3 PHENOMENA INVOLVING ENERGETIC RELEASES FROM MOLTEN METAL COOLANT INTERACTIONS 3.1 Phenomenology of thermal interaction between fluids in liquid metal reactors

33

3

PHENOMENA INVOLVING ENERGETIC RELEASES FROM

MOLTEN METAL COOLANT INTERACTIONS

3.1 Phenomenology of thermal interaction between fluids in

liquid metal reactors

In the frame of the studies concerning liquid metals systems, the analysis of the so called “liquid metal-coolant interaction” is one of the most important concerns. In fact, in case of a steam generator tube rupture accident in HLMRs, there is the likelihood that the liquid metal comes into contact with the coolant of the secondary loop (often water) which removes heat from the primary loop. This kind of interaction is defined as CCI.

In case of SFR there is the likelihood that, due to a loss of coolant accident, the temperature in the core reaches high enough values to melt fuel and steel and this mixture interacts with the surrounding coolant, starting a FCI.

Generally speaking, an amount of high pressured vapor is produced when a hot liquid comes into contact with a colder, more volatile liquid. In fact, the hot liquid transfers its internal energy in a short time to the colder one as a consequence of a fine fragmentation which drastically increases the interfacial area between the two liquids. The temperature and the pressure of the colder fluid increases and it expands affecting the surroundings.

Moreover, the heat transfer between the two fluids may be so rapid that the timescale for heat transfer is shorter than the timescale for pressure relief. This can lead to a vapor explosion with a related formation of shock waves and/or the production of missiles at later time, during the expansion of coolant vapor that may damage the surrounding structures.

It is worth underlining that the evolution of the interaction towards an explosive event (called vapor or steam explosion) are strictly depending on masses and densities of the materials involved. On this basis, one can claim that in case of FCI the interaction evolves frequently into a steam explosion, while in case of CCI the likelihood of occurrence is drastically reduced, as will be explained more extensively later in this chapter (§ 3.6).

However, for studies concerning lead and LBE systems also this rare evolution of the HLM-water interaction must be considered.

Several experimental campaigns have been performed in order to study in depth a phenomenon called “steam explosion”. From the observation of the phenomena involved in this event, the steam explosion can be subdivided into four main phases.

I. premixing II. triggering III. propagation IV. expansion

In the following a short description of each of the above phases will be given. Premixing Phase

Board & Hall [1] described the formation of a premixed or coarsely mixed region as ‘‘... the setting up of a quasi-stable initial configuration.’’ In this phase, as soon as the two liquids meet, high heat transfer from the melt to the water produces a vapor film around the melt droplets. Therefore, three fluid phases, such as molten fuel or heavy liquid metal, liquid coolant and vapor, are present during this period. The

stable vapor film allows large quantities of melt and coolant to intermix owing to density and/or velocity differences, as well as vapor production. In fact, the vapor film reduces the heat transfer between two liquids and delays the quenching of the melt and crust formation on the droplets.

In this stage, the mixture is in quasi-stable or meta-stable state. From a reactor safety point of view, bringing large quantities of already pre-fragmented melt into contact with water and their mixing under limited void provides the largest potential for the most vigorous steam explosions. Therefore, the initial phase of FCI might have significant influence on the subsequent behavior of steam explosions.

Some experimental studies performed in Japan have highlighted the importance of the initial condition, in particular pressure and temperature [2,3]. Tso and Tien [4], through their experiments, have highlighted that it is possible to draw a map of the interaction zone by using the initial temperatures of the interacting hot and cold fluids. They have proposed a multi-mechanism physical model combining several existing mechanisms [5-8].

In particular, the model predicts the existence of an upper limit for the colder liquid above which no interaction is expected, regardless of the hot liquid temperature. The key feature of this model is that interaction will occur when the liquid-to-liquid interface temperature exceeds the superheat limit of the colder liquid at the interface at the instant of contact, if no film boiling occurs. The superheat limit is defined as the maximum temperature limit at which liquid boils explosively. Alternatively, if film boiling of the colder liquid occurs first, the same criterion is applied after the vapor film collapse at the minimum film boiling temperature. Solidification of the hot liquid does not preclude interaction.

Furthermore, also the boundary conditions play an important role in the development of the explosion as it has been noted by Nelson [9], who has underlined the importance of physical and chemical constraints on the occurrence of an explosion for hot and cold fluid systems, and by Taleyarkhan [10], whose experiments have demonstrated the key role of incondensable gas in suppressing explosions.

The timescale of the pre-mixing phase is in the range of few tenths of seconds to several seconds and the length scale is a relatively coarse one, yielding pre-fragmented melt droplets in the range of few millimeters to centimeters (i.e. the melt is progressively fragmented into particles of centimeter-scale size). The complexity of these multiphase, dynamic and multidimensional processes occurring in a very short timescale leads to difficulties in modeling the FCI event, where the premixing phase plays an important role. Several qualitative observations have been taken from experimental and modeling efforts relating to the FCI premixing phase.

First of all it was seen that jet breakup mechanisms, which produces a situation like in Fig. 3.1, are initially driven by relative-velocity–induced hydrodynamic instabilities and directly affect the conditions of the system prior to a vapor explosion establishing. In fact, owing to fluid acceleration the consequent relative velocity between two fluids causes the deformation and the breakup of the particles. Therefore, inertia forces overcome cohesion forces and the process may be connected with the Weber number

2 c rel d dc

v D

We

ρ

σ

=

35 where ρc is the cold fluid density (coolant), Dd is the hot fluid (heavy metal) particle diameter, σdc is the surface tension.

Three of these instabilities are recognized as the main ones. The first one is the Rayleigh-Taylor instability that originates when two fluids are accelerated perpendicularly to the contact plane (see Fig. 3.1). This leads to interface irregularities that grow.

Fig. 3.1 - Rayleigh-Taylor instability and break up

The governing dimensionless parameter is the Bond number

dc d d

aD

Bo

σ

ρ

=

where ρd is the droplet density (coolant), Dd is the hot fluid density (heavy metal) particle diameter, σdc is the surface tension.

The second mechanism is the Kevin-Helmoltz instability that originates when a large relative velocity exist between two fluid streams, thus rippling the surface at the interface. Its governing dimensionless number is again the Weber number.

The third mechanism is called “stripping” (see Fig. 3.2).

