A Proposal for Optimized Design of Multistage Compressors

Lq CS Q THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 47 St., New York, N.Y. 117 The Society shall not be responsible for statements or opinions advanced in papers or in discussion at meetings of the Society or of its Divisions or Sections, or printed in its publications. Discussion is printed only if the paper is published in. an ASME Journal. Papers are available from ASME for fifteen months after the meeting. Printed in USA. Copyright 1989 by ASME 89-GT-34 A Proposal for Optimized Design of Multistage Compressors RAFFAELE TUCCILLO Associate Professor Dipartimento di Ingegneria Meccanica per I'Energetica (D.I.M.E.) Universita' di Napoli Via Claudio 21 8125 Napoli (Italy) ABSTRACT The aim of the paper is to provide a methodology for optimum design of axial flow multistage compressors derived from a flow model, previously defined by the author, that takes into account the effects of several geometrical parameters on blade losses and on the growth of end-wall boundary layers and secondary vorticities. The optimization is performed by means of a "Quasi-Newton" algorithm that controls an "Augmented Lagrangian Function" with the objective of maximizing efficiency for assigned values of total to total head. The paper outlines the theoretical approach and the main practical feature of the methodology, the latter consisting of a performance optimization by imposing even slight variations to the geometry of an existing compressor. NOMENCLATURE c... absolute velocity c p... friction coefficient c L... cascade lift coefficient cp... specific heat at constant pressure Deq... equivalent diffusion factor F... Augmented Lagrangian Function F... axial force defect in boundary layer momentum equation It... enthalpy H... blade height, form factor Ha... modified form factor in entrainment equation Hit... total to total adiabatic head L... compressor length m... mass flow rate p... static pressure, penalty factor in augmented Lagrangian Function r... radial coordinate Re$ Reynolds number based upon momentum thickness s... blade spacing entropy T... static temperature tc... tip clearance u V W x_ x X Xb y peripheral velocity absolute or relative velocity work per unit of mass axial coordinate vector of free variables in optimization vector of geometrical parameters blade loss coefficient relative radial coordinate (i... flow angle (with respect to the axial direction) (sn... blade angle rs,.... total to static pressure ratio r1t t... total to total pressure ratio y... isentropic exponent d... boundary layer thickness 6*... displacement thickness tkt... total to total efficiency 8... momentum thickness *... Lagrange multiplier p... static density a... cascade solidity rx... shear stress in axial direction... blade deflection, objective function rd... angular velocity Subscripts :... total conditions free stream properties, initial geometrical parameters u... conditions upstream of the multistage compressor d... conditions downstream of the multistage compressor h... hub t... tip m... mean radius r, x, u... radial, axial and peripheral components of velocity. 1... station upstream of a blading 2... station downstream of a blading Presented at the Gas Turbine and Aeroengine Congress and Exposition June 4-8, 1989 Toronto, Ontario, Canada Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/22/217 Terms of Use: http://www.asme.org/about-asme/terms-of-use

INTRODUCTION The overall performance of axial flow multistage compressors, expressed in terms of summarizing parameters like total to total head H LL (or compression ratio (t L ) and efficiency Tqi, is the result of complex phenomena that occur in flow through moving and fixed blading and exert their influence on: - meridional and radial flow distribution;. - growth of end-wall boundary layers interacting with the main flow; - development of secondary vorticities also altering the flow pattern and the spanwise flow angle distribution. Several physical and mathematical flow models have been proposed in recent years, that are able to calculate both inviscid flows and flow patterns related to blade and end-wall friction effects, in order to predict the overall perfomance of multistage axial flow compressors with a high level of accuracy. In many cases, even simplified models (based,for example,upon axisymmetric radial equilibrium) satisfactorily work, by employing empirical correlations evaluating blade losses, secondary flow effects and performance reduction due to the end-wall boundary layer. The author himself has recently proposed (1988a, 1988b) a flow model based upon the interaction between a "free stream", external to hub and tip boundary layers, and the end-wall viscous flow; the former is calculated by means of the radial equilibrium equation and the latter is predicted with the integral equations of boundary layer (De Ruick and Hirsch, 1979,1981). The development of secondary vorticities is also taken into account as regards the influence on flow angles (Horlock, 