Optim Eng DOI 10.1007/s11081-013-9242-6

Three-dimensional turbulent optimization of vaned diffusers for centrifugal compressors based on metamodel-assisted genetic algorithms Mattia Olivero · David Pasquale · Antonio Ghidoni · Stefano Rebay

Received: 2 March 2012 / Accepted: 20 May 2013 © Springer Science+Business Media New York 2013

Abstract In this work, the performance of an automotive turbocharger centrifugal compressor, to be used in a microturbine for combined heat and power applications, have been improved through a design optimization procedure for vaned diffusers. This methodology couples a genetic algorithm with a three-dimensional turbulent computational fluid dynamics code. The computational costs have been reduced by using a Kriging metamodel to assist the genetic algorithm. The simulations have been performed by considering both the impeller and vaned diffuser, in order to account for the turbulent, three-dimensional, and non-uniform flow conditions at the diffuser inlet. A multi-objective optimization problem has been solved by minimizing two objective functions, which depend on the compressor stage total-to-static pressure ratio and total-to-total isentropic efficiency. The design variables are the position and inclination of the diffuser vanes leading and trailing edges, the vane number, and the diffuser outlet radius. Three optimized geometries extrapolated from the Pareto front exhibit higher static pressure recovery than the vaneless diffuser, but only one has better efficiency. Nevertheless, the performance of the current compressor can be improved by substituting the vaneless diffuser with a vaned one. Keywords Aerodynamic optimization · Vaned diffuser · Centrifugal compressor · Metamodel · Genetic algorithm · Computational fluid dynamics

B

M. Olivero ( ) · D. Pasquale · A. Ghidoni · S. Rebay Dipartimento di Ingegneria Meccanica e Industriale, Università degli Studi di Brescia, via Branze 38, 25123 Brescia, Italy e-mail: [email protected] D. Pasquale e-mail: [email protected] A. Ghidoni e-mail: [email protected] S. Rebay e-mail: [email protected]

M. Olivero et al.

1 Introduction Centrifugal compressors are widely used in small gas turbines for transport applications and power generation, in automotive turbochargers, and in the process industry for fluids compression or conveyance. Therefore, they can be machined for a wide range of pressure ratios and flow rates, as their size and rotational speed respectively vary from less than 1 cm to more than 2 m, and from a few krpm up to 500 krpm. A centrifugal compressor consists of an impeller, a diffuser, and a volute. A large amount of energy is transferred to the fluid by the rotating impeller, but although the latter is usually designed for very good diffusion, up to 50 % of the energy transferred to the fluid remains available at the impeller outlet as kinetic energy. This energy is then converted into static pressure in the stationary diffuser, which is however responsible for at least 1/3 of the overall fluid dynamic losses occurring in a centrifugal compressor. As a consequence, the diffuser strongly affects the performance of compressor stages. A diffuser can be either vaneless or vaned. The former allows for a wider operating range, but shows lower efficiency and pressure recovery, mainly because of the long logarithmic spiral flow path which causes high friction losses. This flow path cannot be easily reduced, since the desired amount of diffusion depends on the diffuser outlet radius. The most commonly adopted approach to shorten the flow path is to insert diffuser vanes. This would thus increase the compressor stage efficiency and pressure rise, at the expense of a reduction of the operating range. Vaned diffusers (VDs) design usually relies on experimental data and on engineers’ experience. These aspects however represent a limit when a VD has to be employed in small-sized turbocharger centrifugal compressors, since a solid design procedure has not been developed yet. Recently, the coupling of computational fluid dynamics (CFD) codes with optimization techniques has however gained interest in turbomachinery, and proved to be helpful to overcome the weakness of designing a VD on the basis of empirical correlations. Nevertheless, only few studies on this topic can be found in the literature, and even less works about vaned diffusers optimization for improving the performance of turbocharger centrifugal compressors to be used for power generation are available. Benini and Tourlidakis (2001) performed a three-dimensional (3D) design optimization of VDs for centrifugal compressors, by coupling a multi-objective genetic algorithm (MOGA) with a commercial CFD code. The authors ignored the impellerdiffuser interaction in their study, and assumed uniform flow conditions at the diffuser inlet. They evaluated the effectiveness of the optimization method by comparing the efficiency and pressure recovery of an existing VD with those of optimized diffusers. Micheli et al. (2003) redesigned the VD of a centrifugal compressor through a CFD code coupled firstly to a MOGA, and secondly to a non-linear Nelder-Mead Simplex method. In the former case, the diffuser static pressure recovery coefficient and the total-to-total isentropic efficiency were optimized, wherelse in the latter case only the static pressure recovery coefficient was used as objective function. The optimization has been carried out at the compressor design point, and yielded to one optimized geometry having better pressure recovery, and to another with higher efficiency.

