A CATALOG OF RADIATION HEAT
TRANSFER

Derives factors by
contour integration, and presents final analytical expressions. The
resulting expressions contain integrals that must be evaluated numerically.
Numerical integrations are carried out for particular cases, and the
results are correlated and expressions are presented for various ranges of
the geometric parameters. Error ranges and correlation coefficients are
given for each correlation. 
Alexandrov, V.T., 1965,
"Determination of the angular radiation coefficients for a system of two
coaxial cylindrical bodies," Inzh. Fiz. Zh., vol. 8, no. 5, pp.
609612.
Uses numerical
integration of fundamental defining relation between two elements to find
factor from inner surface of outer coaxial cylinder to outer surface of
inner directly opposed cylinder of the same finite length. Closed form is
found for outerouter factor, and outertoinner finite area factor is
found by numerical integration. Configuration factor algebra is then used
to obtain factor from inner cylinder to annular ring end. 
Alciatore, David and Lipp, Stephen, 1989,
"Closed form solution of the general three dimensional radiation configuration
factor problem with microcomputer solution," Proc. 26th National Heat
Transfer Conf., Philadelphia, ASME.
Presents
general algorithm for finding factor between any threedimensional contour
and a differential element. Formulation is based on the unit sphere
technique of Nusselt
(1928). Results of computer implementation of the method are compared
with exact formulation for element to a polygon. 
Alfano, G. and Sarn , A., 1975, "Normal and
hemispherical thermal emittances of cylindrical cavities," J. Heat
Transfer, vol. 97, no. 3, pp. 387390, August.
Gives
factors from a differential element on and normal to the axis to a
differential ring element on the interior of a concentric right circular
cylinder; from a differential element to a circular ring element on a
parallel disk when the element is on the disk axis; from the interior
surface of a circular cylinder to a differential element on and normal to
the cylinder axis; and from a disk to a differential element which is on
and normal to the disk axis. All are in closed form. 
Ameri, A. and Felske, J.D., 1982, "Radiation
configuration factors for obliquely oriented finite length circular
cylinders," Int. J. Heat Mass Transfer, vol. 33, no. 1, pp.
728736.
Numerical
integration is used to compute the factors between the exteriors of two
cylinders of equal radius and length, and oriented to one another in
various ways. Factors between one cylinder and a second of onehalf the
length of the first are also given. Most results are for rotation of
cylinder two about the normal through the center or the end of the axis of
cylinder one. Closed form relations derived by fitting the numerical
results are presented. Graphical and some tabular data are presented. 
Ambirajan, Amrit and Venkateshan, S.P., 1993,
"Accurate determination of diffuse view factors between planar
surfaces," Int. J. Heat Mass Transfer, vol. 36, no. 8, pp. 2203
2208.
Uses
numerical evaluation of general double integral obtained by contour
integration around polygonal surfaces. Special cases of intersecting and
non intersecting surfaces are discussed. Numerical results are presented
for the cases of directly opposed isosceles triangles, squares, and regular
pentagons, hexagons, and octagons, as well as adjoint plates of finite
length at various intersection angles. Points out some errors in similar
results in Feingold
(1966). 
Ballance, J.O. and Donovan, J., 1973, "Radiation
configuration factors for annular rings and hemispherical sectors," J.
Heat Transfer, vol. 95, no. 2, pp. 275276, May.
Monte
Carlo method is used to find the factors to within approximately 5
percent. 
Bartell, F.O. and Wolfe, W.L., 1975, "New
approach for the design of blackbody simulators," Appl. Opt.,
vol. 14, no. 2, pp. 249252, February.
Includes
closedform relations for factors from sphere interior to element on
interior; from circular cone interior to base; and from right circular
cylinder to base. 
Bernard, JeanJoseph and Genot, Jeanne, 1971a,
"Diagrams for computing the radiation of axisymmetric surfaces
(propulsive nozzles)," Office National d' Etudes et de Recherches
Aerospatiales, Paris, France, ONERANT185 (in French).
Gives
diagrams for finding exchange between exterior elements and between
interior elements on various bodies of revolution. Closed form relations
are not given, but auxiliary functions are presented that can be used to
find equivalent configuration factors. For exterior elements, relations are
given for two coaxial cones connected at their apexes; two truncated
coaxial cones connected at the small ends; a cylinder connected to the
small end of a circular cone; and a concentric disk normal to the cone axis
at the cone apex. For interior surfaces, cases treated are two attached
truncated coaxial cones; a cylinder attached to a truncated coaxial cone;
and from any interior element in these assemblies to the end disks. 
Bernard, JeanJoseph and Genot, Jeanne, 1971b,
"Royonnement thermique des surfaces de revolution," Int. J. Heat
Mass Transfer, vol.14, no. 10, pp. 16111619, October.
Contains
abridged information from Bernard
and Genot (1971a). 
Bien, Darl D., 1966, "Configuration factors for
thermal radiation from isothermal inner walls of cones and cylinders," J.
Spacecraft Rockets, vol. 3, no. 1, pp. 155156.
Uses
known disktodisk factors and configuration factor algebra to derive
factors from inside surface of cone, right circular cylinder or frustum of
cone to ends. 
Bobco, R.P., 1966, "Radiation from conical surfaces
with nonuniform radiosity," AIAA J., vol. 4, no. 3, pp. 544546.
Derives
factor from planar element in plane of base of right circular cone to cone
interior in form of integral relation. Cone apex is below the element.
Numerical results are presented for cone halfangles of 10^{o} and
20^{o}. See Edwards
(1969) for discussion of some errors in this reference. 
Boeke, Willem and Wall, Lars, 1976, "Radiative
exchange factors in rectangular spaces for the determination of mean radiant
temperatures," Build. Serv. Engng., vol. 43, pp. 244 253, March.
Derives
analytical expressions for configuration factors between plane rectangles
contained within adjoint and opposed planes. Some tabulated factors are
given. 
Bornside, D.E. and Brown, R.A., 1990, "View
factor between differingdiameter, coaxial disks blocked by a coaxial
cylinder," J. Thermophys. Heat Transfer, vol. 4, no, 3, pp. 414
416, July.
Closedform
solution is presented for specified geometry. 
Brewster, M. Quinn, 1992, Thermal Radiative
Transfer and Properties, John Wiley & Sons, New York.
Comprehensive
radiative transfer text. Appendix B presents algebraic expressions for
thirteen common configurations. 
Brockmann, H., 1994, "Analytic angle factors
for the radiant interchange among the surface elements of two concentric
cylinders," Int. J. Heat Mass Transfer, vol. 37, no. 7, pp.
10951100.
Derives
analytic expressions for factors between concentric right circular
cylinders of finite equal length. Includes factors between inner and outer
cylinders, outer cylinder and itself, ends and inner and outer cylinder,
endtoend, and ends of radius less than outer cylinder radius to other
finite areas. 
Buraczewski, Czeslaw, 1977, "Contribution
to radiation theory configuration factors for rotary combustion
chambers," Pol. Akad. Nauk Pr. Inst. Masz Przeplyw, no. 74, pp.
4773 (in Polish.)
Disktodisk
factors are used with configuration factor algebra to generate all factors
on interior of right circular cone, interior of frustum of right circular
cone, interior of finite right circular cylinder, and combinations of cones
and frustums of cones. 
Buraczewski, Czeslaw, and Stasiek, Jan, 1983,
"Application of generalized Pythagoras theorem to calculation of
configuration factors between surfaces of channels of revolution." Int.
J. Heat & Fluid Flow, vol. 4, no. 3, pp. 157160, Sept.
Derives
closed form relations for coaxial disks of different radii; ring elements
on interior of circular cylinders to coaxial disks of the same diameter;
ringelement to ringelement on interior of circular cylinder; ring element
on interior of cone to coaxial disk; and ringelement to coaxial ring
element, both on interior of cone. 
Buschman, Albert Jr. and Pittman, Claud M., 1961,
"Configuration factors for exchange of radiant energy between
axisymmetrical sections of cylinders, cones, and hemispheres and their
bases," NASA TN D944.
Derives
many relations for factors between combinations of differential and finite
areas on the interior of right circular cylinders, right circular cones and
hemispheres. Straightforward analytical integration is used, resulting in
lengthy expressions in closed form. One typographical error (Eq. A14 of
the reference, where Z^{4} is mistyped as Z^{2}) is
corrected in the present catalog for the factor from an element on the
interior of a right circular cone to a coaxial disk on the base. Some of
the final results are more simply derived using disktodisk factors and
configuration factor algebra, particularly the frustumdisk factors. The
latter are obtained by Buschman and Pittman through the use of elliptic
integrals, and this results in a tedious computation and lengthy
expressions. Results are given in tabular form. 
Byrd, L.W., 1993, "View factor algebra for two
arbitrary sized nonopposing parallel rectangles," J. Heat Transfer,
vol. 115, no. 2, pp. 517518.
Notes
that Hamilton
and Morgan (1952) has an error for this configuration. 
Campbell, James P. and McConnell, Dudley G., 1968,
"Radiantinterchange configuration factors for spherical and conical
surfaces to spheres," NASA TN D4457.
Provides
extensive graphs and factors between spheres of equal radius, between a
sphere and a spherical cap on a sphere of equal radius, and between a
sphere and a coaxial cone with apex toward the sphere. Results are for
sphere separations of 0 to 10 radii in steps of one radius, and for cap
angles of 0 to 90^{o}. Cone results are given for cone semiangles
of 15^{o}, 30^{o}, 45^{o} and 60^{o}; cone
base radii in the range of 0 to1 sphere radius; and for cone apex to sphere
surface separations of 0, 1, 2, 4, 6, 8, and 10 sphere radii. All results
were calculated numerically. 
