spectral and total emissivity of water vapor and carbon dioxide

COMBUSTION AND FLAME 19, 33.-48 (19"/2) 33

Spectral and Total Emissivity of Water Vapor and

Carbon Dioxide

B . L E C K N E R

Department of Steam Engineering, Chalmers University of Technology 402 20 GOteborg 5, ~weden

Data on the infrared radiation characteristics of carbon dioxide and water vapor in the form of absorption coefficients and line spacings averaged over narrow specu~l intervals have been compiled from various sources. These data are to be used in heat transfer calculations from hot gases. In order to investigate the accuracy of the data, the simplest case possible is chosen: a comparison with the total emissivity charts of water vapor and carbon dioxide. It appears however that the charts ate not entirely reliable as standards for comparison: it seems probable that Hottel's chart for water vapor gives too low values at temperatures above 900°C and that the partial pressure correction is temperature dependent. With the exception of some regions where judgment is difficult, the calculations using spectral data seem to represent total emissivities with a maximum error which is estimated to around 10%. Sources of error in the spectral, data and in Hottal's total emissivity charts are discussed. Total emissivity charts, pressure and overlap corrections based on calculations with spectral data are presented.

Introduction The emissivity o f homogeneous gases is usually calculated by means c,f diagrams [1 ] established on the basis o f m~asur¢Tments of total emissivity of hot gases with various path lengths and temperatures. (By " to ta l" is raeant: including the whole spectrum.) These diagrams are also used, in the absence of better m,~,thods of calculation, for estimations of heat transfer from nonhomoge- neous gases such as flames, in such cases, however, one has no conception of the magnitude of' the error involved, in order to investigate the useful- ness of approximations, spectral data on the radiative characteristics of the gases must be em- ployed. However, gas spectra are so complicated that mean-values over spectral regions have to be used together with other approximations, even in calculations which are intended to be highly accurate. Thus, before using available spectral data in complicated calculations, they should be checked for the simplest situation possible. This could be a comparison of total emissivity read from the diagrams and that calculated with spectral data for homogeneous gases. Unfortunately the

diagrams cannot readily be used as a standard, since they may contain errors, especially in the regions which are based on extrapolations. There- fore, the comparison presented in this report can only give indications as to the reasonl~ for dis. crepancy between the results from the two sets o f data in certain eases. In other cases one has still to rely on assumptions.

As a direct result o f the 3oral emissivity calculations from spectral data total emissivity charts, pressure corrections and overlap corrections are given in analogy with Hot ters method of presenta- tion. Since emissivity calculations are often in- eluded in computer programs, the material will also be presented in the form of approximate empirical f'anetions, which are required to be as simple as possible within an error, relative tO the emissivities calculated with spectral data of less than 10%.

Theory The fundamental relationship for the'calculations of total emissivity o f carbon dioxide and wa~er vapor with spectral data is the statistical model

Copyright © 1972 by The Combustion Institute . . . . J=- .a . . . . . . . ~P , "

34

[2] for the emissivity of a spectral interval ~i. The inter~,al, sufficiently large to justify a statis. tical approach, is about 5-25 cm -~ . The strengths and positions of the lines in the interval are considered randomly strengths are assumed tributed.

Then the emissivity "spectral" emissivity) is

oriented and the line- to be exponentially dis-

of the region " i " (the

I_ ~ X 1 '(1) el = 1 - exp (1 + kiX/4al) 1/2 '

where ki: mean v~ue of the absorption coefficient in the ~terval, 1/bar cm, reduced to reference temperature and pressure, To = 273°K and Po = 1 bar

x ~ p A . T o (2) P0 T '

the optical path length, bar cm, reduced to reference temperature and pressure, p: partial pressure, bar, L: geometrical path length, era, and T: temperature of the gas, °K.

The fine structure parameger of the interval, ai, describes the line density

7 o~ = i " (3)

dl ; mean line spacing in the interval cm -l , and 3, rwaan line half-width of the gas at T, p and total pressure PT.

The line half-w:dth varies with pressure, temperature and mixing conditions. Different gas- components contribute somewhat differently to line-broadening. Here it is assumed that the active gas-component (H20 or CO2) is mixed with nitrogen. The influence of other gases or of H20 on CO2 or vice versa is neglected.

The line half-widths are for CO~. [19]

~ ' ~ - - ° ( 1 P ) Yc02 = 0 . 0 7 P r . + 0.28 ~T ' (4)

and for H 2 0 [3]

B. LECKNER

/~o ~ . To y.2o = 0.09PT. 3 / ~ + U.~p ~ - . (S)

TILe coefficients are determined by empirical methods and obtained from various sources.

