# Streamline Rotodynamic pump model for two-phase flow simulation

Simon Martel, Laura Matteo, Gédéon Mauger, Timo Bouvard, Regiane Fortes
Patella

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

## To cite this version:

Simon Martel, Laura Matteo, Gédéon Mauger, Timo Bouvard, Regiane Fortes Patella. Streamline Rotodynamic pump model for two-phase flow simulation. IOP Conference Series: Earth and Environmental Science, 2024, 32nd IAHR Symposium on Hydraulic Machinery and Systems, Roorkee, India, September 11-14, 2024, 1411, pp.012063. 10.1088/1755-1315/1411/1/012063. cea-04922349

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# Streamline rotodynamic pump model for two-phase flow simulation

Simon Martel¹,², Laura Matteo³, Gédéon Mauger⁴, Timo Bouvard¹ and Regiane Fortes-Patella²

¹Service de Thermohydraulique et de Mécanique des Fluides, CEA Paris-Saclay, Gif-sur-Yvette, France

²Laboratoire des Écoulements Géophysiques et Industriels, Grenoble INP, Université Grenoble-Alpes, Saint-Martin-d'Hères, France

³Service d'Etude des Systèmes Innovants, CEA Cadarache, Saint-Paul-Lez-Durance, France

⁴Service d'Etudes Mécaniques et Thermiques, CEA Paris-Saclay, Gif-sur-Yvette, France

E-mail: simon.martel@cea.fr

## Abstract.

This study is concerned with the simulation of water-vapor two-phase flows in pressurized water reactors (PWR) primary pumps. The correct modeling of such flows is essential to predict the consequences of a Loss Of Coolant Accident (LOCA), an accident scenario involving a rupture in the reactor primary circuit. During such accidents, pressure loss leads to important coolant vaporization in the circuits, pumps and other system components.

A review of turbomachinery streamline models, used to represent pumps in system codes, is presented. Particular attention is paid to the model used in the CATHARE-3 system code, as its development is the motivation for the present work.

Numerical simulations were then performed using CATHARE-3 streamline pump model, currently validated for single-phase liquid flows. These simulations included test cases for pure liquid and vapor conditions, as well as one two-phase flow case, with inlet void fraction of approximately 10 %.

For pure liquid water, the agreement is good for discharge between 75% and 150% of nominal discharge. Results are satisfying up to 300% of nominal discharge for the single-phase vapor case. The two-phase results are close to experimental values around 75 and 130 % of nominal discharge.

Further work is required in order to consolidate the physical analysis and to improve and validate CATHARE-3 streamline pump model for two-phase flow conditions.

# 1 Introduction

## 1.1 Industrial context

In order to demonstrate the safety of current and under-design nuclear power plants, a large number of simulations of accident scenarios must be performed using thermal-hydraulic system codes [1]. In France, this task is performed using the CATHARE code, developed by the Atomic Energy Commission (CEA) and its partners. For Pressurised Water Reactors (PWR), the most common reactor technology, one of the most critical accident scenario is the Loss Of Coolant Accident (LOCA). The study of LOCA aims at estimating the consequences of a break in the primary circuit. During such an accident, the primary pressure drops from its nominal value of 15.5 MPa to a value close to atmospheric pressure. This causes the vaporization of the coolant, which greatly reduces its ability to remove residual heat. Consequently,

<figure><img src="image_3.png" /></figure>

Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
a particular effort on pump modeling is required, as their performance is strongly affected by the high void fractions. Depending on the inlet conditions, head and torque degradation, transient phenomena, flow reversals or impeller runaway may be encountered.

Two pump models exist in CATHARE. The first one, currently used for industrial applications, is based on empirical and similarity approaches. While it is satisfactory for current power plant modeling, it has no predictive capacity. This is a serious limitation when trying to model circuits with under-design pumps, or reactors using different fluids for coolant (such as sodium). For this reason, the CEA decided in 2015 to implement a predictive, generic turbomachinery model, capable of simulating transient pump behavior from its general geometry. This was done as part of previous work [2]. The validation was carried out at component and system scale, on several pump geometries, for single-phase conditions.

