# Generalization of a transient mean streamline rotodynamic pump model to radial compressors

Valentin Cottarel, Laura Matteo, Gédéon Mauger

► To cite this version:

Valentin Cottarel, Laura Matteo, Gédéon Mauger. Generalization of a transient mean streamline rotodynamic pump model to radial compressors. ETC15 - 15th European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, Apr 2023, Budapest, Hungary. 10.29008/ETC2023-199. hal-04590224

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Submitted on 28 May 2024

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Generalization of a transient mean streamline rotodynamic pump model to radial compressors

Valentin Cottarel, Laura Matteo, Gédéon Mauger

► To cite this version:

Valentin Cottarel, Laura Matteo, Gédéon Mauger. Generalization of a transient mean streamline rotodynamic pump model to radial compressors. ETC15 - 15th European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, Apr 2023, Budapest, Hungary. 10.29008/ETC2023-199. hal-04590224

HAL Id: hal-04590224

https://hal.science/hal-04590224v1

Submitted on 28 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>
# GENERALIZATION OF A TRANSIENT MEAN STREAMLINE ROTODYNAMIC PUMP MODEL TO RADIAL COMPRESSORS

V. Cottarel¹ - L. Matteo² - G. Mauger¹

¹ Paris-Saclay University, CEA, Thermalhydraulics and Fluid Mechanics Section, DES/ISAS/DM2S, F-91191 Gif-sur-Yvette, France, valentin.cottarel@cea.fr, gedeon.mauger@cea.fr

² CEA, Innovative Systems Studies Section, DES/IRESNE/DER, F-13108 Saint-Paul-Lez-Durance, France, laura.matteo@cea.fr

## ABSTRACT

The French Alternative Energies and Atomic Energy Commission (CEA) works on Brayton cycles modeling and design for energy conversion systems of gas cooled reactors. To provide a solution for the pre-design of turbomachinery and the simulation of transient compressible flows, the thermal-hydraulic simulation section has worked on a one-dimensional rotodynamic pump model. This is a predictive transient two-phase flow model which considers a mean streamline adopted by the fluid on each element of the machine (fixed and rotating parts). Originally dedicated to pumps, the method is applied here to the prediction of the performance of the Low-Speed Centrifugal Compressor from NASA. Direct application of the 1-D pump model on the compressor leads to a relative error to the experimental data of 2,2% on the pressure ratio and 7,0% on the isentropic efficiency at the nominal point, with correct overall trends. A development of an improved mean flow path is carried out, providing a base to qualify the model on other compressors.

## KEYWORDS

COMPRESSOR, RADIAL, LSCC, CATHARE, TRANSIENT, STEAMLINE

## NOMENCLATURE

<table><tr><td>b</td><td>impeller width, m</td></tr><tr><td>h</td><td>specific enthalpy, J/kg</td></tr><tr><td>I</td><td>mesh numer</td></tr><tr><td>L</td><td>Mean flow path length, m</td></tr><tr><td>ṁ</td><td>mass flow rate, kg/s</td></tr><tr><td>N</td><td>rotational speed, rpm</td></tr><tr><td>N<sub>s</sub></td><td>turbomachine specific speed</td></tr><tr><td>P</td><td>pressure, Pa</td></tr></table>

### Greek symbols

<table><tr><td>α</td><td>angle between <b>u</b> and <b>v</b>, rad</td></tr><tr><td>β</td><td>angle between <b>-u</b> and <b>-v</b>, rad</td></tr><tr><td>η</td><td>isentropic efficiency</td></tr><tr><td>γ</td><td>heat capacity ratio</td></tr><tr><td>ω</td><td>rotational speed, rad/s</td></tr><tr><td>Π</td><td>pressure ratio</td></tr><tr><td>σ</td><td>slip factor</td></tr><tr><td>θ</td><td>angle in polar coordinates, rad</td></tr></table>

### Latin letters

<table><tr><td>Q</td><td>flow rate, m³/s</td></tr><tr><td>R</td><td>radial position, m</td></tr><tr><td>S</td><td>flow cross-section, m²</td></tr><tr><td>V</td><td>absolute velocity, m/s</td></tr><tr><td>W</td><td>relative velocity, m/s</td></tr><tr><td>z</td><td>coordinate of the pipe axis, m</td></tr><tr><td>Δz</td><td>mesh size, m</td></tr><tr><td>Z</td><td>impeller blades number</td></tr></table>

### Subscripts

<table><tr><td>1</td><td>impeller inlet</td></tr><tr><td>2</td><td>impeller outlet</td></tr><tr><td>imp</td><td>impeller</td></tr><tr><td>in</td><td>inlet</td></tr><tr><td>is</td><td>isentropic</td></tr><tr><td>m</td><td>meridian</td></tr><tr><td>N</td><td>nominal operating point</td></tr><tr><td>out</td><td>outlet</td></tr></table>
# GENERALIZATION OF A TRANSIENT MEAN STREAMLINE ROTODYNAMIC PUMP MODEL TO RADIAL COMPRESSORS

V. Cottarel¹ - L. Matteo² - G. Mauger¹

¹ Paris-Saclay University, CEA, Thermalhydraulics and Fluid Mechanics Section, DES/ISAS/DM2S, F-91191 Gif-sur-Yvette, France, valentin.cottarel@cea.fr, gedeon.mauger@cea.fr

² CEA, Innovative Systems Studies Section, DES/IRESNE/DER, F-13108 Saint-Paul-Lez-Durance, France, laura.matteo@cea.fr

## ABSTRACT

The French Alternative Energies and Atomic Energy Commission (CEA) works on Brayton cycles modeling and design for energy conversion systems of gas cooled reactors. To provide a solution for the pre-design of turbomachinery and the simulation of transient compressible flows, the thermal-hydraulic simulation section has worked on a one-dimensional rotodynamic pump model. This is a predictive transient two-phase flow model which considers a mean streamline adopted by the fluid on each element of the machine (fixed and rotating parts). Originally dedicated to pumps, the method is applied here to the prediction of the performance of the Low-Speed Centrifugal Compressor from NASA. Direct application of the 1-D pump model on the compressor leads to a relative error to the experimental data of 2,2% on the pressure ratio and 7,0% on the isentropic efficiency at the nominal point, with correct overall trends. A development of an improved mean flow path is carried out, providing a base to qualify the model on other compressors.

## KEYWORDS

COMPRESSOR, RADIAL, LSCC, CATHARE, TRANSIENT, STEAMLINE

## NOMENCLATURE

* b impeller width, m

* h specific enthalpy, J/kg

* I mesh numer

* L Mean flow path length, m

* $\dot{m}$ mass flow rate, kg/s

* N rotational speed, rpm

* $N_s$ turbomachine specific speed

* P pressure, Pa

### Greek symbols

* α angle between $\vec{u}$ and $\vec{v}$, rad

* β angle between $-\vec{u}$ and $\vec{w}$, rad

* η isentropic efficiency

* γ heat capacity ratio

* ω rotational speed, rad/s

* Π pressure ratio

* σ slip factor

* θ angle in polar coordinates, rad

### Latin letters

* Q flow rate, m³/s

* R radial position, m

* S flow cross-section, m²

* V absolute velocity, m/s

* W relative velocity, m/s

* z coordinate of the pipe axis, m

* $\Delta z$ mesh size, m

* Z impeller blades number

### Subscripts

* 1 impeller inlet

* 2 impeller outlet

* imp impeller

* in inlet

* is isentropic

* m meridian

* N nominal operating point

* out outlet
# INTRODUCTION

Turbomachines are energy conversion systems with multiple applications: electricity generation, fluid transport, compression or expansion in thermodynamic cycles. These machines are used in many fields of science and industry. In the nuclear field, these machines are present in all types of reactors, as well as in Pressurized Water Reactor (PWR), Sodium Fast Reactors (SFR), or Gas-cooled Fast Reactor (GFR). They can be pumps, compressors or turbines and play an essential role in the operation and safety of the reactor. In GFR, compressors are key components because they are used in the primary circuit for circulation, safety and electricity production (Bertrand et al., 2012). In some SFR concepts with Brayton cycles as power conversion systems, compressors determine the global cycle efficiency and their transient modeling is therefore essential (see Refs. Bertrand et al., 2016; Mauger et al., 2019). Finally, more and more safety or power generation concepts involve supercritical CO2 cycles. In these cycles, the performance of the compressor is very sensitive to the thermal-hydraulic conditions of the CO2 which is close to the critical point : it is interesting to have more refined models to calculate the transient situations (Hofer et al., 2020). For all these concepts and applications, knowing pumps and compressors performance maps are then critical for normal and accidental operations. Being able to predict the performances of these machines and having an internal representation of it is an asset for the design and study phases of the behavior of these systems in transients.

To predict the performances of a compressor, the historical method is to apply Euler theory on the geometry of the machine. This 0-D input/output model provides the first quadrant characteristics in ideal conditions. In order to obtain the performances in real conditions, various loss models throughout the machine have been developed to refine the predictions (some set of loss models can be found in Oh et al., 1997; Gutiérrez Velásquez E. I., 2017). This method gives stationary first-quadrant off-design characteristics that depends strongly on the chosen correlations. Most modern approaches that predict performances maps usually use numerical computations. Several approaches exist depending on the strategy adopted. 2-D through-flow methods consider the flow to be steady in both the absolute and relative frames of reference and to be axisymmetric (Dixon and Hall, 2014). Stream functions methods, finite difference solutions or time-marching solutions of the stream line curvature equation could be use to solve the problem. This method has been developed since the 80s (Dring and Joslyn, 1985; Dring and Oates, 1989) and are still commonly used for designing and analyzing radial (Casey and Robinson, 2010) or axial (Petrovic et al., 2009; Benichou et al., 2019) turbomachines. A more detailed approach is to use 3-D CFD calculations, like the RANS (Birari et al., 2006; Le Sausse et al., 2013; Cornelius et al., 2013) or LES (Medic et al., 2016; Joo et al., 2014) methods. All these CFD methods have common points: they require detailed geometry of the whole compressor which mean it is already built or in a very advanced design state. The computational time to obtain a first quadrant performance maps is very long (RANS is 40 times longer compared to 2-D through-flow according to Benichou et al., 2019). Reduced order modeling could be envisaged to reduced this computational time (Lucia et al., 2004). Artificial neural network method (Ghorbanian and Gholamrezaei, 2009) to predict the performances could also be found in the literature. Nonetheless, most of the computations mentioned are steady state ones. Performance predictions on abnormal operation point (for example reverse pump mode, see Derakhshan and Nourbakhsh, 2008; Shahin et al., 2013) or under specific conditions (supercritical fluid, Saravi and Tassou, 2019) are recent and very
# INTRODUCTION

Turbomachines are energy conversion systems with multiple applications: electricity generation, fluid transport, compression or expansion in thermodynamic cycles. These machines are used in many fields of science and industry. In the nuclear field, these machines are present in all types of reactors, as well as in Pressurized Water Reactor (PWR), Sodium Fast Reactors (SFR), or Gas-cooled Fast Reactor (GFR). They can be pumps, compressors or turbines and play an essential role in the operation and safety of the reactor. In GFR, compressors are key components because they are used in the primary circuit for circulation, safety and electricity production (Bertrand et al., 2012). In some SFR concepts with Brayton cycles as power conversion systems, compressors determine the global cycle efficiency and their transient modeling is therefore essential (see Refs. Bertrand et al., 2016; Mauger et al., 2019). Finally, more and more safety or power generation concepts involve supercritical CO2 cycles. In these cycles, the performance of the compressor is very sensitive to the thermal-hydraulic conditions of the CO2 which is close to the critical point : it is interesting to have more refined models to calculate the transient situations (Hofer et al., 2020). For all these concepts and applications, knowing pumps and compressors performance maps are then critical for normal and accidental operations. Being able to predict the performances of these machines and having an internal representation of it is an asset for the design and study phases of the behavior of these systems in transients.

