The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 17
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To determine the total emissivity of a mixture it isgenerally necessary to find the spectral absorption coefficient kl of the mixture (i.e., the sum of theabsorption coefficient of all contributors), followed by numerical integration of Equation (4.3.29).However, since the molecular gases tend to absorb only over a small part of the spectrum, to somedegree of accuracye mix @ e gas + e particulates(4.3.39)Equation (4.3.39) gives an upper estimate since overlap effects result in lower emissivity (compareEquation (4.3.31) for gas mixtures).Heat Exchange in the Presence of a Participating MediumThe calculation of radiative heat transfer rates through an enclosure filled with a participating mediumis a challenging task, to say the least.
High-accuracy calculations are rare and a topic of ongoing research.There are, however, several simplistic models available that allow the estimation of radiative heat transferrates, and relatively accurate calculations for some simple cases.Diffusion Approximation. A medium through which a photon can only travel a short distance withoutbeing absorbed is known as optically thick. Mathematically, this implies that kl L @ 1 for a characteristicdimension L across which the temperature does not vary substantially.
For such an optically thick,nonscattering medium the spectral radiative flux may be calculated fromql = -4ÑEbl3k l(4.3.40)similar to Fourier’s diffusion law for heat conduction. Note that a medium may be optically thick atsome wavelengths, but thin (kl L ! 1) at others (e.g., molecular gases!).
For a medium that is opticallythick for all wavelengths, Equation (4.3.40) may be integrated over the spectrum, yielding the totalradiative fluxq=-4416sT 3ÑEb = Ñ sT 4 = ÑT3k R3k R3k R()(4.3.41)where kR is the suitably averaged absorption coefficient, termed the Rosseland-mean absorption coefficient. For a cloud of soot particles, kR @ km from Equation (4.3.37) is a reasonable approximation.Equation (4.3.41) may be rewritten by defining a “radiative conductivity” kR,© 1999 by CRC Press LLC4-78Section 4q = - k R ÑTkR =16sT 33k R(4.3.42)This form shows that the diffusion approximation is mathematically equivalent to conductive heat transferwith a (strongly) temperature-dependent conductivity.Note: More accurate calculations show that, in the absence of other modes of heat transfer (conduction,convection), there is generally a temperature discontinuity near the boundaries (Tsurface ¹ Tadjacent medium),and, unless boundary conditions that allow such temperature discontinuities are chosen, the diffusionapproximation will do very poorly in the vicinity of bounding surfaces.Example 4.3.7A soot cloud is contained between two walls at T1 = 1000 K and T2 = 2000 K, spaced 1 m apart.
Theeffective absorption coefficient of the medium is kR = 10 m–1 and the effective thermal conductivity iskc = 0.1 W/mK. Estimate the total heat flux between the plates (ignoring convection effects).Solution. For simplicity we may want to assume a constant total conductivity k = kc + kR, leading toq = -kT - T1dT=k 2dxLwhere kR must be evaluated at some effective temperature. Choosing, based on its temperature dependence,kR @8s 38 ´ 5.670 ´ 10 -8 W m 2 K 4W1000 3 + 2000 3 K 3 = 136T1 + T23 =3k R3 ´ 10 mmK()()givesq = (0.1 + 136)2000 - 1000 WkW= 136 21m2m KNote that (1) conduction is negligible in this example and (2) the surface emissivities do not enter thediffusion approximation. While a more accurate answer can be obtained by taking the temperaturedependence of kR into account, the method itself should be understood as a relatively crude approximation.Mean Beam Length Method.
Relatively accurate yet simple heat transfer calculations can be carried outif an isothermal, absorbing–emitting, but not scattering medium is contained in an isothermal, blackwalled enclosure. While these conditions are, of course, very restrictive, they are met to some degreeby conditions inside furnaces. For such cases the local heat flux on a point of the surface may becalculated from[]q = 1 - a( Lm ) Ebw - e( Lm ) Ebg(4.3.43)where Ebw and Ebg are blackbody emissive powers for the walls and medium (gas and/or particulates),respectively, and a(Lm) and e(Lm) are the total absorptivity and emissivity of the medium for a pathlength Lm through the medium. The length Lm, known as the average mean beam length, is a directionalaverage of the thickness of the medium as seen from the point on the surface.
On a spectral basisEquation (4.3.43) is exact, provided the above conditions are met and provided an accurate value of the(spectral) mean beam length is known. It has been shown that spectral dependence of the mean beamlength is weak (generally less than ±5% from the mean). Consequently, total radiative heat flux at the© 1999 by CRC Press LLC4-79Heat and Mass Transfersurface may be calculated very accurately from Equation (4.3.43), provided the emissivity and absorptivity of the medium are also known accurately. The mean beam lengths for many important geometrieshave been calculated and are collected in Table 4.3.3.
