RADIATIVE VIEW FACTORS

Transcription

1 RADIATIVE VIEW ACTORS View factor definition... View factor algebra... Wit speres... 3 Patc to a spere... 3 rontal... 3 Level... 3 Tilted... 3 Disc to frontal spere... 3 Cylinder to large spere... Cylinder to its emisperical closing cap... Spere to spere... 5 Small to very large... 5 Equal speres... 5 Concentric speres... 5 Hemisperes... 6 Wit cylinders... 6 Cylinder to large spere... 6 Cylinder to its emisperical closing cap... 6 Concentric very-long cylinders... 6 Concentric very-long cylinder to emi-cylinder... 7 Wire to parallel cylinder, infinite extent... 7 Parallel very-long external cylinders... 7 Base to finite cylinder... 7 Equal finite concentric cylinders... 8 Wit plates and discs... 8 Parallel configurations... 8 Equal square plates... 8 Unequal coaxial square plates... 9 Box inside concentric box... 9 Equal rectangular plates... 0 Equal discs... 0 Unequal discs... Strip to strip... Patc to infinite plate... Patc to disc... Perpendicular configurations... Square plate to rectangular plate... Rectangular plate to equal rectangular plate... Rectangular plate to unequal rectangular plate... Strip to strip... 3 Tilted configurations... 3 Equal adjacent strips... 3 Triangular prism... 3 References... 3

2 VIEW ACTOR DEINITION Te view factor is te fraction of energy exiting an isotermal, opaque, and diffuse surface (by emission or reflection), tat directly impinges on surface (and is absorbed or reflected). Some view factors aving an analytical expression are compiled below. View factors only depend on geometry, and can be computed from te general expression below. Consider two infinitesimal surface patces, da and da (ig. ), in arbitrary position and orientation, defined by teir separation distance r, and teir respective tilting relative to te line of centres, β and β, wit 0 β π/ and 0 β π/ (i.e. seeing eac oter). Te radiation power intercepted by surface da coming directly from a diffuse surface da is te product of its radiance L M /π, times its perpendicular area da, times te solid angle subtended by da, dω ; i.e. d Φ L da dω L (da cos(β ))da cos(β )/r. Tence: d d Φ ( ) ( ) ( ) ( ) ( ) ( ) L dω dα cos β cos β cos β dα cos β dω MdΑ MdΑ π π r cos β cos β cos β cos β d dα da da πr A πr A A ig.. Geometry for view-factor definition. Wen finite surfaces are involved, computing view factors is just a problem of matematical integration (not a trivial one, except in simple cases). Recall tat te emitting surface (exiting, in general) must be isotermal, opaque, and Lambertian (a perfect diffuser), and, to apply view-factor algebra, all surfaces must be isotermal, opaque, and Lambertian. View factor algebra Wen considering all te surfaces under sigt from a given one (enclosure teory), several general relations can be establised among te N possible view factors, wat is known as view factor algebra: Bounding. View factors are bounded to 0 ij by definition (te view factor ij is te fraction of energy exiting surface i, tat impinges on surface j). Closeness. Summing up all view factors from a given surface in an enclosure, including te possible self-view factor for concave surfaces, ij, because te same amount of radiation emitted by a surface must be absorbed. j Reciprocity. Noticing from te above equation tat da i d ij da j d ji (cosβ i cosβ j /(πr ij ))da i da j, it is deduced tat A i ij Ajji. Distribution. Wen two target surfaces are considered at once, i, j+ k ij + ik, based on area additivity in te definition. Composition. Based on reciprocity and distribution, wen two source areas are considered + A + A A + A. togeter, i j, k ( i ik j jk ) ( i j ) or an enclosure formed by N surfaces, tere are N view factors (eac surface wit all te oters and itself). But only N(N )/ of tem are independent, since anoter N(N )/ can be deduced from reciprocity relations, and N more by closeness relations. or instance, for a 3-surface enclosure, we can

3 define 9 possible view factors, 3 of wic must be found independently, anoter 3 can be obtained from A i ij Ajji, and te remaining 3 by ij. WITH SPHERES Patc to a spere rontal j rom a small planar plate facing a spere of radius R, at a distance H from centres, wit H/R. (e.g. for, /) Level rom a small planar plate x level to a spere of radius arctan π x R, at a distance H from centres, wit H/R. wit x ( π ) (e.g. for, 0.09) Tilted -if β <π/ arcsin(/) (i.e. cosβ>), rom a small planar plate cos β tilted to a spere of radius R, at a distance H -if not, from centres, wit H/R; te tilting angle β ( cos βarccos y xsin β y ) is between te normal π and te line of centres. sin β y + arctan π x x, y xcot β Disc to frontal spere wit ( ) (e.g. for and βπ/ (5º), 0.77)

