Multi-mode modeling applied to the Main Feedwater System and Auxiliary Feedwater System switch in a Pressurized Water Reactor

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Multi-mode modeling applied to the Main Feedwater System and Auxiliary Feedwater System switch in a

Pressurized Water Reactor

Luis Corona Mesa-Moles

Master Of Science Thesis

Stockholm 2016

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2 AlbaNova University Center

KTH Royal Institute of Technology Division of Nuclear Power Safety 106 91 Stockholm, Sweden

TRITA-FYS 2016:40 ISSN 0280-316X

ISRN KTH/FYS/–16:40–SE

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Abstract

Electricity generation is a key-challenge for our societies. In this context, power plants have to fulfill a large number of requirements on safety, performance, cost and operational lifetime. It is important to be able to understand and to model correctly the behavior of power plants. Thus, the development of numeric simulation tools is very important. This development can provide more accurate results and also gives the possibility to simulate more complex scenarios. This should contribute to a better and more precise understanding of the behavior of nuclear power plants.

Moreover, this could lead to an increase of the public acceptance of nuclear power as well.

In order to improve the efficiency of its simulation tools (and reduce their costs), EDF decided to replace the LEDA code, developed internally by EDF and used for the static and dynamic simulation of power plants at the system level, by state-of-the-art, already available tools. It was found that Modelica based tools were the most suitable choice to replace the LEDA code. ^[1]

These Modelica based tools are evolving continuously. New features such as multi-mode modeling are currently under development. It consists in modeling systems that have multiple operating modes (switch on/off, dysfunctional behavior, activation of different sub-systems at different moments etc.).

The multi-mode modeling prototype developed by Dassault Systèmes (Dymola) in the framework of the MODRIO project is the most advanced tool of this type that uses the Modelica language ^[2]. EDF is one of the major partners of this European project and it is important for them to know if this new prototype can be used in the modeling of nuclear power plants. Therefore, the scope of this thesis is to know if multi-mode models of thermal-hydraulic components can be included in already existing nuclear power plant models for simulation.

The scenario chosen to test multi-mode modeling was the Main Feedwater System (ARE in French) and Auxiliary Feedwater System (ASG in French) switchover in which three different applications of multi-mode modeling could be tested. These applications are the phase appearance and disappearance in two-phase volumes, the switchover of thermal-hydraulic circuits and the modeling of dysfunctional components.

In this thesis the phase appearance and disappearance aspect of multi-mode modeling has been studied in detail. A multi-mode drum model adapted to the already existing models used by EDF was developed and tested. This drum can be full of liquid water, of vapor, or a mixture of both phases. Each of these three situations correspond to a state or mode of the drum. Results showed that this model was working correctly.

The next step was to use this multi-mode drum in a steam generator model by replacing the previous water/steam drum in which separation of liquid water and steam occurs by the new one.

Results showed that this new model of steam generator was able to simulate correctly phase appearance and disappearance.

The new model of steam generator was incorporated to the model of nuclear power plant used to simulate the Main Feedwater System and Auxiliary Eeedwater System switchover scenario.

Results showed that with this new model it was possible to simulate phase disappearance in a large model which was not possible before.

The switchover aspect of multi-mode was also studied during this thesis. In this case the Main Feedwater System and the Auxiliary Feedwater System are included in different modes or states

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4 of the multi-mode model. For this aspect the thesis showed that it is still necessary to improve the multi-mode modeling prototype itself in order to include large thermal-hydraulic circuits inside the states or modes. Due to these difficulties, mainly related to initialization problems, it was decided to simplify the already mentioned circuits. With this simplification it was then possible to simulate the Main Feedwater System and Auxiliary Feedwater System switchover scenario using a multi-mode model for the circuit switchover. However, these results are not entirely satisfying because of the use of simplified models.

As a conclusion, this study shows that the multi-mode modeling of phase appearance and disappearance works correctly and that it can be used in complex models of nuclear power plant for transients’ simulations. Moreover, this thesis shows that the switchover of circuits can be modeled even if many difficulties still remain. The modeling of dysfunctional components was not studied in this work, but the results obtained in this thesis are encouraging for the study of this aspect of multi- mode modeling.

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[1] Annick Souyri, Daniel Bouskela, Bruno Pentori and Nordine Kerkar. Pressurized Water Reactor Modeling with Modelica, Proceeding of the 5^th International Modelica Conference, September 2006.

[2] Hilding Elmqvist, Sven Erik Mattsson and Martin Otter. Modelica extensions for multi-mode DAE systems. Proceeding of the 10^th International Modelica Conference, March 2014.

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Sammanfattning

Elproduktion är en viktig utmaning för våra samhällen. I detta sammanhang måste kraftverk uppfylla ett stort antal krav på säkerhet, prestanda, kostnad och livslängd. Därför är det viktigt att kunna förstå och korrekt modellera kraftverks beteende. Således är det mycket viktigt att utveckla numeriska simuleringsverktyg. Denna utveckling kan ge mer korrekta resultat och ger också möjlighet att simulera mer komplexa scenarier. Detta bör bidra till att ge en bättre och mer exakt förståelse av kärnkraftverk. Dessutom kan detta också leda till en ökning av allmänhetens acceptans av kärnkraft.

För att förbättra effektiviteten av dessa simuleringsverktyg (och minska dess kostnader) beslutade EDF att ersätta LEDA-koden som används för statiska och dynamiska simulering av kraftverk på systemnivå, genom state-of-the-art redan tillgängliga verktyg. Man fann att Modelica-baserade verktyg är det lämpligaste valet för att ersätta LEDA-koden.^[1]

Dessa Modelica-baserade verktyg utvecklas kontinuerligt. Nya funktioner såsom multi-mode- modellering är för närvarande under utveckling. Den består av modelleringssystem som har flera driftlägen (switch on/off, dysfunktionellt beteende, aktivering av olika delsystem vid olika tidpunkter etc.).

Den multi-mode-modelleringsprototyp som utvecklats av Dassault Systèmes (Dymola) inom ramen för MODRIO-projektet är det mest avancerade verktyg för denna typ som använder Modelica- språket^[2]. EDF är en av de viktigaste partner i detta europeiska projekt och det är viktigt för dem att veta om denna nya prototyp kan användas i modelleringen av kärnkraftverk. Omfattningen av denna avhandling är därför att veta om multi-mode-modeller av termohydraliska komponenter kan ingå i redan befintliga kärnkraftverksmodeller för simulering.

