(23) Harshaw
Chemical Co., Crystal-Solid State Division, Cleveland, Ohio, private communication. (24) Nadeau, J. S.,J . Appl. Phys. 33, 3480 (1962). (25) Natta, G . , Pasquon, I . , ildvan. Catal~yszs12-16 (1959). (26) Ogilvie, G . J.. J.Phys. Chem. Solids 10, 222 (1959). (27) Perkins, T . D.. Sliepcrvich, S. M., Upthegrove, I\'. R., "Effect of Solid State Dislocations upon the Catalytic Activity of Metals." Final Report, Office of Naval Research, Contract Nonr 982-(08), Task No. N R 0501-418. (28) Pozzi, A. L., Jr., Rase, H. F., Ind. Eng. Chem. 53, 813 (1961). (29) Rienacker, G., %. Elektrochem. 46, 369 (1940). (30) Seitz, F.,Adoan. Phys. 1, 43 (1952).
(31) Sosnovsky. H. M. C., .I. Ph,ys. Chem. 10, 304 (1959). (32) Tolstopyatora, A. A,: Vestnik ;tfoskoz* ('nit,. 6 , No. 3, S e r . Fiz. M a t . i Estesiuen AVauk No. 2, 49-57 (1951) : Chem. 4bstr. 46, 3382g. (33) Uhara, Ituro, Yanagino, Sadao, Tani, Kanzuo. Adachi, G.-Y., Teratani. Shosuke, J . Phys. Chem. 6 6 , 2691 (1962). f34) Venables. J. D.. J . A b b l . Phvs. 30. 1503 11960). (355 IVebb, C. R., Dallas,'&. G.,'Carnpbell, \i-. H.: Ind. Eng. Chem. 54, 28 (1962). (36) Wolff, G. A., Bouder. J. D., Acta Cryst. 1 2 , 313 (1959).
RECEIVED for review August 26, 1963 ACCEPTED December 10, 1963
T H E R M A L R A D I A T I O N OF M E T H A N E G A S R I C H A R D H . C .
LEE1 A N D J O H N
N e u ~York Cniversity, ,Veu Yolk. .V.
The infrared absorption of methane in three wavelength regions (2.37, determined a t various temperatures and optical depths.
H A P P E L
Y.
3.31, and 7.65
microns) has been
The experimental results show that at higher
temperatures the absorption of the 2.37-micron band decreases while the absorption of the two remaining bands in general increases with temperature.
The behavior of these bands with respect to temperature
and optical depth variations is well correlated b y the Elsasser band model.
The semiempirical expressions
for the band absorption so obtained are used to calculate the total and band emissivities of methane from 0.01 to 2.0 foot-atm. and from 500" to 3 7 5 0 " R. Expressions relating the absorptivities to the emissivities are given.
The emissivity of carbon monoxide is also determined b y the same method and found to com-
p a r e favorably with other sources.
emissivity and the ahsorptivity of a gas are inilmrtanl physical propertic-s iri hear Transfer analyses where elevated ternl)eraturrs prevail, a n d the consideration of rherrnal radiation f r o m the gas brcomes csseniial. Exaihples o f this type may be found in many areas of application, such as industrial furnace calculations, heat transfer from rocket proprllants, and the re-entry heat ing of military ballistics atid space vehicles. I n i h r past thrre have been three general approaches to the problem of detrrniining the emissivity and the absorptivity of a gas: ihroretical calculation, total energy method, a n d band energy method. Theoretical calculations of gas emissivities were pt,rformrd for carbon monoxide (,?/, ,I.?), nitric oxide ( I ) , hydrogen chloridr (33) by Penner. a n d carbon dioxide by Pennrr 1.3.31 and Plaqs i.j.5). T h e emissivity of high teniperature air has been iinder rxtensive theoretical srtidirs rec.enrly. such as the investigations of h l r y e r o r t (2.4, hleyerott a n d Sokoluff (25), Kivrl t-t al. (19 .?I), a n d Rreerie and Nardone (1). 'l'he absorption band model of randoinly distribiitetl spertral lines was developed by hlayec (1.3) a n d Goody ( 1 7 ) anti was applied b y 'I.hoitison (3') for the correlation of the emissivily of water vapor. In the total enrrgy method, measurements of the total energy radiated from a gas were used to calculate the emissivity, a n d separate 'measurements of the attenuation of blackbody radiation from a n inrervening gas are generally required to determine the absorptivity. Important investigations which followed this approach were the experimenral work of Ilottel (7.3--76). hlangelsdorf (75). Egbert ( 7 4 , Smith (76), Port (A?), and I'llrich (40)for the determination of the emissivities a n d the absorptivities of carbon dioxide, water vapor, ammonia vapor, and carbon monoxide. In the band energy method the energy spectrum is separated into individual bands where the gas is active. T h e band absorption is determined from the spectra of the gas, taking into a r r o u n t the effects of gas temperature and optical depth. HE
' Present address. Aerospace Gorp.. Los Angeles, Calif.
