## Abstract

A novel turbocompressor test facility has been designed for helium–neon gas mixtures, and its specific features are presented. To account for the heat transfer originating from the motor coolant, a surrogate model has been derived. By combining these results with additional numerical ones, a similitude study is conducted by quantifying the individual effects and contributions on efficiency of the Reynolds number, tip Mach number, and specific heat ratio. Decoupling the effect of the different parameters shows that their respective contribution on efficiency variation is highly correlated with the Reynolds number actual value. The negative contribution of the tip Mach number and the positive effect of Reynolds number can be used to explain the efficiency variation with the increasing tip Mach number. Specific heat ratio variation leads to minor changes in polytropic efficiency except at low tip Mach numbers.

## Introduction

As part of a EU-funded H2020 project coordinated by CERN in Geneva, a novel turbocompressor test facility has been designed and recently commissioned at the University of Stuttgart. Within this research project, the focus has been set on the development of a more efficient cryogenic cycle for the future circular collider. From this objective, the idea of using a turbo-Brayton cycle operating with helium–neon mixtures emerged, as presented in the study by Podeur et al. [1]. Therefore, this experimental test rig was developed to better understand the challenges associated with the compression of light gases and is now used to validate the performance of a test case compressor. In addition, this facility provides an opportunity to conduct a similitude study on the effect of Reynolds number, tip Mach number, and specific heat ratio, which is the main focus of the research presented later.

Similitude theory is a powerful and elegant tool to quantify the effect on compressor performance resulting from either scaling, changes in surface roughness, in operating gas or in boundary and operating conditions. More precisely, it tells us that as long as geometrical similitude is maintained, the compressor performance should only vary with its inlet flow coefficient, Reynolds number, tip Mach number, and ratio of specific heat. In the literature, extensive research has been conducted to quantify the effect of Reynolds number such as in the studies by Casey [2], Strub et al. [3], Gülich [4], Kawakubo et al. [5], and Casey and Robinson [6]. The research focus on this specific parameter is commendable since, depending on the application, it can account for a major change in performance. Moreover, since industrial actors often rely on an impeller database used for a wide range of applications and therefore sizes and operating boundary conditions, an accurate model predicting the potential changes in performance is particularly valuable. However, as described in the literature, at high Reynolds number, its effect reaches an asymptote, and thus, no longer plays a role in performance variation. This means that at this stage and when focusing on the highest efficiency point of a speedline, most of the stage performance variation solely depends on a change in tip Mach number or specific heat ratio. The respective impacts on performance of both parameters decoupled from the Reynolds number are however rarely investigated in the literature. This article aims at addressing this gap by quantifying their contribution for a specific stage geometry.

This article is structured as follows: first, the test rig facility and operating procedure is introduced together with an uncertainty analysis on measured performance parameters. Experimental results are then presented and validated numerically for air compression. A surrogate model correcting the heat transfer effect caused by the motor coolant is then provided. Finally, results of a similitude study are presented to quantify the respective effect and contribution of the Reynolds number, tip Mach number, and the specific heat ratio.

## Test Rig Architecture and Operation

### Architecture.

Some of the components of the test rig facility pictured in Fig. 1 are often encountered in cryogenic application. This illustrates the uniqueness of this test rig designed to operate with several gases, some of which have very low molecular weights. A tightness to both helium and low vacuum along the entire test facility is a key to enable this feature. Moreover, the component specifications have to take into account the variety of fluid properties and operating conditions inherent to the diversity of these gases.

The test facility P&ID displayed in Fig. 2 shows the piping and process equipment as well as the instrumentation and control devices. The components shown in Fig. 1 are also referenced.

Two parts of this diagram can be distinguished. First, in the upper left part, buffer tanks are either pressurized with a pure gas or used to prepare a binary gas mixture. The second part, at the bottom of the P&ID, corresponds to the test loop itself and is designed to measure steady compressor stage performance. To that end, two total pressure and temperature probes adjustable in the radial position are located at a same plane, upstream of the compressor. Downstream of the volute, four static pressure taps average the outlet pressure and four resistance thermometers measure the outlet wall temperature. Therefore, total-to-static pressure ratio and efficiency are evaluated on this test loop.

