches zero. Thus, neither polynomial approximations nor the latter model are recommended for CFD modeling of flames requiring very fine computational grids. Although most of the weighted sum of gray gases models tested yield too low or too high predictions of the coefficient k, at low path lengths, they all tend towards a plateau that corresponds to the Planck mean emission coefficient. The total emissivity correlation of Smith, Shen and Friedman was found to outclass all the other WSGGM tested herein. As shown in Table 4, this model exhibits a 5% average error when compared to the EWBM in the path length range 10-4-100m. At high temperatures, the model of Coppale and Vervish14 also appears to be fairly reliable. The good agreement between the results obtained with the latter two WSGGM and the EWBM predictions is not surprising since both models are based on benchmark data generated using the EWBM.

Evaluation of emissivity correlations for H20-CO*-N*/air 557

Table 8. Computing time (in ms) required to calculate the total emissivity of three different gas mixtures. Values in parentheses are the predicted total emissivities

Gas mixtures

H*CP COzb HrO-COrC

EWBM (BA)

5.9 (0.185) 6.9 (0.088)

13.9 (0.25 1)

EWBM (BEN

2.8 (0.185) 2.6 (0.089) 5.5 (0.253)

Leckner [21

0.6 (0.170) 0.8 (0.091) 1.5 (0.236)

Modak Steward and Kocaefe Smith er nl [31 [171 [131

- 2.1 (0.168) 2.7 (0.151) 2.1 (0.086) 1.7 (0.092) 4.2 (0.275) 4.4 (0.225) 0.310 (0.267)

4.5. Runtime Comparison between the EWBM and Emissivity Correlations

Comparison of the CPU time required for the EWBM to calculate the total emissivity of HzO-CO2 gas mixtures with that necessary for some widely applied total emissivity correlations is relevant information for mathematical modelers. A procedure to accelerate radiative property calculations of gas mixtures using the EWBM has been proposed recently.60 The EWBM’s emissivity calculations were performed using both the block approximation (BA) and the band energy approximation (BEA). Discus- sion about the applicability limits of these approximations for total radiative property calculations can be found in Ref. 60. The EWBM’s execution times for the total emissivity calculations of H20-Nz, COz-N2 and an HzO-C02-N2 mixture are compared with those required by the emissivity correlations of Smith and co-workers,‘3 Leckner,3 Modak4 and Steward and Kocaefe.17 The tests were performed on a PC 386 computer running at 25 MHz with a 80387 processor. All the computer subroutines were optimized before testing. Typical run times are listed in Table 8. The computing times required by the EWBM are, in the case of the BEA, comparable to the CPU time required by the polynomial approximations (Leck- ner and Modak) and the hybrid emissivity model (Steward and Kocaefe). For HzO-CO*-N2 mixtures, Smith and co-workers’ sum of gray gases model is about 15 times faster than the BEA method and 45 times faster than the BA method. Though computationally attractive, Smith and co-workers’ model also has a number of shortcomings. As with all weighted sum of gray gases models, it is limited to emissivity calculations at a total pressure of 1 atm, HZ0 to CO2 partial pressure ratios equal to 1 and 2 and cannot easily be adapted to emissivity calculations of mixtures of more than two gases having common overlapping bands. This clearly makes those models unattractive for implementation in CFD computer codes for flame modeling.

The block approximations, although two to ten times slower than the other models tested, still appear competitive. Considering that the EWBM is a general and accurate radiative property model and requires execution times which are about the same order of magnitude as more approximate models, it is

appealing for implementation in CFD codes for flame calculations.

5. COUPLING OF TOTAL EMISSIVITY CORRELATIONS WITH THE RTE

5.1. Spectral Versus Gray Calculations

5.1.1. The spectral and angular dependence of the RTE

Solving of the radiative transfer equation (RTE) in multi-dimensional non-homogeneous media still presents considerable difficulties. The complexity is in its angular and spectral dependencies. The angular dependence may easily be handled using a number of efficient solution methods of the RTE such as the discrete ordinate method (DOM). However, an optimal implementation of spectral calculations with the RTE solution methods is a far more challenging problem. This is all the more critical because, in present computational fluid dynamic (CFD) codes for flame calculations, heat transfer by radiation is only one of the phenomena to be addressed in turbulent combustion problems. In view of the large increase in computing time that is required to perform detailed spectral calculations, it is important to understand where such models may be supertluous and where they are necessary. Important simplifications must be introduced if spectral calculations are to be performed on workstations or desktop computers. On the other hand, there is a large class of industrial problems where the gray gas approximation consistently provides good wall flux predictions.22’23 Therefore, it is useful to classify radiation problems and understand why the gray gas approximation works well in certain cases and not in others.

5.1.2. A simple classification of radiation problems

In modeling radiative heat transfer, two classes of problems may be distinguished.

The first consists of predicting the radiative flux distribution at the surface of a heat sink. In this case, the gray gas approximation may be good enough when the walls of the enclosure are hot and diffusively reflecting. However, care should be taken when cold surfaces represent a large percentage of the total inner surface of the combustion chamber, since in this

558 N. Lallemant er al.

Table 9. Definition of some total mean absorption/emission coefficients appearing in the treatment of the radiative transfer equation

Coefficients Definition Domain of validity

Planck mean emission coefficient

Incident mean absorption coefficient

Patch emission coefficient

k JUVG(T)~d~ Optically thin limit. Limit of small path length

*I,,, = so” k: :;:T4dv

k,,,, = Jr ki T$ efkyL dv

Optically thin limit. Limit of small path length

so” E,” &‘L &, Covers the whole domain of path length, from the optically thin to the optically thick limit

case the absorption of radiation by the cold gas layer 5.1.3. Search for an eficient coupling procedure near the surface of the wall will not be well accounted between existing radiative property models and the for by the gray gas assumption. RTE

The second class consists of predicting the radiative source term within the flame region. This issue is perhaps of even greater importance for burner designers who need to predict the temperature field in the near burner zone as accurately as possible. Simple considerations on radiative heat transfer in non-luminous flames of natural gas suggest that the amount of energy radiated away by a flame can hardly exceed 10-l 5% of the total flame heat release. This quantity is undoubtedly more difficult to compute accurately since it requires that:

The NBM and the EWBM are directed towards spectral transmissivity and total absorptance predictions, respectively. Hence, they cannot easily be coupled to differential solution methods of the RTE such as the DOM since these methods require knowledge of the spectral linear absorption coefficient or its average over a spectral interval. For this reason, methods based on the integral representation of the RTE appear more attractive for treating non-gray calculations using either the EWBM or NBM. However, integral solution methods of the RTE are usually more computationally expensive for radiation calculations in flames and become inefficient when dealing with strongly scattering media.‘8,5’ For these reasons, flux methods are usually favored although the coupling with existing radiative property models proves to be difficult. Similarly to the NBM and EWBM, total emissivity correlations are for predictions of the emissivity of homogeneous gas mixtures. These correlations are difficult to use with differential solution methods of the RTE and the engineer is often compelled to use a single emission-absorption coefficient to solve non-homogeneous non-gray problems. This empirical approach is bound to yield inaccurate total heat flux predictions.

