Modern Engineering Thermodynamics

13.23 Second Law Analysis of Vapor and Gas Power Cycles 523
where _Q fuel is the rate of heat produced by the burning fuel. For
turbojet engines, this is
ðη thrust Þ turbojet
=
_m aðV exhaust − V aircraft ÞV aircraft
_Q fuel
and, for turbofan and turboprop engines, this it
ðη thrust Þ turbofan
= _m aðV exhaust − V aircraft ÞV aircraft
_Q
or turboprop
fuel
We can now maximize the thrust efficiency by differentiating it
with respect to the aircraft’s velocityV while holding all the other
variables constant, then setting the result equal to zero to find the
optimum aircraft velocity as
V optimum =
V
exhaust
2
turbojet
= V
exhaust
2 turbofan
or turboprop
Inserting this result into the turbojet, turbofan, and turboprop
propulsion efficiency equations produces a common optimum
propulsion efficiency for all three engines of
ðη propulsion Þ optimum = 2 = 0:667 = 66:7%
3
The GE90 engine development
In early 1990, the Aircraft Engines Division of General Electric
launched the design of the GE90, a high-thrust, turbofan, gas turbine
engine (Figure 13.64) designed to meet the needs of the emerging
superjumbo wide-body passenger aircraft market. It was clear
at that time that there would be increasing air traffic occurring over
longer distances as world markets continued to evolve in the Pacific
Rim, Eastern Europe, and South America. More passengers traveling
longer distances meant more air congestion or larger planes. Also,
in the 1990s, the aging fleet of over 3000 Boeing 747s could be
replaced with new wide-body, twin-engine passenger aircraft capable
of carrying 300 passengers over 6000 miles. Such an aircraft
would have a wingspan of about 200 ft and a gross weight of about
500,000 lbf. This meant that each engine required a takeoff thrust of
about 100,000 lbf. The resulting GE90 engine specifications were
Bypass ratio ð _m aC / _m aH Þ = 10:0to1
Compressor compression ratio p 2s /p 1 = PR = 50:0to1
Engine thrust = T = 100,000 lbf
Fan diameter = D fan = 123 in: = 10:25 ft
_m a =2150 lbm/s
For a static (takeoff) thrust of 100,000 lbf, the average exit velocity
must be
V exhaust = Tg c
_m a
= ð100,000 lbfÞð32:174 lbm . ft/lbf . s 2 Þ
2150 lbm/s
= 1500 ft/s
and for a cruising speed of 500. mph = 733 ft/s, this exhaust velocity
yields a propulsion efficiency of
η propulsion =
2
1 + 1500: = 0:657 = 65:7%
733
Note that, since V aircraft ≈ V exhaust /2here,thisisveryclosetothe
optimum propulsion efficiency (66.7%) for this engine.
The GE-IA: The first U.S. turbojet engine
In January 1941, the U.S. National Academy of Sciences reported
that gas turbine engines were impractical for aircraft propulsion
because their power to weight ratio was too low. However, on
August 24, 1939, Germany flew its first turbojet-powered aircraft
(the Heinkel-178), and by June 1944, German combat jet aircraft
(Messerschmitt ME-262 Swallow, powered by two Junkers Juno turbojet
engines with an air speed of 540 mph at 20,000 ft) had
entered World War II. Also, on May 15, 1941, the British first flew
aGlosterMeteor jet aircraft powered by a single Whittle W-1X turbojet
engine as part of their war research and development program.
These events prompted the U.S. government to issue a
contract to General Electric in September 1941 to build and test 15
gas turbojet engines based on the designs of the British aircraft
engineer Frank Whittle. On March 18, 1942, the first GE type I-A
(pronounced “eye-A”) turbojet engine was completed and tested at
GE’s River Works facility in Lynn, Massachusetts. It produced about
1250lbfofstaticthrustat15,000rpm.OnOctober1,1942,the
first U.S. turbojet-powered aircraft (the Bell XP-59A Aircomet) was
flown at Muroc Dry Lake (now called Edwards Air Force Base) in
California. It had an air speed of 400 mph, 140 mph less than the
German ME-262.
The tests with the Bell XP-59A did not result in U.S. combat aircraft
during World War II, but it did play a key role in the later development
of the XP-80 Shooting Star, usedintheKoreanWarinthe
early 1950s.
If the air mass flow rate into the GE-IA engine is 18.0 lbm/s and it
produces 1,250 lbf of static thrust, then we can compute the jet
exit velocity from the relation given in the GE90 case study as
V exhaust = T × g c
_m a
= ð1250 lbfÞ½32:174 lbm . ft/ðlbf . s 2 ÞŠ
18:0 lbm/s
= 2230 ft/s
Then, at a flight speed of 400. mph = 587 ft/s, the propulsion efficiency
is
η propulsion =
1 +
2
2230 ft/s
587 ft/s
= 0:417 = 41:7%
FIGURE 13.64
Cross-section of the GE90 engine.
Case study 13.4. Model Stirling engine projects
Though the mechanism and thermodynamic cycle of a Stirling
engine are not easy to understand, you can make a working Stirling
engine at home. Numerous model Stirling engine designs are
available (for example, see Making Stirling Engines, by Andy Ross,
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