A velocity compounded impulse stage consists of an impulse turbine fol-lowed by a series of nozzle and rotor rows which progressively reduce the swirl velocity component. Compared to the impulse stage, it can efficiently manage higher enthalpy drops because it achieves its best efficiency point at a lower peripheral velocity. The optimum values of the design parame-ters for a two row arrangement are X ≈ 0.24 and ν ≈ 0.22. Although the increase of the rows number leads to a reduction in the optimum peripheral velocity, configurations with more than three rows are rarely used.
2.4. VELOCITY COMPOUNDED IMPULSE STAGE 39
2.4.1 Construction features
A velocity compounded impulse stage represents a possible solution when high available enthalpy drop would require too high peripheral ve-locity to operate efficiently. Despite this configuration includes more than one fixed and rotor row (usually two) it is commonly referred as ”stage”.
The steam passes through an impulse stage where occurs the whole pres-sure drop and then enters in a stator row composed of equiangular vanes.
They redirect the fluid towards the second rotor for a further work extrac-tion and so on.
Figure 2.6 shows the expansion in the enthalpy-entropy chart.
C12
Figure 2.6: Enthalpy-entropy diagram for a two stage velocity compounded impulse turbine.
2.4.2 Analytical expressions of work and efficiency versus design parameters
This Section is aimed at evaluating the expressions of the specific work and total-to-static efficiency for a velocity compounded stage with sym-metrical blades
β2 =−β3 α3 =−α4 β4 = β5 (2.40) and real rows, whose loss coefficient for the first rotor (ψ′), second rotor (ψ′′) and stator (ψN) are
ψ′ = W3
W3s = W3
W2 (2.41)
ψ′′ = W5
W5s = W5
W4 (2.42)
ψM = C4 C4s
= C4 C3
(2.43) The velocity triangles are sketched in Fig. 2.7.
U W2
C2
W3=ψ‘W2
U
U U
C3
C4=ψΝC
3
W4
C5
W5=ψ‘‘W4
Figure 2.7: Velocity triangles for a real velocity compounded stage. Angles with the same graphical notation are equal.
The total work W is the sum of the two rotor contributions W12 and W22 and, recalling Eq. (2.24), is
W = W12+W22 = U (C2sin α2− U)(1 + ψ′) + U (C4sin α4− U)(1 + ψ′′) (2.44)
2.4. VELOCITY COMPOUNDED IMPULSE STAGE 41
The expression C4sin α4 can be re-written in terms of loss coefficients and kinematic variables at nozzle outlet
C4sin α4 = ψR(ψ′(C2sin α2− U) − U) (2.45) Thus, Eq. (2.44) becomes
W = U (((1 + ψ′) + ψNψ′(1 + ψ′′)) C2sin α2 − ((1 + ψ′) + ψNψ′(1 + ψ′′) + (1 + ψ′′)(1 + ψN)) U ) (2.46) and including the speed parameter X = U/(C2sin α2) it turns into
W = (X((1+ψ′) + ψRψ′(1 + ψ′′))−X2((1 + ψ′) + ψRψ′(1 + ψ′′) + (1 + ψ′′) (1 + ψR)))C22sin2α2 (2.47) Since for the single stage impulse turbine, the velocity loss coefficients ψ′, ψ′′ and ψN depends on the deflection and, in turn, on the speed ratio X = U/(C2sin α2) the probable values α2 = 76◦, ψ′ = 0.86, ψR= 0.9 and ψ′′= 0.93 are assumed [32]. Accordingly, Eq. (2.47) becomes
W = 3.35X(1 − 2.1X)C22sin α22 (2.48) Consequently, the total-to-static efficiency is recast as
ηts = W
C2/(2φ2) = 6.7X(1− 2.1X)φ2sin2α2 (2.49)
2.4.3 Design guidelines
Specific work and total-to-static efficiency expression (see Section 2.4.2, Eqs. (2.48) and (2.49)) are null for X = 0 and X = 0.48 and are maximum for
Xopt = 0.24 (2.50)
Differently from the single stage configuration, the optimum value of the velocity characteristic ratio X depends on the velocity loss coefficient.