'l'he band absorption, in Turn, generates the band emissivities a n d absorptivities, and rheir slims give the total rmissivily and absorptivity. T h e band energ)- approach provides inure detailed iriformation on the energy- distribution in rhr specrrum a n d eliminates the need to assume gray gas properties i n handling rnulriple reflection problems. Furtherrnore, separate srts of nieasuremeiits of emissivity and at)soiptivity are not required, thris reducing the total experiinental effort. Examples of this approach are the measurernenls of band aLso1.ptioii of h a t e r vapor by Howard, Rurch, and M'illianis a n d carbon dioxide by Howard. Rurch, a n d TVillianis (17) and Edwards (7). Ed\l;ards calculated the toral emissivity of carbon dioxide froin its band absorption arid noted agreement within 1Oyo with the results of Hortel a n d Marigclsdorf ( 1 5 ) obtained from total energy measurements. This paper presents a n evaluation of the emissivity and the absorptivity of ~ n e t h a n ebased on the band energy method. I n particular, band absorption was determined from its infrared spectra taken a t various temperatures and optical depths while keeping the total pressure at 1 a t m . T h e Elsasser model of equally spaced a n d equally intense lines (9. 70) was applied to correlate the experimental d a t a of band absorption a n d to extrapolate rhese results to higher temperatures where excessive decomposition of methane under the present experimental conditions prevented further meaningftil measurements of the specrra. Furthermore. since emissivity d a t a of methane have nor been reported in the literature. it is desirable to carry out similar studies on gases of known emissivity, so rhat the present method can be compared with others. Thus, in addition to methane. the emissivity of carbon monoxide has also been determined a n d compared \vith the results of Oppenheim ( X ) ,Penner ( 3 4 ,a n d Ullrich (40). Theoretical Considerations
General Expressions for Gas Emissivity a n d Absorptivity. T h e total hemispherical emissivity of a gas a t a temperature VOL. 3
NO. 2
M A Y
1964
167
T , and a n optical depth expression :
X
pL is given by the following
=
Within the temperature range considered here. methane has nonzero spectral emissivity only within narro\+ regions of the radiant energy spectrum. so that the blackbody radiancy, R,. does not vary greatly in each region Hence. a n average blackbody radiancy. E,. evaluated a t the band center frequency, Wi. may be used for the entire band width. h i . Equation l can now be written as the sum of the band emissivities. E{
For Equations 7 and 8 to be rigorous the conditions aZdZ*X 2 a b i h i > 1 and 2 a b i 'di* 1 should be met-i.e.. very small optical depths and very weak bands are excluded and the line Lvidth should be much smaller than the line spacing. T h e last condition is satisfied in the fundamental bands of methane ( J 7 ) . T h e parameters b,: Awl. a t . and di* are temperature-dependent quantities and. in general. differ from band to band. For collision-broadened line profiles and a t a constant total pressure. b , is related to the gas temperature as follows according to simple kinetic theory of gases:
Cm -l
~43.31~2
Cm . 31.7 50.3 66.1 75.5 106.3 169.5 215.7 239.6
.
