Research ArticlePressure Waves due to Rapid Evaporation of Water Droplet inLiquid Lead Coolant

S. E. Yakush ,1,2 A. S. Iskhakov ,3,4 V. I. Melikhov ,3,4 and O. I. Melikhov 3,4

1 Ishlinsky Institute for Problems inMechanics of the Russian Academy of Sciences, Ave. Vernadskogo 101, Bldg 1,Moscow 119526, Russia2Bauman Moscow State Technical University, Baumanskaya 2-ya 5, Moscow 105005, Russia3National Research University “Moscow Power Engineering Institute”, Krasnokazarmennaya 14, Moscow 111250, Russia4Joint Stock Company “Electrogorsk Research and Engineering Center on NPP Safety”, Saint Constantine Str. 6, Electrogorsk,Moscow Region 142530, Russia

Correspondence should be addressed to S. E. Yakush; yakush@ipmnet.ru

Received 25 September 2017; Accepted 28 January 2018; Published 1 March 2018

Academic Editor: Michel Giot

Copyright © 2018 S. E. Yakush et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Flash evaporation of a superheated water droplet in heavy liquid metal coolant (lead) is considered, in application to the analysis ofa lead-cooled fast reactor steam generator tube rupture accident. The model is based on thermodynamic equilibrium formulationfor the expanding water-steam mixture and inviscid compressible formulation for the surrounding liquid lead, with the interfaceconditions determined from the solution of the Riemann problem. Numerical solution is performed in the spherically symmetricgeometry using a conservative numerical scheme with a moving sharp interface. Transient pressure and velocity profiles in eachphase are presented for the parameters typical of the steam generator tube rupture accidents, demonstrating the process of boilingwater expansion and pressure wave formation in the coolant. The results obtained are compared with a simplified model whichconsiders the volume-averaged parameters of boiling water droplets and considers coolant as an incompressible liquid. Goodagreement between the full and simplified models is demonstrated. Impacts of coolant flow on structures caused by pressure wavepropagation and subsequent coolant flow are discussed.

1. Introduction

Lead-cooled fast reactors (LFRs) are considered a prospectiveavenue of nuclear industry development towards futuresystems possessing higher efficiency and operational andsafety features in comparison with today’s conventional lightwater reactors (LWRs) [1–4]. Lead and lead alloys have severaladvantages over other liquid metal coolants (e.g., sodium)due to their chemically inert nature. Since no energeticreactions and hydrogen generation occur in molten leadcoolant in the case of steam generator tube rupture (SGTR),reactor system design can be simplified significantly becausethe steam generator can be placed directly in the primaryheat circuit, without the need of any additional heat transportsystems.

The presence of high-pressure water in steam genera-tor tubes raises concerns of possible consequences of tuberupture; therefore, extensive research has been focused on

possible consequences of an SGTR accident. It is recog-nized that numerous physical phenomena accompany SGTR:shock wave propagation in the liquid lead; water flashingand expansion; water displacement from the tube to theintertube space filled with liquid lead; depressurization wavepropagation in the broken tube, lead sloshing in the primaryspace; lead-water thermal interaction and possible steamexplosions; steam bubble entrapment and transport towardsthe core [5]. A number of dedicated experiments on water-lead interaction were carried out, most notably LIFUS exper-imental series [6, 7]; these were supported by numericalsimulations performed by SIMMER III thermohydraulicsystem code [8, 9]. It was found experimentally that waterflashing produced significant pressure peaks in the liquidcoolant (about 0.7–1.1MPa reported in [7]); however, inSIMMER III calculations of these tests the initial pressurepeak was underestimated 2-3 times (0.3–0.35MPa). It shouldbe noted that SIMMER III is a Eulerian multifluid code in

HindawiScience and Technology of Nuclear InstallationsVolume 2018, Article ID 3087051, 10 pageshttps://doi.org/10.1155/2018/3087051

2 Science and Technology of Nuclear Installations

which interactions at the scales of a single water droplet arenot simulated directly but are taken into account via relevantclosures. Such predictions can potentially be improved bymore detailed (time and space-resolved) analysis of the phe-nomena involved in SGTR accidents. Similar experimentalstudies of SGTR accident in BREST reactor were carried outin Russia [10], with the main focus on the possibility of chaintube ruptures. The experimental results have shown that thechain ruptures of the neighboring tubes were unlikely, even ifthe tubes were initially defective.