Chekhovskii, I.R.; Sirotkin, V.V.; ChuDunChu,
Yu. V.; and Chebanov, V.A., 1979, "Determination of radiative view
factors for rectangles of different sizes," High Temp., July
(Trans. of Russian original, vol. 17, no. 1, Jan.Feb., 1979)
Configuration
factor algebra and integration of analytical expressions are used to find
factors between rectangles in parallel planes and in perpendicular planes.
Form is more complex than given by Ehlert and Smith
or Gross,
Spindler and Hahne (1981) 
Chung, B.T.F. and Kermani, M.M., 1989, "Radiation
view factors from a finite rectangular plate," J. Heat Transfer,
vol. 111, no. 4, pp. 11151117, November.
Derives
general relation for configuration factor from tilted differential element
to a nonintersecting rectangle, and then uses integration to obtain
algebraic factor from a tilted differential strip to a nonintersecting
rectangle. (See also Hamilton and Morgan.) The
latter factor is then used to generate factors between a rectangular plate
and other finite objects. Specifically discussed are the factors from a
rectangular plate to a second plate, or to a solid cylinder. These factors
involve an integral that is to be evaluated numerically. Particular
graphical results are presented for factor from rectangular plate a tilted
right triangular plate. 
Chung, B.T.F., Kermani, M.M., and Naraghi, M.H.N.,
1984, "A formulation of radiation view factors from conical
surfaces," AIAA J., vol. 22, no. 3, pp. 429436, March.
Provides
closedform factors between differential elements and cones and frustums of
cones, and between cones and various surfaces of revolution that are on a
common axis with the cone. 
Chung, B.T.F. and Naraghi, M.H.N., 1982, "A
simpler formulation for radiative view factors from spheres to a class of
axisymmetric bodies" J. Heat Transfer, vol. 104, no. 1, pp.
201204, February.
Derives
simple formulation for exchange between exterior of a sphere and exterior
of a coaxial body of revolution. Uses formulation to derive closedform
expressions for a number of such geometries, and provides graphical results
for some ranges of parameters. Receiving bodies include spheres, spherical
caps, cones, ellipsoids and paraboloids. 
Chung, B.T.F. and Naraghi, M.H.N., 1981, "Some
exact solutions for radiation view factors from spheres," AIAA J.
vol. 19, pp. 10771081, August.
Factors
in closed form are derived from the exterior of a sphere to the exterior
surfaces of a cylinder, from a sphere to a coaxial differential ring, and
from a sphere to a coaxial non intersecting or intersecting disk.
Graphical and tabular results are presented for a wide range of parameters. 
Chung, B.T.F. and Sumitra, P.S., 1972, "Radiation
shape factors from plane point sources," J. Heat Transfer, vol.
94, no. 3, pp. 328330, August.
Using
the method of Feingold
and Gupta (1970), authors use idea of surrounding a planar element that
has its projection inscribed on the sphere interior. Factors from a planar
element to a sphere, to the interior of a cylinder lying on the normal to
the element, to an isosceles triangle, to a ring element, and to a disk
segment are presented. Also, the factor from a spherical element to a
sphere is derived. All results are in closed form. Some graphical results
are presented. 
Chung, T.J. and Kim, J.Y., 1982, "Radiation view
factors by finite elements," J. Heat Transfer, vol. 104, pp. 792.
Uses
finite elements plus Gaussian integration to formulate configuration
factors between irregular geometries, and shows accuracy of the method by
comparison of numerical calculation with values for known factors between
opposed squares and between two planes sharing a common edge at various
angles. 
Cox, Richard L., 1976, Radiative heat transfer in
arrays of parallel cylinders, Ph.D. Dissertation, Department of Chemical
Engineering, University of Tennessee, Knoxville.
Crossedstring
method is used to find factors between infinitely long cylinders in
equilateral triangular and square arrays. Results are also given for
factors when tubes are spirally wrapped with cylinders of smaller
diameter. 
Crawford, Martin, 1972, "Configuration factor
between two unequal, parallel, coaxial squares," paper no. ASME
72WA/HT16.
Analytical
closedform expression is derived for the title geometry. Graphical results
and some limiting expressions are given. 
Cunningham, F.G., 1961, "Power input to a
small flat plate from a difffusely reflecting sphere, with application to an
Earth satellite," NASA TN D710 (corrected copy).
Derives
closedform expressions for factor between arbitrarily oriented
differential element and sphere. Some graphs of results are given. Also see
Hauptmann and Modest (1980). 
Currie, I.G. and Martin, W.W., 1980, "Temperature
calculations for shell enclosures subjected to thermal radiation," Computat.
Methods Appl. Mech. Engng, vol. 21, no. 1, pp. 75 79, January.
Presents
factors between a differential element and a ring element on various
combinations of surfaces in an enclosure made up of a coaxial directly
opposed cylinder contained completely within the frustum of a cone; i.e.,
the smallest frustum end is larger than the cylinder diameter. The
expressions given as "view factors" are actually the kernels of
double integrals that must be carried out to get the final configuration
factors between surfaces and ring elements. The integration of the complex
algebraic kernels are not carried out in closed form. 
DiLaura, D.L., 1999, "New procedures for
calculating diffuse and nondiffuse radiative exchange form factors," ASME
Paper C99107, Proc. 33rd. National Heat Transfer Conf., Albuquerque,
August.
Casts
double area integral describing areaarea configuration factors into a
secondorder tensor, which is further transformed into a double contour
integral. Several forms of the integrals are derived, some of which have
superior convergence characteristics in comparison with standard contour
integratoion. Comparison of computed and analytical results is shown for
two squares with a common edge at various enclosed angles. 
Dummer, R.S. and Breckenridge, W.T. Jr., 1963,
"Radiation configuration factors catalog," General
Dynamics/Astronautics Rept. ERRAN224, February.
Dunkle, R.V., 1963, "Configuration factors for
radiant heattransfer calculations involving people," J. Heat
Transfer, vol. 85, no. 1, pp. 7176, February.
Measurements
using a mechanical formfactor integrator are used to derive empirical
relations for factors from points on various surfaces to standing or
sitting persons. These are then integrated to find factors from a person to
various room walls and the ceiling. The empirical relation for the
pointtostandingperson factor has a mean deviation from measured values
of 5.6 percent, and a maximum deviation of 19.4 percent. For the seated
person, the empirical relation differs from the measured factor by a mean
deviation of 6.6 percent, and a maximum deviation of 22 percent.
Surfacetositting person results are given in closed form, but standing
person results could not be integrated in closed form, so graphical results
are presented. 
Edwards, D.K., 1969, "Comment on "Radiation
from conical surfaces with nonuniform radiosity," AIAA J., vol.
7, no. 8, pp. 16561659.
Shows
that graphs given by Bobco
(1966) are in error when planar element is near to cone. Presents
revised graphs for cone halfangles of 10^{o} and 20^{o}
for various spacings of planar element from cone and a range of
dimensionless cone lengths from 1 to 100. 
Eddy, T.L. and Nielsson, G.E., 1988, "Radiation
shape factors for channels with varying cross section," J. Heat
Transfer, vol. 110, no. 1, pp. 264266, February.
Discusses
factors in circular ducts of varying radius r(x) , and formulates the
effects of blockage between differential and finite areas on the duct
surface separated by a distance x. Extends these results to ducts
that transition from circular to rectangular crosssection, and treats
cases of circular to rectangular elements, rectangular to circular
elements, and rectangular to rectangular elements. See also Modest (1988). 
Ehlert, J. R. and Smith, T.F., 1993, "View Factors
for Perpendicular and Parallel, Rectangular Plates," J. Thermophys.
Heat Trans., vol. 7, no. 1, pp. 173174.
Simpler
forms than Gross,
Spindler, and Hahne (1981) for parallel and perpendicular rectangles.
TL 900 J68. 
Eichberger, J.I., 1985, "Calculation of
geometric configuration factors in an enclosure whose boundary is given by an
arbitrary polygon in the plane," Warmeund Stoff bertragung, vol.
19, no. 4, p. 269.
Prescribes
a computer algorithm for applying the crossedstring method in
twodimensional enclosures with blocking and shading. 
Emery, A.F.; Johansson, O.; Lobo, M.; and Abrous, A,
1991, "A comparative study of methods for computing the diffuse
radiation viewfactors for complex structures," J. Heat Transfer,
vol. 113, no. 2, pp. 413422, May.
Paper
is devoted to studying the accuracy and computation time required to
compute configuration factors among various surfaces with and without
obstruction. Comparisons are among Monte Carlo, double area integration, a
modified contour integration, the hemicube method, and a specialized
algorithm. Concludes that Monte Carlo may be the best choice for
computing factors as well as gaining insight into the level of
computational effort required to achieve a given accuracy. In cases with
significant blockage by multiple nonintersecting surfaces, double area
integration was efficient, and other methods showed advantage in particular
situations as well.(Also see Rushmeier et al.) 
Farnbach, J.S., 1967, "Radiant interchange between
spheres: Accuracy of the pointsource approximation," Sandia
Laboratories Tech. Memo. SCTM364, Albuquerque, June.