Figure 3 details the sensors plane location with respect to the compressor. The latter is an ultra-high-speed turbocompressor driven by a high-power density motor. The impeller and shaft are supported by gas bearings directly fed by the operating fluid. This design enables to reach a maximum speed of 120,000 rpm with a rated aerodynamic power of 15 kW. In turn, a cryostat is required to continuously cool the motor and its power supply. A temperature sensor has been positioned on the compressor housing assessing the cooling effectiveness.

Downstream of the compressor, a heat exchanger offsets the enthalpy rise produced by the compression. It is followed by a Venturi flowmeter measuring the mass flowrate controlled with two parallel throttle valves. Finally, before the gas re-enters the compressor, a flow conditioner combined with a settling tank helps removing flow uniformities.

In addition, the choice of the components described earlier was critical to mitigate the inherent risk of helium leakages. Hence, fittings, flanges, and other components have been carefully selected so that leakage tests with pressurized helium show an overall leak rate below 10^{−5} mbar/L.s, and low vacuum can be maintained at about 1 mbar.

### Sensors, Calibration, and Uncertainty.

As mentioned earlier, the test rig is equipped with several sensors at different locations, including one differential and several absolute pressure transducers with adjustable range. Pressure taps of 1 mm allow for static pressure measurement, and pitot probes manufactured in-house are used for the total quantity. Regarding temperature sensors, 4-wires Pt1000 measure static temperature at several wall locations and 3 mm type K thermocouples are embedded into probes to acquire total temperature at the compressor inlet. All of these sensors have been calibrated in situ including the entire measurement chain. The Venturi flowmeter has been designed following the ISO 5167-4 norm. For its calibration, the diversity of gas properties required to split the volume flow operating range in two. Hence, two laminar flow elements were used, and the calibration process was conducted with the upstream and downstream piping, the test rig sensors, and its data acquisition system (DAS).

The uncertainty range of parameters of interest and their associated calibrated range are summarized in Table 1.

Parameter | Uncertainty FS | Calibrated range |
---|---|---|

P_{in} | ±0.03–0.05% | 0.8–1.2 bar |

P_{s,out} | ±0.02–0.04% | 0.8–1.6 bar |

T_{in} | ±0.06–0.08% | 30–90 °C |

T_{s,out} | ±0.03–0.05% | 30–90 °C |

x_{gas} | ±0.04% | 0–1 |

Ω | ±11 rpm | 0–120 krpm |

P | ±17 W | 0–15 kW |

Q | ±0.23% | 0.4–15 m^{3}/min |

Parameter | Uncertainty FS | Calibrated range |
---|---|---|

P_{in} | ±0.03–0.05% | 0.8–1.2 bar |

P_{s,out} | ±0.02–0.04% | 0.8–1.6 bar |

T_{in} | ±0.06–0.08% | 30–90 °C |

T_{s,out} | ±0.03–0.05% | 30–90 °C |

x_{gas} | ±0.04% | 0–1 |

Ω | ±11 rpm | 0–120 krpm |

P | ±17 W | 0–15 kW |

Q | ±0.23% | 0.4–15 m^{3}/min |

On top of the residual measurement uncertainties after calibration, additional sources of errors are taken into account in the overall uncertainty estimation provided in Table 1. First, pressure taps geometries such as size, positioning, and orientation can affect the accuracy of the reading value. In addition, when measuring total quantity, probes geometry together with their recovery factor and angle positioning can also contribute to potential errors. For this reason, additional uncertainties are added depending on the dynamic pressure or temperature. However, it can be noted that low inlet Mach number is inherent to the operation with light gases, and thus, the fluid kinematic energy has only a marginal contribution on total quantities. The piping located downstream of the compressor is thermodynamically isolated to reduce the effect of natural convection. Nevertheless, the remaining heat transfer could influence the outlet temperature measurement. Regarding outlet planes, the limited number of sensors and the choice of their positioning can only give an approximation of the flow field average thermodynamic properties. Similarly, for the inlet plane, the limited number of probes and their chosen position give an estimation of the average total pressure and temperature, as illustrated in the Results section by the inlet pressure distribution measurement. Each of these additional sources of error are then combined using the root-sum square method [7] and can vary depending on the operating point, which explains the range of uncertainties for pressure and temperature displayed in Table 1.

The contributions of these uncertainties on performance parameters can be evaluated at the compressor design point, as shown in Fig. 4.

