The radiation fractional function is the fraction of black body radiation below a given value of λT. Edwards and others have distinguished between the traditional, or “external,” radiation fractional function and an “internal” radiation fractional function. The latter is used for linearization of net radiation from a nongray surface when the temperature of an effectively black environment is not far from the surface's temperature, without calculating a separate total absorptivity. This paper examines the analytical approximation involved in the internal fractional function, with results given in terms of the incomplete zeta function. A rigorous upper bound on the difference between the external and internal emissivity is obtained. Calculations using the internal emissivity are compared to exact calculations for several models and materials. A new approach to calculating the internal emissivity is developed, yielding vastly improved accuracy over a wide range of temperature differences. The internal fractional function should be used for evaluating radiation thermal resistances, in particular.

## Introduction

The use of the radiation fractional function is well known in computing total emissivity from spectral emissivity. Less well known is the use of the *internal* radiation fractional function for approximating net heat flux from one surface to another at a similar temperature. Edwards was a strong proponent of the latter approach, beginning in the 1960s [1], following work published a few years earlier by Czerny and Walther [2]. Edwards' work on radiative property measurements had well acquainted him with the failure of gray-body exchange approximations at even small temperature differences [3], leading him to seek a better approximation that retained the computational convenience of using only a single radiative property.

Edwards distinguished the traditional fractional function by calling it the *external* fractional function. The terms external and internal derive from Edwards' work in the aerospace industry, differentiating between radiation exchange on the exterior of a spacecraft, where temperature differences are large, and on the interior of a spacecraft, where surface temperatures are not far apart [4]. The internal radiation fractional function appears in several textbooks by Edwards and coworkers [5–7] spanning a period of decades. Most other textbooks have not taken it up. In part, the lack of adoption may reflect the limited range of temperature difference over which the internal emissivity provides an accurate estimate of the net heat flux.

The present paper reconsiders the approximation leading to the internal radiation fractional function, with the objective of understanding its accuracy and how that may be improved. The results are given in terms of the incomplete zeta function, and a rigorous upper bound on the difference between the external and internal emissivity is obtained. Calculations based upon the internal radiation fractional function are compared to exact calculations, and a procedure is established that sharply increases domain of accuracy of the internal emissivity. The internal emissivity should be used in computing radiation thermal resistance in multimode heat transfer and for other linearizations of nongray radiation exchange.

## Background

*α*(

*λ*,

*T*

_{1}), of a surface at

*T*

_{1}, is related to the total hemispherical

^{2}emissivity,

*ε*(

*T*

_{1}), by integration over the Planck distribution of blackbody monochromatic emissive power,

*e*

_{λ}_{,}

*(*

_{b}*T*

_{1})

*c*

_{1}and

*c*

_{2}are the first and second radiation constants. If this surface is irradiated by an effectively black environment at

*T*

_{2}(for example, if the surroundings are much larger than the surface), the total hemispherical absorptivity is

## Nongray Errors

For a gray surface, *α*(*λ*, *T*) is independent of wavelength and so *ε*(*T*_{1}) = *α*(*T*_{1}, *T*_{2}). However, some older literature went farther, applying this idea to nongray surfaces for which *T*_{1} is near *T*_{2}. Edwards noted that the fact that $limT2\u2192T1\alpha (T1,T2)=\epsilon (T1)$ is meaningless in calculating heat exchange; and further, working with Nelson, he documented associated “nongray errors” in radiative property measurements that exceeded 60% [3].

*slope*of

*q*

_{net}as a function of temperature difference, which changes if the surface is not truly gray. To see why in a simple way, the last term in Eq. (4) may be linearized

For a gray (or black) surface, *dα*/*dT*_{2} = 0, and we recover a common linearization: $qnet\u22484\sigma \epsilon T13\Delta T$ [11]. Thus, the purpose of introducing the internal fractional function is to account for the difference in slope when *dα*/*dT*_{2} ≠ 0.