The purpose of the current work is to consider anelementary process of SGTR accident, namely, rapid boil-upof a high-pressure water droplet immersed in low-pressureliquid lead. The aim is to obtain the features of two-phasemixture expansion, including the flow field and pressurewaves generated by this expansion in the heavy liquid fillingthe surrounding space. First, a detailed model is presentedrequiring numerical integration of conservation laws in thewater-vapormixture and single-phase liquid lead, after whichmore simplified approaches are discussed relying on thesolution of ordinary differential equations only.

Note that the processes involved are physically similarto those in the well-known boiling liquid expanding vaporexplosions (BLEVEs) regarding one of the major hazardsof accidents with high-pressure vessels [11]. The difference,though, is that in the SGTR accident the ambient space isfilled with a heavy liquid possessing much higher inertia andmuch lower compressibility than air.

2. Problem Formulation

2.1. Scheme of Multiphase Flow. Consider a spherical dropletof saturated water at a high pressure 𝑝𝑤,0 and initial temper-ature equal to the saturation temperature, 𝑇𝑤,0 = 𝑇𝑠(𝑝𝑤,0).At the initial instant, the droplet is immersed in a low-pressure space filled with a heavy liquid coolant (moltedlead) at a lower pressure 𝑝𝑙,0 (water is superheated withrespect to the ambient conditions because 𝑇𝑤,0 = 𝑇𝑠(𝑝𝑤,0) >𝑇𝑠(𝑝𝑙,0)). As the pressure relief wave is propagating throughthe high-pressure towards the droplet center, water boils upand expands due to rapid increase in the specific volumeof the two-phase mixture, generating pressure waves in theambient lead. In this work, we assume that the flow field isspherically symmetric and, therefore, consider the problem inthe spherical coordinates with the origin at the droplet center.

The structure of the flow caused by expanding boilingwater is sketched in Figure 1. Two main zones separated bya sharp interface are considered. Zone 1 corresponds to waterin its single-phase (initial) state and in the form of two-phase liquid-vapor mixture after the passage of boiling frontconverging towards the droplet center. Zone 2 corresponds toliquid coolant (lead). The expanding water works as a piston,causing propagation of a spherical pressure wave in the liquidlead.

2.2. Governing Equations. In Zone 1, thermodynamic equi-librium between the liquid water and vapor is assumed.This assumption, discussed in detail in [12], implies that the

Zone 1Saturated water Zone 2

Boiling water Molten lead

Interface Pressure waveBoiling front

r

Figure 1: Sketch of the main zones for expansion of superheatedwater into liquid coolant.

interphasemass, momentum, and energy exchange processesare fast enough, so that both phases remain on the saturationline following the variation of local pressure; the velocityslip between the phases is also not taken into account (bothphases have the same pressure and velocity). A furtherassumption is that the flow of the two-phase mixture isisentropic (i.e., at any point the sum of entropies of liquidwater and vapor remains equal to the entropy in the initialall-liquid state corresponding to saturated conditions at theinitial water pressure).

Under these assumptions, instead of considering theconservation laws for liquid and vapor phases coupled by theinterphase exchange source terms, we consider themixture ofboth phases characterized by its density𝜌𝑤 and velocity𝑢𝑤 forwhich the conservation equations are derived by summing upthe corresponding conservation laws of both phases. In thisderivation, the interphase exchange source terms (interphasedrag, evaporation rate, etc.) entering the phase equationswithopposite sign are cancelled out. As a result, the flow of boilingwater in Zone 1 is governed by the Euler-type continuity andmomentum conservation equations

𝜕𝜌𝑤𝜕𝑡 + 1𝑟2𝜕 (𝜌𝑤𝑢𝑤𝑟2)𝜕𝑟 = 0,

𝜕 (𝜌𝑤𝑢𝑤)𝜕𝑡 + 1𝑟2𝜕 (𝜌𝑤𝑢𝑤2𝑟2)𝜕𝑟 = −𝜕𝑝𝜕𝑟 ,

(1)

where 𝑝 is the pressure, 𝑡 is the time, and 𝑟 is the radialcoordinate.

