Preliminary study of control systems for MYRRHA plant

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Dipartimento di Ingegneria Civile e Industriale

Corso di Laurea Magistrale in Ingegneria Nucleare

Preliminary study of MYRRHA control system analysis

Relatori Candidato

Prof. Ing. Nicola Forgione Giulia Morresi

Prof. Ing. Walter Ambrosini

Ing. Diego Castelliti

Dr. Gert Van den Eynde

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ABSTRACT

The present work has been carried out during an internship at the research center SCK•CEN in Mol (Belgium). The technical activity has been performed in the framework of MYRRHA project.

MYRRHA is a fast spectrum irradiation facility under development at SCK•CEN. In the context of new nuclear reactor studies, the MYRRHA project contributes to the demonstration of ADS applications and to material testing for GEN IV and fusion reactors, representing an experimental plant for Lead Fast Reactor technology.

The main purpose of this Master Thesis consists in analyzing part of the control systems considered for the MYRRHA plant by the use of both graphical programming languages and system code calculations.

The plant control strategy foresees two main control systems: the power control through the control rod movement, and the secondary pressure control through the tertiary fan air flow rate change.

In order to study the behavior of the plant under controlled action, the development of a dynamic plant model has been necessary.

Two paths have been initially undertaken in order to simulate MYRRHA reactor dynamics: the first one foresaw a very simple and schematic plant model simulated through lumped parameter balance equations in MATLAB – Simulink environment. This approach should have provided fast running transients and easy implementation of the controllers. The components requiring only single phase balance equations to be properly represented have been developed through differential energy balances implementation, while two-phase systems have required a different approach.

The second way followed is instead characterized by a more detailed neutronic and thermal-hydraulic plant model in which computationally heavier (i. e. slower) calculations and less intuitive control system implementation were the prices to pay in order to obtain more accurate results. In fact, the thermal-hydraulics of such a complex plant required the implementation of a model capable to follow the non linearities of the system itself. This is the reason why the second part of the work was devoted to the development of a RELAP5-3D model of MYRRHA.

Even if a detailed model of the plant was already available, there were two main reasons to develop a new one: first of all, the full knowledge of an own-made model and in general the mastery of the code would not have been possible simply by adopting an already existing one. Secondly, some technical choices were made in order to simplify the computational cost of the calculation, i. e. a reduced number of volumes in the discretization of the plant system.

The possibility to couple the system code, to describe the dynamics of the system, and graphical programming languages, to implement the control logics, appears a feasible and interesting future development of the work performed during this analysis.

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1 Introduction

The most advanced research field for nuclear reactor technology is currently represented by Generation IV studies [1.1].

The Heavy Liquid Metal Fast Breeder Reactor (HLMFBR) is one of the most promising solutions proposed to cope with problems such as nuclear waste, long term radioactivity and consumption of nuclear fuel. In particular, fast reactors offer the possibility of burning actinides and breeding more fuel than they consume, so that nuclear can guarantee a sustainable, safe and economic future from the energetic point of view [1.2].

The fast reactor prototype has no moderator and relies on fast neutrons alone to cause fission. It usually uses MOX as its basic fuel because the number of neutrons produced per Pu-239 fission is 25% larger than from uranium (U), and this means that neutrons are enough not only to maintain the chain reaction but also to convert U-238 (blanket) into more Pu-239.

Figure 1.1 shows the increase of the average number of neutrons released per fission ν at high energies and the fact that this number is greater in the case of Pu respect to U fissile isotopes [1.3].

Fig 1.1: Average neutron number per fission as a function of incident particle energy, for different fissile isotopes [1.3]

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8 Figure 1.2: η for common fissile isotopes as a function of energy

Figure 1.2 shows the number of fission neutron produced per absorption in the fuel, η. Its relation with ν is the following [1.4]:

where the values for Pu-239 are higher than the ones for U isotopes at fast neutron energies. In the thermal energy range the most convenient fissile would be U-233.

The coolant is a liquid metal for two main reasons: to avoid neutron moderation, in fact heavy nuclei are less efficient in scattering and slowing down the particles, and to provide a very efficient heat transfer medium in order to cool down the high power density produced in the compact FR core [1.4].

The good heat transfer properties of liquid metals comes from their large thermal diffusivity and volumetric specific heat ρ·cp [1.5]. These two parameters

affect convection through two different processes of conduction and turbulence diffusion, resulting in an effective thermal conductivity which is higher than for normal fluids (e. g. water).

The positive aspect of heavy liquid metals instead of "light" ones (mainly sodium, Na) is related to their low chemical reactivity with the environment (air, water). The choice of lead as heavy liquid metal for nuclear applications appears the only feasible one, even if it presents many problems, first of all the melting temperature (323 °C) that is relatively high compared to other liquid metal coolants used in nuclear technology (e. g., Na, Tmelt = ~80 °C) . This is the reason why Lead

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Bismuth Eutectic (LBE) technology was introduced (melting temperature 127 °C) [1.5].

Given for granted the advantages in developing a fast nuclear reactor, the main limitations and issues still present for LBE technology are briefly presented below:

- Material problems: Corrosion from heavy liquid metal is surely the most significant among the difficulties encountered in developing a HLMR [1.6], even more if the specific environment plays a critical role in terms of high temperature and irradiation effects (embrittlement, creep, swelling) [1.7]. - Chemical problems: Oxygen dilution represents a limit in operation

temperatures for the primary LBE coolant; in particular the oxygen concentration in the coolant must be comprised in a range of values in which the minimum is fixed in order to avoid the clad material dissolution problem and the maximum to avoid the lead oxide precipitation. These limitations become stricter with the increase of temperature (see Figure 1.3) and this is the reason why the range of operating temperatures is relatively narrow [1.8].

Figure 1.3: Oxygen concentration limitation for operational temperatures [1.7]

- Radioprotection problems: Polonium production is caused by neutron capture of bismuth as follows [1.9]:

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_{is a highly radioactive and chemically toxic element.}

- Mechanical problems: maximum velocity of the fluid limited to 2 m/s mainly due to erosion issues [1.7].

These main issues are also the main reasons why HLMRs technology still represents a field of research for nuclear energy.

1.1 MYRRHA

In this background context SCK•CEN is playing an important role through the choice of developing MYRRHA.

MYRRHA stands for Multi-purpose hYbrid Research Reactor for High-tech Applications [1.10]. It is a multifunctional research facility for innovative applications. MYRRHA is the world's first prototype of a subcritical lead-bismuth cooled reactor capable to be driven by a particle accelerator. The main objective of MYRRHA is to be a flexible fast spectrum irradiation sub-critical facility with the ability to operate also in critical mode, thus requiring two different core designs for the two different modes. The fast neutron spectrum offers the perspective of a vastly more efficient use of uranium resources and the ability to burn minor actinides which are otherwise the long-lived component of high-level nuclear wastes.

