Surge Margin Optimization of Centrifugal Compressors Using a New Objective Function Based on Local Flow Parameters
Surge Margin Optimization of Centrifugal Compressors Using a New Objective Function Based on...
Ratz, Johannes;Leichtfuß, Sebastian;Beck, Maximilian;Schiffer, Heinz-Peter;Fröhlig, Friedrich
International Journal of Turbomachinery Propulsion and Power Article Surge Margin Optimization of Centrifugal Compressors Using a New Objective Function Based on Local Flow Parameters 1, 1 1 1 Johannes Ratz , Sebastian Leichtfuß , Maximilian Beck , Heinz-Peter Schiffer and Friedrich Fröhlig Institute of Gas Turbines and Aerospace Propulsion, Technische Universität Darmstadt, Otto-Berndt-Str. 2, 64287 Darmstadt, Germany; firstname.lastname@example.org (S.L.); email@example.com (M.B.); firstname.lastname@example.org (H.-P.S.) MTU Friedrichshafen GmbH, Friedrichshafen 88045, Germany; Friedrich.Froehlig@mtu-online.com * Correspondence: email@example.com † This Paper is an Extended Version of Our Paper Published in the Proceedings of the 13th European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, ETC13, Lausanne, Switzerland, 8–12 April 2019; Paper No. 212. Received: 2 October 2019; Accepted: 10 December 2019; Published: 17 December 2019 Abstract: Currently, 3D-CFD design optimization of centrifugal compressors in terms of the surge margin is one major unresolved issue. On that account, this paper introduces a new kind of objective function. The objective function is based on local ﬂow parameters present at the design point of the centrifugal compressor. A centrifugal compressor with a vaned diffuser is considered to demonstrate the performance of this approach. By means of a variation of the beta angle distribution of the impeller and diffuser blade, 73 design variations are generated, and several local ﬂow parameters are evaluated. Finally, the most promising ﬂow parameter is transferred into an objective function, and an optimization is carried out. It is shown that the new approach delivers similar results as a comparable optimization with a classic objective function using two operating points for surge margin estimation, but with less computational effort since no second operating point near the surge needs to be considered. Keywords: turbocharger; centrifugal compressor; surge margin; CFD; optimization 1. Introduction The forced induction of internal combustion engines by turbochargers is playing an important role. This imposes high requirements on the centrifugal compressor design, which can be achieved with the help of automatic CFD based design optimizations. In centrifugal compressor optimization, many objectives and constraints depending on the requirements and speciﬁcations need to be considered. For many optimizations, the efﬁciency for one or several operating points and the surge margin are crucial values, which shall be increased. While the efﬁciency is straightforward to evaluate, an adequate expression for the surge margin, which can be used for an optimization, is hard to deﬁne. The most common expressions for the surge margin (e.g., see Moore and Reid  or Cumpsty ) are illustrated in Figure 1 (left side and center). Herein, the surge margin is expressed either at constant rotational speed (Figure 1 (left side)) or at constant mass ﬂow (Figure 1 (center)). At constant rotational speed, the surge margin (SM) is expressed by both total pressure ratios and mass ﬂows (Equation (1)), whereas at constant mass ﬂow, only the total pressure ratios are considered (Equation (2)). m ˙ Surge DP SM = (1) m ˙ P Surge DP Int. J. Turbomach. Propuls. Power 2020, 4, 42; doi:10.3390/ijtpp4040042 www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2020, 4, 42 2 of 12 P P Surge DP SM = (2) Surge m ˙ highly throttled highly throttled DP DP DP reduced mass ﬂow reduced mass ﬂow reduced mass ow Figure 1. Left: Surge margin deﬁnition via pressure slope; center: surge margin deﬁnition via pressure ratio; right: numerical determination of surge mass ﬂow (iterative determination). The crucial part with CFD calculations is to determine P and m ˙ . The most appropriate Surge Surge method might be a detailed ﬂow analysis through a compressible large