The equation of state (EOS) for the water-vapor mixtureis

𝜌𝑤 = 1(1 − 𝑥V) V𝑠,𝑙 + 𝑥VV𝑠,V , (2)

where V𝑠,𝑙 and V𝑠,V are the specific volumes of liquid and vaporphases on the saturation line depending on pressure 𝑝. The

Science and Technology of Nuclear Installations 3

mass fraction of vapor, 𝑥V, is determined as a function of localpressure 𝑝 by the isentropic relationship

𝑠𝑤,0 = 𝑠𝑠,𝑙 (𝑝𝑤,0) = (1 − 𝑥V) 𝑠𝑠,𝑙 (𝑝) + 𝑥V𝑠𝑠,V (𝑝) , (3)

where 𝑠𝑤,0 is the specific entropy of saturated water at theinitial water pressure 𝑝𝑤,0, while 𝑠𝑠,𝑙 and 𝑠𝑠,V are the specificentropies of saturated liquid water and vapor at the currentpressure 𝑝, respectively. Therefore,

𝑥V (𝑝, 𝑝𝑤,0) = 𝑠𝑤,0 (𝑝𝑤,0) − 𝑠𝑠,𝑙 (𝑝)𝑠𝑠,V (𝑝) − 𝑠𝑠,𝑙 (𝑝) . (4)

Note that 𝑥V and, therefore, all mixture properties dependparametrically on the initial pressure 𝑝𝑤,0; otherwise allmixture properties are single-values functions of pressure𝑝, that is, the flow in the two-phase zone is barotropic.For such a flow, the continuity and momentum equations(1) are decoupled from the energy equation. In fact, energyconservation was taken into account in the derivation ofthe equations of state (2)–(4), rather than formulated asan independent differential equation. For example, such animportant thermodynamic substance property as the heat ofevaporation Δ𝐻V is contained implicitly in isentropic rela-tionship (4): according to thermodynamics, the denominator(the difference of specific entropies of vapor and liquid on thesaturation line) can be written as 𝑠𝑠,V−𝑠𝑠,𝑙 = Δ𝐻V/𝑇𝑠, where𝑇𝑠is the temperature of phase change (saturation temperature).

In Zone 2, liquid lead is considered as an inviscidcompressible fluid; the flow is assumed to be barotropic(adiabatic) and governed by the continuity and momentumequations, similar to those in Zone 1 (see (1)):

𝜕𝜌𝑙𝜕𝑡 + 1𝑟2𝜕 (𝜌𝑙𝑢𝑙𝑟2)𝜕𝑟 = 0,

𝜕 (𝜌𝑙𝑢𝑙)𝜕𝑡 + 1𝑟2𝜕 (𝜌𝑙𝑢𝑙2𝑟2)𝜕𝑟 = −𝜕𝑝𝜕𝑟 ,

(5)

where 𝑝, 𝜌𝑙, and 𝑢𝑙 are the pressure, density, and velocity ofmolten lead, respectively.

For molten lead, the classic Tait equation of state (e.g.,[13]) is used in the form

𝑝 = 𝑝∗ + 𝐵 ⋅ [( 𝜌𝑙𝜌∗)𝛾 − 1] , (6)

where 𝑝∗ and 𝜌∗ are the pressure and density of lead at thereference conditions, 𝛾 and 𝐵 are EOS parameters specific toeach substance. The Tait equation is widely used to describecompressible liquids in a wide range of pressure variation; forexample, in underwater explosions, its application to moltenlead is discussed below (Section 2.5.2).

To sum up, the mathematical model for high-pressurewater droplet expansion in molten lead is formulated asEuler-type equations in both zones, but with different equa-tions of state reflecting significant differences in the substanceproperties. In order to match the solutions in both zones,conditions on the moving interface between them must beformulated.

2.3. Interface Conditions. Zones 1 and 2 in Figure 1 cor-respond to different substances (water and lead); they aredivided by a moving sharp interface (contact discontinuity).The interface pressure, 𝑝𝐼, as well as its propagation velocity,𝑢𝐼, is obtained as a solution to the Riemann problem. Giventhe instantaneous “left” and “right” states in water andlead, respectively, the following relationships determiningthe interface conditions can be obtained by the method ofcharacteristics (see, e.g., [14, 15]):

𝑢𝐼 = 𝑢𝑤 − ∫𝑝𝐼𝑝𝑤

𝑑𝑝𝜌𝑤𝑐𝑤,𝑠 ,𝑢𝐼 = 𝑢𝑙 + ∫𝑝𝐼

𝑝𝑙

𝑑𝑝𝜌𝑙𝑐𝑙,𝑠 ,(7)

where the top formula corresponds to water having pressure𝑝𝑤 and velocity𝑢𝑤; the bottomone corresponds to lead on theother side of the interface (with similar quantities denoted bysubscript 𝑙); 𝑐𝑤,𝑠 and 𝑐𝑙,𝑠 in (7) are the isentropic sound speedsin the respective substances. For the particular cases of EOSdefined by (2)–(4) and (6), the system of equations (7) forinterface conditions takes the form (see [14])

