The lifecycle of nuclear waste disposal by vitrification depends on the waste loading (the fraction of waste in the glass product) and the processing rate (the rate at which the melters are producing the waste glass). While the former has achieved near perfection over decades of glass formulation development, the latter has received attention relatively recently. For a single waste composition, the glass production rate can be maximized empirically, by trial and error, through optimizing the waste treatment, glass-forming and modifying additives, and melter operation parameters. For the Hanford Waste Vitrification and Immobilization Plant (WTP) that will process hundreds, even thousands, compositions of waste, mathematical modeling is the only realistic method for accomplishing such a task. Yet, the conversion of the material charged into the melter to molten glass is enormously complex; its main features have only recently been identified and mathematically formulated. The current state can be characterized as an expensive and cumbersome empirical method on the one hand and a complex, not yet fully developed mathematical modeling on the other. Fortunately, a middle way exists in the form of an engineering approach in which the relationship between the melting rate and the basic parameters is simplified to the utmost possible degree. This approach has been developed within the recent decade and dubbed the melting rate correlation (MRC).
The MRC is based on the boundary layer theory, according to which the heat flux from the melt pool to the cold cap is Q = h(TM – TB) and the shear stress imposed on the cold cap bottom is s = Vdv/dz. Here h is the heat transfer coefficient, TM is the melter operating temperature, TB is the cold cap bottom temperature, V is the glass melt viscosity, v is the glass melt velocity horizontal component, and z is the vertical coordinate. For a melter in which the melt is pneumatically stirred by gas bubbling, log(h) = a + blog(Re), where a and b are adjustable parameters and Re = yu/k, where y is the characteristic melt pool dimension (depth or width), u is the bubbling gas flow rate, and k is the kinematic viscosity.
For the cold cap to which the heat is delivered from below, the glass production rate is j = Q/H, where H is the melter feed-to-glass melt conversion heat. For a glass produced in a melter operated at a constant temperature with melt stirred by bubbling, the unknown parameters are TB, a, and b. We have developed various methods to estimate TB, none of them fully satisfactory, yet enabling us to determine a and b by fitting the MRC relationship to data reported for reduced scale melters. For high-level waste (HLW) glass processing, the values are a = 2,5 and b = ½.
The reacting zone of the melter feed in the cold cap is separated from the melt below by the primary foam layer. Hence, the cold cap bottom temperature, TB, is a temperature at which the primary foam is collapsing: the foam cell walls are breaking and the cavities with gas released are carried away by the melt circulating under the cold cap.
Glass melt foam stability is determined by viscosity. In the primary foam, viscosity is the lowest at the bottom, where the temperature is highest. Thinning of foam cell walls occurs by three mechanisms: first, the surface curvature gradient drives the melt to the Plateau borders; second, gas phase volume in foam bubbles increases through thermal expansion and gas-evolving reactions; third, the external shear stress imposed by the velocity gradient on the melt side imposes strain on the foal cells. Apart from the transient melt viscosity, the critical thickness at which the cell walls break is a function of the fraction and position of the dispersed solid particles.
Silica particles are the main solid phase that remains undissolved, yet continues to dissolve, at the cold cap bottom. Since the melt around dissolving silica particles has a high concentration of SiO2, its viscosity is increased and can be up to a hundred times higher than in the bulk melt. In the cell walls, silica particles have destabilizing effect: low viscosity melt outside the silica concentration layer are the weak points. Silica particles promote foam stability in Plateau borders, where the high viscosity opposes the melt flow and hinders cell wall thinning. Similar, though less potent effects, are produced by other solids, such as spinel and zircon.
The purely kinetic effect of silica, in the absence of external forces, on the cold cap bottom temperature was expressed, by Ferkl et al., as TB = T0 – pQ, where the T0 and p values were computed for several melter feeds. This linear formula can be extended to express the effect of the shear stress at the cold cap bottom. Provided that the foam strain is proportional to the bubbling rate, we can write TB = T0 – pQ + ru. Although the primary foam deformability was determined by a recent study of the cold cap rheology, the r value is still waiting to be determined through computational fluid dynamics modeling.
Summarizing, when exposed to a higher heat flux (Q) and an enhanced bubbling (u), the way the primary foam layer can withstand collapse is by allowing the low-viscosity bottom foam cells to break until the transient melt viscosity is high enough to resist the wall thinning forces. As a result, the bottom temperature, TB, decreases. With lower TB, the heat transfer driving force, TMO – TB, increases and thus, the melting rate is faster. Ultimately, the melting rate is maximized when primary foam ceases to exist, i.e., when TB = TF, where TF is the foam onset temperature (the temperature at which the transient melt connects. This option, achievable at elevated melting temperatures, was wasted for LAW glass processing by choosing, needlessly, Inconel electrodes that necessitate processing at a low melting temperature.