The k-moment method for modeling the blackbody weighted transmission function for narrow and wide band radiative properties of gases

ARTICLE IN PRESS

Journal of Quantitative Spectroscopy & Radiative Transfer 108 (2007) 1–16 www.elsevier.com/locate/jqsrt

The k-moment method for modeling the blackbody weighted transmission function for narrow and wide band radiative properties of gases Fre´de´ric Andre´, Rodolphe Vaillon Centre de Thermique de Lyon (CETHIL, CNRS-INSA Lyon-UCBL), Baˆt. Sadi Carnot, INSA-Lyon, F-69621, France Received 11 January 2007; accepted 16 February 2007

Abstract This paper presents the k-moment method, based on the calculation of the moments of the absorption coefficient, to calculate the blackbody-weighted transmission function for modeling narrow and wide band radiative properties of gases. The spectral averaging is performed on band widths of the order from 25 to 600 cm1 . The spectrally averaged integral formulation of the radiative transfer equation is established and associated blackbody-weighted transmission function approximate expressions are derived from a series reversion technique. The second and third order k-moment formulations are assessed through comparisons with line-by-line reference results in the case of pure carbon dioxide at atmospheric pressure and for temperatures ranging from 300 up to 2300 K. r 2007 Elsevier Ltd. All rights reserved. Keywords: Radiative transfer; Gas radiative properties; Narrow band model; Wide band model, k-moment

1. Introduction Radiative heat transfer in participating gases is significant in radiative atmospheric cooling and in combustion systems. The line-by-line (LBL) model provides the highest accuracy for the prediction of the radiative properties of spectrally selective gases, but it is too computationally expensive to be used in many complex situations of practical interest. Hence, approximate approaches are unavoidable to optimize computational efficiency while preserving accuracy of the modeling. The existing ones can be put in three main categories: (1) narrow band models, like the statistical (SNB) and correlated-k (CK) ones [1]; (2) wide band models, like the exponential wide band model of Edwards [2]; (3) global models, such as the absorption distribution function (ADF) [3], the spectral-line weighted-sum-of-gray-gases (SLW) [4,5], or the full spectrum k-distributions (FSK) [6]. Each of these approaches has been proven to achieve a quite good accuracy in comparison with LBL calculations, under more or less restricted limits of applicability (see for instance [7,8]). Corresponding author. Tel.: +33 4 72 43 88 16, fax: +33 4 72 43 88 11.

E-mail address: [email protected] (F. Andre´). 0022-4073/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2007.02.013

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Nomenclature In l P x s T

spectral radiative intensity ðW cm2 sr1 =cm1 Þ uniform (isothermal homogeneous) column length (cm) pressure (atm) molar fraction of absorbing gaseous specie coordinate along radiation path (cm) temperature (K)

Greek symbols kn m n O o tn

spectral absorption coefficient ðcm1 atm1 Þ moment of the absorption coefficient wavenumber ðcm1 Þ natural logarithm of the spectrally averaged transmission function coefficients in the series expansion of O and its reciprocal spectral transmission function

Subscripts b ðnÞ OTL

blackbody and blackbody-weighted order of truncation in series expansions optically thin limit

Superscripts ð1Þ a Dn

reciprocal function polynomial equation solution width of the spectral interval for the averaging of spectral properties

Concerning narrow band approaches, the complex spectral dynamics observed in the high resolution (HR) gas spectrum is smoothed by averaging (the associated HR gas spectral property viz. absorption coefficient, transmission function, absorptivity) over small spectral intervals (of typical width equal to few cm1 ), where spectral variations of the Planck function can be assumed negligible. In the case of the SNB approximation, within the framework of randomly distributed Lorentz lines and the Malkmus (inverse-exponential-tailed) distribution function of line intensities, the spectrally averaged transmission function of gas columns is an analytical function of two parameters that are the mean line intensity to spacing ¯ and the line overlapping parameter ðbÞ ¯ [1]. These two parameters are sufficient for the modeling of ratio ðkÞ statistical properties of thousands of spectral lines. This model has been proven to be very accurate and computationally efficient in many situations both for radiative heat transfer simulations [9] and remote sensing applications [10]. However, the model is not exact and its application may introduce some errors if parameter definitions are used [1]. This lack of accuracy can be minimized by performing a leastsquare adjustment of parameter b¯ on transmissivity curves-of-growth (COG), whereas k¯ is obtained at the optically thin limit. For radiative heat transfer applications, these two parameters have to be estimated on each narrow band. For carbon dioxide and water vapor, 926 band model parameters at 25 cm1 are required to cover the spectral interval ½150; 9300 cm1 . The spectral discretization can be optimized to reduce the computational cost by a factor of the order 10–20, while preserving good accuracy compared to LBL calculation results [8]. But mathematical formulations encountered in this context are identical to those of the SNB model and consequently may induce inaccuracies since some physical assumptions, applicable to narrow bands, do not remain valid for wide bands. For example, as it is noticed in Ref. [8], an

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optimized 43 spectral band approach was proven to be efficient for H2 O, but a loss of accuracy compared to narrow band calculations was observed at the same time for CO2 . In this case, errors arise from the fact that the statistical properties of CO2 lines are less spectrally homogeneous than those of H2 O for wide spectral intervals. The present paper is dedicated to the development of a mathematical formulation for the approximate calculation of homogeneous isothermal gas column transmissivities weighted by the blackbody intensity spectral distribution for narrow to wide spectral bands. The theoretical background is first presented. The spectrally averaged integral formulation of the radiative transfer equation (RTE) is established and highlights a blackbody-weighted transmission function (BTF) to be evaluated. Approximate expressions are derived as functions of the moments of the absorption coefficient (k-moment method) through series developments, reversion and truncation. It is shown that at the limit of narrow bands, on which the Planck function can be assumed constant, the second order approach leads to a mathematical formulation which is identical to the one of Malkmus. Analytical expressions are also derived for the third order k-moment formulation likely to be dedicated to wider spectral bands. Validations are presented for various spectral interval widths in the case of pure carbon dioxide at atmospheric pressure and for temperatures representative of situations encountered in combustion environments.

