Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 97
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Oneor more of these surfaces may not be material, such as windows, in which case theyare assigned equivalent properties and equivalent temperatures to account for the radiative energy entering or leaving the enclosure through them.In the following sections the analysis of radiative heat transfer in the absence of aparticipating medium will be considered for different levels of complexity. To makethe analysis tractable it is common practice to make the assumption of an idealizedenclosure and/or ideal surface properties. An enclosure may be idealized in two ways,as indicated in Fig. 8.14: by replacing a complex geometrical shape with a few simplesurfaces, and by assuming surfaces to be isothermal with constant (or average) heatflux values through them.
Obviously, the idealized enclosure approaches the realenclosure for sufficiently small isothermal subsurfaces.Surface properties may be idealized in a number of ways. The greatest simplification arises if all surfaces are assumed black: For such a situation no reflectedradiation needs to be accounted for and all emitted radiation is diffuse. The next levelof difficulty arises if surfaces are assumed to be diffuse gray emitters (and, thus, absorbers) as well as diffuse gray reflectors.
This level of idealization tends to giveresults of acceptable accuracy for the vast majority of engineering problems. If thedirectional reflection behavior of a surface deviates strongly from a diffuse reflector(such a polished metal, which reflects almost like a mirror), one may often approximate the reflectance to consist of a purely diffuse and a purely specular component.However, this greatly complicates the analysis, in particular, if the enclosure includescurved surfaces.
Luckily, the effects of specular (or, indeed, any type of nondiffuse)reflections tend to be very small in most engineering enclosures. Exceptions includelight concentrators and collimators (in solar energy applications), long channels withspecular sidewalls (optical fibers), and others. For the treatment of specular reflectionsthe reader is referred to more detailed textbooks, such as the one by Modest (2003).Figure 8.14 Real and ideal enclosures for radiative transfer calculations.BOOKCOMP, Inc.
— John Wiley & Sons / Page 599 / 2nd Proofs / Heat Transfer Handbook / Bejan[599], (27)Lines: 837 to 846———3.026pt PgVar———Normal PagePgEnds: TEX[599], (27)600123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATIONOf greater engineering importance is the case when the assumption of a gray surfaceis not acceptable. Simple methods to treat nongray behavior are outlined briefly atthe end of this section.8.3.1 View FactorsTo make an energy balance on a surface, the incoming radiative flux, or irradiation,H must be evaluated. In a general enclosure the irradiation has contributions fromall parts of the enclosure surface.
Therefore, one needs to determine how much ofthe energy that leaves any surface of the enclosure travels toward the surface underconsideration. The geometric relations governing this process for diffuse surfaces(which absorb and emit diffusely, and also reflect radiative energy diffusely) areknown as view factors. Other names used in the literature are configuration factor,angle factor, and shape factor.
The view factor between two surfaces Ai and Aj isdefined asFi−j ≡diffuse energy leaving Ai directly toward and intercepted by Ajtotal diffuse energy leaving Ai(8.43)Direct Integration Mathematically, view factors can be expressed in terms of adouble surface integral, that is,Fi−j AiAjcos θi cos θjdAj dAiπSij2(8.44)where Sij is the distance between points on surfaces Ai and Aj , and θi and θj arethe angles between Sij and the local surface normals, as shown in Fig.
8.18. UsingStokes’ theorem, eq. (8.44) can be converted into a double contour integral,Ai Fi−j =12π ΓiΓjln Sij dsj · dsi(8.45)where Γi is the contour of Ai (as also indicated in Fig. 8.18) and si is a vector to apoint on contour Γi .While the integration of eqs. (8.44) and (8.45) may be straightforward for somesimple configurations, it is desirable to have a more generally applicable formula atBOOKCOMP, Inc. — John Wiley & Sons / Page 600 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 846 to 1030———where the word directly is meant to imply “on a straight path, without interveningreflections.” A list of relationships for some common view factors is given in Table8.3, and Figs.