𝑢𝐼 = 𝑢𝑤 − ∫𝑝𝐼𝑝𝑤

𝑑𝑝𝜌𝑤 (𝑝) 𝑐𝑤,𝑠 (𝑝) ,

𝑢𝐼 = 𝑢𝑙 + 2 ⋅ 𝑐𝑙,𝑠𝛾 − 1 [(𝑓 (𝑝𝐼)𝑓 (𝑝𝑙) )

(𝛾−1)/2𝛾 − 1] ,(8)

where 𝑓(𝑝) = 𝑝 − 𝑝𝑙,0 + 𝐵 (see (6)). The upper integralfor water was evaluated numerically due to lack of analyticalexpressions for 𝜌𝑤(𝑝) and 𝑐𝑤,𝑠(𝑝). Solution of equation set (8)gives the interface pressure, 𝑝𝐼, and its propagation velocity𝑢𝐼.2.4. Initial and Boundary Conditions. The initial conditionscorrespond to a water droplet of radius 𝑅0 immersed in aninfinite space filled with liquid lead; both fluids are assumedto be at rest:

𝑝 (𝑡 = 0) = {{{𝑝𝑤,0, 0 ≤ 𝑟 < 𝑅0𝑝𝑙,0, 𝑟 > 𝑅0,

𝑢𝑤 (𝑡 = 0) = 𝑢𝑙 (𝑡 = 0) = 0,(9)

The interface is located initially at 𝑟 = 𝑅0, where pressurediscontinuity exists, dividing saturated water and liquid lead.The boundary conditions are zero water velocity at theorigin (due to spherical symmetry) and zero lead velocity“at infinity.” The pressure in the origin is determined by thecharacteristic relationship (see the top formula in (7)), wherethe velocity on the left-hand side must be set to zero. Atinfinity, pressure tends to its ambient value in the liquid lead.

2.5. Substance Properties. Consider in more detail the prop-erties of each phase described by the respective equations ofstate: (2)–(4) for water and (6) for liquid lead.

4 Science and Technology of Nuclear Installations

(1)

(3)(2)

(4)

50 15 2010

p (MPa)

0.0

0.1

0.2

0.3

0.4

x

(a)

(1)

(3)(2)

(4)

50 15 2010

p (MPa)

0

200

400

600

w(k

g/Ｇ

3)

(b)

WaterVapor

(1)(3)

(2)(4)

50 15 2010

p (MPa)

100

1000

c s,w

(m/s

)

(c)

Figure 2: Calculated pressure dependencies of equilibrium water-vapor mixture properties: (a) mass fraction of vapor 𝑥V(𝑝); (b) mixturedensity 𝜌𝑤(𝑝); (c) isentropic speed of sound 𝑐𝑠,𝑤(𝑝). Curves (1)–(4) correspond to initial pressure 𝑝𝑤,0 = 10, 14, 18, and 20MPa; dashed curvesare speeds of sound in single-phase liquid and vapor.

2.5.1. Saturated Water-Vapor Mixture. In order to close theEOS for water, the pressure dependencies for the specificvolumes, V𝑠,𝑙(𝑝), V𝑠,V(𝑝), and entropies, 𝑠𝑠,𝑙(𝑝), 𝑠𝑠,V(𝑝), of theliquid and vapor phases on the saturation line are required.These propertieswere taken from theNISTdata [16] as appro-priate tables within the pressure range 1–180 atm. Equallyspaced pressure intervals of 0.1 atm were used, with linearinterpolation of all properties between the tabular points.The same procedure was followed in [11] where explosionsof pressure-liquefied gases in air were modeled. A possiblealternative approach would be to implement any suitablecorrelations readily available for water (e.g., the well-knownIAPWS formulations [17]).

Consider now the specific features of a thermodynamicequilibrium two-phase mixture where both phases resideon the saturation curve, with the proportion determinedby the isentropic condition. In Figures 2(a)–2(c), pressuredependencies of the mass fraction of vapor 𝑥V(𝑝), mixture

density 𝜌𝑤(𝑝), and isentropic speed of sound 𝑐𝑠,𝑤(𝑝) =(𝜕𝑝/𝜕𝜌𝑤)1/2𝑠=const are plotted for four initial pressures 𝑝𝑤,0 in therange 10–20MPa; see curves (1)–(4) (note that since the initialstate is assumed to be all-liquid, two-phase mixture existsonly at pressures 𝑝 < 𝑝𝑤,0). Also, in Figure 2(c) the soundspeed in the single-phase water and vapor is plotted by thedashed lines. One can see that density variation with pressureis quite considerable and nonlinear, the mixture possessesmuch higher compressibility than its constituent phases(mixture density variation with pressure mainly occurs dueto phase transition, rather than compression of phases), and,therefore, the speed of sound in the mixture is much lowerthan in each phase.

