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Absorption of radiation by gases from low to high pressures. I. Empirical line-by-line and narrow-band
statistical models
J. Hartmann, J. Bouanich, C. Boulet, M. Sergent
To cite this version:
J. Hartmann, J. Bouanich, C. Boulet, M. Sergent. Absorption of radiation by gases from low to high pressures. I. Empirical line-by-line and narrow-band statistical models. Journal de Physique II, EDP Sciences, 1991, 1 (7), pp.739-762. �10.1051/jp2:1991107�. �jpa-00247553�
Classification Phjsics Abstracts
44 40 33 20E 33 70W
Absorption of radiation by gases from low to high pressures.
I. Empirical fine-by-line and narrow-band statistical models
J M Hartmann ('), J. P Bouamch (2), C Boulet (2) and M Sergent (2)
(') Laboratoire d'Energdtique Moldculaire et Macroscopique-Combustion(*), Ecole Centrale Pans, Grande Voie des Vignes, 92295 Chfitenay-Malabry, France
(2) Laboratoire de Physique Moldculaire et Applications (**), Umversitb de Pans Sud, Campus d'orsay, 91405 Orsay Cedex, France
(Received it January J99J, accepted in final form J0 April J99J)
Rks1Jmk. Nous proposons de nouveaux moddles qui perrnettent un calcul prkcis et peu co0teux de spectres d'absorption infrarouge de gaz des basses aux hautes pressions Le premier est un
calcul raie-par-raie fondb sur des corrections apportbes au profil Lorentzien, celles-ci tiennent compte, de faqon empmque, des effets des couplages de raies et de la durbe fime des collisions Les prbdictions de ce moddle sont en bon accord avec des spectres d'absorption mfrarouge
experimentaux dans un vaste domaine de pressions Sa dbgradation spectrale conduit I un Moddle Statistique I Bande Etroite gbnbrabsb qui perrnet le calcul rapide et pr6cis de propndtes d'absorption h moyenne rdsolution Ce demier, qui est fondd sur des reprdsentations statistiques des paramdtres des raies d'absorption, est une extension de l'approche proposde par Goody ill
dans le cas des basses pressions Des comparaisons entre (es diffbrents moddles et (es rbsultats expdnmentaux ddmontrent la fiabilitd des approches ddveloppdes dans des cas ou (es mod61es proposds prdcddemment sont trdslrnprbcls.
Abstract. We present new models which provide satisfactory and low cost modeling ofmfrared absorption spectra of gases from low to high pressures The first is a line-by-line approach based
on simple and empirical corrections of the Lorent21an shape which account for the effects of line- mixing and finite duration of collisions It leads to satisfactory agreement with expenmental
infrared spectra m a wJde density range. It is degraded for the deduction of a Generalized Narrow-Band Statistical Model (GNBSM) which enables very rapid computation of medium resolution radiative properties The latter is an extension of the low density (i e. when lines are
much narrower than the considered spectral resolution) approach developed by Goody [I], it is
based on a random representation of line-positions and statistical descnption of line integrated-
intensities and leads to a very simple expression of spectrally averaged absorption properties of gases It is consistent with previous low density models and requires computer times several orders of magnitude shorter than line-by-line calculations Compansons between degraded approaches, line-by-line models. and expenmental results demonstrate the accuracy of the present models m cases where previously proposed approaches are very inaccurate
(*) CNRS (UPR 288) (**) CNRS (UPR 136)