4 rom a disc of radius R to a frontal spere of radius R at a distance H between centres (it must be H>R ), wit H/R and r R /R. r + (e.g. for r, 0.586) rom a spere of radius R to a frontal disc of radius R at a distance H between centres (it must be H>R, but does not depend on R ), wit H/R. + (e.g. for R H and R H, 0.6) Cylinder to large spere Coaxial (β0): s arcsin ( s) π + π rom a small cylinder (external lateral area only), at an altitude HR and tilted an angle β, to a large spere of radius R, β is between te cylinder axis and te line of centres). ( ) + wit s + Perpendicular (βπ/): + ( ) xe x dx π 0 x wit elliptic integrals E(x). Tilted cylinder: arcsin + π sin ( θ) z d d π θ 0 φ 0 wit ( ) cos( ) ( θ) ( β) ( φ) z cos θ β + θ φ + sin sin cos (e.g. for and any β, /) Cylinder to its emisperical closing cap

5 rom a finite cylinder (surface ) of radius R and eigt H, to its emisperical closing cap (surface ), wit rr/h. Let surface 3 be te base, and surface te virtual base of te emispere. ρ ρ, 3 ρ ρ,, 3, r r ρ 3, 3 r ρ r, ρ 3 r wit ρ r + r Spere to spere (e.g. for RH, 0.38, 0.3, 0.3, 0.50, 3 0.9, 3 0.6, , ) Small to very large rom a small spere of radius R to a muc larger spere of radius R at a distance H between centres (it must be H>R, but does not depend on R ), wit H/R. (e.g. for HR, /) Equal speres rom a spere of radius R to an equal spere at a distance H between centres (it must be H>R), wit H/R. (e.g. for HR, 0.067) Concentric speres Between concentric speres of radii R and R >R, wit r R /R <. r r (e.g. for r/,, /, 3/)

6 Hemisperes rom a emispere of radius R (surface ) to its base circle (surface ). A /A / / rom a emispere of radius R to a larger concentric emispere of radius R >R, wit R R /R >. Let te closing planar annulus be surface 3. rom a spere of radius R to a larger concentric emispere of radius R >R, wit R R /R >. Let te enclosure be 3. ρ ρ ρ, 3, R, ρ R, ρ 3, R ρ, R 3 ( R ) ρ 3 ( R ) wit ρ R ( R ) arcsin π R (e.g. for R, 0.93, 0.3, , , , , 0.) /, 3 /, /R, ρ 3, R wit ρ R ( R ) arcsin π R WITH CYLINDERS Cylinder to large spere See results under Cases wit speres. (e.g. for R, /, /, 3 /, 3 0.3, 0.) Cylinder to its emisperical closing cap See results under Cases wit speres. Concentric very-long cylinders

7 Between concentric infinite cylinders of radii R and R >R, wit r R /R <. r r (e.g. for r/,, /, /) Concentric very-long cylinder to emi-cylinder Between concentric infinite cylinder of radius R to concentric emicylinder of radius R >R, 3, /, r, 3 /, wit r R /R <. Let te ( r + rarcsin r) enclosure be 3. π (e.g. for r/, /, /, 3 /, 3 0., 0.8) Wire to parallel cylinder, infinite extent rom a small infinite long cylinder to an infinite long parallel cylinder of radius R, wit a distance H between axes, wit H/R. arcsin π (e.g. for HR, /) Parallel very-long external cylinders rom a cylinder of radius R to an equal cylinder at a distance H between + arcsin centres (it must be H>R), wit H/R. π (e.g. for HR, / /π0.8) Base to finite cylinder

8 rom base () to lateral surface () in a cylinder of radius R and eigt H, wit r R/H. Let (3) be te opposite base. ρ ρ, 3, r r ρ ρ ρ,, 3 wit ρ r + r Equal finite concentric cylinders (e.g. for RH, 0.6, 0.3, , 0.38) Between finite concentric cylinders of radius R and R >R and eigt H, wit H/R and RR /R. Let te enclosure be 3. or te inside of, see previous case. f f f arccos π f, 3, f R πr πr R 7 + arctan, wit 3 +, f R f R ( ) f3 A+ R, +, f π f f f arccos + f arcsin, 3 Rf R R +, f 5 R R ( + ) 6 π f7 f5arcsin f6 arcsin f + 5 R, ( ) (e.g. for R R and HR, 0.6, 0.3, , 3 0.3, 0.3) WITH PLATES AND DISCS Parallel configurations Equal square plates