The total emissivity c r is obtained by integration over the entire spectrum

~l ~biei~MO CT - - , (6)

aT 4

where /~biAco is the emitted energy of the spectral interval Ace according to Planek's law, and tr is Stefan-Boltzmann's constant.

The band-emissivity ~n of the nth band is defined ha the same way but the integration includes only the band region.

Water Vapor: Data Measurements of absorption coefficients and fine structure parameters have been performed by General Dynamics/Convair under NASA-contracts [3, 4 ] . Emission and absorption measurements were made with path lengths of 2-20 ft through a uniform volume of high temperature (1200- 2000°K) water vapor, produced above the combustion zone of a flat burner at wave lengths 1-10 /~m. The results are presented in form of values averaged over 25 em-I intervals. Earlier calculations and experimental results [5, 6] complete the set of absorption coefficients in the spectral region 50-9300 cm -1 and the temperature interval 300- 3000°K. The representation of the inverse of the line spacing cover the interval 50-7500 em -1 in the temperature range from 600-30000K, where the values below 1200°K and above 2500°K are extrapolations.

These data are deemed to be of good accuracy in the region 1150-7500 cm -I except in the troughs between the bands. The troughs become important only at path lengths above 600 bar era, however. On the other hand, the accuracy of the rotation baud data and of the line spacing in the extrapo. lated region might contain considerable errors.

Penner and Varanasi [7] have shown that the calculations of absorption coefficients in the rotation band [6] give lesults which are well below

EMISSIVITY OF WATER AND CARBON DIOXIDE

experimental values in the region 700-i000 cm-1 (at 500°K). Thus, at least the absorption coefficients at low temperatures should be replaced. At 300"K, data available from the meteorological literature [2] may be used for this purpose2 Here

line intensi'ty i~ 1 S i era/g, line half-width multi- N

plied with line intensity ~ ~ g - 1 / 2 from i= I

theoretical calculations by Benedict are given for use in band-models. The summation has been done over 20 em -1 intervals. The absorption coefficient averaged over 20 cm -1 is then, according to the definition of line intensity

k = ~ P0 1/har era, (7)

where Po is water vapor density at 273°K and 1 bar, P0 = 7.935 x 10 -4 g/era s, N is the number of lines in the interval, d ~s the average line spacing in the interval, and A~ = N. d, the width of the interval.

The average fine structure parameter is

a a~,.~ s~ ' (8) I=1

since

N 2

and

~s,. S = i=1

N

with an aver~ge line half-width y = 0.09 cm -~ , k and l id may be calculated and the corresponding values for A~0 = 25 em -1 are obtained.

Using ~ e just-overlapping line model Tejwani and Varanasi [8] have extrapolated the fine intensity calculations by Benedict to temperatures

that range 400-1200°K in the wavenumber region 450-1000 cm - t . Their Values at I 1000°K are recalculated to 25 cm -I Lu~arvalsi:a~ standard temperaPare with the corresponding values of [6]° at 1000°K are we n below those 0fLudw!g eta1 : Since it is stated by Tejwani and Varanasi that their absorption coefficients bec0me increasinglyl low at high temperatures (above i200?K)'~and, since there are not sufficient eXperimental'data available to check which of the two sets of dat.~ is most correct, the values of [6] are Used here above 600°K. Thus, the absorption c0efficien~s according to Tejwani and Varanasi are used only at 6 0 0 ° K . .

The line spacings in the rotation band are obtained by fitting the statistica~ model to the experimental data of [6] for temperatures at about 1000°K and higher and from the calculations above at 300~K. The value at 600°K is interpolated between 300 and 1000°K, Fig. 1.

Fil~re 3 indicates that the only spectral region, except the rotation band, that is of importance for tota~ emissivity calculations at low temperatures is the 6.3 /am band. The missing values of line

100.O

10.0

m 1.0

oJ lOOO 2000 3 ~

TEMPE~I'URE ~ Fig. 1. Average line spacing in the water vapor rotation band. The circle indicate!; the calculated value al 300°K.

36 B. LECKNER

10o.c ' t

/ t

'" Ii.i I i - - 1.0 i

0 1 / i I I

1000 2000 3QO0

TEMPERATIJP,~ °K

Fig. 2. Line spacing in th~ ,5.3 prn band of water vapor. The :ireles indicate average values from the data of [3]. The value at 300 is obtained by fitting to band absorption

data.

spacing may be derived by fitting the statistical model to band absorption measurements at 300°K. The experiments by Howard, Butch, and Williams are used [9]. This results in an average line spacing 1/(t for the 6.3/zm band of 0.35-0.40 cm. This value together with values from Ref. 3 are plotted in Fig. 2. It is evident from Fig. 2 that an alternative path of 1/d versus temperature below IO00°K is more probable than that of Ref. 3. The latter suggestion is also supported by fitting to other band absorption data [5]. As a result, in the 6.3/zm band the line spacing 1/d~s chosen to be 0.35 at 300°K and the values of Ref. 3 are increased by 50% at 600°K.