The project was firstly proposed in 1999 as a multi-purpose irradiation facility in order to replace the currently operating BR2 reactor. The design of MYRRHA facility is still under development due to the contemporary advancements in research and design chosen for the project itself; in particular this choice leads to a continuous updating process for the technical design, according to new findings of the ongoing R&D projects.

To introduce the context in which the MYRRHA design idea was born, it must be mentioned the interest for Accelerator Driven Systems (ADS) from the last years of the past century: in fact in 1998 some European countries such as Spain, Italy and France established an European platform aimed at the development of Accelerator Driven System from which it was decided the construction of an Experimental Accelerator Driven system to assess its technical and economic feasibility [1.11]. Among the European research projects, MYRRHA has received particular interest and economic support since 1998.

The flexibility of the design, achieved in the framework of the Central Design Team (CDT) project [1.12], allows two different configurations of the reactor in which the ADS technology is used or not. In the subcritical mode the plant includes a proton accelerator of 600 MeV – 3.5 mA, a spallation target and a multiplying medium (the core itself). The particle accelerator is used as an external neutron source to create and maintain the chain reaction in a reactor where the effective multiplication factor is less than unity, keff ~ 0.9745. This highly safe and controllable nuclear technology

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allows stopping the nuclear reaction automatically by switching off the particle accelerator.

The new research field on ADS technology can present some initial difficulties in the management of such complex and delicate system. This is the main reason why a critical mode of operation has been seen as a solution to possible delays and maintenance issues related to the particle accelerator. The configuration taken into account in this Master Thesis work is the critical one, in which the control rod system assures the reactivity balance.

1.2 SCK•CEN

SCK•CEN (StudieCentrum voor Kernenergie – Centre d'Etudes de l'energie Nucleaire) is one of the largest research institutions in Belgium. It was founded in 1952 and since then it has played a pioneering role in giving the Belgian academic and industrial world access to the worldwide development of nuclear energy [1.13].

Among the nuclear installations developed at SCK•CEN, the following must be mentioned:

- BR1: the oldest research reactor in Belgium, it became critical on 11 May 1956 and is still used for research, silicon production for semiconductor materials and educational purposes. It is a graphite-moderated air-cooled reactor with a maximum thermal power of 4 MW, fueled by natural uranium.

- BR2: from the first start-up in 6 July 1961, this high-flux research reactor is still operating. Fueled by highly enriched uranium in a beryllium matrix, it has been refurbished twice up to now. It is used for the production of medical isotopes and the irradiation of silicon to obtain n-doped silicon by transmutation.

- BR3: started in 19 August 1962, it has been the first PWR in western Europe with the aim of being a demonstration unit for an industrial power station and of constituting a test reactor for prototype nuclear fuel. It has been chosen as a pilot project for the demonstration of the decommissioning of PWR [1.14]. - VENUS: Vulcan Experimental Nuclear Studies facility has been operating since

1964. In 2008 SCK•CEN began the rebuilding of VENUS for the GUINEVERE project (Generator of Uninterrupted Intense NEutrons at the lead VEnus REactor). The project started in 2010 and since then the reactor has been known as VENUS F (First) scale model of a subcritical reactor with a total lead core driven by a particle accelerator.

- HADES: High Activity Disposal Experimental site Storing is an underground laboratory 225 m below the ground built for the storage of high level radioactive waste in deep layers of clay.

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- Nuclear and non-nuclear laboratories: among them the laboratory for high and medium level activity (LHMA) is one of the most important at SCK•CEN.

SCK•CEN is working since several years at the design of MYRRHA, which will replace the BR2 MTR. The Belgian center of research is positioning MYRRHA as one of the corner stones of the European Research Area Experimental Reactors (ERAER). As stated in the Strategic Research Agenda of the Sustainable Nuclear Energy Technology Platform (SNETP), Europe can only retain its worldwide leading position in the field of reactor technology and related future developments, if it provides for the necessary research infrastructure [1.10]. In practice, this will require the complete renewal of the ERAER, which will be based on three pillars:

- Jules Horowitz Reactor (JHR) in Cadarache (France) [1.15] - PALLAS in Petten (the Netherlands) [1.16]

- MYRRHA in Mol (Belgium) [1.10]

MYRRHA will fulfill the role of European Technology Pilot Plant (ETPP) in the roadmap for the development of the lead fast reactor technology.

Among the aims of the MYRRHA project, the research on transmutation of minor actinides, the research on material and nuclear fuel, the production of medical isotopes and semiconductors doping and the demonstrative character of a facility for the Lead Fast Reactor technology are the main ones.

The next chapter will describe the technical choices performed in order to develop a facility capable to cope with these purposes.

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2 MYRRHA general description

In this section, the reactor configuration and the parameters employed in this work are summarized. MYRRHA is a 110 MWth pool-type HLMFR.

Its primary system configuration is depicted in Figure 2.1 [2.1].

All the primary components (e. g. the core, the Primary Heat eXchangers (PHXs) and the Primary Pumps (PPs)) are contained within the reactor vessel. The coolant flow coming from the cold plenum enters the core and, once passed through the latter, is collected in the barrel and from here into the hot plenum to be distributed in the 4 PHXs. After leaving the PHXs, the coolant enters the cold plenum passing through the PPs (1 pump serving 2 PHXs) and returns to the core.

The main components described in this section are summarized below:

 Core

- Neutronic feedbacks - Control Rod system

 Vessel and diaphragm

 Primary cooling system - Primary Heat Exchangers - Primary Pumps

 Secondary cooling system and tertiary cooling system

 Decay Heat Removal (DHR) systems

A B C D E H F G I A. Reactor Vessel B. Reactor Diaphragm C. Reactor Cover

D. Primary Heat Exchanger E. Primary Pump

F. In-Vessel Fuel Handling Machine

G. Core Barrel H. Reactor core

I. Core Restraint System

Figure 2.1: Overview of the MYRRHA Primary System [2.1]

The description of the plant design provided in this Chapter is referred to the last MYRRHA Design Version 1.6 Technical Description [2.1].

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2.1 Core

The MYRRHA core is composed by 108 wrapped hexagonal fuel assemblies with pins arranged on a triangular lattice. This Fuel Assembly (FA) design is similar to the typical design used in fast spectrum reactors cooled by liquid sodium (SFR). Each of the 108 FAs is composed by 127 MOX type fuel elements, with a Plutonium weight fraction of 30%.

In Figure 2.2 a section of the critical core with the complete definition of all the different position channels has been provided.