Numerically
calculates exact factors between sphere exteriors, and compares results
with those obtained by assuming one sphere to be a point source. Range of
computed factors and the differences found are shown graphically as a function
of separation distance to emitting sphere radius ratio D with receiving to
emitting sphere radius ratio R as a parameter. Results are given for R = 1,
2, 5, 10 and 20, with D varying from 2 to 12, 3 to 13, 6 to 16, 11 to 21,
and 21 to 31, respectively. 
Feingold, A., 1978, "A new look at radiation
configuration factors between disks," J. Heat Transfer, vol. 100,
no. 4, pp. 742744, November.
Uses
inscribed nonintersecting circular disks on sphere interior to derive
disktodisk factors in a simple way. Any two such nonintersecting disks
are analyzed. 
Feingold, A., 1966, "Radiantinterchange
configuration factors between various selected plane surfaces," Proc.
Roy. Soc. London, ser. A, vol. 292, no. 1428, pp. 5160.
Tables
of factor values for rectangles with a common edge and at an arbitrary
included angle are presented, and show that the tabulated results of Hamilton
and Morgan (1952) have considerable error, although the equation from
which they are calculated is correct. Discusses effect of truncation and
roundoff errors in factor calculation. Uses configuration factor algebra to
derive factors between opposed regular polygons, and between the surfaces
in a hexagonal honeycomb. Points out that small errors in configuration
factor values can far overshadow the effects of assuming diffuse surface
properties on radiative transfer calculation. (See also Ambirajan
and Venkateshan (1993).) 
Feingold, A. and Gupta, K.G., 1970, "New
analytical approach to the evaluation of configuration factors in radiation
from spheres and infinitely long cylinders," J. Heat Transfer,
vol. 92, no. 1, pp. 6976, February.
Contains
discussion of some previous factors that have errors, and presents
closedform expressions for a number of factors, particularly for surfaces
of revolution, that were previously available only by numerical
integration. Notes many cases where factors are valid even for non diffuse
originating surface, and points out that, for spheretodisk factors, the
solutions are independent of the sphere diameter. Some interesting use of
symmetry in these problems allows bypassing of numerical or difficult
analytical evaluations. 
Felske, J.D., 1981, Personal communication, August
25.
Unpublished
results for the factor between infinite parallel cylinders of unequal
diameters. Simple closedform expression is obtained by curve fit, and is
within 6 percent of the exact analytical result for all ranges of
parameters. 
Felske, J.D., 1978, "Approximate radiation shape
factors between spheres," J. Heat Transfer, vol. 100, no. 3, pp.
547548, August.
Develops
a closedform approximate solution for spheretosphere factors for all
ranges of parameters, accurate to within 5.8 percent at worst, with much
smaller error on average, in comparison with exact numerical
solution. 
Garot, Catherine and Gendre, Patrick, 1979,
"Computation of view factors used in radiant energy exchanges in
axisymmetric geometry," In: Numerical methods in thermal problems;
Proc. First Int. Conf., pp. 99108, July 26, Pineridge Press, Ltd., Swansea,
Wales.
Discusses
numerical evaluation of factors in axisymmetric geometries and methods to
eliminate impossible factors caused by blockage by intervening surfaces or
by orientation of surfaces so their radiating surfaces cannot see
oneanother. Formulates limits for various cases. Results are computed for
concentric spheres, and compare within 1 percent of analytical
result. 
Glicksman, L.R., 1972, "Approximations for
configuration factors between cylinders," unpublished report, MIT.
According
to Ameri
and Felske (1982), this reference contains a closedform approximation
for the factor between cylinders of equal radius and finite length. (This
is the only reference that the compiler of this bibliography did not have
in hand during annotation.) 
Goetze, Dieter and Grosch, Charles B., 1962,
"Earthemitted infrared radiation incident a satellite," J.
Aerospace Sci., vol. 29, no. 5, pp. 521524.
Provides
closedform expressions for configuration factor from exterior of sphere to
arbitrarily oriented planar element. Vector algebra is used to simplify
arguments of integrals, which are then evaluated. Graphical results for the
configuration factor times p are presented for three
spheretoelement distances and for various element tilt angles relative to
the line connecting the element and the sphere center. 
Grier, Norman T., 1969, "Tabulations of
configuration factors between any two spheres and their parts," NASA
SP 3050, (420 pp.)
Extensive
tables of factors between combinations of spherical caps, patches, bands,
and entire spheres. Spheres are of different radii and spacing. Results are
obtained by numerical integration in a bispherical coordinate system. Parts
of spheres are tabulated by areas that subtend angles in increments of 15^{o},
and for radius ratios from 0.01 to 1 in intervals of 0.1 between 0.1 and 1.
Distance between centers of spheres varies from (1.001+r_{2}/r_{1})r_{1}
to 100r_{1}, where r_{1} is the radius of the larger
sphere. 
Grier, Norman T. and Sommers, Ralph D., 1969,
"View factors for toroids and their parts," NASA TN D5006.
Extensive
numerically computed results are presented in tables and graphs for factors
involving various parts of the surface of a toroid. The factors given are
between differential elements and "rim" bands; differential
elements and opposed radial segments; finite bands or segments and the
entire toroid; and between the toroid and itself. Factors are given for
parametric values of bands in increments of 10^{o} width, and of
the ratio (toroidal crosssection radius/toroid radius) for 0.01, 0.1,
0.2,...0.8, 0.9, 0.99. See also Sommers and Grier (1969). 
Gross, U., Spindler, K., and Hahne, E., 1981,
"Shape factor equations for radiation heat transfer between plane
rectangular surfaces of arbitrary position and size with rectangular
boundaries," Lett. Heat Mass Transfer, vol. 8, pp. 219227.
Provides
a closedform solution to the title factor for the cases of rectangles
lying in parallel or perpendicular planes and having parallel or
perpendicular edges. The rectangles may be of arbitrary size and location
within the planes. Solution is also given for the case when the planes
containing the rectangles intersect at an arbitrary angle; however, the
solution contains a single integral that must be evaluated numerically.
These solutions eliminate the tedious configuration factor algebra that
must otherwise be applied to the simple adjacent or opposed rectangle
factors to obtain these results, and which may generate large roundoff
errors [see Feingold
(1966)]. Also see Ehlert and Smith and Byrd. 
Guelzim, A., Souil, J.M., and Vantelon, J.P., 1993,
"Suitable configuration factors for radiation calculation concerning
tilted flames," J. Heat Transfer, vol. 115, no. 2, pp. 489492,
May.
Factors
are given in closed form between differential elements in various
configurations to tilted cylinders with faces parallel to the base
plane. 
Hahne, E. and Bassiouni, M.K., 1980, "The angle
factor for radiant interchange within a constant radius cylindrical
enclosure," Lett. Heat Mass Transfer, vol. 7, pp. 303309.
Derives
factor from onehalf of interior of finitelength right circular cylinder
to the opposite half using contour integration, and presents closedform
expressions and graphical results. 
Haller, Henry C. and Stockman, Norbert O., 1963,
"A note on fintube view factors," J. Heat Transfer, vol.
85, no. 4, pp. 380381, November.
Derives
factor from planar element on longitudinal fin to infinitely long tube, and
corrects errors in derivation in some earlier published works. 
Hamilton, D.C. and Morgan, W.R., 1952,
"Radiantinterchange configuration factors," NASA TN 2836.
One
of the classic compilations of configuration factors. Has a few
typographical errors [see, e.g., Feingold
(1966), Feingold
and Gupta (1970), and Byrd.] Catalogs twelve
different differential area to finite area factors, five differential strip
to finite area factors, and eleven finite area to finite area factors. Some
of the factors are generated by configuration factor algebra from a smaller
set of calculated or derived factors. This is a pioneering work in
cataloguing useful information. 
Hauptmann, E.G., 1968, "Angle factors between a
small flat plate and a diffusely radiating sphere," AIAA J., vol.
6, no. 5, pp. 938939, May.
Provides
simpler derivation than Cunningham
(1961) to find relations for title configuration. 
Holchendler, J. and Laverty, W.F., 1974,
"Configuration factors for radiant heat exchange in cavities bounded at
the ends by parallel disks and having conical centerbodies," J. Heat
Transfer, vol. 96, no. 2, pp. 254257, May.
Closedform
relation for factor from plane element to exterior of truncated right
circular cone with base and element in same plane is derived by contour
integration. Cone apex is above the element. Factor from element to a
concentric annular disk on the exterior of cone is also given. 
Holcomb, R.S. and Lynch, F.E., 1967, "Thermal
radiation performance of a finned tube with a reflector," Rept.
ORNLTM1613, Oak Ridge National Laboratory.
Presents
factors from an infinite strip element on an infinitely long tube to a
parallel infinite fin attached to the tube; from a finite length fin to an
attached parallel tube; and from a parallel finite length fin on a tube to
another parallel fin attached to the tube at 90^{o} from the first
fin. The latter factors are given for a single geometry, and are computed
from the factor for adjoint plates. 
Hollands, K.G.T., 1995. "On the superposition
rule for configuration factors," J. Heat Transfer, vol. 117, no.
1, pp. 241245, Feb.
Uses
the superposition principle to derive factors between differential elements
tilted arbitrarily with respect to various planar and convex finite areas.
An error in Eq. 12 is corrected in factor B17 of this catalog. 
Hooper, F.C. and Juhasz, E.S., 1952, "Graphical
evaluation of radiation interchange factor," ASME Paper 52F19,
ASME Fall Meeting, Chicago.