### Control, Data Acquisition, and Hardware.

The test rig PC, where the human machine interface (HMI) is implemented, interfaces with both the program logic controller (PLC) and the DAS via Ethernet. The PLC is used to ensure a safe rig operation. Hence, all actuators are controlled and their feedback position monitored by the PLC. Specific pressure and temperature are also required by the latter to conduct check lists, keep the compressor operation away from aerodynamic instability regions, or validate the user request command. Moreover, PIDs and additional controllers have been implemented to maintain the desired inlet temperature and pressure as well as to adjust valves positioning.

In parallel, all sensors required for the evaluation of compressor performance and gas properties are connected to the DAS. A sampling rate of 20 Hz is set for all sensors except for the RTDs, for which a sampling rate of 5 Hz is required by the DAS to insure proper data acquisition. All sensors data, averaged values, and compressor performance parameters can be visualized on the HMI in real time.

A moving average over 500 ms is implemented together with the calculation of the relative standard deviation for all measured and calculated parameters. Both of these statistical parameters are monitored during operation to determine if the steady-state condition is reached. Should the user requests to save a specific operating point, a data set is first generated with measured values of 10 s before the request and 30 s after. The user then has the possibility to retain values corresponding to the minimum weighted standard deviation average of several parameters or to take the average over the entire sample.

The PLC is composed of several modules from the Siemens SINAMIC series including a CPU1214 programmed with Tia Portal 15.1. A DAS Expert key 200C from Delphin is used. The HMI visualized on the PC is implemented on a Windows 10 machine and programmed with labview 2016.

### Test Rig Operation and Measurement Campaign.

The following section is dedicated to the description of the test rig operation. Before any operation with a new gas, the entire test rig is first vacuumed to around 4 mbar. Then, in line with the rig architecture, two phases can be distinguished, namely, the gas preparation and the test loop operation.

For a pure gas, the gas preparation consists in filling the buffer volume up to a pressure enabling to ensure a lower pressure inside the test loop than in the buffer volume at all times and for any planned compressor operating point. This filling pressure is estimated depending on the buffer volume, the gas used, as well as the intended compressor boundary conditions and the maximum pressure ratio.

For a binary gas, a final pressure is estimated using the same criteria as for a pure gas. Then, depending on the desired mole fraction of each gas, an intermediary pressure is evaluated for the first gas to prepare the gas mixture by partial pressure. Once the buffer volume has been filled with the two gases, the resulting mixing ratio is measured with the binary gas analyzer tailored to the specific mixture. If the latter does not correspond to the intended mixing ratio, gases are iteratively added until the desired gas mixture is reached. Following this approach, same gas mixtures with ±5.10^{−4} difference in mole fraction can be obtained over various measuring campaigns. Properties of the gas are then obtained using REFPROP [8], and mixing laws are applied for gas mixtures. A minor correction is carried out on these properties to account for the remaining air inside the test facility.

Afterward, the test loop is filled with an estimated pressure based on the test loop volume, the compressor boundary conditions, and its performance at idle speed. The compressor is then started, and the on/off valves between the test loop and the buffer volume are controlled to automatically reach the desired compressor inlet pressure. This pressure adjustment is required for any change in operating point. The water throttle valve automatically adjusts its position to keep the compressor total inlet temperature at the prescribed value. Following this approach, total inlet pressure can be maintained within ±10 mbar, inlet temperature within ±0.05 °C, and mass flowrate within ±0.01 g/s.

For the purpose of the study presented in the Results section, stage performance is measured with air, helium, neon, and a helium–neon mixture at 0.5 mole fraction referred to as Nelium05. They are acquired at various tip Mach numbers and at the stage design boundary conditions of 1 bar inlet pressure and 298.15 K inlet temperature. Additional performances at inlet pressure of 1.44 bar and 0.77 bar for, respectively, neon and air are measured for a comparison at constant Reynolds number. Each measured operating point was acquired several times for repeatability purposes: when ramping up and down as well as when throttling toward surge and toward choke. Moreover, this procedure was repeated on three different days for each gas.

The surge detection during operation depends on the surge line estimated by CFD. The latter is defined at the point where the pressure rise coefficient starts decreasing with any further reduction of mass flow. A safety margin is defined associating for each rotational speed a mass flow below, which the PLC will take over the user control. As soon as this mass flow is reached, both throttle valves open until a safe operating point is reached and the user can re-take control. To mitigate the risk of compressor failure, no performance evaluation is made near the surge line.