To predict the performances of a compressor, the historical method is to apply Euler theory on the geometry of the machine. This 0-D input/output model provides the first quadrant characteristics in ideal conditions. In order to obtain the performances in real conditions, various loss models throughout the machine have been developed to refine the predictions (some set of loss models can be found in Oh et al., 1997; Gutiérrez Velásquez E. I., 2017). This method gives stationary first-quadrant off-design characteristics that depends strongly on the chosen correlations. Most modern approaches that predict performances maps usually use numerical computations. Several approaches exist depending on the strategy adopted. 2-D through-flow methods consider the flow to be steady in both the absolute and relative frames of reference and to be axisymmetric (Dixon and Hall, 2014). Stream functions methods, finite difference solutions or time-marching solutions of the stream line curvature equation could be use to solve the problem. This method has been developed since the 80s (Dring and Joslyn, 1985; Dring and Oates, 1989) and are still commonly used for designing and analyzing radial (Casey and Robinson, 2010) or axial (Petrovic et al., 2009; Benichou et al., 2019) turbomachines. A more detailed approach is to use 3-D CFD calculations, like the RANS (Birari et al., 2006; Le Sausse et al., 2013; Cornelius et al., 2013) or LES (Medic et al., 2016; Joo et al., 2014) methods. All these CFD methods have common points: they require detailed geometry of the whole compressor which mean it is already built or in a very advanced design state. The computational time to obtain a first quadrant performance maps is very long (RANS is 40 times longer compared to 2-D through-flow according to Benichou et al., 2019). Reduced order modeling could be envisaged to reduced this computational time (Lucia et al., 2004). Artificial neural network method (Ghorbanian and Gholamrezaei, 2009) to predict the performances could also be found in the literature. Nonetheless, most of the computations mentioned are steady state ones. Performance predictions on abnormal operation point (for example reverse pump mode, see Derakhshan and Nourbakhsh, 2008; Shahin et al., 2013) or under specific conditions (supercritical fluid, Saravi and Tassou, 2019) are recent and very
specific.

To improve thermal-hydraulic safety and design studies, the thermal-hydraulic simulation section of the French Alternative Energies and Atomic Energy Commission (CEA) has worked on a predictive 1-D pump model in the CATHARE code (see Refs. Matteo et al., 2018, 2019a,b,c,d; Matteo, 2019). A 1-D streamline approach in order to model a turbomachine is rare in the literature, as it lies between the classical input/output 0-D methods and the 2-D through flow and 3-D CFD ones. This 1-D rotodynamic pump model is included in the two-phase flow six-equation model of the CATHARE code. It is generic, predictive and needs a reasonable quantity of geometrical input data. This led to multiple benefits: design studies can be initiated as it is not necessary to have a complete geometry, transient calculations in a wide range of conditions can be easily performed, whole circuit computations including the turbomachine can be achieved with a reasonable uncertainty on predictions, the CPU time is low and an internal representation of the machine is available. This article describes the first application of the model to a compressor.

## The CATHARE Code

The Code for Analysis of Thermal-Hydraulics during an Accident of Reactor and safety Evaluation (CATHARE) is a French thermal-hydraulic modular system code owned and developed since 1979 by CEA, FRAMATOME, Électricité de France (EDF) and the French Radio-protection and Nuclear Safety Institute (IRSN), see Refs. Barre and Bernard, 1990; Emonot et al., 2011; Préa et al., 2020. It is a widely validated code used for nuclear power plant design and safety studies in normal or accidental operation (International Atomic Energy Agency, 2002). Circuit modeling in the CATHARE code is done by assembling hydraulic elements. These elements can be represented and meshed with three modules: volume (0-D), axial (1-D) and threeD (3-D). Several thermal and technological sub-modules can be added such as heat exchangers, valves, fluid source or sink, pumps, compressors and turbines. The code is able to simulate different flow regimes (one-phase or two-phase flow) and fluids. For this, a two-phase flow six-equation model (mass, energy and momentum solved for each phase) closed by physical closure laws is implemented and solved using a Newton-Raphson iterative method (Bestion, 1990). This model allows us to calculate mechanical and thermal disequilibrium between phases (Faydide and Rousseau, 1980). CATHARE is a robust and fast transient code (calculation time is always lower than the simulated duration for the needs of the numerical simulators, Miettinen, 2008).

## The CATHARE 0-D and 1-D Pump Models

The turbomachinery module used industrially in the CATHARE code is a 0-D model (which means nodal, ie. no geometry) which consist in providing the stationary non-dimensional performance curves as input data (see Mauger et al., 2015, 2019). From this data, the model assigns locally the manometric head and torque contributions to the momentum and energy balance equations of the fluid. During a transient, the similarity laws of turbomachinery are used to recalculate the turbomachine performance from the rotational speed and the instantaneous flow rate (application of Buckingham's $\pi$ theorem allows the reduction of the representative variables, see Lakshminarayana, 1995). Although efficient, several limitations can be noticed. To perform computations, the simulated pumps and compressors must be built because the performance maps of the machines must be filled as input data in the 0-D
specific.

To improve thermal-hydraulic safety and design studies, the thermal-hydraulic simulation section of the French Alternative Energies and Atomic Energy Commission (CEA) has worked on a predictive 1-D pump model in the CATHARE code (see Refs. Matteo et al., 2018, 2019a,b,c,d; Matteo, 2019). A 1-D streamline approach in order to model a turbomachine is rare in the literature, as it lies between the classical input/output 0-D methods and the 2-D through flow and 3-D CFD ones. This 1-D rotodynamic pump model is included in the two-phase flow six-equation model of the CATHARE code. It is generic, predictive and needs a reasonable quantity of geometrical input data. This led to multiple benefits: design studies can be initiated as it is not necessary to have a complete geometry, transient calculations in a wide range of conditions can be easily performed, whole circuit computations including the turbomachine can be achieved with a reasonable uncertainty on predictions, the CPU time is low and an internal representation of the machine is available. This article describes the first application of the model to a compressor.

## The CATHARE Code

The Code for Analysis of Thermal-Hydraulics during an Accident of Reactor and safety Evaluation (CATHARE) is a French thermal-hydraulic modular system code owned and developed since 1979 by CEA, FRAMATOME, Électricité de France (EDF) and the French Radio-protection and Nuclear Safety Institute (IRSN), see Refs. Barre and Bernard, 1990; Emonot et al., 2011; Préa et al., 2020. It is a widely validated code used for nuclear power plant design and safety studies in normal or accidental operation (International Atomic Energy Agency, 2002). Circuit modeling in the CATHARE code is done by assembling hydraulic elements. These elements can be represented and meshed with three modules: volume (0-D), axial (1-D) and threeD (3-D). Several thermal and technological sub-modules can be added such as heat exchangers, valves, fluid source or sink, pumps, compressors and turbines. The code is able to simulate different flow regimes (one-phase or two-phase flow) and fluids. For this, a two-phase flow six-equation model (mass, energy and momentum solved for each phase) closed by physical closure laws is implemented and solved using a Newton-Raphson iterative method (Bestion, 1990). This model allows us to calculate mechanical and thermal disequilibrium between phases (Faydide and Rousseau, 1980). CATHARE is a robust and fast transient code (calculation time is always lower than the simulated duration for the needs of the numerical simulators, Miettinen, 2008).

## The CATHARE 0-D and 1-D Pump Models

The turbomachinery module used industrially in the CATHARE code is a 0-D model (which means nodal, ie. no geometry) which consist in providing the stationary non-dimensional performance curves as input data (see Mauger et al., 2015, 2019). From this data, the model assigns locally the manometric head and torque contributions to the momentum and energy balance equations of the fluid. During a transient, the similarity laws of turbomachinery are used to recalculate the turbomachine performance from the rotational speed and the instantaneous flow rate (application of Buckingham's $\pi$ theorem allows the reduction of the representative variables, see Lakshminarayana, 1995). Although efficient, several limitations can be noticed. To perform computations, the simulated pumps and compressors must be built because the performance maps of the machines must be filled as input data in the 0-D
model and therefore measured on a prototype. The two-phase flow characteristics can only be
obtained in some specific well-studied cases. This non-predictive 0-D model is not adapted to
iterations in case of pre-design studies. 3-D CFD calculations can be considered to get around
the problem but computational time is not suitable for pre-design studies (several months to
obtain a single-phase performance map on a standard desktop computer). For this purpose,
CEA developed a predictive 1-D pump model (Matteo, 2019).

The CATHARE 1-D pump model is a predictive transient two-phase flow calculation. It considers a mean streamline adopted by the fluid on each element of the machine : suction, impeller, diffuser and volute. All machines do not necessarily have all these parts, or may conversely have several rotating stages : the structure of the model is intended to be generic and this is made possible thanks to the possibility of assembling elements in the code. The diffuser can be removed and the model can easily evolve towards multi-stage machines. Pump components are meshed perpendicularly to the mean streamline using the 1-D module (see Fig. 1) with typical mesh sizes of the order of magnitude of a centimeter. The equations governing the fluid are solved in each mesh. A change of coordinate system is made in the impeller (change from fixed to rotating coordinate system).

