# Configuration factor

### From Thermal-FluidsPedia

In more complex cases than radiative heat transfer between two areas, a surface is exchanging radiative energy with multiple surfaces, each with a different temperature and different properties. It is instructive to first examine the radiative exchange between a single pair of these surfaces. These are shown in Fig. 1. A line between the two elements has length *S*_{1 − 2}, and the angles between this connecting line and the normals to the elements are θ_{1} and θ_{2}. Both surface elements are assumed to be black.

In the following relations, subscripts such as *A*_{1} or *d**A*_{1} are replaced by *1* or *d1* to avoid two levels of subscripts. The following equation

from Planck distribution gives the radiation leaving element 1 that is incident on element 2, assuming there is no attenuation caused by a medium separating the elements, as

*d*

*e*

_{λ,d1 − d2}=

*I*

_{λb,1}

*d*

*A*

_{1}cosθ

_{1}

*d*Ω

_{2}

*d*λ

Because the element is black, the intensity is independent of θ. The solid angle subtended by surface 2, *d*Ω_{2}, has been replaced by the projected area of element 2 over . Also, the rate of energy incident on element 2 is also the rate of energy absorbed by element 2 because it is black. Similar reasoning gives the rate of energy emitted by element 2 that is absorbed by element 1 as

The net energy exchange between the two elements is then

The final form uses the identity *E*_{λb} = π*I*_{λb}. The net radiative flux on element 1 due to exchange between surfaces 1 and 2 is then

Now, observe that the effects of geometry are all contained in a wavelength- and temperature-independent term defined as

so that eq. (4) becomes

The *d**F*_{d1 − d2} is variously called the configuration factor, shape factor or angle factor. *The configuration factor is interpreted as the fraction of radiation leaving surface 1 that is incident on surface 2*. Equation (5) is the basic definition of the configuration factor; it will now be extended to show some important properties. It will then be applied to configuration factors for areas of finite size. The derivation above also applies to diffuse nonblack surfaces, so that the use of the configuration factor is valid for radiative exchange between black or diffuse surfaces as well as gray or spectrally dependent surfaces because the factor is independent of wavelength.

*Reciprocity*. The rate of energy transfer from element 1 to element 2 must be the negative of the energy transfer rate from element 2 to element 1, or

which results in the identity

This reciprocity relation is valid without restriction for diffuse surfaces of differential area.

*Exchange between a differential area and a finite area*. If surface 2 is finite in extent, (Figure 2), then eq. (1) becomes

Defining

eq. (9) becomes

Observe that in the argument of the integral in eq. (10), cosθ_{1}, cosθ_{2} and *S*_{1 − 2} will all vary as the integration across finite area *A*_{2} is carried out. We might infer some difficulty in evaluating *F*_{d1 − 2} when geometries are complex. Fortunately, there are resources that allow some simplifications.

If the intensity leaving surface 2 is uniform across that surface, then the energy transfer from surface 2 to surface 1 is the emitted energy rate times the fraction leaving surface 2 that reaches surface 1, or

*Reciprocity for element-area factors*: If black element d*A*_{1} and black area *A*_{2} are at the same temperature, then the rate of energy exchange between them must be equal. Equating eqs. (11) and (12) gives a second reciprocity relation

*Exchange between two finite areas*. If both surface 1 and 2 are finite in extent, then eq. (12) can be extended to give

Equation (9) gives

Substituting eq. (13) into eq. (15) results in

Equation (16) also indicates that a general definition of *F*_{1 − 2} is

If *T*_{1} = *T*_{2}, then equating eqs. (14) and (16) gives a final reciprocity relation

*Summation relations*: As an aid to further use of configuration factors, consider surface *j* that is part of an enclosure that completely surrounds surface *A*_{j} (Figure 3). The enclosure is made up of a total of *N* surfaces. Based on the concept that *F*_{j − k} is the fraction of radiation leaving *j* that is incident on *k*, its is clear that

If surface *j* is concave, then it is possible that some radiation leaving *j* may strike another portion of *j*, so that *F*_{i − j} ≠ 0 and must be included in the summation.

A corollary to eq. (18) is that the fraction of energy leaving *j* that is incident on two surfaces *k* and *l* must equal to the fractions incident separately on the two surfaces, or

## References

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