Models MELTS

In contrast to the full equilibrium transport model, melt could be incrementally removed from the melting solid and isolated into channels for melt ascent. This model is the disequilibrium transport model of Spiegelman and Elliott (1993). Instead of substituting Equation (A7) in for Cs, the problem becomes one of separately keeping track of the concentrations of parent and daughter nuclides in the solid and the fluid. In this case, assuming steady state, two equations are used to account for the daughter nuclide ... 

Figure 3. Three spatial scales for modeling melt crystal growth, as exemplified by the vertical Bridgman system.

The governing equations and boundary conditions for modeling melt crystal growth are described for the CZ growth geometry shown in Figure 6. The equations of motion, continuity, and transport of heat and of a dilute solute are as follows ... 

Some conclusions that can be drawn from this simulation of stress relaxation in an atomic model melt are as follows ... 

Figure 2. Three spatial scales for modeling melt crystal growth, as exemplified by the vertical Bridgman process. From Theory of Transport Processes in Single Crystal Growth from the Melt, by R. A. Brown, AJChE Journal, Vol. 34, No. 6, pp. 881-911, 1988, [29]. Reproduced by permission of the American Institute of Chemical Engineers copyright 1988 AIChE.

Figure 7 Cl-normalized Yb/La versus Sc/Sm for mafic differentiated achondrites. Model melt is calculated to be in equilibrium with only olivine. The generally lower Sc/Sm ratios of the achondrites suggest that equilibration with, or fractionation of pyroxene also occurred. Data from sources listed in the text.

Here, the fertile upper-mantle composition derived in the previous section is assumed for oceanic and olf-craton mantle, and the model of Kinzler and Grove (1992a, 1993) is used to model melt extraction at pressures of < 2.5 GPa. 

Eased on two main lines of evidence, Niu et al. (1997) concluded that abyssal peridotites are the end products of melt extraction followed by variable amounts of olivine crystallization. First, in their set of reconstructed compositions they found that model fractional and batch melt extraction trends could not reproduce major and minor element variations in their data set. Most importantly, they found that melt extraction models failed to account for the strong positive correlation between FeO and MgO, as well as incompatible minor-element concentrations. Specifically, at a given Na20 or Ti02 content, abyssal peridotites are enriched in MgO relative to model melt extraction residues. Niu et al. (1997) showed that these compositional anomalies can be reconciled by a model of melt extraction followed by olivine crystallization, with more MgO-enriched samples having more accumulated olivine. If correct, this model has important implications for understanding melt extraction at oceanic ridges, and it has recently been the focus of re-evaluation. 

Systematic Name (Common Name) Structural Formula Condensed Stmctlual Formula Space-filling Model Melting Point (°C) Boiling Point (°C)... 

Alexiades, V. and A.D. Solomon. Mathematical modeling melting and freezing processes. Hemisphere Publishing, Washington, 1993. 

The Phan-Thien/Tanner constitutive equation does not represent the state of the art in modeling melt flow at the time of this writing, but it is adequate to illustrate the response of melts of flexible polymers in complex flows and it has a mathematical structure that does not differ substantively from other equations with a firmer basis in molecular theory. Furthermore, it has been widely used in simulation studies to date. Hence, we will use it for illustrative purposes in this text, recognizing that it is likely to be replaced as the preferred constitutive equation for applications. The minimum rheological information required for simulations is thus the temperature-dependent linear viscoelastic spectrum and the temperature-dependent viscosity as a function of shear rate. Extensional data should be used, but they are often unavailable when the PTT equation is employed it is therefore common to select a reasonable value of to describe the extensional response. 

Frank [2] was the first who assumed that the stmeture of liquid metals could be described on the basis of ieosahedral packing. Much later [3] the assumption was confirmed by the molecular-dynamics simulation of a supercooled model melt within the framework of Lennard-Jones potentials. 

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