Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 92
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8.1. It is seenthat emission is zero at both extreme ends of the spectrum with a maximum at someintermediate wavelength. The general level of emission rises with temperature, andthe important part of the spectrum (the part containing most of the emitted energy)shifts toward shorter wavelengths. Because emission from the sun (“solar spectrum”)is well approximated by blackbody emission at an effective solar temperature ofTsun = 5762 K, this temperature level is also included in the figure. Heat transferproblems generally involve temperature levels between 300 and, say, 2000 K (plus,perhaps, solar radiation).
Therefore, the spectral ranges of interest in heat transferapplications include the ultraviolet (0.1 to 0.4 µm), visible radiation (0.4 to 0.7 µm,as indicated in Figure 8.1 by shading), and the near- and mid-infrared (0.7 to 20 µm).For quick evaluation, a scaled emissive power can be written asEbλC1=n3 T 5(nλT )5 (eC2 /nλT − 1)(n = const)(8.6)whereC1 = 2πhc02 = 3.7419 × 10−16 W · m2C2 =hc0= 14,388 µm · KkEquation (8.6) has its maximum at(nλT )max = C3 = 2898 µm · KBOOKCOMP, Inc. — John Wiley & Sons / Page 576 / 2nd Proofs / Heat Transfer Handbook / Bejan(8.7)———Normal PagePgEnds: TEX[576], (4)FUNDAMENTALS123456789101112131415161718192021222324252627282930313233343536373839404142434445577which is known as Wien’s displacement law.
The constants C1 , C2 , and C3 are knownas the first, second, and third radiation constants, respectively.The total blackbody emissive is found by integrating eq. (8.6) over the entirespectrum, resulting inEb (T ) = n2 σT 4(8.8)where σ = 5.670 × 10−8 W/m2 · K4 is the Stefan–Boltzmann constant. It is oftendesirable to calculate fractional emissive powers, that is, the emissive power contained over a finite wavelength range, say between wavelengths λ1 and λ2 . It is notpossible to integrate eq.
(8.6) between these limits in closed form; instead, one resortsto tabulations of the fractional emissive power, contained between 0 and nλT ,λ nλTEbλ0 Ebλ dλf (nλT ) = ∞d(nλT )(8.9)=3 σT 5n00 Ebλ bλ[577], (5)Lines: 211 to 250so thatλ2———Ebλ dλ = [f (nλ2 T ) − f (nλ1 T )]n σT24(8.10)3.77525pt PgVar———Normal PageAn extensive listing of f (nλT ), as well as of the scaled emissive power, eq. (8.6), * PgEnds: Ejectis given in Table 8.1. Both functions are also shown in Fig. 8.2, together with Wien’sdistribution, which is the short-wavelength limit of eq. (8.5),[577], (5)2πhc02 −hc0 /nλkThc0C1 −C2 /nλTEbλ 2 5 e= 2 5e1(8.11)nλkTn λn λλ1As seen from the figure, Wien’s distribution is actually rather accurate over the entirespectrum, predicting a total emissive power approximately 8% lower than the onegiven by eq.
(8.8). Because Wien’s distribution can be integrated analytically overparts of the spectrum, it is sometimes used in heat transfer applications.8.1.2Solid AnglesRadiation is a directional phenomenon; that is, the radiative flux passing througha point generally varies with direction, such as the sun shining onto Earth fromessentially a single direction. Consider an opaque surface element dAi , as shown inFig. 8.3.
It is customary to describe the direction unit vector ŝ in terms of polar angle θ(measured from the surface normal n̂) and azimuthal angle ψ (measured in the planeof the surface, between an arbitrary axis and the projection of ŝ); for a hemisphere0 ≤ θ ≤ π/2 and 0 ≤ ψ ≤ 2π.The solid angle with which a surface Aj is seen from a certain point P (or dAi inFig.
8.3) is defined as the projection of the surface onto a plane normal to the directionvector, divided by the distance squared, as also shown in Fig. 8.3 for an infinitesimalBOOKCOMP, Inc. — John Wiley & Sons / Page 577 / 2nd Proofs / Heat Transfer Handbook / Bejan578123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATIONTABLE 8.1Blackbody Emissive PowerEbλ /n3 T 5nλT(W/m2 · µm · K 5 )(µm · K)×10−11f (nλT )1,0001,1001,2001,3001,4001,5001,6001,7001,8001,9002,0002,1002,2002,3002,4002,5002,6002,7002,8002,9003,0003,1003,2003,3003,4003,5003,6003,7003,8003,9004,0004,1004,2004,3004,4004,5004,6004,7004,8004,9005,0005,1005,2005,3005,4005,5000.021100.048460.093290.157240.239320.336310.443590.556030.668720.777360.878580.969941.049901.117681.173141.216591.248681.270291.282421.286121.282451.272421.257021.237111.213521.186951.158061.127391.095441.062611.029270.995710.962200.928920.896070.863760.832120.801240.771170.741970.713660.686280.659830.634320.609740.586080.000320.000910.002130.004320.007790.012850.019720.028530.039340.052100.066720.083050.100880.120020.140250.161350.183110.205350.227880.250550.273220.295760.318090.340090.361720.382900.403590.423750.443360.462400.480850.498720.515990.532670.548770.564290.579250.593660.607530.620880.633720.646060.657940.669350.680330.69087BOOKCOMP, Inc.
