4.5 Loss evaluation for the axial turbine
4.5.1 Profile loss (Y p )
Profile (or skin friction) loss is the loss which arises from the growth of the blade surface boundary layers and the attendant surface friction effects. It depends on the blade area in contact with the fluid, surface finish, Reynolds and Mach flow number in the passage and geometry of the airfoil. The profile loss coefficient suggested by Aungier is a modified form of the well-established Ainley-Mathieson [68] model. It can be represented as
Yp = kmodkinckpkRekM (
(Yp1+ ξ2(
Yp2− Yp1
))
(5tmax c
)ξ
− ∆Yte
)
(4.81)
The terms Yp1and Yp2are profile loss coefficients for a stator blade with an axial approach flow and for an impulse blade, respectively. They give the profile loss for a nominal blade thickness (t/c = 0.2), zero incidence, low Mach number (M ≤ 0.6) and a nominal Reynolds number in terms of the pitch-to-chord ratio and exit angle (Re = 2× 105). It is apparent from Yp1 and Yp2 plots (see [68]) that (i) the profile loss is larger for impulse
blades compared to stator blade and (ii) the trend of profile loss versus pitch-to-chord ratio exhibits a minimum which is not too far from the optimum pitch-to-chord ratio according to well-known Zweifel criterion [78]. Item (i) and the increase of the losses with the pitch-to-chord ratio is mainly a consequence of the higher surface area in impulse blades and in high pitch-to-chord ratio blades (i.e., higher deviations or higher chords means higher area) whereas for small chord blades the beneficial effect due to area reduction is compensated by the larger suction surface diffusion losses, resulting in a rise of the overall loss.
Basic terms Yp1 and Yp2 are combined together to get the loss of the reaction blade according to interpolation scheme
Yp1+ ξ2(Yp2− Yp1) (4.82) where
ξ = 90− β1
90− α2
(4.83) The AM model includes in the profile loss term the trailing edge loss assuming t2 = 0.02s, where t2 and s are the trailing edge thickness and blade pitch respectively. Since in Aungier model the trailing edge loss is evaluated separately, the term ∆Yte subtracts this loss accounted with Eq.
(4.98) for t2 = 0.02s. The factor involving tmax/c (tmax is the maximum blade thickness) takes into account the effect of tmax/c values different from the reference one (i.e., 0.2): the higher the ratio, the higher the loss.
Factors ki in Eq. (4.81) are corrections to the basic profile loss for Mach number, Reynolds number etc. different from those on which test data were performed and for off-design incidence angles. In particular:
• kmod is an experience factor suggested by Kacker and Okapuu [70] to account for superior performance of more designs compared to those considered by Ainley and Mathieson [68] in the Fifties. A value of
4.5. LOSS EVALUATION FOR THE AXIAL TURBINE 135
0.67 is suggested by the authors. However, it would be reasonable to use a lower coefficient for contemporary turbines.
• kinc is a correction for off-design incidence effects which depends on the ratio between the actual incidence and the stalling incidence:
the higher the absolute value of this ratio, the higher the corrective factor. The stalling incidence is a function of the exit flow angle (α2), factor ξ (Eq. (4.83)) and pitch-to-chord ratio. However, as in this work only design operation is considered kinc does not influence the expected performance.
• kp is a correction for compressibility effects on channel flow acceler-ation. A consequence of the compressibility of the working fluid is that the flow in the passage between two adjacent blades is subjected to a larger acceleration when the outlet Mach number is increased.
