Fluid and interface dynamics

The thermal separation of material makes use chiefly of four types of column  

Phase flow diagrams for various column types (Stage) 

In packed columns a non-uniform distribution (maldistribution) is to be expected both in the vapour and in the liquid phase. In the liquid it occurs for these reasons 

Wall flow dexjends on both separating length and column diameter. With a ratio of djuldf ss 20 a fixed wall flow of 10 to 20% is established, which in turn is influenced by the ratio of separating length to column diameter. 

The relative effect of maldistribution is the greater the more separating stages a column has. A maldistribution of, say, 10% reduces the plate number of a column with 100 plates to 30 while a column of 10 plates is only reduced to 9. Since also in industry the trend is toward more and more efficient columns the question of the maldistribution of the fluid phases is gaining importance. 

---

The fundamental scheme of arrangement as applied in the first edition has been retained. Section 5.1.3 has been extended to cover pilot plant distillation. Section 4.2 now deals with fluid and interface dynamics. Chapter 8 could be drastically shortened as there are a variety of components of distillation apparatus and the pertaining measuring and control devices commercially available. The nomograms, which were presented separately, have been inserted in the text. The references for the various chapters have been rearranged and important new items added to them. A great number of review articles serve to provide comprehensive lists of references for a longer period. 

We have noticed that most current LBM applications to microfluidics utilize LBM as a differential equation solver, and the true merit of this method - a good representation of the underlying microscopic interactions - has not been well exploited. Solid-fluid interfacial phenomena in microsystems could be particularly suitable for LBM, since it couples the fluid and interface dynamics in a natural way. Future directions for research may include utilizing nonuniform or unstructured lattice meshes for complex microstmctures (e.g., surface roughness), combining LBM with molecular dynamics and CFD (hybrid algorithms), and applying LBM to bio-microfluidic systems. 

In recent years, there has been great interest in developing physically inspired computational models based on the idea that the dynamics of the motion of fluid and interfaces can be represented in terms of the collective behavior of interactions of quasi-particle populations at scales smaller than macroscopic, but larger than molecular scales. These models fall in the class of mesoscopic methods - the LBM [6, 42, 45] being one. The LBM is generally based on minimal discrete kinetic models whose emergent behavior, under appropriate constraints, corresponds to the... 

The extension of generic CA systems to two dimensions is significant for two reasons first, the extension brings with it the appearance of many new phenomena involving behaviors of the boundaries of, and interfaces between, two-dimensional patterns that have no simple analogs in one-dimension. Secondly, two-dimensional dynamics permits easier (sometimes direct) comparison to real physical systems. As we shall see in later sections, models for dendritic crystal growth, chemical reaction-diffusion systems and a direct simulation of turbulent fluid flow patterns are in fact specific instances of 2D CA rules and lattices. 

Miller, R., Fainerman, V.B., Makievski, A.V., Kragel, J., Grigoriev, D.O., Kazakov, V.N., Sinyachenko, O.V. (2000a). Dynamics of protein and mixed protein + surfactant adsorption layers at the water-fluid interface. Advances in Colloid and Interface Science, 86, 39-82. 

Tracqui, P, Perault-Staub, A. M., Milhaud, G., and Staub, J. F. (1987). Theoretical study of a two-dimensional autocatalytic model for calcium dynamics at the extracellular fluid-bone interface. Bull. Math. Biol., 49, 597-613. 

Figure 12. Lagrangian path lines at various stages of a Rayleigh-Taylor collapse for the case of two inviscid, incompressible fluids having a density ratio of 2 1. A free surface is present above the dense fluid and the interface between the fluids is indicated for each stage. The simulation shows how later evolution of the fluid flow is dominated by the strength and dynamics of the vortex pair created during the...