eddy simulation, as was shown by Sundström et al. . However, for an optimization or the analyses of a large number of different impeller designs, such calculations are computationally far too expensive. Considering Equation (1), another method is the iterative evaluation along a speed line (Figure 1, right side), in which the numerical surge limit is deﬁned by the last converged CFD calculation (see Section 5). However, also this method is very time consuming and therefore is inappropriate for an optimization. Hence, the most common approach is to ﬁx the mass ﬂow of a highly throttled operating point for the whole optimization (e.g., see Goinis  or Kim et al. ). Thus, Equation (1) simpliﬁes to: Surge SM = (3) estimation DP and can be used now as the objective function in an optimization. Although those methods are widely used, they have some critical drawbacks. First, the highly throttled operating point might deviate signiﬁcantly from the actual surge limit, which reduces the validity of this expression. Second, the method presumes that the slope of the compressor characteristic correlates with the surge margin, which is not true in all cases. In addition, a highly throttled operating point close to surge is rather unsteady, and conventional steady-state CFD models and boundary conditions used for optimization often fail to predict those precisely. To avoid these issues, Van den Braembussche  and Hiradate et al.  proposed using local ﬂow parameters present at the design point for surge margin estimation. This approach is considered and demonstrated in this paper. A variety of local ﬂow parameters is correlated with the iteratively determined numerical surge limit for 73 designs, and the deceleration of the isentropic Mach number between the leading edge and throat has proven to be the most promising parameter. Based on this knowledge, a new objective function is formulated and used for an optimization with the aim of achieving an increased surge margin by at least constant efﬁciency. 2. General Procedure For the deﬁnition of this new kind of objective function, a high efﬁciency full blade centrifugal compressor with a vaned diffuser is considered. Figure 2 shows the general procedure to formulate the objective function. First, 73 arbitrary design variations are generated through a ULH algorithm by means of a variation of the diffuser width and of the beta angle distribution of the impeller and diffuser blade. Secondly, the surge margin of those designs is iteratively determined. In a third step, several local ﬂow parameters present at the design point are evaluated and correlated with the iteratively Choke Limit Choke Limit Choke Limit Surge Limit Surge Limit Surge Limit total pressure rao total pressure rao total pressure rao camber Int. J. Turbomach. Propuls. Power 2020, 4, 42 3 of 12 determined surge margin. In the last step, the ﬂow parameter with the highest correlation to the surge margin is chosen for surge margin estimation and therefore to formulate the objective function. GENERATION OF DESIGN VARIATIONS ITERATIVE CFD SURGE MARGIN DETERMINATION EVALUATION OF SEVERAL FLOW PHENOMENA AT THE DESIGN POINT CALCULATION OF THE CORRELATION BETWEEN SURGE MARGIN AND FLOW PHENOMENA DEFINITION OF THE NEW OBJECTIVE FUNCTION Figure 2. Flowchart of the procedure shown in this paper. 3. Parametric Model The centrifugal compressor geometry is parametrized within the Software CAESES. Besides the diffuser width b, the blade angle distribution for both the impeller and diffuser is varied at the hub and shroud. The transformation from a deﬁned blade angle distribution to a 3D camber surface is based on Miller et al. . Figure 3 shows the blade-to-blade view of the camber lines of the impeller and diffuser with the respective blade angles b. The camber lines are deﬁned through the blade angle distribution 0 0 b(m ). Figure 4 shows the parametrization of b(m ) through six design parameters. b and b LE TE 0 0 deﬁne the blade angles at LE and TE. b and b deﬁne the slope of the blade angle distribution at LE TE LE and TE. In other words, b describes the blade curvature at LE and TE. Finally, the impact of the slopes on the whole blade angle distribution