Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 93
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Emissive powercan be related to emitted intensity by integrating this intensity over the 2π solid anglesabove a surface, and then realizing that the projection of dA normal to the rays isdA cos θ. Thus,E(r) =02ππ/2I (r, θ, ψ) cos θ sin θ dθ dψ =0(8.16)which is, of course, also valid on a spectral basis. For a black surface it is readilyshown, through a variation of Kirchhoff’s law, that Ibλ is independent of direction, orIbλ = Ibλ (T,λ)(8.17)Using this relation in eq. (8.16), it is observed that the intensity leaving a blackbody(or any surface whose outgoing intensity is independent of direction, or diffuse) maybe evaluated from the blackbody emissive power (or outgoing heat flux) asEbλ (r,λ)π(8.18)In the literature the spectral blackbody intensity is sometimes referred to as the Planckfunction.8.1.4Radiative Heat FluxEmissive power is the total radiative energy streaming away from a surface due toemission.
Therefore, it is a radiative flux, but not the net radiative flux at the surface,BOOKCOMP, Inc. — John Wiley & Sons / Page 581 / 2nd Proofs / Heat Transfer Handbook / Bejan0.30309pt PgVar———Long PagePgEnds: TEX[581], (9)I (r, ŝ)n̂ · ŝ dΩ2πIbλ (r,λ) =Lines: 395 to 443582123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATIONbecause it only accounts for emission and not for incoming radiation and reflectedradiation.
Extending the definition of eq. (8.16) givesIλ (ŝ) cos θ dΩ ≥ 0(8.19)(qλ )out =cos θ>0where Iλ (θ) is now outgoing intensity (due to emission plus reflection). Similarly, forincoming directions (π/2 < θ ≤ π),Iλ (ŝ) cos θ dΩ < 0(8.20)(qλ )in =cos θ<0Combining the incoming and outgoing contributions, the net radiative flux at a surfaceisIλ (ŝ) cos θ dΩ(8.21)(qλ )net = qλ · n̂ = (qλ )in + (qλ )out =4πThe total radiative flux, finally, is obtained by integrating eq. (8.21) over the entirespectrum, or ∞ ∞qλ · n̂ dλ =Iλ (ŝ)n̂ · ŝ dΩ dλ(8.22)q · n̂ =004πOf course, the surface described by the unit vector n̂ may be an imaginary one (locatedsomewhere inside a radiating medium). Thus, removing the n̂ from eq. (8.22) givesthe definition of the radiative heat flux vector inside a participating medium: ∞ ∞qλ dλ =Iλ (ŝ)ŝ dΩ dλ(8.23)q=08.204πRADIATIVE PROPERTIES OF SOLIDS AND LIQUIDSBecause radiative energy arriving at a given point in space can originate from a pointfar away, without interacting with the medium in between, a conservation of energybalance must be performed on an enclosure bounded by opaque walls (i.e., a mediumthick enough that no electromagnetic waves can penetrate through it).
Strictly speaking, the surface of an enclosure wall can only reflect radiative energy or allow a partof it to penetrate into the substrate. A surface cannot absorb or emit photons: Attenuation takes place inside the solid, as does emission of radiative energy (and someof the emitted energy escapes through the surface into the enclosure). In practicalsystems the thickness of the surface layer over which absorption of irradiation frominside the enclosure occurs is very small compared with the overall dimensions of anenclosure—usually, a few angstroms for metals and a few micrometers for most nonmetals. The same may be said about emission from within the walls that escapes intoBOOKCOMP, Inc. — John Wiley & Sons / Page 582 / 2nd Proofs / Heat Transfer Handbook / Bejan[582], (10)Lines: 443 to 480———4.75412pt PgVar———Normal PagePgEnds: TEX[582], (10)RADIATIVE PROPERTIES OF SOLIDS AND LIQUIDS123456789101112131415161718192021222324252627282930313233343536373839404142434445583the enclosure.
Thus, in the case of opaque walls it is customary to speak of absorptionby and emission from a “surface,” although a thin surface layer is implied.If radiation impinging on a solid or liquid layer is considered, a fraction of theenergy will be reflected (reflectance ρ, often also referred to as reflectivity), anotherfraction will be absorbed (absorptance α, often also referred to as absorptivity), andif the layer is thin enough, a fraction may be transmitted (transmittance τ, often alsoreferred to as transmissivity).
Because all radiation must be either reflected, absorbed,or transmitted,ρ+α+τ=1(8.24)If the medium is sufficiently thick to be opaque, then τ = 0 andρ+α=1(8.25)All surfaces also emit thermal radiation (or, rather, radiative energy is emittedwithin the medium, some of which escapes from the surface). The emittance isdefined as the ratio of energy emitted by a surface as compared to that of a blacksurface at the same temperature (the theoretical maximum).All of these four properties may vary in magnitude between the values 0 and 1; for ablack surface, which absorbs all incoming radiation and emits the maximum possible,α = = 1 and ρ = τ = 0.
