Comparison of databases for radiative heat transfer

calculations in combustion applications with the

NBKMcK model

POITOU Damien (poitou@cerfacs.fr)a, ANDRE Frederic(frederic.andre@insa-lyon.fr)b

aCERFACS, 42, Avenue Gaspard Coriolis, 31057 Toulouse Cedex 01, France.bCentre de Thermique de Lyon (CETHIL, CNRS-INSA Lyon-UCBL), Bat. Sadi Carnot,

INSA-Lyon F-69621, France.

Abstract

The NBKMcK (Narrow Band k-moment, i.e. NBKM, with the correlated-k

assumption) model is used to solve the radiative transfer equation (RTE) with

a finite volume method along with the Discrete Ordinate Method (DOM) for

the angular discretization. The parameters of the NBKMcK model can be

derived from high resolution spectra and under several approaches to estimate

them. The influence of the NBKM parameters has been investigated using

three different approaches: (A1) numerically optimized, (A2) optically thin

limit, (A3) optically thick limit. The influence of Line-by-Line (LBL) input

data is also studied by comparing models based on CDSD-1000/HITEMP95

and HITEMP 2010 for each approach. Six sets of band parameters were

tested for several test cases representative of combustion applications in one,

two and three dimensional geometries. Results indicate that the impact of

the different narrow band parameters is very limited. Spectroscopic data used

to build LBL parameters have a more important impact than the technique

Preprint submitted to International Journal of Thermal Sciences September 3, 2012

used to build approximate parameters. Results were also compared to the

Statistical Narrow Band (SNB) database from Soufiani and Taine and were

found to be in satisfactory agreement.

Keywords: Non-gray gas radiation, LBL, NBKM, SNB, correlated-k,

combustion

2

Nomenclature

L0ν Planck function (W.m−2.sr−1.cm−1)

Lν radiation intensity (W.m−2.sr−1.cm−1)

f probability density function

g cumulative density function

h correlation function

k random variable of the absorption coefficient

F any function of the direction u or of the frequency ν

kP mean Planck absorption coefficient

kR mean Rosseland absorption coefficient

x molar fraction of the absorbing gas

P pressure

l length of the gas column

xP l optical path lenght in the gas

∇[.] gradient operator

L−1[.] inverse Laplace transform operator

Greek symbols

3

β mean line-width to spacing ratio

εν wall emissivity

κν absorption coefficient (m−1)

Ω direction of propagation

∆ν wavenumber interval (cm−1)

ν wavenumber (cm−1)

φ composition variable

ρν wall reflectivity

τν spectral transmittance

ωi weights for the angular integration

ωj weight for the spectral integration

µ2 second order k-moment

Subscripts

4

i angular quadrature index

j spectral quadrature index

n mixture gas index

P Planck

R Rosseland

w wall

dir discrete direction or ordinate

gas gas of the mixture

mix gas mixture

quad spectral quadrature point

Notations

COG Curve-of-Growth

DOM Discrete Ordinate Method

DMFS Diamond Mean Flux Scheme

FVM Finite Volume Method

IG Inverse Gaussian

LBL Line by Line

NBKM Narrow Band k-Moment

NBKMcK NBKM with correlated-k model

RTE Radiative Transfer Equation

SNB Statistical Narrow Band

SNBcK SNB with correlated-k model

5

1. Introduction

Radiative heat transfer plays a key role in combustion applications due

to its impact on the energy and temperature distributions in flames. Those

profiles are known to have a strong influence on the production of several

pollutant species as well as on the lifespan of combustion chambers. In such

applications, radiative energy exchanges are mainly due to the burnt gases

that contain several species which emit and absorb radiation in the infrared

spectrum such as H2O, CO2 and CO as well as carbonaceous solid particles

such as soot. For combustors such as engines or turbines, radiative heat losses

represent only a few percent of the total heat released by combustion mech-

anisms. However, several recent works [1, 2, 3] have shown that including

radiation when simulating the energy balance inside such media may change

flame temperature peaks from 50 to 150 K. This difference, although it can

be acceptable for some applications, is too large to ensure precise prediction

of polluting species such as soot or NOx [4]. In bigger combustion devices

such as furnaces, infrared radiation may be the most important heat transfer

process and can not be neglected.

Estimating radiative transfers in combustion media requires dedicated

techniques for the modelling of their radiative properties. However, the Line-

By-Line (LBL) model, which is the most accurate and reliable one, can not

be envisaged for such applications due to its excessive computation cost.

Indeed, in this approach, contributions of all spectral lines of the gas are

taken into account. This requires, at high temperature, to estimate individual

6

contributions of a large number of lines (several millions at 2000 K) with a

high resolution (which means for about 106−7 spectral locations to cover the

infrared spectrum). This approach is usually limited to simple 0D or 1D

problems. Approximate models, based on spectral averaging techniques, are

preferred in more complex geometries or for coupled problems. They are

described, for instance, in Refs [5, 6].

The NBKMcK (Narrow Band k-moment with the correlated-k assump-

tion) model is founded on the k-moment method formulated recently on a

Maximum Entropy basis by Andre and Vaillon in [7]. This approach was

proposed initially to provide accurate approximate models for the band av-

eraged radiative properties of gases over any band width, from a few cm−1

(narrow band modeling) up to the full spectrum. In its original form [8], the

model is restricted to applications in uniform (homogeneous and isothermal)

media and its use in nonuniform ones requires dedicated numerical treat-

ments [9, 10]. NBKMcK model is compatible with any form of the radiative

transfer equation (RTE) but it is written here in a k-distribution form to

perform radiative calculations with the differential form of the RTE. In the

present work, this equation is solved by a finite volume method (FVM) along

with the Discrete Ordinate Method for the angular description in the three

dimensional calculations.

The NBKMcK model relies on Narrow Band parameters that can be de-

rived from high resolution spectra. Several spectral averaging approaches

can be used to estimate them. One of the purposes of the present work is

7

to quantify the impact of those various techniques on the radiative calcula-

tion in a set of test cases representative of actual combustion applications.

