This paper introduces a new approach for the preliminary design and aerothermal analysis of centrifugal impellers using a relative diffusion effectiveness parameter. The relative diffusion effectiveness is defined as the ratio of the achieved diffusion to the maximum available diffusion in an impeller. It represents the quality of the relative diffusion process in an impeller. This parameter is used to evaluate impeller performance by correlating the relative diffusion effectiveness with the impeller isentropic efficiency using the experimental data acquired on a single-stage centrifugal compressor (SSCC). By including slip, which is appropriate considering it is an inviscid effect that should be included in the determination of maximum available diffusion in the impeller, a linear correlation between impeller efficiency and relative diffusion effectiveness resulted for all operating conditions. Additionally, a new method for impeller preliminary design was introduced using the relative diffusion effectiveness parameter, in which the optimal design is selected to maximize relative diffusion effectiveness. While traditional preliminary design methods are based on empirical loss models or empirical knowledge for selection of diffusion factor (DF) in the impeller, the new method does not require any such models, and it also provides an analytical approach for the selection of DF that gives optimal impeller performance. Validation of the method was performed using three classic impeller designs available in the open literature, and very good agreement was achieved. Furthermore, a sensitivity study shows that the method is robust in that the resulting flow angles at the impeller inlet and exit are insensitive to a wide range of blockage factors and various slip models.

## Introduction

High pressure ratio centrifugal compressors have been widely used in turbochargers and turboshaft engines because of their compact size, high efficiency, and wide operating range. Design considerations for high pressure ratio centrifugal compressor have been systematically studied by many researchers for decades. Empirical equations were established for preliminary design of impellers and diffusers, including Stodola [1], Cordier [2], Herbert [3], Rodgers [4], and Wiesner [5]. Rodgers and Sapiro [6,7] performed a detailed parametric study on compressor performance and successfully correlated the efficiency of a single-stage centrifugal compressor (SSCC) with four major parameters: inlet specific speed, impeller tip diameter, inducer tip relative Mach number, and exit discharge Mach number. With the help of these empirical loss correlations and slip estimation, the most essential aerodynamic parameters and geometric dimensions (including stage loading, efficiency, surge margin, inlet shroud radius, impeller exit radius, and blade number) could be determined in the preliminary design process.

Since the impeller is essentially a rotating diffusion system, the diffusion ratio (or reciprocal of the de Haller number) is of great importance and has been investigated by a variety of researchers including Rodgers [8], Young [9], and Benvenuti [10,11]. Rodgers [8] showed diffusion ratios between 1.9 and 2.0 at surge flow rates, while Young [9] provided guidance for the maximum attainable relative diffusion in the discussion of Rodgers of results, as it applies to three-dimensional impellers. Furthermore, Benvenuti [10,11] analyzed the data from industrial centrifugal compressors and provided the guidance for industrial two-dimensional impellers. In addition to the diffusion ratio, reduced static pressure and reduced static pressure coefficient are other parameters that have been used in investigating the secondary flows in rotating diffusion systems [12], and they are particularly useful for low-speed machines and pumps.

One challenge in modeling the impeller flow using one-dimensional (1D) tools is that the actual discharge flow pattern of impellers is different from the ideal pattern predicted by a potential flow solver. The existence of a jet-wake flow pattern at the impeller discharge was introduced by Dean and Senoo [13] and further confirmed in the studies of Eckardt [14], Krain [15], and Skoch et al. [16]. The results from Eckardt [14] on a conventional centrifugal impeller showed a well-conditioned flow in the inducer, but the development of a wake near the suction surface starting at the beginning of the radial turn to the impeller exit. The study performed by Krain [15] and Skoch et al. [16] in modern backswept impellers also showed similar measurements.

Based on the experimental observation of the jet-wake flow pattern, Dean and Senoo [13] developed a model for the impeller discharge flow where the impeller discharge flow was categorized into jet flow and wake flow (a two-zone model). The jet flow is modeled as isentropic and follows the impeller blade, while the wake flow contains all the losses. Furthermore, Japikse [17] assessed the single-zone and jet-wake models in evaluating the component performance in centrifugal compressors. The results showed improved accuracy and advantages for designs optimized using two-zone modeling.

The inlet conditions for impellers studied by Eckardt [14], Krain [15], and Skoch et al. [16] are subsonic and free of shocks. However, modern turbochargers and turboshaft engines are continuously pushing the boundary of pressure ratio and flow capacity. Size limitations on the outer diameter lead to larger rotational speeds and result in transonic flow conditions at the compressor inlet, where additional losses due to the interaction of shock waves and blade surface boundary layers and tip clearance flow are possible. Investigations by Rodgers [18,19] showed a drop in impeller peak efficiency (PE) with the increase of impeller inlet shroud relative Mach number, and this has prompted several investigations on transonic impellers.

The flow inside a centrifugal impeller with transonic inlet conditions has been studied by researchers including Senoo et al. [20,21], Krain et al. [22,23], Ibaraki et al. [24,25], and Hagashimori et al. [26] with conflicting results. The results from Senoo et al. [20,21] showed the existence of two shock waves, a detached wave at the impeller leading edge and a passage shock on the pressure surface, at supersonic flow conditions but with no deterioration in impeller performance. However, the results from Krain et al. [22,23] and Ibaraki et al. [24,25] showed increased loss and size of the wake region from tip leakage flow at the impeller exit due to the interaction between the shock wave and the tip leakage flow in the inducer. The results from Hagashimori et al. [26] showed the presence of an oblique shock at the inducer leading edge together with a passage shock at the inducer throat. Additionally, reversed flow near the shroud in the inducer was present due to the interaction between the shock wave and the tip leakage flow.

