A CATALOG OF RADIATION HEAT
University of Texas at Austin
Abishek, S., Srinivasa Ramanujam,
K. and Katte, S.S., 1995, "View Factors
between Disk/Rectangle and Rectangle in Parallel and Perpendicular Planes,"J. Thermophsics,
vol. 21, no. 1, pp. 236-239.
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
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. 609-612.
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 outer-outer factor, and outer-to-inner finite area factor is
found by numerical integration. Configuration factor algebra is then used
to obtain factor from inner cylinder to annular ring end.
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.
general algorithm for finding factor between any three-dimensional 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. 387-390, August.
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. 728-736.
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 one-half 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.
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.
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
Ballance, J.O. and Donovan, J., 1973, "Radiation
configuration factors for annular rings and hemispherical sectors," J.
Heat Transfer, vol. 95, no. 2, pp. 275-276, May.
Carlo method is used to find the factors to within approximately 5
Bartell, F.O. and Wolfe, W.L., 1975, "New
approach for the design of blackbody simulators," Appl. Opt.,
vol. 14, no. 2, pp. 249-252, February.
closed-form relations for factors from sphere interior to element on
interior; from circular cone interior to base; and from right circular
cylinder to base.
Bernard, Jean-Joseph and Genot,
Jeanne, 1971a, "Diagrams for computing the radiation of axisymmetric
surfaces (propulsive nozzles)," Office National d' Etudes et de Recherches Aerospatiales,
Paris, France, ONERA-NT-185 (in French).
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, Jean-Joseph and Genot,
Jeanne, 1971b, "Royonnement thermique des surfaces de revolution," Int. J.
Heat Mass Transfer, vol.14, no. 10, pp. 1611-1619, October.
Bien, Darl D., 1966,
"Configuration factors for thermal radiation from isothermal inner walls
of cones and cylinders," J. Spacecraft Rockets, vol. 3, no. 1,
known disk-to-disk 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. 544-546.
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 half-angles of 10o and
20o. 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.
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 differing-diameter, coaxial
disks blocked by a coaxial cylinder," J. Thermophys.
Heat Transfer, vol. 4, no, 3, pp. 414- 416, July.
solution is presented for specified geometry.
Brewster, M. Quinn, 1992, Thermal Radiative
Transfer and Properties, John Wiley & Sons, New York.
radiative transfer text. Appendix B presents algebraic expressions for
thirteen common configurations.
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. 1095-1100.
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,
end-to-end, and ends of radius less than outer cylinder radius to other
Buraczewski, Czeslaw, 1977, "Contribution to radiation theory
configuration factors for rotary combustion chambers," Pol. Akad. Nauk Pr. Inst. Masz Przeplyw, no. 74, pp.
47-73 (in Polish.)
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.
closed form relations for coaxial disks of different radii; ring elements
on interior of circular cylinders to coaxial disks of the same diameter;
ring-element to ring-element on interior of circular cylinder; ring element
on interior of cone to coaxial disk; and ring-element 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
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. A-14 of
the reference, where Z4 is mistyped as Z2) 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
disk-to-disk factors and configuration factor algebra, particularly the
frustum-disk 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. 517-518.
Campbell, James P. and McConnell, Dudley G., 1968,
"Radiant-interchange configuration factors for spherical and conical
surfaces to spheres," NASA TN D-4457.
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 90o. Cone results are given for cone semiangles of 15o, 30o, 45o
and 60o; 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.
I.R.; Sirotkin, V.V.; Chu-Dun-Chu, 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.,
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
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. 1115-1117,
general relation for configuration factor from tilted differential element
to a non-intersecting rectangle, and then uses integration to obtain
algebraic factor from a tilted differential strip to a non-intersecting 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. 429-436,
closed-form 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. 201-204,
simple formulation for exchange between exterior of a sphere and exterior
of a coaxial body of revolution. Uses formulation to derive closed-form
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. 1077-1081, August.
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. 328-330, August.
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
Chung, T.J. and Kim, J.Y., 1982, "Radiation view
factors by finite elements," J. Heat Transfer, vol. 104, pp. 792.