Presents
graphical method of computing configuration factors between differential
element and finite area. Method is based on unit sphere method of Nusselt
(1928). Templates are given for easy graphical construction. Method is
largely superseded by computerbased methods, many of which use a similar
technique. 
Hottel, H.C., 1954, "Radiant heat
transmission," in William H. McAdams (ed.), Heat Transmission,
3rd ed., pp. 55125, McGrawHill Book Co., New York.
Among
other things, derives the crossedstring method for computing factors among
surfaces that are infinitely long in one dimension. Presents graphical
results for some common configurations. 
Hottel, H.C., 1931, "Radiant heat transmission
between surfaces separated by nonabsorbing media," Trans. ASME,
vol. 53, FSP53196, pp. 265273.
Includes
derivations of factors from plane element to infinite plane; from plane
element to coaxial parallel disk; element to parallel rectangle normal to
element with normal passing through one corner of rectangle; element to any
parallel rectangle; element to any surface generated by a parallel
generating line; element to a bank of parallel tubes; plane to a bank of
tubes in an equilateral triangular array; plane to bank of tubes in
rectangular array; infinite parallel planes of finite width; one convex
surface enclosed by another; parallel coaxial disks of equal or unequal
radius; parallel opposed equal rectangles; parallel opposed infinitely long
strips; and perpendicular rectangles having a common edge. With a few
exceptions (parallel disks, element to disk), this is the first appearance
of these factors in the literature. 
Hottel, Hoyt C. and Keller, J.D., 1933, "Effect
of reradiation on heat transmission in furnaces and through openings," Trans.
ASME, vol. 55, IS556, pp. 3949.
Uses
derivatives of factors between opposed surfaces to find various factors
(ring on interior of right circular cylinder to similar ring, etc.). Starts
from disks, squares, 1byn rectangles (where n is an integer), and
infinite strips to derive factors, and presents tables of results. 
Hottel, Hoyt C. and Sarofim, A.F., 1967, Radiation
Heat Transfer, McGrawHill Book Co., New York.
Provides
derivation of crossedstring method, details graphical techniques, and
demonstrates contour integration. Generates factors by taking derivatives
of factors for known finite geometries, and derives striptosurface and
stripstrip factors on opposed coaxial disks, opposed squares, opposed
1by2 rectangles, and infinite parallel surfaces. 
Hsu, ChiaJung, 1967, "Shape factor equations for
radiant heat transfer between two arbitrary sizes of rectangular
planes," Can. J. Chem. Eng., vol. 45, no. 1, pp. 5860.
Lengthy
closedform relation is presented for factor between rectangles in parallel
planes. 
Jakob, Max, 1957, Heat Transfer, vol. 2, John
Wiley & Sons, New York.
Complete
treatment of configuration factor properties and relationships. Simple
factors are derived using integration, configuration factor algebra, and
the properties of spherical enclosures. Good survey of early literature is
given. 
Joerg, Pierre and McFarland, B.L., 1962, "Radiation
effects in rocket nozzles", Rept. S62 245, AerojetGeneral
Corporation.
Uses
analytical integration after transforming kernel to complex plane to derive
closedform solution for factor from differential element on the interior
of a right circular cone to cone base. Graphical results are given for cone
halfangles of 15, 20, and 25^{o}. 
Jones, L.R., 1965, "Diffuse radiation view factors
between two spheres," J. Heat Transfer, vol. 87, no. 3, pp.
421422, August.
Gives
numerically computed values in graphical form for title geometry for sphere
radius ratios from 0.1 to 1, and for ratio (distance between sphere
edges/radius) from 0 to 8. 
Juul, N.H., 1982, "View factors in radiation
between two parallel oriented cylinders of finite length" J. Heat Transfer,
vol. 104, no. 2, pp. 384388, May.
Derives
double integral expression for factor between parallel opposed cylinders of
finite length and unequal radius. Numerical results are fitted by
analytical expressions that apply within given ranges of parameters.
Indicates that expression for this geometry in Stevenson
and Grafton (1961) does not give comparable results, and may be in
error. 
Juul, N.H., 1979, "Diffuse radiation view factors
from differential plane sources to spheres," J. Heat Transfer,
vol. 101, no. 3, pp. 558560, August.
Derives
plane element to sphere factors for arbitrary element position and
orientation in space by constructing concave spherical surface that
subtends the same solid angle as the portion of the sphere viewed by the
element. This results in simpler formulation but identical numerical values
with earlier workers. Factors are given for particular cases of element on
the surface of a plane, a sphere, or a cylinder in various orientations to
sphere. 
Juul, N.H., 1976a, "Diffuse radiation
configuration view factors between two spheres and their limits," Lett.
Heat Mass Transfer, vol. 3, no. 3, pp. 205211.
Numerical
results are given for the ratio (spacing between sphere centers/radius of
larger sphere) in the range 17, and for sphere radius ratios of 05. Also,
see Juul
(1976b). 
Juul, N.H., 1976b, "Investigation of approximate
methods for calculation of the diffuse radiation configuration view factors
between two spheres," Lett. Heat Mass Transfer, vol. 3, no. 6,
pp. 513522.
Extends
the results of Juul
(1976a) to ratios (spacing between sphere centers/radius of larger
sphere) up to 12 and to sphere radius ratios up to 10. Compares results
with those from various approximations, and the ranges where each
approximation is within 1 percent are delineated. Also, see the discussion
in Felske
(1978). 
Katte, S.S., 2000, An Integrated Thermal Model for
Analysis of Thermal Protection System of Space Vehicles, PhD Thesis, IIT
Madras, December.
Derives
factor between ring element on interior of frustum of right circular cone
to cone base, including effect of blockage by a coaxial cylinder.
Integrates this factor using Simpson's Rule over interior of frustum to
determine factor from entire interior of frustum to annular disk on base
surrounding blocking cylinder. 
Keene, H.B., 1913, "Calculation of the energy
exchange between two fully radiative coaxial circular apertures at different
temperatures," Proc. Roy. Soc., vol. LXXXVIIIA, pp. 5960.
Contains
first derivation of factor between coaxial disks. This reference is an
appendix to Keene, H.B., 1913, "A determination of the radiation
constant," Proc. Roy. Soc., vol. LXXXVIIIA, pp. 4959. 
Kezios, Stothe P. and Wulff, Wolfgang, 1966,
"Radiative heat transfer through openings of variable
crosssection," Proc. Third Int. Heat Transfer Conf., AIChE, vol.
5, pp. 207 218.
Disktodisk
factors are used to derive ringtoring factors on the interior of a right
circular cone, and ringtodisk factors where ring is on the interior of
the cone and disk is inscribed on cone interior and coaxial with it.
Factors are given in terms of local radius and separation distance rather
than in terms of cone halfangle as in Sparrow
and Jonsson (1963). 
Kobyshev, A.A.; Mastiaeva, I.N.; Surinov, Iu. A.; and
Iakovlev, Iu. P., 1976, "Investigation of the field of radiation
established by conical radiators," Aviats. Tekh., vol. 19, no. 3,
pp. 4349, (in Russian).
Factors
between coaxial disk and cone when disk is centered on cone apex; between
nested coaxial cones; and between various areas on the interior of a cone
and cone frustum are presented. 
Kreith, Frank, 1962, Radiation Heat Transfer for
Spacecraft and Solar Power Plant Design, International Textbook Corp.,
Scranton, Pa.
Catalog
of 33 factors is given, many from Hamilton
and Morgan (1952). Short discussion is included of configuration factor
algebra. See also Stephens
and Haire (1961). 
Krishnaprakas, C.K., 1997, "View Factor
Between Inclined Rectangles," AIAA J. Thermophysics Heat Transfer,
vol. 11, no. 3, pp. 480482.
Uses
configuration factor algebra with factor for two adjacent plates sharing a
common edge to derive the factor between two plates on adjacent inclined
planes sharing a common edge. Indicates better accuracy than that found
using the method of Gross, Spindler and Hahne (1981). 
Kuroda, Z. and Munakata, T., 1979, "Mathematical
evaluation of the configuration factors between a plane and one or two rows
of tubes," Kagaku Sooti (Chemical Apparatus, Japan), pp. 5458,
November (in Japanese).
Uses
crossedstring method to derive factors from infinite plane to first and
second rows of infinitely long parallel equal diameter tubes in an infinite
equilateral triangular array. Also presents factors that include
reradiation from adiabatic plane behind tubes. 
Larsen, Marvin E. and Howell, John R., 1986,
"Leastsquares smoothing of direct exchange factors in zonal
analysis," J. Heat Transfer, vol. 108, no. 1, pp. 239242.
Although
applied to the general problem of factors in an enclosure with
participating medium, the method presented also works for evacuated
enclosure configuration factors. Uses variational calculus subject to
constraints of reciprocity and energy conservation to provide the best set
of factors in the leastsquares sense when the complete set has some
factors that are known to low accuracy. 
Lawson, D.A., 1995, "An improved method for
smoothing approximate exchange areas," Int. J. Heat Mass Transfer, vol.
38, no. 16, pp. 31093110.
Uses
area weighting to modify the smoothing algorithm of van
Leersum, providing better results for a test problem. 
Lebedev, V.A., 1979, "Invariance of radiation
shape factors of certain radiating systems," Akad. Nauk SSSR,
Siberskoe Otdelenie, Izvestiia, Seriia Tekh. Nauk, pp. 7377, October (in
Russian).
The
invariance of a number of factors for certain axisymmetric radiating
systems and for radiating systems containing infinite surfaces is
discussed. Method is proposed for examining invariance of shape factors of
various systems. Approach uses reciprocity and closure properties. 