The influence of the LBL input data is also studied by comparing results

from models based on CDSD-1000 [11] and HITEMP (1995 and 2010 [12])

reference calculations as well as the SNB database provided by Soufiani and

Taine [13].

The present paper is divided into four main parts that are briefly de-

scribed hereafter. In section 2, a short description of radiative heat trans-

fer modelling in multi-dimensional geometries is presented. The radiative

transfer equation is given in both its differential and integral formulations.

Details about the Discrete Ordinate Method are also provided. The spec-

tral modelling based on the NBKM model to estimate the k-distributions

together with the correlated-k approximation are finally discussed. After-

wards, in section 3, several test cases are described to study the impact of

the different LBL reference data, as well as the technique used to evalu-

ate approximate model parameters from them, on the radiation calculation.

This section is divided into two main parts. In the first one, some refer-

ence tests are given to evaluate the quality of the k-moment modelling for

use in simple zero-dimensional (along line-of-sight) calculations. In the sec-

ond one, three situations representative of radiative heat transfer problems

in one- up to three-dimensional geometries are treated. The first case was

proposed recently by Liu et al. in [14, 15] as a benchmark case for combus-

tion application in a one-dimensional geometry. The second one involves a

8

two-dimensional axisymmetric cylindrical geometry and is similar to the one

proposed by Coelho et al. in [16, 17]. The third and last case is representa-

tive of radiative heat transfer in a furnace with a three-dimensional geometry.

Input for the calculation were taken from [18]. Results of the simulations are

then discussed in terms of input spectroscopic databases as well as model

parameter estimation techniques.

2. Radiative heat transfer modelling

2.1. Radiative Transfer Equation (RTE)

The Radiative Transfer Equation can be written in two distinct equivalent

forms: a differential (local, along a ray) or an integral one (formal solution

of the RTE along a radiation path). They are given in the following sections.

More details about them can be found for instance in [5].

2.1.1. Differential formulation

The differential form of the RTE in the direction of propagation Ω, for a

non scattering medium at local thermodynamic equilibrium is given as:

Ω · ∇Lν(x,u) = κν[L0ν(x)− Lν(x,u)

](1)

with the associated boundary conditions:

Lν(xw,u) = εν(xw)L0ν(xw)︸ ︷︷ ︸

Emitted part

+ ρν(xw)Lν,incident(xw,u)︸ ︷︷ ︸Reflected part

(2)

9

where ν is the wavenumber, Lν(x,u) is the radiation intensity at the point x

in the direction u and κν = κν(x) is the absorption coefficient at location x,

εν(xw) is the wall emissivity and ρν(xw) the wall reflectivity with ρν(xw) =

1− εν(xw). L0ν is the equilibrium Planck function that only depends on the

local temperature. Usually, the differential form of the RTE is solved with

the Discrete Ordinate Method, described briefly later in the paper.

2.1.2. Integral formulation

An analytical formal solution of the RTE can be obtained by an integra-

tion of Eq. (1) between points at abscissa x0 and x along the radiation path.

Incident radiative intensity at abscissa x0 is Lν(x0,u), as shown schemati-

cally on Fig. 1 in which cut down notations have been used for simplicity

(Lν(x,u) is noted L(x) on this figure):

Lν(x,u) = Lν(x0,u)τν(x0,x) +

∫ x

x0

L0ν(x′)∂τν(x

′,x)

∂x′dx′ (3)

where τν(x′,x) is the spectral transmission function of the absorbing medium

between x and x′ given by:

τν(x′,x) = exp

[−∫ x

x′κν(x”)dx”

](4)

Notations are the same as those used in the previous section. This formula-

tion of the RTE is generally solved with the Ray Tracing technique.

10

2.1.3. The Discrete Ordinate Method (DOM)

The discrete ordinate method was originally proposed by Chandrasekhar

[19] for astrophysical applications. In this approach, the differential form of

the RTE is solved on a discrete set of directions of the solid angle, or Dis-

crete Ordinates, and the directional integration is performed by the following

approximate formula:

∫4π

F (u)dΩ 'Ndir∑i=1

wai F (ui) (5)

where ui are the discrete ordinates and ωai their associated weights in the

angular quadrature.

Here, the RTE integration (spectral, spatial and directional) is performed

by the CERFACS’ code PRISSMA1 which is described in [1, 2, 3]. The spatial

discretization scheme, which uses unstructured meshes, is based on the Finite

Volume Method (FVM) [20, 21, 22] and the Diamond Mean Flux Scheme

(DMFS) is used for the spatial integration [23]. The angular quadrature is

based on a set ofNdir directions (ordinates) obtained by using a Sn quadrature

(with Ndir = n(n+ 2)) [24] or a LC11 quadrature (where Ndir = 96) [25].

1PRISSMA: Parallel RadIation Solver with Spectral integration on MulticomponentmediA, http://www.cerfacs.fr/prissma

11

2.2. The NBKMcK (Narrow band k-moments with the correlated-k assump-

tion) model

2.2.1. k-distribution and NBKM models

Among the usual models used to estimate the radiative properties of

gases, those based on the so-called ”k-distribution” approach have encoun-

tered a great success during the past decades both in the atmospheric and

combustion communities. Indeed, one of its strongest advantages, when com-

pared with transmission function based models, is that this approach is com-

patible with any technique to solve the Radiative Transfer Equation, both

in differential (Eq. (1)) and integral (Eq. (3)) forms. k-distribution models

are based on the following mathematical result: for a given function F of the

spectral absorption coefficient κν , the spectral mean value of F (κν) over any

band ∆ν can be expressed as:

F∆ν =1

∆ν

∫∆ν

F (κν)dν =

∫ +∞

0

f(k)F (k)dk (6)

where f is the k-distribution defined as:

f(k) =

∫∆ν(k)

dν

∆ν=

∆ν(k)

∆ν(7)

in which ∆ν(k) = ν ∈ ∆ν such that κν = k. In other words, f(k) repre-

sents the fraction of wavenumbers inside ∆ν such that the absorption coef-

ficient of the gas is k. Function f can be obtained directly from LBL data

and by a direct application of its definition (Eq. 7). It can also be derived

12

through the knowledge of the transmission function of the gas as a func-

tion of the optical path length in the gas, xP l (such a function is called the

Curve-of-Growth, COG). In fact, using Eq. (6) with F (k) = exp(−xP lk),

this quantity is given as:

τ∆ν(xP l) =1

∆ν

∫∆ν

exp(−xP lκν)dν =

∫ +∞

0

exp(−xP lk)f(k)dk (8)

that shows that f can be calculated as the inverse Laplace transform of

τ∆ν(xP l). This can be formulated mathematically as:

f(k) = L−1 [τ∆ν(xP l)] (9)

Despite the wide number of numerical techniques available to estimate inverse

Laplace transforms [26], the previous equation Eq. (9) together with band

averaged LBL COG has not, to our knowledge, been used in the literature.

