A CATALOG OF RADIATION HEAT
TRANSFER
CONFIGURATION FACTORS
John
R. Howell
University of Texas at Austin
INTRODUCTION
Standard texts on general heat transfer and on radiative heat transfer
treat the concept of radiation configuration factors in some detail.
The texts discuss various methods of computing these factors and provide examples
of the use of configuration factor algebra for generating factors from
factors that are already available. However, because of obvious space
limitations, only a few of the hundreds of configuration factors that have
been derived and published in the engineering literature are reproduced in
most of these references.
Many of the early catalogs and references that presented common
configuration factors are now out of print or difficult to obtain, and
particular factors are scattered throughout the technical literature dealing
with basic thermal radiative transfer and the engineering design of lighting
systems. In addition, some factors are found in journals concerned with fires
and flame spread, solar energy, industrial furnace design, spacecraft thermal
control, and others.
This catalog is an attempt to gather many of the most useful published
factors into a single source.
Errors identified in earlier publications have been corrected whenever
found, and many factors have been rechecked; however, it was not feasible to
recompute all previously published factors, particularly results given in
graphical form as calculated by numerical integration. The user should
therefore be cautious in using the factors, and should pointcheck factors,
particularly those taken from tables and graphs.
In many cases, errors in original references have been reprinted in
subsequent compilations or references (see, for example, the discussion in Feingold and Gupta, 1970.) Where these
errors are known, only the references with correct values are given in the
reference list provided here. Similarly, some factors have been recomputed by
subsequent researchers because of the need for improved accuracy or the
emergence of better computational or analytical methods. The best data and/or
simplest forms in the opinion of the author are selected for presentation
here.
The configuration factor is defined as the fraction of diffusely radiated
energy leaving surface A that is incident on surface B. No factors are
presented for nondiffuse surfaces, although for a certain subset of
geometries the factors given here are appropriate for nondiffuse surfaces
(particularly certain ones with rotational symmetry).
NOTATION AND CONFIGURATION FACTORS BETWEEN
DIFFERENTIAL ELEMENTS
The notation adopted here is that used in Siegel
and Howell. The configuration factor from a differential area element dA_{1}
to a second element dA_{2} is denoted by dF_{d1d2}. In
general, such a factor is given by
(1)
where
the quantities on the righthandside are shown in Fig. 1.
The differential form dF for elementtoelement configuration factors is
used to maintain the differential order of the equation to agree with the
differential area on the righthandside.
FIGURE 1: DEFINING GEOMETRY FOR
CONFIGURATION FACTOR.
Eq. (1) can also be put in the form
(2)
where
dw is the solid angle
subtended by the projected area dA_{2} to dA_{1}; that is
(3)
It
is easily shown that a reciprocity relation exists:
(4)
CONFIGURATION FACTOR FOR A DIFFERENTIAL
ELEMENT AND A FINITE AREA
If the receiving area is finite, then the configuration factor from
differential surface element dA_{1} to finite receiving area A_{2}
is given by:
(5)
The
reciprocity relation is
(6)
Observe
that the choice of notation again keeps the differential order consistent. If
the receivingarea is a differential element, then the configuration factor
will always be of differential order.
CONFIGURATION FACTOR FOR FINITE AREA TO
FINITE AREA
For
the case of A_{1} and A_{2} both finite, the configuration
factor is
(7)
leading
to the reciprocity relation
(8)
These
basic defining equations, Eqs. (1), (5), and (7) are not in the most useful
form for a given geometry. It is desirable to have an algebraic relation, or
graphical or numerical results that relate the configuration factor to a
simple set of parameters that describe the given geometry. Providing such
relations is the purpose of this catalog.
CONFIGURATION FACTOR ALGEBRA
When
the configuration factor F_{AB} between two surfaces is known, the
reciprocity relation (Eq. (8)) can be used to find F_{BA}. Other
relations can also be developed that allow simple calculations of new factors
from known factors.
If surface B can be subdivided into N nonoverlapping surfaces that
completely cover surface B, then
(9)
because all energy fractions from surface A to parts of surface B must equal
the fraction of the total energy leaving A that is incident on all of B.
Suppose that surface 1 is completely enclosed by a set of M surfaces. In
that case, all energy leaving surface A must strike some other surface
forming the enclosure. In terms of configuration factors,
(10)
Note the term F_{11} in the summation, which must be included if
surface A is concave to account for the fraction of energy leaving surface A that
is incident on itself.
