Phase boundary 32 discontinuity

The theoretical treatment which has been developed in Sections 10.2-10.4 relates to mass transfer within a single phase in which no discontinuities exist. In many important applications of mass transfer, however, material is transferred across a phase boundary. Thus, in distillation a vapour and liquid are brought into contact in the fractionating column and the more volatile material is transferred from the liquid to the vapour while the less volatile constituent is transferred in the opposite direction this is an example of equimolecular counterdiffusion. In gas absorption, the soluble gas diffuses to the surface, dissolves in the liquid, and then passes into the bulk of the liquid, and the carrier gas is not transferred. In both of these examples, one phase is a liquid and the other a gas. In liquid -liquid extraction however, a solute is transferred from one liquid solvent to another across a phase boundary, and in the dissolution of a crystal the solute is transferred from a solid to a liquid. 

When both phases are in turbulent flow, or when one phase is discontinuous as in bubble flow, it is not presently possible to formulate the proper boundary conditions and to solve the equations of motion. Therefore, numerous experimental studies have been conducted where the holdups and/or the pressure drop were measured and then correlated as a function of the operating conditions and system parameters. One of the most widely used correlations is that of Lockhart and Martinelli (L12), who assumed that the pressure drop in each phase could be calculated from the equations... 

M molar mass), where I and III are the tricritical or -regions. Here, the chain molecules exhibit an unperturbed random coil confirmation. In contrast, I and II are the critical or good solvent regimes, which are characterized by structural fluctuations in direction of an expanded coil conformation. According to the underlying concept of critical phenomena, the phase boundaries have to be considered as a continuous crossover and not as discontinuous transitions. 

In the case of a first-order transition, a discontinuity at the phase boundary is seen in Gf, a first derivative of free energy ... 

We have discussed transport in the bulk and transport across interfaces and phase boundaries (i.e., discontinuities). In this section, we shall mainly treat an intermediate transport situation, the so-called junction. At junctions, the atomistic processes that occur under a load have much in common with interface processes, such as the relaxation behavior of the SE s which are swept across them. 

Figure 23.1 is a phase diagram of a system that exhibits precipitation. If cooled along the path indicated, the a phase will become supersaturated with respect to the (3 phase when it crosses the phase boundary, and if there is no intervening spinodal, the /3 phase will then precipitate discontinuously in the a phase (matrix phase) as the system attempts to reach equilibrium. 

T = 7 . above which the liquid and gas phase are no longer distinguishable. Since the liquid can he continuously converted into Ihe gas phase without discontinuous change of properties by any path in the P — T diagram passing above the critical point, there is no definite boundary between liquid and gas. Two liquids ol similar molecules are usually. soluble in all proportions, but very low solubility is sufficiently common to permit the demonstration of as many as seven separate liquid phases in equilibrium at one temperature and pressure (mercury, gallium, phosphorus. perHuoro-kerosene, water, aniline, and heptane at 50 C. I atmosphere). 

Figures 7.11a,b are arbitrary examples of the depths of hydrate phase stability in permafrost and in oceans, respectively. In each figure the dashed lines represent the geothermal gradients as a function of depth. The slopes of the dashed lines are discontinuous both at the base of the permafrost and the water-sediment interface, where changes in thermal conductivity cause new thermal gradients. The solid lines were drawn from the methane hydrate P-T phase equilibrium data, with the pressure converted to depth assuming hydrostatic conditions in both the water and sediment. In each diagram the intersections of the solid (phase boundary) and dashed (geothermal gradient) lines provide the lower depth boundary of the hydrate stability fields.

In a system that contains one or more phases, Equation (14.26) is applicable to each phase. The pressure of the system must be a continuous function of r even in a heterogenous system, but the derivative (8P/8r)T is discontinuous at a phase boundary. Then the pressure must be the same on either side of a phase boundary (neglecting surface effects) when the boundary is wholly within a single region, no matter how small the thickness of the region. The derivative (8P/8r)T however, will have a different value on either side of the boundary. 

Anywhere a chemical potential increment or gradient exists, an elementary separation step can occur. Anywhere random flow currents exist, separation is dissipated. Thus random flow currents are parasitic in regions where incremental chemical potential is used for separation. These currents should thus be eliminated, insofar as possible, in regions where electrical, sedimentation, and other continuous (c) fields are generating separations. Likewise, they should not be allowed to transport matter over discontinuous (d) separative interfaces such as phase boundaries or membrane surfaces. However, they are nonparasitic in bulk phases (removed from the separative interface) where only diffusion occurs. Here, in fact, they aid diffusion and speed the approach to equilibrium. This positive role is recognized in the following category of flow. 

Here, n is a unit normal vector at the interstitial phase boundary Sp, if any, across which the kinematic viscosity v(r) is possibly discontinuous, whereas the generic symbol h denotes either P or j. Equation (8.15) expresses continuity of the pertinent fields across the unit cell faces dr0. In Eq. (8.15), m is to be chosen such that Rm, = Rm - I is a basic lattice vector [cf. Eq. (7.1)]. 

Discussion Note that the concentration of water on a molar basis is 100 percent just beneath the air-water interface and 1.85 percent just above it, even though the air is assumed to be saturated (so this is the highest value at 13°C). Therefore, huge discontinuities can occur in the concentrations of a species across phase boundaries. 

The Two-film theory enables the difficultly accessible chemical potential difference A/z to be replaced by the concentrations of the gas in the gas phase, liquid phase and in the phase boundary c, only then the concentration change from phase to phase is discontinuous but makes a jump, as demonstrated on the left of Fig. 4.1. 

A second similar consequence of the continuum hypothesis is an uncertainty in the boundary conditions to be used in conjunction with the resulting equations for motion and heat transfer. With the continuum hypothesis adopted, the conservation principles of classical physics, listed earlier, will be shown to provide a set of so-called field equations for molecular average variables such as the continuum point velocity u. To solve these equations, however, the values of these variables or their derivatives must be specified at the boundaries of the fluid domain. These boundaries may be solid surfaces, the phase boundary between a liquid and a gas, or the phase boundary between two liquids. In any case, when viewed on the molecular scale, the boundaries are seen to be regions of rapid but continuous variation in fluid properties such as number density. Thus, in a molecular theory, boundary conditions would not be necessary. When viewed with the much coarser resolution of the macroscopic or continuum description, on the other hand, these local variations of density (and other molecular variables) can be distinguished only as discontinuities, and the continuum (or molecular average) variables such as u appear to vary smoothly on the scale L, right up to the boundary where some boundary condition is applied. 

This equation is valid in the entire flow field even if the material properties vary discontinuously across phase boundaries. In Eq. (1), p and p are density and viscosity, v is the velocity field, p is pressure, and /is the body force. The effects of the interfacial tension are accounted for by the last term in Eq. (1). In this term, 5 is two or three dimensional delta function, cr is surface tension coefficient, k is the curvature of two-... 

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