Figure 2.2 - 108 FA-100 MW maximum critical core layout [2.2]

Figure 2.3 summarizes the fuel pin and the FA architecture main features.

Figure 2.3 – Reference FA design [2.2] 15 / 28 17 / 28 19 / 27 20 / 27 21 / 27 21 / 26 23 / 25 26 / 21 27 / 20 28 / 17 28 / 19 27 / 19 28 / 15 27 / 13 27 / 14 27 / 12 26 / 13 25 / 12 21 / 13 19 / 13 17 / 15 16 / 16 15 / 17 15 / 18 14 / 19 13 / 20 12 / 21 13 / 21 12 / 25 12 / 26 14 / 27 13 / 26 13 / 27 20 / 20 20 / 21 21 / 20 21 / 19 19 / 20 20 / 19 19 / 21 18 / 21 18 / 22 19 / 22 18 / 20 21 / 21 20 / 22 22 / 20 22 / 19 22 / 18 21 / 18 20 / 18 19 / 19 18 / 23 19 / 23 20 / 23 21 / 22 22 / 21 22 / 17 23 / 17 23 / 18 23 / 19 23 / 20 17 / 23 21 / 17 20 / 17 19 / 18 18 / 19 17 / 20 17 / 21 17 / 22 17 / 27 18 / 27 18 / 26 16 / 27 17 / 26 15 / 27 19 / 26 19 / 25 18 / 25 17 / 25 16 / 26 19 / 24 20 / 24 20 / 25 20 / 26 18 / 24 17 / 24 16 / 25 15 / 26 13 / 24 13 / 25 14 / 24 14 / 23 13 / 23 14 / 25 15 / 24 15 / 23 15 / 22 14 / 22 13 / 22 14 / 26 15 / 25 15 / 21 16 / 21 16 / 22 16 / 23 16 / 24 14 / 21 16 / 17 16 / 18 17 / 17 17 / 16 15 / 19 17 / 18 16 / 19 18 / 17 18 / 16 18 / 15 14 / 20 15 / 20 16 / 20 17 / 19 18 / 18 19 / 15 19 / 16 19/ 17 23 / 13 23 / 14 22 / 13 22 / 14 21 / 14 21 / 15 22 / 15 24 / 14 23 / 15 25 / 13 21 / 16 22 / 16 23 / 16 24 / 15 25 / 14 20 / 16 20 / 14 20 / 15 27 / 17 26 / 16 27 / 15 26 / 17 25 / 17 25 / 18 26 / 18 25 / 16 27 / 18 26 / 15 25 / 19 26 / 19 24 / 19 26 / 14 25 / 15 24 / 16 24 / 17 24 / 18 24 / 23 25 / 22 23 / 23 24 / 22 23 / 24 22 / 24 22 / 25 22 / 23 25 / 21 24 / 21 23 / 22 26 / 20 25 / 20 24 / 20 23 / 21 22 / 22 21 / 23 21 / 24 21 / 25 24 / 13 27 / 16 13 / 28 12 / 27 14 / 28 12 / 24 12 / 23 12 / 22 13 / 19 14 / 18 18 / 14 19 / 14 20 / 13 21 / 12 22 / 12 24 / 12 23 / 12 26 / 12 28 / 13 28 / 14 28 / 16 28 / 18 27 / 21 26 / 22 25 / 23 24 / 24 22 / 26 19 / 28 18 / 28 16 / 28 108 FA 4 IPS 6 CR (buoyancy) 3 SR (gravity) 48 Dummy (LBE) 42 Reflector (Be) 5.42 5.65 6.55 8.4 mm 1.85 127 Phénix Pins (6 pin rows) 97.55 101.55 104.5*mm Fuel He Clad Wrap LBE 0.45 0.115 VF (Fuel/FA) 31% P/D = 1.28244…* d+0.1mm 1.75 d (D) d/D = 0.267 (P) * 2.95 mm clearance among FA *yields “rounded” d, P Geometrical data at 20 C Fissile zone VF [%] MOX (fuel) 30.9834 He (gap) 2.6854 15/15 Ti SS (clad+wire*) 14.8266 FMS T91 (wrap) 7.2929 LBE (coolant) 44.2117

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Table 2.1 schematically shows the main core pin and FA parameters given by design.

Table 2.1 - Main MYRRHA 1.6 core parameters

Core

Number of positions 211

Core diameter 1900 mm

Layout Centralized around the central position

Maximum core pressure drop 2.5 bar

Fuel

Fuel type MOX, 30 % wt. PuO2

Pellet type Solid pellet

Pellet dimension Outside diameter 5.42 mm

Fuel pin clad 15 – 15 Ti

Fuel pin dimension Outside/Inside diameter 6.55 mm/5.65 mm

Fuel pin length 1500 mm

Fuel active height 65 cm

Fuel assembly

Assembly type Hexagonal fuel bundle with wrapper

P/D 1.28

Number of pins 127

Wrapper material T91

Spacer type Wire spacer in 15-15Ti

Assembly length 2250 mm

Maximum LBE bulk velocity 2 m/s

2.1.1 Neutronic feedbacks

The power variations in a critical reactor depend mainly on reactivity variations and secondly on the effective delayed neutron fraction (βeff) and mean neutron lifetime

(lec). The following definitions apply [2.3]:

- Reactivity is the balance of nuclear reaction rates normalized by the neutron production rate.

- βeff is the contribution of delayed neutrons to reactivity, while β is the fraction of

delayed neutrons in the total amount of fission neutrons. β for Pu239_{is 0.002,}

while the value for U235_{is 0.006; MOX fuel, which is composed by Pu and U,}

(mainly U238_),_{(30% and 70% respectively for the MYRRHA core) has a weighted}

βeff value of 0.0032, meaning a smaller margin to prompt criticality respect with

U fissile material.

- lec is the average time from a neutron emission to a capture that results in

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because the first one only includes neutron absorptions that lead to fission reactions (no other absorption reactions); they are related together through the effective multiplication factor. lec in a fast reactor is of the order of fraction of µs,

almost three orders of magnitude smaller than for a thermal reactor. This means that a fast reactor is faster in response both to positive and to negative power variations.

The first term is related to the particular transient under exam and is equal

to zero at steady state conditions.

The feedback term in the reactivity balance is expressed through the reactivity coefficients as neutronic parameters of the core. Negative coolant/void density, axial expansion and Doppler effects are the major reactivity feedbacks considered. These feedbacks are related to temperature variations of the fuel and the LBE coolant as follows:

where the first term at RHS is the Doppler reactivity feedback, which has a logarithmic trend respect to fuel temperature.

The second term represents the axial expansion of the fuel. The third term is the density variation of the LBE coolant.