1975; Salvage, 1974; Glynn and Marsch, 198). The results of such calculations are in satisfactory agreement with experimental test data (Satta et al., 1988) in terms of both flow pattern and overall performance; besides, are clearly shown the significant differences arising in load coefficient and efficiency prediction when boundary layer effects are neglected or considered. The most complex flow models can be, however, more feasibly applied in "direct problems" (i.e. to verify the flow induced by an assigned compressor configuration) rather than in design procedures for the definition of the geometry enabling the desired performance to be achieved. The latter purpose may be pursued with the decisive aid of optimizing techniques, since they make it possible to control the influence exerted by a great number of variables. For the above reasons, the proposal made in the present paper refers to a method for the design of axial flow multistage compressors based upon a mathematical optimization algorithm applied to the flow model that has been previously defined by the author. The design procedure consists of a sequence of verifications of compressor blade and end-wall geometries by taking into account a number of important aerodynamic phenomena like radial equilibrium and streamline curvature, blade losses, development of end-wall boundary layer and secondary vorticities. The optimizing technique can improve the results of such verifications by modifying the values of the free variables, until maximum efficiency is reached for the required total to total head. It must be stated that, unlike other methods for optimized design of turbomachinery that have been proposed in recent years (Cuan-gang and Yong-miao, 1987; Rao and Gupta, 198; Rizzo and Tuccillo, 1987), the aim of the present method is to directly act on geometrical variables instead of on flow parameters. The method, thus, represents the final step of a design procedure that has previously determined radial and axial flow distribution, and requires the use of a more accurate flow model to verify whether the desired performance has been achieved or not. By modifying the previously defined compressor end-wall meridional shape and blade geometry, the optimizing technique yields an improvement in efficiency and an adjustment of the total head to the assigned value. Similarly, the method may be proposed to make a new compressor design starting from an existing geometry: this appears to be useful if different values are required for design mass flow rate or total head, and hence modifications in compressor geometry are to be imposed in order to achieve the new design point. As will be shown in the following sections, the design strategy seems, also from an industrial point of view, more convenient than initiating a new design, since it allows the preservation of an existing technology by imposing only slight changes in geometrical parameters. THE FLOW MODEL As has been stated above, the flow model assumes that each vane less station between two successive blade rows can be radially divided into three parts: 1) A central region for which the radial equilibrium equation holds, under the hypothesis of axisymmetric flow: c (r c) ac1 ac ahe, as r ar -c, t ax - ar, = ar - T ar (1) Equation (1) may thus be transformed into: ac tg2r 1 a(tg2 fi) (1 + tg2(s) ar + c., ( r + 2 ar ) + 2 Li tg(i + ac,. 1 a as ax c.. (a h> ( 1 - w r 1 c 1 tgoi) - T ar 1 (2) where (i = (1(r) is the discharge angle of either a fixed (w = ) or a moving (i # ) blading prior to the vaneless station and the subscript "1" refers to the flow conditions upstream of this blading. The radial entropy distribution may be calculated starting from the values of blade loss coefficients X t, (Lieblein et al., 1966a, 1966b, 1966c; Koch and Smith, 1976), while the discharge flow angle (3 is obtained by correcting the results of two-dimensional cascade correlations for the influence of the axial velocity and density ratio (Starke, 1981; Stark and Hoisel, 1981) and of secondary vorticities. Streamline curvature effects are taken into account by relating the radial velocity derivative ac,./ax to the radial displacement r(x) of the streamline projected onto the meridional plane (Vavra, 196): ac t, dz r ax = c' dxi (3) It should be noted that the above relationship, while allowing the transformation of equation (2) into an ordinary differential one, establishes a link among the flow properties distributions in the several vaneless stations, thus preserving the principle of uniqueness of the solution for the initial system of equations of motion (Vavra, 196; Katsanis, 1966; Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/22/217 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Marsh, 1971). 