2.5.2.Molten Lead. Unlike the saturatedwater-vapormixture(Section 2.5.1), thermodynamic parameters of molten leadfor the conditions of interest in this work are far from itssaturation line and, therefore, a single-phase formulation is

Science and Technology of Nuclear Installations 5

appropriate. In this work, we use Tait EOS (6), which requiresparameters 𝐵 and 𝛾 to be defined. No explicit data on thesevalues was found for molten lead in the literature; therefore,the following approach was taken. The nominal density andspeed of sound in molten lead at the nominal pressure 𝑝∗ =0.1 MPa and initial temperature 𝑇0 = 800K were evaluatedfrom the experimental correlations [18, 19]:

𝜌∗ = 𝜌𝑙 (𝑝∗, 𝑇0) = 11441 − 1.2795 ⋅ 𝑇0= 10417.4 kg/m3, (10)

𝑐𝑠,∗ (𝑝∗, 𝑇0) = 1953 − 0.246 ⋅ 𝑇0 = 1756m/s. (11)

The adiabatic speed of sound at the nominal conditionscan also be obtained from Tait EOS (6):

𝑐𝑠,𝑙 = ( 𝜕𝑝𝜕𝜌𝑙)1/2

𝑠=const= √𝐵𝛾𝜌𝑙𝛾−1𝜌∗𝛾 . (12)

By equating the speed of sound (12) to (11) at the nominalconditions (𝜌𝑙 = 𝜌∗), we obtain that𝐵𝛾 = 𝜌∗𝑐2𝑠,∗ = 𝐵𝑠 = 3.212⋅104MPa, where 𝐵𝑠 is the adiabatic elastic modulus. Onlythe product of two constants can thus be derived from thedensity and speed of sound; therefore one of themmust be setindependently. Generally, this would require more extensivecompressibility data than is available. However, in the currentproblem the molten lead pressure can only be as high as 18MPa (the initial water pressure). Therefore, by recasting theTait equation in the form

𝜌𝑙𝜌∗ = [1 +𝑝 − 𝑝∗𝐵 ]1/𝛾 (13)

one can see that for typical values of 𝛾 ≃ 10 (for water𝛾 ≃ 7) the second term in the brackets is actually very small,(𝑝 − 𝑝∗)/𝐵 ≤ 5.6 ⋅ 10−3. Therefore, (13) can be presentedin the linearized form 𝜌𝑙/𝜌∗ ≃ 1 + (𝑝 − 𝑝∗)/(𝛾𝐵) = 1 +(𝑝 − 𝑝∗)/𝐵𝑠 independent of particular 𝛾. This confirms thatfor the pressure range of interest in the current problemsensitivity to the parameters of the Tait equation is very low.In the simulations we chose 𝛾 = 12 (accounting for lowcompressibility of molten lead) and 𝐵 = 2677.5MPa.

3. Numerical Method

The coupled system of equations (1)–(6) was solved numeri-cally. The one-dimensional computational domain of radius𝑅max = 0.5m, and a uniform mesh contained 1000 cells. Ineach subdomain, corresponding to Zones 1 and 2, relevantgoverning equations were discretized by an explicit high-order low-dissipation numerical scheme HR-SLAU2 belong-ing to the well-known AUSM family of numerical schemes[20].

In order to describe the sharp interface between Zones 1and 2 (contact discontinuity), a conservative interface track-ing method [14, 15] developed for compressible multifluidswas implemented. The interface was described by the level-set method [21]: a signed distance function was introduced,

Table 1: Parameters of numerical simulation of single-drop water-lead interaction.

Parameter ValueWater dropletPressure 𝑝𝑤,0 18MPaTemperature 𝑇𝑤,0 630.1 KDensity 𝜌𝑤,0 543.6 kg/m3

Initial radius 𝑅0 13mm (25mm)Molten leadPressure 𝑝𝑙,0 0.8MPaTemperature 𝑇𝑙,0 800KDensity 𝜌𝑙,0 10417.4 kg/m3

with the zero value reached on the interface. This functionis transported on the mesh with the interface propagationvelocity obtained from the solution of the Riemann problem(see Section 2.3), and approximations to numerical fluxesare modified appropriately in the cells cut by the interface.Another approach to taking into account the interaction onthe contact discontinuity is the Ghost Fluid Method [22]which was also implemented and applied to test simulations.Both approaches gave very close results in terms of pressureprofiles obtained.