(1)

(2)

the radiant energy exchange between the flame and the rest of the furnace enclosure is modeled accurately; in other words, the solution to the first class of problem must be accurate; the non-homogeneity in the gas concentrations and temperatures is correctly accounted for by the radiation model.

In the case of non-luminous flames, the first class of problems is, indeed, less stringent since even a 100% error in the flame radiative source term will result in an increase of the flame radiant fraction from, say, lo-20%. The additional radiant energy incident upon the load induced by this overshoot cannot be large since the ‘extra’ radiant energy emitted by the flame will be evenly distributed on all surfaces of the enclosure. The larger the volume of the enclosure, the smaller should be the local increase of the incident heat flux at the walls. Consequently, in large furnace enclosures, the flux incident at the walls should not be very much different whether the flame radiant fraction is 10% or 20%.

It should be pointed out that, although the accuracy of the prediction of the radiative source term is closely related to that of the temperature, the accuracy to which the flame radiative source term needs to be calculated is not necessarily high. Indeed, in the regions where intensive combustion takes place, the heat release due to combustion dominates that radiated. Therefore, in the flame region, accurate calculations of the radiative source term should not drastically affect the local temperature predictions.

There is a need to derive efficient and simple methods of coupling the total and spectral radiative property models with the existing solution methods of the RTE. In seeking such a method, it is clear that the coupling procedure should apply to differential as well as integral solution methods of the RTE. Also, the method should maintain the mathematical properties of the basic RTE and should enable calculations in non-gray homogeneous and non-gray non-homogeneous mixtures. Eventually the method should be accurate. The search of a new equation of radiative transfer that would encompass all these features has been the goal of radiation specialists for a number of years. The most successful attempts to develop a simple though general and accurate model are reviewed in the following sections6’-”

Evaluation of emissivity correlations for HSO-COz-Nz/air 559

5.2. Gray and Semi-Gray Gas Approaches

As shown by Viskanta,24 spectral averaging of the radiative transfer equation for non-scattering media:

ar $ = k,” (Iv” - Iv) (8)

gives the semi-gray equation:

ar as = k, I“ - ka,n . I (9)

where I” [W/(m2 sr)] is the blackbody intensity, I[W/(m* sr)] is the total intensity, k,[m-‘1 is Planck’s mean emission coefficient and k,,*[m-‘1 is the Chandrasekhar mean absorption coefficient (see Table 9 for definitions). Although the calculation of the total radiative fluxes using the semi-gray Eqn (9) is possible, the absorption coefficient ka,n requires knowledge of the spectral intensity 1,. Therefore, ka,n can be obtained only if the spectral RTE is solved. For this reason, Eqn (9) is practically useless unless one resorts to an even simpler equation upon invoking the gray gas assumption. This assumption eliminates the wavelength dependence of the spectral absorption coefficient k,U(T,pi,P,,v) --f k,(T,pi,P,) and implies the equality kg = ka,n = k,. Since k, = k,, the Planck mean emission coefficient should be the property coefficient appearing in the gray gas formulation of the RTE. It happens that k, is valid at very small path lengths and is related only to the emission of radiation. This coefficient is not appropriate to describe the emission from a large volume of gas since, for such volumes, the energy emitted by the gas is partly reabsorbed before leaving the volume. In other words, k, neither accounts for the gas self-absorption, nor for the absorption of radiation incoming from remote cells. Hence, the Planck mean emission coefficient is seldom used in the gray equation of radiative transfer.

In CFD modeling of combusting flows, it has become customary to arbitrarily replace the gray gas coefficient kB by an effective absorption coefficient k, defined as:

k, = --A. ln(1 -es)

so that Eqn (9) reads:

(10)

; = k, (I” - I). (11)

It is believed that the main reason for using k, as the mean emission/absorption coefficient in Eqn (11) stems from the existence of a large number of total emissivity correlations for HzO-CO2 mixtures and the importance of the WSGGM in the Hottel zone method. However, the applicability domain of Hottel’s method and models derived from Eqn (11) is not the same. Use of the Hottel zone method together with WSGGM is theoretically sound for non-gray and gray homogeneous problems. Substitu- tion of the coefficients k, and kan in Eqn (9) with the coefficient k, restrains the application of Eqn (11)

3.5

7

g 3.0

E *g 2.5

8 2.0

.E 1.5 Z p 1.0 9 $ 0.5 e 3 0.0

-0.51- -* ‘.a- lE-4 lE-3 0.01 0.1 1 10 100

Pathlength [m]

Fig. 6. Comparison between Patch linear absorption coefficient and linear effective absorption coefficient k, = - 1 /L x 1 n( 1 - EJ for an H20 to CO;! partial pressure

ratio of 2 and temperatures T = 1000, 1500 and 2000 K.

90 -

90-

70 -

w-

50 -

40 -

30 -

20 -

10 - y 1 x”20=2.xco2=o.2 j

4. .b 0.01 0.1 10 1 1

Pathlength [m] 0

Fig. 7. Deviation between Patch linear absorption coefficient and effective coefficient k, = - 1 /L x 1 n( 1 - Q) for pwc : 2,

T = 1000, 1500 and 2000 K.

to calculations in gray homogeneous media only. This arises despite the fact that the total emissivity appearing in Eqn (10) is that of a non-gray homogeneous gas. Beyond the lack of a sound theoretical justification for using Eqn (10) and (1 l), models based on these equations have proved to provide total radiative heat flux predictions which are in surprisingly good agreement with measured fluxes.22,23

5.3. Patch Treatment of the RTE in Homogeneous Non-Gray Media

While the approach outlined by Viskanta14 is limited to homogeneous gray calculations, Patchs9


has shown that the solution of the non-gray homogeneous RTE can be replaced by the solution of the gray homogeneous equation, provided that the influence of the walls is neglected and the gray gas coefficient is given by the Patch coefficient:

k Pa

(L) _ Jr k,” e-kL-L E,” du J;e-k:‘L. E,” &, (12)

Patch’s analysis may be generalized to account for the presence of boundaries. However, in view of the limited practical interest of these equations, it will not be discussed herein.6’ It is interesting to note that Patch’s coefficient may be related to the effective coefficient k, by:

kp,(L)=w=ka+Lz. (13)

Both Patch coefficient, k,,, and the effective absorption coefficient, k,, tend towards the Planck emission coefficient at small path lengths, i.e.:

As L increases, 1 - tg and k, decrease. Therefore, ak,/dL < 0 and after Eqn (13):

k,, I k, (15)

where the equality sign holds only at small path lengths.

