# Opaque Surface Property Definitions

The radiative properties are usually expressed as surface properties, although on a microscopic basis radiation incident on a surface may actually penetrate for some distance into the bulk material. We found that emitted or absorbed intensity is independent of angle for the blackbody; however, the surface properties of real materials are angularly dependent; i.e., the intensity emitted by a real surface may vary with respect to the angle relative to the surface normal, as does the ability of the surface to absorb radiation. However, there is sparse data for the angular dependence of surface radiative properties, so that the surface properties are usually presented for either properties normal to the surface, or averaged over all directions. The properties may also be wavelength dependent; this dependence leads to important effects that can be exploited in some applications such as the design of solar collectors and the control of surface temperatures. Although directional effects can be important for certain types of surfaces (e.g., grooved, patterned, mirrored), most surfaces used in practical engineering applications such as furnaces, ovens, boilers, etc. have weak directional dependence. Because of the dearth of available data on directional properties, emphasis here will be on spectral (wavelength) dependence of the properties.

Figure 1: Experimental blackbody cavity for reference.

For experimental determination of radiative properties, it will often be necessary to compare the radiative characteristics of the real surface with those of a blackbody at the same temperature and possibly wavelength. Remembering the relation between the ability of a blackbody to absorb all incident energy, which implies that the blackbody will emit the greatest possible energy, it can be seen that building a device that absorbs all incident energy means that it will also emit as a blackbody. As mentioned before, no real surface has this characteristic. However, it is possible to closely approach a perfect absorbing surface through the use of a heated cavity. Consider a deep hole in a surface which has the surface of the cavity coated with a highly absorbing coating. The radiation absorbing ability of such a device is similar to shining a flashlight into a narrow cave; very little of the flashlight energy will reflect from the cave. The imagined surface of the mouth of the cave then has radiative characteristics very close to those of a black surface.

A laboratory blackbody standard is constructed similarly (Fig. 1), with a deep cavity with its internal surface coated with a highly absorbing coating. The cavity is drilled into a good thermally conducting material to minimize temperature gradients, and the cavity itself is insulated on its exterior. If the cavity is heated to uniform temperature, then the radiation leaving the cavity will closely approach the Planck blackbody characteristics.

Emissivity: The ability of a real surface to emit radiation at a particular wavelength is expressed by its spectral emissivity, which is the energy emitted by the real surface in a narrow wavelength interval dλ around the wavelength λ divided by the radiation that would be emitted by a blackbody at the same wavelength and temperature. It is given the symbol ε . Emissivity is thus a dimensionless quantity, $0 \le \varepsilon \le 1$. At the direction normal to the surface, the emissivity is

${\varepsilon _\lambda }(\theta = 0,\lambda ,T) \equiv {\varepsilon _{\lambda ,n}}(\lambda ,T) = \frac{{{I_\lambda }\left( {\theta = 0,\lambda ,T} \right)}}{{{I_{\lambda b}}(\lambda ,T)}} = \frac{{{I_{\lambda ,n}}\left( {\lambda ,T} \right)}}{{{I_{\lambda b}}(\lambda ,T)}} \qquad \qquad(1)$

The subscript n indicates a quantity evaluated in a direction normal to the surface (θ = 0). The normal spectral emissivity must be measured for any given material. This can be done by heating the material to temperature T, and then using a radiation detector to measure the emitted radiation normal to the surface. Filters or a diffraction grating can be used to provide a signal at the desired wavelength. This measurement gives the numerator of eq. (1). The radiating surface is then replaced by the reference blackbody cavity at the same temperature, and the detector will read the value to be placed in the denominator of eq. (1); alternatively, the value for denominator may be computed directly from the blackbody relations.