In general, compared to the impeller flow with subsonic inlet conditions, the flow in transonic impellers is more complex due to the presence of shock waves, and flow instabilities start further upstream at the throat of inducer due to the interaction of shock waves and tip leakage flow. The tip leakage flow plays a more important role for these machines, both within the impeller and also at the exit of the impeller. Reversed flow near the shroud in the inducer may occur due to the presence of shock waves.

## Scope of the Paper

Despite the complexity of flow inside centrifugal impellers, this paper demonstrates use of a meanline, 1D approach for impeller preliminary design and aerothermal analysis. In light of this goal, a relative diffusion effectiveness parameter is introduced to evaluate the impeller performance at both design and off-design operating conditions and also to optimize the impeller exit geometry and velocity triangles during the preliminary design phase. The relative diffusion effectiveness is defined as the ratio of the achieved diffusion to the maximum available diffusion in an impeller, and it can be calculated knowing the static pressure and geometry. The application of this parameter in the preliminary design phase and aerothermal analysis of centrifugal impeller performance are discussed.

## Methodology

The purpose of a centrifugal compressor is to raise the static pressure of the working fluid. Rearranging the equation for conservation of rothalpy, the enthalpy rise across the impeller for an adiabatic process is
$h2−h1=U22−U122+W12−W222$
(1)

where $h$ is the static enthalpy, $U$ is the wheel speed, $W$ is the relative velocity, subscript $1$ represents impeller inlet, and subscript $2$ indicates impeller exit.

The thermodynamic relationship for enthalpy gives
$dh=Tds+dP/ρ$
(2)

where $T$ stands for static temperature, $s$ is entropy, $P$ is static pressure, and $ρ$ is density.

Integrating Eq. (2) gives
$h2−h1=∫12Tds+∫12dP/ρ$
(3)
Combining Eqs. (1) and (3), the relationship between static pressure rise and impeller velocity triangles is
$∫12dP/ρ+∫12Tds=U22−U122+W12−W222$
(4)

The first term on the left-hand side represents the static pressure rise achieved in impeller. The second term on the left-hand side represents loss in terms of entropy generation. The first term on the right-hand side is related to the static pressure rise associated with the centrifugal effect. The second term on the right-hand side represents the static enthalpy rise associated with the relative-frame diffusion that occurs in the impeller.

One advantage of an impeller is that there is always a contribution to the static pressure rise from the centrifugal effect regardless of the flow quality inside the impeller. Even though the centrifugal effect contributes to the static pressure rise and impeller efficiency, it also undermines the capability of using impeller efficiency or the traditional diffusion effectiveness as the parameters in evaluating the quality of flow in impeller. As a result, efficiency is not the best indicator for the designer aiming to improve the aerodynamic design of an impeller.

Thus, relative diffusion effectiveness ($ε$) is introduced as a parameter used to describe the quality of the flow inside the impeller. Instead of using reduced static pressure, the relative diffusion effectiveness is defined in terms of diffusion ratio, which is directly related to the velocity triangles and, thus, reduces the complexity of its application within the design process. It is described as
$ε=RMR/RMRi$
(5)
where subscript $i$ stands for the ideal case, and $RMR$ represents the diffusion in terms of relative Mach number ratio to account for compressibility and is described as
$RMR=Mrel_1/Mrel_2$
(6)

where $Mrel$ represents the relative Mach number.

Combining Eqs. (5) and (6), the relative diffusion effectiveness becomes
$ε=Mrel_2i/Mrel_2$
(7)

It is worth noting that the relative diffusion effectiveness is linear with respect to the relative Mach number ratios at the impeller exit, which is different from the traditional diffusion effectiveness obtained in terms of static pressure. Since centrifugal effects contribute to the static pressure rise in radial impellers, the relative diffusion effectiveness isolates the impeller aerodynamic performance from the centrifugal effect and provides a more direct metric for the quality of the impeller aerodynamic design.

### Evaluation of Impeller Exit Relative Mach Number, Mrel_2.

The flow inside the impeller can be categorized as a primary flow zone and secondary flow zone. The primary flow represents the well-diffused isentropic flow, and the secondary flow generates all the entropy and does not contribute to the static pressure recovery. Thus, the primary flow relative Mach number represents the diffusion in the impeller with high fidelity.

The primary flow relative Mach number could be obtained using the temperature and pressure information measured at the impeller inlet with the static pressure measured at impeller exit by
$s2p=s1=s(T1,P1)$and
(8)
$T2p=T(P2,s2p)$
(9)

where subscript $p$ represents the primary flow.

The impeller inlet rothalpy is
$I1=ht1−U1Vθ1$
(10)
in which $I$ is rothalpy, $V$ is the absolute velocity, subscript $t$ stands for the stagnation condition, and $θ$ represents the tangential component. Furthermore, for cases with zero prewhirl, the rothalpy at the impeller inlet equals the stagnation enthalpy at the impeller inlet.
The equation for conservation of rothalpy for an adiabatic process gives
$I2=I1=h2−U222+W222$
(11)
Impeller relative velocity can then be obtained with
$W2=2I1−h2+U222$
(12)

In this and the following procedures, all the fluid properties (including enthalpy, entropy, speed of sound, etc.) are obtained from the National Institute of Standards and Technology reference fluid thermodynamic and transport properties database, REFPROP [27].

### Evaluation of Mrel_2i.

The ideal relative Mach number could be calculated in two different ways depending on the assumptions used. This section discusses the method based on the assumption of isentropic flow, zero slip, and no blockage. This relative Mach number represents the maximum diffusion available for a given geometry. The procedure is iterative and starts with an initial guess of impeller exit static pressure, $P2,j$
$T2,j=T(P2,j,s1)$
(13)
$h2,j=h(T2j,P2,j)$
(14)
$ρ2,j=ρ(T2,j,P2,j)$
(15)

where subscript $j$ stands for the $jth$ iteration.