Vaned diffusers optimization based on metamodel-assisted genetic

Benini and Toffolo (2003) presented the constrained optimization of a diffuser apparatus, consisting of radial and deswirl cascades. The optimization has been performed through a MOGA interfaced to a parametric code for the geometries generation, and to a CFD code for evaluating the candidates. The maximum pressure rise has been achieved with the lower stagger angle of the radial profile, while the total pressure loss has been minimized with the lower camber of the deswirl profile. Wang et al. (2006) performed the optimization of VDs for high-speed centrifugal compressors by means of a 3D CFD turbulent flow analysis, a genetic algorithm (GA), and a Kriging surrogate model, which substituted the time-consuming numerical computations and accelerated the optimization procedure. The objective was the largest decrement of the total pressure loss, while maintaining the static pressure recovery. For the given operating condition, the total pressure loss was reduced by 5.5 %, while the static pressure recovery was increased by 2.6 %. Xi et al. (2008) carried out the design optimization of the VD for a 100-kWmicroturbine centrifugal compressor. A forward-loaded diffuser and a conventional airfoil one have been redesigned by means of 3D CFD calculations and a surrogate model, which considerably accelerated the optimization process. The optimization aimed at the maximum stage total-to-static isentropic efficiency. The CFD predictions showed that the latter was improved at design and off-design conditions. Kim et al. (2010) optimized both the impeller and diffuser of a centrifugal compressor by means of the response surface method. Better performance in terms of total pressure ratio, pressure recovery, and efficiency were obtained with respect to the original design, since the total pressure loss decreased dramatically. This work aims at improving the performance of an automotive turbocharger centrifugal compressor, to be used in a microturbine for combined heat and power applications. Today’s world market for turbochargers is around two millions units per year (Soares 2007), and is characterized by relatively low production prices. Moreover, small gas turbines, of the size and power of microturbines, serve as auxiliary power units on airplanes, such that decades of experience with these applications constitute the basis for the technological development of microturbines. Automotive turbocharger centrifugal compressors represent thus an interesting opportunity for microturbines. However, they usually employ a vaneless diffuser, since a large operating range is needed and manufacturing costs have to be kept low. Conversely, VDs are typically used in high-performance centrifugal compressors mounted on microturbines, as the latter require higher levels of efficiency and pressure recovery, compact size, and narrower operating ranges. Therefore, in this paper a set of empirical correlations have been firstly used to design a preliminary VD. Subsequently, a design optimization procedure for VDs has been developed by coupling a GA with 3D turbulent flow simulations. In order to reduce the computational costs, a metamodel has been used to assist the GA during the optimization process. This procedure takes into account the turbulent, 3D, and non-uniform flow at the diffuser inlet, as the CFD computations are performed over a domain which includes both the impeller and vaned diffuser. As a consequence, the optimization method described here extends the limited literature available on the subject.

M. Olivero et al.

2 The current centrifugal compressor The turbocharger compressor analyzed in this paper is equipped with an unshrouded centrifugal impeller, a vaneless diffuser, and a volute. The impeller has six full and six splitter blades, with a backsweep angle equal to −34◦ . The blade angle at the inducer tip is equal to −64◦ . The blades have constant thickness, from the leading to the trailing edge, equal to 0.5 mm. The impeller tip clearance in running conditions is not known, but a clearance gap equal to 0.375 mm has been measured at the inducer, with the compressor standing still in ambient air, as the difference between the shroud and the blade tip diameters. Figure 1 shows the impeller and diffuser main dimensions. The impeller inlet hub and shroud diameters are equal to 9.6 and 26.2 mm, respectively, while the impeller outlet diameter is equal to 39 mm. The pinched vaneless diffuser is 2.6-mm-high at the inlet and 1.7-mm-high at the outlet, where the diameter is equal to 66 mm. At the design mass flow rate and rotational speed of 50 g/s and 240 krpm, respectively, the compressor delivers at the diffuser outlet a total-to-total pressure ratio and an isentropic efficiency equal to 3.3 and 0.69, respectively. Furthermore, the diffuser static pressure recovery coefficient (CP ) and the total pressure loss coefficient (K) are equal to 0.48 and 0.22, respectively. The two coefficients are defined as p5 − p 2 , p02 − p2 p02 − p05 K= , p02 − p2

CP =

(1) (2)

where subscripts 2 and 5 indicate the diffuser inlet and outlet, respectively, while subscript 0 represents the total condition. Fig. 1 Main dimensions (in mm) of the centrifugal impeller and vaneless diffuser

Vaned diffusers optimization based on metamodel-assisted genetic

According to Kim et al. (2009), vaneless diffusers typically have a static pressure recovery coefficient in the range from 0.45 to 0.6, and vaned diffusers in the range from 0.55 to 0.70. Japikse and Baines (1994) claimed that typical values of pressure recovery can range anywhere from 0.3 to 0.4 up to peaks of 0.8-0.9, although high performance diffusers more commonly exhibit pressure recovery from 0.6 to 0.7. The pressure recovery in this vaneless diffuser is thus located at the low-end of the range, since only about 50 % of the high kinetic energy available at the diffuser inlet is converted into static pressure. Therefore, the employment of a vaned diffuser can be beneficial in order to increase the pressure recovery and to reduce friction losses.

