Page 87 - Advanced Gas Turbine Cycles

Chapter 4. Cycle eficiency with turbine cooling (cooling flow rates specified) 63

B (coolant air)

(To)xc = (Td2C + Q/rb.cF, @o>xc % (po)tc(1 - mzc/2). (4.46)
We can then proceed to determine the changes across control surface C. The final total
temperature (To)sm has already been obtained but the total pressure (po)*,,, has to be
determined. An expression given by Hartsel [ 141 for the mainstream total pressure loss in
this adiabatic mixing process again goes back to the simple one-dimensional momentum
analysis given by Shapiro [ 131 for the flow through control surface C illustrated in Fig. 4.8.
Hartsel developed Shapiro's table of influence coefficients to allow for a difference
between the total temperature of the injected flow (now (To)xc) and the mainstream
(To)xg):

APoIPo = 11 - @o)sm~(Po)xg)l
= -(+YM;,m( I + [(To)xc~(~o)x,)l - 2Y cos 41. (4.47)

Here y is the ratio of the velocity of the injected coolant to that of the free stream
0, = V,/V,), Mx, the Mach number of the free stream and 4 the angle at which the cooling
air enters the mainstream (Fig. 4.8).
The value of y has to be determined; an approximation suggested by Hartsel is to take
= @o)xg, so that Vc/Vg = [(T&c/(To)x,)]'n, since the static pressures must be the
same where the coolant enters. A sufficient approximation might be to take (To)xg as the
exit temperature from the combustion chamber and (To)xc as the exit temperature from
the compressor (Le. again ignoring Q in Eqs. (4.45) and (4.46)).
A more sophisticated approach would not only take account of Eqs. (4.45) and
(4.46) to give the two stagnation temperatures at exit from control surfaces A and B,
but it would also not assume the total pressures of coolant and mainstream to be the
same. For the first nozzle guide vane row these can be derived by accounting for losses
as follows:
(i) in the mainstream (g), the stagnation pressure at delivery from the compressor less
ApKc in the combustion process, and Apo in the nozzle row itself (as in control
surface A, due to friction and the heat transfer away from the mainstream gas if
included);
(ii) in the coolunr air stream (c), the stagnation pressure at extraction from the
compressor less a loss ApoD (in the ducting and disks before coolant enters the
blade itself), and Apes (in the blading heat transfer process in control surface B
due to both friction and heat transfer, if included).
The total pressures at X may thus be determined, as (po)xg and (P~)~,. If, as Hartsel
implies, the mainstream Mach number at X (Mxg) is also known, which means that the
static pressure at the mixing plane ( px) is also known, Mxc may also be determined from
(po)x,. The two different velocities V, and V, are then obtained, together with the required
value of y for Eq. (4.47).
But there is a further subtle point here in determining y, as implied by Young and
Wilcock. With [( p~)~Jp~] known, not only is the Mach number Mx, known but also the
non-dimensional mass flow, { ~R(To)xc]''2/Axc(po)xc }, may be obtained. This means that