Recently, Andre and Vaillon [7] have proposed a technique to estimate this

inverse Laplace transform that uses only partial informations (k-moments)

about the spectra as inputs and was called the k-moment method. Infor-

mations considered in this reference were the Planck kP (first positive k-

moment) and Rosseland kR (first negative k-moment) mean values of k de-

fined as:

13

kP =1

∆ν

∫∆ν

κνdν =

∫ +∞

0

kf(k)dk (10)

1

kR=

1

∆ν

∫∆ν

κ−1ν dν =

∫ +∞

0

k−1f(k)dk (11)

Those quantities can be evaluated from LBL data by applying their re-

spective definitions, Eqs. (10, 11). Assuming that over ∆ν the absorption

coefficient is never null (which means that no transparency region of the

gas spectrum is located inside ∆ν, which is also required for the use of

Eq. (11), the Maximum Entropy estimation technique of statistical densities

[27] enables to build an accurate k-distribution for use in radiative trans-

fer problems. In this case, it was shown [7] that the k-distribution can be

approximated accurately by an inverse Gaussian (IG) function:

f(k) =1

2πk

√2β k

kexp

[β

2π

(2− k

k− k

k

)](12)

withβ

π=

(kP

kR− 1

)−1

(13)

and k = kP (14)

This approach is a generalization of the k-moment method originally pro-

posed in [8], for which similar results were obtained by considering the

asymptotic behaviour of the transmission function. The same formula as

14

Eq. (12) was obtained by Domoto [28] by calculating analytically the inverse

Laplace transform of the transmission function provided by the Statistical

Narrow Band (SNB) model for Lorentz line with the Malkmus distribution of

linestrengths. This transmission function, that can be obtained by combining

Eqs. (12) and (8), is given as:

τ∆ν(xP l) = exp

−βπ

√1 +2πxP lk

β− 1

(15)

The previous k-distribution depends on two variables. The first one, k,

is usually derived directly from LBL data at the optically thin limit and can

be identified with kP . The second one, β, can be obtained by several tech-

niques. The simplest one (A1) is historically the oldest, as it was originally

developed to identify parameters for SNB models and used widely during the

past decades (see for instance [6]). It consists in optimizing numerically β by

a non linear least square technique to minimize the sum of square differences

between Curves-Of-Growths (τ∆ν = τ∆ν(xP l) ) calculated LBL and by the

approximate formula, Eq. (15). It is important to notice that, following [10],

this approach provides exactly the same database as the one that would be

obtained by using an optimization technique to adjust SNB model param-

eters (with the Malkmus distribution for Lorentz lines) to band averaged

LBL COG. The second one (A2) uses the Planck mean absorption coefficient

15

(Eq. (10)) as well as the second order k-moment given by:

µ2 =1

∆ν

∫∆ν

(κν)2dν (16)

Those two moments are the most important ones to describe the radiative

properties of the gas at the optically thin limit, as they are the only infor-

mations required to estimate the COG in this region. This approach was the

one proposed initially to apply the k-moment method [8]. In this case, the

parameter β required in Eq. (15) can be calculated as [7] :

β =π(kP )2

µ2 − (kP )2=

πk

µ2 − k(17)

The third technique (A3) uses a similar approach as A2 but instead of taking

into consideration the optically thin limit, the optically thick one is consid-

ered. In this case, the Rosseland mean absorption coefficient of Eq. (11) is

used and parameter β is estimated from Eq. (13). It should be noticed that

if the actual k-distribution was inverse Gaussian, then the following relation

should hold (it is known as inverse Gaussian symmetry [29]):

kP

kR=

∫ +∞0

k2f(k)dk(kP)2 (18)

In practice, approaches (A2) and (A3) provide results that do not match

exactly Eq. (18) which indicates that the k-distributions are not, in general,

rigorously inverse Gaussian. Accordingly, optimized parameters (A1) should

16

a priori partially correct some sources of errors due to the use of databases

obtained from approaches A2-3 and thus provide more accurate results in

a general frame (viz. in non optically thin nor thick media) for radiative

transfer applications. Moreover, as all models use the same mathematical

formulas (Eqs. (12, 15)) to compute k-distributions or transmission func-

tions, no difference in terms of computational time is observed between the

different approaches (only input data differ, as they are based on distinct

treatments of the same reference LBL data, as detailed in the text).

2.2.2. Correlated k-distribution models

k-distribution models have encountered a considerable interest from the

radiative transfer community during the past decades, both over narrow

bands [6] as well as over the full spectrum [30, 31, 32]. Those models are

known to be very accurate in uniform media but their use in nonuniform

(heterogeneous and/or anisothermal) ones requires additional assumptions.

The most common one assumes that spectra in distinct thermophysical con-

ditions are correlated. It can be explicited as follows.

A thermophysical condition is defined by its composition variable φ, that

is a vector containing all the data required to calculate the absorption co-

efficients: temperature, pressure and species concentrations. If two distinct

thermophysical conditions, represented by their respective composition vari-

ables φ1

and φ2, are considered, the correlation assumption can be mathe-

matically formulated as:

17

κν(φ1) = h

[κν(φ2

)]

(19)

where κν(φ1) and κν(φ2

) are the spectral absorption coefficients associated

with each composition variables. h is the correlating function that depends

both on φ1

and φ2

(and k).