In CFD codes for flame calculation, the linear absorption coefficient appearing in the radiative transfer equation is generally calculated using Eqn (10). The most widely adopted method for radiative transfer is to assume a constant value of the linear absorption coefficient everywhere in the furnace. This value is estimated from the mean effective emission coefficient, k,, calculated using the mean gas temperature and the mean HZ0 and CO2 volumetric fractions in the furnace. The path length L needed to calculate k, is equated to the mean beam length based on the furnace dimension. The justification for applying this procedure is that, in a furnace, the flame is the region where the strongest temperature and concentration gradients exist. Besides, in many industrial combustion applications, the flame volume is often a small fraction of the total volume of the furnace. For instance, the IFRF semi-industrial scale furnace is a parallelepiped furnace of dimension 2 x 2 x 6 m = 24 m3. The volume of a typical 2 MW swirling flame measured in this furnace usually barely exceeds 1 m3, so that the flame represents only 4-5% of the total volume of the radiating gas. Since the average flame temperature is roughly 1.2- 1.5 times higher than the temperature of the downstream and surrounding regions, the flame region contributes up to a maximum of 1.24x 5 s 10% and 1.54 x 5 or 25%

of the wall irradiation, respectively. The remaining 90% (resp. 75%) is due to the radiation from the other walls and that emitted from the outer flame region where the temperature and Hz0 and CO1 concentrations are nearly constant. Therefore, calculation of the total heat fluxes at the walls may be obtained by solving the non-gray homogeneous equation of radiative transfer. According to the analysis presented earlier in this section, the solution of this problem is similar to that of treating the gas as gray-homogeneous, provided that the gray absorption coefficient is given by Eqn (12). Plots of the two coefficients k,, and k, are shown in Fig. 6 for a mixture of Hz0 and CO?, corresponding to complete combustion of methane in air at stoichiometry. As shown in Fig. 7, the deviation between the two coefficients is small at path lengths less than 0.01 m. The deviation increases somewhat as the path length increases (0.01 < L 5 1 m). At path lengths of up to N 1 m, both coefficients remain within a factor two, so that for L < 1 m, substituting k, for k,, does not cause significant errors.

This reasoning partially explains why a good agreement is often found between total flux predictions at the walls and the measurements when a single gray absorption coefficient k, is used to solve the RTE. For the case considered here (combustion of methane in air), Patch’s analysis also points out the limitation of this approximation to furnace enclosures for which the furnace mean beam length does not exceed a few meters. Although Patch’s analysis is developed for open systems, the reasoning applies as well to closed combustion systems with hot walls. Indeed, as the temperature increases, the radiation within a refractory-lined furnace tends towards that of a gray body due to the multiple reflections at the walls. Therefore, the radiation from the walls does not affect the radiation incoming from the gas. Besides, in hot wall enclosures, wall to wall radiative exchanges are often dominant over gas to wall exchanges. This is an additional reason for the good agreement found in Refs 23 and 23.

5.4. Totul Trunsmittance Non-Homogeneous Model

Grosshandler and co-workersh’-64 have developed a simplified model to predict the total radiance in non- homogeneous gas-soot mixtures. The model, called the total transmittance non-homogeneous (TTNH) model, has been devised so as to avoid spectral integration.

The key idea in developing the TTNH model is to replace the spectral integration appearing in the integral solution of Eqn (8) by an effective total transmittance (or transmissivity) times the blackbody intensity of the gas. The major difficulty with this approach lies in the lack of a sound theoretical basis for defining the total transmittance of a non- homogeneous medium. The spectral transmissivity of a gas layer of thickness sN+l - Sk depends on the

Evaluation of emissivity correlations for HrO-CO*-Nr/air 561

1.4 1.4

t; 1.3 1.3

iz

.L = O.OCK? atm.m

.L = 0.02 atm.m t2 l2 1.2

6 (p,+pJ.L = 0.2 atm.m

s!

1.1 1.1

8 1.0 1.0

$ 0.9 0.9

5

E 0.6 0.6

5

B 0.7 0.7

0.6 0.6 0.4 0.6 1.2 1.6 2.0 2.4 2.6 0.4 0.6 1.2 1.6 2.0 2.4

TEMPERATURE RATIO ek TEMPERATURE RATIO ok 2.6

Fig. 8. Comparison between transmittance correction factor calculated using the EWBM (symbols) and Grosshandler’s correlation (solid lines).

gas optical thickness which is defined as:

The optical thickness is a function of the temperatures and concentrations of gases in the cells [+,.rk+r], [sk+r ,~+z],. up to cell [sN, sN+r]. The TTNH model circumvents the difficulty of spectral calculations by assuming that there exists an average temperature, Fk, and an average partial pressure of radiating gas, P&k> over the path [Q,+,+,], such that the non-homogeneous total transmittance can be approximated by a relation similar in form to that defining the total transmittance for a homogeneous mixture. Equation (16) is then approximated by the relation:

&&/v$~+I) = ‘%(~ki,,Pi.k,V) ‘A% (17)

where Ask = sv+, - Sk. This approach is similar to the wide band scaling method proposed by Felske and Tien (see e.g. Ref. 56). However, while in the latter method the mean average temperature and partial pressure of the radiating gases are chosen as direct spatial average, in the TTNH model, they are defined as:

(18) The partial pressure and temperature averaging

are number density weighted averages. The averaging for the partial pressure can also be regarded as a density-weighted average. The above transformation enables one to replace the spectral integration appearing in the integral solution of the RTE

by the non-homogeneous total transmissivity defined as:

so that the discretized solution of the RTE is:

N 6.G +c- .{%h,k+l - %h,k). t20) k=l T

In Eqs (19) and (20), the subscripts k and k + 1 indicate that the temperature and partial pressure are averaged over the cell volume [#k,sN+i] and bk+l! %‘+ll~ respectively. The quantity .!$( Tk) = A. Z,“(T,) is the spectral blackbody emissive power at the temperature Tk. Grosshandler further rewrites Eqn (19) in terms of the total transmittance, rh,k, of a homogeneous mixture at the temperature Fkk, and partial pressure of radiating gas, pi,k, so that Eqn (19) reads:

Tnh,k = ck rh,k (21)

and the transmittance correction factor Ck is defined as:

Similarly, r,,h,k+i = Ck+i ) q,&+l and

ck+l

(23)


. t I 3

“L.il %.il VL,!z %J.P VL.0 vu.l3 VL,i4 %,i4

Wavenumber [cm-‘]

Fig. 9. Schematic representation of the treatment of the spectral dependence of the linear absorption coefficient using

the spectral group model.