Fig. 3.2 - Stripping

In this case, tangential flow components hold a shear force which creates a sort of moving layer. This layer detaches and breaks in case of presence of relative velocity thus creating a sort of droplets fog (also defined as atomization). This phenomenon occurs if 200 < We < 2000.

Several studies have highlighted the dependence of fragmentation from jet entry velocity and therefore from the We number which provides a threshold (generally We>12) for establishing fragmentation. Actually, Dihn [11] stressed the importance of instabilities and concluded that jet breakup processes are dominated by “jet momentum” and “momentum transfer” and film boiling does not contribute significantly to the jet breakup process. Corradini [12] developed mixing models based on dynamic fuel fragmentation that considered a variety of mechanisms for jet breakup, including also boiling effects that may increase local mixing conditions by driving flow instabilities. Burger [13] concluded that the entire vapor generated from the fuel jet entry is accumulated in the film with no detachment and mixing into the coolant pool. Dynamic pressure has limited influence on jet breakup in the coolant.

Concerning the second mechanism, direct liquid-liquid contact allows high heat transfer rates yielding a local rapid pressure rise, thermal film destabilization and coolant entrapment within the melt followed by fine fragmentation of the melt itself. These effects are the cause of the so called thermal fragmentation. In any case, several studies [14-17] have demonstrated that the jet breakup characteristics affect strongly the premixing and triggering phases, which will be described shortly in the following. These two phases are strictly connected through an intermediate phase of fragmentation that therefore can be included in mixing or in triggering phase depending on the “philosophy” chosen.

I. Triggering Phase

Triggering is the phase which leads to local fine fragmentation and then enhanced heat transfer and pressurization. Therefore, it is strictly connected to the previous phase of premixing. Triggering may produce destabilisation of the vapor film, allowing liquid-liquid contacts through the collapse of the film itself. Therefore, this may lead to fragmentation and explosion or, in case the film remains stable for a longer time, to delay the interaction. Several studies have demonstrated that several variables are involved in this phase. In particular, the hot and cold fluid properties, the stability of the system, the amount of liquid metal mass which has been solidified and the liquid particle dimensions and concentration determine the triggering phase as was seen in the KROTOS campaign [18,19].

In addition experimental studies focused on impingement, carried out at CRIEPI [2,3,20], highlighted that the pressure of the environment around the mixture of the fluids plays an important role, because it can suppress vapor explosion when the vapor film density increases. Moreover, the coolant temperature and the respective subcooling can increase or decrease the likelihood of vapor explosion, because steam explosion becomes more likely with a higher subcooling.

II. Propagation Phase

During the propagation phase, there is an escalation process resulting from the coupling between pressure wave propagation, fine fragmentation and heat transfer after the trigger event. Pressurization induced by the trigger destabilizes the surrounding vapor film, leading to a further fine fragmentation of the surrounding melt. The debris produced by such fine fragmentation processes is in a size range of 100 µm. Due to the larger melt surface area, the heat transfer rates strongly increase.

III. Expansion Phase

The expansion of the coolant vapor converts the thermal energy into mechanical work. If there are no chemical reactions involved, this conversion ratio of thermal energy to mechanical work can theoretically be as high as 25 to 50%, although past experiments have only recorded values of a few percent for the injected fuel. A detonation model, which will be described in details later in this chapter, can be applied for evaluating the expansion of the resulting high-pressure mixture.

In this case, the propagation front against the inertial constraints, which are imposed by the surroundings, determines the damage potential of a vapor explosion. In fact, if the local high pressures (the propagating shock front) are quickly relieved, they may not damage the surrounding structures, but the kinetic energy transmitted to the materials around the interaction zone may represent the damaging agent.

37 Different models have been developed in order to evaluate this phase and they may be grouped in four categories, such as thermodynamic, parametric, mechanistic and explosion-expansion models. They will be explained more extensively later.

3.2 Main contact modes

The theoretical subdivision of the steam explosion into four-phases originates from three main contact modes, as can be seen in Fig. 3.3, such as: a) dispersion of hot liquid fluid into cold liquid pool or b) dispersion of cold liquid droplets into a hot liquid pool or c) stratification between the two fluids.

a) Hot liquid droplets inside cold liquid

b) Cold liquid droplets inside hot liquid

c) Stratification of the two fluids

Fig. 3.3 -Contact modes between hot and cold fluid (liquid metal-coolant or fuel-coolant)

The likelihood of having a particular contact mode rather than another one depends on the application, such as reactor type or experiments. For instance, several experiments have been carried out pouring a hot fluid, having higher density, into a cold one in the form of jets (see Fig. 3.3.a). This is because this kind of contact mode represents the most common situation that might occur in LWR or LMR following a Core Disruptive Accident (CDA), such as when the core melts and comes into contact with the coolant.

The injection of cold fluid into a hot fluid pool (see Fig. 3.3.b) is less probable to yield a large mixing considering thermal and density effects. In fact, the melt is denser than the coolant. Nevertheless, such a contact mode is representative for the so called “Coolant-Coolant Interactions” (CCI) which might occur in HLMRs in case of the rupture of one or more tube of the steam generator.

The stratification of the two fluids (see Fig. 3.3.c) might also occur in reactors if the hot dense phase reaches the bottom of the vessel and spreads out without freezing. The vessel now contains two stratified liquid layers, separated by a vapor film. Under these circumstances a trigger wave at the interface can produce a mixing wave that progresses over the surface, causing a kind of planar explosion.

If the first explosion is not too disruptive, and the hot system is not frozen, further explosions can occur. Such behavior has been experimentally observed.

Some experimental studies, as mentioned previously, have highlighted the importance of the initial conditions, in particular pressure and temperature [2, 3,20], and the importance of the presence of incondensable gas in the development of the explosion [10]. For the sake of completeness it must be mentioned the case in which the interaction does not lead to an explosion. The most suitable contact

mode for this case is given by hot and cold fluid dispersed in a vapor continuum, as shown in Fig. 3.4.

Cold and hot fluids dispersed in a vapor continuum

Fig. 3.4 - Contact mode for non-explosive interactions

In similarity to what was observed in the explosive interactions, several phases can be recognized even for the non-explosive interactions, such as: mixing, fragmentation and cooling or quenching. No shock waves, which are typical of steam explosions, are present in this kind of interaction. Therefore, the fragmentation of the hot fluid is not in connection with their propagation. In addition, the rapid boiling phenomena do not propagate spatially in a time scale comparable to the propagation wave time.

