## Abstract

The environmental control system (ECS) is the largest auxiliary power consumer, i.e., around 75% of non-propulsive power, among the aircraft subsystems. The adoption of a novel ECS architecture, based on an electrically-driven vapor compression cycle system, can enable a twofold increase of coefficient of performance at cruise conditions, as compared to the conventional air cycle machine. The core of this technology is a high-speed, miniature centrifugal compressor, consisting of two impellers mounted in back-to-back configuration, and running on gas bearings, operating with refrigerant. The fluid dynamic design optimization of the twin-stage compressor, to be installed in the vapor compression cycle test rig under realization at Delft University of Technology, is documented in this paper. First, the scaling analysis for centrifugal compressor is extended to provide guidelines for the design of twin-stage machines. Then, a multi-objective conceptual design optimization is performed by resorting to an in-house reduced-order model (ROM), coupled to a genetic algorithm. The fluid dynamic performance and the structural integrity of the optimal design are assessed by means of a hybrid framework, encompassing computational fluid dynamics and ROMs, and by finite element analysis. The results show that it is possible to design a twin-stage compressor for the target application, featuring an average efficiency higher than 70%, a maximum compression ratio exceeding 9, and an operating range of 0.27 at the design rotational speed, despite the detrimental effects of motor cooling and miniature size.

## 1 Introduction

The aviation industry is under increasing pressure to improve aircraft fuel efficiency for economic, environmental and societal reasons. Fuel efficiency^{2} has roughly doubled since 1960, albeit at a diminishing rate [1]. The improvements can be mostly attributed to the replacement of traditional turbojets engines with modern high-bypass ratio turbofans. On the opposite, the contribution of aircraft subsystems to fuel efficiency enhancements has been negligible [2]. The environmental control system (ECS) is the largest consumer of non-propulsive power among the aircraft subsystems, accounting for up to 3–5% of the total energy consumption [3]. However, technological innovations in the ECS architectures have been limited since the 1960s, with few exceptions, e.g., the electrically-powered ECS mounted on the Boeing 787. Conventional pneumatic systems extract from the engines more power than is needed in most operating conditions, causing excess energy to be dumped overboard. The use of electrical power is more efficient than engine-generated pneumatic power, leading to a predicted improvement in fuel consumption in the range of 1–2% at cruise conditions [4]. Furthermore, the adoption of an electrically-powered ECS enables a reduction of maintenance costs and an overall increase of system reliability, due to the removal of the maintenance-intensive bleed system.

In addition to the advantages related to ECS electrification, a further reduction of fuel consumption can be arguably achieved by replacing the bleedless air cycle machine, i.e., an inverse Brayton cycle, with a more efficient electrically-driven vapor compression cycle (VCC), leading to a potential twofold increase of coefficient of performance. The technical feasibility and the performance of an electrically-powered VCC system for next-generation ECS will be investigated by means of a new experimental facility, currently under realization at Delft university of Technology. At the core of the inverse Rankine integrated system (IRIS) test rig, there is a high-speed twin-stage centrifugal compressor running on foil bearings, lubricated by the refrigerant vapor. The reduced size, the absence of oil lubricant in the circuit, and the predicted superior performance over traditional scroll compressors make this technology very suitable for airborne applications.

## 2 Objective

The feasibility of a miniature centrifugal compressor, running on gas bearings, for domestic heat pump applications, has been demonstrated by Schiffmann et al. [5,6]. The influence of size and working fluid on various design aspects concerning single-stage high-speed compressors has been systematically investigated in a recent work [7]. However, the design of a twin-stage compressor, featuring two compressor wheels mounted in back-to-back configuration on the same shaft, involves additional considerations regarding the split of the duty among the two stages, the balance of axial thrust produced by the two impellers, the cooling of the bearings, and that of the electric motor.

The objectives of this work are: (i) the extension of the scaling analysis for twin-stage compressors to support the designer in the choice of the non-dimensional parameters characterizing the conceptual design of the two stages; (ii) the multi-objective conceptual design optimization of the twin-stage compressor to be installed in the IRIS test rig; (iii) the performance characterization of the optimal compressor design by means of computational fluid dynamics (CFD), and the preliminary structural assessment by means of finite element analysis (FEA).

