Approximate band absorption and total emissivity calculations for CO2

J. Quant.Spcctrosc.Radtat.

Transfer.

Vol.4, PD.799-806.Pergcunon PressLtd., 1964.Printed in GreatBritain

APPROXIMATE AND

TOTAL

BAND

EMISSIVITY S. S. PENNER

ABSORPTION

CALCULATIONS

FOR

CO2*

and P. VARANASI

Department of the Aerospace and Mechanical Engineering Sciences, University of California, San Diego, La Jolla, California (Received 6 July 1964) Abstract-A highly simplified procedure, using symmetrical band contours and intensity estimates derived from harmonic oscillator approximations, has been used to calculate band and total emissivities for COs. Agreement with experimental data and with more accurate calculations is excellent.

I.

INTRODUCTION

THEORETICAL

calculations of total emissivities for CO2 were performed successfully at low temperatures (300” and 600°K) some years ago.( 192)More recently, a semiempirical procedure has been described@) for correlating spectroscopic data with empirically determined emissivity data.@) Spectral emissivities for the 4-3 and 2.7 p regions have been computed theoretically by MALKMUS@.@ who has demonstrated the utility of the harmonic oscillator intensity formulas for combination bands at elevated temperatures. EDWARDS(~) has used empirical procedures to fit and extrapolate measured results on band absorption. It is the purpose of the present analysis to show how the harmonic oscillator intensity formulas may be used, together with a highly simpliGed prescription of band contours, to calculate band and total emissivities for CO2 from low-temperature data as a function of temperature at pressures that are sufficiently high to justify the use of a just-overlapping line approximation. Aside from yielding results in semianalytic form, our procedure appears to be a reasonable one since the supplementary approximkions introduced by us do not obviously lead to larger errors for band and total emissivities than those already anticipated because of the use of the harmonic oscillator intensity formulas. In this connection, it should be noted particularly that transitions in the 2.7 p region are associated with combination bands that are strictly forbidden in the harmonic oscillator approximation; nevertheless, band intensities are calculated theoretically, in good approximation, by using harmonic oscillator wave functions and an appropriate power series. expansion for the electric dipole moment.@) * Supported by the Physics Branch of the Office of Naval Research under Contract No. Nonr2216(00), NR 015401. Reproduction in whole or in part is permitted for any purpose of the United States Government. 799

800

S. S. PENNER II.

OUTLINE

The integrated a vibration-rotation

OF

P. VARANA~I

and

THEORETICAL

CONSIDERATIONS

intensity CL(in cm-s-atm-l) for the transition ~v&w-wz~vz~J’~~~ of band is given by the expression [see Q*, equation (7-117)J 8Tr3/3aWNT

a(nmzng -+n1'nzfz'n3')=

gZ[exp-

3hcQ v

wv(m2'n3)/kr]

-hcw/kT)]

x [l -exp(

(1)

where p is the square of the electric dipole moment matrix element for the indicated transition, w = w(n1n2zn3~nl’n2’z’1131)denotes the wavenumber at the band center for the indicated transition, NT is the total number of molecules per unit volume per unit pressure, gz represents the statistical weight of the ground state, QV equals the complete vibrational partition function, h is Plan&s constant, c stands for the velocity of light, WV represents the vibrational energy of the lower state involved in the transition, k is the Boltzmann constant, and T identifies the temperature. In the harmonic oscillator approximation, we find for CO2 that [see Q, equation (ll-104)] QVexp[Wv(0000)/kT]

= [l -exp(-hcwl/kT)]-1 x [I -exp(

-hcw2/kT)]-2[1

-exp(

-hcw3/kT)J-1

(2)

where wi = 1388 cm-l, wa = 667 cm-l, and ws = 2349 cm-l. From equations (1) and (2), we obtain the following general expression for the integrated band intensity: 87r3j33cu(n1n2”n3 --+nl’n3”‘n3’)

a(nln3zn3 -+n1’rZ3’z’n3’):=

Net@

3hc

x [exp -(hc/kT)(nl~ + n2w2 +TZ~CO~)] x (1- exp [ - hc+wz2Zn3 -wz~%z~~Z’~~~)/ kT]} x I1 -exp

(-hcw/kT)][l

x [l -exp(

-exp(

-hm2/kT)p

-hcwa/kT)]