Based on the thermo-physical properties of core materials, these neutronic features depend on the core layout under study: in this case the layout considered is the maximum critical core in BoC conditions.

The maximum core entails 108 FA at the equilibrium. To evaluate the reactivity coefficients, the data have been obtained in the framework of FP7 - MAXSIMA project [2.2].

Table 2.2 - Reactivity coefficients for MYRRHA core [2.2]

For the third term in the reactivity balance, the absorber rods come into play.

Dkef f ( pcm ) -1% 1.07017 -471 Dkef f / T +1.5 cm _1.08016 528 [pcm K-1] -10% 1.07638 150 +5% 1.06624 -864 -10% 1.07524 36 +2.5% 1.06816 -672 +2.0% 1.07456 -32 1.07393 -95 1.07393 -95 -0.022 -0.049 -0.241 Coolant LBE Density reduction in the fissile zone -5%

Density reduction overall pin lenght

Wrap T91

Density reduction overall pin lenght External apothem increase

Pin pitch increase

Component Material Elementary Perturbation Variation _k_eff

Fuel MOX Density reduction Height increase

Clad 15-15/Ti Density reduction overall pin lenght External radius increase

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2.1.2 Control Rod system

Two different and independent control rod systems have been foreseen, namely the Control Rods (CRs) and the Safety Rods (SRs). Flux shaping and reactivity compensation during the cycle are performed by the former, while the simultaneous use of both is foreseen for scram purposes, assuring the required reliability for a safe shutdown. The activation of both systems is passively driven by gravity. In the case of the CR, the insertion from the bottom is due to the buoyancy effect in LBE while the SR movement is downward by pneumatic assisted fall in gas-filled driving tube. Due to their different functions the CR are partially inserted in the core for most of the operating time of the reactor, while the SR can be fully extracted (normal condition) or completely inserted (emergency shutdown condition). The CRs are positioned at the periphery of the core in order not to decrease the flux in the center: as MYRRHA is a material testing reactor, the central part of the core contains In Pile Sections (IPSs) for material testing purposes.

Table 2.3 – Absorber data [2.1]

Absorber rods

Absorber Boron-carbide with 90% enriched B-10 Control rod type Buoyancy driven at the periphery Number of control rods In function of core layout, ≤ 6

Safety rod type Gravity driven assisted by forced injection Number of safety rods In function of core layout, ≤ 3

Table 2.4 – Absorber reactivity coefficients [2.3]

2.2 Vessel and diaphragm

In order to house all the primary system components, the vessel dimensions are quite huge (around 10 m in diameter and 13 m of length). The thickness of the vessel itself, 11 cm, is instead lower than in a normal LWR even with such larger dimensions. In fact the LBE is not pressurized and the 10 bar pressure at the bottom of the core is only due to hydrostatic pressure. This is a positive aspect for what concerns the

Dkeff Dkeff / T ( pcm ) [pcm K-1] -10% 1.01401 65 +1.5 cm 1.01146 -190 -10% 1.01345 9 +6 cm 1.01180 -156 -0.030 -0.005 Component Material Elementary Perturbation Variation k_eff

SR B4C

Density reduction

Height increase toward fissile zone

CR B4C

Density reduction

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safety issues in HLMFR. The material of the vessel is AISI316L for a total weight of ~340 tons. Table 2.5 summarizes the main reactor vessel and diaphragm parameters.

Figure 2.4: Vessel cutaway view

The diaphragm structure, which separates the cold, high pressure LBE from the hot, low pressure LBE, is connected, together with the cover, to the reactor vessel. The diaphragm consists of two horizontal plates in AISI316L connected to each other with vertical shells and tubes. In-between the two horizontal plates are housed part of the pumps, the heat e changers and the In- essel Fuel Storages (I FSs where the spent fuel is stored for a period of 420 days in order for the decay heat to reduce. The IVFSs are cooled during operation mainly by the forced circulation of LBE imposed by the pumps, while during shutdown, cooling is ensured by natural circulation.

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19 Table 2.5 - Main MYRRHA reactor vessel and diaphragm parameters

Parameter Unit Value

Reactor vessel internal diameter mm 10200 Reactor vessel thickness mm 110 Diaphragm internal diameter mm 9900

Diaphragm thickness mm 80

Reactor vessel height mm 13035

Core inlet level mm 4900

Fuel active length level mm 5225 NOC1_{Hot LBE level} _mm ₉₅₀₀

NOC Cold LBE level mm 12000

The reactor is located in the reactor pit which features a steel liner able to serve as secondary containment in case of a reactor vessel leakage or break. The reactor cover closes the reactor vessel and supports all the components (due to the buoyancy force which pushes upwards all the vessel internals).

The Primary Cover Gas and Ventilation System (PCGVS) is also implemented in the reactor design for the continuous filtering and monitoring of the cover gas environment in the upper part of the reactor above the LBE free surface.

2.3 Primary cooling system

In normal operation, at 100 MWth core power, the cold LBE (270 °C) at high pressure

(~10 bar) flows through the core channels installed in the core barrel. The LBE flowing through the 108 active fuel assemblies, with a core flow rate of ~7710 kg/s, heats up to an average temperature of 360 °C.

The remaining LBE (about 6090 kg/s) flows through the reflector channels and other bypasses. The pressure drop over the core amounts to ~2.5 bar. Both LBE mass flow rates (core and bypass) will mix in the hot plenum.

An additional conservative power of 10 MW is considered due to additional heat sources such as the polonium decay, the heat dissipation of the PPs, the heat production in the in-vessel fuel storage, the γ-heating and the spallation target2_.

Accounting for all the core power inputs, the temperature of the hot plenum will reach 325 °C.

The temperature difference of 55 °C between the hot plenum and the cold plenum is below the maximal gradient allowed on the diaphragm to limit the thermal stresses on the component (80 °C).

1_{Normal Operation Condition.}

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Because of the lower pressure in the PP boxes, the LBE is aspired through the PHXs. The LBE enters the PHXs at 325 °C and exits at 270 °C, exchanging 27.5 MW per unit.

Then the cold LBE flows into the PP which evacuates the LBE to the cold plenum. The maximal pressure drop from the inlet of the PHXs to the inlet of the pump is estimated at 0.5 bar. Consequently, the pumps have to deliver ~3 bar.

Table 2.6 – General thermal hydraulics parameters for MYRRHA 1.6 reactor design

Power

Maximum core power 100 MWth

Maximum reactor power 110 MWth

Temperatures

Cold shutdown state 200 °C

Maximum core inlet temperature 270 °C Ma imum core ΔT (average channel 90 °C

Average core outlet temperature 360 °C Maximum hot plenum temperature 325 °C

2.3.1 Primary Heat eXchangers

The main thermal connection between the primary and the secondary system is provided by the Primary Heat Exchanger (PHX).