<figure><img src="image_2.png" /><figcaption>Figure 1: Example of the DERAP machine (Matteo et al., 2018) modeled with the 1-D pump model in CATHARE code</figcaption></figure>

The 1-D pump model includes in particular the following modifications on the CATHARE
equations governing the fluid (see Refs. Matteo et al., 2018, 2019b for more details):

1. A frame switch is made at the inlet of the rotating part (the impeller) from the fixed frame to the rotating frame considering relative speed. It requires the introduction of a centrifugal acceleration term to the momentum equations and associated contributions to the energy equations. Flow sections are adapted to ensure mass flow rate conservation with a 2-component velocity. Another frame switch is applied at the outlet of the impeller, when the fluid is entering the downstream fixed part (diffuser or volute);

2. A deviation model is implemented to adjust $\beta_2$ angle at impeller outlet;

3. Two losses source terms are added to the momentum equation: desadaptation head loss and diffusion head loss. They represent the different losses phenomena that take place in the machine due to geometry in normal and off-design conditions;

4. A power dissipation source term is added in the energy equation representing the losses due to recirculations at low flow rates.

The CATHARE code is able to handle various fluids such as water, air, sodium or hydrogen which allows to realize calculations with different fluids and makes it possible to predict the performances of a wide range of turbomachines. As the development of the model was rather
model and therefore measured on a prototype. The two-phase flow characteristics can only be
obtained in some specific well-studied cases. This non-predictive 0-D model is not adapted to
iterations in case of pre-design studies. 3-D CFD calculations can be considered to get around
the problem but computational time is not suitable for pre-design studies (several months to
obtain a single-phase performance map on a standard desktop computer). For this purpose,
CEA developed a predictive 1-D pump model (Matteo, 2019).

The CATHARE 1-D pump model is a predictive transient two-phase flow calculation. It considers a mean streamline adopted by the fluid on each element of the machine : suction, impeller, diffuser and volute. All machines do not necessarily have all these parts, or may conversely have several rotating stages : the structure of the model is intended to be generic and this is made possible thanks to the possibility of assembling elements in the code. The diffuser can be removed and the model can easily evolve towards multi-stage machines. Pump components are meshed perpendicularly to the mean streamline using the 1-D module (see Fig. 1) with typical mesh sizes of the order of magnitude of a centimeter. The equations governing the fluid are solved in each mesh. A change of coordinate system is made in the impeller (change from fixed to rotating coordinate system).

<figure><img src="image_2.png" /><figcaption>Figure 1: Example of the DERAP machine (Matteo et al., 2018) modeled with the 1-D pump model in CATHARE code</figcaption></figure>

The 1-D pump model includes in particular the following modifications on the CATHARE
equations governing the fluid (see Refs. Matteo et al., 2018, 2019b for more details):

1. A frame switch is made at the inlet of the rotating part (the impeller) from the fixed frame to the rotating frame considering relative speed. It requires the introduction of a centrifugal acceleration term to the momentum equations and associated contributions to the energy equations. Flow sections are adapted to ensure mass flow rate conservation with a 2-component velocity. Another frame switch is applied at the outlet of the impeller, when the fluid is entering the downstream fixed part (diffuser or volute);

2. A deviation model is implemented to adjust $\beta_2$ angle at impeller outlet;

3. Two losses source terms are added to the momentum equation: desadaptation head loss and diffusion head loss. They represent the different losses phenomena that take place in the machine due to geometry in normal and off-design conditions;

4. A power dissipation source term is added in the energy equation representing the losses due to recirculations at low flow rates.

The CATHARE code is able to handle various fluids such as water, air, sodium or hydrogen which allows to realize calculations with different fluids and makes it possible to predict the performances of a wide range of turbomachines. As the development of the model was rather
<figure><img src="image_3.png" /><figcaption>a) Mean flow path projection on the $R-H$ plane</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>c) Projection on the DERAP machine (Matteo et al., 2018) modeled with CATHARE 1-D pump model

**Figure 2: Mean flow path projections with CATHARE 1-D pump model**</figcaption></figure>

oriented on pump applications (Matteo et al., 2018, 2019b,c; Matteo, 2019), the purpose of this paper is to extend the applications of the 1-D pump model on compressors. Operate with a compressible fluid has not been done yet and the goal is to validate the theoretical and real performance of the machine. For the LSCC compressor, the air equation of state is available in CATHARE by coupling with the REFPROP library (Lemmon et al., 2013) from the National Institute of Standards and Technology (NIST).

## MEAN STREAMLINE REPRESENTATION IN THE IMPELLER

The CATHARE 1-D pump model is discrete and predictive. The resolution of the equations governing fluid flow in all the 1-D elements of CATHARE code is done along a curvilinear abscissa representing the average direction of flow (known as the "mean streamline" or "mean flow path") and illustrated on Fig. 2. Having an effective method to determine this mean streamline and especially its radial position *R* is fundamental for equation solving considering rotating parts into the machine and 2-component speed projections on 1-D elements. The global method has been introduced in the CATHARE code by Matteo, 2019.

Mean streamline in impeller is obtained by integrating an elementary displacement dependent on two angles: the blade angle $\beta$ and the streamline orientation angle of the impeller $\gamma$. It is necessary to know this streamline as well as possible in order to correctly predict the pressure profile in the impeller, without necessarily being visible on the predicted overall performance. A false pressure profile will not necessary impact the overall performances but will cause discrepancies when looking at local phenomena in the impeller. The calculation is done with a variable $L_{imp}$ (representing the length of the mean streamline in the impeller) which is difficult to know without detailed geometry (moreover, the aim of the model is to predict this length with few input data). Let $z_{3D}$ be the vector carrying the direction of the mean fluid stream along
<figure><img src="image_3.png" /><figcaption>a) Mean flow path projection on the $R-H$ plane</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>c) Projection on the DERAP machine (Matteo et al., 2018) modeled with CATHARE 1-D pump model

**Figure 2: Mean flow path projections with CATHARE 1-D pump model**</figcaption></figure>

oriented on pump applications (Matteo et al., 2018, 2019b,c; Matteo, 2019), the purpose of this paper is to extend the applications of the 1-D pump model on compressors. Operate with a compressible fluid has not been done yet and the goal is to validate the theoretical and real performance of the machine. For the LSCC compressor, the air equation of state is available in CATHARE by coupling with the REFPROP library (Lemmon et al., 2013) from the National Institute of Standards and Technology (NIST).

## MEAN STREAMLINE REPRESENTATION IN THE IMPELLER

The CATHARE 1-D pump model is discrete and predictive. The resolution of the equations governing fluid flow in all the 1-D elements of CATHARE code is done along a curvilinear abscissa representing the average direction of flow (known as the "mean streamline" or "mean flow path") and illustrated on Fig. 2. Having an effective method to determine this mean streamline and especially its radial position *R* is fundamental for equation solving considering rotating parts into the machine and 2-component speed projections on 1-D elements. The global method has been introduced in the CATHARE code by Matteo, 2019.

Mean streamline in impeller is obtained by integrating an elementary displacement dependent on two angles: the blade angle $\beta$ and the streamline orientation angle of the impeller $\gamma$. It is necessary to know this streamline as well as possible in order to correctly predict the pressure profile in the impeller, without necessarily being visible on the predicted overall performance. A false pressure profile will not necessary impact the overall performances but will cause discrepancies when looking at local phenomena in the impeller. The calculation is done with a variable $L_{imp}$ (representing the length of the mean streamline in the impeller) which is difficult to know without detailed geometry (moreover, the aim of the model is to predict this length with few input data). Let $z_{3D}$ be the vector carrying the direction of the mean fluid stream along
the impeller, and $dz_{3D}$ be its infinitesimal displacement. Let's take two planes, one normal to the rotation axis (called $R - \theta$) and the other one radial containing the axis (called $R - H$) represented on Fig. 3. Let $dz_{pH}$ be the projection of $dz_{3D}$ on the $R - H$ plane.

<figure><img src="image_5.png" /><figcaption>Figure 3: Mean flow path projections: a) $R - H$ plane and b) $R - \theta$ plane</figcaption></figure>

By noticing on the $R - H$ plane that $d\vec{R} = \cos(\gamma)dz_{pH}$ and on the $R - \theta$ plane that $dz_{pH} = \sin(\beta)dz_{3D}$, it finally leads to $d\vec{R} = \sin(\beta)\cos(\gamma)dz_{3D}$. As the objective of the model is to be predictive with few geometrical data, the assumption is to know only $\beta$ and $\gamma$ angles at inlet and outlet of the impeller. The hypothesis made in the rotating part is a linear evolution of both $\sin \beta$ and $\cos \gamma$. This gives the following relation for the radial position in a N-meshed 1-D impeller:

$$R_2 = R_1 + \sum_{I=2}^{N} (\sin \beta)|_{z_I}(\cos \gamma)|_{z_I} \Delta z \quad (1)$$

Knowing $R_1$ and $R_2$, the mean flow path length $L_{imp} = \sum \Delta z$ (which is the length of the 1-D element representing the impeller) is then adapted to solve the equation. This method to calculate the impeller length is explained in Matteo, 2019. The fluid cross section in the rotating frame $S(I)$ at the mesh number $I$ is then calculated according to Eq. (2) and the relation between the relative velocity $W$ and the meridian velocity $V_m$. The meridian cross section $S_m$ is an input data of the model at the impeller inlet and outlet and is then interpolated between these points.

$$ \begin{aligned} Q(I) &= V_m(I)S_m(I) = W(I)S(I) = W(I)S_m(I)\sin(\beta(I)) \\ \Leftrightarrow S(I) &= S_m(I) \cdot \sin(\beta(I)) \end{aligned} \quad (2) $$

## LSCC COMPRESSOR

The Low-Speed Centrifugal Compressor (LSCC) was built in the 80s by NASA in order to provide detailed experimental results for the validation of theoretical models (in particular for the first fluid mechanics codes under development at the time). It has already been the subject of publications by Hathaway et al., 1991, 1993, 1995 presenting numerous geometrical elements and experimental data. It is a large scale machine extensively instrumented especially in the rotating part which is rather rare considering technical difficulties.
the impeller, and $dz_{3D}$ be its infinitesimal displacement. Let's take two planes, one normal to the rotation axis (called $R - \theta$) and the other one radial containing the axis (called $R - H$) represented on Fig. 3. Let $dz_{pH}$ be the projection of $dz_{3D}$ on the $R - H$ plane.

