Diffusion moving interface problems

Normally, it is not possible to obtain analytical solutions for this transport problem and so we cannot a priori calculate the reaction path. Kirkaldy [J. S. Kirkaldy, D. J. Young (1985)] did pioneering work on metal systems, based on investigations by C. Wagner and the later work of Mullins and Sekerka. They used the diffusion path concept to formulate a number of stability rules. These rules can explain the facts and are predictive within certain limits if applied properly. One of Kirkaldy s results is this. The moving interface in a ternary system is morphologically stable if... 

Pore Mouth (or Shell Progressive) Poisoning This mechanism occurs when the poisoning of a pore surface begins at the mouth of the pore and moves gradmuly inward. This is a moving boundary problem, and the pseudo-steady-state assumption is made that the boundary moves slowly compared with diffusion of poison and reactants and reaction on the active surface. P is the fraction of the pore that is deactivated. The poison diffuses through the dead zone and deposits at the interface between the dead and active zones. The reactants diffuse across the dead zone without reaction, followed by diffusion-reaction in the active zone. 

Eulerian methods perform weU for a variety of moving boundary problems. However, in these problems, particularly when surface forces are to be included in the flow calculations, the interface is diffused and occupies a few grid cells in practical calculations. This is undesirable in many problems both from an accuracy and physical realizability/modeling standpoint. 

Transport of solute from a fluid phase to a spherical or nearly spherical shape is important in a vari of separation operations such as liquid-liquid extraction, crystallization from solution, and ion exchange. The situation depicted in Fig. 2.3-12 assumes that there is no forced or natural convection in the fluid about the particle so that transport is governed entirely by molecular diffusion. A steady-state solution can be obtained for the case of a sphere of fixed radius with a constant concentration at the interface as well as in the bulk fluid. Such a model will be useful for crystallization from vaqxtrs and dilute solutions (slow-moving boundary) or for ion exchange with rapid irreversible reaction. Bankoff has reviewed moving-boundary problems and Chapters 11 and 12 deal with adsorption and ion exchange. 

The second problem considered in this section is illustrated schematically in Fig. 3.3-3. In this problem, a volatile liquid solute evaporates into a long gas-filled capillary. The solvent gas in the capillary initially contains no solute. As solute evaporates, the interface between the vapor and the liquid solute drops. However, the gas is essentially insoluble in the liquid. We want to calculate the solute s evaporation rate, including the effect of diffusion-induced convection and the effect of the moving interface (Arnold, 1944). 

For a diffusion couple, the definition of Amid requires some thinking because the mid-concentration of the whole diffusion couple is right at the interface, which does not move with time. This is because for a diffusion couple every side is diffusing to the other side. On the other hand, if a diffusion couple is viewed as two half-space diffusion problems with the interface concentration viewed as the fixed surface concentration, then. Amid equals 0.95387(Df), the same as the half-space diffusion problem. 

This problem will be side-stepped for a moment, and a simpler one tackled. Consider two steps that are parallel to each other, the type of steps considered in the analysis of the constant-current transient. As ions transfer across the electrified interface and the adions thus formed surface diffuse and become incorporated in the steps, there is an advance of the steps toward each other (Fig. 7.156). Eventually the two steps approach each other, some closely, so that all one is left with is a one-atom-wide and one-atom-deep chasm. The moment this is filled in, the two steps disappear. The collision of the two steps moving toward each other has resulted in their mutual annihilation. 

The problem is also more complex when heterogeneous catalysed reactions are considered. With porous catalyst pellets, reaction occurs at gas- or liquid-solid interfaces at the outer or inner sphere. When the reactants diffuse only slowly from the bulk phase to the exterior surface of the catalyst, gas or liquid film resistance must be taken into account. Pore diffusion resistance may be involved when the reactants move through the pores into the pellet. 

Solution. Yes. When D varies with concentration we have shown in Section 4.2.2 that the diffusion equation can be scaled (transformed) from zt-space to 77-space by using the variable rj = x/ /4Di (see Eq. 4.19). Also, under diffusion-limited conditions where fixed boundary conditions apply at the interfaces, the boundary conditions can also be transformed to 77-space, as we have also seen. Therefore, when D varies with concentration, the entire layer-growth boundary-value problem can be transformed into 77-space. Since the fixed boundary conditions at the interfaces require constant values of 77 at the interfaces, they will move parabolically. 

The various topics are generally introduced in order of increasing complexity. The text starts with diffusion, a description of the elementary manner in which atoms and molecules move around in solids and liquids. Next, the progressively more complex problems of describing the motion of dislocations and interfaces are addressed. Finally, treatments of still more complex kinetic phenomena—such as morphological evolution and phase transformations—are given, based to a large extent on topics treated in the earlier parts of the text. 

In membrane transport, one-dimensional models are usually used. If the permeates move independently of one another and with the ideal interface permeability, the simple diffusion, described by Fick s law, across the membrane is given by the boundary value problem... 

For completeness we briefly outline in this subsection a generic analytical method of solution for a class of problems in unsteady-state linear diffusion, which involve two phases or regions separated by a moving plane interface. 

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. 

The results of the preceding section allow us now to move on to describe the surfactant transport from the depth of the bulk phase to the interface or in the opposite direction. If any adsorption barriers are absent, this process determines the adsorption and desorption rates. The main step in the solution of this problem consists in the formulation of the surfactant diffusion equations for micellar solutions. The problem of surfactant diffusion to the interface was considered and solved for the first time by Lucassen for small perturbations [94]. He used the simplified model (5.146) where micelles were assumed to be monodisperse and the micellisation process was regarded as consisting of one step. Later Miller solved numerically the problem of adsorption on a fresh liquid surface using the same assumptions [146], Joos and van Hunsel applied also the same model to the interpretation of dynamic surface tension of... 

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