influences of the flow cut and axial lift of the impeller

energies

Article

Influences of the Flow Cut and Axial Lift of theImpeller on the Aerodynamic Performance of aTransonic Centrifugal Compressor

Kun Sung Park 1, In Hyuk Jung 1, Sung Jin You 1, Seung Yeob Lee 1, Ali Zamiri 2 andJin Taek Chung 2,*

1 Department of Mechanical Engineering, Graduate School of Korea University, 02841 Seoul, Korea;[email protected] (K.S.P.); [email protected] (I.H.J.); [email protected] (S.J.Y.);[email protected] (S.Y.L.)

2 Department of Mechanical Engineering, Korea University, 02841 Seoul, Korea; [email protected]* Correspondence: [email protected]; Tel.: +82-2-921-3364

Received: 25 October 2019; Accepted: 26 November 2019; Published: 27 November 2019 ��

Abstract: In this study, the influences of the flow cut and axial lift of the impeller on the aerodynamicperformance of a transonic centrifugal compressor were analyzed. The flow cut is a method to reducethe flow rate by decreasing the impeller passage height. The axial Lift is a method of increasingthe impeller passage height in the axial direction, which increases the impeller exit width (B2) andincreases the total pressure. A NASA CC3 transonic centrifugal compressor with a backswept anglewas used as a base compressor. After applying the flow cut, the total pressure at the target flow ratewas lower than the total pressure at the design point due to the increase in the relative velocity at theimpeller exit. After applying the axial lift, the total pressure at the design flow rate was increased,which was caused by the reduction in the relative velocity as the passage area at the impeller exit wasincreased. By applying the flow cut and axial lift methods, it was shown that the variation in relativevelocity at the impeller exit has a significant effect on the variation in total pressure. In addition,it was found that the relative velocity at the impeller exit of the target flow rate is maintained similarto the base impeller when the flow cut and the axial lift are combined. Therefore, by combining theflow cut and the axial lift, three transonic centrifugal impellers with flow fractions of 0.7, 0.8, and 0.9compared to the design flow rate were newly designed.

Keywords: transonic centrifugal compressor; backswept angle; flow cut; axial lift; relative velocity;aerodynamic design; impeller

1. Introduction

Centrifugal compressors are used in various fields such as power generation, chemistry, andenvironmental plants, and therefore compressors with varying performance (pressure and flow rate)are required [1,2]. Designing the compressors required for each field requires the efforts and timeof engineers [3–5] and significant investments by compressor manufacturers [6,7]. Therefore, it isdesirable to design a new compressor by modifying the impeller for a compressor that has alreadybeen designed and proven in performance. It is also necessary to design compressors as a group withina specific flow range by combining impeller modification methods.

There are two advantages to designing the impeller as a group within a certain range. The first isthat compressors with the performance required for each field be manufactured quickly and at a lowcost, and the design success rate can be increased. This is because only the impeller is modified inthe design of compressors with different performances, so the casings, shafts, and bearings of proven

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Energies 2019, 12, 4503 2 of 19

compressors can be used. Second, all compressors can be operated stably in the operating range with asufficient surge margin because the impeller modification method can shift the performance curve tomatch the operating range of the compressors required for each field.

The most popular impeller modification methods are the flow cut and axial trim methods [8,9].The flow cut method reduces the flow rate by decreasing the impeller passage height. When thismethod is applied, the performance curve is shifted in the direction of decreasing flow rate. The axialtrim method decreases the impeller passage height in the axial direction, which decreases impellerexit width (B2) and the total pressure. This method is generally used to reduce the total pressure inimpellers with a backswept angle. In addition, the flow lift and axial lift methods increase the impellerpassage height, unlike the flow cut and axial trim methods. When the flow lift method is applied,the flow rate is increased because the impeller passage height is increased, and when the axial lift isapplied, the total pressure is increased because B2 is increased. However, when the flow lift and axiallift methods are applied to the impeller, the size and weight of the aerodynamic components of thecompressor are increased, and more power is required. Therefore, these methods are not generallyapplicable because it is necessary to redesign the casings, shafts, and bearings and reselect the motor.

The impeller modification methods such as the flow cut and axial trim methods have longbeen used by most centrifugal compressor manufacturers to design new compressors, and in 2001,Rogers first introduced the flow cut method in the paper [10]. Rogers recommended that the flowcut method should be applied only at a nondimensional specific speed (Ns) between 0.5 and 1.2.Tim David (2006) and Donghui Zhang (2010) applied the flow cut method by up to 75% (a flow fractionof 0.25 compared to the design flow rate of the base compressor) to the impeller of an industrialcompressor with a nondimensional specific speed (Ns) of 0.06. However, Tim David and DonghuiZhang recommended applying only 50% of the flow cut to the impeller [11,12]. Daniel Swain (2014)applied the flow cut and axial trim methods to different types of impellers such as a transonic impellerwith a relative Mach number between 0.8 and 1.2 at the impeller inlet tip, a closed impeller without tipclearance, and an impeller without a splitter [13]. For most impellers, the ratio of the reduced flow rateand the reduced passage area was not 1:1 after applying the flow cut method. Daniel Swain also notedthat applying the axial trim method increases the relative velocity at the impeller exit and reducesthe total pressure. Shin et al. (2015, 2017) designed various industrial compressors using the flow cutand scaling methods. Shin et al. noted that the pressure at the target flow rate was different fromthe pressure at the design flow rate after applying the flow cut method because the work coefficientchanged [14,15].

In previous studies, after applying the flow cut method to the impellers, the ratio of the passagearea reduction and the flow rate reduction was not 1:1. In most impellers, the performance curve wasshifted more to the left, which means that the target flow rate is shifted to near the choke and therelative velocity is increased at the impeller exit of the target flow rate. In the centrifugal impeller witha backswept angle, an increased relative velocity at the impeller exit causes a reduction of the totalpressure [16,17]. The reason for this reduction is that the increased relative velocity reduces the angle ofthe absolute velocity (Meridional angle convention). This reduction means that the tangential velocityis reduced and the work coefficient is reduced, which results in a reduction in the total pressure [18,19].Therefore, the relative velocity at the impeller exit must be maintained at the level of the impeller ofthe base compressor to design an impeller for which the total pressure at the target flow rate is similarto the total pressure at the design flow rate.