Cm.-' 70.3 114.0 139.0 157.2 193.2 268.5 317.0 347.9
I
8:7 -
1
1
80 2
99 2 110 4 124 0 150 5 173 9 184 8
1
I
I
I
I
I
0
0
EXPERIMENTAL
DATA
4
4-
-
3-
-
-
F IO',
7
2b
117.3 107.7 99,7 91 3 89 3 91 0 89 9
Spectral slit width, 31 cm.-'
radiancy, I?,( T,, G t ) in Equation 3, \vas taken from standard radiation tables (22, 27) for the three band centers of 4216, 3019, and 1307 cm.-' They are plotted for the temperature range of 500' to 3750' R . in Figures 6 to 9. .An examination of the individual band emissivity charts indicated that at room temperature the 7.65-micron band accounts for almost all the emitting energy. As the temperature increases, the 3.31micron band becomes progressively more important and finally contributes to more than 70yGof the total radiation above 3000' R. T h e contribution of the 2.37-micron band never exceeds 15TGin the entire temperature range investigated. T h e emissivities above 2000" R. are based on the extrapolated values of the individual band absorption as shotm in Figure 4. Consequently. these emissivities are kn0v.m ivith less certainty than those below 2000" R . Using the band emissivity charts, the total and band absorptivities of methane can easily be obtained by the application of Equations 21 and 22. Thus, after substitution of the parameters [I and TI. the band absorptivities of methane at T , for radiation originated from a blackbody source a t T,, become :
I
-
5-
Effect of Temperature on Band Absorption of Carbon Monoxide at 6.16 Cm.-Atm. T , K. ' 4 r . 5 i p , Cm.-'5
298 378 470 580 666 763 873
57 1
6-
z
Table V.
Effect of Optical Depth on Band Absorption of Methane at 25' C.
as.w(Ts,T,, X ) =
{
~ 2 . 3 7 ~
T,, X =
(2)"'
fiL)
Figure 10. Effect of temperature on band absorption of carbon monoxide a t 6.61 cm.-atm.
----
3-
2-
l&EC FUNDAMENTALS
ON LEES
39 B A S E D
DATA
-
I
174
EQS.3 AND
I
I
I
I
I 1 1 1 1 1
I
I
l
l
I
I
I
l
l
l
2
(35)
10
I
I
I
I
I
I
1
I
4
8-
E!
0
4-
_--
3-
-
5-
OPPENHEIM’S DATA ULLRICH’S DATA PENNEA‘S THEORETICAL CURVE (NON -OVERLAPPING LINES WITH
-EQS. 3
-
-
DATA
-
1 0;:
5
W
LEE’S
-
8-
-
DATA
0
0
765-
-
3 AND 39 BASED ON
I
-
0
4-
4-
>
3-
5 Ln
-EQ
-
8765-
5
0
4-
2-
A N D 39 BASED ON LEE’S
OPPENHEIM’S DATA ULLRICH’S DATA PENNER’S THEORETICAL CURVE(NON-OVERLAPPING LINES WITH b-0.076 CM:’
3-
b-0.076 CM;’
2-
>
-
76-
765-
10;
-
0
0
-
2
3-
Ln
ii
2-
2-
W
-
-
10-2987-
I 0-2
9-
a-
765-
1
65-
4-
3I
2-
1 l0?kO
’ SA0
I
I
1000
I
I
b
I
l
l
I
I500
500
IO00
,OK.
Emissivity of carbon monoxide a t 2 cm.-
Figure 12. atm.
Icr:o
I
(37)
I
I
I
1
0
aT
= a2 a7P
+
a3 U P
+
3fl7.W
(3 8)
where the el’s are read from Figures 7 , 8, and 9 a t T , and optical depths as indicated in Equations 35. 36, and 3’ Carbon Monoxide. A similar investigation of the temperature effect on the band absorption was conducted for the fundamenral band of carbon monoxide centered a t 21 41 cm -1 T h e rupeiimental d a t a are given in Table \. and plortcd in rlgure 10 together with Equation 39, \$hich gave the best hi to the data
where the effective band width of the fundamental band h w 4 s i P was calculated from the following relationship obtained by Penner (34) :
2000
I
I
1
I
ULLRICH ‘S PENNER’S THEORETICAL C U R V E (BOX MODEL) E Q 3 A N D 39 BASED ON LEE‘S DATA
4-
and
I500
Ts ,‘K.
Emissivity of carbon monoxide a t 10 cm.-atm.
Figure 13.
