Page 223 - Advanced Gas Turbine Cycles

Appendix A. Derivation of required cooling Jows I85

where w+ is the 'temperature difference ratio' given by

and eo is the overall cooling effectiveness, defined as
80 = (Tgi - Tbl)/(Tgi - Tci).

Tgi and Tci are usually determined from and/or specified for cycle calculation so that the
cooling effectiveness .zO implicitly becomes a requirement (subject to Tbl which again can
be assumed for a 'level of technology'). If r)cool and C are amalgamated into a single
constant K, then
1
l+b = K&"/( - Eo), (A8)
for convective cooling, as used by El-Masri [3].

A.3. Film cooling

The model used by Holland and Thake [ 11 when film cooling is present is indicated in
Fig. A.lb. Cooling air at temperature Tc, is discharged into the mainstream through the
holes in the blade surface to form a cooling film. The heat transferred is now
Qnet = Asghg(Taw - Tbl) = Wccpc(Tco - Tcih 649)
where Taw is the adiabatic wall temperature and hfg is the heat transfer coefficient under
film cooling conditions. The film cooling effectiveness is defined as
EF = (Tgi - Taw>/(Tgi - Ted. ('410)

Then a new 'temperature difference ratio' W+ may be written as

w+ = (Taw - Tbl)/(Tco - Tci)

= [EO - (1 - r)cool)&F - &O&F~c0011/r)cool(l - EO). ('41 1)
It can be argued that cF should be independent of temperature boundary conditions and
in the subsequent calculations it is taken as 0.4, based on the experimental data.
It follows from Eqs. (A9) and (AlO) that

l+b = (wc/wg> = (c,g/c,)(Asgs~,/A,g>~w+, (A 12)
where p = hfg/[kg( 1 + B)] in which hf, is the heat transfer coefficient under film cooling
conditions and B = hfgt/k is the Biot number, which takes account of a thermal barrier
coating (TBC) of thickness r and conductivity k.
In practice, hfg increases above h,, and (1 + B) is increased as TBC is added. For the
purposes of cycle calculation, p is therefore taken as unity and
l+b = cw+, ('41 3)

where C is the same constant as that used for convective cooling only.