2. RTE formulation Radiative heat transfer for an absorbing, emitting but non-scattering single molecular gaseous specie along a path-length coordinate s0 is governed by the RTE: qI n ðs0 Þ ¼ xðs0 ÞPðs0 Þkn ðs0 ÞI b;n ðs0 Þ  xðs0 ÞPðs0 Þkn ðs0 ÞI n ðs0 Þ, qs0

(1)

where I n ðs0 Þ ðW cm2 sr1 =cm1 Þ is the spectral radiative intensity at wavenumber n ðcm1 Þ and abscissa s0 (cm), kn ðs0 Þ is the spectral absorption coefficient of the medium ðcm1 atm1 Þ, I b;n ðs0 Þ ðW cm2 cm sr1 Þ is the Planck spectral distribution of blackbody intensity at local temperature Tðs0 Þ (K), xðs0 Þ is the molar fraction of absorbing specie and Pðs0 Þ is the total pressure (atm). The RTE can be integrated between two points along the path [11], from an origin at abscissa s0 ¼0, chosen for simplicity to emit blackbody radiation at temperature Tð0Þ, to a point MðsÞ in the gas at abscissa s:  Z s00 ¼s  I n ðsÞ ¼ I b;n ð0Þ exp  xðs00 ÞPðs00 Þkn ðs00 Þ ds00 Z

s00 ¼0

s0 ¼s

þ

 Z I b;n ðs Þxðs ÞPðs Þkn ðs Þ exp  0

0

0

s00 ¼s

0

s0 ¼0

00

00

00

xðs ÞPðs Þkn ðs Þ ds

00



ds0 .

ð2Þ

s00 ¼s0

Exponential terms in the previous equation are the spectral transmissivities of the gas columns:  Z tn ðs ; sÞ ¼ exp 

s00 ¼s

0

 xðs ÞPðs Þkn ðs Þ ds . 00

00

00

00

(3)

s00 ¼s0

An equivalent form of the RTE solution may be written in terms of gas transmissivities as Z

s0 ¼s

I n ðsÞ ¼ I b;n ð0Þtn ð0; sÞ þ s0 ¼0

I b;n ðs0 Þ

qtn ðs0 ; sÞ 0 ds . qs0

(4)

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The averaging of Eq. (4) over a spectral interval Dn yields  Z s0 ¼s Z  0 1 0 qtn ðs ; sÞ 0 ¯I Dn I b;n ð0Þtn ð0; sÞ þ I b;n ðs Þ ds dn n ðsÞ ¼ Dn Dn qs0 s0 ¼0  Z 1 Z Z 1 1 1 ¼ I b;n ð0Þ dn  I b;n ð0Þ dn I b;n ð0Þtn ð0; sÞ dn Dn Dn Dn Dn Dn Dn |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Dn I¯ b;n ð0Þ

Z

s0 ¼s

þ s0 ¼0

(

t¯ Dn ½0;s;Tð0Þ b;n

) 1 Z Z Z 1 q 1 1 I b;n ðs0 Þ dn 0 I b;n ðs0 Þ dn tn ðs0 ; sÞI b;n ðs0 Þ dn ds0 Dn Dn qs Dn Dn Dn Dn |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Dn I¯ b;n ðs0 Þ

ð5Þ

t¯ Dn ½s0 ;s;Tðs0 Þ b;n

or equivalently Dn Dn I¯ n ðsÞ ¼ I¯ b;n ð0Þ¯tDn b;n ½0; s; Tð0Þ þ

Z

s0 ¼s s0 ¼0

Dn I¯ b;n ðs0 Þ

0 0 q¯tDn b;n ½s ; s; Tðs Þ ds0 . qs0

(6)

The detailed mathematical derivation of the previous formulation is provided in Appendix A. Eq. (6) is formally identical to Eq. (4) except that the blackbody intensity is replaced by the spectral average of the Planck function so that an analog transmission function appears through the weighting of the gas spectral transmissivity by the same Planck function:  Z 1 Z  Z s00 ¼s  1 1 Dn 0 0 0 00 00 00 00 t¯ b;n ½s ; s; Tðs Þ ¼ I b;n ðs Þ dn exp  xðs ÞPðs Þkn ðs Þ ds I b;n ðs0 Þ dn Dn Dn Dn Dn s00 ¼s0 ¼ expfO½s0 ; s; Tðs0 Þg,

ð7Þ

where 0 0 O½s0 ; s; Tðs0 Þ ¼  ln t¯ Dn b;n ½s ; s; Tðs Þ.

(8)

0 0 t¯ Dn b;n ½s ; s; Tðs Þ will be called the blackbody-weighted transmission function (BTF) in the following. It corresponds to the transmission function of the gas column between points of abscissa s0 and s, weighted by the Planck function at temperature Tðs0 Þ; which can be different from the gas temperature at point s0 if this point is located at a wall. It must be pointed out that parameter O½s0 ; s; Tðs0 Þ can be identified on narrow bands with the standard mean equivalent black line width divided by the mean line spacing parameter encountered in the SNB approach [1]. If the width Dn of the spectral interval is small enough to assume that the equilibrium radiation intensity is constant over Dn then Z 1 Dn 0 0 0 tn ðs0 ; sÞ dn ¼ t¯ Dn (9) t¯ b;n ½s ; s; Tðs Þ ¼ n ðs ; sÞ, Dn Dn

which does not depend on the temperature at the point of emission but on the gas properties between s0 and s. In this case, physical assumptions are those encountered in narrow band approaches. Then standard SNB or CK formulations can be applied for the radiative properties of gases and the radiation field is given by [1] Z s0 ¼s q¯tDn ðs0 ; sÞ 0 Dn Dn Dn ð0; sÞ þ ds , (10) I¯ b;n ðs0 Þ n 0 I¯ n ðsÞ ¼ I¯ b;n ð0Þ¯tDn n qs s0 ¼0 where in this case Z 1 0 ¯I Dn I b;n ðs0 Þ dn ffi I b;n ðs0 Þ b;n ðs Þ ¼ Dn Dn is a constant on narrow spectral intervals and is only a function of temperature.