The reciprocity relations plus Eqs. (9) and (10) form the basis of what is
called configuration factor algebra. Using these relations, new
factors can be computed from a small set of known factors; sometimes, factors
can be generated from the algebra alone. The procedure is best illustrated by
example.
Consider two right isosceles triangles that are joined along their short
side as shown in Figure 2. The triangles are perpendicular to one another.
FIGURE 2: Perpendicular Right
Isosceles Triangles Joined Along Their Short Sides.
To find F_{12}, note that an enclosure can be formed by first
joining the free corners of the triangles by a line of length l as
shown in Figure 3. This forms a corner cavity with the third congruent
triangle,
FIGURE 3: Construction of Corner
Cavity by Addition of Line Connecting Free Corners of Triangle.
The enclosure is completed by placing an equilateral triangle of side l
(and area A_{4}) over the cavity formed by the three isosceles
triangles, which have equal areas A_{1}, A_{2}, and A_{3}.
This is shown in Figure 4.
FIGURE 4: Completion of
Enclosure by Addition of Equilateral Triangle, Surface 4.
Now, apply configuration factor algebra. Eq. 10 gives
(11)
Because
surface 1 is planar, F_{11} = 0. By symmetry, F_{12}=F_{13}.
Thus, Eq. (11) reduces to
(12)
For
surface 4 of the enclosure, Eq. (10) plus the use of symmetry gives
(13)
Using
reciprocity, Eq. (8), results in
(14)
Substituting
into Eq. (12) results in
(15)
Using
geometry, A_{1}=h^{2}/2 and , giving
(16)
This
is the desired answer. The factors F_{14} = 1/(3^{1/2}) and F_{41} = 1/3 have also been generated.
Siegel and Howell note that for an
Nsurfaced enclosure of all planar or convex surfaces (i.e., F_{ii} =
0 for all i), N(N3)/2 factors must be found from a catalog of factors or by
calculation. The remaining factors can then be determined by configuration
factor algebra. If M of the surfaces (M<N) are concave [i.e., have
F_{ii} > 0], then [N(N3)/2]+M factors must be known
before configuration factor algebra can determine the remaining factors.
The presence of symmetry may reduce the number of factors that must be
known before the rest can be determined.
When the values of certain factors are known approximately, then the
constraints imposed on the factors by reciprocity and conservation in an
enclosure can be used to refine the known values. Methods for this purpose
have been proposed by Sowell and O'Brien, 1972;
Larsen and Howell, 1986; van Leersum, 1989; Lawson, 1995; Loehrke
et al., 1995; and Taylor et al., 1995.
CATALOGING SYSTEM
This
catalog is divided into three sections. Section A contains factors between
differential elements of the form dF_{d1d2}; Section B contains factors
from a differential area to a finite area, F_{d12}; and Section C
provides factors between finite areas, F_{12}.
Generally, the catalog is arranged with factors of increasing complexity
within each section. Factors within each section begin with planar surfaces
to planar surfaces; planar surfaces to cylindrical surfaces; planar surfaces
to conical surfaces, spherical surfaces, etc.. Next, factors from cylindrical
surfaces to cylindrical surfaces, to conical surfaces, spherical surfaces,
etc..
Factors of the form dF_{2d1} are not cataloged, but can be
obtained by reciprocity from the F_{d12} factors.
LIMITATIONS
Many
factors have been cataloged here, but others have had to be omitted for
various reasons. For example, Grier, 1969,
gives about 400 pages of tabular data on configuration factors between
spheres and their parts. Abbreviated portions of the data are reproduced here
from such compilations, along with a listing of the parameter ranges for
which complete data are available in the original reference. The interested
reader can thus go to the original source for complete tabulated data.
Many of the early references present extensive tabular or graphical data
that were calculated from lengthy closedform algebraic relations.
Presentday computer power and availability makes it easy to compute values,
and only the algebraic relations are given in this catalog. Where numerical
integration is necessary to obtain results, graphical or tabular data are
reproduced in this catalog, although the wide availability of
computationallybased engineering programs such as MAPLE, MATHCAD, MATLAB and
MATHEMATICA will probably make even these tabulations of little practical use
in a few years.
Send mail to: John Howell
University of Texas at Austin