The aim of this work is to give this model the capability to perform transient computations for two-phase steam-water flows, such as those encountered in LOCAs scenarios.

## 1.2 The CATHARE code

CATHARE is a thermal-hydraulic system code, that has the capability to model a complete reactor or its primary circuit. It is owned and developed by CEA, Framatome, Electricité de France and IRSN. The circuit is modeled with a set of one-dimensional (1D), point (0D) and three-dimensional (3D) elements, with additional modules representing components such as valves, pumps, heating walls, neutronics, etc. Six local conservation equations (mass, momentum and energy for each of the two phases) are solved on these component meshes. A set of closure laws is used to model the small-scale phenomena which are cancelled out by phase and time averaging.

More details on the CATHARE code and its equations can be found in dedicated publications such as [3].

## 1.3 Phenomenology

The consequences of vapor injection in centrifugal pumps were broadly studied in the 80s, in the context of nuclear energy development [4].

The loss in pump performance is generally represented using a degradation function, which is defined by the two-phase head and specific torque divided by their single-phase values [5, 6]. When plotting such a function against the inlet void fraction, for a given operating point (Q and ω fixed), one generally obtains a U-shaped curve, as shown in figure 1. For low void fractions (up to 20-30 %), the two-phase values are very close to single-phase ones. Close values are also obtained for high void fractions (α > 95%), since the head and specific torque of a pump are fluid-independent. A small performance increase can sometimes be observed for α values around 10-20%. On the other hand, degradation is maximal for intermediate void fractions (α ∈ [30%, 95%]).

Such an effect is explained by a change in flow regime. For low void fractions, the gas is dispersed in the liquid phase, an the pump behaviour close to single-phase flow. Beyond a given threshold, the flow becomes stratified, and shows free-surface behaviour. The energy supplied to the fluid is then transformed into kinetic energy, and no longer into pressure [6]. This results in high desadaptation losses at diffuser inlet, which are responsible for the observed performance losses [7].

Two additional elements affecting the pump behavior are to be considered. First, it was proven [5, 7] that vaporization could occur in the pump, especially in the impeller. Secondly, for high pressure differences, compressible effects such as critical discharge and shocked flow effects have been observed for water-vapor mixtures. This last phenomenon is particularly important since it can, in some configurations, govern by itself the pump behavior [5, 8].

# 2 State-of-the-art of pump modeling with streamline models

## 2.1 Common features

Streamline models are turbomachinery models where the fluid variables are solved along a mean streamline, the values being averaged over a cross-section. They can be seen as an intermediate modeling approach between point models and three-dimensional CFD models. Compared to point models, such as approaches based on Euler theory or characteristic matrices methods, streamline models offer the possibility to have space-dependent results. This allows to get information on local effects inside the pump. Compared to CFD tools, a pump code based on streamline models offers a computational time reduction of several orders of magnitude. This last element makes it an appropriate choice for a system code, where the system size - a full PWR primary circuit - makes the use of CFD tools impossible for two-phase transients [2].
<figure><img src="image_4.png" /><figcaption>Figure 1: Example of the U-shaped curve of a pump two-phase degradation function, from [4]. $H_{2\phi}/H_{1\phi}$ is the degradation function, plotted against inlet void fraction ("taux de vide")</figcaption></figure>

The different streamlines models existing in the literature share many common features.

First, they all solve a set of one-dimensional conservation equations. This is done in order to be able to compute the evolution of the fluid variables all along the pump, which is impossible with point models. A second common feature is the use of a two-fluid model: energy, momentum and mass conservation are solved for each phase. This is done in order to correctly model phase stratification and phase velocity slip. When modeling two-phase flow, a phase change term is required to correctly model vaporization/consensation effects.

Inside the impeller, the equations are solved in a moving frame, for relative velocities. A source term is added to model the centrifugal force. The most common source term for centrifugal acceleration in momentum and energy conservation equations is [5, 9]:

$$a_c = \frac{1}{2} \omega_2 \frac{\partial R^2}{\partial z} \qquad (1)$$

Other source terms have to be added to take into account effects of desadaptation losses, recirculation, and other phenomena proper to turbomachinery. The main difference between streamline models is the set of solved equations and the modeling choices of closure laws.

