Environmental Engineering Reference
In-Depth Information
perfectly absorbing walls when this cavity contains the blackbody radiation. The
blackbody surface emits an isotropic flux cU ω , and hence the photon flux at a fre-
quency
ω
outside the blackbody boundary is
Z d
4
cU ω
4
ω
3
2 c 2 exp(
1
j ω D
cU ω D
D
,
(1.61)
π
4
π
ω
/ T )
where the element of solid angle is d
) and we assumed that the
photon flux is directed perpendicular to the blackbody surface. From this we ob-
tain the Stefan-Boltzmann formula for the total radiative flux leaving the emitting
surface:
Θ D
d
'
d (cos
θ
Z
1
Z
1
c
4
T 4 ,
J
D
j ω d
ω D
U ω d
ω D σ
(1.62)
0
0
where the Stefan-Boltzmann constant
σ
is given by
Z
1
2
1
1
π
10 12 W
1 x 3 dx
σ D
D
D
5.67
cm 2 K 4 .
4
π
2 c 2
3
e x
60 c 2
3
0
One can evaluate the functional dependence of the radiation flux (1.62) in a sim-
ple way on the basis of the dimensional analysis. The result must depend on the
following parameters: T (the radiation temperature),
(the Planck constant), and c
(the light velocity). From these parameters one can compose only one combination
that has the dimension of a flux; it is J
T 4
3 c 2 , consistent with (1.62).
1.2.8
Ionization Equilibrium in a Plasma with Particles
Plasma properties can be influenced by the presence in the plasma of small parti-
cles on a variety of size scales, including atomic and molecular clusters. We shall
refer to all such small particles as aerosols, and the plasma containing them as
an aerosol plasma, although the size of the particles can have an influence on the
plasma properties. One such plasma property is the ionization equilibrium. The
presence of small particles in a hot gas may alter the ionization equilibrium be-
causetheenergyforanelectrontobindtoasurfaceissmallerthantheionization
potential of atoms constituting this surface. For example, the copper ionization po-
tential is 7.73 eV, whereas the copper work function, the energy for an electron to
bind to a copper surface, is 4.40 eV. The corresponding values are 7.58 and 4.3 eV
for silver, and 3.89 and 1.81 eV for cesium. Thus, the presence of such particles
in a hot vapor alters the equilibrium density of charged particles. We assume in
the following that electrons in a hot gas or vapor result from small particles only.
Our goal is to determine the equilibrium charge of these particles and the number
density of electrons in a plasma.
 
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