Designing an impeller for which the total pressure at the target flow rate is similar to the totalpressure at the design flow rate is not easy using the impeller modification method with only a flowcut applied. The reason is that it is very difficult to establish a 1:1 match between the reduced passagearea and the reduced flow rate. However, if the axial lift method is used in combination with the flowcut method, this problem can be solved because after the flow cut is applied, the reduced total pressureat the target flow rate can be increased to a total pressure that is similar to the design flow rate byapplying the axial lift method.

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The main objectives in this study are to analyze the influences of the flow cut and axial lift ofthe impeller on the aerodynamic performance of a transonic centrifugal compressor and to designtransonic impellers by combining these methods. A NASA CC3 transonic centrifugal impeller with abackswept angle of −50◦ (Meridional angle convention) is selected to carry out this study, as shownin Figure 1a. The flow cut, axial lift, and combined flow cut and axial lift methods are applied to theimpeller, as shown in Figure 1b–d. Three-dimensional, compressible, steady Navier–Stokes equationswere solved to investigate the aerodynamic performance of each impeller.

Energies 2019, 12, x FOR PEER REVIEW 3 of 18

backswept angle of −50° (Meridional angle convention) is selected to carry out this study, as shown 95 in Figure 1a. The flow cut, axial lift, and combined flow cut and axial lift methods are applied to the 96 impeller, as shown in Figure 1b–d. Three-dimensional, compressible, steady Navier–Stokes equations 97 were solved to investigate the aerodynamic performance of each impeller. 98

(d)

(a)

(b)

(c)

Figure 1. A base impeller and three impeller modification methods. (a) Base impeller; (b) flow cut; (c) 99 axial lift; (d) combination of flow cut and axial lift method. 100

2. Numerical Method 101

2.1. Compressor Model 102

To carry out this study, the NASA CC3 transonic centrifugal compressor developed by NASA 103 Glenn Research Center was used as a base compressor [20]. This compressor consists of an inlet, an 104 impeller, and a vaneless diffuser. The impeller of the base compressor has a backswept angle of −50° 105 degrees (Meridional angle convention). The total-to-total pressure ratio was approximately 4.17 at 106 the design mass flow rate (4.54 kg/s), and the relative Mach number was approximately 0.85 at the 107 impeller inlet tip of the design point. The impeller consists of 15 main blades and 15 splitter blades. 108 Details about the impeller radius, rotational speed, and non-dimensional specific speed (Ns) are 109

Figure 1. A base impeller and three impeller modification methods. (a) Base impeller; (b) flow cut;(c) axial lift; (d) combination of flow cut and axial lift method.

2. Numerical Method

2.1. Compressor Model

To carry out this study, the NASA CC3 transonic centrifugal compressor developed by NASAGlenn Research Center was used as a base compressor [20]. This compressor consists of an inlet,

Energies 2019, 12, 4503 4 of 19

an impeller, and a vaneless diffuser. The impeller of the base compressor has a backswept angle of−50◦ degrees (Meridional angle convention). The total-to-total pressure ratio was approximately 4.17at the design mass flow rate (4.54 kg/s), and the relative Mach number was approximately 0.85 at theimpeller inlet tip of the design point. The impeller consists of 15 main blades and 15 splitter blades.Details about the impeller radius, rotational speed, and non-dimensional specific speed (Ns) are shownin Table 1. The AxCent aerodynamic design software developed by Concepts NREC was used in thestudy. The 3D shape of the impeller shown in Figure 2 was made by using this software. The inletwas made of a bell-mouth type, and the vaneless diffuser was made sufficiently longer than the datameasuring position (R3 = 255 mm) on the impeller exit.

Table 1. The specifications of the NASA CC3 compressor with a vaneless diffuser.

Specifications

Exit rating station (R3/R2)(Measuring position/Impeller exit radius) 1.18Impeller main blade/splitter blade 15 / 15

Design speed 21,789 RPMDesign flow rate 4.54 kg/s

Tip speed at impeller exit (U2) 492 m/sHub radius at impeller inlet (R1h) 41.4 mm

Shroud radius at impeller inlet (R1s) 105 mmBlade width at impeller exit (B2) 17 mm

Radius at impeller exit (R2) 215.5 mmImpeller backswept angle (β2b) 50◦

Impeller axial length 132.25 mmNondimensional specific speed (Ns) ~ 0.57


shown in Table 1. The AxCent aerodynamic design software developed by Concepts NREC was used 110 in the study. The 3D shape of the impeller shown in Figure 2 was made by using this software. The 111 inlet was made of a bell-mouth type, and the vaneless diffuser was made sufficiently longer than the 112 data measuring position (R3 = 255 mm) on the impeller exit. 113

Table 1. The specifications of the NASA CC3 compressor with a vaneless diffuser. 114

Specifications

Exit rating station (�� ⁄ )

(Measuring position/Impeller exit radius) 1.18

Impeller main blade/splitter blade 15 / 15

Design speed 21,789 RPM

Design flow rate 4.54 kg/s

Tip speed at impeller exit (��) 492 m/s

Hub radius at impeller inlet (��) 41.4 mm

Shroud radius at impeller inlet (��) 105 mm

Blade width at impeller exit (��) 17 mm

Radius at impeller exit (��) 215.5 mm

Impeller backswept angle (��) 50˚

Impeller axial length 132.25 mm

Nondimensional specific speed (Ns) ~ 0.57

Figure 2. The modeling of the NASA CC3 centrifugal compressor using the AxCent design software. 115

2.2. Computational Grid and Validation 116

The computational domain was selected as a single passage, as shown in Figure 3. The multi-117 block hexahedral structured grid was used to carry out CFD simulation, and the ANSYS Turbo Grid 118 (19.3, ANSYS, Canonsburg, PA, USA) was used to generate the mesh, as shown in Figure 3a. The 119 number of grid points for the streamwise, spanwise, and pitchwise directions were 249, 47, and 28, 120 respectively. Thirty grid points were used in the tip clearance to preserve the tip leakage flow 121 accurately. Twenty-one O-grid points were used around the main blade and splitter blade surfaces. 122 To use the SST turbulence model, Y+ was set to almost 2 or less, as shown in Figure 3b [21]. Because 123 the pitch ratio between the rotating and stationary frames was 1, the frozen rotor interface was used. 124 The inlet boundary conditions are the total pressure and the total temperature. The outlet boundary 125

Figure 2. The modeling of the NASA CC3 centrifugal compressor using the AxCent design software.

