## Abstract

Impeller discharge flow plays an important role in centrifugal compressor performance and operability for two reasons. First, it determines the work factor and relative diffusion for the impeller. Second, it sets the flow into the downstream stationary diffusion system. The choice made in the preliminary design phase for the impeller exit velocity triangle is crucial for a successful design. The state-of-the-art design approach for determining the impeller exit velocity triangle in the preliminary design phase relies on several empirical guidelines, i.e., maximum work factor and diffusion ratio for an impeller, the optimal range of absolute flow angle, etc. However, as modern compressors continue pushing toward higher efficiency and higher work factor, this design approach falls short in providing exact guidance for choosing an optimal impeller exit velocity triangles due to its empirical nature as well as the competing mechanism of the two trends. In light of this challenge, this paper introduces a reduced-dimension, deterministic approach for the design of the impeller exit velocity triangle. The method gauges the design of the impeller exit velocity triangle from a different design philosophy using a relative diffusion effectiveness parameter and is validated using six impeller designs, representative of applications in both turbochargers and aero engines. Furthermore, with the deterministic method in place, optimal impeller exit velocity triangles are explored over a broad design space, and a one-to-one mapping from a selection of impeller total-to-total pressure ratios and backsweep angles to a unique optimal impeller exit velocity triangle is provided. This new approach is demonstrated, and discussions regarding the influences of impeller total-to-total pressure ratio, isentropic efficiency, and backsweep angle on the optimal impeller exit velocity triangle are presented.

## Introduction

The performance of a centrifugal compressor is strongly influenced by the impeller exit velocity triangle in two key aspects. First, it determines the work factor and relative diffusion for the impeller. Also, the impeller discharge flow enters the downstream stationary diffusion system and, thus, strongly affects the performance of the downstream diffusion system. The impeller exit velocity triangle is typically set by the design choices of: impeller exit blade angle and blade height. The selection of impeller exit blade angle is determined by considerations of stage efficiency, operating range, and impeller tip diameter. Radial blades were commonly used for impeller designs initially. They reduce the impeller tip diameter, but they also yield lower efficiency due to the large mixing losses from the wake developed at the impeller exit. These machines also feature smaller surge margins because of the reduced gradient of the head-flow characteristic. Modern impellers typically feature backswept blades, as it is now commonplace to implement 25–55 deg backsweep [1], for the benefit of an enhanced stable operating range and improved efficiency. However, large backsweep angles lead to increased impeller tip diameter and mechanical stress, and the maximum backsweep angle is usually limited by impeller material or size constraints. The choice for impeller exit blade angle determines the work factor and overall dimension of the compressor. The choice of impeller exit blade height partly governs both the impeller relative diffusion and impeller exit absolute flow angle. An empirical guideline for the selection of impeller exit blade height (*b*_{2}) is based on impeller outlet width ratio (*b*_{2}/*D*_{2}), the value of which shall not be smaller than a certain limit to obtain good efficiency [1]. In general, the design of the impeller exit velocity triangle using the state-of-the-art approaches are purely empirical following several guidelines for the selection of impeller work factor $(\Delta ht/U22)$, diffusion factor (DF) or ratio (*W*_{1s}/*W*_{2}), and absolute impeller exit flow angle (*α*_{2}). An “optimal” impeller exit velocity triangle is reached when all the parameters fall in the suggested range.

The impeller exit velocity triangle has a strong influence on the relative diffusion in an impeller. Excessive relative diffusion leads to stall, and the internal diffusion limitations for an impeller have been discussed by many researchers including Balje [2], Rodgers [3], Young [4], Dixon [5], and Came and Robinson [6]. Dixon suggests a range for optimal diffusion ratio from 1.43 to 1.82. Came and Robinson find a range between 1.67 and 2.22 is normally chosen. However, Rodgers [3] showed compressor surge occurred at diffusion ratios between 1.9 and 2.0, indicating that impeller diffusion ratios above 1.9 can be concerning. Also, the impeller exit velocity triangle determines the work factor of the impeller. A typical range for centrifugal compressor work factor is between 0.55 and 0.75 [5].

