John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 81
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Some can be “scattered,” or deflected, in variousdirections, and some can be absorbed into the molecules. Scattering isa fairly minor influence in most gases unless they contain foreign particles, such as dust or fog. In cloudless air, for example, we are awareof the scattering of sunlight only when it passes through many miles ofthe atmosphere. Then the shorter wavelengths of sunlight are scattered(short wavelengths, as it happens, are far more susceptible to scatteringby gas molecules than longer wavelengths, through a process known asRayleigh scattering). That scattered light gives the sky its blue hues.At sunset, sunlight passes through the atmosphere at a shallow anglefor hundreds of miles.
Radiation in the blue wavelengths has all beenscattered out before it can be seen. Thus, we see only the unscatteredred hues, just before dark.When particles suspended in a gas have diameters near the wavelength of light, a more complex type of scattering can occur, known as Miescattering. Such scattering occurs from the water droplets in clouds (often making them a brilliant white color). It also occurs in gases that contain soot or in pulverized coal combustion.
Mie scattering has a strongangular variation that changes with wavelength and particle size [10.8].The absorption or emission of radiation by molecules, rather thanparticles, will be our principal focus. The interaction of molecules withradiation — photons, that is — is governed by quantum mechanics. It’shelpful at this point to recall a few facts from molecular physics.
Eachphoton has an energy hco /λ, where h is Planck’s constant, co is the speedof light, and λ is the wavelength of light. Thus, photons of shorter wavelengths have higher energies: ultraviolet photons are more energetic thanvisible photons, which are in turn more energetic than infrared photons.It is not surprising that hotter objects emit more visible photons.§10.5Gaseous radiationFigure 10.19 Vibrational modes of carbon dioxide and water.Molecules can store energy by rotation, by vibration (Fig.
10.19), or intheir electrons. Whereas the possible energy of a photon varies smoothlywith wavelength, the energies of molecules are constrained by quantummechanics to change only in discrete steps between the molecule’s allowable “energy levels.” The available energy levels depend on the molecule’schemical structure.When a molecule emits a photon, its energy drops in a discrete stepfrom a higher energy level to a lower one.
The energy given up is carried away by the photon. As a result, the wavelength of that photon isdetermined by the specific change in molecular energy level that causedit to be emitted. Just the opposite happens when a photon is absorbed:the photon’s wavelength must match a specific energy level change available to that particular molecule.
As a result, each molecular species canabsorb only photons at, or very close to, particular wavelengths! Often,these wavelengths are tightly grouped into so-called absorption bands,outside of which the gas is essentially transparent to photons.The fact that a molecule’s structure determines how it absorbs andemits light has been used extensively by chemists as a tool for deducing565566Radiative heat transfer§10.5molecular structure.
A knowledge of the energy levels in a molecule, inconjunction with quantum theory, allows specific atoms and bonds to beidentified. This is called spectroscopy (see [10.9, Chpt. 18 & 19] for anintroduction; see [10.10] to go overboard).At the wavelengths that correspond to thermal radiation at typicaltemperatures, it happens that transitions in the vibrational and rotationmodes of molecules have the greatest influence on radiative absorptance.Such transitions can be driven by photons only when the molecule hassome asymmetry.4 Thus, for all practical purposes, monatomic and symmetrical diatomic molecules are transparent to thermal radiation.
Themajor components of air—N2 and O2 —are therefore nonabsorbing; so,too, are H2 and such monatomic gases as argon.Asymmetrical molecules like CO2 , H2 O, CH4 , O3 , NH3 , N2 O, and SO2 ,on the other hand, each absorb thermal radiation of certain wavelengths.The first two of these, CO2 and H2 O, are always present in air. To understand how the interaction works, consider the possible vibrations of CO2and H2 O shown in Fig. 10.19. For CO2 , the topmost mode of vibrationis symmetrical and has no interaction with thermal radiation at normalpressures. The other three modes produce asymmetries in the moleculewhen they occur; each is important to thermal radiation.The primary absorption wavelength for the two middle modes of CO2is 15 µm, which lies in the thermal infrared. The wavelength for the bottommost mode is 4.3 µm.
For H2 O, middle mode of vibration interactsstrongly with thermal radiation at 6.3 µm. The other two both affect2.7 µm radiation, although the bottom one does so more strongly. In addition, H2 O has a rotational mode that absorbs thermal radiation havingwavelengths of 14 µm or more. Both of these molecules show additionalabsorption lines at shorter wavelengths, which result from the superposition of two or more vibrations and their harmonics (e.g., at 2.7 µm forCO2 and at 1.9 and 1.4 µm for H2 O). Additional absorption bands canappear at high temperature or high pressure.Absorptance, transmittance, and emittanceFigure 10.20 shows radiant energy passing through an absorbing gas witha monochromatic intensity iλ .
