Research Article 
Corresponding author: Iurii E. Shvetsov ( yshvethome@mail.ru ) Academic editor: Georgy Tikhomirov
© 2023 Yurii S. Khomyakov, Valerii I. Rachkov, Iurii E. Shvetsov.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Khomyakov YuS, Rachkov VI, Shvetsov IE (2023) Coupled modeling of neutronics and thermalhydraulics processes in LFR under SGleakage condition. Nuclear Energy and Technology 9(2): 8592. https://doi.org/10.3897/nucet.9.99154

The lead cooled reactor BRESTOD300 is developing as a part of Russian federal project “PRORYV”. Two circuit scheme is used in the reactor for heat removal. An inherent risk of two circuit reactor is the potential danger of water steam ingression in the core in the case of large leakage in steam generator initiated, for example, Steam Generator Tube Rupture (SGTR).
Reactor power and temperature response on vapor penetration to the core is studied, but pressurization effects are not in the purview of the paper.
The 3D multiphysics (neutronics + thermalhydraulics) UNICO2F code was developed for study of SGTR accident. The code calculates unsteady 3D space dependent distributions of coolant velocity, pressure and temperature, space distributions of vapor concentration and heat release density in the core and 3D temperature distributions in the fuel pins.
Guillotine rupture of one tube in Steam Generator (SG) is considered as initial event of the accident.
It is shown that even with the most conservative assumptions reactivity insertion due to vapor ingress in the core causes small increase of power in level and as a result maximum cladding temperature continue to stay well below safe operation design limit in the entire transient.
Hypothetical option of simultaneous tube rupture in few SG belonging to different loops is also analyzed. It is demonstrated that even in the case of simultaneous large leak in two SG the transient stays mild and temperature in the core after two small oscillations is stabilized at acceptable level.
In the long term the analysis confirmed the high level of reactor selfprotection against SGTR accident.
Lead cooled fast reactors, Nuclear safety, Accident, Steam generator leakage, Multicomponent flows simulation
The lead cooled 700 MW_{th} BRESTOD300 reactor design is under development in the Russian Federation within the framework of “PRORYV” project. Two circuits are used for heat removal from the reactor, without intermediate heat transport system. A consequence of the elimination of such system is the need to address the potential risk of steam ingress into the core in case of large leak in the Steam Generator (SG).
The main concern is caused by the following two aspects of the initial failure:
Until now, attention has been mainly paid to the first one of the above two problems. The review
The results of numerical studies of behavior of parameters of small size (10 MW_{th}) reactor in case of SG tube doubleended guillotine rupture as an initial event are presented in
Detailed numerical studies of the process of steam propagation through the primary circuit of the mediumsize European lead cooled reactor ELSY were made using CFD code
However, the overall goal of the reactor safety analysis is justification for the fact that under this accident conditions no safe operation limits specified for the reactor are exceeded, including max permissible temperature of the fuel element cladding. In order to reach this goal, the following procedures should be performed:
The 3D multiphysics (neutronics + thermalhydraulics) UNICO2F 3D code was developed for studying SGTR accident. The code is capable of calculating transient 3D spatial distributions of coolant velocity, pressure and temperature, steam concentration in the coolant and power density in the core, as well as fuel and cladding temperatures distribution over the core.
С specific heat capacity (J/kg/°K)
С _{ρ} average volumetric heat capacity (J/(m^{3}K)
$\overrightarrow{G}$ average mass flux of mixture (kg/(m^{2}s)
G_{v} mass flux of steam (kg/(m^{2}s)
G_{l} mass flux of lead (kg/(m^{2}s)
I node number in X direction (–)
J node number in Y direction (–)
K node number in Z direction (–)
P mixture pressure (Pa)
q heat release (W/m^{3})
r_{b} bubble radius (m)
T mixture temperature (°C)
U physical velocity of components (steam and liquid) (m/s)
u_{b} rise rate of the bubble in stationary liquid lead (m/s)
X, Y, Z Cartesian coordinates (m)
e porosity (volume fraction occupied by lead and steam mixture), steam and lead volume ratio in mixture (–)
ρ average density of mixture (kg/m^{3})
σ surface tension coefficient for lead (N/m^{2})
$\hat{\Lambda}$ tensor of friction coefficients (1/s)
_{$\hat{\Psi},{\hat{\Psi}}_{E}$} tensors that are taken into account slipping of steam component in the mixture (–)
b bubble
l lead (liquid)
v steam
max maximum
BRESTOD300 Russian Lead Fast Reactor
CFD Computational Fluid Dynamics
ELFR European Lead Fast Reactor
LMR Liquid Metal Reactor
MPPC Main Pump Pressure Chamber
NTC Neutronics and Thermalhydraulics coupled Code
SG Steam Generator
SGTR Steam Generator Tube Rupture
SIMMERIII Twodimensional, multiphase code, multicomponent, fluiddynamics code
UNICO2F Russian code for analysis of transient behavior of power and temperature distributions in the LMR core in the case of SGTR
3D threedimensional
Heat is removed from BRESTOD300 reactor core (Fig.