The paper is structured as follows. First, the methodology is described. Next, the three-dimensional (3D) design of the two stages is constructed from the results of the multi-objective conceptual design optimization. Then, the selected design is characterized by means of FEA and CFD. Finally, concluding remarks summarize the lessons learnt and give a perspective on future work.

## 3 Methodology

### 3.1 Scaling Analysis for Twin-Stage Compressors.

**collects the compressor stage characteristics, e.g., rotational speed, impeller tip radius, and performance metrics, e.g., efficiency, operating range. The vector $\sigma $ groups the non-dimensional geometrical parameters, as listed in Table 1. The stage velocity triangles are univocally defined by the choice of $\varphi t1,\psi $, and**

*y**α*

_{2}. The work input can be expressed as a function of the compression ratio

*β*. Moreover, the influence of the working fluid and of flow non-ideality can be assessed by computing the average value of the isentropic pressure–volume exponent [8,9]

where the subscripts in and out correspond to the initial and final states of the thermodynamic transformation. Finally, the impact of viscous effects is determined by evaluating the average value of the Reynolds number throughout the compressor stage.

Description | Definition |
---|---|

Impeller shape factor | $k=1\u2212(R1,hR1,s)2$ |

Number of blades | $Nbl,\u2009Nsplit$ |

Diffuser radius ratio | $R3/R2$ |

Diffuser blade height ratio | $Hr,pinch=H3\u2212H2H2(R2/Rpinch\u22121)$ |

Diffuser pinch radius ratio | $Rr,pinch=Rpinch\u2212R2R3\u2212R2$ |

Nondimensional length | $Lax/R2$ |

Leading edge thickness ratio | $tle(Nbl+Nsplit)2\pi R1$ |

Trailing edge thickness ratio | $tte(Nbl+Nsplit)2\pi R1$ |

Relative tip clearance | $\u03f5t/H2$ |

Relative back face clearance | $\u03f5b/H2$ |

Relative surface roughness | $Ra/Dhd,imp$ |

Shaft radius ratio | $Rshaft/R1,h$ |

Description | Definition |
---|---|

Impeller shape factor | $k=1\u2212(R1,hR1,s)2$ |

Number of blades | $Nbl,\u2009Nsplit$ |

Diffuser radius ratio | $R3/R2$ |

Diffuser blade height ratio | $Hr,pinch=H3\u2212H2H2(R2/Rpinch\u22121)$ |

Diffuser pinch radius ratio | $Rr,pinch=Rpinch\u2212R2R3\u2212R2$ |

Nondimensional length | $Lax/R2$ |

Leading edge thickness ratio | $tle(Nbl+Nsplit)2\pi R1$ |

Trailing edge thickness ratio | $tte(Nbl+Nsplit)2\pi R1$ |

Relative tip clearance | $\u03f5t/H2$ |

Relative back face clearance | $\u03f5b/H2$ |

Relative surface roughness | $Ra/Dhd,imp$ |

Shaft radius ratio | $Rshaft/R1,h$ |

The main geometrical features of the prescribed compressor configuration are displayed in Fig. 3.

Equation (10) is graphically displayed in Fig. 1, for a prescribed value of compression ratio and average isentropic exponent, resembling the design specifications of the IRIS compressor. By analyzing the trends, one can notice that the selection of an equal split of compression ratio among the two stages, i.e., $\kappa s=1$, combined with the choice of constant flow coefficient, i.e., $\varphi t1|s1/\varphi t1|s2=1$, leads to a strong unbalance in the work coefficients, with the second stage facing a higher non-dimensional duty. On the other hand, the higher is the selected ratio of work coefficients, the larger is the resulting flow coefficients ratio. The disparity in the flow coefficients of the two stages is attenuated by increasing the value of $\kappa s$, i.e., by increasing the compression ratio of the first stage.