2.2. Computational Grid and Validation

The computational domain was selected as a single passage, as shown in Figure 3. The multi-blockhexahedral structured grid was used to carry out CFD simulation, and the ANSYS Turbo Grid (19.3,ANSYS, Canonsburg, PA, USA) was used to generate the mesh, as shown in Figure 3a. The number ofgrid points for the streamwise, spanwise, and pitchwise directions were 249, 47, and 28, respectively.Thirty grid points were used in the tip clearance to preserve the tip leakage flow accurately. Twenty-oneO-grid points were used around the main blade and splitter blade surfaces. To use the SST turbulencemodel, Y+ was set to almost 2 or less, as shown in Figure 3b [21]. Because the pitch ratio betweenthe rotating and stationary frames was 1, the frozen rotor interface was used. The inlet boundary

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conditions are the total pressure and the total temperature. The outlet boundary conditions are themass flow rate and the average static pressure, as shown in Figure 3c. First, the performance curveswere obtained by applying the exit average static pressure conditions, and finally, the performancecurves were obtained by applying the exit mass flow rate condition at the design point and near thesurge. When the exit mass flow rate was applied at the design point and near the surge, the initialvalue was used as the result closest to the target point among the results calculated by the exit averagepressure conditions. The reason is that the calculation using the mass flow condition takes a long timeto converge. Turbulence intensity was set to 5% at the inlet and the length of the inlet domain wasextended long enough to reduce the effects of boundary condition on the flow field. In addition, the tipclearance of the impeller was set to 0.2 mm. In relation to the impellers designed using flow cut andaxial lift, the grid and boundary condition were set in the same method as the base impeller.


conditions are the mass flow rate and the average static pressure, as shown in Figure 3c. First, the 126 performance curves were obtained by applying the exit average static pressure conditions, and 127 finally, the performance curves were obtained by applying the exit mass flow rate condition at the 128 design point and near the surge. When the exit mass flow rate was applied at the design point and 129 near the surge, the initial value was used as the result closest to the target point among the results 130 calculated by the exit average pressure conditions. The reason is that the calculation using the mass 131 flow condition takes a long time to converge. Turbulence intensity was set to 5% at the inlet and the 132 length of the inlet domain was extended long enough to reduce the effects of boundary condition on 133 the flow field. In addition, the tip clearance of the impeller was set to 0.2 mm. In relation to the 134 impellers designed using flow cut and axial lift, the grid and boundary condition were set in the same 135 method as the base impeller. 136

(a)

(b) (c)

Figure 3. Grid generation and computational domain of the base compressor. (a) Grid generation; (b) 137 Y plus; (c) computational domain. 138

The grid independence study is shown in Figure 4a,b, and the total-to-total pressure ratio and 139 the total-to-total isentropic efficiency are compared. As a result, the relative error decreased from 140 more than 2.4 million grids to less than 0.1%. Thus, approximately 2.8 million grids were selected for 141 this study, taking into account some margin. To predict the performance of the impeller, the 3D 142 Steady Reynolds-Averaged Navier–Stokes equations were solved using ANSYS CFX software (19.3, 143 ANSYS, Canonsburg, PA, USA). The SST turbulence model was used as the turbulence model, and 144

Figure 3. Grid generation and computational domain of the base compressor. (a) Grid generation; (b) Yplus; (c) computational domain.

The grid independence study is shown in Figure 4a,b, and the total-to-total pressure ratio andthe total-to-total isentropic efficiency are compared. As a result, the relative error decreased frommore than 2.4 million grids to less than 0.1%. Thus, approximately 2.8 million grids were selected

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for this study, taking into account some margin. To predict the performance of the impeller, the 3DSteady Reynolds-Averaged Navier–Stokes equations were solved using ANSYS CFX software (19.3,ANSYS, Canonsburg, PA, USA). The SST turbulence model was used as the turbulence model, and theCFD results of the impeller of the base compressor were compared with the experimental data tovalidate the analytical method of this study [22]. The performance curves are shown in Figure 5a,b.The total-to-total pressure ratio is in good agreement with the experimental results. For the total-to-totalefficiency, the CFD results were predicted to be approximately 1.5%–2% higher than the experimentalresults. Because the CFD was calculated for adiabatic conditions, the CFD efficiency value was slightlyhigher than the experimental value.


the CFD results of the impeller of the base compressor were compared with the experimental data to 145 validate the analytical method of this study [22]. The performance curves are shown in Figure 5a,b. 146 The total-to-total pressure ratio is in good agreement with the experimental results. For the total-to-147 total efficiency, the CFD results were predicted to be approximately 1.5%–2% higher than the 148 experimental results. Because the CFD was calculated for adiabatic conditions, the CFD efficiency 149 value was slightly higher than the experimental value. 150

(a) (b)

Figure 4. Grid independency test of the base compressor. (a) Total-to-total pressure ratio; (b) total-to-151 total efficiency. 152

(a) (b)

Figure 5. Comparison of the predicted results and the experimental data for validation. (a) Total-to-153 total pressure ratio; (b) total-to-total efficiency. 154

3. Numerical Results 155

3.1. Flow Cut 156

The flow cut method reduces the flow rate by decreasing the impeller passage height, as shown 157 in Figure 1b. When this method is applied, the performance curve is shifted to the left (in the direction 158 of decreasing flow rate) because the passage area is reduced. Thus, this method is used to design 159 compressors with different performances by reducing the flow rate. In this study, the passage area 160 inside the impeller was calculated using the annulus area. The annulus area is calculated using the 161 quasi-orthogonal (QO) lines, as shown in Figure 6. The following Equations (1)–(2) are used to 162 calculate the annulus area. 163

Figure 4. Grid independency test of the base compressor. (a) Total-to-total pressure ratio;(b) total-to-total efficiency.


the CFD results of the impeller of the base compressor were compared with the experimental data to 145 validate the analytical method of this study [22]. The performance curves are shown in Figure 5a,b. 146 The total-to-total pressure ratio is in good agreement with the experimental results. For the total-to-147 total efficiency, the CFD results were predicted to be approximately 1.5%–2% higher than the 148 experimental results. Because the CFD was calculated for adiabatic conditions, the CFD efficiency 149 value was slightly higher than the experimental value. 150

(a) (b)

Figure 4. Grid independency test of the base compressor. (a) Total-to-total pressure ratio; (b) total-to-151 total efficiency. 152

(a) (b)

Figure 5. Comparison of the predicted results and the experimental data for validation. (a) Total-to-153 total pressure ratio; (b) total-to-total efficiency. 154

3. Numerical Results 155

3.1. Flow Cut 156

The flow cut method reduces the flow rate by decreasing the impeller passage height, as shown 157 in Figure 1b. When this method is applied, the performance curve is shifted to the left (in the direction 158 of decreasing flow rate) because the passage area is reduced. Thus, this method is used to design 159 compressors with different performances by reducing the flow rate. In this study, the passage area 160 inside the impeller was calculated using the annulus area. The annulus area is calculated using the 161 quasi-orthogonal (QO) lines, as shown in Figure 6. The following Equations (1)–(2) are used to 162 calculate the annulus area. 163

Figure 5. Comparison of the predicted results and the experimental data for validation. (a) Total-to-totalpressure ratio; (b) total-to-total efficiency.