3 The vaned diffuser geometry A vane profile with a circular arc camberline (Spraker et al. 1994) has been adopted for this optimization work. The choice of such a simple geometry has been based on manufacturing and economic reasons, which are fundamental when considering the very small dimensions of the turbocharger centrifugal compressor investigated here. Furthermore, this configuration can be considered as a first step in the design and optimization of VDs which will later involve more complex vane profiles. Figure 2 shows the geometry of the circular arc VD. A constant thickness distribution equal to 1 mm has been applied to the camberline in order to define the pressure and suction sides, wherelse the vane profile is represented by the use of two NURBS curves. Although the camberline consists of a circular arc, the NURBS approach has been utilized in view of future optimization works involving airfoil-shaped diffuser vanes. The camberline is defined by the radius of curvature rc , while the chord length is c. The vane angles at the leading (α3v ) and trailing (α4v ) edges are connected to each other by the vane turning angle (VTA). The leading and trailing edges radii are respectively r3 and r4 , wherelse the diffuser inlet and outlet are located at r2 and r5 , respectively. The stagger angle is represented by θ . Fig. 2 Details of the circular arc vaned diffuser. The camberline is defined by the radius of curvature rc and its chord length is c. The vane angles at the leading (α3v ) and trailing (α4v ) edges are connected to each other by the vane turning angle VTA. The leading and trailing edges radii are r3 and r4 , respectively, while the diffuser inlet and outlet are located at r2 and r5 , respectively. The stagger angle θ is shown as well

M. Olivero et al. Table 1 Definition ranges for the design variables and corresponding values for the preliminary design RR3

RR4

RR5

α3v [◦ ]

VTA [◦ ]

Z

Minimum

1.05

1.20

1.25

60.0

5

7

Preliminary

1.15

1.67

1.692

74.5

10

13

Maximum

1.15

1.67

1.95

80.0

20

21

The design variables chosen for the optimization problem are the radius ratios at the diffuser vanes leading (RR3 = r3 /r2 ) and trailing (RR4 = r4 /r2 ) edges, the radius ratio at the diffuser outlet (RR5 = r5 /r2 ), the vane angle at the leading edge, the vane turning angle, and the vane number (Z). Table 1 shows the ranges of variation for the design variables, and lists the corresponding values of a preliminary configuration, which has been designed on the basis of empirical correlations found in the literature. The works of Rodgers (1982), Inoue and Cumpsty (1984), and Cumpsty (1989) have been used to define RR3 , while RR4 has been set according to a formula given by Aungier (2000). The latter reference has been utilized as well to choose Z, wherelse α3v and VTA have been specified following the works of Engeda (1997) and Eynon and Whitfield (1997), respectively. The outlet radius ratio RR5 corresponds to that of the vaneless diffuser.

4 The optimization problem The flow at the diffuser inlet is turbulent, 3D, and non-uniform, and affects the diffuser performance. Diffuser inlet flow conditions are influenced by the flow leaving the impeller, whose velocity usually varies due to recirculation and separation from the pressure to the suction sides, as well as from hub to shroud. Furthermore, at high pressure ratios, the tangential component of the flow velocity will be very high, and the flow could be locally transonic, causing shock waves and boundary layer separation. As a consequence, in order to account for those flow phenomena due to the impeller-diffuser interaction, the optimization work has been performed by evaluating the candidate solutions through a numerical domain which includes both the impeller and vaned diffuser. Although the two most important parameters to evaluate the performance of a diffuser are the static pressure recovery coefficient (Eq. (1)) and the total pressure loss coefficient (Eq. (2)), here the performance of the whole compressor stage (i.e., impeller and vaned diffuser) have then been assessed by quantifying the total-to-static pressure ratio πts and the total-to-total isentropic efficiency ηtt,is , respectively. Therefore, two objective functions which respectively depend on πts , in order to quantify the static pressure recovery, and on ηtt,is , in order to estimate the total pressure loss, have been minimized separately and simultaneously. The following multiobjective optimization problem has thus been solved minimize ϕ1 = 1/πts =

p01 , p5

(3)

