Preliminary design of centrifugal compressor using ...

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Mechanics & Industry 20, 628 (2019)© AFM, EDP Sciences 2019https://doi.org/10.1051/meca/2019071

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Preliminary design of centrifugal compressor usingmultidisciplinary optimization methodSun Shouyi, Yue Zhufeng, Li Lei*, Zhang Mengchuang, and Yang Weizhu

Department of Engineering Mechanics, Northwestern Polytechnical University (Chang’an Campus), P.O. BOX 16m,Xi’an 710129, PR China

* e-mail: l

Received: 27 February 2018 / Accepted: 4 October 2019

Abstract. Centrifugal compressor is widely used in turbochargers in which the aerodynamic performance andstrength are invariable among the important design objectives. As high pressure ratio centrifugal compressordevelops, the interaction between multiple disciplines should be involved in the preliminary design process. Astrength prediction method was presented and the prediction error was less than 3% compared with the 3D finiteelement calculation. The preliminary design method was established with consideration of multidisciplinarycouplings. Then, a centrifugal compressor with the lowest pressure ratio of 4.4 was designed based on themethod. The optimal results showed that the aerodynamic efficiency increases by 2.245% compared with theinitial design results. Finally, the 3D validation was carried out including aerodynamic analysis and strengthcalculation, which showed good agreement with the optimal results of the preliminary design.

Keywords: Centrifugal compressor / performance prediction / strength prediction /multidisciplinary optimization / preliminary design

1 Introduction

Centrifugal compressor is widely used in turbochargersbecause of its high reliability, simple structure, high singlestage pressure ratio and other prominent characteristics.At present, the preliminary design still acts as animportant role in the entire design flow, it determineswhether the centrifugal compressor design is successful ornot [1]. Furthermore, the basic geometric parameters of thecentrifugal compressor are mainly obtained by one-dimensional (1D) calculations and analysis during thepreliminary design process [2]. Hence, to get a well-designed centrifugal compressor with high performance,the accurate aero-thermodynamic and structural analysisshould be involved in the preliminary design process.

In the preliminary design process, the aerodynamicperformance prediction should be conducted at both designand off-design points [3]. Although the computational fluiddynamics has been impressively developed, it increases thetime cost. Therefore, the conventional mean-line methodusing empirical loss models is still the most practical way topredict the off-design performance of the centrifugalcompressor [4,5]. In the past few years, 1D aerodynamicperformance analysis method has been developed persis-tently. Oh et al. presented a reliable performance

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prediction method of centrifugal compressor consideringvarious empirical loss models, which agreed fairly well withexperimental data for a variety of centrifugal compressors[5]. Galvas analyzed the off-design performance ofcentrifugal compressor with the introduced 1D model byconsidering surge and choke margins [6]. Muralidharanpresented a 1D analysis method and achieved the design ofa centrifugal compressor, of which the total to totalpressure ratio was up to 8 [2]. Aungier proposed acomprehensive mean streamline performance analysismethod for centrifugal compressor stage, and the detailedvalidation was reliable against experimental data [7]. Liused a new optimum set of loss models to predict theperformance of a low pressure stage centrifugal compressor,the results showed good agreements with the experimentaldata [1]. On the whole, 1D aerodynamic performanceanalysis method is time saving compared with three-dimensional (3D) fluid analysis.

As high pressure ratio centrifugal compressor develops,the blade is becoming thinner and over-backswept.Besides, increased impeller rotation speed, outlet temper-ature and aerodynamic force will promote impeller failure[8]. Xu analyzed the high-stress locations of the centrifugalcompressor impeller by using finite element method, hefound that the impeller was easy to fail at the root of theleading edge area [9]. Lerche and Kang used a fluid-structure interaction (FSI) model to predict stress ofcentrifugal compressor blade, in which pressures were

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Fig. 1. Framework of MDF.

2 S. Shouyi et al.: Mechanics & Industry 20, 628 (2019)

transferred to the structural codes unidirectionally [10,11].In general, strength checks are often carried out during thedetailed design process. However, fewer studies about 1Dstrength prediction method are conducted. When themechanical requirements are not satisfied, the centrifugalcompressor would be redesigned which is time consuming.