Figure 2: Curves of emissive power for a blackbody, Eλb, and a real surface, Eλ, both at 1000 K
Figure 3: Spectral hemispherical emissivity calculated from curves in Fig. 2 using eq. (2)

The hemispherical spectral emissivity, λ, is a measure of the energy emitted into all directions by the real surface in a narrow wavelength interval dλ around the wavelength λ relative to that from a blackbody, and is defined by

${\varepsilon _\lambda }(\lambda ,T) = \frac{{{E_\lambda }\left( {\lambda ,T} \right)}}{{{E_{\lambda b}}\left( {\lambda ,T} \right)}} \qquad \qquad(2)$

Again, the numerator is found experimentally, and the denominator can be measured experimentally or found from the following equation (see Planck distribution):

${E_{\lambda b}}\left( {\lambda ,T} \right) = \frac{{2\pi {C_1}}}{{{\lambda ^5}\left( {{e^{{C_2}/\lambda T}} - 1} \right)}}$

To gain some understanding of the meaning of eq. (2), consider Fig. 2. The top curve is the blackbody emissive power from eq. ${E_{\lambda b}}\left( {\lambda ,T} \right) = \frac{{2\pi {C_1}}}{{{\lambda ^5}\left( {{e^{{C_2}/\lambda T}} - 1} \right)}}$ for a blackbody at 1000 K. The ratio of the values of the Eλ curve to the Eλb curve at the same wavelength is the emissivity of the surface. Taking this ratio at each λ and plotting gives the curve of spectral emissivity. The emissivity calculated for the curves in Fig. 2 is shown in Fig. 3.

For calculation of heat transfer, it is useful to be able to compute the total (all wavelengths) radiation that is emitted by a surface. The total hemispherical emissivity is given by

$\varepsilon (T) = \frac{E}{{{E_b}}} = \frac{E}{{\sigma {T^4}}} \qquad \qquad(3)$
Figure 4: Hemispherical total emissivity as the ratio of total emissive power for the real surface (area under the smaller curve) to blackbody emissive power at the same temperature (area under larger curve).

The numerator of eq. (3) can be either measured directly by a detector that is sensitive to total radiation, or, if spectral data such as in Fig. 2 is available, then that data can be numerically integrated using the definition of eq. (2) to give

$\varepsilon (T) = \frac{{E(T)}}{{\sigma {T^4}}} = \frac{{\int_{\lambda = 0}^\infty {{E_\lambda }(\lambda ,T)d\lambda } }}{{\sigma {T^4}}} = \frac{{\int_{\lambda = 0}^\infty {{\varepsilon _\lambda }(\lambda ,T){E_{\lambda b}}(\lambda ,T)d\lambda } }}{{\sigma {T^4}}} \qquad \qquad(4)$

The meaning of this relation is illustrated in Fig. 4. The ratio of the smaller shaded area to the total larger area (part of which lies beneath the smaller area) is the right-hand term of eq. (4), and the ratio of these areas is the total emissivity ε (T). The total normal emissivity follows a parallel derivation and interpretation. For total normal emissivity, the result is

${\varepsilon _n}(T) = \frac{{{I_n}(T)}}{{{I_b}}} = \frac{{\int_{\lambda = 0}^\infty {{I_{\lambda ,n}}(\lambda ,T)d\lambda } }}{{(\sigma {T^4}/\pi )}} = \frac{{\pi \int_{\lambda = 0}^\infty {{\varepsilon _{\lambda ,n}}(\lambda ,T){I_{\lambda b}}(\lambda ,T)d\lambda } }}{{\sigma {T^4}}}\qquad \qquad(5)$

Equation (1) is substituted to obtain the final term on the right.

Absorptivity: The ability of an opaque material to absorb incident radiation is described by the property of absorptivity, given the symbol α. This property is the amount of incident energy on the surface that is absorbed (converted into internal energy) relative to that which would be absorbed by a blackbody. Because, by definition, all incident energy is absorbed by a blackbody, it is convenient to use the equivalent definition of absorptivity as the incident energy absorbed divided by the incident energy. Absorptivity is thus in the numerical range of $0 \le \alpha \le 1$. As for emissivity, the absorptivity is often measured for incident radiation normal to the surface (useful, for example, for evaluating solar collector materials) and for radiation incident over all directions (common in analyzing materials inside furnaces and ovens).

If radiation is incident normal to the surface, then the rate of absorbed radiation in wavelength interval dλ is ${d^2}{q_{\lambda ,a}}\left( {\theta = 0,\lambda ,T} \right)d\lambda$ and the incident energy can be expressed in terms of the incident radiation intensity as

${d^2}{q_{\lambda ,i}}\left( {\theta = 0,\lambda ,T} \right)d\lambda = {I_{\lambda ,i}}(\theta = 0,\lambda )\cos \theta dAd\omega d\lambda$
.