is expressed through the slope factors SF and SF . LE TE Impeller Diffuser m' Figure 3. Blade-to-blade view of the impeller and diffuser camber line with the respective blade angles ﬁ. camber Int. J. Turbomach. Propuls. Power 2020, 4, 42 4 of 12 m' Figure 4. Parametrization of the blade angle distribution ﬁ(m ) for the impeller and diffuser blade. Table 1 shows the range of the design parameters for the underlying study. This set of parameters has been chosen to express a wide variety of designs by a minimum amount of parameters. Table 1. Variation of the design parameters. b b SF b 5 5 0.1 14% 4. Numerical Setup For all CFD calculations, the commercial CFD code FINE/Turbo provided by Numeca was used. Within FINE/Turbo, the three-dimensional, density based, structured, steady ﬂow solver was used to solve the Reynolds averaged compressible Navier–Stokes equations. FINE/Turbo is based on the ﬁnite volume method and uses the central difference space discretization for spatial discretization. Multigrid methods and local time stepping were used for fast convergence. The closure problem of the Reynolds averaged Navier–Stokes equations was treated by the low Reynolds EARSM (Explicit Algebraic Reynolds Stress Model) model, which also included an additional anisotropy tensor for anisotropic turbulence. Figure 5 illustrates the CFD domain used in this investigation. The interface between rotating and stationary components was treated by the mixing plane approach. 1D non-reﬂecting boundary conditions were applied at the interface since a strong interaction between the rotor and stator was expected. Axial inﬂow, total pressure, and total temperature were used as the inlet boundary condition. The mass ﬂow was used as the outlet boundary condition. The impeller and diffuser were meshed as single passage, and periodic boundaries were applied. The impeller contained approximately 1.3 million and the diffuser approximately 0.6 million cells. The impeller and diffuser were meshed with 69 cells in the spanwise direction, whereof 17 cells were used for the tip gap region in the case of the impeller. Since a low Reynolds turbulence model was used, 6 + the ﬁrst cell width in the impeller and diffuser was set to 3 10 m, which resulted in a y < 5 for all domains. The sensitivity to the grid resolution was evaluated via the grid convergence index according to Celik et al. . The grid reﬁnement factor between the different grids was chosen to be 1.3 in each spatial direction. The resulting grid convergence index for the used grid was around 1% for the efﬁciency. One has to keep in mind that the error based on the grid resolution was similar for all evaluated designs. Therefore, the relative comparison between different designs had a higher accuracy than the comparison to a theoretic grid independent solution, which was considered for the grid convergence index. camber Int. J. Turbomach. Propuls. Power 2020, 4, 42 5 of 12 Outlet Diffuser Interface Inlet Figure 5. CFD domain: meridional view. 5. Iterative Surge Limit Determination Figure 6 shows the procedure of the numeric, iterative surge limit determination. Six operating points (i 2 f0, 1, 2, 3, 4, 5g) were considered. For Operating Points 1 to 5, Operating Point 0 was used as the initial solution. 1 3 4 5 2 DP reduced mass ow Figure 6. Iterative surge margin determination. The bisection algorithm proceeds as follows: 1. Calculation of the operating point i = 0. 2. Check convergence of operating point i = 0. 3. Dependent on the convergence of operating point i = 0, calculation of operating point i = 1 with: m ˙ m ˙ (i) = m ˙ (i 1) Dm ˙ , Dm ˙ = , (4) i 1 ˙ ˙ with m = 0.2 m . 