The remaining parts of the spectrum are less significant for total emissivity calculations and are treated in a more approximate way. The values of 1/d at 300°K are put equal to the ones of Ref. 3 at 600°K and the values at 600°K are increased by 50% since a similar situation prevails in the remaining bands. For band emt;sivity estimates, the line spacings at wavenumbers above 7500 cm -I are needed. Here average values for the 1.38 /zm band are simply extrapolated to 9300 cm -I . A summary of the (H20) spectral data used is presented in Table 1.

1D ¢V 1 5 1 J n ~ ~ ' 00.~ I cm

,,;;: .-

/ / / / ~ ' / / -

;;,:,, I / / , . i , . ,,:';" I I /

TgK

Fig. 3. Relative emissivities of water vapor spectral regions at various temperatures and path-lengths.


TABLE 1 Summary of the Spectral Data Used (H20)

37

Spectral Temperature . region

°K cm- I Sonrce Comment

Absorption 300-3000 1150-9300 [3,4,5] Measurements, or coefficient fitted to measurements.

1000-3000 50-1125 [S ] Calculated.

300 50-1000 [2] Calculated, modified.

600 450-1000 [8] Calculated, modified.

Line spacing 600-3000 1150-7500 [ 3, 4] Measurements and exttapolafious.

600 1150-7500 Modified.

1000-3000 50-1125 [4, 5] Fitted to meast~red data.

300 S0-1125 [2] Calculated, modified.

600 50-112S laterpoJ~ted,

300 1150-7500 Fitted to measured data, av • extrapolated.

300-3000 7525-9300 Ext:apolated.

TABLE 2 Sources, From Which the Water Vapor Emissivity Diagram of Hottal and Egbert [13, 1] is Composed

Temperature Path-length Accuracy stated Source °C bat cm Comments

Hottel and Smith [111 1 ~¢75-1525 1.2-2 Recalculated from measuremen ts on mixtures CO2-H~O

Hottel and Egbert [ 13 ] 143-704 0.5-500 <5%

Hottel and Maagelsdorf [ 12] 30-1000 0.24-51 These data lfave been corrected [13l and the lower limit may be put at 3 bar cm.

Hottel and Mangelsdorf 112] 1200 1 • 0 - 5 0 Absorption measurements. The error could be con- ~iderably above 5%.

Moreover, Hottal has used measurements of Schmidt [ 10], Eckert [14J, and Brooks [ 15].

The total emissivities are obtained from Hot~el's d~iagram [1] of water vapor emissivity at the equivalent partial pressure equal to zero, and from Schmidt's measurements at atmospheric pressure of pure water vapor [101.

Sehmidt's diagram is based on his own measure. ments whereas Hottel compiled data from dif-

ferent sources, among others to a certain extent from Schmidt.

Hot~el's chart is based on the data shown in Table 2.

Water Vapor Emissivity: General ~a order to estimate the influence of possible

38

TABLE 3 Water Vapor Band Regions

Wavenumber n Denomination region

1 P, otation band <1000 2 6.3 ~m band 1025-2500 3 2,7 ~m band 2525.-4500 4 1.~.7 ,am band 4525-6000 5 1.38 ,am band 6025-7700 6 7725-9300

errors and approximations in the total emissivity calculations it is advantageous to know the contd. butions of various spectral regions. Table 3 shows the partition of the water vapor spectrum in 6 band,;.

The relative importance of the hands is shown in Fig. 3 which is calculated with a total pressure of one bar and a line broadening (Eq. (5)) with p = 0 for various pathlengths.

Some conchisions are evident: I. At room temperature the rotation band is

dom!nant. Thus Benedict's calculated values [2] are almost entirely the basis of emissivity calculations in this temperature region.

B. LECKNER

2. The uncertain line spacing w.lues at 600°K are of great importance for total emissivity calculations mostly in the 6.3/~m band and to less extent in the rotation band and the 2.7 /~m band. These line spacing values influence the value of total emissivity at 400.-800°K,

3. Tile short wavelength region (above 7500 cm -~ ) may be omitted in the total emissivity calculations, whereas the long wavelength region (the rotation hand) plays an important role at all temperatures. The intermediate region (1.87 and 1.38 /an bands) is not essential at temperatures below 1000°K, but does contribute with about 10-20% of total emissivity at flame temperatures.

