1、I. _7i ffNATIONAL ADVISORYFOR AERO_AUTICSCOMMITTEERADIANT -iNTERC HANGE C ONFIGUIKATION FAC TORSBy D. C. Hamilton and W. R. MorganPurdue UniversityWashingtonDecember 1952: : :_i _ ,! CLEARINGHOUSEfor Federal Scienlific _A -a. -_- - . 7_5:_ cL, t_Sti.2Y.z : YProvided by IHSNot for ResaleNo reproducti
2、on or networking permitted without license from IHS-,-,-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS_:5.: i71_e:“ r.-7z:r_-u:_A-,:_gq :-72,:TL:2:ii2,12 L_-;:_TECHNICAL NOTE 2836RADIANT-INTERCHAnGE CONFIGURATION FACTORSBy D. C. Hamilton and W. R. MorganSU_A RYThis report is concerned with the geometri
3、c cdnfiguration factorfor computing radiant interchange between opaque surfaces separated bya nonabsorbing medium. The configuration-factor solutions svailable inthe literature have been checked and the more complicated equations arepresented as families of curves. Several new configurations involvi
4、ngrectangles, triangles, and cylinders of finite length have been inte-grated and tabulated. The various methods of determining configurationfactors are discussed and a mechanical integrator is described. Ananalysis is presented, in which configuration factors are employed, ofthe radiant heat transf
5、er to the rotor blades of a typical gas turbineunder different conditions of temperature and pressure.INTRODUCTIONThe many advantages have been evinced that would result fromincreased operating temperatures for gas turbines. This increase wouldrequire a greater amount of cooling if the use of nonstr
6、stegic materialsis continued. At the lower operating temperatures the steady-statecooling requirements for the various internal components of the turbinemay be determined by considering the heat transfer due to convection only.At the higher temperatures presently contemplated and at the even higher
7、itemperatures that will ultimately be envisioned, radiation will cease tobe negligible and may well become the dominant mechanism. Since present itrends indicate cooling nonstrategic materials as the means of increasingoperating temperatures, it is important that the computation of radiantheat trans
8、fer be facilitated.Unless 8 system is intentionally designed to facilitate computstlonof radiant heat transfer, this computation is, in general, a raLherinvolved operation. The engineer desiring to compute the rndlant heattransfer in a system such as a gas turbine is usually discouraged fromperformi
9、ng more than a cursory estimmtlon because of the excessive amountof time involved in obtaining the configuration factors. The absorptivityand emissivity of a surface are dependent upon composition of the surface,nature and thickness of film or oxide layer_ magnitude and form of surfaceProvided by IH
10、SNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Ii_!I_ii_!_-_v,0_ :_ _ _ _ :_“EE=E?NACA TN 2836asperities, and temperatures of the system. Since, in general_ exactknowledge of these properties of the surfaces involved in a particularsystem in practice is not avail
11、able, it is difficult to assign morethan an approximate value to the emissivity or absorptivity.To m_ke the analysis mathematically feasible it is common practiceto divide the system into zones, each of which is assigned a mean tem-perature. In this manner the postulate of isothermal surfaces or zon
12、esmay be made. This latter simplification or idealization of the systemintroduces additional errors in the results.The equation generally used for computing net interchange betweentwo isothermal zones separated by a nonabsorbing medium isql_2 = oAIfI_2(TI 4 - T24 )The v consequently_ these values we
13、re computed by thenumerical integration of L-2. Configurations L-3 and L-4 were alsoobtained by numerical integration.The factor discussed in this paper has been called variously anglefactor, shape factor, and configuration factor. The first name, anglefactor_ does not adequately describe the factor
14、 in question. The second,shap_ factor, has been consistently used in the literature as the name ofthe geometric factor in heat conduction. Because conduction and radiationoften occur simultaneously and because shape factor has a unique meaningin heat conduction it appears that the geometric factor f
15、or radiationshould be differently n_med to avoid confusion. The use of configurationfactor is recommended since it has already been used many times and sinceit is adequately descriptive. The authors wish to acknowledge that theabove discussion and recommendation came to them from Dr. G. A. Hawkinsof
16、 Purdue University. Many of the references were brought to the authorsattention by Dr. W. L. Sibbitt. Dr. J. T. Agnew assisted with the cal-culations in appendix D.This work was conducted at the Purdue University EngineeringExperiment Station under the sponsorship and with the financial assist-ance