## Compressor Design and Numerical Setup

A first radial compressor stage has been designed for a range of gas molecular weight, which until recently was not of interest in the turbomachinery application. This results in a comparatively low design tip Mach number. Table 2 provides additional details on its design and geometrical parameters.

Design parameters | |||
---|---|---|---|

M_{d} | 12.091 g/mol | ϕ_{t1,d} | 0.069 |

P_{1,d} | 1 bar | M_{U2,d} | 0.59 |

T_{1,d} | 298.15 K | Diffuser | Vaneless |

Geometrical parameters | |||

N | 9 + 9 | β_{1s} | −60.0 deg |

D_{2} | 67.7 mm | β_{1h} | −32.4 deg |

D_{1h} | 16.4 mm | β_{2} | −40.3 deg |

D_{1s} | 36.2 mm | h_{2} | 4.5 mm |

Reynolds number | |||

Re_{D2} | 1.7 × 10^{6} | Re_{in} | 6.9 × 10^{4} |

Design parameters | |||
---|---|---|---|

M_{d} | 12.091 g/mol | ϕ_{t1,d} | 0.069 |

P_{1,d} | 1 bar | M_{U2,d} | 0.59 |

T_{1,d} | 298.15 K | Diffuser | Vaneless |

Geometrical parameters | |||

N | 9 + 9 | β_{1s} | −60.0 deg |

D_{2} | 67.7 mm | β_{1h} | −32.4 deg |

D_{1h} | 16.4 mm | β_{2} | −40.3 deg |

D_{1s} | 36.2 mm | h_{2} | 4.5 mm |

Reynolds number | |||

Re_{D2} | 1.7 × 10^{6} | Re_{in} | 6.9 × 10^{4} |

The performance of this compressor stage has been evaluated numerically with various gases and boundary conditions including those available on the test rig. Steady-state calculations are conducted with ansys cfx 2020.1 for one sector and with a frozen rotor approach applied at the rotor–stator interface. No slip boundary conditions are applied at the adiabatic walls. The Reynolds-averaged Navier–Stokes equations are closed with the shear stress transport (SST) turbulence model. While other models have been tested, SST shows the best agreement with experimental results and was retained. Stage efficiency is monitored, and its relative standard deviation is used as a convergence criterion.

The computing domain follows the test rig architecture from the inlet to the outlet measuring plane, as previously shown in Fig. 3. A structured mesh is generated from the inlet plane to the diffuser outlet plane using autogrid 13.1 [9]. The volute domain meshed with ansys icem cfd leads to an unstructured mesh with a refinement near the wall and tongue.

Regarding boundary conditions, inlet total pressure and temperature are set at the compressor inlet plane according to the design parameters given in Table 2. Gases are modeled as ideal with properties provided by REFPROP [8]. Moreover, for all the tested gases, the compressibility factor ranges from 0.995 to 1.0005, validating the reasonable assumption of using an ideal gas approach. Finally, static pressure is set and varied at the outlet plane to evaluate six operating points per speed line.

## Results

### Inlet Pressure Profiles Validation.

Study results of the inlet pressure profile homogeneity are presented below. Three operating points corresponding to different flow velocities, pressure levels, and amount of upstream throttling have been selected and are shown in Fig. 5. In addition, total inlet pressures have been measured at several radial positions and plotted with respect to a reference total pressure acquired with a second probe at the pipe centerline.

For each operating point, experimental total pressure profiles are compared to a theoretical distribution of a fully developed turbulent flow. The good agreement with theoretical distributions confirms the positive effect of the upstream flow conditioner on the flow uniformity at the compressor inlet.

### Stage Performance Map.

The following section presents experimental performance results with air and their validation against numerical calculations. Starting with total-to-static pressure ratios, stage performance is compared at four different speed lines in Fig. 6. The associated tip Mach numbers are chosen to allow for a later performance comparison with other gases of various molecular weights. Even though small differences in performance near surge and choke regions exist, most operating points are in good agreement with numerical results. Based on the error trends, a reasonable assumption for their sources could be due to small differences between the real and CAD geometry.

Regarding the measurement of total-to-static efficiency displayed in Fig. 7, large discrepancies between experimental and numerical results are observed. However, one can notice that they diminish as tip Mach number and mass flowrate increases. This leads to a comparatively good agreement between experimental and numerical results at the highest tip Mach number.