## External and Internal Radiation Fractions

*λ*is

*ζ*(*X*,*s*) is the incomplete zeta function (see derivation in Appendix A). Edwards [1,4,5] refers to *f* as the external radiation fraction. The distinct *internal* radiation fractional function, *f _{i}*(

*λT*), arises from a linearization that allows

*ε*(

*T*) and

_{s}*α*(

*T*,

_{s}*T*) to be combined, as described next.

_{e}*internal*hemispherical emissivity such that

*T*

_{2}is not too much different from

*T*

_{1}

L'Hôpital's rule simply compares the slopes of the numerator and denominator in the limit, so Eqs. (7) and (14) are identical statements, as a change of variables and integration by parts of Eq. (13) shows.

where *X* ≡ *c*_{2}/*λT*. The functions *f*, *f _{i}*, and

*F*are plotted as a function of

*λT*in Fig. 1.

The numerical implementation is described in Appendix B.

## Difference Between External and Internal Emissivities

*X*=

*X*is the finite zero point of

_{z}*dF*/

*dX*,

*X*= 3.92069 (see Appendix C). The function

_{z}*dF*/

*dX*is positive for

*X*<

*X*and negative for

_{z}*X*>

*X*, and $0<\alpha (\lambda ,T)\u2009\u2a7d\u20091$. Thus, we can form two bounds on the difference

_{z}*X*corresponds to

_{z}At 300 K, the wavelength *λ _{z}* = 12.2323

*μ*m.

In the first case, *ε ^{i}* is 44% greater than

*ε*; in the second case,

*ε*is 32% less than

^{i}*ε*.

For a gray surface, Eq. (20) gives *ε* − *ε ^{i}* = 0, consistent with Eqs. (4) and (7). The variation of the total absorptivity for a 300 K surface irradiated by an effectively black source at

*T*

_{2}is shown for both example surfaces in Fig. 2. The total absorptivity varies significantly. The net heat exchange is discussed in detail below.

*α*(

*λ*) switches between zero and one at

*X*. For example, from the preceding development, for the surface black on short wavelengths

_{z}Equation (18) gives *ε ^{i}*, and the results for the long wavelength example may be obtained by subtracting these values from unity.

## Linearization Errors

*q*

_{net}for a black surface. Their ratio is

Equation (32) is plotted in Fig. 3. For *T*_{2} = 320 K and *T*_{1} = 300 K, the linearized flux is 90.5% of the exact flux. The ratio is asymptotic to $1\u22123\Delta T/2T1+5(\Delta T)2/4T12$ as Δ*T* → 0.

*T*

_{1}is considerably less accurate than the traditional approach (i.e., for radiation heat transfer coefficients) of linearizing around

*T*= (

_{m}*T*

_{1}+

*T*

_{2})/2. For that case, the ratio is

and has errors on the order of (Δ*T*/*T _{m}*)

^{2}/4 [11]. As seen in Fig. 3, this linearization is far more accurate. The reason for the greater accuracy is that linearization about

*T*is equivalent to a second-order Runge–Kutta method, with third-order truncation error in heat flux, while linearization about

_{m}*T*

_{1}is simply a forward Euler method (see Appendix D).

*ε*will also have a limited range of accuracy if evaluated at

^{i}*T*

_{1}. Instead,

*ε*should be computed at

^{i}*T*, following the analysis in Appendix D:

_{m}## Comparison of Models

We compare the heat flux predicted by the exact result (Eq. (4)), the gray approximation, *ε ^{i}*(

*T*

_{1}) (Eq. (14)), and

*ε*(

^{i}*T*) (Eq. (35)) in Figs. 4 and 5 for the same two example surfaces at

_{m}*T*

_{1}= 300 K. The charts show

- (1)
The gray surface approximation,

*ε*=*α*, has the wrong slope as*T*_{2}→*T*_{1}, consistent with Eq. (7). This approximation has poor accuracy for even the smallest temperature differences. The ratio of gray slope to exact slope is*ε*/*ε*. Using the values from Eq. (27) or Eq. (28), that ratio is equal to 0.69 and 1.46 for Figs. 4 and 5, respectively. As expected, the gray approximation shows the greatest error for^{i}*T*_{2}>*T*_{1}, where irradiation dominates net exchange. - (2)
The approximation using