3. Numerical Results

3.1. Flow Cut

The flow cut method reduces the flow rate by decreasing the impeller passage height, as shown inFigure 1b. When this method is applied, the performance curve is shifted to the left (in the directionof decreasing flow rate) because the passage area is reduced. Thus, this method is used to design

Energies 2019, 12, 4503 7 of 19

compressors with different performances by reducing the flow rate. In this study, the passage areainside the impeller was calculated using the annulus area. The annulus area is calculated using thequasi-orthogonal (QO) lines, as shown in Figure 6. The following Equations (1)–(2) are used to calculatethe annulus area.

tan(αe f f)= (1− η)× tan(αh)+η× tan(αs) (1)

Annulus area = π×QOlength × (Rh+Rs)× cos(αe f f)

(2)

where, αh and, αs are the angles at which the QO line meets the hub and shroud lines, respectively,and are used to calculate the effective angles (αe f f ). The effective angle, which is calculated usingEquation (1), is the angle of the root mean square (RMS) located between the hub and shroud lines.Finally, the annulus area is calculated by substituting the effective angles, the radius of the hub (Rh),the radius of the shroud (Rs), and the QO line (QOlength) into Equation (2).


164

Figure 6. Q.O. (Quasi-orthogonal) lines in the impeller. 165

tan�αeff� = �1-η�× tan(αh) +η× tan(αs) (1) 166

Annulus area = π×QOlength×(Rh+Rs)× cos�αeff� (2) 167

where, α� and, α� are the angles at which the QO line meets the hub and shroud lines, respectively, 168 and are used to calculate the effective angles (α��). The effective angle, which is calculated using 169 Equation (1), is the angle of the root mean square (RMS) located between the hub and shroud lines. 170 Finally, the annulus area is calculated by substituting the effective angles, the radius of the hub (Rh), 171 the radius of the shroud (Rs), and the QO line (QOlength) into Equation (2). 172

(a) (b)

Figure 7. Aerodynamic design of FC10, 20, and 30 models corresponding to fractions of 0.9, 0.8, and 173 0.7 of the annulus area of the base compressor using the flow cut method. (a) Meridional view; (b) 174 decreased annulus area with respect to the flow cut method. 175

Figure 6. Q.O. (Quasi-orthogonal) lines in the impeller.

To continuously design impellers with different performances by reducing the flow rate,three impellers (FC10, FC20, FC30) corresponding to flow fractions of 0.9, 0.8, and 0.7 compared to thedesign flow rate were designed, as shown in Figure 7a. Table 2 shows the impeller inlet shroud radius(R1s), the impeller exit width (B2), and the annulus area at the impeller inlet and exit. In addition,the annulus area inside the impeller is shown as Meridional (%), as shown in Figure 7b, and theannulus area is decreased in proportion to the flow cut percentage.

Energies 2019, 12, 4503 8 of 19


164

Figure 6. Q.O. (Quasi-orthogonal) lines in the impeller. 165

tan�αeff� = �1-η�× tan(αh) +η× tan(αs) (1) 166

Annulus area = π×QOlength×(Rh+Rs)× cos�αeff� (2) 167

where, α� and, α� are the angles at which the QO line meets the hub and shroud lines, respectively, 168 and are used to calculate the effective angles (α��). The effective angle, which is calculated using 169 Equation (1), is the angle of the root mean square (RMS) located between the hub and shroud lines. 170 Finally, the annulus area is calculated by substituting the effective angles, the radius of the hub (Rh), 171 the radius of the shroud (Rs), and the QO line (QOlength) into Equation (2). 172

(a) (b)

Figure 7. Aerodynamic design of FC10, 20, and 30 models corresponding to fractions of 0.9, 0.8, and 173 0.7 of the annulus area of the base compressor using the flow cut method. (a) Meridional view; (b) 174 decreased annulus area with respect to the flow cut method. 175

Figure 7. Aerodynamic design of FC10, 20, and 30 models corresponding to fractions of 0.9, 0.8,and 0.7 of the annulus area of the base compressor using the flow cut method. (a) Meridional view;(b) decreased annulus area with respect to the flow cut method.

Table 2. The blade height and annulus area at the impeller inlet and exit (flow cut).

-Impeller Inlet Impeller Exit

R1s (mm)Annulus Area

B2 (mm)Annulus Area

Area (m2)Decreased

Ratio Area (m2)Decreased

Ratio

Base 105.00 0.02906 - 17.0 0.02300 -FC 10 100.43 0.02615 0.9 15.3 0.02070 0.9FC 20 95.67 0.02324 0.8 13.6 0.01840 0.8FC 30 90.70 0.02033 0.7 11.9 0.01610 0.7

The performance curves of the FC 10, 20, and 30 are predicted, as shown in Figure 8. The flowrate equal to the total pressure of the design flow rate was smaller than the target flow rate, and thetotal pressure at the target flow rate was less than the total pressure at the design flow rate. This resultshows that the decrease in the flow rate was more than the decrease in the passage area.