A last common feature of these models is what is called *adaptative geometry*. The research of Grison and Lauro [5, 8], as well as many other contributions (see for example [10]) revealed that, for off-design operation of a turbomachine, the effective cross-section of the inter-blade channels is not constant, due to the presence of recirculation regions. When implemented, the different streamline models of the literature have a module to change the cross-section depending on the inlet discharge. The new cross-section value is generally computed using velocity triangles.

## 2.2 Historical approach

The HEXPO model is the first streamline model met in the literature. It was developed in the scope of the EPOPEE campaign, lead under the direction of Grison and Lauro [5].

The HEXPO model had all the main characteristics of streamline models presented in section 2.1 and set the theoretical basis for much of the modeling work that followed. It was based on six steady conservation equations, and aimed at simulating two-phase flows across a PWR primary pump during LOCA accidents. According to the published results [5, 8], this model precision was good in pump mode and turbine mode, including shocked flows, but insufficient in the computation of flow reversals. In addition, the tests were limited to PWR primary pumps geometry.

De Crecy's main contribution to this article bibliography is a set of articles and technical reports proposing a new theoretical approach of stratification process [11] and their use in streamline pump models [12]. The proposed model aimed at taking the best elements of several research programs [6]: CATHARE equations and closure laws, Grison and Lauro modeling approach, and De Crecy's own research on stratification. It was based on CATHARE-2 six unsteady conservation equations (which were very close from CATHARE-3 equations), but used many features from HEXPO model: variable section, loss terms and change of frame.

De Crecy's research on stratification aimed at generalizing the criteria used in CATHARE-2, which were based on Froude number. His approach was to consider each phase pressure as the sum of interfacial pressure and a corrective term:
<figure><img src="image_5.png" /><figcaption>Figure 2: Graphical visualization of a centrifugal pump modeled with CATHARE-3 streamline pump model [16]</figcaption></figure>

$$P_k = P_i + \Delta_k \quad (2)$$

An expression allowing to compute an approximate value of $\Delta_k$ as a function of the problem variables was also proposed. As the $P_i$ terms was already modeled in CATHARE-2, this approach had the advantage to take into account the phase pressure differences while keeping a six-equations system. Although De Crecy's contribution is mainly theoretical, his model can be a strong source of inspiration, in the light of the importance of stratification phenomena in two-phase turbomachinery physics.

From De Crecy's work, a new model was developed and implemented in the CATHARE-2 code [13]. It was based on the code's six equations system, and incorporated the main features of the HEXPO model (adaptive geometry, centrifugal acceleration source term, etc), but used a different set of laws for head losses modeling. The only element adopted from his work on stratification was the use of a generalized Froude number, to take into account the centrifugal acceleration in the stratification effects [14]. In this model, the head loss terms were modeled using pipe-specific loss laws, which led to several difficulties when trying to generalize the model to other geometries [15]. Although this model was tested on a few different pump geometries, no generic prediction capacity could be achieved. It was finally removed from CATHARE-2 last versions.

## 2.3 CATHARE-3 streamline pump model

CATHARE-3 streamline pump model was developed, implemented and validated for monophasic flows and cavitation situations during the PhD work of Laura Matteo [2].

In this model, the pump is represented by three modified CATHARE unidimensionnal elements : suction pipe, impeller and diffuser/volute, as shown in figure 2.

The 1D CATHARE equations, given in [3], are modified as follows [9, 17]:

* The source term $a_c = \frac{1}{2}\omega^2\frac{\partial R^2}{\partial z}$ representing the impact of centrifugal acceleration is integrated to momentum and energy conservation equations in the impeller component.

* Loss terms are added to the momentum equations: secondary flows losses in the volute part of the diffuser/volute 1D element, and desadaptation losses term at impeller inlet.

* A loss term representing recirculation losses is added to the energy equations.

* Impeller outlet blade angle is modified to take into account the deviation phenomenon.