In this table Lo is known as the geometric meanbeam length, which is the mean beam length for the optically thin limit (kl ® 0), and Lm is a spectralaverage of the mean beam length. For geometries not listed in Table 4.3.3, the mean beam length maybe estimated fromLo @ 4TABLE 4.3.3VALm @ 0.9 Lo @ 3.6VA(4.3.44)Mean Beam Lengths for Radiation from a Gas Volume to a Surface on Its BoundaryGeometry of Gas VolumeSphere radiating to its surfaceInfinite circular cylinder to boundingsurfaceSemi-infinite circular cylinder to:Element at center of baseEntire baseCircular cylinder(height/diameter = 1) to:Element at center of baseEntire surfaceCircular cylinder(height/diameter = 2) to:Plane baseConcave surfaceEntire surfaceCircular cylinder(height/diameter = 0.5) to:Plane baseConcave surfaceEntire surfaceInfinite semicircular cylinder to centerof plane rectangular faceInfinite slab to its surfaceCube to a faceRectangular 1 ´ 1 ´ 4 parallelepipeds:To 1 ´ 4 faceTo 1 ´ 1 faceTo all facesCharacterizingDimensionLGeometric MeanBeam LengthL0/LAverage MeanBeam LengthLm/LLm/L0Diameter, L = DDiameter, L = D0.671.000.650.940.970.941.000.810.900.650.900.800.760.670.710.600.920.900.730.820.800.600.760.730.820.930.910.480.530.50—0.430.460.451.260.900.880.90—2.000.671.760.60.880.900.900.860.890.820.710.810.910.830.91Diameter, L = DDiameter, L = DDiameter, L = DDiameter, L = DRadius, L = RSlab thickness, LEdge LShortest edge, Lwhere V is the volume of the participating medium and A is its entire bounding surface area.Example 4.3.8An isothermal mixture of 10% CO2 and 90% nitrogen at 1000 K and 5 bar is contained between twolarge, parallel, black plates, which are both isothermal at 1500 K.
Estimate the net radiative heat lossfrom the surfaces.Solution. The heat loss may be calculated from Equation (4.3.43), after determining the mean beamlength, followed by evaluation of e(Lm) and a(Lm). From Table 4.3.3 it is clear that Lm = 1.76 ´ thicknessof slab = 1.76 m. It turns out that the necessary e(Lm) = 0.15 and a(Lm) = 0.122 have already beencalculated in Example 4.3.6. Thus, the heat flux is immediately calculated from Equation (4.3.43) as© 1999 by CRC Press LLC4-80Section 4q = (1 - 0.122) 5.670 ´ 10 -8 ´ 1500 4 - 0.15 ´ 5.670 ´ 10 -8 ´ 1000 4= 2.44 ´ 10 5W= 244 kW m 2m2Defining TermsAbsorptivity: The ability of a medium to absorb (i.e., trap and convert to other forms of energy)incoming radiation; gives the fraction of incoming radiation that is absorbed by the medium.Absorption coefficient: The ability of a medium to absorb (i.e., trap and convert to other forms ofenergy) over a unit path length; the reciprocal of the mean distance a photon travels before beingabsorbed.Blackbody: Any material or configuration that absorbs all incoming radiation completely.
A blackbodyalso emits the maximum possible amount of radiation as described by Planck’s law.Diffuse surface: A surface that emits and/or reflects equal amounts of radiative energy (photons) intoall directions. Or a surface that absorbs and/or reflects equal amounts of radiation independent ofincoming direction.Emissive power: The rate of radiative energy leaving a surface through emission. The maximum amountof emissive power is emitted by a blackbody with a spectral strength described by Planck’s law.Emissivity: The ability of a medium to emit (i.e., convert internal energy into electromagnetic wavesor photons) thermal radiation; gives the fraction of emission as compared with a blackbody.Gray: A medium whose radiative properties (such as absorptivity, emissivity, reflectivity, absorptioncoefficient) do not vary with wavelength.Irradiation: Incoming radiative flux onto a surface from outside it.Network analogy: Expressing radiative heat exchange between surfaces in terms of an electricalnetwork, with heat flux as “current,” differences in emissive power as “potentials,” and definingradiative resistances.Opaque medium: A medium of sufficient thickness that absorbs all nonreflected irradiation; no radiationis transmitted through the medium.Photon: A massless particle carrying energy in the amount of hn; the quantum mechanical alternativeview of an electromagnetic wave carrying radiative energy.Planck’s law: The law describing the spectral distribution of the radiative energy emitted (emissivepower) of a blackbody.Radiosity: Total radiative flux leaving a surface (diffusely), consisting of emitted as well as reflectedradiation.Reflectivity: The ability of an interface, or of a medium or of a composite with a number of interfaces,to reflect incoming radiation back into the irradiating medium.Semitransparent: See transparent.Spectral value: The value of a quantity that varies with wavelength at a given wavelength; for dimensional quantities the amount per unit wavelength.Transmissivity: The ability of a medium to let incoming radiation pass through it; gives the fractionof incoming radiation that is transmitted through the medium.Transparent: The ability of a medium to let incoming radiation pass through it.
A medium that lets allradiation pass through it is called transparent, a medium that only allows a part to pass throughit is called semitransparent.View factor: The fraction of diffuse radiant energy leaving one surface that is intercepted by anothersurface.© 1999 by CRC Press LLCHeat and Mass Transfer4-81References1. Brewster, M.Q.
1992. Thermal Radiative Transfer & Properties, John Wiley & Sons, New York.2. Buckius, R.O. and Hwang, D.C. 1980. Radiation properties for polydispersions: application tocoal, J. Heat Transfer, 102, 99–103.3. Felske, J.D. and Tien, C.L. 1977. The use of the Milne-Eddington absorption coefficient forradiative heat transfer in combustion systems, J. Heat Transfer, 99(3), 458–465.4.
Hottel, H.C. and Sarofim, A.F. 1967. Radiation Transfer, McGraw-Hill, New York.5. Howell, J.R. 1982. Catalog of Radiation Configuration Factors, McGraw-Hill, New York.6. Leckner, B. 1972. Spectral and total emissivity of water vapor and carbon dioxide, Combust.Flame, 19, 33–48.7. Modest, M.F. 1993. Radiative Heat Transfer, McGraw-Hill, New York.8.
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