2) Two end-wall regions (corresponding to hub and tip of the blade rows) in which the prevailing phenomena are those related to the growth of the wall boundary layers calculated by means of integral equations derived from De Ruick and Hirsch (1979,1981): T. momentum d dck _ F. dx (c 8) + equation c" d dx p + p (4) entrainment 1 d (6-6) =f(hth (5) equation c o dx cost,, a _ a - 6 * form _ factors H $ H ^ 6 In order to calculate the changes in boundary layer parameters after each blade row, equations (4) and (5) are transformed and integrated along the distance ax between two successive stations, 1 and 2, as follows: A6 = A + B + C9 + D8 (6) OH = E + Gil - 's H (7) The A...G coefficients are related to the axial force defect F. and shear stress T,,, to the blade geometrical features in hub and tip regions and to the changes in axial velocities Oc G and in absolute or relative flow angles '3; the latter also determine the values for the lift coefficient c L : c f -.268 A = cos 3.Gx ; c f =.246 Re exp(-1.561 H) z c L B=1 2z Gx?, =.1+.35 c cos (1cm in(2i () C = - 2 cu [1 - exp(-o)] + ku in(2j 1) 1-exp(-o ) - 2 (1+H) Q (tgi -tglsl ) D = - (2+H) Ac ;, r, c,, EcosO A x ; F(H'') =.36 (H* - 3) m Ac d G=.653 Equations (6) and (7) provide values for momentum thickness 6 and form factor H*: further parameters (d, 6*, H) may be determined by assuming a velocity profile law. In this way, total pressure losses due to momentum defect in the end-wall boundary layers may be calculated, thus determining a further entropy increase. In addition to the fundamental equations holding in the central "free-stream" (2) and in the extremity regions (6)(7), secondary vorticities - expressed in terms of crossflow components V, - are taken into account by simultaneously solving equations of the kind (Horlock, 1975; Salvage, 1974; Glynn and Marsch, 198): V1yzc,, = b (1 - - K 8* exp(- Ky)1 (8) c- l cm1 J that relate the secondary flows to the end-wall boundary layer displacement thicknes 6*, to the axial velocity distribution c 1 /c 1 =f(r) and to the induced vorticity parameter "b". The development of this equation and its solution makes it possible, as stated above, to correct the values of relative or absolute discharge flow angles. A full description of the methodology has been presented by author in his previous papers (1988a,1988b). Here it appears sufficient to notice that equations (2),(3), (6),(7) and (8) strongly interact: indeed, equation (2) holds externally to boundary layers, so that its integration limits are determined by the knowledge of the hub and tip boundary layer thicknesses dr, and 6t, the latter reducing the "free-stream" flow area and altering conditions for radial equilibrium and streamline curvature. Values of boundary layer parameters, on the other hand, are influenced by blade loading and changes in freee stream velocities c c, that are evaluated by solving equation (2) and by means of the named correlations. Owing to the described interactions, occurring in each vaneless station, and to the linking established by the streamline equation (3) among the solutions, an iterative procedure is required in order to determine the radial distribution of flow properties; calculations may be carried out by imposing upstream total conditions p,,, T and, hence, by verifying the flow pattern for of a given value of the total mass flow rate m. THE DESIGN METHOD The predicted performance of the multistage compressor by employing the described flow model may be expressed in terms of total to total head H tt and efficiency Tk t as follows: 1k(r) = p-. (r) p (r) rt Htt=2n p c^ Htt (r) r dr m t ry, r t Y^1 ; g Htt(r) = c T,, (ft t (r) - 1) 2n I g Htt W = p c, (} d-11) r dr ; T)tt = W r1-, so demonstrating that such values are the consequence of radial distributions of flow properties associated with the boundary layer growth, secondary vorticities effects and, of course, with their interaction with the main flow through fixed and moving blades. It is obvious that any change in initial geometry will be felt by the flow model as regards both flow pattern and overall performance. The proposed design procedure consists of an optimization of such changes, in order to reach a compressor configuration making it possible to obtain an imposed value of total head with the maximum of efficiency. In other words, naming X the vector of initial geometrical parameters (X 1...IC,-,) and x the vector of the corresponding changes (x1...x,.,), the modifications in values of total to total head and efficiency, with respect to the initial 3 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/22/217 Terms of Use: http://www.asme.org/about-asme/terms-of-use

