Continuum droplet models

The vapor-layer model developed in Section IV.A.2 is based on the continuum assumption of the vapor flow. This assumption, however, needs to be modified by considering the kinetic slip at the boundary when the Knudsen number of the vapor is larger than 0.01 (Bird, 1976). With the assumption that the thickness of the vapor layer is much smaller than the radius of the droplet, the reduced continuity and momentum equations for incompressible vapor flows in the symmetrical coordinates ( ,2) are given as Eqs. (43) and (47). When the Knudsen number of the vapor flow is between 0.01 and 0.1, the flow is in the slip regime. In this regime, the flow can still be considered as a continuum at several mean free paths distance from the boundary, but an effective slip velocity needs to be used to describe the molecular interaction between the gas molecules and the boundary. Based on the simple kinetic analysis of vapor molecules near the interface (Harvie and Fletcher, 2001c), the boundary conditions of the vapor flow at the solid surface can be given by... 

Wiscombe and Welch, 1986). Cloud absorption is related to droplet size in a complex way (Stephens and Tsay, 1990) so that errors in droplet size measurements can alter the model-predicted absorption by a cloud. The treatment of absorption due to water vapor is another possibility. As discussed by Crisp (1997), the treatment of water vapor in models is simplified and may not properly reflect, for example, continuum absorptions between major bands in the near-IR. Model calculations suggest that the presence of a thin, saturated layer of water vapor above the clouds, for example, leads to increased absorption by 2-6% (Davies et al., 1984 Podgorny et al., 1998). However, the Crisp calculations indicate that this cannot account for all of the observed excess absorption. 

There a been a number of interesting applications of the framework developed in the studies of the simple ions were MD simulations of the quadrupolar relaxation has been performed on counterions in heterogeneous systems. Studies of a droplet of aqueous Na embedded in a membrane of carboxyl groups [54], showed that the EFG was strongly effected by the local solvent structure and that continuum models are not sufficient to describe the quadrupolar relaxation. The Stemheimer approximation was employed, which had been shown to be a good approximation for the Na ion. Again, the division into molecular contributions could be employed to rationalize the complex behavior in the EFG tensor. Similar conclusions has been drawn from MD simulation studies of ions solvating DNA... 

Soo and Radke (11) confirmed that the transient permeability reduction observed by McAuliffe (9) mainly arises from the retention of drops in pores, which they termed as straining capture of the oil droplets. They also observed that droplets smaller than pore throats were captured in crevices or pockets and sometimes on the surface of the porous medium. They concluded, on the basis of their experiments in sand packs and visual glass micromodel observations, that stable OAV emulsions do not flow in the porous medium as a continuum viscous liquid, nor do they flow by squeezing through pore constrictions, but rather by the capture of the oil droplets with subsequent permeability reduction. They used deep-bed filtration principles (i2, 13) to model this phenomenon, which is discussed in detail later in this chapter. 

Homogeneous Models. The basic assumption in these models is that the emulsion is a continuum, single-phase liquid that is, its microscopic features are unimportant in describing the physical properties or bulk flow characteristics. It ignores interactions between the droplets in the emulsions and the rock surface. The emulsion is considered to be a single-phase homogeneous fluid, and its flow in a porous medium is modeled by using well-documented concepts of Newtonian and non-Newtonian fluid flow in porous media (26, 38). 

Crowe et al. (1977) proposed an axi-symmetric spray drying model called Particle-Sonrce-In-Cell model (PSI-Cell model). This model includes two-way mass, momentum, and thermal conpling. In this model, the gas phase is regarded as a continuum (Eulerian approach) and is described by pressure, velocity, temperature, and humidity fields. The droplets or particles are treated as discrete phases which are characterized by velocity, temperature, composition, and the size along trajectories (Lagrangian approach). The model incorporates a finite difference scheme for both the continuum and discrete phases. The authors used this PSI-Cell model to simulate a cocurrent spray dryer. But no experimental data were compared with it. More details can be found in the woik by Crowe et al. (1977). 

Whereas in approach 1 lattice models are used, we will work in the continuum, making extensive use of interface thermodynamics. The advantage of such an approach, as it turns out, is that detailed properties such as the size distribution of microemulsion droplets and the interfacial tension of a flat monolayer separating a microemulsion and an excess phase can be predicted. On the other hand, the lattice approaches as summarized in item 1 predict global phase behavior, which is not (yet) possible with the thermodynamic formalism reviewed in the following section. The reason is that currently a realistic model for the middle phase is lacking. A more detailed discussion regarding this issue is presented in Sec. VIII. 

Now Eq. (52) for a droplet containing an ion is subject to all of the criticisms of the drop model itself. It neglects rotation and translation of the drop. It is based on macroscopic continuum thermodynamics and so there is no reason to expect that it should apply to small drops. It contains no consideration of the structure of a small drop. In addition, it does not consider that the ion itself may perturb the configurations of the molecules in the drop. 

Inhomogeneous or multiphase reaction systems are characterised by the presence of macroscopic (in relation to the molecular level) inhomogeneities. Numerical calculations of the hydrodynamics of such flows are extremely complicated. There are two opposite approaches to their characterisation [63, 64] the Euler approach, with consideration of the interfacial interaction (interpenetrating continuums model) and the Lagrange approach, of integration by discrete particle trajectories (droplets, bubbles, and so on). The presence of a substantial amount of discrete particles in real systems makes the Lagrange approach inapplicable to study motion in multicomponent systems. Under the Euler approach, a two-phase flow is described... 

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