Centrifugal compressor is a complicated system that issubjected to multidisciplinary coupling and should meetmulti-objective requirements. Hence, the multidisciplinarydesign optimization (MDO) method is frequently used toimprove accuracy of the compressor design. At present,there are manymultidisciplinary couplingmethods to solvethe MDO problem, including Multi-Disciplinary FeasibleDesign (MDF), Individual Discipline Feasible (IDF),Collaborative Optimization (CO), Concurrent SubspaceOptimization (CSSO), and Bi-Level Integrated SynthesisSystem (BLISS), etc. [12]. And the MDF method waspreferred for its convenience of implementation. Yuoptimized the blade profile of a centrifugal compressorby applying the MDF method [13]. Zhang presented areliability-based multidisciplinary design optimizationframe work for centrifugal compressor to design blade[14]. However, the multidisciplinary optimization processof centrifugal compressor is mostly carried out in thedetailed design process rather than in the preliminarydesign process.

This paper presents a preliminary design method forthe centrifugal compressor based on multidisciplinaryoptimization. First, a performance prediction method isintroduced by considering various loss models. Second, astrength prediction method is proposed to analyze thestress state around the hub of compressor leading edge.Third, the basic structural parameters are optimizedconsidering multiple objectives and multiple disciplines.Final, the optimal results are verified by the 3Daerodynamic analysis and structural finite elementanalysis.

2 Preliminary design method consideringmultidisciplinary coupling

2.1 The structure of MDO system

The MDO method is used to achieve the best design underthe joint action of multiple interacting disciplines.Normally, each disciplinary in a n disciplines couplingsystem is described in the following representation:

yi ¼ fðxi; yj; zÞ; i; j ¼ 1; . . . ;n j≠ i ð1Þ

where n is the total number of coupled disciplines, countedby i. yi represents ith discipline, xi is the local variablesvector. The vector yj corresponds to interdisciplinarycouplings, and z denotes the global or shared variablevector.

As a typical multi-physics model, the design ofcentrifugal compressor involves aero-thermodynamic andmechanical systems. MDF method is used because of smallamounts of design variables. It combines the multidisci-plinary design analysis (MDA) with an optimizer to find

the optimal solution. The MDF method can be formulatedas follows:

find x

min fðz; yiðz; yj;xÞ;xÞ i; j ¼ 1; : : : ;n j≠ i

s:t: gðz; yiðz; yj;xÞ;xÞ � 0

xl � x � xu

8>>>>><>>>>>:

ð2Þ

where f is the objective function and g represent all theequality and inequality system constraints. x, xl and xu arevectors of design variables, variable lower bounds andupper bounds, respectively. z is design parameters. Theframework is shown in Figure 1.

2.2 Preliminary design and analysis method

The 1D aero-thermodynamic calculation is an approximatemethod to solve the complicated flow inside impellermachinery. It is assumed that the internal flow is one-dimensional, which means the physical fluid parametersperpendicular to the direction of flow are equal, whilevariable in other directions. The preliminary design ofcentrifugal compressor is an iterative process, and thedetailed calculation process is described below.

2.2.1 1D calculation2.2.1.1 Impeller inlet

The goal of centrifugal compressor inlet design is to get theminimum inlet relative Mach number at shroud (W1s) for agiven mass flow and structural requirements. The differ-ence of mass flow between choke and stall is significantlyreduced with an unreasonable inlet relative Mach number,which will lead to a bad operation range. For a given inletconditions, there exists a unique absolute inlet velocity thatsatisfies the minimum inlet Mach number. Therefore, it isan optimization process for the centrifugal compressorimpeller inlet design. The nonlinear programming methodis used to solve the optimization problem. As the inletdesign begins, the pressure ratio (pi), work fluid properties,the inlet total pressure (P0), total temperature (T0) as wellas the mass flow ( _m) must be known. Besides, the followingdata should be determined based on the design experience:the impeller rotational speed (n), blockage factor (B),

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Fig. 2. Triangle of velocities.

S. Shouyi et al.: Mechanics & Industry 20, 628 (2019) 3

hub-shroud radius ratio (r1h/r1s) or the hub radius (r1h).The r1h/r1s is general range from 0.3 to 0.5. A greater onewill lead to impeller premature blockage because of reducedinlet flow area, while the inlet Mach number of impellershroud will be too big for a smaller r1h/r1s. The goal of inletdesign is to get a minimum W1s and corresponding designvalues of impeller inlet.