Provides factors from a segment of a ribbon spirally would onto a cylinder of length l and diameter d to the entire interior surface of the ribbon.
Cases for finite ribbon length l and infinite ribbon length are given, along with expressions for reciprocal factors. 
Leuenberger, H. and Person, R.A., 1956,
"Compilation of radiation shape factors for cylindrical
assemblies," paper no. 56A144, ASME, November.
A
concise catalog of many useful factors involving disks, finite length
cylinders, and combinations of disks, cylinders, and rectangles. Most
results are given in closed form, with limiting forms. 
Liebert, Curt H. and Hibbard, Robert R., 1968,
"Theoretical temperatures of thinfilm solar cells in Earth orbit,"
NASA TN D4331.
Presents
results from Cunningham (1961) in a single graph for all orientations and
distances. Also see Hauptmann for this geometry. 
Lin, S.; Lee, P.M.; Wang, J.C.Y.; Dai, Y.L.; and Lou,
Y.S., 1986, "Radiantinterchange configuration factors between disk and
segment of parallel concentric disk," J. Heat Transfer, vol. 29,
no. 3, pp. 501503.
Presents
results of transforming the quadruple integral for this factor into a
double definite integral, and then numerically integrating the result.
Graphical results are presented for disk radius ratios between 0.1 and 5,
and for separation distance/segmented disk radius between 102 and 10.
Three ratios of radius to segment line over segmented disk radius (0.2, 0.6
and 1.0) are graphed. 
Lipps, F.W., 1983, "Geometric configuration factors
for polygonal zones using Nusselt's unit sphere," Solar Energy,
vol. 30, no. 5, pp. 413419.
Compares
numerical results from computer program based on unit sphere method with
analytical and other numerical results based on contour integration.
Contour integration is found to be faster. Some numerical results for
"twisted" adjoint plates are presented. 
Loehrke, R.I., Dolaghan, J.S., and Burns, P.J., 1995,
"Smoothing Monte Carlo exchange factors," J. Heat Transfer, vol.
117, no. 2, pp. 524526, May.
Uses
averaging of reciprocal pairs to achieve reciprocity, and then imposes
leastsquares averaging from Larsen
and Howell to achieve energy conservation. Authors find that
application of this approach can achieve accuracy equivalent to carrying
out Monte Carlo analysis for double the number of samples, i.e., a savings
of a factor of two in computer time. 
Love, Tom J., 1968, Radiative Heat Transfer,
Charles E. Merrill, Columbus, Ohio.
Contains
catalog of factors, mostly from Hamilton
and Morgan (1952), plus elementto sphere factors from Cunningham
(1961) and Buschmann
and Pittman (1961). 
Lovin, J.K. and Lubkowitz, A.W., 1969, "User's
manual for RAVFAC, a radiation view factor digital computer program," Lockheed
Missiles and Space Rept. HREC01541, Huntsville Research Park,
Huntsville, Alabama, LMSC/HREC D148620 (Contract NAS8 30154), November.
Mahbod, B. and Adams, R.L., 1984, "Radiation view
factors between axisymmetric subsurfaces within a cylinder with spherical
centerbody," J. Heat Transfer, vol. 106, no. 1, pp. 244248,
February.
Contour
integration is used to derive factors between a differential band on the
surface of a sphere and a finite strip on the interior of a coaxial
cylinder; between a differential band on the surface of a sphere and
a coaxial annular ring on the cylinder base; and between a coaxial annular
ring on the cylinder base and a finite strip on the interior of a coaxial
cylinder when blocked by a coaxial sphere. Numerical integration is then
used to provide factors between finite areas, and the results are compared
with known factors from Feingold and Gupta and Holchendler and Laverty. 
Masuda, H., 1973, "Radiant heat transfer on
circularfinned cylinders," Rep. Inst. High Speed Mechanics, Tohoku
Univ., vol. 27, no. 225, pp. 6789. (See also Trans. JSME, vol. 38,
pp. 32293234, 1972.)
Factors
between ring elements on tubes to ring elements on coaxial circular fins,
and between rings on adjacent fins, between the tube and one fin, and
between the environment and one fin or the tube are given. Contour
integration is used to develop the finite area factors. Also see Modest (1988). 
Mathiak, F.U., 1985, "Berechnung von
konfigurationsfactoren polygonal berandeter ebener gebiete (Calculation of
formfactors for plane areas with polygonal boundaries)," Warme und
Stoff bertragung, vol. 19, no. 4, pp. 273278.
Proposes
efficient algorithms for using contour integration applied to
elementsurface and surfacesurface configurations where the surfaces are
planar polygons. Derives algebraic relation for factor from differential
element to a right triangle in a parallel plane with the normal to the
element passing through the vertex containing the right angle. 
Maxwell, G.M.; Bailey, M.J.; and Goldschmidt, V.W.,
1986, "Calculations of the radiation configuration factor using ray
casting," Computer Aided Design, vol. 18, no. 7, p. 371.
Mel'man, M.M. and Trayanov, G.G., 1988, "View
factors in a system of contacting cylinders." J. Eng. Phys., vol.
54, no. 4, p. 401.
McAdam, D.W.; Khatry, A.K.; and Iqbal, M., 1971, "Configuration
Factors for Greenhouses," Am. Soc. Ag. Engineers, vol. 14, no. 6,
pp. 10681092, Nov.Dec.
Contour
integration is used to reduce configuration factor formulation to a single
integral, which is evaluated numerically. Various geometries typical of
greenhouses are evaluated, and the results are presented in graphical form.
The abscissas in Figs. 15, 18 and 19 are apparently mislabeled as A, but
should be C. 
Minning, C.P., 1981, Personal communication, Nov.
10.
Provided
information for factor from offaxis planar element to a sphere. 
Minning, C.P., 1979a, "Shape factors between
coaxial annular disks separated by a solid cylinder," AIAA J.,
vol. 17, no. 3, pp. 318320, March.
Contour
integration is used to derive closed form factor between an element on an
annular disk and a second coaxial annular disk separated by a coaxial
cylinder. Result is integrated numerically to find the factor between
coaxial annular disks separated by a coaxial cylinder. Graphical results
for some values of parameters are presented. 
Minning, C.P., 1979b, "Radiation shape factors
between end plane and outer wall of concentric tubular enclosure," AIAA
J., vol. 17, no. 12, pp. 14061408, December.
Contour
integration is used to derive closed form factor between elements and rings
on the annular end plane between concentric coaxial cylinders and the inner
surface of the outer cylinder. Results are integrated numerically to find
the factors between the entire annular end plane and the inside of the
outer cylinder. 
Minning, C.P., 1977, "Calculation of shape
factors between rings and inverted cones sharing a common axis," J.
Heat Transfer, vol. 99, no. 3, pp. 492494, August. (See also discussion
in J. Heat Transfer, vol. 101, no. 1, pp. 189190, August, 1979)
Uses
contour integration to derive closed form factor from planar element to
surface of right circular frustum of cone when element is in plane perpendicular
to cone axis. Element plane may intersect cone or not. Graphs are presented
for cone halfangles of 10 and 20 for a range of cone lengths and spacings
of the element from the cone axis. Sets up but does not carry out
ringtocone factors.
Discussion by D.A. Nelson points out that different variables can be used in the closedform solution, and also notes that Minning's results are valid for elements to frustums that are inverted. 
Minning, C.P., 1976, "Calculation of shape
factors between parallel ring sectors sharing a common centerline," AIAA
J., vol. 14, no. 6, pp. 813815.
Derives
factors by contour integration for element on disk to coaxial ring sector,
disk sector, ring, and disk, and from disk sector to disk sector. Results
are in closed form except for sectorto sector for which graphs of
numerical results are presented. 
Minowa, M., 19961999, "Studies of effective
radiation area and radiation configuration factors of a pig," J. Soc.
Ag. Structures (Japan); "Part 1: Effective radiation area of a pig
based on the surfacemodel," vol. 27, no. 3, (Ser. no. 71), pp. 155161,
December, 1996; "Part 2: Configuration factors of a 27 kg pig to
rectangular planes on the side, front or rear wall," vol. 29, no. 1,
(Ser. no. 77), pp. 18, June, 1998; "Part 3: Configuration factors of a
27 kg pig to rectangular planes on the ceiling or floor," vol. 29, no.
1, (Ser. no. 77), pp. 914, June, 1998; "Part 4: Configuration factors
of a 65 kg pig to rectangular planes and comparisons to a 27 kg pig,"
vol. 29, no. 3, (Ser. no. 79), pp. 137149, December, 1998; "Part 5:
Configuration factors of an 88 kg pig to surrounding rectangular planes and
configuration factor characteristics of fattening pigs," vol. 30, no. 2,
(Ser. no. 82), pp. 145156, Sept., 1999.
This
series of papers uses a threedimensional model of a standing pig using
triangular surface elements to derive the effective radiating area of pigs
of various weights. These areas are used to provide configuration factors
between pigs and various orientations of rectangular areas based on the
unitsphere method. Results are in graphical form. 
Mitalas, G.P. and Stephenson, D.G., 1966,
"FORTRAN IV programs to calculate radiant interchange factors," National
Research Council of Canada, Div. of Building Research Rept. DBR25,
Ottawa, Canada.
Presents
method of analytical evaluation of one of the line integrals arising from
contour integrations giving configuration factor between two finite areas.
