John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 76
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(When the medium is air, we can usually neglect theseeffects.)If surfaces (1) and (2) are black, if they are surrounded by air, and ifno heat flows between them by conduction or convection, then only theThe problem of radiative exchange§10.1527first three considerations are involved in determining Qnet .
We saw someelementary examples of how this could be done in Chapter 1, leading to(10.1)Qnet = A1 F1–2 σ T14 − T24The last three considerations complicate the problem considerably. InChapter 1, we saw that these nonideal factors are sometimes included ina transfer factor F1–2 , such that(10.2)Qnet = A1 F1–2 σ T14 − T24Before we undertake the problem of evaluating heat exchange among realbodies, we need several definitions.Some definitionsEmittance. A real body at temperature T does not emit with the blackbody emissive power eb = σ T 4 but rather with some fraction, ε, of eb .The same is true of the monochromatic emissive power, eλ (T ), which isalways lower for a real body than the black body value given by Planck’slaw, eqn.
(1.30). Thus, we define either the monochromatic emittance, ελ :ελ ≡eλ (λ, T )eλb (λ, T )or the total emittance, ε:∞ε≡e(T )=eb (T )0(10.3)∞eλ (λ, T ) dλσT4=0ελ eλb (λ, T ) dλσT4(10.4)For real bodies, both ε and ελ are greater than zero and less than one;for black bodies, ε = ελ = 1. The emittance is determined entirely by theproperties of the surface of the particular body and its temperature. Itis independent of the environment of the body.Table 10.1 lists typical values of the total emittance for a variety ofsubstances. Notice that most metals have quite low emittances, unlessthey are oxidized.
Most nonmetals have emittances that are quite high—approaching the black body limit of unity.One particular kind of surface behavior is that for which ελ is independent of λ. We call such a surface a gray body. The monochromatic emissive power, eλ (T ), for a gray body is a constant fraction, ε, of ebλ (T ), asindicated in the inset of Fig. 10.2. In other words, for a gray body, ελ = ε.Table 10.1 Total emittances for a variety of surfaces [10.1]MetalsSurfaceNonmetals◦Temp. ( C)AluminumPolished, 98% pure200−600Commercial sheet90Heavily oxidized90−540BrassHighly polished260Dull plate40−260Oxidized40−260CopperHighly polished electrolytic90Slightly polished to dull40Black oxidized40Gold: pure, polished90−600Iron and steelMild steel, polished150−480Steel, polished40−260Sheet steel, rolled40Sheet steel, strong40rough oxideCast iron, oxidized40−260Iron, rusted40Wrought iron, smooth40Wrought iron, dull oxidized20−360Stainless, polished40Stainless, after repeated230−900heatingLeadPolished40−260Oxidized40−200Mercury: pure, clean40−90PlatinumPure, polished plate200−590Oxidized at 590◦ C260−590Drawn wire and strips40−1370Silver200Tin40−90TungstenFilament540−1090Filament2760528ε0.04–0.060.090.20–0.330.030.220.46–0.560.020.12–0.150.760.02–0.0350.14–0.320.07–0.100.660.800.57–0.660.61–0.850.350.940.07–0.170.50–0.700.05–0.080.630.10–0.120.05–0.100.07–0.110.04–0.190.01–0.040.050.11–0.160.39SurfaceAsbestosBrickRed, roughSilicaFireclayOrdinary refractoryMagnesite refractoryWhite refractoryCarbonFilamentLampsootConcrete, roughGlassSmoothQuartz glass (2 mm)PyrexGypsumIceLimestoneMarbleMicaPaintsBlack glossWhite paintLacquerVarious oil paintsRed leadPaperWhiteOther colorsRoofingPlaster, rough limeQuartzRubberSnowWater, thickness ≥0.1 mmWoodOak, planedTemp.
(◦ C)ε400.93–0.9740980980109098010900.930.80–0.850.750.590.380.291040−143040400.530.950.9440260−540260−5404000.940.96–0.660.94–0.740.80–0.900.97–0.98400−26040400.95–0.830.93–0.950.75404040409040404040−260100−10004010−204040200.900.89–0.970.80–0.950.92–0.960.930.95–0.980.92–0.940.910.920.89–0.580.86–0.940.820.960.80–0.900.90§10.1The problem of radiative exchange529Figure 10.2 Comparison of the sun’s energy as typically seenthrough the earth’s atmosphere with that of a black body having the same mean temperature, size, and distance from theearth. (Notice that eλ , just outside the earth’s atmosphere, isfar less than on the surface of the sun because the radiationhas spread out over a much greater area.)No real body is gray, but many exhibit approximately gray behavior.