* A reference frame change is executed at impeller and diffuser inlets, in the same way as it is done with the HEXPO and CATHARE-2 pump models. The equations are solved for the relative velocity in the impeller.

This model uses most of other streamline models features; however, a set of new key elements were developed compared to previous models.

*Deviation*: the Coriolis force influence is modeled through a deviation correlation, that changes the effective outlet blade angle :

$$\beta_2' = \arctan \left( \frac{1}{\frac{\cos(\beta_2)}{\sin(\beta_2)} + F_{dev}} \right) \quad (3)$$
where $\beta_2$ is the geometric outlet blade angle, $\beta'_2$ the fluid effective outlet angle, and $F_{dev}$ a coefficient whose expression depends on the chosen correlation ; in CATHARE-3, four correlations are implemented: Wiesner, Stodola-Matteo (Stodola correlation generalized to helico-axial pumps), Stanitz and Qiu [2, 9, 16]. The Wiesner model is based on empirical data, while Stodola, Stanitz and Qiu are mechanist correlations. Qiu correlation is a more recent model, that asks for more geometrical information than the other three.

**Loss terms:** one of the main difficulties of previous model was to obtain general enough laws for loss terms proper to turbomachinery. In CATHARE-3, the correlations for desadaptation losses, recirculation losses (impeller) and secondary flows (volute) are function of the pump's specific speed. They were obtained using a set of pumps of various geometries. This use of non-dimensional numbers gives the model its predictive capacity on a wide range of geometries [2].

**Validation:** CATHARE-3 streamline pump model was validated on several pumps with various specific speeds, and a compressor, for single-phase conditions, during previous works [2].

**Cavitation:** a cavitation model was developed and tested on a small centrifugal pump. It consists in a modified terms for flashing and condensation in the CATHARE conservation equations [18].

# 3 EVA pump physical model

## 3.1 Motivation

As CATHARE-3 streamline pump model is based on a two-fluid equations system, it is possible, from a numerical point of view, to perform two-phase flow computations with high inlet void fractions. However, as the physical closure laws were developed and validated for single-phase flows only, modeling errors are expected at some point for such conditions.

The present work intends to evaluate the performance of the present pump model representing high two-phase flows, such as are met in LOCA accident conditions. It also aims at understanding which closure laws should be modified or implemented to obtain reliable predictions in such conditions.

In order to do so, a two-phase pump test loop has been modeled using CATHARE-3, and predictions obtained with the streamline pump model compared to experimental results.

## 3.2 Description of test facility

The experimental setup modeled is the EVA loop, a two-phase test loop built in CEA Cadarache in the late 70s [5, 4]. It consisted in a closed loop with vapor injection, and a reduced model of a PWR primary pump at scale 1/3. The experimental campaign was the result of a collaboration between CEA, Framatome and Westinghouse. Test were carried out for a wide range of discharges, rotation speed, inlet pressures and void fractions. The results of the campaign were used to derive some of CATHARE-3 physical models.

A schematic view of the test loop is given in figure 3. The EVA loop allowed the following experimental conditions [4]:

* Pressure difference from 7 to 28 bar

* Temperature from 50 to 232°C

* Liquid water flow rate up to 2840 $m^3/h$

* Steam flow rate up to 103 $t/h$

* Void fraction from 0 to 100 %

* Impeller speed from -100 % to 200 % of nominal value

## 3.3 Numerical model

The first step to build the numerical model is to write the pump geometry description in the CATHARE-3 input file. Sections, diameters, blade angles and blade number are typical value needed for this model. A more exhaustive list of input elements is given in previous publications [9, 17, 19].

The next step is to set the boundary conditions. For monophasic liquid computations, the standard boundary conditions, a discharge at inlet and a pressure at outlet, were used. This comes with no
<figure><img src="image_6.png" /><figcaption>Figure 3: Schematic view of EVA test loop [6]</figcaption></figure>

difficulties, as the fluid can be considered incompressible. For two-phase flows, on the other hand, inlet pressure is much more influent. It is necessary to impose it with precision. The choice of setting a pressure at inlet and a discharge at outlet is not feasible for stability reasons. Therefore, to solve this issue, two possibilities were left :

* Impose two pressure conditions and compare the computed discharges

* Impose discharge at inlet, pressure at outlet, and use a PID-type controller that regulates the outlet pressure to reach the needed inlet pressure

Both approaches were tested during this work, but the use of two pressure conditions raised algorithmic issues that have not yet been solved. Therefore, in the following section, only results obtained through the PID-approach are presented.