IO98765-
,
I
-
-
-
2-
6543-
-
T h e fundamental band emissivity of carbon monoxide calculated from Equations 3 and 39 is shown in Figures 11 to 14 together with the experimental d a t a of Ullrich (40)and Oppenheim ( 2 9 ) and the theoretical curve of Penner (34). Since the overtone bands of carbon monoxide are much weaker than thc fundamental, Equations 3 and 39 should yield essentially the total emissivity for optical depths not extremely
2-
I
I
VOL. 3
I
I
NO. 2
l
l
MAY
I
1964
175
large. Lt’ith reference to Figure 11, our calculations for 298’ K. compares \vel1 with the emissivity extrapolated from Ullrich’s d a t a . - i t higher temperatures and low optical depths, emissivities calculated from Equation 39 compare favorably ivith Penner‘s theoretical results and Oppenheim’s data. \vhile Vllrich‘s d a t a are considerably higher. Finally, a t a large optical depth of 183 cm.-atm.. the present calculations are in essenrial agreement Lvith both Penner‘s and 1Jllrich’s results, even though the overtone bands have been neglected. Acknowledgment
‘[‘he authors appreciate the many fruitful discussions and constructive criticisms of Daniel B. Olfe a t New York University. Salvatore Bartolotta helped in the machining of the infrared cell. T h r fellowship support of K. H. Lee donated by the Phillips Petroleum C o . during a large part of this study is gratefully acknowledged. Nomenclature a,dw
band absorption of ith band, c m . - ’
a, = b a n d absorpticit); of zth band, dimensionless aT = total hemispherical absorptivity. dimensionless
a, B,
=
spectral absorptivity, dimensionless
= h ; 8 ~ ~ c Z ,rotational , constant a t equilibrium positions of
nuclei, c m . -l
I, constant a t vibrational ground state Bo = h , ’ 8 ~ ~ crotational
b.
= spectral half-width in ith band? c m . ?
“
= velocity of light,
DiO
=
c,,
=
d,*
=
h I,
=
I,,
=
J K
=
= =
k,
=
L
=
P
=
R,
R, LTJ
=
a constant defined by Equation 1 9 3.0 X 10’0 cm.,’sec. a constant defined by Equation 20 line spacing in ith band, c m . - ~ 1 Planck’s constant. 6.625 X erg.-sec. intensity of attenuated beam a t wave number w, erg/ cm.-sec. intensity of incident beam a t wave number w, erg/cm.sec. rotational q u a n t u m number, dimensionless integer Boltzmann’s constant, 1.38 X 10-’6 erg/’ K . spectral absorption coefficient, cm. -‘-atm. gas layer thickness, cm. partial pressure of gas. a t m . blackbody radiancy or radiant flux emitted by a blackbody into a spherical angle of 2 7 steradians, for frequency interval of w to w dw, erg/cm.-sec. blackbody radiancy a t center of ith band W f ,erg/cm.sec.
+
=
=
S
k,dw integrated intensity of line, J , cm.-2-atm.-’
liDeJ
T, = T, =
x = ff,
=
temperature of gas. ’ K . temperature of radiating surface, pL = optical depth, cm.-atm.
’ K.
Lw,
kudu integrated intensity of ith band: c m . -2-atm.-’
Cf
= band emissivity of ith band, dimensionless
€T
=
e,
= spectral emissivity, dimensionless = a constant defined by Equation 11 = a constant defined by Equation 10
ct ‘lz
I J =
vi Ei
u
total hemispherical emissivity, dimensionless
micron
= designation of ith fundamental band = a constant defined by Equation 18 = Stefan-Boltzmann constant, 5.67 X 10-6 erglsq. cm.-
sec.-(O K . ) 4
176
l&EC
FUNDAMENTALS
w
=
frequency in wave number units, c m - 1
Awl = effective band width of ith band
Wz
= \\ave number a t center of ith band
sU R S C K I P T s z
= pertaining to ith band
0
=
w