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.
Richard L., 1976, Radiative heat transfer in arrays of parallel cylinders,
Ph.D. Dissertation, Department of Chemical Engineering, University of
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
closed-form 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 D-710
closed-form 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.
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 non-diffuse radiative
exchange form factors," ASME Paper C99-107, Proc. 33rd. National Heat
Transfer Conf., Albuquerque, August.
double area integral describing area-area configuration factors into a
second-order 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
integration. 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. ERR-AN-224,
Dunkle, R.V., 1963,
"Configuration factors for radiant heat-transfer calculations involving
people," J. Heat Transfer, vol. 85, no. 1, pp. 71-76, February.
using a mechanical form-factor 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
point-to-standing-person 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.
Surface-to-sitting 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. 1656-1659.
that graphs given by Bobco (1966) are in error when planar element is
near to cone. Presents revised graphs for cone half-angles of 10o
and 20o for various spacings of planar element from cone and a
range of dimensionless cone lengths from 1 to 100.
T.L. and Nielsson, G.E., 1988, "Radiation
shape factors for channels with varying cross- section," J. Heat
Transfer, vol. 110, no. 1, pp. 264-266, February.
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 cross-section, 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. 173-174.
1985, "Calculation of geometric configuration factors in an enclosure
whose boundary is given by an arbitrary polygon in the plane," Warme-und Stoff bertragung,
vol. 19, no. 4, p. 269.
a computer algorithm for applying the crossed-string method in
two-dimensional 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.
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 hemi-cube 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 non-intersecting 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 point-source approximation," Sandia Laboratories
Tech. Memo. SC-TM-364, Albuquerque, June.
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. 742-744, November.
inscribed nonintersecting circular disks on sphere interior to derive
disk-to-disk factors in a simple way. Any two such non-intersecting disks
Feingold, A., 1966, "Radiant-interchange
configuration factors between various selected plane surfaces," Proc.
Roy. Soc. London, ser. A, vol. 292, no. 1428, pp. 51-60.
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
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. 69-76, February.
discussion of some previous factors that have errors, and
presents closed-form 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 sphere-to-disk factors, the solutions are
independent of the sphere diameter. Some interesting use of symmetry in
these problems allows bypassing of numerical or difficult analytical
Felske, J.D., 1981,
Personal communication, August 25.
results for the factor between infinite parallel cylinders of unequal
diameters. Simple closed-form 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. 547-548, August.
a closed-form approximate solution for sphere-to-sphere 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
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. 99-108, July
2-6, Pineridge Press, Ltd., Swansea, Wales.
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
one-another. Formulates limits for various cases. Results are computed for
concentric spheres, and compare within 1 percent
of analytical result.
1972, "Approximations for configuration factors between cylinders,"
unpublished report, MIT.
and Felske (1982), this reference contains a
closed-form 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, "Earth-emitted infrared radiation incident a
satellite," J. Aerospace Sci., vol. 29, no. 5, pp. 521-524.
closed-form 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
sphere-to-element 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.)
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 15o, 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+r2/r1)r1
to 100r1, where r1 is the radius of the larger
Grier, Norman T. and Sommers, Ralph D., 1969,
"View factors for toroids and their
parts," NASA TN D-5006.
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 10o width, and of
the ratio (toroidal cross-section 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. 219-227.
a closed-form 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 round-off
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.
are given in closed form between differential elements in various
configurations to tilted cylinders with faces parallel to the base
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.
factor from one-half of interior of finite-length right circular cylinder
to the opposite half using contour integration, and
presents closed-form expressions and graphical results.
Haller, Henry C. and Stockman, Norbert O., 1963,
"A note on fin-tube view factors," J. Heat Transfer, vol.
85, no. 4, pp. 380-381, November.
factor from planar element on longitudinal fin to infinitely long tube, and corrects errors in derivation in some earlier
D.C. and Morgan, W.R., 1952, "Radiant-interchange configuration
factors," NASA TN 2836.
of the classic compilations of configuration factors. Has a few
typographical errors [see, e.g., 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. 938-939, May.
simpler derivation than Cunningham
(1961) to find relations for title configuration.