(11)

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In the case of uniform (isothermal homogeneous) media, the transmissivity (Eq. (3)) can be written using the simple expression ðl ¼ s  s0 Þ: tn ðs0 ; sÞ ¼ exp½xPkn ðs  s0 Þ ¼ expðxPkn lÞ ¼ tn ðlÞ

(12)

that also yields 0 0 0 0 O½s0 ; s; Tðs0 Þ ¼  ln t¯ Dn ¯ Dn b;n ½s ; s; Tðs Þ ¼  ln t b;n ½l; Tðs Þ ¼ O½l; Tðs Þ,

where for spectral bands of arbitrary width,  Z 1 Z 1 1 0 0 ½l; Tðs Þ ¼ I ðs Þ dn expðxPkn lÞI b;n ðs0 Þ dn t¯ Dn b;n b;n Dn Dn Dn Dn

(13)

(14)

is the BTF expression for uniform media. 3. Calculation of the BTF The aim of this section is to provide simple approximate formulas for the BTF to be used in the spectrally averaged band formulation of the RTE provided by Eq. (6). The first step of the mathematical development consists in formulating the series expansion of the transmissivity given by Eq. (12): tn ðlÞ ¼ expðxPlkn Þ ¼ 1 

xPl ðxPÞ2 l 2 ð1Þn ðxPÞn l n kn þ ðkn Þ2 þ    þ ðkn Þn þ    . 1! 2! n!

(15)

Weighting with the Planck distribution of blackbody intensity and averaging over the spectral band Dn result in the series expansion:  Z 1 Z l 1 1 Dn 0 0 t¯ b;n ½l; Tðs Þ ¼ 1  I b;n ðs Þ dn xPkn I b;n ðs0 Þ dn 1! Dn Dn Dn Dn  Z 1 Z l2 1 1 þ I b;n ðs0 Þ dn ðxPkn Þ2 I b;n ðs0 Þ dn þ    Dn Dn 2! Dn Dn  Z 1 Z ð1Þn l n 1 1 0 þ I b;n ðs Þ dn ðxPkn Þn I b;n ðs0 Þ dn þ    . ð16Þ Dn Dn Dn Dn n! The BTF can thus be expressed as an infinite series expansion in the variable l involving Planck function weighted powers of the absorption coefficient. From the Taylor–Mac Laurin series formal expression of t¯ Dn b;n , the ith derivative of the BTF in the neighborhood of a point of radiation emission ðl ¼ 0Þ can be derived: )  Z 1 Z di tDn ½l; Tðs0 Þ 1 1 i ¯ b;n 0 ð1Þ ¼ I ðs Þ dn ðxPkn Þi I b;n ðs0 Þ dn ¼ mi;b ðs0 Þ, (17) b;n i Dn Dn Dn Dn dl l¼0

where mi;b is the Planck function weighted ith moment of the absorption coefficient. By using the previous equation in association with the series expansion of its natural logarithm O½l; Tðs0 Þ, whose first terms are given in Appendix B, we obtain " # " # 0 m21;b ðs0 Þ  m2;b ðs0 Þ m1;b ðs0 Þm2;b ðs0 Þ m31;b ðs0 Þ 2 3 m3;b ðs Þ 0 0  þ O½l; Tðs Þ ¼ lm1;b ðs Þ þ l þl þ  6 2 2 3 ¼ lo1;b ðs0 Þ þ l 2 o2;b ðs0 Þ þ l 3 o3;b ðs0 Þ þ   

ð18Þ

Since o1;b ðs0 Þ ¼ m1;b ðs0 Þ, it is obvious that if the spectral absorption coefficient is not identically null over Dn, then o1;b ðs0 Þ is a strictly positive quantity which indicates that O½l; Tðs0 Þ is a strictly increasing function of l (such a strictly monotonous function is mathematically called a diffeomorphism). This proves that the reciprocal of O½l; Tðs0 Þ exists and, by extension, that the series reversion of Eq. (18) is possible. This justifies

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ð1Þ 0 the existence of real coefficients oi;b ðs Þ; i 2 N such that 8 ð1Þ 0 3 ð1Þ 0 0 < O1 ðX Þ ¼ X o1;b ðs Þ þ X 2 oð1Þ 2;b ðs Þ þ X o3;b ðs Þ þ    0 : O1 fO½l; Tðs0 Þg ¼ O1 f ln t¯ Dn b;n ½l; Tðs Þg ¼ l:

(19)

Series reversion formulas can be found in [12]. When applied to Eq. (18), they lead to the following expressions for the first series coefficients of O1 : 8 > oð1Þ ðs0 Þ ¼ 1=o1;b ðs0 Þ; > > < 1;b 0 0 0 3 oð1Þ (20) 2;b ðs Þ ¼ o2;b ðs Þ=½o1;b ðs Þ ; > > > : oð1Þ ðs0 Þ ¼ ½2o2 ðs0 Þ  o ðs0 Þo ðs0 Þ=½o ðs0 Þ5 : 1;b

2;b

3;b

3;b

1;b

The approximate modeling proposed in this work is based on the idea that it is possible to restrict the series expansion of O1 to low orders while preserving a good overall accuracy on the estimation of this function. Consequently, if it is assumed that O1 ðX Þ can be exactly calculated by a series expansion limited to its first n terms, with n a given positive integer, then from Eq. (19), O½l; Tðs0 Þ is a root of the nth order polynomial equation: 0 2 ð1Þ 0 3 ð1Þ 0 n ð1Þ 0 l þ X oð1Þ 1;b ðs Þ þ X o2;b ðs Þ þ X o3;b ðs Þ þ    þ X on;b ðs Þ ¼ 0.