ones Htto and rh t, may be expressed as follows: Htto = g(x) ; Htto + AH = g(x + x) llr.to = f(x) ; Th.ta + AH = f(x + x) The optimization technique operates directly with the x vector, thus dealing with the problem in terms of: Tk t = f'(x) ; Htt = g'(x) (9) Since the compressor configuration must be found that will yield the desired performance level, the mathematical problem has the form of a constrained optimization, strongly influenced by an equality constraint represented by the assigned value Ht for the total head: c(x) = Htt - Htt = (1) Furthermore, several inequality constraints might be imposed, representing limits on blade loading, peripheral velocity and stress conditions, Mach number (for high speed compressors) and so on. At present, since the examples presented in this paper refer to a low speed compressor, whose initial geometry largely satisfies such constraints, only condition (1) was taken into account. On the other hand, it should be noticed that the aim of the proposal is mainly to verify the behaviour of the optimizing method and the equality constraint (1) represents the hardest possible test since it restricts the field of possible solutions that can only lie, in this way, over a surface of the n-dimensional space. In order to solve such a problem, the method of the "Augmented Lagrangian Function" was adopted (Gill et al., 1984): the optimization algorithm for the_function f(x) operates, in reality, on the function F(x) defined as follows: In the above relationships, the vector x* contains the optimal values of (xl,...x r-,) calculated at the (k-l)th optimization; the sequence of admissible errors is obtained by reducing the value of e by an order of magnitude, after each iteration. It is clear that stabilization of values for It and p ensure the convergence of the whole optimizing procedure, that is to say that the vector x* allowing the achievement of thg best efficiency rltt in the assigned design point (ii, Ht t ) has been found. In accordance with the aim of the paper to discuss the capability of the optimizing technique in controlling the complex phenomena that occur in flow through the multistage compressor, a limited number o f geometrical variables was considered in the vector X and, hence, in x; such variables were choosen by taking into account the ones exerting the strongest influence on both blade loading and end-wall boundary layers: - hub and tip radii, rt, and rt, since their variations cause changes in mean flow coefficient and in radial distribution of properties along blade height. The corresponding variables to be optimized are, hence, the changes Ar t and Art in all the vaneless stations. - blade characteristics, in terms of blade inlet angle ri. and deflection ep = rir,l - rst.2, as they strongly influence incidence angles, load conditions and loss coefficients, and, consequently, the flow pattern interacting with the end-wall boundary layers. The free variables for optimization are the variations Arh,,, and A at mean radius for blade inlet angle and deflection: each change in these properties is extended to the whole blade height by assuming a blade twisting law similar to the initial one: rh_, 1 (r.,.,) = f\,1 (r) + A(i, 1 ; it = + A,$ ; F(x) = (x) + X c(x) + p c(x) c(x) (11) r^:,z(r.) = rk,i(r,. r ) - (16) The objective function is (x) = 1 - rk t, while the coefficients X and p represent the Lagrange multiplier bbl ( and the penalty factor for the constraint c(x) r,. ) h, i (r) (1(r) (17) expressed by equation (1). For the function F to be (3 1 (r,, ) - f optimized, the well known "Quasi-Newton" method was r, z ( r,.,) adopted based upon the determination of directions "s" f2(r) = f3t,2(r) given by: (18) (ib2(rr;i) s = 117 F (12) the C coefficient taking into account the hub and tip radii variations: and, consequently, of the step Ax: rt, = rt,o + A rr, rt = rta + A r t (19) Ax = t* s (13) r - r t, r - rh,o where r- is the result of the "one-dimensional" (2) C = rv- rto r - rt research along the direction "s". The "pseudo-hessian" matrix H is updated by means of a "DFP" algorithm (Rao In this way, every change in blade geometry at mean and Gupta, 1981; Gill et al., 1984) and the convergence radius r,.,, will directly influence blade configurations criteria refer to three controls made on stabilization also at hub and tip radii, thus altering the flow of funcion F, moduli of s and of 7 F respectively. conditions in the end-wall zones. In the present case, the optimizing algorithm must Other fundamental variables could, of course, be be iteratively applied since the equality constraint considered such as, for instance, a blade solidity (1) has to be strictly satisfied; an upgrading is parameter or, more directly, the number of blades. At required, at every iteration level "k", for the present the only optimized variables listed above may, lagrangian multiplier X and for the penalty factor p however, produce significant changes in flow behaviour (Beveridge and Sheckter, 197; Gill et al., 1984) : and in compressor performance. It should also be noticed that slight modifications in solidity may be y = X - py i c(x*) (14) caused by the variations Art, and Ar t, since the hub and tip cascade pitches are altered: hence, the effect of Py = (15) solidity will be, indirectly, felt by the algorithm. 4 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/22/217 Terms of Use: http://www.asme.org/about-asme/terms-of-use