0 DP 4. Check convergence, and proceed with i = i + 1 in Step 3. Choke Limit Surge Limit total pressure ra o Int. J. Turbomach. Propuls. Power 2020, 4, 42 6 of 12 This procedure eventually yielded an accuracy of Dm ˙ = 0.013 m ˙ between the last two calculated DP operating points. Convergence was checked for the last 50 iterations on the basis of mass ﬂow error and the standard deviation of total pressure ratio and efﬁciency. Convergence was obtained if the following criteria were fulﬁlled after a maximum number of 600 iterations. std(h ) < 0.0025 last 50 Iterations std(P ) < 0.01 last 50 Iterations max m ˙ m ˙ /m ˙ < 0.005 (j( ) j ) out in out last50Iterations The authors are well aware that the numerical determination of the surge margin was more than critical and that an absolute prediction of the surge mass ﬂow with steady RANS calculations was likely impossible. However, a relative comparison between the considered designs was valid. To paraphrase, if a certain design at a certain mass ﬂow still converged, the ﬂow did not have a major unsteady character. However, if another design did not converge at that certain mass ﬂow, it was assumed that in an experiment, the ﬂow of that design became unsteady earlier and therefore had a smaller surge margin. 6. Experimental Validation of the Predicted Performance Figure 7 shows the comparison between the experimental and the numerical data of the total pressure ratio and isentropic efﬁciency. The experimental data were provided by the MTU Friedrichshafen GmbH. All values were normalized with the respective value taken from the design point (CFD). The choke mass ﬂow and the qualitative trend were in good agreement with the experimental data. One has to keep in mind that the numerical investigation was done without volute, which explains the higher predicted total pressure ratio and efﬁciency, as losses in the volute were neglected. Since only the shown speed line was considered within this paper, the overall comparison indicated that the chosen mesh and the numerical settings were suitable for the numerical evaluation of the surge margin and the subsequently presented ﬂow parameters. 1.125 1 1.05 0.9 0.975 0.8 0.9 0.7 0.825 0.6 Total pressure ratio (CFD) Total pressure ratio (Experiment) 0.75 0.5 Isentropic Ef ciency (CFD) Isentropic Ef ciency (Experiment) 0.675 0.4 0.6 0.3 0.95 1 1.05 1.1 1.15 1.2 normalized reduced mass ow Figure 7. Comparison of the experimental (blue) and CFD (red) data. normalized total pressure ra o normalized e ciency Int. J. Turbomach. Propuls. Power 2020, 4, 42 7 of 12 7. Local Flow Parameters The evaluated ﬂow parameters (FP) can be grouped into physical values based on the non-uniformity of the impeller outlet ﬂow, the impeller and diffuser back ﬂow, the boundary layer thickness, and the isentropic Mach number distribution. It turned out that the parameters deﬁned with the help of the Mach number distribution were the most promising, and therefore, the other values were not considered in detail at this point. Figure 8 shows the characteristic isentropic Mach number distribution on an arbitrary blade surface. Based on this distribution, Ma ,c , PDR, Ma Peak Peak TDR, and DMa were deﬁned. Ma and c denote the magnitude and chord position of the max Peak Ma Peak Mach number peak on the suction side, respectively. The peak deceleration ratio (PDR) denotes the slope of the Mach number curve directly after the Mach number peak. The throat deceleration ratio TDR = Ma /Ma expresses the ﬂow deceleration from the Ma value to the Mach number Throat Peak Peak at the throat. DMa denotes the maximum difference between the Mach number at suction and max pressure side. All values were evaluated at 10%, 50%, and 90% blade span. Ma Ma Peak PDR Suction Side TDR Ma Pressure Side Ma Throat Peak rel. chord length Figure 8. Flow parameters based on the Ma trend. 8. Correlations The evaluation of the ﬂow parameters shown in the last section, concerning their potential for correlating with the surge margin, was done via Spearman’s rank correlation coefﬁcient: cov rg , rg x y r = , (5) s s rg rg x y wherein x and y are random variables, rg denotes the rank of a variable, s is the standard deviation, and cov the covariance. The advantage of Spearman’s rank correlation coefﬁcient is the ability to assess correlations independently of the underlying relation between x and y. Hence, as long as the relation between x and y is monotonically increasing or decreasing, r indicates a high correlation. It does not matter if the relation is linear, quadratic, or anything else. r takes values between 1 and one, whereby 1 and one indicate a pure monotonically decreasing or increasing relation between the two variables, respectively. Zero indicates no