If spectra are correlated, then it can be shown [5] that:

g[φ

1, κν(φ1

)]

= g[φ

2, κν(φ2

)]

(20)

where g is the cumulated distribution function of k defined as:

g(φ, k)

=

∫ k

0

f(k′)dk′ (21)

This function can be derived from function f , given by Eq. (12).

The use of the correlation assumption is appropriate as soon as the

medium remains the seat of small gradients of temperature and species con-

centrations. In some cases, such as for infrared plume signature studies, this

assumption fails to provide accurate results and more sophisticated models

are required [33, 34, 35]. Nevertheless, the uniform medium assumption is

acceptable in many combustion problems. Indeed, due to the flame structure

and to the fact that the absorbing species concentrations are strongly cor-

related with the temperature profiles which are mainly uniform outside the

flame front, the homogeneous assumption is quite reasonable in this case.

18

2.2.3. NBKMcK modelling

NBKMcK modelling is a modified form of the SNBcK model as described

in [36]. It combines two complementary elements: 1/ the NBKM approach

is used to provide the inverse Gaussian k-distribution required to estimate

the radiative properties of the gas in any thermophysical condition (sec-

tion 2.2.1), 2/ gas spectra are assumed correlated (section 2.2.2). The asso-

ciated radiative transfer equation that has to be solved (in differential form,

for an emitting-absorbing but non scattering medium) is:

Ω · Lj(x,u) = g−1[φ (x) , gj

]×[L0

∆ν(x)− Lj(x,u)]

(22)

L∆ν(x,u) =

Nquad∑j=1

ωjLj(x,u) (23)

were wj and gj, j = 1, N , are respectively the weights and abscissa of

any numerical quadrature chosen to integrate spectrally the RTE over the

narrow band. g is the cumulative k-distribution of the mixture of gases,

whose parameters can be obtained from those of the single gases as described

in the next section. In PRISSMA code, this function is inverted by a Newton

method.

2.2.4. Radiative properties of mixtures

In combustion products, the gas is composed by a mixture of several

gaseous species for which spectra can be assumed uncorrelated, which is a

19

usual approach to treat mixtures. In this case, the Planck mean absorption

coefficient of the mixture is simply the sum of the Planck mean absorption

coefficients of all the radiating gases. Similarly, also considering the optically

thin limit, it was shown in [37] that the β parameter of the mixture can be

obtained from those of the single gases as:

kmix =

Ngas∑n=1

kn

k2

mix

βmix

=

Ngas∑n=1

k2

n

βn

(24)

Although several other techniques can be used to estimate mixture pa-

rameters, the previous equation was used in the present work due to its

simplicity.

It can be noticed that if k-distributions of all gases in the mixture are

rigorously IG, then the inverse Gaussian symmetry property holds and the

following equation can be obtained from the previous one in terms of Rosse-

land means (assuming that the k-distribution of the mixture is also IG):

(kmix

)3

kR,mix

=

(kP,mix

)3

kR,mix

=

Ngas∑n=1

(kP,n

)3

kR,n(25)

2.2.5. NBKM model database

NBKM databases are obtained directly from LBL data, whose calculation

was described in [10], in the case of CDSD-1000 [11] for CO2 and HITEMP

1995 [38] for H2O. The same codes as those described in this reference were

20

adapted to be used with HITEMP 2010 [12] database for CO2 and H2O.

The structure of the NBKM databases match exactly those of the SNB data

provided by EM2C laboratory, as also described in the same reference. Thus,

replacing one set of model parameters with another only requires changing its

name while calling the associated file (for example SNBCO2 by NBKMCO2).

Potential users of the NBKM databases should thus refer to the following

reference, [13], associated with the use of the SNB database for more details.

Narrow bands are thus 25 cm−1 wide and parameters are given between 150

cm−1 and 9300 cm−1. This corresponds to 367 narrow bands for H2O and,

due to some transparent spectral regions, to 96 for CO2. No other radiating

species are considered in the present work. For each radiating gas, three

model databases have been generated, based on the use of approaches (A1-

3) described earlier in the paper. Databases based on approach A1 have been

noted A1-1995 or A1-2010 for CDSD-1000/HITEMP 1995 or HITEMP 2010

respectively. Model databases based on approaches A2 and A3 use similar

notation but A1 is replaced by A2 and A3.

3. Application test cases and results

3.1. Preliminary test case: 0D

The zero-dimensional calculations have been resolved using the integral

formulation of the RTE presented in section 2.1.2. Figures 2a and 2b dis-

play transmission function curves of growth calculated using Eq. 15 and

the parameters calculated via approaches A1-3. LBL data are based on

21

HITEMP2010. Gas is pure carbon dioxide at atmospheric pressure. Fig-

ure 2a is for 300 K and 2b is for 2300 K. The narrow band is located

around 625 cm−1. Those figures clearly demonstrate the accuracy that can be

achieved with the k-moment method, whatever optical limit, thin or thick,

is chosen. Results are similar to those obtained by adjustments of model

parameters and errors do not exceed a few percents. Nevertheless, the ad-

justment technique, which partially corrects some limits of the model through

the optimization process, provides the best results over the full range of col-

umn lengths which justifies its use as reference model for multi-dimensional

model comparisons. Preceding results are also illustrated on Figs. 3a and 3b

which where obtained in the same conditions than previously but represent

the spectrum between 350 cm−1 and 4150 cm−1. Again, pure carbon dioxide

is considered and the length of the gas path is 1 cm. Approach A1 (associ-

ated with the narrow band database A1-2010) provides the best results when

assessed against LBL reference data. Nevertheless, methods A2 (A2-2010)

and A3 (A3-2010) both provide very accurate results over a wide range of

wavenumbers and transmission function values.