Note that in the current formulation (thermodynamicallyequilibrium water-vapor mixture) numerical procedure isfree from the difficulties in approximation of stiff sourceterms describing the interphase processes (drag, evaporation,condensation, etc.) typically encountered in the multifluidmodels where conservation equations for each phase aresolved separately.

4. Simulation Results

4.1. Parameters. As the baseline scenario, the parameterslisted in Table 1 were selected. The initial pressures inwater and coolant correspond to the operating conditions ofRussian lead-cooled fast reactor BREST-OD-300 [3, 4]. Thedroplet size in the baseline scenario corresponds to the upperestimate of the volume of liquid water that can be dischargedinto liquid lead (10−5m3) given in [5]. It is comparable withthe inner diameter of the steam generator pipes (13mm)in this reactor. Keeping in mind significant uncertaintiesinvolved in such estimates, we also consider the case wherea larger droplet is formed, of the radius 25mm, which can bea result of a longer release time. Note that detailed study onwater discharge process from a ruptured pipe would requirea multidimensional model, which is beyond the scope ofcurrent study but will be performed in the future.

4.2. PressureWavesGenerated by anExpandingWaterDroplet.Consider the pressure and velocity fields developing duringthe first few milliseconds after droplet boiling and expansionbegin. This stage is featured by quite fast processes ofwave propagation from the high-pressure droplet into low-pressure lead, with the contact discontinuity on the water-lead boundary acting as a spherical piston.

6 Science and Technology of Nuclear Installations

0

5

10

15

p(M

Pa)

0.04 ＧＭ 0.08 ＧＭ 0.12 ＧＭt = 0

10 20 30 400

r (cm)

(a)

t = 0.01 ＧＭ 0.03 ＧＭ 0.05 ＧＭ

3 6 90

r (cm)

0

2

4

6

u(m

/s)

(b)

Figure 3: Propagation of elastic compression wave into liquid lead: (a) pressure; (b) velocity.

0.02 0.04 0.060.00t (Mс)

4

8

12

p (М

Па)

r = 2 ＝Ｇ r = 8 ＝Ｇ

r = 4 ＝Ｇ

r = 6 ＝Ｇ

r = 10 ＝Ｇ

Figure 4: Time histories of local pressure at different distances fromthe droplet center.

In Figures 3(a) and 3(b), radial pressure and velocityprofiles are shown during the first 0.3ms into the process.One can see that an elastic compression wave (generatedby the initial pressure discontinuity) propagates through theliquid coolant at a high velocity, comparable with the speedof sound in liquid lead. This fast pressure wave has quite asharp front; behind this front the velocity and pressure in thelead almost coincide at all times presented in Figure 3. Thisis further illustrated in Figure 4 where pressure time historiesare shown at different distances from the droplet center: afterthe arrival of elastic compression wave, the pressure increasessharply and then remains almost constant.

Figure 5 shows in more detail the pressure profiles inthe water droplet. It can be seen that pressure relief wavepropagates into the saturated liquid converging to the dropletcenter at a time of about 0.1ms, which means that by thistime all water is boiling. As the boiling mixture expands intothe surrounding coolant, the pressure in the droplet is fallinggradually, with no significant radial gradients.

0ＧＭ

2

4

6

8

10

12

14

16

18

p(M

Pa)

1 2 3 4 5 6 7 80

r (cm)

0.1 ＧＭ

0.2 ＧＭ

0.3 ＧＭ0.4 ＧＭ

0.6 ＧＭ

0.5 ＧＭ

0.7 ＧＭ

Figure 5: Pressure profile near the expanding water droplet.

In Figure 6, radial velocity profiles in are plotted nearthe expanding droplet. In the two-phase water mixture, theradial distribution of velocity is nearly linear, which meansthat expansion occurs more or less uniformly throughoutthe droplet. In the surrounding lead, the velocity profile ishyperbolic, as will be shown later.

A distinct feature of the processes considered, also men-tioned in [5], is that, despite the high-pressure differencebetween the water droplet and surrounding liquid lead, theexpansion velocity is rather low, reaching just few m/s. Thisis attributed to high inertia of the heavy liquid metal, insharp contrast with explosive boil-up of superheated liquidsin air (BLEVE phenomenon) where the interface expands atvelocities comparable, or even exceeding the speed of soundin the air (hundreds of meters per second). This rather slowexpansion is the main reason for the absence of significantpressure gradients in the two-phase water mixture, despitethe low sound speed in it. This important finding can justifythe application of a more simple approach based on thevolume-averaged parameters for the water droplet.

