## PROPERTIES OF REAL SURFACES

* Following from: *The blackbody

* Leading to: *Configuration factors for radiation transfer between diffuse surfaces

The characteristics of the blackbody are independent of material properties and its
emission properties depend only on the independent parameters *T* and λ or their
product. Real materials absorb (and therefore emit) less thermal radiation than
predicted for a blackbody. To account for the differences between the ideal blackbody
and the performance of real materials, the radiative properties of opaque surfaces are
used.

### 1 RADIATIVE PROPERTIES OF OPAQUE SURFACES

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. There is sparse data for the angular dependence of surface radiative properties, so that the available surface properties are usually for properties either 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. Because of the lack of data for directional properties, emphasis here is on spectral (wavelength) dependence of the properties.

#### 1.1 Emissivity

The energy emitted by a 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 is expressed by its spectral emissivity. It is given the symbol ε. Emissivity is thus a dimensionless quantity, 0 ≤ ε ≤ 1.

At the direction normal to the surface, the emissivity is

(1) |

The subscript *n* indicates a quantity evaluated in a direction normal to the
surface (θ = 0).

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

(2) |

The numerator is found experimentally, and the denominator can be measured experimentally or found from the Planck blackbody relation.

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

(3) |

The numerator can be either measured directly by a detector that is sensitive to total radiation, or, if spectral data is available, then that data can be numerically integrated to give

(4) |

For total normal emissivity, the result is

(5) |

#### 1.2 Absorptivity

The ability of an opaque material to absorb incident radiation is described by the
absorptivity, given the symbol α. This property defines the amount of incident energy
on the surface d*A* 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 ≤ α ≤ 1. As for
emissivity, the absorptivity is often measured for incident radiation normal to the
surface (useful, i.e., 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*_{λ,a}(θ, λ, *T*)dλ and the incident
energy can be expressed in terms of the incident radiation intensity as
*d*^{2}*q*_{λ,i}(θ, λ, *T*)dλ = *I*_{λ,i}(θ = 0,λ)cosθd*A*dΩdλ = I_{λ,i}(θ = 0,λ)d*A*dΩdλ, where
dΩ is the solid angle. Taking the ratio of absorbed to incident radiation for the
normal direction gives the normal spectral absorptivity as

(6) |

Observe that the incident intensity *I*_{λ,i}(λ) depends on spectral characteristics of
the source of the incident radiation and in general will not have a blackbody spectral
distribution. The *T* dependence in the equation indicates that the absorbed energy
rate may depend on the *T* of the absorbing surface.

Consider the case when radiation is incident on the absorbing surface from many
directions. The absorbed energy is d*q*_{λ,a}(λ, *T*)dλ. For convenience, give the symbol
*G*_{λ} 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

(7) |

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

(8) |

and the hemispherical total absorptivity is

(9) |

#### 1.3 Kirchhoff’s Law

Because the blackbody is at once the perfect absorber and best possible
emitter of radiative energy, there is a relationship between the properties of
emissivity and absorptivity. 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 α_{λn} at wavelength λ (see Fig.
1).

**Figure 1. Equal temperature normal surfaces exchanging radiation.**

The radiant energy absorbed in the wavelength range dλ by element 1 from element 2 placed normal to surface 1 is then

(10) |

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

(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
ε_{λ,n} = α_{λ,n}. This form of Kirchhoff’s law for spectral properties in a particular
direction is taken as correct in engineering situations. We have not treated
directionally dependent emissivities and absorptivities, but a similar argument 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 Kirchhoff’s Law to replace the spectral normal absorptivity gives the total normal absorptivity as

(12) |

This 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, the total absorptivity can be modified using directionally independent spectral properties to give the total hemispherical absorptivity as

(13) |

The total hemispherical properties α(*T*) and ε(*T*) are equal only if the
irradiation *G*_{λ} 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.

#### 1.4 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 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, 2010).

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

(14) |

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

(15) |

The reflectivity of a surface depends on the orientation of incident radiation to the surface. The incident radiation 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. 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. For highly polished surfaces, polarization effects are very important, since they are at the nanoscale. Property relations for the two polarization components are given in Siegel and Howell (2002) and Modest (2003).

#### 1.5 Further Property Relations

Consider one unit of spectral radiation incident per unit area on a surface, i.e., *G*_{λ} =
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

(16) |

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

(17) |

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

(18) |

where the restrictions on applications of Kirchhoff’s law for total properties must be observed for the substitution of ε to be valid.

#### 1.6 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 made because of the major increase in effort that is necessary to treat the complete property variations in wavelength and direction for real surfaces.

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 α_{λn} = α_{λ},
and ε_{λn} = ε_{λ}. 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 second idealization is to assume that the properties do not vary with
wavelength, so that α_{λ} = α, and ε_{λ} = ε. Such a surface is called a gray surface. For a
gray surface, integrations over wavelength are not required, again simplifying
radiative heat transfer analysis at the expense of accuracy. 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.

It is possible to predict the properties of dielectric solids and metals through electromagnetic theory, and relations for such prediction are in Siegel and Howell (2010) and Modest (2003).

#### REFERENCES

Siegel, R. and Howell, J. R., Thermal Radiation Heat Transfer, 5th ed., Taylor and Francis, New York, 2010.

Modest, M. M., Radiation Heat Transfer, 2nd ed., Academic Press, New York, 2003.

#### References

- Siegel, R. and Howell, J. R., Thermal Radiation Heat Transfer, 5th ed., Taylor and Francis, New York, 2010.
- Modest, M. M., Radiation Heat Transfer, 2nd ed., Academic Press, New York, 2003.