The optimum design parameter Xopt = 0.24 can be converted in terms of velocity ratio by means of Eq. (2.30), where φ = 0.95 and α2 = 80◦ are supposed
νopt= 0.22 (2.51)
Figure 2.8 shows the total-to-static efficiency for an impulse (Eq. (2.26)) and velocity compounded impulse stage (Eq. (2.49)) with equiangular blades for different velocity ratios (X) and nozzle outlet angles (α2). The velocity coefficient for the single stage arrangements ψ is set equal to the stator velocity coefficient of the two stage arrangement ψN. With these assumptions the efficiency ratio of the two configurations at the respective best efficiency points is
(ηts,max)Z=2
(ηts,max)Z=1 = 0.8φ2sin2α2
0.5φ2sin2α2(1 + ψ) = 0.842 (2.52) The efficiency of the velocity compounded turbine is lower than the pure impulse arrangement because (i) losses in the first rotor are higher due to the higher deflection and (ii) there are also the additional losses in the stator and in the second rotor.
To further investigate the effect of the velocity compounding staging on the optimum design parameter when more that two row are put in series, ideal rows (i.e., ψ = φ = 1) are considered hereafter for the sake of sim-plicity. As the discharge kinetic energy is the only considered loss source, the condition of maximum total-to-static efficiency is achieved for no swirl discharge component at rotor outlet. Accordingly, velocity triangles in-dicate that the optimum peripheral velocity Uopt needed for no discharge swirl is
2.4. VELOCITY COMPOUNDED IMPULSE STAGE 43
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
X η ts
α2=80°
70°
65°
α2=80°
70°
65°
Figure 2.8: Total-to-static efficiency versus X velocity ratio for different nozzle
outlet angles (65, 71, 80) for a single (right) and double stage (left) impulse turbine elaborating the same enthalpy drop. The assumed loss coefficients for the single impulse turbine are ϕ = 0.95 (nozzle) and ψ = 0.9 (rotor); for the double impulse turbine they are ϕ = 0.95 (nozzle), ψ′ = 0.86 (first rotor), ψs= 0.9 (stator) and ψ′′= 0.93 (second rotor).
Uopt = C2sin α2
4 (2.53)
which corresponds to an optimum velocity ratio Xopt= U
C2sin α2 = 1
4 (2.54)
The work for the maximum efficiency operation (which is also the max-imum work) is
Wmax =W|X=0.25 = 1
2C22sin2α2 = 8U2 (2.55) (6U2 from the first stage and 2U2 from the second).
Extending Eqs. (2.53), (2.54) and (2.55) to Z stages arrangement Uopt = C2sin α2
2Z (2.56)
Xopt = U C2sin α2
= 1
2Z (2.57)
Wmax = 1
2C22sin2α2 = 2Z2U2 (2.58) Thus, Eq. (2.58) indicates that the ratio between (i) the maximum work extracted by a turbine with Z ideal and equiangular rows (WZvd) and (ii) the maximum work extracted by an ideal and equiangular single row impulse turbine (W1vd) is
• Z2, when U is the same;
• unitary, when the nozzle outlet velocity C2sin α2 is the same.
A velocity compounded arrangement is advantageous because requires a lower optimum peripheral velocity Uopt in accordance with Eq. (2.56).
Figure 2.9 compares the ideal velocity triangles and the work of an impulse turbine (part a) to a two row velocity compounded stage for equal
2.4. VELOCITY COMPOUNDED IMPULSE STAGE 45
C2REF UREF UREF
W2
W3 C3
α2REF
W=W11=2UREF2
C2=C2REF W2
U=0.5UREF U
U U W3
C3W5 C5 W4 C4
W12=6U2=1.5UREF2
W22=2U2=0.5UREF2
W=W12+W22 =8U2=2UREF2 α2=α2REF
C2= 4UREF sin α2REF W2
U=UREF C4
U U U
W3
C3 W5 C5 W4
W12=6U2=6UREF2
W22=2U2=2UREF2
W=W12+W22 =8U2=8UREF2
(a)
(b)
(c)
Figure 2.9: Comparison between single stage impulse turbine velocity triangles
(a) and two stage velocity compounded impulse turbine with equal C2sin α2(b) and equal U (c). Ideal and equiangular rows and maximum work operation are assumed.
Table 2.1: Main parameters for velocity compounded ideal turbines at opti-mum efficiency point for fixed C2sin α2.
w1max w2max w3max wratio uopt Xopt
Z = 1 12C22sin2α2 − − 1 12C2sin α2 1