relation at all, and everything in between indicates a more or less strong monotonic relation. To decide if the correlation between two variables is signiﬁcant, the T-test was used. For the 73 designs, the T-test revealed that for r > 0.3 and r < 0.3, the correlation between two variables had 99.5% signiﬁcance. Table 2 lists the correlations between the selected ﬂow parameters and the iteratively determined surge mass ﬂow (m ˙ ). The considered ﬂow parameters were expressed through the sum of the Surge respective ﬂow parameter at all three span positions (10%, 50%, and 90%) in the impeller and diffuser (cf. Equation (6)). The TDR showed with r = 0.82 the highest correlation and therefore TDR,m ˙ Surge Int. J. Turbomach. Propuls. Power 2020, 4, 42 8 of 12 was used for the formulation of the new objective function. Funabashi et al.  reported that a high deceleration between LE and the throat, which corresponded to a low TDR, caused an increased boundary layer thickness, which was more likely to separate and therefore led to surge. Table 2. Spearman’s correlation between the ﬂow parameter and surge mass ﬂow. FP TDR PDR Ma Ma DMa Peak Pos max r 0.82 0.74 0.73 0.71 0.76 FP,m ˙ Surge 9. New Objective Function Equation (6) shows the new objective function based on the TDR. TDR denotes the xx% Span, ref values of the basis design. As the correlation is negative and the surge mass ﬂow shall be minimized, the objective of the optimization is to maximize TDR. TDR TDR TDR 1 10% Span 50% Span 90% Span TDR = + + 6 TDR TDR TDR 10% Span, ref 50% Span, ref 90% Span, ref Impeller (6) ! # TDR TDR TDR 10% Span 50% Span 90% Span + + + TDR TDR TDR 10% Span, ref 50% Span, ref 90% Span, ref Diffuser 10. Optimization Example For the validation of the new objective function, two optimizations were carried out. One was with the classic objective function SM (Equation (3)), referred to as SM-Optimization, estimation and another one with the new objective function TDR (Equation (6)), referred to as TDR-Optimization. Both optimizations had the aim to increase the surge margin by nearly constant efﬁciency at the design point. During the optimization, only the design speed was considered. Table 3 shows the operating points and constraints used in the optimizations. For the SM-Optimization, an additional highly throttled operating point was needed to evaluate SM , estimation which was chosen to be 94% of m ˙ and corresponded to the surge mass ﬂow of the basis design. DP For the TDR-Optimization, only the design point was needed to evaluate the objective function, which saved one computation for each design iteration. Table 3. Operating points with boundary conditions and constraints. OP surge is only used for the SM-Optimization. OP BC Constraint ˙ ˙ Choke p = 200 kPa m > 1.17 m DP P > 0.975P DP DP m ˙ = m ˙ DP h > 0.99 h DP Surge m ˙ = 0.94 m ˙ - DP Both optimizations were started with the same DOE database of 255 designs, created by a ULH algorithm. Then, the optimizations were run using a genetic algorithm combined with meta surfaces for faster convergence. The last optimization iterations were executed by combining a genetic and a gradient based algorithm. In addition to the DOE database, 210 designs were evaluated during the optimization runs, whereof the TDR-Optimization resulted in 100 and the SM-Optimization in 57 valid designs. From each optimization, the best design regarding the objective function and efﬁciency was chosen for comparison. Figure 9 shows the performance map (total pressure ratio and efﬁciency (total-total)) for the optimized designs and the basis design. Both optimizations resulted in a very similar output. The efﬁciency was slightly increased by 0.64 (TDR) and 0.71 (SM) percentage points Int. J. Turbomach. Propuls. Power 2020, 4, 42 9 of 12 and the surge margin m ˙ m ˙ /m ˙ by ﬁve percentage points. It has to be mentioned that DP Surge DP due to the chosen constraints, the speed line was shifted to a slightly smaller mass ﬂow. However, still, the total width of the speed line m ˙ m ˙ /m ˙ was increased by 2.5 percentage points. Surge Choke Choke 1.044 1.008 0.986 0.941 0.928 0.874 0.87 0.807 Basis design 0.812 SM-Optimization TDR-Optimization 0.74 