The influence of impeller discharge flow on the performance of the downstream diffusion system was addressed by Whitfield [7] who suggested that the two most important aerodynamic parameters are the absolute Mach number, M_{2}, and absolute flow angle, *α*_{2}. An impeller discharge flow with large M_{2} will impose an increased burden upon the downstream diffusion system, in terms of static pressure recovery, and can also result in high frictional losses and possibly shock losses near the diffuser leading edge. Also, the direction of impeller discharge flow affects both the mixing loss and the development of the end wall boundary layers in the vaneless space.

For losses in the vaneless space, the theory of Johnston and Dean [8] predict a mixing loss dependent on the value of tan *α*_{2}. For impeller discharge flow with small *α*_{2}, the large mixing loss in the vaneless space is dominated by the mixing of the radial velocity component between the primary (jet) flow and secondary (wake) flow. However, a large *α*_{2} results in a shallow spiral flow through the vanless space, increased frictional losses, thicker boundary development, and possible stalling. After taking both factors into consideration, Johnston and Dean [8] suggested an optimum impeller absolute flow angle between 63 deg and 68 deg at the design condition. More recently, Came and Robinson [6] suggest a range between 69 deg and 73 deg considering vaneless mixing loss and pressure recovery.

Rodgers and Sapiro [9] investigated the influence of impeller discharge flow angle on the performance of a vaned diffuser and recommended an optimum range between 60 deg and 70 deg for high-pressure ratio centrifugal compressors in gas turbine engines. Recently, Filipenco et al. [10] and Deniz et al. [11] examined the effects of diffuser inlet Mach number, absolute flow angle, blockage, and axial flow non-uniformity on the performance of a discrete passage diffuser and straight-channel diffuser. The influences of these parameters on diffuser pressure recovery and stable operating range were investigated. Results showed that, for both types of diffusers, the pressure recovery of the diffuser depends on the inlet flow angle, and in fact, it is not sensitive to variations in inlet Mach number or non-uniformity. Also, the operating range for both types of diffusers was limited by the onset of the rotating stall at a fixed momentum-averaged flow angle into the diffuser, which is around 70 deg for the straight-channel diffuser.

In summary, empirical guidelines for the design of impeller exit velocity triangles based on the findings from the earlier research include:

An optimal range for impeller diffusion factor is from 1.43 to 1.9.

A typical range of work factor is between 0.55 and 0.75 [5].

An optimal range of impeller exit absolute flow angle is from 60 deg to 73 deg.

## Challenges in the State-of-the-Art Design Approach

As modern compressor designs aim for higher efficiency and higher work factor, the state-of-the-art design approach falls short in providing precise guidance for the design of optimal impeller exit velocity triangle due to its empirical nature plus the inconsistency among these guidelines. For instance, the optimal ranges for *α*_{2}, as indicated by the shaded bands in Fig. 1, are varied amongst the different researchers. Thus, it is quite challenging to design an optimal impeller exit velocity triangle following these empirical guidelines alone.

To better illustrate these challenges, Fig. 2 compares the impeller exit velocity triangle in three cases (shown in Fig. 1) with a selection of different values for *α*_{2} using the well-known NASA CC3 impeller [12]. In addition to the original design, two additional designs (designs A and B) represent the cases of a lower $(60deg)$ and upper $(73deg)$ band for *α*_{2}. In all the cases, the exit blade angle, as well as the primary aerodynamic parameters of the impeller, including total-to-total pressure ratio, isentropic efficiency, and slip factor, are kept the same. As shown in the figure, the impeller exit velocity triangle is fairly sensitive to changes in impeller exit absolute flow angle. A variation of *α*_{2} from 60 deg to 73 deg results in a 11% change in impeller tip speed, a 24% change in work factor, and a 61% change in diffusion factor. Thus, the search for an optimal impeller exit velocity triangle using available empirical guidelines can be challenging.

In the meantime, it is worth noting that these challenges can be remedied by the experience of the designer or by an exhaustive database of the performance of past designs. When these are not available, a simple but elegant optimization algorithm for the design of the impeller exit velocity triangles in the preliminary design phase for centrifugal compressors is desired, which motivates the work presented in this paper.