The pressure broadening o f spectral lines makes total emissivities pressure dependent. Only at high temperatures, where the lines saturate the spectrum, the pressure dependence vanishes and the emissivity, which in this case is called maAmum emissivity, remains a function o f temperature and path-length. Ho~tel and Egbert [13] chose to reduce the total emissivity data at atmospheric pressure and various partial pressures to emissivities at zero partial pressure, E 0 (equal to an

'7 ~dh;

Fig. 4.

° - _ 11 Iltl

°./ I . "

!1 t -----,........

10QD 5000 PATH LENGTH B ~ G M

Total emissivities of water vapor calculated from spectral data at PT = I bar and p = l bar, q compared with ¢1rlissivities a~ ":'r = I bar and p = 0 (equivalent press.urn of 1 bar) ¢o.


equivalent pressure of 1 bar; Eq. (9)). By means of a pressure correction correlation, the actual emissivity et~ at a partial pressure p is obtained. The same procedure will be followed here," since the correction in combustion systems at atmospheric total pressure will be small, alth6ugh important. An example of the magnitude of the correction for pure water vapor at atmospheric pressure is given in Fig. 4. Considering the fact that Hottel and Egbert used measured values from various somces in the temperature range 420-980°K and path- lengths 1.5-300 bar cm one may claim that their correlation agrees relatively well with Fig. 4. (It is interesting to note that the temperature dependence is particularly small in lhe region where most of the data of Hottel and Egbert lie.)

The general fo~rrn of the pressure correction relationship is ~r involved function of total pressure, partial pressure, temperature, path- lengtla, and gas components. It will be somewhat simplified by introducing an equivalent pressure P c , based on Eq. (5) valid for water vapor- nitrogen.

p ~K~/7). (9) P~ : P r ( l + 4 . 9

Then the pressure correction correlation is

~ = f(T,15L,PE). (10) ¢o

Wtmn the equivalent pressure is different from those of Fig. 4 the curves are merely displaced as a function of Pe. The position of the maxima depends with good accuracy only on temperature and may be represented with the empirical relationship

Ama~ = log(13.2r2), (11)

where the parameters

k = logpL, (12)

r = T / I O 0 0 ,

will be used throughout in the empirical representation. T is temperature in °K and pL is

39

1.O

0.5

A 0.0

-0.e J

~m'K /"g"

ler~

o.1 1.o lO.O lOO. Equlvarlnt preasum I~ (HIO)

Fig. 5. Maxima of craves similar to those of Fig. 4 for various equivalent pressures compared with ernissit, ities at Pe = t, co. The values of Fig. 4 are indicated with circles and the dasl:ed part of the curves represent the

case with PT = I and 0 < P < 1.

path-length in b~ir cm. The corresponding maximum values [~p/e0 - 1],-aax are plotted in Fig. 5 as a function of equivalent pressure. The dashed portion of Fig. 5 is valid for atmospheric pressure. This part of the curves may be represented by the empirical relationship

(e~-~O - l 1 max= log P~ (l .149 - 0.412r).03 )

A general representation (q" Fig. 5 must satisfy the asymptotic behavior at high pressures and the condition

i PE = 1

ep ~=1. It should thus be of the form

~ A ' P E +/3 (14) max P~ + A + I 3 - 1

where

40

A = 1.888 - 2.053 logr,

(r = 0.75 if T <750°K) ,

represents the asymptotic limit and

B = 1.10r -z '4

fits Eq. (14) to the curves at points not too far from the intersection. Now the pressure correction relationship may be normalized according to

[ (~ /~o ) - 1] f ( p L ) . ( 1 5 )

L(~,/,~o) -Z] , , ,~x

Neglecting t.he very weak temperature and pressure dependence left and assuming that the function of path-length,/'(pL) may be represanted by an exponential term

f(pL) = expl-~F(Ama x - A)2J, (16)

the description of the pressure correction is complete. By choosing the value ~ = 0.5 the error is smallest at atmospheric pressure and increases as the pressure departs from that value. The error is small at intermediate and high temperatures and increases towards lower temperatures.

It should be noted that the curves are calculated based on measurements at atmospheric pressure and are extended to higher or lower pressures by means of a model of line broadening valid for Lorentz.shaped lines. In the low pressure range of Fig. 5, Doppler broadening gradually becomes irr.por.tant and extrapolations are not possible ~thout considering the influence of Doppler broadening. In tho high pressure region physical changes in the behavior uf water vapor molecules may be of some importa~,ce.