Energies 2019, 12, 4503 9 of 19


Table 2. The blade height and annulus area at the impeller inlet and exit (flow cut). 176

-

Impeller Inlet Impeller Exit

R1s

(mm)

Annulus Area B2

(mm)

Annulus Area

Area

(m2)

Decreased

Ratio

Area

(m2)

Decreased

Ratio

Base 105.00 0.02906 - 17.0 0.02300 -

FC 10 100.43 0.02615 0.9 15.3 0.02070 0.9

FC 20 95.67 0.02324 0.8 13.6 0.01840 0.8

FC 30 90.70 0.02033 0.7 11.9 0.01610 0.7

To continuously design impellers with different performances by reducing the flow rate, three 177 impellers (FC10, FC20, FC30) corresponding to flow fractions of 0.9, 0.8, and 0.7 compared to the 178 design flow rate were designed, as shown in Figure 7a. Table 2 shows the impeller inlet shroud radius 179 (R1s), the impeller exit width (B2), and the annulus area at the impeller inlet and exit. In addition, the 180 annulus area inside the impeller is shown as Meridional (%), as shown in Figure 7b, and the annulus 181 area is decreased in proportion to the flow cut percentage. 182

The performance curves of the FC 10, 20, and 30 are predicted, as shown in Figure 8. The flow 183 rate equal to the total pressure of the design flow rate was smaller than the target flow rate, and the 184 total pressure at the target flow rate was less than the total pressure at the design flow rate. This result 185 shows that the decrease in the flow rate was more than the decrease in the passage area. 186

Figure 8. Reduced total pressure at the target flow rate after applying the flow cut method. 187

3.2. Axial Lift 188

The axial lift method increases the impeller passage height in the axial direction, as shown in 189 Figure 1c, which increases B2 and the total pressure. When B2 is increased, the passage area at the 190 impeller exit increases, and the relative velocity (�� ) decreases. Because the blade speed at the 191 impeller exit (��) is constant, tangential velocity (��) increases. As a result, the work coefficient is 192 increased and the total pressure is increased. 193

To analyze the influence of the axial lift, B2 was set to 18 mm (AL-1) and 19 mm (AL-2) and 194 increased by 1 mm and 2 mm from the base impeller, as shown in Figure 9a. Table 3 shows the 195 impeller inlet shroud radius (R1s), the impeller exit width (B2), and the annulus area at the impeller 196 inlet and exit. In addition, the annulus area inside the impeller is shown in Meridional (%), which can 197 be seen to increase proportionally with respect to the axial lift, as shown in Figure 9b. 198

Figure 8. Reduced total pressure at the target flow rate after applying the flow cut method.

3.2. Axial Lift

The axial lift method increases the impeller passage height in the axial direction, as shown inFigure 1c, which increases B2 and the total pressure. When B2 is increased, the passage area at theimpeller exit increases, and the relative velocity (W2) decreases. Because the blade speed at the impellerexit (U2) is constant, tangential velocity (Cθ2) increases. As a result, the work coefficient is increasedand the total pressure is increased.

To analyze the influence of the axial lift, B2 was set to 18 mm (AL-1) and 19 mm (AL-2) andincreased by 1 mm and 2 mm from the base impeller, as shown in Figure 9a. Table 3 shows the impellerinlet shroud radius (R1s), the impeller exit width (B2), and the annulus area at the impeller inlet andexit. In addition, the annulus area inside the impeller is shown in Meridional (%), which can be seen toincrease proportionally with respect to the axial lift, as shown in Figure 9b.Energies 2019, 12, x FOR PEER REVIEW 9 of 18

(a) (b)

Figure 9. Aerodynamic design of AL-1 and AL-2 with the B2 value increased by 1 mm and 2 mm over 199 the B2 value of the base impeller using the axial lift method. (a) Meridional view; (b) increased 200 annulus area with respect to the axial lift method. 201

Table 3. The blade height and annulus area at the impeller inlet and exit (axial lift). 202

Figure 10. Increased total pressure at the target flow rate after applying the axial lift method. 203

The performance curves of AL-1 and AL-2 are predicted, as shown in Figure 10. After the axial 204 lift was applied, there was no significant change in the choking flow rate because there is no change 205 in the passage area at the impeller inlet. However, the total-to-total pressure ratio increased at all 206

-


R1s

(mm)

Annulus Area B2

(mm)

Annulus Area

Area

(m2)

Increased

Ratio

Area

(m2)

Increased

Ratio

Base 105.00 0.02906 - 17.0 0.02300 -

AL 1 105.00 0.02906 1.0 18.0 0.02436 1.06

AL 2 105.00 0.02906 1.0 19.0 0.02571 1.12

Figure 9. Aerodynamic design of AL-1 and AL-2 with the B2 value increased by 1 mm and 2 mm overthe B2 value of the base impeller using the axial lift method. (a) Meridional view; (b) increased annulusarea with respect to the axial lift method.

Energies 2019, 12, 4503 10 of 19

Table 3. The blade height and annulus area at the impeller inlet and exit (axial lift).

-Impeller Inlet Impeller Exit

R1s (mm)Annulus Area

B2 (mm)Annulus Area

Area (m2)Increased

Ratio Area (m2)Increased

Ratio

Base 105.00 0.02906 - 17.0 0.02300 -AL 1 105.00 0.02906 1.0 18.0 0.02436 1.06AL 2 105.00 0.02906 1.0 19.0 0.02571 1.12

The performance curves of AL-1 and AL-2 are predicted, as shown in Figure 10. After the axiallift was applied, there was no significant change in the choking flow rate because there is no change inthe passage area at the impeller inlet. However, the total-to-total pressure ratio increased at all flowrates except the choking flow rate. At the design flow rate, the total-to-total pressure ratio increased byapproximately 3% as B2 increased by 1 mm.


(a) (b)

Figure 9. Aerodynamic design of AL-1 and AL-2 with the B2 value increased by 1 mm and 2 mm over 199 the B2 value of the base impeller using the axial lift method. (a) Meridional view; (b) increased 200 annulus area with respect to the axial lift method. 201

Table 3. The blade height and annulus area at the impeller inlet and exit (axial lift). 202

Figure 10. Increased total pressure at the target flow rate after applying the axial lift method. 203

The performance curves of AL-1 and AL-2 are predicted, as shown in Figure 10. After the axial 204 lift was applied, there was no significant change in the choking flow rate because there is no change 205 in the passage area at the impeller inlet. However, the total-to-total pressure ratio increased at all 206

-


R1s

(mm)

Annulus Area B2

(mm)

Annulus Area

Area

(m2)

Increased

Ratio

Area

(m2)

Increased

Ratio

Base 105.00 0.02906 - 17.0 0.02300 -

AL 1 105.00 0.02906 1.0 18.0 0.02436 1.06

AL 2 105.00 0.02906 1.0 19.0 0.02571 1.12

Figure 10. Increased total pressure at the target flow rate after applying the axial lift method.

3.3. Combination of Flow Cut and Axial Lift

The combined flow cut and axial lift method, which uses the flow cut and axial lift methodstogether, is proposed in this study to increase the reduced total pressure to the design flow rate afterthe flow cut is applied, as shown in Figure 1d. In particular, this method can be used for centrifugalimpellers with a backswept angle such as the base impeller of this study. If only the axial lift method isapplied to the centrifugal compressor, components such as the shafts, gears, and bearings of the basecompressor may not be used due to the increase in the weight of the aerodynamic parts. However, if theaxial lift method is applied after applying the flow cut method, as shown in Figure 1d, the componentsof the base compressor can be used because the weight of the aerodynamic parts is not increasedcompared with the weight of the base compressor.