He, F., Shi, J., Zhou, L.,
and Li, X., 2018, “Monte
Carlo calculation of view factors between some complex surfaces: Rectangular
plane and parallel cylinder, rectangular plane and torus, especially
cold-rolled strip and W-shaped radiant tube in continuous annealing furnace,”
Int. J. Thermal Sci., vol. 134, pp.
Verifies Monte Carlo by reproducing known plate to parallel finite-cylinder factor, then uses Monte Carlo to evaluate factors for geometries described in the title.
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. 254-257, May.
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.
R.S. and Lynch, F.E., 1967, "Thermal radiation performance of a finned
tube with a reflector," Rept. ORNL-TM-1613, Oak Ridge National
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 90o 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. 241-245, Feb.
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 B-17 of this catalog.
Hooper, F.C. and Juhasz,
E.S., 1952, "Graphical evaluation of radiation interchange factor,"
ASME Paper 52-F-19, ASME Fall Meeting, Chicago.
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 computer-based methods, many of which use a similar
Hottel, H.C., 1954,
"Radiant heat transmission," in William H. McAdams (ed.), Heat
Transmission, 3rd ed., pp. 55-125, McGraw-Hill Book Co., New York.
other things, derives the crossed-string 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 non-absorbing media," Trans. ASME, vol. 53,
FSP-53-196, pp. 265-273.
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, IS-55-6,
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, 1-by-n 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, McGraw-Hill Book Co.,
derivation of crossed-string method, details graphical techniques, and
demonstrates contour integration. Generates factors by taking derivatives
of factors for known finite geometries, and
derives strip-to-surface and strip-strip factors on opposed coaxial disks,
opposed squares, opposed 1-by-2 rectangles, and infinite parallel
Hsu, Chia-Jung, 1967, "Shape factor equations for
radiant heat transfer between two arbitrary sizes of rectangular
planes," Can. J. Chem. Eng., vol. 45, no. 1, pp. 58-60.
closed-form relation is presented for factor between rectangles in parallel
Max, 1957, Heat Transfer, vol. 2, John Wiley & Sons, New York.
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
Joerg, Pierre and McFarland, B.L., 1962, "Radiation
effects in rocket nozzles", Rept. S62- 245, Aerojet-General
analytical integration after transforming kernel to complex plane to derive
closed-form solution for factor from differential element on the interior
of a right circular cone to cone base. Graphical results are given for cone
half-angles of 15, 20, and 25o.
Jones, L.R., 1965, "Diffuse radiation view factors
between two spheres," J. Heat Transfer, vol. 87, no. 3, pp.
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. 384-388,
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
Juul, N.H., 1979,
"Diffuse radiation view factors from differential plane sources to
spheres," J. Heat Transfer, vol. 101, no. 3, pp. 558-560, August.
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.
results are given for the ratio (spacing between sphere centers/radius of
larger sphere) in the range 1-7, and for sphere radius ratios of 0-5. 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. 513-522.
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.
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
Keene, H.B., 1913, "Calculation of the energy
exchange between two fully radiative coaxial circular apertures at different
temperatures," Proc. Roy. Soc., vol. LXXXVIII-A, pp. 59-60.
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. LXXXVIII-A, pp. 49-59.
Kezios, Stothe P. and Wulff, Wolfgang,
1966, "Radiative heat transfer through openings of variable
cross-section," Proc. Third Int. Heat Transfer Conf., AIChE, vol. 5, pp. 207- 218.
factors are used to derive ring-to-ring factors on the interior of a right
circular cone, and ring-to-disk 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 half-angle 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. 43-49, (in Russian).
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.
C.K., 1997, "View Factor Between Inclined Rectangles," AIAA J.
Thermophysics Heat Transfer, vol. 11, no. 3, pp. 480-482.
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).
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. 54-58, November (in Japanese).
crossed-string 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,
"Least-squares smoothing of direct exchange factors in zonal
analysis," J. Heat Transfer, vol. 108, no. 1, pp. 239-242.
