ic (h)/c-nbe-65-0'^Ö space science laboratory

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(H)/C-nBE-65-0'^Ö Space Science Laboratory

EMPIRICAL INFRARED ABSORPTION COEFFICIENTS /

OF H20 FROM 3000K TO 30000K

C. C. Ferriso C. B. Ludwig

and A. L. Thomson

DECEMBER 1965

D D C

k, m b i9bb j 1

DDCIRA B

Technical Report

Research and Advanced Technology Department

C I E A ^ i ?' C ,J ^ r

FOTl P- -

UJ-jJ o.7<r 0>?- 0*

(;IIIIIIIII) dv-t&yl (1VNBRAL. DYNAMICS CONVAIR

r1/? c^d'

(a)/c-riBE-65-028 Space Science Laboratory

EMPIRICAL IBFRARED ABSORPTION COEFFIdEWTS

OF H20 FROM 300 "K TO 3000 0K

C. C. Ferriso C. B. Ludwig

and A. L. Thomson

DECEMBER I965

Technical Report

This work was supported by Project DEFENDER, Advanced Research Projects Agency, AO 237, through the Office of Naval Research Nonr 3902(00)./

TABLE OF CONTENTS

Rjge

UST OF ILLUSTRATIONS iii

ABSTRACT vi

urmoDUCTioN i

DETERMIHATION OF ABSQRPTICH COEFFICIENTS FROM NON-TEEN GAS SPECTRA 2

Table I: Integrated Intensities Used in Present Analyses . . 6

RESULTS 7

COMPARISON WITH OTHER THEORETICAL AND EHERIMENTAL RESULTS .... 9

CONCLUDING REMARKS 10

ACKNOWLEDGEMENTS 11

REFERENCES 12

ILLUSTRATIONS ill-

Table II: Absorption Coefficients of RJ) hi

ii

LIST OF ILLUSTRATIOMS

Figure Caption ft^e

1 Ttxe integrated intensity or of the 2.7-ii band as function of ä

for various temperatures. Ik

2 Plot of absorption coefficients at 300oK and 3000oK. 15

3 Compilation of average absorption coefficients versus temper-

ature at 1250 cm as obtained from spectra taken by several

investigators. Solid line is the average value. l6

U Compilation of average absorption coefficients versus temper-

ature at 3750 cm' as obtained from spectra taken by several

investigators. Solid line is the average value. 17

5 Compilation of band averaged line spacings d versus temperature

as obtained from spectra taken by several investigators. 18

6 The ratio of line width to line spacing (2rrY /d) as function

of wavenumber, obtained by Oppenheim of the 2.7-M. band at

1200oK. 19

7 Comparison of absorption coefficients, obtained by Benedict

at 3000K, averaged over 25 cm' . 20

8 Comparison of p^sorption coefficients, obtained by Benedict

at 6000K, averaged over 25 en' . 21

9 Comparison of absorption coefficients, obtained by Benedict

at 3000K, averaged over 25 cm . 22

10 Comparison of absorption coefficients, obtained by Oppenheim. 23

11 Comparison of absorption coefficients, obtained by Goldstein

of the 6.3-M, band at U730K (solid line). Present values are

given as points. 2h

111

LIST OF ILLÜSTRATIOWS (CON'T)

Figure Caption ftige


of the 2.7-ii band at 12730K (solid line). Present values are

given as points. 25


of the 1.9-^ band at Vf30K (solid line). Present values are

given as points. 26

Ik Comparison of absorption coefficients, obtained by Goldstein

of the 1.9-^ band at 8730K (solid line). Present values are

given as points. 27


of the 1.4-p, band at 4730K (solid line). Present values are

given a* points. 28


of the l.U-p, band at 8730K (solid line). Present values are

given as points. 29

17 Comparison of measured and calculated emissivlty of the 2.7-^

band. Solid lines are the measured spectra taken from Ref. 26,

points ai« the present values. 30

18 Comparison of measured and calculated emlssivity of the 2.7-n


points are the present values. 31

19 Comparison of measured and calculated emissivity of the 2.7-u,

band. Solid Dines are the measured spectra taken from Ref. 26


IV

UST OF ILLOSTRATIOHS (CON'T)