Method allows application of contour integration to factors for planar
surfaces sharing a common edge, which have a numerical singularity if
standard contour integration is used. 
Modest, Michael F., 2003, Radiation Heat Transfer, 2nd ed.,
Academic Press, London.
Comprehensive
text on radiative transfer presents configuration factors for 51 geometries
in Appendix D. All of the factors are included in this catalog. 
Modest, M.F., 1988, "Radiative shape factors
between differential ring elements in concentric axisymmetric bodies," J.
Thermophys. Heat Trans., vol. 2, no. 1, pp. 8688.
Derives
factors between a ring element or band on the interior of an axisymmetric
body and a ring element or band on the exterior of an inner concentric
axisymmetric body, and between two bands that both lie on either body. Also
presents the integration limits that result for general bodies of
revolution. Shows that general relations reduce to the correct result for
the case of a ring element on a cylinder to a ring element on a perpendicular
circular fin as derived by Masuda. See also Eddy and Nielson. 
Modest, M.F., 1980, "Solar flux incident on an
orbiting surface after reflection from a planet," AIAA J., vol. 18,
no. 6, pp. 727730.
Provides
exact numerical results for solar radiation reflected from a spherical
planet to an orbiting element at arbitrary orientation, and also gives a
simple approximate relation. 
Moon, Parry, 1936, The Scientific Basis of
Illuminating Engineering, McGrawHill Book Co., New York, (Reprinted 1961
by Dover Publications, New York.)
A
standard reference, this book contains much material "rediscovered"
in later work. Careful and complete discussions are included of the use of
the unit sphere method, various earlier work to obtain factors by contour
integration, configuration factor algebra, Yamouti's reciprocity relations,
and the invariance properties of factors on the interior of spheres.
Notation in this text is a problem as the symbol F is used for radiant
flux, and no explicit symbol is defined for the configuration factor. 
Morizumi, S.J., 1964, "Analytical determination
of shape factors from a surface element to an axisymmetric surface," AIAA
J., vol. 2, no. 11, pp. 20282030.
Geometrical
relations for subtended solid angle and distance between elements are
derived for use in integrals that define factors between an elemental area
and a paraboloidal surface, a conical surface, and a cylindrical surface.
Factor values are not computed. The effect of blockage is discussed. 
Mudan,
K.S., 1987 "Geometric View Factors for Thermal Radiation Hazard
Assessment," Fire Safety J., vol. 12, pp. 8996.
Algebraic
relations for factors from horizontal and vertical planar elements to a
tilted cylinder are given. Elements are in upwind, downwind, or crosswind
orientations relative to the cylinder tilt, and are in the plane of the
cylinder base. See also Guelzim et al. (1993). 
Naraghi, M.H.N., 1988a, "Radiation view factors
from differential plane sources to disks a general formulation," J.
Thermophys. Heat Trans., vol. 2, no. 3, pp. 271 274.
Derives
general relation for factor from a tilted planar element to a disk in an
intersecting or nonintersecting plane using contour integration. Results
are given as algebraic equations, and some graphical results are presented
for limiting cases. 
Naraghi, M.H.N., 1988b, "Radiation view factors
from spherical segments to planar surfaces," J. Thermophys. Heat
Trans., vol. 2, no. 4, pp. 373375.
Derives
factor between a spherical segment and a planar element that lies in a
plane parallel to the segment faces. Various cases are presented for when
the plane containing the element intersects or does not intersect the
segment, and for when the element can view the entire segment or only a
portion of it. This factor is then integrated in general form to obtain the
factor from a spherical segment to a planar surface parallel to a segment
face. No applications of this latter factor (which is in the form of an
integral) are given. See also Naraghi and Chung (1982). 
Naraghi, M.H.N., 1981, Radiation configuration
factors between disks and axisymmetric bodies, Master of Science Thesis,
Department of Mechanical Engineering, The University of Akron.
Gives
complete derivations and analysis of results given by Naraghi
and Chung (1982) and Chung
and Naraghi (1980, 1981). 
Naraghi, M.H.N. and Chung, B.T.F., 1982,
"Radiation configuration between disks and a class of axisymmetric
bodies," J. Heat Transfer, vol. 104, no. 3, pp. 426431, August.
Contour
integration is used to derive the factor between an arbitrarily oriented
differential element and a disk. This factor is used to develop the factor
from the disk to a coaxial differential ring. The latter expression is then
integrated to find the factor from a disk or an annular ring to various
axisymmetric bodies, including a cylinder, a cone, an ellipsoid, and a
paraboloid. Some results are in closed form, others require a single
numerical integration. Limited graphical results are presented for each
factor. 
Naraghi, M.H.N. and Warna, J.P., 1988,
"Radiation configuration factors from axisymmetric bodies to plane
surfaces," Int. J. Heat Mass Transfer, vol. 31, no. 7, pp. 1537
1539.
Derives
factors between bodies of revolution and plane surfaces lying in planes
perpendicular to the axis of revolution. Derives and presents relations for
noncoaxial parallel disks of differing radius; a disk and a noncoaxial
disk segment in parallel planes; a sphere and a noncoaxial disk lying in a
plane that intersects the sphere; a finitelength cylinder and a disk lying
in a plane perpendicular to the cylinder axis ( plane intersecting the
cylinder or not); and a right circular cone and a disk lying in a plane
perpendicular to the cone axis (plane intersecting the cone or not). 
Nichols, Lester D., 1961, "Surfacetemperature
distribution on thinwalled bodies subjected to solar radiation in
interplanetary space," NASA TN D584.
Derives
factor between any two elements on the interior of a sphere. 
Nusselt, W., 1928, "Graphische bestimmung des
winkelverhaltnisses bei der warmestrahlung," VDI Z., vol. 72, p.
673.
Original
presentation of the unit sphere method, which is presented more accessibly
in Alciatore
and Lipp (1989) and Siegel
and Howell (2001). 
O'Brien, P.F. and Luning, R.B., 1970,
"Experimental study of luminous transfer in architectural systems,"
Illum. Engng, vol. 65, no. 4, pp. 193198, April.
Comparison
is made of the factors between a parallel differential area on the normal
to a disk or rectangle and the disk or rectangle as determined by three
methods; analytically, experimentally, and numerically by the use of the
program CONFAC II. Accuracy of the measurements was within 5 percent. 
Perry, R.L., and Speck, E.P., 1962, "Geometric
factors for thermal radiation exchange between cows and their
surroundings," Trans. Am. Soc. Ag. Engnrs., General Ed., vol. 5,
no. 1, pp. 3137.
Used
mechanical integrator to measure factors from various wall elements to a
cow, and presents some results for size of equivalent sphere that gives
same factor as cow. It is found that the sphere origin should be placed at
onefourth of the withers to pinbone distance behind the withers, at a
height above the floor of twothirds of the height at the withers, and that
the equivalent sphere radius should be 1.8, 2.08, or 1.78 times the heart
girth for exchange with the floor and ceiling, sidewalls, or front and back
walls, respectively. Also discusses exchange between cows and entire
bounding walls, floor and ceiling, and between parallel cows. 
Plamondon, Joseph A., 1961, "Numerical
determination of radiation configuration factors for some common geometrical
situations," Jet Propulsion Laboratory Tech. Rept. 32127,
California Institute of Technology, July 7.
Derives
numerical relations for finding configuration factors in arbitrary
geometries. Gives specific integral forms and limits for the cases of two
arbitrarily oriented plates; arbitrarily oriented cylinder and plate;
arbitrarily oriented cone and plate; arbitrarily oriented sphere and plate;
coaxial cylinders of unequal length with midpoints at the same axial
position; and parallel cylinders of unequal radius and length. No results
are given. 
Rao,V.R. and Sastri, V.M.K., 1996, "Efficient
evaluation of diffuse view factors for radiation," Int. J. Heat Mass
Transfer, vol.19, no. 6, pp. 12811286.
Uses
various degrees of accuracy in integration of the line integrals in contour
integration to determine finitearea to finite area factors. Presents
results for some simple geometries to evaluate relative accuracy of
different integration schemes. Presents simple way of numerically treating
case when two areas have a coincident side or otherwise touch. (See also Mitalas
and Stephenson (1966)) 
Rea, Samuel N., 1975, "Rapid method for determining
concentric cylinder radiation view factors," AIAA J., vol. 13,
no. 8, pp. 11221123.
Derives
closedform factor from a cylinder to an annular ring at the end of the
cylinder. Configuration factor algebra is then used to find factors for a
variety of configurations involving coaxial cylinders of different finite
lengths that are displaced axially from one another. 
Reid, R.L. and Tennant, J.S., 1973, "Annular ring
view factors," AIAA J., vol. 11, no. 3, pp. 14461448.
Quadruple
area integral for factor between finite length segments on the surfaces of
coaxial cylinders is analytically integrated three times, and the remaining
integral is then integrated numerically. Shelltoshell and shelltotube
factors between areas that are axially displaced are given in graphs.
Discussion is given of the use of configuration factor algebra for finding
other factors such as between annular disks. 
Rein, R.G., Jr., Sliepcevich, C.M., and Welker, J.R.,
1970, "Radiation view factors for tilted cylinders," J. Fire
Flammability, vol. 1, pp. 140153.
Factors
given from a vertical differential element normal to line through the base
of a tilted circular cylinder for various cylinder length/radius ratios,
and for various distances of the element from the cylinder. Discusses use
of configuration factor algebra for cases when the element is above or
below the cylinder base plane, and effects of cases when element lies under
tilted cylinder or far from the cylinder. 
