SURGE MARGIN OPTIMIZATION OF CENTRIFUGAL COMPRESSORS USING A NEW OBJECTIVE FUNCTION BASED ON LOCAL FLOW PARAMETERS

Nowadays, 3D-CFD design optimization of centrifugal compressors in terms of 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 vaned diffuser is considered to demonstrate the performance of this approach. By means of a variation of the beta angle distribution of 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 surge needs to be considered.


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 point(s) and the surge margin are crucial values which shall be increased.While the efficiency is straightforward to evaluate, an adequate expression for surge margin, which can be used for an optimization, is hard to define.
The most common expressions for surge margin (e.g.see Moore and Reid [1980] or Cumpsty [2004]) are illustrated in fig. 1 (left side and center).Herein, the surge margin is expressed either at constant rotational speed (fig. 1 (left side)) or at constant mass flow (fig. 1 (center)).At constant rotational speed the surge margin (SM) is expressed by both total pressure ratios and mass flows (eq.( 1)), whereas at constant mass flow only the total pressure ratios are considered (eq.( 2)).
The crucial part with CFD calculations is to determine Π Surge and ṁSurge .The most appropriate method might be a detailed flow analysis through a compressible Large Eddy Simulation like it is shown by Sundström et al. [2018] et al.But, for an optimization or the analyses of a large number of different impeller designs such calculations are computationally far too expensive.Considering eq. ( 1), another method is the iterative evaluation along a speed line (fig.1, right side), in which the numerical surge limit is defined by the last converged CFD calculation (see section ITERATIVE SURGE LIMIT DETERMINATION).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 [2013] or Kim et al. [2013]).Thus, eq. ( 1) simplifies to and now can be used as 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 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 [2006] and Hiradate et al. [2010] 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 to the iteratively determined numerical surge limit for 73 designs and the deceleration of the isentropic Mach number between 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.

GENERAL PROCEDURE
For the definition of this new kind of objective function a high efficiency full blade centrifugal compressor with vaned diffuser is considered.Figure 2 shows the general procedure to formulate the objective function.through a ULH algorithm by means of a variation of the diffuser width and of the beta angle distribution of 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 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.

PARAMETRIC MODEL
The centrifugal compressor geometry is parametrized within the Software CAESES.Besides the diffuser width b, the blade angle distribution for both impeller and diffuser is varied at hub and shroud.The transformation from a defined blade angle distribution to a 3D camber surface is based on Miller et al. [1996].Figure 3 shows the blade to blade view of the camber lines of impeller and diffuser with the respective blade angles β.The camber lines are defined through the blade angle distribution β(m ).Table 1 shows the range of the design parameter 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 design parameters

NUMERICAL SETUP
For all CFD calculations the commercial CFD code FINE/Turbo provided by Numeca is used.Within FINE/Turbo the three-dimensional, density based, structured, steady flow solver is 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 are used for fast convergence.The closure problem of the Reynolds-averaged Navier-Stokes equations is treated by the low-Reynolds EARSM model which as well includes an additional anisotropy tensor for anisotropic turbulence.
Fig. 5 illustrates the CFD domain used in this investigation.The interface between rotating and stationary components is treated by the mixing plane approach.1D non-reflecting boundary conditions are applied at the interface since a strong interaction between rotor and stator is expected.Axial inflow, total pressure and total temperature are used as inlet boundary condition.The mass flow is used as outlet boundary condition.
Impeller and diffuser are meshed as single passage and periodic boundaries are applied.The impeller contains approximately 1.3 million and the diffuser approximately 0.6 million cells.Impeller and diffuser are meshed with 69 cells in spanwise direction whereof 17 cells are used for the tip gap region in case of the impeller.Since a low-Reynolds turbulence model is used, the first cell width in impeller and diffuser is set to 3e-06 m which results in a y + < 5 for all domains.The sensitivity to the grid resolution is evaluated via the grid convergence index according to Celik et al. [2008].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 is around 1% for the efficiency.One has to keep in mind that the error based on the grid resolution is similar for all evaluated designs.Therefore, the relative comparison between different designs has a higher accuracy than the comparison to a theoretic grid independent solution, which is considered for the grid convergence index.