Figure Caption p^ge

20 Comparison of measured and calculated emissivity of the 2.7-n



21 Comparison of measured and calculated emissivity of the 2.7-ti


points are the present values. 3^-

22 Comparison of measured and calculated emissivity of the 2.7-M.



23 Comparison of measured and calculated emissivity. 36

2k Comparison of measured and calculated emissivity of the

rotational band from 500 cm~ to 1050 cm" , Solid line is

the measured spectrum by the present authors, points are the

present values, (p., = 1 atm, i = 3.12 cm, T = 2200% u = 0.3

cm at STP). 37

25 Comparison of measured and calculated emissivity of the 1.9-^-

bard. Solid line is the measured spectrum by the present

authors, points are the present values. (p_ = 1 atm, i = 3*12

cm, T = 2200% u = 0.3 cm at STP). 38

26 Comparison of measured and calculated emissivity of the lA-p,

band. Solid line is the measured spectrum by the present

authors, points pre the present values. (p_ = 1 atm, I = 3«12 cm,

T = 2200% u = 0.3 cm at STP). 39

27 Comparison of measured and calculated emissivity of the 2.7-^

band. Solid line is the measured spectrum by Burch and Gryvnak.

(pT = 1 atm, T = 1200% u = I.76 cm at STP, 100^ HgO). ho

ABSTRACT

A set of absorption coefficients for water vapor as

functions of temperature has been empirically dedecued from

existing quantitative absorption and emission spectra between

1 and 22 ^i. The basic assumption was made that, for optically

thick gases, the curve of growth is given by a statistical

model. Bie absorption coefficients were obtained for each

vibration-rotation band by adjusting the band averaged line

half widths and line spacings such that the integral of the

absorption coefficients was equal to the known integrated

intensity of the particular band at a given temperature. The

effect of foreign gas broadening and non-resonant water molecule

broadening was included. The present results are compared with

independent theoretical and experimental results and the agreement

is satisfactory.

vi

IRERODÜCTION

For the purpose of analyzing and predicting rocket plume radiation,

it is necessary to have available a representation of the spectral prop-

erties of all the major radiating species as functions of the relevant

parameters. This representation should be eis simple as possible yet

detailed enough to allow for determining the essential features of the

spectrum.

ülie major infrared molecular radiators in present rocket exhausts

are carbon dioxide, water vapor, and several diatomic molecules. The

spectral properties of carbon dioxide and many of the diatomic molecules

have been reported in the literature. However, there are at present no

theoretical predictions of the complete water vapor spectra in the infrared

region at high temperatures. It is the purpose of this paper to derive

-1 a set of absorption coefficients for water vapor between 50 and 7500 cm

in the temperature range from 300° to 30C)0oK from available experimental

data.

During the last few years, a large number of spectra of high tempera-

1-12 ture water vapor between 1 and 22 microns have been published. These

spectra have been measured by different techniques and cover the tempera-

ture range from 300° to 2700oK. The optical depths range between about

0.2 to 100 cm-atm, total pressures range between 50 mmHg and 10 atm, but

the range of parameters covered at any one temperature is rather limited,

especially at higher temperatures. These measurements show that the

spectral emissivity depends on the following independent parameters:

absorption coefficient, wavelength, temperature, partial pressure, total

pressure, line broadening ability of foreign gases, and pathlength. The

most important parameter is the absorption coefficient as a function of

wavelength and temperature. If sufficient thin-gas spectra at all tempera-

tures were available, a set of absorption coefficients could be derived

directly. However, this is not the case. In order to obtain absorption

coefficients from measured spectra in which the gas was not thin, the

functional relationship between emissivity or absorptivity and optical

depth (curve of growth) must be known. A number of theoretical models

13 Ik Ik for the curves of growth have been developed. ' Goody has treated

the absorptivity of a gas whose spectral lines are distributed statistically

both in position and intensity. In the same paper the applicability of

the statistical model to water vapor was established. Also, Howard, Burch,

and Williams * have applied this model to water at room temperature

and 1 atm pressure and found good agreement over a wide range of optical

depths.

The expression for the spectral curve of growth contains the mean

spectral absorption coefficient, the optical deptfc, and a rotational fine

structure term which is dependent on a collision parameter and the mean

line spacing.

