Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 41, 4, pp. 789-803, Warsaw 2003 THE INVERSE PROBLEM IN A HYPERBOLIC-SECTION RADIAL COMPRESSOR Romuald Puzyrewski Paweł Flaszyński Turbomachinery and Fluid Mechanics Faculty, Gdańsk University of Technology e-mail: rpuzyrew@pg.gda.pl; pflaszyn@pg.gda.pl Acomparisonbetween the inversemethod, leadingwithin the framework of the 2Dmodel to prediction of the optimal rotor shape and the direct approach for evaluating flow through a preexisting rotor shape utilizing a 3Dmodel is presented in this paper. The principle of shaping the rotor envelope and blading within the 2D model is illustrated, followed by subsequent computation of 3D flow through the resulting model. The design goal is to obtain uniform distributions of flow parameters within the rotorwhile avoiding separatedflow. It is also shownhow the altering of the overall shape of the rotor from conical to hyperboloidal affects the uniformity of velocity distribution upstream of the rotor inlet. Key words: compressor, inverse problem, 2Dmodel Important symbols e0 – total energy f – shape function of stream surface S2 fx1,fx2,fx3 – body force components p – pressure T – temperature Ux2,Ux3 – velocity components x1,x2,x3 – curvlinear coordinates ζ – isentropic loss coefficient τ – flow-area reduction factor 790 R.Puzyrewski. P.Flaszyński 1. Introduction The evolution of the outline of the compressor stage is shown in Fig.1, according to publications bySiemens company (Tosza andMagdalinski, 2002). Fig. 1. Evolution of compressormeridional shape As may be observed, strict radial shape delimited by straight lines has gradually evolved towards a curved hyperbola-delimited cross-section, with marked tendency toward reducing the diameter and extending the blade in the axial direction. The present work can serve to illustrate this progress from the traditional cone-bounded rotor towards the hyperboloid-of-revolution- bounded rotor. The starting point is the rotor shape illustrated in Fig.2, for which 3D computations have been carried out. The results have shown a distinct region of separated flow localized at the inlet (Fig.3.). Fig. 2. Radial compressor (meridional section) The inverse problem in a hyperbolic-section... 791 Fig. 3. Separation zone upstream to the leading edge Three distinct vortices can bemade out in Fig.3, which indicates that the primary vortex is of intensity high enough to generate and feed the subse- quent two vortices. The large extent of the separation-affected region is also quite noticeable. There are two incentives that make the elimination of this region desirable. Firstly, unwanted dissipation effects are invariably exacerba- ted through the presence of separation zones. Secondly, the existing separation adversely affects the uniformity of flow distribution at the entrance to the ro- tor. Such non-uniform inflow requires complex special modifications of blade shape; if such design modifications are not made and the flow is treated as uniform, additional losses will be incurred within the rotor. 2. Hyperbolic outline of rotor cross-section The cross-section outline of the modified rotor is shown in Fig.4. The casing has the form of a hyperboloid of revolution generated by the hyperbola fd = d1+d2x −nd 3 (2.1) The particular hyperbola was chosen so as to avoid the occurrence of separa- tion similar to that shown in Fig.3. The remaining curves enclosing the rotor cross-section are described analytically by the equations fb = b1+ b2x −nb 3 finl = a+ bx3 fout = rk (2.2) 792 R.Puzyrewski. P.Flaszyński Thus, the configuration of the rotor envelope is fully described by nine para- meters: d1, d2, nd, b1, b2, nb, a, b, rk. The