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 least-squares 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. 3109-3110.
area weighting to modify the smoothing algorithm of van
Leersum, providing better results for a test
Lebedev, V.A., 1979, "Invariance of radiation shape
factors of certain radiating systems," Akad.
Nauk SSSR, Siberskoe Otdelenie, Izvestiia, Seriia Tekh. Nauk, pp. 73-77, October (in Russian).
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
and Person, R.A., 1956, "Compilation of radiation shape factors for
cylindrical assemblies," paper no. 56-A-144, ASME, November.
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 thin-film solar cells in Earth orbit,"
NASA TN D-4331.
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, "Radiant-interchange configuration factors between disk and
segment of parallel concentric disk," J. Heat Transfer, vol. 29,
no. 3, pp. 501-503.
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 10-2 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. 413-419.
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. 524-526, May.
averaging of reciprocal pairs to achieve reciprocity, and then imposes
least-squares 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.
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. HREC-0154-1,
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. 244-248, February.
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
circular-finned cylinders," Rep. Inst. High Speed Mechanics, Tohoku
Univ., vol. 27, no. 225, pp. 67-89. (See also Trans. JSME, vol. 38,
pp. 3229-3234, 1972.)
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 form-factors for plane areas with
polygonal boundaries)," Warme- und
Stoff bertragung, vol. 19, no. 4, pp. 273-278.
efficient algorithms for using contour integration applied to
element-surface and surface-surface 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. 1068-1092, Nov.-Dec.
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.
1981, Personal communication, Nov. 10.
information for factor from off-axis planar element to a sphere.
1979a, "Shape factors between coaxial annular disks separated by a solid
cylinder," AIAA J., vol. 17, no. 3, pp. 318-320, March.
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.
1979b, "Radiation shape factors between end plane and outer wall of
concentric tubular enclosure," AIAA J., vol. 17, no. 12, pp.
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
1977, "Calculation of shape factors between rings and inverted cones
sharing a common axis," J. Heat Transfer, vol. 99, no. 3, pp.
492-494, August. (See also discussion in J. Heat Transfer, vol. 101,
no. 1, pp. 189-190, August, 1979)
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 half-angles 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 ring-to-cone factors.
Discussion by D.A. Nelson points out that different variables can be
used in the closed-form solution, and also notes
that Minning's results are valid for elements to
frustums that are inverted.
1976, "Calculation of shape factors between parallel ring sectors
sharing a common centerline," AIAA J., vol. 14, no. 6, pp.
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 sector-to- sector for which graphs of
numerical results are presented.
Minowa, M., 1996-1999, "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
surface-model," vol. 27, no. 3, (Ser. no. 71), pp. 155-161, 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. 1-8, 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. 9-14, 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. 137-149, 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. 145-156, Sept., 1999.
series of papers uses a three-dimensional 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
unit-sphere method. Results are in graphical form.
Mirhosseini, M. and Saboonchi, A., 2011; “View factor calculation using the
Monte Carlo method for a 3D strip element to a circular cylinder,” Int. Communications Heat Mass Transfer,
vol. 38, pp. 821-826.
Invokes Monte Carlo method to solve for factor B-80, strip of finite length to parallel cylinder of equal length.
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. DBR-25, Ottawa, Canada.
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., 1991, Radiative Heat Transfer,
McGraw-Hill Book Company, New York.
text on radiative transfer presents configuration factors for 51 geometries
in Appendix D. All of the factors are included in
Modest, M.F., 1988, "Radiative shape factors
between differential ring elements in concentric axisymmetric bodies," J.
Thermophys. Heat Trans., vol. 2, no. 1, pp.
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,
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, McGraw-Hill Book Co., New York, (Reprinted 1961
by Dover Publications, New York.)
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.
1964, "Analytical determination of shape factors from a surface element
to an axisymmetric surface," AIAA J., vol. 2, no. 11, pp.
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. 89-96.
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.
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. 373-375.
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.
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. 426-431, August.