Total Emissivity of Water Vapor: Result of Cow, parison Carves from the total emissivity charts of I-;ottel and from the mefisured emissivities of Schmidt are shown in Figs. 6 and 7 as solid lines compared with the corresponding values calculated with spectral data represented by dashed lines. A comparison between these total emissivity values

B. LECKNER

o~o o rmTlnL rm:ou~ o ms

o.:~o . . . . . - . . ,

~o.o~o • - - • •

tom tz~l 114OO t ~ :e¢o ~

TEMPERRTURE QEG. Fig. 6. Comparison betweeen calculated emissivities of water vapor (daubed lines) and emissivities from Hottel's

chart (solid lines).

o.~oo

o.soo o . ~

o.soo

o,~oo 1 >. ~-o.lo0

~o.¢m

o.o~o

o.oso

o.~o

o.o~o o.o~s o,ooe

~o ~oo eoo coo zor~ i~oo leoo leoo ~ooo I TEHPERATUBE DEG. C

Fig, 7, Compar ison between calculated emissivities o f water vapor (dashed l incs) and SchmJdt's measured valves

(sol id lines1.


and those calculated with GD/C's data have recently been presented by Boynton and Ludwig [16] and here only a few points will be discussed in order to analyze the reasons for divergency.

The agreement with Sehrnidt's emissivities is good except in some parts of the region of values extrapolated by Schmidt (incidently tlae 0.7 bar cm curve is extrapolated in Fig. 7 somewhat beyond the original curve). One notes that Sehmidt's curve at 0.7 bar cm is above the dashed curve whereas Hottel's curves in the same region fall considerably below. The disagreement above 10 bar cm at temperatures below 2000C can be explained considering the fact that Schmidt em- ployed band contours when calculating emissivities from spectral measurements by ttettner, thus ignoring the influence of line structure. These values are actually maximum emissivities and do agree with the corresponding maximum emissivities calculated with spectral data.

Figure 6 shows that there is a reasonable agreement between the calculated emissivities and Hottel's values in the region which is best covered by measured total emissivities, particularly those of Hottel and Egbert. At temperatures below and above this region there is a considerable discrepancy.

The original measurements at 1200°C by Hottel and Mangelsdorf [12] and by Eekert [14] must have been of decisive importance in supporting the extrapolations in the high temperature region of the final emissivity chart of Hottel and Egbert [13]. Comparing the original measurements with the corresponding calculated emissivities one finds a discrepancy of less than 10% at path-lengths above about 5 bar era. This discrepancy increases towards shorter path-lengths where it becomes about 30%. The measurements have been performed in a constant length cell, which means large partial pressure correction at large path. lengths and increasingly smaller corrections towards the smaller path-lengths. As pointed out above, the temperature independent partial pressure correction of Hottel and Egbert is essen- tially in agreement with the spectral partial pressure correction in the region for which it was established. However, at higher temperatures the temperature dependence of the spectral partial

41

pressure correction makes Hottel's correction an overestimate. This contributes to half of the discrepancy shown in Fig. 6 at larger path-lengths, but is insignificant at smaller path-lengths where the partial pressure is close to zero. The second reason for discrepancy comes from an unexplained ur'~derestimate of .*he points of Hottel and Mangels- dorf by Hottel and Egbert in preparing the final working chart.

Measurements made at path-lengths less than 3 bar cm by Hottel and Mangelsdorf and by Eckert have been discarded by Hottel and Egbert who showed that the presence of hurnid air between the radiometer ,~nd the cell could have contributed to making the results too low. This has been considered in the range below 700°C but not at 1200°C.

Some of these reasons for discrepancy can be applied to the measurements by Hottel and Smith, but it seems impossible to explain the entire difference between the two sets of emissivity data in the lowe~ path4ength range of the diagram, it should be noted however, that in the high temperature low path.length range the calculated emissivities are almost entirely independent oi line structure (Fig. 4) and possible errors should be found in the absorption coefficients. The absorption coefficients are in this region obtained by measurements whereas Hottel's emissivities are extrapolations.

The disagreement at long and short path.lengths at room temperature is of a certain importance since it could be due to an error in the description of the line structure in the rotation band. This could affect also calculated emissivities at intermediate temperatures (Fig. 3) since the room temperature value of line spacing serves as a basis of interpolation (Fig. 1). A comparison with available total emissivities at room temperature, those of Brooks (1950) and of Cowling, both sets described by Goody [2] and those of Staley and Jurica [17] shows good agreement with the emissivities calculated from spectral d~ta at path- lengths above 1 bar cm, whereas emissivities from the various sources differ considerably as the path-length decreases below I bar era.