Vaned diffusers optimization based on metamodel-assisted genetic

ϕ2 = 1 − ηtt,is = 1 −  subject to ψ =

Aout

) ( pp05 01

sgn(v · n)dA Aout

 1 where sgn(x) = 0

γ −1 γ

T05 T01

−1

−1

≥ 0.95,

if x > 0, otherwise,

,

(4)

(5) (6)

and where p01 and T01 are the total pressure and temperature at the compressor inlet, p05 , T05 , and p5 are the total pressure and temperature and the static pressure at the diffuser outlet, γ is the specific heats ratio, v is the velocity vector, and n is a vector normal to the diffuser outlet area Aout . The constraint ψ has been introduced in order to minimize the backflow at the diffuser outlet, since flow separation on the vane suction side, and consequent recirculation, are expected for a large number of geometries. As a consequence, according to Eq. (5), all the individuals for which backflow occurred on more than 5 % of the diffuser outlet area have been considered as not fulfilling the constraint ψ , and thus have been discarded. The optimization is driven by a GA which is an implementation of the Nondominated Sorting Genetic Algorithm II (NSGA-II) introduced by Deb et al. (2002). In order to reduce the computational costs due to the fluid dynamics computations, metamodels can be usually employed to assist GAs during the optimization process without altering the quality of the final result (Giannakoglou 2002). Here, a Kriging metamodel (Krige 1951) with a quadratic regression function and a Gaussian correlation function has been selected to assist the GA. For each candidate analyzed during the optimization process, ϕ1 , ϕ2 , and ψ have been estimated by a dedicated metamodel. The optimization presented in this paper has been performed through the so-called “off-line trained metamodel” technique (Pierret and Van Den Braembussche 1999; Verstraete et al. 2007), which utilizes the metamodel as an evaluation tool during all the evolutionary process. After several generations the evolution is stopped and the best individual is analyzed by the CFD tool (i.e., grid generator, CFD solver, and post-processing). The difference between the value calculated with the latter and that predicted by the metamodel is a direct measure of the metamodel accuracy. At the beginning of the optimization process, the difference can be expectedly large, but as soon as the new evaluated individual is added to the training database, the metamodel accuracy will improve in the region of the design space where it was previously predicting the best individual. This feedback is the most valuable part of the algorithm, as it makes the system self-learning, such that it somehow behaves like a human being which learns from his previous mistakes. Figure 3 illustrates the layout of the implemented optimization procedure. The main components are the geometry generator, the CFD tool, and the optimization blocks (i.e., the GA, the metamodel, the initial database, and the design of experiments (DOE)). The initial database was made of 48 geometries and has been generated by means of a Latin hypercube sampling strategy (Tang 1993; Ye 1998). By minimizing a potential energy function, this technique is able to spread the generated

M. Olivero et al. Fig. 3 Layout of the metamodel-assisted GA optimization procedure

points as uniformly as possible within the design space, ensuring that all its portions are represented. The number of initial individuals has been selected on the basis of the number of design variables of the problem (Van Den Braembussche 2010; Rao 2009). For every optimization cycle, the NSGA-II computed the evolution of 48 individuals over 2,000 generations. The population size has been chosen as a function of the number of design variables, as suggested by Rao (2009) and Van Den Braembussche (2010). The probabilities of cross-over and mutation have been specified equal to 0.75 and 0.015, respectively, while elitism has been used. The multi-objective optimization problem has been solved in the optimization environment Nexus (2010d), where the GA has been coupled to an in-house built parametric tool for generating the VDs geometry, and to the commercial CFD code ANSYS CFX 13.0 (2010b), which solves the 3D Reynolds-averaged Navier-Stokes (RANS) equations using a finite-volume-based method.

5 The computational fluid dynamics tool 5.1 Grid generator The 3D impeller geometry has been created with ANSYS BladeGen 13.0 (2010a). The impeller has been modeled without considering the tip clearance, while a 13.45-mmlong stationary inlet duct has been placed upstream of the impeller, in order to reduce the effects of the inlet boundary condition on the potential field of the blade. At the diffuser outlet, the computational domain has been extended further downstream to ensure that the outlet boundary condition does not affect the flow field around the diffuser vane trailing edge. The volute has not been included in the computational domain. The structured grids of an impeller passage containing a full and a splitter blade, and of the vaned diffuser containing one vane have been generated with ANSYS TurboGrid 13.0 (2010c), while the structured grid of the inlet duct has been generated using GAMBIT 2.4.6 (2007). A relatively coarse grid of about 42,000 elements has been generated for the impeller (Fig. 4(a)), according to Tsuei et al. (1999), who investigated the influence of a

