Page 513 - Bird R.B. Transport phenomena

§16.3 Planck's Distribution Law, Wien's Displacement Law, and the Stefan-Boltzmann Law 493

in which e must be evaluated at temperature T. The use of Eqs. 16.2-10 and 11 to calcu-
late radiant heat transfer rates between heated surfaces is discussed in §§16.4 and 5.
We have mentioned that the Stefan-Boltzmann constant has been experimentally
determined. This implies that we have a true black body at our disposal. Solids with
perfectly black surfaces do not exist. However, we can get an excellent approximation
to a black surface by piercing a very small hole in the wall of an isothermal cavity. The
hole itself is then very nearly a black surface. The extent to which this is a good ap-
proximation may be seen from the following relation, which gives the effective emis-
sivity of the hole, e ho[e , in a rough-walled enclosure in terms of the actual emissivity e
of the cavity walls and the fraction / of the total internal cavity area that is cut away
by the hole:

( 1 6 2 4 2 )
-
If e = 0.8 and / = 0.001, then е = 0.99975. Therefore, 99.975% of the radiation that falls
Ше
on the hole will be absorbed. The radiation that emerges from the hole will then be very
nearly black-body radiation.

§163 PLANCK'S DISTRIBUTION LAW, WIEN'S DISPLACEMENT
LAW, AND THE STEFAN-BOLTZMANN LAW ' '
1 2 3
The Stefan-Boltzmann law may be deduced from thermodynamics, provided that cer-
tain results of the theory of electromagnetic fields are known. Specifically, it can be
shown that for cavity radiation the energy density (that is, the energy per unit volume)
within the cavity is

^
u i r ) = q ( e ) (16.3-1)
b
Since the radiant energy emitted by a black body depends on temperature alone, the
energy density u {r) must also be a function of temperature only. It can further be
shown that the electromagnetic radiation exerts a pressure p on the walls of the cav-
(r)
ity given by
p (r) = \u {r) (16.3-2)
The preceding results for cavity radiation can also be obtained by considering the cavity
to be filled with a gas made up of photons, each endowed with an energy hv and mo-
mentum hv/c. We now apply the thermodynamic formula

->

to the photon gas or radiation in the cavity. Insertion of U 00 = Vu {r) and p {r) = \u {r) into
this relation gives the following ordinary differential equation for u (T):
(r)
(r) i du^__yr) _
u = T ( 1 6 3 4 )

1
J. de Boer, Chapter VII in Leerboek der Natuurkunde, 3rd edition, (R. Kronig, ed.), Scheltema and
Holkema, Amsterdam (1951).
H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd edition, Wiley, New York
2
(1985), pp. 78-79.
M. Planck, Vorlesungen iiber die Theorie der Warmestrahlung, 5th edition, Barth, Leipzig (1923); Ann.
3
Phys., 4, 553-563, 564-566 (1901).