CONFIGURATION FACTORS FOR RADIATION TRANSFER BETWEEN DIFFUSE SURFACES
Following from: Properties of real surfaces
Leading to: Net radiation method in radiative transfer
1. INTRODUCTION
Configuration factors (also often called shape factors, form factors, view factors, or angle factors) define the fraction of uniformly distributed radiative energy leaving diffuse surface j that is incident on a second surface k. The configuration factor from a differential area element dA_{j} to a second element dA_{k} is denoted by dF_{dj-dk}. In general, such a factor is given by
(1) |
where the quantities on the right-hand side are shown in Fig. 1.
Figure 1. Defining geometry for configuration factor.
The notation adopted here is that used in Siegel and Howell (2010). The differential form dF for element-to-element configuration factors is used to maintain the differential order of the equation to agree with the differential area on the right-hand side.
Equation (1) can also be set in the form
(2) |
where dΩ_{j} is the solid angle subtended by the projected area dA_{k} to dA_{j}, that is
(3) |
It is easily shown that a reciprocity relation exists,
(4) |
2. 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_{j} to finite receiving area A_{k} is given by
(5) |
The reciprocity relation is
(6) |
Equation (5) has the implicit assumptions that surface k is diffuse, and that the intensity leaving surface k is uniform across the surface. Observe that the choice of notation again keeps the differential order consistent. If the receiving area is a differential element, then the configuration factor will always be of differential order.
3. CONFIGURATION FACTOR FOR FINITE AREA TO FINITE AREA
For the case of A_{j} and A_{k} both finite, the configuration factor is
(7) |
leading to the reciprocity relation
(8) |
Equation (7) has the implicit assumptions that both surfaces j and k are diffuse, and that the intensity leaving the surfaces is uniform across the surface.
These basic defining equations, (1), (4), and (6), 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.
4. CONFIGURATION FACTOR ALGEBRA
When the configuration factor F_{j-k} between two surfaces is known, the reciprocity relation (7) can be used to find F_{k-j}. Other relations can also be developed that allow simple calculations of new factors from known factors.
If surface k can be subdivided into N nonoverlapping surfaces that completely cover surface k, then
(9) |
because all energy fractions from surface j to parts of surface k must equal the fraction of the total energy leaving j that is incident on all of k.
Suppose that surface j is completely enclosed by a set of M surfaces. In that case, all energy leaving surface j must strike some other surface forming the enclosure. In terms of configuration factors,
(10) |
Note the term F_{j-j} 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 Fig. 2. The triangles are perpendicular to one another.
Figure 2. Perpendicular right isosceles triangles joined along their short sides.
To find F_{1-2}, 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 Fig. 3. This forms a corner cavity with the third congruent triangle, A_{3}.
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 Fig. 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_{1-1} = 0. By symmetry, F_{1-2} = F_{1-3}. Thus, Eq. (11) reduces to
(12) |
For surface 4 of the enclosure, Eq. (10) plus the use of symmetry gives
(13) |
Using reciprocity, Eq. (7) results in
(14) |
Substituting into Eq. (12) results in
(15) |
Using geometry, A_{1} = h^{2}/2 and A_{4} = √3 l^{2}/4 = √3 h^{2}/4, giving
(16) |
This is the desired answer. The factors F_{1-4} = 1/√3 and F_{4-1} = 1/3 have also been generated.
Siegel and Howell note that for an N-surfaced enclosure of all planar or convex surfaces (i.e., F_{j-j} = 0 for all j), N(N-3)/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_{j-j} ≥ 0, then N(N-3)/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), Taylor and Luck (1995), and Daun et al. (2005).
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. Over 300 factors are gathered in an online catalog available at http://www.engr.uky.edu/rtl/Catalog/
REFERENCES
Daun, K. J., Morton, D. P., and Howell, J. R., Smoothing Monte Carlo exchange factors through constrained maximum likelihood estimation, J. Heat Transfer, vol. 127, no. 10, pp. 1124-1128, 2005.
Larsen, M. E. and Howell, J. R., Least squares smoothing of direct exchange areas in zonal analysis, J. Heat Transfer, vol. 18, no. 1, pp. 239-242, 1986.
Lawson, D. A., An improved method for smoothing approximate exchange areas, Int. J. Heat Mass Transfer, vol. 38, no. 16, 3109-3110, 1995.
Loehrke, R. I., Dolaghan, J. S., and Burns, P. J., Smoothing Monte Carlo exchange factors, J. Heat Transfer, vol. 117, no. 2, pp. 524-526, 1995.
Siegel, R. and Howell, J. R., Thermal Radiation Heat Transfer, 5th ed., Taylor and Francis, New York, 2010.
Sowell, E. F. and O’Brien, P. F., Efficient computation of radiant-interchange configuration factors within an enclosure, J. Heat Transfer, vol. 94, no. 3, pp. 326-328, 1972.
Taylor, R. P. and Luck, R., Comparison of reciprocity and closure enforcement methods for radiation view factors, J. Thermophys. Heat Transfer, vol. 9, no. 4, pp. 660-666, 1995.
van Leersum, J., 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, pp. 83-85, 1989.
References
- Daun, K. J., Morton, D. P., and Howell, J. R., Smoothing Monte Carlo exchange factors through constrained maximum likelihood estimation, J. Heat Transfer, vol. 127, no. 10, pp. 1124-1128, 2005.
- Larsen, M. E. and Howell, J. R., Least squares smoothing of direct exchange areas in zonal analysis, J. Heat Transfer, vol. 18, no. 1, pp. 239-242, 1986.
- Lawson, D. A., An improved method for smoothing approximate exchange areas, Int. J. Heat Mass Transfer, vol. 38, no. 16, 3109-3110, 1995.
- Loehrke, R. I., Dolaghan, J. S., and Burns, P. J., Smoothing Monte Carlo exchange factors, J. Heat Transfer, vol. 117, no. 2, pp. 524-526, 1995.
- Siegel, R. and Howell, J. R., Thermal Radiation Heat Transfer, 5th ed., Taylor and Francis, New York, 2010.
- Sowell, E. F. and Oâ€™Brien, P. F., Efficient computation of radiant-interchange configuration factors within an enclosure, J. Heat Transfer, vol. 94, no. 3, pp. 326-328, 1972.
- Taylor, R. P. and Luck, R., Comparison of reciprocity and closure enforcement methods for radiation view factors, J. Thermophys. Heat Transfer, vol. 9, no. 4, pp. 660-666, 1995.
- van Leersum, J., 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, pp. 83-85, 1989.