Vaned diffusers optimization based on metamodel-assisted genetic

Fig. 4 Grids of the computational domain for a generic vaned diffuser

large number of computing options by comparing experimental and numerical results for seven turbomachines, including centrifugal compressors. By using two commercial CFD codes, and following a hypothesis formulated by Denton (1994), the authors concluded that most of the important effects in a turbomachinery blade row may be captured using a coarse grid of only 30,000 nodes, since almost all their results regarding the efficiency and pressure ratio agreed well with the test data. Therefore, in the early phases of turbomachinery design and optimization, where a significant number of repeated CFD computations have to be performed, the use of very complicated and fine grids seems reasonably useless. On the other hand, no guidelines can be found in the literature about the grid generation for VDs. At the same time, various cell numbers have been used in the works listed in Sect. 1. Here, a grid of about 164,000 elements has been created for the VD (Fig. 4(b)), on the basis of a grid sensitivity analysis performed at the compressor design point. Total and static pressures and temperatures, absolute velocity, and velocities components, have been calculated at the diffuser inlet and outlet and at the vanes leading and trailing edges, and compared for three grid sizes. The differences between the coarse (∼80,000 elements) and the baseline (∼164,000 elements) grids are negligible, wherelse they are slightly larger between the baseline and fine (∼328,000 elements) grids. Furthermore, the VD grid has been created through the automatic topology and meshing (ATM) optimized feature available in ANSYS TurboGrid 13.0. The ATM optimized topology is an alternative to the traditional topologies, enables the creation of high-quality grids with minimal effort, since there is no need for control points adjustment, and automatically computes a default grid by setting the base grid dimensions. Each unique grid dimension has an edge refinement factor that is multiplied by the base grid dimension, and a global size factor to determine the final size of the grid dimension. In order to resolve the boundary layer on the diffuser vanes and end-walls, the dimensionless wall distance y + of the first grid node off the wall has been set equal to 1.

M. Olivero et al. Fig. 5 Boundaries definition of the computational domain for a generic vaned diffuser

5.2 CFD solver The mass, momentum, energy, and turbulence equations have been solved with a high-resolution advection scheme. The total energy model has been switched on to include the heat transfer, and the shear stress transport k − ω (Menter 1993, 1994) turbulence model has been adopted. The convergence of the steady-state simulations has been set to 1 × 10−5 for the root mean square residuals of all the equations, and has been controlled through a physical timescale equal to 1/ω, where ω is the angular velocity expressed in rad/s. The working fluid is air, with a molar mass equal to 28.96 kg/ kmol. The thermodynamic properties have been obtained with the ideal gas equation of state, and with a four-coefficient zero-pressure polynomial equation for the specific heat capacity. Figure 5 shows the boundaries definition of the computational domain. At the inlet, total pressure and temperature equal to 101,325 Pa and 293.15 K have been specified, respectively. The inflow direction is normal to the inlet plane, and the turbulence intensity has been kept at 5 %, which is the recommended value when no information about the inlet turbulence is available. At the outlet, a mass flow rate has been imposed. The lateral faces of the domain have been specified as rotational periodic boundaries, and the walls have been modeled as adiabatic and smooth. The impeller shroud has been specified as counter-rotating, such that the relative motion between the rotating impeller and the stationary shroud is captured. The no-slip condition has been imposed on all walls, apart from the inlet duct shroud and hub, where the free-slip condition has been used, in order not to account for losses in this portion of the domain. For the steady-state simulations, a mixing plane approach has been used at the interface between the rotating impeller and the stationary vaned diffuser. This method performs a circumferential averaging of the fluxes through bands on the interface, and steady-state solutions are then obtained in each reference frame. As a consequence, the transient interaction effects are not accounted for.

Vaned diffusers optimization based on metamodel-assisted genetic

6 Results and discussion This section presents the results of the 3D RANS multi-objective optimization, which has been carried out at the compressor design point. The behaviour of the optimization technique is firstly analyzed, then the preliminary and optimized designs are compared with each other. At every optimization cycle, the metamodel-assisted GA creates a Pareto front based on the values predicted by the metamodels. This front is made of several Paretooptimal individuals, among which the five most representative designs of the front, in terms of the objective functions ϕ1 and ϕ2 , are extracted, analyzed by means of the CFD tool, and added to the training database for the successive cycles. In particular, the five extrapolated geometries have been selected such that one exhibits the lowest ϕ1 and the highest ϕ2 , one shows the highest ϕ1 and the lowest ϕ2 , while three designs are equally spaced along the front, between the first two individuals, thus having intermediate objective functions ϕ1 and ϕ2 with respect to the other two designs. The optimization process has been stopped after 45 optimization cycles. Figure 6 shows the deviations between the computed and estimated values for ϕ1 , ϕ2 , and ψ , for one individual (i.e., the one exhibiting the lowest ϕ1 and the highest ϕ2 ) among the five extrapolated from the Pareto front. The differences are high in the early stages of the optimization process, as the predicted values underestimate the calculated ones, then they significantly decrease as the number of the individuals in the database increases, leading to more accurate predictions by the metamodels. Only about 10 cycles for ϕ1 and about 20 cycles for ϕ2 and ψ were necessary to obtain a good agreement between the estimated and computed values. Figure 7(a) shows all the individuals evaluated with the CFD tool, and the Pareto front which has been created at the end of the optimization process. Two different symbols have been given to the individuals which satisfy the constraint (ψ ≥ 0.95), and to those which do not (ψ < 0.95). The latter represent about 40 % of all the computed individuals. All the individuals are distributed within wide ranges for both objective functions, as a consequence of the broad ranges assigned to the design variables. The preliminary (DP ) and vaneless (DVNL ) designs are visible as well. The former exhibits better pressure recovery than the latter, but has poor performance in terms of efficiency. Furthermore, most of the computed individuals perform better