When the axial lift method is applied after applying the flow cut method, B2 was increased,as shown in Table 4. This reduces the relative velocity at the impeller exit and increases the totalpressure. The impellers for which a combination of the flow cut and axial lift methods were used arerepresented in the order of 1, 2, and 3. The B2 values of FC 10, 20, and 30 were 15.3 mm, 13.6 mm,and 11.9 mm, respectively. When applying the combined flow cut and axial lift method, the B2 value

Energies 2019, 12, 4503 11 of 19

was increased by approximately 2%, as shown in Table 4. After applying the combined flow cut andaxial lift method, the modified annulus area inside the impeller is shown in Figure 11.

Table 4. The variation of B2 with respect to the combined flow cut and axial lift method.

R1s (mm) B2

Width (mm) Increased Percentage (%)

FC10100.43

15.3 -FC10-1 15.6 ~ 2%FC10-2 15.9 ~ 4%

FC20

95.67

13.6 -FC20-1 13.9 ~ 2%FC20-2 14.1 ~ 4%FC20-3 14.4 ~ 6%FC20-4 14.7 ~ 8%FC20-5 14.9 ~ 9.5%

FC30

90.70

11.9 -FC30-1 12.1 ~ 2%FC30-2 12.4 ~ 4%FC30-3 12.6 ~ 6%FC30-4 12.9 ~ 8%FC30-5 13.1 ~ 10%FC30-6 13.3 ~ 12%FC30-7 13.6 ~ 14%FC30-8 13.8 ~ 16%


(a) (b)

Figure 11. Increased annulus area in the impeller by using the combined flow cut and axial lift 227 method. (a) Base, FC 10, and FC 20; (b) base and FC 30. 228

(a) (b)

Figure 12. Increased total-to-total pressure at the target flow rate with respect to the combined flow 229 cut and axial lift method. (a) Total-to-total pressure ratio versus mass flow rate (kg/s); (b) total-to-total 230 pressure ratio versus B2. 231

232

Figure 13. Comparison of performance curves of the flow cut method only and the combined flow 233 cut and axial lift method. 234

Figure 11. Increased annulus area in the impeller by using the combined flow cut and axial lift method.(a) Base, FC 10, and FC 20; (b) base and FC 30.

Figure 12 shows the CFD results at the target flow rate after applying the combined flow cut andaxial lift method. The total-to-total pressure ratio was constantly increased by increasing B2, as shownin Figure 12b, and the impellers with a total-to-total pressure ratio that was approximately the same asthat of the base impeller were FC10-2, FC20-5, and FC30-8. Figure 13 shows the performance curvesof FC10-2, FC20-5, and FC30-8 using the combined flow cut and axial lift method. The total-to-totalpressure ratio at the target flow rate is almost the same as the total-to-total pressure ratio at the designpoint of the base impeller.

Energies 2019, 12, 4503 12 of 19


(a) (b)


(a) (b)


232


Figure 12. Increased total-to-total pressure at the target flow rate with respect to the combined flow cutand axial lift method. (a) Total-to-total pressure ratio versus mass flow rate (kg/s); (b) total-to-totalpressure ratio versus B2.


(a) (b)


(a) (b)


232


Figure 13. Comparison of performance curves of the flow cut method only and the combined flow cutand axial lift method.

4. Discussion

4.1. Characteristic of Base Impeller

Figure 14 shows the flow fields at five operating points in the performance curve: Near choke,between near choke and design point, design point, between design point and near surge, and nearsurge. Figure 14a shows the relative Mach number at the 50% blade-to-blade position. The relativeMach number of the impeller inlet tip is approximately 0.8 to 0.85 at the design point and approximately1.1 to 1.2 near the choke, which means that the base impeller shows the general characteristics ofa transonic compressor with a relative Mach number between 0.8 and 1.2 at the impeller inlet tip.In addition, Figure 14b shows the flow fields at 80% span from point 1 to point 5, and it can be seenthat the total-to-total pressure ratio decreases sharply when the shock wave occurs between point 1and point 2. At point 5, a flow blockage occurred on the suction surface of the main blade and on thepressure surface of the splitter blade, and as a result, the total pressure decreases from point 4 to point5. In this impeller, shock waves are generated only near the choke, not between the design point andnear the surge. Therefore, the transonic centrifugal compressor used in this study is a compressor that

Energies 2019, 12, 4503 13 of 19

is very useful for industrial applications if it can be controlled properly within this operating range sothat shock waves are not generated.


Figure 12 shows the CFD results at the target flow rate after applying the combined flow cut and 235 axial lift method. The total-to-total pressure ratio was constantly increased by increasing B2, as shown 236 in Figure 12b, and the impellers with a total-to-total pressure ratio that was approximately the same 237 as that of the base impeller were FC10-2, FC20-5, and FC30-8. Figure 13 shows the performance curves 238 of FC10-2, FC20-5, and FC30-8 using the combined flow cut and axial lift method. The total-to-total 239 pressure ratio at the target flow rate is almost the same as the total-to-total pressure ratio at the design 240 point of the base impeller. 241

(a)

(b)

Figure 14. Flow fields in the base impeller. (a) Contour of relative Mach number (Mrel) on meridional 242 view; (b) contour of relative Mach number (Mrel) at 80% span. 243

4. Discussion 244

4.1. Characteristic of Base Impeller 245

Figure 14. Flow fields in the base impeller. (a) Contour of relative Mach number (Mrel) on meridionalview; (b) contour of relative Mach number (Mrel) at 80% span.

4.2. Correlation of Velocity and Total Pressure at Impeller Exit

To analyze the correlation between the velocity and the total pressure at the impeller exit, relativevelocity, tangential velocity (Cθ), meridional velocity (Cm), and total pressure were analyzed.

Energies 2019, 12, 4503 14 of 19

4.2.1. Relative Velocity

Figure 15 shows the variation of relative velocity at the impeller exit with respect to the flow cut.After the flow cut was applied, the relative velocity at the impeller exit was increased; as the flow cutpercentage increased, the relative velocity increased even more.