3.2. Radiative Transfer cases (RTC)

3.2.1. RTC1: one-dimensional planar enclosure (Liu et al.)

This test case was proposed recently by Liu and co-workers [14, 15] as a

benchmark. It is representative of an air combustion situation in a one-

dimensional planar enclosure at atmospheric pressure. This geometry is

22

treated with a three-dimensional mesh (see Fig. 4a). The domain size in

X, Y, Z directions is respectively 0.5, 1.0 and 1.0 m and is approximated

by an unstructured mesh composed of 198k2 tetrahedral cells. Boundary

conditions in x = 0 and x = 0.5 are defined as cold black walls, ε = 1 at

temperature Tw = 300K. In the Y and Z directions, surfaces are defined as

purely reflecting ones, ε = 0, to simulate an infinite symmetry in those direc-

tions. The temperature and the molar fractions of H2O and CO2 along the

x-direction are provided on Fig. 4b. The corresponding results are plotted

on Figs. 5 to 6.

3.3. RTC2: 2D axisymmetric cylinder (Coelho et al.)

This test case was proposed by Coelho and co-workers [16, 17] as a

benchmark to investigate the influence of non homogeneous temperature

and species concentration profiles of H2O and CO2 on radiative transfer in

a two dimensional axisymmetric enclosure. The geometry is a cylinder with

a length L = 1.2 m and a radius R = 0.3 m. This geometry is approxi-

mated by an unstructured mesh with 161k tetrahedral cells. Boundaries are

black surfaces ε = 1 at temperature Tw = 800K except at x = 1.2 m where

Tw = 300K, as shown on Fig.7a. The temperature and concentration fields

2k=1000

23

are given by analytical functions:

T (x, r) = 800 + 1200(1− r/R)(x/L) (26)

XH2O(x, r) = 0.05[1− 2(x/L− 0.5)2

](2− r/R) (27)

XCO2(x, r) = 0.04[1− 3(x/L− 0.5)2

](2.5− r/R) (28)

Those profiles are schematized on Fig.7b. In the present work, there is no

soot, as in the original case. This choice is justified by the fact that our

aim is mainly to compare several techniques for the modelling of the spectral

properties of gases alone. Including soot would alter our conclusions.

Four test cases are derived from those geometry and profiles. Case RTC2a

is at atmospheric pressure. Nevertheless, aeronautical burners usually oper-

ate under high pressure. For this reason, two test cases representative of

such thermophysical conditions have be chosen: cases RTC2b and RTC2c

are respectively at a pressure of 10 atm and 40 atm. Other parameters are

the same as RTC2a. Finally, case RTC2d is at atmospheric pressure but

with a scaling (multiplication by a factor 10) of the domain size: the length

is L = 12 m and the radius R = 3 m. The corresponding results are plotted

on Figs. 8 to 15.

3.4. RTC3: 2D axisymmetric furnace (Pedot et al.)

This test case is based on the recent work of Pedot et al. [18]. It is repre-

sentative of a combustion situation in a furnace at atmospheric pressure. The

geometry is a two dimensional axisymmetric cylinder with a length L = 15 m

24

and a radius R = 3 m, as shown on Fig.16a. This geometry is approximated

by a mesh with 350k tetrahedral cells. The boundary conditions are repre-

sented as black walls ε = 1 at temperature Tw = 1000 K.

The initial solution for the combustion was obtained with a simple model

of steady diffusion jet flame, i.e. a fuel jet located at the centre of the cylin-

drical enclosure surrounded by a co-flowing air stream (see [18] for details).

Turbulence mainly impacts the flame length, which is used as a parameter

in the diffusion flame model. Temperature and species concentration profiles

are assumed two-dimensional axisymmetric, where a fuel jet (fuel mass frac-

tion Y 0F , density ρ0

F and inlet velocity u0f ) is injected parallel to the x axis

with an oxidizer co-flow (fuel mass fraction Y 0O, density ρ0

0 and inlet veloc-

ity u0O). The physical solution for the temperature and molar fractions of

H2O, CO2 and CO (not used here) are given on Fig. 16b. The corresponding

results are plotted on Figs. 17 to 18.

3.4.1. Results and discussion

The impact of NBKM parameters has been investigated in 3D cases (cases

RTC1 to RTC3) using approaches A1 to A3 to built the databases from

LBL data calculated with CDSD-1000/HITEMP 1995 (notations for those

databases are the same as those for HITEMP 2010 but ”2010” is replaced

by”1995”) or HITEMP 2010. Results were also compared to radiative trans-

fer simulations with the SNB parameters from Soufiani et al. [13] which are

frequently used as reference for such calculations. Corresponding results

25

were noted ”EM2C”. For model comparison, A1-2010 model was chosen as

the reference model as already discussed in section 3.1.

Radiative source terms are plotted along the central axis in cases RTC1,

RTC2a to RTC2d ( Figs. 5a, 8a, 10a, 12a, 14a) and along a vertical axis for

z = 4 m in case RTC3 (Fig. 17a). The relative error for the various narrow

band databases were calculated along the profiles as:

ε(x) =Sr,model(x)− Sr,A1-2010(x)

Sr,A1-2010(x)(29)

The profiles of relative errors are plotted for cases RTC1, RTC2a to 2d and

RTC3 on Figs. 5b, 8b, 10b, 12b, 14b and 17b respectively.

Profiles of radiative source terms give a local information on differences

between the models. As a complementary indicator of the quality of the

various model databases when compared to the reference one, total radiative

heat losses are considered. It is calculated by integrating local radiative

power values so as to provide a global information on heat transfers between

the gaseous medium and its neighbourhood. Total radiative heat losses are

reported on Table 1.

In each case the mean relative error along the profile is estimated as:

εProfile =

∫Profile

ε(x)dx (30)

26

and similarly the relative error on total radiative heat losses is computed as:

εTotal =Sr,Tot. model(x)− Sr,Tot. A1-2010(x)

Sr,Tot. A1-2010(x)(31)

where Sr,Tot. model =∫

VolumeSr(x)dx.

As the radiative source terms are very close to each others in most cases,

a detailed view is given close to the maximal value of Sr on Figs. 6, 9, 11,

13, 15 and 18.

First of all, it can be noticed that the impact of the different narrow band

model parameters is weak both on local profiles and on total radiative heat

losses. Indeed, in all cases, relative errors show that the use of any database

provide comparable trends although results are different. Moreover, they are

more sensitive to the spectroscopic database (CDSD-1000/HITEMP 1995

or HITEMP 2010) used to calculate the reference LBL data than to the

technique used to derive model parameters from them.