Nevertheless, even in the case of the absence of explosions, the amount of vapor produced is enough to lead to increase pressure, thus representing a threat for structural integrity. In any case, only the explosive interaction will be considered in this work.

3.3 Theoretical Models

The previous overview does not consider the behavior of the interaction process in its complexity, but was mainly focused on analyzing its particular aspects. In order to explain the whole steam explosion behavior, two different theories have been developed. The first one is based on the spontaneous nucleation theory, therefore examining those initial conditions which lead to an explosion. The second one is focused on explosion propagation, caused by hydrodynamic fragmentation behind pressure wave, on the basis of the possible analogy between a chemical detonation and a vapor explosion. In the following these two theories are shortly summarized.

3.3.1 Spontaneous nucleation theory

The spontaneous nucleation theory starts from the fact that a vapor cavity with critical dimension is formed due to density fluctuations in a liquid or at the liquid – gas (or vapor) interface.

It is firstly necessary introducing the definition of interface temperature between two semi-infinite masses of fluids, which may be evaluated as

C H C C H H I

k

k

k

T

k

T

T













+

























+













=

κ

κ

κ

κ

39 where k is the thermal conductivity, κ the thermal diffusivity and the suffix H and C are for hot and cold fluid, respectively.

According to the kinetic gas theory, some vapor bubbles may be formed because of molecular fluctuations. Vapor may originate nucleation at the moment in which is possible to reach the formation of a vapor embryo and therefore a vapor nucleus with a radius equal or larger than the so called critical radius given by

C g crit

P

P

r

−

=

2

σ

where Pg is the vapor pressure of the nucleus, PC is the liquid pressure corresponding to saturation temperature and σ is the surface tension. Therefore, spontaneous nucleation temperature can be defined as the temperature at which it is possible to reach this vapor embryo (and vapor nucleus) formation at the interface of two liquids.

A bubble with a given internal pressure equal to Pgis generally unstable and consequently the liquid around is in a metastable state.

If the radius of the bubble is smaller than the critical radius rcrit the bubble collapses, otherwise it grows. The work needed for starting is therefore equal to the energy required to form the spherical vapor bubble’s nucleus in the liquid environment, that is

)

(

3

4

4

r

r

P

P

W

=

π

σ

−

π

−

where the first term represents the energy loss due to surface tension σ and the second term represents the energy gained in creating a new volume.

The energy needed for achieving rcrit is

(

)

3

16

P

P

W

−

=

πσ

Applying kinetic gas theory, it is possible to evaluate the nucleation rate per unit volume and time as

        −

=

wNe

J

where N is a constant approximately equal to the number of molecules per unit volume, w is the collision frequency of the liquid’s molecules (~1010s-1) and

W

/K

T

required for nucleation and the molecule’s kinetic energy due to the thermal motion.

Beyond a certain value of the nucleation rate depending on temperature, the number of bubbles formed is so high to destabilize the metastable liquid state. As the interface temperature TI approaches the spontaneous nucleation temperature TSN, the bubble nucleation rate increases exponentially, as shown in

Fig. 3.5.a, to the point at which vapor production occurs at explosive rates. In

It must be pointed out that this model requires that a period of film boiling should be established prior to explosion, allowing the insulation of the hot liquid from the colder one and the gradual fragmentation and mixing of the two liquids by the instabilities arising from film boiling.

Fig. 3.5 - Spontaneous nucleation model

When the nucleation process takes place at the interface between two liquids, the work needed for the bubble formation decreases together with diminishing surface wettability. In particular, the contact wetting angle influences the spontaneous nucleation temperature.

In fact, as can be seen also in Fig. 3.6, in case of completely wetted surface, the spontaneous nucleation rate corresponds to the homogeneous nucleation rate and consequently spontaneous and homogeneous nucleation rates are equal. Just to remind the reader, the homogeneous nucleation takes place if nucleation process has no preferential nucleation sites, which means that the creation of vapor bubble nuclei occurs in a superheated liquid without any impurities and boundaries. The temperature at which the homogeneous nucleation starts is the homogeneous nucleation temperature.

In case of not wetted surface, the spontaneous nucleation temperature is equal to saturation temperature.

Finally, in case of partially wetted surface spontaneous nucleation temperature ranges from saturation temperature to homogeneous temperature.

a)

41

Fig. 3.6 - Nucleation temperature between two liquids

Fluids like water and LBE have weak wettability and consequently the spontaneous nucleation temperature is very close to the homogeneous nucleation temperature that for water is about 583 K.

The validity of the theory for LBE-water systems criterion was confirmed by the observations collected by Kurata [20] which are summarized in Fig. 3.7 and in Fig.

3.8 .

Fig. 3.7 - Vapor explosion curves for interaction between different molten metals and water

As can be seen in Fig. 3.7, a steam explosion is possible over an interface temperature range from about 300 °C to the water cr itical temperature.

Fig. 3.8 - Vapor explosion map for the LBE-water system

More detailed results can be deduced from Fig. 3.8. As can be seen, for a LBE-water system, vapor explosion can occur for melting temperatures ranging from about 400°C and 500°C at pressures under 0.1 M Pa. Furthermore explosions seem to be more likely if the contact temperature is higher than the spontaneous bubble-nucleation temperature. In addition, vapor explosions seem to be affected by water subcooling, because decreasing the subcooling decreases the explosion intensity and from the pressure of the enviroment. In fact, no explosions occur for pressures above 0.2 MPa.

In case of no explosion occurrence, the expansion of released water in LBE happens at a slower rate and simpler thermodynamic models can describe the behavior.

Summarizing, in the spontaneous nucleation theory a sequence of events and criteria must be satisfied in order to have a steam explosion.

First of all, there must be a stable film boiling aimed at separating the two fluids and allowing their mixing without excessive heat transfer.

Then, due to the vapor film collapse, a direct contact between fluids must occur. Eventually, there must be “inertial constraints” which sustain the pressure wave during the time needed for developing an explosion on large scale.

The achievement of all these conditions to get an explosion is difficult. This constitutes a limitation of this theory, which has been criticized in the light of some experimental results. In fact, it was observed that in low temperature experiments

43 explosions are possible, even though the interface temperature does not exceed spontaneous nucleation temperature.

