Configuration factor

Figure 1: Radiative exchange between two differential surface elements.

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 S1 − 2, and the angles between this connecting line and the normals to the elements are θ1 and θ2. Both surface elements are assumed to be black.

In the following relations, subscripts such as A1 or dA1 are replaced by 1 or d1 to avoid two levels of subscripts. The following equation

$d{e_{\lambda ,d{A_s} - d{A_{}}}} = {I_{\lambda b}}({\theta _s} = 0)d{A_s}d\Omega d\lambda = {I_{\lambda b}}({\theta _s} = 0)d{A_s}\left( {\frac{{dA \cdot \cos \theta }}{{{R^2}}}} \right)d\lambda$

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

deλ,d1 − d2 = Iλb,1dA1cosθ1dΩ2dλ
$= {I_{\lambda b,1}}d{A_1}\cos {\theta _1}\left( {\frac{{d{A_2}\cos {\theta _2}}}{{S_{1 - 2}^2}}} \right)d\lambda \qquad \qquad(1)$

Because the element is black, the intensity is independent of θ. The solid angle subtended by surface 2, dΩ2, has been replaced by the projected area of element 2 over ${S_{1-2}}^2$. 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

$d{e_{\lambda ,d2 - d1}} = {I_{\lambda b,2}}d{A_2}\cos {\theta _2}\left( {\frac{{d{A_1}\cos {\theta _1}}}{{S_{1 - 2}^2}}} \right)d\lambda \qquad \qquad(2)$

The net energy exchange between the two elements is then

$\begin{array}{l} {d^2}{q_{\lambda ,d1 - d2}} = d{e_{\lambda ,d1 - d2}} - d{e_{\lambda ,d2 - d1}} \\ = {I_{\lambda b,1}}d{A_1}\cos {\theta _1}\left( {\frac{{d{A_2}\cos {\theta _2}}}{{S_{1 - 2}^2}}} \right)d\lambda - {I_{\lambda b,2}}d{A_2}\cos {\theta _2}\left( {\frac{{d{A_1}\cos {\theta _1}}}{{S_{1 - 2}^2}}} \right)d\lambda \\ = \left( {{I_{\lambda b,1}} - {I_{\lambda b,2}}} \right)d{A_1}\left( {\frac{{\cos {\theta _1}\cos {\theta _2}d{A_2}}}{{S_{1 - 2}^2}}} \right)d\lambda \\ = \left( {{E_{\lambda b,1}} - {E_{\lambda b,2}}} \right)d{A_1}\left( {\frac{{\cos {\theta _1}\cos {\theta _2}d{A_2}}}{{\pi S_{1 - 2}^2}}} \right)d\lambda \\ \end{array}\qquad \qquad(3)$

The final form uses the identity Eλb = πIλb. The net radiative flux on element 1 due to exchange between surfaces 1 and 2 is then

${d^2}{{q''}_{\lambda, d1-d2}}= {d^2}{q_{\lambda, d1-d2}}/ d{A_1} = ({E_{\lambda b,1}} - {E_{\lambda b,2}})(cos{{\theta}_1}cos{{\theta}_2}d{A_2}/\pi {S_{1-2}}^2)d \lambda \qquad \qquad(4)$

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

$d{F_{d1 - d2}} \equiv \left( {\frac{{\cos {\theta _1}\cos {\theta _2}d{A_2}}}{{\pi S_{1 - 2}^2}}} \right)\qquad \qquad(5)$

so that eq. (4) becomes

${d^2}{{q^{‘’}}_{\lambda, d1 – d2}} = ({E_{\lambda b, 1}} – {E_{\lambda b, 2}}) d{F_{d1 – d2}} d \lambda \qquad \qquad (6)$

