# Configuration factor algebra

## Basis

The reciprocity equations

dFd1 − d2dA1 = dFd2 − d1dA2

dA1Fd1 − 2 = A2dF2 − d1

or

A1F1 − 2 = A2F2 − 1

and the summation relation, $\sum\limits_{k = 1}^N {{F_{j - k}}} = 1$

and the associative relation

Fj − (k + l) = Fjk + Fjl

from configuration factor are the basis of configuration factor algebra.

Configuration factors for simple geometries can be found directly by applying configuration factor algebra, eliminating the integrations that may otherwise be necessary. The method is shown in the following two examples.

## Configuration factors for the case of infinitely long concentric circular cylinders

The inner cylinder has outer diameter D1 and the outer cylinder has inside diameter D2. By definition, all radiation leaving surface 1 is incident on surface 2, so F1 − 2 = 1. Using reciprocity, ${F_{2 - 1}} = \frac{{{A_1}}}{{{A_2}}}{F_{1 - 2}} = \frac{{\pi {D_1}L}}{{\pi {D_2}L}} = \frac{{{D_1}}}{{{D_2}}}$. Finally, using the summation relation, $\sum\limits_{k = 1}^N {{F_{2 - k}}} = {F_{2 - 1}} + {F_{2 - 2}} = 1$. In this case, surface 2 is concave, so F2 − 2 must be included in the summation. Thus, ${F_{2 - 2}} = 1 - {F_{2 - 1}} = 1 - \frac{{{D_1}}}{{{D_2}}}$. The full set of configuration factors was found for this geometry without recourse to integration.

## An infinitely long enclosure has a cross-section made up of a scalene triangle with planar sides L1,L2,, and L3

We can find an expression for the configuration factor F1 − 2 in terms of the side lengths.

Equation $\sum\limits_{k = 1}^N {{F_{j - k}}} = 1$ from configuration factor can be written for each surface. In this case, for an enclosure of all planar surfaces, each term Fij = 0, so the summation rule, eq. $\sum\limits_{k = 1}^N {{F_{j - k}}} = 1$ from configuration factor, gives $\begin{array}{l} {F_{1 - 2}} + {F_{1 - 3}} = 1 \\ {F_{2 - 1}} + {F_{2 - 3}} = 1 \\ {F_{3 - 1}} + {F_{3 - 2}} = 1 \\ \end{array}$

This results in three equations involving six unknowns: not a good portent! However, three of the unknowns may be eliminated by using reciprocity. Multiply each equation by its respective area (assuming an equal length of channel), resulting in $\begin{array}{l} {L_1}{F_{1 - 2}} + {L_1}{F_{1 - 3}} = {L_1} \\ {L_2}{F_{2 - 1}} + {L_2}{F_{2 - 3}} = {L_2} \\ {L_3}{F_{3 - 1}} + {L_3}{F_{3 - 2}} = {L_3} \\ \end{array}$

Using reciprocity to eliminate three of the unknowns gives $\begin{array}{l} {L_1}{F_{1 - 2}} + {L_1}{F_{1 - 3}} = {L_1} \\ {L_1}{F_{1 - 2}} + {L_2}{F_{2 - 3}} = {L_2} \\ {L_1}{F_{1 - 3}} + {L_2}{F_{2 - 3}} = {L_3} \\ \end{array}$

Solving this set results in ${F_{1 - 2}} = \frac{{{L_1} + {L_2} - {L_3}}}{{2{L_1}}}$.

Once the factor for the triangular enclosure is known, factors for other geometries can be derived. For example, if L1 = L2, then the factor between two sides of a wedge is given by ${F_{1 - 2}} = \frac{{2{L_1} - {L_3}}}{{2{L_1}}} = 1 - \frac{{{L_3}}}{{2{L_1}}} = 1 - \sin \alpha \qquad \qquad(1)$

where α is the half-angle between the sides 1 and 2. Clearly, a number of other factors can also be generated. However, all factors can be generated from configuration factor algebra only for two-sided enclosures if one of the sides is concave as in Example 1, or for a three-sided enclosure if all sides are planar as in Example 2. There are exceptions to this general rule if symmetry allows some of the factors to be equal.

In all cases, configuration factor algebra allows reduction in the number of factors that must be determined independently. The number of independent factors U that must be found in order to determine all factors in an N-sided enclosure made up of planar or convex surfaces is (Siegel and Howell, 2002): $U = N(N - 3)/2\qquad \qquad(2)$

and if M of the N surfaces are concave, then $U = \left[ {N(N - 3)/2} \right] + M\qquad \qquad(3)$

For the case of concentric cylinders in Example 1, N = 2 and M = 1, so from eq. (3), $U = \left[ {2(2 - 3)/2} \right] + 1 = - 1 + 1 = 0$ and, as found in the example, no independent factors must be found; all are determined by configuration factor algebra.

For the triangular enclosure of planar surfaces in Example 2, N = 3 and M = 0, so $U = 3\left( {3 - 3} \right)/2 = 0$, and again no independent factors must be found.

## The crossed-string method

We now derive an extremely useful extension of the factor for the triangular enclosure derived in the above (Hottel, 1954). The restriction still applies that the result will be for an enclosure that is infinitely long in one dimension. Observe an enclosure cross-section made up of four surfaces shown in Fig. 3(a). We seek F1 − 2.

