John R. Howell
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 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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

Closed-form solution is presented for specified geometry. 

Brewster, M. Quinn, 1992, Thermal Radiative Transfer and Properties, John Wiley & Sons, New York.

Comprehensive radiative transfer text. Appendix B presents algebraic expressions for thirteen common configurations. 

Brockmann, H., 1994, "Analytic angle factors for the radiant interchange among the surface elements of two concentric cylinders," Int. J. Heat Mass Transfer, vol. 37, no. 7, pp. 1095-1100.

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.)

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.

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.

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.

Notes that Hamilton and Morgan (1952) has an error for this configuration.

Campbell, James P. and McConnell, Dudley G., 1968, "Radiant-interchange configuration factors for spherical and conical surfaces to spheres," NASA TN D-4457.

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)

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.


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.

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.

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.

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.

Using the method of Feingold and Gupta (1970), authors use idea of surrounding a planar element that has its projection inscribed on the sphere interior. Factors from a planar element to a sphere, to the interior of a cylinder lying on the normal to the element, to an isosceles triangle, to a ring element, and to a disk segment are presented. Also, the factor from a spherical element to a sphere is derived. All results are in closed form. Some graphical results are presented. 

Chung, T.J. and Kim, J.Y., 1982, "Radiation view factors by finite elements," J. Heat Transfer, vol. 104, pp. 792.

Uses finite elements plus Gaussian integration to formulate configuration factors between irregular geometries, and shows accuracy of the method by comparison of numerical calculation with values for known factors between opposed squares and between two planes sharing a common edge at various angles.

Cox, Richard L., 1976, Radiative heat transfer in arrays of parallel cylinders, Ph.D. Dissertation, Department of Chemical Engineering, University of Tennessee, Knoxville.

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.

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).

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.

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.

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 integratoion. Comparison of computed and analytical results is shown for two squares with a common edge at various enclosed angles. 

Dummer, R.S. and Breckenridge, W.T. Jr., 1963, "Radiation configuration factors catalog," General Dynamics/Astronautics Rept. 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.

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.


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.

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.


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.


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.

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.

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.

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. 

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. 

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).

The invariance of a number of factors for certain axisymmetric radiating systems and for radiating systems containing infinite surfaces is discussed. Method is proposed for examining invariance of shape factors of various systems. Approach uses reciprocity and closure properties. 

Lebedev, V.A. and Solovjov, V.P.: View Factors of Cylindrical Spiral Surfaces, J. Quant. Spectroscopy Radiative Transfer, 2015.

Provides factors from a segment of a ribbon spirally would onto a cylinder of length l and diameter d to the entire interior surface of the ribbon. Cases for finite ribbon length l and infinite ribbon length are given, along with expressions for reciprocal factors. 

Leuenberger, H. and Person, R.A., 1956, "Compilation of radiation shape factors for cylindrical assemblies," paper no. 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.

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 Stephenson (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.

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. 

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.


Derives factors among all surfaces in an enclosure composed of a closed finite length cylinder contained entirely within a longer coaxial closed cylinder


Siegel, Robert and Howell, John R., 2001, Thermal Radiation Heat Transfer, 4th ed., Taylor and Francis-Hemisphere, Washington.

Comprehensive text on radiative transfer. Provides this entire catalog of configuration factors on compact disk .

Sommers, Ralph D. and Grier, Norman T., 1969, "Radiation view factors for a toroid: comparison of Eckert's technique and direct computation." J. Heat Transfer, vol. 91, no. 3, pp. 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. 

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.

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. 

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.

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. 

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. 

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.

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. 

Sparrow, E.M. and Cess, R.C., 1978, Radiation Heat Transfer, Augmented edition, Hemisphere, Washington, D.C.

Contains a complete discussion of configuration factor algebra with examples, and a catalog of 15 factors. 

Sparrow, E.M. and Eckert, E.R.G., 1962, "Radiant interaction between fin and base surfaces," J. Heat Transfer, vol. 84, no. 1, pp. 12-18.

Uses 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.


Uses derivatives of disk-to-disk factors to obtain ring-to-ring factors on parallel circular disks. 

Sparrow, E.M. and Heinisch, R.P., 1970, "The normal emittance of circular cylindrical cavities," Appl. Opt., vol. 9, no. 11, pp. 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. 

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.

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. 

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.

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. 

Sparrow, E.M. and Jonsson, V.K., 1963c, "Thermal radiation absorption in rectangular- groove cavities," J. Appl. Mechs., vol. E30, pp. 237-244.


Derives relation for parallel but not directly opposed infinite strips in parallel planes

Sparrow, E.M. and Jonsson, V.K., 1962a, "Absorption and emission characteristics of diffuse spherical enclosures," NASA TN D-1289.

Derives factors between any two differential elements or between any element and any finite area on the interior of a sphere. 

Sparrow, E.M. and Jonsson, V.K., 1962b, "Absorption and emission characteristics of diffuse spherical enclosures," J. Heat Transfer, vol. 84, pp. 188-189.

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. 

Stasenko, A.L., 1967, "Self-irradiation coefficient of a Moebius strip of given shape," Akad. Nauk, SSSR, Izv. Energetika Transport, pp. 104-107, July-August.