*ε*(^{i}*T*_{1}) has the correct slope as*T*_{2}→*T*_{1}, but is accurate for only a limited range of temperatures. For the surface black on short wavelengths, the magnitude of the linearized flux from Eq. (14) at*T*_{2}= 320 K is 13% less than the exact value, Eq. (4). This case corresponds to the seemly small temperature difference Δ*T*/*T*_{1}= −0.067. - (3)
The approximation using

*ε*(^{i}*T*) remains accurate over a very broad range of temperature variation, to the degree that the curves for this equation and the exact equation are difficult to distinguish. The charts in Figs. 4 and 5 include values as high as Δ_{m}*T*/*T*_{1}= ±0.33.

Although Figs. 4 and 5 use 300 K as a reference point, Eqs. (29)–(31) show that, for a surface at *T*_{1} = *T _{z}* that switches from black to reflective behavior at

*λ*, both

_{z}*ε*(

*T*) and

_{z}*ε*(

^{i}*T*) are fixed. The values of $\epsilon i(Tm)=\epsilon i(Tz\u2212\Delta T/2)$ and

_{z}*α*(

*T*,

_{z}*T*− Δ

_{z}*T*) depend additionally only upon Δ

*T*/

*T*. Hence, the trends seen will be the same at any other temperature level. These comparisons are put into nondimensional form in Appendix F.

_{z}Edwards [1,4,5] and, later, Mills [7] were both very specific in recommending to evaluate *ε ^{i}* at

*T*

_{1}. Edwards et al. [6] did not specify the temperature, but without comment included an example worked using

*T*. Likewise, Czerny and Walther [2] provided an example worked using

_{m}*T*, again without comment. The present results show clearly that only

_{m}*T*should be used when evaluating

_{m}*ε*.

^{i}## Application to Representative Materials

The previous example surfaces were contrived to maximize the differences between *ε* and *ε ^{i}*. In this section, we examine the differences for representative nongray materials.

### Alumina.

Experimental data for polycrystalline alumina (99.5% Al_{2}O_{3}, 6 mm thick, 1 *μ*m roughness) have been reported by Teodorescu and Jones [12] for *T *=* *823 K and $2\u2009\u2a7d\u2009\lambda \u2009\u2a7d\u200925\u2009\mu m$. The experimental uncertainty is 3.5%. These data are available in tabular form on 1 *μ*m intervals and as a chart (Fig. 6). These results are generally consistent with other reports on polycrystalline alumina.

*ε*. One additional data point is extrapolated,

_{n}*ε*(1

*μ*m, 823 K) ≈ 0.128, because

*f*at 2

_{i}*μ*m still has the finite value 0.059; the extrapolation extends the range to

*f*= 0.00012. At the other end of this spectrum,

_{i}*f*=

*0.987 at 25*

*μ*m, but no extrapolation was done. Thus, the value for total normal emissivity

*ε*is thus likely to be low by ∼1%. Integration of the discrete data by a simple trapezoidal rule yields the following values, both at 823 K:

_{n}Our interest, ultimately, lies with estimating heat exchange, so we require the spectral hemispherical emissivity. Teodorescu and Jones [12] also measured the spectral directional emissivity. We have integrated those data, as described in Appendix E, to obtain the spectral hemispherical values from 1 to 25 *μ*m.

We consider exchange with an effectively black environment at temperatures down to 625 K. At 625 K, *f *=* *0.972 at 25 *μ*m so that the absorptivity calculation may be low by 2% or so. The results are shown in Fig. 7. Once again, Eq. (35) using *ε ^{i}*(

*T*) is in excellent agreement with the exact result, Eq. (4).

_{m}### Drude/Hagens-Ruben Metal.

*r*(

_{e}*T*) is the electrical resistivity in Ω cm.