We propose to use these curves of growth to determine the mean spectral

absorption coefficients from non-thin water spectra by adjusting the fine

structure term in puch a way that the integral of the absorption coefficients

over a given vibration-rotation band results in the known value of the

band intensity.

DETERMINATION OF ABSORPTION COEFFICIEHTS FROM NON-THIN GAS SPECTRA

In the statistical model the spectral emissivity is given by

where W(u)) is the average equivalent width and d(a)) is the average line

Ik spacing. Goody has shown that the curves of growth are quite insensitive

to the choice of the line intensity distribution functions. When the

smoothed absorption coefficient and mean line width to line spacing ratio

are chosen so that the curves of growth agree in the optically thin and

W in the square root limits, the values of -r corresponding to a delta func-

tion, an exponential or a l/S distribution differ by at most 25^.

In the present reduction of the experimental data, we have used the

intermediate form corresponding to an exponential distribution of line

intensities:

Ü'T^F (2)

f ̂(u)) vhere k(u)) is the mean absorption coefficient, u the optical depth (at

STP), and a(u>) the fine structure parameter. The fine structure term

a(u)) is proportional to the local mean value of the ratio of the collision

half-width to the line spacing, y/d. It is to be noted that the value

of d, deduced from experimental curves of growth, is only meaningful when

interpreted in terms of a particular intensity distribution function.

In the present analysis, we have chosen to normalize the value of d

so that it is appropriate for a delta function distribution (i.e., a =

Y/d).

The line width is expected to be proportional to the number of

collisions experienced by the water molecule per unit time. Since

collisicns with other molecules of water vapor are more effective than

17 those with molecules of a different species, it has been found that

good agreement with the experimental data at room temperature has been

obtained by assuming that the line width is proportional to (YTT QPB 0 +

"YyP )j where y = line width and the subscript x designates the foreign

gas. Since we want to extend this treatment to higher temperatures, the

effect of interactions between resonating and non-resonating water molecules

10.19 and their temperature dependency should be taken into account, dus,

we obtain a three term expression,

a(«) d(«) = P^Y^W ^ * y£,0M /I) + PXV>) /£ . (3)

where y„ is the line width dxie to non-resonating dipole interactions of H20

HpO at T . The following assumptions are now made: The line widths Y0(u))

due to Interactions with resonating and non-resonating water molecules and

with foreign gas species, which are functions of frequency, can be approxi-

mated by a band averaged value, so that the frequency dependence of the

broadening term a (tu) appear? only through the mean line spacing d(u)).

Thus, Eq. (3) becomes

of water. where u = \' ^Jy 0 and a = Y^Yn 0 a^^L c is the mole fraction —^2 *2_ "'• 2 17_20

Values of y- ,. and a are given in the literature and the following V X — -L ^

values will be used in the present study: y- _ = 0.5 cm atm ; aw = 0.18; H20 N2

a0 = 0,09. Since no value for a' is given, we vill assume initially that _ 2 a is approximately 0.1. As a next step we assume that a band averaged

mean line spacing d can be used so that a (a)) beccmes independent of the

frequency and we can write Sq. (2) as

«(.) = *("" (5)

Tae frequency-Independent broadening term a' can then be evaluated from

W existing spectra, where T(W) = - fa (l-e[«n]), by using as the second of

the two equations required to define k(aj) and a

J k(u))da) = aO0{T), (6)

bamd

where a is the known integrated intensity at STP of the band in question

and 0(T) is a known function of the temperature. It can be shown that,

in the temperature range of interest here (up to ~ 30000K), the function

0(T) = 1 for fundamental bands in general. However, as pointed out recently

22 by W. S. Benedict, the 6.3-ti fundamental band of ILO may not obey this rule

because of strong rotation-vibration interaction. Room temperature measure-

ments of the 6.3-^ band indicate a value of 250 (cm per cm at STP), while

measurements at temperatures of 2700 0K indicate a value of 3Ö0 (cm per cm

at STP). Therefore, an empirical function 0(T) = 250 + 0.05T has been used

to describe the temperature dependence of the 6.3-n band between 3000K and

3000CK. For combination and overtone bands, 0(T) was given in Ref. 21 for

the 1.87- and 1.30-u bands. The values of a in (cm" per cm at STP) for

the four major vibration-rotation bands used in the present analysis are

shown in Table I.

