Heat transfer is governed by three distinct mechanisms: convection, conduction, and radiation. Unlike convection or conduction, heat transfer through radiation does not occur through a particular medium. To understand this phenomenon one must enter into the atomic or quantum realm. All atoms, at finite temperatures, are continuously in motion. Consequently, it may be understood that the mechanism of radiation is derived from the energetic vibrations and oscillations of these atomic particles, namely electrons.
At finite temperatures, conditions exist in which electrons are in a thermally excited state. These conditions are sustained by internal energy and thus, are directly associated with temperature. In these thermally excited states electrons emit energy in the form of quanta or photons, resulting in the propagation of electro-magnetic waves. Accordingly, the emission of electromagnetic waves from these thermally excited electrons make up the thermal radiation portion of the electromagnetic spectrum, which occurs between 0.1μm and 100μm. Thermal radiation thus encompasses the near UV, and the entire VIS and IR portion of the electromagnetic spectrum (Figure 1).
Thermal radiation propagates from a radiating surface in all possible directions and is emitted over a range of wavelengths. Thus, the magnitude of emitted radiation must be described such that it is defined both by its wavelength and its direction — its spectral and its directional components — respectively. As a result, radiative terms are used to treat these components. Spectral intensity Iλ,e is defined as the energy flux, at a specific wavelength λ, per unit area, in a direction normal to that area, per unit solid angle about that direction, per unit wavelength (Figure 2). It has units of (W/m2*sr*μm) and is given by the equation:
Alternatively, spectral emissive power Eλ, is defined as energy flux in all direction, at a wavelength λ, per unit area, per unit wavelength; and is in units of (W/m2*μm) and is given by the equation:
Eλ is the energy flux based on the surface area of the actual radiating surface while Iλ,e is the energy flux through a projected area. It is also necessary to account for incident radiation on a surface due to emission and reflection of radiation from other surfaces. The incident radiation from all surfaces is defined as the irradiation (Γλ); Γλ is the energy flux at a wavelength λ that is incident on a surface, per unit area, per unit wavelength. It has units of (W/m2*μm) and is given by the equation:
Finally, the radiosity (Ρ) can be introduced. The radiosity is similar in definition to the emissive power, but also accounts for irradiation. As a result, Ρ is associated with the radiant energy from both direct emission and reflection, and Ρλ is thus defined as the radiant energy, at a wavelength λ, (in all directions), per unit area, per unit wavelength. It has units of (W/m2*μm) and is given by the equation:
(Note, “e+r” refers to the total intensity due to emission and reflection).
In order to connect these terms and their associations with real surfaces, they must relate to something of theoretical measure. This relation, of course, is the concept of a blackbody. A blackbody is a theoretical object that is both the perfect emitter and absorber of radiation; it is an ideal surface. The characteristics of a blackbody are as follows:
- Absorbs all incident radiation independent of wavelength and direction.
- For a given wavelength and finite, nonzero temperature, no object can emit more energy at the same temperature.
- It is a diffuse emitter.
The Planck Distribution approximates a blackbody where the spectral intensity of a blackbody, at a given temperature, is given by the equation:
The spectral emissive power is given by the equation:
Wien’s displacement law prescribes a peak wavelength to a given temperature, and is given by the equation:
Where C3 = 2897.8 μm.K
For example, the sun, which can be approximated as a blackbody at 5800 K, has a max spectral distribution at about .5 μm using Wien’s displacement law (Figure 3) This peak is in the visible spectrum. Alternatively, a blackbody at 1450 K, would have a max spectral distribution at about 2.0 μm; corresponding to SWIR portion of the electromagnetic spectrum.
The total emissive power of a blackbody may be found using the Stefan-Boltzmann law, which expresses the total emissive power of a blackbody, as:
Where C1 = 3.742x108 W.μm4/m2 and C2 = 1.439x104 μm.K
A blackbody is both the perfect absorber and emitter of radiation. Any real body can never emit or absorb more energy than a blackbody at the same temperature. However, it is convenient to analyze real surfaces in reference to blackbodies. Thus, any real radiating surface can be described with the dimensionless parameter, ε, known as the emissivity; which may be defined as the ratio of the radiation emitted by a real surface to that radiated by a blackbody at the same temperature. The total emissivity, emissivity averaged over all wavelengths and in a hemispherical direction, is given by the total emissive power of the real surface at a given temperature over the total emissive power of a blackbody at the same temperature:
It is, however, important to realize that spectral radiation by a real surface differs from the Planck distribution and additionally, is not necessarily diffuse. For instance, a real surface may have a preferential distribution of radiation in certain directions or wavelengths. Therefore, analogous wavelength dependent and directional dependent emissivities are considered. The spectral directional emissivity is, thus, the ratio of the intensity of the energy radiated by a surface at a wavelength λ in the direction θ, φover the intensity of the energy radiated by a blackbody at the same temperature and wavelength:
Hence, the total directional emissivity is defined as the ratio of the spectral average of the intensity of the radiation emitted by a surface in the direction θ, φ over the intensity of the radiation emitted by a blackbody at the same temperature, and is given by:
Conversely, the spectral hemispherical emissivity is defined as the ratio of the radiation emitted by a surface at a particular wavelength λ in a hemispherical direction, over the radiation of a blackbody at the same temperature and wavelength. It is given by Equation 12.
