4. INCORPORATION OF FSW IN PRELIMINARY CVD 1. Introduction
4.3. Outline of blade design
4.3.1. Relationships describing rotor flows
Let us consider a Q2D (cylindrical) and a Q3D (conical) elementary cascade of equal mean radius r12 within the same rotor (Fig. 4.1). The optimum solidities will be approximated for both of them on the basis of 2D cascade data. Characteristics in the Q3D and Q2D approach will be denoted with subscripts for a clear distinction.
The same functions will be used for both approaches for describing the radial distributions of ˆ2
ψ and ϕˆ2. R12 is equal, but the inlet and outlet radii, R1 and R2, are different for the two approaches. Accordingly, ϕˆ2
( )
R2 ,ϕˆ12( )
R12 , and ψˆ2( )
R2 are also different for a cylindrical and for a conical elementary cascade of equal R12.Relationships valid in both Q2D and Q3D approach. As suggested by e.g. reference [1], the axial velocity through an elementary cascade is represented by an inlet-to-outlet arithmetic mean value, i.e.
( ) ( ) ( )
2 ˆ ˆ12 12 ˆ1 R1 2 R2
R ϕ ϕ
ϕ = + (4.7)
As an approximation for the fulfilment of conservation of mass through the elementary cascade, ϕˆ12R12dR12 ≈ϕˆ2R2dR2 is taken into account.
The z-wise component of the angular momentum equation applied in the absolute frame of reference, to the control volume enclosing the elementary cascade, takes the following approximate form (conf. e.g. [1, 74]). Comments on obtaining Eq. (4.8) are given in Appendix M.
z u
z v dM
v dr
r22 2 ˆ 2ˆ 2 =
2 π ρ (4.8)
The spanwise distribution of outlet axial velocity appearing in Eq. (4.8) is approximated with numerical integration of the simplified radial equilibrium equation, the use of which is suggested by references [1, 74, 185] for general whirl distribution, i.e. including CVD:
( )
22 2 22 2 2
t 2 ˆ ˆ
2
ˆ ψ ϕ
η ψ
dR d dR d R =
− (4.9)
In the above equation, the kinetic energy related to the outlet radial velocity has been neglected in comparison to those related to the other outlet velocity components. Furthermore, the local total efficiency ηt [164] is taken herein as constant along the span. This simplifying assumption, suggested in [74], and supported by references [27, 38, 122, 164], is reasonable in the region away from the endwalls, to which the approach presented herein is anyway restricted.
Various styles for prescribing the radial distribution of the isentropic total pressure rise
( )
2ˆ2 R
ψ incorporated in Eq. (9), being a main feature of the CVD concept, are summarized e.g. in
dc_242_11
[1, 74]. The power function ψˆ2
( )
R2 distribution, utilized e.g. in [26, 59, 79-81, 84-86, 144-145], and specified in the following equation is taken herein as example. This choice means there is no restriction to introducing other ψˆ2( )
R2 distributions in the presented concept.( )
R( )
R mAppendix N delivers information on the way of obtaining Eq. (4.9), the integral conditions related to the ψˆ2
( )
R2 and ϕˆ2( )
R2 distributions, as well as expression of global total efficiency ηt.Relationships valid for the Q3D approach. The flow angles are obtained using the following equations, which consider that the meridional mean velocity is approximated as
β
R (swirl-free inlet) (4.11a)
The free-stream relative velocity is as follows:
( ) { ( ) [
12 2Q3D(
2Q3D) ]
2}
12The free-stream flow angle is calculated as
( )
The elementary z-wise torque reacting on the elementary cascade, dMz, included in Eq. (4.8), can be expressed in another way as follows (upper part of Fig. 4.1).
( )
Q3DWhere the tangential component of elementary blade force is approximated as
( ) ( ) ( ) ( )
radius of the elementary blade section, r12. In Eq. (4.15), it has been considered that the surface ofdc_242_11
an elementary blade section between the conical surfaces of half-angle β and radial distance dr12 is
“fundamental cascade equation” of the elementary cascade (conf. [74]):
Q3D cascade equation in [74, 87]) are obtained. These formulae are presented in Appendix P.