— John Wiley & Sons / Page 578 / 2nd Proofs / Heat Transfer Handbook / BejanEbλ /n3 T 5nλT(W/m2 · µm · K 5 )(µm · K)×10−115,6005,7005,8005,9006,0006,1006,2006,3006,4006,5006,6006,7006,8006,9007,0007,1007,2007,3007,4007,5007,6007,7007,8007,9008,0008,2008,4008,6008,8009,0009,2009,4009,6009,80010,00010,20010,40010,60010,80011,00011,20011,40011,60011,80012,0000.563320.541460.520460.500300.480960.462420.444640.427600.411280.395640.380660.366310.352560.339400.326790.314710.303150.292070.281460.271290.261550.252210.243260.234680.226460.211010.196790.183700.171640.160510.150240.140750.131970.123840.116320.109340.102870.096850.091260.086060.081210.076700.072490.068560.06488f (nλT )0.701010.710760.720120.729130.737780.746100.754100.761800.769200.776310.783160.789750.796090.802190.808070.813730.819180.824430.829490.834360.839060.843590.847960.852180.856250.863960.871150.877860.884130.889990.895470.900600.905410.909920.914150.918130.921880.925400.928720.931840.934790.937580.940210.942700.94505(continued)[578], (6)Lines: 250 to 309———-1.53975pt PgVar———Normal PagePgEnds: TEX[578], (6)FUNDAMENTALS123456789101112131415161718192021222324252627282930313233343536373839404142434445TABLE 8.1Blackbody Emissive Power (Continued)Ebλ /n3 T 5nλT(W/m2 · µm · K 5 )(µm · K)×10−11f (nλT )12,20012,40012,60012,80013,00013,20013,40013,60013,80014,00014,20014,40014,60014,80015,00015,20015,40015,60015,80016,00016,20016,40016,60016,80017,00017,20017,40017,60017,80018,00018,20018,40018,60018,80019,0005790.061450.058230.055220.052400.049760.047280.044940.042750.040690.038750.036930.035200.033580.032050.030600.029230.027940.026720.025560.024470.023430.022450.021520.020630.019790.018990.018230.017510.016820.016170.015550.014960.014390.013850.013340.947280.949390.951390.953290.955090.956800.958430.959980.961450.962850.964180.965460.966670.967830.968930.969990.971000.971960.972880.973770.974610.975420.976200.976940.977650.978340.978990.979620.980230.980810.981370.981910.982430.982930.98340Ebλ /n3 T 5nλT(W/m2 · µm · K 5 )(µm · K)×10−11f (nλT )19,20019,40019,60019,80020,00021,00022,00023,00024,00025,00026,00027,00028,00029,00030,00031,00032,00033,00034,00035,00036,00037,00038,00039,00040,00041,00042,00043,00044,00045,00046,00047,00048,00049,00050,0000.012850.012380.011930.011510.011100.009310.007860.006690.005720.004920.004260.003700.003240.002840.002500.002210.001960.001750.001560.001400.001260.001130.001030.000930.000840.000770.000700.000640.000590.000540.000490.000460.000420.000390.000360.983870.984310.984740.985150.985550.987350.988860.990140.991230.992170.992970.993670.994290.994820.995290.995710.996070.996400.996690.996950.997190.997400.997590.997760.997920.998060.998190.998310.998420.998510.998610.998690.998770.998840.99890element dAj .
If the surface is projected onto a unit sphere above point P , the solidangle becomes equal to the projected area, ordAjpcos θ0 dAjΩ=== Ajp(8.12)2S2Ajp SAjwhere S is the distance between P and dAj . Thus, an infinitesimal solid angle issimply an infinitesimal area on a unit sphere, orBOOKCOMP, Inc. — John Wiley & Sons / Page 579 / 2nd Proofs / Heat Transfer Handbook / Bejan[579], (7)Lines: 309 to 369———0.47249pt PgVar———Normal Page* PgEnds: Eject[579], (7)580123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATIONdΩ = dAjp= (1 × sin θ dψ)(1 × dθ) = sin θ dθ dψ(8.13)Integrating over all possible directions yields 2π π/2sin θ dθ dψ = 2πψ=0(8.14)θ=0[580], (8)Lines: 369 to 395———0.84001pt PgVar———Long Page* PgEnds: Eject[580], (8)Figure 8.2 Normalized blackbody emissive power spectrum.dAjddAj⬙1ndAin00ssin dPdFigure 8.3 Definitions of direction vectors and solid angles.BOOKCOMP, Inc.
— John Wiley & Sons / Page 580 / 2nd Proofs / Heat Transfer Handbook / BejandAjpFUNDAMENTALS123456789101112131415161718192021222324252627282930313233343536373839404142434445581for the total solid angle above the surface. If a point inside a medium removed fromthe surface is considered, radiation passing through that point can strike any pointof an imaginary unit sphere surrounding it; that is, the total solid angle here is 4π,with 0 ≤ θ ≤ π, 0 ≤ ψ ≤ 2π. Similarly, at a surface one can talk about an upperhemisphere (outgoing directions, 0 ≤ θ < π/2), and a lower hemisphere (incomingdirections, π/2 < θ ≤ π).8.1.3Radiative IntensityThe directional behavior of radiative energy traveling through a medium is characterized by the radiative intensity I , which is defined asI ≡ radiative energy flow/time/area normal to the rays/solid angle[581], (9)Like emissive power, intensity is defined on both spectral and total bases, related by ∞Iλ (ŝ,λ) dλ(8.15)I (ŝ) =0———However, unlike emissive power, which depends only on position (and wavelength),the radiative intensity depends, in addition, on the direction vector ŝ.
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