In recent years a great many studies have reported on the dynamic systems where a drop of liquid is placed on a smooth solid surface. ° The system liquid drop-solid is a very important system in everyday life, for example, rain drops on tree leaves or other surfaces. It is also significant in all kinds of systems where a spray of fluid is involved, such as in sprays or combustion engines. The dynamics of liquid drop evaporation rate is of much interest in many phenomena. The liquid-solid interface can be considered as follows. Real solid surfaces are, of course, made up of molecules not essentially different in their nature from the molecules of the fluid. The interaction between a molecule of the fluid and a molecule of the boundary wall can be regarded as follows. The molecules in the solid state are not as mobile as those of the fluid. It is therefore permissible for most purposes to regard the molecules in the solid state as stationary. However, complexity arises in those liquid-solid systems where a layer of fluid might be adsorbed on the solid surface, such as in the case of water-glass. 

The oil-water interface is one of the most important systems. The liqnid-liquid interface constitutes a phase separation where two different molecules meet. We can directly measure the magnitude of the surface tension, with rather high precision. It wonld thns seem that much useful information can be obtained if we could measure a dynamic parameter of the interface, such as the freezing phenomenon. It is widely known that liqnids can be cooled below their freezing temperature without solidification (snpercooled fluid) and that they can be heated above their boiling temperature without vaporization (snperheated liquid). 

In summary, we have so far seen that there are two types of boundary conditions that apply at any solid surface or fluid interface the kinematic condition, (2-117), deriving from mass conservation and the dynamic boundary condition, normally in the form of (2-122), but sometimes also in the form of a Navier-slip condition, (2-124) or (2-125). When the boundary surface is a solid wall, then u is known and the conditions (2-117) and (2-122) provide a sufficient number of boundary conditions, along with conditions at other boundaries, to completely determine a solution to the equations of motion and continuity when the fluid can be treated as Newtonian. 

Here, ptot and ptot represent the actual total pressure in the exterior and interior fluids, including both dynamic and hydrostatic contributions. In crossing an interface, we see that the normal component of the total stress undergoes a jump equal to y(V n). In the limiting case of no motion in the fluids, this implies that... 

Orientational structure at a liquid vapor-interface of diatomic interaction site fluids has been studied extensively by Gubbins and Thompson using both thermodynamic perturbation theory and molecular dynamics simulation, and by Tarazona and Navascues using perturbation theory. Chacon et al. have applied density-functional theories to these systems. The theoretical methodology and results are reviewed in a comprehensive article by Gubbins, to which the reader is directed for more complete details. 

The equilibrium and dynamic behaviour of mixed monolayers of soluble and insoluble amphiphiles at fluid/liquid interfaces plays an important role in various technological and biological processes, which was studied in numerous publications [139-157], However, even for very simple systems, say, gaseous mixed monolayers, the thermodynamic analysis is not trivial. For more complicated systems (the formation of two-dimensional domains) such an analysis is very cumbersome due to mathematical difficulties. 

The total force transmitted across the fluid-solid interface requires fluid pressure, not dynamic pressure. Since dynamic pressure is a combination of gravitational potential energy per unit volume and actual fluid pressure, it is rather simple to use equation (8-147) and calculate fluid pressure. The rectangular Cartesian coordinate that increases in the direction opposite to gravity is z = r cos 0. Hence,... 

Interphase momentum transfer is the focus of this section. Macroscopic correlations are based on dynamic forces due to momentum flux that act across the fluid-solid interface, similar to terms of type 2, 3, and 4 in the equation of motion. Gravity enters into this discussion via the hydrostatic contribution to fluid pressure, because volumetric body forces are not operative across an interface. The outward-directed unit normal vector from the solid surface into the fluid is n. As discussed earlier, forces due to total momentum flux, transmitted in the —n direction from the flnid to the solid across the interface at r = / , are (i.e., see equation 8-20) ... 

Generalized Interpretation of f vs. Re. When the characteristic velocity and the Reynolds number increase, the fi iction factor for flow around solid spheres decreases if Re < 500, and / remains approximately constant at 0.44 if Re > 500. However, the dynamic force transmitted across the fluid-solid interface increases at higher Reynolds numbers in all flow regimes. The generalized correlations are... 

---