The dFd1 − 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

$\begin{array}{l} {q_{\lambda ,d1 - d2}} = \left( {{E_{\lambda b,1}} - {E_{\lambda b,2}}} \right)d{F_{d1 - d2}}d{A_1}d\lambda = - {q_{\lambda ,d2 - d1}} \\ = - \left( {{E_{\lambda b,2}} - {E_{\lambda b,1}}} \right)d{F_{d2 - d1}}d{A_2}d\lambda \\ \end{array}\qquad \qquad(7)$

which results in the identity

$d{F_{d1 - d2}}d{A_1} = d{F_{d2 - d1}}d{A_2}\qquad \qquad(8)$

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

$d{e_{\lambda ,d1 - 2}} = \pi {I_{\lambda b,1}}d\lambda d{A_1}\int_{{A_2}} {\left( {\frac{{\cos {\theta _1}\cos {\theta _2}}}{{\pi S_{1 - 2}^2}}} \right)} d{A_2} = {E_{\lambda b,1}}d\lambda d{A_1}\int_{{A_2}} {d{F_{d1 - d2}}} \qquad \qquad(9)$
Figure 2 Exchange between surface element dA1 and finite surface A2.

Defining

${F_{d1 - 2}} = \int_{{A_2}} {\left( {\frac{{\cos {\theta _1}\cos {\theta _2}}}{{\pi S_{1 - 2}^2}}} \right)} d{A_2} = \int_{{A_2}} {d{F_{d1 - d2}}} \qquad \qquad(10)$

eq. (9) becomes

$d{e_{\lambda ,d1 - 2}} = {E_{\lambda b,1}}d\lambda d{A_1}{F_{d1 - 2}}\qquad \qquad(11)$

Observe that in the argument of the integral in eq. (10), cosθ1, cosθ2 and S1 − 2 will all vary as the integration across finite area A2 is carried out. We might infer some difficulty in evaluating Fd1 − 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

$d{e_{\lambda ,2 - d1}} = {E_{\lambda b,2}}{A_2}d\lambda d{F_{2 - d1}}\qquad \qquad(12)$

Reciprocity for element-area factors: If black element dA1 and black area A2 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

$d{A_1}{F_{d1 - 2}} = {A_2}d{F_{2 - d1}}\qquad \qquad(13)$

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

$d{e_{\lambda ,2 - 1}} = {E_{\lambda b,2}}{A_2}d\lambda \int_{{A_1}} {d{F_{2 - d1}}} = {E_{\lambda b,2}}{A_2}d\lambda {F_{2 - 1}}\qquad \qquad(14)$

Equation (9) gives

$d{e_{\lambda ,d1 - 2}} = {E_{\lambda b,1}}d\lambda \int_{{A_1}} {\left[ {\int_{{A_2}} {\left( {\frac{{\cos {\theta _1}\cos {\theta _2}}}{{\pi S_{1 - 2}^2}}} \right)} d{A_2}} \right]} d{A_1} = {E_{\lambda b,1}}d\lambda \int_{{A_1}} {\left[ {\int_{{A_2}} {d{F_{d1 - d2}}} } \right]} d{A_1}\qquad \qquad(15)$

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

$d{e_{\lambda ,d1 - 2}} = {E_{\lambda b,1}}d\lambda \int_{{A_1}} {\left[ {\int_{{A_2}} {d{F_{d1 - d2}}} } \right]} d{A_1} = {E_{\lambda b,1}}d\lambda \int_{{A_1}} {{F_{d1 - 2}}} d{A_1} = {E_{\lambda b,1}}{A_1}{F_{1 - 2}}d\lambda \qquad \qquad(16)$

Equation (16) also indicates that a general definition of F1 − 2 is

${F_{1 - 2}} = \int_{{A_1}} {{F_{d1 - 2}}} d{A_1} = \int_{{A_1}} {\left[ {\int_{{A_2}} {d{F_{d1 - d2}}} } \right]} d{A_1} = \int_{{A_1}} {\left[ {\int_{{A_2}} {\frac{{\cos {\theta _1}\cos {\theta _2}d{A_2}}}{{S_{1 - 2}^2}}} } \right]} d{A_1}\qquad \qquad(17)$
Figure 3 Configuration factors in an enclosure of N surfaces

If T1 = T2, then equating eqs. (14) and (16) gives a final reciprocity relation

${A_1}{F_{1 - 2}} = {A_2}{F_{2 - 1}}\qquad \qquad(18)$

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

$\sum\limits_{k = 1}^N {{F_{j - k}}} = 1\qquad \qquad(19)$

If surface j is concave, then it is possible that some radiation leaving j may strike another portion of j, so that Fij ≠ 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

${F_{j - (k + l)}} = {F_{j - k}} + {F_{j - l}}\qquad \qquad(20)$

References

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