0.754 0.672 0.914 1.029 1.143 1.257 0.914 1.029 1.143 1.257 normalized reduced mass ow normalized reduced mass ow Figure 9. Comparison of total pressure ratio (left) and efﬁciency (total-total) (right) between the optimized designs and the basis design. Mass ﬂow, total pressure ratio, and efﬁciency (total-total) are normalized with the respective value of the DP (basis design). Figure 10 compares the original impeller and diffuser blade design with the ﬁnal blade shape of both optimizations. Regarding the impeller (right side), the optimization mainly led to an increase of the rake angle, which in turn led to a redistribution of the impeller outﬂow. More precisely, due to the higher rake angle, the impeller outﬂow between the hub and 70% span was more radially directed compared to the original impeller. Regarding the diffuser (left side), both optimized blades were more twisted compared to the original diffuser. The higher twist led to a better adaptation to the different impeller outﬂow angles at the hub and shroud. Figure 10. Comparison of the original (blue) and both optimized impeller and diffuser blade designs (SM-Optimization in red and TDR-Optimization in green). Besides, at least one third of the computation time was saved as no surge point needed to be considered, the TDR-Optimization revealed two further beneﬁts. First, the TDR-Optimization contained almost twice as many valid designs as the SM-Optimization. The larger number of “error ” designs in the SM-Optimization was due to the non-convergence of many designs at the operating point normalized total pressure rao normalized e ciency Int. J. Turbomach. Propuls. Power 2020, 4, 42 10 of 12 0.94m ˙ . Even though those designs were considerably bad in terms of surge margin, the optimizer ’s DP database was increased by every new valid design, though the optimization converged faster. Second, the TDR-Optimization considered both the impeller and the diffuser design regarding the surge margin. For illustration, Figure 11 shows the Mach number distribution for the impeller and diffuser at 90% span at the DP. Table 4 lists the percentage change of the TDRs compared to the basis design for the impeller and diffuser. Considering the impeller and diffuser, the TDR (cf. Equation (6)) was increased in both optimizations. However, considering the single components, in the SM-Optimization, only the diffuser was improved in terms of surge margin, whereas the impeller declined. This was due to the fact that the basis design and both optimized designs hit the surge limit due to stall in the diffuser. Hence, while trying to increase the surge margin, the SM-Optimization tried to improve the diffuser as only integral information between the impeller inlet and diffuser outlet was available. Let us assume that during an optimization, the critical component changes. For example, the diffuser had such a good shape that now the impeller was responsible for surge. Then, the TDR-Optimization clearly had the advantage over the SM-Optimization as the impeller already was improved in terms of the surge margin. 1.07 Impeller - 90% Span Diffuser - 90% Span 0.92 0.95 0.76 0.71 0.61 0.46 0.48 0.31 Basis design 0.24 SM-Optimization TDR-Optimization 0.15 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 rel. chord length rel. chord length Figure 11. Comparison of the isentropic Mach number distribution between the optimized designs and the basis design for the impeller (left) and diffuser (right). The Mach number is normalized with the Ma of the impeller and diffuser, respectively. Peak Table 4. Percentage change of TDR compared to the basis. Flow Parameter TDR-Optimization SM-Optimization TDR +4.5% 0.4% Impeller TDR +7.2% +9.2% Diffuser TDR +5.8% +4.4% 11. Conclusions Seventy-three centrifugal compressor designs were generated by means of a variation of the beta angle distribution and the diffuser width. For all 73 design variations, several local ﬂow parameters present at the design point were correlated with the respective iteratively determined surge margin of the design. Values based on the isentropic Mach number distribution showed the highest correlation with the surge margin. Therefore, the so-called