Page 396 - Advanced thermodynamics for engineers

386 CHAPTER 17 GAS TURBINES

Thus the cycle efficiency,
w net w 34 þ w 12 ðT 3 T 4 Þ ðT 2 T 1 Þ
h ¼ ¼ ¼ (17.4)
q 23 q 23 ðT 3 T 2 Þ
Assuming that the processes 1–2 and 3–4 in Fig. 17.2 are isentropic then the temperatures can be
related to the pressure ratios. Let
p 2 p 3
r p ¼ ¼ (17.5)
p 1 p 4
Then, for isentropic processes,
k 1 k 1
p 2 k k
T 2 ¼ T 1 ¼ T 1 r p (17.6)
p 1
and
k 1
p 4 k T 3
T 4 ¼ T 3 ¼ k 1 (17.7)
p 3
k
r p
Substituting in Eqn (17.4) gives
1
h ¼ 1 (17.8)
k 1
k
r p
The work ratio of the cycle, which defines the sensitivity of the engine to irreversibilities,is
w net w 34 þ w 12 ðT 3 T 4 Þ ðT 2 T 1 Þ
r w ¼ ¼ ¼ (17.9)
w 34 w 34 T 3 T 4
which can be written as
k 1
T 2 T 1 k
r w ¼ 1 ¼ 1 r p : (17.10)
T 3 T 3
Equation (17.8) shows that the efficiency of a gas turbine is dependent on the ratio of maximum and
minimum pressures, whereas in the reciprocating engine it is dependent on the ratio of maximum and
minimum volumes. Equation (17.10) shows that the work ratio, which is a measure of the sensitivity of
the engine to irreversibilities, is a function of both the pressure ratio and the temperature ratio. To get
maximum values of efficiency and work ratio it is necessary to operate the turbine at the highest values
of r p and T 3 and the minimum value of T 1 . Normally T 1 is limited by atmospheric conditions, say
around 300 K at sea level. T 3 will be limited by the metallurgical properties of the blade materials:
attempts to increase T 3 are brought about by film cooling of the blades of the turbine or coating them
with ceramic materials.
Note that it is possible to define the maximum pressure ratio that can be used by relating this to the
temperature ratio. In this case
k
T 3 k 1
b r p ¼ (17.11)
T 1