Moving boundary systems

Surprisingly enough, it is possible for the steady-state assumption in a moving-boundary system to be essentially exact. This occurs in the deposition of a filter cake under the influence of a constant pressure difference. An interesting example is given by Brenner (B13), in a study of the unconfined growth of a filter cake on a circular cloth-covered aperture in a plane wall. It is assumed that the velocity vector of the filtrate within the cake is everywhere and at all times proportional to the pressure gradient (Darcy s law). 

The Ornstein-Davis system was developed for separation of serum proteins, and unfortunately there is no "universal" system suitable for all separations. Computational treatments have been developed to establish the constituents of moving boundary systems with the desired properties. The most powerful of these is that of Jovin (18) which has generated in excess of 4,000 buffer systems the so-caTTed "extensive buffer system output". A selection of 19 of these systems has been published to aid the investigator to identify the most suitable system for his particular separation problem (8, 1 9). 

It was not until the work of Tiselius in the 1930s that the potential of electrophoresis as a biochemical tool was realized. Tiselius developed the use of electrophoresis for separating proteins in suspension on the basis of their charge. He optimized the geometry and temperature of the system in an apparatus known as the Tiselius moving boundary system. Detection of the separated components was carried out by detecting concentration changes in optical refraction. 

These two processes provide examples of the moving boundary problem in diffusing systems in which a solid solution precedes the formation of a compound. The diickness of the separate phase of the product, carbide or... 

Zone electrophoresis is defined as the differential migration of a molecule having a net charge through a medium under the influence of an electric field (1). This technique was first used in the 1930s, when it was discovered that moving boundary electrophoresis yielded incomplete separations of analytes (2). The separations were incomplete due to Joule heating within the system, which caused convection which was detrimental to the separation. 

A feature of the phase diagram in Fig. 8.12 is that the liquid-vapor boundary comes to an end at point C. To see what happens at that point, suppose that a vessel like the one shown in Fig. 8.13 contains liquid water and water vapor at 25°C and 24 Torr (the vapor pressure of water at 25°C). The two phases are in equilibrium, and the system lies at point A on the liquid-vapor curve in Fig. 8.12. Now let s raise the temperature, which moves the system from left to right along the phase boundary. At 100.°C, the vapor pressure is 760. Torr and, at 200.°C, it has reached 11.7 kTorr (15.4 atm, point B). The liquid and vapor are still in dynamic equilibrium, but now the vapor is very dense because it is at such a high pressure. 

The primary difference between the two equations is the unsteady term in equation (E2.2.2) and the convective term in equation (E5.3.2). Now, let s convert our coordinate system of Example 2.2 to a moving coordinate system, moving at the bulk velocity, U, which suddenly experiences a pulse in concentration as it moves downstream. This is likened to assuming that we are sitting in a boat, moving at a velocity U, with a concentration measuring device in the water. The measurements would be changing with time, as we moved downstream with the flow, and the pulse in concentration would occur at x = 0. We can therefore convert our variables and boundary conditions as follows ... 

Proceeding systematically, diffusion controlled a-fi transformations of binary A-B systems should be discussed next when a and / are phases with extended ranges of homogeneity. Again, defect relaxations at the moving boundary and in the adjacent bulk phases are essential for their understanding (see, for example, [F. J. J. van Loo (1990)]). The morphological aspects of this reaction type are dealt within the next chapter. 

Let us briefly outline the main concepts of a (linear) stability analysis and refer to the situation illustrated in Figure 11-7. If we artificially keep the moving boundary morphologically stable, we can immediately calculate the steady state vacancy flux, /v, across the crystal. The boundary velocity relative to the laboratory reference system (crystal lattice) is... 

In the context of the morphological evolution of non-equilibrium systems, let us then ask whether the reaction path, when constructed for a system with stable interfaces, can tell us something about the instability of moving boundaries. For this we... 

Atoms are diffusing into the boundary laterally from its edges and can diffuse out through its front face into the forward grain. At the same time, atoms will be deposited in the backward grain in the wake of the boundary. In the quasi-steady state in a coordinate system fixed to the moving boundary, the diffusion flux in the forward grain is J = — DXL(dc/dx) — vc and the diffusion equation is... 

The moving boundary has now been eliminated and the problem reduced to that of solving the heat conduction equation for a nonmelting solid with internal heat absorption, corresponding to the last term of Eq. (284). This system is then differenced with respect to the space variable and solved on a passive analog system. 

For the technique to be applicable, a solution of (37) moving the system from the CA to Sff must exist, and one has to be able to identify the boundary conditions for this solution on the CA. 

This hypothesis can be elaborated further using a statistical analysis of the trajectories arriving a small tube around S3 with the noise intensity reduced by a few orders of magnitude up to D = 1.5 x 10 6, see Fig. 17 [173]. The analysis reveals that the energetically favorable way to move the system from the CA to the stable limit cycle starts at the saddle cycle of period 5 (S5) embedded in the CA, passes through saddle cycle S3 and finishes at the saddle cycle SI at the boundary of the basin of attraction of the CA. Subsequent motion of the system towards the stable limit cycle does not require external action. 

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