JOURNAL DE PHYSIQUE ii T w 7 JUILLET 1991
740 JOURNAL DE PHYSIQUE II N 7
1. Introduction. Aim and principle of approximate models.
Many industnal applications leg furnaces, nuclear safety, plume signature, etc require accurate modeling of radiative transfers in gas media for wide temperature and density
ranges. The problem is generally to predict radiative fluxes at walls and dissipated powers
within the semi-transparent medium The basic quantity of interest is then the wavelength and direction of propagation dependent intensity at all points in the system Knowledge of its
spectral dependence is untractable in realistic 3-dimensional systems when gases contnbute to radiation
,
indeed, modeling of the fine spectral structure of their contnbutions to absorption
and emission may require very high resolution and lead to prohibitive computer costs
Fortunately, the problem can be greatly simplified; indeed, when integrated over all
wavelengths, the intensity emitted by an homogeneous and isothermal gas column, takes the
typical form :
It °~
= j~ °' ii rim)] I~IT, ~r) d~r, ii)
where r(«) is the transmission coefficient of the path at wavenumber wand I~(T, « ) is the
spectral black-body intensity at the considered temperature T. For practical applications, the integration range in equation (I) can be reduced to the finite and discrete intervals in which the gas absorbs significantly. When dividing the spectral range into intervals of width Am in which I~(T, «) is practically constant, II °'can be approximated by
+~ Ii ha +~
Ii °'= jj [I rim)] I~(T, « d«
= Am jj [I f~"(i Am )] f("(T,
i Am ), (2)
,=1 0-1)A« ,=1
where f~"
i Am) and f(" T,
i Am are averages of
T « and I~ T, « ) on the
Ii I) Am, i Am interval The problem then reduces to the calculation of averaged values of r (« ) The approximation in equation (2) requires spectral intervals typically 25 cm- ' wide
(=0.04~m at 4~m); on the other hand, accounting for the spectral dependence of
rim) in the case of gases requires, at latm pressure, a step of typically 0.01cm-' (4 x10-~ ~m at 4~m); much computer-time can thus be saved by using approximate
models which enable direct calculations of f~" Ii Am ).
Various approaches have been proposed for rapid computations of low resolution
absorption properties of gases at densities near the ambient (reviews are gJven in Refs [1-6]).
They include the simple but inaccurate gray-gas and box models as well as more accurate
approaches such as the Narrow-Band Statistical Models (NBSM) The latter have been studied in details at low pressures (near I atm) previously (see Refs [6-8] and works quoted therein) and widely used for radiative heat-transfer calculations [9-11] The NBSM start from
a line-by-line approach (Starting Model, SM) which is degraded through statistical modehngs of gas absorption-lines some have been developed from SM based on the addition of
individual line-contnbutions with Lorentz, Doppler, and Voigt shapes [3-6, 12, 13]. They lead
[6-8] to satisfactory results at moderate densities At elevated densities, these line-shapes are
very inaccurate [14, 15] so that building of a simple model first requires determination of an
accurate SM
In the present paper, we propose simple approaches for modeling of gas absorption spectra from low to elevated densities Empirical corrections to individual line-shapes are proposed in
section 2, they lead to satisfactory agreement between the corresponding line-by-line
calculations and experimental spectra In section 3 an extension of the low density NBSM
proposed by Goody [I] is developed. It uses corrected line-shapes and statistical modehngs of
positions and intensities of absorption lines It is consistent with previous low density models and requires little CPU time. Comparisons between degraded and line-by-line calculations and experimental results are presented in section 4 They show that the approaches proposed
are accurate from low to high pressures. The interest of the present approaches when
compared with others is discussed in section 5.