2.2.1.2 Impeller outlet

The impeller outlet flow angle, blade height and theimpeller diameter are concluded from the impeller outletdesign. According to the 1D steady flow equations, therelationship between the impeller pressure ratioand impeller inlet and outlet velocity is as shown inequation (3)

P 02

P 01

� �k�1k

¼ 1þ k� 1

kRT 01hl U2Cu2 � U1Cu1ð Þ: ð3Þ

The impeller pressure ratio is only related to the outletparameters and impeller efficiency if the inlet flow directionis axial. The back sweep angle (b2) is of great benefit toimprove compressor performance. Figure 2 shows thetriangle of velocities of two types impeller with twodifferent back sweep angle of 0° and �30°, which arecorresponding to radial impeller and backswept impeller,respectively.

When the total work of the radial impeller andbackswept impeller is equal, i.e. they have the sameU2Cu2, the absolute flow angle of the backswept impellerexit is smaller than that of radial impeller. The velocity atimpeller exit decreases as the back sweep angle increases,which will lead to a wide surge margin. However, a biggerback sweep angle would give rise to the stress levels [15].Besides, the back sweep angle increases the radius ofcurvature of streamline at S1 stream surface. Thecentrifugal force of the blade would counteract the pressuredifference between pressure side and suction side, andfurther reducing the flow loss of the secondary flow. Hence,an appropriate design of back sweep angle is the key to highperformance centrifugal compressor.

In order to reflect the phenomenon of jet-wake, the slipfactor is used to model the flow deviation and calculate thetangential absolute velocity component at the impelleroutlet. Stanitz [16], Weisner [17], Qiu [18] have proposedmany different forms of the model, and Weisner slip factoris frequently used for its good correlation with a lot of

experiments:

m ¼ 1�ffiffiffiffiffiffiffiffiffiffiffifficosb2

p=Z0:7: ð4Þ

The impeller exit absolute flow angle (a2) is the keyparameter which influences both the mixing loss at theimpeller exit and the development of the wall boundarylayers in the vaneless space. A range of 69° < a2 < 73° isrecommended with acceptable mixing loss and vanelessspace pressure recovery [15].

2.2.1.3 Diffuser

Diffuser is the device converting kinetic energy intopotential energy that fixed on the outlet of centrifugalimpeller. The performance of diffuser has an importanteffect on the aerodynamic efficiency, pressure ratio andstable operation range of centrifugal compressor. And theflow loss of diffuser in the centrifugal compressor is about30% of the total loss [19,20].

Vaneless diffusers have a wider operating rangecompared to the vaned diffusers, and there are no fatiguefailure problems caused by vibration between blade andimpeller. The flow choke phenomenon of diffuser willhardly happen since there is nearly no throat area invaneless diffuser. But the flow channel length of diffusershould be long enough to get an appropriate diffusingcapacity. The long flow channel will increase the frictionloss and decrease the aerodynamic efficiency of centrifugalcompressor.

2.2.2 Performance prediction2.2.2.1 Efficiency

According to the mean line approach, a compressor stagecan be treated as several characteristic parts, each definedby inlet and outlet surface. Losses occurring in differentparts are modeled in the past several decades. Themechanisms of different loss types are various, thecorresponding contributions to the entropy increase inturbomachines were described by Whitfield and Dentonet al. [21,22]. A poor loss model will cause problems in thedesign process. Hence, the appropriate loss models ofcentrifugal compressor should be selected to predict theefficiency of centrifugal compressor accurately.

Furthermore, an optimum loss model set should bechosen during the preliminary design of centrifugal compres-sor. Recent years, many empirical loss models are availablefor each respective loss mechanism. Oh [5], Galvas [6],Aungier [7] and others researchers put forward some sets ofloss models that agree well with the experimental results.Table 1 presents loss correlations used in the present study.

2.2.2.2 Surge margin

At surge condition, the airflow oscillates at low frequenciesthroughout the compressor system. Hence, the stagestability is seriously limited by the surge margin.Generally, the backswept angle of centrifugal compressorcould lead to a low surge flow and hence larger surgemargin, making the locus of peak efficiency away from thesurge line [28].

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Table 1. Loss correlations.