The transmittance correction factor may easily be computed using the EWBM or the NBM. Calcula- tions of the coefficient C, have been performed using the EWBM for mixtures of HZ0 and COZ. As shown in Fig. 8, the correction factor is a fairly smooth function of the temperature ratio 8k = TklTk and combined partial pressures (p,,, + p,). This suggests the possibility of fitting the correction factor by a simple analytical expression. A polynomial correlation has been proposed by Grosshandler.

Equations (19)-(23) together with Modak’s correlation’ for calculation of the homogeneous transmissivity make up the total transmittance non-homogeneous model. Total radiance predictions in non-homogeneous media have been shown to be comparable with results obtained using NBM calculations. The gain in CPU time has been estimated to be around 500 when compared to NBM calculations.bl Although the TTNH model is an important improvement over both the gray gas and the Patch method, it has a number of shortcomings. First, the model is limited to integral solution methods of the RTE and cannot be used with flux models. Second, the correlation developed to compute the transmittance correction factor is limited to total atmospheric pressure, temperature range: 800 < T (resp. T> < 18OOK, 0.4 < 0, < 2.4 and partial pressure range 2 5 (PC +p,+,) L < 4Oatmcm. Therefore, it is not optimal for calculations in small computational cells. Third, the model cannot account for the non-gray radiative properties of the walls.

*The authors of Ref. 65 employ the terminology of the spectral group model to describe a variant of the weighted sum of gray gases model. Hereinafter, the term spectral group model is used to describe the solution method of the RTRrather than the radiative property model itself. Indeed, basically both existing WSGGM and the radiative nronertv model of Song and ?iskanta6’ may be recast in t-he more general frame of the k-distribution method.” By analogy with the wording used to define the SLW model, the authors find it more appropriate to refer to Song and Viskanta’s radiative property model as the exponential wide band model based WSGGM.

5.5. Spectral Group Model*

Song and Viskanta6jm6’ proposed a simple solution method to account for the non-gray nature of radiative heat transfer in non-homogeneous media. Their approach is based on the observations that a spectral average of the RTE over an emission/ absorption band yields an equation in which the fitting coefficients of the WSGGM appear. Basically, the idea of solving the RTE using the weighted sum of gray gases models goes back to the work of Hottel and Sarofim,4’5 Johnson6 Truelove” and Bartelds et &.9*‘o Despite the remarkable agreement the authors of Refs. 6,9, 10 and 15 obtained between their model predictions and the total radiance measurements in luminous and non-luminous flames, none of them considered studying the soundness of the method they used. This step appears to have been undertaken for the first time by Song65.67 and to a lesser extent by Modest.(‘* The spectral group model (SGM) of Song and Viskanta is recalled in the next section. The presentation mainly follows that of Song.65,67 Infor- mation complements may be found in the works of Denison,‘* Lacis and Oinas” and Domoto.56

5.5.1. Model,formulation

5.5.1.1. Spectrally averaged RTE

Assume that the emission-absorption spectrum can be divided into M absorption bands over which the absorption coefficient k,” is taken to be constant. In order to minimize the number of gray bands, the same value of the absorption coefficient is used to characterize emission and absorption over a group of bands located at different positions in the infrared spectrum. As schematized in Fig. 9, since the value of ki is fixed, the lower and upper band limits vL,ij and Vu,ij must vary with temperature and composition. In the spectral intervals [vL,,j, vu,i,] where the spectral absorption coefficient is regarded as constant, the RTE becomes:

al,_ ds

- -&j (I, - Z,o). (24)

Integration of this equation over the spectral interval [VL,ij, v”,ij] yields the spectrally averaged equation”5:

al,_ &Cl ij

as - - k, . I, + ki Wi, I0 - T. I,(LJ~;,~,)

where I, E s;:;y I, du and Wij z $f;y Zz dull’ are the band intensity and fraction of biackbody radiation in the spectral interval [VL,ij, Vn,ij], respectively. I0 is the blackbody intensity. Equation (25) applies to the group of spectral bands for which the spectral absorption coefficient k,” is equal to ki (see Fig. 9). Summing Eqn (25) over all bands ‘j’ belonging to the same spectral group yields the weighted sum of gray


Table 10. Some features of the radiative property models which can be combined with the spectral group solution method of the RTE

Radiative property model

k-distribution model Arking and Grossman [1968] Domoto [ 19741 Denison and Webb [ 19941, etc.

ki

Specified as input parameter

W,

Fqn (4)

Possibility to account for non-gray walls

-

Yes

EWBM-based WSGGM Song and Viskanta [1986]

WSGGM Hottel and Sarofim [1967] Johnson [1971] Smith et al. [1982], etc.

Fitting coefficient

Fitting coefficient

Fitting coefficient

Fitting coefficient

Yes

No

SLW Denison and Webb [1993, 19941 Specified as input parameter Eon (6) No

s,=0 s, s, s. s, s.5 s, s, ”

Fig. 10. Discretization of the gas layer for total radiance calculations across the BERL flames.

gases RTE:

d’l_ as - - k; . Zi + k; wi .I0

aY, ij +x--i

j a3 zv(vL,ij). (26)

The Leibnitz terms involving av”,ij/& and aV,,i$?r are very difficult to estimate since the intensities at uv,ij and VL,ij are not known unless an LBL calculation is performed. In non-homogeneous media, these terms are non-zero because the line intensities and line half-widths vary with position, along with the model histogram schematized in Fig. 9. In order to use existing solution methods of the RTE, these terms are omitted. Thus Eqn (26) simplifies to:

aZi -P -kj. Zi+ ki. wi. I“. as

5.5.1.2. Domain of validity of spectral group model

Physically, this approximation is rigorously exact in the three following cases only:

(1) the medium is gray; (2) the medium is non-gray and homogeneous. In

*The soot volume fraction is defined asfv = cI/ps, where ps is the soot density and @g/m’] is the mass soot concentration. Typical soot volume fractions in hydrocarbon flames range from 10m6 to lo-*. For gas and oil flames, the constant k, typically lies between 4 and 10.

(3)

this case, the same spectrum representation applies everywhere within the medium and the Leibnitz’s terms appearing in Eqn (26) cancel; and the medium is non-gray and non-homogeneous and the spectral linear absorption coefficient obeys the Milne-Eddington approx- imationk,V(T,p,,v) = al(v) .&(Tlpi,F’,).This approximation is valid for clouds of small hydrocarbon soot particles. Indeed, the spectral absorption law for small soot particles is of the Milne-Eddington type with (Ye = v and P,(T,c,) =f, ,k,, where f, is the soot volumetric fraction and k, is a constant* function of the soot composition and temperature.69

The verification that the SGM is nearly an exact solution in gas-sooty flames is well illustrated by the remarkable agreement found between the total radiance measurements reported in Johnson’s thesis6 and his radiance calculations. As pointed out by Song:’ although the Milne-Eddington approximation gives reliable results when applied to gas layers with smooth temperature and concentrations gradients, it should be applied with caution to radiative heat transfer calculations in strongly non-homogeneous gases. It is interesting to note that the mathematical signs of the two Leibnitz terms are opposite. Con- sequently, Eqn (27) could well prove to be a better approximation of Eqn (26) than might be expected, if the effect of these two terms cancel each other.