The conclusion, then, would be that the comparison made provides a justification to use the

42 B. LECKNER

0. 780 I I I I f I i - ~ ~ ~ WIqTEB VRPOB -,

0.500 4o0. TOTRL PRESSURE t 8,I:IR ~ - z ~ P~R,R, ~ESSURE 0 BR. o.,oo -

~0.080 \ ~ ~ ~ ~. 0.0,0 % --.2

O. 020 ~ ~ ~ "~"~.~

o.o~o "-- " , o.oo~ ~ ~ O, 008 I I I " ~ -

200 qO0 500 800 lOOO 1~00 lqO0 1600 1800 2000 2200

TEHPERATU6E DEG. C Fig, 8, Calculated total emissivities of water vapor.

emissivities and the line spacings calculated from Benedict's data at room temperatures. It seems probable that Hottel's data of water vapor total emissivities are too low at temperatures above about 900°C. On the other hand, in the range covered by measurements by Hottel and Egbert, Hottel's curves are likely to be more reliable. Only in the small path-length range no conclusion can be made.

An emissivity chart calculated with the spectral data is presented in Fig. 8. This diagram may be repre.oented with two polynomials of 2nd order (M = 2,N = 2) with a maximum error of less than ~'5% for temperatures greater than 100°C:

kl

In ct~ = fro + ~ o iA i , i=[

tz i = co l + Ciir~. i=1

(l 7)

The coefficients c i i are given in the Appendix.

Total Emissivity of Carbon Dioxide Figure 9 shows the total emissivity of carbon dioxide calculated with spectral data (dashed lines) compared with emissivities from Hottel's chart (solid lines). The spectral data used and the


o, Io~

,

. . . . " . . . . . . .

TEHPEflRTURE OEG. C

Fig. 9. Comparison between calculated emissivities of carbon dioxide (dashed lines) and emissivities fJtom

Hottel's chart (solid lines).

reasons for discrepancy have been discussed in Ref. 18. In summary, it should be mentioned that spectral data are defined within discrete band regions. At path-lengths approaching 1000 bar cm and above the neglected ~:ery weak bands outside these regions may make the calculated total emissivities increasingly too low. At high pressures and relatively large path-lengths contributions from wings of spectral lines that penetrate outside of the band limits used, may likewise contribute to making the calculated total emissivities low.

It has been shown [18] that a large part of the discrepancy at short path-lengths may be explained if the absorption in the air-layer that could have been present between the radiometer and gas container in the measurements of Hottei and Mangelsdorf, is taken into consideration. At temperatures below about 600°K the line structure of carbon dioxide gradually becomes ordered and the statistical model is less suitable.. However, the fine structure parameter has been obtained by fitting the statistical model to measured band absorption data and the error should be of the same order as the rest of the diagram. The only evident source of error at low temperatures comes from the 4.3 pm

43

band where the contribution of a weak vibration- rotation band has been neglected.

The carbon dioxide emissivities of Fig. 9 are calculated for line broadening at 1 bar total pressure and zero partial pressure (whereas Hottel's diagram includes emissivities at i bar total pressure at various partial pressures). However, at atmospheric pressure the influence o f pressure on total emissivities amounts to a few percent and may be neglected altogether. Also at high pressures the pressure correction is small, especially at intermediate and high temperatures. The pressure correction behaves similar to ~.at of water vapor (Figs. 4 and 5) except for a reversal in the position of the maxima at temperatures 300-700°K. This depends on a change over in the main contribution of emissive power from the 4.3/ lrn and 2.7/ira bands zLt higher temperatures to the 15 pm band at lower temperatures. The functions, which describe the pressure correction analogous to those of water vapor but based on an eqnivalent pressure from Eq. (4),

are given in the Appendix. The empirical ~epresen- tation underestima:es ~be correction somewhat on the long path-length side far from the maxima. Since the correction ~ere is small in itself, the error in the resulting total emissivity is small. Coefficients to approximations of carbon dioxide total emissivities of Fig. 9, according to Eq. (17) are also given in the Appehdix.

Overlapping Finally some comparisom of the overlapping correlations will be made. When carbon dioxide and water vapor are both present in a gas-mixture, the total emissivity is equal to the sum of emissivities of the two gases minus a correction term A~ due to overlap in some ~pectral regions

= ~"2o + ~co2 - A~. (18)

In a spectral interval Aco at the wave number oa, the resulting transmission r is

,14

415~K

0,o4

o ~

~ . ~

B. LECKNER

Offt

800"K 0 ~

o 004 o

o~°~ ~ ° .,.,s o It

OD 112 0.4 Q6 118 ~

t.~06

0~)4

OD3

CL01

OCO O~ 02 0.4 O~ 0.8 tO

Fig. 10. Overlap correction according to Hottel (solid lines) compared with calculated overlap corrections.