(3)

since Wv(WZzZ?Z3)-

Wv(OO"O)

=

hc(nlwl+nzw+

n3w3)

to the order of approximation used in the present studies. Equation (3) is identical with equation (19) of reference 6. Using harmonic oscillator wave functions and a power series expansion for the electric dipole moment, it is found(sss+J’) that, for example, p =

(nS+l)

for

/?2 x(n3+1)(n3+2) ,@ a

?Zl?Z2z?Z3+wZ2z(~3-b1),

for

(n2+2)(?Z2+3)@3+

?'Zl?Z2"n3+?Zl?Z2z(?Z3+2) 1)

for

m2"n3+n@2+

2j"(n3+ 1)

after averaging over all allowed values of I, etc. The listed values(sSs) of 8s are sufficient for first-order calculations since the intense normal and combination bands that contribute at temperatures up to about 2500°K are contained within the indicated group of * Q refers to reference 2.

801

Approximate band absorption and total emissivitycalculations for COa

transitions. Introduction of the specified data( %Q) for $ into equation (3) allows immediate evaluation of the sums of harmonic bands in the ideal gas approximation. The following generalized relation is consistent with the results obtained in this manner (provided symmetric bands of zero intensity are eliminated): .l,;a,~..pL(nr&s R,,n?.. ,

=--

+ (n1+ &)(nz + &)c+Yns + &I);T)

la(nm2Ens~(nl+$)(ne+B2)(~~~)(na+Ss);To) *

1 - exp( -hc/kT)(%~

To

+&w + 830~)

T [l -exp( -hcwr/kT)]61[1 -exp(-hcwa/k7’)]8$1 where TO is a low reference temperature

-exp(

-hcq/kT)]68 (4)

such that

[l -exp(-hc/kTo)(Glwl+62w2fssos)]

cz 1

and 8r (i = 1, 2, 3) is zero or a small integer. Various special cases of equation (4) have been given before. For example,@*@ 2 a(nln&rs --t nrns*(ns + 1) ; T) 2 a(nm&zs -tnd(ns+

TO

(W

1); TO)= r

or(s) 2 a(nr&s

+ (nr + l)nsr(ns + 1) ; T)

TO

11-ew(-hc/kT)(ol+41

2 a(n~~s~s-,(m+l)na’(ns+l);T~)=

-T [l-exp(-hcwl/kT)][l-exp(-hc~~~/kT)1

c a(nlns %zs--t nr(ns+ 2)Yns + 1) ; T)

TO

[l -exp(

--,(4b)

-hc/kT)(2w2+w)]

2 a(nrna rns+ nr(ns + 2)91s + 1) ; TO)= -T [l - exp( -Zzcws/kT)Js[l - exp( -hcw/kT)] where it is understood

that the summations extend over all quantum numbers.

A. Approximate calculations of band absorption We shall describe the band contours by a symmetric profile, approximation@) that the band absorption of the i-th band is A{=

’ (4c)

s

[I - ew(

-p,X)ldw =

[4&,d2/(ydlWt)

starting from the

(5)

t-thband

where I(&)

= mIl- exp( - K&e-f’)]dt s 0

(6)

(see Fig. 1 for a plot of Z(K) as a function of K), I(&) 21 Z&/2 for small Kc (this approximation is good to 2 per cent for Kt < O-05))

@a)

I(&) N 1.11 q(2.303 log (1.21 Kf)) for large and intermediate values of

Kf.

(6b)

802

S. S.

PENNER

and P. VARANASI

In equation (5), P, denotes the spectral absorption coefficient (in cm-r-atm-1) at the wave number w, X is the optical depth (in cm-atm), at = 1.44 B$,r/Tif Be,t is the rotational constant in cm-l of the i-th band and T represents the temperature in “K, K = aX(dy)/ 2Be with c(identifying the integrated intensity for the band (in cm-2-atm-l) at the temperature T. Equation (5) yields an upper bound for At if the i-th band is an isolated band because the rotational fine structure has been assumed to be completely smeared out. For two or more partially overlapping bands in the j-th spectral region, we may obtain both upper and lower bounds for A$ = J[l -exp( fth

-P,x)]dw

(7)

spectral region

by noting that

[4&/hhW(

2 JAI < Al G 2 ~4&,t/(~hrW(Kd t f

(8)

where the summation extends over all of the bands contributing to the j-th region and 3, and j? are (arithmetic) average values for the contributing bands in the j-th region.

