Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 99
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In these situations any two shields Ai and Aj often encloseone another, or are very close together, such that Fi−j 1. The radiative heat transferbetween two diffusely reflecting plates is then, from eq. (8.69),Ebi − Ebj111Q=Rij =+−1(8.70)Riji A ijAjwhere Rij is termed the radiative resistance. Equation (8.70) is seen to be analogousto an electrical circuit with current Q and voltage potential Ebi − Ebj .
Therefore,expressing radiative fluxes in terms of radiative resistances is commonly known asthe network analogy (Oppenheim, 1956). The network analogy is a very powerfulmethod for solving one-dimensional problems (e.g., whenever only two isothermalsurfaces see each other, such as infinite parallel plates, or when one surface totallyencloses another).
Consider, for example, two large parallel, or concentric, plates,A1 and AN , separated by N − 2 radiation shields, as shown in Fig. 8.21. Let eachshield j have an emittance j on both sides. Then by applying eq. (8.70) to anytwo consecutive surfaces and using the fact that Q remains constant throughoutthe gap,Q=Eb1 − Eb2Ebk−1 − EbkEbN −1 − EbNEb1 − EbN= ··· == ··· == #NR12Rk−1,kRN −1,Nj =2 Rj −1,jBOOKCOMP, Inc.
— John Wiley & Sons / Page 612 / 2nd Proofs / Heat Transfer Handbook / Bejan(8.71)———Normal PagePgEnds: TEX[612], (40)RADIATIVE EXCHANGE BETWEEN SURFACES123456789101112131415161718192021222324252627282930313233343536373839404142434445613[613], (41)Figure 8.21 Arrangement of parallel or concentric radiation shields.Lines: 1329 to 1352———2.94913pt PgVarwhereRj −1,j1=+j −1 Aj −111−1jAj(8.72)The analysis of radiation shields is one of the few applications where analysis of specularly reflecting surfaces is relatively simple and may lead to substantially differentanswers for concentric shields with strongly varying radii.
For a specularly reflectingshield Aj (with Aj −1 being specular or diffuse), the radiative resistance becomesRj −1,j =1j −111+ −1jAj −1(Aj specular)(8.73)Note that it is desirable to make shields highly reflective (low ), and this tendsto make them specularly reflecting (also desirable, because it also increases theresistance).Further simplifications arise if all shields are of identical material (2 = 3 =· · · = N−1 ); on the other hand, eqs. (8.71) through (8.73) remain valid for shieldswith different emittances on both of its sides (different values for j in Rj −1,j andRj,j +1 ).While the network analogy can (and has been) applied to configurations with morethan two surfaces seeing each other, this leads to very complicated circuits (becausethere is only one resistance between any two surfaces). For such problems the networkanalogy is not recommended, and the net radiation method, eq.
(8.68), should beemployed.BOOKCOMP, Inc. — John Wiley & Sons / Page 613 / 2nd Proofs / Heat Transfer Handbook / Bejan———Normal PagePgEnds: TEX[613], (41)123456789101112131415161718192021222324252627282930313233343536373839404142434445614THERMAL RADIATION8.3.5Radiative Exchange between Diffuse Nongray SurfacesIn a number of important engineering problems the assumption of gray surface properties may not provide adequate accuracy (when properties exhibit strong spectralvariations across the important range of the spectrum). To deal with such effects, twosimple models, known as the semigray approximation and the band approximation,will be described.Semigray Approximation Method This method employs the principle of superposition: The radiative flux at any given point is the sum of the contributions fromthe various emitters in the enclosure, each one acting independently.
In some applications there is a natural division of the radiative energy within an enclosure into two ormore distinct spectral regions. For example, in a solar collector the incoming energycomes from a high-temperature source with most of its energy below 3 µm, whereasradiation losses for typical collector temperatures are at wavelengths above 3 µm.In the case of laser heating and processing, the incoming energy is monochromatic(at the laser wavelength); reradiation takes place over the entire near- to midinfrared(depending on the workpiece temperature). In such a situation, eq. (8.68) may besplit into two sets of N equations each, one set for each spectral range, and withdifferent radiative properties for each set. For example, consider an enclosure subjectto external irradiation, which is confined to a certain spectral range (1).
The surfacesin the enclosure, owing to their temperature, emit over spectral range (2). * Then fromeq. (8.68),NQ(1)1 Q(1)1ii−−1F+ Hoi = 0(8.74a)i−j(1)AAjj(1)ijj =1NNQj(2)1 Q(2)1i−−1F=E−Fi−j Ebji−jbiAjAj(2)j(2)ij =1j =1Q(1)Q(2)Qi= i + iAiAiAii = 1, 2, . . . , N(8.74b)(8.74c)where j(1) is the average emittance for surface j over spectral interval (1), and soon. The semigray approximation is not limited to two distinct spectral regions. Eachsurface of the enclosure may be given a set of absorptances and reflectances, one valuefor each different emission temperature (with its different emission spectra).
However, while simple and straightforward, the method can never become exact no matterhow many different values of absorptance and reflectance are chosen for each surface.Band Approximation Method Another commonly used method to deal withnongray surfaces is the band approximation method. This method employs the fact*Note that spectral ranges (1) and (2) do not need to cover the entire spectrum, and indeed, they mayoverlap.BOOKCOMP, Inc. — John Wiley & Sons / Page 614 / 2nd Proofs / Heat Transfer Handbook / Bejan[614], (42)Lines: 1352 to 1374———-4.44827pt PgVar———Normal PagePgEnds: TEX[614], (42)RADIATIVE PROPERTIES OF PARTICIPATING MEDIA123456789101112131415161718192021222324252627282930313233343536373839404142434445615that even for nongray materials, eq.