Accordingly, the boundary layer becomes thinner and the risk of flow separation reduces. Since the original Ainley and Mathieson [68] loss model is based on low speed cascade test (M2 ≤ 0.2), the estimated performance might be too pessimistic when high expan-sion ratio turbines are analyzed. Kacker and Okapuu [70] suggest an expression of kp which depends on inlet-to-outlet Mach number ratio (X = M1/M2) and on the outlet Mach number (M2): losses decreases (i.e., kp decreases) as both these parameters increase. Aungier [45]
uses the same model structure but replaces M1, M2and X = M1/M2 with
M˜1 = (M1+ 0.566− |0.566 − M1|)/2 (4.84)
M˜2 = (M2+ 1− |M2− 1|)/2 (4.85)
X = 2 ˜˜ M1/( ˜M1+ ˜M2+ ˜M2− ˜M1 ) (4.86) This modification to Kacker and Okapuu [70] model imposes an up-per limit to these variables (i.e., M1 ≤ 0.566, M2 ≤ 1 and X ≤ 1) to avoid very low unrealistic kp values in severe operating conditions, while maintaining quite similar results in the remaining cases. The correction factor for compressibility effects is
kp = 1− (1 − k1) ˜X2 (4.87)
where
k1 = 1− 0.625( ˜M2− 0.2 + ˜M2− 0.2 ) (4.88)
• kReis used to account for Reynolds number effects when the Reynolds numbers are different from those on which the experimental cascade tests are based. Reynolds number correction is based on friction models which estimate the variation of the skin friction coefficient with the blade chord Reynolds number Recevaluated at the discharge flow conditions.
Three flow regimes are identified: laminar (Rec ≤ 105), transition (105 ≤ Rec≤ 5 × 105) and turbulent (Rec≥ 5 × 105).
In the transition regime it is assumed kRe = 1 because friction models cannot estimate the complex variations of the skin friction coefficient and, in turn, kRe without a detailed knowledge of the blade shape.
Furthermore, experimental cascade tests refer to Reynolds number levels (i.e., Rec = 200000) which lie in this transitional regime.
In the laminar region kReis higher than unit to account for a general thickening of the laminar boundary layers and a gradual increase of separated laminar flow regions. It is based on the laminar skin
4.5. LOSS EVALUATION FOR THE AXIAL TURBINE 137
friction model
kRe=
(1× 105 Rec
)0.5
(4.89) In the turbulent region the evaluation of kRe is less straightforward because the effect of surface roughness e has to be considered. To this end, a critical blade Reynolds number Rer is introduced which is defined as the ratio between chord and surface roughness multiplied by a factor of 100
Rer = 100c
e (4.90)
If Rec≤ Rer the surface roughness is not significant and the correc-tion factor is dependent only on Rec
kRe=
(log10(5× 105) log10Rec
)2.58
(4.91)
If Rec≥ Rer and Rer ≥ 5 × 105 kRe=
(log10(5× 105) log10Rer
)2.58
(4.92)
whereas, if Rec≥ Rer and Rer ≤ 5 × 105 kRe= 1 +
((log10(5× 105) log10Rer
)2.58
− 1 ) (
1−5× 105 Rec
)
(4.93)
This model is quite similar to that of Kacker and Okapuu [70], al-though the latter considers slightly different intervals for the flow regimes, suggests a more severe correction at low Reynolds number and especially does not consider the influence of the surface roughness in the turbulent flow. Reynolds correction of Kacker and Okapuu [70]
model are
kRe=
( Rec 2× 105
)−0.4
Re≤ 2 × 105 (4.94)
kRe = 1 2 < ×105Re≤ 106 (4.95)
kRe= (Rec
106 )−0.2
Re > 106 (4.96) The AMDC model simply suggests to use the multiplicative factor (Re/2× 105)−0.2 to penalize small turbines where low Reynolds flow regime could occur and disregards the influence of the surface rough-ness. Instead, the method of Craig and Cox [72] is one of the few which considers the effects of surface roughness on the Reynolds cor-rection.
Note that, according to Aungier model the influence of Reynolds number on kRe for very high Reynolds number (i.e., Rec ≥ 107) is weaker than in KO and AMDC models but in accordance with Craig and Cox findings. Thus, Reynolds number correction which disregard the influence of surface roughness might lead to unreliable results, especially when dealing with high-pressure ratio turbines and when dealing with a variety of sizes.
• kM is the Mach number correction as originally proposed by the AM model and it depends on the ratio of the pitch to the suction surface radius of curvature s/Rc and the discharge Mach number M2: the higher s/Rc and M2, the higher kM, i.e., the higher the loss
kM = 1 + (1.65(M2− 0.6) + 240(M2− 0.6)4)(s/Rc)3M2−0.6 (4.97) Note that the Mach number range is limited to 0.6≤ M2 ≤ 1 when applying Eq. (4.97).