parameters d1, d2, nd have been chosen so as to eliminate the separated region where the flow is turned from the axial to radial direction. The resulting line is characterized by continuous curvature, as opposed to the conventional jump in the curvature at the junc- ture of the straight line and arc (Fig.2). The parameters b1, b2, a, b follow from the choice of the points P2, R1, R2, as shown in Fig.5; these points, as well as the quantity rk, follow in turn from the assumed dimensions of the compressor stage.The remaining free parameter nb can be obtainedby stating the requirement of uniform flow parameter distribution at the outlet from the rotor, along the line P1 -P2.With this end inmind, let us define the following family of curves f = f(x1,x3)=x1fb+(1−x1)fd (2.3) Fig. 4. Blade channel boundaries Note that with x1 = 0 one obtains the profile of the hub disc, and with x1 =1theprofileof theouter casing.Thecurves x1 = const follow thepattern shown in Fig.5, forming the basis of a family of hyperboloids of revolution which can be used as the basis for defining the curvilinear coordinate system (x1,x2,x3). The conversion rule with regard to the cartesian system can bewritten as x= f(x1,x3)cosx2 y= f(x1,x3)cosx2 z=x3 (2.4) The inverse problem in a hyperbolic-section... 793 Fig. 5. Streamsurfaces S1 in meridional cross-section If the surfaces x1 = const are takenas streamsurfaces, then theproblemof designing the corresponding blade configuration within the R1R2P2P1 region reduces to the solution of the following system of equations: • the mass conservation (continuity) equation [1− τ(x1,x3)]ρUx3 f ∂f ∂x1 √ 1+ ( ∂f ∂x3 )2 =m(x1) (2.5) • momentum conservation equations: – in the x1 direction − ρU2x2 f + ρU2x3 ∂2f ∂x2 3 1+ ( ∂f ∂x3 )2 =− ∂p ∂x1 1+ ( ∂f ∂x3 )2 ∂f ∂x1 + ∂p ∂x3 ∂f ∂x3 +ρfx1 (2.6) – in the x2 direction ρUx3 f √ 1+ ( ∂f ∂x3 )2 ∂(fUx3) ∂x3 = ρfx2 (2.7) 794 R.Puzyrewski. P.Flaszyński – in the x3 direction ρUx3 √ 1+ ( ∂f ∂x3 )2 ( ∂Ux3 ∂x3 − Ux3 1+ ( ∂f ∂x3 )2 ∂f ∂x3 ∂2f ∂x23 ) = (2.8) = ∂p ∂x1 ∂f ∂x3 ∂f ∂x1 √ 1+ ( ∂f ∂x3 )2 − ∂p ∂x3 √ 1+ ( ∂f ∂x3 )2 +ρfx3 • the energy conservation equation (ur-rotor velocity) U2x2 +U 2 x3 2 −urUx2 + k k−1 p ρ = e0(x1) (2.9) • the process equation (an integral of the Gibbs equation with the defini- tion of the loss coefficient) ρ= ρ1 ( p p1 ) 1 k exp [ ζ [( p1 p ) k−1 k −1 ] √ (1− ζ) [ 1− ζ ( p1 p ) k−1 k ] ] (2.10) This system of 6 equations, however, contains 9 unknowns Ux2, Ux3, fx1, fx2, fx3, ρ, p, τ, ζ Its closure thus calls for additional assumptionswith regard to three of the above quantities. Let us focus for the moment on continuity equation (2.5). Transforming it, one obtains the formula ρUx3 = m(x1) [1− τ(x1,x3)] f ∂f ∂x1 √ 1+ ( ∂f ∂x3 )2 = m(x1) [1− τ(x1,x3)]Pu(x1,x3) (2.11) where the parameter Pu depends solely on the geometric configuration Pu(x1,x3)= f ∂f ∂x1 √ 1+ ( ∂f ∂x3 )2 (2.12) If one were to aim at uniformity of ρUx3 distribution at the outlet stage, then all the functions taking part in relation (2.11) should approach constant The inverse problem in a hyperbolic-section... 795 values. The function m(x1) is dependent on conditions at the outlet stage (line P1 -P2). Any non-uniformity with regard to the parameters making up m(x1) is reflected in a corresponding non-uniformity at the outlet. This can be compensated by varying Pu. Let us introduce a quantitative measure of the non-uniformity in the parameter Pu, based on deviation from the mean value mPu = 1 x3R2 −x3P2 x3R2 ∫ x3P2 Pu dx3 (2.13) The non-uniformity factor can be then expressed as follows NPu = 1 mpu √ √ √ √ √ √ 1 x3R2 −x3P2 x3R2 ∫ x3P2 (Pu−mPu)2 dx3 (2.14) In relation (2.12) for theoutlet, the coordinate x1 