Integrating Eqn (27) over path length and discretizing the new equation over the non-homogeneous path schematized in Fig. 10 gives:

k=l

- Tii(sk, s?v’+l)] (28)

where the spectral transmissivity Ti(s&, sv+1 ) is

N. Lallemant et al.

p Flue gas

Water-cooled

Measurement

Fig. 11. BERL experimental furnace.

approximated as:

Ti(S&,SN+l) r exp (-$ki,k.(Sk+~ -ski). (29)

Once Eqs (28) and (29) are solved for each group of bands, the total intensity is found by adding the intensity of each spectral group:

5.5.1.3. Computation of parameters wi and ki

The parameters Wi and ki may be calculated using various correlations. These include the weighted sum of gray gases models (WSGGM),6-‘5 the EWBM based WSGGM,6s@ the single line based sum of gray gases (SLW) mode118-20 and the k-distribution model.18*5’ Most of these correlations coupled with the SGM have been assessed by a number of authors. Johnson,6 Bartelds et al.931o Truelove,15 Soufiani and Djavdan” and Ramamurthy et al.” applied WSGGM. Song and Viskanta6svti developed and used the EWBM-based WSGGM, while Denison validated the SLW and k-distribution models.

Basically, the k-distribution model, SLW model

and WSGGM treat non-gray radiative properties in a similar way. Originally, use of the WSGGM coefficients ki and wi was driven by the need to extend gray gas to non-gray calculations. Thus, it may be fair to say that it is only since the development of the EWBM- based WSGGM and the k-distribution model, in particular, that the limits of the weighted sum of gray gases model have become well understood.

5.5.1.4. Advantages and limitations of spectral group model

In the k-distribution and SLW models, the number and value of absorption coefficients ki are selected arbitrarily. In the WSGGM and EWBM-based WSGGM, both k, and M’, are fixed values determined by the fitting procedure. As summarized in Table 10, the coupling of the k-distribution and EWBM-based weighted sum of gray gases models with the SGM can account for the non-gray radiative properties of the walls. The SLW and WSGGM models require gray walls.

The superiority of the spectral group model over other methods discussed in this section is over- whelming. The SGM can be applied to differential as well as to integral solution methods of the RTE. The model formulation is sound and the approximation introduced to derive Eqn (27) is well understood. However, this equation remains rigorously exact in non-gray homogeneous media and in non-gray non- homogeneous media that are loaded with small soot particles. The method may also be applied to gray gas calculations. The computational load invoked by the necessity to solve this equation over several spectral groups is small compared to NBM or/and EWBM calculations. Indeed, the use of fitting coefficients developed for WSGGM require solving Eqn (27) around a maximum of three to four times (see Table 2). Using the spectral models implies solving the RTE a minimum of eleven times for the EWBM and few hundred times for the NBM.

6. ASSESSMENTS OF RADIATIVE PROPERTY MODELS FOR NON-HOMOGENEOUS CALCULATIONS

Predictions of the total radiance have been compared against total radiance measurements. Measurements were taken at two axial locations in two non-sooty turbulent swirling diffusion flames of natural gas. The radiance calculations used four groups of models. The first group is based on the gray non-homogeneous solution of the RTE with different definitions of the linear absorption coefficient, k,. The second makes use of the TTNH mode1,62-64 while the third uses the EWBM2’ coupled with the Chan and Tien72 scaling methods for non-homogeneous calculations. The fourth group of models consists of the SGM6’ together with various weighted sum of gray gases models, the SLW and the k-distribution models.

Evaluation of emissivity correlations for H20-CO*--N*/air 565

-8 -6 -4 -2 0 2 4 6 8 RADIAL POSITION @D/Do)

Fig. 12. Total radiance measurements for hot and cold wall conditions 2 t AD/D, = 0.3 1 (full symbols) and AD/D, = 1.25 (open symbols) for two 300 kW baseline flames.

6.1. The BERL Experiment73374 stands for the axial distance measured from the quarl outlet (see Fig. 1 l), while D,(= 87mm) is the

6.1.1. Experimental furnace diameter of the combustion air duct.

The experimental furnace used for the measurements is the vertically-fired chamber (see Fig. 11) of the Burner Engineering Research Laboratory

6.1.2. Natural gas flames

(BERL) at the Livermore Laboratories. For the The burner for the BERL experiment is the swirl- in-flame measurements, panels were constructed stabilized natural gas burner used for the Scaling 400 with ports to allow access at five traverses located at study (a sketch of the burner is given in Ref. 74). The AD/D, of 0.31, 1.25, 2.19, 3.94 and 4.96, where AD burner has generated symmetrical, non-sooty flames.

0.025 3.0

‘;‘ 2.8

; 0.020 2.6 0 i= 0

2.4

$ 0.015

k

2.2

E 2.0

ki 0.010

f

1.8

6 0.005 1.6

> 1.4

0.0001’ -* 11.2 -1.6 -1.4 -1.0 -0.6 -0.2 0.2 0.6 1.0 1.4 1.8

RADIAL DISTANCE (RD/Do)

Fig. 13. (0) and (A) designate hydrogen and carbon monoxide measurements for flame 2 of Ref. 74 at AD/ D, = 0.31 (D, = 87mm). The solid tine shows the water vapor to carbon dioxide partial pressure ratio computed from detailed HS, CO, CO*, 0s and CH4 measurements at this traverse. RD/D, = 0.0

corresponds to the axis of the burner.


: *_._

--._ 5 : --__ -. 5 15 : --. -

f : : Measured Radiance

-6 -4 -2 0 2 4 6

RADIAL POSITION (RD/DJ

Fig. 14. Comparison between total radiance measurements and predictions using non-homogeneous gray models in Flame 2 (cold walls) at AD/D, = 0.3 1. RD/D, corresponds to burner centerline.

E 35 - ,’ Measured Radiance

-7 -5 -3 -1 1 3 5 7

RADIAL POSITION (RD/Do)

Fig. 15. Comparison between total radiance measurements and predictions using gray and semi-gray models in Flame 1 (hot walls) at AD/D, = 0.31. RD/D, corresponds to burner centerline.

Two flames are relevant to this study: a baseline hot walls flame (Flame 1) and a baseline cold walls flame (Flame 2).73,74 The first flame was generated in the BERL furnace configured out of the refractory-lined segments, while the second flame was fired in the furnace of water-cooled (steel) segments. Both flames were of 300 kW thermal input.