Fig. 6 Comparison between computed and estimated objective functions (ϕ1 and ϕ2 ) and constraint (ψ ) for the individual of the Pareto front with lowest ϕ1 and highest ϕ2

M. Olivero et al.

Fig. 7 All the computed individuals, the Pareto front, and the vaneless (DVNL ), preliminary (DP ), and optimized (D1 , D2 , D3 ) designs Table 2 Comparison between objective functions (ϕ1 , ϕ2 ) and constraint (ψ ) for the vaneless (DVNL ), preliminary (DP ), and optimized (D1 , D2 , D3 ) designs

ϕ1

ϕ2

ψ

DVNL

0.346

0.308

1.000

DP

0.313

0.324

0.923

D1

0.297

0.324

VNL [%]

P [%] D2

VNL [%]

P [%] D3

−14.2 −5.1 0.309

0.972

+5.2

−2.8

0.0

+5.3

0.314

0.992

−10.5

+1.9

−0.8

−1.1

−3.2

+7.4

0.322

0.296

0.975

VNL [%]

−6.8

−4.1

−2.5

P [%]

+3.0

−8.6

+5.6

than the vaneless configuration in terms of pressure recovery, but only a few have better efficiency. Figure 7(b) illustrates a close-up view of the Pareto front. The front is made of eleven individuals, among which three representative configurations can be identified. D1 is the individual with the lowest ϕ1 , D3 is the one with the lowest ϕ2 , and D2 exhibits intermediate performance. Five individuals located on the left-hand side of the Pareto front are also noticeble in Fig. 7(b). They have been discarded because did not fulfill the constraint. Table 2 shows the comparison between the two objective functions and the constraint calculated for the vaneless, preliminary, and optimized designs. D1 , D2 , D3 show higher static pressure recovery than DVNL , as confirmed by the values of ϕ1 , which decreased the most for D1 (−14.2 %). Conversely, the use of a vaned diffuser can be disadvantageous in terms of stage efficiency. D3 is the only configuration that performs better than DVNL (4.1 %-decrease for ϕ2 ), but D1 and D2 have lower effi-

Vaned diffusers optimization based on metamodel-assisted genetic Table 3 Comparison between compressor (πts , ηtt,is ) and diffuser (CP, K) performance for the vaneless (DVNL ), preliminary (DP ), and optimized (D1 , D2 , D3 ) designs

πts

ηtt,is

CP

K

DVNL

2.892

0.692

0.482

0.216

DP

3.196

0.676

0.554

0.327

D1

3.368

0.676

0.612

VNL [%]

P [%] D2

VNL [%]

P [%] D3

+16.5 +5.4 3.232

0.319

−2.3

+27.0

+47.7

0.0

+10.5

−2.4

0.686

0.596

0.278

+11.8

−0.8

+23.7

+28.7

+1.1

+1.6

+7.6

−15.0

3.103

0.704

0.549

0.235

VNL [%]

+7.3

+1.8

+13.9

+8.8

P [%]