A possible explanation for this discrepancy (not fully accepted) might be in the dynamic change of surface wettability characteristics which affect the spontaneous nucleation temperature.

Moreover, some spontaneous and self-triggered steam explosions have been observed in experiments concerning contacts between water and corium. During these events, the critical temperature, which is the upper limit for self-triggered explosions, is widely exceeded [21-23]. For this reason, it is not possible to apply this temperature limit to nuclear reactor and conventional industry applications because the contact temperatures of molten material and water would exceed the critic temperature of the water.

Furthermore, some experimental studies have highlighted the spontaneous triggering inhibition caused by high pressure environment, which represent a parameter not considered in this theory [24-28]

In addition, neither the lack of information concerning the two fluids masses involved in the process nor the lack of solidification effects are taken into account by the nucleation theory, despite of the fact that they represents important parameters in evaluating steam explosion efficiency.

Hence, spontaneous nucleation theory, which is fundamentally a microscopic model based on physical phenomena occurring nearby the interface between the two fluids, can only partially explain explosion mechanisms. Thus, the use of this theory, which explains every steam explosion as a product of a spontaneous nucleation following an extreme heating, is nowadays limited.

3.3.2 Thermal detonation theory

Thermal detonation theory starts from Board’s team observations of the explosions obtained in the THERMIR facility [23]. They proposed an analogy between a chemical detonation and a vapor explosion, which was then called “thermal detonation”.

In a chemical detonation [29], a shock wave goes through the inert reactants, compresses them and increases their temperature. This causes sudden chemical reactions to occur in a short region behind the shock front, which is the reaction zone.

For a 1D steady-state detonation, the classical jump conditions for shock waves are applied. For example, in the shock-front frame of reference, it is possible to write

G

V

V

=

=

ρ

ρ

V

P

V

P

+

ρ

=

+

ρ

(

)(

)

2

1

v

v

P

P

U

U

−

=

+

−

where indexes 1 and 2 represents, respectively, the material before the shock front that has not still reacted and the material that has reacted completely. Moreover, ρ, P, V are density, pressure and velocity of the mixture while U (p, v) is the internal energy of the mixture.

In the case of 1D steady-state detonation, Eq.(3.10) gives all the possible states of detonation and is called “Adiabatic curve of detonation” or “Crussard curve” (Fig. 3.9.b, thick curve).

It must be pointed out that in case the jump conditions would be applied just through the shock front, and not in the zones behind and in front of the shock front as mentioned before, the index 2 corresponds to the adiabatic compression of the reactants without any taking into account their reactions. Thus, in this case Eq.(3.10) would give all the possible states after a shock in an inert medium. These states are located on the shock adiabatic or Hugoniot curve (Fig. 3.9.b, dashed curve).

Fig. 3.9 - (a) Geometry and schematic pressure and velocity profiles of a 1-D explosion (b). Schematic shock adiabatic (solid curve, reacted material; dashed curve, unreacted material) [29].

From Equations (3.8) and (3.9), it is possible to find

(

)

(

)













−

−

=

v

v

p

p

G

This equation determines the shock front velocity that is given by the slope of the line of the possible states of detonation (Rayleigh line), passing through the pole X (p1, v1).

The only possible stable steady state is obtained when the Rayleigh line is tangent to the adiabatic curve of detonation [29]. This determines the Chapman-Jouguet (CJ) point (Fig. 3.9.b, point O).

Considering the shock-front frame of reference, the velocity V2 of the

material is equal to the sound velocity in the CJ point. So, rarefaction waves from the expansion region after the CJ point cannot get into the reaction zone.

A steady-state chemical detonation can be described as in the following. The pressure of a shock wave going through the inert material increases up to the so called ‘Von Neumann spike’ (point N in Fig. 3.9.b). Then, the chemical reactions

45 start, the fluid behind the shock front begins to expand and the pressure falls more or less rapidly, depending on the reaction rate, until the CJ point.

In the shock front frame of reference the relative velocity between V1 and V2

increases during the expansion. Thus, the relative velocity is smaller than the sound velocity between the points N and O (see Fig. 3.9.b) ,in O is equal to the sound velocity of the material coming out from the front and finally between O and A increases up to the shock front velocity.

The liquid metal–coolant analogy is based on the hypothesis that fragmentation and mixing are due to shock wave propagation. Board [30] described the metastable premixture as the non-reacted material in which a shock wave propagates, increasing pressure up to the Von Neumann point for a chemical detonation. This pressure wave makes vapor film collapse and, owing to the density difference, induces large relative velocities between the fuel drops and the surrounding liquid. These velocity differences allow fine fragmentation and rapid heat transfer. This occurs in the fragmentation region which corresponds to the reaction zone for an explosive. Then, if the energy release is large enough to sustain the shock propagation, a steady state may be reached.

So, Board applied Eq. (3.10) to build adiabatic detonation curves for a UO2-Na system assuming complete reaction. This means that, the melt and the

coolant are in thermal and mechanical equilibrium at the end of the reaction zone. Making this, they obtained very high pressures as can be seen in Fig. 3.10.b. The concept was then revised considering that only a partial amount of melt may be fragmented coming to the equilibrium with coolant [31, 32], and only a partial amount of coolant may be involved in the rapid heat transfer process [33].

To test this concept, firstly Scott and Berthoud [31,32] then Sharon and Bankoff [34], provided the first multiphase-flow modeling of a steady-state thermal detonation. In their model, they described the fragmentation induced by the shock wave assuming that fragments come instantaneously into thermal and mechanical equilibrium with coolant, because of fragment size and turbulence in the interaction zone and then they looked for the existence of a CJ plane.

An example of such a mechanistic calculation is given in Fig. 3.9.a, which shows the behavior of the two components (large melt drops and debris coolant mixture) in the reaction zone after the shock wave in a self-sustained thermal detonation. It must be pointed out that there is only one shock wave that can lead to steady-state propagation for a given premixture (see Fig. 3.9.b, X point) and for a given fragmentation model. To check the validity of the two-component model, Sharon and Bankoff also performed thermodynamic calculations of the CJ states from the adiabatic detonation curves built through the classical detonation relations (Eq.(3.8),(3.9) and (3.10)), using the percentage of fragmented melt included in the interaction as a parameter.