Rearranging the equation for conservation of rothalpy gives
$W2i_cor,j=2I1−h2,j+U222and$
(16)
rearranging the equation for conservation of mass gives
$W2i_com,j=m˙ρ2,jA2cosβ2b$
(17)

where $A$ is the effective area at the impeller exit (considering blade thickness), $β2b$ is the blade angle at the impeller exit, subscript $cor$ represents the value obtained from the equation for conservation of rothalpy, and subscript $com$ represents the value calculated from the equation for conservation of mass.

In each iteration, two values of relative velocity, $W2i$, were obtained from equations for conservation of rothalpy and conservation of mass. The iteration of impeller exit static pressure occurs until both the equation for conservation of rothalpy and the equation for conservation of mass are satisfied. Thus, the ideal relative Mach number at the impeller exit is calculated using the converged relative velocity and speed of sound at the impeller exit:
$Mrel_2i=W2i/a2i$
(18)

### Evaluation of Mrel_2i_slip.

The parameter $Mrel_2i$ represents the maximum diffusion available for a given geometry. However, due to the inviscid nature of slip, the maximum diffusion obtained based on the no-slip assumption provides an overestimation to maximum diffusion. Additionally, the work input by the impeller is overestimated using the assumption of zero slip. To mitigate the inconsistency in the work input calculation, a corrected ideal relative Mach number, $Mrel_2i_slip$ is introduced. This method removes the no-slip assumption. The corrected relative Mach number represents the diffusion available for a given geometry, together with a known work input based on the assumptions of isentropic flow and zero blockage. The procedure is iterative and includes an outer iteration and an inner iteration. It starts with initial guess of impeller exit static pressure, $P2,j$ (inner loop) and slip angle, $βslip,k$ (outer loop)
$β2f,k=β2b+βslip,k$
(19)

where $β2f$ is the relative flow angle at the impeller exit, and $βslip$ is the slip angle.

The relative velocity at the impeller exit considering slip can be found both by using conservation of rothalpy (as shown in Eq. (20)) and conservation of mass (as shown in Eq. (21))
$W2i_slip_cor,j=2I1−h2,j+U222$
(20)
$W2i_slip_com,j=m˙ρ2,jA2cosβ2f,j$
(21)
The static pressure is iteratively adjusted until the relative velocity is matched from both equations. Then, the total energy is used to iterate on slip angle. This is done by calculating the absolute velocity at impeller exit
$V2,k=W2i_slipcosβ2f,k2+U2−W2i_slipsinβ2f,k2$
(22)
The impeller exit stagnation enthalpy is calculated from static enthalpy and kinetic energy
$ht2,k=h2+V2,k2/2$
(23)
The impeller exit stagnation enthalpy obtained from the total pressure and total temperature is
$ht2=h(Tt2,Pt2)$
(24)
The slip angle is adjusted until the calculated impeller stagnation enthalpy matches the measured stagnation enthalpy. Thus, the process essentially adjusts the impeller exit static pressure and slip angle until all the equations for conservation of rothalpy, conservation of mass, and conservation of energy are satisfied. The corrected ideal relative Mach number at impeller exit is calculated as
$Mrel_2i_slip=W2i_slip/a2i_slip$
(25)
There will be two different values for the relative diffusion effectiveness depending on the slip assumptions used in calculating the minimum relative Mach number at impeller exit. The relative diffusion effectiveness calculated with $Mrel_2i$ is defined as the relative diffusion effectiveness without slip
$εno_slip=Mrel_2i/Mrel_2$
(26)
The relative diffusion effectiveness calculated with $Mrel_2i_slip$ is defined as the relative diffusion effectiveness with slip
$εslip=Mrel_2i_slip/Mrel_2$
(27)

Since the maximum diffusion derived from the assumption of zero slip is an overestimate of what can actually be achieved, $εslip$ is greater than $εno_slip$.

## Performance Evaluation

The relative diffusion effectiveness is correlated with impeller isentropic efficiency using experimental data acquired on a single-stage centrifugal compressor. The compressor features a transonic impeller with backswept blades. The impeller has 17 main blades and 17 splitter blades. The design operating speed for the compressor is about 45,000 rpm, and the total pressure ratio for the entire stage is on the order of 6.5. The details of SSCC facility were documented by Lou et al. [28].

The steady performance is characterized by the total pressure ratio, total temperature ratio, and efficiency. The performance of the entire compressor stage is calculated from the area-averaged measurements acquired at the compressor inlet and exit. The impeller-only performance is evaluated by using the area-averaged measurements at the compressor inlet and the impeller exit condition determined using static pressure measurements at that location. The total temperature at the impeller exit is assumed to be the same as that measured at the deswirl exit (stage exit) based on the adiabatic assumption. The impeller exit total pressure is derived from the measured total temperature at the deswirl exit, the inlet mass flow rate, and the area-averaged static pressure measured at the impeller trailing edge using the continuity equation and the turbomachinery Euler equation [29]. The compressor corrected conditions (speed and mass flow rate) [30] and efficiency [31] are calculated using properties for humid air retrieved from REPROP. The results presented in this section are normalized using the operating condition at design point. The compressor inlet pressure is measured using highly accurate 2.5 psid modules with an uncertainty less than 0.12%. Compressor exit pressure is measured using 100 psid modules with an uncertainty less than 0.05% full scale. This renders the uncertainty in the total pressure ratio less than 0.2% from 80% to 100% corrected speed. The mass flow rate is measured using a calibrated bellmouth with uncertainty less than 0.5%.