For high-speed miniature centrifugal compressors, the choice of the splitting factor is primarily driven by manufacturing constraints. Given that the second stage inherently features lower volumetric flow rate, its efficiency and operating range are penalized by larger values of relative tip clearance, relative surface roughness, and blade blockage. A design approach to limit the impact of such detrimental effects consists in selecting $\kappa s\u22651$, and maximizing the efficiency of the first stage, i.e., the one responsible for most of the compression ratio. Figure 2 shows the contours of the total-to-total efficiency and operating range, i.e., $OR=(m\u02d9max\u2212m\u02d9min)/m\u02d9des$ at $\Omega des$, of a miniature compressor stage operating with R1233zd(E) [7]. The efficiency of the first stage can be maximized by selecting a design point lying on the locus of optimal $\varphi t1$, namely, the dash-dotted black line, located on the right. By selecting $\kappa s\u22651,\u2009\psi |s1/\psi |s2=1$, and resorting to Eq. (5), the flow coefficient of the second stage results to be located on the dash-dotted red line, located on the left, which also corresponds to the region of optimal operating range. The consequence thereof is that the efficiency of the entire compression process is enhanced, without penalizing the operating range of the twin-stage machine.

### 3.2 Conceptual Design Model.

A detailed description of the method used for the conceptual design of single-stage centrifugal compressors can be found in Ref. [7]. In this section, only the fundamental building blocks are reviewed, and emphasis is given to the additional procedures needed to cope with the design of twin-stage compressors, as depicted in Fig. 3.

The in-house reduced-order compressor model relies on the lumped parameters approach. However, the flow quantities are evaluated at five different span-wise locations at the inducer section, to capture the variation of the centrifugal potential over the blade span. The inlet velocity triangle is determined by the choice of the swallowing capacity, and by the minimization of the relative Mach number at the inducer shroud [10]. The first guess of the outlet velocity triangle is set by the selection of *α*_{2} and of the isentropic work coefficient. The actual work coefficient is then iteratively adjusted to match the target compression ratio, accounting for slip and losses, by means of semi-empirical correlations. The flow in the vaneless diffuser is modeled by integrating the system of two-dimensional differential equations derived by Stanitz [11].

The stage off-design performance is computed according to the following method. First, the choking point at the selected rotational speed is estimated by solving the isentropic mass, momentum, and energy balances between the inlet and throat sections, until the choking conditions are established at each span-wise station. Then, the mass flow rate is progressively reduced and the compressor performance is computed, until either rotating stall conditions are met [12], or the stage efficiency falls below a predefinied threshold, e.g., $\eta tt\u226450%$. This procedure is repeated over different speed-lines to compute the compressor operating map. In Fig. 4, the predictions of the compressor model are compared with the experimental data of three reference test cases available in literature [5,6,13–16], and with the CFD simulation results of the first stage of the IRIS compressor, whose design is documented in the following. The outcome of the validation study is that more than 95% of the experimental and CFD data fall within the $\xb15%$ uncertainty bands of the values predicted by the reduced-order model (ROM), except for the data of the EPFL compressor in the close proximity of the choking point, at 210 krpm.

The design of a twin-stage compressor involves additional steps. The two compressor wheels are mounted on the same shaft, thus the rotational speed of the second impeller is constrained by the design of the first stage. In other words, once fixed the design of the first impeller, the choice of the work coefficient of the second stage univocally determines also its flow coefficient, or viceversa. As a consequence, an ad hoc design strategy is needed to comply with both the requirements of high efficiency and wide operating range, as discussed in Sec. 3.1.

In addition to that, the use of gas bearings imposes stringent constraints on the maximum allowable axial thrust generated by the two compressor wheels. To cope with this limitation, it is necessary to adopt a back-to-back configuration and to carefully design the two impellers to balance their axial thrust. In the reduced-order compressor model, this is achieved by estimating the axial thrust of the two stages with the method proposed in Ref. [10], and by imposing a constraint on the maximum unbalance, computed over the entire operating range.