canbeeliminatedbymaking use of the formula x1 = rk−fd fb−fd (2.15) where the expressions rP2 = rk = b1+ b2x −nb 3P2 (2.16) rR2 = rk = b1+ b2x −nb 3R2 serve to determine the parameters b1, b2. Thus, the parameter Pu becomes a function of the arguments (x3,b1,b2,nb) only, while NPu is the function of nb alone. For actual values of all remaining parameters as listed below d1 =0.195789 rk =0.176 d2 =0.186336 a=0.0900253 nd =0.1 b=0.472992 (2.17) the non-uniformity factor NPu varies with nb as shown in Fig.6. It should be noted that the minimum of NPu falls in the vicinity of 0.50-0.55. The values of the remaining parameters are summarized inTable 1. Table 1 nb b1 b2 Nf 0.45 −0.05020536 0.040771974 0.035485 0.50 −0.03115712 0.030865716 0.012092 0.55 −0.01560754 0.023599713 0.019275 0.60 −0.00268185 0.018192493 0.047338 796 R.Puzyrewski. P.Flaszyński Fig. 6. Non-uniformity factor NPu The behavior of the parameter Pu =Pu(x3) at the outlet is illustrated in Fig.7 for nb = 0.5. This is the value for which NPu reaches its minimum of approximately NPu ≈ 0.012. Fig. 7. Parameter Pu as function of x3 3. Results of 3D calculations for the hyperboloid-walled rotor Solving the above system of equations describing the 2Dmodel yields the shape of the rotor blade surface. The solution uses additional closing assump- tions in the formof the following relations describing the loss coefficientwithin the rotor region (r,x1) r1 = r−rin1 rk−rin1 (3.1) ζ = [w0r1+(1−w0)r n0 1 ][w1+w2(1−x1) n1]+w3x n2 1 The inverse problem in a hyperbolic-section... 797 with the constants set to w0 =0.5 w1 =0.05 w2 =0.0 w3 =0.0 n0 =15 n1 =40 n2 =30 (3.2) so that the level of losses should closely mimic what can be expected in the rotor region. Two additional functions τ(x1,x3) = 0 and fx1 = 0 were for- mulated. These are preliminary simplifications which can be prospectively re- placed by other closure functionsmatching better the expected values, of fx1 in particular. The resulting surface is shown inFig.8.After allowing for the profile thick- ness (approx. 5mm) the blade can bemodelled along it.The curvesmakingup the surface are fluid element trajectories passing along axisymmetric stream surfaces described by the function f as defined in relation (2.3). Fig. 8. Stream surface S2 (a) and final blade shape (b) The following illustrations compare the calculation results fromthe 2Dand 3D models. The boundary conditions for the 3D model are as formulated by Flaszyński and Puzyrewski (2001). Figure 9 shows the distribution of static pressure at the inlet to the bladed region of the rotor. Figure 10 shows the corresponding velocity distribution, andFig.11 the distribution of pressure at the rotor disc from the inlet to outlet. As may be observed, the principal effect of passing into the three- dimensional regimen ismanifested bynoticeably lower pressure ratio along the disc, as seen in Fig.11. This effect is likewise evident in Fig.12 and Fig.13. As it turns out, the outlet angle in the 3D calculations differs from the 2D results by some 6◦. This, in turn, influences the velocity distribution at the outlet (Fig.13). All these effectsmust be taken into account in the design pro- cess, if the 2D inverse method is to be used to its full potential. Specifically, 798 R.Puzyrewski. P.Flaszyński Fig. 9. Pressure distribution at the inlet section Fig. 10. Velocity distribution at the inlet section Fig. 11. Pressure distribution along the disc in anticipation of these minor discrepancies due to 3D effects, the designer should specify a slightly higher pressure ratio, while realizing that the outlet angle obtained from a finite number of blades is likely to be lower than the theoretically obtained value. The inverse problem in a hyperbolic-section... 