Taking the ratio of absorbed to incident radiation for the normal direction gives the normal spectral absorptivity as

${\alpha _\lambda }(\theta = 0,\lambda ,T) \equiv {\alpha _{\lambda ,n}}(\lambda ,T) = \frac{{{d^2}{q_{\lambda ,a}}\left( {\theta = 0,\lambda ,T} \right)}}{{{I_{\lambda ,i}}(\lambda ,\theta = 0)dAd\omega }} \qquad \qquad(6)$

Observe that the incident intensity Iλ,i (λ) depends on the temperature and spectral characteristics of the source of the incident radiation. In general, it will not have a blackbody spectral distribution. The T dependence in the equation indicates that the absorbed energy rate by the absorbing surface may depend on the absorbing surface temperature, so the normal spectral absorptivity may depend on the T of the absorbing surface.

Consider the case when radiation is incident upon the absorbing surface from many directions. The absorbed energy is $d{q_{\lambda ,a}}\left( {\lambda ,T} \right)d\lambda$. For convenience, give the symbol Gλ to the spectral energy incident per unit area per wavelength interval dλ from all directions; G is called the irradiation. The hemispherical spectral absorptivity is then

${\alpha _\lambda }(\lambda ,T) \equiv \frac{{d{q_{\lambda ,a}}\left( {\lambda ,T} \right)}}{{{G_\lambda }dA}} \qquad \qquad(7)$

For radiative heat transfer calculations, total (integrated over all wavelengths) absorptivities are most useful. The normal total absorptivity αn is then

${\alpha _n}(T) \equiv \frac{{d{q_a}\left( {\theta = 0,T} \right)}}{{{I_i}\left( {\theta = 0} \right)dAd\omega }} = \frac{{\int_{\lambda = 0}^\infty {{\alpha _{\lambda ,n}}{I_{\lambda ,i}}\left( {\theta = 0} \right)dAd\omega d\lambda } }}{{{I_i}\left( {\theta = 0} \right)dAd\omega }} = \frac{{\int_{\lambda = 0}^\infty {{\alpha _{\lambda ,n}}{I_{\lambda ,n}}d\lambda } }}{{{I_n}}} \qquad \qquad(8)$

and the hemispherical total absorptivity is

$\alpha (T) \equiv \frac{{\int_{\lambda = 0}^\infty {d{q_{\lambda ,a}}\left( {\lambda ,T} \right)} d\lambda }}{{dA\int_{\lambda = 0}^\infty {{G_\lambda }d\lambda } }} = \frac{{\int_{\lambda = 0}^\infty {{a_\lambda }} {G_\lambda }d\lambda }}{{\int_{\lambda = 0}^\infty {{G_\lambda }d\lambda } }} = \frac{{\int_{\lambda = 0}^\infty {{a_\lambda }} {G_\lambda }d\lambda }}{G} \qquad \qquad(9)$
Figure 5: Equal temperature normal surfaces exchanging radiation

The final two forms in eq. (9) are found by substituting eq. (7) to eliminate dqλ,a.

Kirchhoff's Law: Because the blackbody is at once the perfect absorber and best possible emitter of radiative energy, it would seem that there might also be a relationship between the properties of emissivity and absorptivity. This is indeed the case, but care must be taken in applying this relationship.

Consider two surfaces at the same temperature T. Surface 2 is a blackbody, and is placed normal to surface 1, which has normal spectral emissivity ε λn and spectral absorptivity αλn at wavelength λ (Fig. 5). The radiant energy absorbed in the wavelength range dλ by element 1 from element 2 placed normal to surface 1 is then

${\rm{d}}{e_{\lambda ,a}}{\rm{ = }}{\alpha _{\lambda ,n}}{{\rm{I}}_{\lambda ,i}}{\rm{(}}\theta {\rm{ = 0)d}}{{\rm{A}}_2}{\rm{d}}{\Omega _1}{\rm{d}}\lambda {\rm{ = }}{\alpha _{\lambda ,n}}{{\rm{I}}_{\lambda ,i}}{\rm{(}}\theta {\rm{ = 0)}}d{A_2}\frac{{{\rm{d}}{{\rm{A}}_1}}}{{{S^2}}}{\rm{d}}\lambda \qquad \qquad(10)$