<figure><img src="image_5.png" /><figcaption>Figure 3: Mean flow path projections: a) $R - H$ plane and b) $R - \theta$ plane</figcaption></figure>

By noticing on the $R - H$ plane that $d\vec{R} = \cos(\gamma)dz_{pH}$ and on the $R - \theta$ plane that $dz_{pH} = \sin(\beta)dz_{3D}$, it finally leads to $d\vec{R} = \sin(\beta)\cos(\gamma)dz_{3D}$. As the objective of the model is to be predictive with few geometrical data, the assumption is to know only $\beta$ and $\gamma$ angles at inlet and outlet of the impeller. The hypothesis made in the rotating part is a linear evolution of both $\sin \beta$ and $\cos \gamma$. This gives the following relation for the radial position in a N-meshed 1-D impeller:

$$R_2 = R_1 + \sum_{I=2}^{N} (\sin \beta)|_{z_I}(\cos \gamma)|_{z_I} \Delta z \quad (1)$$

Knowing $R_1$ and $R_2$, the mean flow path length $L_{imp} = \sum \Delta z$ (which is the length of the 1-D element representing the impeller) is then adapted to solve the equation. This method to calculate the impeller length is explained in Matteo, 2019. The fluid cross section in the rotating frame $S(I)$ at the mesh number $I$ is then calculated according to Eq. (2) and the relation between the relative velocity $W$ and the meridian velocity $V_m$. The meridian cross section $S_m$ is an input data of the model at the impeller inlet and outlet and is then interpolated between these points.

$$ \begin{aligned} Q(I) &= V_m(I)S_m(I) = W(I)S(I) = W(I)S_m(I) \sin(\beta(I)) \\ \Leftrightarrow S(I) &= S_m(I) \cdot \sin(\beta(I)) \end{aligned} \quad (2) $$

## LSCC COMPRESSOR

The Low-Speed Centrifugal Compressor (LSCC) was built in the 80s by NASA in order to provide detailed experimental results for the validation of theoretical models (in particular for the first fluid mechanics codes under development at the time). It has already been the subject of publications by Hathaway et al., 1991, 1993, 1995 presenting numerous geometrical elements and experimental data. It is a large scale machine extensively instrumented especially in the rotating part which is rather rare considering technical difficulties.
## Description of the Studied Turbomachine

The LSCC is a centrifugal compressor with a 20-vane semi-open impeller and 23-vane diffuser. The fluid is air, the flow enters the impeller axially at standard sea-level conditions ($T_{in} = 15^{\circ}C, P_{in} = 101,325 Pa$) end exits radially. The design nominal point is $\Pi_{N} = 1,14, N_{N} = 1920$ rpm and $\dot{m}_{N} = 30$ kg/s. Its specific speed according to the European definition is $N_{S} = 45,79$. The LSCC impeller illustration is presented in Fig. 4 and the main characteristics are shown in Table 1.

<figure><img src="image_6.png" /><figcaption>Figure 4: LSCC impeller illustration from Hathaway et al., 1995</figcaption></figure>

<table><caption>Table 1: LSCC geometric specifications from Hathaway et al., 1995</caption><thead><tr><th>Geometric specifications</th><th>Symbol</th><th>Value</th><th>Unit</th></tr></thead><tbody><tr><td>Impeller hub</td><td>$R_{1min}$</td><td>0,2147011</td><td>m</td></tr><tr><td>Impeller tip</td><td>$R_{1max}$</td><td>0,4325804</td><td>m</td></tr><tr><td>Outlet radius</td><td>$R_{2}$</td><td>0,7620</td><td>m</td></tr><tr><td>Outlet relative angle</td><td>$\beta_{2}$</td><td>35</td><td>°</td></tr><tr><td>Number of vanes</td><td>$Z$</td><td>20</td><td>—</td></tr><tr><td>Outlet width</td><td>$b_{2}$</td><td>0.1431</td><td>m</td></tr></tbody></table>

## First Computations

A first step of verification is to build an ideal input deck in order to verify the behavior of the 1-D pump model on the LSCC geometry and with a compressible working fluid, to check if the code handles the calculation correctly. Therefore, the desadaptation, diffusion and dissipation losses models of the CATHARE 1-D pump model are deactivated, as well as the regular wall friction losses (linked to the surface roughness) are removed on the elements. It is also considered that there is no deviation angle (no slip factor) of the fluid at the impeller outlet. In this ideal case, CATHARE calculations must fit with the curves of Euler's theory of turbomachines. A pressure boundary condition is set at the discharge outlet and the variation of the flow rate boundary condition at the suction inlet allows to obtain the results for the impeller presented on Figs. 5 and 6. The 1-D pump model curves are almost superimposed with those of Euler's theory (the maximum relative error over the gas flow rate range [27%, 173%]) is 0,75% for the ideal pressure ratio and 0,79% for the ideal torque and obtained for both curves at low flow rates). These differences remain very small and are not concerning as these operating conditions are very difficult to reach experimentally (very low flow rate and nominal rotational speed, the energy given to the fluid is high while its passage time through the machine is long). This work has already been done on several pumps (quasi-incompressible fluid) by Matteo, 2019 and therefore the model correctly manage different fluids. This step ensures that the modeling has been well constructed.

Then, real conditions calculations with 1-D pump model have been performed and compared to experimental data (experimental rotational speed is $N = 1862$ rpm). Figs. 7 and 8 show pressure ratio and isentropic efficiency results in the impeller for the 3 implemented deviation angle models: Stodola, 1927, Stanitz, 1952 and Wiesner, 1967. The Stanitz model Stanitz, 1952 is not recommended for $\beta_{2}$ angles lower than 45° (here $\beta_{2} = 35^{\circ}$) but has been studied to evaluate its behavior out of its validity range. The pressure ratio predictions gives satisfactory results as the maximum relative error over the gas flow rate range [50%, 142%] is under 3%
## Description of the Studied Turbomachine

The LSCC is a centrifugal compressor with a 20-vane semi-open impeller and 23-vane diffuser. The fluid is air, the flow enters the impeller axially at standard sea-level conditions ($T_{in} = 15^{\circ}C, P_{in} = 101,325 Pa$) end exits radially. The design nominal point is $\Pi_{N} = 1,14, N_{N} = 1920$ rpm and $\dot{m}_{N} = 30$ kg/s. Its specific speed according to the European definition is $N_{S} = 45,79$. The LSCC impeller illustration is presented in Fig. 4 and the main characteristics are shown in Table 1.

<figure><img src="image_6.png" /><figcaption>Figure 4: LSCC impeller illustration from Hathaway et al., 1995</figcaption></figure>

<table><caption>Table 1: LSCC geometric specifications from Hathaway et al., 1995</caption><thead><tr><th>Geometric specifications</th><th>Symbol</th><th>Value</th><th>Unit</th></tr></thead><tbody><tr><td>Impeller hub</td><td>$R_{1min}$</td><td>0,2147011</td><td>m</td></tr><tr><td>Impeller tip</td><td>$R_{1max}$</td><td>0,4325804</td><td>m</td></tr><tr><td>Outlet radius</td><td>$R_{2}$</td><td>0,7620</td><td>m</td></tr><tr><td>Outlet relative angle</td><td>$\beta_{2}$</td><td>35</td><td>°</td></tr><tr><td>Number of vanes</td><td>$Z$</td><td>20</td><td>—</td></tr><tr><td>Outlet width</td><td>$b_{2}$</td><td>0.1431</td><td>m</td></tr></tbody></table>

## First Computations

A first step of verification is to build an ideal input deck in order to verify the behavior of the 1-D pump model on the LSCC geometry and with a compressible working fluid, to check if the code handles the calculation correctly. Therefore, the desadaptation, diffusion and dissipation losses models of the CATHARE 1-D pump model are deactivated, as well as the regular wall friction losses (linked to the surface roughness) are removed on the elements. It is also considered that there is no deviation angle (no slip factor) of the fluid at the impeller outlet. In this ideal case, CATHARE calculations must fit with the curves of Euler's theory of turbomachines. A pressure boundary condition is set at the discharge outlet and the variation of the flow rate boundary condition at the suction inlet allows to obtain the results for the impeller presented on Figs. 5 and 6. The 1-D pump model curves are almost superimposed with those of Euler's theory (the maximum relative error over the gas flow rate range [27%, 173%]) is 0,75% for the ideal pressure ratio and 0,79% for the ideal torque and obtained for both curves at low flow rates). These differences remain very small and are not concerning as these operating conditions are very difficult to reach experimentally (very low flow rate and nominal rotational speed, the energy given to the fluid is high while its passage time through the machine is long). This work has already been done on several pumps (quasi-incompressible fluid) by Matteo, 2019 and therefore the model correctly manage different fluids. This step ensures that the modeling has been well constructed.

Then, real conditions calculations with 1-D pump model have been performed and compared to experimental data (experimental rotational speed is $N = 1862$ rpm). Figs. 7 and 8 show pressure ratio and isentropic efficiency results in the impeller for the 3 implemented deviation angle models: Stodola, 1927, Stanitz, 1952 and Wiesner, 1967. The Stanitz model Stanitz, 1952 is not recommended for $\beta_{2}$ angles lower than 45° (here $\beta_{2} = 35^{\circ}$) but has been studied to evaluate its behavior out of its validity range. The pressure ratio predictions gives satisfactory results as the maximum relative error over the gas flow rate range [50%, 142%] is under 3%
<figure><img src="image_7.png" /><figcaption>Figure 5: Pressure ratio - ideal conditions</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 6: Torque - ideal conditions</figcaption></figure>

and obtained for high flow rates (relative error at nominal point is 2,2%). The different fluid deviation angle correlations at impeller outlet have no major impact on predictions, although Stanitz model gives slightly better results for the LSCC compressor. This angle deviation model remains satisfactory and consistent with $\beta_2 = 35^\circ$. In contrast, the isentropic efficiency defined as $\eta = (h_{out,is} - h_{in})/(h_{out} - h_{in})$ is at first sight overestimated (some points are over 0, 98) even if the maximum relative error remains below 13,5% for the studied flow rate range (relative error at nominal point is 7,0%). This means that in the impeller, the 1-D pump model does not reflect enough losses. One possible reason may be a not precise enough description of the flow into the machine. Indeed, the advantage of the 1-D pump model is that it requires few geometrical input data (impeller radius, $\alpha$ and $\beta$ angles at inlet and outlet, no detailed geometry is given) but it may poorly reflect the "S-shaped" profile of the blades, which is common for compressor but not for pumps. In the following parts a work on this flow path throughout the turbomachine is done in order to improve the model. However, it can be noticed that the overall trend of the isentropic efficiency prediction is correct.