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.
factors between bodies of revolution and plane surfaces lying in planes
perpendicular to the axis of revolution. Derives and presents relations for
non-coaxial 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 finite-length 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, "Surface-temperature
distribution on thin-walled bodies subjected to solar radiation in
interplanetary space," NASA TN D-584.
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.
P.F. and Luning, R.B., 1970, "Experimental
study of luminous transfer in architectural systems," Illum. Engng, vol. 65, no. 4, pp. 193-198, April.
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. 31-37.
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 one-fourth of the withers to pin-bone distance behind
the withers, at a height above the floor of two-thirds 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
A., 1961, "Numerical determination of radiation configuration factors
for some common geometrical situations," Jet Propulsion Laboratory
Tech. Rept. 32-127, California Institute of Technology, July 7.
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
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. 1281-1286.
various degrees of accuracy in integration of the line integrals in contour
integration to determine finite-area 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. 1122-1123.
closed-form 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. 1446-1448.
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. Shell-to-shell and shell-to-tube
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. 140-153.
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
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 wall-temperature distribution of a
radiation-cooled nuclear rocket nozzle," NASA TN D-878.
form of integral and limits of integration for converging-diverging
surfaces of revolution, including blockage due to throat.
Interior-to-interior and exterior-exterior elements are included. No
numerical results are given.
Baum, D.R.; and Hall, D.E., 1991, "Accelerating the hemi-cube algorithm
for calculating radiation form factors," J. Heat Transfer, vol.
113, no. 4, pp 1044-1047.
both hardware and software enhancements to speed the hemi-cube 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 hemi-cube
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.
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. 215-218.
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
Sauer, Harry J., Jr., 1974, "Configuration factors
for radiant energy interchange with triangular areas," ASHRAE Trans.,
vol. 80, part 2, no. 2322, pp. 268-279.
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 closed-form relations, and the program was
checked against results for perpendicular rectangles with "excellent
Schröder, Peter and
Hanrahan, Pat, 1993, "On the form factor between two polygons," Computer
Graphics, Proc., Ann. Conf. Series, SIGGRAPH 93, pp. 163-164.
factors calculated using contour integration. Polygons can be planar,
convex, or concave. Provides closed-form 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. 730-732, August.
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. 244-246, July.
factors among all surfaces in an enclosure composed of a closed finite
length cylinder contained entirely within a longer coaxial closed cylinder
Robert and Howell, John R., 2010, Thermal Radiation Heat Transfer, 6th
ed., Taylor and Francis-Hemisphere, Washington.
text on radiative transfer, including basic definitions of and relations
among configuration factors.
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.
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
Sotos, Carol J. and
Stockman, Norbert O., 1964, "Radiant-interchange view factors and limits
of visibility for differential cylindrical surfaces with parallel generating
lines," NASA TN D-2556.
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 element-element factors with limits of integration. Discusses
effect of finite length on the error involved in using factors for infinite
Sowell, E.F. and O'Brien, P.F., 1972, "Efficient
computation of radiant-interchange factors within an enclosure," J.
Heat Transfer, vol. 49, no. 3, pp. 326-328.
matrix-algebra based method for computing remaining factors for an
N-surfaced 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
E.M., 1962, "A new and simpler formulation for radiative angle
factors," J. Heat Transfer, vol. 85, no. 2, pp. 81-88, May.
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. 73-81.
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 disk-disk factors. Steps to find final ring-ring
factor for rings on cylinder interior are outlined.
Sparrow, E.M. and Cess,
R.C., 1978, Radiation Heat Transfer, Augmented edition, Hemisphere,
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. 12-18.
derivatives of disk-disk factors to obtain ring-ring 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. 494-502, Nov.
derivatives of disk-to-disk factors to obtain ring-to-ring factors on parallel circular disks.
E.M. and Heinisch, R.P., 1970, "The normal
emittance of circular cylindrical cavities," Appl. Opt., vol. 9,
no. 11, pp. 2569-2572, November.
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 parallel-oriented tubes," J.
Heat Transfer, vol. 85, no. 4, pp. 382-384, Nov.