The CJ states are obtained through the classical tangency condition and are represented by the thick line in Fig. 3.10.b. In Fig. 3.10.b, CJ point calculated thermodynamically in connection with the conditions given in Fig. 3.10.a corresponds to a degree of fragmentation between 0.35 and 0.40, which is in agreement with the value obtained by the model (see Fig. 3.9.a).

Fig. 3.10 - (a) Evolution of components velocities and temperature, of pressure and degree of fragmentation for a UO2-Na mixture behind a 1.4-kbar

shock wave (Q)= mass ratio between melt and coolant, XV =initial void fraction, TS = breakup time constant, CD = drag coefficient, D = shock wave propagation velocity; (b) possible states of detonation in a PV diagram for the above premixture based on the fragmentation degree FR

Several objections have been made also to this model [35].

First of all, it does not consider the different mass component’s quantities involved in the mixture, which gives different velocities. Secondly, the hydrodynamic fragmentation, according to the interface layer’s stripping, cannot be so fast to sustain thermal detonation because of the relative velocity’s reduction. Therefore, thermal detonation cannot estimate with a good accuracy without quantitative information on hydrodynamic fragmentation. Furthermore, other thermal phenomena such as spontaneous nucleation must be taken into account. Additionally, some doubts arise on the influence of geometrical features in developing a detonation wave. Finally, there are some difficulties concerning the extension of the 1-D chemical detonation to a multiphase thermal detonation. In fact, thermal detonation theory is based on the thin wave approximation while actually shock waves are not thin.

Despite of these criticisms, several experiments have highlighted that vapor explosions produce a shock wave similar to that coming from chemical detonation. It must be pointed out that this is a macroscopic model and it is not suitable to explain the microscopic mechanisms like fragmentation, triggering and heat transfer. Nevertheless, thermal detonation theory represents a sound

47 contribution to steam explosion modeling, because it gives an overall (and macroscopic) picture of the phenomenon. Since some aspects like, for instance, fragmentation are explained in a more comprehensive manner with the spontaneous nucleation theory, the two models can be considered as complementary theories.

3.4 Vapor Explosion Modeling

As mentioned above, the interaction between two fluids, one at higher temperature and with higher volatility with respect to the other, produces high pressure vapor which may expand making work on the surroundings. Considering a LMFR, vapor explosion can occur following a CDA while in HLMR it may happen because of a SGTR.

Despite the fact that the likelihood of these events is very low (CDA is considered as a part of the category of beyond-design basis events), it is clear that this topic is one of the most important issues for LMFR safety. In fact, the work done by the expanding vapor may damage structures and generate missiles thus representing a threat for vessel and internals integrity. The evaluation of the expansion phase is aimed at assessing the energy potential and giving information concerning possible damages, debris size and gas release rate.

Several models have been developed for assessing the consequent energy releases. They may be grouped in different categories, namely TNT equivalence model, superheat limit model, thermo-dynamical models, parametrical models and mechanistic models.

A brief survey of these models is given in the following.

3.4.1 TNT equivalence model

One of the most widely used methods for the prediction of blast effects from explosion sources has been the TNT equivalence method. A principal explosion parameter, such as overpressure peak, can be related to the scaled distance for an explosion of TNT using a cube root relation [36].

Therefore the energy released from explosion sources must be compared with the one of an exploding mass of TNT [37], which means the TNT energy release must be evaluated. The following values are generally used for this evaluation.

• 4.187 MJ·kg-1 (Baker and Tang [38]);

• 4.6 MJ·kg-1 (IChemE [37]);

• 4.689 MJ·kg-1 (Lees [36] quoting Kinney [39]);

• 4.187 MJ·kg-1 (Genco and Lemmon [40]);

• 3.643 MJ·kg-1 (Cook [41]);

• 4.517 MJ·kg-1 (Strehlow and Baker [42]).

These energy values are typically applied to the estimation of the energy available in explosion processes in pipelines and process plant. It has been noted by Nelson [43] that physical explosions have a sharper pulse than chemical deflagrations because of the “harder” (with respect to TNT) molten metal capability of contraction during the cooling.

Lipsett [44], as well as Anderson and Armstrong [45], has analysed explosions from molten materials and water starting from an accident occurred in a Quebec foundry where 30 lb (~ 13.6 kg) of water took part in the explosion with

steel. He has evaluated that the total thermodynamic work done by the air shock wave (detonation energy) from 1 lb of TNT is 8.1·104 calories. By converting directly into SI units, a value of 748 kJ·kg-1 of TNT is obtained, to which an amount of 11.3 lb (~ 5.13 kg) of TNT corresponds. From these results one can calculate the “equivalent detonation energy” of the 30 lb of expanding water at 620°F (~ 326 °C) is 91.5 ·104 calories. A ratio of 0.38 kg TNT per kgof H2O is obtained applying

directly Lipsett’s finding.

It must be pointed out that the value of 748 kJ·kg-1 found by Lipsett as well as by Anderson and Armstrong is quite different from the value, of about 4.6 MJ·kg

-1

of TNT, which is usually applied.. Therefore, the ratio of 0.38 kg TNT per kgof H2O together with the values of the detonation energy of TNT must be used with

care because the results have been based on different energy content.

For example, Lipsett has estimated that the energy available is 0.282 MJ·kg-1H2O, while using the actual energy content of TNT this value increases up

to 1.73 MJ·kg-1H2O. This is due to the application of Cook’s method [41], which can

be summarized as in the following. Cook has noted that only a fraction of the total explosive energy Q should radiate from the explosion in the air pressure wave. Hence, from the data supplied by him, the total explosive energy is 3643 kJ·kg-1 TNT and the pressure wave energy is 718 kJ·kg-1. These data suggest an efficiency factor of about 20% for TNT.

Other studies performed again by Lipsett and Anderson and Armstrong on the basis of Stoner and Bleakney’s experiments [46] has highlighted the possibility of evaluating the air shock wave pressure corresponding to a given weight of TNT from the damages observed in the experimental facilities at a certain distance. The value of 748 kJ·kg-11of TNT have been used for this evaluation. However, even using the detonation energy value of 4.6 MJ·kg-1of TNT, which is approximately 6 times larger than the Lipsett’s value, the effect on a scaled distance has been reduced only of a factor of less than 2. Despite this, further investigations of the appropriate TNT detonation energy to apply in the TNT equivalence method are needed to reduce uncertainties.