Loss mechanism Loss model Reference

Incidence loss Dhinc ¼ 0:4 Wm1 � Cm1

sinb1

� �2

Aungier [7]

Blade loading loss Dhbl ¼ 0:05D2fU

22

where: Df ¼ 1� W2

W1sþ 0:75DhEuler=U

22

ðW1s=W2Þ½ðZ=pÞð1�D1s=D2Þ þ 2D1s=D2�DhEuler =Cu2U2�Cu1U1

Coppage et al. [23]

Skin friction loss Dhsf ¼ 2CfLB

DhydW

2

W ¼ 2W2 þW1s þW1h

4

LB ¼ p

8D2 �D1s þD1h

2� b2 þ 2Lz

� �2

cosb1s þ cosb1h

2þ cosb2

0B@

1CA

Dhyd

D2¼ cosb2�

Z

pþ d2cosb2

b2

�þ1

2

D1s

D2þD1h

D2

� �cosb1s þ cosb1h

2

� �Z

pþ D1s þD1h

D1s �D1h

� �cosb1s þ cosb1h

2

� �

Jansen [24]

Clearance loss Dhcl ¼ 0:6Dncl

b2

� �Cu2

U2

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4p

b2Z

r21s � r21hr2 � r1sð Þ 1þ r2=r1ð Þ

� �Cu2

U2

� �Cm2

U2

� �sJansen [24]

Mixing loss Dhmix ¼ 1

1þ tan2a2

1� ewake � b�

1� ewake

� �2 C22

2

where: ewake= 0.25, b*= 1

Johnston and Dean [25]

Disc friction lossDhdf ¼ fdfrr

22U

32

4 _m

where: r ¼ r1 þ r22

fdf ¼

2:67

Re0:5df

; Redf < 3�105

0:0622

Re0:2df

; Redf ≥ 3� 105

8>>>><>>>>:

Redf ¼ U2r2n2

Daily and Nece [26]

Vanelss diffuser loss Dhvlf ¼ 2CfdðLd=DhydÞC2m

Cfd ¼ k1:8 � 105

Red

� �0:2

; k ¼ 0:015

Japikse [27]

Recirculation loss Dhrc ¼ 0:02D2fU

22

ffiffiffiffiffiffiffiffiffiffiffiffiffiffitana2r

pJansen [24]

4 S. Shouyi et al.: Mechanics & Industry 20, 628 (2019)

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Fig. 4. Equivalent model for strength prediction.Fig. 3. Impeller topology block structure.

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As the number of vanes increases, the distribution ofthe flow near vanes will be unsteady, which results in a kindof obstruction for the incoming flow to the diffuser [29].Besides, the diffuser with few vanes contributes to the widesurge margin [30].

The surge point can be defined as the point of maximumtotal pressure on the compressor characteristic curve.Usually the expression of the surge margin is as follows:

SM ¼ _mdesign � _msurge

_mdesign

� �: ð5Þ

2.3 Strength prediction of centrifugal compressorimpeller2.3.1 Strength prediction method

Centrifugal impeller is rotating with the rotor assembly athigh rotational speeds. Typically, the tip speed and thematerial are used to assess structural strength. However,the range of the tip speed is too large to get the precisestrength evaluation for the same material [15]. Simulta-neously, the high temperature and high speed gas flowthrough the impeller will affect the stress state of theimpeller.

Figure 3 shows a 3D model of a sector of impeller with asingle blade. A relatively large stress region is located at thehub of the centrifugal impeller blade near the leading edge,as shown in the red region in Figure 3, which is resultedfrom the centrifugal force, aerodynamic force and thermaleffect.

To calculate the stress state of the impeller blade hubnear the leading edge, a strength prediction (SP) methodwas proposed. The leading edge region of impeller bladebefore the dotted line in Figure 3 is simplified to anequivalent model, as shown in Figure 4. The root of the

model is a square area with the same width as the blade hubthickness. As the blade thickness gradually decreases alongthe leading edge from root to tip, the equivalent model isalso designed with gradient thickness. A concentrate forceis applied on the tip of the equivalent model to simulate thecentrifugal force. The moment is to produce bending stressinduced by the blade twist, aerodynamic force and thermaleffect.

The bladeheight,flowangle, impeller hub and tip radius,blade thickness can be obtained through the preliminarydesign. Because the stress at the hub is a superposition ofdifferent stress component, which is resulted from differentforce, the maximum stress at the hub of centrifugal impellernear the leading edge could be described through a linearexpression as below:

smax ¼ KFsF þKMsM ð6Þwhere

sF ¼ 1

t21h

Z r1s

r1h

rDVv2rdr ð7Þ

sM ¼ P0t1hðr1s � r1hÞ22Wb

ð8Þ

whereV represents the volume of the equivalent model.KFand KM is the stress coefficient for tensile stress andbending stress, respectively. Wb ¼ 1

6 t31h is the section

modulus.