ITERATIVE SURGE LIMIT DETERMINATION
Fig. 6 shows the procedure of the numeric, iterative surge limit determination.Six operating points (i ∈ {0, 1, 2, 3, 4, 5}) are considered.For operating points 1 to 5 the operating point 0 is used as initial solution.The bisection algorithm proceeds as follows: 3. Dependent on the convergence of operating point i = 0, calculation of operating point i = 1 with

Check convergence and proceed with
This procedure eventually yields in an accuracy of ∆ ṁ = 0.013 ṁDP between the last two calculated operating points.Convergence is checked for the last 50 iterations on the basis of mass flow error and the standard deviation of total pressure ratio and efficiency.Convergence is obtained if after a maximum number of 600 iterations the following criteria are fulfilled.
• std(η last 50 Iterations ) < 0.0025 The authors are well aware that the numerical determination of the surge margin is more than critical and that an absolute prediction of the surge mass flow with steady RANS calculations is likely impossible.However, a relative comparison between the considered designs is valid.To paraphrase, if a certain design at a certain mass flow still converges, it means that the flow does not have a major unsteady character.But if another design does not converge at that certain mass flow, it is assumed that in an experiment the flow of that design becomes unsteady earlier and therefore has a smaller surge margin.

EXPERIMENTAL VALIDATION OF THE PREDICTED PERFORMANCE
Fig. 7 shows the comparison between the experimental and the numerical data of total pressure ratio and isentropic efficiency.The experimental data is provided by the MTU Friedrichshafen GmbH.All values are normalized with the respective value taken from the design point (CFD).The choke mass flow and the qualitative trend is in good agreement with the experimental data.One has to keep in mind that the numerical investigation is done without volute, which explains the higher predicted total pressure ratio and efficiency as losses in the volute are neglected.
Since only the shown speed line is considered within this paper, the overall comparison indicates that the chosen mesh and the numerical settings are suitable for the numerical evaluation of the surge margin and the subsequently presented flow parameters.

CORRELATIONS
The evaluation of the flow parameters shown in the last section, concerning their potential for correlating with the surge margin, is done via the Spearman's rank correlation coefficient wherein x and y are random variables, rg denotes the rank of a variable, σ 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 in-or decreasing, ρ indicates a high correlation.It does not matter if the relation is linear, quadratic or anything else.ρ takes values between -1 and 1, whereby -1 and 1 indicate a pure monotonically de-or increasing relation between the two variables, respectively.0 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 is used.For a number of 73 designs the T-test reveals, that for ρ > 0.3 and ρ < -0.3 the correlation between two variables is of 99.5 % significance.
Table 2 lists the correlations between the selected flow parameters and the iteratively determined surge mass flow ( ṁSurge ).The considered flow parameters are expressed through the sum of the respective flow parameter at all three span positions (10 %, 50 % and 90 %) in impeller and diffuser (cf.eq. ( 6)).The TDR shows with ρ TDR, ṁSurge = −0.82 the highest correlation and therefore is used for the formulation of the new objective function.Funabashi et al. [2012] reported that a high deceleration between LE and throat, which corresponds to a low TDR, causes an increased boundary layer thickness which is more likely to separate and therefore leads to surge.
OPTIMIZATION EXAMPLE For validation of the new objective function two optimizations are carried out.One with the classic objective function SM estimation (eq.( 3)), hereinafter referred to as SM-Optimization, and another one with the new objective function TDR (eq.( 6)), hereinafter referred to as TDR-Optimization.Both optimizations have the aim to increase the surge margin by nearly constant efficiency at the design point.During the optimization only the design speed is considered.Table 3 shows the operating points and constraints used in the optimizations.For the SM-Optimization an additional highly throttled operating point is needed to evaluate SM estimation , which is chosen to be 94 % of ṁDP and corresponds to the surge mass flow of the basis design.For the TDR-Optimization only the design point is needed to evaluate the objective function, which saves one computation for each design iteration.
Both optimizations are started with the same DOE-database of 255 designs, created by a ULH algorithm.Then, the optimizations are run using a genetic algorithm combined with meta surfaces for faster convergence.The last optimization iterations are executed by combining a genetic and a gradient based algorithm.In Addition to the DOE-database, 210 Designs are evaluated during the optimization runs, whereof the TDR-Optimization results in 100 and the SM-Optimization in 57 valid designs.From each optimization the best design regarding objective function and efficiency is chosen for comparison.Figure 9 shows the performance map (total pressure ratio and efficiency) for the optimized designs and the basis design.Both optimizations result in a very similar output.The efficiency is slightly increased by 0.64 (TDR) and 0.71 (SM) percentage points and the surge margin (( ṁDP − ṁSurge ) / ṁDP ) by 5 percentage points.It has to be mentioned that due to the chosen constraints the speed line is shifted to a slightly smaller mass flow.But still, the total width of the speed line (( ṁChoke − ṁSurge ) / ṁChoke ) is increased by 2.5 percentage points.
Besides, saving at least one third of computation time as no surge point needs to be considered, the TDR-Optimization reveals two further benefits.First, the TDR-Optimization contains almost twice as many valid designs as the SM-Optimization.The larger number of "error"designs in the SM-Optimization is due to the non-convergence of many designs at the operating point 0.94 ṁDP .Even though those designs are considerably bad in terms of surge margin, the optimizer's database is increased by every new valid design and though the optimization converges faster.Second, the TDR-Optimization considers both the impeller and the diffuser design regarding surge margin.Figure 10 shows the Mach number distribution for impeller and diffuser at 90 % span at the DP.Table 4 list the percentage change of the TDR's compared to the basis design for impeller and diffuser.Considering impeller and diffuser, the TDR (cf.eq. ( 6)) is increased in both optimizations.But considering the single components, in the SM-Optimization only the diffuser is improved in terms of surge margin, whereas the impeller is declined.This is due to the fact that basis design and both optimized designs hit the surge limit due to stall in the diffuser.Hence, trying to increase the surge margin, the SM-Optimization tries to improve the diffuser as only integral information between impeller inlet and diffuser outlet are available.Let us assume that during an optimization the critical component changes.For example, that the diffuser has such a good shape that now the impeller is responsible for surge.Then, the TDR-Optimization clearly has the advantage over the SM-Optimization as the impeller already is improved in terms of surge margin.