−2.9

+4.5

−0.9

−28.1

ciencies. Nevertheless, the efficiency of the current centrifugal compressor featuring a vaneless diffuser can be increased by using a vaned diffuser instead. Further on, D1 and D2 have better pressure recovery than DP , since ϕ1 is 5.1 % and 1.1 % lower, respectively, while the value for D3 is 3.0 % higher. D2 and D3 have higher efficiencies than DP (ϕ2 is 3.2 % and 8.6 % lower, respectively), while D1 exhibits the same efficiency. The three optimized geometries also have higher values of the constraint in comparison with DP , as a consequence of lower backflow at the diffuser outlet. Table 3 shows the values of the total-to-static pressure ratio, total-to-total isentropic effiency, static pressure recovery coefficient, and total pressure loss coefficient for the vaneless, preliminary, and optimized designs. This comparison allows for a distinction upon the diffuser and stage performance. When comparing the vaneless diffuser to the optimized designs, one can note that the latter have higher values of πts than the former, with decreasing deviations from D1 (+16.5 %) to D3 (+7.3 %), as a consequence of higher static pressure recovery coefficients (+27.0 % for to D1 , +23.7 % for to D1 , and +13.9 % for D3 ). However, another conclusion can be drawn in terms of K and ηtt,is . The optimized geometries show very high values of the total pressure loss coefficient in comparison to the value of DVNL (+47.7 %, +28.7 %, and +8.8 % for D1 , D2 , and D3 , respectively), but the differences in terms of totalto-total isentropic efficiency are small. In particular, while D1 and D2 have slightly lower efficiencies than DVNL , D3 shows a 1.8 %-higher ηtt,is , although K is 8.8 % higher than that obtained with the vaneless diffuser. Furthermore, in accordance with the previous results, D1 and D2 have higher values of πts than DP , while D3 has a lower total-to-static pressure ratio. This is reflected as well by looking at the CP -values, which are higher for D1 and D2 and lower for D3 , in comparison to that of DP . On the contrary, some differences can be noticed about the relationship between the total pressure loss coefficient and the efficiency. The three optimized configurations exhibit lower K-values than the preliminary diffuser, with the highest difference calculated for D3 (−28.1 %), but the differences in terms of ηtt,is are one order of magnitude lower. In general, a lower total pressure loss coefficient leads to a higher efficiency, although this does not hold for D1 .

M. Olivero et al. Fig. 8 Comparison between preliminary (DP ) and optimized (D1 , D2 , D3 ) designs

Table 4 Comparison between design variables for the preliminary (DP ) and optimized (D1 , D2 , D3 ) designs

RR3

RR4

RR5

α3v [◦ ]

VTA [◦ ]

Z 13

DP

1.150

1.670

1.692

74.5

10.0

D1

1.139

1.611

1.825

72.9

10.9

9

D2

1.110

1.489

1.621

75.2

10.4

11

D3

1.149

1.441

1.511

78.4

10.2

8

Figure 8 shows a graphical comparison between the preliminary and the three optimized designs, while the corresponding design variables are summarized in Table 4. The optimized designs have less vanes than DP , but while D3 has eight vanes according to the minimization of total pressure loss, D2 has eleven vanes, in contrast to the nine vanes of D1 . Increasing the vane number is therefore not a guarantee of a higher pressure recovery. The latter is however achieved with longer vanes, as in the case of D1 . This configuration exhibits as well the largest vaneless space downstream of the vanes trailing edge, such that the highest pressure recovery is a combination of the velocity decrease both in the vaned passages and in the second vaneless space. On the contrary, the efficiency is maximized with a large first vaneless space to allow for a velocity reduction upstream of the vanes leading edge, short vanes, and a small second vaneless space. Further on, the optimized designs exhibit different vane angles at the leading edge, wherelse the values of VTA are very similar. As a consequence, at the trailing edge D1 show more radial vanes, while D3 has more tangential vanes. In the rest of the section, the flow field of the preliminary and optimized designs will be assessed by studying the contours of the Mach number, static pressure, and total pressure on a diffuser section located at mid-span.

Vaned diffusers optimization based on metamodel-assisted genetic

Fig. 9 Comparison between Mach number contours at mid-span for the preliminary (DP ) and optimized (D1 , D2 , D3 ) designs

Figure 9 shows the Mach number contours. A supersonic Mach number of about 1.27 is visible for D1 , which exhibits a shock on the vane pressure side, very close to the leading edge. This leads to a velocity decrease on the vane suction side, where D1 has the lowest average velocities in comparison to the other geometries. However, the lowest peak Mach numbers are noticeable in DP and D3 , whose vanes are located the furthest from the diffuser inlet. Furthermore, D1 does not exhibit the low-velocity region which is visible on the vane suction side of the preliminary design. This area can still be noticed for D2 at about mid-chord, and for D3 to a very small extent. However, in the two optimized geometries flow separation and recirculation on the vanes suction side does not occur. Figure 10 shows the contours of the static pressure, normalized by the total pressure at the compressor inlet. In comparison with the preliminary design, D1 has the highest pressure rise in the vaned passage. On the vane pressure side, the isobars are very close to each others, and rapidly increase in magnitude from the leading to the trailing edges. On the other hand, the worst static pressure recovery has been obtained

M. Olivero et al.

Fig. 10 Comparison between static pressure contours at mid-span for the preliminary (DP ) and optimized (D1 , D2 , D3 ) designs

with D3 , because of very short vanes. D2 is better than D3 in terms of static pressure recovery, but worse than D1 . Figure 11 shows the contours of the total pressure, normalized by the total pressure at the compressor inlet. In all the four cases, the total pressure is minimum approximately in correspondance of the low-velocity pools seen in Fig. 9. As a consequence, the wakes which are visible on the vanes suction side are the source of the losses within the diffuser. D3 has the highest total pressure in the proximity of vanes leading edge, which in turn gives the lowest losses in the downsteam vaned passages, minimizing thus ϕ2 . The highest total pressure loss in the vaned passage is visible in D1 , whose efficiency is equal to that of the preliminary diffuser.