We consider a surface at *T*_{1} = 373 K with *r _{e}* = 13.1 × 10

^{−6}Ω ⋅ cm, a value representative of platinum. At this temperature, $0.01\u2009\u2a7d\u2009f\u2009\u2a7d\u20090.99$ for $3.88\u2009\u2a7d\u2009\lambda \u2009\u2a7d\u200961.3\u2009\mu m$ and

*f*= 0.01 at

_{i}*λ*= 3.36

*μ*m. Over this range of wavelengths,

*ε*(

*λ*) varies from 0.086 down to 0.022 according to Eq. (37). We may consider heat exchange with a surface at

*T*

_{2}down to 275 K; at 275 K,

*f*>

*0.99 for*

*λ*> 83.1

*μ*m. The integrations are, therefore, done from 3 to 100

*μ*m. The results are shown in Fig. 8.

*ε*(

^{i}*T*) closely tracks the exact solution over a wide range (down to at least 300 K or Δ

_{m}*T*/

*T*

_{1}= 20%). In this instance, nongray effects are present but less pronounced; the gray approximation, while not the best alternative for even small Δ

*T*, shows less divergence than in previous examples. In addition, at 373 K

The internal and external emissivities are closer, as would be expected when nongray effects are smaller.

### Spectrally Selective Materials.

*λ*from a short-wavelength absorptivity,

_{c}*α*

_{sw}, to a long-wavelength absorptivity,

*α*

_{lw}

*X*

_{c}_{,}

*=*

_{m}*c*

_{2}/

*λ*. Finally, from Eq. (3)

_{c}T_{m}*X*

_{c}_{,2}=

*c*

_{2}/

*λ*

_{c}T_{2}. The previous results show that the impact of selectivity will be greatest when

*X*and

_{c}*X*are close.

_{z}As an example, we may consider the following crude approximation to the soft-anodized aluminum described in Ref. [5]: *α*_{sw} = 0.10, *α*_{lw} = 0.85, and *λ _{c}* = 7

*μ*m (Fig. 9). If this selective solar reflector is at

*T*

_{1}= 360 K and exchanges radiation with sky at

*T*

_{2}= 290 K, we obtain the results in Table 1. At these temperatures, most of the radiant energy is on wavelengths above 7

*μ*m, and nongrayness is not pronounced. Nonetheless, as in previous examples, the agreement between Eq. (4), 371.8 W/m

^{2}, and Eq. (35), 371.0 W/m

^{2}, is excellent, whereas Eq. (14), 462.1 W/m

^{2}, performs quite poorly.

X_{z} | X_{c} | X_{c}_{,}_{m} | X_{c}_{,2} |

3.92069 | 5.70943 | 6.32429 | 7.08757 |

ε(T_{1}) | ε(^{i}T_{1}) | ε(^{i}T)_{m} | α(T_{1}, T_{2}) |

0.7258 | 0.6237 | 0.6807 | 0.7964 |

q_{gray} | $qint,\u2009T1$ | $qint,\u2009Tm$ | q_{exact} |

400.2 | 462.1 | 371.0 | 371.8 |

X_{z} | X_{c} | X_{c}_{,}_{m} | X_{c}_{,2} |

3.92069 | 5.70943 | 6.32429 | 7.08757 |

ε(T_{1}) | ε(^{i}T_{1}) | ε(^{i}T)_{m} | α(T_{1}, T_{2}) |

0.7258 | 0.6237 | 0.6807 | 0.7964 |

q_{gray} | $qint,\u2009T1$ | $qint,\u2009Tm$ | q_{exact} |

400.2 | 462.1 | 371.0 | 371.8 |

## Accuracy Versus Effort

One rationale for introducing *ε ^{i}* is simply to avoid computing the total hemispherical emissivity and absorptivity separately when the spectral absorptivity has a significant wavelength dependence. The same labor-saving rationale was used in some older literature to justify a gray body approximation,

*ε*(

*T*

_{1}) ≈

*α*(

*T*

_{1},

*T*

_{2}), when the temperatures are close. When the internal fractional function was first developed, personal computers (and pocket calculators, spreadsheets, symbolic integration, etc.) did not exist. Routine analyses required either tedious hand calculations or cumbersome trips to the mainframe computer center (to type up Hollerith cards and then wait for the SysOp to run the job!). So, one attraction of the method was clearly to reduce labor.