3.4.2 Superheat limit explosion

Another model for molten metal-water explosions may be obtained considering the superheat limit temperature for evaluating the explosion energy.

This model is based on three main points. First of all, a rapid phase transition (RPT) arises when a hot liquid is able to heat up a cooler liquid up to its superheat limit temperature (above its boiling point). Secondly, a RPT can generate blast waves similar to those produced by high explosives except that the maximum energy yield is limited by the superheat energy of the cold fluid involved. Only a small fraction of the available superheat energy is transferred into the shock wave; so, in energy terms, RPTs are less efficient than high explosive in generating blasts.Finally, the TNT equivalent mass for a RPT is proportional to the amount of energy in the superheated fluid at the time of the explosion.

This scheme agrees with observations and comments of Katz and Sliepcevich [5] and Nelson [9] regarding LNG explosions and with Lees’ viewpoint [36] in which the energy available for such an explosion is that corresponding to the superheat degree between the normal boiling point and the superheat limit. From these observations it emerges that the initial condition of the expansion is given by the superheat limit temperature. Katz [5] suggests that the pressure in the superheat limit layer converts from atmospheric pressure at the limit of superheat

49 to vapor pressure at the limit of superheat. This internal pressure generation causes the distinct explosive effect defined as “superheat-limit explosion”.

Aimed at having an approximate maximum for the available energy, at the beginning the quantitative examination of the energy released in a steam explosion was based on saturated steam tables, referring to a temperature of 500 K. According to the assumption of saturated vapor at 500 K, the available energy has been evaluated as 0.117 MJ·kg-1H2O.

The next use of the quadratic Brinkley’s equation [47], which takes into account ionic equilibrium in mechanical compression, has given a value approximately 3 times higher but without taking into accounts the non ideal character of steam.

Wakeshima and Takata [48] pioneered the field of superheat’s limit on the basis of Kenrick’s work [49], mainly focusing on saturated hydrocarbons, establishing the superheat limit temperatures as percentage of critical temperature, as shown in Table 3.1.

Table 3.1. Superheat limit temperatures as percentage of critical temperature

Katz and Sliepcevich [5] and Blander [50] focused their attention on water thus evaluating its superheat limit temperature as 84% and 89% of critical temperature, respectively. The last value is commonly used. The enthalpy available for the expansion from 576 K to the atmospheric boiling point is 0.069 MJ·kg-1 and the fluid expansion calculated for these conditions is 0.056 MJ·kg-1. This approach calculates the energy released during the expansion of the vapor from the superheat limit rather than the energy absorbed by the liquid as it rises to the limit of superheat.

These estimated values were considerably lower than estimated by other methods available in the literature which suggests that the rapid phase transition model as described in Lees [36] and IChemE [37] requires further work.

3.4.3 Thermodynamic models

All the thermodynamic models have been based on the adiabatic expansion’s theory formulated by Hicks e Menzies [51].

This model was aimed at estimating the conservative upper limit of the vapor explosion work potential for postulated fast reactor meltdown accidents. This work potential is taken equal to the change in fuel internal energy during an isentropic expansion from a compressed state to an expanded state. The

Superheat Limit [K] Temperature [K] %

n-Hexane 383 507 76 n-Heptane 386 539 72 lsopentane 383 461 83 Cyclohexane 408 554 74 Ethyl ether 385 468 82 n-Pentane 383 470 81 Cyclopentane 404 512 79 Methanol 453 513 88 Ethanol 473 516 92 Acetone 447 508 88 Benzene 477 563 85

expansion work might be represented as a function of sodium-to-fuel mass ratio and of the final pressure.

Since thermodynamic calculations do not take into account the heat transfer specific path or fragmentation rate, peak pressure and pressure history caused by time delay between various rates processes cannot be obtained. Anyway, a transient analysis is essential in evaluating in detail safety problems, because some explosion damages depend on the severity of pressure pulse, generally.

It is possible to consider a thermodynamic explosion model with two different final conditions. The first one is the expansion of the coolant, which is initially at high pressure, to a specific final ambient pressure. The second one is coolant expansion to a specific volume the expansion.

In the following these two models are presented because they represent a basis for a comparison with other models.

3.4.3.1 Specific final ambient pressure expansion

It is possible to consider the vapor explosion as an idealized process made of two stages. At first, a constant volume fuel – coolant thermal equilibration and then, an isentropic products expansion. Of course, several assumptions must be made.

First of all, all the heat in the process is transferred from fuel to coolant; therefore, it is possible to assume system’s thermal isolation. Secondly, liquids are considered incompressible and the specific volume of liquid is negligible with respect to vapor specific volume. Then, specific and latent heat are assumed constant and the vapor behaves as a perfect gas. Concerning the first of the two stage process, it is possible to suppose that the coolant mass mc at temperature Tc mixes with fuel mass mf at temperature Tf and thermal equilibrium is established

between the two fluids. The equilibrium temperature will be

pf f pv v pc c f pf f cv pv v c pc c e

c

m

c

m

c

m

T

c

m

T

c

m

T

c

m

T

+

+

+

+

=

where cpc, cpv and cpf are the specific heat of coolant, vapor coolant and fuel, respectively.

From the thermodynamic equation of state it is possible to obtain the entropy change to this equilibrium state during the constant volume process and then it is possible to obtain quality and pressure of the equilibrium state.

Then it is possible to consider two kinds of expansion in the second stage. The first one is the expansion of the overall mixture while the second one is the expansion of coolant only. It must be pointed out that the first evaluation is more conservative than the second one because thermal equilibrium between fuel and coolant is maintained during the expansion for the mixture case.