CONCLUSIONS
73 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 and the so called TDR (TDR = Ma Throat /Ma Peak ), which represents the flow deceleration from the Ma Peak value near the LE to the Mach number at the throat, was defined as new objective function.
For validation of the new objective function, two optimizations were carried out.One optimization 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 surge margin and efficiency.However, the optimization with the TDR showed several advantages.First, as no operating point near surge needs to be considered, the optimization time is at least reduced by one third.Second, the TDR optimization led to more valid designs, whereby the optimizer learns faster.Third, as not only integral values between inlet and outlet are considered, the TDR-Optimization improves impeller and diffuser regarding surge margin at the same time.
In conclusion, it can be stated that the approach using local flow parameters is suited for surge margin optimization and even has advantages compared to the classic objective function using two operation points surge margin estimation.

Figure 1 :
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) First, 73 arbitrary design variations are generated 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 :
Figure 2: Flow chart of the procedure shown in this paper Figure4shows the parametrization of β(m ) through 6 design parameters.β LE and β T E define the blade angles at LE and TE.β LE and β T E define the slope of the blade angle distribution at LE and TE.In other words, β 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 LE and SF T E .

Figure 3 :Figure 4 :
Figure 3: Blade-to-Blade view of impeller and diffuser camber line with the respective blade angles β

Figure 7 :
Figure 7: Comparison of experimental (blue) and CFD (red) data

Figure 8 :
Figure 8: Flow parameters based on the Ma trend c Ma Peak , PDR, TDR and ∆Ma max are defined.Ma Peak and c Ma Peak denote the magnitude and chord position of the 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 Throat /Ma Peak expresses the flow deceleration from the Ma Peak value to the Mach number at the throat.∆Ma max denotes the maximum difference between the Mach number at suction and pressure side.All values are evaluated at 10 %, 50 % and 90 % blade span.

Table 2 :
Spearman's correlation between flow parameter and surge mass flow FP TDR PDR Ma Peak Ma Pos ∆Ma max ) shows the new objective function based on the TDR.TDR xx% Span, ref denotes the 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.

Table 3 :
Operating points with boundary conditions and constraints.OP Surge is only used for the SM-Optimization.

Table 4 :
Percentage change of TDR compared to the basis

normalized reduced mass ¡ow normalized reduced mass ¡ow normalized e¢ ciency normalized total pressure ra o
Comparison of total pressure ratio (left) and efficiency (right) between the optimized designs and the basis design.Mass flow, total pressure ratio and efficiency are normalized with the respective value of the DP (basis design)

rel. chord length rel. chord length normalized Ma normalized Ma
Figure 10: Comparison of isentropic Mach number distribution between the optimized designs and the basis design for impeller (left) and diffuser (right).The Mach number is normalized with the Ma Peak of impeller and diffuser respectively.