## EXTENSION OF THE 1-D PUMP MODEL TO COMPRESSORS

### First use of the Qiu Deviation Angle Model

In order to improve the real conditions prediction, a first work has been done on the deviation angle model used. The Qiu model (Qiu et al., 2011) has been implemented in the CATHARE code and presented in Matteo et al., 2019c but it has never been used before this study. It is an "unified" model valid for pumps and compressors whether there are radial, mixed or axial and provides good results (Dixon and Hall, 2014). It is also validated for at design and off-design conditions. Its formula given in Eq. (3) shows that multiple phenomena are taken into account : the radial rotation effect, the blade turning and the passage width variation. The model only requires one new input data : the blade width at impeller outlet. It can be noticed that under certain conditions (if the slip factor decrement due to the radial rotation effect $\Delta\sigma_{radial}$ is only considered with $F = 1$ and $\gamma_2 = 0$), the Qiu model is equal to the Stodola model.

$$ \sigma_{Qiu} = 1 - \Delta\sigma_{radial} - \Delta\sigma_{turn} - \Delta\sigma_{passage} \quad (3) $$

The results on the prediction of the performances are shown on Fig. 7 for the pressure ratio and on Fig. 8 for the isentropic efficiency. This model has a small impact on the predictions of the performances and does not give the better results for the LSCC compressor. Nevertheless, the Qiu model is now available in the CATHARE code and can be used to perform sensitivity
<figure><img src="image_7.png" /><figcaption>Figure 5: Pressure ratio - ideal conditions</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 6: Torque - ideal conditions</figcaption></figure>

and obtained for high flow rates (relative error at nominal point is 2,2%). The different fluid deviation angle correlations at impeller outlet have no major impact on predictions, although Stanitz model gives slightly better results for the LSCC compressor. This angle deviation model remains satisfactory and consistent with $\beta_2 = 35^\circ$. In contrast, the isentropic efficiency defined as $\eta = (h_{out,is} - h_{in})/(h_{out} - h_{in})$ is at first sight overestimated (some points are over 0, 98) even if the maximum relative error remains below 13,5% for the studied flow rate range (relative error at nominal point is 7,0%). This means that in the impeller, the 1-D pump model does not reflect enough losses. One possible reason may be a not precise enough description of the flow into the machine. Indeed, the advantage of the 1-D pump model is that it requires few geometrical input data (impeller radius, $\alpha$ and $\beta$ angles at inlet and outlet, no detailed geometry is given) but it may poorly reflect the "S-shaped" profile of the blades, which is common for compressor but not for pumps. In the following parts a work on this flow path throughout the turbomachine is done in order to improve the model. However, it can be noticed that the overall trend of the isentropic efficiency prediction is correct.

## EXTENSION OF THE 1-D PUMP MODEL TO COMPRESSORS

### First use of the Qiu Deviation Angle Model

In order to improve the real conditions prediction, a first work has been done on the deviation angle model used. The Qiu model (Qiu et al., 2011) has been implemented in the CATHARE code and presented in Matteo et al., 2019c but it has never been used before this study. It is an "unified" model valid for pumps and compressors whether there are radial, mixed or axial and provides good results (Dixon and Hall, 2014). It is also validated for at design and off-design conditions. Its formula given in Eq. (3) shows that multiple phenomena are taken into account : the radial rotation effect, the blade turning and the passage width variation. The model only requires one new input data : the blade width at impeller outlet. It can be noticed that under certain conditions (if the slip factor decrement due to the radial rotation effect $\Delta\sigma_{radial}$ is only considered with $F = 1$ and $\gamma_2 = 0$), the Qiu model is equal to the Stodola model.

$$ \sigma_{Qiu} = 1 - \Delta\sigma_{radial} - \Delta\sigma_{turn} - \Delta\sigma_{passage} \quad (3) $$

The results on the prediction of the performances are shown on Fig. 7 for the pressure ratio and on Fig. 8 for the isentropic efficiency. This model has a small impact on the predictions of the performances and does not give the better results for the LSCC compressor. Nevertheless, the Qiu model is now available in the CATHARE code and can be used to perform sensitivity
analysis. The Stanitz model will be used for the calculations on the LSCC. In the end, the various deviation angle models have a minor impact on the performance predictions and do not explain the discrepancies observed.

<figure><img src="image_9.png" /><figcaption>Figure 7: LSCC pressure ratio</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 8: LSCC isentropic efficiency</figcaption></figure>

## Improvements on the Mean StreamLine in the Impeller

The linear evolution hypothesis of both $\sin \beta$ and $\cos \gamma$ considered in the impeller can lead to a final value of $\mathcal{L}_{imp}$ far from reality especially for compressors and their "S-shaped" blades. A work has been performed to add optional angles $\beta_{middle}$ and $\gamma_{middle}$ that represent the angles at mean streamline half length. These angles were determined manually with precision thanks to the detailed geometry available for the LSCC. If not possible, they can be interpolated from impeller schematics or estimated from compressors with similar applications and operating points. The interpolation of the mean streamline is then done twice from 0 to $\mathcal{L}_{imp}/2$ and from $\mathcal{L}_{imp}/2$ to $\mathcal{L}_{imp}$. The same work has been done on the DERAP radial pump (see Matteo et al., 2018, 2019b,c). This two steps interpolation gives satisfactory results, which are shown graphically (mean streamline $R-H$ and $R-\theta$ plane projections) on Figs. 9-10 and numerically on Table 2 (real length is deduced from the detailed geometry available). The improved mean flow path is closer to the geometric one. For the LSCC compressor and the DERAP pump the relative error on $\mathcal{L}_{imp}$ is divided by about 3. This result is expected because more data is given at the input (the model is more constrained), however it requires to have these two angles at mean streamline half length which is not necessarily easy. This work could have been done thanks to the detailed geometrical data available for the LSCC compressor. Nevertheless, without this kind of 3-D geometrical data or for a pump that does not exist, the two angles can be estimated based on existing machines designed for similar applications.

<table><caption>Table 2: Comparison of LSCC and DERAP impeller lengths for both models</caption><thead><tr><th>Studied impeller</th><th>Real length, m</th><th>LC<sub>imp</sub>, m</th><th>Relative error, %</th><th>Improved LC<sub>imp</sub>, m</th><th>Relative error, %</th></tr></thead><tbody><tr><td>LSCC</td><td>0, 849</td><td>1, 397</td><td>64, 54</td><td>1, 071</td><td>26, 15</td></tr><tr><td>DERAP</td><td>0, 198</td><td>0, 157</td><td>20, 71</td><td>0, 184</td><td>7, 01</td></tr></tbody></table>
analysis. The Stanitz model will be used for the calculations on the LSCC. In the end, the various deviation angle models have a minor impact on the performance predictions and do not explain the discrepancies observed.

<figure><img src="image_9.png" /><figcaption>Figure 7: LSCC pressure ratio</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 8: LSCC isentropic efficiency</figcaption></figure>

## Improvements on the Mean StreamLine in the Impeller

The linear evolution hypothesis of both $\sin \beta$ and $\cos \gamma$ considered in the impeller can lead to a final value of $\mathcal{L}_{imp}$ far from reality especially for compressors and their "S-shaped" blades. A work has been performed to add optional angles $\beta_{middle}$ and $\gamma_{middle}$ that represent the angles at mean streamline half length. These angles were determined manually with precision thanks to the detailed geometry available for the LSCC. If not possible, they can be interpolated from impeller schematics or estimated from compressors with similar applications and operating points. The interpolation of the mean streamline is then done twice from 0 to $\mathcal{L}_{imp}/2$ and from $\mathcal{L}_{imp}/2$ to $\mathcal{L}_{imp}$. The same work has been done on the DERAP radial pump (see Matteo et al., 2018, 2019b,c). This two steps interpolation gives satisfactory results, which are shown graphically (mean streamline $R-H$ and $R-\theta$ plane projections) on Figs. 9-10 and numerically on Table 2 (real length is deduced from the detailed geometry available). The improved mean flow path is closer to the geometric one. For the LSCC compressor and the DERAP pump the relative error on $\mathcal{L}_{imp}$ is divided by about 3. This result is expected because more data is given at the input (the model is more constrained), however it requires to have these two angles at mean streamline half length which is not necessarily easy. This work could have been done thanks to the detailed geometrical data available for the LSCC compressor. Nevertheless, without this kind of 3-D geometrical data or for a pump that does not exist, the two angles can be estimated based on existing machines designed for similar applications.

<table><caption>Table 2: Comparison of LSCC and DERAP impeller lengths for both models</caption><thead><tr><th>Studied impeller</th><th>Real length, m</th><th>LC<sub>imp</sub>, m</th><th>Relative error, %</th><th>Improved LC<sub>imp</sub>, m</th><th>Relative error, %</th></tr></thead><tbody><tr><td>LSCC</td><td>0, 849</td><td>1, 397</td><td>64, 54</td><td>1, 071</td><td>26, 15</td></tr><tr><td>DERAP</td><td>0, 198</td><td>0, 157</td><td>20, 71</td><td>0, 184</td><td>7, 01</td></tr></tbody></table>
<figure><img src="image_11.png" /><figcaption>Figure 9: LSCC mean flow path (R - H Figure 10: LSCC mean flow path (R - θ plane)</figcaption></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>Figure 11: LSCC pressure ratio</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 12: LSCC isentropic efficiency</figcaption></figure>

## Impact of the New Mean Streamline Model on Performance Predictions

By taking the data set built in paragraph and adjusting the mean streamline length in the impeller according to the new model, a significant difference is obtained on the total curvilinear abscissa of the LSCC impeller which is more realistic. Lengths and projections on each mesh are closer to the real geometry of LSCC compressor. The results on the performance predictions in the impeller are presented on figures 11 and 12. The performance curves with and without the new mean flow path model are very similar (see Figs. 7 and 8). The new model does not degrade overall performance and allows more realistic element lengths to be entered (see Table 2). It can be noted that the isentropic efficiency (and with a minor impact the pressure ratio) is increased with the new model, which is not a desired outcome as the performances were already overestimated. Indeed the reduction of the previously overestimated length $L_{imp}$ of the 1-D element representing the impeller reduces friction and therefore losses, which results in better performance. As in Fig. 8, the isentropic efficiency is still too high compared to the experimental data. This means there are not enough losses in the 1-D pump model impeller (there are not enough dissipation effects in the model). Some leads are investigated to predict this efficiency better. A promising work to be done is on the effect of the temperature, as the efficiency can be expressed with it [see Eq. (4)].