Derives relation for factor from Moebius strip to itself by numerical integration. Strip is of constant width and has constant radius between strip axis and centerline. 

Stefanizzi, P., 1986, "Reliability of the Monte Carlo method in black body view factor determination," Termotechnica, vol. 40, no. 6, p. 29.

Stephens, Charles W. and Haire, Alan M., 1961, "Internal design considerations for cavity- type solar absorbers," ARS J., vol. 31, no. 7, pp. 896-901.

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. 

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.

Fairly complete treatment of the fundamentals of radiative transfer as applied to spacecraft design. Details energy exchange between spacecraft and nearby planets, and thus presents relations for factors between spheres and various other solid bodies. Full chapter is devoted to "Configuration factor studies and data." Work of Hamilton and Morgan (1952) and Leuenberger and Person (1956) is reproduced and discussed, as are other factors from the literature. Other factors are presented in a form suitable for numerical integration. See Juul 1982b for discussion of a possible error. 

Sydnor, C.L., 1970, "A numerical study of cavity radiometer emissivities," NASA Contractor Rept. 32-1462, Jet Propulsion Lab., Feb. 15.

Appendix presents closed form factors for ring element to ring element on interior of enclosure composed of right circular cylinder closed at both ends by coaxial cones. One cone is truncated. Some typographical errors exist, particularly in 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.

Uses uncertainty analysis to determine effects on enclosure analysis accuracy when configuration factors are independently determined and are not forced to meet closure, reciprocity, or both. 

Toups, K.A., 1965, "A general computer program for the determination of radiant interchange configuration and form factors- CONFAC-I," North American Aviation, Inc. Rept. SID-65- 1043-1 (NASA CR-65256), October.

Tripp, W.; Hwang, C.; and Crank, R.E., 1962, "Radiation shape factors for plane surfaces and spheres, circles, or cylinders" (Spec. Rept. 16) Kansas State Univ. Bull., vol. 46, no. 4.

Derives 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. 

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).

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.

Uses configuration factor algebra with the factors of Brockmann to provide factors among rings on the interior surface of an outer cylinder in the presence of a central concentric cylinder. 

Usiskin, C.M. and Siegel, R., 1960, "Thermal radiation from a cylindrical enclosure with specified wall heat flux," J. Heat Transfer, vol. 82, no. 4, pp. 369-374.

Uses factor from ring element on inside of cylinder to disk in thermal analysis. 

van Leersum, J., 1989, "A method for determining a consistent set of radiation view factors from a set generated by a nonexact method," Int. J. Heat Fluid Flow, vol. 10, no. 1, p 83.

Presents compatibility requirements for set of factors needed for enclosure analysis that will meet the requirement of overall energy conservation for the enclosure. 

Wakao, Noriaki; Kato, Koichi; and Furuya, Nobuo, 1969, "View factor between two hemispheres in contact and radiation heat transfer coefficient in packed beds," Int. J. Heat Mass Transfer, vol. 12, pp. 118-120.

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. 

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.

Presents numerically computed figures for factors between segments on parallel disks of different radii and between an isosceles trapezoid and the segment of a disk that intersects the trapezoid at right angle. 

Watts, R.G., 1965, "Radiant heat transfer to Earth satellites," J. Heat Transfer, vol. 87, no. 3, pp. 369-373, August.

Derives relations for factors from large sphere to small sphere, small hemisphere, small cylinder, or small ellipsoid. "Small" means that the angle between the line connecting any point on the small body and the sphere center and the line from the same point to an arbitrary point on the sphere can be considered invariant over the receiving body. Closed forms are given for the receiving body being a sphere or hemisphere. Numerical evaluation is used in other cases. All results are a factor of 4 times larger than for the configuration factor as used in this catalog because the sphere surface area is taken as p r2 rather than 4p r2

Wiebelt, John A., 1966, Engineering Radiation Heat Transfer, Holt, Rinehart & Winston, New York.

Contains catalog of factors excerpted from Hamilton and Morgan (1952), and a chapter on configuration factors. 

Wiebelt, J.A. and Ruo, S.Y., 1963, "Radiant-interchange configuration factors for finite right circular cylinder to rectangular plane," Int. J. Heat Mass Transfer, vol. 6, no. 2, pp. 143-146.

Numerically computed factors are presented as graphs for various parametric values of rectangle size and spacing. Factors are judged by authors to have possible errors of approximately 5 percent. 

Wong, H.Y., 1977, Handbook of Essential Formulae and Data on Heat Transfer for Engineers, Longman Group, London.

Catalog of 33 factors for common geometries given mostly as closed-form expressions. 

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.

Uses Monte Carlo to compute factors for various simple geometries, and compares with analytical solutions. Presents original results for strip on finite length rectangular fin to parallel cylinder and from cylinder of finite length placed at focus of parallel paraboloid. All results are calculated to be within +/- 5 percent.

Yuen, W.W., 1980, "A simplified approach to shape-factor calculation between three-dimensional planar objects," AIAA J. Heat Transfer, vol. 102, no. 2, pp. 386-388.

Derives general relation for factor between arbitrarily arranged general polygons based on contour integration. Presents some numerical results.

Send mail to: John Howell
University of Texas at Austin