EMISSIVITY OF WATER AND CARBON DIOXIDE 45

0.15

0.10

~06

OgD

P~o %.--~cq "°'e

o

~70o

# / 2azo

/ - y,o/O

r,O 11:10 100. 1000.

I ~ o " ~:~1 L barcm

Fig. 1 1. Overlap correction at ¢~ = 0.6 as: a fimcIion of path-length at vaxio¢.~ temperatures.

;-(co) = ra20 (col. rco 2 (co).

Since r(co) = 1 - ~(co),

(co) = ~"2o (co) + ~co2(co)

- ¢nzo (co).6c02 (co), (19)

and Eq. (18) is obtained through integration of Eq. (19).

The overlap correction is then

= " J ~co2 (co)" ~.2o (col ~bicol ~ . (20) At w CrT 4

The overlap diagrams of Hottel, based on cross- plotting of data calculated by Eckert from available absorption spectra of carbon dioxide and water vapor are not accurate enough to permit any conclusions to be drawn from comparisons with calculated A~ from Eq. (20). The agreement, however, is relatively good (Fig. I0). Only at path-lengths exceeding 90 bar em are dis- crepancies considerable. Figure 11 illustrates the

temperature and path4ength relationships at = (PHzo/pa~o + pco2)= 0.6. The overlap

correction has a broad maximum at temperatures from 1200-1700°K and decreases onIy slowly with increasing temperateres, whereas the decreese is rather rapid at temperatures below 1000°K. Her e, overlapping becomes important only inthe 15#m region of the spectrum. The low temperature curve of Fig. I 1 is approximately a minimum value and may be used down to room-temperatures. The crosses in Figs. 10 ~ 11 indicates values calculated with pH2o + Pco 2 = 0.30 and pN2 = 0.7. The cirale~ belonging to the temperatures 415, 800, and IIP0°K show the result of the case Pn2o + Pco2 = L0. (The total pressure is 1 bar and the influence of the two active gas components on the line broadening of each other has been neglected. The values of one active gas versus inactive gas consisting of nitrogen have been used throughout.) An increase of pressure till saturation of the spectra would mean an increase of A¢ with 20-30% at ~ = 0.6 and somewhat m o r e at lower since the composition curves become more symmetric around ~ = 0.5 when the line structure

46 B. LECKNER

sensitMty of water vapor diminishes. It is also seen in Fig. 11 that the overlap correction is negligibly small at path-lengths below about 20 bar cm.

Considering the relatively small influence of A~ on the total emissivity of the mixture, a descrip- tlou of the overlap coriection may be rather approximative. Thus the overlap .correction at temperatures of 1000-2200°K at all pressures is represented by one curve, which is drawn through the values of 1 bar. In this, way the empirical correction correlation remain.~ a function of path- length and composition. The empirical function is given i~. Appendix.

Conclusion It has been shown that the spectral data used with the statistical model, although consisting pm'tly of interpolated values and of regions where accuracy is difficult to check, do represent total emissivities of carbon dioxide and water vapor in homoge-

aeous gases with an 3ccuracy which is sufficient for heat transfer calculations. Only in some parts covered by the diagrams, tile maximum error may be larger than 10%; around 0.1 bar cm in carbon dioxide and in the low path-length region of the water vapor diagram. Furthermore the low temperature part of the diagrams is less well defined than the high temperature regions.

It remains to be shown that the data are valid also in noahomogeneous gases. At present, however, there are no suitable measurements available for comparison.

The accuracy of the empirical description of the diagrams of total emissivity and corrections is around a few percent in most of the regior~ covered by the diagrams. Only in a fcw tow temperature regions it amount~ to 10%. Since the description is not based on theoretical relationship, extrapol,~tions outside the diagrams should be done with c,~re.

