Under conditions of thermodynamic equilibrium the Maxwell- Boltzmann probability density function characterizes the velocities of ideal gas molecules. Assuming isotropic scattering [58], the mass flux of ideal gas molecules traversing one way through a conceptual plane sur­face is derived [210] on this basis from classical kinetic theory. Ignoring the significant inter-molecular attractions in the liquid and vapor states (i. e., the Joule-Kelvin effect), applications of this classical analysis to the liquid-vapor interface gives the net mass flux in the liquid as [241]

where

GB — interfacial mass flux (kg/m2 — s); R — specific gas constant PGB, TGB — interfacial vapor pressure and temperature, respectively TLB — interfacial liquid temperature s — condensation coefficient (0 < 0 < 1)

It is recommended [229] that the molecular structure function F(s) takes the form

where:

g — specific heat ratio of the vapor (Cp/Cv)

Equation (5.27) correctly implies a net mass flux of zero when saturated liquid and vapor co-exist at an interface under conditions of thermodynamic equilibrium. However, under non-equilibrium condi­tions it loses some accuracy. Specifically, apart from the omission of inter-molecular forces, vapor molecules near an interface originate as

i. Those having just emerged from the liquid

ii. Those having diffused from a higher temperature near the melt

iii. Those reflected from the liquid surface

This motley ensemble is unlikely to be characterized by a Maxwell- Boltzmann distribution as required for the validity of equation (5.27). Moreover, the rigorous definition [3] of temperature is in the context of thermal equilibrium, so with net interfacial mass transport neither TGB nor TLB strictly exists. Nevertheless there appears no alternative to equation (5.27), and for simulation purposes they are taken as extrap­olations of heat diffusion calculations in the two media. Under net evaporation or condensation equation (5.27) predicts an interfacial temperature jump (discontinuity) that is confirmed by experiments with liquid metals [230,231]. An early simulation of vapor film destabilization by Corradini [198] assumes both interfacial fluids to be saturated, but his later corrected model [226] reveals the marked effect of this discontinuity on computed transients.

A condensation coefficient is in essence the probability that a molecule impinging the interface enters the other phase. Mills and Saban [232] comprehensively review many published analytical deri­vations, but conclude that reliance is best placed on experimental data. They consider the most reliable measurements for water to be those by Nabavian [233], Berman [234] or themselves which together give

0.35 < a < 1.0 (5.29)

After reviewing 11 independent publications on the condensation coefficient for various liquid metals over the pressure range 0.001 to 1 bar, Fedorovich and Rosenhow [231] conclude that

0.1 < a < 1.0 (5.30)

Results are tightly clustered for pressures no greater than 0.1 bar, but thereafter their dispersion increases. The wide uncertainties in
equations (5.29) and (5.30) are exceptional by twentieth century standards, and might well reflect the purity of the coolant. In fact experiments with a liquid metal indicate that the condensation coefficient decreases with increasing contamination [242]. Later labo­ratory measurements show that the coefficient assumes its maximum value of unity when the water or liquid metal is exceptionally pure [255], but in Severe Accidents a reactor coolant would be heavily contaminated. Consequently, because decreased interfacial condensa­tion saps less energy from an expanding MFCI bubble, the least of the above values for a condensation coefficient should be used in safety simulations.

Permanent gas molecules in sufficient numbers can seriously reduce the mass flow rate from industrial steam condensers by restricting access to heat transfer surfaces [219]. As a preventative measure, deaerators are installed in the feed-trains of power station boilers, where they also provide emergency supplies (see Section 3.4). In Severe Accidents hydrogen or fission product gases might similarly be expected to reduce interfacial condensation rates, and thereby conserve the energy of an expanding MFCI vapor bubble. However, fast reactor tests [228] show that collapse times of sodium bubbles are largely unaffected when contaminated with Xenon concentrations representative of spent fuel. Other laboratory experiments [236,237] with steam bubbles establish that condensation rates are reduced by less than 10% with the introduction of 15% molar concentrations of nitrogen. In both cases, the authors conclude that efficient turbulent mixing must exist within a collapsing bubble. No experiments concerning the effect of permanent gases on triggered vapor — film destabilization were found in the literature. From Table 5.2 the principal effect of hydrogen is seen to increase a film’s thermal conduc­tivity and therefore to a degree its stability.15

The above discussion describes the considerable uncertainties in predicted interfacial condensation rates. Nevertheless, Section 5.8 shows that conservative interfacial condensation rates largely account for the experimentally observed reduction in MFCI energies from the idealized isentropic Hicks-Menzies yields [85]. From the viewpoint of reactor safety assessments, the identification of a physical process to justify the extension of experimental 4 to 5% MFCI efficiencies to reactor-scale tonne-quantities is highly significant.

See Section 5.6.