## Design Philosophy and Its Validation

In light of challenges associated with the state-of-the-art approach in choosing the optimal impeller exit velocity triangles, this paper considers how to design the optimal impeller exit velocity triangle from a different design philosophy. The method is based on the concept that an impeller is a rotating diffuser and gauges the design of the impeller exit velocity triangle using a relative diffusion effectiveness parameter. Validation of the method is conducted using six different impeller designs representative of both turbocharger and aero engine applications.

### Gauging the Design of the Impeller Exit Velocity Triangle Using a Relative Diffusion Effectiveness Parameter.

*ɛ*) is [13]

A significant portion of the static pressure rise in the impeller is associated with the centrifugal effect and is essentially loss-free. The relative diffusion effectiveness parameter isolates the impeller aerodynamic performance from the centrifugal effect and provides a more direct metric for the quality of the impeller aerodynamic design. The calculation of relative diffusion effectiveness is an iterative procedure using equations for the conservation of rothalpy and conservation of mass. A detailed procedure for the calculation of relative diffusion effectiveness can be found in Ref. [13].

In the present study, the impeller exit velocity is characterized using three primary parameters: work factor (*µ*), machine Mach number (M_{U2}), and impeller exit blade angle (*β*_{2b}). Of these, *µ* indicates loading, and *M*_{U2} is connected to the size of the compressor. Though not presented here, the equations detailing the relationship between the impeller exit velocity triangle and the parameters *µ*, M_{U2}, and *β*_{2b} are provided in the Appendix. In the new design philosophy, for a given set of design requirements (impeller total-to-total pressure ratio, Π_{imp}, and isentropic efficiency, *η*_{imp}) and design choice for *β*_{2b}, the search for an optimal impeller exit velocity triangle is to find a combination of work factor and machine Mach number that yields the maximum relative diffusion effectiveness.

The workflow of the new method is shown in Fig. 3. For a given set of input parameters including, Π_{imp}, *η*_{imp}, and *β*_{2b}, the procedure starts with a wide range of work factor and calculates the associated values of the machine Mach number and relative diffusion effectiveness. The optimal preliminary design is achieved when the combination of work factor and machine Mach number gives the maximum relative diffusion effectiveness. Then, the parameters for the impeller exit velocity triangle can be obtained using the procedure included in the Appendix. Compared to the 1D model presented in Ref. [13], the current method's breakthrough is the significant dimension reduction in the design space. For instance, the 1D model in Ref. [13] includes 11 input parameters, corresponding to 11 dimensions. Though the model is doing a good job finding an optimal impeller preliminary design with the specific design requirements, the relatively large number of dimensions makes it challenging to map a broad scope of design space or understand the interdependent relations of the input parameters and the resulting optimal design. In the present paper, by focusing on the design of the impeller exit velocity triangle, the number of input parameters is reduced from 11 to 4, shown in Fig. 3, which significantly improves the method's simplicity, allowing the designer to quickly synthesize the design space.

### Validation of the Design Philosophy.

Validation of the design philosophy was performed using the preliminary design information of six impellers as follows:

Represent a variety of applications such as turbochargers and aero engines.

Cover a broad range of specific speed, total-to-total pressure ratio, work factor, inlet relative Mach number, machine Mach number, and impeller exit blade angle;

Represent designs achieved by either well-experienced designers or designers from an institute with a good reputation for the design of centrifugal compressors.

Figure 4 shows the selected impeller designs in terms of specific speed described as $\epsilon =2\u2205t1/(\eta \mu )0.75$. The impellers are representative of applications of centrifugal compressors in gas turbines and turbochargers, and their dimensionless specific speed varies between 0.53 and 0.81. The values of the impeller total-to-total pressure ratio vary from 4.1 to 7.6, and the impeller backsweep angle ranges from 30 deg to 50 deg. Overall, the selected impeller designs cover a wide design space and, thus, serve as good validation cases of the new method.