The source of these discrepancies can be explained mainly by the presence of a coolant flowing through the motor housing as depicted in Fig. 8. In fact, while this coolant is required to maintain the high-power density motor at a reasonable temperature for operation at ultra-high speed, it also removes heat from the compressor housing. In turns, force convection between the operating gas and the volute walls leads to a reduction of the outlet temperature and an artificial increase in compressor efficiency.

Consequences of this heat transfer support the previous observations on efficiency discrepancies. In fact, reducing the mass of gas being cooled by almost a factor 2 on each speedline considerably helps to bring down its temperature. Moreover, as the rotational speed and power increases, the coolant temperature after having cooled the power supply and motor also increases. This contributes to a reduction of the heat transfer down to a point where its sign changes and leads to warming the operating gas. Moreover, at low tip Mach number, the enthalpy generation being comparatively low, small differences in temperature due to heat removal lead to major difference in efficiencies.

In the following section, an attempt is made to quantify the overall heat transfer with the final objective to correct the outlet temperature and retrieve the compressor efficiency.

### Heat Transfer Model.

As introduced earlier, a model has been developed to predict the heat removal from the operating gas. This heat transfer, which is responsible for the overestimation of the compressor efficiency, is affected by many factors including the heat conduction inside the compressor housing, the heat transfer area, the gas turbulence intensities, the heat convection around the motor or the coolant temperature, and volume flowrate. Moreover, these factors potentially vary from one operating point to the other. Thus, given the complexity of deriving a theoretical expression for the overall heat transfer, it was preferred to develop an empirical expression instead.

To do so, 288 experimental data points obtained at different operating conditions with different gases and cryostat temperatures have been combined. From this set, 12 points have been selected as a result of an optimization to serve as control points of a surrogate model. Figure 9 shows three of these control points in the case of air compression where the difference in temperature between the measured and the CFD prediction is also indicated. Moreover, the heat transfer effect is illustrated further by varying the cryostat temperature and comparing the measured and CFD-estimated stage efficiency.

The model with the highest COP was retained and includes four inputs. Two of them are often encountered in convective heat transfer application, namely, the Prandtl number, which compares heat transport to momentum transport, and the Reynolds number, which quantifies the turbulence intensities. The characteristic length for the Reynolds number is here chosen as the sum between the diffuser and volute length, taking into account the reduced length traveled by the fluid as the mass flowrate increases at the constant rotational speed. The two other inputs are the housing temperature, which gives an estimate of the heat transfer surface temperature and the inlet flow coefficient, which relates to the gas velocities and correlates strongly with the compressor work input. Figure 11 provides two quantities of interest for the different inputs of the obtained model. The first one is the Sobol index, which is obtained after a variance-based sensitivity analysis and thus shows how much of the output variance can be attributed to each input. The second parameter is the coefficient of importance (COI) [10]. It indicates for each input how much the model would lose in accuracy without it. The housing temperature is thus crucial to an accurate model and contribute to most of the output value. The Prandtl and Reynolds numbers, and in turn the Nusselt number, play a lesser role in the model accuracy but still contributes heavily on the heat transfer prediction.

Finally, to further validate the veracity of this model, an overall convection heat transfer coefficient is estimated for all operating points using the whole compressor housing area and the temperature. This leads to values up to 86 W/m^{2} K, which is in the typical range of such applications of forced convection with gases.

This model is then applied to all measured operating points to correct the outlet temperature and in turns the stage efficiency. Results of this correction are presented in Fig. 12 for the compression of air and illustrates the very good agreement with numerical results.

### Normalized Performance Parameter Evaluation.

As previously mentioned, static flow conditions are measured at the volute outlet. However, normalized performance coefficients required for a later study often rely on total quantities. Therefore, the procedure described below is implemented to retrieve total quantities from static ones at the volute outlet.

In Fig. 13, fac_{kin} quantifies the loss of kinematic energy in both the diffuser and volute domain. This factor enables to determine the ratio of density between the impeller outlet and the diffuser outlet plane. fac_{kin} is thus calibrated to a value of 0.39 using numerical results as depicted in Fig. 14. The model prediction at the design point for various values of fac_{kin} and the tip Mach number are presented and compared with CFD results.

The effect of this transformation and its accuracy when applied to numerical results with air at various rotational speeds are illustrated in the two graphs provided in Fig. 15.