Robbins, William H., 1961, "An analysis of
thermal radiation heat transfer in a nuclear rocket nozzle," NASA TN
D586.
Derives
general expression for factors between any two differential elements on the
interior of an arbitrary surface of revolution, and between a differential
element on a plane normal to the axis of revolution and any element on the
interior. Factors from elements to entire interior surface of revolution
are derived for specific nozzle geometries, including limits of integrals
to account for possible blockage by concave portions of the surface. No
computed results are given. 
Robbins, William H. and Todd, Carroll A., 1962,
"Analysis, feasibility, and walltemperature distribution of a
radiationcooled nuclear rocket nozzle," NASA TN D878.
Presents
form of integral and limits of integration for convergingdiverging
surfaces of revolution, including blockage due to throat.
Interiortointerior and exteriorexterior elements are included. No
numerical results are given. 
Rushmeier, H.E.; Baum, D.R.; and Hall, D.E., 1991,
"Accelerating the hemicube algorithm for calculating radiation form
factors," J. Heat Transfer, vol. 113, no. 4, pp 10441047.
Investigates
both hardware and software enhancements to speed the hemicube method for
calculation of configuration factors. Additionally uses spatial or
geometric coherence, i.e. the fact that both a primary area and a second
area that blocks or shades the primary area need not both be processed in
finding the factor to the primary area using hemicube algorithms. Some
simple checks can reduce this redundancy, resulting in considerable savings
in computation time. Implementing all three enhancements simultaneously
resulted in speedups of a factor greater than 6 in many cases. 
Sabet, M. and Chung, B.T.F., 1988, "Radiation view
factors from a sphere to nonintersecting planar surfaces," J.
Thermophysics Heat Transfer, vol. 2, no. 3, pp. 286 288.
Presents
algebraic expressions for factor from sphere to noncoaxial disk sector;
sphere to noncoaxial disk segment; sphere to noncoaxial rectangle; and
sphere to noncoaxial ellipse. All factors require numerical integration for
evaluation, and graphical results are given for some parameter sets for
each geometry. All results reduce to correct limits for coaxial cases. 
Saltiel, C. and Naraghi, M.H.N., 1990, "Radiative
configuration factors from cylinders to coaxial axisymmetric bodies," Int.
J. Heat Mass Transfer, vol. 33, no. 1, pp. 215218.
Derives
factor from tilted differential element to a cylinder in closed algebraic
form. Uses this factor to generate factors between a cylinder and coaxial
bodies, including a coaxial differential conical ring, a cylinder to a
coaxial paraboloid attached to the cylinder base, and from a cylinder to a
coaxial axisymmetric body of revolution described by a power law attached
to the cylinder base. 
Sauer, Harry J., Jr., 1974, "Configuration factors
for radiant energy interchange with triangular areas," ASHRAE Trans.,
vol. 80, part 2, no. 2322, pp. 268279.
Numerical
integration is used to find factors for nine arrangements of triangles and
rectangles that lie in perpendicular planes. Configuration factor algebra
is used to show the relations for an additional 13 arrangements. Results
were checked against available closedform relations, and the program was
checked against results for perpendicular rectangles with "excellent
agreement." 
Schröder, Peter and Hanrahan, Pat, 1993, "On the
form factor between two polygons," Computer Graphics, Proc., Ann.
Conf. Series, SIGGRAPH 93, pp. 163164.
Configuration
factors calculated using contour integration. Polygons can be planar,
convex, or concave. Provides closedform but complicated expression for
factor between polygons in arbitrary configuration. 
Shapiro, A.B., 1985, "Computer implementation,
accuracy and timing of radiation view factor algorithms," J. Heat
Transfer, vol. 107, no. 3, pp. 730732, August.
Compares
execution time and accuracy of numerical computation of factors between
finite areas by using double area integration; line integration after
applying contour integration; and transformed line integrals using the
method of Mitalas
and Stephenson (1966). Calculations are for directly opposed
rectangles. Mitalas and Stephenson method is found to be most accurate, but
line integration formulation is more accurate and faster when the boundary
is divided into seven or fewer elements. 
Shukla, K.N. and Ghosh, D., 1985, "Radiation
configuration factors for concentric cylinder bodies in enclosure," Indian
J. Technology, vol. 23, pp. 244246, July.
Derives
factors among all surfaces in an enclosure composed of a closed finite
length cylinder contained entirely within a longer coaxial closed cylinder 
Siegel, Robert and Howell, John R., 2001, Thermal
Radiation Heat Transfer, 4th ed., Taylor and FrancisHemisphere,
Washington.
Comprehensive
text on radiative transfer. Provides this entire catalog of configuration
factors on compact disk . 
Sommers, Ralph D. and Grier, Norman T., 1969,
"Radiation view factors for a toroid: comparison of Eckert's technique
and direct computation." J. Heat Transfer, vol. 91, no. 3, pp.
459461, August.
Compares
results of experimental determination of configuration factor for
differential element on surface of toroid to entire toroid by use of
translucent hemisphere to numerical results in Grier
and Sommers (1969). For a particular case, integrated results and
experiment for toroid toroid factor agreed within 6 percent. 
Sotos, Carol J. and Stockman, Norbert O., 1964,
"Radiantinterchange view factors and limits of visibility for
differential cylindrical surfaces with parallel generating lines," NASA
TN D2556.
Treats
factors for many geometries involving long cylinders with external parallel
fins, for fins in rectangular enclosures, and for adjacent parallel long
cylinders connected by fins inside enclosures. Presents results in terms of
integrals of elementelement factors with limits of integration. Discusses
effect of finite length on the error involved in using factors for infinite
length geometries. 
Sowell, E.F. and O'Brien, P.F., 1972, "Efficient
computation of radiantinterchange factors within an enclosure," J.
Heat Transfer, vol. 49, no. 3, pp. 326328.
Presents
matrixalgebra based method for computing remaining factors for an
Nsurfaced enclosure with all planar or convex surfaces once the minimal
set is computed separately. Discusses accuracy of technique when some of
the factors are numerically small in value, so that direct application of
reciprocity and conservation are insufficient to provide desired
accuracy. 
Sparrow, E.M., 1962, "A new and simpler
formulation for radiative angle factors," J. Heat Transfer, vol. 85,
no. 2, pp. 8188, May.
Gives
careful and concise exposition of contour integration for determining
configuration factors. Derives factor from planar element to parallel
rectangle, planar element to parallel coaxial disk, planar element to
segment of disk, between parallel opposed rectangles, and between parallel
coaxial disks. Notes the superposition properties of the method and the
considerable simplifications available over direct area integration. 
Sparrow, E.M., Albers, L.U., and Eckert, E.R.G.,
1962, "Thermal radiation characteristics of cylindrical heat
transfer," J. Heat Transfer, vol. 84, no. 1, pp. 7381.
Appendix
to paper provides derivation of closed form factor from ring element inside
right circular cylinder to ring on cylinder base. Derivation is based on
taking derivative of diskdisk factors. Steps to find final ringring
factor for rings on cylinder interior are outlined. 
Sparrow, E.M. and Cess, R.C., 1978, Radiation
Heat Transfer, Augmented edition, Hemisphere, Washington, D.C.
Contains
a complete discussion of configuration factor algebra with examples, and a
catalog of 15 factors. 
Sparrow, E.M. and Eckert, E.R.G., 1962,
"Radiant interaction between fin and base surfaces," J. Heat
Transfer, vol. 84, no. 1, pp. 1218.
Uses
derivatives of diskdisk factors to obtain ringring factors on parallel
coaxial circular disks. 
Sparrow, E.M. and Gregg, J.L., 1961, "Radiant
interchange between circular disks having arbitrarily different
temperatures," J. Heat Transfer, vol. 83, no.4, pp. 494502, Nov.
Uses
derivatives of disktodisk factors to obtain ringtoring factors on parallel
circular disks. 
Sparrow, E.M. and Heinisch, R.P., 1970, "The
normal emittance of circular cylindrical cavities," Appl. Opt.,
vol. 9, no. 11, pp. 25692572, November.
Presents
without derivation the factors from the inside of a cylinder ring element
to a planar element on the cylinder axis or to a coaxial disk, and from a
disk to a coaxial disk or to a normal planar element on the disk
axis. 
Sparrow, E.M. and Jonsson, V.K., 1963a, "Angle
factors for radiant interchange between paralleloriented tubes," J.
Heat Transfer, vol. 85, no. 4, pp. 382384, Nov.
Factors
for exchange between ring elements on parallel tubes are used to
numerically find the factors from ring elements to tubes of finite length.
Case of tubes connected by thin plane through cylinder axes is also
presented. Results are given for separationto cylinder radius ratios of
0.01 to 10. 
Sparrow, E.M. and Jonsson, V.K., 1963b,
"Radiation emission characteristics of diffuse conical cavities," J.
Opt. Soc. Am., vol. 53, no. 7, pp. 816821.
Derives
closedform expressions for factors between parallel coaxial disks
contained within a cone, and between coaxial ring elements on cone
interior. Uses derivatives of diskdisk factors for the latter case. Also
sets up but does not carry out diskring factors. 
Sparrow, E.M. and Jonsson, V.K., 1963c,
"Thermal radiation absorption in rectangular groove cavities," J.
Appl. Mechs., vol. E30, pp. 237244.
Derives
relation for parallel but not directly opposed infinite strips in parallel
planes 
Sparrow, E.M. and Jonsson, V.K., 1962a,
"Absorption and emission characteristics of diffuse spherical
enclosures," NASA TN D1289.