Figure 14 shows the flow fields at five operating points in the performance curve: Near choke, 246 between near choke and design point, design point, between design point and near surge, and near 247 surge. Figure 14a shows the relative Mach number at the 50% blade-to-blade position. The relative 248 Mach number of the impeller inlet tip is approximately 0.8 to 0.85 at the design point and 249 approximately 1.1 to 1.2 near the choke, which means that the base impeller shows the general 250 characteristics of a transonic compressor with a relative Mach number between 0.8 and 1.2 at the 251 impeller inlet tip. In addition, Figure 14b shows the flow fields at 80% span from point 1 to point 5, 252 and it can be seen that the total-to-total pressure ratio decreases sharply when the shock wave occurs 253 between point 1 and point 2. At point 5, a flow blockage occurred on the suction surface of the main 254 blade and on the pressure surface of the splitter blade, and as a result, the total pressure decreases 255 from point 4 to point 5. In this impeller, shock waves are generated only near the choke, not between 256 the design point and near the surge. Therefore, the transonic centrifugal compressor used in this 257 study is a compressor that is very useful for industrial applications if it can be controlled properly 258 within this operating range so that shock waves are not generated. 259

4.2. Correlation of Velocity and Total Pressure at Impeller Exit 260

To analyze the correlation between the velocity and the total pressure at the impeller exit, 261 relative velocity, tangential velocity (��), meridional velocity (��), and total pressure were analyzed. 262

4.2.1. Relative Velocity 263

Figure 15 shows the variation of relative velocity at the impeller exit with respect to the flow cut. 264 After the flow cut was applied, the relative velocity at the impeller exit was increased; as the flow cut 265 percentage increased, the relative velocity increased even more. 266

Figure 15. Increased relative velocity after applying flow cut at impeller exit (at Mid-span). 267

In the centrifugal impeller with a backswept angle, increasing the relative velocity at the impeller 268 exit causes a problem. Because the increased relative velocity reduces the angle of the velocity 269 (Meridional angle convention), as shown in Figure 16a, the tangential velocity is reduced, as shown 270 in Figure 16b. Eventually, the work coefficient is reduced based on Equation (3). 271

work coefficient = �� ⁄ (3) 272

where �� and �� are the tangential velocity and the blade speed at the impeller exit, respectively. 273 The head decreases when the work coefficient is reduced, and the total pressure is reduced. 274

Figure 15. Increased relative velocity after applying flow cut at impeller exit (at Mid-span).

In the centrifugal impeller with a backswept angle, increasing the relative velocity at the impellerexit causes a problem. Because the increased relative velocity reduces the angle of the velocity(Meridional angle convention), as shown in Figure 16a, the tangential velocity is reduced, as shown inFigure 16b. Eventually, the work coefficient is reduced based on Equation (3).

work coefficient = Cθ2/U2 (3)

where Cθ2 and U2 are the tangential velocity and the blade speed at the impeller exit, respectively. Thehead decreases when the work coefficient is reduced, and the total pressure is reduced.Energies 2019, 12, x FOR PEER REVIEW 14 of 18

(a) (b)

Figure 16. Velocity triangle change of the centrifugal impeller with a backswept angle at the impeller 275 exit. (a) Correlation between the increased relative velocity and the changed absolute velocity angle; 276 (b) correlation between the changed absolute velocity angle and the decreased magnitude of the 277 tangential velocity. 278

In Figure 17, as axial lift was applied, B2 was increased and relative velocity was decreased. In 279 Figure 18, the relative velocity of FC10, FC20, and FC30 were higher than that of base impeller at the 280 impeller exit, while the relative velocity of FC10-2, FC20-5, and FC30-8 were similar to that of base 281 impeller. 282

283

Figure 17. Decreased relative velocity after applying axial lift at impeller exit (at Mid-span). 284

285

Figure 18. The same relative velocity of the target flow rate and design flow rate of base impeller after 286 applying the combination of flow cut and axial lift at impeller exit (at Mid-span). 287

4.2.2. Tangential/meridional Velocities at the Impeller Exit 288

Figure 19a shows the tangential velocity (��) at the trailing edge of the impeller. When the flow 289 cut was applied, the tangential velocity was reduced due to the higher relative velocity at the impeller 290

Figure 16. Velocity triangle change of the centrifugal impeller with a backswept angle at the impellerexit. (a) Correlation between the increased relative velocity and the changed absolute velocity angle;(b) correlation between the changed absolute velocity angle and the decreased magnitude of thetangential velocity.

In Figure 17, as axial lift was applied, B2 was increased and relative velocity was decreased.In Figure 18, the relative velocity of FC10, FC20, and FC30 were higher than that of base impeller atthe impeller exit, while the relative velocity of FC10-2, FC20-5, and FC30-8 were similar to that ofbase impeller.

Energies 2019, 12, 4503 15 of 19


(a) (b)



283


285




Figure 17. Decreased relative velocity after applying axial lift at impeller exit (at Mid-span).


(a) (b)



283


285




Figure 18. The same relative velocity of the target flow rate and design flow rate of base impeller afterapplying the combination of flow cut and axial lift at impeller exit (at Mid-span).

4.2.2. Tangential/meridional Velocities at the Impeller Exit

Figure 19a shows the tangential velocity (Cθ) at the trailing edge of the impeller. When the flowcut was applied, the tangential velocity was reduced due to the higher relative velocity at the impellerexit. However, by applying the axial lift after the flow cut, the tangential velocity was increased to thesimilar level as the base.

Figure 19b shows the meridional velocity (Cm) at the trailing edge of the impeller. The meridionalvelocity was increased after applying the flow cut because the relative velocity was also increasedat the impeller exit. However, when axial lift was used after the flow cut, the meridional velocitywas decreased. Therefore, it is confirmed that the flow phenomenon at the impeller exit could bemaintained similar to that of the base impeller, when the flow cut and axial lift are applied together.