In the following section, numerical results are combined with experimental ones obtained with light gases following the same procedure as with air. This data base is then used to analyze the effect of Reynolds number, specific heat ratio, and tip Mach number on the compressor stage performance.

### Similitude Study.

*h*

_{2}/

*D*

_{2}quantifies the geometrical similitude. The total inlet flow coefficient

*ϕ*

_{t1}and a chord-based Reynolds number Re

_{c}are used to determine the fluid dynamic similitude. Finally, thermodynamic similitude is defined using the tip Mach number

*M*

_{U2}and specific heat ratio

*γ*. Hereafter, a study on the effect of these different parameters is conducted combining experimental and numerical results. To do so, compressor performance is compared at their maximum efficiency point for the various gases listed in Table 3.

#### Reynolds Number Effect.

The model presented by Casey and Robinson [6] is first implemented to study the effect of the Reynolds number. It uses the analogy of fluid flow in pipes associating a friction coefficient and thus pressure loss for each characteristic flow Reynolds numbers. Here, the Reynolds number uses the impeller chord as the characteristic length and the relative flow velocity as the characteristic speed. In the compressor application, the model enables to associate a change in efficiency as well as an inlet flow coefficient and a work input coefficient for each change in Reynolds number and friction coefficient.

First, to illustrate the Reynolds number range of interest in this study, Fig. 16 shows the derived friction coefficient *f* for all operating points measured on the test rig as well as the ones obtained numerically for a same tip Mach number of 0.36. In fact, while the gas bearings were designed to allow for the compression of light gases as well as to ensure gas tightness, such architecture limits the range of allowable inlet pressure and gas molecular weight. For this reason, numerical performance obtained at operating points far from the gas bearing design point is used to extend the range of studied Reynolds number as well as to compare performance along iso-Reynolds number curves with the varying specific heat ratio. The friction coefficient is derived following Gülich [4], which provides laminar and turbulent friction factors for the skin friction of a flat plate.

Once the peak stage efficiency is plotted against friction coefficient at constant specific heat ratio as depicted in Fig. 17, consistent trends are observed. These trends are obtained by linear regression on the experimental points except for the case of CO_{2}, where only numerical results are available. Minor differences easily attributed to small differences in Reynolds number, interpolation error, or numerical and experimental uncertainties can be observed in this same figure. These curves, described in great details in the study by Casey and Robinson [6], allow to distinguish losses attributed to Reynolds number effect from Reynolds-independent ones. The *B*_{ref} coefficient quantifying the rate of change in efficiency can be derived from these curves. In this specific case, one can note that this coefficient remains constant and equal to 0.0697. Moreover, when comparing this value with the prediction of Casey and Robinson [6] by associating specific speed with *B*_{ref} coefficient, one finds that it falls within the predicted range for this specific compressor stage.

#### Tip Mach Number Effect.

The second parameter of interest in this study to characterize compressor performance is the tip Mach number. A similar analysis as for the Reynolds number effect is conducted using Nelium mixtures and different tip Mach numbers. Results are presented in Fig. 18 with tip Mach numbers ranging from 0.24 up to the design value (0.59) using solely experimental results.

It can be observed that varying the tip Mach number leads to a shift of the efficiency versus friction coefficient curves. Thus, as the tip Mach number increases, the *B*_{ref} coefficient remains constant but the A coefficient, which is independent from Reynolds losses, increases. The bottom graph on Fig. 18 shows the evolution of efficiency with increasing tip Mach numbers for the three gases. These trends are common in compressor map performance but can also be explained in terms of tip Mach number and Reynolds number effects. Representing the stage performance as shown earlier helps decoupling the contribution of the Reynolds and tip Mach numbers on efficiency when increasing the rotational speed. In fact, varying the rotational speed for a specific gas inherently affects the Reynolds number. Moreover, comparing the efficiency between two tip Mach numbers for a same gas and along a Reynolds number contour already indicates a major contribution of the Reynolds number in comparison to the tip Mach number.

Hence, based on these results for the different gases and changes in tip Mach number, Fig. 19 provides the contribution of both parameters on the efficiency gain. First, it can be observed that for all gases and increases in tip Mach number, the Reynolds number effect contributes positively to the efficiency gain, whereas the tip Mach number contributes negatively. Second, when comparing gases undergoing the same change in the tip Mach number, the difference in efficiency solely depends on the Reynolds number effect. However, for a given gas, a change in the tip Mach number leads to a reduction of the absolute contributions of both the Reynolds and tip Mach numbers. The difference between the two contributions always remains positive, but decreases as the tip Mach increases further.