Since few companies publish the details of their compressor designs for proprietary reasons, results from three well-known impellers available in the open literature are included. The three impellers are the NASA CC3 impeller [12], Came's impeller [14], and the SRV2-O impeller [15]. The NASA CC3 impeller is a medium-pressure ratio wheel representative of the application of gas turbine engines with a large backsweep angle (50 deg). The impeller produces a total-to-total pressure ratio of 4.1 at the design speed, and the estimated isentropic efficiency of the impeller is approximately 92%. The work factor of the impeller at the design condition is 0.65, and the machine Mach number of the compressor is 1.45. Came's impeller is a high-pressure ratio wheel representative of the application of gas turbine engines with a small backsweep angle (30 deg). The impeller produces a total-to-total pressure ratio of 7.65 at the design speed, and the estimated isentropic efficiency of the impeller is approximately 87%. The impeller has a high-work factor (0.8), and the machine Mach number of the compressor is 1.69. The SRV2-O impeller is a medium-to-high-pressure ratio wheel with a backsweep angle of 38 deg. The impeller total-to-total pressure ratio is 6.1, and the estimated impeller isentropic efficiency is 0.84. The work factor of the impeller at the design condition is 0.68, and the machine Mach number is 1.73. Table 1 lists the primary parameters of the three open literature impellers.

Description | Parameter | CC3 impeller | Came impeller | SRV2-O impeller |
---|---|---|---|---|

Specific speed | N_{s} | 0.58 | 0.46 | 0.81 |

Total pressure ratio | П | 4.1 | 7.65 | 6.1 |

Isentropic efficiency | η | 0.92 | 0.87 | 0.84 |

Work factor | µ | 0.65 | 0.8 | 0.68 |

Machine Mach number | M_{U}_{2} | 1.45 | 1.69 | 1.73 |

Exit blade angle | ß_{2b} | 50 deg | 30 deg | 38 deg |

Exit wheel speed | U_{2} | 492 m/s | 575 m/s | 586 m/s |

Description | Parameter | CC3 impeller | Came impeller | SRV2-O impeller |
---|---|---|---|---|

Specific speed | N_{s} | 0.58 | 0.46 | 0.81 |

Total pressure ratio | П | 4.1 | 7.65 | 6.1 |

Isentropic efficiency | η | 0.92 | 0.87 | 0.84 |

Work factor | µ | 0.65 | 0.8 | 0.68 |

Machine Mach number | M_{U}_{2} | 1.45 | 1.69 | 1.73 |

Exit blade angle | ß_{2b} | 50 deg | 30 deg | 38 deg |

Exit wheel speed | U_{2} | 492 m/s | 575 m/s | 586 m/s |

A detailed investigation of the three well-known impeller preliminary designs using the relative diffusion effectiveness parameter can be found in Ref. [13]. Thus, only a brief discussion was included in the present paper for the purpose of completeness. In the process of searching for the optimal impeller exit velocity triangle using the relative diffusion effectiveness parameter, the value of the impeller work factor, *μ*, is varied while parameters including *β*_{2b}, Π, *η*, and *σ* are kept constant. The changes in relative diffusion effectiveness, machine Mach number, and parameters related to the impeller exit velocity triangle, with respect to the changes in work factor, are shown in Fig. 5. For comparison, the original design choice and the optimal design choice predicted using the new method are indicated by the star and square symbols, respectively. Also, sketches of the impeller exit velocity triangles comparing the original design to the current method are also shown. In each case, the new method yields an optimal impeller work factor, machine Mach number, and impeller exit velocity triangle almost identical to the original design choice. The maximum differences in the impeller work factor and machine Mach number are within 3% and 1.5%, respectively. The maximum differences in the absolute and relative flow angle at the impeller exit are less than 2.8 deg and 1.5 deg, respectively.

Additionally, it is worth noting that the method used in the present study is different from the design consideration of minimizing the absolute Mach number at impeller exit. The optimal absolute Mach number obtained using the new method does not coincide with the minimum absolute Mach number at the impeller exit. However, in all the validation cases, the new method does yield an optimal impeller exit Mach number in proximity to the minimum value. Though not included in the present paper in detail, good agreements were also achieved for the other three impeller designs (indicated by the star symbols in Fig. 4), and the new method predicts very similar impeller exit velocity triangles to the ones obtained using the state-of-the-art approaches.

