The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 14
Текст из файла (страница 14)
Some selective surfaces (as compared with a common steel) aredepicted in Figure 4.3.4. For a solar collector it is desirable to have a high spectral emissivity for shortwavelengths l < 2.5 mm (strong absorption of solar irradiation), and a low value for l > 2.5 mm (tominimize re-emission from the collector). The opposite is true for a spacecraft radiator panel used toreject heat into space.It is clear that (1) values of spectral surface emissivity are subject to great uncertainty and (2) onlya relatively small range of infrared wavelengths are of importance.
Therefore, it is often assumed thatthe surfaces are “gray”, i.e., the emissivity is constant across (the important fraction of) the spectrum,el ¹ el (l), since this assumption also vastly simplifies analysis. Table 4.3.1 gives a fairly detailed listingof total emissivities of various materials, defined ase( T ) =1Eb ( T )¥ò e ( l, T ) E0lbl( T ) dl(4.3.9)which may be enlisted for a gray analysis.View FactorsIn many engineering applications the exchange of radiative energy between surfaces is virtually unaffected by the medium that separates them.
Such (radiatively) nonparticipating media include vacuumas well as monatomic and most diatomic gases (including air) at low to moderate temperature levels(i.e., before ionization and dissociation occurs). Examples include spacecraft heat rejection systems,solar collector systems, radiative space heaters, illumination problems, and so on. It is common practice© 1999 by CRC Press LLC4-61Heat and Mass TransferFIGURE 4.3.4 Spectral, hemispherical reflectivities of several spectrally selective surfaces.to simplify the analysis by making the assumption of an idealized enclosure and/or of ideal surfaceproperties. The greatest simplification arises if all surfaces are black: for such a situation no reflectedradiation needs to be accounted for, and all emitted radiation is diffuse (i.e., the radiative energy leavinga surface does not depend on direction). The next level of difficulty arises if surfaces are assumed to begray, diffuse emitters (and, thus, absorbers) as well as gray, diffuse reflectors.
The vast majority ofengineering calculations are limited to such ideal surfaces, since, particularly, the effects of nondiffusereflections are usually weak (see discussion in previous section).Thermal radiation is generally a long-range phenomenon. This is always the case in the absence ofa participating medium, since photons will travel unimpeded from surface to surface.
Therefore, performing a thermal radiation analysis for one surface implies that all surfaces, no matter how far removed,that can exchange radiative energy with one another must be considered simultaneously. How muchenergy any two surfaces exchange depends in part on their size, separation, distance, and orientation,leading to geometric functions known as view factors, defined asFi - j =diffuse energy leaving Ai directly toward and intercepted by A jtotal diffuse energy leaving Ai(4.3.10)In order to make a radiative energy balance we always need to consider an entire enclosure rather thanand infinitesimal control volume (as is normally done for other modes of heat transfer, i.e., conductionor convection).
The enclosure must be closed so that irradiation from all possible directions can beaccounted for, and the enclosure surfaces must be opaque so that all irradiation is accounted for, foreach direction. In practice, an incomplete enclosure may be closed by introducing artificial surfaces. Anenclosure may be idealized in two ways, as indicated in Figure 4.3.5: by replacing a complex geometricshape with a few simple surfaces, and by assuming surfaces to be isothermal with constant (i.e., average)heat flux values across them. Obviously, the idealized enclosure approaches the real enclosure forsufficiently small isothermal subsurfaces.Mathematically, the view factor needs to be determined from a double integral, i.e.,Fi - j =© 1999 by CRC Press LLC1AiòòAiAjcos q i cos q jpSij2dA j dAi(4.3.11)4-62TABLE 4.3.1Section 4Total Emissivity and Solar Absorptivity of Selected SurfacesTemperature (°C)Alumina, flame-sprayedAluminum foilAs receivedBright dippedAluminum, vacuum-depositedHard-anodizedHighly polished plate, 98.3%pureCommercial sheetRough polishRough plateOxidized at 600°CHeavily oxidizedAntimony, polishedAsbestosBerylliumBeryllium, anodizedBismuth, brightBlack paintParson’s optical blackBlack siliconeBlack epoxy paintBlack enamel