= spcctral property from w to w
1-
=
at reference ternperature total
To
+ dw
literature Cited
(1) Benitez. L. E.. Penner, S. S., ,J. Appl. Phys. 21, 907 (1950). (2) Breene, K . G.: Jr., Nardone, M., ”Radiant Emission from High Temperature Air.” General Electric Co., MSVD TIS R61SD020 (May 1961). (3) Burgess, J. S.. Bell. E. E.. Nielsen, €I. H., J . Opt. Soc. ,4m. 43, 1058 (1953). (4) Childs, \V. H . J.: Proc. Ruy. Sac. London 153A, 555 (1936). (5) Childs, \I-. H . J.. Jahn, H . A . , Zbid., 169A, 451 (1939). (6) Cooley, J. P.. AJtrophys. J . 62, 73 (1925). (7) Edwards, D. K., J . Opt. Soc. Ani. 50, 617 (1960). (8) Ellis, J . \V.; Proc. .Vatf. h a d . Sci. 13, 202 (1927). (9) Clsasser, \V, M.. “Heat Transfer by Infrared Radiation in the Atmosphere,‘’ Harvard Meteorological Studies No. 6, Harvard University. 1942. (10) Elsasser, \.V. M., Phys. Reu. 54, 126 (1938). (11) Goody, R . M.. Quart. .J. Roj. M e t e o r . Soc. 78, 165 (1952). (12) Herzberg, G.: “Molecular Spectra and Molecular Structure,” Vol. 11, “Infrared and Raman Spectra of Polyatomic hlolecules.” Van Nostrand. Princeton, N. J., 1945. (13) Hottel; H . C., Trans. A.Z.Ch.E. 19, 173 (1927); Znd. Eng. Chem. 19, 888 (1927). (14) Hottel. H. C.. Egbert, K. B., Trans. A.Z.Ch.E. 38, 531 (1942). (15) Hottel, H . C . , Mangelsdorf, H . C., Zbid.,31, 517 (1935). (16) Hottel, H. C., Smith, V. C., 7rans. A . S . M . E . 57, 463 (1935). (17) Howard, J. N., Burch, D. E.; Williams, D., J . Opt. Suc. Am. 46, 237 (1956). (18) Zbid., p. 242. (19) Keck, J. C , Camm, J. C., Ki\el, B., LVentink, T., Jr., Ann. Phys. 7, 1 (1959). (20) Kivel, B.: Bailey, K., “Tables of Radiation from High ’Temperature Air,” AVCO Research Lab., Rept. 21 (December
I
21) Kivel. B., Mayer, H., Bethe, H . , Ann. Phys. 2, 57 (1957). 22) 1957)’ Lowan, A N.,Blanch, G., J . Opt. Sic. Ani. 30, 70 (1940). 23) Maver. H., Los Alamos Sci. Lab., Rept. LA-647 (1947). 124) Meyerott, R . E.: “Radiant Heat Transfer to Hypersonic Vehicles,” Lockheed Aircraft Corp. LMSD-2264-R1 (September 1958). (25) hleyerott, R. E., Sokuloff, J., “Absorption Coefficient of A i r , “ Lockheed Aircraft Corp. LMSD-288052 (September 1959).
(26) Moorehead. J. G., Phys. Reu. 39, 83 (1932). (27) Natl. Bur. Standards, “Planck’s Radiation Functions and lilectronic Functions,’‘ 1941, (28) Norris. \V, V., Unger, H . I . ?Phys. Reu. 43, 467 (1933). (29) Oppenheini, U. P., J . Appl. Phys. 30, 803 (1959). (30) Paschen, F.. Wiedernanns Ann. 51, 1 (1894). (31) Penner: S. S., J . Afipl. Phys. 21, 685 (1950); J . Appl. Mech. 18, 33 (1951). (32) Penner, S. S., J . Appl. Phy’i. 23, 825 (i952). (33) Zbid.. 25, 660 (1954). (34) Penner. S. S., “Quantitative Molecular Spectroscopy and Gas Emissivities,’‘ Addison-LVesley Co., Reading, Mass., 1959. (35) Plass. G. N.; J . Opt. SOC.A m . 48, 690 (1958); 49, 921 (1959). (36) Port, F. ,J., Sc. D. thesis, Massachusetts Institute of Technology, 1940. (37) Schmidt. H., .4nn. Physrk 42, 415 (1913). (38) Thomson, A , , “Approximate Analytical Expression for the I’ngineerinp Emissivity of LVater Vapor. I,” Gruen Applied Scientific Labs., Inc., Pasadena, Calif.. ‘Tech. Note 4, Contr. AF 04(645)-24 (1957). (39) Thorndike, A . hf,, J . Chein Phys. 15, 868 (1947). (40) Ullrich. \ V . , Sc. D. thesis, .Massachusetts Institute of Technology, 1935. (41) \’incent-Geisse, J . , Ann. Phys. 10, 693 (1955). R E C E I V ~for D review October 9. 1963 ACCEPTEDFebruary 26, 1964