E EXAMPLES OF RESULTS In order to show the optimizing method possibility to increase the performance level of a multistage compressor, three examples of optimization are presented, referring to an existing geometry of a two stage compressor (Satta et al., 1988). For this machine the initial estimated performance is summed up as follows: operating fluid : air revolution speed = 1 rpm = 1.16 bar T= 298 K A [kg/s] Htt [m] ntt a) 6. 1.6.893 b) 7. 93.8.948 c) 8. 77.5.921 The new design specifications assign the value Htt =1 m to all the mass flow rates, thus proposing three different problems. Indeed, changes in geometry for case (a) should have the main purpose of increasing efficiency, since the load level is already close to the assigned one. In case (b) - and more remarkably in (c) - the new optimized design, while preserving the purpose of a high value for efficiency T) Lt, must provide a higher blade loading. Figs. 1-15 show, from different points of view, the results of the three optimizing procedures using the above described algorithm. The evolution of the results during optimizations is summarized in figs. 1-3, clearly demonstrating the typical tendency of both the objective function (i.e. Tk t ) and the equality constraint (on total to total head) to reach a stable value after the same number Nf of function evaluations. It must be remembered that the whole optimization is carried out through a number of iterations in order to update the values of the Lagrange multiplier and the penalty factor. The sudden initial variation of the functions and of the free variables occurs during the first iteration, while the smoother behaviour of the several curves refers to the subsequent ones it should be noticed that the first iteration of the procedure makes it possible to reach, in a reduced number of function evaluations, a performance level near to the imposed one, so that the aim of the other iterations essentially consists of an adjustment of these results so as to achieve convergence; cases (a),(b) and (c) required 6,7 and 5 iterations of the optimizing procedure respectively. As regards the optimized geometrical variables, figs. 1-3 also show the calculated changes for both blade inlet angle (i, and deflection for all the moving and fixed rows; furthermore an example is presented for the optimized changes in blade height H, referring to the second stator. An examination of the trend of the variables highlights the fact that the algorithm correctly feels their tendency to cause changes in flow conditions and in all phenomena affecting overall performance; hence, variation in blade deflection allows the blade loading adjustment to the new assigned design specifications, while the optimized values of (s,, q are more suitable to improve incidence conditions and the blade loading itself. The modifications proposed by the optimized design technique for blade height produce favourable variations in mean flow coefficients and in radial distribution of properties: indeed, i case (a) the decrease in blade height makes it possible to lower the blade loading level, while its increase in cases (b) and (c) reduces mean flow coefficients, thus yielding higher load conditions as required. A more detailed investigation of the new flow pattern obtained by means of the optimizing technique is provided by figs. 4-6. The level curves of the local flow coefficient (c /u,.;,), total and static pressure (in terms of load coefficient) display the development of the end-wall boundary layers and the conditions of loading and total pressure losses.it can be seen that in case (a) a clear reduction in boundary layer thickness is obtained, while in the other cases the increased blade loading necessarily enlarges the regions involved in viscous phenomena. Level curves of total and static pressure also show the new conditions of energy transfer from rotors to operating fluid and the more feasible load repartition between the two stages. All groups of diagrams in figs. 7-15 show the radial distribution of properties as regards both local performance level (total to total and total to static pressure ratio visit and (sty total head and efficiency evaluated on streamlines) and blade parameters (loss coefficient Xb and equivalent diffusion factor Deq). Dotted, dashed and continuous lines refer to the initial compressor geometry, to the changes after the first optimizing iteration and to the final one, when convergence on total head is reached, respectively, thus confirming the considerable improvements already obtained through the first iteration and the following adjustments. These representations (being y a relative radial coordinate : y = r-r,) point out the extents of the free stream regions and of the end-wall flow conditions and allow the improvements in local values, obtained by employing the proposed method, to be appreciated. Furthermore, it is confirmed that the control carried out by the algorithm on the flow model operates correctly since, in the various cases, it makes it possible to achieve a better uniformity in flow conditions and in performance parameters, and, when possible, a reduction in the influence of the end-wall phenomena. For examples, diagrams of case (a) (figs. 7-9) present strong reductions in blade loss coefficients