A CATALOG OF RADIATION HEAT TRANSFER
CONFIGURATION FACTORS

John R. Howell
University of Texas at Austin

REFERENCES

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. Also, see Srinivasa Ramanujam, K., Abishek, S. and Kette, S., 2006, "Differential View Factor for a Rectangle with Intervening Parallelepiped or Sphere, " J. Thermophysics, vol. 20, no. 3, pp. 604-607 for extension to cases with intervening bodies.

 

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. 


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