17、of the National Advisory Committee for Aeronautics. oDEFINITION OF CONFIGURATION FACTORThe configuration factor from A I to A 2 written FI2 is hereindefined as the fraction of the total radiant flux leaving A I that isincident upon A2. The configuration factor from e plane point source(point configu
18、ration factor) is obtained by integration over A2 whilethe mean configuration factor from a line source or a finite source isan average of the point configuration factor over the line source orfinite source, respectively. The configuration factor is a fractionthat is a function of the geometry of th
19、e two surfaces; it also dependson the directional distribution of the radiation from the source. Forthe present discussion let a directional distribution function D()be defined as follows:f?i!Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4 NACA TN
20、2836where l(e) is the intensity at the angle e (measured from the normalto the plane point source) and Im represents the mean value of theintensity defined by Im = W/_. Experiments (see reference 25) indi-cate that most engineering materials do not exactly follow Lambertscosine principle (reference
21、26); this principle gives values that aretoo low for polished conductors and too high for insulators at largevalues of the angle e. Lamberts cosine principle states that thefunction D(e) is a constant equal to unity and invariable with e.The error introduced by using Lsmberts form of D in the calcu-
22、lation of radiant heat trsnsfer has been assumed too small (in com-parison with other calculation errors tolerated in practice) to warrantthe complications introduced by the use of a more accurate form of D.Eckert (reference 25) gives a method for determining the configurationfactor when a non-Lambe
23、rtian distribution is postulated. One couldintegrate and tabulate configuration factors for the D functionstypical of nonconductors and conductors this may become desirable atsome distant future date.The configuration factor may be defined as the ratio of the radiantflux leaving a source that is inc
24、ident on another surface to the totalflux leaving the source. The limiting values are then zero and unity.After a form of the distribution function D is postulated, the con-figuration factor becomes a purely geometric function.From figure 1 and the definition of the configuration factor fora radiati
25、ng point source dAI and an intercepting area dA2, theconfiguration-factor equation is derived as:where d_l, the solid angle subtended by dA2 at dAl, is_2 cos e 2 do-,Z= (2)r 2:z “ 2, L L,f.: : : “Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2az2 A
26、 .,-_NACA .TN 2836 5From equations (I) and (2) obtaindA 1 dFdAl-dA2 = DI(6)1)cos eI cos e2 dA IdA 2_r2(la)cos eI cos e2 dAIdA 2dA2 dFdA2-dAl = D2(02) _r 2(lb)For any two areas dAIthat, if DI(91) andequation (3), holds:and dA2 it follows from equations (la) and (Ib)D2(02) are identical, the reciproci
27、ty theorem,dA2 dFdA2_dA I = dA I dFdAl_dA2(3)It is interesting to note thatthe reciprocity theorem fails when thedistribution functions of the two surfaces are not identical. Thus,the reciprocity theorem is not true by definition, as some referenceshave implied.For the present work a Lambert/an dist
28、ribution will be postulatedin which case D disappears from equations (is) and (ib) and equa-tion (3) applfes.The configuration factor FdAI_A2 from dAl to A 2 is obtainedby integrating equation (la) over A2.idAIFdAI_A2 = dAI JA 2 dFdAl-dA2p cos eI cos e2 dA2= dA I JA 2 “_r2_ -(4)Provided by IHSNot fo
29、r ResaleNo reproduction or networking permitted without license from IHS-,-,-_:_:._x.- NACA TN 2836!/iSimilarly, integration of equation (4) over AI gives FAI_A 2 from AIto A2:- “ AIFAI-A2 = _A FdAI-A2 dAl1=_ _ cos 01 cos 02 dAIdA 2 r is the lengthof the straight line connecting the two differential
30、 areas; and 81 and_2 are the angles between r and the respective normsls.Fr0m-fig6_re 2-th6-equatibns for _cos 01,- c6s 02,- and- r- can begiven in terms of the coordinates x and y and the constants C-and a. These, when substituted in equation (4), give equation (6):sin20 om oh dy, _r 2 2 (6)E_)2+ (
31、y,)2 + a - (2aycos_Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-! N_A TN 2836 ! 7_ A change to dimensionless variables will, in general, simplify theintegration. _t x = x/a, y = y/a, M = m/a, and H = h/a. ThenI _i sin2 H y dyPF“ /!iL_:). , ,:al%1-
32、N“_ 2“ -_:“C_“.7_ ,7Integration with respect to y yields.dAIFdAI_A2 =dA I sin2- cos 2 + sin21 +(i + H2 + x2 - 2H cosx_o_0 I:on-_(_-_os0_+(7)and integration with respect to x givesA2 n-IM“I,- a +I + H2 - 2H cos 1 + H2 - 2H cos Z liM cos tEsn_l/ H - c_os h + /an_l( cos ._1(8)Provided by IHSNot for Res