Connect the corners a and b, and corners b and c with lines that follow around any convex areas as shown in Fig. 3(b). Do the same with lines connecting points a and d, and b and c. Then connect points a and c, forming triangles (abc) and (acd). These triangles are made up of all convex surfaces composed of the constructed lines. Similarly, connect points b and d with a dotted line, making a second pair of triangles (bcd) and (abd) also made up of convex or planar surfaces. Now, applying the result for a triangle of planar or convex surfaces to the two constructed triangles (abd) and (I) in Fig. 3(b): $\begin{array}{l} {F_{cd - bc}} = \frac{{{L_{cd}} + {L_{bc}} - {L_{bd}}}}{{2{L_{cd}}}} \\ {F_{cd - ad}} = \frac{{{L_{cd}} + {L_{ad}} - {L_{ac}}}}{{2{L_{cd}}}} \\ \end{array}\qquad \qquad(4)$

where the L values are the lengths of the constructed lines connecting the end points. The summation relation $\sum\limits_{k = 1}^N {{F_{j - k}}} = 1$ applied to this geometry gives ${F_{cd - ad}} + {F_{cd - ab}} + {F_{cd - bc}} = 1\qquad \qquad(5)$

Noting that all radiation reaching the constructed boundary ab will also be incident on L1, eq. (5) becomes ${F_{cd - ab}} = {F_{cd - 1}} = 1 - \left( {{F_{cd - ad}} + {F_{cd - bc}}} \right)\qquad \qquad(6)$

Applying reciprocity gives ${F_{1 - cd}} = \frac{{cd}}{{{L_1}}}{F_{cd - 1}} = \frac{{cd}}{{{L_1}}}\left[ {1 - \left( {{F_{cd - ad}} + {F_{cd - bc}}} \right)} \right]\qquad \qquad(7)$

Again, observe that F1 − cd = F1 − 2 because all radiation incident on cd will finally be incident on surface 2. Substituting eq. (4) into (7) gives the final result $\begin{array}{l} {F_{1 - 2}} = \frac{{{L_{cd}}}}{{{L_1}}}{F_{cd - 1}} = \frac{{{L_{cd}}}}{{{L_1}}}\left[ {1 - \left( {\frac{{{L_{cd}} + {L_{ad}} - {L_{ac}}}}{{2{L_{cd}}}} + \frac{{{L_{cd}} + {L_{bc}} - {L_{bd}}}}{{2{L_{cd}}}}} \right)} \right] \\ = \frac{{\left( {{L_{bd}} + {L_{ac}}} \right) - \left( {{L_{ad}} + {L_{bc}}} \right)}}{{2{L_1}}} \\ \end{array}\qquad \qquad(8)$

The final form of the equation can be interpreted as follows; consider the values of L as the length of taut strings that connect the corners a,b,c,d. The final form shows that the factor F1 − 2 is equal to the sum of the lengths of the strings connecting the corners of surface 1 and 2 that cross one-another, minus the sum of the lengths of the uncrossed strings that connect the corners, all divided by the length of surface 1.

Now, we can apply cross-string method to find the configuration factor between two infinitely long directly opposed plates as shown Fig. 4(a).  Figure 4: Two infinitely long directly opposed plates

Connect the corners with the solid (uncrossed) and dotted (crossed) lines shown in Fig. 4(b). Equation (8) then gives $\begin{array}{l} {F_{1 - 2}} = \frac{{\left( {crossed} \right) - \left( {uncrossed} \right)}}{{2{L_1}}} = \frac{{2{{({H^2} + {W^2})}^{1/2}} - 2H}}{{2L}} \\ = {\left[ {{{\left( {\frac{H}{W}} \right)}^2} + 1} \right]^{1/2}} - \left( {\frac{H}{W}} \right) \\ \end{array}$

Two methods have now been examined for finding values of configuration factors in terms of the particular variables that describe a given geometry: Configuration factor algebra, and the crossed-string method (remembering that the latter method is restricted to geometries that are infinitely long in one dimension.) For more general geometries, there are two additional possibilities for determining the configuration factors that are necessary in calculating radiative energy transfer. First, configuration factors for many geometries have been derived, and the results published in the open literature. These are collected in a web-based catalog (Howell, 2003) that includes well over 300 geometries with element-element, element-area, and area-area factors. Some of these are given in Configuration factor relations. Second, methods for carrying out the required integrations in the general definitions of ${F_{d1 - 2}} = \int_{{A_2}} {\left( {\frac{{\cos {\theta _1}\cos {\theta _2}}}{{\pi S_{1 - 2}^2}}} \right)} d{A_2} = \int_{{A_2}} {d{F_{d1 - d2}}}$ or ${F_{1 - 2}} = \int_{{A_1}} {{F_{d1 - 2}}} d{A_1} = \int_{{A_1}} {\left[ {\int_{{A_2}} {d{F_{d1 - d2}}} } \right]} d{A_1} = \int_{{A_1}} {\left[ {\int_{{A_2}} {\frac{{\cos {\theta _1}\cos {\theta _2}d{A_2}}}{{S_{1 - 2}^2}}} } \right]} d{A_1}$

can be implemented. These methods include direct numerical integration using commercial codes, contour integration (which replaces integrations over the receiving area with integrations around the area boundary), and differentiation of known finite-area factors to find element-area factors. Detailed information on these methods is in standard radiative transfer texts (Siegel and Howell, 2002; Modest, 2003).

## References

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Hottel, H.C, 1954, “Radiant Heat Transmission,” Chap. 4, in W.H. McAdams (Ed.), Heat Transmission, 3rd ed., McGraw-Hill, New York, NY.

Howell, J.R., 2003, A Catalog of Radiation Configuration Factors, 2nd ed., http://www.me.utexas.edu/~howell/

Modest, M.M., 2003, Radiation Heat Transfer, 2nd ed., Academic Press, New York, NY.

Siegel, R. and Howell, J.R., 2002, Thermal Radiation Heat Transfer, 4th ed., Taylor and Francis, New York, NY.