TDR (TDR = Ma /Ma ), which represents the Throat Peak ﬂow deceleration from the Ma value near the LE to the Mach number at the throat, was deﬁned as Peak the new objective function. For validation of the new objective function, two optimizations were carried out. One optimization was with the new objective function TDR and another one with a classic objective function using two operating points for surge margin estimation. Both optimizations led to a similar result regarding the normalized Ma normalized Ma Int. J. Turbomach. Propuls. Power 2020, 4, 42 11 of 12 surge margin and efﬁciency. However, the optimization with the TDR showed several advantages. First, as no operating point near surge needed to be considered, the optimization time was at least reduced by one third. Second, the TDR optimization led to more valid designs, whereby the optimizer learned faster. Third, as not only integral values between the inlet and outlet were considered, the TDR-Optimization improved the impeller and diffuser regarding the surge margin at the same time. In conclusion, it can be stated that the approach using local ﬂow parameters was suited for surge margin optimization and even had advantages compared to the classic objective function using two operation points for surge margin estimation. Author Contributions: Conceptualization, J.R. and F.F.; data curation, M.B.; funding acquisition, S.L. and H.-P.S.; investigation, J.R.; methodology, J.R.; project administration, J.R.; supervision, S.L. and H.-P.S.; visualization, J.R.; writing, original draft, J.R.; writing, review and editing, S.L., H.-P.S., and F.F. Funding: The authors acknowledge the ﬁnancial support by the Federal Ministry for Economic Affairs and Energy of Germany (BMWi) in the project GAMMA (Project Number 03ET1469). The APC of this paper was funded by Euroturbo. Acknowledgments: Calculations for this research were conducted on the Lichtenberg high performance computer of the TU Darmstadt. Moreover, we give thanks to Friendship Systems AG and the NUMECA Ingenieurbüro for the great support regarding parametrization and CFD. Last but not least, we want to thank MTU Friedrichshafen GmbH for providing the geometries and the permission to publish the results. Conﬂicts of Interest: The authors declare no conﬂicts of interest. Nomenclature Latin c Chord length [ ] cov Covariance D Diameter [m] th i i operating point m Normalized arc length [ ] m ˙ Mass ﬂow [kg/s] Ma Mach number [ ] rg Rank Abbreviations BC Boundary condition CFD Computational ﬂuid dynamics DOE Design of experiments FP Flow parameter OP Operating point PDR Peak deceleration ratio SF Slope factor SM Stall margin TDR Throat deceleration ratio ULH Uniform Latin hypercube Greek b Blade angle [ ] h Efﬁciency [ ] P Total pressure ratio [ ] q Angular coordinate [ ] Subscripts DP Design point LE Leading edge Surge Surge point TE Trailing edge Int. J. Turbomach. Propuls. Power 2020, 4, 42 12 of 12 References 1. Moore, D.; Reid, L. Performance of Single-Stage Axial-Flow Transonic Compressor with Rotor and Stator Aspect Ratios of 1.19 and 1.26, Respectively, and with Design Pressure Ratio of 2.05; NASA Technical Paper 1659; Lewis Research Center: Cleveland, Ohio, USA, 1980. 2. Cumpsty, N.A. 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Fluid Mach. Syst. 2010, 3, 379–385. [CrossRef] 8. Miller, P.L.; Oliver, J.H.; Miller, D.P.; Tweedt, D.L. BladeCAD: An Interactive Geometric Design Tool for Turbomachinery Blades. In Proceedings of the 41st Gas Turbine and Aeroengine Congress, Birmingham, UK, 10–13 June 1996. 9. Celik, I.B.; Ghia, U.; Roache, P.J.; Freitas, C.J.; Coleman, H.; Raad, P.E. Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications. J. Fluids Eng. 2008, 130, 078001. 10. Funabashi, S.; Iwase, T.; Hiradate, K.; Fukaya, M. Platform Technology for Computational Fluid Dynamics Supporting Design of System Products. Hitachi Rev. 2012, 61, 244. 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY-NC-ND) license (https://creativecommons.org/licenses/by-nc-nd/4.0/).
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Surge Margin Optimization of Centrifugal Compressors Using a New Objective Function Based on Local Flow Parameters