61.3. In-fame measurements

Detailed in flame velocity, temperature and composition measurements were taken at the five traverses located at AD/D, of 0.31, 1.25, 2.19, 3.94 and 4.96. Non-intrusive velocity measurements were performed by applying the LDA technique on the combusting flow seeded with finely sized zirconia particles in the combustion air. Miniaturized versions of standard semi-industrial IFRF probes were used to

intrusively sample gas composition and measure gas temperature by suction pyrometry. The permanent gases CO, Nz, CO*, Hz, 02, NO, and total unburned hydrocarbons were measured using continuous gas analyzers. Nitrogen concentrations were measured only for the baseline cold wall flame at traverse location AD/D, = 0.3 1. As discussed in Ref. 61, these measurements were a valuable check on the calculated water-vapor mole fraction, estimated from the fuel composition.

6.1.4. Total radiance measurements

Both flames were traversed at AD/D, = 0.31 and 1.25 using a narrow angle radiometer with a viewing angle of 0.4” according to the technique described by Beer and Claus.” As shown in Fig. 12, the total radiance measured in Flame 1 at AD/D, = 0.31 and


Fig.

, Hot Walls Flame

Cold Walls Flame

-6 -4 -2 0 2 4 6

RADIAL POSITION @D/Do)

16. Comparison between total radiance measurements and predictions using TTNH model in Flames 1 (hot walls) and 2 (cold walls) at AD/D, = 0.31. RD/D, corresponds to the burner centerline.

AD/D, = 1.25, asymptotically reach a zero flux condition at RD/D, = -7.36 (-640mm) which is at the opposite wall to the measurement port when the refractory panel is replaced with a quartz window. The total radiance measurements are seen to be significantly lower for the flame with the water-cooled panels due to the lower wall temperature, non- emissive nature of the walls and lower temperature of the gas layer near the wall.

6.2. Water Vapor to Carbon Dioxide Partial Pressure Ratio

The detailed chemistry (CO*, CO, unburned hydrocarbons, 02, N2, HZ, NO,) measurements in the two natural gas flames allow for calculations of the water vapor to carbon dioxide partial pressure ratios, pwc . Figure 13 shows the pw values computed for Flame 2 at 3cm downstream of the quarl outlet (AD/D, = 0.3 1, D, = 87 mm). The measured CO and HZ volumetric fractions (wet basis) are also shown on this graph.

In the flame region (IRD/D,I < l), the Hz0 to CO2 partial pressure ratio is not constant. In the flame shear layer (0.61 IRD/D,I < 1.2), it varies from 1.4 to 2.7. The ratio is practically constant in the swirl- induced internal recirculation zone (IRZ) (IRD/D,I 5 0.6) where it takes a value of 2.3. For IRD/D,I 1 1, pwc is constant and nearly equal to two. The ratio pwc varies significantly across the flame region due to incomplete combustion. In the core region (IRZ) of the flame, pwc is larger than the value of two, expected for complete combustion of methane in air. This is because a substantial amount of the carbon is still in the form of CO. In the flame shear layer, pw reaches a peak value of 2.7 very close to the location where the CO volumetric fraction reaches its maximum Vahe, and xn2 is at its

minimum. Thus the deficit in CO2 is more pro-

nounced than that of Hz0 which explains why pwc > 2. Conversely, pwc reaches its minimum value of around 1.4 at RD/D, = 1 where there is no CO but some hydrogen is still present. Therefore, from the chemistry point of view, one expects a depletion in the amount of HZ0 andp, < 2. In the outer flame region (IRD/D,I 2 I), the ratio pwc nearly equals two since this region mainly recirculates products of complete combustion. The calculated Hz0 to CO* partial pressure ratios at the two axial traverses for the hot walls and cold walls baseline flames can be found in Ref. 61.

6.3. Evaluation of Gray Gas and Patch Model

It is clear from the results shown in Figs 14 and 15 that non-homogeneous gray models poorly reproduced the measured intensity gradients. In the cold walls flame, the homogeneous gray model with k, = 0.3 m-l gives a reasonable estimate of the peak radiance at the first axial traverse. However, the model fails to predict the fall-off of the intensity due to gas re-absorption in the external recirculation zone (ERZ). As expected, this effect is more pronounced for the ‘cold walls’ flame than for the flame generated in the refractory line furnace (hot walls) due to the lower ERZ temperature. Although the ‘locally homogeneous’ model largely overestimates the total intensity, this model shows a decrease in the intensity past the peak radiance. The latter must be attributed to the decrease of temperature in the ERZ rather than to the capability of the model to properly treat gas absorption. In the cold walls flame, Patch’s model neither predicts the total peak radiance nor the intensity variation across the gas layer. However, the model’s predictions are in excellent agreement with the measured radiance in the hot walls case. The latter results are not too surprising since the temperature and concentration


0.0

Measured radiance

-7 -5 -3 -1 1 3 5 7

RADIAL DISTANCE (RD/DJ

Fig. 17. Comparison between total radiance measurements in Flame 2 and predictions using the spectral group model together with six HzO-CO2 sum of gray gases models.

Measured radiance

-6 -4 0 RA;;AL DISTANCE (&Do)

4 6

Fig. 18. Comparison between total radiance measurements in Flame I and predictions using the spectral group model together with six H,O-CO2 sum of gray gases models.

gradients in the hot wall flames are less important than in the flames generated in the bare metal wall furnace.

The first two models (see Figs 14 and 15) which are representative of the models commonly used world- wide in CFD codes for flame calculations, poorly predict the total radiance. Although Patch’s models appears to be less reliable for in-flame radiance calculation in strongly non-homogeneous media (Flame 2), it yields surprisingly good predictions of the wall total radiance (AD/D, = +6.5) in the two cases presented here. For the cold walls flame, at AD/D, = 0.31, all three models overestimate the measured wall radiance by 50%, 112% and 5%, respectively. These results clearly indicate that it is

not sufficient to compare total radiance predictions at the walls with single point measurements to obtain a thorough assessment of radiation models’ performances within the flame region.

6.4. Evaluation of TTNH Model

The TTNH equations were solved for the two flames described previously. The homogeneous transmissivity was calculated using Modak’s modeL3 Comparisons between the measured total radiance and the predictions using Grosshandler’s TTNH models are presented in Fig. 16.

The total radiance predictions based on the TTNH model are in good agreement with the measured

Evaluation of emissivity correlations for HrO-CO*-Nz/air 569

Measured radian

-3 -1 1 3

RADIAL DISTANCE (RD/DJ

Fig. 19. Comparison between total radiance measurements in Flame 2 and predictions using the exponential wide band model and the spectral group model.

Measured radiance

_____.._._.-.--.--

I I I I1 I1 1.1.1 1 I. I, I. I.1 1

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 RADIAL DISTANCE (RD/Do)

Fig. 20. Comparison between total radiance measurements in Flame 1 and the predictions using the exponential wide band model and the spectral group model.

values. In the cold wall flame, the TTNH model overestimates the peak values by around 25%. As a consequence of this overshoot the calculated intensities in the ERZ (IRD/D,I 2 1) remain higher than the measured values.