Since it is well known that iso-efficiency curves on a compressor maps form islands around the design point, it is expected that the efficiency decreases at higher tip Mach numbers than the ones available for this study. Increasing the tip Mach number further would keep increasing the Reynolds number, but its contribution to the efficiency gain would remain marginal. Hence, the only explanation would be the inversion of the tip Mach number trend. Hence, this would lead to an increase of the negative tip Mach number contribution resulting in a negative difference of both contributions. As a consequence, efficiency would be lower.

#### Ratio of Specific Heat Effect.

To conclude this similitude study, a focus is put on certainly the least documented parameter and its effect, namely, the ratio of specific heat. In addition to illustrating the Reynolds number effect, the previously shown Fig. 17 also provides a hint on the effect of varying specific heat ratio. In fact, similar to the tip Mach number effect, shifts of these efficiency curves can also be observed corresponding to a variation of losses not imputable to Reynolds number effect and defined with coefficient A.

In the works of Roberts and Sjolander [11], authors derived a correlation between efficiency change and varying specific heat ratios. However, their prediction greatly overpredicts the preliminary observation of this study. As it will be discussed hereafter, a first explanation lies in the difference of efficiency definition used by the two studies, namely, isentropic versus polytropic efficiency. Furthermore, the proposed relation by Roberts and Sjolander [11] is derived for a compressor stage operating at significantly higher tip Mach numbers and thus higher flow velocities than the one understudy. In fact, when looking at numerical results presented by the same author, the impact on efficiency at lower tip Mach number seems to be indeed drastically lower than what the suggested relation predicts. Therefore, a complementary study has been conducted and is presented below for a reduced computing domain as the one previously described. For this purpose, the inlet is shortened and the outlet plane is moved at the diffuser exit plane. Inlet pressure are then varied for the case of air, neon and CO_{2} to insure a constant Reynolds number at different tip Mach numbers. Hence, with the intent of highlighting a possible coupling effect of tip Mach number, Reynolds number, and specific heat ratio, five different tip Mach numbers are investigated at three different Reynolds numbers. For each speed line, five operating points around the highest efficiency are calculated.

As previously introduced, the aerodynamic performance is assessed here using the polytropic efficiency instead of the isentropic efficiency. This choice enables to avoid the effect of varying pressure ratio on efficiency. Figure 20 illustrates this point by showing the variation in efficiency for the case of neon and CO_{2} in comparison to air using both the isentropic and polytropic efficiency definition at a constant Reynolds number and various tip Mach numbers.

Comparing compressor performance at different specific heat ratios and deriving correlation solely based on isentropic efficiency can be misleading. Efficiency change could be attributed to specific heat ratio effect even though it would mainly come from comparing performance at different pressure ratios. This would result in an overestimation of specific heat ratio effect and tip Mach number contribution. In fact, when focusing on the polytropic efficiency for this compressor stage, it can be seen that the specific heat ratio has only a minor influence on the efficiency for most of the tip Mach number range investigated. Going even further, Fig. 21 provides the polytropic efficiency variation of both neon and CO_{2} with respect to air for three chord-based Reynolds number ranging from 3.10^{5} to 7.10^{6}.

First, it can be observed that for each tip Mach number, the relative change in efficiency due to a change in specific heat ratio is independent from the Reynolds number, confirming the previous observation of Fig. 17. In fact, even though the absolute change in efficiency can vary by more than 5% for a same tip Mach number, the relative difference in efficiency to air does not vary by more than 0.1%. Then, for most of the tip Mach number range, i.e., *M*_{U2} > 0.4, varying the specific heat ratio only lead to minor changes in efficiency, i.e., between 0.1% and 0.25%. In addition, a surprising trend can be observed in the very low range of tip Mach numbers. In fact, for both neon and CO_{2}, the efficiency decreases as the tip Mach number is reduced leading to differences of up to 1.5%. More investigation on the flow field would be required to determine the underlying reasons. Nevertheless, this trend is particularly interesting for the case of very light gas application, since high gas speed of sound constrained the compressor operation in this low tip Mach number range.

