Boundary, moving

In the case of small ions, Hittorf transference cell measurements may be combined with conductivity data to give the mobility of the ion, that is, the velocity per unit potential gradient in solution, or its equivalent conductance. Alternatively, these may be measured more directly by the moving boundary method. 

There are a number of complications in the experimental measurement of the electrophoretic mobility of colloidal particles and its interpretation see Section V-6F. TTie experiment itself may involve a moving boundary type of apparatus, direct microscopic observation of the velocity of a particle in an applied field (the zeta-meter), or measurement of the conductivity of a colloidal suspension. 

Modelling of steady-state free surface flow corresponds to the solution of a boundary value problem while moving boundary tracking is, in general, viewed as an initial value problem. Therefore, classification of existing methods on the basis of their suitability for boundary value or initial value problems has also been advocated. 

Elliot C.M., Ockendon J.R. (1982) Weak and variational methods for moving boundary problems. Pitman, Research Notes Math. 59. 

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... 

A. Boesch, H. Miiller-Krumbhaar, O. Shochet. Phase field models for moving boundary problems Controlling metastability and anisotropy. Z Physik B 97 161, 1995. 

W. Shyy, H. S. Udaykumar, M. M. Rao, R. W. Smith. Computational Fluid Dynamics with Moving Boundaries in Series in Computational and Physical Processes in Mechanics and Thermal Sciences. Washington, DC Taylor Francis, 1995 W. Shyy. Computational Modeling for Fluid Flow and Interfacial Transport. Amsterdam Elsevier, 1994. 

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. 

Transport numbers are intended to measure the fraction of the total ionic current carried by an ion in an electrolyte as it migrates under the influence of an applied electric field. In essence, transport numbers are an indication of the relative ability of an ion to carry charge. The classical way to measure transport numbers is to pass a current between two electrodes contained in separate compartments of a two-compartment cell These two compartments are separated by a barrier that only allows the passage of ions. After a known amount of charge has passed, the composition and/or mass of the electrolytes in the two compartments are analyzed. Erom these data the fraction of the charge transported by the cation and the anion can be calculated. Transport numbers obtained by this method are measured with respect to an external reference point (i.e., the separator), and, therefore, are often referred to as external transport numbers. Two variations of the above method, the Moving Boundary method [66] and the Eiittorff method [66-69], have been used to measure cation (tR+) and anion (tx ) transport numbers in ionic liquids, and these data are listed in Table 3.6-7. 

Another approach to the problem of curved edges is based on a solution of Frank s equations in the case of moving boundaries by Mansfield [139], Figure 3.14 shows the ellipitical profile which would arise if the sides of a crystal sector move outwards with a constant velocity, h, which is of comparable size to the spreading rate, g. The magnitude of h is supposed to be determined by the growth rate of the adjoining dominant sector.10... 

Fig. 3.14. The profile of a lamellar growth face which may be obtained by using moving boundary conditions. The boundaries move with a constant velocity, h...

The analysis of fluid-solid reactions is easier when the particle geometry is independent of the extent of reaction. Table 11.6 lists some situations where this assumption is reasonable. However, even when the reaction geometry is fixed, moving boundary problems and sharp reaction fronts are the general rule for fluid-solid reactions. The next few examples explore this point. 

Douglas et al. [98] have measured protein (serum albumin, ovalbumin, and hemoglobin) mobilities over a range of pH values using a free-flow electrophoresis apparatus and a particle electrophoresis apparatus. They found good agreement between the two measurements however, they also found some differences between their measurements and those reported in the older literature. They attributed the differences to the use of moving-boundary electrophoresis methods in the early experimental work and to differences in... 

Chrambach, A, Unified View of Moving Boundary Electrophoresis Practical Implications, Journal of Chromatography 320, 1, 1985. 

For obtaining internal or external mobilities, the corresponding transport numbers are usually measured. There are several methods for determining transport numbers in molten salts that is, the Kleimn method (countercurrent electromigration method or column method), the Hittorf method (disk method), the zone electromigration method (layer method), the emf method, and the moving boundary method. These are described in a comprehensive review. ... 

The root surface at r is usually taken to be a fixed point. The effect of root thickening on uptake, where the root surface became a moving boundary has also been modeled (24). 

The transference or transport number of an ion can be determined by (i) the analytical method (ii) the moving boundary method and (iii) the emf method. The first two methods will be dealt with here, but the third will figure in a later section. 

The second method, namely the moving boundary method, is based on the direct observation of migration of ions under the influence of an applied potential - unlike Hittorf s method in which the changes in concentration at the electrodes are measured. 

The apparatus used to determine the transport number by the moving boundary method is shown in Figure 6.4. It consists of a long vertical tube of uniform cross-section which is fitted with two electrodes at the two ends. Let the electrolyte, the transport number of whose... 

Figure 6.4 Moving boundary experimental set-up for determining the transport number of E+ ions.

The solutions for moisture uptake presented in this section are based on the experimental condition of a pure water vapor atmosphere. In the next section a derivation of moisture uptake equations is based on both heat and mass transport that are characteristic of moisture uptake in air. The final section of this chapter presents the results of studies where heat transport is unimportant and mass transport dominates the process. Thus, we will have a collection of solutions covering models that are (1) heat transport limited, (2) mass transport limited, (3) heat and mass transport limited, and (4) mass transport limited with a moving boundary for the uptake of water by water-soluble substances. 

The mathematical solution to moving boundary problem involves setting up a pseudo-steady-state model. The pseudo-steady-state assumption is valid as long as the boundary moves ponderously slowly compared with the time required to reach steady state. Thus, we are assuming that the boundary between the salt solution and the solid salt moves slowly in the tablet compared to the diffusion... 

The boundary conditions are that (1) at the moving boundary ( ) the solution is saturated with salt with a corresponding concentration of water (Cs) and (2) at the disk/atmosphere surface the concentration of water is governed by the equilibrium vapor pressure in the chamber to give a water concentration of C0. 

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