As it passes through an element of thick4The asymmetry required is in the distribution of electric charge — the dipole moment. A vibration of the molecule must create a fluctuating dipole moment in orderto interact with photons. A rotation interacts with photons only if the molecule has apermanent dipole moment.Gaseous radiation§10.5567Figure 10.20 The attenuation ofradiation through an absorbing (and/orscattering) gas.ness dx, the intensity will be reduced by an amount diλ :diλ = −ρκλ iλ dx(10.43)where ρ is the gas density and κλ is called the monochromatic absorption coefficient. If the gas scatters radiation, we replace κλ with γλ , themonochromatic scattering coefficient. If it both absorbs and scatters radiation, we replace κλ with βλ ≡ κλ + γλ , the monochromatic extinctioncoefficient.5 The dimensions of κλ , βλ , and γλ are all m2/kg.If ρκλ is constant through the gas, eqn.
(10.43) can be integrated froman initial intensity iλ0 at x = 0 to obtainiλ (x) = iλ0 e−ρκλ x(10.44)This result is called Beer’s law (pronounced “Bayr’s” law). For a gas layerof a given depth x = L, the ratio of final to initial intensity defines thatlayer’s monochromatic transmittance, τλ :τλ ≡iλ (L)= e−ρκλ Liλ0(10.45)Further, since gases do not reflect radiant energy, τλ + αλ = 1.
Thus, themonochromatic absorptance, αλ , isαλ = 1 − e−ρκλ L(10.46)Both τλ and αλ depend on the density and thickness of the gas layer.The product ρκλ L is sometimes called the optical depth of the gas. Forvery small values of ρκλ L, the gas is transparent to the wavelength λ.5All three coefficients, κλ , γλ , and βλ , are expressed on a mass basis. They could,alternatively, have been expressed on a volumetric basis.568Radiative heat transfer§10.5Figure 10.21 The monochromatic absorptance of a 1.09 mthick layer of steam at 127◦ C.The dependence of αλ on λ is normally very strong. As we have seen,a given molecule will absorb radiation in certain wavelength bands, whileallowing radiation with somewhat higher or lower wavelengths to passalmost unhindered. Figure 10.21 shows the absorptance of water vaporas a function of wavelength for a fixed depth.
We can see the absorptionbands at wavelengths of 6.3, 2.7, 1.9, and 1.4 µm that were mentionedbefore.A comparison of Fig. 10.21 with Fig. 10.2 readily shows why radiation from the sun, as viewed from the earth’s surface, shows a numberof spikey indentations at certain wavelengths. Several of those indentations occur in bands where atmospheric water vapor absorbs incomingsolar radiation, in accordance with Fig. 10.21. The other indentations inFig.
10.2 occur where ozone and CO2 absorb radiation. The sun itselfdoes not have these regions of low emittance; it is just that much of theradiation in these bands is absorbed by gases in the atmosphere beforeit can reach the ground.Just as αλ and ελ are equal to one another for a diffuse solid surface,they are equal for a gas. We may demonstrate this by considering anisothermal gas that is in thermal equilibrium with a black enclosure thatcontains it. The radiant intensity within the enclosure is that of a blackbody, iλb , at the temperature of the gas and enclosure.
Equation (10.43)shows that a small section of gas absorbs radiation, reducing the intensity by an amount ρκλ iλb dx. To maintain equilibrium, the gas musttherefore emit an equal amount of radiation:diλ = ρκλ iλb dx(10.47)Now, if radiation from some other source is transmitted througha nonscattering isothermal gas, we can combine the absorption fromGaseous radiation§10.5569eqn. (10.43) with the emission from eqn.
(10.47) to form an energy balance called the equation of transferdiλ= −ρκλ iλ + ρκλ iλbdx(10.48)Integration of this equation yields a result similar to eqn. (10.44):iλ (L) = iλ0 e−ρκλ L +iλb 1 − e−ρκλ L =τλ(10.49)≡ελThe first righthand term represents the transmission of the incomingintensity, as in eqn. (10.44), and the second is the radiation emitted bythe gas itself.
The coefficient of the second righthand term defines themonochromatic emittance, ελ , of the gas layer. Finally, comparison toeqn. (10.46) shows thatελ = αλ = 1 − e−ρκλ L(10.50)Again, we see that for very small ρκλ L the gas will neither absorb noremit radiation of wavelength λ.Heat transfer from gases to wallsWe now see that predicting the total emissivity, εg , of a gas layer will becomplex. We have to take account of the gases’ absorption bands as wellas the layer’s thickness and density. Such predictions can be done [10.11],but they are laborious. For making simpler (but less accurate) estimates,correlations of εg have been developed.Such correlations are based on the following model: An isothermalgas of temperature Tg and thickness L, is bounded by walls at the singletemperature Tw .
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