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Theory of thermography
34
Figure 34.7 Josef Stefan (1835–1893), and Ludwig Boltzmann (1844–1906)
Using the Stefan-Boltzmann formula to calculate the power radiated by the human body,
at a temperature of 300 K and an external surface area of approx. 2 m
2
, we obtain 1 kW.
This power loss could not be sustained if it were not for the compensating absorption of
radiation from surrounding surfaces, at room temperatures which do not vary too drasti-
cally from the temperature of the body – or, of course, the addition of clothing.
34.3.4 Non-blackbody emitters
So far, only blackbody radiators and blackbody radiation have been discussed. However,
real objects almost never comply with these laws over an extended wavelength region –
although they may approach the blackbody behavior in certain spectral intervals. For ex-
ample, a certain type of white paint may appear perfectly white in the visible light spec-
trum, but becomes distinctly gray at about 2 μm, and beyond 3 μm it is almost black.
There are three processes which can occur that prevent a real object from acting like a
blackbody: a fraction of the incident radiation α may be absorbed, a fraction ρ may be re-
flected, and a fraction τ may be transmitted. Since all of these factors are more or less
wavelength dependent, the subscript λ is used to imply the spectral dependence of their
definitions. Thus:
• The spectral absorptance α
λ
= the ratio of the spectral radiant power absorbed by an
object to that incident upon it.
• The spectral reflectance ρ
λ
= the ratio of the spectral radiant power reflected by an ob-
ject to that incident upon it.
• The spectral transmittance τ
λ
= the ratio of the spectral radiant power transmitted
through an object to that incident upon it.
The sum of these three factors must always add up to the whole at any wavelength, so
we have the relation:
For opaque materials τ
λ
= 0 and the relation simplifies to:
Another factor, called the emissivity, is required to describe the fraction ε of the radiant
emittance of a blackbody produced by an object at a specific temperature. Thus, we
have the definition:
The spectral emissivity ε
λ
= the ratio of the spectral radiant power from an object to that
from a blackbody at the same temperature and wavelength.
Expressed mathematically, this can be written as the ratio of the spectral emittance of
the object to that of a blackbody as follows:
Generally speaking, there are three types of radiation source, distinguished by the ways
in which the spectral emittance of each varies with wavelength.
• A blackbody, for which ε
λ
= ε = 1
• A graybody, for which ε
λ
= ε = constant less than 1
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