The PID controller approach works as follows : a target inlet pressure $P_{target}$ is given to the controller, and the outlet pressure is modified until the inlet pressure is close enough from the target value. The applied correction takes the following form :

$$e(t) = K_p \cdot \epsilon(t) + K_I \cdot \int_0^t \epsilon(\tau)d\tau + K_d \cdot \frac{d\epsilon}{dt}(t) \quad (4)$$

with $\epsilon = P_{inlet} - P_{target}$ and $K_p, K_I$ and $K_d$ are coefficients to be determined through try-and-error.

The last element is the choice between the implemented deviation correlations (see section 2.3). In the present case, single-phase liquid pump characteristics are available. It was then decided to test the available correlations against these data points, in order to select the most appropriate to our case. In the absence of information, the use of default correlation (Wiesner) is recommended, except for impellers with a high number of blades, where the Stanitz correlation is often more precise [2].

# 4 Results

## 4.1 Computed physical variables

In the following parts, we plot two physical quantities : the piezometric elevation $H_p$, and the hydraulic torque $C_h$.

The piezometric elevation corresponds to the pressure difference between inlet and outlet, changed to a length dimension in order to be density-independent. It is defined as :

$$H_p = \frac{\Delta P}{\rho_{inlet} \cdot g} \quad (5)$$
The hydraulic torque $C_h$ is defined as the power transferred from the impeller blades to the fluid. It is divided by the pump rotation speed to make it homogenous to a torque. It is supposed that the volumetric losses are negligible, and that the pump is adiabatic. The power dissipated by hydraulic losses is therefore transformed into temperature.

Hydraulic torque can be then computed as follows :

$$C_h = \frac{P_t}{\omega} = \frac{\dot{m}}{\omega} (h_{outlet}^{tot} - h_{inlet}^{tot}) \quad (6)$$

with $h^{tot} = (\alpha_G\rho_G + \alpha_L\rho_L)^{-1} \cdot (\alpha_L\rho_L \cdot (h_L + \frac{1}{2}V_L^2) + \alpha_G\rho_G \cdot (h_G + \frac{1}{2}V_G^2))$ the local (mass) total enthalpy, averaged on the two fluids [2]. The use of enthalpy instead of energy elevation allows to take into account the energy dissipated through hydraulic losses. Computing $C_h$ with CATHARE-3 is easy, since phase enthalpy and velocity are state variables.

It is more complex to obtain from experimental data. In our dataset, it is computed as follows :

$$C_h = C_m - C_f \quad (7)$$

$C_m$ is the torque measured on the shaft, and $C_f$ a term modeling the mechanical losses: disk friction, bearings and labyrinth seals. $C_f$ is given by an empirical correlation, function of inlet pressure and rotation speed. Its value is coherent with other existing laws, such as proposed by Stepanoff [20].

It is to be noted, however, that it was derived for monophasic liquid flows, and might not be correct enough for the case of two-phase or vapor flows.

## 4.2 Monophasic results

First, single-phase computations have been carried out both for pure liquid and pure vapor inlet conditions.

<figure><img src="image_7.png" /><figcaption>Figure 4: $H_p - Q$ and $C - Q$ curves for EVA pump at 1500 rpm, with single-phase liquid water flow</figcaption></figure>

Results for pure liquid are plotted on figure 4. Piezometric elevation and hydraulic torque obtained with CATHARE-3, for the three possible deviation laws, are compared to experimental results. The agreement is good for the flows between 75 and 150% of nominal discharge. Below this value, the head is overestimated, an error attributed to the pre-rotation, that is not modeled in CATHARE-3. Above the value of 150%$Q_N$, the pressure difference and torque are underestimated. Such difficulties were already met in previous works [2], and could be tackled by adapting the volute loss term. For the next computations, we will use the Stodolla-Matteo deviation correlation, that seems the most appropriate.

