Modern Engineering Thermodynamics

498 CHAPTER 13: Vapor and Gas Power Cycles
The optimum value of T 4s that causes the net isentropic output work to be a maximum can be found by
differentiating ð _W out Þ net isentropic with respect to T 4s and setting the result equal to zero, or
Then, solving for T 4s = (T 4s ) opt gives
dð _W out Þ net isentropic
dT 4s
= _mc p 0 + T 1 T 3 /T 2 4s − 1 + 0 = 0
p
ðT 4s Þ opt =
ffiffiffiffiffiffiffiffiffiffi
T 1 T 3
(13.24)
The corresponding optimum pressure and compression ratios are
and
PR opt =
h i k/ðk−1Þ
ðT 4s Þ opt /T 3 = ð T1 /T 3 Þ k/½2ðk−1ÞŠ (13.25)
h i 1/ðk−1Þ
CR opt = ðT 4s Þ opt /T 3 = ð T1 /T 3 Þ 1/½2ðk−1ÞŠ (13.26)
while the thermal efficiency at the maximum net isentropic work output is
ðη T Þ max work
Brayton
cold ASC
= 1 − T 3 / ðT 4s Þ opt = 1 − ðT 3 /T 1 Þ 1/2 = 1 − ðT L /T H Þ 1/2 (13.27)
HOW DID THE BRAYTON CYCLE BECOME A GAS TURBINE ENGINE?
The reciprocating piston-cylinder Brayton cycle engine, while more efficient than the Lenoir cycle engine, was at the same
time mechanically more complex and costly. Its relatively low compressor pressure ratio limited its efficiency and its ability
to compete effectively with existing reciprocating steam engine economics. These factors stifled the development of the
reciprocating Brayton cycle engine, and the cycle might have quickly become obsolete if it had not been for a new prime
mover technology being developed for steam, the turbine. By replacing steam with gas, a new type of gas-powered prime
mover, the gas turbine, was produced.
Because gas and steam turbines have many characteristics in common, several gas turbine engines were under development
at the same time steam turbines were being developed. One characteristic that they do not have in common, however, is
that gas turbine power plants require gas compressors, while vapor turbine power plants condense the working fluid to the
liquid phase before compressing it with pumps. Early liquid pumps were fairly efficient, but early gas compressors were
very inefficient due to a lack of understanding of the dynamics of high-speed compressible flow. This single fact proved to
be a major stumbling block in the development of gas turbine engine technology.
This problem is illustrated by Eq. (13.29). For a gas turbine to have a net work output, its thermal efficiency obviously has to
be a positive number. This means that both the turbine (the prime mover) and the compressor need high enough isentropic
efficiencies for Eq. (13.28) to be obeyed. One of the early major problems in compressible fluid mechanics was to understand
how to compress a gas efficiently in a rotary compressor. Turbine prime movers, on the other hand, had already undergone
considerable development within the steam power industry and were already 70 to 90% isentropically efficient.
A gas compressor is not simply a turbine running backward, and since compressible flow theory had not yet been completely
developed, gas compressor development was carried out largely by trial and error. By 1900, most compressors had
isentropic efficiencies of less than 50%, so that the product (η s ) pm (η s ) c in Eq. (13.28) was on the order of 0.4. Since typical
early gas turbines operated with very small compressor pressure ratios, say, PR = 1.5, and relatively small combustion
chamber temperatures, say, 700°F = 1150 R, for an ambient inlet temperature of 70°F = 530 R, Eq. (13.28) requires that
(η s ) pm (η s ) c ≥ (530/1160)(1.5) 0.286 = 0.513. But this was impossible for early units, because while they may have had good
turbine isentropic efficiencies of around 90%, they also had very poor compressor isentropic efficiencies of around 50% or
less. Thus, many early prototype gas turbine test engines failed to operate under their own power.
The first Brayton cycle gas turbine unit to produce a net power output (11 hp) was built in 1903. It had a very low actual
thermal efficiency (about 3%) and could not compete economically with the other prime movers of its time. Compressor
efficiency problems continued to plague gas turbine technology, and many new prototype engines were designed and built
as late as the 1930s that still could not produce a net power output. Since the thrust produced by an aircraft engine is not
considered to be part of the engine’s work output (thrust is force, not work), aircraft engines do not necessarily need high
thermal efficiencies to be effective. It was in this industry that the gas turbine engine first became successful.