Page 88 - High Power Laser Handbook

58 G a s , C h e m i c a l , a n d F r e e - E l e c t r o n L a s e r s Chemical Lasers 59

When considering the evolution of kinetic processes, one simply
uses the velocity U to relate position and time, using dx = Udt. At high
velocities, where compressibility of the gas becomes significant, the
flow behavior becomes complicated. This regime is usually defined to
occur when the Mach number, M = U/c, becomes greater than ~0.3. For
the case of a nonreacting flow with neither friction nor heat addition
(isentropic), the flow is characterized by its stagnation properties,
which correspond to flow conditions after the flow is isentropically
brought to rest, given by:

2
T /T = 1 + 0.5 (γ – 1)M
0
2 γ/(γ−1)
P /P = [1 + 0.5 (γ – 1)M ] (3.18)
0
2 1/(γ−1)
ρ 0 /ρ = [1 + 0.5 (γ – 1)M ]
where P, T, and ρ are the static properties and P , T , and ρ are the
0
0
0
stagnation properties.
Gas flows that travel isentropically through a duct with variable
cross section A satisfy Eq. (3.19):
dU/U = (dA/A)/(M – 1) (3.19)
2
This expression illustrates the principle of operation behind the con-
verging-diverging nozzle that is widely used in laser applications. In
the converging section, the flow accelerates until it reaches the mini-
mum area throat location, where the flow reaches M = 1. It then con-
tinues to accelerate beyond the throat in the expanding region, where
M continues to increase to supersonic values, resulting in much lower
pressure, static temperature, and density.
In parallel, one also flows the secondary flow of hydrogen that
reacts with the fluorine atoms to produce the vibrationally excited HF
and the associated heat of reaction. The addition of heat tends to
drive the flow toward Mach 1 conditions, or the so-called thermal
choking case. Avoiding this condition is a major concern in chemical
laser designs. Thermal choking of supersonic flows leads to a variety
of unfavorable behaviors, such as reduced velocity, increased density
and pressure, higher temperatures, large optical path difference
(OPD) effects associated with density variations, and feedback of
flow behavior into upstream flow regions. To avoid thermal choking,
an inert, diluent gas, such as helium or, more infrequently nitrogen, is
used to increase the flow mixture’s heat capacity, thus minimizing the
effects of heat release. Alternatively, one can mitigate heat release
through area expansion; however, this increases vacuum pumping
demands. Figures 3.10 to 3.12 show the Mach number, temperature,
and pressure dependence of the gas mixture as a function of position
in a typical laser cavity with and without the addition of heat due to
the secondary flow.