33、aleNo reproduction or networking permitted without license from IHS-,-,-!i .!8 - , NACA TN 2836Refer now to figure 3(a) for the third integration (over the x-axisof AI):FdAI-A2 = (FdAI-A2 + FdAI-A2“ ) (9)Combining equations (8) and (9) and integrating with respect to x get:1 _ sin2FdAI-A2 = _ sn-iM
34、+ 234 E 1 + E 2 + M2 - 2H-cos _- Io ge L( _ _I M_ _ H2 - _ _ o s _ JJ,n.co. cos.,n2 1 + H2 - 2H cos _ + H2 - 2H cos Ics- _M2+M- sin2 JLltan-l_I_f“H- cos_iA./_+ tan-lf _(N2cs+ Jn90 _S(io)It should be noted that, in equation (i0), dA I refers to a line sourceof width dz and of length M, whereas in all
35、 preceding equations dA1has pertained to s point source, both dimensions of which were of dif- ferential order. .To accomplish the final integration it is desirable to change toa new set of dimensionless variables: Let N = h/m 3 L = s/m, andz = z/m:z sin cos _-_ - ; tan-l( N -zzsincos _ + cos ._I +
36、z2sln2lan-iI_l- z _2cs_in_/_ +tan_l!_l z cos _ + N co_s.- z tan_l/ 1 I(_o_)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-qD_:=_ _ i_-_ACA T_ 2836 90 0r-iv! %UIo %!_3-ta4-%r_ .r-t0! +r-I!I%0100Provided by IHSNot for ResaleNo reproduction or networki
37、ng permitted without license from IHS-,-,-_!1?tr_i _Z._lO . _ACA TN 2836Equation (ii) is too complex algebraically to make numerical solutionof it practicable; for this reason the last integral was not solved andthe values for FAI_A 2 (configurstion A-2) were obtained by the numer-ical integration o
38、f the values obtained for configuration L-2 fromequation (10a).For the particular case ofto:FA1-A2 .= _l I tan-l(1) + IN tan-l(1)- IN2+ L2 cot-1 _N2+ L2 +i _ F(1. +. L2)(1 + N2)_+ L2 loge S2(i+N 2+ U2)equal to 90 the equation simplifies(lls)The labor involved in the first two integrations can genera
39、lly be reducedby use of Stokes theorem and vector calculus.Methods of Determining “Point“ Configuration FactorsBased on Double-Projection Principle of NusseltThe following methods apply Nusselts geometric interpretation ofthe configuration-factor equation (reference 5) in one way or another toget co
40、nfiguration factors in which the source is a plane point source.The “point“ configuration-factor equation is:_A cos eI d_1= - - (_a)FdAI-A 2 _ 1In figure 4 it is seen that the solid angle d_ 1at dA 1 iscos e 2 dA 2 _ dA 2d_ 1 = R2 r 2subtended by dA2Provided by IHSNot for ResaleNo reproduction or ne
41、tworking permitted without license from IHS-,-,-:2; i:/_. _,247- L|NACA TN 2836 llwhere dA2 is the radial projection of dA2 on the surface of thesphere of radius r. Also_ dA2“ = dA2 cos 81; therefore, the con-figuration factor is interpreted as:_2“- (12)FdAI-A2 _r2To recepitulate_ let a hemisphere o
42、f radius r be constructed aboutthe plane point source dA1 with the point source in the plane of thebase of the hemisphere. Let each point on the perimeter of A 2 be pro-jected radially to the surface of the hemisphere_ thence vertically(parallel to the normal to dA1) down to the base of the hemisphe
43、re.The ratio of this projected ares A2“ on the base of the hemisphere tothe area of the entire base of the hemisphere is the desired “point“configuration factor from dA1 to A2.Drawing-board solution.- By the methods of elementary descriptivegeometry the point configuration factor can be obtained by
44、projecting thevarious points on A2 to the base of a hemisphere constructed about theradiating point source. Simple geometry such as in configurations P-3and P-4 (appendix B) _y be solved quickly by drawing-board projection. Transit method.- If the desired configuration factor is essqciatedwith a rel
45、atively large structure already built_ the point configurationfactors can be obtained by using a surveyors transit with an elbow-telescope attachment to permit vertical shots. If 8 represents theangle in a vertical plane (measured from the horizon) and representsthe angle in the horizontal plane (me
46、asured from some arbitrary place)and readings are taken of these angles for strategic points on thep_im_er of the intercepting area, then e plot of cos28 against # .will give an.area that is proportional to the configuration factor.Optical projection method of Eckert.- The optical projection method
47、of Eckert (reference 27) employed a point light source situated in thecenter of the base of a milk-glass hemisphere as shown in figure 5.Paper models were suspended by small wires from the base. The modelscast a shadow on the milk-glass hemisphere. Photographs were taken at_reat distance from the he
48、misphere. The area of the image of theshadow on the film divided by the area of the image of the hemispherewas a measure of the configuration factor from a point source at thecenter of the base of the hemisphere to the paper model. Excellentresults were obtained with this apparatus.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-i 12 NACA TN 2836Mechanical integrator.- The first mechan