Science and Technology of Nuclear Installations 7

0.1 ＧＭ

0.2 ＧＭ

0.3 ＧＭ

0.4 ＧＭ 0.7 ＧＭ

0.5 ＧＭ

0.6 ＧＭ

0

5

10

15

20

25

u(m

/s)

1 2 3 4 5 6 7 80

r (cm)

Figure 6: Radial velocity profiles at different times.

Simulations performed for a larger water droplet with theinitial radius of 𝑅 = 25mm gave qualitatively similar results.Expectedly, amplitude of the pressure waves was higher dueto larger energy yield for larger mass of water. In more detail,the pressure profileswill be compared in the following sectionwhere predictionswill also be comparedwith a simplermodelfor water droplet expansion.

5. Comparison with a Simplified Model

Analysis of the results obtained above shows that, due tohigh inertia of the heavy liquid surrounding the waterdrop, its expansion proceeds much slower than propagationof the elastic compression waves through the molten leadand of pressure relief wave propagating through the two-phase mixture in the boiling droplet. Water expansion canbe considered as a relatively slow process with essentiallysubsonic characteristic velocities, superimposed by fast wavepropagation at sonic velocities. Therefore, a simplified modelcan be derived for the “slow” expansion under assumptionsthat (i) in the two-phase mixture all parameters are dis-tributed uniformly over the droplet volume, so that only thevolume-averaged parameters of water-vapor mixture can beconsidered and (ii) molten lead behaves as an incompressibleliquid. Such a model was recently developed in [23]; here weoutline its main equations and focus on the comparison ofthe results with those obtained in Section 4 on the basis ofcomplete problem formulation.

Under the above assumptions, the droplet is characterizedby its radius 𝑅 and mixture density 𝜌𝑤(𝑝𝑤) which dependson the current water pressure 𝑝𝑤 according to the equationof state (2)–(4) (note that, unlike Sections 3 and 4, bothquantities here are scalars rather than functions of radius).Water mass conservation implies that 𝜌𝑤𝑅3 = const, whichgives a nonlinear relationship between the droplet radius andits pressure:

𝑅𝑑𝜌𝑤𝑑𝑝𝑤𝑑𝑝𝑤𝑑𝑡 + 3𝜌𝑤 𝑑𝑅𝑑𝑡 = 0. (14)

Simplified model

10 15 20 255r (cm)

1

2

3

4

5

p (M

Pa)

0.04 ＧＭ 0.08 ＧＭ 0.12 ＧＭ

0.04 ＧＭ

0.08 ＧＭ

0.12 ＧＭ

(a)

2

4

6

8p

(MPa

)

Simplified model

10 15 20 255r (cm)

0.04 ＧＭ

0.08 ＧＭ

0.12 ＧＭ

0.04 ＧＭ 0.08 ＧＭ 0.12 ＧＭ

(b)

Figure 7: Comparison of pressure profiles predicted by the full(curves with times indicated) and simplified (curves denoted bysymbols) models: (a) droplet radius 𝑅 = 13mm; (b) 𝑅 = 25mm.

Coolant is described as an incompressible liquid ofdensity 𝜌𝑙 = const, implying that in the ambient space (𝑟 ≥ 𝑅)the velocity 𝑢𝑙(𝑟, 𝑡) and pressure 𝑝𝑙(𝑟, 𝑡) satisfy the continuityand momentum equations

𝜕𝑢𝑙𝜕𝑟 + 2𝑢𝑙𝑟 = 0,𝜕𝑢𝑙𝜕𝑡 + 𝑢𝑙 𝜕𝑢𝑙𝜕𝑟 + 1𝜌𝑙

𝜕𝑝𝑙𝜕𝑟 = 0.(15)

By matching the pressure (𝑝𝑤 = 𝑝) and velocity(𝑢𝑙|𝑟=𝑅 = 𝑑𝑅/𝑑𝑡) on the water-lead boundary and satisfyingthe boundary conditions “at infinity” (𝑢𝑙 = 0, 𝑝𝑙 = 𝑝𝑙,0as 𝑟 → ∞), we obtain a second-order ordinary differential

8 Science and Technology of Nuclear Installations

5 10 15 20 250t (ms)

0

2

4

6

8

R (c

m)

5 10 15 20 250t (ms)

0

5

10

15

p (M

Pa)

pl,0

5 10 15 20 250t (ms)

−20

−10

0

10

20

V (m

/s)

(a)

5 10 15 20 25 30 35 40 45 500t (ms)

0

5

10

15

R (c

m)

5 10 15 20 25 30 35 40 45 500t (ms)