51

Fig. 3.11 - Possible expansion’s paths

where

Path 1: equilibrium state ‘e’ in saturation region final state 2 in saturation region Path 2: equilibrium state ‘e’ in saturation region

final state 2 in superheated region Path 3: equilibrium state ‘e’ in superheated region

final state 2 in saturation region Path 4: equilibrium state ‘e’ in superheated region

final state 2 in superheated region

Considering first the process in which the expansion takes place in the saturation region, it is possible to get the final state quality x2 given the final ambient pressure P2 and its saturation temperature T2 by differentiating and integrating the thermodynamic mixture state and combining everything with Clausius-Clapeyron equation. Therefore:

0

ln

)

(

=













−













+













+

T

x

h

m

T

x

h

m

T

T

c

m

c

m

where T2 and x2are temperature and quality of final state, respectively, xequality of the equilibrium state and hfgccoolant latent heat of vaporization

Then, the final temperature of the portion of the expansion process lying in the superheated state can be obtained by the equation of the saturated vapor (x=1) and known superheated state or by the equation of the superheated state and again the known superheated state. Thus,

n

P

P

T

T













=

R

m

c

m

c

m

n

=

+

By combining the above equations it is possible to express all the possible process’s paths.

3.4.3.2 Specific final volume expansion

The vapor explosion can take place within a fixed volume, also. Even in this case it is possible to consider two ways in which the explosion can affect the surroundings. In fact, the expansion work can be calculated in correspondence to the final volume (and the calculation procedure will be the same of the one explained in the previous paragraph with the only difference being a known final volume instead of a known pressure) or through the calculation of the pressure in the fixed volume after the expansion which will be explained briefly in the following. For this case the same assumption considered previously are still valid. The system has no heat transfer with the environment and in this case no work is done on the environment.

When the final state is in the saturation region, the first law gives

[

]

)

(

)

(

c

T

T

m

c

T

T

x

U

x

U

m

−

=

−

+

−

where Ufg1and Ufg2 are the coolant internal energy of the initial and final state. If the final state lies in the superheated region the Eq.(3.16) becomes

[

]

)

(

)

(

c

T

T

m

c

T

c

T

x

U

x

U

m

−

=

−

+

−

Temperature and pressure of the final state are obtained from the above equations by trial and error iterative method using coolant equation of state.

3.4.3.3 Hall model

While Hicks and Menzies type models provide an estimation of the maximum thermodynamic work, Hall [55] proposed a model that gives a lower limit to the estimation of the work.

In this model, compression of the fluid around the interaction zone in a fixed volume occurs reversibly and adiabatically and the expansion of the two fluids’ mixture is irreversible. The process can be shared in two stages, as well as in the Hicks and Menzies model. The first stage is the fuel-coolant mixing to the end state pressure and the second one is the constant pressure expansion up to the pressure is uniform over the whole system. The end state temperature of fuel and coolant is assumed to be uniform. The surrounding fluid, such as a cover gas, is compressed in a reversible and adiabatic manner. The minimum work required for isentropic compression of the surrounding fluid, assuming negligible the liquids’ compression work and considering gas as an ideal gas is

53





















−













−

=

1

1

γ

p

p

V

p

W

where p1and p3 are initial and end state pressure of cover gas, respectively and

Vsurr,1 is the cover gas volume.

This work is claimed to be a FCI lower limit and a comparison with Hicks and Menzies model is shown in Fig. 3.12.

Fig. 3.12 - Thermodynamic path of a vapor explosion on P-v diagram for Hicks and Menzies and Hall model

Due to the short time scale of the interaction it is possible to assume the adiabaticity of the system therefore the energy balance is

0

=

∆

+

U

W

)

(

)

(

m

u

m

u

m

u

m

u

U

=

+

−

+

∆

Since the internal energy of a thermodynamic system is a state function its variation between two different states does not depend on the kind of path made. Therefore the choice of the transformation performed is arbitrary and it is allowed to choose the simplest one that in this case is an isobaric expansion of the mixture. Thus

)

(

P

V

V

W

=

−

This is the work done during the irreversible expansion of the mixture at constant pressure.

However it must be pointed out that this work is the minimum work that might be done by mixture if the pressure of the mixture is in equilibrium with the surrounding pressure (cover gas pressure) during the whole expansion process. In fact, in this case Wsurr=Wm, with Wsurr and Wm given by Eq.(3.18) and (3.21), respectively. This model represents a simple quasi-static process that can connect initial and final system’s states. It can be applied just to a fixed control volume system because it considers the minimum work in terms of surrounding fluids.

3.4.4 Parametric models

The main objection that has been made to the thermodynamic models is that they do not take into account any kinetic rate process that may be involved, considering the overall mass and energy balance of the interaction, only. Therefore, this approach gives a conservative upper bound of the work potential from an explosion. For a more realistic assessment of damage potential the explosion analysis must take into account the kinetic energy of the process involved in the interaction.

Thus, parametric models have been developed in order to evaluate pressure and the work done on the surrounding considering the uncertainties deriving from contact’s modes, fragmentation and heat transfer that are given as empirical input data. The importance of these parameters has been assessed through sensitivity studies that have quantified them in order to make them available as explosion data.

Basically, these models schematize the expansion in two stages. The first stage is dominated by heat conduction in liquid. In this part, a peak pressure is caused by the sudden heat transfer and by the limited coolant ability to expand. The second stage is characterized by rapid vapor generation and pressure release due to the expansion depending on constraints of the system. The most significant parametric models are Cho-Wright and Caldarola’s models [52-54], which will be explained briefly in the following, because they represent a sort of reference of a huge model’s series in which fragmentation rate and characteristic fragment’s dimension are input data.

3.4.4.1 Cho-Wright model

The Cho-Wright model [52] was developed for FCI interaction deriving from postulated accidents in LMFBR. In this model, the coolant transient evolution of the in the interaction zone is described. Coolant receives heat from the fuel and expands against acoustic and inertial constraints (see Fig. 3.13).

55

Fig. 3.13 - System used by Cho and Wright to develop their model

Three kinds of one dimensional acoustic constraint have considered: first, an acoustic constraint of infinite extent; second, a finite inertial constraint in a single reactor subassembly over the all time and third an acoustic constraint and a finite inertial constraint for expansion at longer times.

Furthermore, coolant and fuel are considered as lumped parameter masses, each one at same pressure but different temperatures. Thus, instantaneous thermal equilibrium is not considered. In any case a heat transfer time constant has been used and mixing of the fuel within the coolant is described through progressive fragmentation using another time constant tm. The final heat transfer rate from fuel to coolant becomes

(

)

T

T

t

t

nt

t

t

M

C

dt

dQ

−

























₋

−

















+

=

3

1

1

exp

R

t

α

3

=

and represents the heat transfer time constant, which is controlled by conduction within the fuel), and Rf is of fuel particles radius after fine fragmentation and Cf and αf are fuel’s thermal capacity and diffusivity, respectively.