799 Fig. 12. Velocity angle at the impeller outlet Fig. 13. Velocity at the impeller outlet Fig. 14. Velocity vectors close to the disc upstream the leading edge 800 R.Puzyrewski. P.Flaszyński On the other hand, it is encouraging to note that the qualitative pattern of parameter variation exhibits close correspondence between the 2D and 3D models. A major improvement with regard to the existing design is evident with regard to the velocity distribution at the inlet, where the extensive separation zone has been entirely eliminated, as shown in Fig.14. This translates to a marked improvement in the flow conditions at the inlet to the blading region. Fig. 15. Velocity vectors close to the disc Fig. 16. Velocity vectors in the vicinity of the trailing edge Figures 15 and 16 illustrate the velocity distributions in the neighborhood of the rotor disc. Again, no separation zones are noted, in contrast to what is typical for the majority of existing compressors. The inverse problem in a hyperbolic-section... 801 Figures 17 and 18 show the corresponding velocity patterns in the vicinity of the cover plate. Here the extensive separation zones are absent as well. At the trailing edge a small separation region occurs, but it is caused by the abrupt profiling of the trailing edge. Fig. 17. Velocity vectors close to the shroud Fig. 18. Velocity vectors in the vicinity of the trailing edge 802 R.Puzyrewski. P.Flaszyński 4. Conclusions The designmethod presented in this paper hasmade it possible to unifor- mize substantially the velocity fieldwithin the blading of a compressor. It still remains to optimize the flowwith regard to the dissipative losses generated at the washed surfaces as well as within the recirculation zones. Acknowledgment The research for this paper has been carried out as a part of the StateCommittee for Scientific Research (KBN) project No. 8T10B00419. References 1. Flaszyński P., Puzyrewski R., 2001, Obliczenia 3D (FLUENT) dla róż- nychmodeli turbulencji.Analizawyników.Wnioski, ProjektKBN8T10B00419, Etap 6, Gdańsk, September 2. PuzyrewskiR., 1998,14wykładów teorii stopniamaszynywirnikowej –model dwuwymiarowy (3D), Wydawnictwo Politechniki Gdańskiej, Gdańsk 3. Puzyrewski R., Flaszyński P., 2001, Projekt wirnika metodą zadania od- wrotnego 2D, Projekt KBN 8T10B00419, Etap 7, Gdańsk 4. Puzyrewski R., Flaszyński P., 2002, Modyfikacja I (pierwsza) kształtu wirnika w strefie oderwań. (Sprawdzenie efektów obliczeniami), Projekt KBN 8T10B00419, Etap 10, Gdańsk 5. Tosza T., Magdalinski Ch., 2002, Modernizaciya kompressorov. Vozmo- zhnosti i koniechnye rezultaty adaptacii rabochikh kharakteristik kompresso- ra k izmeneniyuuslowǐı,Mezhdunarodnǐı Sympozyum ”Potrebitieli-Proizvoditeli kompressorow i kompressornogo oborudovaniya”, Sankt Peterburg Izdatelstvo SPGTU Zadanie odwrotne dla sprężarki promieniowej o hiperbolicznym kształcie Streszczenie Porównano rozwiązanie zadania odwrotnegow ramachmodelu dwuwymiarowego, prowadzącego do kształtu koła wirnikowego, z zadaniem prostym rozwiązanymw ra- machmodelu trójwymiarowego dla skonstruowanegowirnika. Przedstawiono elemen- ty algorytmu kształtowania łopatek przy wykorzystaniu modelu dwuwymiarowego. The inverse problem in a hyperbolic-section... 803 W podanym przykładzie wskazanomożliwość optymalizacji kształtu jednego z ogra- niczeń kanału w celu uzyskania równomiernego rozkładu parametrów w przekroju wylotowym. Pokazano jak zmiana obrysu wirnika z klasycznego kształtu stożkowe- go na hiperboliczny może wpłynąć na równomierność rozkładu pól prędkości przed układem łopatkowym. Manuscript received November 21, 2002; accepted for print February 27, 2003