[9.3.3 — The Stefan-Boltzmann law]
© FLIR Systems AB – Publ. No. 557 369 – Ed. A
55
9.3.3 The Stefan-Boltzmann law
By integrating Planck’s formula from
λ = 0 to λ = ∞, we obtain the total radiant
emittance (W
b
) of a blackbody:
where
σ = the Stefan-Boltzmann constant = 5.7 x 10
-8
Watt/m
2
.
This is the Stefan-Boltzmann formula, which states that the total emissive power of
a blackbody is proportional to the fourth power of its absolute temperature.
Graphically, W
b
represents the area under the Planck curve for a particular tem-
perature. It can be shown that the radiant emittance 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.
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 compen-
sating absorption of radiation 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.
9.3.4 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 behaviour in cer-
tain spectral intervals. For example, white paint appears perfectly »white« in the
visible light spectrum, but becomes distinctly »grey« 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 reflected, 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 spec-
tral 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 object to that incident upon it.
W
b
σT
4
Wattm
2
⁄[]=
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