These observations in combination with the ones previously made on the Reynolds number and tip Mach number enable to determine their respective contribution and more importantly the variation of these contributions. In fact, in the low Reynolds number range, i.e., small machines or low inlet pressure, most of the performance variation can be attributed to a change in the Reynolds number. Therefore, a model quantifying the latter would enable to capture most of the performance change when varying the rotational speed or the operating gas. However, for higher Reynolds number, the contribution of both a change in the specific heat ratio and the tip Mach number increases. Models predicting their respective effect would then become even more relevant. In particular, it would allow to quantify the efficiency decrease above the design tip Mach number as well as the specific heat ratio effect in the low tip Mach number range.

Finally, it should be emphasized that all results derived and observation made earlier have been obtained for a specific stage geometry with a certain design inlet flow coefficient and specific speed. The effect of the respective parameters understudy would certainly vary from one machine type to the other. Hence, even if extensive studies have been conducted on the Reynolds number effect allowing to derive a uniform correlation for all machine types, such conclusion cannot be made for the effect of tip Mach number and specific heat ratio yet.

## Conclusions

In this article, a novel closed-loop test facility was presented. The latter was developed to validate compressor performance for very light gases. Stage performance of a test case compressor designed for these low-molecular-weight gases and thus low tip Mach number was measured experimentally and estimated numerically for various gases and boundary conditions. The measured efficiency required a correction to mitigate the effect of heat transfer caused by the motor coolant.

By using these different results, a similitude study was conducted on the Reynolds number, tip Mach number, and specific heat ratio. By relating the Reynolds number effect with a relation of efficiency versus friction coefficient, it was possible to isolate and identify the effect of tip Mach number and specific heat ratio. For the stage of interest, results show that the respective effect and contribution of these dimensionless parameters depend heavily on the absolute value and change in the Reynolds number. Therefore, for low Reynolds number and large variation in its absolute value, the change in polytropic efficiency is strongly dominated by the Reynolds number effect. At higher Reynolds number or small changes in its value, the effect of tip Mach number and specific heat ratio dominate. For the tip Mach number effect, a negative contribution of the latter on the efficiency is observed, but its absolute value decreases with increasing rotational speed. It is assumed that it would then increase at higher tip Mach number than the design value to explain the efficiency decrease at the tip Mach number above the design value. Regarding the specific heat ratio, a complementary study with an extended range of tip Mach number has shown that the polytropic efficiency vary in a small manner when changing from air to CO_{2} or neon. This observation is valid for most of the tip Mach number range understudy but not at the lowest tip Mach number where changing gas could lead to an efficiency decrease of up to $1.5%$.

To pursue this research, one could also look at the specific heat ratio effect on other performance parameters such as pressure rise coefficient, inlet flow coefficient, as well as on the surge or choke margin. Finally, as highlighted in the previous section, a more comprehensive study would be required on multiple machine types to derive a unified relation and quantify how the already observed trends would vary with machine specific speed or design inlet flow coefficient. Both of these points are currently being investigated.

## Acknowledgment

The author would also like to thank the ITSM for providing the computing tools and the Marie Skodolwska-Curie Action (MSCA) for its financial support, which made this work possible. EASITrain - European Advanced Superconductivity Innovation and Training. This Marie Sklodowska-Curie Action (MSCA) Innovative Training Networks (ITN) has received funding from the European Union’s H2020 Framework Programme under Grant Agreement no. 764879.

## 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. The authors attest that all data for this study are included in the paper.

## Nomenclature

*f*=friction coefficient (–)

*x*=mole fraction (–)

*C*=gas absolute velocity (m/s)

*M*=molecular weight (g/mol)

*N*=number of blades (–)

*P*=pressure (bar), power (W)

*Q*=volume flowrate (m

^{3}/s), heat transfer (W)*T*=temperature (K)

- $m\u02d9$ =
mass flowrate (kg/s)

*h*_{2}=blade exit width (m)

*Cp*=specific heat capacity (J/kgK)

*D*_{2}=impeller tip diameter (m)

*D*_{1s}=impeller inlet shroud diameter (m)

*D*_{1h}=impeller inlet hub diameter (m)

- $MU2$ =
impeller tip Mach number $(U2/\gamma RTt1)$ (–)

*U*_{2}=impeller tip speed (m/s)

*A*,*B*_{ref}=coefficient in Reynolds effect model (–)

- Pr =
Prandtl number (–)

- Re =
Reynolds number (–)