Appendix

Coefficients to be tired in Eq. (17)

Watervapor, T :~400°K,M = 2,N= 2

i COl Cli C2i

0 -2.2118 -1.1987 0.035596 1 0.85667 0.93048 -0.14391 2 -0.Ie838 -0.17156 0.045915

Deflation fiom v~uescalculatedwith spectr~ datais max ,5%

C~bon dioxide, T >400 ~ K, M = 2, N = 3

i C01 Cli C21 C3t

0 - 3.9893 2.7669 - 2.1081 0.39163 I 1.2710 - 1.1090 1.0195 - 0.21897 2 -0.23678 0.19731 -0.19544 0.044644

Deviation from values c~culated with spectral data is max +10%, pL > 0.03

EMISSIVITY OF (:IATER AND CARBON DIOXIDE

Carbon dioxide, T > 400 °K, M = 3, N = 4

i Col Cu 021 C31 C41

47

0 -3.9781 2.7313 -1.9882 0.31054 0.015719 1 1.9326 -3.5932 3.7247 -1A535 0.20132 2 -0.35366 0.61766 -0.84207 0.39859 -0.063356 3 --0.080181 0.31466 -0.19973 0.046532 -0.0033086

Deviation from values calculated with spectral data is max -+5%

Carbon dioxide, Hottel's diagzam, M = 2, N -~,

i col cli c2i CM

0 -3.3390 1.1996 -1.0604 0.16454 1 0.90786 0.086726 0.13797 -0.035144 2 -0.15563 -0.10292 0.064443 -0.014128

and M = 3 , N = 4

i Co~ Cll C2i C3i e4i

0 -3.0380 0.0S7994 0A4952 -0.62679 0.14030 1.1288 -1.0822 i.5792 -0.74749 0.12207

2 -0.25513 0.045499 -0.22845 0.16615 -0.034597 3 0.036827 0.040937 0.018056 -0.031075 ' 0.0076346

The accuracy is of the same order as above.

Pressure CorteeUon, Carbon Dioxide Equivalent pressure

P c = PT (l + 0.28 P ) •

Position of maxima

Amax = log(0.225r 2) when T > 700°K

Ama~ = log(0.054r -2) when T < 700°K.

Functions in Eq. 0 4 )

A = 0.10r -L4~ + 1.0

B = 0.23.

Coefficient in Eq. (16)

= 1 . 4 7 .

Overlap Correction A¢

= P . 2 o

Pu:o + Pco2

~e {10.7 + - ~ 0 1 - ~ - 0"0089~"°'4"~'hz'° " ,

This work has been supported in pare by the Swedish Board o f Technical De~elopment.

48

P~ferences 1. Hottel, H. C., and Sarof'm3, A. F., Radiative Transfer,

McGraw-Hill, New York (1967). 2. Goody, R. M., Atmospheric Radiation, Part I,

TheoreticalBasis, Oxford, London (196g). 3. Study on exhaust plume radiation predictions, Gen-

eral Dynamics]Convair, NASA CR-61233, 1968. 4. Study on exhaust plume radiation predictions, Final

report, General Dynamics/Convair, GD/C-DBE-66- 017,1966.

5. Feniso, C. C., Ludwig, C. B., and Thomson, A. L.,J~ Quans. Spectrosc. Radiat. Transfer 6, 241 (1966).

6. Ludwig, C. B., Ferriso, C. C., Ma[kmus, W., and Boyuton, F. P., 3". Quant. Spectros¢. Radiat. Tmnsfa" 5, 697 (1965).

7. Penner, S. C., and Varaaasi, P., Z Quant. Spectrosc. Radiat. Transfer 7, 687 (1967).

8. TejwanJ, G. D. T., and Vatanasi, p., J. Quant. Speetrose. Radiat. Transfer 10, 373 (1970).

9. Howard, J. N., Butch, D. E., and WilIiaras, D., Geophys. Res. Paper No. 40, Air Force Cambridge Research Center, 1955.

B. LECKNER

10. Sclunidt, E., Forsehung Geb. Ing. Wes. 3, 57 (1932). I1. Hottel, H. C., and Smith, V. C., Trans. A,~dffE 57, 463

(1935). I2. Hottel, H. C., and Mangelsdorf, H. C., Trans. Am.

Inst. Chem. Engrs. 31,517 (1935). 13. Hottel, H. C., and Egbert, R. B., Trans. Av~ Inst.

Chem. Eng. 38, 531 (1942). 14. Eckett, E., YDI.Forsehungsheft 387 (1937). 15. Brooks, F. A.~Pap. Phys. Ocean. Met. 8, 2 (1941). 16. Boynton, F. P., and Ludwig, C. B., Int, J. HeatMass

Tramfer 14, 963 0971). 17. Staley, D. O., and Jurica, G. M.,J. ,~ppl. MeteoroL 9,

365 (1970). 18. Leclmer, B., Combus•'on and Flame 17, 37 (1971). 19. Study on exhaust plume rac~ation predictions,

Intetfim Progress Report, Parts I and II, General Dynamics/Convair, GD/C-DBE-66-001 and GD/C-DBE-66-001a, 1966.

(Received November 19 71 )

spectral and total emissivity of water vapor and carbon dioxide

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