As pointed out earlier (see Sections 4.2.-4.4.), Modak’s model is not accurate for cell sizes less than about 1 cm. Similarly, Grosshandler’s correlation to compute the transmittance correction factor is applicable in the combined partial pressure path length range 2 I (PC + pW) . L 5 40 atm e cm. In order to compute the total radiance in the flame shear layer (0.6 5 IRD/D,I I 1.2), it was assumed that both correlations were applicable down to cell sizes as small as 0.5 mm. This yields combined partial

pressure-path lengths N low4 atm. m, outside the applicability domain of both Modak emissivity correlation and the Ck correction factor. This partially explains the discrepancies observed between the TTNH-Modak models and the measured radiance in Flame 2. It is interesting to note that the error introduced by the use of these correlations tends to disappear in the total radiance predictions of the hot wall flame. The difference between the radiance predictions and the measured data in both flames is related to the importance of the temperature gradients in these flames. In the region of the flame shear layer, the temperature gradient is approximately 750 K/cm in Flame 2, while it is only 500K/cm in Flame 1. At AD/D, = 1.25, these


gradients decrease to 250 K/cm and 80 K/cm for the cold and hot wall flames, respectively. This observation strongly suggests that the main reason for the discrepancies between the TTNH model predictions and the measurements are due to the inaccurate treatment of the temperature gradients. This identi- fies the total transmittance correction factor as the main source of discrepancy. Therefore, it appears that a new total transmittance correlation should be developed if accurate calculations of the total radiance within the flame region is to be accomplished using Grosshandler’s TTNH model. In view of the temperature measurements in the cold wall flames, the new correlation should be extended to cover at least the range of temperature ratio 0.2 I ek < 5.

6.5. Evaluation of Spectral Group Model

The total radiances in the cold wall and hot wall flames have been predicted using the spectral group model (SGM) in combination with the following radiative property models:

(1) The Weighted Sum of Gray Gases Models of: -Johnson6 -Taylor and Foster7 -Truelove’ (two models) -Smith et aLI3 -Soufiani and Djavdan7’. For all the models, the Wi and ki parameters corresponding to a H20/C02 partial pressure ratio of 2 were used in the calculations.

(2) The Single Line Based Sum of Gray Gases Model of Denison and Webb18m20.

(3) The k-distribution Model. The correlations developed by Denison” for the cumulative k-distributions of Hz0 and CO2 were used in all the calculations.

The results of the spectral group model calculations are presented in Figs 17-20. The predictions using the weighted sum of gray gases models are in reasonable agreement with the experimental data. In the cold wall flame, the Truelove models and the emissivity correlation of Taylor and Foster WSGGM approximation overestimate the peak radiance at RD/D, = + 1. These models also overrate the attenuation of the radiation beam by the cold gas layer past the peak radiance. For Flame 1, the WSGGM of Johnson6 and Souflani and Djavdan” are the closest to the experimental data. However, both models substantially underpredict the measured total radiance in Flame 2. It should be noted that similar discrepancies were found in other tests6’ Therefore, the good agreement between the model predictions and the measured data in Flame 1 may be seen as coincidental. The poor performance of Johnson’s model is expected since the model is used outside its range of applicability for temperature and partial pressure-path length. The main reason for the

poor quality predictions of Soufiani and Djavdan’s WSGGM approximation is more ambiguous. Use of the model outside its applicability domain in partial pressure-path length ( 10e4 5 pw L I: 1.22 atm . m) and temperature (300 5 T 5 2500K) is an unlikely reason for the bad performance of the model since, for the same tests, the WSGGM of Smith and co-workers, which has a similar applicability domain, yields good total radiance predictions. It may be surmised that the poor quality prediction of Soufiani and Djavdan’s model is related to the slight modifi- cation to the classical WSGGM mathematical formulation which these authors adopted*.@ However, as shown by some recent calculations by the authors,6’ this assumption cannot explain the observed discrepancy. Invoking the effect of turbulence of radiation is not a tenable argument either in view of the good quality predictions obtained with the other models. The origin of this discrepancy remains, as yet, unclear. In comparison, the SGM predictions based on the WSGGM of Taylor and Foster,7 Truelove” and Smith and co-workersI are in reasonable agreement with the measurements. Although the latter model yields somewhat higher predictions than the k-distribution and EWBM models, it reproduces reasonably well the magnitude of the radiance absorption in the region RD/D, > 1.

Because total radiance calculations using the k-distribution model are time-consuming, only the simplest procedure neglecting the effect of overlapping has been tested. As shown in Figs 19 and 20, the total radiance predictions based on the k- distribution model are in good agreement with both the experimental data and the EWBM. In testing the SLW model, two models have been considered. In the first model, the total radiance is computed assuming that overlapping effects are negligible. In the second model, band overlapping is accounted for using the double integration method proposed by Denison.”

As shown by Eqn (7) the blackbody absorption

* In the classical formulation of the WSGGM approximation, the partial pressure p appearing in the factor, (I- exp(-4, ‘P L)) is generally equated with the sum of the partial pressure of Hz0 and COz, regardless of the fact that the H20/C02 partial pressure ratio is pw/pC = 1 or 2. The latter assumption strongly suggests replacement of p with pw in the WSGGM formulation. Indeed, if p,,,/pw = constant (say K,), andp=p,+p,,pmayaswellhewrittenasp=(l+K,)~p, = K2 pw In other words, p and kig may be replaced in Eqn (3) byp, and K2 kin, tqectively. This is the approach adopted in the WSGGM of Soufiani and Djavdan. In spite of the soundness of this approach, the choice p = pw + pC is a much less restrictive assumption than p = pw, since it allows a better account of the variation in the total amount of absorber when the model is used outside the ratiosp,/p, = 1 or 2. By replacing P=PwfPc Ah P’Pw9 the authors of Ref. 70 constrain pw/pC and @,., +p,) to always remain constant. Therefore, it may be conjectured that the choice p = pw + pC is better suited for radiative heat transfer calculations in non-homogeneous media since it enables the use of the WSGGM in regions where the p,/p, ratio deviates from the constant.

Evaluation of emissivity correlations for HSO-CO*-Nz/air 571

distribution function, F,, depends on the gas temperature, T, and the background temperature Tb. Herewith, Tb was set to the gas temperature. Although the authors of Refs 18-20 have suggested a procedure to estimate the temperature T,, their approach was not used in the tests shown in this study. It is believed that beyond the lack of a sound theoretical background, such an approach introduces a computational load that makes the SLW unattractive. Results of the predictions are shown in Figs 19 and 20. For the two traverses considered herein, the SLW predictions based on the double integration technique are significantly lower than both the measurements, and the SLW predictions neglecting the effect of band overlapping. Though low, the latter predictions are in reasonable agreement with the measurements.