8.15 through 8.17 give convenient graphical representations of the threemost important view factors.Radiation view factors may be determined by a variety of methods, such as directintegration (analytical or numerical integration), statistical evaluation [through statistical sampling using a Monte Carlo method (Modest, 2003), or through a varietyof special methods, some of which are described briefly in what follows.1=Ai[600], (28)3.66231pt PgVar———Normal PagePgEnds: TEX[600], (28)RADIATIVE EXCHANGE BETWEEN SURFACES123456789101112131415161718192021222324252627282930313233343536373839404142434445TABLE 8.3601Important View Factors1. Two infinitely long, directly opposed parallel plates ofthe same finite width:F1−2hH =w= F2−1 = 1 + H 2 − H2. Two infinitely long plates of unequal widths h and w,having one common edge, and at an angle of 90° toeach other:H =F1−2 =hw[601], (29)11 + H − 1 + H22Lines: 1030 to 1030———wA2A1␣w3.
Two infinitely long plates of equal finite width w,having one common edge, forming a wedgelike groovewith opening angle a:αF1−2 = F2−1 = 1 − sin24.08975pt PgVar———Normal PagePgEnds: TEX[601], (29)rrsA2r1A14. Infinitely long parallel cylinders of the same diameter:sX =1+2r1−1 1sin+ X2 − 1 − XF1−2 =πXA1r2sA25.
Two infinite parallel cylinders of different radius:sr2S=C =1+R+SR=r1r11π + C 2 − (R + 1)2 − C 2 − (R − 1)2F1−2 =2π−1 R − 1−1 R + 1+(R − 1) cos− (R + 1) cosCCBOOKCOMP, Inc. — John Wiley & Sons / Page 601 / 2nd Proofs / Heat Transfer Handbook / Bejan(continued)602123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATIONTABLE 8.3Important View Factors (Continued)6. Exterior of infinitely long cylinder to unsymmetricallyplaced, infinitely long parallel rectangle; r ≤ a:b1aB1 =F1−2 =b2aB2 =1(tan−1 B1 − tan−1 B2 )2π7.
Identical, parallel, directly opposed rectangles:X=F1−2 =2πXYln(1 + X2 )(1 + Y 2 )1 + X2 + Y 2ac1/2Y =bcX+ X 1 + Y 2 tan−1 √1 + Y2Y+Y 1 + X 2 tan−1 √− X tan−1 X − Y tan−1 Y1 + X28. Two finite rectangles of same length, having one commonedge, and at an angle of 90° to each other:H =F1−2hlW =wl1111W tan−1+ H tan−1− H 2 + W 2 tan−1 √=πWWHH2 + W2W 2 2H 2 1(1 + W 2 )(1 + H 2 ) W 2 (1 + W 2 + H 2 )H (1 + H 2 + W 2 )+ ln41 + W2 + H2(1 + W 2 )(W 2 + H 2 )(1 + H 2 )(H 2 + W 2 )9.
Disk to parallel coaxial disk of unequal radius:r1r2R2 =R1 =aaX =1+F1−2BOOKCOMP, Inc. — John Wiley & Sons / Page 602 / 2nd Proofs / Heat Transfer Handbook / Bejan1 + R22R12 21R2 2=X− X −42R1[602], (30)Lines: 1030 to 1030———5.43091pt PgVar———Normal PagePgEnds: TEX[602], (30)RADIATIVE EXCHANGE BETWEEN SURFACES123456789101112131415161718192021222324252627282930313233343536373839404142434445TABLE 8.3603Important View Factors (Continued)10.
