676 ENGINEERING THERMODYNAMICSdharm
\M-therm\Th13-5.pm5∴ ηair-standard = 1 –
TTTrp Trp4141
11−−−−
() ()γ
γγ
γ= 1 –
1
1
()rpγ
γ− ...(13.24)The expression shows that the efficiency of the ideal joule cycle increases with the pressure
ratio. The absolute limit of pressure is determined by the limiting temperature of the material of
the turbine at the point at which this temperature is reached by the compression process alone,
no further heating of the gas in the combustion chamber would be permissible and the work of
expansion would ideally just balance the work of compression so that no excess work would be
available for external use.
Now we shall prove that the pressure ratio for maximum work is a function of the limiting
temperature ratio.
Work output during the cycle
= Heat received/cycle – Heat rejected/cycle
= mcp (T 3 – T 2 ) – mcp (T 4 – T 1 ) = mcp (T 3 – T 4 ) – mcp (T 2 – T 1 )= mcpT 3 1
4
3−
F
HGI
KJT
T – T^1T
T2
1− 1
F
HGI
KJ
In case of a given turbine the minimum temperature T 1 and the maximum temperature T 3
are prescribed, T 1 being the temperature of the atmosphere and T 3 the maximum temperature
which the metals of turbine would withstand. Consider the specific heat at constant pressure cp to
be constant. Then,Since,
T
T
r
T
p T
3
41
2
1==−
()γ
γUsing the constant ‘z’ =
γ
γ− 1
,we have, work output/cycle W = K T
r
Tr
pz p
31 F^1 −^1 z^1
HGG
I
KJJ−−
L
N
M
MO
Q
P
P
()Differentiating with respect to rpdW
drp = K^T z
rzTzr
p
p
z
311
1×
+−L
NM
MO
QP
P−
()()
= 0 for a maximum∴
zT
rpz3
()+ 1 = T 1 z(rp)
(z–1)∴ rp^2 z =
T
T3
1∴ rp = (T 3 /T 1 )1/2z i.e., rp = ()T/T 3121 ()γ
γ−
Thus the pressure ratio for maximum work is a function of the limiting temperature ratio.
Fig. 13.50 shows an arrangement of closed cycle stationary gas turbine plant in which air is
continuously circulated. This ensures that the air is not polluted by the addition of combustion
waste product, since the heating of air is carried out in the form of heat exchanger shown in the