LBE from the hot plenum (~325 °C) enters one of the four PHXs from the inlet openings in the external shroud. The flow is then directed downwards, through the tube bundle, where the actual heat transfer takes place.

Outlet openings, directing the LBE flow towards the Primary Pumps, provides the exit path for the cold (~270 °C) LBE.

On the secondary side, water at a pressure of 16 bar at nearly saturated conditions (~200 °C) flows down the central down-comer pipe into the PHX bottom head and then upwards through the tube bundles where it is heated by the counter-current flowing LBE, thus producing a water steam mixture with a final quality of ~0.3.

A summary of the main geometrical and thermal-hydraulical PHX parameters is shown in Tables 2.7 and 2.8.

As shown in Figure 2.6, the tube bundle is extended from the bottom tube plate to the top tube plate, as normal for shell-and-tube HXs. But, in MYRRHA design, the LBE inlet is placed at ~ 2.35 m from the bottom plate, instead of being located at top of the component (~11 m).

This configuration defines an "active length" for the tube bundle of ~2.1 m, where the LBE flow actually takes place.

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21 Table 2.7 – MYRRHA PHX main geometrical parameters

Power in one PHX MW 27.5

Shroud external diameter mm 850

Shroud internal diameter mm 820

Feed water pipe external diameter mm 200

Water tubes number - 684

Water tubes pitch mm 26

Water tubes external diameter mm 16

Water tubes internal diameter mm 14

Thickness of water tubes mm 1

Total length of water tubes mm 10920 Active length of water tubes mm 2100

Table 2.8 - MYRRHA PHX main thermal - hydraulics parameters

PHX LBE inlet temperature °C 325

PHX LBE outlet temperature °C 270

LBE safe shutdown temperature °C 200

PHX LBE mass flow rate kg/s 3450

PHX water inlet temperature °C 200

PHX water outlet temperature °C 201.4

PHX water mass flow rate kg/s 47

PHX water pressure bar 16

PHX water outlet quality - 0.3

PHX water outlet void fraction - 0.9

LBE velocity m/s 0.93

Water outlet velocity m/s 3.3

Steam outlet velocity m/s 18.63

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22 Figure 2.6: MYRRHA PHX design overview

2.3.2 Primary Pumps

In the following tables (2.9 and 2.10) the MYRRHA Primary Pumps (PPs) main geometrical and functional data are provided.

Table 2.9: MYRRHA pump main design data

MYRRHA pump main geometrical data for simulation codes

Rated pump velocity ωR 185.6 rpm = 19.436 rad/s

Rated flow QR 0.65839 m3/s

Rated head given by the rotor HR 2.98 m

Rated torque τR 10379 N⋅m

Constant friction torque coefficient τFr0 ~100 N⋅m Constant friction torque coefficient τFr2 ~100 N⋅m Moment of inertia (pump + motor) Ipn 2600 kg⋅ m2

Rated density ρR (at 270 °C) 10480 kg/m3 Rated Efficiency based on the Torque ηR 0.8753

Rated Hydraulic efficiency ηHyd-R 0.4957

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23 Table 2.10: MYRRHA pump head loss coefficients and flow area data

MYRRHA pump pressure loss coefficients and flow area data

Global kforward at 0 rpm 481 Global kreverse at 0 rpm 307

kinlet 0.5

koutlet 1.0

Inlet area 1.649 m2

Flow area at impeller 0.597 m2

Outlet area 0.597 m2

Length of volume ~1.60 m

Inlet hydraulic diameter 0.518 m Outlet hydraulic diameter 1.113 m

The complete set of characteristic curves defines, for a wide range of rotational velocities, the pump head and the pump torque in function of the mass flow rate [2.4].

2.4 Secondary and tertiary cooling systems

The Secondary Cooling System (SCS) has been developed as four parallel loops wor ing independently, each one connected with one P , so the ma imum power e tracted by each loop in nominal conditions is 27.5 MWth. The main operational

parameters for each SCS loop are enlisted in table 2.11.

The SCS design foresees four different water/steam two-phase loops operating in forced circulation [2.5]:

 Water enters into the heat exchanger, the coolant flow is maintained constant in nominal conditions, and leaves it as a water/steam mixture (quality ~0.3 for maximum reactor power)

 This two-phase fluid is then separated in a steam drum  Steam flow is directed to the air cooled condensers  Condensed water returns to the drum

Table 2.11 – Main design requirements for one single secondary system loop

Power Maximum 27.5 MWth

Pressure Operation 16 bar

Temperature Operation 200 °C

PHX outlet steam quality Forced circulation: 0.3 (max. reactor power) PHX mass flow Forced circulation: 47.05 kg/s Steam mass flow Natural circulation: 14 kg/s

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24 Figure 2.7 - Secondary and tertiary cooling systems – schematics of one single loop.

The secondary cooling pumps introduce the water into the heat exchangers with a design head of 2.15 bar.

The steam drum is the component where water/steam mixture is separated, water flows back to the P ’s and steam is sent to air cooled condensers.

The steam drum has been dimensioned as follows (Table 2.12):

Table 2.12: Steam drum design requirements Liquid water mass (normal level) 14115 kg

Drum volume 32.62 m3

Drum length 8.73 m

Drum diameter 2.2 m

Operating pressure 15 bar

The steam produced is directed through natural circulation towards the aero-condensers, where it is condensed and the heat is released to the atmosphere.

A series of fans guarantees the forced circulation of the air, tuning the heat transfer coefficient through the change of velocity according to the conditions of both plant and environment.

In particular, as it will better explained in chapter 3, the air mass flow, directly connected to the fan speed, is the parameter that controls the secondary pressure constancy.

The very preliminary design of this component foresees a counter-current vertical flow (condensing water flowing downwards in tubes). Each condenser unit is divided in 8 banks. The design data are summarized in following table 2.13.

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25 Table 2.13 – Condenser design requirements

n° tubes per bank 230

n° banks 8

Tube internal

diameter 2 cm

Tube length 8 m Tube thickness 1.46 m

The Tertiary Cooling System (TCS) design is still flexible to change according to the design revision of the plant, in particular of the condenser itself.

A preliminary design for the TCS requires an air mass flow rate of 410 kg/s in nominal conditions.

The condensed steam is then collected in a condensate tank where the condensate water is stored: if the water level of the drum decreases, the system will direct water from the condensate tank to fill the drum until its normal level. In nominal conditions, the condensed water, flows in the bottom part of the drum, mixes with the liquid phase separated before and goes back to the PHX through the secondary pump.

