Page 58 - Advanced Gas Turbine Cycles

Chapter 3. Basic gas turbine cycles 35

The specific heat supplied is
q = cp(T3 - T2) = CpT1K8 - 1) - (x - 1)/77cl, (3.12)
or

~
4 = cPr3 - ~ 1 = - 1) - (XC - 1)1,
C,Tw
so that the thermal efficiency is given by
7 = w/q = (a - x)[l - (l/x)]/(p - x), (3.13)

where p = 1 + vC(8 - l), or
77 = w/q = [e(l - (IIXT)) - (xc - l)l/[(8 - 1) - (xc - 1)l. (3.13P)
The important point here is that the efficiency is a function of the temperature ratio 8 as
well as the pressure ratio T (and x), whereas it is a function of pressure ratio only for the
reversible cycle, [CJ4"IR.

Optimum conditions and graphical plot
The isentropic temperature rise for maximum specific work (x,) is obtained by
differentiating Eq. (3.1 1) with respect to x and equating the differential to zero, giving
x,=a . (3.14)
li2
By differentiating Eq. (3.13) with respect to x and equating the differential to zero, it
may be shown that the isentropic temperature ratio for maximum thermal efficiency (x,) is
given by the equation

Ax: +BX, + c = 0, (3.15)
where A = (a - p - l), B = -2a, C = ap.
Solution of this equation gives

x, = ap/{a + [a(p - a)(p - l)lln}. (3.16)
In their graphical interpretations, using isentropic rather than polytropic efficiencies,
Hawthorne and Davis plotted the following non-dimensional quantities, all against the
parameter x = T("-')'" :
Non-dimensional compressor work,
NDCW = wc/cp(T3 - TI) = (x - 1y(p - 1); (3.17)
Non-dimensional turbine work,
NDTW = WT/CP(T3 - TI) = a(x - l)/X(P - 1); (3.18)

Non-dimensional net work,
NDNW = w/cp(T3 - TI) = (a[l - (l/x)] - {x - l)}/(P - 1); (3.19)
Non-dimensional heat transferred,

NDHT = q/cp(T3 - TI) = (p - x)/@ - 1); (3.20)