Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 100
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First, the individual emissivities for water vapor and carbon dioxide, respectively, are calculated separately from(pa L,p,Tg ) = 0 (pa L,Tg )(pa L,p,Tg )0(8.77a) (pa L)m 2(a − 1)(1 − PE )(pa L,p,Tg ) = 1 −exp −c log100a + b − 1 + Pepa L j iM NTgpa L0 (pa L,Tg ) = exp log10cj iT(p0a L)0i=0 j =0(8.77b)(8.77c)[617], (45)Here 0 is the total emissivity at a reference state, which is p = 1 bar total pressure andpa → 0 (but pa L > 0). The correlation constants a, b, c, cj i , PE , (pa L)0 , (pa L)m ,Lines: 1417 to 1466and T0 are given in Table 8.4 for water vapor and carbon dioxide (for convenience,plots of 0 are given in Fig. 8.23 for CO2 and Fig.
8.24 for H2O). The total emissivity———of a mixture of nitrogen with both water vapor and carbon dioxide is calculated from * 28.55127pt PgVar———CO2 +H2 O = CO2 + H2 O − ∆(8.78)Long Page2.76* PgEnds: PageBreakζ(pH2 O + pCO2 )L∆ =(8.79)− 0.0089ζ10.4 log1010.7 + 101ζ(pa L)0[617], (45)TABLE 8.4 Correlation Constants for the Determination of the Total Emissivityfor Water Vapor and Carbon DioxideM, Nc00 · · · cN0.. . . ...
. .c0M · · · CNMPE−2.2118H2OCO22,22,3−1.1987−0.035596−3.9893−2.7669−2.1081−0.39163−0.85667 −0.93048 −0.14391 −1.2710 −1.1090 −1.0195 −0.21897−0.10838 −0.17156 −0.045915 −0.23678 −0.19731 −0.19544 −0.044644√(p + 2.56pa / t)/p0(p + 0.28pa )/p0(pa L)m /(pa L)013.2t 20.054/t 2 , t < 0.70.225t 2 , t > 0.7a2.144,t < 0.751.888 − 2.053 log10 t, t > 0.751 + 0.1/t 1.45b1.10/t 1.40.23c0.51.47T0 = 1000 K,p0 = 1 bar, t = T /T0 , (pa L)0 = 1 bar · cmSource: Leckner (1972).BOOKCOMP, Inc. — John Wiley & Sons / Page 617 / 2nd Proofs / Heat Transfer Handbook / Bejan618123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATION[618], (46)Lines: 1466 to 1478———5.854pt PgVarFigure 8.23 Total emissivity of carbon dioxide at a total pressure of 1 bar and zero partialpressure. (From Leckner, 1972.)———Normal Page* PgEnds: Eject[618], (46)Figure 8.24 Total emissivity of water vapor at a total pressure of 1 bar and zero partialpressure. (From Leckner, 1972.)BOOKCOMP, Inc.
— John Wiley & Sons / Page 618 / 2nd Proofs / Heat Transfer Handbook / BejanRADIATIVE PROPERTIES OF PARTICIPATING MEDIA123456789101112131415161718192021222324252627282930313233343536373839404142434445619whereζ=pH 2 OpH2 O + pCO2and where the ∆ compensates for overlap effects between H2O and CO2 bands, andthe CO2 and H2O are calculated from eq. (8.77).If radiation emitted externally to the gas (e.g., by emission from an adjacent wallat temperature Tw ) travels through the gas, the total amount absorbed by the gas is ofinterest.
This leads to the absorptivity of a gas path at Tg with a source at Tw : ∞1α(pa L, p, Tg , Tw ) =[1 − e−κλ (T )g L ]Ebλ (Tw ) dλ(8.80)Eb (Tw ) 0[619], (47)which for water vapor or carbon dioxide may be estimated fromα(pa L, p, Tg , Tw ) =TgTw1/2Tw pa L , p, TwTg(8.81)where is the emissivity calculated from eq. (8.77) evaluated at the temperature ofthe surface, Tw , and using an adjusted pressure path length, pa LTw /Tg . For mixturesof water vapor and carbon dioxide, band overlap is again accounted for by takingαCO2 +H2 O = αCO2 + αH2 O − ∆(8.82)with ∆ evaluated for a pressure path length of pa LTw /Tg .8.4.2Particle CloudsNearly all flames are visible to the human eye and are therefore called luminous(sending out light). Apparently, there is some radiative emission from within the flameat wavelengths where there are no vibration–rotation bands for any combustion gases.This luminous emission is today known to come from tiny char (almost pure carbon)particles, called soot, which are generated during the combustion process.
The dirtierthe flame, the higher the soot content and the more luminous the flame.Soot Soot particles are produced in fuel-rich flames, or fuel-rich parts of flames,as a result of incomplete combustion of hydrocarbon fuels. As shown by electronmicroscopy, soot particles are generally small and spherical, ranging in size betweenapproximately 5 and 80 nm and up to about 300 nm in extreme cases. Althoughmostly spherical in shape, soot particles may also appear in agglomerated chunks andeven as long agglomerated filaments.