To conclude this section, a reduced-dimension, deterministic method to select the optimal impeller exit velocity triangle was presented. It is based on the earlier developed 1D model for impeller preliminary design and aero-thermal analysis [13] that optimizes relative diffusion effectiveness. Furthermore, validation of the new method was performed using the preliminary design information of six impellers developed at the highest level using the state-of-the-art design approaches. For all the cases, results show that the new method provides very similar impeller exit velocity triangles to the original design choices.

## Results and Discussions

With the new method in place, the optimal impeller exit velocity triangles are explored over a broad design space in terms of impeller total-to-total pressure ratio and impeller exit blade angle. The mapping of the optimal impeller work factor, machine Mach number, as well as the velocity triangles at the impeller exit, in terms of impeller total-to-total pressure ratio and backsweep angle, are presented. Also, the influence of impeller total-to-total pressure ratio, isentropic efficiency, and backsweep angle on the optimal impeller exit velocity triangle are discussed in detail.

### Mapping the Optimal Impeller Exit Velocity Triangle Within the Design Space.

For the analysis and results presented in the following sections, a constant slip factor of 0.9 was utilized. As a result, the mapping of the optimal impeller exit velocity triangle in the design space is performed using a triple-loop. The variables are the impeller efficiency, impeller total-to-total pressure ratio, and impeller exit blade angle, respectively. In the inner-loop (1st loop), the value of the impeller isentropic efficiency varies from 0.88 to 0.92 at a resolution of 0.002. In the middle-loop (2nd loop), the search for the optimal impeller exit velocity triangle in terms of impeller total pressure ratio over a range from 2.5 to 10 at a resolution of 0.25 is carried out. In the outer-loop (3rd loop), the optimal impeller exit velocity triangle is explored over a range of impeller exit blade angles from 25 deg to 55 deg with a 1 deg resolution. For each specific set of impeller efficiency, total pressure ratio, and backsweep angle, the search for the corresponding optimal impeller exit velocity triangle is conducted following the procedure highlighted in Fig. 3. In the present study, a total of 20,181 calls of the new optimization algorithm was conducted to cover the full search space. The mapping of the optimal impeller exit velocity triangle is shown in Figs. 6–9. It is worth noting that the influences of impeller isentropic efficiency on optimal impeller exit velocity triangle are indicated by the shaded bands in the figures, with its width indicating a ±2 *point* variation in impeller isentropic efficiency.

Figure 6 shows the results of the optimal work factor (*μ*) and machine Mach number (M_{U2}) in terms of impeller total-to-total pressure (Π_{imp}) and impeller exit backsweep angle (*β*_{2b}). Four important takeaways can be drawn here. First, the optimal set of *μ* and M_{U2} varies with respect to changes in Π_{imp} and *β*_{2b}. Second, the selection of *β*_{2b} has a strong influence on the optimal impeller work factor. An increase in *β*_{2b} results in a decrease in optimal work factor and an increase in machine Mach number. Third, at a fixed design choice of *β*_{2b}, the optimal work factor for an impeller of higher total pressure ratio is larger than that for a low-pressure ratio impeller, which indicates that it is more challenging to design a high-work factor impeller at low-to-medium total-to-total pressure ratio. In such cases, despite being less favored, a reduced backsweep angle at the impeller may be necessary. Otherwise, the design choice can end up quite far away from the optimal case. Finally, the influence of impeller isentropic efficiency on optimal impeller work factor and machine Mach number (indicated by the shaded band in the figure) is less significant compared to the influence of impeller total-to-total pressure ratio, particularly for low-to-medium (Π_{imp} < 5) pressure ratio impellers. On the other hand, for high-pressure ratio (Π_{imp} > 5) impellers, the influence from impeller isentropic efficiency is non-trivial and should be taken into consideration.

