Page 81 - Advanced Gas Turbine Cycles

Chapter 4. Cycle eficiency with turbine cooling (cooling flow rates specified) 57

than the efficiency of the uncooled turbine (q)121 at the same T3 (point B), as given in
at
Eq. (4.24). But it is the same as the efficiency of the uncooled turbine (q)~ point C, at a
maximum temperature T5 (the rotor inlet temperature of the cooled turbine). Here the
analysis of Section 4.2.2.1, for a/s cycles with constant specific heats, is developed further,
to find the slopes of the curves (aq/a/ae), at all the three points A, B and C; the slopes are
then used to determine the relations between the expressions for (q)~cl and (q)1~.
An approximate relation for the cooling fraction $ obtained by El-Masri [lo], and
derived in Appendix A, is also used,
$= KIT3 - Tbll/[Tbl - T217 (4.25)
where Tbl is the allowable (constant) blade temperature, T2 is the compressor delivery
(coolant) temperature and K is a constant (approximately 0.05). Differentiation of
Eq. (4.24) at constant x (and T2), and using Eq. (4.25), yields

Xi)~{ - [(p - XYV~IM~ - x12,
ra(~)lcl/ae]x = EW(X - w(p - X? + E ~ - 1
(4.26)
where T = (0, - &), with = Tbl/TI assumed constant. The first term on the right hand
side gives the rate of increase of thermal efficiency in the absence of cooling, [a(q),,/a 01,.
After some algebra [5], it follows that
(4.27)
[a(7))ICl/a e], )A = ( - K, { [a(q)lU/a e], B *
So in Fig. 4.6a, the slope of the curve at A is (1 - K) times the slope of the (q)rU
curve at B. (q)Icl thus increases with T3 at a smaller rate than (7)~. (4.27) may then be
Eq.
integrated to (non-dimensional) temperature 6, from @.,I where = 0, and the uncooled
and cooled efficiencies are the same, [(q)IU]bl = [(q)I(-llbl = (&I.
Thus
(q)ICl - (q)b1 = (1 - K)[(r))lU - (q)bll, (4.28)
or

(q)ICI = (1 - K)(q)lU + K(?l)hl- (4.29)
as illustrated in Fig. 4.6b. Hence
(4.30)
= (v)lU - (rl)ICl = K[(q)lU - (q)bl]*
An alternative approach is shown in Fig. 4.612. The cooled efficiency (q)lcl may be
presented as a unique function of the rotor inlet temperature (T5), for a given x and
component efficiencies. But from the El-Masri expression, Q. (4.25), it can be deduced
that the cooling air quantity $is a function of the combustion temperature T3, for a given x
(and T2) and a selected blade temperature Tbl, so that from the steady flow energy equation
for the mixing process, there is a value of T3 corresponding to the rotor inlet temperature
Ts. Analytically, we may therefore state that ( q)IcI = f(Ts) and ( ~ 7 ) ~ ~f(T3), but taking
=
note that Ts =f(T3). Thus, uncooled and cooled efficiencies may be plotted against two
horizontal scales, T3 and Ts, as indicated in Fig. 4.6c, which results in a single line. This
point is further discussed in Section 4.4.