2.3.2 Validation of the strength prediction method

To validate the strength prediction method, the FEM wasapplied to carry out the mechanical analysis of the impellerblade. The geometric parameters obtained through the 1Daero thermodynamic calculation were used to generate a

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Fig. 5. Geometry of impeller blade.

Table 2. Comparisons between the FEM and SP.

Num. pdesign v (rpm) _m (kg/s) Strength prediction(MPa)

Validationof 3D (MPa)

Error

1 5.2 32000 6.0 346.186 345.539 0.187%2 5.2 28500 6.5 316.811 315.471 0.425%3 5.2 25000 7.0 281.518 284.435 1.025%4 5.2 22000 7.5 250.781 244.239 2.678%5 4.2 28500 6.0 299.889 297.813 0.697%6 4.2 25000 6.5 265.506 272.541 2.581%7 4.2 22000 7.0 239.356 239.427 0.030%8 4.2 20000 7.5 222.617 223.548 0.412%

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model of 3D impeller blade is as shown in Figure 5. The huband shroud profiles were generated by using Bezier curves.Suction surface and pressure surface were modeled throughruled surface in NX 8.0.

The material of the impeller blade was TC4(E=109GPa, and Poisson’s ratio y=0.34) for its stableperformance under high temperature. The impeller bladewas subjected to a centrifugal force with the hub surfacefixed. Thermal effect was neglected due to less impact

on the stress near the leading edge. Figure 6 shows thestress contour of the blade. The maximum stress occurs atthe hub of impeller exit which may be resulted from theexcessively strong constraint.

Stress concentration occurs at the hub of the leadingedge area. Figure 7 shows the stress values along path 1from left to right that is labeled in the red ellipse area ofFigure 6. The stress distribution at the hub of the impellernear the leading edge exhibits a single peak shape.

Table 2 lists 8 different design conditions to ensure theuniversality of the strength prediction method. And theresults of the FEM analysis were compared with thoseobtained by the strength prediction method. It can be seenthat the strength prediction method shows good accuracy,the maximum errors are less than 3%.

3 Multidisciplinary optimization designof centrifugal compressor

3.1 Multidisciplinary optimization model

Combining the 1D aero-thermodynamic calculation andstrength prediction method, the multidisciplinary designoptimization method of centrifugal compressor duringpreliminary design stage was proposed. Figure 8 showsthe diagram of optimization procedure of centrifugalcompressor. A centrifugal impeller with lowest pressureratio of 4.4 was designed by means of this optimizationmethod. At the beginning of optimization, the designvariables initialization can be conducted based on the 1Daero-thermodynamic calculation. Then the initial designvariable are simultaneously imported into aerodynamicanalysis and mechanical analysis. Subsequently, the aero-dynamic analysis results will be also imported to themechanical analysis to calculate the maximum stress atimpeller hub. And aerodynamic analysis and mechanicalanalysis are carried out in sequence. If the convergence issatisfied, the optimization process will stop. Otherwise, thecalculated performance parameters in MDA system will bereturned to the Isight optimizer and new design parameterswill be evaluated.

The model of the centrifugal compressor optimizationincludes objective function, design variables and con-straints. The design variables are shown in Table 3. The

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Fig. 6. Stress contour of the blade.

Fig. 7. Stress values along path 1.

Fig. 8. Diagram of optimization procedure.

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variable bounds were determined based on the designexperience in the literature that published in the pastseveral decades. The given values of design parameters areobtained through the initial design results of 1D aerothermodynamic calculations.

In this paper, constraints of the preliminary optimiza-tion are set as follows:

– maximum stress at the hub near the leading edge s; – pressure ratio of the centrifugal compressor p;

–

the relative Mach number at the shroud of impeller inletMa1s.

The details of the design constraints and the safetyfactors are shown in Table 4.

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Table 3. Design variables of centrifugal compressor.

Parameters Givenvalues

Lowerbounds

Upperbounds

r1h (mm) 32.0 30.0 35.0r1s (mm) 91.34 85.0 98.0r2 (mm) 164.50 160.0 170.0b2 (mm) 22.90 20.0 25.0LZ (mm) 115.10 110.0 120.0b1h (°) 26.34 20.0 30.0b1s (°) 50.90 45.0 52.0b2 (°) –30.00 –35.0 –25.0

Table 5. Parameters comparison between the optimiza-tion design and the original design.