While numerical integration is very easy today, *ε ^{i}*(

*T*) can still save some time in doing basic calculations. The greater challenge may be to obtain complete spectral, hemispherical data spanning the necessary range of wavelengths. If, instead, electromagnetic theory is used, working from the complex refractive index and calculating spectral directional emissivity, much coding has to be undertaken, and the single integration saved by

_{m}*ε*will not be at all significant.

^{i}*A*, such a resistance has the form

*R*

_{rad}is routinely evaluated using a gray body approximation. For small temperature differences, using the internal emissivity,

*ε*(

^{i}*T*

_{1}), in Eq. (47) will provide a more accurate value than a gray body approximation. The present work has shown

*ε*(

^{i}*T*) to be far more robust, with excellent accuracy for $|\Delta T/T1|$ of 20–30%. Hence, the radiation thermal resistance is best calculated as

_{m}Despite modern computing power, the need for an accurate radiation resistance remains as important today as it was 60 years ago. Thus, we conclude that the internal fractional function has an ongoing value when surfaces do not politely behave as “gray bodies.”

## Summary

The internal radiation fractional function has been in the literature since the late 1950s, although not widely adopted in textbooks covering thermal radiation. The most visible recommendations have been from Edwards and coworkers, generally advising the use of *ε ^{i}*(

*T*

_{1}) to linearize nongray exchange for enclosures with modest temperature differences. In this study, the following new findings are reported:

- (1)
The maximum absolute difference between

*ε*(*T*_{1}) and*ε*(^{i}*T*_{1}) is 0.18400, Eq. (25). - (2)
The internal emissivity should be evaluated at the mean temperature,

*T*, not_{m}*T*_{1}as has often been suggested. The differences in accuracy are shown to be significant because the evaluation at*T*using Eq. (35) is effectively a second-order accurate numerical method, with truncation error in the heat flux of $O(\Delta T3)$. In contrast, evaluation at_{m}*T*_{1}is a first-order approximation. - (3)
Equation (35) with

*ε*(^{i}*T*) has excellent accuracy for $|\Delta T/T1|$ up to 20–30%. Beyond this range, the linearization error of Fig. 3 will cause increasing error for any surface._{m} - (4)
Theory and examples for several nongray materials show that the gray-body approximation gives the wrong slope for heat flux as

*T*_{2}→*T*_{1}. - (5)
Calculations involving both the internal and external fractional functions can be conveniently implemented using the incomplete zeta function.

- (6)
*ε*(^{i}*T*) should be used in calculating radiation thermal resistances for nongray surfaces, with Eq. (48)._{m}

## Acknowledgment

Don Edwards was on the faculty of UCLA's CNTE Department when I studied there, although he was not my instructor. Nevertheless, I felt his positive influence through the curriculum and the other faculty in the heat transfer program.

## Nomenclature

*c*_{1}=first radiation constant, 3.741772 × 10

^{−16}(W m^{2})*c*_{2}=second radiation constant, 14387.77 (

*μ*m K)*e*_{λ}_{,}=_{b}blackbody monochromatic emissive power (W m

^{−3})*f*=(external) radiation fractional function

*F*=*f*−_{i}*f*, Eq. (18)*f*=_{i}internal radiation fractional function

*q*_{net}=net radiant heat flux from surface 1 to 2 (W m

^{−2})*r*=_{e}electrical resistivity (Ω cm)

*R*_{rad}=radiation thermal resistance, Eq. (47) (K W

^{−1})*T*=temperature (K)

*T*=_{m}mean temperature, (

*T*_{2}+*T*_{1})/2 (K)*T*_{1}=temperature of surface (K)

*T*_{2}=temperature of environment (K)

