By comparing the various cooling techniques， it becomes obvious thata blade with trailing edge slots is thermodynamically the mostefficient， as shown in Figure 9-25. The porous stator blades decrease the stage efficiency considerably. This efficiency indicates losses in the turbine but does not takeinto account cooling effectiveness. As indicated earlier， the porous blades are more effective for cooling.
Turbine Losses
The primary cause of efficiency losses in an axial-flow turbine is the build-up of boundary layer on the blade and end walls. The losses associated witha boundary layer are viscouslosses， mixinglosses， and trailing edge losses.To calculate these losses， the growth of the boundary layer on a blade must be known so that the displacement thickness and momentum thickness can be computed. A typical distribution of the displacement and momentum thickness is shown in Figure 9-26. The profile loss from this type of bound-ary-layer build-up is due to a loss of stagnation pressure， which in turn is caused by a loss of momentum in the viscous fluid. The blade shape and the pressure gradient to which the flow is subjected are major factors in this typeof loss. The endwall losses are also due to a loss of momentumand， althoughthey are also dependent on the profile and pressure gradient， the profile shape and pressure gradient are considerably different. Endwall losses areoften combined with secondary losses， since adjacent blade profiles cause a pressure gradient from the pressure surface to the suction surface. The blade loading is thus produced by the different pressures on the opposite side of the same blade. The pressure gradient across the blade passage induces flow from the higher to the lower-pressure regions. This secondary flow causes losses and results in vorticity in the exit flow.

Figure 9-2.. Growth of displacement and momentum thickness on an airfoil
Table 9-2 Turbine Loss .alues in t.e Overall Stage
Loss .ec.anics Loss (%)
Profile 2-4 Endwall 1 1/2-4 Secondary flow 1-2
.otor incidence 1-3 Tip clearance 1 1/2-3 Wheel disc 1-2
Tip clearance loss occurs when the blade tip is mechanically free of theshroud casing， and the pressure gradient across the blade thickness induces flow leakage through the clearance space. This flow across the tip causesturbulence， a pressuredrop， and interferes with the main stream flow. All of these effects contribute to tip clearance loss. Another loss is caused by flowincidence when the gas angle and the blade angle of the flow do not coincide， resulting in a disruption of the flow at the blade leading edge. Disc friction loss occurs in an axial-flow compressor because of the close clearances between the casing and the rotor disc. The entrapped fluid causes a viscous power dissipation when the fluid is dragged by the rotor. Table 9-2 shows the approximate value of these losses in the overall stage.
A simple but effective technique for calculating the loss in an axial-flowturbine has been developed. In the loss computation， the blade geometry， thespacing between the blades， the aspect ratio， the thickness ratio， and the effect of the .eynolds number are taken into account.However， thosefactors not taken into account are the staggerangle， the trailing edge thick-ness， and the effects of Mach number. Neglecting Mach number effects causes a problem highly loaded stages. The optimum solidity (σ二吵/s)ofthebladesinisctheomputed from
σ二 2.5 (cot α2 + cot α1) sin2 α2 (9-18)
The loss coefficient can now be computed
 1/4
ω二 105 [(1 + ωe川R)-1]ωi

)(0.975 + 0.075/(9-19)
.e
where川R is the aspect ratio (h/吵)， ωe is the loss from blade geometry seen inFigure9-27， ωiis the loss due to the incidence angle seen in Figure9-28， and .e二 V3Dn/V3 where Dn二 (2川R s sin α2)/(σ sin α2 +川R)
The change in enthalpy is given by
h2a二 h2s + ωV32/2 (9-20)
　
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