2. Empirical tine-by-line approach for pressure broadened absorption bands.
When the line-shape is dominated by molecular collisions ~pressure broadening, when Doppler effects are neglected) it can, under some assumptions that are discussed thereafter, be modeled m the infrared by using the Lorentz profile, the absorption coefficient
a at
wavenumber «, for a pure absorbing gas at temperature T and density D, is then gJven by
~~,~~~
DS,iT) y,iD, T)
~ ~~~ ~~ ~
all
ir "11" ", ~riD, T))~ + Y,iD, T)~i ~ ~~~
where S,, y,, and A, are the integrated-intensity, self-broadened collisional half-width, and colhsional shift of line r, respectively. Let us note that in the far infrared or millimeter-wave regions, the Van-Vleck Weisskopf line-shape is to be used Equation (3) is the basic equation of many line-by-line calculations [7, 16-19] and was used previously to build the NBSM of
references [4, 7, 8, 19, 20] Typical comparisons between expenmental pure CO~ transmission spectra and predictions of the Lorentzian model are plotted in figure I II) (details on the data used for calculations are gJven m Appendix A) ; they show that this approach is accurate at
moderate density but fails when density is high
As widely discussed in previous papers [14, 21] the Lorentzian approach is based on two
approximations which are the absorption results from the addition of individual line- contributions Ii e line-mixing is neglected, this is accurate near line-centers when half-widths of lines are much smaller than the separation between lines of a gJven vibrational transition,
fi
,"'
E ', /
~ ' i
c
a '
# /
o
,"h /~ l
i I I I
j ( '
/
'j I
I
c700 c800 ~ ?oo
Fig Transmission coefficient m the 2 ~m region, for a 5 29 cm long path. pure COz at 293 K, and
the densities a) 7 70 Am (79 atm). b) 91.4 Am (5? atm) (~) expenmental spectra from
reference [21]
,
(- -) Lorentzian calculation
(1) At pressure P and temperature T, the density D(P, T) in Amagat is given by D(P, T) = v(I atm, 273 K)/v(P, T) where v(P, T) is the molar volume This umt~ which is widely used m the
considered pressure range, is proportional to the number density (molec cm-3) and accounts for deviations from the perfect gas law
742 JOURNAL DE PHYSIQUE II bt 7
since colhsional half-widths are proportional to density, this approximation is accurate at low
pressures). The line-shape is Lorentzian (thJs results from the impact approximation which
is accurate within a few wavenumbers around line-center). Both approximations fail at
elevated density since absorption m the far-rung and line-mixing become important [14, 21]
the resulting spectrum is, m the case of CO~ for instance, super-Lorentzian near band-centers and strongly sub-Lorentzian in the wings (see Fig. I and Refs. [14, 15, 21-23]) Correct
modeling must then account for both line-mixing and the break-down of the impact approximation Such approaches [24] have been developed recently and applied with quite
success to C02-Ar [25] and H~O [26, 27] unfortunately, they are still approximate and
involve very large calculations whJch are untractable for complex molecular systems when a
large number of rotational levels must be accounted for Furthermore, the absorption
coefficient takes a complicated form from which deduction of degraded models is untractable.
In order to correct for the failure of the Lorentzlan profile in the far wing, an empirical line-
shape correction factor X (« «,) has been introduced by many authors [14, 22, 23, 25, 26]
~~ ~~
DS,(T) y,(D, T) " ~~~~ ~
a ' ~ '(«, D, T)
= jj
~ ~
~~~
x an i,wr "I(" "r 6r(D, T)) + y,(D, T)
~ ~~~~
hC"r
' 2 k~ T
X Xl" ", 6,lD, T)) j4)
c is the velocity of light, h and k~ are the Planck and Boltzmann constants, respectively Let us note that some authors omit the hyperbolic sines and include them in the X factor X is either determined from fits of expenmental spectra as shown in references [14, 22] or calculated from first principles [25-27j
,
it depends on temperature and on the collision-partner but it is line and density independent. Since x is equal to one near line-centers where the impact
approximation is valid, a~'~~""'reduces to a~°'C~~
in the near wing. Equation (4) enables correct modeling of absorption in the far-wing but remains inaccurate near line-centers at elevated density due to line-mixing (see, for instance Refs [14, 21]). Moreover the absorption coefficient of equation (4) generally does not verify the fundamental equality
+ OJ
a j«, D, T d«
= z Ds,jT) (5)
o an i,nmr
(for practical applications, the integral in Eq (5) can be restncted to spectral ranges in which the absorption coefficient takes significant values). In order to correct for this discrepancy, we
introduce a near-wing high-density correction function t (« «,, D ), i e
~
~~~~~~ ~~~~~ ~~ « sinh
)
~~~'~~~~~ ~~~ ~~ ~ lli~~,
" II" "r 6,lD, T))~ + Y,lD, T)~l
~ ~~~~
h~"r ~
' 2k~ T
x x1« «, ~,iD, T)) i1« «, ~,iD, T), D
= jj a)°'~**~(« «,, D, T) (6)
au i>nmr
t is temperature, perturber, and density dependent and such that equation (5) is verified. In order to be consistent with the preceding models, it must satisfy
hm t 3 «, D)
=
I and lim ii 3 «, D
= (7)
D 0 3«
- + eo
a(i values of 3« ail values ofD
We have retained the following expression, which is convenient for further analytical developments :
~~2 t(3«, D
= I A (D) 1
~ ~ ,
(8)
&« + Y(D, T)
where its an average colhsional half-width. The unknown parameter A is calculated by using
equations (5), (6), it depends on density, temperature and, through x and the line- parameters y and A, on the molecular system At low densities, t reduces to one, since
A(D) is practically zero, and equation (6) leads to the Lorentzian profile near line-centers
(where X
= I). It is important to note that correction of the Lorentzian profile through the introduction of the x and t factors is strictly empirical and only roughly models the physical
mechanisms In particular, the empirical shape will fail in modeling Q-branches which are very affected by line-mixing even at low densities.
