Page 115 - Fluid Mechanics and Thermodynamics of Turbomachinery

96 Fluid Mechanics, Thermodynamics of Turbomachinery
Stage losses and efficiency

In Chapter 2 various definitions of efficiency for complete turbomachines were
given. For a turbine stage the total-to-total efficiency is,
actual work output
tt D
ideal work output when operating to same back pressure
D .h 01 h 03 //.h 01 h 03ss/ .
At the entry and exit of a normal stage the flow conditions (absolute velocity and
flow angle) are identical, i.e. c 1 D c 3 and a 1 D a 3 . If it is assumed that c 3ss D c 3 ,
which is a reasonable approximation, the total-to-total efficiency becomes.
h 3ss /
tt D .h 1 h 3 //.h 1
D .h 1 h 3 //f.h 1 h 3 / C .h 3 h 3s / C .h 3s h 3ss /g. .4.5/
Now the slope of a constant pressure line on a Mollier diagram is .∂h/∂s/ p D T,
obtained from eqn. (2.18). Thus, for a finite change of enthaply in a constant pressure
process, h + Ts and, therefore,
s 3ss /, .4.6a/
h 3s h 3ss + T 3 .s 3s
s 2s /. .4.6b/
h 2 h 2s + T 2 .s 2
s 2s , the last two equations can be
Noting, from Figure 4.2, that s 3s s 3ss D s 2
combined to give
h 2s /. .4.7/
h 3s h 3ss D .T 3 /T 2 /.h 2
The effects of irreversibility through the stator and rotor are expressed by
the differences in static enthalpies, .h 2 h 2s / and .h 3 h 3s / respectively. Non-
dimensional enthalpy “loss” coefficients can be defined in terms of the exit kinetic
energy from each blade row. Thus, for the nozzle row,
1 2
h 2s D c N . .4.8a/
h 2
2 2
For the rotor row,
1 2
h 3 h 3s D w R . .4.8b/
3
2
Combining eqns. (4.7) and (4.8) with eqn. (4.5) gives
1
2 2
R w C N c T 3 /T 2
3
2
tt D 1 C . .4.9/
2.h 1 h 3 /
When the exit velocity is not recovered (in Chapter 2, examples of such cases are
quoted) a total-to-static efficiency for the stage is used.
h 3ss /
ts D .h 01 h 03 //.h 01
2 2 2 1
R w C N c T 3 /T 2 C c 1
2
3
D 1 C , .4.10/
2.h 1 h 3 /
where, as before, it is assumed that c 1 D c 3 .