and diffusion factors, and, as a consequence of the more suitable distribution of blade loading, an extension of the region external to end-wall boundary layers with appreciable improvements in efficiency; case (b) (figs. 1-12), which was the nearest to the initial design point, is characterized by a more uniform load sharing between the two-stages, since a remarkable increase in blade loading is proposed for the second stage while the first rotor undergoes a reduction in flow deflection thus operating far from stall conditions; case (c) (figs. 13-15) presents the more important changes in blade geometry and flow condition, owing to the required higher increase in total head: also for this case, a good uniformity in stage loading is obtained and it has to be observed that the performance level is completely changed by the optimizing method as is clearly shown, for example, by the discharge static pressure distribution (figs. 6 and 13), that undergoes a considerable increase. Finally, it can be observed that, even in the presence of more appreciable boundary layers growth induced by the increase in load conditions, the optimized design provides improved values of total head and efficiency in the free stream region thus confirming that a correct choice for blade parameters was made. Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/22/217 Terms of Use: http://www.asme.org/about-asme/terms-of-use

11 Htt [m] 1 12 Htt C 98 99. 96 94 98. 2 4 6 8 1 12 92 2 4 6 8 1 12 n1 n I 2 4 6 8 1 12 2 4 6 8 1 12 1.5 c lr 4. 5 W _ 2r b1 1. 2r ir 4.^^ deg. ^ -- --2s. 5 ` s // lr 2r 3e 5^ - 2 -.5-1.5 off lr = 1st rotor ris = 1st stator 2r = 2nd rotor 2s = 2nd stator 2 4 6 8 1 12 2 4 6 8 1 12-2. t -2. 5 2 4 6 8 1 12 Nf 2 4 6 8 1 12 Nf Fig. 1 - Total to total head Htt and efficiency rk t, optimized variables Ar,,'I (continuous lines)) and (dashed lines), and optimized percent changes in blade Height OH% versus number of function evaluations Nf (case (a), its= 6 kg/s). Fig. 2 - Total to total head H, and efficieucv Th. optimized variables (i (continuous lines)) and ^ (dashed lines), and optimized percent changes in blade Height GH% versus number of function evaluations Nf (case (b), iii= 7 kg/s). 6 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/22/217 Terms of Use: http://www.asme.org/about-asme/terms-of-use

15 1 Htt Em] 95 9 1. y/ H.8 level curves of c /u x m 85 8.6 75. 95 ntt.94 2 4 6 8 1.4.2 X/L level curves of (p ou )/( pu 2 /2) m 2 4 6 8 1 2_r 1 Apb1 Acs 8 ^r--------- 2s Jl-------- r is deg. 6 4 --- -- --? s _II T r T^--r ^--^-- 2 4 6 8 1 22,. A H % ^-- 16 1 4-2 4 6 8 1 Nf Fig. 3 - Total to total head H tt and efficiency rt«, optimized variables 4 (continuous lines)) and 114 (dashed lines), and optimized percent changes in blade Height AH% versus number of function evaluations Nf (case (c), m= 8 kg/s). I "'-..17.33.5.67 X/L.63 1. 1..8.6.4.2 level curves of (p - p )/(p u 2 /2) ou m.8 ( X/ L Fig. 4.1 - Level curves of axial velocity, total pressure and static pressure for the initial compressor geometry, for the case (a) (m= 6 kg/s). 7 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/22/217 Terms of Use: http://www.asme.org/about-asme/terms-of-use

1. F.8 J 1..8 JII ^IIII M.6 i N N vi fp.6 ^o U) //7A Q^.4.4.2.2 ' o ^ o d.^.17.33.5.67 x/ L.83 1..8 (.17.33.5.67 x/l.83 1. level curves of 1. ^J z (p -pou)ll gum/2) 1.8 1..8 2.6f' F.4 r.2 1.9.6.4.2.8 1.7.9.17.33.5.67 x/l.83 1..g 1 x/l 1. level curves of (p - p ou )/( pu2/2) 1. level curves of (p - p )/( pu 2 /2) ou m.8l o a r.8 J.6.6 o.4 A N.4 V.2 ^ ^.^.17.33.5.67 x/l.83 1. Fig. 4.2 - Level curves of axial velocity, total pressure and static pressure for the optimized compressor geometry after the last iteration, for the case (a) (m= 6 kg/s). o.2.^^\ O.Oa.17.33.5.67x/f.83 1. Fig. 5.1 - Level curves of axial velocity, total pressure and static pressure for the initial compressor geometry, for the case (b) (rim= 7 kg/s). 8 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/22/217 Terms of Use: http://www.asme.org/about-asme/terms-of-use

a 1. level curves of c /u x m level curves of c x /um.8 o '.6 ' o.4 I..2.46 '^.17.33.5.67x/L.83 1.(.. 2.. x/l 2 1. 1. y/ H.5.8.8 1.25.6.6 1.45.4.4.2.2 B------------ x/ L.^.17.33.5.67/L.83 1. level curves of (p - p )/( ou pu2/2) m 1. level curves of (p - p )/(p u 2 /2) ou m N.8 n O 'UI o.6 NCA M) NM tn N O o N.4.2 1 LP ^ 4s -..17.33.5.67x/L.83 1. Fig. 5.2 - Level curves of axial velocity, total pressure and static pressure for the optimized compressor geometry after the last iteration, for the case (b) (m= 7 kg/s)..8.17.33.5.67.83 1. x/l Fig. 6.1 - Level curves of axial velocity, total pressure and static pressure for the initial compressor geometry, for the case (c) (m= 8 kg/s). 9 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/22/217 Terms of Use: http://www.asme.org/about-asme/terms-of-use