Figure 1 shows the normalized total pressure ratio versus the normalized corrected mass flow rate. The results for both the impeller and the entire stage from 80% to 100% corrected speed are presented. The PE conditions are represented by the solid green symbols. The low loading conditions are represented by the solid blue symbols. The choke conditions are shown as solid red symbols. Those color schemes are consistent with the rest of figures presented in this section. Regardless of the variation in the total pressure ratio for the entire compressor stage relative to the changes in the loading conditions, the total pressure ratio for the impeller stays very consistent along the choke line due to the choked flow in the diffuser.

Fig. 1
Fig. 1
Close modal

Figure 2 shows the performance of the impeller and entire compressor stage in terms of isentropic efficiency. Compared to the entire stage, the impeller operates more efficiently over the entire operating range, from choke to near surge. There is no obvious deterioration in impeller efficiency along the choke line as loading decreases. In fact, at subsonic inlet conditions from 80% to 95% corrected speed, the impeller efficiency increases as loading decreases. At design speed with supersonic inlet conditions, the trend in impeller efficiency lines up with the trend of the entire compressor stage in that they both increase from choke to PE condition.

Fig. 2
Fig. 2
Close modal

The effect of inlet tip relative Mach number on impeller efficiency was investigated, and the results are shown in Fig. 3. The inlet tip relative Mach number is mainly determined by the inlet mass flow rate and prewhirl angle. At supersonic inlet conditions, the impeller peak efficiency drops with the increase of inlet tip relative Mach number, and there is about a 0.7 point drop in the peak efficiency from 95% to 100% corrected speed. This agrees with the observation from Rodgers [18,19]. However, at subsonic inlet conditions, the impeller peak efficiency increases as the inlet tip relative Mach number increases, and there is a 3.2 point improvement in the impeller peak efficiency from 80% to 90% corrected speed. Additionally, the impeller efficiency is closely related to loading condition. At design speed, there is about a 1.0 point change in the impeller efficiency from choke to near surge. At 80% corrected speed, the variation in the impeller efficiency with loading is about 1.5 points.

Fig. 3
Fig. 3
Close modal

The relationship between impeller peak efficiency and inlet tip relative Mach number is very helpful in optimizing the inducer size during preliminary design. However, its utility is limited in correlating the impeller performance to the entire operating range. Figure 4 shows the relationship between the impeller isentropic efficiency and the relative diffusion effectiveness without slip. The impeller efficiency is proportional to the relative diffusion effectiveness, and this applies to all the operating points from 80% corrected speed to 100% corrected speed. The variation in impeller efficiency with respect to the changes in loading condition for each speed line observed in Fig. 3 is associated with the relative diffusion in the impeller. Despite the differences in the peak efficiency at various corrected speeds, the slope between the impeller efficiency and the relative diffusion effectiveness is similar from 80% to 100% corrected speed. Additionally, for each 0.2-point change in the relative diffusion effectiveness, there is about a 1.0-point change in the efficiency, which indicates that the relative diffusion effectiveness without slip is a reliable parameter in comparing the performance of various impellers.

Fig. 4
Fig. 4
Close modal

The correlation between the impeller isentropic efficiency and the relative diffusion effectiveness with slip is shown in Fig. 5. There is also a similar trend between impeller efficiency and the relative diffusion effectiveness with slip. However, the advantage of using the relative diffusion effectiveness that incorporates slip is that all the operating points from 80% to 100% corrected speed are correlated by a single linear fit. This greatly reduces the complexity of using this parameter in analyzing impeller performance.

Fig. 5
Fig. 5
Close modal

## Preliminary Design

In addition to the aerothermal analysis that has been presented, the relative diffusion effectiveness parameter also offers a new approach for optimizing the geometry and velocity triangles at impeller exit during the preliminary design phase. The objective of the impeller preliminary design is to determine the principle aerodynamic and geometrical parameters for a required design duty. Traditional preliminary design methods rely on empirical loss models, and the selection of diffusion factor (DF) in the impeller is usually subject to the designer's experience. The new method of using relative diffusion effectiveness has zero dependence on empirical loss models, and thus, it reduces the empirical input parameters needed to simplify the slip model and blockage factors at the impeller inlet and exit. Additionally, the new method provides an analytical approach for the selection of diffusion factor for the optimal design. The workflow for impeller preliminary design using relative diffusion effectiveness is shown in Fig. 6. For a given set of design requirements and preselected parameters, the procedure starts with a wide range of diffusion factors and calculates the associated value of relative diffusion effectiveness for each diffusion factor. The optimal preliminary design is achieved as the diffusion factor gives the maximum relative diffusion effectiveness. With completion of the preliminary design, the detailed impeller geometry, including the meridional contour and blade profile, are obtained in the following two-dimensional and three-dimensional design loop. With the impeller geometry now available, detailed aerodynamic analyses can be performed to check if the design meets the requirements.

Fig. 6
Fig. 6
Close modal

This new approach was applied to three classic impellers available in the open literature with the objective being to compare the optimum designs obtained from the new method based on relative diffusion effectiveness to the original design choices of three classic impellers. Furthermore, a sensitivity analysis of the new method for different slip models and different blockage factors was performed.