Wesee in Fig. 10.2, for example, that the sun appears to us on earth as anapproximately gray body with an emittance of approximately 0.6. Somematerials—for example, copper, aluminum oxide, and certain paints—areactually pretty close to being gray surfaces at normal temperatures.Yet the emittance of most common materials and coatings varies withwavelength in the thermal range. The total emittance accounts for thisbehavior at a particular temperature. By using it, we can write the emissive power as if the body were gray, without integrating over wavelength:e(T ) = ε σ T 4(10.5)We shall use this type of “gray body approximation” often in this chapter.530Radiative heat transferSpecular or mirror-likereflection of incoming ray.Reflection which isbetween diffuse andspecular (a real surface).§10.1Diffuse radiation in whichdirections of departure areuninfluenced by incomingray angle, θ.Figure 10.3 Specular and diffuse reflection of radiation.(Arrows indicate magnitude of the heat flux in the directionsindicated.)In situations where surfaces at very different temperatures are involved, the wavelength dependence of ελ must be dealt with explicitly.This occurs, for example, when sunlight heats objects here on earth.
Solar radiation (from a high temperature source) is on visible wavelengths,whereas radiation from low temperature objects on earth is mainly in theinfrared range. We look at this issue further in the next section.Diffuse and specular emittance and reflection. The energy emitted bya non-black surface, together with that portion of an incoming ray ofenergy that is reflected by the surface, may leave the body diffusely orspecularly, as shown in Fig. 10.3.
That energy may also be emitted orreflected in a way that lies between these limits. A mirror reflects visibleradiation in an almost perfectly specular fashion. (The “reflection” of abilliard ball as it rebounds from the side of a pool table is also specular.)When reflection or emission is diffuse, there is no preferred direction foroutgoing rays. Black body emission is always diffuse.The character of the emittance or reflectance of a surface will normally change with the wavelength of the radiation. If we take account ofboth directional and spectral characteristics, then properties like emittance and reflectance depend on wavelength, temperature, and anglesof incidence and/or departure.
In this chapter, we shall assume diffuse§10.1The problem of radiative exchange531behavior for most surfaces. This approximation works well for manyproblems in engineering, in part because most tabulated spectral and total emittances have been averaged over all angles (in which case they areproperly called hemispherical properties).Experiment 10.1Obtain a flashlight with as narrow a spot focus as you can find. Directit at an angle onto a mirror, onto the surface of a bowl filled with sugar,and onto a variety of other surfaces, all in a darkened room. In each case,move the palm of your hand around the surface of an imaginary hemisphere centered on the point where the spot touches the surface.
Noticehow your palm is illuminated, and categorize the kind of reflectance ofeach surface—at least in the range of visible wavelengths.Intensity of radiation. To account for the effects of geometry on radiant exchange, we must think about how angles of orientation affect theradiation between surfaces. Consider radiation from a circular surfaceelement, dA, as shown at the top of Fig.
10.4. If the element is black,the radiation that it emits is indistinguishable from that which would beemitted from a black cavity at the same temperature, and that radiationis diffuse — the same in all directions. If it were non-black but diffuse,the heat flux leaving the surface would again be independent of direction. Thus, the rate at which energy is emitted in any direction from thisdiffuse element is proportional to the projected area of dA normal to thedirection of view, as shown in the upper right side of Fig.
10.4.If an aperture of area dAa is placed at a radius r and angle θ fromdA and is normal to the radius, it will see dA as having an area cos θ dA.The energy dAa receives will depend on the solid angle,1 dω, it subtends. Radiation that leaves dA within the solid angle dω stays withindω as it travels to dAa . Hence, we define a quantity called the intensityof radiation, i (W/m2 ·steradian) using an energy conservation statement:radiant energy from dAdQoutgoing = (i dω)(cos θ dA) = that is intercepted by dAa1(10.6)The unit of solid angle is the steradian. One steradian is the solid angle subtendedby a spherical segment whose area equals the square of its radius. A full sphere therefore subtends 4π r 2 /r 2 = 4π steradians. The aperture dAa subtends dω = dAa r 2 .532Radiative heat transfer§10.1Figure 10.4 Radiation intensity through a unit sphere.Notice that while the heat flux from dA decreases with θ (as indicatedon the right side of Fig. 10.4), the intensity of radiation from a diffusesurface is uniform in all directions.Finally, we compute i in terms of the heat flux from dA by dividingeqn.
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