Outer surface of cylinder to annular disk at endof cylinder:A1A2R=lr2r1r2L=lr2A = L2 + R 2 − 1r1F1−211BA+cos−1 −=8RL 2πB2LB = L2 − R 2 + 1(A + 2)2AAR−sin−1 R− 4 cos−1R2B2RLA2r1r2hF2−2[603], (31)11. Interior of finite-length, right-circular coaxial cylinderto itself:R=r2r1H =hr1Lines: 1030 to 1030———√√2H 2 + 4R 2 − H1 21−1 2 R − 1+tan=1− −R4Rπ RHH−2R√*4R 2 + H 2H 2 + 4(R 2 − 1) − 2H 2 /R 2R2 − 2sin−1− sin−122HH + 4(R − 1)R232.5553pt PgVar———Normal PagePgEnds: TEX[603], (31)12.
Sphere to rectangle, r < d:rD1 =A1dA2F1−2l21tan−1=4πl1A1dl1D2 =dl21D12 + D22 + D12 D2213. Sphere to coaxial disk:aF1−2rA2BOOKCOMP, Inc. — John Wiley & Sons / Page 603 / 2nd Proofs / Heat Transfer Handbook / BejanrR=a111− √=21 + R2604123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATION[604], (32)Lines: 1030 to 1030———*32.50198pt PgVar———Normal PagePgEnds: TEXFigure 8.15 View factor between identical, parallel, directly opposed rectangles (configuration 7).[604], (32)Figure 8.16 View factor between perpendicular rectangles with common edge (configuration 8).BOOKCOMP, Inc.
— John Wiley & Sons / Page 604 / 2nd Proofs / Heat Transfer Handbook / BejanRADIATIVE EXCHANGE BETWEEN SURFACES123456789101112131415161718192021222324252627282930313233343536373839404142434445605[605], (33)Lines: 1030 to 1030———*21.953pt PgVar———Normal PagePgEnds: TEXFigure 8.17 View factor between parallel coaxial disks of unequal radius (configuration 9).[605], (33)Figure 8.18 View factor evaluation between two surfaces.BOOKCOMP, Inc.
— John Wiley & Sons / Page 605 / 2nd Proofs / Heat Transfer Handbook / Bejan606123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATIONone’s disposal. Using an arbitrary coordinate origin, a vector pointing from the originto a point on a surface may be written asr = xı̂ + ŷ + zk̂(8.46)where ı̂, ̂, and k̂ are unit vectors pointing into the x, y, and z directions, respectively.Thus the vector from dAi going to dAj is determined assij = −sj i = rj − ri = (xj − xi )ı̂ + (yj − yi )̂ + (zj − zi )k̂(8.47)The length of this vector is determined as 2 2sj i = sij = S 2 = (xj − xi )2 + (yj − yi )2 + (zj − zi )2ij(8.48)It will now be assumed that the local surface normals are known in terms of the unitvectors ı̂, ̂, and k̂, orn̂ = lı̂ + m̂ + nk̂(8.49)[606], (34)Lines: 1030 to 1081———9.58319pt PgVar———where l, m, and n are the direction cosines for the unit vector n̂.
For example, l = n̂ ·ı̂Normal Pageis the cosine of the angle between n̂ and the x axis, and m and n can be representedin a similar manner. Thus cos θi and cos θj may be evaluated as* PgEnds: Ejectcos θi =n̂i · sij1 =(xj − xi )li + (yj − yi )mi + (zj − zi )niSijSij(8.50)cos θj =n̂j · sj i1 =(xi − xj )lj + (yi − yj )mj + (zi − zj )njSijSij(8.51)For contour integration the tangential vectors ds are found from eq. (8.46), that is,ds = dx ı̂ + dy ̂ + dz k̂(8.52)where dx is the change in the x coordinate along the contour Γ, and dy and dz followin a similar fashion. Examples for the application of area and contour integration toview factor evaluation may be found in textbooks (Modest, 2003; Siegel and Howell,2002).Special Methods The mathematics of view factors follow certain rules, whichmay be exploited to simplify their evaluation. The two most important ones are thesummation rule for a closed configuration consisting of N surfaces,NFi−j = 1j =1BOOKCOMP, Inc.
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