J-LI 0

IC

100

FIG. 1. The quantity Z(K) as a function of K for 0 Q K < 10,000.

The lower bound in equation (8) corresponds to the case in which the various contributing bands are practically coincident; the upper bound corresponds to the case where strong contributions from different bands fall in different spectral regions. The upper bound in equation (8) is generally a reasonable approximation for molecular bands provided harmonic bands are treated as coincident (see below) and the band centres are separated by more than about 100 3, cm-l. As an illustration of the utility of equation (8), we present a detailed comparison between theoretical and experimental results obtained for the 2.7 p region of COs. The required spectroscopic data are listed in Table 1 and the results of the calculations are

Approximate band absorption and total emissivity calculations for COs

803

TABLE 1. SPECTROSCOPIC DATA RBQUIRBD FOR THE THFsORETK!AL CALCULATION OF BAND ABSORPTION OF Con IN m 2.7~ REGION (Be = 0*3906cm-1)

a* (cm-*-atm-I)

Band center (cm-l) 3716 (n1O%+(nr + l)OO(ns+ 1) 3609 (Ona’na+O(nr+ 2)rQrna + 1) 3716 3609 3716 3609 3716 3609 3716 3609 3716 3609 3716 3609

42.3

300

285 21.2 195 14.1 21.8 13.8 24.2 125 28.3 11.65 33.5 10.1 38.8

600 900 1200 1500 2000 2500

* The values of a have been calculated from the measured room temperature values according to equation (4b) for the 3716 cm-l band and according to equation (4~) for the 3609 cm-l band, respectively.

summarized in Table 2. Following MALKMUS,@) we assume that the only important contributing bands are the On&zs+O(ns + 2)Qs + 1) and n10%s-+ + l)Oo(~~s+ 1) transitions for which we use the room-temperature intensity measurements of WEBER, HOLM and PEWNER.(~@

AND

TABLE CALCULATED

2. COMPARISON

X

(cm-atm) 1200 1273 1500 2000 2500

15.5 7.75 11.7 15.5 7.75 15.5 7.75 15.5 7.75

OF MEASURED VALUES OF A FOR THE 2.7 /A REGION

id observed* (cm-l)

z[4B,,d(~rt)lWG) 1 (cm-l)

197 121

216 151

169 208 128 -

178 219 153 280 189 303 209

* Literature citations to the experimental studies, as well as a comparison of experimental data with results derived from extensive numerical calculations, may be found in reference 6.

804

S. S;

PBNNBR

and P. VARANMI

Reference to the data listed in Table 2 shows that the upper bound specified in equation (8) is in agreement with the results of direct measurements of A at elevated temperatures, well within the quoted limits of accuracy of the room-temperature estimates for the integrated intensities (+20 per cent). Hence the theoretical predictions of band absorption must be considered to have been accomplished successfully. The fact that the theoretical predictions appear to become too large for the smaller optical depths at elevated temperatures is probably associated with the use of a just-overlapping line model under conditions where the rotational fine structure is not completely smeared out. However, no precise corrections for the line structure are possible in any case in view of the lack of adequate measurements or adequate theoretical relations concerning the prediction of line widths for polyatomic molecules at elevated temperatures. B. Approximate calculations of total emissivities We evaluate the separate contributions made by all known bands according to our simplified procedure by starting with the data listed in Table 3, which has been reproduced from an earlier paper.(s) TABLE OF

THE

STRONGER

3. INTEGRATED INTENSITIES VIBRATION-ROTATION

BANDS OF c&t

Band canter (cm-l)

Transition

648* 667.3 720.5 (740.8) (960.8) (10636) (1886) (1932.5) (2076.5) (2094) (2137) 2284*5* 2349.3 z

0110 -+ 0000 011o+OOW 1000 -+ 0110 1110 + 02’Lo ooo1+1OW 0001 -+ 0200 0400 -+ 0110 0310 + 0000 11W + 0000 12% -+014-J 2000 + 0110 0001 -+ 0000 0001 -+ 0000 0201 1001 + +OOoo OOOO

z2 0.0219 0.0532 oG415t 0*0415t 0.12 0*020t oGO5t 30.0 2676 42.3 28.5

4860.5 4983.5 (5109)

