Liquids possess elasticity as well as mass, so the interfacial liquid in Figure 5.4 does not move in unison with the application of a trigger pulse at its far end. Such lack of concomitance is often irrelevant, but here a typical experimental film destabilization period [207] of 20 ms is comparable with the 67 ms transit time of a pressure pulse. Conse­quently as illustrated in Figure 5.5, a distributed model of shock propagation is necessary, though some simulations [21,225,226] feature only point models. Moving boundaries often pose additional degrees of analytical difficulty, which are aggravated here by heat and mass transport phenomena. However, according to equation (5.23), heat diffuses only 3.5 mm during film collapse whilst the pressure shock travels16 about 0.3 m. Moreover, the mass of a 100 mm of steam film is totally negligible in relation to that of the 100 mm liquid slug. Thus heat diffusion and shock dynamics in the liquid can be advantageously decoupled (i. e., solved independently). Like other moving boundary problems, one-dimensional shock propagation is best formulated in terms of the “constant mass packet” Lagrangian equations (momentum) (5.31)

(mass) (5.32)

V = V

@I p dv dt dt

(energy) (5.33)

16 The isentropic sonic speed of 1500 m/s is slightly slower than the weak anisentropic shock propagation of a trigger.

P = P(I, v) (thermodynamic state) (5.35)

where

z — particle displacement along the x-coordinate axis (m)

V — particle velocity (m/s); P — pressure (Pa) v — specific volume (m3/kg); I — specific internal energy (J/kg) vo — unshocked specific volume

An experimental correlation relates the wave speed Xs to particle velocity and for water this takes the form

Xs = bo + biV + &V2 (5.36)

Published data [208,209] shows that the sonic speed c with

@P

c2 = — : S — entropy (5.37)

@P S constant

is strongly dependent on temperature but only weakly so on pressure, and therefore similarly for bo, bi, and b2. Because thermodynamic equilibrium is rapidly regained across a shock front, the Rankine — Hugoniot equations [203] closely approximate pressure and energy changes. Accordingly, close to a Rankine-Hugoniot curve the relation­ship between pressure and internal energy is found to be represented [237] by

P(y)= Po + G(v)(I — I °)/v (5.38)

with the appropriate constant value for the Gruneisen function Q.

Integration over the moving mesh points in the liquid and vapor is effected by Leibnitz’s theorem [112] to provide first of all ordinary differential equations, and then the required difference equations [206]. A necessary condition for the von Neumann Stability of the adopted explicit solution scheme is [208] where Ncfl is the Courant-Friedrichs-Levy Number. Because triggers evolve as relatively weak shocks their propagation is approximately sonic, so for water around STP

Ncfl = 1500 (dt/dx) < 1 (5.40)

The water slug approaches the (assumed) rigid melt according to the calculation

zn+1 = Z + Vn+1 dt; zn = z(k; ndt) (5.41)

and numerical breakdown is prevented by progressively reducing the time step with decreasing film thickness. Though explicit solutions [203,238] of the Lagrangian equations without added viscosity and thermal conductivity[86] lead to the progressive sharpening of a shock front, and then often to computational failure, the adopted solution scheme appears stable.

Neglecting relaxation effects [210,211], Fourier’s heat conduction equation and the first law of thermodynamics describe one-dimensional heat diffusion in an isotropic semi-infinite slab by [224]

@ T d2T

= a with a — thermal diffusivity = к/рЄр (5.42)

ot ox2

Corresponding central-space and backward-time linear difference equa­tions for a fixed or moving mesh have tridiagonal structures which are solvable by Gaussian elimination, but preserving second-order spatial accuracy at the boundaries requires special care [117,206]. Necessary and sufficient conditions for this solution procedure to be stable with a fixed Eulerian mesh are [238]

adt/(dx)2 <x/2 (5.43)

Low liquid compressibility and the relatively small mass of vapor ensure that matrix terms for the moving mesh are largely those for a fixed mesh.

Consequently, equation (5.43) is adjudged apposite for present pur­poses. During the time step of a shock-wave calculation, the thermal penetration distance in water evaluates from equation (5.22) as only 0.8 mm. Using this guide, sensibly converged thermal diffusion calculations are obtained with the lattice parameters

St = 1ms ; Sz = 1mm for which aSt/(Sx)2 = 0.16

Attempted shock calculations with these values would have Ncfl = 1500, so justifying the suggested decoupling of shock and heat diffusion calculations.

