Theories for Solid-Fluid Interfaces

While mass transfer most commonly takes place across fluid-fluid interfaces, many well-developed theories are based on fluid-solid interfaces. There are two reasons for this. First, fluid-solid interfaces are more easily specified they normally do not wave or ripple but sit right where they were. Second, fluid-solid interfaces are common for heat transfer, so results calculated and verified for heat transfer can be confidently converted to mass transfer. 

In this section we develop two such theories. The first is for mass transfer out of a solution in laminar flow in a short tube. This Graetz-Nusselt problem finds some application for blood oxygenators and artificial kidneys. The second example is the 

This theory calculates how the mass transfer coefficient varies with the fluid s flow and the solute s diffusion. In other words, it finds the mass transfer as a function of quantities like Reynolds and Schmidt numbers. 

The problem, shown schematically in Fig. 9.4-1, again assumes a dilute solute, so that the velocity profile is parabolic, as expected for laminar flow. The detailed solution depends on the exact boundary conditions involved. The most important case assumes fixed solute concentration at the wall of a short tube. In this case, a mass balance on the solute gives 

Sparlngly soluble walls dissolve into fluid 

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To find these connections, we will develop theories of mass transfer. These theories are rarely predictive, but they clarify the chemistry and physics which are involved. They are less predictive because they are most often for fluid-fluid interfaces whose geometry is not well known. They are much more successful for solid fluid interfaces, which are much better defined. Unfortunately, fluid fluid interfaces are much more important for mass transfer than fluid-solid interfaces are. 

The double integral represents the nonzero terms of the dissipation rate tensor as adapted by Middleman [61] and Bernhardt and McKelvey for adiabatic extrusion [62]. The nontensorial approach was adopted by Tadmor and Klein in their classical text on extrusion [9]. In essence these are the nonzero terms of the dissipation rate tensor when it is applied to the boundary of the fluid at the solid-fluid interface. In the following development this historic analysis was adopted for energy dissipation for a rotating screw. In this case the velocities Ui are evaluated at the screw surface s and calculated in relation to screw rotation theory. The work in the flight clearance was previously described in the literature [9]. The shear... 

The chemical reaction between a solid and a reactive fluid is of interest in many areas of chemical engineering. The kinetics of the phenomenon is dependent on two factors, namely, the diffusion rate of the reactants toward the solid/fluid interface and the heterogenous reaction rate at the interface. Reactions can also take place within particles, which have accessible porosity. The behavior will depend on the relative importance of the reaction outside and inside the particle. Fractal analysis has been applied to several cases of dissolution and etching in such natural occurring caves, petroleum reservoirs, corrosion, and fractures. In these cases fractal theory has found usefulness for quantifying the shape (line or surface) with only a few parameters the fractal dimension and the cutoffs. There have been some attempts to use a fractal dimension for reactivity as a global parameter. Finally, fractal concepts have been used to aid in the interpretation of experimental results, if patterns quantitatively similar to DLA are obtained. 

A major drawback of the gradient theory is that the number of components that can be included in the mixture is limited. In the present sturty a maximum number of two components has been used, while in real tystems more than twenty components may be present. Therefore, fi om this theory the tools as developed merely can serve as a tool for the study of model systems, and to describe several qrecial phenomena that can occur at fluid and solid-fluid interfaces. 

We can extend the two-scale HA theory to a multiscale porous medium, which is schematically shown in Fig. 8.21 where 2/ is the fluid phase, 2 is the solid phase, sf is the mixture phase of fluid and solid (i.e., a mixture of stacks of beidellite and fluid) in the meso-domain, F / is the solid/fluid interface, and F/ is the periodic boundary for the fluid phase. 

The purpose of this chapter is to introduce the effect of surfaces and interfaces on the thermodynamics of materials. While interface is a general term used for solid-solid, solid-liquid, liquid-liquid, solid-gas and liquid-gas boundaries, surface is the term normally used for the two latter types of phase boundary. The thermodynamic theory of interfaces between isotropic phases were first formulated by Gibbs [1], The treatment of such systems is based on the definition of an isotropic surface tension, cr, which is an excess surface stress per unit surface area. The Gibbs surface model for fluid surfaces is presented in Section 6.1 along with the derivation of the equilibrium conditions for curved interfaces, the Laplace equation. 

