With a strong enough trigger and a large enough inertial mass of coolant (>0.1m), the liquid’s final oscillations in Figure 5.8 become large enough for localized contact(s) with the melt. At such instants
enormous heat fluxes into the liquid occur. If a relatively low thermal- conductivity blanket were then to re-form, a passive equilibration of melt and coolant would continue. However, the existence of MFCI implies that such vapor blanketing is somehow transiently suppressed, and research into possible mechanisms was initiated by Derewnicki and Hall [243]. Their analysis supported by visualization studies using a platinum wire indicates that acoustic loading (local pressure increases) and Marangoni flows20 inhibit vapor production during ultra-rapid boiling. It is now suggested that the violent return to thermodynamic equilibrium of locally superheated liquid at contact areas with a melt launches shock waves that create local melt fragmentation [244]. During propagation21 shock intensity is escalated by further melt fragmentation across its steep frontal pressure gradient, and direct coolant contact is sustained by viscous forces that strip embryonic bubbles from the fragments. Though the above description is in part conjecture the following analysis establishes that the creation of fine debris (< 250 mm) and a highly efficient heat transfer mechanism are necessary in order to match experimental MFCI time scales.

If an isotropic sphere at a uniform temperature TM is abruptly immersed in an infinite sea of perfectly stirred coolant, its spatially one­dimensional temperatures thereafter satisfy [224]

(5.51)

with the boundary condition

h[T (R; t)- TL ] =-k — 4 f(t)

dr R

where

r — radial coordinate; R — radius of sphere

k — thermal conductivity of the sphere; TL — coolant temperature

0 A surface tension gradient in a fluid pulls liquid toward the greater value to create a Marangoni flow.

21 At around 350 km/h in tin-water experiments [248].

h — external heat transfer coefficient; f — surface heat flux from the sphere

The above partial differential equation has a countably infinite spec­trum of eigenvalues, and its Laplace transform solution is the corre­sponding infinite series [224]. The thermal energy released from the sphere is

t

Surface area x j f(Z)dZ (5.53)

0

and when the external heat transfer process is highly efficient it is largely represented by the smallest eigenvalue term of the series. Under these conditions the reciprocal of the smallest eigenvalue is called the dominant time constant t*, and the heat released per unit mass is approximately

E(t) =Ei[1 — exp (-t/t*)]

where

Ei = CpDT; DT = Tm — TL lim t* = R2/p2d

hn

Because the dominant thermal time constants of square prisms are insignificantly different from those of spheres [224], heat transfer from irregular MFCI debris is taken as that from spheres. Dominant thermal time constants for uranium spheres as a function of external heat transfer coefficient are shown in Figure 5.9 for diameters of 30, 100, 250, and 500 mm. Heat transfer in experimental MFCI with sodium or water is completed within 2-4 ms. Consequently dominant time con­stants no greater than about 1 ms are involved, and to match these values Figure 5.9 shows that spherical diameters below 250 mm and heat transfer coefficients exceeding 100kW/m2 are necessary. Thermal radiation is immaterial because Section 5.4 shows that water is broadly transparent to infrared over these length scales, and even the heat absorbed by a black body from a source at 3500 K corresponds to a coefficient[88] of less than 2.5 kW/m2K.

After separation from the coolant experimental MFCI debris is graded using a cascaded nest of precision sieves. The size or equiv­alent diameter d of an individual particle is then taken as the arithmetic mean of the smallest and largest sieve sizes through which it can and then cannot pass. Particle diameters for both water and sodium coolant scan be characterized [246] by a Log Normal proba­bility density function

P(d’) = ■ P exp

‘ s’ 2p 2 V s’

where

d’ = log d; m’ = e(d’) and (s’)2 = e(d’ — m’)2

However, validation of an MFCI simulation code against experiment is best achieved using the particular a posteriori measured debris sizes. The energetics of an MFCI are clearly influenced by the contact rate of
melt fragments and coolant[89] as well as by their sizes. Accordingly if dM of a coarse mixture becomes finely fragmented, (< 250 mm) to N(0) particles over the time interval [0,0 + d0], then assuming spherical debris

N(0) 4 3

dM = — РрмГк(0) (5.56)

k=1 3

where

pM — density of solidified thermite mix

rk (0) — spherical radius corresponding the kth sieve-size at 0[90] The average of {rk(0)|1 < k < N(0)} is by definition

N(0)

N(0)Av. [r3(0)] = £ r — (0) k=1

which substituted into equation (5.56) gives

dM = 4ppMN(0)Av. [r3(0)] (5.57)

If each particle is assumed to liberate its heat independently, then the power released from those newly created during 0 to 0 + d0 is similarly derived from equation (5.54) as

where

E(t — 0; rk(o)) = energy released per unit mass of a spherical
particle of radius rk(o) at time 0 = 0 as in equation (5.54)

Expressing equation (5.57) in terms of the mass creation rate of fine debris W in an interaction

4

W(6)86 = 3ppMN(6)Av. [r3(6)] (5.59)

In the limit as 86 ! 0, substitution of the above into equation (5.58) yields

§ = W(6)Av. jr3(6) d6E[t — 6; rk(6)]}/Av. [r|(6)]

so

P(6) 4Av.<| r3(6) — E[6; rk(6)^ /Av. /1(6)]