If the outer surface of a semi-infinite slab of an isotropic conductor is abruptly changed by T*, then according to equation (5.42), a temperature wave diffuses into its interior as

T(x, t) = T*erfc(yX/4<xt^ (5.44)

In order that T/T* ‘ 0.01, published tables [239] give

x/4a t = (1.83)2

Molten urania has a thermal diffusivity of order[87] 2 x 10“6 m2/s, so the penetration distance is 26.8t microns. Experiments [207] show that steam film destabilization occurs in far less than 1 s, so that negligible temperature change occurs at the far end of a simulated 30 mm thickness of melt. Furthermore, during the time step of a shock wave calculation, the thermal penetration in the urania is about 2.8 mm. Accordingly, satisfactory heat diffusion calculations in the molten urania are accom­plished with the lattice parameters

St = 1ms ; Sx = 3 mm for which aSt/(Sx)2 = 0.22

The previously justified uniform pressure in a simulated vapor film renders momentum conservation unnecessary, but mass and energy conservation still require formulation due to interfacial transport. Intuitively or formally from the Rankine-Hugoniot mass conservation equation, vapor particles in contact with the melt have zero velocity. Just one intermediate mesh point between liquid and vapor is recommended in order to ease numerical problems as the film approaches collapse. Now with respect to mass conservation for example, a linear spatial variation of particle velocity V across the thin film is reasonable so

V = vgb(z — Zb)/(zmb — Zb) (5.45)

where

VGb — interfacial vapor particle velocity

zB — interfacial position; zMB — melt position (fixed)

The Rankine-Hugoniot mass conservation equation then yields the interfacial velocity as

t

Zb = VLb — vlbGb with zb = j Zb (Z)dZ (5.46)

0

and mass conservation for the film as a whole in terms of the mid-point density ~G is evidently

d [(zmb — Zb)~g] — Gb = 0 with vg = 1/~g (5.47)

at

Reference [233] details all the required finite difference equations, physical processes and the flow chart for a digital simulation. It also demonstrates that in this situation the Knudsen effect [244] does not compromise Fourier’s heat-conduction equation. Though physical pro­cesses are usually described for both sodium and water, simulations concern only the latter. This bias occurs because steam film destabi­lization experiments are far more tractable, less expensive and were more immediate to the safety case for Sizewell B.

Cine recordings at AEEW of molten urania poured into water at 0.1 MPa depict an agglomerate of melt and steam descending in the coolant. Prior to a trigger, a state of disequilibrium exists in which latent heat transfer diffuses into the surrounding liquid that is continuously replenished and cooled by the induced turbulence. To replicate something of this situation in a simulation, the temperature of the surrounding mass
of water is taken as a uniform 20 °C at 0.1 MPa except for an initially saturated value at the interface. Initial temperatures of a 100 mm thick steam film are assumed saturated throughout, and the melt temperature is taken as 3000 °C. A simulation begins with a 50 ms-5 MPa trigger at the remote end of a 100 mm water column, and the resulting interfacial kinetics and temperatures are shown in Figure 5.8. Prior the shock front’s arrival at the interface, a state of quasi-equilibrium appears after initially strong interfacial condensation. Despite the subsequently weaker evap­oration and the absence of lateral mass convection in the model, the essentially constant film thickness in this period can be justified by a straightforward energy balance using Table 5.2. After a delay of 66 ms, the shock front arrives at the interface whose displacement to some 8 mm from the melt in 20 ms is largely unresisted. Due to the water column’s inertia, small oscillations occur but actual liquid-melt contact is pre­vented by interfacial evaporation from molecular conduction across the now much thinner film. The absence of a material increase in steam pressure during film collapse is supported by experiments [207] that involve an independent calculation of initial film thickness and the following analysis [206] of its collapse time as a function of trigger pressure.

A simulated trigger pulse appears as a weak shock with an almost (isentropic) sonic propagation speed, so the linear wave equation approximates water slug kinetics by

@2z _ 2 @2z dt2 C dx2

where

Under experimental conditions, the Acoustic Impedances19 of water and steam are respectively

ZL = 1.5 x 106 kg/m2 — s and ZG = 2.8 x 102 kg/m2 — s

19 In general Z = pc.

-10 • -20 —

Figure 5.8 Typical Simulation Results for a Melt Temperature of 3000 °C which yield the interfacial reflection coefficient

Г 4 (ZG — ZL)/(ZG + ZL)’-1 (5.49)

and this matches the essentially unresisted simulated motion of a liquid-vapor interface. Laplace transform analysis of equation (5.48) for a step function trigger of amplitude PT and the boundary condition in equation (5.49) gives the transformed interfacial velocity

VB = PT[(2/sZL)exp — 2st]exp — 2snt

n=0

where

t — one sonic transit time along the liquid slug = slug length/c n — number of forward and backward transits

In the simulated and experimental situations, film collapse occurs after just one transmit time so the relevant interfacial velocity is

VB = 2PT/ZL for t > t = 0 otherwise

Hence the predicted time for film collapse is

Tc = (Initial film thickness)/(2PT/ZL) (5.50)

Experimental measurements [207] using an electrical probe and an initial film thickness derived from an independent model confirm the above for a solid heat transfer surface below 500 °C.

Computer simulations predict that a steam film over molten urania largely fails to oppose its collapse by a weak shock wave. This prediction is supported by specific steam-film collapse experiments, and MFCI urania tests with sodium or water which are triggered by just the modes of contact. Reactor safety assessments must therefore assume a priori that a contact between molten corium and coolant would produce an MFCI. Moot questions remain concerning the quantity of participating melt and the mechanical energy released (yield). With specific regard to the cited aluminum-water tests in Rig A, Table 5.2 and the discussion of the results in Figure 5.8 suggest that hydrogen generation enhances the thermal conductivity of the steam film, and thereby its stability to the point where a military explosive trigger becomes necessary.