Scenariot valt för att testa multi-mode-modellering var huvudmatarvattensystem (ARE på franska) och nödmatarvattensystemövergången (ASG på franska) i vilken tre olika tillämpningar av multi- mode-modellering kan testas. Dessa program är fasframträdande och fasförsvinnande i tvåfasvolymer, övergången av termohydraliska kretsar och modellering av dysfunktionella komponenter.

I detta arbete har aspekter av fasframträdande och fasförsvinnande av multi-mode-modellering studerats i detalj. I denna avhandling har en multi-mode-trummodell anpassad till de redan befintliga modeller som används av EDF utvecklats och testats. Denna trumma kan vara full av flytande vatten, ånga eller kan innehålla en blandning av båda faserna. Var och en av dessa tre situationer motsvarar ett tillstånd eller läge av trumman. Resultaten visade att denna modell fungerar.

Nästa steg var att använda detta multi-mode-trumma i en ånggeneratormodell genom att ersätta den tidigare vatten/ång-trumma där separation av flytande vatten och ånga sker genom den nya.

Resultaten visade att denna nya modell av ånggeneratorn kunde simulera korrekt fasframträdande och fasförsvinnande.

Den nya modellen av ånggeneratorn inkorporerades till modellen av kärnkraftverk som används för att simulera huvudmatarvattensystemets och nödmatarvattensystemets övergångsscenario.

Resultaten visade att det var möjligt att med denna nya modell simulera fasförsvinnande i en stor modell som inte var möjligt tidigare.

Övergångsaspekten av multi-mode studerades också under denna avhandling. I detta fall inkluderades huvudmatarvattensystemet och hjälpmatarvattensystemet i olika lägen eller tillstånd hos multi-mode- modellen. För denna aspekt visade avhandlingen att det fortfarande är nödvändigt att förbättra multi-

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6 mode-modelleringsprototypen för att inkludera stora termohydraliska kretsar inuti lägen eller moder.

På grund av dessa svårigheter, i huvudsak relaterade till initieringsproblem, beslutades det att förenkla de redan nämnda kretsarna. Med denna förenkling var det då möjligt att simulera huvudmatarvattensystemets och nödmatarvattensystemets övergångsscenario med ²användning av en multi-mode-modell för kretsövergången. Dessa resultat är dock inte helt tillfredsställande på grund av användningen av förenklade modeller.

Som en slutsats, visar detta arbete att multi-mode-modellering av fasframträdandet och fasförsvinnandet fungerar ordentligt och att den kan användas i komplexa modeller av kärnkraftverk för transienta simuleringar. Denna avhandling visade dessutom att kretsövergångar kan modelleras även om många svårigheter kvarstår. Den modelleringen av dysfunktionella komponenter har inte studerats i detta arbete, men de resultat som uppnåtts i denna avhandling är uppmuntrande för att studera denna aspekt av multi-mode-modellering.

[1] Annick Souyri, Daniel Bouskela, Bruno Pentori and Nordine Kerkar. Pressurized Water Reactor Modeling with Modelica, Proceeding of the 5^th International Modelica Conference, September 2006.

[2] Hilding Elmqvist, Sven Erik Mattsson and Martin Otter. Modelica extensions for multi-mode DAE systems. Proceeding of the 10^th International Modelica Conference, March 2014.

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Acknowledgements

I would like first to thank my supervisors Mme. Audrey JARDIN and M. Daniel BOUSKELA for giving me the opportunity to work on this topic and for welcoming me within the group “Fonctionnment et conduite” of STEP department.

I would like to thank M. Baligh EL HEFNI for all the good advices and for the help provided during the whole internship.

I would like also to address my gratitude to all the engineers and researchers of the group that helped me during my internship.

I would like to thank M. Sean ROSHAN GHIAS for his help and advices.

I would like to thank Prof. Sevostian BECHTA for accepting to be my supervisor and for his help. I would like to thank Prof. Waclaw GUDOWSKI for his help and advices.

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List of Figures

Figure 1.1 State machine concept applied to the modeling of phase appearance and disappearance

... 18

Figure 1.2 DynamicDrum ThermoSysPro component ... 19

Figure 1.3 Multi-mode mass and energy balance equations ^[13] ... 20

Figure 2.1 ThermoSysPro version of the multi-mode drum ... 31

Figure 2.2 Test scenario for the multi-mode drum ... 33

Figure 2.3 Liquid volume evolution in the multi-mode drum ... 38

Figure 2.4 Detail of the liquid volume evolution in the multi-mode drum ... 38

Figure 2.5 Pressure evolution in the multi-mode drum ... 39

Figure 2.6 Liquid specific enthalpy evolution in the multi-mode drum ... 39

Figure 2.7 State evolution in the multi-mode drum ... 40

Figure 2.8 Liquid mass fraction evolution in the multi-mode drum ... 40

Figure 2.9 Steam generator Modelica model ... 42

Figure 2.10 Multi-mode steam generator Modelica model ... 43

Figure 2.11 Modelica model of a 4 loop Nuclear Power Plant ... 44

Figure 2.12 Primary circuit ... 45

Figure 2.13 Pressurizer in the primary circuit ... 46

Figure 2.14 Pressurizer replaced in the primary circuit ... 46

Figure 2.15 Simplified primary circuit ... 47

Figure 2.16 Simplified representation of the ARE circuit ^[8] ... 48

Figure 2.17 Modelica representation of the ARE circuit (liquid water circulates from the right to the left) ... 48

Figure 2.18 Simplified representation of the ASG system ^[8]... 49

Figure 2.19 Modelica representation of the ASG system (feedwater circulates from the right to the left) ... 50

Figure 2.20 Modelica representation of the VVP system and other connected sub-systems ... 51

Figure 2.21 Modelica representation of the APP system ... 51

Figure 2.22 Modelica representation of the secondary circuit ... 52

Figure 2.23 Modelica representation of the reconstructed model ... 53

Figure 2.24 Modelica representation of the ARE/ASG switchover condition ... 55

Figure 2.25 ARE and ASG systems localization in the nuclear power plant ... 55

Figure 2.26 Liquid water volume in steams generator ... 56

Figure 2.27 Detail of the liquid water volume in steams generator ... 57

Figure 2.28 State machine for the ARE/ASG switchover multi-mode model ... 58

Figure 2.29 Modelica representation of the ARE/ASG switchover multi-mode model ... 59

