Page 80 - Advanced Gas Turbine Cycles

56 Advanced gas turbine cycles

of gas flow + from T2 to T7. However, the work input to compress the fraction + of the
mainstream compressor flow is not now effectively cancelled by the latter expansion. The
cycle [ 1,2,3,5,6,1] is thus equivalent to a combination of two cycles: one of unit mass flow
following the original uncooled cycle state points [1,2,3,4,1] (and with the same efficiency
(v)~); another of mass flow +following the state points [1,2,2,7,1]. The second cycle
and
effectively has a negative work output and a heat supply which in the limit is zero.
Analytically, the efficiency of this combination of the two cycles may be expressed as

(V)lCl = I(T3 - T4) + rcr(T2 - T7) - (1 + *v2 - TdW, - T2)

= (V)IIJ - 4V7 - TiY(T3 - T2) = (V)IU - $4~ 1)/(B - x), (4.24)
-
where E = [l - (%vc/x) - % + (%/x)I.
Thus the efficiency of the cooled cycle is now less than that of the uncooled cycle by an
amount which is directly proportional to the cooling air used ($1. The magnitude of the
correction term to the uncooled efficiency is small for a cycle with compressors and
turbines of high isentropic efficiency. For a cooled version of the uncooled cycle
considered earlier, with + = 0.15 and x = 2.79, the second term on the right hand side of
Eq. (4.24) is 0.0200, the efficiency dropping from (T)~ 0.4442 to (q)lcl = 0.4242.
=
Thus cooling apparently has a relatively small effect on cycle efficiency, even when the
amount of cooling flow needed becomes quite large. But Eq. (4.24) indicates that for a
given + the reduction in efficiency should also decrease as the maximum temperature
increases, for a given pressure ratio.

4.2.2.2. Eficiency as a function of combustion temperature or rotor inlet temperature
(for single-step cooling)
An important point needs to be re-emphasised, that the cooled efficiency (q),cl with
'combustion' temperature (T, = Tat) is the same as the uncooled efficiency (7)m at the
'rotor inlet temperature' (T5 = T~,)
and
Fig. 4.6 shows diagrammatically both (v)~ (v)~c~ plotted against maximum
temperature (in Fig. 4.6a). The efficiency of the cooled gas turbine (V)~C, (point A) is less

rl

T
5 l3
a t
Fig. 4.6. Efficiency plots for irreversible uncooled and single-step cooled cycles (after Ref. [5]). (a) Efficiency
against maximum temperature. (b) Eficiency against nondimensional maximum temperature. (c) Efficiency
against combustion temperature (T3) and rotor inlet temperature (T5).