Table 2.14 – Condensate tank design requirements Liquid water mass 4230 kg

Tank volume 9.78 m3 Tank length 5.84 m Tank diameter 1.46 m

2.5 Decay Heat Removal systems

MYRRHA SCS serves also as Decay Heat Removal 1 (DHR1) in case of accidental event. In such events, the SCS must be able to fulfill the DHR function operating in natural circulation (passive mode), while the active cooling is not available anymore.

The DHR2 function is performed by a tank filled with liquid water lying above the reactor vessel, isolated through a set of valves which opens in accidental conditions and floods the reactor pit, thus cooling the external vessel wall. The DHR function is then performed by pool boiling and natural circulation of steam towards a condenser. Figure 2.8 represents DHR2 configuration in MYRRHA design revision 1.6 [2.1].

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LBE

_~15 m

Liquid water level

~ 50 m Silicon doping pool Refilling line

Pressure relief line towards reactor hall

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3 MYRRHA Control strategy

As a preliminary reactor control strategy developed for the MYRRHA plant, two main control systems are foreseen:

- Reactivity control through the control rod system

- Secondary pressure control through the air fan velocity control

The power control through the movement of the control rods guarantees that the reactor power follows the set point value. The control rod system has both the functions of controller, to maintain zero reactivity balance during operational conditions, and of servomechanism, when changing the power set point, e. g. for the start-up from zero to full power operation mode.

The secondary pressure control guarantees the constancy of steam pressure which has to follow the steady state set-point value of ~15 bar. The pressure of the water side is the highest in the system: limiting this pressure assures a safety margin for what concerns tube rupture failures and high pressures in maintenance or low power conditions [3.1].

It is important to point out that as MYRRHA will be an irradiation facility, it will not produce any electrical power; therefore no turbine group will be coupled with the nuclear installation and, from the point of view of plant control, this represents a considerable simplification. Frequency - power control with respect to the electrical network user is not in this case a constraint. This leads to a simplified plant control design [3.2].

In the following paragraphs, a brief introduction about the general feed-back control loop and the Proportional Integral Derivative (PID) controller will be given; secondly, the description of the two control systems is performed.

3.1 PID controllers

In general, the objective of control is to obtain a system output parameter, as close as possible to its desired set point value. This can be done through two main logics: feed-forward or feed-back. While the first one acts directly on the input signal in order to obtain the required output, the second one is based on the error between the set point value and the output of the system: the error enters the controller which attempts to minimize it by adjusting the process through the use of a manipulated variable (output or control variable) (see Figure 3.1).

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The PID controller algorithm involves three separate parameters referred as "gain constants": the proportional (P), the integral (I) and the derivative (D) constants. These values can be interpreted in terms of time: P depends on the present error, I on the accumulation of past errors and D is a prediction of future errors based on current rate of change. The weighted sum, through the gain constants, of these actions is used to adjust the process via a control loop (see Figure 3.2).

Figure 3.2: PID controller [3.3]

- The proportional term produces an output value that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant Kp called the proportional gain constant. A high

proportional gain results in a large change in the output for a given change in the error. If the proportional gain is too high, the system can show permanent oscillations around the set point value. In contrast, a small gain results in a small output response to a large input error, and a less responsive or less sensitive controller. If the proportional gain is too low, the control action may be too limited when responding to system disturbances. Because a non-zero error is required to drive it, a proportional controller generally operates with a steady state error.

- The contribution from the integral term is proportional to both the magnitude of the error and the duration of the error. The integral in a PID controller is the sum of the instantaneous error over time and gives the accumulated effect that should have been corrected previously. The accumulated error is then multiplied by the integral gain and added to the controller output. The integral term accelerates the movement of the process towards set-point and eliminates the residual steady-state error that occurs with a pure proportional controller. However, since the integral term responds to accumulated errors from the past, it can cause the present value to overshoot the set-point value.

- The derivative of the process error is calculated by determining the slope of the error over time and multiplying this rate of change by the derivative gain. The

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magnitude of the contribution of the derivative term to the overall control action is determined by the derivative gain. The derivative action predicts system behavior and thus improves settling time and stability of the system. The implementation of PID controllers includes an additional low pass filtering for the derivative term, to limit the high frequency gain and noise. Derivative action is seldom used in practice though because of its variable impact on system stability.

Tuning a control loop consists in the adjustment of its control parameters to the optimum values for the desired control response.

PID design and tuning is not an easy task because usually multiple and often conflicting objectives such as short transient and high stability are to be achieved. Nowadays PID tuning and loop optimization software are used to ensure consistent results. These software packages will gather the data, develop process models, and suggest optimal tuning.

Mathematical PID loop tuning induces an impulse in the system, and then uses the controlled system's frequency response to design the PID loop values.

One of these software, namely PidTuner MATLAB toolbox [3.4], has been used for the analysis on secondary pressure PI controller (see below).

3.2 Power control system

The power control in critical mode is performed through the movement of the control rods. In particular, the power error is defined as the difference between the set point value and the measured value.

This error is related to the control rod velocity: a positive velocity means an insertion of the control rod, which happens in case of negative power errors (sensed variable higher than the set point value). Vice versa, a negative velocity represents an extraction of the control rod, needed in case of positive power errors.

This velocity is related to the actual position of the control rod through the relation

where “shim speed” is referred to the fact that the rod moves in regulation mode (in comparison with scram speed).

The position is calculated iteratively at each time step at which the power error is updated. The initial position is evaluated at the initial steady state conditions (null reactivity), when the rod is partially inserted. This partial insertion required to have the initial zero reactivity, is related to the hypotheses of Beginning of Cycle (BoC) conditions, i. e. absence of fuel burn-up, and of hot zero power conditions, i. e. a condition in which the weight of intrinsic feedbacks is weaker due to the 200 °C average temperature.

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The rod position is related to the reactivity inserted through the control rod worth curve, given by design (see figure 3.3).

Figure 3.3: Control rod worth curve [3.5]

The power controller can be considered a “physical” PI regulator, in fact the relation that stands between the input and the output variables is of integral type, namely, from velocity to position variables.

Figures 3.4 and 3.5 illustrate the power control logic followed during the analysis.

Figure 3.4: Reactivity control logic

Figure 3.5: Feed-back control loop for the power control system

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 0 10 20 30 40 50 60 70

k

_eff

vs. CR insertion (cm)

CR worth CR insertion [cm] 0 1.5 14 22 34 42 50 56 64 66.5 68 Dkeff (pcm) -88 -1397 -2653 -4943 -6502 -7835 -8553 -9078 -9114 -9112

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3.3 Secondary pressure control

The secondary pressure control is performed through the tertiary air fan movement, which changes its flow rate according to the pressure error. In particular, the pressure error is defined as the difference between the set point value of pressure and the pressure measured in the secondary loop (for each loop the steam pressure is measured in the steam line).

