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.
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 , following work published a few years earlier by Czerny and Walther . Edwards' work on radiative property measurements had well acquainted him with the failure of gray-body exchange approximations at even small temperature differences , 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 . 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.
For a gray surface, α(λ, T) is independent of wavelength and so ε(T1) = α(T1, T2). However, some older literature went farther, applying this idea to nongray surfaces for which T1 is near T2. Edwards noted that the fact that is meaningless in calculating heat exchange; and further, working with Nelson, he documented associated “nongray errors” in radiative property measurements that exceeded 60% .
For a gray (or black) surface, dα/dT2 = 0, and we recover a common linearization: . Thus, the purpose of introducing the internal fractional function is to account for the difference in slope when dα/dT2 ≠ 0.
External and Internal Radiation Fractions
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, fi(λT), arises from a linearization that allows ε(Ts) and α(Ts,Te) to be combined, as described next.
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 ≡ c2/λT. The functions f, fi, 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
At 300 K, the wavelength λz = 12.2323 μm.
In the first case, εi is 44% greater than ε; in the second case, εi is 32% less than ε.
For a gray surface, Eq. (20) gives ε − εi = 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 T2 is shown for both example surfaces in Fig. 2. The total absorptivity varies significantly. The net heat exchange is discussed in detail below.
Equation (18) gives εi, and the results for the long wavelength example may be obtained by subtracting these values from unity.
and has errors on the order of (ΔT/Tm)2/4 . As seen in Fig. 3, this linearization is far more accurate. The reason for the greater accuracy is that linearization about Tm is equivalent to a second-order Runge–Kutta method, with third-order truncation error in heat flux, while linearization about T1 is simply a forward Euler method (see Appendix D).
Comparison of Models
We compare the heat flux predicted by the exact result (Eq. (4)), the gray approximation, εi(T1) (Eq. (14)), and εi(Tm) (Eq. (35)) in Figs. 4 and 5 for the same two example surfaces at T1 = 300 K. The charts show
The gray surface approximation, ε = α, has the wrong slope as T2 → T1, 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/εi. 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 T2 > T1, where irradiation dominates net exchange.
The approximation using εi(T1) has the correct slope as T2 → T1, 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 T2 = 320 K is 13% less than the exact value, Eq. (4). This case corresponds to the seemly small temperature difference ΔT/T1 = −0.067.
The approximation using εi(Tm) 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 ΔT/T1 = ±0.33.
Although Figs. 4 and 5 use 300 K as a reference point, Eqs. (29)–(31) show that, for a surface at T1 = Tz that switches from black to reflective behavior at λz, both ε(Tz) and εi(Tz) are fixed. The values of and α(Tz, Tz − ΔT) depend additionally only upon ΔT/Tz. Hence, the trends seen will be the same at any other temperature level. These comparisons are put into nondimensional form in Appendix F.
Edwards [1,4,5] and, later, Mills  were both very specific in recommending to evaluate εi at T1. Edwards et al.  did not specify the temperature, but without comment included an example worked using Tm. Likewise, Czerny and Walther  provided an example worked using Tm, again without comment. The present results show clearly that only Tm should be used when evaluating εi.
Application to Representative Materials
The previous example surfaces were contrived to maximize the differences between ε and εi. In this section, we examine the differences for representative nongray materials.
Experimental data for polycrystalline alumina (99.5% Al2O3, 6 mm thick, 1 μm roughness) have been reported by Teodorescu and Jones  for T = 823 K and . The experimental uncertainty is 3.5%. These data are available in tabular form on 1 μm intervals and as a chart (Fig. 6). These results are generally consistent with other reports on polycrystalline alumina.
Our interest, ultimately, lies with estimating heat exchange, so we require the spectral hemispherical emissivity. Teodorescu and Jones  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 μm.
We consider exchange with an effectively black environment at temperatures down to 625 K. At 625 K, f = 0.972 at 25 μm so that the absorptivity calculation may be low by 2% or so. The results are shown in Fig. 7. Once again, Eq. (35) using εi(Tm) is in excellent agreement with the exact result, Eq. (4).
We consider a surface at T1 = 373 K with re = 13.1 × 10−6 Ω ⋅ cm, a value representative of platinum. At this temperature, for and fi = 0.01 at λ = 3.36 μm. 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 T2 down to 275 K; at 275 K, f > 0.99 for λ > 83.1 μm. The integrations are, therefore, done from 3 to 100 μm. The results are shown in Fig. 8.
The internal and external emissivities are closer, as would be expected when nongray effects are smaller.
Spectrally Selective Materials.