In case of SG leakage, steam from damaged tube first enters SG shell side, and then the generated bubbles carried away by the coolant flow are moving along the path “main pump suction chamber – main pump – Main Pump Pressure Chamber (MPPC) – inlet ring duct – reactor downcomer section – core diagrid – core”. Steam bubbles have an opportunity to separate to the reactor gas system moving to the core through the primary circuit parts that have free levels of the coolant. These are gas cavities in SG, MPPC and reactor downcomer section.
Steam fraction supplied to the core inlet depends on several factors and, primarily, on the steam bubble size, since this parameter governs the effectiveness of bubbles separation into gas plenum of the reactor on their way to the reactor core. The initial bubble size is determined by the conditions in the rupture point, in particular, by the rupture configuration and actual pressure difference between the primary and the secondary circuits. Moreover, steam bubbles formed after SG tube rupture have different sizes, and bubble size spectrum depends on the outflow conditions. SG tube doubleended guillotine rupture is usually taken as an initial event for safety analysis.
In this case, steam flows out from two ends of ruptured tube and total flow rate of steam entering the primary circuit are determined by hydraulic resistance of the ruptured tube sections and the secondary circuit pressure at the inlet and outlet of the steam generator. The most conservative estimates made for the steam generator of BRESTOD300 reactor plant give steam flow rate Gv = 0.73 kg/s.
One more limitation of steam flow rate is related to limited conveyance capacity of MPPC due to inevitable and drastic bubble coalescence after reaching critical value of steam fraction. This critical value corresponds to packing factor of the balls when the surfaces of adjacent balls will inevitably contact to each other. Maximum packing factor 0.74046 (Gausse) corresponds to the case when the balls are placed in the angles of triangular pyramid. However, the realization of such regular and dense structure in twophase flow looks extremely improbable. More realistic is the flow structure when the bubbles are positioned in the angles of cube. The maximum packing factor for the case is 0.52. As evaluation shows the conveyance capacity of MPPC of BRESTOD300 reactor is limited by value – 0.58кg/с
The possibility of simultaneous failure of SG tubes in more than one loop can be considered theoretically, however, it’s clear that in view of extremely low probability of such initial event it should be treated as hypothetical.
It should be noted that the least favorable consequences would be expected not as a result of the guillotine tube rupture but in case of appearance of the long narrow crack in the tube wall, since in the latter case there is high probability of formation of the small size bubbles, which can be hardly separated. In the course of their movement the bubbles can break down or, vice versa, merge forming larger bubbles depending on the flow mode and separate into the reactor gas plena.
Analytically one can assume the possibility of chain mechanism causing rupture of many SG tubes, when a tube adjacent to that already failed is broken by the jet coming from the breach. However, it should be taken into account that impossibility of a single tube rupture escalation to multiple ruptures of the tube bundle has been confirmed experimentally for conditions of BRESTOD300 steam generator
The primary circuit of BRESTOD300 reactor was intentionally designed to assure to max extent the separation of the steam into the cover gas reactor system though gaslewd interface, namely the cavities with free lead levels are arranged. Separation goes sequentially in the following three elements: steam generator itself, main pump pressure chamber and reactor downcomer.
While moving in liquid lead, steam bubbles would either break down into smaller bubbles or, vice versa, coalesce into larger bubbles, depending on flow characteristics, ultimately reaching the reactor gas plenum.
Many publications have been devoted to determination of raising velocity of bubbles in liquid. In particular, relationships worked out by Peables and Garber are considered the most universal
The primary circuit of BRESTOD300 reactor was intentionally designed to assure to max extent separation of the steam into the cover gas though free surfaces while it passes through MPPC.