9 Between two identical parallel square plates of side L and separation H, wit ww/h. x ln + wy π w + w wit x + w and w y xarctan arctan w x (e.g. for WH, 0.998) Unequal coaxial square plates ln p + s t, wit π w q p rom a square plate of ( w + w + ) side W to a coaxial q ( x + )( y + ) square plate of side W at separation H, wit x w w, y w + w w W /H and w W /H. x y s u xarctan yarctan u u x y t v xarctan yarctan v v u x +, v y + (e.g. for W W H, 0.998) Box inside concentric box rom face to te oters: Between all faces in te enclosure formed by te internal side of a cube box (faces ), and te external side of a concentric cubic box (faces ( ) of size ratio a. (A generic outer-box face #, and its corresponding face #7 in te inner box, ave been cosen.) rom an external-box face: 0, x, 3 y, x, 5 x, 6 x, 7 za, 8 r, 9 0,,0 r,, r,, r rom an internal-box face: 7 z, 7 ( z), 73 0, 7 ( z), 75 ( z), 76 ( z), 77 0, 78 0, 79 0, 7,0 0, 7, 0, 7, 0 wit z given by: rom face 7 to te oters:

10 ( a) p z 7 ln + s+ t π a q 3 a+ 3a p ( a) 8 + a+ 8a q ( a) w s u arctan warctan u u w t v arctan warctan v v 8 ( + a ) + a u 8, v, w a a and: r a ( z) y 0.( a) x ( y za r) (e.g. for a0.5, 0, 0.6, 3 0.0, 0.6, 5 0.6, 6 0.6, 7 0.0, 8 0.0, 9 0,,0 0.0,, 0.0,, 0.0), and ( , , 73 0, , , , 77 0, 78 0, 79 0, 7,0 0, 7, 0, 7, 0). Notice tat a simple interpolation is proposed for y 3 because no analytical solution as been found. Equal rectangular plates Between parallel equal rectangular plates of size W W separated a distance H, wit xw /H and yw /H. xy + ln π xy x y wit x + x y arctan arctan x y y y x arctan arctan y + x x + x and y + y (e.g. for xy, 0.998) Equal discs

11 Between two identical coaxial discs of radius R and separation H, wit rr/h. r + + r (e.g. for r, 0.38) Unequal discs rom a disc of radius R to a coaxial parallel disc of radius R at separation H, wit r R /H and r R /H. wit x y x + r + r r and y x r r (e.g. for r r, 0.38) Strip to strip Between two identical parallel strips of widt W and separation H, wit H/W. + (e.g. for, 0.) Patc to infinite plate rom a finite planar plate at a distance H to an + cos β infinite plane, tilted an ront side: angle β. cos β Back side: (e.g. for β π/ (5º),,front 0.85,,back 0.6) Patc to disc

12 rom a patc to a parallel and concentric disc of radius R at distance H, wit H/R. + (e.g. for, 0.5) Perpendicular configurations Square plate to rectangular plate rom a square plate of wit W to an adjacent + arctan arctan ln π rectangles at 90º, of eigt H, wit H/W. wit + and + ( ) (e.g. for, /, for, 0.000, for /, 0.6) Rectangular plate to equal rectangular plate Between adjacent equal rectangles at 90º, of arctan arctan π eigt H and widt L, wit H/L. ln + wit ( ) + and (e.g. for, 0.000) Rectangular plate to unequal rectangular plate rom a orizontal rectangle of W L to adjacent vertical rectangle of H L, wit H/L and ww/l. arctan warctan π w + w b arctan + + ln wit w w ( ab c ) a w ( + + w ) ( + w )( + w ) ( + )( + w ) + w, + + w + + w, c + + w ( ) ( )( )

13 rom non-adjacent rectangles, te solution can be found wit viewfactor algebra as sown ere (e.g. for w, 0.000) A A + ' ' + ' ' + ' ' A A A A ( ) ( ) + ' ' + ' + ' + ' ' ' + ' ' ' A A Strip to strip Adjacent long strips at 90º, te first () of widt W and te second () of widt H, wit H/W. + + (e.g. 0.93) H W Tilted configurations Equal adjacent strips Adjacent equal long strips at an angle α. α sin (e.g. π 0.93) Triangular prism Between two sides, and, of an infinite long triangular prism of sides L+ L L3 L, L and L 3, wit L L /L and φ being te angle between sides + + cosφ and. (e.g. for and φπ/, 0.93) References Howell, J.R., A catalog of radiation configuration factors, McGraw-Hill, 98. Back to Spacecraft Termal Control