Figure 2.30 Modelica representation of state1 (ARE only) ... 60

Figure 2.31 Modelica representation of state3 (ARE and ASG) ... 60

Figure 2.32 Modelica representation of state2 (ASG only) ... 61

Figure 2.33 Modelica representation of the secondary circuit of the PWR model ... 62

Figure 2.34 Modelica representation of the model used for testing the ARE/ASG switchover multi- mode model ... 63

Figure 3.1 Modelica representation of the multi-mode drum steady-state initialization... 65

Figure 3.2 Modelica representation of a model with the multi-mode drum ... 66

Figure 3.3 Heat flux applied in the vaporization loop ... 66

Figure 3.4 Liquid water and vapor volume in the multi-mode drum... 67

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10 Figure 3.5 Multi-mode drum state (3 corresponds to the two-phase state, 2 to the only vapor state

and 1 to the only liquid state) ... 67

Figure 3.6 Liquid specific enthalpy in the multi-mode drum ... 68

Figure 3.7 Vapor specific enthalpy in the multi-mode drum ... 69

Figure 3.8 Pressure in the multi-mode drum ... 70

Figure 3.9 Modelica representation of a model with the multi-mode drum ... 71

Figure 3.10 Vapor outlet mass flow rate in the multi-mode drum ... 71

Figure 3.11 Heating flux in the multi-mode drum ... 72

Figure 3.12 Liquid outlet mass flow rate in the multi-mode drum... 72

Figure 3.13 Inlet mass flow rate in the multi-mode drum ... 73

Figure 3.14 Liquid water and vapor volume in the multi-mode drum ... 74

Figure 3.15 Multi-mode drum state (3 corresponds to the two-phase state, 2 to the only vapor state and 1 to the only liquid state) ... 74

Figure 3.16 Liquid specific enthalpy in the multi-mode drum ... 75

Figure 3.17 Vapor specific enthalpy in the multi-mode drum ... 76

Figure 3.18 Pressure in the multi-mode drum ... 77

Figure 3.19 Modelica representation of the steam generator steady-state initialization ... 78

Figure 3.20 Modelica representation of a model with a multi-mode steam generator ... 80

Figure 3.21 Liquid water and vapor volume in the multi-mode water/steam drum of the steam generator ... 81

Figure 3.23 Pressure in the multi-mode water/steam drum of the steam generator ... 82

Figure 3.26 Pressure in the multi-mode water/steam drum of the steam generator ... 84

Figure 3.29 Coolant specific enthalpy ... 86

Figure 3.32 Pressure at the inlet of the turbines ... 89

Figure 3.33 Average water level in the steam generators ... 89

Figure 3.34 Secondary inlet SG1 ... 90

Figure 3.35 Secondary inlet SG2 ... 90

Figure 3.36 Normal control rods position... 91

Figure 3.37 Fine adjustment control rods position ... 92

Figure 3.38 Total thermal power of the reactor ... 92

Figure 3.39 Decay heat power ... 93

Figure 3.40 Total reactivity in the core ... 93

Figure 3.41 Power delivered by the steam turbine ... 94

Figure 3.42 Specific enthalpy of the coolant ... 95

Figure 3.43 Coolant mass flow rate ... 95

Figure 3.44 Pressure in the primary circuit (outlet of the core) ... 96

Figure 3.45 Fine adjustment control rods position ... 98

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Figure 3.46 Normal control rods position... 98

Figure 3.47 Total reactivity ... 99

Figure 3.48 Total reactor power and decay heat power ... 99

Figure 3.49 Specific enthalpy of the coolant ... 100

Figure 3.50 Fuel cladding maximal temperature ... 100

Figure 3.51 Angular velocity of the main feedwater pumps ... 101

Figure 3.52 Mass flow rate at the inlet of steam generators ... 101

Figure 3.53 Detailed view of the mass flow rate at the inlet of steam generators ... 102

Figure 3.54 Condition for the switching from the ARE to the ASG system ... 102

Figure 3.55 Mass flow rate in the Auxiliary Feedwater System... 103

Figure 3.56 Steam turbine mechanical power delivered ... 103

Figure 3.57 Mass flow rate through the steam turbine ... 104

Figure 3.58 Mass flow rate through the turbine bypass circuit ... 104

Figure 3.59 Liquid water volume inside the steam generator dome ... 105

Figure 3.60 Detailed view of the liquid water volume inside the steam generator dome ... 106

Figure 3.61 Vapor volume inside the steam generator dome ... 106

Figure 3.62 Detailed view of the vapor volume in the steam generator dome ... 107

Figure 3.63 Average water volume in the steam generators ... 107

Figure 3.64 State of the multi-mode water/steam drum inside steam generators (3 corresponds to the two-phase state, 2 to the only vapor state and 1 to the only liquid state) ... 108

Figure 3.65 Pressure in the steam generator dome ... 108

Figure 3.66 Liquid water volume in the dome of the steam generator ... 109

Figure 3.67 Vapor volume in the drum of the steam generator ... 110

Figure 3.68 State of the drum in the steam generator (3 corresponds to the two-phase state, 2 to the only vapor state and 1 to the only liquid state) ... 110

Figure 3.69 State of the ARE/ASG switchover model (1 corresponds to the only ARE system, 2 corresponds to the only ASG system and 3 corresponds to the both ARE and ASG systems) ... 111

Figure 3.70 Detailed view of the state of the ARE/ASG switchover model (1 corresponds to the only ARE system, 2 corresponds to the only ASG system and 3 corresponds to the both ARE and ASG systems) ... 111

Figure 3.71 Boolean transition conditions in the ARE/ASG switchover multi-mode model ... 112

Figure 5.1 Pressure in the dome of the steam generator ... 117

Figure 5.2 Liquid specific enthalpy in the dome of the steam generator ... 118

Figure 5.3 Liquid water density in the dome of the steam generator... 119

Figure 5.4 Inlet coolant specific enthalpy ... 120

Figure 5.5 Feedwater specific enthalpy ... 121

Figure 5.6 Feedwater mass flow rate ... 121

Figure 5.7 Pressure at the outlet of the steam generator (secondary circuit) ... 122

Figure 5.8 Mass vapor fraction of the liquid phase ... 123

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List of Tables

Table 2.1 Re-initialization of state variables choice ... 32 Table 3.1. Coolant Properties ... 78 Table 3.2 Secondary circuit propertie ... 79