1. level curves of cx/u m do.8. 8 y /H 6.6 C.4-9 o q^.2 4 ) 2-7-- T--r-rte 6 7 8 9 1 11 12 Htt [ml.8 \.17.33.5.67 x/ L.83 1. \ 1. level curves of (po - pou)/(pun/2) 2.8. 6 4 \.6.85.87.89 n.91.93.95 tt.4.2 8. 6 4.8..17.33.5.67.83 1. x/l. 2 1. Y/H.8 level curves of (p - p )/( pu 2 /2) ou m.s \(_\ a o a \\.6 o.4 O 1.6 1.8 1.1 1.12 1.14 Rt t 1.. 8 y /H 6 4 2.2 O o^ 1. 1.2 1.4 1.6 1.8 ats C1lif1L^.8.17.33.5.67.83 1. x/l Fig. 6.2 - Level curves of axial velocity, total pressure and static pressure fon the optimized compressor geometry after the last iteration, for the case (c) (it= 8 kg/s). Fig. 7 - Radial distributions of total to total head Htl, efficiency 'rh t and pressure ratio (stt and total-static pressure ratio f t., for the case (a) (m = 6 kg/s). '[I Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/22/217 Terms of Use: http://www.asme.org/about-asme/terms-of-use

1. ---...- 1. 8 5 4 2.8 \.5 4. 2 1 2. 2. 1 2.2 2. 3 2. 4 4. 5. 6. 7. 8. 9. Xb x 1 1st. stator 1..8 \. 8 \ 6 \ 6 4 2 1.3 1. 4 1. 5 1. 6 1.7 1.8 2nd. rotor 4.2 1. 1. 4. 5. 6. Xb x 1 2nd. rotor.8.6 4. 2 BO r 6 4. 2 1.4 1.5 1.6 1.7 1.8 1.9 2nd. s tator 1. 2. 3. 4. 5. 5. Xb x 1 2nd. stator 1. 8 y /H.6 41 2^ INITIAL GEOMETRY... FIRST ITERATION - - - LAST ITERATION. 6. 4. 2 O T T-T-T T- T-TTT 1. 2 1.3 1. 4 1.5 1.5 1.7. 1. 2. 3. 4. 5. Xb x 1 Fig. 8 - Radial distributions of equivalent diffusion factor Deq for the case (a) (m = 6 kg/s(. Fig. 9 - Radial distributions of blade loss coefficient Xb for the case (a) (ii = 6 kg/s). Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/22/217 Terms of Use: http://www.asme.org/about-asme/terms-of-use

U 1.. 8 \.6 INITIAL GEOMETRY... \ \ FIRST ITERATION - - -.4 LAST ITERATION.2 1. 5 6 7 8 9 1 11 Htt [nil. 8 \ \. 6 \. 4 1.. 8.6 4. 2 1.. 8 6. 4 1st. rotor 1.7 1.8 1.9 2. 2.1. 2 T-T-^ 9. 92. 94. 95. 98 1. ntt 1.... \. 8 \ \. 6 \. 4 \.2 1.6 1.8 1.1 1.12 1.14 att 1.. 8. 6. 4 1. 2 }. 2 1..8. 6 4. 2 1. 8 6. 4 1 1. 4 1.5 1.6 1. 7 1.8 2nd. rotor 1.5 1.6 1.7 1.8 1.9 jo. 2 1. 1.2 1.4 1.6 1.8 Sts 1.1 1.2 1.3 1.4 1.5 1.6 Fig. 1 - Radial distributions of total to total head Htt, efficiency Tk t and pressure ratio ('sit and Fig. total-static pressure 11 ratio f for - the case (b) Radial distributions of equivalent (m = 7 kg/s). diffusion factor Deq for the case (b) (rim = 7 kg/s). 12 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/22/217 Terms of Use: http://www.asme.org/about-asme/terms-of-use

1st. rotor 1.... /H 1 6 \ J 2 / 1 3. 4. 5. 6. Xb x 1 1st. stator 1..8' y /H'. 6 1. 4 1.2' 1 1. 2. 3. 4. Xb x 1 2nd. rotor 1..8.6 ITIAL GEOMETRY... CFIRST ITERATION - - - ST ITERATION.4.2 2.5 3.5 4.5 5.5 Xb x 1 2nd. stator 1. '. 8 \.6 1. 4.2. 1. 2. 3. Xb x 1 Fig. 12 - Radial distributions of blade loss coefficient Xb for the case (b) (m = 7 kg/s).. 8. 6 \. 4 \. 2 1...... 5 6 7 8 9 1 11 Htt [mj. 8 \. 6 4 i.2'... 9.92.94.96.98 1. lit 1.. 8 \ \. 6 \. 4. 2 iii 1.6 1.8 1.1 1.12 1.14 Dtt 1.. 8. 6. 4....998 1. 1.2 1.4 1.6 ats Fig. 13 - Radial distributions of total to total head Htt, efficiency 11 t, and pressure ratio Os^ tand total-static pressure ratio (i for the case (c) lm = 8 kg/s). 13 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/22/217 Terms of Use: http://www.asme.org/about-asme/terms-of-use

1. 1. 8 8 6 6. 4 4 2 2 1.3 1.4 1.5 1.6 1.7 1.8 2. 3. 4. 5. Xb x 1 1. 1. 8. 8 6. 6 4 4 2 2 1.3 1.4 1.5 1.6 1.7 1.8 1. 2. 3. 4. Decd Xb x 1 1. 1.. 8 8 6. 6. 4. 4 2.2 1. 1.1 1.2 1.3 1.4 1.5 3. 5. 7. 9. Decd Xb x 1 2nd. st ator 1. 1.. 8. 6. 6. 4 4. 2.2 1.2 1.4 1.6 U. 1.8 1. d. J. 2. 2.2 Xb x 1 Fig. 14 - Radial distributions of equivalent diffusion Fig. 15 - Radial distributions of blade loss factor Deq for the case (c) (m = 3 kg/s). coefficient Xb for the case (c) (fig _ 5 kg/s 1. 14 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/22/217 Terms of Use: http://www.asme.org/about-asme/terms-of-use

CONCLUSIONS The results obtained by applying the proposed method for optimized design show its capability in making a choice of geometrical parameters of multistage compressors that appears particularly suitable to improve both flow pattern and overall performance. It must be remarked that the original feature of such a methodology lies in the possibilty of adopting an optimization technique to control a quite complex flow model simulating numerous of the main phenomena occurring in a multistage turbomachine: at the same time, since the design procedure starts from an assigned geometry, it seems possible to improve performance by minimizing the entity of changes in meridionaal shape and blade features, so exploiting an already acquired technology. Finally, it should be noted that the method can be employed with more sofisticated flow models and with an increased number of free variables and constraints, thus pursuing the same objective as the one illustrated in the present paper with an improved accuracy and a higher performance prediction level. AKNOWLEDGMENTS The research was carried out with the financial contribution of Italian Government (MPI 6% fund). REFERENCES Beveridge, G.S.G., and Sheckter R.S., 197, Optimization in Theory and Practice, ed., Mc Graw Hill, New York. Cuan-gang Gu, and Yong-miao Miao, 1987,"Blade Design of Axial-Flow Compressors by the Method of Optimal Control