As an example, we may consider the following crude approximation to the soft-anodized aluminum described in Ref. : αsw = 0.10, αlw = 0.85, and λc = 7 μm (Fig. 9). If this selective solar reflector is at T1 = 360 K and exchanges radiation with sky at T2 = 290 K, we obtain the results in Table 1. At these temperatures, most of the radiant energy is on wavelengths above 7 μm, and nongrayness is not pronounced. Nonetheless, as in previous examples, the agreement between Eq. (4), 371.8 W/m2, and Eq. (35), 371.0 W/m2, is excellent, whereas Eq. (14), 462.1 W/m2, performs quite poorly.
Accuracy Versus Effort
One rationale for introducing εi 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(T1) ≈ α(T1, T2), 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, εi(Tm) 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 εi will not be at all significant.
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.”
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 εi(T1) to linearize nongray exchange for enclosures with modest temperature differences. In this study, the following new findings are reported:
The maximum absolute difference between ε(T1) and εi(T1) is 0.18400, Eq. (25).
The internal emissivity should be evaluated at the mean temperature, Tm, not T1 as has often been suggested. The differences in accuracy are shown to be significant because the evaluation at Tm using Eq. (35) is effectively a second-order accurate numerical method, with truncation error in the heat flux of . In contrast, evaluation at T1 is a first-order approximation.
Theory and examples for several nongray materials show that the gray-body approximation gives the wrong slope for heat flux as T2 → T1.
Calculations involving both the internal and external fractional functions can be conveniently implemented using the incomplete zeta function.
εi(Tm) should be used in calculating radiation thermal resistances for nongray surfaces, with Eq. (48).
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.
- c1 =
first radiation constant, 3.741772 × 10−16 (W m2)
- c2 =
second radiation constant, 14387.77 (μm K)
- eλ,b =
blackbody monochromatic emissive power (W m−3)
- f =
(external) radiation fractional function
- F =
fi − f, Eq. (18)
- fi =
internal radiation fractional function
- qnet =
net radiant heat flux from surface 1 to 2 (W m−2)
- re =
electrical resistivity (Ω cm)
- Rrad =
radiation thermal resistance, Eq. (47) (K W−1)
- T =
- Tm =
mean temperature, (T2 + T1)/2 (K)
- T1 =
temperature of surface (K)
- T2 =
temperature of environment (K)
- X =
- Xz =
finite value of X for which dF/dX = 0, 3.92069
- Y =
- αlw, αsw =
see Eq. (39)
- α(T1, T2) =
total hemispherical absorptivity
- α(λ, T) =
spectral hemispherical absorptivity
- δT =
T2 − T1 (K)
- ΔT =
T1 − T2 (K)
- Δα =
αlw − αsw
- ε(T) =
total hemispherical emissivity (external)
- εn(T) =
total normal emissivity (external)
- εn(λ, T) =
spectral normal emissivity
- ε(λ, T) =
spectral hemispherical emissivity
spectral directional emissivity
- εi(T) =
internal total hemispherical emissivity
internal total normal emissivity
- ζ(s) =
Riemann zeta function
- ζ(X, s) =
incomplete zeta function
- θ =
polar angle (rad)
- λ =
- 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 [30, Sec. 9.3].
Appendix A: Incomplete Zeta Function
Appendix B: Numerical Implementation
The integrals were computed using the GNU Scientific Library  with FFI bindings  to Lua code  under LuaLaTeX  using TEXShop over TEX Live . 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, D3(X) = 3 ζ(X, 4) Γ(4)/X3, and computed using GSL's built-in routine. These routines were supplied to PGFPLOTS  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
We may show by contradiction that Xz is irrational. Assuming Xz is rational, Xz = a/b for nonzero integers a and b. Then, the left-hand side of Eq. (C1) is 4 − 4e−a/b; however, ea/b is itself irrational (e.g., see Ref. [28, Sec. 4.7]). Hence, 4 − 4e−a/b is an irrational number and cannot equal the assumed right-hand side, a/b.
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
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. , 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 ; and, in particular, the data at 17 and 23 μm 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
In all cases, , since Tm/T1 and ΔT/T1 may both be written in terms of T2/T1.
Surface Black Below λz.
where and .
The curves for this case are plotted in Fig. 10. Accuracy of the linearization-based εi(Tm) is excellent over the range T2/T1 ≈ 1 ± 0.30 in even this extreme case of nongray behavior. The gray approximation strongly overpredicts qnet when T2 > T1 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 εi(Tm) is very accurate over the range T2/T1 = 1 ± 0.30. The gray approximation strongly underpredicts qnet when T2 > T1 because it overestimates the absorption of irradiation.
The internal emissivity at Tm is shown for both surfaces in Fig. 12.