Parameter Originaldesign

Optimaldesign 1

Optimaldesign 2

r1h (mm) 32.0 30.00 30.01r1s (mm) 91.34 93.56 97.99r2 (mm) 164.50 169.92 165.65b2 (mm) 22.90 21.80 20.94LZ (mm) 115.10 112.12 112.08b1h (°) 26.34 29.72 29.75b1s (°) 50.90 47.33 47.34b2 (°) –30.00 –34.82 –34.99h 86.261 87.923 88.506SM 0.3273 0.3455 0.3727PR 4.467 4.82 4.406Ma1s 1.05 0.912 0.821s (MPa) 287.529 307.13 339.273

Table 4. Constraints and safety factors of the centrifugalcompressor.

Constrainedparameters

Symbolsof safetyfactors

Lowerbounds

Upperbounds

PR PRdesign 4.4 –

s [s] – 650 (MPa)Ma1s [Ma1s] – 1.2

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The objective of the present optimization is to find themaximum efficiency with sufficient surge margin under theconstraints of strength and aerodynamic requirements.The optimization can be expressed in a mathematic modelas equation (9).

Maximize Eff;SM

s:t:

s < ½s�PR≥PRdesign

Ma1s < ½Ma1s�:

8><>:

ð9Þ

3.2 Optimization results

The optimization was completed based on Isight commer-cial software using Non-dominated Sorting GeneticAlgorithm-II (NSGA-II), which is excellent as a multi-object optimization method. Table 5 shows the parameterscomparison between the optimal design and the originaldesign. Optimal design 1 set the maximum of pressure ratioas the third optimization target rather than constraint.While Optimal design 2 were obtained following theoptimization model of equation (9).

The results of optimal design 2 show that the efficiencyof the impeller improves by 2.24%. And the impellerpressure ratio is slightly decreased. It can be seen that themaximum stress value at the impeller hub near the leadingedge becomes larger, which is because of the increasedinlet blade height. In addition, the surge margin of the

impeller improves significantly. When the pressure ratiowas set as the third optimization target, the optimizedpressure ratio was up to 4.82. The impeller efficiency andsurge margin decreased compared to Optimal design 2,however, the impeller efficiency and surge margin are stillhigher than the original design. It is noticed that themaximum stress value of optimal design 1 is less than thatof optimal design 2, but larger than the original design.This is because of the difference of the inlet blade height.Besides, it can be inferred that the strength requirementwill limits the development of high performance ofcentrifugal compressor with high aerodynamic efficiencyand large surge margin. During the optimization process,8 core CPU was used, and the overall time foroptimization is about 20 minutes after 600 optimizediterations, which is very time-saving.

4 3D validation of optimal design

The blademodeling method is described in Section 2.3. Theblade thickness at the shroud is normally designed as thinas practicable to get an optimum efficiency. Splitters areadded to the impeller for high pressure ratio, and it is just acut-back main blade.

4.1 3D validation of aerodynamic performance

The optimal design 1 was verified by a 3D CFD codeNumeca, and S-A turbulence models was applied for its finestability. For the near-wall grid, the y+ of the mesh wereless than 5 and the corresponding distribution is shown inFigure 9. The impeller flow field was using H&I meshtopology with about 3,100,000 grid nodes. The inlet totaltemperature 287.15K, total pressure 1,01,325Pa and axialflow direction were given as the compressor inlet boundarycondition.

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Fig. 10. Meridional view of static pressure through the impeller.

Fig. 9. y+ distribution of the blade mesh.

Table 6. Comparison between the results of 3D validationand Optimal design 1.

3D validation Optimal design 1

h (%) 89.48 87.923SM 0.3291 0.3455PR 5.255 4.82

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Figure 10 presents contour plot of static pressure on themeridional plane of the impeller. The results show that thestatic pressure distributes smoothly through the impelleras expected. The aerodynamic performance results of the3D aerodynamic calculation are listed in Table 6, whichagree well with the results of Optimal design 1. Thepressure ratio calculated by 3D CFD is higher than thepreliminary results, which is mainly induced by the addedsplitter blades.