4.3.2. Preliminary CVD procedure
Design based on the Q2D approach. For correction, the design process may return from a given step to a previous one, within an iterative procedure. 1a) Values for Φ, Ψ, ηt[164], and νht
are taken as initial input data. 1b) An inlet condition ϕˆ1
( )
R1 is prescribed. 1c) An initial ψˆ2( )
R2 distribution is prescribed, e.g. on the basis of Eq. (4.10). 1d) An initial ϕˆ2( )
R2 distribution is calculated on the basis of Eq. (4.9), for which a representative ηt value is chosen. 1e) The ψˆ2( )
R2 and ϕˆ2( )
R2 distributions are refined, in order to match with the integral conditions represented by Φ, Ψ, and ηt (conf. [74]), as well as to control the diffusion through the blading. The ϕˆ2( )
R2 distribution can be corrected for endwall blockage consideration. An empirical method for predicting blockage in radially stacked rotor bladings of CVD is outlined in [139], see Chapter 2.dc_242_11
1f) The imaginary rotor is subdivided into Q2D elementary cascades along the span.
Each elementary cascade under consideration has a mean radius R12. For each elementary cascade, the following characteristics are calculated in steps 1g) to 1i), with consideration of R1 = R2 = R12, and β = 0 (corresponding to the Q2D approach):
1g) ϕˆ12Q2D, based on Eq. (4.7); and CLoptQ2D⋅
(
c sb12)
optQ2D, based on Eq. (4.16b).1h) α1Q2D and α2Q2D are calculated, on the basis of Eqs. (4.11a) and (4.11b). CLoptQ2D is estimated using 2D cascade correlations, e.g. Howell’s correlation represented by Eq. (4.4).
1i) Dividing CLoptQ2D⋅
(
c sb12)
optQ2D, obtained in step 1g), by CLoptQ2D, obtained in step 1h), results in the optimum solidity(
c sb12)
optQ2D for the elementary cascade under consideration.The solidity can be expressed as c/sb12 = cNb/(2r12π). Therefore, calculating the optimum solidity controls cNb, related to the suction side flow path length summed for all of the blades of number Nb. By selecting an appropriate Nb value, and with knowledge of r12, the blade spacing sb12 is calculated, and the optimum chord length assigned to the elementary cylindrical cascade, coptQ2D, is obtained.
1j) The camber geometry as well as the stagger angle γ12 are designed in order to match the designed performance at moderate losses, taking 2D cascade correlations as starting point.
Appropriate ways of cambering and staggering are beyond the scope of the thesis.
Design based on the Q3D approach. Besides the Q2D approach, the blade characteristics of the same rotor can be obtained via a Q3D-based process as well.
2a) The flow field in the rotor designed with the Q2D approach is to be tested using CFD. The distribution of mean yaw angle ε~L along the radius is estimated. See Eq. (4.5) for details.
2b) With knowledge of the designed Q2D geometry incorporating the stagger angle γ12, the spanwise distribution of cone half-angle β for the Q3D approach is estimated along the span, using Eq. (4.6).
2c) The ψˆ2
( )
R2 and ϕˆ2( )
R2 distributions established in Q2D design are used as basis also for the Q3D approach. For realisation of these distributions also in the Q3D approach, and for further improvement of efficiency, re-cambering and re-staggering [4] techniques may iteratively be used.2d) Steps from 1g) can be carried out in the Q3D approach, for the same elementary rotors of radii R12: ϕˆ12Q3D [Eq. (4.7)], CLoptQ3D⋅
(
c sb12)
optQ3D [Eq. (4.16b)], α1Q3D[Eq. (4.11a)],Q3D
α2 [Eq. (4.11b)], CLoptQ3D [Eq. (4.4)],