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TRANSCRIPT
Comparison of databases for radiative heat transfer
calculations in combustion applications with the
NBKMcK model
POITOU Damien ([email protected])a, ANDRE Frederic([email protected])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