In the commercial market, there are blackbodies available in a number of configurations. Technically, all commercial blackbodies are “graybodies” since their emissivity is less than one. In industry, this distinction is rarely applied and these products are referred to generically as blackbodies.
The geometry of a blackbody can produce higher emissivity but at a cost. In the preliminary discussion, the equations apply to a surface at a specific temperature. The most common industrial blackbody geometries are cavities and plates.
Radiation emitted from a source plate follows the equations to the first order. The reflectance and absorbance of source plates are derived from their surface treatments. The coating applied to the surface enhances the emissivity. Unfortunately, the surface coating behavior is not always ideal.
The data shown in Figure 4 shows that the emissivity of the coating on a surface is not uniform with respect to wavelength and temperature. In this particular example, the emissivity performance in the MWIR is considerably better than in the LWIR. Also, the emissivity becomes more ideal as the sample is heated. Knowledge of the coating performance properties can allow the user to calculate the expected output from a source plate for a given wavelength range. Using a single emissivity value will lead to errors in expected radiance.
There exists surface treatments that improve the emissivity of the source plate for a given coating (Figure 5).
A radiometric calibration of the source plate can be done by comparing the output of the plate to a known primary standard. However, this technique introduces additional sources of error. The calibration device must be sensitive in the same wavelength range as the UUT that will be used with the blackbody. Otherwise, for instance, if the radiometric calibration was done with a detector active in the 8-12 μm range but the UUT is sensitive in the 3-5 μm range, the data will be skewed.
The other major blackbody geometry is cavities. Cavities essentially force the majority of the photons emitted from the surface to bounce off other surfaces within the cavity before the photon finds its way out of the cavity. This randomization improves the uniformity, but at the same time creates a directed beam of photons instead of full Lambertian output. Common cavity shapes are cylindrical, conical, and spherical. The emissivity of a cavity blackbody depends upon both its geometry and surface treatment.
The radiant output of a 1 inch (25.4 mm) diameter cavity diverges in a cone of 11°; 0.2 steradian, instead of the 2π steradian from a flat plate. Typically, the cavity is used to uniformly illuminate a target next to the cavity exit. This target is usually positioned at the focal point of an optical collimator to project an image of the target at a specific temperature. The other way the target is used is to focus the UUT on the target.
As in the case of the source plates, the emissivity of the surface or surface coating changes with spectrum and temperature. In this case, as long as the emissivity of the surface is nominally the same, it is a small effect compared to the geometry. So the effects due to the properties of the spectral variation of surface emissivity do not have to be calculated beyond the first order to determine the emissivity of a cavity.
There are two ways to calibrate cavity blackbodies. One can measure the temperature of the surface with a contact thermocouple. Using this method the output of the cavity can be calculated using Planck’s equation. Using a radiometric method introduces other potential sources of error. If the detector used for radiometric calibration of the blackbody does not have the same optical properties as the UUT (FOV, spectral sensitivity, etc.), this could produce a calibration that is inferior to the thermometric calibration.
With the knowledge of the emissivity value one may calculate its radiometric equivalent based on the thermometric calibration, or one may perform a radiometric calibration. The thermometric calibration can be NIST traceable to 0.01°C for 0-100°C and 0.28°C for values up to 1400°C. Radiometric calibrations tend to have lower accuracy in the lower temperature range. Radiometric calibrations are affected by detector wavelength response and the coating spectral emissivity leading to potential misrepresentation of the expected photons for the UUT if the spectral dependencies of the UUT and calibration instrument do not match. A similar argument applies to mismatching the FOV of the UUT and the calibration instrument. Making a transfer measurement against a standard to claim effective 0.99 emissivity for a blackbody with true emissivity of significantly less is error prone due to spectral and FOV differences.
In summary, choosing a blackbody that is designed for high emissivity will provide superior results. The closer the emissivity of the blackbody is to 1.0 by design, the less spectral differences will affect the UUT’s readings. The closer the design in emissivity is to 1.0 the more reliable the prediction of the output is by thermometric calibration. Even with lower emissivity values (i.e. ε = 0.9), a thermometric calibration is able to predict output with precision as a function of wavelength.