Theory," ASME Journal of Turbomachinery, vol. 19, pp. 99-13. De Ruick,J., Hirsch,C., and Kool, P., 1979, "An Axial Compressor End-Wall Boundary Layer Calculation Method," ASME Journal of Engineering for Power, vol. 11, pp. 233-245. De Ruick,J., and Hirsch, C., 1981, "Investigation of an Axial Flow Boundary Layer Prediction Method ", ASME Journal of Engineering for Power, vol. 13, pp. 2-33. Dring, R.P. and Josylin, H.D., 1986, "Through flow Modeling of Axial Turbomachinery," ASME Journal of Engineering for Gas Turbines and Power, vol. 18, pp. 246-253. Gill,P.H., Murray, W., and Wright, M.H., 1984, Practical Optimization, ed.,academic Press, London. Glynn, D.R., and Marsh, H., 198, "Secondary flows in Annular Cascades," International Journal of Heat and Fluid Flow, vol.2, No. 1, pp.29-33. Hirsch, C., 1974, "Flow Prediction in Axial Flow Compressors Including End-Wall Boundary Layer," ASME Journal of Engineering for Power, vol.96, pp. 413-426. Horlock, J.H., 1971, "On Entropy Production in Adiabatic Flow in Turbomachines," ASME Journal of Basic Engineering, Dec., pp. 587-593. Horlock, J.H., 1975, "Secondary Flows," VKI Lecture Series 72, Bruxelles. Horlock, J.H., and Lakshminarayana, B., 1975, "Secondary Flows: Theory, Experiment and Application in Turbomachinery Aerodynamics," Annual Review of Fluid Mechanics, vol. 51, p. 247. Horlock, J.H., 1962, "Flow in Meridional Plane in Relation with Stage and Compressor Design," VKI Course Note 27, Bruxelles. Katsanis, T., 1966, "Use of Arbitrary Quasi-Orthogonals for Calculating Flow Distribution in a Turbomachine ASME Journal of Engineering for Power, vol. 88, pp. 197-22. Koch, C.C, and Smith, L.H., 1976, "Loss Sources and Magnitudes in Axial Flow Compressors," ASME Journal of Engineering for Power, July, pp. 411-424. Koch, C.C., 1981, "Stalling Pressure Rise Capability of Axial Compressor Stages," ASME Journal of Engineering for Power, vol. 13, pp. 645-656. Lakshminarayana, B., and Popovsky, P., 1987, "Three Dimensional Boundary Layer on a Compressor Rotor Blade at Peak Pressure Rise Coefficient," ASME Journal of Turbomachinery, vol. 19, pp. 91-98. Lieblein,S., 1965a, "Experimental Flow in Two Dimensional Cascades," NASA SP 36, pp. 181-226. Lieblein, S., Jackson,R.J., and Robbins W.H., 1965b, "Blade Element Flow in Annular Cascades," NASA SP 36, pp. 227-252. Lieblein, S., and Roudebush, W.H., 1965c, "Viscous Flow in Two Dimensional Cascades," NASA SP 36, p. 151. Marsh,H., 1971, "The Uniqueness of Turbomachinery Flow Calculations Using the Streamline Curvature and Matrix Through Flow Methods," Journal of Mechanical Engineering Science, vol.13, n. 6, pp. 376-379. Mc Kenzie, A.B., 198, "The Design of Axial Compressor Blading Based on Tests of a Low Speed Compressor," Proceedings of Institutions of Mechanical Engineering, vol. 194, pp. 13-111. Mellor, M., and Wood, G.L., 1971,"An Axial Compressor end-wall Boundary Layer Theory," ASME Journal of Basic Engineering, Vol. 93, pp. 3-316. Pittaluga, F., 1985, "Una Metodologia Generale per L'analisi del Flusso e it Progetto di Compressori Assiali", La Termotecnica, Nov., pp. 27-35. Rao, S.S., and Gupta,R.S., 198, "Optimum Design of Axial Flow Turbine Stage (Part I and II)," ASME Journal of Engineering for Power, vol. 12, pp. 782-797. Rizzo, G., and Tuccillo, R., 1983,"Sul Calcolo nel Piano Meridiano di Turbomacchine Operatrici Assiali," Test-Cases di Fluidodinamica delle Macchine, ed., CLUP Milano, pp. 257-288. Rizzo,G., and Tuccillo, R., 1987,"Tecniche per la Progettazione Ottimale di Turbomacchine Multistadio," Proceedings of XLII ATI Congress, ed. CLEUP, Padova, pp. III- 29-224. Roberts, W.B., Serovy, K., and Andercock, D.M., 1986, "Modeling of the 3-D Flow Effects on Deviation Angle for Axial Compressor Middle Stage," ASME Journal of Engineering for Gas Turbines and Power, vol. 18, pp. 131-137. Salvage, J.W., 1974, "A Review of the Current Concept of Cascade Secondary Flows Effects", VKI Technical Note 95, Bruxelles. Satta, A. et al., 1988, Metodi di Calcolo Fluidodinamico a Confronto con un Caso Sperimentale, ed., CLEUP, Padova. Starke J.,1981, "The Effect of the Axial Velocity Density Ratio on the Aerodynamic Coefficients of Compressor Cascades", ASME Journal of Engineering for Power, vol. 13, pp.21-219. Stark, U., and Hoisel H., 1981,"The Combined Effect of Axial Velocity Density Ratio and Aspect Ratio on Compressor Cascade Performance," ASME Journal of Engineering for Power, vol. 13, pp. 247-255. Tuccillo, R., 1988a, "Una Metodologia di Analisi del Flusso nel Piano Meridiano per Compressori Assiali Multistadio," paper published in (Satta et a!., 1988), pp. 135-175. Tuccillo, R., 1988b, "Analysis of the Influence of the End-Wall Boundary Layer Growth on the Performance of Multistage Compressor, paper to be published in the International Journal of Turbo & Jet Engines. Vavra M.H., 196, Aero-Thermodynamics and Flow in Turbomachines, ed., John Wiley & Sons, New York. 15 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/22/217 Terms of Use: http://www.asme.org/about-asme/terms-of-use