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List of acronyms

APP Main Feedwater Pump

ARE Main Feedwater System

AREVA NP Areva Nuclear Power, ex-Framatome ASG Auxiliary Feedwater system

CATHARE The Code for Analysis of Thermalhydraulics during an Accident of Reactor and safety Evaluation

CEA Commissariat à l’Energie Atomique et aux Energies Renouvelables Dymola Dynamic Modeling Laboratory

EDF Electricité de France

IAEA International Atomic Energy Agency

IRSN Institut de Radioprotection et de Sûreté Nucléaire LEDA FORTRAN thermal hydraulic code developed by EDF MODRIO Model Driven Physical Systems Operation

Modelica A Unified Object-Oriented Language for Systems Modeling MPS Motopump of the Auxiliary Feedwater System

PWR Pressurized Water Reactor RHR Residual Heat Removal

SG Steam Generator

ThermoSysPro Library for the modeling and simulation of power plants and energy systems

TPA Main Feedwater Pump

TPS Turbopump of the Auxiliary Feedwater System

VVP Steam lines

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Introduction

Background

Electricity produced by nuclear power plants represents 10.6% of the electricity produced worldwide ^[3]. Nuclear power plants must fulfill a large number of requirements and how they do so must be studied and analyzed. Simulation tools are an important way to study this kind of requirements. These simulation tools can be used for different purposes such as safety analyses, improvement of efficiency, better understanding of the global behavior of nuclear power plants, etc.

In this context, electric utilities such as EDF have contributed to the development of some of these simulation tools. For example, EDF developed the LEDA code in the early 80’s. This code was maintained only by EDF and it was used to perform static and dynamic simulation at the system level. However, due to its ageing structure and EDF’s desire to improve the efficiency of its simulation tools, EDF decided to replace LEDA by Modelica based tools. The goal was to modernize this simulation tool, keeping up with the latest trend in modeling and simulation technology, while reducing the costs. Moreover, by using Modelica, which is a non-proprietary language, EDF encourages the development of open source software tools ^[1]. This should lead to an increase of these tools’ durability in the long run, and should contribute to foster partnership with other research institutions working on similar topics. Since this decision was made, EDF developed a Modelica open-source library for the modeling and simulation of its nuclear and conventional power plants called ThermoSyspro that has been validated against several test-cases ^[4]. Several simulation tools that implement Modelica language exist, but Dymola is the most mature one and is therefore preferred. ^[5]

For EDF, the aim of using Modelica based tools is to perform simulations at the system level in order to optimize and validate the design and the operation of its nuclear power plants facilities ^[1]. With this tool, all the phases of the plant lifecycle (from basic design to plant operation) can be studied and analyzed. It is a way for EDF to model and simulate easily and rapidly the physical behavior of its power plants. Therefore, its aim is not to replace other computer codes used for safety analyses such as CATHARE. This advanced safety code used for the studies of incidental or accidental transients on PWRs was developed jointly by CEA, EDF, AREVA NP and IRSN ^[6].

In partnership with other companies and research institutions, EDF has contributed to and encouraged the development of Modelica based tools. One of the latest features developed for Modelica based tools is multi-mode modeling. The most advanced prototype for multi-mode modeling has recently been developed by Dymola. This feature is a key challenge for the European project MODRIO in which EDF and Dymola participate ^[7]. Multi-mode modeling consists in modeling systems having multiple operating modes or states. For example, this can be used for turning components or subsystems on or off, modeling and simulating appearance and disappearance of phases (liquid and vapor), or switching of a component from normal operation to a dysfunctional mode (pump cavitation).

Even before this new prototype developed by Dymola was available, EDF started working on a scenario to test the potential applications of multi-mode modeling in their Modelica models of nuclear and conventional power plants. The chosen scenario was the Main Feedwater System (ARE in French) and Auxiliary Feedwater System (ASG in French) switchover as a consequence of the break of the two feedwater pumps ^[8]. This scenario was chosen because several aspects of multi-

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15 mode could be studied: switchover of thermal-hydraulic circuits (ARE and ASG), modeling of the dysfunctional behavior of a component (pump trip), and the phase appearance and disappearance (vapor and liquid). The study of the appearance and disappearance was particularly important since in previous simulations the disappearance of liquid water in the steam generators during the simulation of this transient was responsible for computational errors and negative volumes were obtained.

Thesis objectives

From a general perspective, the purpose of the thesis is to contribute to the development of multi-mode modeling and simulation in the framework of Modelica based tools. More precisely, the objective is to contribute to its evolution in the field of nuclear power plants by developing and testing new multi-mode models. As was mentioned in the previous section, Dymola has developed the most advanced prototype for this kind of modeling and simulation. Therefore, it is the simulation tool that this thesis is based on.

According to what has been indicated in the previous section, the main aspect of multi-mode to be studied in this thesis is its capability to model and simulate phase appearance and disappearance.

Therefore, the main objective of this thesis is to develop a multi-mode model able to correctly handle phase appearance and disappearance and to include this model in large models of nuclear power plants to simulate the ARE and ASG switchover scenario previously described.

The objective of this thesis is to study other aspects of multi-mode modeling based on Modelica tools as well. Therefore, the study of multi-mode modeling applied to the switchover of thermal-hydraulic circuits is also an important aspect of this thesis. More precisely, the objective is to develop a multi-mode model able of simulating the switchover of the Main Feedwater System to the Auxiliary Feedwater System.

Outline of the thesis

In order to clarify the modeling context, Chapter 1 presents the Dymola and ThermoSyspro environment. In Chapter 2, the different multi-mode models developed and its inclusion in the large model of nuclear power plant are presented. In Chapter 3 the simulation results and the verification of the multi-mode models is addressed. Finally, Chapter 4 summarizes the present work and states its main conclusions.

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1. Context and literature review

1.1 Modelica, Dymola and ThermoSysPro

Modelica

Modelica is an object-oriented, equation based language to model complex models in which mechanical, electrical, hydraulic, thermal or control subcomponents can be included. Moreover, it is a non-causal language which means that the equations included in Modelica models correspond to the physical equations, there is no an input-output logic. ^[9]

It is an open source language. A program is needed to simulate Modelica models, some of them are free like OpenModelica developed at Linköping University and others are paying like Dymola developed at Lund University (now bought by Dassault Systèmes).

In a typical Modelica model the basic structure used is very intuitive: the parameters and variables of the model are declared and then the physical equations are written.