(21)

ðX a ðnÞ Þ

Once solved and the appropriate root chosen according to physical constraints (the transmissivity value is between 0 and 1, is a decreasing function of l; . . .), the BTF can be calculated by 0 0 a t¯ Dn b;n ½l; Tðs ÞðnÞ ¼ expfO½l; Tðs Þg ¼ expðX ðnÞ Þ,

(22)

where subscript ðnÞ indicates the order of truncation in the series expansion to get an approximate value of the BTF. The following analytical expression of the second-order approximation of the BTF (BTF2) is obtained from the adequate solution X a ð2Þ of the second-order polynomial (Eq. (21)): 8vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9+ * ð1Þ 0
pm21;b ðs0 Þ m2;b

ðs0 Þ



m21;b ðs0 Þ

¼

pm21 ðs0 Þ . m2 ðs0 Þ  m21 ðs0 Þ

(26)

(27)

Mathematically, the approach yields formulas which are identical to those of the Malkmus transmission function [1]. The same result, obtained by using some mathematical properties of the inverse Gaussian

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distribution function, was reported previously in Ref. [13]. The present reasoning offers some advantages in comparison with the standard one since the Malkmus transmission function and associated parameters have been derived without making any assumption on the line profile or line strength probability distribution function. It can be noticed that this extended use of the Malkmus approximation is frequently performed without any justification. Equivalently, the present result is in agreement with a previous work which demonstrated that for CO2 and H2 O, among all the available two-parameter SNB approaches, the formulation of Malkmus is found to be the most accurate [1]. The case of third-order approximation and the corresponding formulation of the transmission function resulting from the solution of the third order polynomial (Eq. (21)) can also be treated analytically. Let us first introduce three parameters: k¯ b ðs0 Þ ¼

b¯ b ðs0 Þ ¼

Z¯ b ðs0 Þ ¼ ¼

1 0 oð1Þ 1;b ðs Þ

¼ m1;b ðs0 Þ,

p 0 2k¯ b ðs0 Þoð1Þ 2;b ðs Þ

¼

(28)

pm21;b ðs0 Þ m2;b ðs0 Þ  m21;b ðs0 Þ

,

(29)

p 0 6k¯ b ðs0 Þb¯ b ðs0 Þoð1Þ 3;b ðs Þ

m21;b ðs0 Þ½m2;b ðs0 Þ  m21;b ðs0 Þ m41;b ðs0 Þ  3m21;b ðs0 Þm2;b ðs0 Þ þ 3m22;b ðs0 Þ  m1;b ðs0 Þm3;b ðs0 Þ

,

ð30Þ

which are functions of the first, second and third Planck-weighted moments of the absorption coefficient. Using these notations, the writing of the polynomial (Eq. (21)) limited to the third order is, after rearranging, 

6lkb ðs0 Þb¯ b ðs0 Þ¯Zb ðs0 Þ 6b¯ b ðs0 Þ¯Zb ðs0 Þ þ X þ 3¯Zb ðs0 ÞX 2 þ X 3 ¼ 0. p p

(31)

This equation has between one and three real roots. In all investigated cases, only one real solution X a ð3Þ was found so that the third-order approximation of the BTF (BTF3) can be expressed as   p½Tðs0 Þ 0 a 0 0 t¯ Dn þ u½l; Tðs ½l; Tðs Þ ¼ expðX Þ ¼ exp  Þ þ Z ðs Þ , (32) ¯ ð3Þ b b;n ð3Þ u½l; Tðs0 Þ with u½l; Tðs0 Þ ¼ h

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fq½l; Tðs0 Þg2 þ fp½Tðs0 Þg3 þ q½l; Tðs0 Þi1=3 ,

 ¯ 0  2bb ðs Þ 0  Z¯ b ðs Þ , p½Tðs Þ ¼ Z¯ b ðs Þ p 0

0

q½l; Tðs0 Þ ¼ ½¯Zb ðs0 Þ3 

3b¯ b ðs0 Þ¯Zb ðs0 Þ ½¯Zb ðs0 Þ þ kb ðs0 Þl. p

(33)

(34)

(35)

Although it may a priori look a little bit more complicated than the BTF2 expression (Eqs. (24)–(27)), implementation of Eqs. (28)–(30), (32)–(35) is still simple and is likely to remain computation cost effective. The second- and third-order approximations of the BTF (BTF2 and BTF3) have been assessed. Results are given and discussed in the next section.