5

Table I: Integrated intensities used in present analyses.

Bands 6.3 y. 2.7 y. 1.8? y. 1.38 y.

a° 2^0 ±20$ 230 ± 15^ 26 ± 15$ 21 ± 15^

The pure rotational band could not be evaluated in this fashion since

no measurements for the total band exist. However, mean absorption coef-

ficients vere calculated in Hef. 11; these have been included in the

present tabulation.

Approximately 80 published emission and absolution spectra were

used in the present analysis. The spectral emissivity at intervals of

25 cm' , together with the optical depth, temperature and total pressure

served as inputs in a computer program. When the experimentally determined

spectra were given with a higher resolution than 25 cm , the cxv-^z * re

smoothed by hand. The absorption coefficient was calculated fror.

oau oau

by choosing a such that Eq, (6) was satisfied (see Fig. l). If

- \ Sn-. 1/ . dtu «a , (8) u J I-C(UJ) ' v '

the spectrum was from a thin gas sample and the absorption coefficients

could be detennined directly.

Two sources of error in the determination of a(T) and k(cj,T) are

recognized:

l) Error in the measurements of the state parameters of the gas

or of the spectral emissivity or absorptivity, and

2) Uncertainty in the integrated intensities of the vibration-

rotation bands.

An analysis was performed in which reasonable error limits for the state

parameters and the integrated intensities were selected. The state param-

eters T, u, c, p , and the emissivity were varied by ± 5$, while a was

varied by ± 20^. It is believed that these variations constitute upper

limits, since many experimenters quote smaller error limits. It is

found that the uncertainty of ± 5^ in the measured emissivity has the

greatest influence on ä(T) and k(u),T) (Aa es ± 100^, Ak a= ± 20^). An

uncertainty of ± 20^ in or gives Aa ä ± 20^ and Ak « i 15^. All other

parameters influence a and k by ± 5^ or less.

RESUUS

The average absorption coefficients obtained by the method described

in the previous section are tabulated in Table II, and are plotted in Fig. 2

for the two extreme temperatures, 300oK and 3000oK. In general, the mean

deviation of the absorption coefficients taken from the individual spectra

is within ± 20^. As an example, in Fig. 3 k at 1250 cm" is plotted versus

temperature. The solid line represents the mean of the individual points.

From this curve, the absorption coefficients at 300°, 600°, 1000°, 1300°,

2000°, 2500°, and 3000oK are selected and entered in Table II. In some

portions of the spectra, where the absorption coefficient changes rapidly,

a much greater spread in the individual absorption coefficients is

observed. In Fig. K k at 3750 cm , the Q-branch region oi the 2.7-^

band, is plotted versus temperatuie. "nie large spread here is probably

introduced by small errors in the wavenumber calibration. Also, the

absorption coefficients in the spectral regions between the individual

bands ("troughs") and in the region u) > 7500 cm" are not very certain ,, !

because of lack of sufficient data. Since, at the present time, there

are no data for temperat'ires below 2000oK in this region, no absorption

coefficients are listed in Table II.

The average line spacings d(T) obtained frcm the values of a(T),

using Bq. (U), are plotted in Pig. 5« Data fron approximately 80 spectra

of the 2.7~ and 6.3-^ band are represented. Each point has about a factor

of 2 uncertainty. The overall spread of points is mostly within a factor

of 2 to 3, but can be as large as a factor of 10 (at 12000K for instance).

The straight line is a least square fit to the points and is given by

d(T) = exp[- .00106T + 1.21] (9)

One point at 22000K is obtained fron the rotational band. The point from

Oppenheim at 1200oK is a simple average taken from his measured {2ny /d)

values (see Fig. 6). The majority of points in Fig. 5 are restricted to

temperatures < l800oK because the high-temperature measurements made with

the rocket motor in this laboratory were in the thin gas region where

Eq. (8) applies.

The two combination bands at 1.67 ^ and I.38 p. are much weaker than

the fundamental bands. Room temperature absorption spectra of the 1.87-IJ.

band indicate that the value of d is practically the same as that obtained

frcm the two fundamental bands. Values of d derived from the data presented

3 — by Nelson are generally higher than values of d obtained from his data

for the two fundamental bands, but the experimental uncertainty in his

measurements does not permit any definite conclusions. Our own data were

always obtained at low optical depths, so that no d could be obtained.