Dymola

As already mentioned Dymola is the simulation environment in which the thesis has taken place. Nowadays it is the most powerful Modelica simulation tool ^[5]. Moreover, Dymola has developed the only multi-mode prototype than can handle differential algebraic equations. This is of course necessary when modeling the dynamic behavior of thermal-hydraulic circuits: energy, mass and momentum balance equations must be included in the model. That is why this tool was chosen for the thesis.

As indicated in the introduction, Dymola is not used by EDF to perform safety analyses. It is used to perform simulations of its power plants at the system level in order to understand better its physical behavior. These simulations can be static or dynamic. In the first case simulations are used for plant sizing and in the second case they are used for the study of normal or incidental plant transients. In both cases simulations results’ aim is to provide a better understanding of the global behavior of the power plant.

With Dymola, the different components of a given model can easily been added and/or modified in order to get the desired level of complexity in the model. Moreover, the simulations can simply been performed. In Dymola, a graphical view and a text view can be used to develop models.

Usually the text view is used to create new components. The graphical view is usually used to include and put together several components through a “drag and drop” action. This leads to a simulation environment having a high flexibility.

The use of Dymola by EDF does not replace the use of codes dedicated to safety analyses or to quantify conservative analysis margins such as CATHARE (Code Avancé de THermohydraulique pour les Accidents des Reacteurs à Eau (REP), Advanced Safety Corde for Pressurized Water Reactors (PWR)).

Its goal is to replace the aging code LEDA that was developed internally by EDF.

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17 ThermoSysPro

ThermoSysPro is a library of thermal-hydraulic components developed in Modelica language by Daniel Bouskela and Baligh El Hefni. It contains over 100 0D/1D model components (heat exchangers, pipes, drums, tanks, combustion chambers etc.). This library is used at EDF for the modeling of all kind of power plants. In in particular it is used for the modeling of nuclear power plants (components for modeling the reactor core are also included in the library). In this thesis, the model of nuclear power plant used for simulations is constructed with components of this library.

The components of this library are able to describe satisfactory single mass flow and two- phase flow. This library uses a finite volume approach, based on the staggered grid scheme for space discretization. Discretization is only made along the main direction, resulting in a 1D approach. In this library, there are two basic types of components: nodes and edges. Nodes represent mixing volumes such as tanks, splitters, mergers or boilers. In the nodes are implemented the mass and energy balance equations. Edges represent flow resistant elements such as pipes or valves. In the edges are implemented the momentum balance equations. A given model is constructed by connecting edges to nodes and nodes to edges in order to get a complete set of mass, energy and momentum equations. That way, the numerical scheme requirements are automatically fulfilled. ^[1]

The components of the library are able to deal with flow reversals in the approximation of convective flow only. It is considered that convection is the main mechanism responsible for energy transportation ^[10]. The study of diffusive transfers is now under development, the goal being to get a more robust computation of flow reversal near zero-flow. ^[11]

In the different components of the library, up-to-date pressure and heat exchange correlations are used. For the properties of water and steam, a Modelica implementation of the IAPWS-IF97 standard by H.Tummescheit is employed. All the components of the library have been validated against several test-cases belonging to all the main domains of power plant modeling, namely the nuclear, thermal, biomass and solar domains. ^[4]

It is important to notice that this library is designed for 0D/1D modeling. Therefore, when a model is developed with this library the goal is to understand the global behavior of the system modeled and not to determine precisely the evolution of all the parameters of the system.

Consequently the purpose of the models developed with this library is to understand what is physically happening in the system modeled and not to quantify precisely the evolution of the physical magnitudes of the model.

1.2 Multi-mode modeling and Dymola

In Dymola, multi-mode modeling is supported through the implementation of continuous- time state machines. In Modelica, a state machine corresponds to a model in which at least two different states or modes are included. These modes are used to describe different behaviors of a given model, and only one mode is active at a given time. The number of variables defined in each state or made may vary. Transitions between the different states or modes are possible when the pre- defined transition conditions are reached. In general, Modelica state machines can only include discrete variables. ^[12]

A continuous-time state machine is a state machine in which differential-algebraic-equations can be included in the different states of the state machine. This means that continuous-time

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18 variables, such as pressure or enthalpy, can be included in the states or modes. This is the key point that gives the possibility to use the multi-mode modeling prototype developed by Dymola for thermal-hydraulic purposes.^[2]

Figure 1.1 illustrates the concept of state machine. The example of a multi-mode component to model phase appearance and disappearance is represented here.

Figure 1.1 State machine concept applied to the modeling of phase appearance and disappearance

In Figure 1.1, three different states can be observed. They correspond to the circles and they describe the different situations that can be encountered: either the component is full of liquid water, either there is a mixture of liquid water and steam or the component is full of steam. As previously mentioned, these states are linked through transitions (corresponding to the blue arrows in Figure 1.1). Transitions represent a very important aspect of state machines since they define the behavior of the multi-mode model. The way variables are initialized or reinitialized when a transition from one state to another state occurs is also an essential element of state machines, it is particularly important for the variables that appear in two or more states of the state machine.

1.3 First version of the drum multi-mode model

Motivation

The DynamicDrum model is widely used in power plants modeled with ThermoSysPro library. It is a “node” component of the ThermoSysPro library, as defined in Section 1.1. This two- phase volume has several inlet and outlet ports (corresponding to the blue and red squares respectively in Figure 1.2). This means that this drum contains a mixture of two phases, a homogeneous vapor phase and a homogeneous liquid water phase. Therefore, two mass balance equations and two energy balance equations are included in the model (a couple of mass and energy balance equations for each phase). Figure 1.2 illustrates this ThermoSysPro component where blue connectors are inlet connectors and red connectors are outlet ones.

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19

Figure 1.2 DynamicDrum ThermoSysPro component

If steady-state is simulated there is by definition no time variation of variables, thus the volumes of each phase do not change. However, if a transient is simulated, variations of the volume of each phase can occur. Important variations can lead to a phase disappearance.

The current model is not able to handle this behavior in a proper way, either a solution with a negative volume for the disappearing phase is given, either the behavior of a very small volume of the disappearing phase still has to be described by its mass and energy balance equations. In the first case the solution is unacceptable since it is unphysical and in the second one a certain number of useless equations that can lead to simulation errors are kept in the model. This issue illustrates the need to develop a more satisfactory way to model this kind of situations. Naturally multi-mode modeling appears as a more adapted way to model this behavior: when one phase disappears, there is a mode or state change from a two-phase state to a one-phase state in which only one phase is modelled, the opposite when a phase appears.