Finally, the high compactness requirement for the entire compressor assembly poses additional challenges with regards to bearings and electric motor cooling. In the proposed twin-stage configuration, a small portion of the refrigerant mass flow is bled at the outlet of the first stage, and is recirculated at the inlet of the machine, after being used for bearings cooling. On the other hand, the main portion of the pressurized refrigerant vapor leaving the first stage is used to cool the electric motor, before entering the second stage. In this way it is possible to minimize the size, thus the weight, of the compressor assembly, at the expenses of a fluid dynamic efficiency penalty, due to interstage flow heating. A summary of the design procedure implemented in the ROM is presented in Fig. 5.

### 3.3 Multi-Objective Optimization Framework.

The twin-stage compressor model has been integrated within a multi-objective optimization framework. The objective functions selected in this work are the operating range, i.e., $OR=(m\u02d9max\u2212m\u02d9min)/m\u02d9des$, and the compressor efficiency, both evaluated at the design rotational speed. To account for multiple operating points, the compressor efficiency is computed as the weighted average of the total-to-total efficiency, evaluated over the entire design speed-line. The weights used for this calculation are inversely proportional to the distance of each operating point from the selected design point.

In addition to that, a set of non-linear inequality constraints are imposed to ensure compressor manufacturability. To cope with the choice of the electric motor and the design of the gas bearings, additional inequality constraints are imposed on the rotational speed, the axial thrust, the power, and the torque produced by the two stages. The thresholds defining the maximum allowable axial thrust, power and torque are compared with the maximum values evaluated over the design speed-line. Overall, the optimization problem comprises 16 design variables, 2 objectives and 16 inequality constraints, as summarized in Table 2.

Variable | Type | Value |
---|---|---|

$\beta tt,target$ | Design specification | 7 |

$m\u02d9$ | Design specification | 0.114 kg/s |

Fluid | Design specification | R1233zd(E) |

$Pt1|s1$ | Design specification | 47.79 kPa |

$Tt1|s1$ | Design specification | 283.19 K |

$m\u02d9b$ | Design specification | 0.0044 kg/s |

$Q\u02d9b$ | Design specification | 360 W |

$\eta el$ | Design specification | 95% |

$\kappa s$ | Optimization variable | 0.5–2 |

$\varphi t1|s1\u2212s2$ | Optimization variable | 0.06–0.2 |

$\psi is|s1$ | Optimization variable | 0.6–1.0 |

$\alpha 2|s1\u2212s2$ | Optimization variable | 60–75 deg |

$k|s1\u2212s2$ | Optimization variable | 0.65–0.95 |

$Nbl|s1\u2212s2$ | Optimization variable | 12–20 |

$R3/R2|s1\u2212s2$ | Optimization variable | 1.3–2 |

$Rr,pinch|s1\u2212s2$ | Optimization variable | 0–1 |

$Hr,pinch|s1\u2212s2$ | Optimization variable | 0–1 |

$min\u2009(R1,h)|s1\u2212s2$ | Inequality constraint | 3.25 mm |

$min\u2009(a)|s1\u2212s2$ | Inequality constraint | 1 mm |

$min\u2009(H2)|s1\u2212s2$ | Inequality constraint | 1.35 mm |

$max\u2009(R4)|s1\u2212s2$ | Inequality constraint | 50 mm |

$max\u2009(M3)|s1\u2212s2$ | Inequality constraint | 0.6 |

$min\u2009(\Omega )\u2212max\u2009(\Omega )$ | Inequality constraint | 50–112 krpm |