(8.68) remains valid on a spectral basis (replacingemissive power ∞ Eb by spectral emissive power Ebλ , related to the total emissive powerby Eb = 0 Ebλ dλ). In this method the spectrum is broken up into M bands, overwhich the radiative properties of all surfaces in the enclosure are constant. Therefore,NNQj(m)1 Q(m)1(m)(m)(m)i−−1F+H=E−Fi−j Ebji−joibi(m)AAij(m)iij =1j =1i = 1, 2, .
. . , N,Ebj =M(m)Ebjm=1m = 1, 2, . . . , MM(m) QjQj=AjAjm=1Hoi =(8.75a)MHoi(m)(8.75b)m=1Here, Eb(m) is the fractional emissive power contained in band m and so on. Equations(8.75) are, of course, nothing but a simple numerical integration of the spectral versionof eq. (8.68), using the trapezoidal rule with varying steps. This method has theadvantage that the widths of the bands can be tailored to the spectral variation ofproperties, resulting in good accuracy with relatively few bands. For very few bandsthe accuracy of this method is similar to that of the semigray approximation but is alittle more cumbersome to apply. On the other hand, the band approximation methodcan achieve any desired accuracy by using many bands.8.4[615], (43)Lines: 1374 to 1403———-0.01006pt PgVar———Normal PagePgEnds: TEXRADIATIVE PROPERTIES OF PARTICIPATING MEDIA[615], (43)In many high-temperature applications, when radiative heat transfer is important,the medium between surfaces is not transparent but is “participating”; that is, itabsorbs, emits, and (possibly) scatters radiation.
In a typical combustion processthis interaction results in (1) continuum radiation due to tiny, burning soot particles(of dimension < 1 µm) and also due to larger suspended particles, such as coalparticles, oil droplets, and fly ash; (2) banded radiation in the infrared due to emissionand absorption by molecular gaseous combustion products, mostly water vapor andcarbon dioxide; and (3) chemiluminescence due to the combustion reaction itself.While chemiluminescence may normally be neglected, particulates as well as gasradiation generally must be accounted for.8.4.1Molecular GasesWhen a photon (or an electromagnetic wave) interacts with a gas molecule, it maybe absorbed, raising the energy level of the molecule.
Conversely, a gas moleculemay spontaneously lower its energy level by the emission of an appropriate photon.This leads to large numbers of narrow spectral lines, which partially overlap andtogether form vibration–rotation bands. As such, gases tend to be transparent overmost of the spectrum but may be almost opaque over the spectral range of a band.The absorption coefficient κλ is defined as a measure of how strongly radiation isBOOKCOMP, Inc. — John Wiley & Sons / Page 615 / 2nd Proofs / Heat Transfer Handbook / Bejan616400T = 300 K, p = 1 bar, pCO2 = 0 bar300 (cm⫺1/bar)123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATION200100[616], (44)Lines: 1403 to 141702300232523502375 (cm⫺1)Figure 8.22 Absorption coefficient spectrum for the CO2 4.3-µm band.absorbed or emitted along a path in a participating medium.
Figure 8.22 shows theabsorption coefficient of the important 4.3-µm vibration–rotation band of CO2 (perpartial pressure of CO2), for small amounts of CO2 contained in nitrogen, with atemperature of 300 K and a mixture pressure of 1 bar, generated from the HITRANdatabase (Rothman et al., 1998). The figure shows that the band consists of a largenumber of strong spectral lines, and a number of weak lines can also be observed. Inreality, there are many more spectral lines than appear in the figure. However, at therelatively high total pressure of 1 bar, the lines strongly overlap, giving a relativelysmooth appearance. Lowering the pressure would decrease line overlap, and moreand more of the ≈ 12,500 lines contained in the HITRAN database for this bandwould become distinguishable. Similarly, with increasing temperature, lines becomenarrower (less overlap), and many additional “hot lines” must be considered, whichare negligible at room temperature.
The new HITEMP database (Rothman et al.,2000), which is designed for temperatures up to 1000 K, includes ≈ 185,000 linesfor the 4.3-µm CO2 band alone!Fortunately, for many engineering problems, for simple heat transfer calculations,it is sufficient to determine the total emissivity for an isothermal, homogeneous pathof length L, ∞1=(1 − e−κλ L )Ebλ (Tg ) dλ(8.76)Eb 0For a mixture of gases the total emissivity is a function of path length L, gas temperature Tg , partial pressure(s) of the absorbing gas(es) pa , and total pressure p.BOOKCOMP, Inc. — John Wiley & Sons / Page 616 / 2nd Proofs / Heat Transfer Handbook / Bejan———0.98203pt PgVar———Long PagePgEnds: TEX[616], (44)617RADIATIVE PROPERTIES OF PARTICIPATING MEDIA123456789101112131415161718192021222324252627282930313233343536373839404142434445Especially important in combustion application, the total emissivity in mixtures ofnitrogen with water vapor and/or carbon dioxide may be calculated from Leckner(1972).
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