Flash evaporation of a superheated water droplet in heavy liquid metal coolant (lead) is considered, in application to the analysis of a leadcooled fast reactor steam generator tube rupture accident. The model is based on thermodynamic equilibrium formulation for the expanding watersteam mixture and inviscid compressible formulation for the surrounding liquid lead, with the interface conditions determined from the solution of the Riemann problem. Numerical solution is performed in the spherically symmetric geometry using a conservative numerical scheme with a moving sharp interface. Transient pressure and velocity profiles in each phase are presented for the parameters typical of the steam generator tube rupture accidents, demonstrating the process of boiling water expansion and pressure wave formation in the coolant. The results obtained are compared with a simplified model which considers the volumeaveraged parameters of boiling water droplets and considers coolant as an incompressible liquid. Good agreement between the full and simplified models is demonstrated. Impacts of coolant flow on structures caused by pressure wave propagation and subsequent coolant flow are discussed.
Leadcooled fast reactors (LFRs) are considered a prospective avenue of nuclear industry development towards future systems possessing higher efficiency and operational and safety features in comparison with today’s conventional light water reactors (LWRs) [
The presence of highpressure water in steam generator tubes raises concerns of possible consequences of tube rupture; therefore, extensive research has been focused on possible consequences of an SGTR accident. It is recognized that numerous physical phenomena accompany SGTR: shock wave propagation in the liquid lead; water flashing and expansion; water displacement from the tube to the intertube space filled with liquid lead; depressurization wave propagation in the broken tube, lead sloshing in the primary space; leadwater thermal interaction and possible steam explosions; steam bubble entrapment and transport towards the core [
The purpose of the current work is to consider an elementary process of SGTR accident, namely, rapid boilup of a highpressure water droplet immersed in lowpressure liquid lead. The aim is to obtain the features of twophase mixture expansion, including the flow field and pressure waves generated by this expansion in the heavy liquid filling the surrounding space. First, a detailed model is presented requiring numerical integration of conservation laws in the watervapor mixture and singlephase liquid lead, after which more simplified approaches are discussed relying on the solution of ordinary differential equations only.
Note that the processes involved are physically similar to those in the wellknown boiling liquid expanding vapor explosions (BLEVEs) regarding one of the major hazards of accidents with highpressure vessels [
Consider a spherical droplet of saturated water at a high pressure
The structure of the flow caused by expanding boiling water is sketched in Figure
Sketch of the main zones for expansion of superheated water into liquid coolant.
In
Under these assumptions, instead of considering the conservation laws for liquid and vapor phases coupled by the interphase exchange source terms, we consider the mixture of both phases characterized by its density
The equation of state (EOS) for the watervapor mixture is
In
For molten lead, the classic Tait equation of state (e.g., [
To sum up, the mathematical model for highpressure water droplet expansion in molten lead is formulated as Eulertype equations in both zones, but with different equations of state reflecting significant differences in the substance properties. In order to match the solutions in both zones, conditions on the moving interface between them must be formulated.
Zones 1 and 2 in Figure
The initial conditions correspond to a water droplet of radius
Consider in more detail the properties of each phase described by the respective equations of state: (
In order to close the EOS for water, the pressure dependencies for the specific volumes,
Consider now the specific features of a thermodynamic equilibrium twophase mixture where both phases reside on the saturation curve, with the proportion determined by the isentropic condition. In Figures
Calculated pressure dependencies of equilibrium watervapor mixture properties: (a) mass fraction of vapor
Unlike the saturated watervapor mixture (Section
The adiabatic speed of sound at the nominal conditions can also be obtained from Tait EOS (
By equating the speed of sound (
The coupled system of equations (
In order to describe the sharp interface between Zones 1 and 2 (contact discontinuity), a conservative interface tracking method [
Note that in the current formulation (thermodynamically equilibrium watervapor mixture) numerical procedure is free from the difficulties in approximation of stiff source terms describing the interphase processes (drag, evaporation, condensation, etc.) typically encountered in the multifluid models where conservation equations for each phase are solved separately.
As the baseline scenario, the parameters listed in Table
Parameters of numerical simulation of singledrop waterlead interaction.
Parameter  Value 



Pressure 
18 MPa 
Temperature 
630.1 K 
Density 
543.6 kg/m^{3} 
Initial radius 
13 mm (25 mm) 


Pressure 
0.8 MPa 
Temperature 
800 K 
Density 
10417.4 kg/m^{3} 
Consider the pressure and velocity fields developing during the first few milliseconds after droplet boiling and expansion begin. This stage is featured by quite fast processes of wave propagation from the highpressure droplet into lowpressure lead, with the contact discontinuity on the waterlead boundary acting as a spherical piston.
In Figures
Propagation of elastic compression wave into liquid lead: (a) pressure; (b) velocity.
Time histories of local pressure at different distances from the droplet center.
Figure
Pressure profile near the expanding water droplet.
In Figure
Radial velocity profiles at different times.
A distinct feature of the processes considered, also mentioned in [
Simulations performed for a larger water droplet with the initial radius of
Analysis of the results obtained above shows that, due to high inertia of the heavy liquid surrounding the water drop, its expansion proceeds much slower than propagation of the elastic compression waves through the molten lead and of pressure relief wave propagating through the twophase mixture in the boiling droplet. Water expansion can be considered as a relatively slow process with essentially subsonic characteristic velocities, superimposed by fast wave propagation at sonic velocities. Therefore, a simplified model can be derived for the “slow” expansion under assumptions that (i) in the twophase mixture all parameters are distributed uniformly over the droplet volume, so that only the volumeaveraged parameters of watervapor mixture can be considered and (ii) molten lead behaves as an incompressible liquid. Such a model was recently developed in [
Under the above assumptions, the droplet is characterized by its radius
Coolant is described as an incompressible liquid of density
By matching the pressure
In Figures
Comparison of pressure profiles predicted by the full (curves with times indicated) and simplified (curves denoted by symbols) models: (a) droplet radius
Thus, the full model (Section
Time histories of water droplet pressure, radius, and expansion velocity upon oscillations in liquid lead for different initial radii: (a)
The studies of singledroplet waterlead interaction show that effects on the surrounding structures (say, nearby tubes of steam generator) of highpressure water expansion in molten lead can be of two types, differing in the characteristic time scales. The first type is a rapid (shock) impact caused by propagation of an elastic compression wave travelling at a speed close to the speed of sound in liquid coolant. Say, for a water droplet of 13 mm initial radius, the characteristic arrival time to a point located at 10 cm distance from the droplet is of the order of 0.05 ms, and pressure jump can be as high as 3.5 MPa.
Another type of interaction is the effect of slowly changing radial flow past the structures. This stage of interaction is featured by much longer characteristic times, of the order of water droplet oscillation time which can be as long as dozens of milliseconds. At this stage, liquid lead behaves as an incompressible fluid, and the force is mainly attributed to viscous drag. Evaluation and quantitative comparison of the effects of pressure waves and coolant flows on steam generator pipes will be the subject of future research.
While the 1D spherically symmetric statement of the problem used in the current work is the geometrically simplest, the proposed model takes into account the basic physical and fluid dynamics processes. In the future research, the model will be extended to more complicated 2D and 3D models in order to study various modes of water discharge from broken pipes into molten lead pool.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The research was supported by the Russian Foundation for Basic Research (RFBR) (Projects no. 170800308, no. 170800287, and no. 160800239).