7 Conclusions The work described in this paper aimed at the improvement of the performance of an automotive turbocharger centrifugal compressor, currently featuring a vaneless

Vaned diffusers optimization based on metamodel-assisted genetic

Fig. 11 Comparison between total pressure contours at mid-span for the preliminary (DP ) and optimized (D1 , D2 , D3 ) designs

diffuser, to be used in a microturbine for combined heat and power applications. To carry out this task, a design optimization procedure for vaned diffusers has been developed by coupling a genetic algorithm with a commercial 3D CFD code which solves the RANS equations. In order to reduce the computational costs, a Kriging metamodel has been used to assist the genetic algorithm. The optimization procedure takes into account the turbulent, 3D, and non-uniform flow conditions at the diffuser inlet, as the CFD computations have been performed over a numerical domain which includes both the impeller and vaned diffuser. A multi-objective optimization problem has been solved by simultaneously minimizing two objective functions, which respectively depend on the compressor stage total-to-static pressure ratio and total-to-total isentropic efficiency. The position of the vanes between the diffuser inlet and outlet, their inclination with respect to the radial direction at the leading and trailing edges, the vane number, and the diffuser outlet radius have been selected as design variables.

M. Olivero et al.

Furthermore, a preliminary vaned diffuser (DP ) has been designed on the basis of empirical correlations found in the literature. It has better static pressure recovery than the vaneless configuration, but lower efficiency. The following conclusions have been drawn at the end of the optimization process: – The solution of the multi-objective problem led to a Pareto front made of eleven individuals, among which three representative designs have been identified. D1 is the individual with the highest total-to-static pressure ratio, D3 is the one with the highest total-to-total isentropic efficiency, and D2 exhibits intermediate performance. – D1 and D2 perform better than DP in terms of static pressure recovery, while D3 does not. The three optimized geometry exhibit the same efficiency, or better, compared to DP . The ideal best vaned diffuser would couple the pressure recovery level of D1 (5.1 %-decrease for ϕ1 ), with the efficiency level of D3 (8.6 %-decrease for ϕ2 ). – The three optimized geometry show better pressure recovery than the vaneless configuration as well, but differences are noticed in terms of total pressure loss, since they all have higher total pressure loss coefficients. However, D3 has a better efficiency (+1.8 %) than the vaneless diffuser. The performance of this turbocharger centrifugal compressor featuring a vaneless diffuser can thus be improved by the addition of diffuser vanes. Furthermore, in this case the diffuser outlet radius is 12 % shorter than that of the vaneless diffuser, leading thus to a more compact configuration. – Longer vanes increase the static pressure recovery, while shorter ones, coupled with small vaneless spaces upstream and dowstream of the vanes, have a positive effect on the efficiency. On the other hand, the influence of the vane number on the pressure recovery and efficiency could not be properly assessed. – The optimized geometries have less backflow than DP , as a consequence of higher average Mach numbers and due to the absence of flow recirculation on the vanes suction side. In all cases, losses are generated in correspondance of the lowvelocity regions present in the flow field. The work described in this paper left same open issues, which will be addressed in future vaned diffusers optimization works: – The tip clearance effects in small turbocharger centrifugal impellers are very strong, because the manufacturing tolerances cause the relative tip clearance (i.e., the ratio of the tip clearance to the blade height at the impeller outlet) to be very large, leading to higher total pressure loss and lower efficiencies. Future optimization procedures will therefore consider the impeller tip clearance, which has been currently neglected, in order to assess its influence on vaned diffusers and compressor stages performance. – Since the optimization has been performed at the compressor design point only, a so-called “multi-point” optimization will be carried out, in order to optimize the vaned diffusers for a wider range of operating conditions. – Variations of the mass flow rate and/or rotational speed lead to different flow velocities at the diffuser inlet, and thus to different incidence angles at the vanes leading edge. The shape of the latter will then be optimized in the spanwise direction, to account for the flow angle variations from hub to shroud.

Vaned diffusers optimization based on metamodel-assisted genetic

– Furthermore, although the use of a relative simple vane profile resulted in high levels of static pressure recovery, airfoil shapes (e.g., the NACA-series airfoils) will be considered to further increase the pressure recovery.

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