### Validation of the Method.

Validation of the present method was performed using the preliminary design information of three impellers available in the open literature. The impellers are representative applications of centrifugal compressors in gas turbines and turbochargers, and their dimensionless specific speed varies between 0.53 and 0.81. The first impeller was designed by Came [32]. The second impeller is the CC3 impeller scaled up from an Allison Engine Design [33]. The third impeller is the SRV2-O impeller [34] designed at the German Aerospace Center (DLR). Came's impeller and the SRV2-O impeller are high-speed and high-pressure-ratio machines, while the CC3 impeller is an intermediate-pressure ratio machine. The inlet condition is subsonic for Came's impeller and the CC3 impeller, and it is transonic for the SRV2-O impeller. The principle design parameters for the three impellers are listed in Table 1.

Table 1

Principle design parameters for the three selected impellers

DescriptionParameterCame's impellerCC3 impellerSRV2-O impeller
Rotational speedN (rpm)40,00021,78950,000
Mass flow rate$m˙$ (kg/s)1.814.542.55
Total-total pressure ratioπ127.654.16.1
Isentropic Efficiencyηt120.870.920.84
Inlet prewhirlα1 (deg)000
DescriptionParameterCame's impellerCC3 impellerSRV2-O impeller
Rotational speedN (rpm)40,00021,78950,000
Mass flow rate$m˙$ (kg/s)1.814.542.55
Total-total pressure ratioπ127.654.16.1
Isentropic Efficiencyηt120.870.920.84
Inlet prewhirlα1 (deg)000

Came's impeller has a design mass flow rate of 1.81 kg/s and rotational speed of 40,000 rpm. The dimensionless specific speed of the impeller is 0.53, and the total-to-total pressure ratio is 7.65 at design point. The estimated impeller isentropic efficiency is 0.87. The impeller inlet hub radius was set at 30.48 mm. The CC3 impeller was designed to produce a stage pressure ratio of 4.0 at a corrected mass flow rate of 4.54 kg/s and corrected speed of 21,789 rpm when coupled with a vaned diffuser. The dimensionless specific speed of the impeller is 0.6. The impeller total pressure ratio at design point is 4.1, and the estimated isentropic efficiency is 92%. The impeller total pressure ratio and isentropic efficiency were tabulated from the validated CFD results [35]. As to the SRV2-O impeller, it produces a total-to-total pressure ratio of 6.1 at the design speed of 50,000 rpm. The dimensionless specific speed of the impeller is 0.81, and its estimated efficiency is 0.84. The impeller inlet hub radius was 30 mm. All of the impellers are of advanced design with backswept trailing edges and splitter blades. A blade count of 17 full blades and 17 splitter blades was selected in Came's design. A blade count of 13 full blades and 13 splitter blades was selected for SRV2-O impeller, and CC3 impeller features 15 full blades and 15 splitter blades. The impeller exit blade angles were provided for Came's impeller (30 deg backsweep) and the CC3 impeller (50 deg backsweep). However, instead of blade angle, an impeller exit radius of 112 mm was selected for SRV2-O impeller. There is zero prewhirl for all the three impellers.

Based upon the design duty listed in Table 1, selection of Wiesner slip model [5], and chosen values of blockage factor, the preliminary design program using the method based on relative diffusion effectiveness calculated the remaining preliminary parameters necessary for the basic overall design. The output obtained using the new method is compared to the original design choices, and the results are listed in Table 2. The impeller inlet calculations utilized in the new method follow the standard practice for centrifugal compressor design [36]: minimize the inlet tip relative Mach number at the design mass flow rate and inlet conditions. The values of the impeller inlet blockage were adjusted to match the inlet tip relative Mach number provided in the original designs, and this renders an inlet blockage of 0.095 for Came's impeller and an inlet blockage of 0.09 for CC3 and SRV2-O impeller. The geometry and velocity triangles at the impeller exit were optimized based on the relative diffusion effectiveness parameter, and an exit blockage factor of 0.19 was selected for all three impellers.

Table 2

Comparison between results from the new method and the original design selections

Came's impellerCC3 impellerSRV2-O impeller
DescriptionParameterOriginal designNew methodOriginal designNew methodOriginal designNew method
Diffusion factorDF1.61.61.41.4N/A1.7
Inlet tip relative Mach numberMrel_1t0.980.980.850.851.301.30
Exit wheel speedU2 (m/s)575.4583.6492.2489.2586586
Exit absolute flow angleα2 (deg)71.270.767.1
Came's impellerCC3 impellerSRV2-O impeller
DescriptionParameterOriginal designNew methodOriginal designNew methodOriginal designNew method
Diffusion factorDF1.61.61.41.4N/A1.7
Inlet tip relative Mach numberMrel_1t0.980.980.850.851.301.30
Exit wheel speedU2 (m/s)575.4583.6492.2489.2586586
Exit absolute flow angleα2 (deg)71.270.767.1
a

Represents preselected parameters.

At the impeller inlet, the new method calculates a tip radius of 68.9 mm and tip blade angle of 55.6 deg for Came's impeller, which are very close to the values of 67.3 mm and 53.8 deg adopted in the original design. In the case of CC3 impeller, the new method calculates a tip radius of 109.2 mm which is 4.2 mm different from the value adopted in CC3 impeller. As to the SRV2-O impeller, the new method calculates the tip radius of 74.4 mm and blade angle of 65 deg. Those values are similar to the design choices from DLR, which are 78 mm for inlet tip radius and 63.5 deg for the tip blade angle.

As to the impeller exit geometry and flow properties, the new method gives an optimum diffusion factor of 1.6, which is also the choice in Came's design. Additionally, the new method calculates an impeller exit radius of 139.3 mm, with less than 2.0 mm difference from the selection in Came's design. In the case of the CC3 impeller, the method based on relative diffusion effectiveness calculated the same diffusion factor (1.40) as the design choice of the CC3 impeller. Additionally, the new method gives an optimum exit radius of 214.4 mm, with less than a 1.2 mm difference from the CC3 impeller design. As to the SRV2-O impeller, the new method gives an optimum diffusion factor of 1.7, leading to an exit blade angle of 39 deg. The value of diffusion factor was not presented in the literature for the SRV2-O impeller. However, the backswept angle was 38 deg, which is very close to the value optimized with the new approach.