The next step was to launch monophasic vapor computations (see figure 5). We tried to reproduce the EVA test points with a void fraction of 100 %. Since the vapor was close to saturation values, the condensation models were still activated.

The piezometric head and hydraulic torque prediction is satisfying up to 300% of nominal discharge, after which both quantities show an important prediction error. Evaluation of torque prediction performance is ambiguous for low flow rates, since the measured torque is of the same order of magnitude than the estimated friction losses.
<figure><img src="image_8.png" /><figcaption>Figure 5: $H_p - Q$ and $C - Q$ curves for EVA pump at 1500 rpm, with single-phase steam flow</figcaption></figure>

### 4.3 Two-phase results

For our first two-phase computation, we chose to simulate the set of experimental test points with the following properties: $P_{\text{inlet}} \in [28.4 \text{ bar}, 29.6 \text{ bar}]$, $\alpha_{\text{inlet}} \in [8.6\%, 13.6\%]$, $N \in [1494 \text{ rpm}, 1505 \text{ rpm}]$, $T_{\text{inlet}} \in [228.7^\circ\text{C}, 230.8^\circ\text{C}]$.

The vapor and liquid flows are such that the volumic discharges are taken between approximately 50 % and 200 % of the nominal discharge. The variations of $P_{\text{inlet}}$, $\alpha_{\text{inlet}}$, $T_{\text{inlet}}$ and $N$ come from the difficulties to reach the exact set value with the test loop.

Figure 6 shows the result of the simulation. Two CATHARE computations were run : a first one trying to follow as close as possible the experimental inlet conditions; and a second one, aiming at plotting a curve with no other inlet change than discharge. This second curve was plotted with the following values: $P_{\text{inlet}} = 29 \text{ bar}$, $\alpha_{\text{inlet}} = 10\%$, $N = 1500 \text{ rpm}$, $T_{\text{inlet}} = 229.5^\circ\text{C}$ These two first curves are respectively identified as "C3, CST inlet" and "C3, exp inlet".

For comparison, experimental values corresponding to the single-phase liquid curve obtained for $N = 1500 \text{ rpm}$ are also plotted.

<figure><img src="image_9.png" /><figcaption>Figure 6: $H_p - Q$ curves obtained for EVA pump with CATHARE computations and experimental measurements for inlet void fractions of approximately 10 %</figcaption></figure>

The computations could not be run beyond 160 % due to numerical problems. Appart from this difficulty, we still observe, as for the monophasic CATHARE simulation, an error for high and low flow rates. As observed on the experimental case, the presence of vapor at pump inlet does not affect much its behavior in the CATHARE simulation. The data points close to nominal flow rate are well simulated by the numerical model. It seems, however, that the code does not grasp the small performance increase due to the vapor presence. This effect shall be investigated in further computations. At this point, this result seems rather positive, even though many additional computations will have to be runned before a conclusion can be drawn. For these first points, the model seems rather capable of executing two-phase computations. Many other configurations will have to be tested before a conclusion can be drawn.
# 5 Conclusion and perspectives

The present paper dealt with the question of simulation of two-phase flow in pumps, in the context of nuclear power plant accidents modeling. Many studies were conducted in the 70s and 80s in order to have a better understanding of these very complex phenomena. They revealed the importance of a correct modeling of stratification, vaporization and sonic blockage effects.

In order to increase the predictive capacity and generality of pumps modeling, many streamline modelizations were proposed to complement the traditional empirical models. Their main shared characteristics were the use of a unidimensional mesh, a two-phase six equations modeling and an adaptive geometry. The analysis of the research lead on these models highlighted the necessity to take into account physical effects proper to turbomachinery, such as deviation and recirculation, which was done in more recent studies. The last of the presented models was implemented in CATHARE-3 and validated in 2019. The target of present research, is to develop and validate its ability to realize two-phase computations.