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 separation-to 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. 816-821.
closed-form expressions for factors between parallel coaxial disks
contained within a cone, and between coaxial ring elements on cone
interior. Uses derivatives of disk-disk factors for the latter case. Also
sets up but does not carry out disk-ring factors.
Sparrow, E.M. and Jonsson, V.K., 1963c,
"Thermal radiation absorption in rectangular- groove cavities," J.
Appl. Mechs., vol. E30, pp. 237-244.
relation for parallel but not directly opposed infinite strips in parallel
Sparrow, E.M. and Jonsson, V.K., 1962a,
"Absorption and emission characteristics of diffuse spherical
enclosures," NASA TN D-1289.
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. 188-189.
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.
contour integration and configuration factor algebra to find closed-form
factors between all combinations of surfaces in an enclosure formed by
opposed coaxial cylinders of finite length and the annular ends.
1967, "Self-irradiation coefficient of a Moebius strip of given
shape," Akad. Nauk,
SSSR, Izv. Energetika
Transport, pp. 104-107, July-August.
relation for factor from Moebius strip to itself by numerical integration.
Strip is of constant width and has constant radius between strip axis and
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. 896-901.
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 Fcavity
surface-aperture = Aaperture/Asurface for all geometries. The results for
q are reproduced in Kreith (1962) with an error of a factor of 10 in
Stevenson, J.A. and Grafton, J.C., 1961,
"Radiation heat transfer analysis for space vehicles," Rept.
SID-61-91, North American Aviation (AFASD TR 61-119, pt. 1), Sept. 9.
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
Sydnor, C.L., 1970, "A numerical study of cavity
radiometer emissivities," NASA Contractor
Rept. 32-1462, Jet Propulsion Lab., Feb. 15.
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 dF2-1, where h1 has apparently been
substituted in error for y1, and in dF3-1, 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 diffuse-gray radiation
enclosure problems," J. Thermophys. Heat
Trans., vol. 9, no. 1, pp. 63-69, Jan.-March.
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-
CONFAC-I," North American Aviation, Inc. Rept. SID-65- 1043-1 (NASA
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,
closed-form 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.
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. 816-819.
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
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. 375-379.
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
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.
factor from ring element on inside of cylinder to disk in thermal
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.
compatibility requirements for set of factors needed for enclosure analysis
that will meet the requirement of overall energy conservation for the
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. 118-120.
integration of analytical relation for factor between differential element
on the surface of one hemisphere to a second coaxial hemisphere is used to
find hemisphere-hemisphere factors. Hemisphere bases are parallel. Results
are presented for radius ratios of 1 to 10.
Wang, Joseph C.Y.; Lin, Sui; Lee, Pai-Mow; Dai,
Wei-Liang; and Lou, You-Shi, 1986, "Radiant-interchange configuration
factors inside segments of frustum enclosures of right circular cones," Int.
Comm. Heat Mass Transfer, vol. 13, pp. 423-432.
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. 369-373,
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 r2 rather than 4p r2.
Wiebelt, John A.,
1966, Engineering Radiation Heat Transfer, Holt, Rinehart &
Winston, New York.
Wiebelt, J.A. and Ruo, S.Y., 1963, "Radiant-interchange configuration
factors for finite right circular cylinder to rectangular plane," Int.
J. Heat Mass Transfer, vol. 6, no. 2, pp. 143-146.
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.
of 33 factors for common geometries given mostly as closed-form
Yang, L., Chen, W., Luo, L., and Zhao, X., 2014, "Calculation of Radiation Heat Transfer View Factors Among Fuel Rod Bundles Based on CFD Method," Annals of Nuclear Energy, vol. 71, pp. 462-466.
Computes factors using both the discrete ordinates and discrete transfer methods in a commercial CFD code. Compares results with previous analytical expressions for simple parallel cylinder arrays and presents numerical results for one case of a control rod surrounded by fuel rods of different diameter.
Yarbrough, David W. and Lee, Chon-Lin, 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. 563-574.
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
shape-factor calculation between three-dimensional planar objects," AIAA
J. Heat Transfer, vol. 102, no. 2, pp. 386-388.
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