First version

The first version of a multi-mode drum was developed by Daniel BOUSKELA. In this first version dimensionless quantities were used and simplified steam/water properties were used. Figure 1.3 illustrates the multi-mode approach implemented in this model. Three states are represented in Figure 1.3, for each one the most significant equations are indicated. One state represents the two- phase state and the two other states represent the single vapor or liquid phase. In this first version these equations were simplified when implemented. Moreover, in this first model there were four states instead of three (the 4^th state is not included in Figure 1.3). An initial state was first introduced in order to have a better control of some key-variables’ initialization since initialization mechanisms for multi-mode models were not correctly understood.

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20

Figure 1.3 Multi-mode mass and energy balance equations ^[13]

where:

• and are the liquid and vapor densities respectively,

• and are the liquid and vapor volumes,

• and liquid and vapor mass flow rates

• and are the liquid and vapor specific internal energies,

• and are the liquid and vapor specific enthalpies,

• is the other type of energy brought to the system.

As already mentioned, transition conditions are also an important part of multi-mode modeling since they must be carefully chosen to avoid for example unwanted state changes or unlimited oscillations between states. For the multi-mode drum the transition conditions were on the volumes of vapor and liquid water. For the transition from the two-phase state to the one-phase state, a lower limit (identified as in Figure 1.3) is set to fire the transition when the disappearing phase reaches this limit. For the transition from the one-phase state to the two-phase state, an upper limit (identified as in Figure 1.3) is set to fire the transition when the appearing phase reaches this limit.

It is of course necessary to have to avoid oscillations between states. Moreover, it is important to choose correctly these limits in order to get a good behavior of the model (for example

and cannot be too high).

It is interesting to notice that in the one-phase states three equations are written. One could think that only the mass and energy balance equations of the existing phase would be necessary to describe this state. However, it is necessary to introduce an extra equation corresponding to the mass balance equation of the non-existing phase in order to consider the possibility of its appearance and the coming back to the two-phase state.

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21 The results of these simulations were encouraging: phase changes could be observed using the simplified water/steam properties. However, this model was still far away from being a ThermoSysPro component and it could not be included in other models.

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22

2 Methodology

2.1 ThermoSysPro version of the multi-mode drum

The first version showed that multi-mode modeling could be used to model phase appearance and disappearance. However, this model could not be included in any other model composed of ThermoSysPro components. This version could not even be connected to other ThermoSysPro components because of the lack of connectors. Connectors are (by definition) necessary to connect the components of the ThermoSysPro library between them. Moreover, the IAPWS-IF97 standard computation of the water and steam properties contained in the ThermoSysPro library was not used. Finally, the simplified equations used for energy balances in the previous version were also modified in order to make the model more realistic.

It is important to note that the model chosen to describe the two-phase state is the same as for the other ThermoSysPro components in which these two phases (liquid water and vapor) are present. In these components, two separate regions are defined: a vapor region and a liquid region.

The physical model is based on a non-equilibrium formulation of the fluid energy and mass balance equations for each region. From a mathematical point of view, the model is based on the mass and energy balance equations for both phases, plus a heat balance equation at the wall of the drum (one heat balance equation between the wall and the vapor and another one between the wall and the liquid). Finally, two closure equations are used for the evaporation and the condensation flow rates at the interface between the vapor region and the liquid region (these mass flow rates are defined below). ^[1]

Below the main equations included in the model for the different states are presented. It is reminded that this multi-mode drum is a node component in the ThermoSysPro library structure.

Therefore, the mass and energy balance equations are implemented in a 0D approach. As indicated in Section 1.2, only one state is active at a given time. Therefore, the equations included in each state are independent of the ones included in the other states.

Two-phase state

Liquid mass balance equation:

where:

• is the liquid density,

• is the liquid volume,

• are the liquid related mass flow rates.

When developed and implemented in the Modelica model, Equation (2.1) becomes:

∗ ∗ + ℎ ∗ ℎ + ∗ =

where:

• ℎ is the liquid specific enthalpy,

(2.2) ( ∗ )

= (2.1)

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23

• is the pressure,

Liquid energy balance equation:

( ∗ ∗ )= ℎ ∗ + (2.3)

where:

• is the liquid specific internal energy,

• are the liquid related mass flows,

• ℎ are the liquid related specific enthalpies,

• is the other type of energy brought to the liquid phase.

It is reminded that the link between specific internal energy and specific enthalpy is, by definition:

= ℎ − (2.4)

where:

• is the liquid specific internal energy,

• is the pressure.

Therefore, when developed and implemented in the Modelica model, Equation (2.3) becomes:

∗ ∗ − 1 ∗ + ∗ ℎ + ∗ ℎ

= [ℎ − ℎ − ] ∗ + (2.5)

where:

• are the liquid related mass flow rates,

• ℎ are the liquid related specific enthalpies of the incoming mass flows,

• is the other type of energy brought to the liquid phase.

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24 Vapor mass balance equation:

( ∗ )

= (2.6)

where:

• is the vapor density,

• is the vapor volume,

• are the vapor related mass flows.

∗ ∗ + ℎ ∗ ℎ

+ ∗ = (2.7)

where:

• ℎ is the vapor specific enthalpy,

• are the vapor related mass flow rates.

Vapor energy balance equation:

( ∗ ∗ )

= ℎ ∗ + (2.8)

where:

• is the vapor specific internal energy,

• are the vapor related mass flows,

• ℎ are the vapor related specific enthalpies,

• is the other type of energy brought to the vapor phase.

The link between specific internal energy and specific enthalpy, for the vapor phase is:

= ℎ − (2.9)

where:

• is the vapor specific internal energy,

• is the pressure.

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25 Therefore, when developed and implemented in the Modelica model, Equation (2.8) becomes:

∗ ∗ − 1 ∗ + ∗ ℎ + ∗ ℎ = [ℎ − ℎ − ] ∗ + (2.10)

where:

• are the vapor related mass flow rates,

• ℎ are the vapor related specific enthalpies of the incoming mass flows,

• is the other type of energy brought to the vapor phase.

The condensation and evaporation flows are included among the mass flows defined in the previous equations. They act as regulators for the transitions between liquid and vapor phases and they correspond to the two closure equations previously mentioned. These two equations use empirical coefficients which are related to the bubble rising time in the liquid and the droplet falling time in the vapor.

