A CATALOG OF RADIATION HEAT
TRANSFER
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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. 609-612.
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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 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. |
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.
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Presents
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. |
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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.
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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. 728-736.
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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 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. |
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.
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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. 275-276, May.
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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. 249-252, February.
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Includes
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).
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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, 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.
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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. 155-156.
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Uses
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.
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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 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.
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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 differing-diameter, coaxial
disks blocked by a coaxial cylinder," J. Thermophys.
Heat Transfer, vol. 4, no, 3, pp. 414- 416, July.
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Closed-form
solution is presented for specified geometry. |
Brewster, M. Quinn, 1992, Thermal Radiative
Transfer and Properties, John Wiley & Sons, New York.
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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. 1095-1100.
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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, end-to-end, 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.
47-73 (in Polish.)
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Disk-to-disk
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.
157-160, Sept.
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Derives
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
D-944.
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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. 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.
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Notes
that Hamilton
and Morgan (1952) has an error for this configuration. |
Cabeza Lainez, J.M., Pulido Arcas, J.
A., Castilla, M.-V., Bellido, C. R., and Bonilla Martínez, J. M. , 2015,
“Radiative Heat Transfer for Curvilinear Surfaces,” Chapter 1 in Solar Radiation Applications,
InTech, London.
Derives many factors for areas
on the interior surface of a sphere.
Campbell,
James P. and McConnell, Dudley G., 1968, "Radiant-interchange
configuration factors for spherical and conical surfaces to spheres," NASA
TN D-4457.
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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 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. |
Chekhovskii,
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.,
1979)
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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. 1115-1117,
November.
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Derives
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, March.
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Provides
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, February.
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Derives
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.
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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. 328-330, August.
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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.
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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.
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Crossed-string
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
72-WA/HT-16.
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Analytical
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
(corrected copy).
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Derives
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.
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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 non-diffuse radiative
exchange form factors," ASME Paper C99-107, Proc. 33rd. National Heat
Transfer Conf., Albuquerque, August.
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Casts
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,
February.
Dunkle, R.V., 1963, "Configuration factors for radiant
heat-transfer calculations involving people," J. Heat Transfer,
vol. 85, no. 1, pp. 71-76, February.
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Measurements
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.
|
Shows
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. |
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. 264-266, February.
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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 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.
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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," Warme-und Stoff bertragung,
vol. 19, no. 4, p. 269.
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Prescribes
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.
413-422, May.
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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 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.
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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. 742-744, November.
|
Uses
inscribed nonintersecting circular disks on sphere interior to derive
disk-to-disk factors in a simple way. Any two such non-intersecting disks
are analyzed. |
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.
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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. 69-76, February.
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Contains
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 evaluations. |
Felske, J.D., 1981,
Personal communication, August 25.
|
Unpublished
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.
|
Develops
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
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. 99-108, July
2-6, 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
one-another. 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
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.
|
Provides
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.)
|
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 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 sphere. |
Grier, Norman T. and Sommers, Ralph D., 1969,
"View factors for toroids and their
parts," NASA TN D-5006.
|
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 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.
|
Provides
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.
489-492, 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.
303-309.
|
Derives
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.
|
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, "Radiant-interchange 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. 938-939, May.
|
Provides
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.
465-474.
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. 254-257, May.
|
Closed-form
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. ORNL-TM-1613, 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 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.
|
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 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.
|
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 computer-based 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. 55-125, McGraw-Hill Book Co., New York.
|
Among
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.
|
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, IS-55-6, pp. 39-49.
|
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, 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., New York.
|
Provides
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
surfaces. |
Howell,
John R., Menguc, Pinar, Daun,
Kyle, and Siegel, Robert, 2021, Thermal Radiation Heat Transfer, 7th
ed., Taylor and Francis-Hemisphere, Washington.
|
Comprehensive
text on radiative transfer, including basic definitions of and relations
among configuration factors. |
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.
|
Lengthy
closed-form 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, Aerojet-General
Corporation.
|
Uses
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.
421-422, 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. 384-388, 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. 558-560, 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. 205-211.
|
Numerical
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.
|
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. LXXXVIII-A, pp. 59-60.
|
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. 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.
|
Disk-to-disk 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).
|
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. 480-482.
|
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. 54-58, November (in Japanese).
|
Uses
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.
|
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 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.
|
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. 73-77, October (in Russian).
Lebedev V.A., and Solovjov V.P.,
2016, “View Factors of Cylindrical Spiral Surfaces,” J. Quant. Spectr.
Rad. Transfer, 171, pp. 1–3.
Lebedev, V.A., Solovjev, V. P.,
and Webb, B.W., 2022, "View Factors of Spherical, Conic, and Cylindrical Spiral
Surfaces,"
J. Quant. Spectr. Rad. Transfer, Accepted.
The view factors of a family of spiral surfaces are derived
using a parameterization and differential geometry.