The energy emitted by surface 1 that is incident on surface 2 is

${\rm{d}}{{\rm{e}}_{\lambda ,e}}{\rm{ = }}{\varepsilon _{\lambda ,n}}{{\rm{I}}_{\lambda b}}{\rm{(}}\lambda {\rm{,T)d}}\Omega {\rm{d}}{{\rm{A}}_1}{\rm{d}}\lambda {\rm{ = }}{\varepsilon _{\lambda ,n}}{{\rm{I}}_{\lambda b}}{\rm{(}}\lambda {\rm{,T)}}\frac{{d{A_2}}}{{{S^2}}}{\rm{d}}{{\rm{A}}_1}{\rm{d}}\lambda \qquad \qquad(11)$

Using the Second Law argument that no energy can be transferred between surfaces at the same temperature, these two equations can be set equal, resulting in the relation

${\varepsilon _{\lambda ,n}} = {\alpha _{\lambda ,n}} \qquad \qquad(12)$

This is a very general relation. There are some restrictions, such as an inherent assumption that the surfaces are in thermodynamic equilibrium (in both the macroscopic sense and the microscopic sense of having an equilibrium distribution of energy states described by the single parameter T), but this form of Kirchhoff's Law for spectral properties in a particular direction is taken as correct in engineering situations. We have not treated directional emissivities and absorptivities, but an argument similar to that used in deriving eq. (12) shows that it is correct that the spectral directional emissivity of a surface is equal to the spectral directional absorptivity at the same wavelength and for the same direction.

Using eq. (12) to replace the spectral normal absorptivity in eq. (8) gives the total normal absorptivity as

${\alpha _n}(T) = \frac{{\int_{\lambda = 0}^\infty {{\varepsilon _{\lambda ,n}}(T){I_{\lambda ,i}}\left( {\theta = 0} \right)d\lambda } }}{{{I_i}\left( {\theta = 0} \right)}} \qquad \qquad(13)$

Comparing eq. (13) with eq. (5) shows that the total normal emissivity is equal to the total normal absorptivity only in a special case. The normal incoming intensity must come from a blackbody at the same temperature as the absorbing surface. Thus, the simple statement that emissivity equals absorptivity is not valid except for the fundamental spectral-directional properties.

Similarly, eq. (9) can be modified using directionally-independent spectral properties to give the total hemispherical absorptivity as,

$\alpha (T) = \frac{{\int_{\lambda = 0}^\infty {{\alpha _\lambda }} {G_\lambda }d\lambda }}{G} = \frac{{\int_{\lambda = 0}^\infty {{\varepsilon _\lambda }} {G_\lambda }d\lambda }}{G} \qquad \qquad(14)$

and comparing with eq. (4) shows that the total hemispherical properties α (T) and ε (T) are equal only if the irradiation Gλ has the same spectral distribution as a blackbody at the same temperature T as the absorbing surface. In addition, if the surface has significant directional property variations, then the directional distribution incident on the surface must be uniform from all directions; the effect of directional characteristics has been ignored here except for the normal case.

The use of Kirchhoff's Law with the appropriate restrictions allows finding some of the radiative properties for opaque surfaces through measurement of the corollary property.

Reflectivity: The final property of interest for opaque surfaces is the fraction of incident radiation that is reflected from the surface, called the reflectivity and given the symbol ρ . This property in its most fundamental form depends on surface temperature, wavelength, and the direction of incidence of the radiation and the direction of reflection. The most useful of these for engineering calculations are defined here; complete derivations for each of the more detailed reflectivities are defined in detail elsewhere (Siegel and Howell, 2002).