It follows from presented data that rise rate rapidly decreases with the decrease of bubble size and it becomes low with respect to typical flow velocity in the primary circuit elements (1 – 2 m/s) already for the bubbles of less than 10^{4} m diameter. Hence it appears that in conservative analysis rise rate should be neglected, as well as the possibility of steam separation in the course of its transport from tube rupture point to the reactor core.
The inhouse UNICO2F was developed for analysis of transient behavior of power and temperature distributions in the LMR core. Its structure is shown on Fig.
The code consists of three main modules, their functions being described below.
SVIR – calculation of velocity, pressure and temperature of coolant within computational domain, which, in this case, includes reactor downcomer section and reactor core, and temperature of structural elements, fuel and cladding (heat and mass transfer equations set is solved to approximation of the model of viscous ideal liquid flowing in porous body in Cartesian or cylindrical coordinate system; there is a model of thermal conductivity of multilayer cylindrical element bounded by the coolant flow, which can be used for evaluation of 3D temperature distribution in the fuel elements).
DENS_V – calculation of steam concentration within computational domain.
MAGTDP – calculation of reactor core power profile taking into account steam concentration variable (MAG neutronics module was designed on the basis of similarlynamed code
When formulating twophase thermal hydraulics equations set and related twophase flow pattern diagram the following simplifying assumptions are used:
It is generally clear that such an approach is applicable, first, to relatively slow processes and, second, when steam volume fraction in lead is still not too big.
Taking into account the above listed simplifications the following equation set describing movement of liquid metal and steam phases mixture in nuclear reactor in porous body approximation has been stated:
Mass balance equation for mixture:
$\frac{\partial \rho}{\partial \tau}+\frac{1}{\epsilon}\left(\overrightarrow{\nabla}\epsilon \overrightarrow{G}\right)=0$ (1)
Momentum equation for mixture:
$\frac{\partial \overrightarrow{G}}{\partial \tau}+\frac{1}{\epsilon}\left(\overrightarrow{\nabla}\hat{\Psi}\frac{\epsilon}{\rho}\overrightarrow{G}\right)\overrightarrow{G}=\overrightarrow{\nabla}P\hat{\Lambda}\overrightarrow{G}+\overrightarrow{F}$ (2)
Energy equation for mixture:
$\frac{\partial \left({C}_{p}T\right)}{\partial \tau}+\frac{1}{\epsilon}\left(\overrightarrow{\nabla}{\hat{\Psi}}_{E}{C}_{p}\frac{\epsilon}{\rho}\overrightarrow{G}\right)T=q$ (3)
Steam mass balance equation:
$\frac{\partial {\rho}_{v}{\epsilon}_{v}}{\partial \tau}+\frac{1}{\epsilon}\overrightarrow{\nabla}\left(\epsilon {\epsilon}_{v}{\rho}_{v}{\overrightarrow{U}}_{v}\right)=0$ (4)
Simplified momentum equation for steam:
${\overrightarrow{U}}_{v}={\overrightarrow{U}}_{l}+{u}_{b}\left({r}_{b}\right)\frac{\overrightarrow{\nabla}P}{\left\overrightarrow{\nabla}{P}_{0}\right}$, (5)
where P_{0} is pressure distribution in the reactor for normal conditions (without steam injection).
Equation of volume fraction balance:
${\epsilon}_{v}+{\epsilon}_{l}=1$ (6)
Equation of state
for lead: ${\rho}_{l}={\rho}_{l}\left(T\right)$, C_{l} = const
for steam: ${\rho}_{v}={\rho}_{v}(T,{P}_{b})$, C_{v} = const
where pressure in the bubble P_{b} is composed of circumjacent lead pressure Р and addition caused by surface tension effect P_{b} = P + 2σ/r_{b}
In the equations, average density ρ, average volumetric heat capacity Сρ and average mass velocity $\overrightarrow{G}$ of mixture are calculated by the following relationships:
$\rho ={\rho}_{v}{\epsilon}_{v}+{\rho}_{l}{\epsilon}_{l}$ (7)
$\overrightarrow{G}={\rho}_{v}{\epsilon}_{v}{\overrightarrow{U}}_{v}+{\rho}_{l}{\epsilon}_{l}{\overrightarrow{U}}_{l}$ (8)
$C\rho ={C}_{v}{\rho}_{v}{\epsilon}_{v}+{C}_{l}{\rho}_{l}{\epsilon}_{l}$ (9)
Heat conductivity equation is solved for calculation of 3D fuel and cladding temperature in the pins. Heat transfer coefficient on clad outer surface is calculating taking into account deterioration of heat transfer due to decrease of the heat capacity.