By using this law, the progressive evolution of the interfacial area between fuel and coolant is described as

























₋

−

=

t

t

A

t

A

(

)

1

exp

f f

M

A

ρ

3

=

that represents the final interfacial area.

It must be pointed out that this is not representative of any particular fragmentation mechanism, but it is an easy way to describe the progressive fragmentation and mixing between fuel and coolant. With such a model, it is then possible to calculate the pressure time history and the mechanical-energy delivery law.

3.4.4.2 Caldarola model

An extensive parametric study has been carried out by Caldarola for LMBFR [54]. One difference from Cho’s model is that the coolant, in particular for these studies sodium, placed in the interaction zone is distinguished in three zones, such as liquid, vapor film and mixed vapor which are all at the same pressure. The ratio between mixed vapor in liquid and mixed vapor and vapor film around fuel particles is obtained through the force balance in the moment in which the vapor flows away along the space between the fuel surface and the liquid coolant fuel surface. The results have shown that the small particles have a vapor layer thickness larger than the large particles. It is worth noting that there is a competing effect between the amount of vapor generation and due to the energy transfer and the insulating effect of the vapor film.

In this model the assumption of thermal equilibrium between sodium vapor and its liquid has been made. As a consequence, the mixed vapor is the saturated vapor and it has the same temperature of the liquid. The fuel and coolant masses taking part in the interaction have been considered as linearly dependent on time, even though in the early development of the model no dependence from time has been considered. The following main conclusions have been made from based on this model.

First of all, the work done by the interaction decreases with the increasing fragmentation and mixing time constant. It must be noted that for values larger than 5 ms the reduction is not so important. However, this result is similar to Cho-Wright’s findings. Secondly the vapor blanketing effects during the second stage (vapor generation and pressure release) is effective only if it is accompanied with a relatively slow fragmentation and mixing. Moreover, the effect of fuel fragmentation size distribution and gas constant are important for a very rapid fragmentation and mixing. Finally, by using a higher vapor film time constant it is possible to decrease the work. In fact, this time constant determines, as previously mentioned, the ratio between vapor film and mixed vapor.

The vapor film constant is defined by

v f f b

k

R

c

t

3

ρ

=

ρf, R, Cf are, respectively, density, average radius and thermal capacity of fuel, while kv is the fuel conductivity.

Finally, the increase of work depends on the initial height of sodium piston acting as an inertial constrain.

57 In this model the criterion for establishing the changing between first and second stage is given by the overtaking of the coolant bulk temperature with respect to saturation temperature, but the acoustic unloading time used in Cho-Wright’s model seems to be more appropriate. In addition the vapor in the second stage may be superheated, therefore thermal conductivity is lower than that corresponding to saturated vapor, thus increasing vapor thermal inertia. This might decrease the work calculated. Furthermore, the energy transfer from fuel to coolant in the early reaction is used for increasing coolant mass temperature up to saturation temperature instead of superheat temperature of the vapor film around particles. Thus, the pressure peak and the time delay are larger than in a non-equilibrium model.

In any case lots of the Caldarola’s model findings are in agreement with Cho-Wright’s model. It must be noted that parametric models do not give temperature’s limits needed for having explosions, as highlighted from experiments. This is due to the use of equilibrium models which consider coolant as a homogeneous component. It is clear that a non-equilibrium model is needed but, in any case, parametric models have the merit of focusing the main parameter for vapor explosions.

3.4.5 Mechanistic propagation models

The importance of the contribution of the detonation theory on the steam explosion study was previously underlined. On the basis of observations performed on the propagation mechanism, the so called “vapor detonation model”, which takes into account non-equilibrium thermal effects due to the production of vapor inside the reaction zone, has been developed.

Board and Hall [1] have been the first who has applied the steady-state shock adiabatic model to steam explosion. For this reason, it is worth explaining briefly this model, which constitutes the basis of several other models developed.

3.4.5.1 Board and Hall shock adiabatic model

This model has been developed applying the classical detonation theory in chemical reactive flows to the one dimensional case of propagating explosion front through a coarsely mixed region of fuel and coolant.

A planar explosion front, which steadily proceeds through a uniform fuel coolant mixture initially at rest, leaves a mixture of materials, deriving from explosion, behind. It is possible to apply the equation of mass, energy and momentum conservation together with Hugoniot relationship in order to have a relationship between the values P2 and v2

(

)(

)

2

1

u

u

v

v

P

P

+

−

=

−

where the subscripts 1 and 2 indicate, respectively, the initial and high pressure equilibrium state, u and v are the specific internal energy and the specific volume. The velocities U1 and U2in the moving frame of shock front maybe calculated as

2 1 1 2 1 1

v

v

P

P

v

U

−

−

=

2 1 1 2 2 2

v

v

P

P

v

U

−

−

=

U1 is the shock propagation velocity and the velocity behind the shock front in the laboratory frame of reference is U1-U2 .

The materials behind the shock front moves in the front’s direction. This material is followed by a region of expansion which acts as the driving power of the explosion. The leading edge of the expansion region is a rarefied wave which travels at the local fuel-coolant mixture speed of sound c2. In the shock front frame of reference the velocity U2 is equal to the sound velocity c2 (Chapman-Joguet condition, see §3.2). This condition determines a single state on the Hugoniot curve (Eq. 3.67). In addition, the detonation condition, that is P2>P1, implies that the material behind the shock front must be compressed (v2<v1) as shown in Fig. 3.14.

Fig. 3.14 - Comparison between Hicks & Menzies, Hall and Board and Hall models

The work in the isentropic expansion of fuel coolant mixture from the Chapman-Joguet condition to a specific may be calculated similarly to Hicks and Menzies model but including the kinetic energy term at Chapman-Joguet condition. Thus

(

) (

)

(

)(

)

2

U

U

m

m

u

u

m

u

u

m

W

−

+

+

−

+

−

=

(

It must be pointed out that W2-3 results to be larger than Hicks and Menzies evaluation (see Fig. 3.14) because Chapman-Joguet state has the minimum specific entropy along the shock adiabatic curve.

The travelling wave is self-sustaining in this model, which is the mixture’s expansion provides the driving power for explosions. Therefore, the work done by the shock wave for the initial compression of materials initially at rest must be taken