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4. Parameter generation and model assessment in the case of pure carbon dioxide 4.1. Parameter generation from LBL data Radiative properties of gaseous CO2 are often required to treat energetic and environmental science issues. That is why spectroscopic data for this molecule are available for a wide range of thermophysical conditions. Databases gather all the information required for a LBL modeling of each of the transition line whose parameters (spectral position, line strength, energy of the lower level of the transition, half width at half maximum,..) are registered. The most widely used databanks for CO2 are HITRAN (HIgh resolution TRANsmission) [14] for atmospheric applications, its high temperature extension, HITEMP and CDSD (Carbon Dioxide Spectroscopic Database), which is distributed in three packages, CDSD-296, CDSD-1000 [15] and CDSD-3000, depending on the reference temperature. It must be noticed that this parameter strongly influences the number of lines that are above a line strength cut-off, viz. the number of spectral lines which have to be taken into account for an accurate estimation of the molecular gas spectra at a given temperature. In this work, the CDSD-1000 spectroscopic database is selected. It contains the data for more than three billion lines for the four most abundant isotopic species of CO2 in the ½26028310 cm1 spectral range. HR ð102 cm1 Þ reference spectra were obtained by the LBL code available at http://cdsd.iao.ru/en/range/specparm/. A line strength cut-off ð1028 cm1 =ðmolecule cm2 ÞÞ was applied. LBL absorption coefficient spectra were computed for pure carbon dioxide at atmospheric pressure and for four temperature values (300, 700, 1100 and 2300 K). In these conditions, Voigt line shapes were assumed and the wings of the spectral profiles were computed up to 50 halfwidths away from the line center. The spectral averaging on wide bands was performed numerically applying the Riemann integration formula. LBL results were used as reference data for the assessment of narrow and wide band blackbody-weighted transmission function curve-of-growth (BTF-COG) calculations. Beforehand, parameters associated with BTF2 and BTF3 expressions were generated, at atmospheric pressure for each temperature and spectral interval investigated in the present work. Parameter k¯ (or k¯ b ) was obtained by forcing ¯ (or k¯ b l) values lower than the k-moment model behavior to follow the LBL data at the optically thin limit viz. for kl 0.0001. The following method was applied: an estimate of the first-order (or Planck function weighted) moment of the absorption coefficient, m~ 1 ðs0 Þ (or m~ 1;b ðs0 Þ), was calculated using LBL absorption coefficient data and through application of Eq. (17) with i ¼ 1. A length l OTL was then defined by m~ 1 ðs0 Þl OTL ¼ 104 (or m~ 1;b ðs0 Þl OTL ¼ 104 ). Parameter k¯ (or k¯ b ) was then obtained from an LBL estimate of the BTF given by Eq. (14) using the gas Dn ¯ k¯ b Þ ¼ f1  t~¯ Dn ½l OTL ; Tðs0 Þg=l OTL . It column length chosen equal to l OTL ðt~¯ b;n ½l OTL ; Tðs0 ÞÞ and by application of kðor b;n ¯ OTL (or k¯ b l OTL ) obtained using this approach was lower than 0.0001 in all cases was checked that values of kl investigated in the present work. Small differences, lower than 0.4%, were observed between k¯ (or k¯ b ) and m~ 1 ðs0 Þ (or m~ 1;b ðs0 Þ): the differences are caused by the application of the discrete sum formula used for the approximation of the spectral integration of the first-order moment and are directly related to the spectral resolution chosen for the LBL calculation. Application of the previous method for determination of k¯ (or k¯ b ) was thus chosen instead of a direct application of the k-moment method at order 1 since it ensures that the model reproduces perfectly the LBL data at the optically thin limit. For similar reasons, although the other parameters ðb¯ b ; Z¯ b Þ are clearly defined in terms of the moments of the absorption coefficient weighted by the Planck function, direct application of the definition was found to be not accurate enough to perform reliable comparisons with LBL data. As a matter of fact, when orders higher than one are involved, it was found that the implementation of the discrete sum integration formulas has to be performed with high care to be robust, to account for the strong dynamics of the absorption coefficient power functions and to provide reliable integral estimates. For this reason, for the sake of simplicity and in the frame of first applications of the k-moment method, it was also chosen to use the LBL spectra for an initial estimation (without using any specific numerical technique) from a direct application of the definition of the statistical moments ðmi;b ðs0 Þ; iX2Þ. Then values of these moments were iteratively adjusted on LBL calculations of COG, similarly to what is done for the usual reckoning of standard SNB parameters [1]. The nonlinear least-square fitting approach used for the calculations is based on the evolutionary programming technique described in Ref. [16]. For the optimization of parameters, lengths of the gas columns were selected so that the Planck-weighted average transmissivities obtained from the LBL approach are equal to 0.02, and from 0.05 to 0.95 with 0.05 steps [10]. Once the first-, second- and third-order moments of the absorption coefficient weighted by the blackbody intensity were evaluated, all parameters associated with BTF2 and BTF3 approximations were definitively generated.

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Various spectral interval widths, from 25 cm1 , corresponding to a typical width in SNB modeling, up to 600 cm1 , more frequently encountered in wide band approaches, were examined. The specific spectral discretization given in Appendix C, already proposed in [8] in the frame of a high-pressure approximate model, was also used. Both BTF2 and BTF3 approximations were assessed.

4.2. Sample calculation results For the comparisons presented hereafter, isothermal homogeneous columns of pure CO2 at atmospheric pressure are considered and temperature used for the blackbody weighting is assumed to be the gas temperature.

Fig. 1. Transmission function curves-of-growth for the 2:7 mm band of pure CO2 at room temperature and atmospheric pressure; the spectral interval is 25 cm1 wide and is centered at 3550 cm1 .

Fig. 2. Transmission function curves-of-growth for the 2:7 mm band of pure CO2 at room temperature and atmospheric pressure; the spectral interval is 50 cm1 wide and is centered at 3550 cm1 .

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Fig. 3. Transmission function curves-of-growth for the 2:7 mm band of pure CO2 at room temperature and atmospheric pressure; the spectral interval is 100 cm1 wide and is centered at 3550 cm1 .

Fig. 4. Transmission function curves-of-growth for the 4:3 mm band of pure CO2 at room temperature and atmospheric pressure; the spectral interval is 100 cm1 wide and is centered at 2300 cm1 .