$max\u2009(\Delta Fax)$ | Inequality constraint | 26 N |

$max\u2009(Wel)$ | Inequality constraint | 6.1 kW |

$max\u2009(T)$ | Inequality constraint | 0.52 Nm |

Variable | Type | Value |
---|---|---|

$\beta tt,target$ | Design specification | 7 |

$m\u02d9$ | Design specification | 0.114 kg/s |

Fluid | Design specification | R1233zd(E) |

$Pt1|s1$ | Design specification | 47.79 kPa |

$Tt1|s1$ | Design specification | 283.19 K |

$m\u02d9b$ | Design specification | 0.0044 kg/s |

$Q\u02d9b$ | Design specification | 360 W |

$\eta el$ | Design specification | 95% |

$\kappa s$ | Optimization variable | 0.5–2 |

$\varphi t1|s1\u2212s2$ | Optimization variable | 0.06–0.2 |

$\psi is|s1$ | Optimization variable | 0.6–1.0 |

$\alpha 2|s1\u2212s2$ | Optimization variable | 60–75 deg |

$k|s1\u2212s2$ | Optimization variable | 0.65–0.95 |

$Nbl|s1\u2212s2$ | Optimization variable | 12–20 |

$R3/R2|s1\u2212s2$ | Optimization variable | 1.3–2 |

$Rr,pinch|s1\u2212s2$ | Optimization variable | 0–1 |

$Hr,pinch|s1\u2212s2$ | Optimization variable | 0–1 |

$min\u2009(R1,h)|s1\u2212s2$ | Inequality constraint | 3.25 mm |

$min\u2009(a)|s1\u2212s2$ | Inequality constraint | 1 mm |

$min\u2009(H2)|s1\u2212s2$ | Inequality constraint | 1.35 mm |

$max\u2009(R4)|s1\u2212s2$ | Inequality constraint | 50 mm |

$max\u2009(M3)|s1\u2212s2$ | Inequality constraint | 0.6 |

$min\u2009(\Omega )\u2212max\u2009(\Omega )$ | Inequality constraint | 50–112 krpm |

$max\u2009(\Delta Fax)$ | Inequality constraint | 26 N |

$max\u2009(Wel)$ | Inequality constraint | 6.1 kW |

$max\u2009(T)$ | Inequality constraint | 0.52 Nm |

## 4 Case Study

The in-house compressor model, coupled to the multi-objective optimization framework, has been used to design the twin-stage compressor to be installed in the IRIS facility. The simplified process flow diagram of the test rig is shown in Fig. 6. It consists of a two pressure levels refrigeration cycle, resembling the configuration of an ECS for large helicopters, i.e., 19 passengers and 2 pilots. The condenser is an air-to-refrigerant microchannel heat exchanger, operated to reproduce the actual atmospheric conditions of the system, while the evaporator is designed as a water-cooled plate heat exchanger, in order to tightly control the system loading. At a later stage, a second water-cooled evaporator will be integrated in the test rig, to explore the possibility of separately cooling the avionics at a higher temperature.

### 4.1 Conceptual Design.

The compressor design specifications, as well as the values set for the constraints, and the bounds selected for the design variables, are listed in Table 2. The values of the additional geometrical parameters set by manufacturing constraints are reported in Table 3. The Pareto front of the optimal designs is computed by resorting to the NSGA-II algorithm described in Ref. [17] and implemented in the open-source library Pymoo [18]. All the optimization variables are floating point, except for the number of blades of the two impellers, which are treated as integers. The initial population comprises ten individuals for each design variable, and is sampled according to the latin hypercube methodology along the floating point directions, and randomly along the integer axes. The population is evolved for 120 generations, leading to a total of 19,200 function evaluations.

Variable | Value | Variable | Value |
---|---|---|---|

$\u03f5b$ | 0.15 mm | Ra | 3.2 μm |

$tbl,h$ | 0.6 mm | $tbl,s$ | 0.3 mm |

$\u03f5t,le$ | 0.2 mm | $\u03f5t,te$ | 0.15 mm |

Variable | Value | Variable | Value |
---|---|---|---|

$\u03f5b$ | 0.15 mm | Ra | 3.2 μm |

$tbl,h$ | 0.6 mm | $tbl,s$ | 0.3 mm |

$\u03f5t,le$ | 0.2 mm | $\u03f5t,te$ | 0.15 mm |

The optimal conceptual design has been chosen as a trade-off between compressor efficiency and operating range, by giving priority to the second objective, as required by the target application. The values of the design variables corresponding to the selected design are listed in Table 4. The velocity triangles computed at the outlet of the two optimized impellers are sketched in Fig. 7. With the purpose of showcasing the actual size of the prototype, the 3D printed mockup of the two optimized impellers is shown in Fig. 8.