$$ \eta = \frac{\frac{P_{out}}{P_{in}}^{\frac{\gamma-1}{\gamma}} - 1}{\frac{T_{out}}{T_{in}} - 1} = \frac{\prod^{\frac{\gamma-1}{\gamma}} - 1}{\frac{T_{out}}{T_{in}} - 1} \quad (4) $$
<figure><img src="image_11.png" /><figcaption>Figure 9: LSCC mean flow path (R - H Figure 10: LSCC mean flow path (R - θ plane)</figcaption></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>Figure 11: LSCC pressure ratio</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 12: LSCC isentropic efficiency</figcaption></figure>

## Impact of the New Mean Streamline Model on Performance Predictions

By taking the data set built in paragraph and adjusting the mean streamline length in the impeller according to the new model, a significant difference is obtained on the total curvilinear abscissa of the LSCC impeller which is more realistic. Lengths and projections on each mesh are closer to the real geometry of LSCC compressor. The results on the performance predictions in the impeller are presented on figures 11 and 12. The performance curves with and without the new mean flow path model are very similar (see Figs. 7 and 8). The new model does not degrade overall performance and allows more realistic element lengths to be entered (see Table 2). It can be noted that the isentropic efficiency (and with a minor impact the pressure ratio) is increased with the new model, which is not a desired outcome as the performances were already overestimated. Indeed the reduction of the previously overestimated length $L_{imp}$ of the 1-D element representing the impeller reduces friction and therefore losses, which results in better performance. As in Fig. 8, the isentropic efficiency is still too high compared to the experimental data. This means there are not enough losses in the 1-D pump model impeller (there are not enough dissipation effects in the model). Some leads are investigated to predict this efficiency better. A promising work to be done is on the effect of the temperature, as the efficiency can be expressed with it [see Eq. (4)].

$$ \eta = \frac{\frac{P_{out}}{P_{in}}^{\frac{\gamma-1}{\gamma}} - 1}{\frac{T_{out}}{T_{in}} - 1} = \frac{\prod^{\frac{\gamma-1}{\gamma}} - 1}{\frac{T_{out}}{T_{in}} - 1} \quad (4) $$
<figure><img src="image_15.png" /><figcaption>Figure 13: Axial velocity in impeller</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure 14: Relative Tangential velocity in impeller</figcaption></figure>

For the LSCC data, a small increase of the temperature at impeller exit $T_{out}$ of 1°C (which would mean that CATHARE model more losses in this part) can bring the efficiency from 0,98 to about 0,91, very close to the experimental data. According to Hathaway et al., 1995, the temperature ratio across the impeller is equal to 1,042. Thanks to coefficients that have been implemented on each of the CATHARE 1-D pump model losses, a sensitivity analysis could be carried out in the future to calibrate these losses on the LSCC compressor and get closer to the experimental data. Nevertheless, this solution does not seem to be appropriate as the aim of the 1-D pump model is to be generic. These discrepancies may be due to a compressibility phenomenon, which for the meantime has not been observed on the validation base composed only of pumps. Investigations on this point will be conducted in the future.

Detailed data on velocities in the impeller are available and the CATHARE code is able to compute physical variables on each mesh of an element, thus figures 13 and 14 show two velocity profiles inside the impeller. The results of the calculations are satisfactory, and adding the two optional angles $\beta_{middle}$ and $\gamma_{middle}$ allows to better correspond to the trends. For the axial velocity, the new mean streamline model provides results closer to the experimental data. For the relative tangential velocity, it can be noticed that the new model allows to better capture the rebound in the velocity profile, which was not the case before, even if the relative error at the outlet remains high (30%). These results on the velocity profiles were expected as more data is given (the two angles at the middle of the impeller). Another computed variable that can be displayed on each mesh is pressure along the 1-D elements inside the compressor: figure 15 show the total pressure profiles in suction, impeller and diffuser and volute. The pressure ratio in impeller $\Pi_N$ = 1,14 is well respected for both mean flow path models. The new mean flow path model does not degrade the profile trends. These profiles can't be compared to experimental data because they are not available for the LSCC, but seem to respect the fluid mechanics. According to the profile, all the pressure increase throughout the machine takes place in the impeller (the increase is even higher than the nominal pressure ratio), and a slight decrease is observed in the diffuser and the volute. In the future, 3-D CFD calculations could be considered to validate this profile.
<figure><img src="image_15.png" /><figcaption>Figure 13: Axial velocity in impeller</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure 14: Relative Tangential velocity in impeller</figcaption></figure>

For the LSCC data, a small increase of the temperature at impeller exit $T_{out}$ of 1°C (which would mean that CATHARE model more losses in this part) can bring the efficiency from 0.98 to about 0.91, very close to the experimental data. According to Hathaway et al., 1995, the temperature ratio across the impeller is equal to 1.042. Thanks to coefficients that have been implemented on each of the CATHARE 1-D pump model losses, a sensitivity analysis could be carried out in the future to calibrate these losses on the LSCC compressor and get closer to the experimental data. Nevertheless, this solution does not seem to be appropriate as the aim of the 1-D pump model is to be generic. These discrepancies may be due to a compressibility phenomenon, which for the meantime has not been observed on the validation base composed only of pumps. Investigations on this point will be conducted in the future.

Detailed data on velocities in the impeller are available and the CATHARE code is able to compute physical variables on each mesh of an element, thus figures 13 and 14 show two velocity profiles inside the impeller. The results of the calculations are satisfactory, and adding the two optional angles $\beta_{middle}$ and $\gamma_{middle}$ allows to better correspond to the trends. For the axial velocity, the new mean streamline model provides results closer to the experimental data. For the relative tangential velocity, it can be noticed that the new model allows to better capture the rebound in the velocity profile, which was not the case before, even if the relative error at the outlet remains high (30%). These results on the velocity profiles were expected as more data is given (the two angles at the middle of the impeller). Another computed variable that can be displayed on each mesh is pressure along the 1-D elements inside the compressor: figure 15 show the total pressure profiles in suction, impeller and diffuser and volute. The pressure ratio in impeller $\Pi_N$ = 1.14 is well respected for both mean flow path models. The new mean flow path model does not degrade the profile trends. These profiles can't be compared to experimental data because they are not available for the LSCC, but seem to respect the fluid mechanics. According to the profile, all the pressure increase throughout the machine takes place in the impeller (the increase is even higher than the nominal pressure ratio), and a slight decrease is observed in the diffuser and the volute. In the future, 3-D CFD calculations could be considered to validate this profile.
<figure><img src="image_17.png" /><figcaption>Figure 15: Total pressure along elements at nominal point for LSCC 1-D pump model</figcaption></figure>

## CONCLUSION

This paper describes the first application of CEA's recently developed 1-D pump model on a radial air compressor. It is the first step on the process of qualification of a compressible fluid turbomachine, the model having been tested previously only on pumps operating with a quasi-incompressible liquid fluid. The studied compressor is the NASA Low-Speed Centrifugal Compressor, especially built and tested for the validation of computational codes. A detailed geometry and data allowed us to realize a rigorous analysis of the machine, first using Euler's theory and then with the CATHARE code. The first computations of the 1-D pump model on the compressor yield a relative error to the experimental data of 2,2% on the pressure ratio and 7,0% on the isentropic efficiency at the nominal point, and the first-quadrant performances trends are correctly predicted. In particular, a work on the mean streamline (i.e. the mean flow path) of a compressor has been performed, by improving its description throughout the machine. The possibility to enter additional intermediate data has been developed, which is particularly useful in the case of compressors due to the S-shaped blades. Speed profiles are thus more accurate when comparing the available experimental data with the computational results. Moreover, this work does not degrade overall performance of the machine, although some work is underway to improve the isentropic efficiency computation.

The 1-D pump model can therefore be applied to a compressor. The CATHARE code correctly manages the compressible flow in the rotating part. However, flow blockage or shock waves phenomenon are not encountered as the pressure ratio remains low. It will be necessary to carry out the study of other compressors, with different geometries, flow rates and rotation speeds in order to validate the model in a more complete way. The examination of transient and entire systems containing compressors is also possible. Numerous applications can be considered: study of transients on gas-cooled reactors, modeling of multi-stage compressors, as well as operation in turbine mode.

## ACKNOWLEDGMENTS

The authors thank the French Alternative Energies and Atomic Energy Commission (CEA) for the support of this research work in the framework of the Gen IV program of the Energy Division.
<figure><img src="image_17.png" /><figcaption>Figure 15: Total pressure along elements at nominal point for LSCC 1-D pump model</figcaption></figure>

## CONCLUSION

This paper describes the first application of CEA's recently developed 1-D pump model on a radial air compressor. It is the first step on the process of qualification of a compressible fluid turbomachine, the model having been tested previously only on pumps operating with a quasi-incompressible liquid fluid. The studied compressor is the NASA Low-Speed Centrifugal Compressor, especially built and tested for the validation of computational codes. A detailed geometry and data allowed us to realize a rigorous analysis of the machine, first using Euler's theory and then with the CATHARE code. The first computations of the 1-D pump model on the compressor yield a relative error to the experimental data of 2,2% on the pressure ratio and 7,0% on the isentropic efficiency at the nominal point, and the first-quadrant performances trends are correctly predicted. In particular, a work on the mean streamline (i.e. the mean flow path) of a compressor has been performed, by improving its description throughout the machine. The possibility to enter additional intermediate data has been developed, which is particularly useful in the case of compressors due to the S-shaped blades. Speed profiles are thus more accurate when comparing the available experimental data with the computational results. Moreover, this work does not degrade overall performance of the machine, although some work is underway to improve the isentropic efficiency computation.

The 1-D pump model can therefore be applied to a compressor. The CATHARE code correctly manages the compressible flow in the rotating part. However, flow blockage or shock waves phenomenon are not encountered as the pressure ratio remains low. It will be necessary to carry out the study of other compressors, with different geometries, flow rates and rotation speeds in order to validate the model in a more complete way. The examination of transient and entire systems containing compressors is also possible. Numerous applications can be considered: study of transients on gas-cooled reactors, modeling of multi-stage compressors, as well as operation in turbine mode.