Computations were carried on using this model, in order to examine its capacity to deal with high void fractions, conditions for which it was not yet adapted. First results, obtained with single-phase vapor flows, showed that the model could handle successfully such flows for discharge under 300 % of nominal value. A two-phase flow was also simulated, with an inlet void fraction of approximately 10 %. It showed a good capability of the model to reproduce the pump piezometric head near nominal point.

To extend CATHARE-3 streamline pump model to simulate two-phase flows characterized by high void ratios, several tasks will be undertaken. First, investigations on current streamline model capacity to simulate two-phase flow will be carried out for other void fractions and pump speeds. Careful inventory and further post-processing of the available experimental data are also in progress. These data will be valuable in order to validate the implemented models. The improvement of the streamline modeling will be based on the analyzes of the 3D CFD results obtained with the neptune_cfd software [21] thanks to an upscaling approach.

## Nomenclature

## Index

G Gas phase

L Liquid phase

N Nominal conditions

## Physical quantities

$\alpha$ Volumic void fraction (-)

$\alpha_k$ Phase volumic fraction

$\beta_2$ Outlet impeller blade angle (°)

$\beta'_2$ Fluid effective outlet angle (°)

$\dot{m}$ Mass flow rate (kg.s⁻¹)

$\omega$ Pump speed (rad.s⁻¹)

$\rho$ Fluid density (kg.m⁻³)

$a_c$ Centrifugal acceleration

$C_f$ Bearings, seals and disc friction losses torque (N.m)

$C_h$ Hydraulic torque (N.m)

$C_m$ Measured torque (N.m)

$h$ Static enthalpy (J.kg⁻¹)

$h^{tot}$ Total enthalpy (J.kg⁻¹)

$H_p$ Piezometric head (m)

$K_p, K_I, K_d$ PID controller coefficients

$N$ Pump speed (rpm)

$P$ Pressure (Pa)

$P_i$ Interfacial pressure (Pa)

$P_k$ Phase pressure (Pa)

$V_k$ Velocity of phase k (m.s⁻¹)

$Q$ Volumetric flow rate

$R$ Impeller mean line radius

$z$ Mean line coordinate

## Abbreviations

CEA Atomic Energy Commission

LOCA Loss Of Coolant Accident

PWR Pressurized Water Reactor

# References

[1] D. Bestion, "11 - The structure of system thermal-hydraulic (SYS-TH) code for nuclear energy applications," in *Thermal-Hydraulics of Water Cooled Nuclear Reactors* (F. D'Auria, ed.), pp. 639-727, Woodhead Publishing, Jan. 2017.

[2] L. Matteo, *Modélisation unidimensionnelle du comportement d'une pompe rotodynamique en fonctionnement normal et accidentel*. PhD thesis, Ecole Nationale Supérieure d'Arts et Métiers, Oct. 2019.
[3] D. Bestion, “The physical closure laws in the CATHARE code,” *Nuclear Engineering and Design*, vol. 124, pp. 229–245, 1990.

[4] P. Grison and J. Lauro, “Fonctionnement des pompes en écoulement diphasique eau-vapeur,” *Entropie*, no. 83, pp. 26–32, 1978.

[5] P. Grison and J. Lauro, “Bilan des études de thermohydraulique des pompes primaires de réacteurs PWR,” *La Houille Blanche*, vol. 8, no. 7, pp. 655–663, 1982.

[6] M. De Crécy, “Modélisation des pompes en régime diphasique,” in *La Houille Blanche*, vol. 4, pp. 261–265, 1984.

[7] P. Grison and J.-F. Lauro, “Comportement des pompes en régime diphasique,” *La Houille Blanche*, vol. 7, no. 6, pp. 405–411, 1979.

[8] P. Grison, “Exposé de M. Grison,” in *La Houille Blanche*, vol. 4, pp. 269–273, 1984.

[9] 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*, vol. 12, no. 2, pp. 147–158, 2019.

[10] P. Harley, S. Spence, D. Filsinger, M. Dietrich, and J. Early, “Meanline modeling of inlet recirculation in automotive turbocharger centrifugal compressors,” *Journal of Turbomachinery*, vol. 137, 2015.

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