They may also be functions of temperature as wellas wavelength and direction (incoming and/or outgoing). One distinguishes betweenspectral and total properties (an average value over the spectrum) and also betweendirectional and hemispherical properties (an average value over all directions).It may be shown (through another variation of Kirchhoff’s law) that, at least on aspectral, directional basis,α(T,λ,ŝ) = (T,λ,ŝ)(8.26)This is also true for hemispherical values if either the directional emittance or theincoming radiation are diffuse (they do not depend on direction). It is also true fortotal values if either the spectral emittance does not depend on wavelength or if thespectral behavior of the incoming radiation is similar to blackbody radiation at thesame temperature.Typical directional behavior is shown in Fig. 8.4a (for nonmetals) and b (metals).In these figures the total, directional emittance, a value averaged over all wavelengths,is shown.
For nonmetals the directional emittance varies little over a large range ofpolar angles but decreases rapidly at grazing angles until a value of zero is reached atθ = π/2. Similar trends hold for metals, except that at grazing angles, the emittancefirst increases sharply before dropping back to zero (not shown). Note that emittancelevels are considerably higher for nonmetals.A surface whose emittance is the same for all directions is called a diffuse emitter,or a Lambert surface. No real surface can be a diffuse emitter because electromagnetic wave theory predicts a zero emittance at θ = π/2 for all materials.
However,BOOKCOMP, Inc. — John Wiley & Sons / Page 583 / 2nd Proofs / Heat Transfer Handbook / Bejan[583], (11)Lines: 480 to 515———0.0pt PgVar———Normal PagePgEnds: TEX[583], (11)584123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATIONa[584], (12)Lines: 515 to 529———0.70102pt PgVar———Normal Page* PgEnds: Ejectb[584], (12)Figure 8.4 Directional variation of surface emittances: (a) for several nonmetals; (b) forseveral metals. (From Schmidt and Eckert, 1935.)little energy is emitted into grazing directions, as seen from eq. (8.16), so that theassumption of diffuse emission is often a good one.Typical spectral behavior of surface emittances is shown in Fig.
8.5 for a fewmaterials, as collected by White (1984). Shown are values for directional emittancesin the direction normal to the surface. However, the spectral behavior is the samefor hemispherical emittances. In general, nonmetals have relatively high emittances,which may vary erratically across the spectrum, and metals behave similarly for shortwavelengths but tend to have lower emittances with more regular spectral dependencein the infrared.Mathematically, the spectral, hemispherical emittance is defined in terms of emissive power asλ (T,λ) ≡BOOKCOMP, Inc.
— John Wiley & Sons / Page 584 / 2nd Proofs / Heat Transfer Handbook / BejanEλ (T,λ)Ebλ (T,λ)(8.27)RADIATIVE PROPERTIES OF SOLIDS AND LIQUIDS1.0Carbon0.8Normal emissivity, nλ123456789101112131415161718192021222324252627282930313233343536373839404142434445White enamelSilicon carbideCopper0.6585AluminumGoldMagnesium oxideAluminum oxide0.4TungstenNickel0.20Aluminum01234Wavelength (m)56[585], (13)78Figure 8.5 Normal, spectral emittances for selected materials. (From White, 1984.)This property may be extracted from the spectral, directional emittance λ by integrating over all directions,1λ (T,λ) =λ (T,λ,θ,ψ) cos θ dΩπ 1 2π π/2 =λ (T,λ,θ,ψ) cos θ sin θ dθ dψ(8.28)π 00and finally, the total, hemispherical emittance may be related to the spectral hemispherical emittance through∞ ∞Eλ (T,λ) dλE(T )1(T ) =λ (T,λ)Ebλ (T,λ) dλ(8.29)= 0= 2 4Eb (T )Eb (T )n σT 0Here a prime and subscript λ have been added temporarily to distinguish directionalfrom hemispherical properties, and spectral from total (spectrally averaged) values.If the spectral emittance is the same for all wavelengths, eq.
(8.29) reduces to(T ) = λ (T )(8.30)Such surfaces are termed gray, and for the very special case of a gray, diffuse surface,this implies that(T ) = λ = = λBOOKCOMP, Inc. — John Wiley & Sons / Page 585 / 2nd Proofs / Heat Transfer Handbook / Bejan(8.31)Lines: 529 to 564———2.31412pt PgVar———Normal PagePgEnds: TEX[585], (13)586123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATIONAlthough no real surface is truly gray, it often happens that λ is relatively constantover that part of the spectrum where Ebλ is substantial, making the simplifyingassumption of a gray surface warranted.8.2.1Radiative Properties of MetalsWavelength Dependence Electromagnetic theory states that the radiative properties of interfaces are strong functions of the material’s electrical conductivity.
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