Figure 7 shows the optimal absolute flow angle at the impeller exit (*α*_{2}) in terms of Π_{imp} and *β*_{2b}. Most of the optimal *α*_{2} values fall within the range from 60 deg to 73 deg, as recommended by earlier researchers. Four important takeaways can be concluded here. First, a design choice of a larger *β*_{2b} leads to a larger optimal absolute flow angle at the impeller exit. For instance, at an Π_{imp} = 5.0, the optimal *α*_{2} increases from 66 deg to 73 deg as the *β*_{2b} increases from 25 deg to 55 deg. Second, at a fixed design choice for *β*_{2b}, the optimal impeller exit velocity triangle migrates toward a larger *α*_{2} as the impeller total-to-total pressure ratio increases. For instance, with a selection of *β*_{2b} = 45 deg, the optimal *α*_{2} for an impeller of Π_{imp} = 2.5 is 64 deg, and its value increases to 73 deg for an impeller that produces a total-to-total pressure ratio of ten. Third, there is a considerable impact from the impeller isentropic efficiency on the optimal absolute flow angle at impeller exit, indicated by the shaded band in the figure. For instance, at an Π_{imp} = 5.0, a variation of ±2 points in impeller isentropic efficiency introduces a maximum of 1.5 deg variation in the optimal absolute flow angle at the impeller exit. Finally, due to the significant influences from impeller total-to-total pressure ratio and backsweep angle on the optimal absolute flow angle at the impeller exit, the differences in the suggested optimal range for *α*_{2} (60 deg to 70 deg by Rodgers and Sapiro, 63 deg to 68 deg by Johnston and Dean, and 69 deg to 73 deg by Came and Robinson) are likely associated with the differences, such as impeller backsweep angle or total-to-total pressure ratio, in the designs used in the different studies.

Figure 8 shows the optimal absolute Mach number at the impeller exit (M_{2}) in terms of Π_{imp} and *β*_{2b}. Two takeaways can be drawn here. First, both *β*_{2b} and Π_{imp} exert strong influences on the optimal absolute Mach number at the impeller exit. A design choice of larger *β*_{2b} or lower Π_{imp} renders lower absolute Mach number at impeller exit, and vice versa. For instance, with a selection of *β*_{2b} = 45 deg, the optimal *M*_{2} for a low-pressure ratio impeller (Π_{imp} = 2.5) is 0.72 and is 1.09 for a high-pressure ratio impeller (Π_{imp} = 10). Second, the influence of a ±2 points swing in the impeller isentropic efficiency on the optimal M_{2} is negligible for impellers of low- to-high pressure ratios from 2.5 to 10, as indicated by the slim shaded bands in the figure.

Figure 9 shows the optimal relative flow angle at the impeller exit (*β*_{2}) in terms of Π_{imp} and *β*_{2b}. First of the three key results is that the primary driver for the optimal *β*_{2} is the design choice of blade angle at the impeller exit (*β*_{2b}). Second, besides *β*_{2b}, the impeller total-to-total pressure ratio has considerable influence on the optimal *β*_{2}, and an increase in the impeller total-to-total pressure ratio results in a larger *β*_{2}. For instance, with a selection of *β*_{2b} = 45 deg, the optimal *β*_{2} for an impeller of Π_{imp} = 2.5 is 53 deg, and its value increases to 56 deg as the Π_{imp} increases to ten. Finally, there is a minor impact from the impeller isentropic efficiency on the optimal *β*_{2}. For an impeller of a medium total-to-total pressure ratio of 5, a ±2 points swing in impeller isentropic efficiency introduces a maximum of 0.6 deg variation in the optimal *β*_{2}.

To conclude, the optimal velocity triangle at the impeller exit is explored over a broad design space covering impeller designs of low-to-high pressure ratio as well as lightly-to-heavily backswept blades. A one-to-one mapping from a selection of impeller total-to-total pressure ratios and backsweep angles to an optimal impeller work factor, machine Mach number, and impeller exit velocity triangle is provided. These one-to-one mappings allow quick screening of the optimal velocity triangle at the impeller exit with a given design requirement in the preliminary design phase. However, questions also arise related to the trend of the optimal impeller work factor concerning the changes in impeller total-to-total pressure ratio, isentropic efficiency, and backsweep angle. For instance:

Why is it inherently challenging to design a low-to-medium-pressure ratio, high-work factor centrifugal compressor?