Variable | Stage 1 | Stage 2 |
---|---|---|

$\beta tt,target$ | 2.95 | 2.37 |

$\varphi t1$ | 0.151 | 0.067 |

$\psi is\u2212\psi $ | 0.83–0.66 | 0.76–0.65 |

$\alpha 2$ | 64.2 deg | 67.4 deg |

k | 0.948 | 0.897 |

$Nbl\u2212Nsplit$ | 7–7 | 9–9 |

$R3/R2$ | 1.521 | 1.442 |

$Rpinch/R2$ | 1.464 | 1.383 |

$H3/H2$ | 0.717 | 0.748 |

Variable | Stage 1 | Stage 2 |
---|---|---|

$\beta tt,target$ | 2.95 | 2.37 |

$\varphi t1$ | 0.151 | 0.067 |

$\psi is\u2212\psi $ | 0.83–0.66 | 0.76–0.65 |

$\alpha 2$ | 64.2 deg | 67.4 deg |

k | 0.948 | 0.897 |

$Nbl\u2212Nsplit$ | 7–7 | 9–9 |

$R3/R2$ | 1.521 | 1.442 |

$Rpinch/R2$ | 1.464 | 1.383 |

$H3/H2$ | 0.717 | 0.748 |

By analyzing the results, it is possible to formulate the following considerations. The first impeller is characterized by a higher compression ratio than the second, while the work coefficient is nearly constant for the two stages. Moreover, the first impeller features a value swallowing capacity 2.25 times larger than the one of the second stage. These trends confirm the validity of the design guidelines derived from the extended scaling analysis for twin-stage compressors.

Furthermore, one can notice that the second impeller is characterized by a higher backsweep angle, as compared to the first one. The reason of that is inherently related to the small scale of the target application. On one hand, the minimum tip clearance gap is set by manufacturing limitations. On the other hand, the second impeller systematically features a lower blade height, as a result of the lower volumetric flow rate. In turn, the second impeller is always characterized by a higher relative clearance gap, leading to a significant efficiency drop. A viable strategy to limit this decay in efficiency is to decrease the meridional component of the velocity at impeller outlet, aiming to increase the outlet blade span. This is achieved by designing impeller blades featuring a higher backsweep angle, as depicted in Fig. 7.

### 4.2 Detailed Design.

The three-dimensional geometry of the two stages has been constructed from the results of the conceptual design optimization, using a commercial software [19]. The shape of the main and the splitter blades is controlled by specifying the hub, mid, and shroud profiles and by stacking them along the radial direction at leading edge. To enhance the structural integrity, the wrap angles have been chosen to ensure a rake angle of 18° and 23.5° for the first and second stage, respectively. The meridional location of the splitter blades leading edge has been selected to ensure that the geometrical throat is located in the main blade passage. The blades feature constant thickness distribution in the blade-to-blade plane, and linear tapering in the span-wise direction.

Despite of the described commonalities in the design of the two stages, the impellers exhibit major geometrical discrepancies, that can be attributed to the different regime of operation, namely, $Mw1,s|s1=1.17,\u2009Mw1,s|s2=0.77$. Given that the inducer of the first stage is transonic, the blades are designed to be aft-loaded at shroud, to improve the choke margin and reduce the shock losses [20]. Conversely, the blades of the second stage are shaped such to obtain a mid-loaded shroud and an aft-loaded hub, to mitigate the impact of secondary flows, while delaying the formation of the tip leakage vortex in presence of large clearance gap [21,22]. The vaneless diffuser is characterized by a linear pinch up to 146% and 138% of the outlet impeller radius for stage 1 and stage 2, respectively.

A static structural analysis has been carried out on the preliminary impeller-shaft assemblies to assess their mechanical integrity. Only the centrifugal load at the maximum rotational speed, i.e., 100 krpm, has been considered in the simulations. The properties of the selected material are listed in Table 5. As shown in Fig. 9, the maximum von-Mises stress is located at the leading edge of the splitter blades for the two impellers, whereas the blades trailing edge is relatively unloaded, due to the beneficial effect of the adopted rake angles. The maximum stresses computed for stage 1 and stage 2 are approximately 108 Mpa and 106 Mpa, leading to a safety factor of about 3.36 with the respect to the yielding stress of the selected material. The combined effect of the centrifugal and the aerodynamic loads will be investigated in the following design iteration, together with the characterization of the dynamic response of the system.