Tests of the present approach (Eqs (6), (8)) are plotted in figures 2-4 in the cases of CO and C02 (the data and computation procedure used are described in Appendix A). They show that a great improvement is achieved by using the present approach Remaining discrepancies result from the simplicity of the model which accounts approximately for the
influence of line-mixing near line-centers, this phenomenon is in particular responsible for the shift between expenmental and calculated spectra remarked in figure 2 ; indeed, this shift
is not due to pressure shifts of individual lines A, since these parameters [29] were accounted for m the present calculation Fortunately, a shift of a few wavenumbers can be neglected for low resolution radiative transfer applications
[d/)>Ja] (cmj
/
_- -,
, ~ ,.,"
"- ;"
5 ,.-- .---,-z..-'
"' ~,_
"..,
__
~
(ill "''~""
~i50 c200 c250
Fig 2 Normalized absorption coefficient (divided by its integrated value) in the 2
- 0 CO band for a
CO-N2 mixture at 296 K, and the density 291.6 Am (395 atm) (~) experimental spectra from reference [29], calculated vqth the (-- --) Lorentzian model; (...) corrected line-shape of equaUon (6)
744 JOURNAL DE PHYSIQUE II bt 7
Transmission ,-,
, ,"
i
,'
3500
Fig 3. TranslrJlssion coefficient m the 2 7 ~m region, for a 5 02 cm long path, pure C02 at 303 K,
and the densities- a) lo 8 Am Ill atm), b)95 Am (61atm) ( expenmental spectra from
reference [15], calculated wJth the (----) Lorentzian model, (...) corrected line-shape m equation (6)
Transmiss>on ~,,:-"""
,"'
,
~ j~~ij
oo
Fig 4 -Transmlssion coefficient in the 4 3 ~m region, for a 440 cm long path, a 8 8 x 10~~ N~- 0 999912 CO~ mixture at 296 K, and the density 54 4 Am (59 atm) ( ) expenmental spectra from
reference [14], calculated with the (----) Lorentzian model, (..,..) corrected line-shape in equation (6)
3. Narrow-band statistical models.
The Narrow-Band Statistical Model was first proposed by Goody [I] It is based on statistical
representation of the positions (regularly placed in Elsasser's model [28], randomly placed in Goody's model [I]) and integrated,intensities (statistical distribution) of absorption lines.
Previous tests under low density conditions (near I atm pressure) have shown [5, 7, 8] that for the active molecules of interest in most radiative transfer applications (C02, H~O and CO)
above room temperature, the random approximation of line-positions is the more accurate In the following we recall the basic steps of the random statistical approach developed previously (Low Density Narrow Band Statistical Model, LDNBSM)
,
we use the Lorentzian
shape and treat the case of a pure and absorbing gas under constant density and temperature conditions (inhomogeneous, anisothermal mixtures will be seen later on) An extension of this approach for high densities, starting from equation (6), is then proposed.
3.I ABSORPTION AT LOW PRESSURE. GOODY MODEL (LDNBSM) The spectral range is
divided into A«-wide intervals quoted I~
= [k ha, (k + ha ], centered at wave-number
(k+1/2) ha. The I; interval contains N~ lines, of positions, integrated,intensities, and
pressure-broadened half-widths noted a~,, S~,, and y;,, (i
= I, N~) respectively, pressure-
shifts A~,, which are small and of negligible influence on medium resolution spectra, are
neglected The problem is then to calculate the averaged transmission coefficient f~ on
interval1;
(k+I)A~ (k+1)3~
f~ = ha ' r (a) da
= ha ' exp [- La (a, D, T)] da, (9)
kA~
kA~
where L is the length of the gas column and a is, in the present case, given by equation (3) without the shifts A which are neglected. In the LDNBSM approach, f~ is calculated by making four approximations
Approximation1 The spectrum is composed of an infinite number of intervals
I~ which contain lines identical (i e., N~ = N~, S~
, = S~
,
and y~
, =
y~,) to those centered
in I~.