A more complete analysis (see Ref. 61) has shown that the predictions based on a five group division are systematically off from the results obtained using either 10 or 20 spectral groups. Differences between the results obtained using 10 and 20 groups were found to be comparatively small. These results suggest that a 10 group division using either the SLW or k-distribution models should be suthcient for most radiative heat transfer calculations in combustion chambers.

7. CONCLUSION

The benchmark emissivity data for water vapor, carbon dioxide and mixtures of these gases has been evaluated. The accuracy of total emissivity correlations for H20-CO*-Nr/air mixtures was assessed against the predictions of the EWBM over a range of temperatures, path lengths and total pressures that cover a wide range of industrial applications. The models have been validated both in homogeneous and non-homogeneous situations. The non-homogeneous calculations have been compared with in situ total radiance measurements in two non-luminous gas flames of natural gas. Various coupling methods between existing total emissivity correlations and the radiative transfer equation have been examined. The findings of the present review are summarized as follows:

(1) The accuracy attributed to the total emissivity predictions of Hz0 and CO2 using either the line-by-line, the narrow band or the exponential wide band models are not better than lo-20% at temperatures above 1200- 1500 K. For CO*, the Fara study has shown that very good agreement exists between the two model predictions. Below 2400 K and a partial pressure-path length less than about 1 atm G m, the departure between the EWBM and the NBM predictions of the Hz0 standard emissivity does not exceed f 15%. In comparison, the LBL predictions are up to 40% lower than the EWBM or NBM

predictions in the temperature range 1200- 2400K. Therefore, Farag data which are based on an extensive comparison of measured total emissivities, NBM and EWBM predictions, are recommended for validating total emissivity predictions using LBL or total emissivity correlations.

(2) Both homogeneous and non-homogeneous tests indicate the advantage of using the Smith, Shen and Friedman weighted sum of gray gases model over the polynomial and SLW models. Polynomial correlations, though interesting because of the possibility of their use at HZ0 to CO1 partial pressure ratios other than one and two, cannot be coupled with differential solution methods of the RTE to perform non-gray non-homogeneous radiation calculations. The wide applicability domain and the simplicity of the SLW model makes it very attractive for implementation in CFD codes. However, the lack of an accurate and simple procedure to account for the effect of band overlapping in mixtures of gases under- mines the model’s appeal.

(3) The non-homogeneous k-distribution model calculations do not show significant differences with the predictions obtained using the WSGGM. The benefit of using such a model in CFD codes for flame calculation appears whenever account has to be taken of the spectral properties of walls.

(4) It is believed that the generality of the EWBM, together with its computational times comparable to that required by polynomial emissivity correlations, makes it an attractive radiative property model. The model should be used a.s a benchmark model to generate EWBM based weighted sum of gray gases models (WSGGM).

(5) The spectral group model (SGM) provides an elegant and accurate method of coupling WSGGM, k-distribution and single line weighted (SLW) property models to the equation of radiative transfer. The SGM may easily be applied to integral (e.g. discrete transfer) and differential (e.g. flux methods) solution methods of the radiative transfer equation. The model formulation and the approximation introduced to derive the spectral group equation (Eqn (27)) are mathematically sound. This equation is rigorously exact in gray media, non-gray homogeneous media and non-gray, non-homogeneous media which are loaded with small soot particles. In addition, as shown by the results of this study, it provides a very good approximation for non-gray, non- homogeneous calculations in gases.

Eventually, the implementation of the SGM into a CFD code for flame calculation is a fairly straightforward procedure if the existing code already has a ‘gray radiation model’ running.

N. Lallemant et al. 572

(6)

The transformation from the gray treatment of the RTE into its SGM analogue may be described in terms of the following interchange of variables:

gray gas coefficient k, + ki absorption coefficient of spectral group ‘i’; total intensity I - Zi intensity of spectral group ‘i’. blackbody intensity I0 + U‘i I0 product of the fraction of blackbody radiation and blackbody intensity emitted in the spectral group ‘i’.

The weight coefficient M?i and the blackbody intensity 1’ are evaluated at the same temperature. It should also be clear that the above transformations apply at the boundaries, and that the RTE with the SGM has to be solved as many times as there are spectral groups. It has been shown that in the near burner region of a natural gas diffusion flame, the water vapor to carbon dioxide partial pressure ratio departs significantly from the value expected for the complete combustion of methane in air. This finding emphasizes the limitation of existing WSGGM to Hz0 to COZ partial pressure ratios of one and two only.

8. SOME UNSOLVED PROBLEMS

The analysis of Song and Viskantab5 on the spectral group model and the introduction of the k-distribution model for radiative heat transfer calculations are perhaps the most significant advances in the field of infrared radiation for a number of years. The k- distribution model reveals the interrelations which exist between the line-by-line model, the band models (NBM and EWBM),” and the weighted sum of gray gases models.” The spectral group mode16’ bridges the gap between the radiative transfer equation and weighted sum of gray gases mode1.6s In this respect, the latter development places a calculation procedure initiated about 30 years ago in engineering on a sound theoretical basis.6’9~‘0~7’

For many years, the study of radiative heat transfer in flames has benefited from a close interaction between theory and computer simulation. Despite the progress which has been achieved, a few fundamental and practical issues are as yet unsolved.

From a theoretical viewpoint, a complete and consistent treatment of the RTE in highly non- homogeneous media is not yet at hand. Much research is still needed to incorporate existing infrared radiative property models for soot, coal and ash into a band or spectral group model of the RTE. As for gas radiative property models, it is interesting to note that the origin of the path length dependence of the spectrally averaged absorption coefficients appearing in the statistical band models

has not yet been thoroughly investigated. The unphysical nature of this dependence is the major impediment preventing the coupling of the statistical narrow band model with differential solution methods of the RTE. It is worth noting that the concept of path length dependent physical properties appears in numerous physical problems whenever long-range interaction phenomena are important, e.g. turbulence, transition phenomena near the critical point, etc.

From a practical viewpoint, the k-distribution model, SLW model and WSGGM are undermined by not accounting for the effect of pressure variation on the infrared spectra. The treatment of band overlapping using these models is not entirely satisfactory and a soundly based model for gas-soot mixtures remains to be developed. The lack of data on infrared radiative properties of soot, coal and ash call for more experimental work.

Acknowledgmenrs Financial support for this work was provided by the IFRF, the Institut Francais du Petrole. Electricity of France and Gaz de France. The authors wish to express their gratitude to Henk Horsman for his contribution to the total radiance measurements during the BERL experiment. The authors also acknowledge Pr. M.K. Webb and Dr M.K. Dension, from Brigham Young University, who kindly provided the extensive documentation about their work on the k-distribution and SLW models. The critical reading of the document and constructive sugges- tions of William L. Grosshandler during the review of this article are particularly appreciated.

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