Theory of thermography
20
20.3.3 Stefan-Boltzmann's law
By integrating Planck’s formula from λ = 0 to λ = ∞, we obtain the total radiant emittance
(W
b
) of a blackbody:
This is the Stefan-Boltzmann formula (after Josef Stefan, 1835–1893, and Ludwig Boltz-
mann, 1844–1906), which states that the total emissive power of a blackbody is propor-
tional to the fourth power of its absolute temperature. Graphically, W
b
represents the area
below the Planck curve for a particular temperature. It can be shown that the radiant emit-
tance in the interval λ = 0 to λ
max
is only 25% of the total, which represents about the
amount of the sun’s radiation which lies inside the visible light spectrum.
Figure 20.7 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 ra-
diation from surrounding surfaces, at room temperatures which do not vary too drastically
from the temperature of the body – or, of course, the addition of clothing.
20.3.4 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 ob-
ject 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:
#T559770; r. AF/30726/30726; en-US
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