Page 526 - Bird R.B. Transport phenomena

506 Chapter 16 Energy Transport by Radiation

§16.6 RADIANT ENERGY TRANSPORT IN ABSORBING MEDIA 1
The methods given in the preceding sections are applicable only to materials that are
completely transparent or completely opaque. To describe energy transport in nontrans-
parent media, we write differential equations for the local rate of change of energy as
viewed from both the material and radiation standpoint. That is, we regard a material
medium traversed by electromagnetic radiation as two coexisting "phases": a "material
phase," consisting of all the mass in the system, and a "photon phase," consisting of the
electromagnetic radiation.
In Chapter 11 we have already given an energy balance equation for a system con-
taining no radiation. Here we extend Eq. 11.2-1 for the material phase to take into ac-
count the energy that is being interchanged with the photon phase by emission and
absorption processes:

• pUv) - (V • q) - (V • pv) - (T:VV) - {% - (16.6-1)

Here we have introduced % and si, which are the local rates of photon emission and ab-
sorption per unit volume, respectively. That is, % represents the energy lost by the mate-
rial phase resulting from the emission of photons by molecules, and su represents the
local gain of energy by the material phase resulting from photon absorption by the mole-
cules (see Fig. 16.6-1). The q in Eq. 16.6-1 is the conduction heat flux given by Fourier's
law.
For the "photon phase," we may write an equation describing the local rate of
(r)
change of radiant energy density u :
-si) (16.6-2)

(r)
in which q is the radiant energy flux. This equation may be obtained by writing a radi-
ant energy balance on an element of volume fixed in space. Note that there is no convec-

Fig. 16.6=1. Volume element over which energy
balances are made; circles represent molecules.

1
G. C. Pomraning, Radiation Hydrodynamics, Pergamon Press, New York (1973); R. Siegel and
J. R. Howell, Thermal Radiation Heat Transfer, 3rd edition, Hemisphere Publishing Co., New York (1992).