where a is the cone halfangle. A 15° halfangle yields a
geometric nozzle efficiency equal to 0.983 and is typically
twothirds the length of an ideal nozzle. For lowarearatio
nozzles, where simple fabrication methods are desired, 15°
cones have become an accepted standard. The length of any nozzle
type is commonly referenced to the length of a 15° cone having
the same nozzle area ratio.
It had long been assumed that for a given
nozzle area ratio and length, there existed a unique nozzle
contour that would yield the maximum geometric nozzle efficiency
or maximum thrust. In the late 1950s, Dr. G.V.R. Rao derived a
method for analytically defining this unique contour. His method
is widely accepted by the propulsion industry, and any nozzle
contour designed for maximum thrust for a given nozzle area
ratio and length is referred to as a Rao optimum contour. By
shaping the nozzle wall according to Rao's method, a shorter
nozzle and an improvement of over one percent in nozzle
efficiency can be obtained relative to a 15° cone.
Nozzle contours can also be designed for
reasons other than for maximum thrust. For example, contours can
be tailored to yield certain desired pressures or pressure
gradients to minimize flow separation concerns at sea level.
Rocketdyne has a parabolic curvefit program, generally used to
approximate Rao optimum contours, which can also be used to
generate desired nozzle wall pressures.
Early booster engines typically incorporated
conical nozzles to simplify fabrication. Since booster engines
perform only at low altitude and are then jettisoned, peak
nozzle efficiency has less of an impact on the total mission.
The SSME, however, is used from sea level to orbit insertion.
Its nominal thrust time is eight minutes for each mission, and
efficient use of the propellants is a prime consideration.
The SSME nozzle is 10.3 inches in diameter at
the throat, increasing to 90.7 inches at the nozzle exit over a
length of 121 inches. At 100 percent power level, propellants
flow through the nozzle at a rate of 1,035 pounds per second.
The nozzle accelerates the combustion products to 17,000 feet
per second at the nozzle exit, generating 470,000 pounds of
thrust at vacuum. Because the last one percent of SSME thrust at
a fixed mass flow rate translates to about 5,000 pounds of
shuttle payload, high priority was placed on nozzle design and
performance.
The SSME nozzle configuration was the result
of a number of design iterations. Various system studies and
mission optimizations showed that high nozzle area ratio was
critical and the nozzle was configured with an area ratio of
77.5:1, and a length equal to 80 percent of a 15° conical
nozzle. The task left to the nozzle designers was to specify the
shape of the nozzle contour from the throat to the nozzle end
point which was dictated by the area ratio and nozzle length.
The first choice for an SSME nozzle contour
would obviously be one that maximized nozzle thrust; that is, a
Rao optimum contour. However, if a Rao optimum contour was used,
the wall pressure at the nozzle exit, (pw(exit)) would be much
lower than the ambient pressure at sea level. Even at 100
percent power levels, corresponding to a chamber pressure equal
to 3,000 psia, (pw(exit)) would be 4.6 psia or 31 percent of the
ambient pressure at sea level. Past experience showed that
nozzle flow separation would likely occur if the wall pressure
approached this level. Since nozzle flow separation is dependent
upon a number of variables (boundarylayer thickness, pressure
gradient, Mach number, etc.) and is thus difficult to accurately
predict, additional margin in exit pressure was sought. Some
margin was also required to permit sealevel testing of engine
throttling capability.
A Rao design resulted in a wall angle of
7.5° at the nozzle exit. By reducing this angle, additional
flow turning is produced, and then, an increase in nozzle wall
pressure is created. A study was performed by Rocketdyne
engineers in which a large number of parabolicshaped contours,
with a variety of different initial wall angles (qmax) and exit
wall angles (qe), were analyzed. After careful analysis of these
contours, it was determined that a parabolic contour with qmax=37°
and qe=5.3° would produce the desired wall pressure increase
with the least amount of performance loss. The wall exit
pressure was raised 24 percent (from 4.6 psia to 5.7 psia) at a
cost of only 0.1 percent in nozzle efficiency. Validation of the
design approach was provided by subsequent testing of the SSME
which demonstrated that the engine can be throttled to below 80
percent power level at sea level without nozzle flow separation.
Since more of the SSME operation is at high
rather than low altitude, vacuum performance is the overriding
factor relating to mission performance and high nozzle area
ratio is therefore desirable. However, nozzle overexpansion at
sea level does result in a thrust loss because the wall pressure
near the nozzle exit is below ambient pressure. If the nozzle
exit area could somehow be reduced for launch and then gradually
increased during ascent, overall mission performance would be
improved. The ideal rocket engine would make use of a
continuously changing "rubber" or variablegeometry
nozzle that adjusted contour, area ratio and length to match the
varying altitude conditions encountered during ascent. This
feature is referred to as altitude compensation.
For singlestagetoorbit (SSTO)
applications, where performance margins are even more stringent
than for the SSME, some form of altitude compensation in the
nozzle is a must. An SSTO vehicle relies on a single propulsion
system that operates from sea level to orbit. The aerospike
engine, built and tested by Rocketdyne in the 1960s, is
currently being evaluated for potential use with an SSTO vehicle
because of its builtin altitude compensation features and the
beneficial manner in which it "packages" or integrates
with the vehicle.
In the annular aerospike nozzle, flow issues
from an annulus at a diameter located some radial distance from
the nozzle axis. Flow is directed radially inward toward the
nozzle axis. This concept is the opposite of a bell nozzle which
expands the flow away from the axis along diverging nozzle
walls. In an aerospike, the nozzle expansion process originates
at a point on the outer edge of the annulus which is referred to
as the "cowllip." Because this point is also exposed
to ambient pressure, the flow turning or