When the mass vapor fraction of the liquid phase, which depends on the enthalpy and pressure, becomes higher than the upper limit an evaporation flux is defined. In a similar way, when the mass vapor fraction in the vapor phase becomes lower than the under limit a condensation flow is defined. Equations (2.11) and (2.12) give respectively the expression of the evaporation and condensation flows.

! "#= $ 0 &' ( ≤ (*

+! "#∗ ∗ ∗ (( − (*) &' ( > (_* (2.11)

where:

• +! "# is the constant related to the bubble raising time,

• ( are the vapor mass fraction of the vapor phase,

• (_* is the upper limit chose for the condensation mass flow.

,-./= $ 0 &' ( ≥ (*

+,-./∗ ∗ ∗ (( *− ( ) &' ( < (*

(2.12)

where:

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26

• +_,-./ is the constant related to the droplet falling time,

• ( are the vapor mass fraction of the vapor phase,

• ( * is the under limit chose for the condensation mass flow.

In addition of the equations previously presented, the volumes of the liquid and of the vapor phase are linked through Equation (2.13) to the total volume of the drum.

where:

• _2-2 is the total volume of the drum,

• is the liquid water volume.

One-phase liquid state

In this case the mass balance equations are simplified. For the remaining phase, here the liquid one, it is considered that the volume is constant and equal to the total one. Therefore, the mass balance equation is simplified and it becomes:

where:

• are the liquid related mass flows.

∗ ∗ + ℎ ∗ ℎ

= (2.15)

where:

For the disappearing phase, here the vapor phase, the mass balance equation must be conserved in order to give the possibility to the system to come back to the two-phase state. The only mass flow rate considered for this “non-existing” phase is the evaporation one (since only the liquid

2-2= + (2.13)

∙ = (2.14)

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27 phase is theoretically present in this phase). The mass balance equation is therefore simplified for the vapor phase (now called “onset of bubble”) and becomes (for = 0):

where:

• _"# is the vaporization mass flow.

Only the energy balance equation for the liquid phase is maintained. The energy balance equation for the vapor phase is not included anymore. That is why the number of state variables changes from one state to another and that is why multi-mode modeling was necessary. The energy balance equation for the liquid phase is now:

∗ ∗ − 1 ∗ + ∗ ℎ + ∗ ℎ = [ℎ − ℎ − ] ∗ + (2.17)

where:

• are the liquid related mass flow rates,

• ℎ are the liquid related specific enthalpies of the incoming mass flows,

All the mass flow rates are in this state related to the liquid phase (of course the condensation flow is not considered for this state because there is no vapor in the drum for this state).

One-phase vapor state

For this state the resulting equations are the same as for the one-phase liquid state, adapted to the vapor phase. The mass balance equations are simplified. The total volume of the vapor is constant and equal to the total volume of the drum. Therefore, the mass energy balance for the vapor phase becomes:

where:

∙ = "# (2.16)

∙ = (2.18)

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28 When developed and implemented in the Modelica model, Equation (2.18) becomes:

∗ ∗ + ℎ ∗ ℎ = (2.19)

where:

For the disappearing phase, here the liquid phase, the mass balance equation must be conserved in order to give the possibility to the system to come back to the two-phase state. The only mass flow rate considered for this “non-existing” phase is the condensation one (since only the vapor phase is theoretically present in this phase). The mass balance equation is therefore simplified for the liquid phase (now called “onset of droplet”) and becomes (for = 0):

where:

• is the liquid volume.

• _,-./ is the condensation mass flow.

Only the energy balance equation for the vapor phase is maintained. The energy balance equation for the liquid phase is not included anymore. That is why the number of state variables changes from one state to another and that is why multi-mode modeling was necessary, as for the only liquid phase. The energy balance equation for the vapor phase is now:

∗ ∗ − 1 ∗ + ∗ ℎ + ∗ ℎ = [ℎ − ℎ − ] ∗ + (2.21)

where:

• are the vapor related mass flow rates,

• ℎ are the vapor related specific enthalpies of the incoming mass flows,

All the mass flow rates are in this state related to the liquid phase (of course the evaporation flow is not considered for this state because there is no liquid water in the drum for this state).

∙ = ,-./ (2.20)

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29 Heat balance equation at the drum wall

In this section are presented the heat balance equations at the wall of the drum. The wall of the drum is in contact with the fluids present in the drum (liquid water, vapor or both) from one side, and with the outside from the other side.

At the wall the heat balance equation is:

where:

• 45 is the mass of the wall,

• 6₅ is the wall specific heat capacity,

• 75 is the wall temperature ,

• ₅ is the thermal power exchanged between the wall and the liquid phase,

• ₅ is the thermal power exchanged between the wall and the vapor phase,

• _5- is the thermal power exchanged between the wall and the outside.

In this equation appear several thermal fluxes related to the thermal exchanges between the wall and the surrounding elements or fluids in contact with the wall. Equations (2.23), (2.24) and (2.25) give the expression of the heat exchange between the wall and the liquid phase, between the wall and the vapor phase and between the wall and the outside respectively.

where:

• 85 is the heat exchange coefficient between the wall and the liquid phase,

• 95 is the exchange area between the wall and the liquid phase,

• 7 is the liquid phase temperature,

• ₅ is the thermal power exchanged between the wall and the liquid phase.

where:

• 85 is the heat exchange coefficient between the wall and the vapor phase,

• 95 is the exchange area between the wall and the vapor phase,

• 7 is the vapor phase temperature,

• 7₅ is the wall temperature ,

• ₅ is the thermal power exchanged between the wall and the vapor phase.

45∙ 65∙ 75= 5 + 5 + 5- (2.22)

5 = 85 ∙ 95 ∙ (7 − 75) (2.23)

5 = 85 ∙ 95 ∙ (7 − 75) (2.24)

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30 where:

• 85- is the heat exchange coefficient between the wall and the outside,

• 95- is the exchange area between the wall and the outside,

• 7- is the outside temperature,

• _5- is the thermal power exchanged between the wall and the outside.

Further adaptations

The introduction of connectors in the multi-mode model was a very important step to adapt the multi-mode drum to the ThermoSysPro standards. This step is fundamental since without these connectors the drum multi-mode model cannot not be included in other models containing ThermoSysPro components. The connectors are used to share boundary conditions (pressure, enthalpy and mass flow rate) between the components used to construct the 1D models of power plants and they correspond to the blue and red squares that appear in Figure 1.2.