Furthermore, in all the three cases, the values for the optimum absolute flow angle at the impeller exit stay fairly close, within a tight range between 67.1 deg and 71.2 deg. This agrees well with the empirical knowledge that the optimal impeller exit flow angles are between 60 deg and 70 deg, as recommended by Rodgers and Sapiro [6]. In summary, the new method based on relative diffusion effectiveness provides designs similar to those achieved via traditional preliminary design methods, as indicated by the similar impeller geometries and velocity triangles calculated for the three classic impellers available in the open literature.

### Sensitivity Study.

A sensitivity analysis of the present method to different slip models and to different blockage factors was performed. The effect of slip models was investigated using three different models: Stodola [1], Stanitz [37], and Wiesner [5]. The sensitivity study of impeller inlet and exit blockage factors was performed over a wide range of blockage values, from 0 to 0.25 in increments of 0.05.

Figure 7 shows the effect of the slip models on screening the optimum preliminary design for Came's impeller. The optimum diffusion factor and the corresponding relative diffusion effectiveness from each slip model are represented by symbols. In these calculations using different slip models, the blockage factors were set the same, with a value of 0.095 at impeller inlet and 0.19 at impeller exit. The optimum diffusion factors and maximum diffusion effectiveness calculated from Wiesner's and Stodolar's slip models are very close to each other. However, the slip model from Stanitz gave significantly higher values for both the optimum diffusion factor and maximum relative diffusion effectiveness. The optimum diffusion factor and the corresponding relative diffusion effectiveness from Stanitz's slip model are 12% and 5% higher than the values from Wiesner's slip model. Recalling that the slip model from Stanitz was derived from an impeller with radial blades (zero backsweep), this is likely the reason for the differences.

Fig. 7
Fig. 7
Close modal

In addition to the diffusion factor and relative diffusion effectiveness, other preliminary design parameters from each slip model are listed in Table 3. Since slip model does not affect the calculations at impeller inlet, the inlet properties and geometry are the same for all three slip models and are not included. The geometry and flow properties at the impeller exit vary slightly with respect to the slip model. The impeller exit radius, blade height, and absolute flow angle from Stodola's model are almost identical to those obtained from Wiesner's model, with less than a 0.2% difference in magnitude. In contrast, the impeller exit blade height calculated from Stanitz's model is about 12% larger than the value calculated using Wiesner's model, which is similar to the difference in diffusion factor. Additionally, the differences in the impeller exit radius and absolute flow angle between Stanitz's and Wiesner's models are much smaller, with a 1.9% reduction in impeller exit radius and 2.3% increase in the exit absolute flow angle using Stanitz's model.

Table 3

Sensitivity study of slip models for Came's impeller

DescriptionStodolaStanitzWiesner
Diffusion factor1.591.791.60
Exit absolute flow angle (deg)71.172.971.2
Slip factor0.910.940.92
Relative diffusion effectiveness0.640.670.64
DescriptionStodolaStanitzWiesner
Diffusion factor1.591.791.60
Exit absolute flow angle (deg)71.172.971.2
Slip factor0.910.940.92
Relative diffusion effectiveness0.640.670.64

A sensitivity study of the new method to different slip models for the CC3 impeller was performed, and the results are shown in Fig. 8. The blockage factors in the calculations were set to be 0.09 at the impeller inlet and 0.19 at the impeller exit. As indicated by symbols in the figure, the slip models from Stodola and Stanitz give very similar values in the optimum diffusion factor and peak relative diffusion effectiveness. Comparing to the model from Stodola, the slip model from Wiesner gave a reduced optimum diffusion factor by 2.7% and reduced relative diffusion effectiveness by 1.9%. In addition, the detailed data output using various slip models is listed in Table 4. The geometry and flow properties at the impeller exit vary slightly for the different slip models, with less than a 2.0 mm variation in exit radius, less than 0.5 mm variation in blade height, and less than a 1 deg variation in the absolute flow angle.

Fig. 8
Fig. 8
Close modal
Table 4

Sensitivity study of slip models for CC3 impeller

DescriptionStodolaStanitzWiesner
Diffusion factor1.441.451.40
Exit absolute flow angle (deg)71.171.270.7
Slip factor0.930.930.93
Relative diffusion effectiveness0.740.740.72
DescriptionStodolaStanitzWiesner
Diffusion factor1.441.451.40
Exit absolute flow angle (deg)71.171.270.7
Slip factor0.930.930.93
Relative diffusion effectiveness0.740.740.72

Figure 9 shows the effect of the slip model on screening the optimum preliminary design for SRV2-O impeller, with the optimum design from each slip model indicated by the symbols. The blockage factors in the calculations were set the same as CC3 impeller, with a value of 0.09 at the impeller inlet and 0.19 at the impeller exit. For the case of the SRV2-O impeller, slip models from Stanitz and Wiesner give peak values (local maxima) in the relative diffusion effectiveness, which are considered as the optimum design case. However, the slip model from Stodola does not give such a peak value but an inflection point.

Fig. 9
Fig. 9
Close modal

The relative diffusion effectiveness obtained from Stodola's slip model increases quickly for diffusion factors between 1.2 and 1.5, flattens out around 1.6, and then starts to increase sharply. Since it is impossible for relative diffusion effectiveness to continue increasing with the increase of relative diffusion, the inflection point in the case of Stodola's slip model is considered as the optimum design case. The slip models from Wiesner and Stodola give very similar values for the optimum diffusion factor and relative diffusion effectiveness. However, the Stanitz model gives a slightly lower (1%) value for the optimum diffusion factor and a higher (3%) value of the relative diffusion effectiveness, compared to the values obtained using Wiesner's slip model.