paintBrass, polishedRolled plate, natural surfaceDull plateOxidized by heating at 600°CCarbon, graphitizedCandle sootGraphite, pressed, filed surfaceChromium, polishedCopper, electroplatedCarefully polished electrolyticcopperPolishedPlate heated at 600°CCuprous oxideMolten copperGlass, Pyrex, lead, and sodaGypsumGold, pure, highly polishedInconel X, oxidizedLead, pure (99.96%), unoxidizedGray oxidizedOxidized at 150°CMagnesium oxideMagnesium, polishedMercuryMolybdenum, polished© 1999 by CRC Press LLC–25Total Normal EmissivityExtraterrestrial SolarAbsorptivity0.800.28202020–25225–5750.040.0250.0250.840.039–0.0570.100.100.9210010040200–60095–50035–26035–370150370600150370600750.090.180.055–0.070.11–0.190.20–0.310.28–0.310.93–0.940.180.210.300.900.880.820.34–25–25–750–2595–42540–3152250–350200–600100–320320–50095–270250–51040–110020800.950.930.890.81–0.800.100.060.220.61–0.590.76–0.750.75–0.710.9520.980.08–0.360.030.018115200–600800–11001075–1275260–54020225–625–25125–22525200275–825900–170535–2600–10035–260540–13700.0230.570.66–0.540.16–0.130.95–0.850.9030.018–0.0350.710.057–0.0750.280.630.55–0.200.200.07–0.130.09–0.120.05–0.080.10–0.180.770.9750.940.950.470.904-63Heat and Mass TransferTABLE 4.3.1(continued)Total Emissivity and Solar Absorptivity of Selected SurfacesTemperature (°C)Nickel, electroplatedPolishedPlatinum, pure, polishedSilica, sintered, powdered, fusedsilicaSilicon carbideSilver, polished, pureStainless steelType 312, heated 300 hr at260°CType 301 with Armco blackoxideType 410, heated to 700°C inairType 303, sandblastedTitanium, 75A75A, oxidized 300 hr at 450°CAnodizedTungsten, filament, agedZinc, pure, polishedGalvanized sheetTotal Normal Emissivity275020100225–625350.290.030.0720.054–0.1040.84150–65040–6250.83–0.960.020–0.03295–425Extraterrestrial SolarAbsorptivity0.220.080.27–0.32–250.750.89350.130.760.420.10–0.190.21–0.250.730.032–0.350.045–0.0530.210.689595–42535–425–2527–3300225–3251000.800.51FIGURE 4.3.5 Real and ideal enclosures for radiative transfer calculations.where qi and qj are the angles between the surface normals on Ai and Aj, respectively, and the line (oflength Sij) connecting two points on the two surfaces.
Analytical solutions to Equation (4.3.11) may befound for relatively simple geometries. A few graphical results for important geometries are shown inFigures 4.3.6 to 4.3.8. More-extensive tabulations as well as analytical expressions may be found intextbooks on the subject area (Modest, 1993; Siegel and Howell, 1992) as well as view factor catalogs(Howell, 1982). For nontrivial geometries view factors must be calculated numerically, either (1) bynumerical quadrature of the double integral in Equation (4.3.11), or (2) by converting Equation (4.3.11)into a double-line integral, followed by numerical quadrature, or (3) by a Monte Carlo method (statisticalsampling and tracing of selected light rays).View Factor Algebra.
For simple geometries analytical values can often be found by expressing thedesired view factor in terms of other, known ones. This method is known as view factor algebra, bymanipulating the two relations,Reciprocity rule:© 1999 by CRC Press LLCAi Fi - j = A j Fj -i( 4.3.12)4-64FIGURE 4.3.6 View factor between parallel, coaxial disks of unequal radius.FIGURE 4.3.7 View factor between identical, parallel, directly opposed rectangles.© 1999 by CRC Press LLCSection 44-65Heat and Mass TransferFIGURE 4.3.8 View factor between perpendicular rectangles with common edge.NSummation rule:åFi- j= 1,i = 1, N( 4.3.13)j =1assuming that the (closed) configuration consists of N surfaces.
The reciprocity rule follows immediatelyfrom Equation (4.3.11), while the summation rule simply states that the fractional energies leavingsurface Ai must add up to a whole.Example 4.3.2Assuming the view factor for a finite corner, as shown in Figure 4.3.8 is known, determine the viewfactor F3–4, between the two perpendicular strips as shown in Figure 4.3.9.FIGURE 4.3.9 Configuration for Example 4.3.2 (strips on a corner piece).Solution. From the definition of the view factor, and since the energy traveling to A4 is the energy goingto A2 and A4 minus the one going to A2, it follows thatF3- 4 = F3-( 2 + 4 ) - F3-2and, using reciprocity,© 1999 by CRC Press LLC4-66Section 4F3- 4 =1( A2 + A4 ) F(2+4)-3 - A2 F2-3A3[]Similarly, we findF3- 4 =A2 + A4AF( 2 + 4 )-(1+3) - F( 2 + 4 )-1 - 2 F2-(1+3) - F2-1A3A3()()All view factors on the right-hand side are corner pieces and, thus, are known from Figure 4.3.8.Crossed-Strings Method.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
