Pressure compounded impulse stage

Z = 2 ³₈C₂²sin²α2 1

8C₂²sin²α2 − 3 : 1 ¹₄C2sin α2 1 4

Z = 3 ₁₈⁵C₂²sin²α2 3

18C₂²sin²α2 1

18C₂²sin²α2 5 : 3 : 1 ¹₆C2sin α2 1 6

nozzle outlet velocity C₂sin α₂ (part b) and peripheral velocity U (part c).

All the designs refer to the maximum work (or efficiency) operation.

Velocity compounded stages composed by more than three rows are not convenient because the relative contribution to the total work provided by the farthest rows is rather low. In fact, after some tedious algebra, the work extracted by the n^thideal and equiangular rotor wheel (that is independent of the turbine stage number) can be expressed by

Wn = 2U (C₂sin α₂− U) − 4(n − 1)U² (2.59)

whereas the maximum work WnZmax extracted by the n^th ideal and equiangular rotor wheel of a Z row stage turbine is deduced by substituting Eq. (2.57) in Eq. (2.59)

WnZmax = C₂²sin²α₂

(2(Z− n) + 1 2Z²

)

(2.60) Table 2.1 summarizes the effect of the row number Z on the work distribution and optimum velocity ratio.

2.5 Pressure compounded impulse stage

A sequence of impulse stages put in series constitutes the pressure com-pounded impulse stage. Each fixed row elaborates a fraction of the avail-able enthalpy drop. The optimum design parameters values are the same of

2.5. PRESSURE COMPOUNDED IMPULSE STAGE 47

the impulse stage (X_opt = 0.5), but, for fixed enthalpy drop, the optimum peripheral velocity is Z^−0.5 times smaller. Instead, for fixed peripheral ve-locity the pressure compounded stage allows for an efficient exploitation of an enthalpy drop Z times greater than the impulse stage.

2.5.1 Construction features

A pressure compounded impulse stage represents a further viable option when facing with available enthalpy drop which would require an impulse stage arrangement with an excessive optimum tangential speed.

Two or more impulse rows are arranged in series to reduce the optimum peripheral speed and the steam is expanded partially in each of the nozzles until the reaching of the discharge pressure. No pressure drop occurs across each rotor.

2.5.2 Analytical expressions of work and efficiency versus design parameters

The total-to-total efficiency is the proper metric for the performance evaluation, independently of the effective recovery of the discharge kinetic energy

η_tt = W h₀₁− h02s

(2.61) It has already been shown in Eq. (2.24) that the work extracted by an impulse stage with equiangular rotor blades (i.e., β₂ =−β3) is

W = C2²sin²α₂X(1− X)(1 + ψ) (2.62) By using the definition of nozzle velocity coefficient, the denominator of Eq. (2.61) can be conveniently re-written as

h₀₁− h02s= C₂² 2φ² − C₃²

2 (2.63)

Thus, the stage efficiency is updated

η_tt = 2φ²sin²α₂X(1− X)(1 + ψ) 1− φ²(

C3

C2

)2 (2.64)

The velocity ratio C₃²/C₂² is then rearranged C₃²

C₂² = 1

C₂²(U²+ W₃²+ 2W₃U sin β₃) = 1

C₂²(U²+ ψ²w₂²+ 2ψW₂U sin β₃) (2.65) and including in Eq. (2.65) the expression W₂² = U²+ C₂²−2C2U sin α₂ and W₂ = (C₂sin α₂ − U)/ sin β2 it becomes

C₃²

C₂² = ψ²+ X sin²α2((1 + ψ)²X− 2ψ(1 + ψ)) (2.66) Accordingly, the final expression of the total-to-total efficiency results in

ηtt = 2φ²sin²α₂X(1− X)(1 + ψ)

1− φ²(ψ²+ X sin²α2((1 + ψ)²X− 2ψ(1 + ψ))) (2.67) To avoid any ambiguity, it is useful to point out that Eq. (2.67) ex-presses the performance of a single couple nozzle-rotor of the pressure compounded arrangement. The derivation of an analytical formulation of the efficiency of the entire pressure compounded stage consisting in more than one row, would be impractical and it is therefore omitted.

2.5.3 Design guidelines

The considerations on the optimum design parameters values of the pressure compounded stage are the same of the impulse stage (see Sec-tion 2.3) because the pressure compounded arrangement is essentially a sequence of impulse stages. Obviously, for equal enthalpy drop, the nozzle outlet velocity of the pressure compounded stage is lower than the impulse stage, because the nozzle converts into kinetic energy only a fraction of the available enthalpy drop.

As the total-to-total stage efficiency do not account for the kinetic energy losses, the optimum velocity ratio X_opt in Eq. (2.67) is slightly

2.5. PRESSURE COMPOUNDED IMPULSE STAGE 49

higher than the one maximizing the total-to-static efficiency, leading to X_opt > 0.5. Nevertheless,

X ≈ 0.5 (2.68)

is chosen in practice because it guarantees lower parasitic losses (e.g.

disk friction loss which rapidly increases with X) in the face of a very limited decrease of the efficiency.

In the following the pressure compounded stage with Z rows (subscript Zpd, pd stands for pressure drops) is compared both with the impulse stage and the velocity compounded stage. It is assumed that (i) the available enthalpy drop is equally divided among the Z rows, (ii) rotor rows are equiangular and (iii) working conditions are optimum (X_opt=0.5).

For the pressure compounded impulse stage

UZpd = C2sin α2

2 = φsin α2

2

(2(h01− h2s)Zpd

Z

)0.5

(2.69) where (h₀₁− h2s)_Zpd is the isoentropic enthalpy drop inlet/outlet of the turbine and C₂ is the nozzle outlet velocity.

Similarly, for the impulse stage U_1pd = φsin α₂

2 (2(h₀₁− h2s)_1pd)^0.5 (2.70) Accordingly (see Eqs. (2.69) and (2.70)),

• for equal nozzle outlet angle (α2), nozzle loss coefficient (φ) and peripheral velocity (U ) it is

(h₀₁− h2t)_Zpd

(h₀₁− h2t)_1pd = Z (2.71) that is the pressure compounded stage allows an efficient exploitation of an enthalpy drop Z times greater than the impulse stage.

• for equal available enthalpy drop ((h01− h2s)_Zpd = (h₀₁− h2s)_1pd)

UZpd

U_1pd = 1

Z^0.5 (2.72)

that is the optimum peripheral velocity of the pressure compounded stage is Z^−0.5 times the impulse stage.

When the velocity triangles of pressure compounded stage are similar to those of the impulse stage, Eq. (2.72) can be extended in the form

U_Zpd

whereas in the pressure compounded stage they are Z times those occurring in the impulse stage

(Ln+ Lr)Zpd =

Assuming equal velocity coefficients and considering Eq. (2.73), it results

(L_n+ L_r)_1pd

(L_n+ L_r)_Zpd = 1 (2.76) Thus, in spite of the higher number of rows in a pressure compounded stage, the total loss amount is almost the same being velocities pro-portionally lower. On the other hand, in the pressure compounded stage the discharge kinetic energy loss is 1/Z times that of the im-pulse stage.