In Figs. 1–5, five different ways to calculate COG are analyzed. The spectrally averaged transmission function is calculated either without Planck function weighting ð¯tDn n ½l; TÞ: (1) LBL; (2) by applying the standard SNB approximation (Eqs. (25)–(27)) or with blackbody intensity weighting ð¯tDn b;n ½l; TÞ: (3) LBL; (4) using the second order k-moment method (BTF2, Eqs. (25), (28)–(29)); (5) using the third order k-moment method (BTF3, Eqs. (28)–(30), (32)–(35)). Figs. 1–3 depict these transmission functions in the case of the 2:7 mm band at room temperature (300 K) for increasing spectral interval widths (25, 50, and 100 cm1 ) centered at a fixed wavenumber ð3550 cm1 Þ. They reveal that: for narrow bands, the second-order approximation (BTF2), provides slightly better results than the third order one (BTF3); as the width of the spectral interval increases, the BTF3 approach enables to achieve a better accuracy than BTF2. This indicates that for wide band computations, the BTF3 approach should thus be considered as a good alternative to the standard Malkmus SNB formulation.

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Fig. 5. Transmission function curves-of-growth for the 4:3 mm band of pure CO2 at 2300 K and atmospheric pressure; the spectral interval is 100 cm1 wide and is centered at 2300 cm1 .

Fig. 6. Low-resolution ð150 cm1 Þ BTF spectra of pure CO2 at 700 K and atmospheric pressure; column length l ¼ 3:25 cm.

Figs. 4 and 5 represent the COG for atmospheric (300 K) and high-temperature (2300 K) conditions for a 100 cm1 wide spectral interval centered at 2300 cm1 (4:3 mm spectral region). Results obtained by the standard SNB and BTF2 approaches are very close to each other. At 300 K, results given by BTF3 are clearly in better agreement with LBL calculations than those of the BTF2 approximation. However, this advantage totally disappears at high temperature (2300 K). Figs. 6 and 7 depict low ð150 cm1 Þ resolution spectra at 700 K and 1100 K, respectively. In both cases, the column length was arbitrarily chosen ðl ¼ 3:25 cmÞ. The agreement between BTF2, BTF3 and LBL is very good, with maximum absolute errors lower than 0.02–0.03, similarly to what is generally encountered in 25 cm1 narrow band approaches [1].

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Fig. 7. Low-resolution ð150 cm1 Þ BTF spectra of pure CO2 at 1100 K and atmospheric pressure; column length l ¼ 3:25 cm.

Table 1 Mean absolute discrepancy between second and third order approximate model estimations and LBL calculations of the BTF for pure CO2 at atmospheric pressure, a discretization of the spectral interval ½450; 5062:5 cm1  given in Appendix C, 100 column length values between 104 and 105 cm and four gas temperatures Gas temperature (K)

Mean absolute error between LBL and BTF2

Mean absolute error between LBL and BTF3

300 700 1100 2300

3.40 1.96 1.80 2.31

1.92 1.57 1.57 1.56

Fig. 8. BTF-COG for Dn ¼ ½650; 750 cm1  [ ½2150; 2250 cm1  of pure CO2 at room temperature and atmospheric pressure.

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In Table 1, the mean absolute errors between BTF results computed by applying the approximate models and LBL calculations for 20 spectral intervals associated with the spectral discretization given in Appendix C on the ½450; 5062:5 cm1  interval and 100 column length values between 104 and 105 cm are reported for four gas temperatures. These results are representative of radiative transfer situations and illustrate the high accuracy the BTF approach can achieve within this context, since it must be noticed that at the same time it is very likely to reduce drastically computation costs thanks to the wide band spectral integration of the RTE. Finally, Fig. 8 illustrates a situation for which the spectral interval used in the BTF calculation at 300 K is obtained by regrouping two arbitrarily chosen non-contiguous spectral intervals: Dn ¼ ½650; 750 cm1  [ ½2150; 2250 cm1 . According to Section 2, the present approach remains rigorous while using this kind of spectral discretization, which is a priori not the case for standard narrow or wide band modeling. In this situation, it can be noticed that BTF3 approach greatly improves accuracy compared to BTF2 since the absolute difference between LBL and approximate model results does not exceed 0.03 with BTF3 whereas it reaches 0.08 with BTF2.

5. Conclusion In this work, an approximate modeling of narrow and wide band radiative properties of molecular gases has been presented: the k-moment method. It is based on the calculation of the moments of the absorption coefficient to determine the blackbody-weighted transmission function (BTF) of uniform gas columns. In this case, it has been shown that the wide band formulation of the RTE is similar to the one encountered for standard transmissivity-based models. Afterwards, it has been demonstrated that the BTF can be approximated as a solution of a polynomial equation whose coefficients are closely related to the moments of the absorption coefficient distribution function. Analytical solutions have been derived for second- and third-order equations, respectively, called BTF2 and BTF3. For narrow band intervals over which the Planck function is nearly constant, BTF2 has been shown to yield exactly the Malkmus transmission function. The BTF3 approach requires the estimation of three parameters closely related to the first three moments of the absorption coefficient distribution weighted by the Planck function. These parameters have been estimated by non-linear least square adjustments on LBL COG and the theoretical description of the BTFs has been assessed numerically in the case of pure carbon dioxide. When wide ð100 cm1 Þ spectral intervals have been considered in sample calculations, BTF3 has been found to achieve a better accuracy than BTF2 compared to LBL, on a large domain of temperatures for pure CO2 at atmospheric pressure. It has also been shown that the model can be applied with non-contiguous spectral intervals. However, if the various comparisons clearly point out the quality of the mathematical approach, they also suggest optimizations: for the choice of the boundaries of the spectral intervals for a given order of application of the k-moment method; for the development of numerical methods to perform a direct calculation of moments to generate model parameters. Strategies for such optimizations should to be investigated to provide the most reliable approximate model for a given set of application constraints. Future work will also be devoted to applications of the approach to nonhomogeneous non-isothermal columns of mixtures of radiating gases and particles.