Additionally, the rest of the output data at the impeller exit obtained from different slip models are listed in Table 5. The variations in the geometry and flow properties at the impeller exit associated with the change of slip model are very small. There is less than a 3% variation in the impeller exit blade height, and the variations in the impeller exit blade angle and absolute flow angle are less than 1 deg, showing that the approach is relatively insensitive to the choice of slip model.

Table 5

Sensitivity study of slip models for SRV-O impeller

DescriptionStodolaStanitzWiesner
Diffusion factor1.711.681.70
Exit absolute flow angle (deg)67.466.567.1
Slip factor0.910.920.91
Relative diffusion effectiveness0.570.590.57
DescriptionStodolaStanitzWiesner
Diffusion factor1.711.681.70
Exit absolute flow angle (deg)67.466.567.1
Slip factor0.910.920.91
Relative diffusion effectiveness0.570.590.57

Furthermore, a sensitivity study of impeller inlet blockage factor was performed for a wide range of blockage values, from 0 to 0.25 in increments of 0.05, and the results are listed in Table 6 (Came's impeller), Table 7 (CC3 impeller), and Table 8 (SRV2-O impeller). A constant blockage factor of 0.19 at the impeller exit and the Wiesner slip model are used for all these calculations.

Table 6

Sensitivity study of B1 for Came's impeller

ParameterValues
B100.050.10.150.20.25
DF1.551.571.61.631.671.7
R1t (mm)66.867.969.170.371.773.2
ß1t (deg)57.557.557.657.557.757.7
Mrel,1t0.950.960.980.991.011.04
b2 (mm)6.06.06.06.06.16.0
α2 (deg)71.271.171.271.271.371.3
R2 (mm)139.3139.4139.3139.3139.3139.3
ParameterValues
B100.050.10.150.20.25
DF1.551.571.61.631.671.7
R1t (mm)66.867.969.170.371.773.2
ß1t (deg)57.557.557.657.557.757.7
Mrel,1t0.950.960.980.991.011.04
b2 (mm)6.06.06.06.06.16.0
α2 (deg)71.271.171.271.271.371.3
R2 (mm)139.3139.4139.3139.3139.3139.3

Note: Boldface rows represent the parameters that change with respect to the change of inlet blockage factor.

Table 7

Sensitivity study of B1 for CC3 impeller

ParameterValues
B100.050.10.150.20.25
DF1.361.381.41.431.451.49
R1t (mm)105.7107.5109.4111.6113.7116.4
ß1t (deg)60.860.860.961.061.061.2
Mrel,1t0.830.840.850.870.890.91
b2 (mm)16.116.016.016.015.916.1
α2 (deg)70.770.770.670.770.570.7
R2 (mm)214.3214.4214.5214.4214.6214.4
ParameterValues
B100.050.10.150.20.25
DF1.361.381.41.431.451.49
R1t (mm)105.7107.5109.4111.6113.7116.4
ß1t (deg)60.860.860.961.061.061.2
Mrel,1t0.830.840.850.870.890.91
b2 (mm)16.116.016.016.015.916.1
α2 (deg)70.770.770.670.770.570.7
R2 (mm)214.3214.4214.5214.4214.6214.4
Table 8

Sensitivity study of B1 for SRV2-O impeller

ParameterValues
B100.050.10.150.20.25
DF1.641.681.721.741.781.82
R1t (mm)72.073.374.776.277.979.7
ß1t (deg)64.764.965.165.165.465.5
Mrel,1t1.261.281.311.331.361.39
b2 (mm)10.410.610.710.510.610.6
ß2b (deg)38.739.239.639.139.239.3
α2 (deg)66.967.267.567.167.267.2
ParameterValues
B100.050.10.150.20.25
DF1.641.681.721.741.781.82
R1t (mm)72.073.374.776.277.979.7
ß1t (deg)64.764.965.165.165.465.5
Mrel,1t1.261.281.311.331.361.39
b2 (mm)10.410.610.710.510.610.6
ß2b (deg)38.739.239.639.139.239.3
α2 (deg)66.967.267.567.167.267.2

The results show that the impeller inlet blockage factor has a negligible effect on the geometry and flow properties at the impeller exit. The impeller exit radius, blade height, and absolute flow angle remain fairly constant over the entire range of impeller inlet blockage, with less than a 0.3 mm variation in impeller exit radius and blade height, and less than 1 deg variation in impeller exit blade angle and absolute flow angle. The blade angle at the impeller inlet also remains fairly constant, with less than a 1.0 deg variation in magnitude.

The variation in impeller inlet blockage factor primarily affects the value of the inlet tip radius and relative Mach number. There is an approximately 10% increase in the impeller inlet tip radius as the inlet blockage factor increases from 0 to 0.25. This increase in the impeller tip radius renders a higher inlet tip relative Mach number and a higher relative diffusion ratio across the impeller. The increase in the impeller inlet tip relative Mach number and diffusion factor is about the same as the increase in the impeller inlet tip radius.

At last, a sensitivity study on the blockage factor at the impeller exit was performed over the same range (from 0 to 0.25), and the results from the three impellers are listed in Table 9. The same slip model from Wiesner was selected, and a constant inlet blockage factor of 0.09 was used. The impeller exit blockage factor only influences the blade height at the impeller exit. As the impeller exit blockage increases, the percentage of effective flow area decreases, and thus, requires an increased blade height at the impeller exit. There is an approximately 35% (Came's impeller) and 33% (CC3 and SRV2-O impeller) increase in the blade height as the blockage factor increases from 0 to 0.25 at the impeller exit.