Cases RTC1 and RTC2a are representatives of intermediate optical thick-

nesses encountered in actual combustion configurations. In these cases, rel-

ative errors along the axis close to the maximal emission are around a few

percents in case RTC1 and decrease to zero in case RTC2a. Total radiative

heat losses are respectively 49.95 kW and 12.50 kW for the same situations.

Errors on total heat losses differ between these two calculations and exhibits

slightly higher differences. The maximal error on total heat losses is 5.74% for

the A2-1995 set of parameters. There are no significant differences between

27

the results provided by the databases based on the optically thin (A2) and

the optically thick (A3) approaches. Error profiles for the radiative transfer

simulations based on EM2C parameters oscillate between those using approx-

imate models founded on CDSD-1000/HITEMP 1995 and HITEMP 2010. In

terms of total heat losses, the relative difference between EM2C and A1-2010

calculations is 2.21 and 4.81% in cases RTC1 and RTC2a respectively.

Cases RTC2b and RTC2c are representative of high pressure combus-

tion applications, such as those encountered in most aeronautical engines.

Radiative source term profiles are very close to each other whatever set of

parameters is used. All databases based on HITEMP 2010 produce relative

errors close to zero along the profiles. Higher error values are obtained for

the radiative source term with parameters based on CDSD-1000/HITEMP

1995, with a maximum close to 4%. The technique used to build the narrow

band database parameters does not have a significant influence on total heat

losses, at fixed spectroscopic bank input. The error on this quantity com-

puted with EM2C parameters decreases when gas pressure increases from 1

(case RTC2a) to 40 atm (case RTC2c).

Cases RTC2d and RTC3 are representative of large scale configurations

for which large optical thicknesses can be encountered. In these situations,

the impact of the technique used to build narrow band model parameters

is comparable to that of the spectroscopic database. In case RTC3, errors

along the profiles attain values larger than in the previous tests, with a

maximum value close to 10% (for A2-1995) near the highest value of Sr.

28

Error on total radiative heat losses is also more important and the optically

thin parameters (A2) give the worst accuracy in case RTC3. However, this

error remains bounded and does not exceed 5% when using HITEMP 2010.

As previously, EM2C parameters give a correct value for the maximum of

the radiative source term, with errors that are not rigorously null but remain

around 2% in any case. The difference on total radiative heat losses between

EM2C and A1-2010 parameters is small and reaches 4.85 and 2.88% in cases

RTC2d and RTC3 respectively.

Conclusions of the present work are thus in agreement with those of [15]

in which it was found, from similar tests but restricted to 1D geometries,

that radiative transfer simulations based on EM2C parameters were in good

agreement with HITEMP 2010 ones.

4. Conclusions

The impact of different spectrosopic databases for the building of LBL ref-

erence data as well as of techniques to derive narrow band model parameters

from them is investigated on different benchmarks representative of combus-

tion applications. Six sets of band model parameters were developed from

LBL data obtained with CDSD-1000/HITEMP 1995 and HITEMP 2010 and

using three techniques to derive narrow band parameters from them: opti-

misation on Curves-of-Growth (A1), optically thin limit (A2) and optically

thick limit (A3). Results have been compared to computations using EM2C

parameters provided by Soufiani et al. that were based on a proprietary high

29

temperature spectroscopic dataset.

Line-of-sight and curves-of-growth calculations have been conducted and

have shown that narrow band databases based on the most up-to-date spec-

troscopic database HITEMP 2010 can be used as a reference for higher di-

mensional tests. Those comparisons also confirm that using techniques A2

or A3 provides less accurate results than those based on optimization (A1).

In one-, two- and three-dimensional geometries, results in terms of ra-

diative source profiles and total radiative heat losses indicate that the im-

pact of the different narrow band parameters is very limited. Spectroscopic

databases used to build reference LBL data have a more important influence

on the results of the simulations than the technique used to build approxi-

mate model parameters from them.

EM2C databases give results intermediate between those based on the

use of CDSD-1000/HITEMP 1995 and HITEMP 2010. In all cases, EM2C

parameters were found satisfactory to recover the correct value of the maxi-

mal radiative source term. There may be an impact on total radiative losses

compared to the optimized approach based on HITEMP 2010 (A1-2010) but

the relative difference remains small with a maximum value lower than 5%.

Band model parameters used in the present work are available on request

(contact Dr. F. Andre).

30

Acknowledgement

This work was supported by the GDR ACCORT (GDR 3438 CNRS).

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34

List of Figures

1 Integral form of the radiative transfer equation in a gas layer

from x0 to x. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2 Band averaged Curve-Of-Growth (COG) for pure carbon diox-

ide at 300 K (a) and 2300 K (b). Spectral band centered at

625 cm−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Band averaged transmision function for pure carbon dioxide

at 300 K (a) and 2300 K (b). Gas path is 1 cm long. Erri is

the absolute difference LBL – (Ai-2010). . . . . . . . . . . . . . 38

4 Case RTC1 from Liu et al. [14, 15]: (a) Three-dimensional

domain with temperature field, (b) Profiles for T/4000 K, H2O

and CO2 molar fractions. . . . . . . . . . . . . . . . . . . . . . 39

5 Case RTC1 for the seven tested narrow bands databases : (a)

Radiative source term Sr (W/m3) along the central axis x, (b)

relative error compared to the database A1-2010. . . . . . . . 39

6 Radiative source term Sr (W/m3) for the case RTC1 detail

close to the temperature peak. . . . . . . . . . . . . . . . . . . 40

7 Case RTC2 from Coelho et al. [16, 17]: (a) Three dimensional

domain for the cylindrical enclosure with the wall tempera-

ture; (b) Fields for a plane along the central axis for the gas

temperature, CO2 and H2O molar fractions. . . . . . . . . . . 41

35

8 Case RTC2a for the seven tested narrow bands databases : (a)

Radiative source term Sr (W/m3) along the central axis x, (b)