0401 -+ OOOO 1201 + 0000 2001 + 0000

0.272 1.01 0.426

-

a (3OO’K) (cm-a-atm-1)

23:;

* Transition of C180,‘8 (Cl3 Of’ ia assumad to represent 1.1 per cent of the total COP). t These numerical values have been chosen to equal the listed band intensities (Q, pp. 310,314). $ Reproduced from reference 3. Deletion of contributions by bands listed in parentheses will introduce errors smaller than 2 per cent into the total emissivity for T < 2000°K and X < 4 ft-atm. Use of the

value(r0) a = 172 cm-s-atm-1 for the 0110 --f 0000 bands will lead to slightly improved agreement for small values of X but will not affect the curves for larger X appreciably.

Approximate

band absorption

and total emissivity calculations

for COa

805

Using the data listed in Table 3, the generalized intensity formula given in equation (4), and the procedure for computing band absorption described in Section IIA, we have computed the total emissivities of CO2 as a function of temperature according to the defining relation -

cf RO,,At

c=--

CT4 where 2‘ is the blackbody radiancy evaluated at the center of the i-th vibrationrotation band and G denotes the Stefan-Boltzmann constant. The results of the calculations are compared in Fig. 2 with Hottel’s data. In view of the approximations made in the analysis, the agreement with the experimental data must be considered to be excellent. For the smallest optical depth (O-01 ft-atm), the theoretical values tend to be too large because of the previously noted failure of the assumption that the rotational fine structure has been completely smeared out. At the highest temperature, the calculated emissivity values tend to be relatively large compared with the experi-

0.24 r 0.20

0.16

:gzE$z+g;;;

0.12

0.08

e---

0.04

c----L-_ i 900

I

I

600

900

1000

T,

1500

2000

“K

FIG. 2. Comparison of observed@) (solid lines) and calculated (dotted through 0) emissivities for COa as a function of temperature.

lines passing

mental values, possibly because the contributions of the 0401 + 0000 and 1201 + 0000 transitions have been overestimated by the simplified harmonic oscillator approximation. The present calculations verify a well-known “theorem” in opacity calculations: band and total emissivities are usually predicted adequately with the use of a highly simplified model, whereas accurate prediction of spectral data is often beyond our capabilities of analysis or of our knowledge of basic spectroscopic constants. We expect that the procedure which we have used for the theoretical calculation of band and total emissivities can be applied successfully to other polyatomic molecules whenever the optical depths are sufficiently large to justify the assumption that the rotational fine structure is smeared out.

806

S. S. PENNER aud P. VABANASI REFERENCES S. S. PENNER, J. Appl. Phys. 25, 660 (1954).

: : S. S. &NNER, Quantitative Molecular Spectroscopy and Gas &is&ties, 3. 4. 2 7. 8. 9. 10.

Chapters 7 and 11. AddisonWesley, Reading, Mass. This book is referred to as Q. M. LAPP, L. D. GRAY and S. S. PENNER, International Developments in Heat Transfer, Vol. 4, pp. 812-819. Am. Sot. Mech. Eng., New York (1962). H. C. HOITEL, Chapter IV in W. H. MCADNNS, Heat Transmission (3rd ed.). McGraw-Hill, New York (1954). W. MALKMUS,J. Opt. Sot. Amer. 53, 951 (1963). W. MALKMUS,J. Opt. Sot. Amer. 54,751 (1964). D. K. EDWARDS, and W. A. MEN-, Appl. Opt. 3,621,847 (1964), aud earlier papers referred to in these articles. D. M. DENNISON, Rev. Mod. Phys. 3,280 (1931). W. S. Beuedict, High-temperature thermodynamics, Nat. Bur. Stand. Rapt. 1123, Appendix 1A (1951); K. H. IU~ER and D. E. FREEMAN,J. Mol. Spectrosc. 9, 191 (1962); W. H. SHAFPBR, Rev. Mod. Phys. 16,245 (1944). D. Warma, R. J. HOLM and S. S. PENNER,J. Chem. Phys; 20,182O (1952).