Table 9

Sensitivity study of B2 for the three selected impellers

ParameterCameCC3SRV2-O
B2b2 (mm)b2 (mm)b2 (mm)
04.813.08.5
0.055.113.79.0
0.15.414.59.5
0.155.715.310.0
0.26.116.310.6
0.256.517.311.3
ParameterCameCC3SRV2-O
B2b2 (mm)b2 (mm)b2 (mm)
04.813.08.5
0.055.113.79.0
0.15.414.59.5
0.155.715.310.0
0.26.116.310.6
0.256.517.311.3

In summary, the slip model does not affect the calculations at the impeller inlet, and its effect on the calculations at impeller exit is also minor. In all three cases, the method provides similar results of flow angles at the impeller inlet and exit regardless of the choice of slip models. As to the effect of blockage factors, the dominant effect from inlet blockage is on the inlet tip radius and associated relative Mach number, and the blockage factor at the impeller exit is the primary driver for impeller exit blade height. Thus, some intelligent choice of blockage must be exercised, but this choice will not affect the velocity triangles, and thus, the optimization of the velocity triangles is not affected by blockage either.

## Conclusions

This paper introduces a new approach for the preliminary design and aerothermal analysis of centrifugal impellers using a relative diffusion effectiveness parameter. The relative diffusion effectiveness is defined as the ratio of the achieved diffusion to the maximum available diffusion in an impeller. The parameter allows for a new optimization method for centrifugal impeller 1D preliminary design, in which the optimum design is selected to achieve the maximum relative diffusion effectiveness. Additionally, it could also be used to compare the performance of various impellers.

The relative diffusion effectiveness was correlated with impeller isentropic efficiency using experimental data acquired on a single-stage centrifugal compressor. The results show positive correlation between impeller efficiency and relative diffusion effectiveness at both design and off design conditions. Furthermore, after incorporating slip, the correlation between impeller efficiency and relative diffusion effectiveness could be represented by a single linear fit over the entire operating range from 80% to 100% corrected speed, which greatly reduces the complexity of its application in analyzing impeller performance. The positive correlation between relative diffusion effectiveness and isentropic efficiency at both design and off-design conditions indicates that relative diffusion effectiveness could be used as an alternate parameter of impeller isentropic efficiency in evaluating impeller performance. Compared with isentropic efficiency, the relative diffusion effectiveness does not include the benefits associated with the centrifugal effect and thus, provides a direct indication of the impeller aerodynamic performance. Additionally, relative diffusion effectiveness could be calculated from static pressure measurements acquired at the impeller exit and thus, greatly reduces the instrumentation cost for total pressure/temperature rakes required for the calculation of isentropic efficiency.

Relative diffusion effectiveness is used in a new approach to impeller preliminary design, and it was applied to three classic impellers available in the open literature. The selected impellers [3234] are representative applications of centrifugal compressors in gas turbines and turbochargers. The results obtained from the new method based on relative diffusion effectiveness agree very well with the original design selections obtained from traditional methods based on well-tuned empirical correlations. Furthermore, sensitivity studies show that the new approach is robust and provides similar results of flow angles at impeller inlet and exit despite a wide range of choices for impeller inlet and exit blockage factors, as well as different slip models.

This new approach is useful since it does not require empirical loss models. It provides an analytical approach for selection of the optimal diffusion factor. Since it can be challenging to establish a tuned set of empirical loss models or explore designs outside of past design envelopes, the analytical approach of the new method greatly reduces the complexity of its application and provides a useful tool for centrifugal impeller preliminary design.

## Acknowledgment

This research has been sponsored by Honeywell, Inc., and this support is most gratefully acknowledged. The authors also wish to thank Honeywell for granting permission to publish this work. Additionally the guidance and advice offered by Mr. Darrell James and Dr. Rakesh Srivastava of Honeywell were very valuable to this project. Assistance from Herbert Harrison and Amelia Brooks at the Purdue Compressor Research Laboratory during data acquisition was also very much appreciated.

## Nomenclature

• a =

speed of sound

•
• A =

effective area

•
• b =

•
• B =

blockage factor

•
• DF =

diffusion factor, $DF=W1t/W2$

•
• h =

enthalpy

•
• I =

rothalpy

•
• $m˙$ =

mass flow rate

•
• M =

Mach number

•
• N =

rotational speed in rpm

•
• P =

pressure

•
• R =

•
• RMR =

relative Mach number ratio

•
• s =

entropy

•
• T =

temperature

•
• U =

wheel velocity

•
• V =

absolute velocity

•
• W =

relative velocity

•
• Z =

•
• α =

•
• β =

•
• ε =

relative diffusion effectiveness

•
• η =

isentropic efficiency, total to total

•
• π =

total-total pressure ratio

•
• ρ =

density

### Subscripts

Subscripts

• b =

•
• com =

properties obtained from equation for conservation of mass

•
• cor =

properties obtained from equation for conservation of rothalpy

•
• f =

properties associated with flow, full blade

•
• h =

hub

•
• i =

ideal

•
• j =

jth iteration

•
• k =

kth iteration

•
• m =

properties derived from conservation of mass

•
• no_slip =

properties derived assuming zero slip

•
• p =

primary flow

•
• rel =

properties in relative frame coordinate

•
• s =

•
• slip =

properties incorporating slip

•
• t =

stagnation properties, tip

•
• θ =

tangential

•
• 1 =

impeller inlet

•
• 2 =

Impeller exit

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