Radiation Exchange Between Black and Gray Surfaces

RADEXCHGWEB.TEXECE 309 M.M. YovanovichRadiation Exchange Between Black and Gray SurfacesRadiation View FactorWhen radiation leaves a black convex surface whose area is A 1 at absolutetemperature T 1 , a certain fraction F 12 will be absorbed by a second convexsurface whose area is A 2 at absolute temperature T 2 < T 1 . The radiationview factor is given by the following relation:A 1 F 12 = A 2 F 21 =ZZA2ZZA1cos 1 cos 2r 2 dA 1 dA 2where r is the radial distance between the centroids of the arbitrary dierentialareas dA 1 and dA 2 which are located in the surfaces A 1 and A 2 respectively.The angle 1 is subtended by the radius r and the outward-directed normal tothe dierential area dA 1 at its centroid. The angle 2 is dened in a similarmanner. In the previous relation F 21 is the fraction of radiation which leavessurface A 2 and is intercepted by surface A 1 . The view factors are dimensionlessradiation parameters whereReciprocity RelationThe relation0 F 12 1 and 0 F 21 1A 1 F 12 = A 2 F 21is called the reciprocity relation which isvalid for any two convex surfaces. Theview factors for a simple system consisting of two isothermal surfaces the viewfactors have the relations:F 11 + F 12 =1 and F 21 + F 22 =1If the surfaces are convex or at F 11 = F 22 = 0. If the surfaces are concave,then 0

The radiation view factors depend on the geometry of the surfaces and theirspatial relation to each other. The calculations of the view factors by the aboverelation are usually dicult to do. There are simple techniques available todetermine the view factors for certain axisymmetric systems and some innitelylong two-dimensional systems.Radiant Exchange Between Two Isothermal Black SurfacesThe net radiant exchange between two isothermal black surfaces is given by_Q 12= E b1 , E b2R 12where E b1 = T 4 1 and E b2 = T 4 2 are the blackbody radiative nodes and R 12 =1=A 1 F 12 =1=A 2 F 21 = R 21 is the spatial radiative resistance between the twosurfaces. The units of R 12 and R 21 are m ,2 .The previous relation can be expressed in the following form:_Q 12 = A 1 F 12hT , 4 T 4 1 2iwhich clearly reveals the non-linear nature of radiative heat exchange. Theradiative exchange between two isothermal black surfaces can be representedby a simple thermal circuit consisting of two nodes: E b1 ;E b2 separated by asingle radiative resistance R 12 . The throughput is _Q 12 .Radiant Exchange Between Two Isothermal Gray SurfacesThe net radiant exchange between two isothermal gray surfaces whose areas areA 1 ;A 2 respectively and whose emissivities are 1 ; 2 respectively is given by_Q 12= E b1 , E b2R totalwhere the total radiative resistance now consists of three resistances in series:R totalThe gray surface resistances are given by:= R s1 + R 12 + R s2R s1 = 1 , 1A 1 1and R s2 = 1 , 2A 2 2with 0 1 1 and 0 2 1. When 1 = 1 and 2 = 1 the gray surfacerelation goes to the blackbody relation given above. The equivalent thermal2

circuit consists of two blackbody nodes E b1 ;E b2 and two internal nodes denotedJ 1 and J 2 which are called the radiosity. The units of radiosity J are identicalto the units of E b . The surface resistance R s1 connects the two nodes E b1 andJ 1 , and the surface resistance R s2 connects the two nodes E b2 and J 2 . The tworadiosity nodes are connected by the spatial resistance R 12 . The throughput isthe net radiant exchange _Q 12 .The general gray surface radiant exchange expression gives the following expressionsfor two innite parallel planes, two long concentric circular cylinders, andtwo concentric spheres.Two Innite Parallel PlanesIn this case A 1 = A 2 = A and F 12 = 1. Therefore we get_Q 12 = A (E b1 , E b2 )1 1+ 1 2, 1Long Concentric Gray Circular CylindersIn this case the inner isothermal cylinder at absolute temperature T 1 and whosearea is A 1 =2r 1 L is placed inside the outer cylinder at absolute temperatureT 2 >r 2 >r 1 . The emissivities of the two cylinders are 1 and 2 respectively.The net radiative exchange is given by_Q 12 = 2 r 1 L (E b1 , E b2 )1+ r 1 1, 1 1 r 2 2Concentric Gray SpheresIn this case the inner isothermal sphere at absolute temperature T 1 and whosearea is A 1 =4r 2 is placed inside the outer sphere at absolute temperature1T 2

The expression for two gray spheres gives a simple relation when the outersphere is much larger than the inner sphere, i.e. r 2 >> r 1 . For this case theprevious expression goes to_Q 12 = 1 4 r 2 1(E b1 , E b2 )which is independent of the emissivity and surface area of the larger sphere.The emissivity and the surface area of the smaller sphere control the radiativeexchange between the two surfaces.4