Sharp-interface models

M. Benes. On a computational comparison of phase-field and sharp-interface model of microstructure growth in solidification. Acta Technica CSAV 41 591,... 

Uniform motion of the solid pellets with constant voidage. Pellets are spherical in shape and a shrinking core,sharp interface model is assumed for the pellet reduction(8 9 ). For the gas species, the dimensionless continuity eqtns are ... 

Sharp Interface Model For a first-order reaction in gas reactant,... 

Sharp interface model (SIM) Reacting solid is nonporous. Reacted solid ash is porous. Reaction occurs at the ash-unreacted solid... 

FIG. 7-11 Sharp interface model—concentration profiles. [From Wen, Noncatalytic Heterogeneous Solid-Fluid Reaction Models Ind. Eng. Chem. 60(9) 34-54 (1968), Fig. L]... 

The first model, the shrinking core model (SCM) or the sharp interface model (SIM), was proposed about half a century ago. Other models also describe the behavior of the solid as it undergoes... 

Figure 15.7a shows reaction at the interface between a solid reactant (B) and a product (5) after the fluid has diffused through an inwardly advancing shell of the product (ash). There is no reaction in either the ash layer or in the body of the reactant B (the core), but only at the surface of B. This is the shrinking core or sharp interface model and represents perhaps the most common mechanism of gas-solid reactions for nonporous solids. The overall reaction can be controlled... 

As in any solid-liquid reaction, when the solid is sparingly soluble, reaction occurs within the solid by diffusion of the liquid-phase reactant into it across the liquid film surrounding the solid. Thus two diffusion parameters are operative, the solid-liquid mass transfer coefficient sl and the effective diffusivity D. of the reactant in the solid. A reaction in the solid can occur by any of several mechanisms. The simpler and more common of these were briefly explained in Chapter 15. For reactions following the sharp interface model, ultrasound can enhance either or both these constants. Indeed, in a typical solid-liquid reaction such as the synthesis of dibenzyl sulfide from benzyl chloride and sodium sulfide ultrasound enhances SL by a factor of 2 and by a factor of 3.3 (Hagenson and Doraiswamy, 1998). Similar enhancement in was found for a Michael addition reaction (Ratoarinoro et al., 1995) and for another mass transfer-limited reaction (Worsley and Mills, 1996). 

B. Li and Y. Zhao. Variational implicit solvation with solute molecular mechanics From Diffuse-Interface to Sharp-Interface models. SIAM J. Appl Math, 73Cl) l-23, 2013. 

Nevertheless, a comparison of the band intensities calculated on the basis of the typical model (assuming a sharp interface) and experimental results [76] show that in the experimental spectra of layers, the band intensities are several times lower than predicted. Therefore, absorption indices of layers determined fi om experimental spectra using the relationships derived with the sharp interface model (Sections 1.4-1.7) include a systematic error. The root of this problem hes not only in the difference between the optical properties of the thin-fihn materials and solid materials and the error introduced by an uncertainty in the... 

Thus, the difference between the band intensities in the experimental spectra measured by IRRAS and the calculated ones based on the sharp interface model can be connected with the existence of the optical property gradient in real optical systems, which is most clearly manifested in IRRAS of strong absorbers on the surface of transparent and weakly reflecting substrates. 

The integral can be computed numerically using here f p) given by the second expression in Eq. (6). The result (Fig. 4) is more informative than the sharp-interface computation, as it also gives the dependence of surface tension on temperature. The surface tension decreases with growing temperature and vanishes at the critical point a/ bTc) = bpc = The low-temperature limit 7 oc pfy/aK/b oc pfAi/(fi is qualitatively the same as in the sharp interface model, and the interface thickness reduces to y/Kja a d away from the critical point. The density profile and surface energy can be computed analytically in the vicinity of a critical point [at Tc-T = 0(e ) and p - Pc = 0( )] where g p) can be approximated by a cubic polynomial. 

For a sharp interface model, the additional interfacial energy of a curved interface can be obtained by repeating the computation in the beginning of the preceding subsection with distances recomputed with the help of the metric tensor of the coordinate frame aligned with the interface. One can also consider directly a spherical interface with the radius equal to the inverse mean curvature. The result is the same as in Eq. (23) or (29), and is applicable when far exceeds the molecular diameter d. 

With these we enlist the two fundamental approaches to the noncatalytic gas-solid reaction systems The shrinking core model and volume reaction model. In the volnme reaction model, the solid is porous, the fluid easily diffuses in or ont of the solid, such that the reaction can take place homogeneously everywhere in the solid. On the other hand, with the shrinking core model (SCM), also called the sharp interface model (SIM), there is a sharp interface between the unreacted core and reacted shell of the particles. 

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