4.7 Model resolution for axial turbine
The design and performance prediction of the axial turbine is carried out by a Matlab⃝R script developed by the author coupled with Refprop fluid properties database.
The model assumes the three requirements of a “normal” stage (named also “repeating” stage) [29]:
cm = constant d = constant α1 = α3 (4.113) where cm is the axial velocity, d is the root mean square diameter (√
2(r2h+ rt2)), α1 and α3 are the absolute flow angles at stations 1 and 3. The hypothesis of constant axial velocity is consistent with the design practice commonly accepted in multistage turbines. Since the inlet flow angle does not markedly affect the stage performance [79], the choice α1 = α3 is not a particularly restrictive constraint. So, the stage calculated by the model can be interpreted as the generic stage of maximum efficiency of a multistage axial turbine at constant axial velocity.
A schematic of the circumferential and meridional section of the blade row is drawn in Figures 4.13 and 4.14 to highlight the main geometrical parameters calculated by the design procedure (see the flowchart in Fig.
4.15).
The flow chart in Fig. 4.15 shows the model architecture, the main equations and how they are implemented and solved sequentially. A de-tailed explanation of all steps for the design of the axial turbine is listed below.
1. The values of the mass flow rate ( ˙m), the inlet state (p1, T1) and condensation pressure (p3) are known from the thermodynamic cycle analysis, and represent design specifications, so that states 1 and the corresponding isentropic one at turbine outlet 3ss are defined.
bN cN
sN oN
tN
sR cR
bR
tR γN
γR
oR
α1 V1
α2 α’2
α3 α’3 V2
V3
W2 U W3 U
Figure 4.13: Main geometrical blade parameters.
D
h1h
1b
Nh
2h
3b
RD
mFL
NtFL
NhFL
RtFL
RhD
h3D
h2D
h2h
2CL
Figure 4.14: Schematic representation of turbine stage meridional channel.
4.7. MODEL RESOLUTION FOR AXIAL TURBINE 169
Figure 4.15: Model flowchart for the axial turbine.
2. The enthalpy h3 is calculated by assuming a guess value of the total-to-total efficiency ηtt. So, the rotor outlet state is known; h2, u and cm are calculated from the design specifications reaction (R), loading coefficient (ψ) and flow coefficient (ϕ), respectively.
3. Loading coefficient ψ, flow coefficient ϕ and reaction R allow to cal-culate all absolute and relative flow angles by means of Eqs. (4.114) and (4.115):
tan α3 = 1−R−Ψ/2ϕ tan α2 = 1−R+Ψ/2ϕ
(4.114)
tan β3 =−R+Ψ/2ϕ
tan β2 =−R−Ψ/2ϕ (4.115) 4. All the velocity triangles can be obtained by means of trivial trigono-metric calculations. So, the total states 01, 03, 03R, 03ss are defined and the resulting h3 value can be used to update the calculation of h2.
5. A first guess value for the pressure p2 varying linearly with the reac-tion is chosen in order to define state 2.
6. The throat-to-pitch (o/esse) is calculated by means of empirical cor-relations as a function of the outlet flow angle and Mach number (see Aungier [45] for further details); the sweep angle (γ) is estimated as a function of the inlet blade angle and the throat-to-pitch ratio; the pitch-to-chord ratio (esse/c) is selected from the flow angles in order to minimize the basic profile loss coefficient (Eq. 4.82).
7. The passage area A1 at the inlet of the stator is calculated from the mass flow rate definition. A set of couples “stator inlet hub-to-tip radius ratios (λ1 = r1h/r1t - see Fig. 4.14) - nozzle axial
4.7. MODEL RESOLUTION FOR AXIAL TURBINE 171
chord (bN)” is considered to calculate a wide spectrum of the follow-ing geometrical parameters: chord (cN), root mean square diameter (d = 2A1/π√
(1− λ21)/(1 + λ21)), blade span (hN) (from continuity equation), blade pitch (sN), number of blades (zN) and throat open-ing (oN). When necessary, converging-diverging nozzle vanes are con-sidered. Unfeasible solutions in the set (λ1 − bN) are eliminated ac-cording to the constraints summarized in Table 4.2, among which the most binding are those associated with the flaring angle (FLN < 20◦) and the hub-to-tip radius ratio at rotor outlet (λ3 ≥ 0.30). The set of the remaining stator geometries are compared on the basis of the as-sociated losses to select the best performing one: the couple (λ1−bN) yielding the lowest total nozzle pressure loss coefficient (YN) is cho-sen. The blade spans (h) along the stator and rotor blade channels are then found from the continuity equation (see Fig. 4.14);
8. At this point the stator outlet pressure p2 (previously assumed as a guess value at point 5) can be calculated from the definition of total pressure loss coefficient, being now YN known:
YN = p01− p02
p02− p2
= p01− f(h02, s2) f (h02, s2)− p2
(4.116)
where, in turn, s2 = f (h2, p2). So, state 2 is defined.
9. The calculation procedure of the rotor geometry and losses is similar but simpler than the stator one. In fact, the only free variable is bR, being λ3 (λ3 = r3h/r3t in Figure 4.14) evaluated from λ1 (that is known from the calculations in the stator). Indeed, it can be demon-strated from the continuity equation that the hub-to-tip radius ratios of two consecutive sections i and j in a normal axial turbine stage can be linked to each other by means of the volumetric expansion ratio VR:
λj =
(1 + λ2i − VR(1 − λ2i) 1 + λ2i + VR(1− λ2i)
)0.5
(4.117) For each value of bR the rotor geometrical parameters (sR, zR, oR, etc.) are calculated and, likewise for the nozzle, the rotor design yielding the lowest YR and fulfilling all the constraints is selected.
10. Starting from the definition of YR, p03R is then calculated and used to find a new value of h3, which updates the h3 calculated from the ηtt guess at step 2;
11. A new estimate of the total-to-total efficiency is calculated using the current h3 value. This estimate along with the current value of p2 is used as input for the next iteration until convergence.
4.7. MODEL RESOLUTION FOR AXIAL TURBINE 173
Table 4.2: Upper and lower bounds of the free variables in the optimization procedure.
min value max value
λ1 0.30 0.95
λ3 0.30 1
F LN (◦) -20 20
F LR (◦) -25 25
(h/d)3 0 0.25
arcsin(o/s)N (◦) 13 60 arcsin(o/s)R (◦) 13 60
(b/d)N 0 0.25
(b/d)R 0 0.25
bN ( mm) 3 100
bR( mm) 3 100
oN ( mm) 1.5 100
oR ( mm) 1.5 100
zN 10 100
zR 10 100
Table 4.3: Geometric parameters in (m) the rigorous similarity of which cannot be maintained at low actual turbine dimensions.
e 2× 10−6
δ max(0.001 or 5×10−4d) t max(0.001 or 0.05 o)