relative error compared to the database A1-2010. . . . . . . . 41

9 Radiative source term Sr (W/m3) for the case RTC2a: detail

close to the temperature peak. . . . . . . . . . . . . . . . . . . 42

10 Case RTC2b for the seven tested narrow bands databases :

(a) Radiative source term Sr (W/m3) along the central axis x,

(b) relative error compared to the database A1-2010. . . . . . 43

11 Radiative source term Sr (W/m3) for the case RTC2b: detail

close to the temperature peak. . . . . . . . . . . . . . . . . . . 44

12 Case RTC2c for the seven tested narrow bands databases : (a)

Radiative source term Sr (W/m3) along the central axis x, (b)

relative error compared to the database A1-2010. . . . . . . . 45

13 Radiative source term Sr (W/m3) for the case RTC2c: detail

close to the temperature peak. . . . . . . . . . . . . . . . . . . 46

14 Case RTC2d for the seven tested narrow bands databases :

(a) Radiative source term Sr (W/m3) along the central axis x,

(b) relative error compared to the database A1-2010. . . . . . 47

15 Radiative source term Sr (W/m3) for the case RTC2d: detail

close to the temperature peak. . . . . . . . . . . . . . . . . . . 48

16 Case RTC3 from Pedot et al. [18]: (a) Three-dimensionnal

domain (b) fields for a plane along the central axis for the

temperature, H2O, CO2 and CO molar fractions. . . . . . . . . 49

36

17 Case RTC3 for the seven tested narrow bands databases : (a)

Radiative source term Sr (W/m3) along the central axis x, (b)

relative error compared to the database A1-2010. . . . . . . . 49

18 Radiative source term Sr (W/m3) for the case RTC3: detail

close to the temperature peak. . . . . . . . . . . . . . . . . . . 50

37

x0

x

L(x0) L(x)

x'

L0(x')

dx'

Figure 1: Integral form of the radiative transfer equation in a gas layer from x0 to x.

(a) (b)

Figure 2: Band averaged Curve-Of-Growth (COG) for pure carbon dioxide at 300 K (a)and 2300 K (b). Spectral band centered at 625 cm−1.

(a) (b)

Figure 3: Band averaged transmision function for pure carbon dioxide at 300 K (a) and2300 K (b). Gas path is 1 cm long. Erri is the absolute difference LBL – (Ai-2010).

38

x axis

(a) 0 0.1 0.2 0.3 0.4 0.5x axis (m)

0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5T/4000KXH20XCO2

(b)

Figure 4: Case RTC1 from Liu et al. [14, 15]: (a) Three-dimensional domain with tem-perature field, (b) Profiles for T/4000 K, H2O and CO2 molar fractions.

0 0.1 0.2 0.3 0.4 0.5x axis (m)

0

5e+05

1e+06

Sr (W

.m-3

)

A1-1995A1-2010A2-1995A2-2010A3-1995A3-2010EM2C

(a)

0 0.1 0.2 0.3 0.4 0.5x axis (m)

-20

-10

0

10

20

Rel

ativ

e er

ror (

%)

A1-1995A2-1995A2-2010A3-1995A3-2010EM2C

(b)

Figure 5: Case RTC1 for the seven tested narrow bands databases : (a) Radiative sourceterm Sr (W/m3) along the central axis x, (b) relative error compared to the databaseA1-2010.

39

0.12 0.13 0.14 0.15 0.16 0.17x axis (m)

7e+05

8e+05

9e+05

1e+06

1.1e+06

Rad

aitiv

e so

urce

term

Sr (

W.m

-3)

A1-1995A1-2010A2-1995A2-2010A3-1995A3-2010EM2C

Figure 6: Radiative source term Sr (W/m3) for the case RTC1 detail close to the temper-ature peak.

40

x axis

(a)

(b)

Figure 7: Case RTC2 from Coelho et al. [16, 17]: (a) Three dimensional domain for thecylindrical enclosure with the wall temperature; (b) Fields for a plane along the centralaxis for the gas temperature, CO2 and H2O molar fractions.

0 0.5 1x axis (m)

0

1e+05

2e+05

3e+05

4e+05

5e+05

Sr (W

.m-3

)

A1-1995A1-2010A2-1995A2-2010A3-1995A3-2010EM2C

(a)

0 0.5 1x axis (m)

-20

-10

0

10

20

Rel

ativ

e er

ror (

%)

A1-1995A2-1995A2-2010A3-1995A3-2010EM2C

(b)

Figure 8: Case RTC2a for the seven tested narrow bands databases : (a) Radiative sourceterm Sr (W/m3) along the central axis x, (b) relative error compared to the databaseA1-2010.

41

0.8 0.9 1 1.1x axis (m)

3e+05

3.5e+05

4e+05

4.5e+05

Rad

aitiv

e so

urce

term

Sr (

W.m

-3)

A1-1995A1-2010A2-1995A2-2010A3-1995A3-2010EM2C

Figure 9: Radiative source term Sr (W/m3) for the case RTC2a: detail close to thetemperature peak.

42

0 0.5 1x axis (m)

0

5e+05

1e+06

1.5e+06

2e+06

2.5e+06

3e+06

Sr (W

.m-3

)

A1-1995A1-2010A2-1995A2-2010A3-1995A3-2010EM2C

(a)

0 0.5 1x axis (m)

-8

-6

-4

-2

0

2

4R

elat

ive

erro

r (%

)

A1-1995A2-1995A2-2010A3-1995A3-2010EM2C

(b)

Figure 10: Case RTC2b for the seven tested narrow bands databases : (a) Radiative sourceterm Sr (W/m3) along the central axis x, (b) relative error compared to the database A1-2010.

43

0.8 0.9 1 1.1x axis (m)

1e+06

1.5e+06

2e+06

2.5e+06

Rad

aitiv

e so

urce

term

Sr (

W.m

-3)

NBKMNBKM 2010NBKM RNBKM R2010NBKM optNBKM opt2010SNB EM2C

Figure 11: Radiative source term Sr (W/m3) for the case RTC2b: detail close to thetemperature peak.

