Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 98
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— John Wiley & Sons / Page 606 / 2nd Proofs / Heat Transfer Handbook / Bejan(8.53)[606], (34)RADIATIVE EXCHANGE BETWEEN SURFACES123456789101112131415161718192021222324252627282930313233343536373839404142434445607stating that the sum of fractions must total unity, and the reciprocity rule,Ai Fi−j = Aj Fj −i(8.54)which follows directly from eq. (8.44). The methods known as view factor algebraand the crossed-strings method are discussed briefly next.View Factor Algebra Many view factors for fairly complex configurations maybe calculated without any integration simply by using the rules of reciprocity andsummation and perhaps the known view factor for a more basic geometry. For example, suppose that the view factor for a corner piece from configuration 8 in Table 8.3is given.
Using this knowledge, the view factor F3−4 between the two perpendicularstrips shown in Fig. 8.19 can be evaluated.From the definition of the view factor and because the energy traveling to A4 isthe energy going to A2 and A4 minus the energy going to A2 , it follows thatF3−4 = F3−(2+4) − F3−21 (A2 + A4 )F(2+4)−3 − A2 F2−3A3Similarly,F3−4 =Lines: 1081 to 1127———and using reciprocity,F3−4 =[607], (35) A2 A2 + A4 F(2+4)−(1+3) − F(2+4)−1 −F2−(1+3) − F2−1A3A3All view factors on the right-hand side are for corner pieces and may be found byevaluating view factor 8 with appropriate dimensions.As a second example, the view factor from the inside surface of a finite-lengthcylinder to itself will be determined (as shown in the figure for view factor 11 for theA4A2A1A3Figure 8.19 View factor between two strips on a corner.BOOKCOMP, Inc.
— John Wiley & Sons / Page 607 / 2nd Proofs / Heat Transfer Handbook / Bejan4.5711pt PgVar———Normal PagePgEnds: TEX[607], (35)608123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATIONlimiting case of r1 → 0). Calling the cylinder A1 , and top and bottom openings ofthe cylinder A2 and A3 , respectively, the summation rule givesF1−1 + F1−2 + F1−3 = 1orF1−1 = 1 − 2F1−2because F1−2 = F1−3 .
Now, using reciprocity, or A1 F1−2 = A2 F2−1 , and summationonce again,F1−1 = 1 − 2A2A2F2−1 = 1 − 2 (1 − F2−3 )A1A1because F2−2 = 0. Then F2−3 can be evaluated from configuration 9, with R1 =R2 = r/ h = R, X = 2 + 1/R 2 , or1411F2−3 =2+ 2 − 4+ 2 + 4 −42RRR[608], (36)Lines: 1127 to 1178Finally, with A2 /A1 = πr 2 /2πrh = r/2h = R/2,11111F1−1 = 1 − R − 2 +=1+1+− 1+2RR4R 22R4R 2———0.12416pt PgVar———Long PagePgEnds: TEXCrossed-Strings Method View factor algebra may be used to determine all theview factors in long enclosures with constant cross section. The method is called thecrossed-strings method because the view factors can be determined experimentallyby a person armed with four pins, a roll of string, and a yardstick. Consider the configuration in Fig.
8.20, which shows the cross section of an infinitely long enclosure,continuing into and out of the plane of the figure: The determination of F1−2 is sought.Obviously, the surfaces shown are rather irregular (partly convex, partly concave), andthe view between them may be obstructed. For such geometries, integration is out ofthe question; however, repeated application of the reciprocity and summation rulesallows the evaluation of F1−2 asF1−2 =(Abc + Aad ) − (Aac + Abd )2A1(8.55)where, in general, Aab is the area (per unit depth) defined by the length of the stringbetween points a and b.
This relationship is easily memorized by looking at theconfiguration between any two surfaces as a generalized “rectangle,” consisting ofA1 , A2 , and the two sides of Aac and Abd . ThenF1−2 =diagonals − sides2 × originating area(8.56)As an example, suppose that the calculation of F1−2 for configuration 1 in Table 8.3via the crossed-strings method is desired. For that geometry, both “sides” would haveBOOKCOMP, Inc. — John Wiley & Sons / Page 608 / 2nd Proofs / Heat Transfer Handbook / Bejan[608], (36)RADIATIVE EXCHANGE BETWEEN SURFACES123456789101112131415161718192021222324252627282930313233343536373839404142434445609[609], (37)Figure 8.20 Crossed strings method for two-dimensional configurations.Lines: 1178 to 1199√length h, both diagonals would have length h2 + w 2 , and the originating area wouldhave width w. Thus,√ 22 h2 + w 2 − 2hhhF1−2 =−= 1+(8.57)2wwwas given in the table.8.3.2Radiative Exchange between Black SurfacesConsider an enclosure consisting of N opaque, black, isothermal surfaces.
One ormore of these surfaces may not be actual material but a hole through which radiationescapes, and through which external radiation may enter the enclosure. A hole isusually best modeled as a cold black surface, because a hole does not reflect internalradiation (black) and because usually no diffuse energy is entering through the hole(no emission or T = 0 K).
If external radiation is entering through an opening, thisenergy tends to be directional and is best accounted for in the energy balances forindividual surfaces that receive it.An energy balance for any isothermal surface Ai in the enclosure yieldsQi= Ebi − HiAiHi =NEbj Fi−j + Hoii = 1, 2, . . . , N(8.58)j =1where Qi /Ai is the average radiative heat flux on Ai , Ebi = σTi4 is the surface’semissive power, and Hi is the total irradiation onto Ai , consisting of the fractions ofemitted radiation from all surfaces in the enclosure (including itself if Ai is concave)BOOKCOMP, Inc. — John Wiley & Sons / Page 609 / 2nd Proofs / Heat Transfer Handbook / Bejan———4.68pt PgVar———Long PagePgEnds: TEX[609], (37)610123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATIONthat are intercepted by Ai , plus, possibly, external radiation Hoi that enters through ahole and hits Ai (per unit area of Ai ).