If the creation of fine debris occurs at a constant rate W0 over a prescribed period [0, f] and thereafter is zero, then

W(t) = W0 [U(t) — U(t — tf)]

where U(t) is the unit step function. By effecting the variable change

6′ = t — 6 d6 = —d6

the convolution (5.60) expands into

t

Q(t) = W0 P(6)d6 for t < tf

0

Г t t—tf

00

which is still intractable unless further assumptions are made. Accord­ingly, it is conjectured that fine fragmentation by a shock wave is a

statistically stationary process [247], so the fraction of each debris size remains statistically constant throughout an interaction. Thus by virtue of the large sample sizes Av. [r|(0)] and Av. [r|(U) E[U; r(U)]] are largely independent of U, and can be represented by averages calculated a posteriori from the recovered debris, as

t

P(U)dU=(5.62)

0

In the BUBEX simulation of an MFCI test, a one-dimensional look-up table of the above integral (5.62) is first computed separately for each time step using equation (5.54) and the recovered debris spectrum. The mass creation rate of W0 of fine fragments is estimated from experi­ment. Alternatively, shock propagation at 100-150 m/s over the small volume of experimental coarse mixtures creates fine debris much faster than their energy release rates, which implies the simpler approximation

s,

Q(t)= dMn-E(t, rn) (5.63)

n=1

where dMn is the recovered mass from the nth of S sieve sizes.

The yield of an MFCI is defined as the mechanical work delivered by the expansion of its vapor bubble. Experiments at AEEW measure Yield in terms of the assumed isentropic pressurization of a coolant’s argon cover gas, and the efficiency of an MFCI is defined as:

MFCI Efficiency = Yield/Heat content of participating mass (i. e., debris < 250 mm)

(5.64)

However, even if the extrapolation of experimental MFCI efficiency to reactor-scale masses is valid, the resulting Yield alone does not repre­sent the potential damage to a reactor structure. Specifically, the containment vessel of a fast reactor in Figure 5.10 includes primary circuit pumps and intermediate heat exchangers which can focus explosively displaced coolant to exacerbate damage: particularly to the rotating shield. Analytical and corroborative experimental investi­gations of this phenomenon for a fast reactor are described later in

A. C. Induction motor incorporating flywheel

Fluid coupling -L™I

■■■ — rr

Figure 5.10 Vertical Section of PFR [60]

Section 6.1. In a PWR, the lower core-plate and support casting in Figure 1.4 would constrain the explosion to increase the mechanical loading at the base of its pressure vessel.

To place the explosive violence of an MFCI in perspective, consider a typical Rig A experiment in which 0.5 kg of molten urania thermite at 3500K reacts with 50 kg of water at 293K. If passive thermal equi­libration were to occur with the whole coolant mass, its temperature would increase to just a modest 299K. However, the experimental MFCI yield is about 0.16MJ which corresponds to the kinetic energy of a 1V2 tonne vehicle travelling at 55km/h. Because water reactors and fast reactors have typical fuel inventories of 100 and 20 tonne respec­tively, the very localized heat transfer in an MFCI appears as potentially catastrophic especially as a 1 GJ yield gives cause for concern with
regard to the failure of either reactor vessel. Specifically, granted a Hicks-Menzies isentropic efficiency of 16% and a participating mass of 20% of a fuel inventory,[91] the resulting yields for a molten fuel temperature of 3500 K are

Water reactor yield = 4.8 GJ Fast reactor yield = 1.4 GJ

Computer simulations involving equation (5.62) are clearly unlikely to precisely replicate experimental measurements of an MFCI yield. Nevertheless by also marrying the condensation mass flux equation (5.27) into apposite fluid dynamics, the BUBEX code in Section 5.8 confirms interfacial condensation as the principal thermo­dynamic irreversibility that saps material amounts of energy from an MFCI vapor bubble. Consequently, justification is derived for extrap­olating the 4 to 5% experimental MFCI efficiency to reactor-scale and thereby significantly enhancing a reactor safety case. Specifically, granted a 4 to 5% efficiency and a molten corium temperature of 3500 K for a Severe Accident in a PWR, then the required participating mass for a 1 GJ event evaluates as about 17 tonne or 17% of the entire fuel inventory. Because experiments indicate that just some 20% of a total melt mass has[92] particle sizes less than 250 mm, a 1 GJ yield corresponds to the non-credible event of the entire PWR fuel inventory in a molten state and in contact with enough water. On the same basis for a fast reactor, but with the highest predicted melt temperature of 5000 K, the required participating mass for a 1 GJ yield is 11 tonne or nearly three times the entire fuel inventory in a molten state! In this context, simula­tions of Severe Accidents [213,269,270] with distributed neutronics and thermal hydraulics indicate a progressive degradation of a reactor core. Confirmation is provided by the post-accident inspection of the Three Mile Island reactor in which just 8 to 16 tonne of its 100 tonne fuel inventory lay below the lower core-support plate [69]. Furthermore, Section 5.2 outlines the different molten fuel-coolant contact modes as

additional factors that would materially restrict the participating mass. For all these reasons the creation of the participating mass for a single coherent 1GJ event is considered very improbable especially with present safety systems, operational experience and legislation.