*X*=*c*_{2}/*λT**X*=_{z}finite value of

*X*for which*dF*/*dX*= 0, 3.92069*α*_{lw},*α*_{sw}=see Eq. (39)

*α*(*T*_{1},*T*_{2}) =total hemispherical absorptivity

*α*(*λ*,*T*) =spectral hemispherical absorptivity

*δT*=*T*_{2}−*T*_{1}(K)- Δ
*T*= *T*_{1}−*T*_{2}(K)- Δ
*α*= *α*_{lw}−*α*_{sw}*ε*(*T*) =total hemispherical emissivity (external)

*ε*(_{n}*T*) =total normal emissivity (external)

*ε*(_{n}*λ*,*T*) =spectral normal emissivity

*ε*(*λ*,*T*) =spectral hemispherical emissivity

- $\epsilon \u2032(\theta ,\lambda ,T)$ =
spectral directional emissivity

*ε*(^{i}*T*) =internal total hemispherical emissivity

- $\epsilon ni(T)$ =
internal total normal emissivity

*ζ*(*s*) =Riemann zeta function

*ζ*(*X*,*s*) =incomplete zeta function

*θ*=polar angle (rad)

*λ*=wavelength (m)

*σ*=Stefan–Boltzmann constant, 5.670367 × 10

^{−8}(W m^{−2}K^{−4})

Standard theory shows that for multistep integration, the global error of Euler's method is $O(\Delta T)$ [30, Sec. 9.3].

### Appendix A: Incomplete Zeta Function

*λT*→

*∞*,

*f*=

*1 and the last equation yields the well-known result*

*ζ*(4), has the indicated integral representation [14, Sec. 13.12]. A classical result due to Euler [15] gives

*ζ*(4) =

*π*

^{4}/90 (see also Ref. [16, Sec. 167]), from which we recover the usual definition of the Stefan–Boltzmann constant,

*σ*. Returning to Eq. (A2) with this information, we have

*X*=

*c*

_{2}/

*λT*and we have identified the incomplete zeta function,

*ζ*(

*X*,

*s*) [17, Sec. 8.22]. Hence

### Appendix B: Numerical Implementation

The integrals were computed using the GNU Scientific Library [18] with FFI bindings [19] to Lua code [20] under LuaLaTeX [21] using TEXShop over TEX Live [22]. The C code was compiled using XCode under Mac OS X. Integration was performed using GSL's QAG adaptive integration with 61 point Gauss–Kronrod rules. Convergence was checked, and the numerical values given in the text are believed to be accurate to the number of digits shown.

The incomplete zeta function was rendered in terms of the third-order Debye function, *D*_{3}(*X*) = 3 *ζ*(*X*, 4) Γ(4)/*X*^{3}, and computed using GSL's built-in routine. These routines were supplied to PGFPLOTS [23] to generate the charts.

The integrations of the discrete data for alumina were handled outside GSL, as described in the main text and Appendix E.

### Appendix C: The Constant Xz

*X*is the finite zero point of

_{z}*dF*/

*dX*, specifically the nonzero root of

*y*=

*4 −*

*X*, Eq. (C1) becomes

_{z}*W*function

We may show by contradiction that *X _{z}* is irrational. Assuming

*X*is rational,

_{z}*X*=

_{z}*a*/

*b*for nonzero integers

*a*and

*b*. Then, the left-hand side of Eq. (C1) is 4 − 4

*e*

^{−}

*; however,*

^{a/b}*e*is itself irrational (e.g., see Ref. [28, Sec. 4.7]). Hence, 4 − 4

^{a/b}*e*

^{−}

*is an irrational number and cannot equal the assumed right-hand side,*

^{a/b}*a*/

*b*.

*X*may be constructed by calculating a continued fraction representation of

_{z}*X*through the usual process

_{z}*X*

_{z}The second of these is within 0.1% of the exact value; the last agrees to six digits.