If the temperatures for all N surfaces makingup the enclosure are known, eq. (8.58) constitutes a set of N explicit equations forthe unknown radiative fluxes (Qi /Ai ).Suppose that for surfaces i = 1, 2, . . . , n the heat fluxes are prescribed (andtemperatures are unknown), whereas for surfaces i = n + 1, . . . , N the temperaturesare prescribed (heat fluxes unknown).
Unlike for the heat fluxes, no explicit relationsfor the unknown temperatures exist. Placing all unknown temperatures on one side,eq. (8.58) may be written asEbi −nFi−j Ebj =j =1NQi+ Hoi +Fi−j EbjAij =n+1i = 1, 2, . . . , n(8.59)where everything on the right-hand side of the equation is known. In matrix form thisis written asA · eb = b(8.60)Lines: 1199 to 1252———where1 − F1−1 −F2−1A=...−Fn−1E b1 Eb2 eb = ... Ebn−F1−2···−F1−n1 − F2−2···...−F2−n...−Fn−2Q1A 1 Q2b= A2 QnAn· · · 1 − Fn−n+ Ho1 +N-1.86508pt PgVar(8.61)F1−j Ebj N+ Ho2 +F2−j Ebj j =n+1...N+ Hon +Fn−j Ebjj =n+1(8.62)j =n+1eb = A−1 · b(8.63)Radiative Exchange between Diffuse Gray SurfacesIt will now be assumed that all surfaces are gray and that they are diffuse emitters,absorbers, and reflectors.
Under these conditions, = λ = αλ = α = 1 − ρ. TheBOOKCOMP, Inc. — John Wiley & Sons / Page 610 / 2nd Proofs / Heat Transfer Handbook / Bejan———Normal PagePgEnds: TEX[610], (38)The n × n matrix A is readily inverted on a computer (generally with the aid of asoftware library subroutine), and the unknown temperatures are calculated as8.3.3[610], (38)RADIATIVE EXCHANGE BETWEEN SURFACES123456789101112131415161718192021222324252627282930313233343536373839404142434445611radiative heat flux leaving surface Ai now consists of emission plus the reflection ofincoming radiation,Ji = i Ebi + ρi Hi(8.64)which is called the surface radiosity.
In the same way as eq. (8.58) was formulated,one can make an energy balance at surface Ai , but there are now two ways to makethis balance,Qi= Ji − Hi = i Ebi − αi HiAii = 1, 2, . . . , N(8.65)the first stating that the net flux is equal to outgoing minus incoming radiation,the other stating that net flux is equal to the emitted minus the absorbed radiation.Eliminating Hi between them gives an expression for radiosity in terms of localtemperature and flux,1QiJi = Ebi −−1(8.66)iAiReplacing emissive power by radiosities in eq.
(8.58) and using eq. (8.65) leads to Nsimultaneous equations for the N unknown radiosities,NJi = i Ebi + ρi (8.67a)Fi−j Jj + Hoi j =1orNJi =Qi +Fi−j Jj + HoiAij =1(8.67b)dependent on whether temperature (Ebi ) or flux (Qi /Ai ) is known for surface i.While commonly solved for in the older literature, there is rarely ever any need todetermine radiosities. Eliminating them through the use of eq.
(8.66) yields a set ofN simultaneous algebraic equations,N NQj1 Qi 1−− 1 Fi−j+ Hoi = Ebi −Fi−j Ebji AijAjj =1j =1i = 1, 2, . . . , N(8.68)Equation (8.68) contains N different Ebi and N different Qi /Ai . Therefore, if thetemperatures are specified for all N surfaces, all fluxes can be calculated. It is alsopossible to specify an arbitrary mix of N surface temperatures and heat fluxes, and eq.(8.68) allows determination of the remaining N unknowns (the one exception beingBOOKCOMP, Inc. — John Wiley & Sons / Page 611 / 2nd Proofs / Heat Transfer Handbook / Bejan[611], (39)Lines: 1252 to 1293———7.37305pt PgVar———Normal PagePgEnds: TEX[611], (39)612123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATIONthat it is not proper to specify all N heat fluxes: because from conservation of energy,#Ni=1 Qi = 0, this amounts to only N − 1 specifications).
As for black enclosures,holes are modeled as cold black surfaces; because for such surfaces Ebi = Ji = 0,they do not appear in eq. (8.68) (except for external irradiation, entering throughholes, accounted for at the surfaces receiving this irradiation).Convex Surface Exposed to Large Isothermal Enclosure In many important engineering applications a flat or convex surface (i.e., a surface that cannot “seeitself”) is radiating into (and receiving radiation from) a large isothermal enclosure.In such a case, N = 2 (Ai being the convex surface and Ae the large enclosure), and,if Fi−e = 1, eq.
(8.68) reduces toσ(Ti4 − Te4 )Qi i σ Ti4 − Te4=Ai(1/i ) + (Ai /Ae )[(1/e ) − 1](8.69)because for a large enclosure, Ai Ae . Equation (8.69) provides a simple, yetaccurate set of radiation boundary conditions for engineering problems, which aredominated by conduction and/or convection.[612], (40)Lines: 1293 to 1329———2.73325pt PgVar8.3.4Radiation ShieldsIf it is desired to minimize radiative heat transfer between two surfaces, it is commonpractice to place one or more radiation shields between them (usually, thin metallicsheets of low emittance).
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