How does a change in impeller isentropic efficiency affect the optimal impeller work factor?

How does the design choice of impeller backsweep angle affect the optimal impeller work factor?

To answer these questions, the influences of the impeller total-to-total pressure ratio, isentropic efficiency, and backsweep angle on the optimal impeller work factor are analyzed in the upcoming sections.

### Influence of Π_{imp} and *η*_{imp}.

To illustrate the influence of the impeller total-to-total pressure ratio on the optimal impeller work factor, an analysis was conducted at three levels of impeller total-to-total pressure ratio from 2.5 to 10. In addition, the influence of the impeller isentropic efficiency on the optimal impeller work factor was investigated at 88%, 90%, and 92% efficiency. All analyses in this section were conducted with a design choice of 45 deg backsweep angle, and the results are shown in Fig. 10. The abscissa is the impeller work factor, and the ordinate is the relative diffusion effectiveness. As shown in the figure, the optimal impeller design (indicated by the symbols in Fig. 10) tends to move towards a higher work factor with increases in impeller total-to-total pressure ratio. For instance, at an impeller isentropic efficiency of 0.9, the optimal work factor for a low- (Π_{imp} = 2.5) and high- pressure ratio impeller (Π_{imp} = 10) is 0.60 and 0.69, respectively. As modern designs aim for higher work factors, this will push the design away from its optimal case. Besides, due to a smaller optimal work factor associated with a low-pressure ratio impeller, it is inherently more challenging to push the design of a low-pressure ratio impeller toward the same goal of high-work factor compared to that of a high-pressure ratio impeller.

The impeller isentropic efficiency also influences the optimal impeller work factor. With the same design requirement of impeller total-to-total pressure ratio, an impeller of a higher efficiency can accommodate a slightly higher work factor compared to a less efficient impeller. The impact of the impeller isentropic efficiency is significantly less when compared to the influence of the impeller total-to-total pressure ratio. For instance, for a medium-pressure ratio impeller (Π_{imp} = 5), a ±2 *point* swing in impeller isentropic efficiency yields less than 0.2% variation in the value of the impeller optimal work factor.

### Influence of *β*_{2b}.

Modern impeller designs typically feature backswept blades. The advantages of backswept impeller blades are well documented in turbomachinery books and literature [16–18]. In brief, backswept blades increase the streamlined curvature in the blade-to-blade plane and, thus, reduce losses associated with secondary flows. They also reduce the impeller discharge absolute Mach number and, thereby, reduce the diffusion requirements of the downstream stationary diffusion system. However, an increase in backsweep angle can lead to either a decrease in work factor (and, thus, an increase in machine Mach number for a given design requirement) or an increase in absolute flow angle at the impeller exit. As compressor designers continue pushing centrifugal compressor designs to higher efficiency and higher work factors, the balance of all factors that affect efficiency, operating range, and machine size becomes quite challenging. For example, in the effort to redesign the NASA CC3 compressor to suit the needs for a final stage centrifugal compressor in rotorcraft applications, the redesign of the impeller utilized a backsweep angle of 37 deg compared to an original design of 50 deg [19]. While this reduction in backsweep angle provides a much higher work factor for the redesigned impeller, experimental results showed an insufficient surge margin for the new design [20].

To address this topic, the influence of *β*_{2b} on the optimal design choices of the impeller work factor and machine Mach number was investigated. The analysis was performed over a backsweep angle from 25 deg to 55 deg, with a fixed design requirement for the impeller total-to-total pressure ratio (Π_{imp} = 5.0) and isentropic efficiency (*η*_{imp} = 0.9). The results are shown in Fig. 11 with the optimal design choices of the impeller work factor and machine Mach number indicated by the square symbols. As shown in the figure, at a given design requirement for the impeller total-to-total pressure ratio and isentropic efficiency, a design choice of a smaller backsweep angle locates the optimal impeller design toward higher work factor and smaller machine Mach number. For instance, the optimal impeller work factor increases from 0.64 to 0.77, as the impeller backsweep angle reduces from 55 deg to 25 deg. In the meantime, the optimal machine Mach number decreases from 1.45 to 1.33.