In the absence of further information, it is assumed that the d(T) for

the two combination bands is the same as that deduced for the fundamental

bands.

The rotational band is very strong and fine structure affects the

measurements even at the highest tenperatures measured (~ 2700oK).

However, since the measurements vere carried out only up to 22 u, no

total band strength could be obtained. Thus, the procedure used before

for the vibration-rotation bands could not be employed. Additional know-

ledge about the integrated intensity of this band up to 22 n was needed

which presupposed the knowledge of the spectral absorption coefficient.

The theoretical relationships of Ref. 11 were used here in a semi-empirical

fashion by adjusting the effective mean rotational constant slightly to fit

the observed spectra in the far wing of the rotational band. The values of

k thus determined were used to calculate

1000 cm"1

] k(u),T) duj = c/(T) (10)

U50 cm'

which then replaced Eq. (7)«

COMPARISON WITH OTHER THEORETICAL AHD KXPERIMEI1TAL RESULTS

A check of the absorption coefficients derived here against indepen-

2k 25 dent work is possible for some selected temperatures. Benedict * has

obtained absorption coefficients for individual lines in the 6.3-M. band

at »s 3000K and as 600oK and in the 2.7-H band for s» 3000K. We have averaged

Benedict's values and compared them with the absorption coefficients taken

2^ from this work (see Figs. 7, 8, and 9)- Oppenheim's " detailed measure-

ment of the absorption coefficients in the 2.7-^ band at 1200oK is

7 compared in Fig. 10. Goldstein's determination of the absorption coef-

ficients of the I.38-, I.87-, 2.7-, and 6.3-|i bands at « 500oK and 1200oK

are compared with the present values in Figs. 11 through l6. One set of

experimental data has recently become available. Simmons, Arnold, and

26 Smith have measured the 2.7-ji bands between 700° and 12000K at various

optical paths. Since their data were not included in our derivation of

the absorption coefficients and band averaged line spacing, they provide

us with an independent check for calculating thick gas spectra of the

2.7-^ bands at around 1000oK. Tixe result of this comparison for some of

the spectra is given in Figs. 17-22. Similar agreement is found with the

other spectra from Ref. 26.

In a consistency check, seme of the measured spectra, which were

used to obtain a band-averaged line spacing and average absorption

coefficients, are recalculated using the d from Fig. 5 (dashed line)

and the k's from Table II. In Fig. 23 we compare Nelson's measure-

cents of the 6,3-\L band at 11I10K with the present representation of

k(tu) and T. In Figs. 2k, 25, and 26, the spectra obtained in this

laboratory are compared at 2200oK. In Fig. 27, one spectrum obtained

by Burch and Gryvnak of the 2.7-^1 band at 12000K is compared with the

present results. In general the agreement is adequate for the present

representation of HpO at high temperatures.

CONCLUDING REMARKS

The absorption coefficients averaged over 25 cm intervals given

in this study for temperatures greater than 300"K represent the "best"

values available at the present time. (The values at 3000K were included

here only for completeness. More detailed tabulations are available in

the literature, for example, References 2k and 27). The absorption

coefficients derived here have been based on nearly all the available

published spectra, using a statistical model for the curve of growth

and a frequency-independent fine structure term to extrapolate to zero

pathlength. In the slowly varying portions of the spectra, the uncer-

10

tainties are believed to be within ± 20it, while in portions of the spectra

where steep changes occur, the uncertainties may be higher. Also the

values in the troughs between the vibration-rotation bands are not very

certain because of the lack of sufficient data.

•Hie different frequency-independent fine structure terms obtained

from the analysis of the optically thick spectra were used to calculate

band-averaged line spacings as functions of temperature. These individual

points are uncertain by a factor of 2 and although the spread can be

as high as a factor of 10 (at 1200oK for instance), the mean values

(dashed curve in Fig. 5) are believed to be uncertain by a factor of 2

only. That these mean values may be used for first approximations in

the calculation of moderately thick gases was shown in the comparison

26 with some of the spectra by Simmons, et.al., in Figs. 17-22. More

accurate calculations for very thick gases will require a detailed

evaluation of the dependence of the fine structure parameters on both

temperature and wavelength.