A distinction between solid/fluid and solid/solid boundaries is irrelevant from the point of view of transport theory. Solid/fluid boundaries in reacting systems are, for example, (A,B)/A, B, X (aq) or (A,B)/X2(g). More important is the distinction according to the number of components. In isothermal binary systems, the boundary is invariant if local equilibrium prevails. In higher than binary systems, the state of the a/fi interface is, in principle, variable and will be determined by the reaction kinetics, including the diffusion in the adjacent bulk phases. 

Soft biological structures exhibit finite strains and nonlinear anisotropic material response. The hydrated tissue can be viewed as a fluid-saturated porous medium or a continuum mixture of incompressible solid (s), mobile incompressible fluid (f), and three (or an arbitrary number) mobile charged species a, (3 = p,m, b). A mixed Electro-Mechano-Chemical-Porous-Media-Transport or EMCPMT theory (previously denoted as the LMPHETS theory) is presented with (a) primary fields (continuous at material interfaces) displacements, Ui and generalized potentials, ifi ( , r/ = /, e, to, b) and (b) secondary fields (discontinuous) pore fluid pressure, pf electrical potential, /7e and species concentration (molarity), ca = dna/dVf or apparent concentration, ca = nca and c = Jnca = dna/dVo. The porosity, n = 1 — J-1(l — no) and no = no(Xi) = dVj/dVo for a fluid-saturated solid. Fixed charge density (FCD) in the solid is defined as cF = dnF/dV , cF = ncF, and cF = cF (Xf = JncF = dnF/d o. 

The weak point in the method is that is a sensitive, though esoteric quantity. We discussed its value for fluid interfaces in sec. 2.8, where we concluded that the order of magnitude is well established but that a precise vedue is difficult to assign. For SG interfaces we did not develop a theory in those systems represents the depth over which the atoms near the surface assume mutual positions differing from those in the bulk of the solid. However, we shall show below that for contact angles the choice of is less critical than it is for interfacial tensions. 

The mechanism of dissolution was proposed by Nernst (1904) using a film-model theory. Under the influence of non-reactive chemical forces, a solid particle immersed in a liquid experiences two consecutive processes. The first of these is solvation of the solid at the solid-liquid interface, which causes the formation of a thin stagnant layer of saturated solution around the particle. The second step in the dissolution process consists of diffusion of dissolved molecules from this boundary layer into the bulk fluid. In principle, one may control the dissolution through manipulation of the saturated solution at the surface. For example, one might generate a thin layer of saturated solution at the solid surface by a surface reaction with a high energy barrier (Mooney et al., 1981), but this application is not commonly employed in pharmaceutical applications. 

Apart from the trivial inclusion of the gravitational body-force terms in (6-2) and (6-3), the governing equations, and the analysis leading to them, are identical to the governing equations for the lubrication theory of the previous chapter. The primary difference in the formulation is in the boundary conditions, and the related changes in the physics of the thin-film flows, that arise because the upper surface is now a fluid interface rather than solid surface of known shape. The boundary conditions at the lower bounding surface are ... 

ABSTRACT In the present paper a multiphase model including a hypoplastic formulation of the solid phase is presented and its application to earthquake engineering problems discussed. The macroscopic soil model, which is based on the theory of porous media, comprises three distinct phases namely, solid, fluid and gas phase. For each of these the compressibility of the respective medium is taken into account in the mathematical formulation of the model. The solid phase is modelled using the hypoplastic constitutive equation including intergranular strain to allow for a realistic description of material behaviour of cohesionless soils even under cyclic loading. The model was implemented into the finite element package ANSYS via the user interface and also allows the simulation of soil-structure interaction problems. 

Bennema et al. [47] introduce a modeling concept for the prediction of crystal morphology grown in the presence of additives, which is based on the knowledge of the internal crystal structure. The method employs the theory of the roughening temperature [48]. Preliminary investigations on the solid-liquid interface revealed an intensive structuring of molecules in the interfacial fluid phase and none in the bulk fluid phase [49]. 

The liquid wets the solid completely (0 -> 0) when Agps > Aipi. A similar computation can be carried out in the diffuse interface theory, but is somewhat more problematic, since one has to define a boundary condition for the fluid density on the solid surface. 

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