As centrifugal compressor designers typically favor a larger backsweep angle at the impeller exit for the benefits in stage efficiency and surge margin, it is also very important to understand that an increase in backsweep angle migrates the optimal design towards lower work factor. Therefore, on the path toward high-work factor compressor designs, a reduced backsweep angle at the impeller exit may be necessary in many cases. Otherwise, the design choice can be quite far away from the optimal case and result in a very tangential absolute flow at the impeller exit.

## Conclusions

Impeller discharge flow plays an important role in centrifugal compressor performance and operability. The choice of the impeller exit velocity triangle is crucial for design success, and shortcomings in the preliminary design phase can not always be rectified in later design phases. The state-of-the-art design approach for the impeller exit velocity triangle in the preliminary design phase relies on several empirical guidelines, i.e., maximum work factor and diffusion ratio for an impeller, the optimal range of absolute flow angle, etc. However, as modern compressors continue pushing toward higher efficiency and higher work factor, this design approach falls short in providing exact guidance to choose an optimal impeller exit velocity triangles because of its ambiguity and inconsistency among the empirical guidelines. In light of this challenge, the work presented in this paper addresses this problem through a two-step effort.

First, this paper introduces a reduced-dimension, deterministic approach to designing the impeller exit velocity triangle based on the previously developed 1D model for impeller preliminary design that focuses on optimization of relative diffusion effectiveness [13]. The method is based on the concept that the impeller is essentially a rotating diffuser, and sets the design of the impeller exit velocity triangle using a relative diffusion effectiveness parameter. The method was validated using six different impeller designs representative of the applications in both turbochargers and aero engines.

With the deterministic method in place, the optimal impeller exit velocity triangles are explored over a broad design space, and a one-to-one mapping from a selection of impeller total-to-total pressure ratio and backsweep angle to a unique optimal impeller exit velocity triangle is provided. Based on the knowledge of the authors, this is the first time these results become available in the open literature.

Also, discussions regarding the influences of impeller total-to-total pressure ratio, isentropic efficiency, and backsweep angle on the optimal work factor of the impeller were presented. Below are a few important takeaways for designers:

The optimal impeller work factor varies with respect to changes in impeller total-to-total pressure ratio. An impeller of high-pressure ratio accommodates a larger work factor and vice versa. Therefore, this makes it inherently challenging to design a high-work factor impeller with a low-to-medium total-to-total pressure ratio.

The design choice of the backsweep angle has a strong influence on the optimal set of impeller work factor and machine Mach number, and an increase in backsweep angle moves the optimal impeller design toward lower work factor and higher machine Mach number. Therefore, on the path toward high-work factor compressor designs, a reduced backsweep angle at the impeller exit may be necessary in many cases.

The optimal impeller exit velocity triangle varies with respect to changes in the impeller total-to-total pressure ratio and exit blade angle. For instance, an impeller with higher pressure ratio and larger backsweep angle results in a more tangential flow at the impeller exit (larger

*α*_{2}). As a result, the influences of impeller total-to-total pressure ratio, isentropic efficiency, and backsweep angle must be taken into consideration in selecting the optimal impeller exit velocity triangle.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

### Greek Symbols

### Subscripts

### Appendix

The impeller exit velocity triangle, shown in Fig. 11, can be characterized using the three parameters including work factor (*µ*), machine Mach number (M_{U2}), and impeller exit blade angle (*β*_{2b}). *µ* represents loading, and M_{U2} reflects the size of the compressor.

*µ*,

*β*

_{2b}, and slip factor (

*σ*)

*σ*is defined

*µ*,

*β*

_{2b}, and

*σ*

*M*

_{U2}), work factor (

*µ*), and impeller isentropic efficiency (

*η*)