44

0 0.5 1x axis (m)

0

1e+06

2e+06

3e+06

4e+06

5e+06

6e+06

Sr (W

.m-3

)

A1-1995A1-2010A2-1995A2-2010A3-1995A3-2010EM2C

(a)

0 0.5 1x axis (m)

-6

-4

-2

0

2

Rel

ativ

e er

ror (

%)

A1-1995A2-1995A2-2010A3-1995A3-2010EM2C

(b)

Figure 12: Case RTC2c for the seven tested narrow bands databases : (a) Radiative sourceterm Sr (W/m3) along the central axis x, (b) relative error compared to the database A1-2010.

45

0.8 0.9 1 1.1x axis (m)

2e+06

3e+06

4e+06

5e+06

6e+06

Rad

aitiv

e so

urce

term

Sr (

W.m

-3)

A1-1995A1-2010A2-1995A2-2010A3-1995A3-2010EM2C

Figure 13: Radiative source term Sr (W/m3) for the case RTC2c: detail close to thetemperature peak.

46

0 5 10x axis (m)

0

50000

1e+05

1.5e+05

2e+05

2.5e+05

3e+05

Sr (W

.m-3

)

A1-1995A1-2010A2-1995A2-2010A3-1995A3-2010EM2C

(a)

0 5 10x axis (m)

-30

-20

-10

0

10

20

Rel

ativ

e er

ror (

%)

A1-1995A2-1995A2-2010A3-1995A3-2010EM2C

(b)

Figure 14: Case RTC2d for the seven tested narrow bands databases : (a) Radiative sourceterm Sr (W/m3) along the central axis x, (b) relative error compared to the database A1-2010.

47

8 9 10 11 12x axis (m)

50000

1e+05

1.5e+05

2e+05

2.5e+05

Rad

aitiv

e so

urce

term

Sr (

W.m

-3)

A1-1995A1-2010A2-1995A2-2010A3-1995A3-2010EM2C

Figure 15: Radiative source term Sr (W/m3) for the case RTC2d: detail close to thetemperature peak.

48

15 m

6 m

y ax

is

(a)

(b)

Figure 16: Case RTC3 from Pedot et al. [18]: (a) Three-dimensionnal domain (b) fieldsfor a plane along the central axis for the temperature, H2O, CO2 and CO molar fractions.

-3 -2 -1 0 1 2 3x axis (m)

0

1e+05

2e+05

3e+05

4e+05

5e+05

Sr (W

.m-3

)

A1-1995A1-2010A2-1995A2-2010A3-1995A3-2010EM2C

(a)

-2 -1 0 1 2x axis (m)

-30

-20

-10

0

10

20

Rel

ativ

e er

ror (

%)

A1-1995A2-1995A2-2010A3-1995A3-2010EM2C

(b)

Figure 17: Case RTC3 for the seven tested narrow bands databases : (a) Radiative sourceterm Sr (W/m3) along the central axis x, (b) relative error compared to the databaseA1-2010.

49

-0.4 -0.2 0 0.2 0.4x axis (m)

3e+05

4e+05

5e+05

Rad

aitiv

e so

urce

term

Sr (

W.m

-3)

A1-1995A1-2010A2-1995A2-2010A3-1995A3-2010EM2C

Figure 18: Radiative source term Sr (W/m3) for the case RTC3: detail close to thetemperature peak.

50

List of Tables

1 Total heat losses for differents databases for the cases RTC1

to RTC3. εTotal represents the relative error to the A1-2010

database on total heat losses. εProfile represents the relative

error to the A1-2010 database averaged on the axis used to

plot Sr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

51

Bas

eR

TC

1R

TC

2aR

TC

2bR

TC

2cR

TC

2dR

TC

3A

1-20

10T

otal

loss

es(k

W)

49.9

512

.50

37.7

750

.34

3119

.58

1553

.13

Tot

allo

sses

(kW

)47

.08

12.2

236

.84

49.1

130

07.0

713

95.8

2A

1-19

95ε T

ota

l(%

)-3

.32

-2.0

6-2

.03

-2.3

5-2

.39

-4.0

9ε P

rofi

le(%

)-3

.99

-2.2

4-2

.61

-2.5

4-2

.85

-6.2

8T

otal

loss

es(k

W)

47.8

611

.90

36.9

849

.59

2968

.41

1508

.33

A2-

2010

ε Tota

l(%

)-2

.13

-0.5

1-0

.56

-0.0

9-1

.56

4.92

ε Pro

file

(%)

1.71

-0.5

4-0

.48

-0.0

9-1

.34

1.61

Tot

allo

sess

(kW

)48

.88

12.4

337

.26

49.2

231

34.5

915

09.9

3A

2-19

95ε T

ota

l(%

)-5

.74

-2.2

8-2

.47

-2.4

6-3

.61

-10.

13ε P

rofi

le(%

)-1

.47

-2.5

6-2

.96

-2.6

1-3

.80

1.67

Tot

allo

sess

(kW

)48

.88

12.4

437

.56

50.3

030

70.8

214

76.6

8A

3-19

95ε T

ota

l(%

)-2

.15

-0.5

9-1

.37

-2.2

30.

48-2

.78

ε Pro

file

(%)

-2.1

9-1

.14

-2.0

8-2

.43

-0.1

21.

87T

otal

loss

es(k

W)

50.9

712

.71

38.0

650

.43

3218

.20

1609

.81

A3-

2010

ε Tota

l(%

)2.

041.

680.

770.

163.

163.

65ε P

rofi

le(%

)1.

441.

270.

640.

163.

032.

61T

otal

loss

es(k

W)

47.9

612

.24

37.0

149

.16

3045

.01

1455

.64

EM

2Cε T

ota

l(%

)-2

.21

-4.8

1-2

.10

-1.5

0-4

.85

2.88

ε Pro

file

(%)

-4.1

9-3

.53

-1.9

8-1

.41

-4.5

2-7

.09

Table 1: Total heat losses for differents databases for the cases RTC1 to RTC3. εTotal

represents the relative error to the A1-2010 database on total heat losses. εProfile representsthe relative error to the A1-2010 database averaged on the axis used to plot Sr.

52