## 5 Computational Fluid Dynamics Analysis

### 5.1 Hybrid Computational Method.

The fluid dynamic performance of the compressor prototype has been characterized by resorting to a hybrid framework encompassing CFD and reduced-order models, as schematically described in Fig. 10. The flow field throughout the impeller and the diffuser of the two stages is computed with CFD. The fluid-dynamic performance of the remaining components along the main flow path, namely, the volute, the exit cone, and the return channel, is determined by means of lumped-parameter models [7]. The choice of this modeling approach is motivated as follows. First, the use of a single CFD model to compute the flow field throughout the entire machine would entail full annulus simulations, at least within the volutes, the exit cones, and the return channel. However, this would lead to a sizeable computational overhead, hindering the simulation of multiple compressor operating points. Moreover, the adoption of a single CFD model would require the modeling of the motor and the bearings cooling channels, whose detailed design has not been addressed yet. Finally, in the compressor configuration under study, the flow field at the inlet of the second stage is only weakly coupled to the one at the outlet of the first stage. Hence, the benefits deriving from the use of a single CFD model for the entire compressor are not deemed as sufficient to justify the considerable increase of computational cost.

Single passage RANS computations [23] with a frozen-rotor interface are used to assess the fluid dynamic performance of the impeller and the diffuser of the two stages. The boundary conditions are imposed in terms of flow direction, total pressure, and total temperature at the inlet, whereas the mass flow rate is assigned at the outlet. The flow direction is assumed to be normal at the inlet of the two stages. The $k\u2212\omega $ SST turbulence model is employed, together with adequate cell clustering near the walls to guarantee $y+\u22641$. Turbulence boundary conditions are set in terms of inlet turbulence intensity ($k=5%$), and eddy viscosity ratio ($\mu /\mu t=10$), while the turbulent Prandtl number is set to $Prt=1$, in accordance with what documented in Ref. [24]. The advective and turbulent fluxes are discretized with a total variation diminishing scheme [25]. A look-up table method is employed to speed-up the evaluation of the thermo-physical fluid properties. The property values are calculated using the multi-parameter equation of state model available in Ref. [26]. Figure 11 shows the tabulated region, together with the saturation curve, the spinodal line, and the target isentropic compression processes in the *P*–*T* thermodynamic plane. The vapor properties are extended up to the spinodal line to improve solver robustness in the initial phase of the calculation, without affecting the accuracy of the converged solution. After performing a sensitivity analysis on the first compressor stage, simulated at design point, see Fig. 12, a grid size of approximately 3.7 × 10^{6} cells, and a thermodynamic mesh of 1 × 10^{6} elements are set as optimal trade-off between accuracy and computational cost. The computational grid of the second stage has been scaled, accounting for the variation of the average Reynolds number, and of the number of blade passages, leading to a grid size of approximately 3.3 × 10^{6} nodes.

The mass flow averaged flow properties computed by CFD at the outlet of the first diffuser are passed to the ROMs of the volute, exit cone, and return channel, leading to a first estimate of $Pt1|s2$ and $Tt1|s2$, in absence of electric motor cooling. Then, the power required by the second impeller is estimated with the compresssor reduced-order model [7], and it is used to compute the overall power input of the electric motor, at the prescribed operating point. Next, the total thermodynamic state at the inlet of the second stage is recomputed, by accounting for electric motor cooling, determined by assuming a constant $\eta el=95%$. Finally, the flow conditions at the outlet of the second stage are evaluated by resorting to the same hybrid CFD-ROM model used to compute the flow within the first stage.