The integral action needed to have a null pressure error at regime conditions, is given through a PI controller.

The proportional and integral gain constants of the controller have been calculated through the use of MATLAB PidTuner MATLAB R2014b toolbox [3.4]. PidTuner is capable to extrapolate the behavior of the system from imported input-output data, finding its transfer function and designing its optimal controller. In order to do this, a step to the fan flow rate was given as input in the RELAP5-3D calculation and the secondary pressure output variation was recorded. The air flow rate and the secondary pressure data as a function of time have been then imported into the MATLAB environment.

Some flexibility is given through the choice of the controller type (P, PI, PID), the number of poles, possible delays. From the results it was seen that the optimal choice was a PI controller with three real poles. This is related to the energy sink and sources of the system, e. g. the volumes with huge thermal inertia that can be represented as different transfer function in series (namely, the lower plenum, the upper plenum and the water loop).

The output of the PI controller has been lagged for 300 seconds, in order to take into consideration the fan inertia, i. e. a delay of response to the system. This inertia can be both a positive and negative aspect, according to the situation analyzed: if the system has rapidly to follow pressure variations, an high inertia is clearly a negative aspect; considering instead safety issues, a high inertia is seen as a benefit during particular transients in which the power removal from the secondary system is needed (e. g., for the decay heat removal).

Figure 3.6: Feed-back control loop for the secondary pressure control system

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3.4 Future developments for the MYRRHA control strategy

The preliminary control strategy analyzed during this study does not consider any primary temperature control. A primary temperature control may be anyway required in order to avoid temperature errors; in particular, one main issue for heavy liquid metal coolants is related to the minimum temperature limitation to avoid the solidification of the coolant itself. This temperature is ~123 °C for LBE coolant, and the safety margin requires a minimum temperature of 200 °C.

The secondary side of the system is constituted by water at saturation conditions: a pressure control is translated in a temperature control. This means that the heat transfer from the primary system to the secondary cooling loops is controlled by the water side (considering almost constant the heat transfer coefficient in the two-phase operating range). In that sense there is no possibility to control the average primary temperature of the LBE in the PHX. In fact, the dynamics of the system needs a degree of freedom to evolve, which in this case is represented by the LBE heat transfer properties.

Fixing the secondary temperature, the LBE primary temperatures are changing according to the power level, as shown in figure 3.7.

Figure 3.7: Primary temperatures change as a function of the power level

A power control only assures a power error equal to zero, but the required power level may not correspond to the desired temperature level. The relationship between power and temperature is of integral type, because the temperature is directly related to the internal energy stored in the system (this is true for a single phase system as the LBE primary side is).

100 150 200 250 300 350 400 0 20 40 60 80 100 Te m p e ra tu re [ °C] Power [%]

Reactor control strategy

Average core outlet

temperature Secondary system water temperature

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The primary temperature control may refer or to the hot plenum temperature (inlet of PHX) or to the cold plenum one (exit of PHX).

Controlling the LBE PHX exit temperature means having a control over the minimum primary system temperature, the nearest to the 200 °C limit, therefore it assures the respect of the safety margin.

Anyway, also the temperature at the inlet of the PHX can be controlled; in fact, it is related to the minimum temperature by the temperature difference across the PHX (~55°C in normal operating conditions), controlled by the secondary side heat transfer conditions.

This control over one of the two temperatures may be coupled to the power control, for example sending a temperature error coupled3_{with the power error to}

define the control rod velocity.

Some considerations about the absence of a primary temperature control in the preliminary study performed are presented below:

- In the case of MYRRHA, the high thermal inertia of the primary LBE pool is playing an important role in dumping temperature errors. Qualitatively speaking, this can be explained through the time constants of the system: power variations are much faster than the internal energy variations due to the thermal inertia, i. e. the delay, of the system.

- The temperature control can be considered as related to a protection system that plays a role only when temperature errors exceed the established margins. - As already mentioned, the relation between power and temperature is of

integral type. The power control logic already foresees an integral action through the physical relation between the velocity and the position of the control rod [3.6].

- The absence of a power-frequency control contributes to narrow the range of temperature variations situations. The temperature set point has not to be related to the electrical grid requirements.

These observations are confirmed by the results of the RELAP5 – 3D calculations illustrated in chapter 5. Among all the transients analyzed, the primary temperature limits are always respected even without a primary temperature control system.

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4 MYRRHA MATLAB - Simulink model

In this chapter, the developed mathematical model of MYRRHA in MATLAB – Simulink [4.1] environment will be described.

Simulink is a graphical block diagramming tool, which allows an intuitive and modular representation of the reactor components [4.1]. A distinct advantage of using Simulink programming language for modeling a complex plant such as MYRRHA is found in the availability of control system design packages. This allows a very easy and visible control system implementation, with respect to the more complicated RELAP5 solutions.

The analytical model is composed of different subsystems describing the core dynamics (i. e., neutronics, thermal hydraulics and reactivity), and dedicated blocks representing the PHX, SD and condenser. In the following, each block with the respective equations implemented has been briefly described.

Figure 4.1 represents the simplified plant model adopted for MYRRHA Simulink description. As can be seen, both the secondary and tertiary cooling loops are here collapsed into two single loops.

Figure 4.1 - MYRRHA Simulink model

A brief overview is given to illustrate how the different coolant flow paths are organized.

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In the primary cooling system, almost half (7710 of the total 13800 kg/s) of the LBE coolant passes through the reactor core and heats up, while the other half flows along the bypass channel with a small increase of the average temperature (10 MW exchanged vs. 100 MW in the core). The two flows mix together in the hot plenum. Next, the coolant enters the PHX, where heat is transferred towards the secondary cooling system. Once cold again, the LBE enters the cold plenum closing the primary loop. Concerning the secondary water flow, the fluid enters from the bottom of the PHX in slightly subcooled conditions and exchanges almost 55 °C in terms of enthalpy jump. In normal operation the water exits the PHX with 0.3 flow quality. This two-phase flow is directed to the steam drum component where the steam is separated from the liquid. The steam is directed through the steam line to the aerocondenser where it exits in saturated liquid conditions and then comes back to the SD trough the feedwater line. The liquid part of the two-phase mixture mixes together with the feedwater and enters again the PHX. The air side exchanges 110 MW in normal operation, with an increase of 70 °C for the air.

4.1 Neutronics

Point neutron kinetics model with six delayed neutron precursor groups has been employed for the core neutronics, in which the total power is considered as generated only by fission events while the contribution of decay heat being neglected [4.2]. Table 4.1 shows the delayed neutron constants, namely, the delayed neutron fraction and the delayed neutron decay constant, for MYRRHA reactor design.