Energies 2019, 12, 4503 16 of 19


exit. However, by applying the axial lift after the flow cut, the tangential velocity was increased to 291 the similar level as the base. 292

Figure 19b shows the meridional velocity (��) at the trailing edge of the impeller. The meridional 293 velocity was increased after applying the flow cut because the relative velocity was also increased at 294 the impeller exit. However, when axial lift was used after the flow cut, the meridional velocity was 295 decreased. Therefore, it is confirmed that the flow phenomenon at the impeller exit could be 296 maintained similar to that of the base impeller, when the flow cut and axial lift are applied together. 297

298 (a) 299

300 (b) 301

Figure 19. Tangential/meridional velocities at impeller exit. (a) Tangential velocity ( �� ); (b) 302 Meridional velocity (��) 303

4.2.3. Total Pressure at Impeller Exit 304

Figure 20 shows the total pressure at the trailing edge of the impeller. Applying the flow cut 305 showed that the total pressure is decreased at the impeller exit. However, when axial lift was 306 combined with the flow cut, the total pressure was increased to the similar value as that of the base. 307

Increasing the relative velocity after applying the flow cut reduced the tangential velocity, which 308 results in total pressure reduction. However, as the axial lift was applied after the flow cut, the 309 tangential velocity and total pressure were increased to those of the base level. Therefore, it was 310 proved that the relative velocity at the impeller exit should be kept almost constant compared to 311 reference impeller for designing a new impeller by the impeller modification methods. 312

313

Figure 20. Total pressure at impeller exit. 314

Figure 19. Tangential/meridional velocities at impeller exit. (a) Tangential velocity (Cθ); (b) Meridionalvelocity (Cm ).

4.2.3. Total Pressure at Impeller Exit

Figure 20 shows the total pressure at the trailing edge of the impeller. Applying the flow cutshowed that the total pressure is decreased at the impeller exit. However, when axial lift was combinedwith the flow cut, the total pressure was increased to the similar value as that of the base.


exit. However, by applying the axial lift after the flow cut, the tangential velocity was increased to 291 the similar level as the base. 292

Figure 19b shows the meridional velocity (��) at the trailing edge of the impeller. The meridional 293 velocity was increased after applying the flow cut because the relative velocity was also increased at 294 the impeller exit. However, when axial lift was used after the flow cut, the meridional velocity was 295 decreased. Therefore, it is confirmed that the flow phenomenon at the impeller exit could be 296 maintained similar to that of the base impeller, when the flow cut and axial lift are applied together. 297

298 (a) 299

300 (b) 301

Figure 19. Tangential/meridional velocities at impeller exit. (a) Tangential velocity ( �� ); (b) 302 Meridional velocity (��) 303

4.2.3. Total Pressure at Impeller Exit 304

Figure 20 shows the total pressure at the trailing edge of the impeller. Applying the flow cut 305 showed that the total pressure is decreased at the impeller exit. However, when axial lift was 306 combined with the flow cut, the total pressure was increased to the similar value as that of the base. 307

Increasing the relative velocity after applying the flow cut reduced the tangential velocity, which 308 results in total pressure reduction. However, as the axial lift was applied after the flow cut, the 309 tangential velocity and total pressure were increased to those of the base level. Therefore, it was 310 proved that the relative velocity at the impeller exit should be kept almost constant compared to 311 reference impeller for designing a new impeller by the impeller modification methods. 312

313

Figure 20. Total pressure at impeller exit. 314 Figure 20. Total pressure at impeller exit.

Increasing the relative velocity after applying the flow cut reduced the tangential velocity,which results in total pressure reduction. However, as the axial lift was applied after the flow cut,the tangential velocity and total pressure were increased to those of the base level. Therefore, it wasproved that the relative velocity at the impeller exit should be kept almost constant compared toreference impeller for designing a new impeller by the impeller modification methods.

5. Conclusions

In this study, the influences of the flow cut and axial lift of the impeller on the aerodynamicperformance of a transonic centrifugal compressor were analyzed, and three transonic centrifugalimpellers with flow fractions of 0.7, 0.8, and 0.9 compared to the design flow rate were newly designedby applying the combined flow cut and axial lift methods.

Energies 2019, 12, 4503 17 of 19

• The NASA CC3 transonic centrifugal compressor with a backswept angle of −50◦ was used as thebase compressor to carry out the study. The aerodynamic performance of the impeller of the basecompressor was analyzed. There was a strong shock wave at the impeller inlet tip near the choke;however, in the range of the design point to the surge, no shock wave was observed. Therefore,the high-pressure transonic centrifugal impeller used in this study will be useful for industrialapplications when they operate properly within the operating range without shock waves.

• To analyze the influence of the flow cut method, the target flow rate of the flow cut was set at flowfractions of 0.7, 0.8, and 0.9 compared to the design flow rate. After applying the flow cut, the totalpressure at the target flow rate was lower than the total pressure at the design point due to theincrease in the relative velocity at the impeller exit. In the centrifugal impeller with a backsweptangle, the increased relative velocity reduces the angle of the absolute velocity (Meridional angleconvention), which results in a reduction of the tangential velocity and the work coefficient.

• To analyze the influence of the axial lift, the B2 value was increased by 1 mm and 2 mm comparedto the value of the base impeller. After applying the axial lift method, the total pressure at thedesign flow rate was increased due to the decrease in the relative velocity caused by increasingthe passage area at the impeller exit.

• Applying the flow cut and axial lift methods showed that the variation in the relative velocityat the impeller exit has a significant effect on the variation of the total pressure. In addition,it was found that the relative velocity at the impeller exit of the target flow rate is similar to thatof the base impeller when the flow cut and axial lift methods are applied together. Therefore,by combining these methods, three transonic centrifugal impellers with flow fractions of 0.7, 0.8,and 0.9 compared to the design flow rate were newly designed. These conclusions are expected tobe guidelines to design impellers within a specific flow range by applying impeller modificationmethods to transonic centrifugal impellers with a backswept angle.

Author Contributions: All authors contributed to this work by collaboration. K.S.P. is the main author of thismanuscript. I.H.J., S.J.Y., S.Y.L., A.Z., and J.T.C. assisted and provided useful suggestions in the contents ofresearch. All authors approved the publication.

Funding: This research received no external funding.

Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature

AL Axial liftB Blade heightC Absolute velocityCFD Computational fluid dynamicsEfftt Total-to-total efficiencyFC Flow cutM Mach numberNs Nondimensional specific speedPR Pressure ratioPRtt Total-to-total pressure ratioR Blade radiusRMS Root mean squareQO Quasi-orthogonalU Blade speedW Relative velocity

Energies 2019, 12, 4503 18 of 19

Mathematical Symbols

αe f f Angle between QO lines and RMS contourαh Angle between QO lines and hub contourαs Angle between QO lines and shroud contourβ2b Backswept angleη Fraction between the shroud and RMS contour

Subscripts

1 Impeller inlet2 Impeller exit3 Measuring positionb Bladeh Hubs Shroudt Tipm Meridional directionθ Tangential direction

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© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution(CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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influences of the flow cut and axial lift of the impeller

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