## ACKNOWLEDGMENTS

The authors thank the French Alternative Energies and Atomic Energy Commission (CEA) for the support of this research work in the framework of the Gen IV program of the Energy Division.
REFERENCES

F. Barre and M. Bernard. The CATHARE code strategy and assessment. *Nuclear Engineering and Design*, 124(3):257-284, 1990.
ISSN 0029-5493.

E. Benichou, G. Dufour, Y. Bousquet, N. Binder, A. Ortolan, and X. Carbonneau. Body Force Modeling of the Aerodynamics of a Low-Speed Fan under Distorted Inflow. *International Journal of Turbomachinery*, *Propulsion and Power*, 4(3), 2019.

F. Bertrand, C. Bassi, F. Bentivoglio, F. Audubert, C. Guéneau, G. Rimpault, and C. Journeau. Synthesis of the safety studies carried out on the GFR2400. *Nuclear Engineering and Design*, 253: 161-182, 2012.

F. Bertrand, G. Mauger, M. Bensalah, and P. Gauthé. Transient behavior of ASTRID with a gas power conversion system. *Nuclear Engineering and Design*, 308:20-29, 2016.

D. Bestion. The physical closure laws in the CATHARE code. *Nuclear Engineering and Design*, 124(3):229-245, 1990. ISSN 0029-5493.

Y. Birari, S. Gosavi, and P. Jorwekar. Use of cfd in design and development of r404a reciprocating compressor. In *Proceedings of the International Compressor Engineering Conference*, Purdue, IN, 07 2006.

M. Casey and C. Robinson. A New Streamline Curvature Through-flow Method for Radial Turbomachinery. *Journal of Turbomachinery*, 132(3), 04 2010.

C. Cornelius, T. Biesinger, P. Galpin, and A. Braune. Experimental and Computational Analysis of a Multistage Axial Compressor Including Stall Prediction by Steady and Transient CFD Methods. *Journal of Turbomachinery*, 136(6), 11 2013. ISSN 0889-504X.

S. Derakhshan and A. Nourbakhsh. Theoretical, numerical and experimental investigation of centrifugal pumps in reverse operation. *Experimental Thermal and Fluid Science*, 32(8):1620-1627, 2008. ISSN 0894-1777.

S. Dixon and C. Hall. Chapter 7 - Centrifugal Pumps, Fans, and Compressors. In *Fluid Mechanics and Thermodynamics of Turbomachinery (Seventh Edition)*, pages 265-317. Butterworth-Heinemann, Boston, Seventh Edition edition, 2014. ISBN 978-0-12-415954-9.

R. P. Dring and H. D. Joslyn. Through-Flow Modeling of Axial Turbomachinery. In *Turbo Expo: Power for Land, Sea, and Air*, volume 1, 09 1985.

R. P. Dring and G. C. Oates. Through Flow Theory for Nonaxisymmetric Turbomachinery Flow. In *Turbo Expo: Power for Land, Sea, and Air*, volume 1, 06 1989.

P. Emonot, A. Souyri, J. Gandrille, and F. Barré. CATHARE-3: A new system code for thermal-hydraulics in the context of the NEP-TUNE project. *Nuclear Engineering and Design*, 241(11):4476-4481, 2011. ISSN 0029-5493.

B. Faydide and J. Rousseau. Two-phase flow modeling with thermal and mechanical non-equilibrium. In *European Two Phase Flow Group Meeting*, Glasgow, UK, 1980.

K. Ghorbanian and M. Gholamrezaei. An artificial neural network approach to compressor performance prediction. *Applied Energy*, 86 (7):1210-1221, 2009. ISSN 0306-2619.

Gutiérrez Velásquez E. I. Determination of a suitable set of loss models for centrifugal compressor performance prediction. *Chinese Journal of Aeronautics*, 30(5):1644-1650, 2017. ISSN 1000-9361.

M. Hathaway, J. Wood, and C. Wasserbaueur. NASA Low-Speed Centrifugal Compressor for 3-D viscous code assessment and fundamental flow physics research. In *Gas Turbine and Aeroengine Congress and Exposition*, Orlando, Florida USA, 1991.

M. Hathaway, R. Chriss, A. Strazisar, and J. Wood. Experimental and computational investigation of the NASA Low-Speed Centrifugal Compressor flow field. *NASA Technical Memorandum 4481*, 1993.

M. Hathaway, R. Chriss, A. Strazisar, and J. Wood. Laser anemometer measurements of the three-dimensional rotro flow field in the NASA Low-Speed Centrifugal Compressor. *NASA Technical Paper TP-3527*, 1995.

M. Hofer, M. Buck, A. Cagnac, T. Prusek, N. Sobecki, P. Vlcek, D. Kriz, F. Hecker, and A. Hacks. Report on the validation status of codes and models for simulation of sCO2-HeRo loop. Technical report, sCO2-4-NPP project, Deliverable 1.2, 2020.

International Atomic Energy Agency. *Accident Analysis for Nuclear Power Plants*. Number 23 in Safety Reports Series. Vienna, 2002. ISBN 92-0-115602-2.

J. Joo, G. Medic, D. Philips, and S. Bose. Large-eddy simulation of a compressor rotor. In *Proceedings of the Summer Program, Center for Turbulence Research*, page 467. Center for Turbulence Research Stanford, CA, USA, 2014.

B. Lakshminarayana. Fundamental Principles, Analysis, and Performance of Turbomachinery. In *Fluid Dynamics and Heat Transfer of Turbomachinery*, chapter 2, pages 43-188. John Wiley & Sons, Ltd, 1995. ISBN 9780470172629.

P. Le Sausse, P. Fabrie, Arnou, D., and Clunet, F. CFD comparison with centrifugal compressor measurements on a wide operating range. *EPJ Web of Conferences*, 45:01059, 2013.

E. Lemmon, M. Huber, and M. McLinden. NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 9.1, 2013.

D. J. Lucia, P. S. Beran, and W. A. Silva. Reduced-order modeling: new approaches for computational physics. *Progress in Aerospace Sciences*, 40(1):51-117, 2004. ISSN 0376-0421.

L. Matteo. *Modelisation unidimensionnelle du comportement d'une pompe rotodynamique en fonctionnement normal et accidentel*. PhD Thesis, Ecole nation supérieure d'arts et métiers - EN-SAM, Oct 2019.

L. Matteo, A. Dazin, and N. Tauveron. Development and validation of a one-dimensional transient rotodynamic pump model at component scale. In *29th IAHR Symposium on Hydraulic Machinery and Systems*, volume 240, page 11, Kyoto, Japan, Sept. 2018. IOP Publishing.

L. Matteo, F. Cerru, A. Dazin, and N. Tauveron. Investigation of the Pump, Dissipation and Inverse Turbine Operating Modes Using the CATHARE-3 One-Dimensional Rotodynamic Pump Model. In *The ASME - JSME - KSME Joint Fluids Engineering conference 2019. AJKFLUIDS2019*, San Francisco, United States, 07 2019a.

L. Matteo, A. Dazin, and N. Tauveron. Modelling of a Centrifugal Pump Using the CATHARE-3 One-dimensional Transient Rotodynamic Pump Model. *International Journal of Fluid Machinery and Systems*, 12, 03 2019b.

L. Matteo, G. Mauger, A. Dazin, and N. Tauveron. Simulation of a Radial Pump Fast Startup and Analysis of the Loop Response Using a Transient 1D Mean Stream Line Based Model. *International Journal of Turbomachinery, Propulsion and Power*, 4:38, 11 2019c.

L. Matteo, G. Mauger, P. Gyomlai, A. Dazin, and N. Tauveron. A Rotodynamic Pump Seizure Transient Simulated Using the CATHARE-3 One-Dimensional Pump Model. In *NURETH-16*, Portland, OR, USA, 08 2019d.

G. Mauger, F. Bentivoglio, and N. Tauveron. Description of an improved turbomachinery model to be developed in the CATHARE-3 code for ASTRID power conversion system application. In *NURETH-16*, Chicago USA, 2015.

G. Mauger, N. Tauveron, F. Bentivoglio, and A. Ruby. On the dynamic modeling of Brayton cycle power conversion systems with the CATHARE-3 code. *Energy*, 168:1002-1016, 2019.

G. Medic, V. Zhang, G. Wang, J. Joo, and O. P. Sharma. Prediction of Transition and Losses in Compressor Cascades Using Large-Eddy Simulation. *Journal of Turbomachinery*, 138(12), 06 2016. ISSN 0889-504X.

J. Miettinen. Nuclear power plant simulators: Goals and evolution. In *Seminar on Transfer of Competence, Knowledge and Experience Gained through CSNI Activities in the Field of Thermal-hydraulics (THICKET)*, Pisa, Italy, Jul 2008.
REFERENCES

F. Barre and M. Bernard. The CATHARE code strategy and assessment. *Nuclear Engineering and Design*, 124(3):257-284, 1990.
ISSN 0029-5493.

E. Benichou, G. Dufour, Y. Bousquet, N. Binder, A. Ortolan, and X. Carbonneau. Body Force Modeling of the Aerodynamics of a Low-Speed Fan under Distorted Inflow. *International Journal of Turbomachinery*, *Propulsion and Power*, 4(3), 2019.

F. Bertrand, C. Bassi, F. Bentivoglio, F. Audubert, C. Guéneau, G. Rimpault, and C. Journeau. Synthesis of the safety studies carried out on the GFR2400. *Nuclear Engineering and Design*, 253: 161-182, 2012.

F. Bertrand, G. Mauger, M. Bensalah, and P. Gauthé. Transient behavior of ASTRID with a gas power conversion system. *Nuclear Engineering and Design*, 308:20-29, 2016.

D. Bestion. The physical closure laws in the CATHARE code. *Nuclear Engineering and Design*, 124(3):229-245, 1990. ISSN 0029-5493.

Y. Birari, S. Gosavi, and P. Jorwekar. Use of cfd in design and development of r404a reciprocating compressor. In *Proceedings of the International Compressor Engineering Conference*, Purdue, IN, 07 2006.

M. Casey and C. Robinson. A New Streamline Curvature Through-flow Method for Radial Turbomachinery. *Journal of Turbomachinery*, 132(3), 04 2010.

C. Cornelius, T. Biesinger, P. Galpin, and A. Braune. Experimental and Computational Analysis of a Multistage Axial Compressor Including Stall Prediction by Steady and Transient CFD Methods. *Journal of Turbomachinery*, 136(6), 11 2013. ISSN 0889-504X.