In addition, in the ThermoSysPro version of the multi-mode drum several inlets and outlets were also included (contrary to the previous version in which only one inlet and one outlet ports were modeled). Inlet ports are represented by blue squares and outlet ones are represented by red squares in Figure2.1.

Moreover, the thermal connectors were also included. They correspond to the orange/brown squares in Figures 1.2 and 2.1. They can be used to introduce boundary conditions from the thermal point of view. One of these connectors can be used to add a heat flux to the liquid phase inside the drum and the other one can be used to define the outside temperature or a heat flux from the outside to the wall of the drum.

The introduction of water/steam properties tables included in the ThermoSysPro library was also an important and necessary step of the adaptation of the multi-mode drum to the ThermoSysPro standards. These tables correspond to the IF97 water/steam tables and they must be included in the model in order to make the multi-mode drum compatible with the other ThermoSysPro components.

It is necessary to compute the water/steam properties in a consistent way, in all the components of the ThermoSysPro library.

Figure 2.1 shows the view of the multi-mode DynamicDrum in Dymola. Tables correspond to the physical properties of the different fluids taken into account in the model (inlet flows, vapor phase, liquid phase, and saturation properties). Red squares correspond to outlet ports, blue ones correspond to inlet ports and orange one correspond to thermal ones. The blue triangle corresponds to a real output representing the water level inside the drum. The way the water level is calculated is presented in Appendix 1.

Boxes named “state1”, “state2” and “state3” represent the state machine used for multi- mode modeling. State3 corresponds to the two-phase state and it is set as the initial state. State1 corresponds to the liquid phase and State2 corresponds to the vapor one. One state must be set as

“initial state”. For this multi-mode model, the initial state corresponds to the two-phase state (State3).

Arrows between states represent possible transitions, and the conditions for transition between states can be written directly in the graphical view of Dymola. Transition conditions are the same as for the previous multi-mode model.

5-= 85-∙ 95-∙ (7"− 75) (2.25)

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31

Figure 2.1 ThermoSysPro version of the multi-mode drum

As mentioned in Section 1.2, variables re-initialization is an important aspect of multi- models. The state variables that are re-initialized when a new state is entered, for the multi-mode drum, are the liquid and vapor volume, the liquid and vapor enthalpy and the pressure of the system.

The way these state variables are re-initialize is given below:

• When state1 (liquid state) is re-initialized pressure and liquid enthalpy are initialized in a continuous way, liquid volume is the same as the total volume of the drum and vapor volume is equal to 0.

• When state2 (vapor state) is re-initialized pressure and vapor enthalpy are initialized in a continuous way, vapor volume is the same as the total volume of the drum and liquid volume is equal to 0

• When state3 (two-phase state) is re-initialized all the state variables are initialized in a continuous way.

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32 This choice is illustrated in Table 2.1.

Table 2.1 Re-initialization of state variables choice

State entered State variable Re-initialization choice

Only liquid phase Liquid specific enthalpy Continuous

Liquid volume Total volume of the drum Vapor specific enthalpy Vapor saturation specific

enthalpy

Vapor volume Equal to 0

Pressure Continuous

Only vapor phase Liquid specific enthalpy Liquid saturation specific enthalpy

Liquid volume Equal to 0

Vapor specific enthalpy Continuous

Vapor volume Total volume of the drum

Pressure Continuous

Two-phase Liquid specific enthalpy Continuous

Liquid volume Continuous

Vapor specific enthalpy Continuous

Vapor volume Continuous

Pressure Continuous

In this ThermoSysPro version of the multi-mode drum variables’ units have been included as well. Some other minor modifications were made in order to improve the model like the definition of a bottom pressure (taking into account hydrostatic pressure) for connectors located at the bottom of the drum. These modifications are not crucial for the physical behavior of the drum and are therefore not detailed in this paper.

Improvement of the model - Problems due to oscillation between modes

For the first ThermoSysPro version of the multi-mode DynamicDrum multiple oscillations between states were observed in some test simulations.

For example in the case presented in Figure 2.2 some oscillations were observed. In this simulation, the drum is initially at equilibrium filled with vapor and liquid water (in the same proportions) and after a few seconds the outlet liquid mass flow increases. Consequently the liquid volume in the drum decreases and the vapor one increases.

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33

Figure 2.2 Test scenario for the multi-mode drum

When the liquid phase disappears the first time then it appears again rapidly. Then liquid disappears and appears again and again. This behavior is not satisfactory and often causes calculation problems leading to simulation crashes. The reason for this oscillations to happen is related to the way outlet mass flows are taken into account in energy and mass balance equations from one state to another. For example in the example previously mentioned, in the two-phase state the so-called outlet liquid port corresponds to an outlet mass flow included in the liquid mass and energy balance equation. However, in the only vapor state this outlet mass flow is included in vapor energy and mass balance equations. Therefore, if this mass flow is too high vapor is removed rapidly from the drum and system goes back to the two-phase state. This process is then repeated and oscillations between states appear.

It is not clear if these oscillations are due to numerical problems (it must kept in mind that a new prototype for multi-mode modeling is being used) or to modeling choices (the way the system is described in the Modelica model). In the framework of this thesis, the only possibility to face this difficulty was to modify the modeling choices. Therefore, it was decided to model the behavior of the multi-mode drum when a phase change occurs in a more accurate way. The idea developed to solve this problem was to introduce a kind of mass fraction in mass and energy balance equations in the one single phase states. This mass fraction is calculated from the vaporization and condensation fluxes.

The idea is that if the drum is full of liquid water and some vapor bubbles appear in the drum (as consequence of the vaporization mass flow rate), these vapor bubbles will not lead to a switch to the two-phase state unless the vaporization mass flow rate is high enough to ensure the appearance of a stable vapor phase. Therefore, while the vaporization flow is not high enough, it is considered that these vapor bubbles will leave the drum before the liquid water. This is achieved by introducing the made-up mass fraction defined in Equation (2.27). In a similar way, if the drum is full of vapor the same and some liquid droplets appear in the drum (as a consequence of the condensation mass flow rate), these droplets will not lead to a switch to the two-phase state unless the condensation mass flow rate is high enough to ensure the appearance of a stable liquid phase. Therefore, while the condensation flow is not high enough, it is considered that these liquid droplets will leave the drum

Multi-mode modeling applied to the Main Feedwater System and Auxiliary Feedwater System switch in a Pressurized Water Reactor