Analytical results are presented for the view factors of
the conic, spherical, and cylindrical spiral surfaces.
Leuenberger, H.
and Person, R.A., 1956, "Compilation of radiation shape factors for
cylindrical assemblies," paper no. 56-A-144, 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 thin-film solar cells in Earth orbit,"
NASA TN D-4331.
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, "Radiant-interchange configuration factors between disk and
segment of parallel concentric disk," J. Heat Transfer, vol. 29,
no. 3, pp. 501-503.
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 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.
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. 524-526, May.
Uses 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.
Contains catalog of factors, mostly from Hamilton
and Morgan (1952), plus element-to- 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. 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.
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
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.)
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 form-factors for plane areas with
polygonal boundaries)," Warme- und
Stoff bertragung, vol. 19, no. 4, pp. 273-278.
Proposes 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.
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 off-axis 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. 318-320, 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.
1406-1408, 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.
492-494, August. (See also discussion in J. Heat Transfer, vol. 101,
no. 1, pp. 189-190, 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 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.
Minning, C.P.,
1976, "Calculation of shape factors between parallel ring sectors
sharing a common centerline," AIAA J., vol. 14, no. 6, pp.
813-815.
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 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.
This 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.
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.
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., 1991, Radiative Heat Transfer,
McGraw-Hill Book Company, New York.
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.
86-88.
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. 727-730.
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, McGraw-Hill 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.
2028-2030.
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. 89-96.
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. 373-375.
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. 426-431, 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 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.
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. 193-198, 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. 31-37.
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 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 cows.
Plamondon, Joseph
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.
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. 1281-1286.
Uses 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 Stephen son (1966))
Rea, Samuel N., 1975, "Rapid method for determining
concentric cylinder radiation view factors," AIAA J., vol. 13,
no. 8, pp. 1122-1123.
Derives 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.
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.
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.
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
D-586.
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 wall-temperature distribution of a
radiation-cooled nuclear rocket nozzle," NASA TN D-878.
Presents 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.
Rushmeier, H.E.;
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.
Investigates 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.
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. 215-218.
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.
Sasaki, Kaname and Sznajder, Maciej,
2020, "Analytical view factor
solutions of a spherical cap from an infinitesimal surface," Int. J. Heat Mass
Trans, vol. 163, 120477.
Factors from a tilted differential element to a spherical
cap are derived by contour integration. Some 21 cases are analyzed,
considering shading and blockage of part of the cap. Results are verified by
Monte Carlo calculation.
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.
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 closed-form
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. 163-164.
Configuration 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.
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.
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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. 459-461, 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. |
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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 element-element factors with limits
of integration. Discusses effect of finite length on the error involved in
using factors for infinite length geometries. |
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Presents 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 accuracy. |
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Sparrow,
E.M., 1962, "A new and simpler formulation for radiative angle
factors," J. Heat Transfer, vol. 85, no. 2, pp. 81-88, 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. |
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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 disk-disk factors. Steps to
find final ring-ring factor for rings on cylinder interior are
outlined. |
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Contains a complete discussion of configuration factor
algebra with examples, and a catalog of 15 factors. |
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Uses derivatives of disk-disk factors to obtain ring-ring
factors on parallel coaxial circular disks. |
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Uses derivatives of disk-to-disk factors to obtain
ring-to-ring factors on parallel
circular disks. |
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Sparrow,
E.M. and Heinisch, R.P., 1970, "The normal emittance of circular
cylindrical cavities," Appl. Opt., vol. 9, no. 11, pp. 2569-2572,
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. |
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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 separation-to cylinder radius ratios of
0.01 to 10. |
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Derives 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. |
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Derives relation for parallel but not directly opposed
infinite strips in parallel planes. |
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Derives factors between any two differential elements or
between any element and any finite area on the interior of a sphere. |
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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.
1291-1299.
Uses 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. |
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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. |
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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 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 the
abscissa. |
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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. |
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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 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. |
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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. |
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Derives 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. |
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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. 816-819.
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). |
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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. |
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Uses factor from ring element on inside of cylinder to disk
in thermal analysis. |
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Presents compatibility requirements for set of factors
needed for enclosure analysis that will meet the requirement of overall
energy conservation for the enclosure. |
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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 hemisphere-hemisphere factors. Hemisphere
bases are parallel. Results are presented for radius ratios of 1 to 10. |
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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. |
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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 r2 rather than 4p r2. |
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Contains catalog of factors excerpted from Hamilton
and Morgan (1952), and a chapter on configuration factors. |
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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. |
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Catalog of 33 factors for common geometries given mostly
as closed-form expressions. |
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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. |
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Derives general relation for factor between arbitrarily
arranged general polygons based on contour integration. Presents some
numerical results. |
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Send mail to: John Howell
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