S. Derakhshan and A. Nourbakhsh. Theoretical, numerical and experimental investigation of centrifugal pumps in reverse operation. *Experimental Thermal and Fluid Science*, 32(8):1620-1627, 2008. ISSN 0894-1777.

S. Dixon and C. Hall. Chapter 7 - Centrifugal Pumps, Fans, and Compressors. In *Fluid Mechanics and Thermodynamics of Turbomachinery (Seventh Edition)*, pages 265-317. Butterworth-Heinemann, Boston, Seventh Edition edition, 2014. ISBN 978-0-12-415954-9.

R. P. Dring and H. D. Joslyn. Through-Flow Modeling of Axial Turbomachinery. In *Turbo Expo: Power for Land, Sea, and Air*, volume 1, 09 1985.

R. P. Dring and G. C. Oates. Through Flow Theory for Nonaxisymmetric Turbomachinery Flow. In *Turbo Expo: Power for Land, Sea, and Air*, volume 1, 06 1989.

P. Emonot, A. Souyri, J. Gandrille, and F. Barré. CATHARE-3: A new system code for thermal-hydraulics in the context of the NEPTUNE project. *Nuclear Engineering and Design*, 241(11):4476-4481, 2011. ISSN 0029-5493.

B. Faydide and J. Rousseau. Two-phase flow modeling with thermal and mechanical non-equilibrium. In *European Two Phase Flow Group Meeting*, Glasgow, UK, 1980.

K. Ghorbanian and M. Gholamrezaei. An artificial neural network approach to compressor performance prediction. *Applied Energy*, 86 (7):1210-1221, 2009. ISSN 0306-2619.

Gutiérrez Velásquez E. I. Determination of a suitable set of loss models for centrifugal compressor performance prediction. *Chinese Journal of Aeronautics*, 30(5):1644-1650, 2017. ISSN 1000-9361.

M. Hathaway, J. Wood, and C. Wasserbaueur. NASA Low-Speed Centrifugal Compressor for 3-D viscous code assessment and fundamental flow physics research. In *Gas Turbine and Aeroengine Congress and Exposition*, Orlando, Florida USA, 1991.

M. Hathaway, R. Chriss, A. Strazisar, and J. Wood. Experimental and computational investigation of the NASA Low-Speed Centrifugal Compressor flow field. *NASA Technical Memorandum 4481*, 1993.

M. Hathaway, R. Chriss, A. Strazisar, and J. Wood. Laser anemometer measurements of the three-dimensional rotro flow field in the NASA Low-Speed Centrifugal Compressor. *NASA Technical Paper TP-3527*, 1995.

M. Hofer, M. Buck, A. Cagnac, T. Prusek, N. Sobecki, P. Vlcek, D. Kriz, F. Hecker, and A. Hacks. Report on the validation status of codes and models for simulation of sCO2-HeRo loop. Technical report, sCO2-4-NPP project, Deliverable 1.2, 2020.

International Atomic Energy Agency. *Accident Analysis for Nuclear Power Plants*. Number 23 in Safety Reports Series. Vienna, 2002. ISBN 92-0-115602-2.

J. Joo, G. Medic, D. Philips, and S. Bose. Large-eddy simulation of a compressor rotor. In *Proceedings of the Summer Program, Center for Turbulence Research*, page 467. Center for Turbulence Research Stanford, CA, USA, 2014.

B. Lakshminarayana. Fundamental Principles, Analysis, and Performance of Turbomachinery. In *Fluid Dynamics and Heat Transfer of Turbomachinery*, chapter 2, pages 43-188. John Wiley & Sons, Ltd, 1995. ISBN 9780470172629.

P. Le Sausse, P. Fabrie, Arnou, D., and Clunet, F. CFD comparison with centrifugal compressor measurements on a wide operating range. *EPJ Web of Conferences*, 45:01059, 2013.

E. Lemmon, M. Huber, and M. McLinden. NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 9.1, 2013.

D. J. Lucia, P. S. Beran, and W. A. Silva. Reduced-order modeling: new approaches for computational physics. *Progress in Aerospace Sciences*, 40(1):51-117, 2004. ISSN 0376-0421.

L. Matteo. *Modelisation unidimensionnelle du comportement d'une pompe rotodynamique en fonctionnement normal et accidentel*. PhD Thesis, Ecole nation supérieure d'arts et métiers - EN-SAM, Oct 2019.

L. Matteo, A. Dazin, and N. Tauveron. Development and validation of a one-dimensional transient rotodynamic pump model at component scale. In *29th IAHR Symposium on Hydraulic Machinery and Systems*, volume 240, page 11, Kyoto, Japan, Sept. 2018. IOP Publishing.

L. Matteo, F. Cerru, A. Dazin, and N. Tauveron. Investigation of the Pump, Dissipation and Inverse Turbine Operating Modes Using the CATHARE-3 One-Dimensional Rotodynamic Pump Model. In *The ASME - JSME - KSME Joint Fluids Engineering Conference 2019. AJKFLUIDS2019*, San Francisco, United States, 07 2019a.

L. Matteo, A. Dazin, and N. Tauveron. Modelling of a Centrifugal Pump Using the CATHARE-3 One-dimensional Transient Rotodynamic Pump Model. *International Journal of Fluid Machinery and Systems*, 12, 03 2019b.

L. Matteo, G. Mauger, A. Dazin, and N. Tauveron. Simulation of a Radial Pump Fast Startup and Analysis of the Loop Response Using a Transient 1D Mean Stream Line Based Model. *International Journal of Turbomachinery, Propulsion and Power*, 4:38, 11 2019c.

L. Matteo, G. Mauger, P. Gyomlai, A. Dazin, and N. Tauveron. A Rotodynamic Pump Seizure Transient Simulated Using the CATHARE-3 One-Dimensional Pump Model. In *NURETH-16*, Portland, OR, USA, 08 2019d.

G. Mauger, F. Bentivoglio, and N. Tauveron. Description of an improved turbomachinery model to be developed in the CATHARE-3 code for ASTRID power conversion system application. In *NURETH-16*, Chicago USA, 2015.

G. Mauger, N. Tauveron, F. Bentivoglio, and A. Ruby. On the dynamic modeling of Brayton cycle power conversion systems with the CATHARE-3 code. *Energy*, 168:1002-1016, 2019.

G. Medic, V. Zhang, G. Wang, J. Joo, and O. P. Sharma. Prediction of Transition and Losses in Compressor Cascades Using Large-Eddy Simulation. *Journal of Turbomachinery*, 138(12), 06 2016. ISSN 0889-504X.

J. Miettinen. Nuclear power plant simulators: Goals and evolution. In *Seminar on Transfer of Competence, Knowledge and Experience Gained through CSNI Activities in the Field of Thermal-hydraulics (THICKET)*, Pisa, Italy, Jul 2008.
H. W. Oh, E. S. Yoon, and M. K. Chung. An optimum set of loss models for performance prediction of centrifugal compressors. *Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy*, 211(4):331–338, 1997.

M. V. Petrovic, A. Wiedermann, and M. B. Banjac. Development and Validation of a New Universal Through Flow Method for Axial Compressors. 7: Turbomachinery, Parts A and B:579–588, 06 2009.

R. Préa, P. Fillion, L. Matteo, G. Mauger, and A. Mekkas. CATHARE-3 V2.1: The New Industrial Version of the CATHARE Code. In *Advances in Thermal Hydraulics 2020*, pages 730–742, October 2020.

X. Qiu, D. Japikse, J. Zhao, and M. R. Anderson. Analysis and Validation of a Unified Slip Factor Model for Impellers at Design and

Off-Design Conditions. *Journal of Turbomachinery*, 133(4), 042011. ISSN 0889-504X.

S. S. Saravi and S. A. Tassou. An investigation into sCO2 compressor performance prediction in the supercritical region for power systems. *Energy Procedia*, 161:403–411, 2019. ISSN 1876-6102.

I. Shahin, M. Gadala, M. Alqaradawi, and O. Badr. Centrifugal Compressor Spiral Dry Gas Seal Simulation Working at Reverse Rotation. *Procedia Engineering*, 68:285–292, 2013. ISSN 1877-7058.

J. D. Stanitz. Some theoretical aerodynamic investigations of impellers in radial and mixed-flow centrifugal compressors. *Transactions of the ASME*, 74:473–497, 1952.

A. Stodola. *Steam and Gas Turbines*. McGraw-Hill New York, 1927.

F. J. Wiesner. A Review of Slip Factors for Centrifugal Impellers. *Journal of Engineering for Power*, 89(4):558–566, 10/1967. ISSN 0022-0825.
H. W. Oh, E. S. Yoon, and M. K. Chung. An optimum set of loss models for performance prediction of centrifugal compressors. *Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy*, 211(4):331–338, 1997.

M. V. Petrovic, A. Wiedermann, and M. B. Banjac. Development and Validation of a New Universal Through Flow Method for Axial Compressors. 7: Turbomachinery, Parts A and B:579–588, 06 2009.

R. Préa, P. Fillion, L. Matteo, G. Mauger, and A. Mekkas. CATHARE-3 V2.1: The New Industrial Version of the CATHARE Code. In *Advances in Thermal Hydraulics 2020*, pages 730–742, October 2020.

X. Qiu, D. Japikse, J. Zhao, and M. R. Anderson. Analysis and Validation of a Unified Slip Factor Model for Impellers at Design and

Off-Design Conditions. *Journal of Turbomachinery*, 133(4), 042011. ISSN 0889-504X.

S. S. Saravi and S. A. Tassou. An investigation into sCO2 compressor performance prediction in the supercritical region for power systems. *Energy Procedia*, 161:403–411, 2019. ISSN 1876-6102.

I. Shahin, M. Gadala, M. Alqaradawi, and O. Badr. Centrifugal Compressor Spiral Dry Gas Seal Simulation Working at Reverse Rotation. *Procedia Engineering*, 68:285–292, 2013. ISSN 1877-7058.

J. D. Stanitz. Some theoretical aerodynamic investigations of impellers in radial and mixed-flow centrifugal compressors. *Transactions of the ASME*, 74:473–497, 1952.

A. Stodola. *Steam and Gas Turbines*. McGraw-Hill New York, 1927.

F. J. Wiesner. A Review of Slip Factors for Centrifugal Impellers. *Journal of Engineering for Power*, 89(4):558–566, 10/1967. ISSN 0022-0825.