Appendix A. Rigorous development of Eqs. (5)–(7) The RHS of Eq. (2) can be rigorously written as Z lim 0

ds !0

s0 ¼s

s0 ¼0

 Z I b;n ðs0 Þxðs0 ÞPðs0 Þkn ðs0 Þds0 exp 

s00 ¼s

 xðs00 ÞPðs00 Þkn ðs00 Þ ds00 ,

s00 ¼s0

~0 such that where s0 is a point in the medium infinitely close to s0 in the direction of radiation propagation O ! ~0 ¼ ds0 O ~0 , lims0 !s0 ds0 ¼ ds0 and I b;n ðs0 Þ ¼ I b;n ðs0 Þ. Mðs0 ÞMðs0 Þ ¼ ðs0  s0 ÞO  

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With these notations, Eq. (4) rigorously yields  Z  Z s0 ¼s 1 qtn ðs0 ; sÞ 0 Dn I b;n ð0Þtn ð0; sÞ þ I b;n ðs0 Þ ds I¯ n ðsÞ ¼ dn Dn Dn qs0 s0 ¼0  Z    Z s0 ¼s 1 qtn ðs0 ; s0 Þ 0 0 I b;n ð0Þtn ð0; sÞ þ I b;n ðs0 Þ t ðs ; sÞds ¼ 0lim 0 dn n s !s Dn Dn qs0 s0 ¼0 Z    

Z s0 ¼s 0 0 1 0 1  tn ðs ; s Þ 0 0  0lim 0 I b;n ð0Þtn ð0; sÞ þ I b;n ðs Þ tn ðs ; sÞds dn s !s Dn Dn ds0 s0 ¼0 8 2 3 9 R s0 ¼s <1 Z = I b;n ð0Þtn ð0; sÞ þ s0 ¼0 ds1 0 I b;n ðs0 Þtn ðs0 ; sÞds0 4 5 dn . ¼ 0lim 0 R 0 s ¼s s !s :Dn Dn ;  0 1 0 I b;n ðs0 Þtn ðs0 ; sÞds0 s ¼0 ds



If we introduce the notations Z 1 Z Dn 0 00 00 0 00 t¯ b;n ½s ; s; Tðs Þ ¼ t¯ Dn I b;n ðs Þ dn n ðs ; sÞI b;n ðs Þ dn Dn

Dn

and 1 Dn I¯ b;n ðsÞ ¼ Dn

Z I b;n ðsÞ dn, Dn

then Dn I¯ n ðsÞ

8 Dn 9 R s0 ¼s 1 Dn 0 Dn 0 ¯ tb;n ½s ; s; Tðs0 Þds0 = < I¯ b;n ð0Þ¯tDn b;n ½0; s; Tð0Þ þ s0 ¼0 ds0 I b;n ðs Þ¯ ¼ 0lim 0 R s0 ¼s Dn s !s : ; 0 0 0  s0 ¼0 ds1 0 I¯ b;n ðs0 Þ¯tDn b;n ½s ; s; Tðs Þds ( ) Z s0 ¼s tDn ½s0 ; s; Tðs0 Þ 0 Dn Dn 0 q¯ b;n Dn ¼ 0lim 0 I¯ b;n ð0Þ¯tb;n ½0; s; Tð0Þ þ ds I¯ b;n ðs Þ s !s qs0 s0 ¼0 Z s0 ¼s q¯tDn ½s0 ; s; Tðs0 Þ 0 Dn Dn ¼ I¯ b;n ð0Þ¯tDn ½0; s; Tð0Þ þ ds . I¯ b;n ðs0 Þ b;n b;n qs0 s0 ¼0

These last equations are formally equivalent to Eqs. (5)–(7). 0 Appendix B. Calculation of the first series coefficients of X½l; Tðs0 Þ ¼  lnf¯sDm b;m ½l; Tðs Þg 0 Mathematically, it can be shown that the nth derivative of O½l; Tðs0 Þ ¼  lnf¯tDn b;n ½l; Tðs Þg can be expressed as Dn a function of the n first derivatives of the natural logarithm function and those of t¯ b;n ½l; Tðs0 Þ: 0 d¯tDn dO½l; Tðs0 Þ 1 b;n ½l; Tðs Þ 40, ¼  Dn dl dl t¯ b;n ½l; Tðs0 Þ ( )2 0 0 d2 t¯ Dn d¯tDn d2 O½l; Tðs0 Þ 1 1 b;n ½l; Tðs Þ b;n ½l; Tðs Þ ¼  Dn þ Dn , dl t¯ b;n ½l; Tðs0 Þ t¯ b;n ½l; Tðs0 Þ dl 2 dl 2 ( )2 0 2 Dn 0 d3 t¯ Dn d¯tDn ¯ b;n ½l; Tðs0 Þ d3 O½l; Tðs0 Þ 1 1 b;n ½l; Tðs Þ b;n ½l; Tð sÞ d t ¼  Dn þ 3 3 3 0 dl t¯ b;n ½l; Tðs0 Þ t¯ Dn dl dl 2 dl b;n ½l; Tðs Þ ( )3 0 d¯tDn 1 b;n ½l; Tðs Þ  2 Dn . dl t¯ b;n ½l; Tðs0 Þ