Onephase 3D thermalhydraulics module of the code UNICO2F named GRIF was validated against a few inpile reactor tests including analyses of Sodium Natural Convection in the Upper Plenum of the MONJU Reactor Vessel
Computational domain simulating the central part of BRESTOD300 reactor (Fig.
Conditions at the inlet of the reactor downcomer section are used as input data. These include flow rate and temperature of lead and steam, as well as steam bubble size (it is assumed that all incoming bubbles have the same given size, which then changes in the process of their movement depending on external temperature and pressure).
The following assumption were put into the base of the reference scenario:
All those assumptions are conservative.
Results of analysis of steam velocity and concentration patterns behavior for the reference option are shown in Fig.
There are two specific stages of the process of steam propagation through the reactor. The first stage starts when the steam enters downcomer section of the reactor. Then steamlead mixture “plume” moves downward in the downcomer section and on 22nd second the steam enters the left part of the core. This stage is characterized by strongly pronounced azimuthal nonsymmetry of steam distribution in the downcomer and in the core. As a result, the first sudden change of steam content occurs in the core (Fig.
In the second stage (after first 50 seconds of transient), the downcomer is gradually filled with steamlead mixture, because it also starts floating up and fills in steps the whole downcomer volume. Steam accumulated in the downcomer section is taken up by the lead flowing from the other (intact) loops and brought to the reactor core. From this moment (approximately after 40th second) the amount of steam entering the core starts increasing again, and now pattern of steam supply to the core is azimuthally symmetric. The second stage is finished by steady state onset (Figs
Max reactor power achieved during the transient exceeds rated value by 25%, and max fuel element cladding temperature is 680 °С, this being significantly lower than the corresponding safe operation limit Moreover, safe operation limit in terms of max fuel element cladding temperature (it is allowed to expose cladding to 900 °С no longer than for 10 minutes) is not reached.
Flow pattern in the downcomer section and effectiveness of steam separation to the gas plenum of the reactor depends on the size of bubbles. Large bubbles are floating up immediately after their arrival from the inlet nozzles forming upward flow plume, and entering reactor gas plenum. Small bubbles are brought downward by the flow and they reach the reactor core mostly in its periphery. Ratio of numbers of separated steam bubbles and those brought to the reactor core strongly depends on the bubble size (Fig.
Effect of vapor penetration to the reactor core has two main components:
It can be seen from comparison of curves in Fig.
It is clear that the probability of large leakages occurring simultaneously in two or more loops is extremely low, and common cause failure is excluded by the independence of each primary loop. Nevertheless, for the sake of deep insight into the accidental processes and reactor selfprotection margins numerical studies were carried out on the accident with simultaneous SG tube doubleended guillotine ruptures in two or more loops as an initial event. It is also conservatively supposed that all steam bubbles generated in the leakage location are transported to the core inlet without separation.
It has been found that in case of a sudden emergence of large leaks simultaneously in two adjacent SGs transient nature is the same and, as usual, after two fluctuations core power and temperature become stable (Fig.
Moreover, safe operation limit in terms of max fuel element cladding temperature (to expose cladding to 900 °С no longer than for 10 minutes) is not exceeded (Table
In the context of analysis of accident caused by large leak in the steam generator in twocircuit lead cooled reactor plant, it has been demonstrated that 2F 3D UNICO2F code is capable of modeling core thermohydraulics and neutronics taking into account their mutual influence.
Numerical studies have been performed showing high level of selfprotection of BRESTOD300 reactor in case of SG leak, even with the most conservative assumptions (negligibly small gas bubble raising velocity due to small specified diameter of incoming steam bubbles, neglecting of possible decrease of vapor flow rate through the breakup during the transient, total failure of safety system). Max cladding temperature in the entire transient is well below safe operation design limit, and transient is ended with safe and stable reactor condition. Moreover, even in hypothetic accident with simultaneous tube ruptures in two steam generators in two different loops safety limit on clad temperature is not exceeded and therefore and core saves its integrity.
The authors acknowledge helpful discussions with Dr. I.R. Suslov of JSC Proryv.