It has been determined experimentally in typicaldiffusion flames of hydrocarbon fuels that the volume percentage of soot generallylies in the range 10−4 to 10−6 %.Because soot particles are very small, they are generally at the same temperature asthe flame and therefore strongly emit thermal radiation in a continuous spectrum overBOOKCOMP, Inc. — John Wiley & Sons / Page 619 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 1478 to 1521———0.88713pt PgVar———Normal PagePgEnds: TEX[619], (47)620123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATIONthe infrared region. Experiments have shown that soot emission often is considerablystronger than the emission from the combustion gases.For a simplified heat transfer analysis it is desirable to use suitably defined meanabsorption coefficients and emissivities.
If the soot volume fraction fv is known aswell as an appropriate√ spectral average of the complex index of refraction of the soot,m = n − ık(ı = −1), one may approximate the spectral absorption coefficientfrom Felske and Tien (1977) asκλ = C0fvλC0 =(n2−k236πnk+ 2)2 + 4n2 k 2(8.83)and a total or spectral-average value may be taken asκm =3.72fv C0 TC2(8.84)where C2 = 1.4388 cm · K is the second Planck function constant. Substituting eq.(8.84) into eq. (8.76) gives a total soot cloud emissivity of(fv TL) = 1 − e−κm L = 1 − e−3.72C0 fv T L/C2[620], (48)Lines: 1521 to 1552———(8.85)4.96216pt PgVar———Normal PagePulverized Coal and Fly Ash Dispersions To calculate the radiative proper- * PgEnds: Ejectties of arbitrary size distributions of coal and ash particles, one must have knowledgeof their complex index of refraction as a function of wavelength and temperature.Data for carbon and different types of coal indicate that its real part, n, varies little[620], (48)over the infrared and is relatively insensitive to the type of coal (anthracite, lignite,bituminous), while the absorptive index, k, may vary strongly across the spectrumand from coal to coal.
If the number and sizes of particles are known and if a suitableaverage value for the complex index of refraction can be found, the spectral absorptioncoefficient of the dispersion may be estimated by a correlation given by Buckius andHwang (1980). They observed spectral behavior to be weak (similar to that of smallsoot particles) and that spectrally averaged properties do not depend appreciably onthe optical properties of the coal. However, due to their larger size, coal particles tendto scatter radiation as well as absorb and emit radiation, leading to the definition ofthe scattering coefficient σs and extinction coefficient β = κ + σs .
Interpolating thedata of Buckius and Hwang, crude approximations for spectrally averaged absorptionand extinction coefficients may be determined from−6/5 −5/61.8 −6/5 φ10.99κm= 0.0032 1 ++fA425φ0.02−5/4 −4/52.0 −5/4 βmφ13.75= 0.0032 1 ++fA650φ0.13BOOKCOMP, Inc. — John Wiley & Sons / Page 620 / 2nd Proofs / Heat Transfer Handbook / Bejan(8.86)(8.87)RADIATIVE EXCHANGE WITHIN PARTICIPATING MEDIA123456789101112131415161718192021222324252627282930313233343536373839404142434445621where fA is the total projected area of particles per unit volume (e.g., fA = πa 2 N foruniform spheres of radius a and a particle density of N particles/unit volume), and φis a size parameter defined asφ = āTā =3fv4fAa in µm, T in K(8.88)where ā is an average particle size.
This leads to total coal cloud emissivity andabsorptivity:α(φ) = (φ) = 1 − e−κm L(8.89)On the other hand, if one is interested in transmitted radiation (i.e., radiation notabsorbed or scattered away), the cloud transmissivity becomesτ(φ) = e−βm L(8.90)If both soot as well as larger particles are present in the dispersion, the absorptioncoefficients of all constituents must be added before applying eqs. (8.89) and (8.90).Mixtures of Molecular Gases and Particulates To determine the total emissivity of a mixture, it is generally necessary to find the spectral absorption coefficientκλ of the mixture (the sum of the absorption coefficient of all contributors), followedby numerical integration of eqs.
(8.89) and (8.90). However, because molecular gasestend to absorb only over a small part of the spectrum, to some degree of accuracymix gas + particulates(8.91)Equation (8.91) gives an upper estimate because overlap effects result in lower emissivity [compare eq. (8.78) for gas mixtures].8.5RADIATIVE EXCHANGE WITHIN PARTICIPATING MEDIATo calculate the radiative heat transfer rates within—and to the bounding wall of—aparticipating medium, it is necessary to solve the radiative transfer equation (RTE),dIλσsλ= ŝ · ∇Iλ = κλ Ibλ − βλ Iλ +Iλ (ŝi )Φλ (ŝi , ŝ)dΩi(8.92)ds4π 4πto some degree of accuracy, followed by integration over all directions and all wavelengths, to obtain the radiative heat flux desired.
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