Immobile interface,

Let us consider the influence of a solid-liquid interface advancing at a constant velocity on the solid-liquid fractionation of an element i. In the case of unidirectional solidification, it is convenient to consider that liquid crosses the immobile interface with an absolute constant velocity v, while a solid-liquid fractionation coefficient K is applied to the fractionation of element i. Let us assume that the interface is at x=0, the medium being solid for x<0. Liquid fills the half-space 0[Pg.442]

Elimination of dll between this equation and Eq. (33) above gives the damping group requirement for the condition of a totally immobile interface ... 

Separations in two-phase systems with one immobilized interface(s) are much newer. The first paper on membrane-based solvent extraction (MBSE) published Kim [4] in 1984. However, the inventions of new methods of contacting two and three liquid phases and new types of liquid membranes have led to a significant progress in the last forty years. Separations in systems with immobilized interfaces have begun to be employed in industry. New separation processes in two- and three-phase systems with one or two immobilized L/L interfaces realized with the help of microporous hydrophobic wall(s) (support) are alternatives to classical L/L extraction and are schematically shown in Figure 23.1. Membrane-based solvent extraction (MBSE) in a two-phase system with one immobilized interface feed/solvent at the mouth of microspores of hydrophobic support is depicted in Figure 23.1a and will be discussed... 

PT through SLM 3 One immobilizing wall for two immobilized interfaces - Volume ratio of phases can be varied without limitations - Very small volume of membrane phase - Limited stability of SLM... 

Schlosser, . (2000) Membrane based processes with immobilized interface, in Integration of Membrane Processes into Bioconversions (eds K. Bako, L. Gubicza and M. Mulder), Kluwer Academic, p. 55. 

For "thick films" where disjoining pressure effects are negligible, the constant B in Equation 6 ranges from 1.337 for the perfectly mobile interface case to 2.123 for the immobile interface case. 

For the case of immobile interfaces and long bubbles (L- 0(2R) or greater), the principal contribution to the pressure drop is the viscous drag in the uniform film region. The pressure drop relation is then ... 

In comparison. Equation 17 indicates that the total contact length Lf of the films is the important parameter for immobile interface systems. Foaming systems of this type would show no dependence on foam texture (bubble size and bubble size distribution), but would exhibit very large apparent viscosities in porous media if the bubble trains were sufficiently long. 

This expression is valid for a partially mobile interface analogous expressions hold for fully mobile and immobile interfaces (Minale et al. 1997). Here, he is the critical thickness of the hquid gap between droplets at which coalescence occurs. Theoretically, one expects he (AHa/STrr), where Ah is the Hamaker constant (Chesters 1991). A value he = 0.2pm gives the solid line in Fig. 9-10, this value of he is an order of magnitude larger than predicted, no doubt because he is used as a fitting parameter to accomodate rough approximations used in the theory. 

Sirkar KK, Immobilized-interface solute transfer process, US Patent 4,997,569, 1991. 

Kiani A, Bhave RR, Sirkar KK, Further studies on solvent extraction with immobilized interfaces in a microporous hydrophobic membrane. J. Membr. Sci. 1986 26, 79. 

The final thickness, hp may coincide with the critical thickness of film rupture. Equation 5.273 is derived for tangentially immobile interfaces from Equation 5.259 at a fixed driving force (no disjoining pressure). 

There are various cases of particle-interface interactions, which require separate theoretical treatment. The simpler case is the hydrodynamic interaction of a solid particle with a solid interface. Other cases are the interactions of fluid particles (of tangentially mobile or immobile interfaces) with a solid surface in these cases, the hydrodynamic interaction is accompanied by deformation of the particle. On the other hand, the colloidal particles (both solid and fluid) may hydrodynamically interact with a fluid interface, which thereby undergoes a deformation. In the case of fluid interfaces, the effects of surfactant adsorption, surface diffusivity, and viscosity affect the hydrodynamic interactions. A special class of problems concerns particles attached to an interface, which are moving throughout the interface. Another class of problems is related to the case when colloidal particles are confined in a restricted space within a narrow cylindrical channel or between two parallel interfaces (solid and/or fluid) in the latter case, the particles interact simultaneously with both film surfaces. 

Osseointegration A term developed by RI. Branemark and his colleagues indicating the ability of host bone tissues to form a functional, mechanically immobile interface with the implant. Originally described for titanium only, several other materials are capable of forming this interface, which presumes a lack of connective tissue (foreign body) layer. 

As usual, the superscript (e) denotes that the respective quantity should be estimated for the equilibrium state the dimensionless parameter b accounts for the effect of bulk diffusion, whereas has a dimension of length and takes into account the effect of surface diffusion. In the limiting case of very large Gibbs elasticity Eq (tangentially immobile interface) the parameter d tends to zero and then Eq. (52) yields f —should be expected (121,135,136). 

The function accounts for the effect of the surface mobility. For large interfacial elasticity one has d —>0, see Eq. (53) then T —>1 and Eq. (61) acquires a simpler form, corresponding to drops of tangentially immobile interfaces. In the other limit, small interfacial elasticity, one has d 1 and in such a case T oc 1/ln J, i.e., T decreases with the increase in d, that is, with the decrease in Eq. A numerical solution to this problem is reported in Ref 24. The effect of the interfacial viscosity on the transitional distance, which is neglected in Eq. (61), is examined in Ref 141. It is established therein that the critical distance, h, can be with in about 10% smaller than hp... 

Equation (82) shows that the disjoining pressure significantly influences the transitional thickness The effect of surface mobility is characterized by the parameter d, see Eq. (53) in particular, d = 0 for tangentially immobile interfaces. Equation (82) is valid for H < 2o/u, i.e., when the film thins and ruptures before reaehing its equilibrium thickness, eorresponding to H = 2o/a [cf Eqs (42), (43), and (59)]. 

In the speeial ease of tangentially immobile interfaces and large film (negligible effeet of die transition zone) one has Oy(h) = 1, and the integration in Eq. (84) ean be earried out... 

Note, that the problem (46) and (47) describes the case of tangentially immobile interfaces. One proves that the mobility of interfaces changes only slightly the values of her - it affects the value of the critical wave number, ker. 

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