The fraction of the radiation incident from all directions in the wavelength interval dλ that is reflected into all directions in the same wavelength interval is called the hemispherical-hemispherical spectral reflectivity, ρ λ (λ,T), and is defined by

${\rho _\lambda }\left( {\lambda ,T} \right) = \frac{{d{q_{\lambda ,r}}}}{{{G_\lambda }dA}} \qquad \qquad(15)$

The total (hemispherical-hemispherical) reflectivity is the fraction of the total incident energy from all directions that is reflected into all directions, and is

$\rho \left( T \right) = \frac{{d{Q_r}}}{{GdA}} = \frac{{\int_{\lambda = 0}^\infty {d{q_{\lambda r}}d\lambda } }}{{dA\int_{\lambda = 0}^\infty {{G_\lambda }} d\lambda }} = \frac{{\int_{\lambda = 0}^\infty {{\rho _\lambda }(\lambda ,T){G_\lambda }d\lambda } }}{G} \qquad \qquad(16)$

Because radiation is in the form of a propagating wave, the reflectivity of a surface depends on the orientation of the wave relative to the surface. The wave can be resolved into components that vibrate perpendicular to the surface and parallel to the surface, and each of these components has a different reflectivity. The different reflectivities are discussed in Section 9.3.2. For many engineering surfaces (furnace refractory surfaces, soot-covered and oxidized surfaces) the emitted and reflected radiation is unpolarized, and the differing reflectivities are simply averaged to give a single value. However, for highly polished surfaces, polarization effects are very important, as they are at the nanoscale.

Further property relations: Consider one unit of spectral radiation incident per unit area on a surface; i.e., Gλ = 1. Because for an opaque surface this radiation must either be reflected or absorbed, it follows that the fractions absorbed and reflected must sum to unity, or Fraction absorbed at λ + fraction reflected at λ = 1

${\alpha _\lambda } + {\rho _\lambda } = 1\qquad \qquad(17)$

where the properties are spectral hemispherical values. If the surface has minimal directional variation in properties, then Kirchhoff's Law can be invoked to give

${\varepsilon _\lambda } + {\rho _\lambda } = 1 \qquad \qquad(18)$

Equations (17) and (18) show that measuring any one of the three spectral hemispherical properties allows evaluation of the others.

If one unit of total energy is incident on the surface (G = 1), then a similar analysis gives Total fraction absorbed + total fraction reflected = 1

$\alpha + \rho = \varepsilon + \rho = 1\qquad \qquad(19)$

where the stringent restrictions on applications of Kirchhoff's Law for total properties must be observed for the substitution of ε to be valid. The restrictions can be imposed when evaluation is done experimentally, so again measurement of any one of the three total hemispherical properties is sufficient to evaluate all three.

Idealizations for properties: To simplify radiative heat transfer analysis, surface properties are often idealized. These idealizations make analysis of radiative transfer among multiple surfaces tractable, but can introduce serious errors in some cases. They are commonly made because of the major increase in computational effort that is necessary to treat the complete property variations in wavelength and direction for real surfaces.

Figure 6: Comparison of measured normal spectral emissivity for platinum at 1400 K compared with properties using the gray assumption. Experimental values from Harrison et al. (1963).

The first simplification is to assume that all of the properties for opaque surfaces are independent of direction. Such a surface is said to be diffuse; for a diffuse surface, for example, the normal and hemispherical properties are equal, so that αλn = αλ, and ελn = ελ. For a diffuse surface, emitted intensity will be uniform into each direction; reflected energy from any direction is also reflected with equal intensity into each direction. The characteristics of a diffuse surface allow great simplification in analysis.

The second property idealization is to assume that the properties do not vary with wavelength, so that α λ α and ε λε. Such a surface is called a gray surface. For a gray surface, integrations of emitted, absorbed or reflected radiation over wavelength are not required, again simplifying radiative heat transfer analysis at the expense of accuracy. Figure 6 shows the measured spectral emissivity for platinum at a temperature of 1400 K, with an averaged gray emissivity for this temperature of 0.166 computed using eq. (4). At 1400 K, Wien's Law [eq. ${\left( {\lambda T} \right)_{\max }} = {C_3} = 2897.8(\mu m \cdot K)$ from Planck distribution and its consequences] predicts that the peak blackbody emission will be at

${\lambda _{\max }} = \frac{{{C_3}}}{T} = \frac{{2897.8(um \cdot K)}}{{1400K}} = 2.07\mu m$

so the averaged emissivity is heavily weighted by the values in that region of the spectrum.

A diffuse-gray surface automatically obeys Kirchhoff’s Law, so that equations for radiative heat transfer among such surfaces can be written in terms of a single radiative property for each measured total properties for a number of opaque materials are given radiation properties.

## References

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

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