### Appendix D: Linearizations as Finite Difference Approximations

*q*

_{net}to be a function of

*T*that we wish to determine approximately at

*T*=

*T*

_{2}by evaluating at other temperatures, i.e.,

*T*=

*T*

_{1}and

*T*=

*T*. With

_{m}*T*=

*T*

_{1}would approximate

*Y*(

*T*

_{2}) based only on conditions at

*T*

_{1}as follows:

*δT*=

*T*

_{2}−

*T*

_{1}(= −Δ

*T*), which gives Eq. (7) or Eq. (14). The local truncation error of this

*single-step*approximation is $O(Y\u2033\Delta T2)$ [29].

^{3}

*T*with expansions toward both

_{m}*T*

_{1}and

*T*

_{2}, subtracting the former from the latter

*α*(

*λ*,

*T*

_{1}) is evaluated at the surface temperature,

*T*

_{1}. With Eqs. (3) and (13)

### Appendix E: Integration of Directional Emissivity for Alumina

The spectral directional emissivity data are in 12 deg increments of polar angle *θ* from 0 deg to 72 deg. In all cases, the data are essentially constant from 0 to 36 deg, and this range was integrated analytically. From 36 deg to 84 deg, a five-point trapezoidal rule was used, and the integral from 84 deg to 90 deg was approximated as a trapezoid. The value at 90 deg was set to zero, in line with theory. This procedure was found to have a numerical truncation error of 1.0% for a gray surface.

The data for the lowest emissivities showed angular behavior consistent with a dielectric medium, as noted in Ref. [12], decreasing at higher angles. On this basis, the point at 84 deg was interpolated, using a value representative of variations at large angle for a dielectric: *ε*(84 deg, *λ*) ≈ 0.75 *ε*(72 deg, *λ*). Without large angle measurements or the complex refractive index, we cannot exclude the presence of the kind of peak emissivity above 80 deg predicted by Drude's model for metals [13]; and, in particular, the data at 17 and 23 *μ*m do show a 30–50% increase in emissivity by 72 deg. Even so, a sensitivity analysis letting *ε*(84 deg, *λ*) ≈ 2.5 *ε*(72 deg, *λ*) increases the hemispherical emissivity by only about 5% of the previous estimate. Absent further data, there is not much basis upon which to adjust the calculation further.

### Appendix F: Nondimensional Formulation of Model Surface Heat Exchange

*T*

_{1}=

*T*that switches between perfectly black and perfectly reflective behavior at wavelength

_{z}*λ*, where

_{z}*X*=

_{z}*c*

_{2}/

*λ*= 3.92069, for the case

_{z}T_{z}*T*

_{1}=

*T*= 300 K. The four equations used may be nondimensionalized as follows:

_{z}In all cases, $qnet/\sigma T14=fn(T2/T1)$, since *T _{m}*/

*T*

_{1}and Δ

*T*/

*T*

_{1}may both be written in terms of

*T*

_{2}/

*T*

_{1}.

##### Surface Black Below λz.

where $Xc,m=c2/\lambda zTm=Xz(T1/Tm)=3.92069(T1/Tm)$ and $Xc,2=c2/\lambda zT2=Xz(T1/T2)=3.92069(T1/T2)$.

The curves for this case are plotted in Fig. 10. Accuracy of the linearization-based *ε ^{i}*(

*T*) is excellent over the range

_{m}*T*

_{2}/

*T*

_{1}≈ 1 ± 0.30 in even this extreme case of nongray behavior. The gray approximation strongly overpredicts

*q*

_{net}when

*T*

_{2}>

*T*

_{1}because it underestimates the absorption of irradiation.

##### Surface Black Above λz.

The curves for this case are plotted in Fig. 11. As before, the linearization-based *ε ^{i}*(

*T*) is very accurate over the range

_{m}*T*

_{2}/

*T*

_{1}= 1 ± 0.30. The gray approximation strongly underpredicts

*q*

_{net}when

*T*

_{2}>

*T*

_{1}because it overestimates the absorption of irradiation.

The internal emissivity at *T _{m}* is shown for both surfaces in Fig. 12.