Table 4.1 - Delayed neutron constants

Delayed neutron constants

GROUP [/] [s-1_] 1 0.0246 0.0129 2 0.2086 0.0313 3 0.1871 0.1346 4 0.362 0.3443 5 0.1657 1.3764 6 0.052 3.7425

For sufficiently long irradiation times at a constant power level, the heat produced in the core can be attributed to prompt and delayed fissions. The effects of actinide decay only come into play when a transient in the power level occurs, where it introduces a small time delay between the power output and the neutron flux: the error made neglecting the contribution of decay heat is in the order of few percents.

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The system equations 4.1 represents 7 ODEs, i. e., one nonlinear equation for neutron density and six linear ones for precursor densities. In the present model, a further simplified version has been adopted, in which all the precursor groups have been collapsed into a unique one by means of an abundance-weighted average decay constant [4.2].

4.2 Core thermal - hydraulics

A zero-dimensional approach has been adopted to describe the system thermal hydraulics as well. Some simplifying hypotheses have been assumed and a single node (for each material) heat transfer model has been implemented by accounting for four distinct temperature regions – corresponding to fuel, gap, cladding and coolant – enabling the reactivity feedbacks to include all the major contributions as well as the margin against technological and safety limits to be monitored. A steady-state temperature distribution model, based on the evaluation of the successive thermal resistances, has been employed to evaluate the global heat transfer coefficients from the fuel centerline to the coolant bulk, on the basis of thermal inertia (table 4.2) of the single zones.

Table 4.2 - Thermal inertia data for the primary system nodes

Node Mass [kg] Heat capacity _[J/(kgK)] Thermal inertia _[J/K]

FUEL 2119 323.4 685348 CLAD 608 554.2 336954 LBE CORE 25000 145.4 3635000 LBE BYPASS 30000 145.4 4362000 LBE COLD PLENUM 4500000 145.4 654300000 LBE HOT PLENUM 1100000 145.4 159940000

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Table 4.2 gives a quantitative demonstration of the acceptability of a heat conduction steady state approach in which the fuel, gap and cladding are supposed "non accumulating" or better, quite fast in the transport of heat, with respect to the LBE coolant flowing through the core.

The thermal resistances between fuel and coolant can be easily defined in cylindrical geometry (the data refer to the fuel pin dimensions) as:

Where is the thermal conductivity of the MOX fuel taken as a constant as a first approximation.

Where the heat transfer coefficient is defined as

treating the heat transfer across the gap as a convective condition.

is the resistance across the cladding material where the radii are referred to the pin dimensions (see chapter 2).

is the convective heat transfer between the fuel clad and the LBE in the core. The heat transfer coefficient of the primary coolant in the core is related to the Nusselt number as follows:

Where the Nusselt number for LBE has been performed through Ushakov correlation for liquid metal heat transfer [4.3]:

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The fission power generated within the fuel is calculated according to the following equation, in which the subscript 0 refers to steady-state values, and is treated as an input for the heat transfer dynamic model:

The linear power is defined as the total power per unit length. The total length is calculated as the active pin height times the total number of pins:

Given the linear power and the maximum temperature of the fuel (at the center of the pin), it is possible to calculate the radial temperature trend from the centerline to the bulk of LBE, known the heat transfer properties of the materials in between, i. e. the thermal resistances defined above.

In particular, the clad outlet temperature is given by the following relation:

The bulk LBE temperature is analogously calculated:

The physical properties of the fuel, gap and cladding have been assumed to be constant with temperature as a first approximation. In particular, calculations of material properties have been performed in correspondence with the average nominal steady-state temperatures. The thermal conduction in the axial direction within the fuel pin has been neglected [4.2].

4.3 Reactivity feedbacks

Consistently with the lumped-parameter model employed, the reactivity feedbacks have been expressed as functions of the average values of the fuel and coolant temperature fields. Moreover, externally induced reactivity has been simulated through a dedicated coefficient associated with the insertion length of a representative control rod, which has been treated as a simple input parameter.

Reactivity effects by coolant density variations, axial expansion, and control rod motion have been accounted for by adopting a linear equation with constant coefficients. In particular, axial expansion has been related to the fuel thermal conditions, whereas the coolant density feedback has been considered as governed by the average core coolant temperature. As far as the Doppler coefficient determination is concerned, an effective average fuel temperature that accounts for resonances broadening has been calculated [4.2].

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In this work, a linear relation for axial core expansions and coolant density reactivity effects has been adopted, leading to the following expression with constant reactivity coefficients:

Where steady state average temperatures have been calculated in correspondence with the power level considered. The terms in the equation above indicate the reactivity margin stored in the core ( , the feedbacks induced by fuel temperature changes (i. e., Doppler effect through the Doppler coefficient that has a logarithmic dependence on the fuel temperature, and axial expansion though the respectively), the effect due to the coolant temperature variations (expansion through the coefficient ) and the last term is the control rod induced reactivity ( . The temperature subscript 0 refers to the hot zero power conditions (200 °C) in which the coefficient have been calculated [4.7]. Table 4.3 gives the values of the feedback coefficients used in the calculation.

Table 4.3 – Reactivity coefficients for MYRRHA fuel and LBE [4.4]

Material (fissile zone) FUEL LBE Temperature [°C] 1000 315

Effect Doppler Axial exp. Expansion Feedback [pcm] 482 328 67

Coefficient [pcm/K] -0.410 -0.583

The neutronics equations implemented required a numerical solution model different from the ode45 used for thermal balances (following paragraph). The option ode23s or ode15s [4.1] were adopted to cope with such stiff problem. A stiff equation is a differential equation for which certain numerical methods for solving the equation are unstable, unless the step size is taken to be extremely small [4.5]. The main reason is that the equation includes some terms that can lead to rapid variation in the solution. The system of ODEs is described as a stiff system when the variables change according to very different time scales, in this particular case the neutronic and the thermal hydraulics time constants. Both ode45 and ode15s/23s are variable time steps method in which the time step varies according to the accuracy required by the solution.

4.4 Primary circuit thermal - hydraulics

For the energy balance equation within the coolant, the respective temperature at the end of the channel has been assumed as a state variable. The thermal inertias expressed in the equations are given in table 4.2. Since the coolant inlet temperature has been considered as a fixed input, the energy balance is:

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Where the average fluid temperature and the core outlet temperature can be related together through the relation

which means assuming a temperature trend symmetric with respect to the center of the channel. This same approach was applied for the other thermal balances.

The by-pass loop thermal balance is

The hot plenum thermal balance is

The cold plenum thermal balance is expressed as:

And the primary Heat Exchanger energy balance is written as follows:

Figure 4.2 shows the MYRRHA primary system Simulink model with the equations implemented.