0

5

10

15

p (M

Pa)

pl,0

5 10 15 20 25 30 35 40 45 500t (ms)

−20

−15

−10

−5

05

101520

V (m

/s)

(b)

Figure 8: Time histories of water droplet pressure, radius, and expansion velocity upon oscillations in liquid lead for different initial radii:(a) 𝑅 = 13mm; (b) 𝑅 = 25mm.

equation of Rayleigh-Plesset type, widely applied for bubbleoscillation and cavitation modeling [12]:

𝑅𝑑2𝑅𝑑𝑡2 + 32 (𝑑𝑅𝑑𝑡 )2 + 𝑝𝑙,0 − 𝑝𝜌𝑙 = 0. (16)

Due to strong nonlinearity of 𝜌𝑤(𝑝𝑤) function (see Figures2(b) and 2(c)), the system of coupled equations (14) and(16) must be solved numerically to provide the water dropletpressure 𝑝𝑤(𝑡), radius 𝑅(𝑡), and expansion velocity 𝑉(𝑡) =

𝑑𝑅/𝑑𝑡. After that, the velocity and pressure distributions inmolten lead are obtained as

𝑢𝑙 (𝑟, 𝑡) = (𝑅 (𝑡)𝑟 )2 𝑉 (𝑡) , (17)

𝑝𝑙 (𝑟, 𝑡) = 𝑝𝑙,0 − 𝜌𝑙 [𝑉 (𝑡)]22 (𝑅 (𝑡)𝑟 )4

+ 𝜌𝑙𝑅 (𝑡)𝑟 (2 [𝑉 (𝑡)]2 + 𝑅 (𝑡) 𝑑𝑉 (𝑡)𝑑𝑡 ) .(18)

Science and Technology of Nuclear Installations 9

In Figures 7(a) and 7(b), the pressure profiles obtained fortwo initial droplet sizes from the full model (see Sections 3and 4) are compared with the corresponding solutions (17)and (18) at three time instants. The assumption of incom-pressible liquid adopted in the simplifiedmodelmeans that assoon as the interface starts to move, nonzero velocity 𝑢𝑙 andnonconstant pressure 𝑝𝑙 appear instantaneously in the wholespace. The full model predicts finite-velocity propagationof compression wave; however, behind the wave front thepressure distribution practically coincides with that given by(18). Also note that during the short time interval for whichresults are plotted in Figure 7 the pressure profiles (18) arevery close to each other, which means that the characteristictime scales of the processes described by the simplifiedmodel(radial outflow of coolant with no significant compressibilityeffects) are much longer than the characteristic times ofcompression wave propagation.

Thus, the full model (Section 3) must be applied in orderto evaluate the effects of fast compression waves travelling inliquid lead at a speed of the order of speed of sound (about1700m/s). In such waves, sharp increase in flow pressureand velocity occurs; however, the duration of this transientimpact is very short. Model (17), (18) is much simpler inimplementation than the full model, and it can be used tostudy longer-term effects of the liquid lead flow caused bywater droplet expansion. An example of such simulation isgiven in Figures 8(a) and 8(b) where time histories of thewater droplet pressure and radius are shown for two initialradii, 13 and 25mm (all other parameters were the same asthose in Table 1).

6. Conclusions

The studies of single-droplet water-lead interaction show thateffects on the surrounding structures (say, nearby tubes ofsteam generator) of high-pressure water expansion in moltenlead can be of two types, differing in the characteristic timescales. The first type is a rapid (shock) impact caused bypropagation of an elastic compression wave travelling at aspeed close to the speed of sound in liquid coolant. Say, for awater droplet of 13mm initial radius, the characteristic arrivaltime to a point located at 10 cm distance from the droplet isof the order of 0.05ms, and pressure jump can be as high as3.5MPa.

Another type of interaction is the effect of slowly chang-ing radial flow past the structures. This stage of interactionis featured by much longer characteristic times, of the orderof water droplet oscillation time which can be as long asdozens of milliseconds. At this stage, liquid lead behaves asan incompressible fluid, and the force is mainly attributed toviscous drag. Evaluation and quantitative comparison of theeffects of pressurewaves and coolant flows on steamgeneratorpipes will be the subject of future research.

While the 1D spherically symmetric statement of theproblem used in the current work is the geometricallysimplest, the proposed model takes into account the basicphysical 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 dischargefrom broken pipes into molten lead pool.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

Acknowledgments

The research was supported by the Russian Foundation forBasic Research (RFBR) (Projects no. 17-08-00308, no. 17-08-00287, and no. 16-08-00239).

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