Derives
factors between any two differential elements or between any element and
any finite area on the interior of a sphere. 
Sparrow, E.M. and Jonsson, V.K., 1962b,
"Absorption and emission characteristics of diffuse spherical enclosures,"
J. Heat Transfer, vol. 84, pp. 188189.
Sparrow, E.M.; Miller, G.B.; and Jonsson, V.K.,
1962, "Radiative effectiveness of annular finned space radiators,
including mutual irradiation between radiator elements," J. Aerospace
Sci., vol. 29, no. 11, pp. 12911299.
Uses
contour integration and configuration factor algebra to find closedform
factors between all combinations of surfaces in an enclosure formed by
opposed coaxial cylinders of finite length and the annular ends. 
Stasenko, A.L., 1967, "Selfirradiation
coefficient of a Moebius strip of given shape," Akad. Nauk, SSSR,
Izv. Energetika Transport, pp. 104107, JulyAugust.
Derives
relation for factor from Moebius strip to itself by numerical integration.
Strip is of constant width and has constant radius between strip axis and
centerline. 
Stefanizzi, P., 1986, "Reliability of the
Monte Carlo method in black body view factor determination," Termotechnica,
vol. 40, no. 6, p. 29.
Stephens, Charles W. and Haire, Alan M., 1961,
"Internal design considerations for cavity type solar absorbers," ARS
J., vol. 31, no. 7, pp. 896901.
Presents
in Fig. 6 of the paper a factor q defined as the "average fraction of
light passing directly out opening" for various cavities (hemisphere,
cylinder, cone and sphere) as a function of aperture to interior area.
However, q cannot be the configuration factor, because configuration factor
algebra shows that F_{cavity surfaceaperture} = A_{aperture}/A_{surface}
for all geometries. The results for q are reproduced in Kreith
(1962) with an error of a factor of 10 in the abscissa. 
Stevenson, J.A. and Grafton, J.C., 1961,
"Radiation heat transfer analysis for space vehicles," Rept.
SID6191, North American Aviation (AFASD TR 61119, pt. 1), Sept. 9.
Fairly
complete treatment of the fundamentals of radiative transfer as applied to
spacecraft design. Details energy exchange between spacecraft and nearby
planets, and thus presents relations for factors between spheres and
various other solid bodies. Full chapter is devoted to "Configuration
factor studies and data." Work of Hamilton
and Morgan (1952) and Leuenberger
and Person (1956) is reproduced and discussed, as are other factors
from the literature. Other factors are presented in a form suitable for
numerical integration. See Juul 1982b for discussion
of a possible error. 
Sydnor, C.L., 1970, "A numerical study of cavity
radiometer emissivities," NASA Contractor Rept. 321462, Jet
Propulsion Lab., Feb. 15.
Appendix
presents closed form factors for ring element to ring element on interior
of enclosure composed of right circular cylinder closed at both ends by
coaxial cones. One cone is truncated. Some typographical errors exist,
particularly in dF_{21}, where h_{1} has apparently been
substituted in error for y_{1}, and in dF_{31}, which has
a dimensional inconsistency in the last term. The brief descriptions in the
paper make the notation and definitions difficult to follow. 
Taylor, Robert P.; Luck, Rogelio; Hodge, B.K.; and
Steele, W. Glenn, 1995,"Uncertainty analysis of diffusegray radiation
enclosure problems," J. Thermophys. Heat Trans., vol. 9, no. 1,
pp. 6369, Jan.March.
Uses
uncertainty analysis to determine effects on enclosure analysis accuracy
when configuration factors are independently determined and are not forced
to meet closure, reciprocity, or both. 
Toups, K.A., 1965, "A general computer program for
the determination of radiant interchange configuration and form factors
CONFACI," North American Aviation, Inc. Rept. SID65 10431 (NASA
CR65256), October.
Tripp, W.; Hwang, C.; and Crank, R.E., 1962, "Radiation
shape factors for plane surfaces and spheres, circles, or cylinders"
(Spec. Rept. 16) Kansas State Univ. Bull., vol. 46, no. 4.
Derives
closedform solution for factor from sphere to rectangle with one corner on
and normal to sphere axis. Derives relationships for factor from outside of
right circular cylinder to right triangle in base plane with one vertex on
axis, and from disk to right triangle in parallel plane with one vertex on
disk axis. The latter two relations contain one integral that is evaluated
numerically. Graphs are presented of all results. Examples of using
configuration factor algebra to generate factors from spheres, cylinders,
and disks to displaced planar areas are presented. 
Tseng, J.W.C. and Strieder, W., 1990, "View factors
from wall to random dispersed solid bed transport," J. Heat Transfer,
vol. 112, no. 3, pp. 816819.
Derives
relations in integral form for the configuration factor from a plane
surface to a randomly packed bed of spheres of uniform diameter as a
function of bed thickness and void fraction. Provide similar results for
factor from a plane wall to a bed of randomly packed cylinders of equal
diameter that are parallel to the wall and to each other. Results for the
latter case are compared with results for a plane wall to cylinders
arranged in staggered rows with equal spacing between cylinder spacing
(pitch). 
Tso, C.P. and Mahulikar, S.P., 1999, "View factors
between finite length rings on an interior cylindrical shell," AIAA
J. Thermophysics Heat Transfer, vol. 13, no. 3, pp. 375379.
Uses
configuration factor algebra with the factors of Brockmann
to provide factors among rings on the interior surface of an outer cylinder
in the presence of a central concentric cylinder. 
Usiskin, C.M. and Siegel, R., 1960, "Thermal
radiation from a cylindrical enclosure with specified wall heat flux," J.
Heat Transfer, vol. 82, no. 4, pp. 369374.
Uses
factor from ring element on inside of cylinder to disk in thermal
analysis. 
van Leersum, J., 1989, "A method for
determining a consistent set of radiation view factors from a set generated
by a nonexact method," Int. J. Heat Fluid Flow, vol. 10, no. 1, p
83.
Presents
compatibility requirements for set of factors needed for enclosure analysis
that will meet the requirement of overall energy conservation for the
enclosure. 
Wakao, Noriaki; Kato, Koichi; and Furuya, Nobuo, 1969,
"View factor between two hemispheres in contact and radiation heat
transfer coefficient in packed beds," Int. J. Heat Mass Transfer, vol.
12, pp. 118120.
Numerical
integration of analytical relation for factor between differential element
on the surface of one hemisphere to a second coaxial hemisphere is used to
find hemispherehemisphere factors. Hemisphere bases are parallel. Results
are presented for radius ratios of 1 to 10. 
Wang, Joseph C.Y.; Lin, Sui; Lee, PaiMow; Dai,
WeiLiang; and Lou, YouShi, 1986, "Radiantinterchange configuration
factors inside segments of frustum enclosures of right circular cones," Int.
Comm. Heat Mass Transfer, vol. 13, pp. 423432.
Presents
numerically computed figures for factors between segments on parallel disks
of different radii and between an isosceles trapezoid and the segment of a
disk that intersects the trapezoid at right angle. 
Watts, R.G., 1965, "Radiant heat transfer to Earth
satellites," J. Heat Transfer, vol. 87, no. 3, pp. 369373,
August.
Derives
relations for factors from large sphere to small sphere, small hemisphere,
small cylinder, or small ellipsoid. "Small" means that the angle
between the line connecting any point on the small body and the sphere
center and the line from the same point to an arbitrary point on the sphere
can be considered invariant over the receiving body. Closed forms are given
for the receiving body being a sphere or hemisphere. Numerical evaluation
is used in other cases. All results are a factor of 4 times larger than for
the configuration factor as used in this catalog because the sphere surface
area is taken as p r^{2} rather than 4p r^{2}. 
Wiebelt, John A., 1966, Engineering Radiation
Heat Transfer, Holt, Rinehart & Winston, New York.
Contains
catalog of factors excerpted from Hamilton
and Morgan (1952), and a chapter on configuration factors. 
Wiebelt, J.A. and Ruo, S.Y., 1963,
"Radiantinterchange configuration factors for finite right circular
cylinder to rectangular plane," Int. J. Heat Mass Transfer, vol.
6, no. 2, pp. 143146.
Numerically
computed factors are presented as graphs for various parametric values of
rectangle size and spacing. Factors are judged by authors to have possible
errors of approximately 5 percent. 
Wong, H.Y., 1977, Handbook of Essential Formulae and
Data on Heat Transfer for Engineers, Longman Group, London.
Catalog
of 33 factors for common geometries given mostly as closedform
expressions. 
Yarbrough, David W. and Lee, ChonLin, 1984,
"Monte Carlo calculation of radiation view factors," in Integral
Methods in Sciences and Engineering, Payne, Fred R. et al., eds, Harper
and Rowe/Hemisphere, New York, pp. 563574.
Uses
Monte Carlo to compute factors for various simple geometries, and compares
with analytical solutions. Presents original results for strip on finite
length rectangular fin to parallel cylinder and from cylinder of finite
length placed at focus of parallel paraboloid. All results are calculated
to be within +/ 5 percent. 
Yuen, W.W., 1980, "A simplified approach to
shapefactor calculation between threedimensional planar objects," AIAA
J. Heat Transfer, vol. 102, no. 2, pp. 386388.
Derives
general relation for factor between arbitrarily arranged general polygons
based on contour integration. Presents some numerical results. 
Send
mail to: John Howell
University of Texas at Austin