### 5.2 Results.

The performance of the compressor prototype has been characterized by means of a total of 54 CFD simulations, ranging from 90% to 110% of the design rotational speed, i.e., 89.7 krpm. The left and right boundaries of each speed-line are defined by the numerical prediction of the onset of unsteady flow phenomena, and by the establishment of choking conditions, respectively. To get a more accurate estimation of the inception of rotating stall, it is necessary to resort to higher order calculations, e.g., full annulus URANS. However, this is beyond the scope of the present work. The resulting stable operating points are displayed in Fig. 13, along with the corresponding $\eta tt$ predictions. The efficiency values are referred to the entire compressor, including the influence of motor cooling, which accounts for $\Delta \eta tt\u22484%$. One can notice that, despite the detrimental effects of motor cooling and small size, it is possible to achieve an average efficiency higher than 70%, with a maximum compression ratio exceeding 9, and an operating range of 0.27 at the design rotational speed. The flow field computed at the design point is displayed in Fig. 14.

## 6 Conclusions

The multi-objective design of the high-speed twin-stage compressor, to be installed in the IRIS test rig at Delft University of Technology, has been documented. The scaling analysis for single-stage centrifugal compressors has been extended to address the optimal design of twin-stage machines. The conceptual design has been performed by means of an in-house ROM, coupled to a multi-objective optimization framework. The fluid dynamic performance and the structural integrity of the optimal design have been assessed by means of a hybrid framework, coupling 3D RANS and ROMs, and by FEA. The outcomes of this study can be summarized as follows.

When dealing with the design of miniature twin-stage compressors, the efficiency and the operating range of the second stage are inherently penalized by lower values of volumetric flow rate, thus by larger values of relative tip clearance, relative surface roughness, and blade blockage. As a result, the optimal set of non-dimensional parameters is such that the first stage features a larger compression ratio, and a value of swallowing capacity in the proximity of the locus of optimal efficiency. In turn, by selecting the same value of work coefficient for the two stages, the flow coefficient of the second impeller lies in the region of optimal operating range. Consequently, the efficiency of the entire compression process is enhanced, without penalizing the operating range of the twin-stage machine.

The results of the multi-objective optimization corroborate the outcomes of the extended scaling analysis. This finding confirms both the suitability of the proposed methodology for the conceptual design optimization of twin-stage compressors, and the validity of the design guidelines derived from similarity principles.

It is possible to design a twin-stage compressor for the target application featuring an average efficiency higher than 70%, a maximum compression ratio exceeding 9, and an operating range of 0.27 at the design rotational speed, despite the detrimental effects of motor cooling and miniature size.

The presence of a large relative clearance gap leads not only to a reduction of $\eta tt$, but also to a decrease of $\beta tt$, that must be accounted for preliminary design, in order to achieve the target compression ratio. In the present work, an empirical correction of the compression ratio, expressed as a function of the impeller work and the relative clearance gap, has been adopted to match the predictions of the in-house compressor model with the CFD results.

## Funding Data

Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Award No. 17091; Funder ID: 10.13039/501100003246).

## Nomenclature

*E*=Young modulus

*D*=diameter

- $Fax$ =
axial thrust

*h*=specific enthalpy

*H*=blade height

*k*=impeller shape factor

- $Lax$ =
axial length

- $m\u02d9$ =
mass flow rate

- Ma =
Mach number

- $Nbl$ =
number of blades

- $Nsplit$ =
number of splitter blades

- OR =
operating range

*P*=pressure

- $Q\u02d9$ =
heat flow rate

*R*=radius

- Ra =
surface roughness

- Re =
Reynolds number

*t*=thickness

*T*=temperature–torque

- $Tm$ =
melting point temperature

*V*=absolute velocity

- $V\u02d9$ =
volumetric flow rate

*U*=peripheral speed

- $weul$ =
Euler work

*W*=power - relative velocity

=*x*design variables

*α*=absolute flow angle

*β*=compression ratio

- $\gamma Pv$ =
isentropic pressure–volume exponent

- $\u03f5b$ =
back face clearance

- $\u03f5t$ =
tip clearance gap

*η*=efficiency

- $\kappa s$ =
compression ratio splitting factor

*ν*=Poisson's ratio

*ρ*=density

- $\sigma $ =
stage dimensionless geometrical characteristics

- $\sigma uts$ =
ultimate tensile strength

- $\sigma y$ =
yielding stress

- $\varphi t1$ =
swallowing capacity

*ψ*=work coefficient

- Ω =
rotational speed

## Footnotes

Defined by the number of passengers times flight distance per unit fuel burn.