Approximation 2 f; can be approximated by computing the average of r(a) over all
randomly distributed line,positions in the whole wavenumber range
Using the Lorentzian shape, these approximations lead to
~~ ~~~~~~~~~~ J~~
oJ
p M,~i ~~~~
(2 fi~~l
) h" ~~~ ~~'~~k~~'~~~~~
(lo)
i e
~~~~
~k W~,, (D, T) 2 M + Nk w~,, (D, T)
~~'~'~ ~f~ 0~ ~
(2 M + I) ha ~~~ ~j~
ha
= exp (- ~~ lll~(D, T)j ,
(I I)
A"
where the equivalent,width W~, (D, T) of the k, i line has been introduced :
j+O~
DLS;, (T) y~,(D, T) W;
,
(D, T)
= I exp
~
d« (12)
m
"I? + Yk,, (D, T)
The next step concerns the calculation of the averaged value lil~(D, T), of W on the
N; lines as shown in equation (11), it is based on the following approximations
Approximation 3 W~,, (D, T) only depends on S~,(T) and a conveniently chosen
« averaged » half-width j7~(D, T), whose expression will be determined latter
Approximation 4 The distribution of line intensities can be modeled by a statistical law
P~(S)
Within this frame, lf~~(D, T) can be approximated by lil~(D, T)
=
lil~(D, T)]~~~~
=
j+ x
j+ O~ DLS j7~(D, T)
= P~(S) I-exp
~ ~
d« dS. (13)
o m W [Y + Yk(D, T)
746 JOURNAL DE PHYSIQUE II N 7
Various probability functions have been proposed and tested [8]. We have retained the
exponential distribution [I] which is not the more accurate [8], but is convenient for further calculations :
Pk(S)
=
xP1- ~
(14)
S~(T~ S~(T)
Equations (13), (14) lead to
~~~~ ~~~~~~~
/ ~~~~DL ~~~~
l +
"Yk(D, T)
The parameters Y~(D, T) and £~(T) are determined from the equality of $il~(D, T) and
[$il~(D, T)]~~~~ in the limiting cases of strong and weak absorption at line-centers. This
procedure is described in appendix Bl and leads to.
&,(T)
=
I ,,,(T)j/N~,
(16) Y~(D, T)
=
( ( ~/s~_, (T)
y~_, (D, )/N~j~/£~(T)
j7;(D, T) depends linearly on density and j7,(D, T) and £~(T) involve the number of accounted lines N;, it is thus more convenient to use the following only temperature dependent parameters which have been introduced by most authors
Kk(T~
=
I Sk, T)j/A«,
~,
l~~ ~
3; T)
= by r
; T) ~ s;,
,
T) z ~/s~,
,
T) y
~,, (D, T ID
,
, (17)
i
,
r~ T) is an averaged broadening coefficient whose definition is of no importance at this step (only 3;(T)/r,(T) is involved) and will be determined later
«~(T), 3,(T), and r;(T~, are respectively proportional to an average intensity, average line-spacing, and average broadening coefficient f, is then approximated by
f)~~~~'~
= exp (- K, T) DL/ 11 + ~~j~fi~/j~~
~
(18)
k
Various sets of parameters «, 3, r have been proposed which are reviewed in reference [6]
Some have been deduced from compilations of line spectroscopic parameters and the use of
equation (17) (see for instance Ref [8])
,
others result from fits of measured or calculated (by
using an accurate line-by-line model, for instance) transmissivities by using equation (18) (for
instance Ref. [13]) The accuracy of equations (17), (18) at low densities has been widely tested [7,8] The example plotted in figure 5 shows that good agreement is obtained with both
Transmission
@