4.2 Strength check

In order to verify the structural strength accuratelypredicted by the preliminary design based on themultidisciplinary method, a 3D model of a sector ofcentrifugal impeller with a single blade was established.Considering the effects of pressure and temperature onthe structural analysis, the aerodynamic results wereinvolved in the strength check process. Thanks to the meshdifference between aerodynamic analysis and structuralanalysis, the inverse distance weighted (IDW) method wasused as the interpolation method to transfer the physicalinformation (pressure, temperature and mesh property,etc.) from aerodynamic analysis to mechanical analysis. A3D IDW method can be formulated as:

Fðx; y; zÞ ¼PN

i¼1 wiðx; y; zÞfiPNi¼1 wiðx; y; zÞ

ð10Þ

where

wiðx; y; zÞ ¼ 1=dmi

di ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� xiÞ2 þ ðy� yiÞ2 þ ðz� ziÞ2

q:

The information of pressure field and temperature fieldobtained byNUMECAwas interpolated into the structuralcomputational nodes using IDW method. Figure 11 showsthe temperature distribution on blade surface calculated byNUMECA and the interpolation results, which exhibitsgood agreement.

The mechanical analysis was conducted using ABA-QUS. The calculated stress distribution of the impeller is asshown in Figure 12, and themaximum stress of the impellerhub near the leading edge is 313.082MPa, which is nearlyequal to the predicted results of Optimal design 1. Besides,it can be seen that the maximum stress occurs at the centerof the impeller, which is resulted from the large centrifugalforce at impeller outlet area.

5 Conclusions

This paper presents a preliminary design method byconsidering multidisciplinary coupling. An 1D strengthprediction method was proposed to evaluate the maximumstress at the impeller hub near the leading edge. Themultidisciplinary design optimization system of preliminary

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Fig. 12. Stress distribution of compressor impeller.

Fig. 11. Temperature distribution of centrifugal impeller blade (a) after interpolation; and (b) before interpolation.

10 S. Shouyi et al.: Mechanics & Industry 20, 628 (2019)

design was established and used to design a centrifugalcompressor with a lowest pressure ratio of 4.4. Theconclusions of the study are as follows:

– An equivalent strength prediction model was establishedto simulate the stress state of impeller hub near theleading edge area. The strength prediction formulationwas deduced by considering tensile stress induced bycentrifugal force and moment induced by blade twist,aerodynamic force and thermal effect. Then the 1Dstrength method was verified by 3D finite element data,and the maximum errors were less than 3%.

–

The preliminary design of a centrifugal compressor withlowest pressure ratio of 4.4 is executed based onmultidisciplinary design optimization method. Theresults show that the structural strength is closelyrelated to the blade inlet height. The maximumaerodynamic efficiency of the optimized impeller is88.506% with pressure ratio of 4.406. Another optimaldesign with higher pressure ratio up to 4.82 sacrificesaerodynamic efficiency and surge margin.

–

The 3D CFD was applied to analyze the aerodynamicperformance of the optimized centrifugal compressor.The calculated results agreed well with the preliminarydesign results. The efficiency was up to 89.48% with apressure ratio of 5.255. The higher pressure ratio wasproduced by the added splitter blades. Besides, thecomplete mechanical analysis of centrifugal impeller wascarried out, the maximum stress at the impeller hub nearthe leading edge was 313.082MPa, which was consistentwith the strength prediction results.

Nomenclature

P

Pressure T Temperature k Ratio of specific heats R Gas constant h Efficiency U Impeller blade tip speed C Absolute gas velocity W Relative gas velocity a Absolute air angle

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S. Shouyi et al.: Mechanics & Industry 20, 628 (2019) 11

b

Relative air angle Dh Enthalpy change (J/kg) m Slip factor Z Blade number Df Diffuser factor D Diameter LB Impeller flow length Cf Skin friction coefficient Dhyd Impeller average hydraulic diameter Lz Axial length of impeller b Blade height r Radius Dncl Clearance r Density ewake Wake fraction of blade-to-blade space _m Mass flowrate Re Reynolds number n Kinematic viscosity DV Volume of blade element t Blade thickness PR Stage total-to-total pressure ratio Ma Mach number SM Surge margin s Max stress of impeller blade

Subscripts

0

Total condition 1 Impeller inlet 2 Impeller exit l Impeller m Meridional direction u Tangential direction s Shroud h Hub d Diffuser passage length

Acknowledgements. National Science and Technology MajorProject (2017-II-0006-0019) and National Natural ScienceFoundation of China (11902259) support this work.

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Cite this article as: S. Shouyi, Y. Zhufeng, L. Lei, Z. Mengchuang, Y.Weizhu, Preliminary design of centrifugal compressor usingmultidisciplinary optimization method, Mechanics & Industry 20, 628 (2019)