ACKHWLEIXgaffiWTS

The authors wish to thank C. N. Abeyta and D. Brewer for assisting

in obtaining and analyzing the data.

11

RKKEHKIICES

1. R. H. Tourin aud P. M. Beary, "Infrared Spectral Bnissivl+ies and

Internal Energy Distributions of Carbon Dioxide and Water Vapor at

Higji Temperatures," APCRC-TR-60-203, The Warner and Swasey Company,

Control Instrument Division (December 1959)«

2. D. E. Durch and D. A. Gryvmüi, "Infrared Radiation Bmitted by Hot

Gases and. Its Transmission through Synthetic Atmospheres," Report

No. U-I929, Ford Motor Co., Aeronutronics Division (31 Oct. 1962).

3. K. E. Nelson, "Experimental Determination of the Band Absorptivities

of Water Vapor at Elevated Pressures and Temperatures," M.S. Kiesis,

University of Calif. (1959)-

k. J. N. Howard, D. E. Burch, and D. Williams, J. Opt. Soc. Am. 46, 186

(1956).

5. R. Goldstein, J. Quant. Spectrosc. Rad. Transfer 3, 91 (1963).

6. R. Goldstein, J. (fciant. Spectrosc. Rad. Transfer k, 3^3 (I96IO.

7. R. Goldstein, "Quantitative Spectroscopic Studies on the Infrared

Absorption by Water Vapor and Liquid Water," Ri.D. Hiesis, California

Institute of Technology (196U).

8. D. E. Burch, W. L. France, and D. Williams, Appl. Opt. 2, 505 (1963)-

9. C. C. Ferriso and C. B. Ludwig, J. Quant. Spectrosc. Rad. Transfer k,

215 (1964).

10. C. C. Ferriso, C. B. Ludwig, and C. N. Abeyta, J. Quant. Spectrosc.

Rad. Transfer 5, 28l (1965).

11. C. B. Ludwig, C. C. Ferriso, W. Jfelkmus, and F. P. Boynton, J. Quant.

Spectrosc. Rad. Transfer 5, 697 (1965).

12. C. C. Ferriso and C. B. Ludwig, J. Chem. Riys. hi, 1668 (1964).

12

13- G. N. EUSB, J. Opt. Soc. An. k&, 69O (195Ö).

Ik. R. M. Goody, Quart. J. Roy. Jfeteorol. Soc. 78, 165 (1952).

15. J. N. Howard, D. E. Burch, and D. Willians, J. Opt. Soc. Am. US,

2k2 (1956).

16. J. N. Howard, D. E. Burch, and D. Williams, J. Opt. Soc. Am. US,

33k (1956).

IT- D. E. Burch, E. B. Singleton, and D. Williams, Appl. Opt. 1, 359 (1962).

18. W. S. Benedict and L. D. Kaplan, J. Chem. Riys. 30, 388 (1959).

19. W. S. Benedict and L. D. Kiaplan, J. Quant. Spectrosc. Rad. Transfer

h, ^53 (1964).

20. K. P. Vasilevsky and B. S. Reporent, Opt. Spectrosc. (translation) 7,

353 (1959).

21. J. C. Breeze, C. C. Ferriso, C. B. Ludwig, and W. Malkmus, J. Chem.

Hiys. k2, k02 (1965).

22. W. S. Benedict, private coamunication.

23. U. P. Oppenheim and A. Goldman, Tenth Symposium on Combustion, 105 (1965)'

2k. D. M. Gates, R. F. Calfee, D. W. Hansen, and W. S. Benedict, NBS

Monograph 71 (196U).

25- W. S. Benedict, private communication, to be published.

26. F. S. Simmons, C. B. Arnold, and D. H. Smith, "Studies of Infrared

Radiative Transfer in Hot Gases. I: Spectral Absorptance Measurements

in the 2.7-U HgO Bands," BAMIRAC Report it6l3-91-T, Infrared Physics

Laboratory, Willow Run Laboratories (August 1965).

27. V. R. Stull, P. J. Wyatt, and G. N. Plass, Appl. Opt. 3, 229 (1964).

13

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