During Severe Accidents quantities of corium as a melt or slurry might slump into the lower head of the pressure vessel.[114] Its sensible and decay heat would then be transferred to the structure whose creep strength progressively decreases with temperature. To mitigate this potential cause of a catastrophic rupture, some PWRs have their lower structures submerged in water [315]. Temperature increases in the pressure vessel are clearly dependent on a debris-bed’s morphology, as voidage for example decreases molecular conduction and thereby extends the margin-to or time-to failure. The Three Mile Island incident in 1979 has created the (presently) one and only authentic water reactor debris-bed whose importance to safety assessments motivated an OECD-funded investigation into its relevant character­istics [316].

For a margin-to failure calculation data are required on the physical, metallurgical and radiochemical composition of the debris. Its density, porosity and particle-size statistics broadly characterize morphology, while its metallurgy indicates initial temperatures, melting points, cooling rates and oxidation levels. Radiochemical data completes the required information with regard to decay heat production. “Loose” debris with sizes exceeding 150 mm were found to be broadly sur­rounded by denser material with sizes less than 75 mm. Measured porosities of samples varied markedly between 5.7 to 32% with an average11 of 18 ± 11%. Chemical analyses reveal that 97% of the debris had the broad composition (U; 70%), (Zr; 13.75%) and (O; 13%) by weight of mixed uranium-zirconium oxides, which supports simula­tions having a chemically homogeneous melt pool [93,319]. Some samples of these oxide mixtures contained stratified layers of “pores” which are suggested to result from steam or metallic vapors locked in situ as the corium became more viscous prior to solidification. Debris metallurgy indicates gradual cooling rather than rapid quenching, and that single component regions solidified first. The lowest temperature for uranium to dissolve in zirconium is predicted to be around 1760 ° C, which is about 1000 °C below the melting point of urania. Metallo — graphic and scanning electron microscope examinations reveal that the maximum temperature for a well-mixed solid solution of uranium- zirconium oxides is between 2600 to 2850 °C. Accordingly, the simu­lated initial temperature of corium entering the lower head is taken as 2600 °C. Finally, chemical analyses of the debris samples enable the decay heat to be calculated as 130 W/kg and 96 W/kg after 224 and 600 min respectively.[115] [116]

If no MFCIs occur during a Severe Accident, an increasing mass of molten corium would first vaporize any residual water as it slumps into the lower head. Sensible and decay heat would then be transferred by turbulent natural convection into the pressure vessel and by thermal radiation into the degraded structure above [318]. A likely scenario is a multi component liquid pool encased in a growing solidified crust which forms on its free surface and on the vessel’s wall. Sohal et al. [315] provide a thorough assessment of available experimental corre­lations for turbulent natural convective heat transfer between molten corium and solid surfaces. With internal heat generation the recom­mended correlation takes the form

Nu = a(Ra’)b; Ra’ = gbqvL5/avk (7.5)

where Ra’ is the Internal Rayleigh Number and

g — gravitational acceleration; b — thermal expansion coefficient qv — volumetric heat generation rate; L — a “characteristic” length a — molecular thermal diffusivity; y — kinematic viscosity k-molecular thermal conductivity; a, b — parameters to match the data

Unlike turbulent flow in pipes and laminar flow aerodynamics, the characteristic length L here is open to more subjective interpretations such as

i. The radius of a pool’s top surface

ii. The maximum depth of a pool

iii. The average of 1 and 2 above

iv. The radius of a hemispherical pool equivalent to the volume of the cylindrical one

If the exponent b in equation (7.5) equals 0.2, the characteristic length exactly cancels out. Because the experimental data is best correlated with b ‘ 0.2 predicted heat transfer coefficients are quite insensitive to the actual choice. Table 7.1 depicts the recommended values of (a; b) with option 3 to give a regression within about13 ± 10% of the data from six water or Freon experiments. Though these experiments

This author’s own estimate from graphs in Ref. [315].

involve radically different fluids compared to corium, it should be noted that the burn-out margin for the SGHWR derived from a low- pressure replica-scaled Freon rig accurately matched later measure­ments derived from a prototype water-cooled rig at 62 bar [63,297]. For situations with Internal Rayleigh Numbers outside those in Table 7.1, the nearest correlation should be used.

An alternative situation to a developing homogenous melt pool is one with a purely metallic layer floating on its upper surface. No internal heat generation occurs within this superficial layer for which turbulent natural convective heat transfer is correlated by

Nu = a(Ra)b; Ra=gfi(AT )L3/av (7.6)

where Ra is the External Rayleigh Number and

AT — local temperature difference between the bulk liquid and its boundary

Table 7.2 depicts the recommended values for (a; b) with option 3 as the characteristic length. No well-tested correlation for horizontal flow is apparently available, but as the contact area with a pressure vessel is relatively small, an average of the upward and downward flow coef­ficients is suggested [315] as adequate.14

Molecular heat diffusion into the particulates of a debris bed or in the reactor pressure vessel can be characterized by the axisymmetric form of

@ T

— = a [V2T + s/k (7.7)

Two terms from equation (7.6) added together with these (a ; b)s.

where

a — thermal diffusivity; к — thermal conductivity s — volumetric heat generation rate

Flow through porous media was first quantified experimentally by Henri Darcy in 1856 in connection with aquifers that supplied the civic fountains of Dijon. Under the essentially laminar flow conditions the usually intractable Navier-Stokes equations [256,268] become analyti­cally solvable to generalize Darcy’s empirical formula as

Z

qv = — VP (7.8)

m

where

qv — volumetric flow rate (m/s); P — pressure (Pa) m — dynamic viscosity (kg/ms); Z — permeability

The corresponding flow velocity is given by

V = qv/h; h-porosity (7.9)

with the Reynolds Number

Re = pVD/m (7.10)

Here the characteristic dimension D is taken as the smallest sieve-size to allow free-passage for 30% of all particles, and equation (7.8) is valid for Re <10.

In addition to the above equations and correlations, simulations of debris-bed cooling by the SCDAP/REALP5 code [319] involve

i. The usual two-phase fluid conservation equations

ii. A range of possible debris-bed permeabilities

iii. The oxidation of intact and slumped cladding under re-flooded conditions

iv. The penetration of melted core-plate into existing porous debris, and its effect on heating up the lower head

v. The re-slumping of previously frozen fuel-cladding

vi. The up-take of oxygen and hydrogen under conditions of steam starvation or rapid changes of temperature, etc.

The “stand-alone” development of SCDAP began early in the 1970s to assess the progressive oxidation or melt-down of fuel elements and control rods. Two-phase fluid dynamics patently interacts with these processes, so in 1979 it was merged with the RELAP5 code whose own development had started previously in 1975. Thereafter evaluation and validation of the combination’s phenomenological-based modules have been ongoing. It should be recalled that the accuracy of degraded core dynamics is not the usual ± 10% for engineering design purposes, but it is required only to be demonstrably conservative and bounded. In addition to assessing the margin-to failure of a reactor’s pressure vessel, degraded core calculations bound the amount of liquid corium and water in the lower head during a Severe Accident, and thereby the yield of potential MFCIs.