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In the neighborhood of a point of radiation emission ðl ! 0Þ, the series coefficients of 0 0 O½l; Tðs0 Þ ¼  lnf¯tDn ¯ Dn b;n ½l; Tðs Þg can be written as functions of those of t b;n ½l; Tðs Þ, viz. the blackbody-weighted moments of the absorption coefficient:  0 d¯tDn m1;b ðs0 Þ dO½l; Tðs0 Þ 1 b;n ½l ! 0; Tðs Þ ¼ ¼ m1;b ðs0 Þ, ¼ o1;b ðs0 Þ ¼  Dn 0Þ 0 Þ m dl dl ðs t ½l ! 0; Tðs ¯ 0 0;b s!s b;n ( )2  2 Dn 0 0 2 d t¯ b;n ½l ! 0; Tðs Þ d¯tDn d O½l; Tðs0 Þ 1 1 b;n ½l ! 0; Tðs Þ ¼  Dn þ Dn dl t¯ b;n ½l ! 0; Tðs0 Þ t¯ b;n ½l ! 0; Tðs0 Þ dl 2 dl 2 s!s0   m2;b ðs0 Þ m1;b ðs0 Þ 2 þ ¼ 2!o2;b ðs0 Þ ¼  ¼ m21;b ðs0 Þ  m2;b ðs0 Þ, 0 m0;b ðs Þ m0;b ðs0 Þ d3 O½l; Tðs0 Þ dl 3



0 d3 t¯ Dn 1 b;n ½l ! 0; Tðs Þ 0 t¯ Dn dl 3 b;n ½l ! 0; Tðs Þ ( )2 2 Dn 0 d¯tDn ¯ b;n ½l ! 0; Tðs0 Þ 1 b;n ½l ! 0; Tðs Þ d t þ 3 Dn dl t¯ b;n ½l ! 0; Tðs0 Þ dl 2 ( )3 0 d¯tDn 1 b;n ½l ! 0; Tðs Þ  2 Dn dl t¯ b;n ½l ! 0; Tðs0 Þ

¼  s!s0

¼ 3!o3;b ðs0 Þ ¼ m3;b ðs0 Þ  3m1;b ðs0 Þm2;b ðs0 Þ þ 2m31;b ðs0 Þ. Appendix C. Spectral discretization of the ½450; 5062:5 cm1  interval extracted from Ref. [8]

Band number

Band center ðcm1 Þ

Band width ðcm1 Þ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

562.5 712.5 887.5 1087.5 1312.5 1562.5 1812.5 2000.0 2112.5 2212.5 2337.5 2612.5 3012.5 3312.5 3487.5 3662.5 3862.5 4062.5 4462.5 4912.5

150 150 200 200 250 250 250 125 100 100 150 400 400 200 150 200 200 200 600 300

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References [1] Taine J, Soufiani A. Gas IR radiative properties: from spectroscopic data to approximate models. Adv Heat Transfer 1999;33:295–414. [2] Edwards DK, Menard WA. Comparison of models for correlation of total band absorption. Appl Opt 1964;3:621–5. [3] Pierrot L, Rivie`re P, Soufiani A, Taine J. A fictitious-gas-based absorption distribution function global model for radiative transfer in hot gases. JQSRT 1999;62:609–24. [4] Denison MK, Webb BW. The spectral line weighted-sum-of-gray-gases model—a review. In: Mengu¨c- MP, editor. Radiative transfer—I, Proceedings of the first international symposium on radiative transfer. New York, Boca Raton, FL: Begell House Publishers; 1996. p. 193–208. [5] Solovjov VP, Webb BW. SLW modeling of radiative transfer in multicomponent gas mixtures. JQSRT 2000;65:655–72. [6] Modest MF, Zhang H. The full-spectrum correlated-k distribution function for thermal radiation for molecular gas–particles mixtures. J Heat Transfer 2002;124:30–8. [7] Goutiere V, Liu F, Charette A. An assessment of real-gas modelling in 2D enclosures. JQSRT 2000;64:299–326. [8] Pierrot L. De´veloppement, e´tude critique et validation de mode`les de proprie´te´s radiatives infrarouges de CO2 et H2 O a` haute tempe´rature. Application au calcul des transferts dans des chambres ae´ronautiques et a` la te´le´de´tection. PhD thesis. Ecole Centrale de Paris. Paris. France; 1997 (in french). [9] Hartmann JM, Levi di Leon R, Taine J. Line-by-line and narrow-band statistical model calculations for H2 O. JQSRT 1984;32:119–27. [10] Soufiani A, Andre´ F, Taine J. A fictitious-gas based statistical narrow-band model for IR long-range sensing of H2 O at high temperature. JQSRT 2002;73:339–47. [11] Modest MF. Radiative heat transfer, 2nd ed. New York: Academic Press; 2003 (ISBN 0-12-503163-7). [12] Abramowitz M, Stegun IA. Handbook of mathematical functions. New York: Dover; 1965. [13] Andre´ F, Vaillon R. A simple and accurate method to derive SNB parameters for the radiative properties of gases. In: Lemonnier D, Selc- uk N, Lyabaert P, editors. Proceedings of Eurotherm78-computational thermal radiation in participating media II, Poitiers. Lavoisier, Paris; 5–7 April, 2006. p. 167–76. [14] Rothman LS, Jacquemart D, Barbe A, Benner DC, Birk M, Brown LR, et al. The HITRAN 2004 molecular spectroscopic database. JQSRT 2005;96:139–204. [15] Tashkun SA, Perevalov VI, Teffo J-L, Bykov AD, Lavrentieva NN. CDSD-1000, the high-temperature carbon dioxide spectroscopic databank. JQSRT 2003;82:165–96. [16] Andre´ F, Vaillon R, Ayranci I, Escudie´ D, Selc- uk N. Radiative transfer diagnostic technique of sooting flames from emission spectroscopy. In: Mengu¨c MP, Selc- uk N, editors. Proceedings of the fourth international symposium on radiative transfer. New York: Begell House; 2004. p. 173–80.